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[ [ "Geometric Median in Nearly Linear Time" ], [ "Abstract In this paper we provide faster algorithms for solving the geometric median problem: given $n$ points in $\\mathbb{R}^{d}$ compute a point that minimizes the sum of Euclidean distances to the points.", "This is one of the oldest non-trivial problems in computational geometry yet despite an abundance of research the previous fastest algorithms for computing a $(1+\\epsilon)$-approximate geometric median were $O(d\\cdot n^{4/3}\\epsilon^{-8/3})$ by Chin et.", "al, $\\tilde{O}(d\\exp{\\epsilon^{-4}\\log\\epsilon^{-1}})$ by Badoiu et.", "al, $O(nd+\\mathrm{poly}(d,\\epsilon^{-1})$ by Feldman and Langberg, and $O((nd)^{O(1)}\\log\\frac{1}{\\epsilon})$ by Parrilo and Sturmfels and Xue and Ye.", "In this paper we show how to compute a $(1+\\epsilon)$-approximate geometric median in time $O(nd\\log^{3}\\frac{1}{\\epsilon})$ and $O(d\\epsilon^{-2})$.", "While our $O(d\\epsilon^{-2})$ is a fairly straightforward application of stochastic subgradient descent, our $O(nd\\log^{3}\\frac{1}{\\epsilon})$ time algorithm is a novel long step interior point method.", "To achieve this running time we start with a simple $O((nd)^{O(1)}\\log\\frac{1}{\\epsilon})$ time interior point method and show how to improve it, ultimately building an algorithm that is quite non-standard from the perspective of interior point literature.", "Our result is one of very few cases we are aware of outperforming traditional interior point theory and the only we are aware of using interior point methods to obtain a nearly linear time algorithm for a canonical optimization problem that traditionally requires superlinear time.", "We hope our work leads to further improvements in this line of research." ], [ "Nearly Linear Time Geometric Median", "Here we show how to use the structural results from the previous section to obtain a nearly linear time algorithm for computing the geometric median.", "Our algorithm follows a simple structure (See Algorithm ).", "First we use simply average the $a^{(i)}$ to compute a 2-approximate median, denoted $x^{(0)}$ .", "Then for a number of iterations we repeatedly move closer to $x_{t}$ for some path parameter $t$ , compute the minimum eigenvector of the Hessian, and line search in that direction to find an approximation to a point further along the central path.", "Ultimately, this yields a point $x^{(k)}$ that is precise enough approximation to a point along the central path with large enough $t$ that we can simply out $x^{(k)}$ as our $(1+\\epsilon )$ -approximate geometric median.", "[H] $\\mathtt {AccurateMedian}(\\epsilon )$ Input: points $a^{(1)},...,a^{(n)}\\in \\mathbb {R}^{d}$ Input: desired accuracy $\\epsilon \\in (0,1)$ Compute a 2-approximate geometric median and use it to center Compute $x^{(0)}:=\\frac{1}{n}\\sum _{i\\in [n]}a^{(i)}$ and $\\widetilde{f}_{*}:=f(x^{(0)})$ Note $\\tilde{f}_{*}\\le 2f(x_{*})$ by Lemma  Let $t_{i}=\\frac{1}{400\\widetilde{f}_{*}}(1+\\frac{1}{600})^{i-1}$ , $\\tilde{\\epsilon }_{*}=\\frac{1}{3}\\epsilon $ , and $\\tilde{t}_{*}=\\frac{2n}{\\tilde{\\epsilon }_{*}\\cdot \\tilde{f}_{*}}$ .", "Let $\\epsilon _{v}=\\frac{1}{8}(\\frac{\\tilde{\\epsilon }_{*}}{7n})^{2}$ and let $\\epsilon _{c}=(\\frac{\\epsilon _{v}}{36})^{\\frac{3}{2}}$ .", "$x^{(1)}=\\mathtt {LineSearch}(x^{(0)},t_{1},t_{1},0,\\epsilon _{c})$ .", "Iteratively improve quality of approximation Let $k=\\max _{i\\in \\mathbb {Z}}t_{i}\\le \\tilde{t}_{*}$ $i \\in [1,k]$ Compute $\\epsilon _{v}$ -approximate minimum eigenvalue and eigenvector of $\\mathbb {\\nabla }^{2}f_{t_{i}}(x^{(i)})$ $(\\lambda ^{(i)},u^{(i)})=\\mathtt {ApproxMinEig}(x^{(i)},t_{i},\\epsilon _{v})$ .", "Line search to find $x^{(i+1)}$ such that $\\Vert x^{(i+1)}-x_{t_{i+1}}\\Vert _{2}\\le \\frac{\\epsilon _{c}}{t_{i+1}}$ $x^{(i+1)}=\\mathtt {LineSearch}(x^{(i)},t_{i},t_{i+1},u^{(i)},\\epsilon _{c})$ .", "Output: $\\epsilon $ -approximate geometric median $x^{(k+1)}$ .", "We split the remainder of the algorithm specification and its analysis into several parts.", "First in Section REF we show how to compute an approximate minimum eigenvector and eigenvalue of the Hessian of the penalized objective function.", "Then in Section REF we show how to use this eigenvector to line search for the next central path point.", "Finally, in Section REF we put these results together to obtain our nearly linear time algorithm.", "Throughout this section we will want an upper bound to $f(x_{*})$ and a slight lower bound on $\\epsilon $ , the geometric median accuracy we are aiming for.", "We use an easily computed $\\tilde{f}_{*}\\le 2f(x_{*})$ for the former and $\\tilde{\\epsilon }_{*}=\\frac{1}{3}\\epsilon $ throughout the section." ], [ "Eigenvector Computation and Hessian Approximation", "Here we show how to compute the minimum eigenvector of $\\mathbb {\\nabla }^{2}f_{t}(x)$ and thereby obtain a concise approximation to $\\mathbb {\\nabla }^{2}f_{t}(x)$ .", "Our main algorithmic tool is the well known power method and the fact that it converges quickly on a matrix with a large eigenvalue gap.", "To improve our logarithmic terms we need a slightly non-standard analysis of the method and therefore we provide and analyze this method for completeness in Section .", "Using this tool we estimate the top eigenvector as follows.", "$\\texttt {ApproxMinEig}(x,t,\\epsilon )$ Input: Point $x\\in \\mathbb {R}^{d}$ , path parameter $t$ , and target accuracy $\\epsilon $ .", "Let $\\mathbf {A}=\\sum _{i\\in [n]}\\frac{t^{4}(x-a^{(i)})(x-a^{(i)})^{\\top }}{(1+g_{t}^{(i)}(x))^{2}g_{t}^{(i)}(x)}$ Let $u:=\\mathtt {PowerMethod(}\\mathbf {A},\\Theta (\\log \\left(\\frac{n}{\\epsilon }\\right)))$ Let $\\lambda =u^{\\top }\\mathbb {\\nabla }^{2}f_{t}(x)u$ Output: $(\\lambda ,u)$ [Computing Hessian Approximation]lemmaevecandhess Let $x\\in \\mathbb {R}^{d}$ , $t>0$ , and $\\epsilon \\in (0,\\frac{1}{4})$ .", "The algorithm $\\mathtt {ApproxMinEig}(x,t,\\epsilon )$ outputs $(\\lambda ,u)$ in $O(nd\\log \\frac{n}{\\epsilon })$ time such that if $\\mu _{t}(x)\\le \\frac{1}{4}t^{2}w_{t}(x)$ then $\\langle v_{t}(x),u\\rangle ^{2}\\ge 1-\\epsilon $ with high probability in $n/\\epsilon $ .", "Furthermore, if $\\epsilon \\le \\left(\\frac{\\mu _{t}(x)}{8t^{2}\\cdot w_{t}(x)}\\right)^{2}$ then $\\frac{1}{4}\\mathbf {Q}\\preceq \\mathbb {\\nabla }^{2}f_{t}(x)\\preceq 4\\mathbf {Q}$ with high probability in $n/\\epsilon $ where $\\mathbf {Q}\\stackrel{\\mathrm {{\\scriptscriptstyle def}}}{=}t^{2}\\cdot w_{t}(x)-\\left(t^{2}\\cdot w_{t}(x)-\\lambda \\right)uu^{\\top }$ .", "Furthermore, we show that the $v^{(i)}$ computed by this algorithm is sufficiently close to the bad direction.", "Combining REF with the structural results from the previous section and Lemma REF , a minor technical lemma regarding the transitivity of large inner products,we provide the following lemma.", "lemmabaddirectionstable Let $(\\lambda ,u)=\\mathtt {ApproxMinEig}(x,t,\\epsilon _{v})$ for $\\epsilon _{v}<\\frac{1}{8}$ and $\\Vert x-x_{t}\\Vert _{2}\\le \\frac{\\epsilon _{c}}{t}$ for $\\epsilon _{c}\\le (\\frac{\\epsilon _{v}}{36})^{\\frac{3}{2}}$ .", "If $\\mu _{t}\\le \\frac{1}{4}t^{2}\\cdot w_{t}$ then with high probability in $n/\\epsilon _{v}$ for all unit vectors $y\\perp u$ , we have $\\langle y,v_{t}\\rangle ^{2}\\le 8\\epsilon _{v}$ .", "Note that this lemma assumes $\\mu _{t}$ is small.", "When $\\mu _{t}$ is large, we instead show that the next central path point is close to the current point and hence we do not need to compute the bad direction to center quickly.", "lemmalargestep Suppose $\\mu _{t}\\ge \\frac{1}{4}t^{2}\\cdot w_{t}$ and let $t^{\\prime }\\in [t,(1+\\frac{1}{600})t]$ then $\\Vert x_{t^{\\prime }}-x_{t}\\Vert _{2}\\le \\frac{1}{100t}$ ." ], [ "Line Searching", "Here we show how to line search along the bad direction to find the next point on the central path.", "Unfortunately, simply performing binary search on objective function directly may not suffice.", "If we search over $\\alpha $ to minimize $f_{t_{i+1}}(y^{(i)}+\\alpha v^{(i)})$ it is unclear if we actually obtain a point close to $x_{t+1}$ .", "It might be the case that even after minimizing $\\alpha $ we would be unable to move towards $x_{t+1}$ efficiently.", "To overcome this difficulty, we use the fact that over the region $\\Vert x-y\\Vert _{2}=O(\\frac{1}{t})$ the Hessian changes by at most a constant and therefore we can minimize $f_{t}(x)$ over this region extremely quickly.", "Therefore, we instead line search on the following function $g_{t,y,v}(\\alpha )\\stackrel{\\mathrm {{\\scriptscriptstyle def}}}{=}\\min _{\\Vert x-(y+\\alpha v)\\Vert _{2}\\le \\frac{1}{49t}}f_{t}(x)$ and use that we can evaluate $g_{t,y,v}(\\alpha )$ approximately by using an appropriate centering procedure.", "We can show (See Lemma REF ) that $g_{t,y,v}(\\alpha )$ is convex and therefore we can minimize it efficiently just by doing an appropriate binary search.", "By finding the approximately minimizing $\\alpha $ and outputting the corresponding approximately minimizing $x$ , we can obtain $x^{(i+1)}$ that is close enough to $x_{t_{i+1}}$ .", "For notational convenience, we simply write $g(\\alpha )$ if $t,y,v$ is clear from the context.", "First, we show how we can locally center and provide error analysis for that algorithm.", "$\\texttt {LocalCenter}(y,t,\\epsilon )$ Input: Point $y\\in \\mathbb {R}^{d}$ , path parameter $t>0$ , target accuracy $\\epsilon >0$ .", "Let $(\\lambda ,v):=\\mathtt {ApproxMinEig}(x,t,\\epsilon ).$ Let $\\mathbf {Q}=t^{2}\\cdot w_{t}(y)\\mathbf {I}-\\left(t^{2}\\cdot w_{t}(y)-\\lambda \\right)vv^{\\top }$ Let $x^{(0)}=y$ $i=1,...,k=64\\log \\frac{1}{\\epsilon }$ Let $x^{(i)}=\\min _{\\Vert x-y\\Vert _{2}\\le \\frac{1}{49t}}f(x^{(i-1)})+\\langle \\mathbb {\\nabla }f_{t}(x^{(i-1)}),x-x^{(i-1)}\\rangle +4\\Vert x-x^{(i-1)}\\Vert _{\\mathbf {Q}}^{2}$ .", "Output: $x^{(k)}$ lemmalocalcenter Given some $y\\in \\mathbb {R}^{d}$ , $t>0$ and $0\\le \\epsilon \\le \\left(\\frac{\\mu _{t}(x)}{8t^{2}\\cdot w_{t}(x)}\\right)^{2}$ .", "In $O(nd\\log (\\frac{n}{\\epsilon }))$ time $\\mathtt {LocalCenter}(y,t,\\epsilon )$ computes $x^{(k)}$ such that with high probability in $n/\\epsilon $ .", "$f_{t}(x^{(k)})-\\min _{\\Vert x-y\\Vert _{2}\\le \\frac{1}{49t}}f_{t}(x)\\le \\epsilon \\left(f_{t}(y)-\\min _{\\Vert x-y\\Vert _{2}\\le \\frac{1}{49t}}f_{t}(x)\\right)\\,.$ Using this local centering algorithm as well as a general result for minimizing one dimensional convex functions using a noisy oracle (See Section ) we obtain our line search algorithm.", "$\\texttt {LineSearch}(y,t,t^{\\prime },u,\\epsilon )$ Input: Point $y\\in \\mathbb {R}^{d}$ , current path parameter $t$ , next path parameter $t^{\\prime }$ , bad direction $u$ , target accuracy $\\epsilon $ Let $\\epsilon _{O}=\\left(\\frac{\\epsilon \\tilde{\\epsilon }_{*}}{160n^{2}}\\right)^{2}$ , $\\ell =-6\\widetilde{f}_{*}$ , $u=6\\widetilde{f}_{*}$ .", "Define the oracle $q:\\mathbb {R}\\rightarrow \\mathbb {R}$ by $q(\\alpha )=f_{t^{\\prime }}\\left(\\mathtt {LocalCenter}\\left(y+\\alpha u,t^{\\prime },\\epsilon _{O}\\right)\\right)$ Let $\\alpha ^{\\prime }=\\mathtt {OneDimMinimizer}(\\ell ,u,\\epsilon _{O},q,t^{\\prime }n$ ) Output: $x^{\\prime }=\\mathtt {LocalCenter}\\left(y+\\alpha u,t^{\\prime },\\epsilon _{O}\\right)$ lemmalinesearch Let $\\frac{1}{400f(x_{*})}\\le t\\le t^{\\prime }\\le (1+\\frac{1}{600})t\\le \\frac{2n}{\\tilde{\\epsilon }_{*}\\cdot \\tilde{f}_{*}}$ and let $(\\lambda ,u)=\\mathtt {ApproxMinEig}(y,t,\\epsilon _{v})$ for $\\epsilon _{v}\\le \\frac{1}{8}(\\frac{\\tilde{\\epsilon }_{*}}{3n})^{2}$ and $y\\in \\mathbb {R}^{d}$ such that $\\Vert y-x_{t}\\Vert _{2}\\le \\frac{1}{t}(\\frac{\\epsilon _{v}}{36})^{\\frac{3}{2}}$ .", "In $O(nd\\log ^{2}(\\frac{n}{\\tilde{\\epsilon }_{*}\\cdot \\epsilon \\cdot \\epsilon _{v}}))$ time and $O(\\log (\\frac{n}{\\tilde{\\epsilon }_{*}\\cdot \\epsilon }))$ calls to the $\\mathtt {LocalCenter}$ , $\\mathtt {LineSearch}(y,t,t^{\\prime },u,\\epsilon )$ outputs $x^{\\prime }$ such that $\\Vert x^{\\prime }-x_{t^{\\prime }}\\Vert _{2}\\le \\frac{\\epsilon }{t^{\\prime }}$ with high probability in $n/\\epsilon $ .", "We also provide the following lemma useful for finding the first center.", "[]lemmalinesearchtwo Let $\\frac{1}{400f(x_{*})}\\le t\\le t^{\\prime }\\le (1+\\frac{1}{600})t\\le \\frac{2n}{\\tilde{\\epsilon }_{*}\\cdot \\tilde{f}_{*}}$ and let $x\\in \\mathbb {R}^{d}$ satisfy $\\Vert x-x_{t}\\Vert _{2}\\le \\frac{1}{100t}$ .", "Then, in $O(nd\\log ^{2}(\\frac{n}{\\epsilon \\cdot \\tilde{\\epsilon }_{*}}))$ time, $\\mathtt {LineSearch}(x,t,t,u,\\epsilon )$ outputs $y$ such that $\\Vert y-x_{t}\\Vert _{2}\\le \\frac{\\epsilon }{t}$ for any vector $u\\in \\mathbb {R}^{d}$ ." ], [ "Putting It All Together", "Combining the results of the previous sections, we prove our main theorem.", "[]theoremmetaalgorithm In $O(nd\\log ^{3}(\\frac{n}{\\epsilon }))$ time, Algorithm  outputs an $(1+\\epsilon )$ -approximate geometric median with constant probability." ], [ "Acknowledgments", "We thank Yan Kit Chim, Ravi Kannan, and Jonathan A. Kelner for many helpful conversations.", "We thank the reviewers for their help in completing the previous work table.", "This work was partially supported by NSF awards 0843915, 1065106 and 1111109, NSF Graduate Research Fellowship (grant no.", "1122374) and Sansom Graduate Fellowship in Computer Science.", "Part of this work was done while authors were visiting the Simons Institute for the Theory of Computing, UC Berkeley." ], [ "Derivation of Penalty Function", "Here we derive our penalized objective function.", "Consider the following optimization problem: $\\min _{x\\in \\mathbb {R}^{d},\\alpha \\ge 0\\in \\mathbb {R}^{n}}f_{t}(x,\\alpha )\\hspace{5.0pt}\\text{ where }\\hspace{5.0pt}t\\cdot 1^{T}\\alpha +\\sum _{i\\in [n]}-\\ln \\left(\\alpha _{i}^{2}-\\Vert x-a^{(i)}\\Vert _{2}^{2}\\right)\\hspace{5.0pt}.$ Since $p_{i}(\\alpha ,x)\\stackrel{\\mathrm {{\\scriptscriptstyle def}}}{=}-\\ln \\left(\\alpha _{i}^{2}-\\Vert x-a^{(i)}\\Vert _{2}^{2}\\right)$ is a barrier function for the set $\\alpha _{i}^{2}\\ge \\Vert x-a^{(i)}\\Vert _{2}^{2}$ , i.e.", "as $\\alpha _{i}\\rightarrow \\Vert x-a^{(i)}\\Vert _{2}$ we have $p_{i}(\\alpha ,x)\\rightarrow \\infty $ , we see that as we minimize $f_{t}(x,\\alpha )$ for increasing values of $t$ the $x$ values converge to a solution to the geometric median problem.", "Our penalized objective function, $f_{t}(x)$ , is obtain simply by minimizing the $\\alpha _{i}$ in the above formula and dropping terms that do not affect the minimizing $x$ .", "In the remainder of this section we show this formally.", "Fix some $x\\in \\mathbb {R}^{d}$ and $t>0$ .", "Note that for all $j\\in [n]$ we have $\\frac{\\partial }{\\partial \\alpha _{j}}f_{t}(x,\\alpha )=t-\\left(\\frac{1}{\\alpha _{j}^{2}-\\Vert x-a^{(i)}\\Vert _{2}^{2}}\\right)2\\alpha _{j}\\,.$ Since $f(x,\\alpha )$ is convex in $\\alpha $ , the minimum $\\alpha _{j}^{*}$ must satisfy $t\\left(\\left(\\alpha _{j}^{*}\\right)^{2}-\\Vert x-a^{(i)}\\Vert _{2}^{2}\\right)-2\\alpha _{j}^{*}=0\\hspace{5.0pt}.$ Solving for such $\\alpha _{j}^{*}$ under the restriction $\\alpha _{j}^{*}\\ge 0$ we obtain $\\alpha _{j}^{*}=\\frac{2+\\sqrt{4+4t^{2}\\Vert x-a^{(i)}\\Vert _{2}^{2}}}{2t}=\\frac{1}{t}\\left[1+\\sqrt{1+t^{2}\\Vert x-a^{(i)}\\Vert _{2}^{2}}\\right]\\hspace{5.0pt}.$ Using (REF ) and (REF ) we have that $\\min _{\\alpha \\ge 0\\in \\mathbb {R}^{n}}f_{t}(x,\\alpha )=\\sum _{i\\in [n]}\\left[1+\\sqrt{1+t^{2}\\Vert x-a^{(i)}\\Vert _{2}^{2}}-\\ln \\left[\\frac{2}{t^{2}}\\left(1+\\sqrt{1+t^{2}\\Vert x-a^{(i)}\\Vert _{2}^{2}}\\right)\\right]\\right]\\hspace{5.0pt}.$ If we drop the terms that do not affect the minimizing $x$ we obtain our penalty function $f_{t}$ : $f_{t}(x)=\\sum _{i\\in [n]}\\left[\\sqrt{1+t^{2}\\Vert x-a^{(i)}\\Vert _{2}^{2}}-\\ln \\left(1+\\sqrt{1+t^{2}\\Vert x-a^{(i)}\\Vert _{2}^{2}}\\right)\\right]\\hspace{5.0pt}.$" ], [ "Technical Facts", "Here we provide various technical lemmas we use through the paper." ], [ "Linear Algebra", "First we provide the following lemma that shows that any matrix obtained as a non-negative linear combination of the identity minus a rank 1 matrix less than the identity results in a matrix that is well approximated spectrally by the identity minus a rank 1 matrix.", "We use this lemma to characterize the Hessian of our penalized objective function and thereby imply that it is possible to apply the inverse of the Hessian to a vector with high precision.", "Lemma 1 Let $\\mathbf {A}=\\sum _{i}\\left(\\alpha _{i}\\mathbf {I}-\\beta _{i}a_{i}a_{i}^{\\top }\\right)\\in \\mathbb {R}^{d\\times d}$ where the $a_{i}$ are unit vectors and $0\\le \\beta _{i}\\le \\alpha _{i}$ for all $i$ .", "Let $v$ denote a unit vector that is the maximum eigenvector of $\\sum _{i}\\beta _{i}a_{i}a_{i}^{\\top }$ and let $\\lambda $ denote the corresponding eigenvalue.", "Then, $\\frac{1}{2}\\left(\\sum _{i}\\alpha _{i}\\mathbf {I}-\\lambda vv^{\\top }\\right)\\preceq \\mathbf {A}\\preceq \\sum _{i}\\alpha _{i}\\mathbf {I}-\\lambda vv^{\\top }\\,.$ Let $\\alpha \\stackrel{\\mathrm {{\\scriptscriptstyle def}}}{=}\\sum _{i}\\alpha _{i}$ .", "Since clearly $v^{\\top }\\mathbf {A}v=v^{\\top }\\left(\\sum _{i}\\alpha _{i}\\mathbf {I}-\\lambda vv^{\\top }\\right)v$ it suffices to show that for $w\\perp v$ it is the case that $\\frac{1}{2}\\alpha \\Vert w\\Vert _{2}^{2}\\preceq w^{\\top }\\mathbf {A}w\\preceq \\alpha \\Vert w\\Vert _{2}^{2}$ or equivalently, that $\\lambda _{i}(\\mathbf {A})\\in [\\frac{1}{2}\\alpha ,\\alpha ]$ for $i\\ne d$ .", "However we know that $\\sum _{i\\in [d]}\\lambda _{i}(\\mathbf {A})=\\mathrm {tr}(\\mathbf {A})=d\\alpha -\\sum _{i}\\beta _{i}\\ge (d-1)\\alpha $ and $\\lambda _{i}(\\mathbf {A})\\le \\alpha $ for all $i\\in [d]$ .", "Consequently, since $\\lambda _{d}(\\mathbf {A})\\in [0,\\lambda _{d-1}(\\mathbf {A})]$ we have $2\\cdot \\lambda _{d-1}(\\mathbf {A})\\ge (d-1)\\alpha -\\sum _{i=1}^{d-2}\\lambda _{i}(\\mathbf {A})\\ge (d-1)\\alpha -(d-2)\\alpha =\\alpha \\,.$ Consequently, $\\lambda _{d-1}(\\mathbf {A})\\in [\\frac{\\alpha }{2},\\alpha ]$ and the result holds by the monotonicity of $\\lambda _{i}$ .", "Next we bound the spectral difference between the outer product of two unit vectors by their inner product.", "We use this lemma to bound the amount of precision required in our eigenvector computations.", "Lemma 2 For unit vectors $u_{1}$ and $u_{2}$ we have $\\Vert u_{1}u_{1}^{\\top }-u_{2}u_{2}^{\\top }\\Vert _{2}^{2}=1-(u_{1}^{\\top }u_{2})^{2}$ Consequently if $\\left(u_{1}^{\\top }u_{2}\\right)^{2}\\ge 1-\\epsilon $ for $\\epsilon \\le 1$ we have that $-\\sqrt{\\epsilon }\\mathbf {I}\\preceq u_{1}u_{1}^{\\top }-u_{2}u_{2}^{\\top }\\preceq \\sqrt{\\epsilon }\\mathbf {I}\\,$ Note that $u_{1}u_{1}^{\\top }-u_{2}u_{2}^{\\top }$ is a symmetric matrix and all eigenvectors are either orthogonal to both $u_{1}$ and $u_{2}$ (with eigenvalue 0) or are of the form $v=\\alpha u_{1}+\\beta u_{2}$ where $\\alpha $ and $\\beta $ are real numbers that are not both 0.", "Thus, if $v$ is an eigenvector of non-zero eigenvalue $\\lambda $ it must be that $\\lambda \\left(\\alpha u_{1}+\\beta u_{2}\\right) & =\\left(u_{1}u_{1}^{\\top }-u_{2}u_{2}^{\\top }\\right)(\\alpha u_{1}+\\beta u_{2})\\\\& =(\\alpha +\\beta (u_{1}^{\\top }u_{2}))u_{1}-(\\alpha (u_{1}^{\\top }u_{2})+\\beta )u_{2}$ or equivalently $\\left(\\begin{array}{cc}(1-\\lambda ) & u_{1}^{\\top }u_{2}\\\\-(u_{1}^{\\top }u_{2}) & -(1+\\lambda )\\end{array}\\right)\\left(\\begin{array}{c}\\alpha \\\\\\beta \\end{array}\\right)=\\left(\\begin{array}{c}0\\\\0\\end{array}\\right)\\,.$ By computing the determinant we see this has a solution only when $-(1-\\lambda ^{2})+(u_{1}^{\\top }u_{2})^{2}=0$ Solving for $\\lambda $ then yields (REF ) and completes the proof.", "Next we show how the top eigenvectors of two spectrally similar matrices are related.", "We use this to bound the amount of spectral approximation we need to obtain accurate eigenvector approximations.", "Lemma 3 Let $\\mathbf {A}$ and $\\mathbf {B}$ be symmetric PSD matrices such that $(1-\\epsilon )\\mathbf {A}\\preceq \\mathbf {B}\\preceq (1+\\epsilon )\\mathbf {A}$ .", "Then if $g\\stackrel{\\mathrm {{\\scriptscriptstyle def}}}{=}\\frac{\\lambda _{1}(\\mathbf {A})-\\lambda _{2}(\\mathbf {A})}{\\lambda _{1}(\\mathbf {A})}$ satisfies $g>0$ we have $[v_{1}(\\mathbf {A})^{\\top }v_{1}(\\mathbf {B})]^{2}\\ge 1-2(\\epsilon /g)$ .", "Without loss of generality $v_{1}(\\mathbf {B})=\\alpha v_{1}(\\mathbf {A})+\\beta v$ for some unit vector $v\\perp v_{1}(\\mathbf {A})$ and $\\alpha ,\\beta \\in \\mathbb {R}$ such that $\\alpha ^{2}+\\beta ^{2}=1$ .", "Now we know that $v_{1}(\\mathbf {B})^{\\top }\\mathbf {B}v_{1}(\\mathbf {B})\\le (1+\\epsilon )v_{1}(\\mathbf {B})^{\\top }\\mathbf {A}v_{1}(\\mathbf {B})\\le (1+\\epsilon )\\left[\\alpha ^{2}\\lambda _{1}(\\mathbf {A})+\\beta ^{2}\\lambda _{2}(\\mathbf {A})\\right]$ Furthermore, by the optimality of $v_{1}(\\mathbf {B})$ we have that $v_{1}(\\mathbf {B})^{\\top }\\mathbf {B}v_{1}(\\mathbf {B})\\ge (1-\\epsilon )v_{1}(\\mathbf {A})^{\\top }\\mathbf {A}v_{1}(\\mathbf {A})\\ge (1-\\epsilon )\\lambda _{1}(\\mathbf {A})\\,.$ Now since $\\beta ^{2}=1-\\alpha ^{2}$ combining these inequalities yields $(1-\\epsilon )\\lambda _{1}(\\mathbf {A})\\le (1+\\epsilon )\\alpha ^{2}\\left(\\lambda _{1}(\\mathbf {A})-\\lambda _{2}(\\mathbf {A})\\right)+(1+\\epsilon )\\lambda _{2}(\\mathbf {A})\\,.$ Rearranging terms, using the definition of $g$ , and that $g\\in (0,1]$ and $\\epsilon \\ge 0$ yields $\\alpha ^{2} & \\ge \\frac{\\lambda _{1}(\\mathbf {A})-\\lambda _{2}(\\mathbf {A})-\\epsilon (\\lambda _{1}(\\mathbf {A})+\\lambda _{2}(\\mathbf {A}))}{(1+\\epsilon )(\\lambda _{1}(\\mathbf {A})-\\lambda _{2}(\\mathbf {A}))}=1-\\frac{2\\epsilon \\lambda _{1}(\\mathbf {A})}{(1+\\epsilon )(\\lambda _{1}(\\mathbf {A})-\\lambda _{2}(\\mathbf {A}))}\\ge 1-2(\\epsilon /g)\\,.$ Here we prove a an approximate transitivity lemma for inner products of vectors.", "We use this to bound the accuracy need for certain eigenvector computations.", "Lemma 4 Suppose that we have vectors $v_{1},v_{2},v_{3}\\in \\mathbb {R}^{n}$ such that $\\langle v_{1},v_{2}\\rangle ^{2}\\ge 1-\\epsilon $ and $\\langle v_{2},v_{3}\\rangle ^{2}\\ge 1-\\epsilon $ for $0<\\epsilon \\le \\frac{1}{2}$ then $\\langle v_{1},v_{3}\\rangle ^{2}\\ge 1-4\\epsilon $ .", "Without loss of generality, we can write $v_{1}=\\alpha _{1}v_{2}+\\beta _{1}w_{1}$ for $\\alpha _{1}^{2}+\\beta _{1}^{2}=1$ and unit vector $w_{1}\\perp v_{2}$ .", "Similarly we can write $v_{3}=\\alpha _{3}v_{2}+\\beta _{3}w_{3}$ for $\\alpha _{3}^{2}+\\beta _{3}^{2}=1$ and unit vector $w_{3}\\perp v_{2}$ .", "Now, by the inner products we know that $\\alpha _{1}^{2}\\ge 1-\\epsilon $ and $\\alpha _{3}^{2}\\ge 1-\\epsilon $ and therefore $|\\beta _{1}|\\le \\sqrt{\\epsilon }$ and $\\left|\\beta _{3}\\right|\\le \\sqrt{\\epsilon }$ .", "Consequently, since $\\epsilon \\le \\frac{1}{2}$ , $|\\beta _{1}\\beta _{3}|\\le \\epsilon \\le 1-\\epsilon \\le |\\alpha _{1}\\alpha _{3}|$ , and we have $\\langle v_{1},v_{3}\\rangle ^{2} & \\ge \\langle \\alpha _{1}v_{2}+\\beta _{1}w_{1},\\alpha _{3}v_{2}+\\beta _{3}w_{3}\\rangle ^{2}\\ge \\left(\\left|\\alpha _{1}\\alpha _{3}\\right|-\\left|\\beta _{1}\\beta _{3}\\right|\\right)^{2}\\\\& \\ge \\left(1-\\epsilon -\\epsilon \\right)^{2}=(1-2\\epsilon )^{2}\\ge 1-4\\epsilon .$" ], [ "Convex Optimization", "First we provide a single general lemma about about first order methods for convex optimization.", "We use this lemma for multiple purposes including bounding errors and quickly compute approximations to the central path.", "Lemma 5 ([21]) Let $f\\,:\\,\\mathbb {R}^{n}\\rightarrow \\mathbb {R}$ be a twice differentiable function, let $B\\subseteq \\mathbb {R}$ be a convex set, and let $x_{*}$ be a point that achieves the minimum value of $f$ restricted to $B$ .", "Further suppose that for a symmetric positive definite matrix $\\mathbf {H}\\in \\mathbb {R}^{n\\times n}$ we have that $\\mu \\mathbf {H}\\preceq \\mathbb {\\nabla }^{2}f(y)\\preceq L\\mathbf {H}$ for all $y\\in B$ .Then for all $x\\in B$ we have $\\frac{\\mu }{2}\\Vert x-x_{*}\\Vert _{\\mathbf {H}}^{2}\\le f(x)-f(x_{*})\\le \\frac{L}{2}\\Vert x-x_{*}\\Vert _{\\mathbf {H}}^{2}$ and $\\frac{1}{2L}\\Vert \\mathbb {\\nabla }f(x)\\Vert _{\\mathbf {H}^{-1}}^{2}\\le f(x)-f(x_{*})\\le \\frac{1}{2\\mu }\\Vert \\mathbb {\\nabla }f(x)\\Vert _{\\mathbf {H}^{-1}}^{2}\\,.$ Furthermore, if $x^{(1)}=\\operatornamewithlimits{arg\\,min}_{x\\in B}\\left[f(x^{(0)})+\\langle \\mathbb {\\nabla }f(x^{(0)}),x-x^{(0)}\\rangle +\\frac{L}{2}\\Vert x^{(0)}-x\\Vert _{\\mathbf {H}}^{2}\\right]$ then $f(x^{(1)})-f(x_{*})\\le \\left(1-\\frac{\\mu }{L}\\right)\\left(f(x^{(0)})-f(x_{*})\\right)\\,.$ Next we provide a short technical lemma about the convexity of functions that arises naturally in our line searching procedure.", "Lemma 6 Let $f:\\mathbb {R}^{n}\\rightarrow \\mathbb {R}\\cup \\lbrace \\infty \\rbrace $ be a convex function and and let $g(\\alpha )\\stackrel{\\mathrm {{\\scriptscriptstyle def}}}{=}\\min _{x\\in S}f(x+\\alpha d)$ for any convex set $S$ and $d\\in \\mathbb {R}^{n}$ .", "Then $g$ is convex.", "Let $\\alpha ,\\beta \\in \\mathbb {R}$ and define $x_{\\alpha }=\\operatornamewithlimits{arg\\,min}_{x\\in S}f(x+\\alpha d)$ and $x_{\\beta }=\\operatornamewithlimits{arg\\,min}_{x\\in S}f(x+\\beta d)$ .", "For any $t\\in [0,1]$ we have $g\\left(t\\alpha +(1-t)\\beta \\right) & =\\min _{x\\in S}f\\left(x+(t\\alpha +(1-t)\\beta \\right)\\\\& \\le f(tx_{\\alpha }+(1-t)x_{\\beta }+(t\\alpha +(1-t)\\beta )d)\\\\& \\le t\\cdot f(x_{\\alpha }+\\alpha d)+(1-t)\\cdot f(x_{\\beta }+\\beta \\cdot d)\\\\& =t\\cdot g(\\alpha )+(1-t)\\cdot g(\\beta )$ Lemma 7 For any vectors $y,z,v\\in \\mathbb {R}^{d}$ and scalar $\\alpha $ , we can compute $\\operatornamewithlimits{arg\\,min}_{\\Vert x-y\\Vert _{2}^{2}\\le \\alpha }\\Vert x-z\\Vert _{\\mathbf {I}-vv^{\\top }}^{2}$ exactly in time $O(d)$ .", "Let $x^{*}$ be the solution of this problem.", "If $\\Vert x^{*}-y\\Vert _{2}^{2}<\\alpha $ , then $x^{*}=z$ .", "Otherwise, there is $\\lambda >0$ such that $x^{*}$ is the minimizer of $\\min _{x\\in \\mathbb {R}^{d}}\\Vert x-z\\Vert _{\\mathbf {I}-vv^{\\top }}^{2}+\\lambda \\Vert x-y\\Vert _{2}^{2}\\,.$ Let $\\mathbf {Q}=\\mathbf {I}-vv^{\\top }$ .", "Then, the optimality condition of the above equation shows that $\\mathbf {Q}(x^{*}-z)+\\lambda (x^{*}-y)=0\\,.$ Therefore, $x^{*}=(\\mathbf {Q}+\\lambda \\mathbf {I})^{-1}(\\mathbf {Q}z+\\lambda y)\\,.$ Hence, $\\alpha =\\Vert x^{*}-y\\Vert _{2}^{2}=(z-y)^{\\top }\\mathbf {Q}(\\mathbf {Q}+\\lambda \\mathbf {I})^{-2}\\mathbf {Q}(z-y).$ Let $\\eta =1+\\lambda $ , then we have $(\\mathbf {Q}+\\lambda \\mathbf {I})=\\eta \\mathbf {I}-vv^{\\top }$ and hence Sherman–Morrison formula shows that $(\\mathbf {Q}+\\lambda \\mathbf {I})^{-1}=\\eta ^{-1}\\mathbf {I}+\\frac{\\eta ^{-2}vv^{\\top }}{1-\\Vert v\\Vert ^{2}\\eta ^{-1}}=\\eta ^{-1}\\left(\\mathbf {I}+\\frac{vv^{\\top }}{\\eta -\\Vert v\\Vert ^{2}}\\right)\\,.$ Hence, we have $(\\mathbf {Q}+\\lambda \\mathbf {I})^{-2}=\\eta ^{-2}\\left(\\mathbf {I}+\\frac{2vv^{\\top }}{\\eta -\\Vert v\\Vert ^{2}}+\\frac{vv^{\\top }\\Vert v\\Vert ^{2}}{\\left(\\eta -\\Vert v\\Vert ^{2}\\right)^{2}}\\right)=\\eta ^{-2}\\left(\\mathbf {I}+\\frac{2\\eta -\\Vert v\\Vert ^{2}}{\\left(\\eta -\\Vert v\\Vert ^{2}\\right)^{2}}vv^{\\top }\\right)\\,.$ Let $c_{1}=\\Vert \\mathbf {Q}(z-y)\\Vert _{2}^{2}$ and $c_{2}=\\left(v^{\\top }\\mathbf {Q}(z-y)\\right)^{2}$ , then we have $\\alpha & = & \\eta ^{-2}\\left(c_{1}+\\frac{2\\eta -\\Vert v\\Vert ^{2}}{\\left(\\eta -\\Vert v\\Vert ^{2}\\right)^{2}}c_{2}\\right).$ Hence, we have $\\alpha \\eta ^{2}\\left(\\eta -\\Vert v\\Vert ^{2}\\right)^{2}=c_{1}\\left(\\eta -\\Vert v\\Vert ^{2}\\right)^{2}+c_{2}\\left(2\\eta -\\Vert v\\Vert ^{2}\\right).$ Note that this is a polynomial of degree 4 in $\\eta $ and all coefficients can be computed in $O(d)$ time.", "Solving this by explicit formula, one can test all 4 possible $\\eta $ 's into the formula (REF ) of $x$ .", "Together with trivial case $x^{*}=z$ , we simply need to check among 5 cases to check which is the solution." ], [ "Weighted Geometric Median", "In this section, we show how to extend our results to the weighted geometric median problem, also known as the Weber problem: given a set of $n$ points in $d$ dimensions, $a^{(1)},\\ldots ,a^{(n)}\\in \\mathbb {R}^{d}$ , with corresponding weights $w^{(1)},\\ldots ,w^{(n)}\\in \\mathbb {R}_{>0}$ , find a point $x_{*}\\in \\mathbb {R}^{d}$ that minimizes the weighted sum of Euclidean distances to them: $x_{*}\\in \\operatornamewithlimits{arg\\,min}_{x\\in \\mathbb {R}^{d}}f(x)\\hspace{5.0pt}\\text{ where }\\hspace{5.0pt}f(x)\\stackrel{\\mathrm {{\\scriptscriptstyle def}}}{=}\\sum _{i\\in [n]}w^{(i)}\\Vert x-a^{(i)}\\Vert _{2}.$ As in the unweighted problem, our goal is to compute $(1+\\epsilon )$ -approximate solution, i.e.", "$x\\in \\mathbb {R}^{d}$ with $f(x)\\le (1+\\epsilon )f(x_{*})$ .", "First, we show that it suffices to consider the case where the weights are integers with bounded sum (Lemma REF ).", "Then, we show that such an instance of the weighted geometric median problem can be solved using the algorithms developed for the unweighted problem.", "Lemma 8 Given points $a^{(1)},a^{(2)},\\ldots ,a^{(n)}\\in \\mathbb {R}^{d}$ , non-negative weights $w^{(1)},w^{(2)},\\ldots ,w^{(n)}\\in \\mathbb {R}_{>0}$ , and $\\epsilon \\in (0,1)$ , we can compute in linear time weights $w_{1}^{(1)},w_{1}^{(2)},\\ldots ,w_{1}^{(n)}$ such that: Any $(1+\\epsilon /5)$ -approximate weighted geometric median of $a^{(1)},\\ldots ,a^{(n)}$ with the weights $w_{1}^{(1)},\\ldots ,w_{1}^{(n)}$ is also a $(1+\\epsilon )$ -approximate weighted geometric median of $a^{(1)},\\ldots ,a^{(n)}$ with the weights $w^{(1)},\\ldots ,w^{(n)}$ , and $w_{1}^{(1)},\\ldots ,w_{1}^{(n)}$ are nonnegative integers and $\\sum _{i\\in [n]}w_{1}^{(i)}\\le 5n\\epsilon ^{-1}$ .", "Let $f(x)=\\sum _{i\\in [n]}w^{(i)}\\Vert a^{(i)}-x\\Vert $ and $W=\\sum _{i\\in [n]}w^{(i)}$ .", "Furthermore, let $\\epsilon ^{\\prime }=\\epsilon /5$ and for each $i\\in [n]$ , define $w_{0}^{(i)}=\\frac{n}{\\epsilon ^{\\prime }W}w^{(i)}$ , $w_{1}^{(i)}=\\left\\lfloor w_{0}^{(i)}\\right\\rfloor $ and $w_{2}^{(i)}=w_{0}^{(i)}-w_{1}^{(i)}$ .", "We also define $f_{0},f_{1},f_{2},W_{0},W_{1},W_{2}$ analogously to $f$ and $W$ .", "Now, assume $f_{1}(x)\\le (1+\\epsilon ^{\\prime })f_{1}(x_{*})$ , where $x_{*}$ is the minimizer of $f$ and $f_{0}$ .", "Then: $f_{0}(x)=f_{1}(x)+f_{2}(x)\\le f_{1}(x)+f_{2}(x_{*})+W_{2}\\Vert x-x_{*}\\Vert _{2}$ and $W_{2}\\Vert x-x_{*}\\Vert _{2} & =\\frac{W_{2}}{W_{1}}\\sum _{i\\in [n]}w_{1}^{(i)}\\Vert x-x_{*}\\Vert _{2}\\le \\frac{W_{2}}{W_{1}}\\sum _{i\\in [n]}w_{1}^{(i)}\\left(\\Vert x-a^{(i)}\\Vert _{2}+\\Vert a^{(i)}-x_{*}\\Vert \\right)\\\\& \\le \\frac{W_{2}}{W_{1}}(f_{1}(x)+f_{1}(x_{*}))\\,.$ Now, since $W_{0}=\\frac{n}{\\epsilon ^{\\prime }}$ and $W_{1}\\ge W_{0}-n$ we have $\\frac{W_{2}}{W_{1}}=\\frac{W_{0}-W_{1}}{W_{1}}=\\frac{W_{0}}{W_{1}}-1\\le \\frac{W_{0}}{W_{0}-n}-\\frac{W_{0}-n}{W_{0}-n}=\\frac{n}{\\frac{n}{\\epsilon ^{\\prime }}-n}=\\frac{\\epsilon ^{\\prime }}{1-\\epsilon ^{\\prime }}\\,.$ Combining these yields that $f_{0}(x) & \\le f_{1}(x)+f_{2}(x_{*})+\\frac{\\epsilon ^{\\prime }}{1-\\epsilon ^{\\prime }}(f_{1}(x)+f_{1}(x_{*}))\\\\& \\le \\left(1+\\frac{\\epsilon ^{\\prime }}{1-\\epsilon ^{\\prime }}\\right)(1+\\epsilon ^{\\prime })f_{1}(x^{*})+\\frac{\\epsilon ^{\\prime }}{1-\\epsilon ^{\\prime }}f_{1}(x_{*})+f_{2}(x_{*})\\\\& \\le (1+5\\epsilon ^{\\prime })f_{0}(x_{*})=(1+\\epsilon )f_{0}(x_{*})\\,.$ We now proceed to show the main result of this section.", "Lemma 9 A $(1+\\epsilon )$ -approximate weighted geometric median of $n$ points in $\\mathbb {R}^{d}$ can be computed in $O(nd\\log ^{3}\\epsilon ^{-1})$ time.", "By applying Lemma REF , we can assume that the weights are integer and their sum does not exceed $n\\epsilon ^{-1}.$ Note that computing the weighted geometric median with such weights is equivalent to computing an unweighted geometric median of $O(n\\epsilon ^{-1})$ points (where each point of the original input is repeated with the appropriate multiplicity).", "We now show how to simulate the behavior of our unweighted geometric median algorithms on such a set of points without computing it explicitly.", "If $\\epsilon >n^{-1/2}$ , we will apply the algorithm $\\mathtt {ApproximateMedian}$ , achieving a runtime of $O(d\\epsilon ^{-2})=O(nd)$ .", "It is only necessary to check that we can implement weighted sampling from our points with $O(n)$ preprocessing and $O(1)$ time per sample.", "This is achieved by the alias method [15].", "Now assume $\\epsilon <n^{-1/2}$ .", "We will employ the algorithm $\\mathtt {AccurateMedian}$ .", "Note that we can implement the subroutines $\\mathtt {LineSearch}$ and $\\mathtt {ApproxMinEig}$ on the implicitly represented multiset of $O(n\\epsilon ^{-1})$ points.", "It is enough to observe only $n$ of the points are distinct, and all computations performed by these subroutines are identical for identical points.", "The total runtime will thus be $O(nd\\log ^{3}(n/\\epsilon ^{2}))=O(nd\\log ^{3}\\epsilon ^{-1})$ ." ] ]
1606.05225
[ [ "Polarization shaping for control of nonlinear propagation" ], [ "Abstract We study the nonlinear optical propagation of two different classes of space-varying polarized light beams -- radially symmetric vector beams and Poincar\\'e beams with lemon and star topologies -- in a rubidium vapour cell.", "Unlike Laguerre-Gauss and other types of beams that experience modulational instabilities, we observe that their propagation is not marked by beam breakup while still exhibiting traits such as nonlinear confinement and self-focusing.", "Our results suggest that by tailoring the spatial structure of the polarization, the effects of nonlinear propagation can be effectively controlled.", "These findings provide a novel approach to transport high-power light beams in nonlinear media with controllable distortions to their spatial structure and polarization properties." ], [ "lightblue Polarization shaping for control of nonlinear propagation Frédéric Bouchard The Max Planck Centre for Extreme and Quantum Photonics, Department of Physics, University of Ottawa, 25 Templeton, Ottawa, Ontario, K1N 6N5 Canada Hugo Larocque The Max Planck Centre for Extreme and Quantum Photonics, Department of Physics, University of Ottawa, 25 Templeton, Ottawa, Ontario, K1N 6N5 Canada Alison M. Yao SUPA and Department of Physics, University of Strathclyde, 107 Rottenrow, Glasgow G4 0NG, Scotland, U.K. Christopher Travis SUPA and Department of Physics, University of Strathclyde, 107 Rottenrow, Glasgow G4 0NG, Scotland, U.K. Israel De Leon The Max Planck Centre for Extreme and Quantum Photonics, Department of Physics, University of Ottawa, 25 Templeton, Ottawa, Ontario, K1N 6N5 Canada School of Engineering and Sciences, Tecnológico de Monterrey, Monterrey, Nuevo Leon 64849, Mexico.", "Andrea Rubano Dipartimento di Fisica, Università di Napoli Federico II, Compl.", "Univ.", "di Monte S. Angelo, via Cintia, 80126 Napoli, Italy Ebrahim Karimi [email protected] The Max Planck Centre for Extreme and Quantum Photonics, Department of Physics, University of Ottawa, 25 Templeton, Ottawa, Ontario, K1N 6N5 Canada Department of Physics, Institute for Advanced Studies in Basic Sciences, 45137-66731 Zanjan, Iran Gian-Luca Oppo SUPA and Department of Physics, University of Strathclyde, 107 Rottenrow, Glasgow G4 0NG, Scotland, U.K. Robert W. Boyd The Max Planck Centre for Extreme and Quantum Photonics, Department of Physics, University of Ottawa, 25 Templeton, Ottawa, Ontario, K1N 6N5 Canada Institute of Optics, University of Rochester, Rochester, New York, 14627, USA We study the nonlinear optical propagation of two different classes of space-varying polarized light beams – radially symmetric vector beams and Poincaré beams with lemon and star topologies – in a rubidium vapour cell.", "Unlike Laguerre-Gauss and other types of beams that experience modulational instabilities, we observe that their propagation is not marked by beam breakup while still exhibiting traits such as nonlinear confinement and self-focusing.", "Our results suggest that by tailoring the spatial structure of the polarization, the effects of nonlinear propagation can be effectively controlled.", "These findings provide a novel approach to transport high-power light beams in nonlinear media with controllable distortions to their spatial structure and polarization properties.", "Valid PACS appear here Introduction: Light beams that can propagate without significant change to their spatial profile are of interest for modern optical technologies and high-power laser systems.", "Self-trapped light filaments, or spatial solitons, are formed when their spreading due to linear diffraction is carefully balanced by a self-focusing (Kerr) nonlinearity that causes the beam to narrow.", "Due to their potential to carry an increased information content, there has been significant interest in the formation of spatial solitons carrying orbital angular momentum (OAM) [1], [2], [3], [4], [5], [6].", "OAM-carrying beams are characterized by an azimuthal phase dependence of the form $\\exp {(i\\ell \\varphi )}$  [7], where the integer $\\ell $ corresponds to the topological charge of the phase singularity present at the beam centre (e.g.", "see [8] and references therein).", "Such beams include the Laguerre-Gauss (LG) modes [9], which are solutions to the paraxial wave equation.", "It is well known that in (2+1) dimensions, spatial solitons are unstable in homogeneous Kerr media [10].", "One way to increase their stability is to use a saturable self-focusing medium to prevent the catastrophic collapse due to self-focusing.", "By using an intensity dependent nonlinear refractive index, $n_2$ , it becomes possible to balance out the effects of self-focusing and diffraction.", "However, even in the case of saturable self-focusing media, it is known that optical beams carrying OAM will fragment into several solitons possessing particle-like attributes [2], [11].", "In particular, a scalar beam carrying an OAM value of $\\ell $ is predicted to break up into $2\\ell $ daughter solitons [1], [6].", "In comparison with scalar ring solitons that carry a definite non-zero OAM, it has been suggested that stability can be increased by using two beams with opposite OAM to produce a beam with a net zero OAM [4], [5].", "For example, “necklace” (petal) beams, which consist of a scalar superposition of two modes with equal and opposite OAM, have been shown to exhibit quasi-stable propagation in a self-focusing medium although they expand upon propagation [12].", "For vectorial superpositions of beams carrying OAM, however, the resulting vector solitons have been shown theoretically to exhibit quasi-stable propagation for much larger distances than the corresponding scalar vortex solitons and necklace beams [2], [5].", "Vector vortex beams are fully correlated solutions to the vector paraxial wave equation that have space-varying polarization distributions.", "Cylindrical vector (CV) beams are a subclass of vector beams with an axially symmetric polarization profile about the beam's propagation axis [13], [14].", "Examples, include radial, azimuthal and spiral polarization distributions.", "Another class of light beams with non-uniform polarization structure is that of full Poincaré beams [15], [14].", "These are of intrinsic interest because they carry polarization singularities.", "Such beams typically consist of a superposition of two orthogonally polarized LG modes of different orbital angular momenta [15], [14] and thus carry a net value of angular momenta.", "In this Letter, we demonstrate, both experimentally and numerically, the stable propagation of space-varying polarized light beams in a saturable self-focusing nonlinear medium.", "More specifically, our study focuses on vector vortex and Poincaré beams traveling through rubidium vapour.", "We compare the intensity and polarization distributions of the beams at the entrance and exit of the nonlinear cell.", "This allows us to see how beam-break up is affected both by the net OAM of the beam and by its polarization distribution.", "Theory: Light beams with spatially inhomogeneous polarization distributions can be obtained by a superposition of two spatial transverse modes, $E_1$ and $E_2$ , with orthogonal polarizations $\\mathbf {E}(r,\\varphi ,z)=E_1(r,\\varphi ,z)\\,\\mathbf {e}_1+E_2(r,\\varphi ,z)\\,\\mathbf {e}_2,$ where $\\mathbf {e}_1$ and $\\mathbf {e}_2$ are orthonormal polarization vectors, and $r,\\varphi ,z$ are the cylindrical coordinates.", "Here we adopt the circular polarization basis, i.e.", "$\\mathbf {e}_1=\\mathbf {e}_\\text{L}$ and $\\mathbf {e}_2=\\mathbf {e}_\\text{R}$ , and the LG basis for the spatial transverse modes [15].", "If the two beams have equal but opposite OAM, the polarization of the beam varies along the azimuthal coordinate and, if the beams are equally weighted, spans the equator of the Poincaré sphere.", "These radially symmetric vector vortex beams [13] can have polarization distributions that are radial, azimuthal (see Fig.", "REF -(a), (b)) or spiral.", "If $E_1$ and $E_2$ carry a zero and a non-zero OAM value, respectively, the resulting full Poincaré beam [15] has a polarization that varies in both the angular and radial coordinates [14] and covers all polarization states on the Poincaré sphere [15].", "The state of elliptic polarization varies with position [16] and polarization singularities occur at C-points where the azimuth is not defined and the polarization is circular [17], and along L-lines where the polarization is linear and its handedness is not defined [18].", "C-points can have three fundamental polarization topologies classified by the index $\\eta $ and the number of polarization lines that terminate at the singularity: these topologies are known as “star” ($\\eta =-1/2$ , three lines), “lemon” ($\\eta =1/2$ , one line) and “monstar” ($\\eta =1/2$ , infinitely many, with three straight, lines) [19], [16].", "Examples of lemon and star topologies are shown in Fig.REF (c) and (d).", "Figure: (a) Radial and (b) azimuthal vector beams.", "(c) Lemon and (d) star Poincaré beams.", "Red (blue) ellipses correspond to left (right) circular polarization.We simulate propagation through the medium using a two-dimensional nonlinear Schrödinger equation with saturable self-focusing nonlinerity, derivable from the two-level model, under the slowly varying envelope and paraxial approximations and normalized to dimensionless quantities, $\\rho =r/w_0$ and $\\zeta ={z}/({2z_R})$ , where $w_0$ is the beam waist and $z_R$ is the Rayleigh range of the beam [1], [5], [2].", "As we are dealing with vector beams, our model consists of two coupled equations that interact through the cross phase modulation (XPM) term characterized by the parameter $\\nu $ , which takes the value of 2 for circularly polarized beams [20] $\\frac{\\partial E_1}{\\partial \\zeta } = \\frac{i}{2}\\nabla ^2_\\bot E_1 +i\\mu \\; \\frac{|E_1|^2+\\nu \\, |E_2|^2}{1+\\sigma (|E_1|^2+ \\nu \\, |E_2|^2)} E_1, \\\\\\frac{\\partial E_2}{\\partial \\zeta } = \\frac{i}{2}\\nabla ^2_\\bot E_2 +i\\mu \\; \\frac{|E_2|^2+\\nu \\, |E_1|^2}{1+\\sigma (|E_2|^2+ \\nu \\, |E_1|^2)} E_2.", "\\nonumber $ The parameters of importance are the nonlinear parameter, $\\mu $ , and the saturation parameter, $\\sigma $ , given by: $\\mu = \\frac{k_0^2 n_2 P_0}{n_0} \\; ; \\qquad \\quad \\sigma = \\frac{2P_0}{I_{sat} w_0^2} \\, ,$ where $k_0$ is the free-space wavenumber, $n_0$ and $n_2$ are the linear and nonlinear refractive indices ($n_2 > 0$ for self-focusing), $I_{sat}$ is the saturation intensity, and $P_0$ is the power of the incident laser beam.", "In the simulations reported below, we have selected $\\mu =386$ , $\\sigma =51.7$ that reproduce the experimental configuration of the natural rubidium (Rb) cell.", "We performed numerical integrations of the propagation Eqs.", "(REF ), using the split-step method with fast Fourier transforms and parameters corresponding to the experiments performed.", "Figure: Experimental set-up.", "A CW wavelength-tunable (760 nm - 790 nm) Toptica diode laser is coupled to a polarization-maintaining single-mode optical fiber.", "The power of the laser beam is adjusted by means of a half-wave (λ/2\\lambda /2) plate followed by a polarizing beam splitter (PBS) in order to reach the saturation threshold (I sat =5I_\\text{sat}=5 Wcm -2 {}^{-2}) for the D 2 D_2 resonance line of Rb.", "A combination of a λ/2\\lambda /2 and a λ/4\\lambda /4 plate is used to generate an arbitrary polarization state, which is then sent to a qq-plate (QP), with a topical charge of q=±1/2q=\\pm 1/2, to generate vector vortex and Poincaré beams.", "This is then focused (w 0 =60μw_0=60~\\mu m) into a thermally controlled Rb atomic vapour cell by means of a lens (f=150f=150 mm).", "The transmitted beam is analyzed with a combination of a λ/4\\lambda /4 plate, a λ/2\\lambda /2 plate and a PBS (polarization tomography), and attenuated using neutral density filters before being recorded by a CCD camera.Figure: Simulated (upper row) and experimentally reconstructed (lower row) intensity and polarization distributions of scalar and vector superposition beams after propagating through the Rb cell.", "The incident beams are cos(γ)LG 0,-1 (r,ϕ,z)𝐞 L +sin(γ)exp(iβ)LG 0,1 (r,ϕ,z)𝐞 R \\cos {(\\gamma )}\\, \\text{LG}_{0,-1}(r,\\varphi ,z)\\, \\mathbf {e}_\\text{L} + \\sin {(\\gamma )} \\,\\exp {(i \\beta )}\\,\\text{LG}_{0,1}(r,\\varphi ,z)\\,\\mathbf {e}_\\text{R}, with β=-π/2\\beta =-\\pi /2 and γ\\gamma specified at the top.", "Red (blue) ellipses correspond to left (right) circular polarization.Experiment: We use a spatially filtered, linearly polarized, tunable CW single-mode diode laser (Toptica DL pro 780, 760 nm – 790 nm) together with a set of half- and quarter-wave plates to generate a Gaussian beam with an arbitrary polarization state, $\\cos {\\left(\\theta /2\\right)}\\,\\mathbf {e}_\\text{L}+\\exp {(i\\chi )}\\,\\sin {\\left(\\theta /2\\right)}\\,\\mathbf {e}_\\text{R}$ , where $\\theta $ and $\\chi $ are set by the orientation of the waveplates.", "This beam is converted into a space-varying polarized light beam using a $q$ -plate – a slab of patterned liquid crystal – that couples optical spin to OAM [23].", "The unitary action of a $q$ -plate in the circular polarization basis is described by $\\hat{\\text{U}}_q\\cdot \\left[\\begin{matrix}\\mathbf {e}_\\text{L} \\\\\\mathbf {e}_\\text{R} \\\\\\end{matrix} \\right] = \\cos {\\left(\\frac{\\delta }{2} \\right)}\\left[\\begin{matrix}\\mathbf {e}_\\text{L} \\\\\\mathbf {e}_\\text{R} \\\\\\end{matrix} \\right]+ i \\sin {\\left(\\frac{\\delta }{2}\\right)} \\left[\\begin{matrix}\\mathbf {e}_\\text{R} e^{+2 i \\left( q \\varphi + \\alpha _0 \\right)} \\\\\\mathbf {e}_\\text{L} e^{-2 i \\left( q \\varphi + \\alpha _0 \\right)} \\\\\\end{matrix} \\right],$ where $q$ is a half-integer number corresponding to the topological charge of the liquid crystal pattern, $\\alpha _0$ is the azimuthal orientation of the liquid crystal elements at $\\varphi =0$ (laboratory frame) and $\\delta $ is the optical retardation of the $q$ -plate.", "The parameter $\\delta $ can be experimentally adjusted by applying an electric field onto the plate in such a way that the resulting optical retardation corresponds to a half ($\\delta =\\pi $ ) or quarter ($\\delta =\\pi /2$ ) wavelength [24].", "The generated beam is then focused by a 150-mm-focal-length lens into a 9-cm-long cell containing Rb atomic vapour.", "A detailed depiction of this experimental apparatus is provided in Fig.", "REF .", "The intensity of the incident beam and the temperature of the atomic vapour are set so that the medium exhibits saturable Kerr nonlinearities (powers in the vicinity of $7.44$  mW and a temperature of $95^\\circ $ C).", "In order to observe beam breakup of OAM-carrying beams, the laser's output wavelength was tuned near the $D_2$ transition line of Rb.", "Moreover, the laser was blue-detuned by less than $0.7$  GHz from the $5S_{1/2}(F=3) \\rightarrow 5 P_{3/2} (F=4)$ hyperfine transition.", "The beam exiting the Rb cell is then imaged using a lens with a focal length of 200 mm (not shown in the experimental setup) and its polarization distribution is reconstructed using polarization tomography.", "The tomography is performed using an appropriate sequence of a quarter-wave plate, a half-wave plate, a polarizer and a spatially resolving detector (CCD camera) set at an exposure time of $0.1$  s. In order to generate vector vortex beams (radial, azimuthal and spiral), a linearly polarized Gaussian beam is sent to a tuned ($\\delta =\\pi $ ) $q$ -plate of topological charge $q=1/2$  [25].", "Apart from a global phase, the generated beam will be given by $\\left(\\text{LG}_{0,-1}(r,\\varphi ,z)\\, \\mathbf {e}_\\text{L} + \\exp {\\left(i \\beta \\right)}\\,\\text{LG}_{0,1}(r,\\varphi ,z)\\,\\mathbf {e}_\\text{R} \\right)/\\sqrt{2}$ , where $\\beta \\, \\equiv 4 \\alpha _0$ depends on the orientation of the $q$ -plate with respect to the input polarization.", "The generated beams correspond to radial ($\\beta =0$ ), azimuthal ($\\beta =\\pi $ ) and spiral ($\\beta =\\pm \\pi /2$ ) vector vortex beams.", "To generate the lemon and star Poincaré beams, a circularly polarized Gaussian beam is sent to a perfectly detuned ($\\delta =\\pi /2$ ) $q$ -plate with $q=1/2$ and $q=-1/2$ , respectively [26].", "Here, $\\beta =2\\alpha _0$ and does not affect the polarization topology but does cause a rotation in the polarization pattern.", "Thus, we choose $\\beta = 0$ , resulting in an output beam of the form $(\\text{LG}_{0,0}(r,\\varphi ,z)\\, \\mathbf {e}_\\text{L} +\\text{LG}_{0,2q}(r,\\varphi ,z)\\, \\mathbf {e}_\\text{R})/\\sqrt{2}$ , again omitting any global phase factors.", "Note also that monstar topologies cannot be readily generated in the laboratory.", "This scheme allows us to switch between different structured light beams without altering their intensities.", "Analysis: It is well-known that azimuthal modulational instabilities associated with the helical phase structure found in beams carrying OAM result in their filamentation as they propagate through self-focusing media [1], [2], [5].", "It has been shown both analytically and numerically, however, that the dominant low-frequency perturbations that typically disrupt ring solitons are inhibited for vector solitons with no net OAM [2], [5].", "Here we confirm this experimentally and numerically by propagating a beam of the form $\\cos {\\gamma }\\;\\text{LG}_{0,-1}(r,\\varphi ,z)\\; \\mathbf {e}_\\text{L}+\\sin {\\gamma }\\; \\exp {(i\\beta )}\\;\\text{LG}_{0,1}(r,\\varphi ,z)\\;\\mathbf {e}_\\text{R}$ through a self-focusing medium.", "When $\\gamma = 0,\\pi /2$ , the beam is a scalar LG mode of $\\ell =-1 , +1$ with left, right-hand circular polarization.", "For $\\gamma = \\pi /4$ we have a vector vortex beam with linear polarization and for $\\gamma = \\pi /8, 3 \\pi /8$ , a vector vortex beam of left, right-hand elliptical polarization, respectively, following the topology of the vector beam with $\\gamma = \\pi /4$ .", "Note that each beam has the same total intensity.", "A comparison of the experimental results with the simulations based on Eqs.", "(REF ) is shown in Fig.", "REF for a vector beam defined by $\\beta = -\\pi /2$ .", "From our results we can see that the nonlinearity counterbalances diffraction up until the point at which the beams fragment.", "As expected, we see that the scalar OAM-carrying beams ($\\gamma =0, \\, \\pi /2$ ) are starting to break-up at the exit of the nonlinear cell.", "The vector beam ($\\gamma = \\pi /4$ ), on the other hand, seems almost unperturbed, both in terms of its amplitude and polarization distribution.", "Indeed, we can see numerically that the fragmentation point occurs much later for vector beams, $\\sim 16$  cm, than for scalar beams, $\\sim 9$  cm.", "For the elliptically polarized vector beams ($\\gamma = \\pi /8, 3 \\pi /8$ ) the stability length is between the cases of scalar and linear vector vortex beams.", "In this case the effect of the nonlinear propagation is also evident in the change of polarization distributions: the initial spiral distribution has now become almost azimuthal or radial, respectively (see Fig.", "REF ).", "This suggests that vector beams allow for greater control over the nonlinearity.", "In spite of the approximations used in the derivation of Eqs.", "(REF ), small differences in polarization distributions between the experiment and the numerical simulation are only really evident in the biased mode cases of $\\gamma = \\pi /8, 3 \\pi /8$ , thus demonstrating the robustness of the vector vortex beam ($\\gamma = \\pi /4$ ).", "It has been proposed that the increase in stability seen in vector beams is due to the fact that they carry no net OAM [5].", "We therefore repeated our analysis using lemon and star polarization topologies which have a net OAM.", "Figure: Intensity and polarization distributions of lemon and star topologies after propagating through the Rb cell, numerical (upper row) and experimental (lower row).", "Red (blue) ellipses correspond to left (right) circular polarization.The experimental and numerical results are shown in Fig.", "REF where we have plotted the intensity and the polarization distributions for the lemon and star topologies after propagating through the Rb cell.", "These results again show that beam break-up has been inhibited when compared to scalar OAM-carrying beams.", "This result demonstrates that the increased stability is not simply due to a net OAM of zero.", "As in the case of linear vector vortex beams ($\\gamma =\\pi /4$ ) above, the polarization distribution remains unaltered after nonlinear propagation and even preserves the number and kind of polarization singularities.", "Conclusion: We have compared the propagation of scalar OAM-carrying beams with two different classes of beams with non-uniform transverse polarization distributions in a saturable self-focusing nonlinear medium of Rb vapour.", "With respect to scalar vortex beams (LG modes), we found that beam break up can be inhibited while nonlinear confinement, self-focusing and polarization distributions are not altered for specific cases of non-uniform spatial polarization, both with and without net OAM.", "This suggests that the spatial structure of the polarization plays an important role in preventing beam fragmentation.", "These findings provide a novel approach to transport high-power light beams in nonlinear media with controllable distortions to their spatial structure and polarization properties.", "Acknowledgment: F.B., H.L., E.K.", "and R.W.B.", "acknowledge the support of the Canada Excellence Research Chairs (CERC) program.", "A. R. acknowledges funding from the European Union (FP7-PEOPLE-2012-CIG, PCIG12-GA-2012-326499-FOXIDUET).", "E.K.", "acknowledges the support of the Canada Research Chairs (CRC) program and Canada Foundation for Innovation (CFI)." ] ]
1606.05010
[ [ "Leapfrogging vortex rings for the three dimensional Gross-Pitaevskii\n equation" ], [ "Abstract Leapfrogging motion of vortex rings sharing the same axis of symmetry was first predicted by Helmholtz in his famous work on the Euler equation for incompressible fluids.", "Its justification in that framework remains an open question to date.", "In this paper, we rigorously derive the corresponding leapfrogging motion for the axially symmetric three-dimensional Gross-Pitaevskii equation." ], [ "Introduction", "The goal of this paper is to describe a class of cylindrically symmetric solutions to the three-dimensional Gross-Pitaevskii equation $i\\partial _t u - \\Delta u = \\frac{1}{\\varepsilon ^2}u(1-|u|^2)$ for a complex-valued function $u:\\:\\mathbb {R}^3\\times \\mathbb {R}\\rightarrow .", "In the regimewhich we shall describe, it turns out that the Gross-Pitaevskiiequation bears someresemblance with the Euler equation forflowsofincompressible fluids$$\\left\\lbrace \\begin{array}{l}\\partial _t v + (v\\cdot \\nabla ) v = -\\nabla p\\\\{\\rm div}\\, v = 0,\\end{array}\\right.$$where $ v:R3RR3$ is the velocity field and $ p: R3RR$ is the pressure field.", "In this analogy,the role of the velocity $ v$ is played by the current\\footnote {For y\\in and z=(z_1,\\cdots ,z_k) \\in k we write (y,z):=\\big ({\\rm Re}(y\\bar{z}_1),\\cdots ,{\\rm Re}(y\\bar{z}_k)\\big ) \\in \\mathbb {R}^k and y\\times z :=(iy,z).", "}$$j(u) := u \\times \\nabla u = (iu,\\nabla u) = {\\rm Re} (u \\nabla \\bar{u} )$$and the vorticity field $ :=curl  v$ therefore corresponds, up to afactor of two, to the Jacobian$$J(u) := \\frac{1}{2}{\\rm curl}\\, j(u) = \\Big ( \\partial _2 u \\times \\partial _3 u,\\partial _3 u \\times \\partial _2 u, \\partial _1 u \\times \\partial _2u\\Big ).$$$ In his celebrated work [7], [8] on the Euler equation, Helmholtz considered with great attention the situation where the vorticity field $\\omega $ is concentrated in a “circular vortex-filament of very small section”, a thin vortex ring.", "A central question in Helmholtz's work, as far as dynamics is concerned, is related to the possible forms of stability of the family of such vortex rings, allowing a change in time of cross-section, radius, position or even possibly of inner profile, and a description of these evolutions.", "When only one vortex-filament is present, Helmholtz's conclusions are : Hence in a circular vortex-filament of very small section in an indefinitely extended fluid, the center of gravity of the section has, from the commencement, an approximately constant and very great velocity parallel to the axis of the vortex-ring, and this is directed towards the side to which the fluid flows through the ring.", "Instead, when two vortex-filaments interact, Helmholtz predicts the following : We can now see generally how two ring-formed vortex-filaments having the same axis would mutually affect each other, since each, in addition to its proper motion, has that of its elements of fluid as produced by the other.", "If they have the same direction of rotation, they travel in the same direction; the foremost widens and travels more slowly, the pursuer shrinks and travels faster till finally, if their velocities are not too different, it overtakes the first and penetrates it.", "Then the same game goes on in the opposite order, so that the rings pass through each other alternately.", "The motion described by Helmholtz, and illustrated in Figure REF below, is often termed leapfrogging in the fluid mechanics community.", "Even though it has been widely studied since Helmholtz, as far as we know it has not been mathematically justified in the context of the Euler equation, even in the axi-symmetric case without swirlWe refer to [5], [17] for some attempts in that direction, and an account of the difficulties.. As a matter of fact, the interaction leading to the leapfrogging motion is somehow borderline in strength compared to the stability of isolated vortex rings.", "Figure: ©T.T.", "Lim, Phys.", "of Fluids, Vol.", "9Our main results in this paper, Theorem REF and REF below, provide a mathematical justification to the leapfrogging motion of two or more vortex rings in the context of the axi-symmetric three-dimensional Gross-Pitaevskii equation." ], [ "Reference vortex rings", "A well-known particularity of the Gross-Pitaevskii equation is that vortex ring intensities are necessarily quantized.", "For stability reasons, we only consider simply quantized rings.", "Let $\\mathcal {C}$ be a smooth oriented closed curve in $\\mathbb {R}^3$ and let $\\vec{\\mathcal {J}}$ be the vector distribution corresponding to $2\\pi $ times the circulation along $\\mathcal {C}$ , namely $\\langle \\vec{\\mathcal {J}},\\vec{X}\\rangle = 2\\pi \\int _{\\mathcal {C}} \\vec{X} \\cdot \\vec{\\tau }\\qquad \\forall \\vec{X} \\in \\mathcal {D}(\\mathbb {R}^3,\\mathbb {R}^3),$ where $\\vec{\\tau }$ is the tangent vector to $\\mathcal {C}.$ To the “current density” $\\vec{\\mathcal {J}}$ is associated the “induction” $\\vec{B}$ , which satisfies the equations ${\\rm div }(\\vec{B})=0,\\qquad {\\rm curl} (\\vec{B}) = \\vec{\\mathcal {J}} \\qquad \\text{in }\\mathbb {R}^3,$ and is obtained from $\\vec{\\mathcal {J}}$ by the Biot-Savart law.", "To $\\vec{B}$ is then associated a vector potential $\\vec{A}$ , which satisfies ${\\rm div }(\\vec{A})=0,\\qquad {\\rm curl }(\\vec{A}) = \\vec{B} \\qquad \\text{in } \\mathbb {R}^3,$ so that $-\\Delta \\vec{A} = {\\rm curl}\\, {\\rm curl} (\\vec{A}) = \\vec{\\mathcal {J}} \\qquad \\text{in} \\mathbb {R}^3.$ Since we only consider axi-symmetric configurations in this paper, we let $\\mathbb {H}$ to be the half-space $\\lbrace (r,z)\\ | \\ r>0 , z \\in \\mathbb {R}\\rbrace $ and we denote by $r(\\cdot )$ and $z(\\cdot )$ the coordinate functions in $\\mathbb {H}.$ For $a\\in \\mathbb {H}$ , let $\\mathcal {C}_a$ be the circle of radius $r(a)$ parallel to the $xy$ -plane in $\\mathbb {R}^3$ , centered at the point $(0,0,z(a))$ , and oriented so that its binormal vector points towards the positive $z$ -axis.", "By cylindrical symmetry, we may write the corresponding vector potential as $\\vec{A}_a \\equiv A_a(r,z) \\vec{e}_\\theta .$ The expression of the vector Laplacian in cylindrical coordinates yields the equation for the scalar function $A_a$ : $\\left\\lbrace \\begin{array}{ll}\\displaystyle -\\left(\\partial ^2_r +\\frac{1}{r}\\partial _r - \\frac{1}{r^2} +\\partial ^2_z\\right) A_a = 2\\pi \\delta _{a} &\\qquad \\text{in } \\mathbb {H}\\\\A_a = 0 & \\qquad \\text{on } \\partial \\mathbb {H},\\end{array}\\right.$ or equivalently $\\left\\lbrace \\begin{array}{ll}\\displaystyle -{\\rm div} \\left(\\frac{1}{r} \\nabla \\left( r A_a\\right)\\right) = 2\\pi \\delta _{a} &\\qquad \\text{in } \\mathbb {H}\\\\A_a = 0 & \\qquad \\text{on } \\partial \\mathbb {H},\\end{array}\\right.$ which can be integrated explicitly in terms of complete elliptic integralsThe integration is actually simpler in the original cartesian coordinates.", "A classical reference is the book of Jackson [10], an extended analysis can be found in the 1893 paper of Dyson [6].", "See Appendix A for some details.. Up to a constant phase factor, there exists a unique unimodular map $u^*_a\\in \\mathcal {C}^\\infty (\\mathbb {H}\\setminus \\lbrace a\\rbrace ,S^1)\\cap W^{1,1}_{\\rm loc}(\\mathbb {H},S^1)$ such that $r (iu^*_a,\\nabla u^*_a) = rj(u^*_a) = -\\nabla ^\\perp (rA_a).$ In the sense of distributions in $\\mathbb {H}$ , we have $\\left\\lbrace \\begin{array}{ll}\\displaystyle {\\rm div}(rj(u^*_a)) & = 0\\\\\\displaystyle {\\rm curl}(j(u^*_a)) & = 2\\pi \\delta _{a},\\end{array}\\right.$ and the function $u^*_a$ corresponds therefore to a singular vortex ring.", "In order to describe a reference vortex ring for the Gross-Pitaevskii equation, we shall make the notion of core more precise.", "In $\\mathbb {R}^2$ , the Gross-Pitaevskii equation possesses a distinguished stationary solution called vortex : in polar coordinates, it has the special form $u_{\\varepsilon }(r,\\theta ) = f_{\\varepsilon }(r)\\exp (i\\theta )$ where the profile $f_{\\varepsilon } : \\mathbb {R}^+ \\rightarrow [0,1]$ satisfies $f_{\\varepsilon }(0)=0$ , $f_{\\varepsilon }(+\\infty )=1,$ and $\\partial _{rr}f_{\\varepsilon } + \\frac{1}{r}\\partial _r f_{\\varepsilon } -\\frac{1}{r^2}f_{ \\varepsilon } +\\frac{1}{\\varepsilon ^2} f_{\\varepsilon } (1-f_{\\varepsilon }^2) = 0.$ Notice that $\\varepsilon $ has the dimension of a length, and since by scaling $f_{\\varepsilon } (r)=f_{1}(\\tfrac{r}{\\varepsilon })$ it is the characteristic length of the core.", "The reference vortex ring associated to the point $a \\in \\mathbb {H}$ is defined to be $u_{\\varepsilon ,a}^*(r,z) = f_\\varepsilon \\big (\\Vert (r,z)-a\\Vert \\big ) u^*_a(r,z).$ More generally, when $a = \\lbrace a_1,\\cdots ,a_n\\rbrace $ is a family of $n$ distinct points in $\\mathbb {H}$ , we set $u_{a}^*(r,z) := \\prod _{k=1}^n u^*_{a_k}(r,z),\\quad \\text{and}\\quad u_{\\varepsilon ,a}^*(r,z) := \\prod _{k=1}^n u^*_{\\varepsilon ,a_k}(r,z),$ where the products are meant in $$ The field $u_{\\varepsilon ,a}^*$ hence corresponds to a collection of $n$ reference vortex rings (sharing the same axis and oriented in the same direction), and is the typical kind of object which we shall study the evolution of.", "It can be shown that $\\Big \\Vert J u_{\\varepsilon ,a}^* - \\pi \\sum _{i=1}^n\\delta _{a_{i}}\\Big \\Vert _{\\dot{W}^{-1,1}(\\mathbb {H})} = \\ O(\\varepsilon )\\quad \\text{as } \\varepsilon \\rightarrow 0,$ where here and in the sequel, for a complex function $u$ on $\\mathbb {H}$ we denote by $Ju$ its jacobian function $Ju= \\partial _ru \\times \\partial _z u.$" ], [ "The system of leapfrogging", "Being an exact collection of (or even a single) reference vortex rings is not a property which is preserved by the flow of the Gross-Pitaevskii equationExact traveling waves having the form of vortex rings have been constructed in [4], these are very similar in shape but not exactly equal to reference vortex rings.. To carry out our analysis, we rely mainly on the energy density and the current density.", "For cylindrically symmetric solutions $u\\equiv u(r,z,t)$ , the Gross-Pitaevskii equation writes $\\left\\lbrace \\begin{array}{lll}\\displaystyle ir\\partial _t u - {\\rm div}(r\\nabla u) =\\frac{1}{\\varepsilon ^2}ru(1-|u|^2) &\\text{in } &\\mathbb {H}\\times \\mathbb {R},\\\\\\displaystyle \\partial _r u = 0 & \\text{on }&\\partial \\mathbb {H}\\times \\mathbb {R}.\\end{array}\\right.\\qquad \\mathrm {{(\\text{GP})_\\varepsilon ^{c}}}$ Equation $(\\text{GP})_\\varepsilon ^{c}$ is an hamiltonian flow for the (weighted) Ginzburg-Landau energy ${\\mathcal {E}}_\\varepsilon ^{\\rm w}(u) := \\int _{\\mathbb {H}} \\left( \\frac{|\\nabla u|^2}{2} +\\frac{(1-|u|^2)}{4\\varepsilon ^2}\\right) r\\, dr dz,$ and the Cauchy problem is known to be well-posed for initial data with finite energy.", "Classical computations leads to the estimate : Lemma 1 It holds ${\\mathcal {E}}_\\varepsilon ^{\\rm w}\\big (u^*_{\\varepsilon ,a}\\big ) = \\sum _{i=1}^n r(a_i) \\Big [ \\pi \\log \\big (\\tfrac{r(a_i)}{\\varepsilon }\\big ) + \\gamma + \\pi \\big (3\\log (2)-2\\big )+ \\pi \\sum _{j\\ne i}A_{a_j}(a_i) +O\\big ((\\tfrac{\\varepsilon }{\\rho _a})^{\\tfrac{2}{3}}\\log ^2(\\tfrac{\\varepsilon }{\\rho _a})\\big )\\Big ],$ where $\\rho _a :=\\frac{1}{4}\\min \\Big ( \\min _{i\\ne j}|a_i-a_j|, \\min _ir(a_i)\\Big ).$ In Lemma REF , the constant $\\gamma $ is defined by (see [2]) $\\gamma := \\liminf _{\\varepsilon \\rightarrow 0} \\Big [ {\\mathcal {E}}_{\\varepsilon }(v_\\varepsilon ,B_1) - \\pi |\\!\\log \\varepsilon |\\Big ] \\text{ with } v_\\varepsilon \\in H^1(B_1, \\text{ and } v_\\varepsilon (z)=z \\text{ on} \\partial B_1,$ where $B_1$ is the unit disk in $\\mathbb {R}^2$ and where for an open subset $\\Omega \\subset \\mathbb {R}^2$ and $u \\in H^1_{\\rm loc}(\\Omega ,$ we denote the unweighted two-dimensional Ginzburg-Landau energy of $u$ in $\\Omega $ by ${\\mathcal {E}}_{\\varepsilon }(u,\\Omega ) =\\int _\\Omega e_\\varepsilon (u) d\\mathcal {L}^2 := \\int _{\\Omega } \\left(\\frac{|\\nabla u|^2}{2} +\\frac{(1-|u|^2)}{4\\varepsilon ^2}\\right) d\\mathcal {L}^2.$ In light of Lemma REF , we define the quantity $H_\\varepsilon (a_1,\\cdots ,a_n) := \\sum _{i=1}^n r(a_i) \\Big [ \\pi \\log \\big (\\tfrac{r(a_i)}{\\varepsilon }\\big ) + \\gamma + \\pi \\big (3\\log (2)-2\\big )+ \\pi \\sum _{j\\ne i}A_{a_j}(a_i)\\Big ],$ and we consider the associated hamiltonian system $\\hspace{42.67912pt} \\dot{a}_i(s) = \\ \\frac{1}{\\pi |\\!\\log \\varepsilon |}\\mathbb {J}\\nabla _{a_i}H_\\varepsilon \\big (a_1(s),\\cdots ,a_n(s)\\big ), \\hspace{71.13188pt}{i=1,\\cdots ,n,} \\qquad \\mathrm {{(\\text{LF})_\\varepsilon }}$ where, with a slight abuse of notation, $\\mathbb {J} := \\begin{pmatrix}0 & -\\frac{1}{r(a_i)} \\\\\\frac{1}{r(a_i)} & 0\\end{pmatrix}.$ In addition to the hamiltonian $H_\\varepsilon ,$ the system $(\\text{LF})_\\varepsilon $ also conserves the momentum $P(a_1,\\cdots ,a_n) := \\pi \\sum _{k=1}^n r^2(a_k),$ which may be interpreted as the total area of the disks determined by the vortex rings.", "As a matter of fact, note also that $\\mathcal {P}(u_{\\varepsilon ,a}^*) := \\int _\\mathbb {H}Ju_{\\varepsilon ,a}^*\\,r\\,drdz = \\pi \\sum _{k=1}^n r^2(a_k) + o(1),$ as $\\varepsilon \\rightarrow 0,$ and that, at least formally, the momentum $\\mathcal {P}$ is a conserved quantity for $(\\text{GP})_\\varepsilon ^{c}$ .", "When $n=2$ , the system $(\\text{LF})_\\varepsilon $ may be analyzed in great details.", "Since $P$ is conserved and since $H_\\varepsilon $ is invariant by a joint translation of both rings in the $z$ direction, it is classical to introduce the variables $(\\eta ,\\xi )$ by $\\left\\lbrace \\begin{array}{l}r^2(a_1)=\\frac{P}{2} - \\eta \\\\r^2(a_2)=\\frac{P}{2} + \\eta \\end{array}\\right.,\\qquad \\xi =z(a_1)-z(a_2),$ and to draw the level curves of the function $H_\\varepsilon $ in those two real variables, the momentum $P$ being considered as a parameter.", "The next figure illustrates the global behavior of the phase portrait, with three distinct regions which we have called “pass through”, “attract then repel” and “leapfrogging”.", "The leapfrogging region corresponds to the central part, where all solutions are periodic in time; its interpretation was discussed earlier in this introduction.", "In the pass through region, the first vortex ring always remains the smallest, hence quickest, of the two vortex rings : being initially located below the second vortex ring on the z-axis it first catches up, then passes inside the second and finally gets away in front of itA similar situation is described by Hicks [9] for a simplified vortex model introduced by Love [16] in 1894..", "Instead, in the attract then repel region the first vortex ring initially starts to catch up, but doing so its circular radius increases whereas the one of the second vortex ring decreases, up to a point where both vortex rings have the same radius and the first still lag behind the second.", "From that point on, the first one has a larger radius than the second, and therefore the second increases its lead indefinitely.", "The behavior in those last two regions is actually very much reminiscent of two-solitons interactions in the Korteweg - de Vries equation, in particular the speeds at plus and minus infinity in time are equal or exchanged.", "Notice also that the two points at the common boundary of the three regions correspond, up to labeling, to the same situation : two vortex rings travel with the same constant speed at a special mutual distanceWe stress that this holds at the level of the system $(\\text{LF})_\\varepsilon $ , we do not know whether such special solutions exist at the level of equation $(\\text{GP})_\\varepsilon ^{c}$ ..", "Figure: Phase portrait of the system (LF) ε (\\text{LF})_\\varepsilon for two vortex ringsThe typical size of the leapfrogging region is also described in the figure.", "In particular, it shrinks and becomes more flat as $\\varepsilon $ decreases towards zero." ], [ "Statement of the main results", "We present two results in this section.", "The first one follows rather easily from the second, but its statement has the advantage of being somewhat simpler.", "On the other hand, it involves a limiting procedure $\\varepsilon \\rightarrow 0,$ whereas the second one is valid for small but fixed values of $\\varepsilon .$ In order to state those results, and in view of the size of the leapfrogging region mentioned at the end of the previous subsection, we fix some $(r_0,z_0)\\in \\mathbb {H}$ , an integer $n\\ge 1$ , and $n$ distinct points $b_1^0,\\cdots ,b_n^0$ in $\\mathbb {R}^2$ .", "The initial positions of the cores of the vortex rings are then set to be $a_{i,\\varepsilon }^0 := \\Big (r_0+ \\frac{r(b_i^0)}{\\sqrt{|\\!\\log \\varepsilon |}} ,z_0 + \\frac{z(b_i^0)}{\\sqrt{|\\!\\log \\varepsilon |}} \\Big ),\\qquad i=1,\\cdots ,n.$ As a matter of fact, this is the appropriate scaling for which relative self-motion and interactions between vortex-rings are of the same magnitude.", "In any scaling in which $a^0_{i,\\varepsilon } - (r_0,z_0) = o(1)$ as $\\varepsilon \\rightarrow 0$ for all i, the “leading-order” vortex motion is expected to be a translation with constant velocity $1/r_0$ in the vertical direction and in the rescaled time.", "The above scaling is the appropriate one for which, in the next-order correction, the difference in the self-motion speeds (due to different values of the radii at the next order) and interaction between vortices are of the same magnitude.", "In the case of two vortices, for example, this will give rise to small-scale periodic corrections to a leading-order translation, which is the signature of \"leapfrogging\".", "Note that $a_{i,\\varepsilon }^0 \\in \\mathbb {H}$ provided $\\varepsilon $ is sufficiently small, which we assume throughout.", "Concerning their evolution, we consider the solution to the Cauchy problem for the system of ordinary differential equations $\\qquad \\quad \\left\\lbrace \\begin{array}{rl}\\displaystyle \\dot{b}_i(s) &= \\ \\sum _{j\\ne i}\\frac{(b_i(s)-b_j(s))^\\perp }{\\Vert b_i(s)-b_j(s)\\Vert ^2} -\\frac{r(b_i(s))}{r_0^2}\\begin{pmatrix}0\\\\1\\end{pmatrix}\\\\\\displaystyle b_i(0) &= \\ b_i^0\\end{array}\\right.", "\\qquad \\qquad i=1,\\cdots ,n,\\qquad \\mathrm {{(\\text{LF})}}$ and we finally set $a_{i,\\varepsilon }(s) := \\Big (r_0+ \\frac{r(b_i(s))}{\\sqrt{|\\!\\log \\varepsilon |}} ,z_0 + \\frac{s}{r_0} + \\frac{z(b_i(s))}{\\sqrt{|\\!\\log \\varepsilon |}} \\Big ).$ System (LF) and (REF ) describe the main order asymptotic of $(\\text{LF})_\\varepsilon $ in the leapfrogging region, after a proper rescaling in time.", "We will prove Theorem 1 Let $(u_\\varepsilon ^0)_{\\varepsilon >0}$ be a family of initial data for $(\\text{GP})_\\varepsilon ^{c}$ such that $\\Big \\Vert J u_\\varepsilon ^0 - \\pi \\sum _{i=1}^n\\delta _{a_{i,\\varepsilon }^0}\\Big \\Vert _{\\dot{W}^{-1,1}(\\Omega )} = \\ o\\Big (\\frac{1}{|\\!\\log \\varepsilon |}\\Big ),$ as $\\varepsilon \\rightarrow 0$ , for any open subset $\\Omega $ strongly included in $\\mathbb {H}$ .", "Assume also that ${\\mathcal {E}}_\\varepsilon ^{\\rm w}(u_\\varepsilon ^0) \\le H_\\varepsilon (a_{1,\\varepsilon }^0,\\cdots ,a_{n,\\varepsilon }^0) +o(1), \\text{ as } \\varepsilon \\rightarrow 0.$ Then, for every $s \\in \\mathbb {R}$ and every open subset $\\Omega $ strongly included in $\\mathbb {H}$ we have $\\Big \\Vert J u_\\varepsilon ^s - \\pi \\sum _{i=1}^n\\delta _{ a_{i,\\varepsilon }(s)}\\Big \\Vert _{\\dot{W}^{-1,1}(\\Omega )} =o\\Big (\\frac{1}{\\sqrt{|\\!\\log \\varepsilon |}}\\Big )$ where we denote by $u_\\varepsilon ^s$ the solution of $(\\text{GP})_\\varepsilon ^{c}$ with initial datum $u_\\varepsilon ^0$ and evaluated at time $t=s/|\\!\\log \\varepsilon |,$ and where the points $a_{i,\\varepsilon }(s)$ are defined in (REF ) through the solution of the system (LF).", "In the statement of Theorem REF , the $\\dot{W}^{-1,1}$ norm is defined by $\\Vert \\mu \\Vert _{\\dot{W}^{-1,1}(\\Omega )} = {\\rm sup}\\left\\lbrace \\int \\varphi \\, d\\mu , \\ \\varphi \\in W^{1,\\infty }_0(\\Omega ),\\ \\Vert \\nabla \\varphi \\Vert _\\infty \\le 1\\right\\rbrace .$ Remark 1 Asymptotic formulas for the potential vectors $A_{a_i}$ (see Appendix A) lead to the equivalence $H_\\varepsilon (a_{1,\\varepsilon },\\cdots ,a_{n,\\varepsilon }) = \\Gamma _\\varepsilon (r_0,n) +W_{\\varepsilon ,r_0}(b_1,\\cdots ,b_n) + o(1) \\text{ as } \\varepsilon \\rightarrow 0,$ where $\\Gamma _\\varepsilon (r_0,n) = nr_0( \\pi |\\log \\varepsilon | + \\gamma + \\pi n\\log r_0 + \\pi n(3\\log 2-2) + \\pi \\tfrac{n-1}{2}\\log |\\log \\varepsilon |)$ and $W_{\\varepsilon ,r_0}(b_1,\\cdots ,b_n) = \\pi \\sum _{i=1}^n r(b_i) \\sqrt{|\\!\\log \\varepsilon |} - \\pi r_0\\sum _{i\\ne j}\\log |b_i-b_j|.$ Also, expansion of the squares leads directly to $P(a_{1,\\varepsilon },\\cdots ,a_{n,\\varepsilon }) = \\pi n r_0^2 + 2\\pi r_0\\sum _{i=1}^n\\frac{r(b_i)}{\\sqrt{|\\!\\log \\varepsilon |}} + \\pi \\sum _{i=1}^n \\frac{r(b_i)^2}{|\\!\\log \\varepsilon |},$ and therefore $\\Big (H_\\varepsilon - \\frac{|\\!\\log \\varepsilon |}{2r_0}P\\Big )(a_{1,\\varepsilon },\\cdots ,a_{n,\\varepsilon }) =-\\frac{\\pi }{2} nr_0 |\\!\\log \\varepsilon |+\\Gamma _\\varepsilon (r_0,n)+ \\pi r_0 W(b_1,\\cdots ,b_n) + o(1),\\quad \\text{as }\\varepsilon \\rightarrow 0,$ where $W(b_1,\\cdots ,b_n) := -\\sum _{i\\ne j}\\log |b_i-b_j|- \\frac{1}{2r_0^2} \\sum _{i=1}^n r(b_i)^2.$ The function $W$ , which does not depend upon $\\varepsilon $ , is precisely the hamiltonian for the system (LF).", "A second quantity preserved by (LF) is given by $Q(b_1,\\cdots ,b_n):=\\sum _{i=1}^n r(b_i).$ When $n=2$ , all the solutions are (LF) are periodic in time.", "We will now state a quantitative version of Theorem REF which holds for small but fixed values of $\\varepsilon ,$ not just asymptotically as $\\varepsilon \\rightarrow 0.$ We fix positive constants $K_0$ and $r_0$ and we consider an arbitrary solution $a_\\varepsilon (s)\\equiv \\lbrace a_{i,\\varepsilon }(s)\\rbrace _{1\\le i\\le n}$ of the system $(\\text{LF})_\\varepsilon $ on some time interval $[0,S_0]$ , $S_0\\ge 0,$ which we assume to satisfy $\\begin{array}{c}\\displaystyle \\frac{K_0^{-1} }{\\sqrt{|\\!\\log \\varepsilon |}}\\le \\min _{s\\in [0,S_0]}\\min _{i\\ne j}|a_{i,\\varepsilon }(s)-a_{j,\\varepsilon }(s)|\\le \\max _{s\\in [0,S_0]}\\max _{i\\ne j} |a_{i,\\varepsilon }(s)-a_{j,\\varepsilon }(s)| \\le \\frac{K_0}{\\sqrt{|\\!\\log \\varepsilon |}}\\\\\\displaystyle \\frac{r_0}{2} \\le \\min _{s\\in [0,S_0]}\\min _{i} r(a_{i,\\varepsilon }(s))\\le \\max _{s\\in [0,S_0]}\\max _{i} r(a_{i,\\varepsilon }(s))\\le 2r_0.\\end{array}$ We define the localization scale $r_a^0 := \\Big \\Vert J u_\\varepsilon ^0 - \\pi \\sum _{i=1}^n\\delta _{a_{i,\\varepsilon }^0}\\Big \\Vert _{\\dot{W}^{-1,1}(\\Omega _0)},$ where $\\Omega _0:= \\lbrace r\\ge \\frac{r_0}{4}\\rbrace $ , and the excess energy $\\Sigma ^0 := \\left[{\\mathcal {E}}_\\varepsilon ^{\\rm w}(u_\\varepsilon ^0) -H_{\\varepsilon }(a_{1,\\varepsilon }^0,\\cdots ,a_{n,\\varepsilon }^0)\\right]^+$ at the initial time.", "Theorem 2 Let $a_\\varepsilon (s)\\equiv \\lbrace a_{i,\\varepsilon }(s)\\rbrace _{1\\le i\\le n}$ be a solution of the system $(\\text{LF})_\\varepsilon $ on some time interval $[0,S_0]$ , $S_0\\ge 0,$ which satisfies (REF ).", "There exist positive numbers $\\varepsilon _0$ , $\\sigma _0$ and $C_0$ , depending only on $r_0$ , $n$ , $K_0$ and $S_0$ with the following properties.", "Assume that $0 < \\varepsilon \\le \\varepsilon _0$ and that $r_a^0|\\!\\log \\varepsilon |+ \\Sigma ^0 \\le \\sigma _0,$ then $\\Big \\Vert J u_\\varepsilon ^s - \\pi \\sum _{i=1}^n\\delta _{a_{i,\\varepsilon }(s)}\\Big \\Vert _{\\dot{W}^{-1,1}(\\Omega _0)} \\le C_0\\left( r_a0 + \\frac{\\Sigma ^0}{\\sqrt{|\\!\\log \\varepsilon |}}+ \\frac{C_\\delta }{|\\!\\log \\varepsilon |^{1-\\delta }}\\right)e^{C_0s},$ for every $s \\in [0,S_0],$ where $\\delta >0$ can be chosen arbitrarily small.", "To finish this introduction, let us mention that we have not analyzed the convergence of $(\\text{GP})_\\varepsilon ^{c}$ towards $(\\text{LF})_\\varepsilon $ in the “pass through” and “attract then repel” regions.", "It is conceivable, yet probably difficult, to obtain closeness estimates valid for all times in those cases, reminiscent of what is sometimes called orbital stability of multi-solitons, e.g.", "in the Korteweg - de Vries equation [18] or the 1D Gross-Pitaevskii equation [3].", "One would have to deal with algebraic rather than exponential interaction estimates.", "Also, having in mind the initial question related to the Euler equation, let us mention that one crucial advantage in the analysis of the Gross-Pitaevskii equation is that it has an inherent core localization scale $\\varepsilon .$ On the other hand, Euler velocity fields are divergence free, whereas Gross-Pitaevskii ones only have small divergence when averaged in time.", "Analysis of leapfrogging for the Euler equation would therefore probably require a different strategy.", "Acknowledgements.", "The research of RLJ was partially supported by the National Science and Engineering Council of Canada under operating grant 261955.", "The research of DS was partially supported by the Agence Nationale de la Recherche through the project ANR-14-CE25-0009-01." ], [ "Strategy for the proofs", "The overall strategy follows many of the lines which we adopted in our prior work [11] on the inhomogeneous Gross-Pitaevskii equationAnother work on the 2D inhomogeneous GP equation is a recent preprint of Kurzke et al [15], which studies a situation where the inhomogeneity and its derivatives are of order $|\\!\\log \\varepsilon |^{-1}.$ This is critical in the sense that interaction of vortices with the background potential and with each other are of the same order of magnitude.", "In the present work, by contrast, critical coupling occurs in hard-to-resolve corrections to the leading-order dynamics.", "The effort is actually focused on Theorem REF first, Theorem REF can be deduced from it rather directly.", "The essential new ingredients with respect to [11] are refined approximation estimates (mainly Proposition REF ) and the key observation in Proposition REF ." ], [ "Localisation, excess energy and approximation by a reference field", "In this section we present arguments which are not directly related to the time evolution but only to some assumptions on the energy density and on the Jacobian of a function $u$ .", "In rough terms, we assume that $u$ is known a priori to satisfy some localisation estimates and some energy upper bounds, and we will show, by combining them together, that under a certain approximation threshold this can be improved by a large amount, without any further assumption.", "In order to state quantitative results, we assume here that $\\lbrace a_i\\rbrace _{1\\le i \\le n}$ is a collection of points in $\\mathbb {H}$ such that $\\begin{array}{c}\\displaystyle \\frac{8}{|\\!\\log \\varepsilon |} \\le \\min _{i\\ne j} |a_{i}-a_{j}|\\le \\max _{i\\ne j} |a_{i}-a_{j}| \\le K_1\\\\\\displaystyle \\frac{r_0}{2} \\le r(a_{i})\\le \\max _{i} r(a_{i})\\le 2r_0.\\end{array}\\qquad \\mathrm {({H_1})}$ We assume next that $u \\in H^1_{\\rm loc}(\\mathbb {H},$ is such that its Jacobian $Ju$ satisfies the rough localisation estimate $r_a := \\Vert Ju-\\pi \\sum _{i=1}^n \\delta _{a_i}\\Vert _{\\dot{W}^{-1,1}(\\Omega _0)}< \\frac{\\rho _a}{4},$ where $\\rho _a$ is defined in (REF ).", "We finally define the excess energy relative to those points, $\\Sigma _a:= \\left[ {\\mathcal {E}}_\\varepsilon ^{\\rm w}(u)-H_\\varepsilon (a_1,\\cdots ,a_n)\\right]^+.$ We will show that if $r_a$ and $\\Sigma _a$ are not too large then actually a much better form of localisation holds.", "Proposition 1 Under the assumption $(H_1)$ and (REF ), there exist constants $\\varepsilon _1,\\sigma _1,C_1>0,$ depending only on $n$ , $r_0$ and $K_1$ , with the following properties.", "If $\\varepsilon \\le \\varepsilon _1$ and $\\Sigma _a^r := \\Sigma _a +r_a|\\!\\log \\varepsilon |\\le \\sigma _1|\\!\\log \\varepsilon |,$ then there exist $\\xi _1,\\cdots ,\\xi _n$ in $\\mathbb {H}$ such that $\\Vert Ju-\\pi \\sum _{i=1}^n \\delta _{\\xi _i}\\Vert _{\\dot{W}^{-1,1}\\left(\\left\\lbrace r\\ge C_1(\\Sigma _\\xi +1)/|\\!\\log \\varepsilon |\\right\\rbrace \\right)} \\le C_1 \\varepsilon |\\!\\log \\varepsilon |^{C_1}e^{C_1\\Sigma _a^r},$ and $\\int _{\\mathbb {H}\\setminus \\cup _i B(\\xi _i,\\varepsilon ^\\frac{2}{3})} \\!\\!r \\left[ e_\\varepsilon (|u|) +\\big |\\frac{j(u)}{|u|}-j(u_\\xi ^*)\\big |^2\\right] \\le C_1\\big (\\Sigma _\\xi +\\varepsilon ^\\frac{1}{3}|\\!\\log \\varepsilon |^{C_1}e^{C_1\\Sigma _a^r }\\big ),$ where we have written $\\Sigma _\\xi := \\left[ {\\mathcal {E}}_\\varepsilon ^{\\rm w}(u)-H_\\varepsilon (\\xi _1,\\cdots ,\\xi _n)\\right]^+.$ Moreover, $\\Sigma _\\xi \\le \\Sigma _a + C_1r_a|\\!\\log \\varepsilon |+ C_1 \\varepsilon |\\!\\log \\varepsilon |^{C_1}e^{C_1\\Sigma _a^r},$ and the values of $\\varepsilon _1$ and $\\sigma _1$ are chosen sufficiently small so that $C_1|\\!\\log \\varepsilon |^{C_1}e^{C_1\\sigma _1|\\!\\log \\varepsilon |} \\le \\varepsilon ^{-\\frac{1}{6}},\\qquad \\text{and}\\qquad C_1(\\Sigma _\\xi +1)/|\\!\\log \\varepsilon |\\le \\frac{r_0}{4}$ whenever $\\varepsilon \\le \\varepsilon _1.$ Remark 2 It is tempting to simplify somewhat the statement of Proposition REF by replacing the term $\\Sigma _\\xi $ in the right-hand side of (REF ) by $\\Sigma _a^r$ (in view of (REF ) this would be correct up to a possible change of $C_1$ ), and hence obtain error bounds that only depend on the input data.", "Yet, it turns out that (REF ) is not optimal in all cases and the key step of our subsequent analysis will make use of that difference.", "We will now focus on estimates that are valid up to and including the cores.", "By definition (see Appendix REF ), we have $r j(u_\\xi ^*) = -\\nabla ^\\perp \\big (r\\Psi ^*_\\xi \\big ).$ Since the latter is singular at the points $a_i$ and not in $L^2_{\\rm loc}$ , there is no hope that estimate (REF ) in Proposition REF could be extended to the whole of $\\mathbb {H}.$ For that purpose, we have to replace $j(u_\\xi ^*)$ by some mollified version.", "The function $j(u^*_{\\xi ,\\varepsilon })$ would be a natural candidate, but that would require that the vortex locations $\\xi _i$ are known to a precision at least as good as $\\varepsilon ,$ which is not the case in view of (REF ).", "For that reason, instead we modify the function $\\Psi _\\xi ^*$ to a function $\\Psi _\\xi ^\\natural $ in the following way (truncate $r\\Psi _\\xi ^*$ ): We write $r_\\xi := C_1 \\varepsilon |\\!\\log \\varepsilon |^{C_1}e^{C_1\\Sigma _a^r}$ and for each $i=1,\\cdots ,n$ we consider the connected component $\\mathcal {C}_i$ of the superlevel set $\\lbrace r\\Psi _\\xi ^* \\ge r\\Psi ^*_\\xi (\\xi _i+(r_\\xi ,0))\\rbrace $ (by convention we include $\\xi _i$ , where $\\Psi ^*_\\xi $ is in principle not defined, in this set) which contains the point $\\xi _i+(r_\\xi ,0)$ , and we set $r\\Psi ^\\natural _\\xi = r\\Psi ^*_\\xi (\\xi _i+(r_\\xi ,0))$ inside $\\mathcal {C}_i$ .", "Next, we set $\\Psi ^\\natural _\\xi =\\Psi ^*_\\xi $ on $\\mathbb {H}\\setminus \\cup _{i=1}^n \\mathcal {C}_i$ and finally we define $r j^\\natural (u_\\xi ^*) = -\\nabla ^\\perp \\big (r\\Psi ^\\natural _\\xi \\big ).$ Remark 3 Note that by construction $j^\\natural (u_\\xi ^*)$ and $j(u_\\xi ^*)$ coincide everywhere outside $\\cup _i \\mathcal {C}_i$ , that is everywhere except on a neighborhood of order $r_\\xi $ of the points $\\xi _i,$ and that $j^\\natural (u_\\xi ^*)\\equiv 0$ inside each $\\mathcal {C}_i.$ In the sense of distributions, ${\\rm div}(rj^\\natural (u_\\xi ^*)) = 0$ and ${\\rm curl}(j^\\natural (u_\\xi ^*)) =\\sum _{i=1}^n|j(u_\\xi ^*)|d\\mathcal {H}^1\\hspace{1.0pt}{\\hbox{t}o 10.8pt{\\hfill \\vrule height 7pt width 0.4pt depth 0pt\\hbox{\\vrule height 0.4ptwidth 7.6pt depth 0pt}\\hfill }}_{\\partial \\mathcal {C}_i}.$ Proposition 2 In addition to the statements of Proposition REF , there exists $\\varepsilon _2 \\le \\varepsilon _1$ such that if $\\varepsilon \\le \\varepsilon _2$ then we have $\\int _{\\mathbb {H}} \\!\\!r \\left[ e_\\varepsilon (|u|) +\\big |\\frac{j(u)}{|u|}-j^\\natural (u_\\xi ^*)\\big |^2\\right] \\le C_2\\big (\\Sigma _a^r + \\log |\\!\\log \\varepsilon |\\big ),$ where $C_2$ depends only on $n$ , $K_1$ and $r_0.$ The term $\\log |\\!\\log \\varepsilon |$ is not small and even diverging as $\\varepsilon \\rightarrow 0,$ but since the main order for the energy in the core region is of size $|\\!\\log \\varepsilon |$ that estimate will be sufficient for our needs.", "Away from the cores we will of course stick to estimate (REF )." ], [ "Time evolution of the Jacobian and conservation of momentum", "For sufficiently regular solutions of $(\\text{GP})_\\varepsilon ^{c}$ we have $\\begin{split}\\partial _t (iv,\\nabla v) &=(i\\partial _t v,\\nabla v) - (v,\\nabla i \\partial _t v)\\\\&=(\\frac{1}{r}{\\rm div}(r\\nabla v)+\\frac{1}{\\varepsilon ^2}v(1-|v|^2),\\nabla v) -(v,\\nabla (\\frac{1}{r}{\\rm div}(r\\nabla v)+\\frac{1}{\\varepsilon ^2}v(1-|v|^2)))\\\\&= \\frac{2}{r}({\\rm div}(r\\nabla v),\\nabla v) -\\nabla \\left( \\frac{1}{r}(v,{\\rm div} (r\\nabla v)) + \\frac{1-|v|^4}{2\\varepsilon ^2}\\right).\\end{split}$ Taking the curl of the previous identy and integrating against a test function $\\varphi $ with bounded support and which vanishes at $r=0$ we obtain $\\begin{split}\\frac{d}{dt}\\int _\\mathbb {H}Jv\\, \\varphi \\, drdz&= -\\int _\\mathbb {H}\\varepsilon _{ij} \\frac{1}{r} (\\partial _k(r\\partial _k v),\\partial _j v) \\partial _i\\varphi \\\\&= -\\int _\\mathbb {H}\\varepsilon _{ij} \\frac{\\partial _k r}{r} (\\partial _j v ,\\partial _k v)\\partial _i\\varphi +\\int _\\mathbb {H}\\varepsilon _{ij}(\\partial _jv,\\partial _k v)\\partial _{ik}\\varphi \\\\& \\qquad \\ +\\int _\\mathbb {H}\\varepsilon _{ij}\\partial _j(\\frac{\\sum _k|\\partial _k v|^2}{2})\\partial _i \\varphi \\\\&= -\\int _\\mathbb {H}\\varepsilon _{ij} \\frac{\\partial _k r}{r} (\\partial _j v ,\\partial _k v)\\partial _i\\varphi +\\int _\\mathbb {H}\\varepsilon _{ij}(\\partial _jv,\\partial _k v)\\partial _{ik}\\varphi \\\\\\end{split}$ where we sum over repeated indices and since $\\int _\\mathbb {H}\\varepsilon _{ij}\\partial _j(\\sum _k\\frac{|\\partial _k v|^2}{2})\\partial _i \\varphi = \\int _\\mathbb {H}\\varepsilon _{ij}(\\sum _k\\frac{|\\partial _k v|^2}{2})\\partial _{ij} \\varphi = 0$ by anti-symmetry.", "In the sequel we will write $\\mathcal {F}(\\nabla v, \\varphi ):= -\\int _\\mathbb {H}\\varepsilon _{ij} \\frac{\\partial _k r}{r} (\\partial _j v ,\\partial _k v)\\partial _i\\varphi +\\int _\\mathbb {H}\\varepsilon _{ij}(\\partial _jv,\\partial _k v)\\partial _{ik}\\varphi ,$ so that (REF ) is also rewritten as $\\frac{d}{dt}\\int _\\mathbb {H}Jv\\, \\varphi \\, drdz = \\mathcal {F}(\\nabla v, \\varphi ),$ and is the equation from which the dynamical law for the vortex cores will be deduced.", "For a real Lipschitz vector field $X=(X_r,X_z)$ , we expand $\\mathcal {F}(X,\\varphi )= \\int _\\mathbb {H}-\\frac{1}{r}X_rX_z\\partial _r \\varphi + \\frac{1}{r} X_r^2 \\partial _z \\varphi + X_rX_z \\partial _{rr}\\varphi +X_z^2 \\partial _{rz}\\varphi -X_r^2\\partial _{rz}\\varphi - X_rX_z \\partial _{zz}\\varphi .$ Integrating by parts, we have $\\begin{split}\\int _\\mathbb {H}X_rX_z\\partial _{rr}\\varphi &= \\int _\\mathbb {H}-\\partial _r X_z X_r \\partial _r \\varphi - X_z\\partial _r X_r \\partial _r \\varphi \\\\&=\\int _\\mathbb {H}(-\\partial _zX_r - {\\rm curl} X)X_r\\partial _r \\varphi + (\\frac{1}{r}X_r+\\partial _z X_z -\\frac{1}{r}{\\rm div}(rX))X_z\\partial _r\\varphi ,\\\\\\int _\\mathbb {H}X_z^2\\partial _{rz}\\varphi &= \\frac{1}{2} \\int _\\mathbb {H}X_z^2\\partial _{rz}\\varphi +\\frac{1}{2} \\int _\\mathbb {H}X_z^2\\partial _{rz}\\varphi \\\\&= \\int _\\mathbb {H}-\\frac{1}{2} \\partial _z(X_z^2)\\partial _r\\varphi -\\frac{1}{2} \\partial _r(X_z^2)\\partial _z \\varphi ,\\\\\\int _\\mathbb {H}-X_r^2 \\partial _{rz}\\varphi &= \\int _\\mathbb {H}\\frac{1}{2} \\partial _r(X_r^2)\\partial _z\\varphi + \\frac{1}{2} \\partial _z(X_r^2)\\partial _r \\varphi ,\\end{split}$ and $\\int _\\mathbb {H}-X_rX_z\\partial _{zz}\\varphi = \\int _\\mathbb {H}(\\partial _rX_z-{\\rm curl}X)X_z\\partial _z \\varphi + X_r(\\frac{1}{r}{\\rm div}(rX)-\\frac{1}{r}X_r-\\partial _rX_r)\\partial _z\\varphi ,$ so that after summation and simplification $\\mathcal {F}(X,\\varphi )= \\int _\\mathbb {H}-({\\rm curl}X)X\\cdot \\nabla \\varphi +\\frac{1}{r}{\\rm div}(rX)X\\times \\nabla \\varphi .$ Formally, the choice $\\varphi = r^2$ in (REF ) leads to the conservation of the momentum along the z-axis $\\frac{d}{dt}\\int _\\Omega Jv\\, r^2 drdz = 0,$ but its justification would require additional arguments at infinity.", "In the next section we shall consider a version of the momentum localized on some large but finite part of $\\mathbb {H}.$" ], [ "Expansion of the main terms in the dynamics", "In this section we strengthen assumption $(H_1)$ into $\\begin{array}{c}\\displaystyle \\frac{K_0^{-1}}{\\sqrt{|\\!\\log \\varepsilon |}} \\le \\min _{i\\ne j} |a_{i}-a_{j}|\\le \\max _{i\\ne j} |a_{i}-a_{j}| \\le \\frac{K_0}{\\sqrt{|\\!\\log \\varepsilon |}}\\\\\\displaystyle \\frac{r_0}{2} \\le \\min _{i} r(a_{i})\\le \\max _{i} r(a_{i})\\le 2r_0\\\\\\end{array}\\qquad \\mathrm {({H_0})}$ which is nothing but the time independent version of (REF ), and we define $r_a$ and $\\Sigma _a$ as in (REF ) and (REF ).", "We shall also always implicitly assume that $\\max _i |z(a_i)| \\le K_0.$ Since the problem is invariant under translation along the z-axis, and since we have already assumed that all the points are close to each other (as expressed by the first line in $(H_0)$ ), it is clear that this is not really an assumption but just a convenient way to avoid the necessity for various translations along the z-axis in some our subsequent claims.", "Note that for sufficiently small $\\varepsilon $ , and adapting the constant $K_0$ if necessary, the situation described by $(H_0)$ indeed implies $(H_1)$ , and therefore in the sequel we shall refer freely to the improved approximation points $\\xi _i$ whose existence was established in Proposition REF .", "Our analysis in the next sections will make rigorous the fact that the main contribution in the dynamical law for the vortex cores is obtained from (REF ), with a suitable choice of test function $\\varphi $ , by replacing in the expression $\\mathcal {F}(\\nabla u,\\varphi )$ the term $\\nabla u$ by $j^\\natural (u_\\xi ^*).$ Regarding $\\varphi $ , we assume that it satisfies $\\varphi $ is affine on each ball $B(\\xi _i,\\frac{1}{|\\!\\log \\varepsilon |}),$ $\\varphi $ is compactly supported in the union of disjoint balls $\\cup _i B(\\xi _i,1/(2K_0\\sqrt{|\\!\\log \\varepsilon |})),$ $|\\nabla \\varphi |\\le C$ and $|D^2\\varphi | \\le CK_0\\sqrt{|\\!\\log \\varepsilon |}$ , where $C$ is a universal constant for such a test function to exist.", "We will refer to the above requirement as condition $(H_\\varphi ).$ Proposition 3 Under the assumptions $(H_0)$ , (REF ), (REF ) and $(H_\\varphi )$ , there exist $\\varepsilon _3 \\le \\varepsilon _2$ and $C_3$ depending only on $n$ and $K_0$ and $r_0$ such that if $\\varepsilon \\le \\varepsilon _3$ we have $\\left| \\mathcal {F}(j^\\natural (u_\\xi ^*),\\varphi ) - \\sum _{i=1}^n\\mathbb {J}\\nabla _{a_i}H_\\varepsilon (\\xi _1,\\cdots ,\\xi _n)\\cdot \\nabla \\varphi (\\xi _i) \\right| \\le C_3\\left(\\Sigma _a^r + \\log |\\!\\log \\varepsilon |\\right).$ The main task in the remaining sections will be to control the discrepandcy between $\\mathcal {F}(\\nabla u,\\varphi )$ and $\\mathcal {F}(j^\\natural (u_\\xi ^*),\\varphi )$ ; for that purpose we will have to use the evolution equation to a larger extent (up to now our analysis was constrained on fixed time slices)." ], [ "Approximation of the momentum", "As remarked earlier, the choice $\\varphi =r^2$ in (REF ) formally leads to the conservation of the momentum $\\int Ju r^2 \\, drdz.$ Yet, giving a clear meaning to the previous integral and proving its conservation in time is presumably not an easy task.", "Instead, we will localise the function $r^2$ by cutting-it off sufficiently far away from the origin and derive an approximate conservation law.", "More precisely, we set $R_\\varepsilon := |\\!\\log \\varepsilon |^2$ and we let $0\\le \\chi _\\varepsilon \\ \\le 1$ be a smooth cut-off function with compact support in $[0,2R_\\varepsilon ]\\times [-2R_\\varepsilon ,2R_\\varepsilon ]$ and such that $\\chi _\\varepsilon \\equiv 1$ on $[0,R_\\varepsilon ]\\times [-R_\\varepsilon ,R_\\varepsilon ]$ and $|\\nabla \\chi _\\varepsilon | \\le C/R_\\varepsilon .$ In the sequel we write $P_\\varepsilon (u) := \\int _\\mathbb {H}Ju r^2 \\chi _\\varepsilon \\, drdz.$ Proposition 4 Under the assumption $(H_0)$ and (REF ), there exist $\\varepsilon _4 \\le \\varepsilon _3$ such that if $\\varepsilon \\le \\varepsilon _4$ then we have : $\\left|P_\\varepsilon (u) - P(\\xi _1,\\cdots ,\\xi _n)\\right| \\le C_4 \\frac{(1+\\Sigma _\\xi )^2}{|\\!\\log \\varepsilon |^2},$ and $\\left| \\partial _t P_\\varepsilon (u)\\right| \\le C_4 \\frac{1+\\Sigma _\\xi }{R_\\varepsilon } = C_4\\frac{1+\\Sigma _\\xi }{|\\!\\log \\varepsilon |^2},$ where $C_4$ depends only on $n$ , $K_0$ and $r_0.$" ], [ "A key argument", "Coming back to Remark REF and Remark REF we now state Proposition 5 Under the assumptions $(H_0)$ and (REF ), there exists $\\varepsilon _5\\le \\varepsilon _4$ and $\\sigma _5>0$ , depending only on $K_0$ and $n$ , such that if $\\varepsilon \\le \\varepsilon _5$ and if $\\Sigma _a + |H_\\varepsilon (a_1,\\cdots ,a_n)-H_\\varepsilon (\\xi ,\\cdots ,\\xi _n)| \\le \\sigma _5 |\\!\\log \\varepsilon |,$ then $\\Sigma _\\xi \\le 2\\Sigma _a + C_5\\left[ r_a\\sqrt{|\\!\\log \\varepsilon |} +\\frac{1}{|\\!\\log \\varepsilon |}+ |\\!\\log \\varepsilon |\\,|P_\\varepsilon (u)-P(a_1,\\cdots ,a_n)|\\right]$ where $C_5$ depends only on $n$ , $K_0$ and $r_0.$ For a quantity $f$ we temporarily write $\\Delta f := |f(a_1,\\cdots ,a_n)-f(\\xi _1,\\cdots ,\\xi _n)|$ when the latter has a well defined meaning.", "By the triangle inequality we have $\\Delta H_\\varepsilon \\le \\Delta \\left( H_\\varepsilon -\\frac{|\\!\\log \\varepsilon |}{2r_0}P\\right)+ \\frac{|\\!\\log \\varepsilon |}{2r_0} \\Delta P,$ and also $\\Delta P \\le \\left| P_\\varepsilon (u)-P(\\xi _1,\\cdots ,\\xi _n)\\right| + \\left|P_\\varepsilon (u)-P(a_1,\\cdots ,a_n)\\right|.$ In view of the expansion in Remark REF (the $o(1)$ holds in particular in $\\mathcal {C}^1$ norm under assumption $(H_0)$ ), we have $\\Delta \\left( H_\\varepsilon -\\frac{|\\!\\log \\varepsilon |}{2r_0}P\\right) \\le C|(\\xi _1-a_1,\\cdots ,\\xi _n-a_n)|\\sqrt{|\\!\\log \\varepsilon |}$ and by (REF ) and (REF ) $|(\\xi _1-a_1,\\cdots ,\\xi _n-a_n)|\\le C (r_a+ \\varepsilon |\\!\\log \\varepsilon |^{C_1}e^{C_1\\Sigma _a^r}).$ By (REF ) we also have $\\frac{|\\!\\log \\varepsilon |}{2r_0}\\left| P_\\varepsilon (u)-P(\\xi _1,\\cdots ,\\xi _n)\\right|\\le \\frac{C_4}{2r_0}\\frac{(1+\\Sigma _\\xi )^2}{|\\!\\log \\varepsilon |} \\le \\frac{C_4}{2r_0} \\frac{(1+\\Sigma _a + \\Delta H_\\varepsilon )^2}{|\\!\\log \\varepsilon |}$ and $\\frac{C_4}{2r_0} \\frac{(1+\\Sigma _a + \\Delta H_\\varepsilon )^2}{|\\!\\log \\varepsilon |} \\le \\frac{3C_4}{2r_0} \\frac{1+\\Sigma _a^2 +(\\Delta H_\\varepsilon )^2}{|\\!\\log \\varepsilon |} \\le \\frac{3C_4}{2r_0}\\left( \\frac{1}{|\\!\\log \\varepsilon |} +\\sigma _5 (\\Sigma _a+\\Delta H_\\varepsilon )\\right).$ By summation of all the inequalities gathered so far we obtain $\\begin{split}\\Delta H_\\varepsilon &\\le C\\left((r_a+\\varepsilon |\\!\\log \\varepsilon |^{C_1}e^{C_1\\Sigma _a^r})\\sqrt{|\\!\\log \\varepsilon |} +\\frac{1}{|\\!\\log \\varepsilon |} + \\Sigma _a + |\\!\\log \\varepsilon |\\,|P_\\varepsilon (u)-P(a_1,\\cdots ,a_n)|\\right)\\\\& \\qquad + \\frac{3C_4}{2r_0}\\sigma _5 \\Delta H_\\varepsilon .\\end{split}$ We therefore choose $\\sigma _5$ in such a way that $\\frac{3C_4}{2r_0}\\sigma _5 \\le \\frac{1}{2}$ , and we may then absorb the last term of the previous inequality in its left-hand side.", "Combined with the fact that $\\Sigma _\\xi \\le \\Sigma _a+\\Delta H_\\varepsilon $ the conclusion (REF ) follows.", "Remark 4 The main gain in (REF ) is related to the fact that in the right-hand side we have a term of the form $r_a\\sqrt{|\\!\\log \\varepsilon |}$ rather than (the easier) $r_a|\\!\\log \\varepsilon |$ which would have followed from a crude gradient bound on $H_\\varepsilon .$ Note however that we have exploited here the assumption $(H_0)$ , that is the fact that all the cores are of order $\\sqrt{|\\!\\log \\varepsilon |}$ apart from each other, whereas Proposition REF holds under the weaker assumption $(H_1)$ .", "The right-hand side of (REF ) also contains a term involving $P_\\varepsilon (u)$ and $P(a_1,\\cdots ,a_n).$ When introducing time dependence in the next sections, we will take advantage of the fact that $P$ is preserved by the ODE flow $(\\text{LF})_\\varepsilon $ and that $P_\\varepsilon $ is almost preserved by the PDE flow $(\\text{GP})_\\varepsilon ^{c}$ , as already expressed in (REF )." ], [ "Time dependence and Stopping time", "In this section we introduce time dependence and go back to the setting of Theorem REF , that is we assume (REF ) and (REF ).", "For $s \\in [0,S_0],$ we define the localization scales $r_a^s := \\Big \\Vert J u_\\varepsilon ^s - \\pi \\sum _{i=1}^n\\delta _{a_{i,\\varepsilon }(s)}\\Big \\Vert _{\\dot{W}^{-1,1}(\\Omega _0)},$ and the excess energy $\\Sigma ^s := \\left[{\\mathcal {E}}_\\varepsilon ^{\\rm w}(u_\\varepsilon ^s) -H_{\\varepsilon }(a_{1,\\varepsilon }(s),\\cdots ,a_{n,\\varepsilon }(s))\\right]^+$ where we recall that $\\Omega _0= \\lbrace r\\ge \\frac{r_0}{4}\\rbrace $ and $u_\\varepsilon ^s$ is the solution of $(\\text{GP})_\\varepsilon ^{c}$ evaluated at time $t=s/|\\!\\log \\varepsilon |.$ Since ${\\mathcal {E}}_\\varepsilon ^{\\rm w}$ is preserved by the flow of $(\\text{GP})_\\varepsilon ^{c}$ and since $H_\\varepsilon $ is preserved by $(\\text{LF})_\\varepsilon ,$ we have $\\Sigma ^s = \\Sigma ^0 \\qquad \\forall s \\in [0,S_0].$ We introduce the stopping time $S_{\\rm stop} := \\inf \\left\\lbrace S \\in [0,S_0],\\ r_a^s \\le \\frac{\\rho _{\\rm min}}{8}, \\quad \\forall s \\in [0,S]\\right\\rbrace ,$ where we have set, in view of (REF ), $\\rho _{\\rm min} := \\frac{K_0^{-1}}{\\sqrt{|\\!\\log \\varepsilon |}}.$ By (REF ) and continuity it is clear that $S_{\\rm stop}>0,$ at least provided $\\varepsilon _0$ and $\\sigma _0$ are chosen small enough.", "By construction, we also have $r_a^s < \\frac{\\rho _{a_\\varepsilon (s)}}{4} \\qquad \\forall s \\in [0,S_{\\rm stop}],$ and likewise by (REF ) $\\Sigma ^s + r_a^s|\\!\\log \\varepsilon |\\le \\sigma _1 |\\!\\log \\varepsilon |,$ where $\\sigma _1$ is given in Proposition REF .", "Applying Proposition REF for each $s \\in [0,S_{\\rm stop}]$ , we get functions $s\\mapsto \\xi _i(s) \\equiv \\xi _i^s$ , $i=1,\\cdots ,n.$ By continuity of the flow map for $(\\text{GP})_\\varepsilon ^{c}$ , and doubling $C_1$ if necessary, we may further assume that these maps are piecewise constant and hence measurable on $[0,S_{\\rm stop}].$ In the sequel, in view of (REF ), we set $r_\\xi ^s = C_1\\varepsilon |\\!\\log \\varepsilon |^{C_1}e^{C_1(\\Sigma ^0+r_a^s|\\!\\log \\varepsilon |)} \\stackrel{(\\ref {eq:securitybounds})}{\\le } \\varepsilon ^\\frac{5}{6},$ for each $s \\in [0,S_{\\rm stop}].$ The following Proposition yields a first estimate on the time evolution of the vortex cores.", "At this stage it does not contain any information about the actual motion law, but only a rough (but essential) Lipschitz bound.", "Proposition 6 For sufficiently small values of $\\sigma _0$ and $\\varepsilon _0,$ whose threshold may be chosen depending only on $n$ , $K_0$ and $r_0,$ the following holds: There exist $C_6>0,$ also depending only on $n$ , $K_0$ and $r_0,$ such that for all $s_1,s_2 \\in [0,S_{\\rm stop}]$ such that $s_1 \\le s_2 \\le s_1 + {|\\!\\log \\varepsilon |}^{-1}$ we have $&\\Vert Ju_\\varepsilon ^{s_1} - Ju_\\varepsilon ^{s_2}\\Vert _{\\dot{W}^{-1,1}(\\Omega _0)} \\le C_6(|s_1-s_2|+r_\\xi ^{s_1}),\\\\&r_\\xi ^{s_2} \\le r_\\xi ^{s_1}\\left(1+ C_6\\left(|s_2-s_1|+\\varepsilon ^\\frac{5}{6}\\right)|\\!\\log \\varepsilon |\\right),\\\\&\\left\\lbrace a_{i,\\varepsilon }(s_2),\\xi _i(s_2)\\right\\rbrace \\subset B(a_{i,\\varepsilon }(s_1),\\frac{\\rho _{\\rm min}}{4}).$ Moreover, if $r_a^{s_1} \\le \\rho _{\\rm min}/16,$ then $S_{\\rm stop} \\ge s_1+ (C_6\\sqrt{|\\!\\log \\varepsilon |})^{-1}.$" ], [ "Control of the discrepancy", "The following proposition is the final ingredient leading to the proof of Theorem REF , it can be regarded as a discrete version of the Gronwall inequality for the quantity $r_a^s.$ Proposition 7 Assume that $s< S_{\\rm stop}$ and that $r_a^s \\le \\rho _{\\min }/16$ and set $S := s + \\frac{(r_\\xi ^s)^2}{\\varepsilon }.$ Then $S < S_{\\rm stop}$ and $\\frac{r_a^S-r_a^s}{S-s} \\le C_0\\left(r_a^s + \\frac{\\Sigma ^0+r_a^0|\\!\\log \\varepsilon |}{\\sqrt{|\\!\\log \\varepsilon |}} +\\frac{C_\\delta }{|\\!\\log \\varepsilon |^{1-\\delta }}\\right),$ where $C_0$ depends only on $n$ , $K_0$ and $r_0$ , $\\delta >0$ is arbitrary and $C_\\delta $ depends only on $\\delta .$ Remark 5 The time step $S-s$ on which the differential inequality (REF ) holds is not arbitrary, in view of (REF ) it satisfies $S-s = C_1^2 \\varepsilon |\\!\\log \\varepsilon |^{2C_1}e^{2C_1(\\Sigma ^0+r_a^s|\\!\\log \\varepsilon |)}$ which, for $\\varepsilon $ sufficiently small, is both large with respect to $\\varepsilon $ and small with respect to lower powers of $\\varepsilon .$ The fact that it is large with respect to $\\varepsilon $ , as the proof of Proposition REF will show, is essential in order to allow the averaging effects of the continuity equation (see (REF )) to act.", "On the other hand, the fact that it is small with respect to lower powers of $\\varepsilon $ will allow us, when using it iteratively, to rely on the softer estimates of Proposition REF to bridge the gaps between the discrete set of times so obtained and the full time interval $[0,S_0]$ which appears in the statement of Theorem REF ." ], [ "Proofs", "Proof of Lemma REF It suffices to combine the expansion of Lemma (REF ) with those (see e.g.", "[2]) for the optimal Ginzburg-Landau profile $f_\\varepsilon .$$\\Box $ Proof of Proposition REF We divide the proof in several steps.", "We first set $\\ell _{\\epsilon ,a}= 4\\max (\\frac{1}{|\\!\\log \\varepsilon |},r_a).$ Step 1 : rough lower energy bounds on $B(a_i,\\ell _{\\epsilon ,a})$ .", "In view of our assumptions and the fact that the $\\dot{W}^{-1,1}$ is decreasing with respect to the domain, we are in position, provided $\\varepsilon _1$ and $\\sigma _1$ are sufficiently small (depending only on $r_0$ and $K_1$ ), to apply Theorem REF after translation to the balls $B(a_i,\\ell _{\\epsilon ,a}).$ This yields the lower bounds $\\begin{split}{\\mathcal {E}}_{\\varepsilon }(u_\\varepsilon ^s,B(a_i,\\ell _{\\epsilon ,a})) &\\ge \\pi \\log \\frac{\\ell _{\\epsilon ,a}}{\\varepsilon } + \\gamma - \\frac{C}{\\ell _{\\epsilon ,a}} \\left(\\varepsilon \\sqrt{\\log (\\ell _{\\epsilon ,a}/\\varepsilon )} + r_a\\right)\\\\& \\ge \\pi |\\!\\log \\varepsilon |-C \\left(r_a|\\!\\log \\varepsilon |+ \\log |\\!\\log \\varepsilon |\\right),\\end{split}$ for any $1\\le i \\le n,$ where $C$ is universal provided we require that $\\varepsilon _1$ is also sufficiently small so that $\\log |\\!\\log \\varepsilon |\\ge 1$ for $\\varepsilon \\le \\varepsilon _1.$ From (REF ) and the global energy bound given by the assumption of $\\Sigma _a$ it follows, comparing the weight function $r$ with its value $r(a_i)$ , that ${\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,B(a_i,\\ell _{\\epsilon ,a})) \\ge \\pi r(a_i)|\\!\\log \\varepsilon |- C \\left( r_a|\\!\\log \\varepsilon |+ \\log |\\!\\log \\varepsilon |\\right)$ for any $1\\le i \\le n,$ and for a possibly larger constant $C$ depending only on $K_1$ , $r_0$ and $n.$ Step 2 : rough upper energy bounds on $\\mathbb {H}\\setminus \\cup _{i=1}^n B(a_i,\\ell _{\\epsilon ,a}/2)$ and $B(a_i,2\\ell _{\\epsilon ,a})$ .", "The equivalent of (REF ) with $\\ell _{\\epsilon ,a}$ replaced by $\\ell _{\\epsilon ,a}/2$ , combined with the global upper energy bound given by the definition of $\\Sigma _a$ , yields the upper bound ${\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,\\mathbb {H}\\setminus \\cup _{i=1}^n B(a_i,\\frac{\\ell _{\\epsilon ,a}}{2}) \\le C \\left(\\Sigma _a^r + \\log |\\!\\log \\varepsilon |\\right),$ where $C$ depends only $K_0$ and $n$ .", "Also, combining (REF ) (for all but one $i$ ) with the definition of $\\Sigma _a$ , we obtain the upper bound ${\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,B(a_i,2\\ell _{\\epsilon ,a})) \\le \\pi r(a_i)|\\!\\log \\varepsilon |+ C \\left(\\Sigma _a^r + \\log |\\!\\log \\varepsilon |\\right),$ for any $1\\le i \\le n,$ and therefore ${\\mathcal {E}}_{\\varepsilon }(u,B(a_i,2\\ell _{\\epsilon ,a})) \\le \\pi \\log \\frac{2\\ell _{\\epsilon ,a}}{\\varepsilon } + C \\left(\\Sigma _a^r + \\log |\\!\\log \\varepsilon |\\right),$ for any $1\\le i \\le n,$ where $C$ depends only on $K_0$ and $n.$ Step 3 : first localisation estimates.", "We apply Theorem REF , after translation, to each of the balls $B(a_i,2\\ell _{\\epsilon ,a})$ , and we denote by $\\xi _i$ the corresponding points.", "In view of (REF ), this yields $\\sum _{i=1}^n\\Vert Ju-\\pi \\delta _{\\xi _i}\\Vert _{\\dot{W}^{-1,1}(B(a_i,2\\ell _{\\epsilon ,a}))} \\le \\varepsilon e^{C(\\Sigma _a^r+\\log |\\!\\log \\varepsilon |)}\\le \\varepsilon |\\!\\log \\varepsilon |^C e^{C\\Sigma _a^r}.$ Note that from (REF ) and the definition of $r_a$ in (REF ) we have the bound $\\max _{i=1,\\cdots ,n} | a_i-\\xi _i| \\le \\frac{1}{\\pi }\\left( r_a + \\varepsilon |\\!\\log \\varepsilon |^C e^{C\\Sigma _a^r}\\right).$ Provided $\\varepsilon _1$ and $\\sigma _1$ are sufficiently small, this also implies that $B(\\xi _i,\\ell _\\epsilon ) \\subset B(a_i,\\ell _{\\epsilon ,a}) \\qquad \\forall i=1\\cdots ,n,$ where we have set $\\ell _\\epsilon := \\frac{1}{|\\!\\log \\varepsilon |}.$ From now on we will rely entirely on the points $\\xi _i$ rather than on the $a_i$ for our constructions.", "Step 4 : improved lower energy bounds close to the cores.", "We apply Theorem REF , after translation, to each of the balls $B(\\xi _i,\\rho ),$ where $\\ell _\\epsilon /2 \\ge \\rho \\ge \\varepsilon ^\\frac{4}{5}$ is some free parameter which we will fix later.", "Since $\\dot{W}^{-1,1}$ norms are monotone functions of the domain and since $B(\\xi _i,\\rho )\\subset B(a_i,\\ell _{\\epsilon ,a})$ by (REF ), in view of (REF ) we obtain ${\\mathcal {E}}_{\\varepsilon }(u,B(\\xi _{i},\\rho )) \\ge \\pi \\log \\frac{\\rho }{\\varepsilon } + \\gamma - C\\varepsilon |\\!\\log \\varepsilon |^Ce^{C\\Sigma _a^r}\\rho ^{-1},$ and therefore ${\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,B(\\xi _{i},\\rho )) \\ge r(\\xi _{i})\\Big (\\pi \\log \\frac{\\rho }{\\varepsilon } + \\gamma \\Big ) - C(\\varepsilon |\\!\\log \\varepsilon |^Ce^{C\\Sigma _a^r}\\rho ^{-1}+\\rho |\\!\\log \\varepsilon |),$ for $i=1,\\cdots ,n.$ Note that taking $\\rho =\\ell _\\epsilon /2$ and then arguing exactly as in Step 2 yields the slight variant of (REF ): ${\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,\\mathbb {H}\\setminus \\cup _{i=1}^n B(\\xi _i,\\frac{\\ell _\\epsilon }{2}) \\le C \\left(\\Sigma _a^r + \\log |\\!\\log \\varepsilon |\\right).$ Yet at this point we wish to keep $\\rho $ as a free parameter.", "Step 5 : towards lower energy bounds away from the cores.", "In this step we compare $u$ , away from the cores, with the singular vortex ring $u_\\xi ^*$ .", "For convenience, we simply denote $j(u^*_\\xi )$ by $j^*$ , we let $0 \\le \\chi \\le 1$ be a lipschitz function on $\\mathbb {H}$ , and we set $\\mathbb {H}_{\\xi ,\\rho } :=\\mathbb {H}\\setminus \\cup _{i=1}^n B(\\xi _i,\\rho ).$ The starting point is the pointwise equality $e_\\varepsilon (u) = \\frac{1}{2} |j_*|^2 +j_*\\big (\\frac{j(u)}{|u|}-j_*\\big ) +e_\\varepsilon (|u|) + \\frac{1}{2} \\big | \\frac{j(u)}{|u|}-j_*\\big |^2,$ which holds almost everywhere in $\\mathbb {H}.$ Notice that all the terms in the right-hand side of (REF ) are pointwise non-negative except possibly the second one.", "We integrate (REF ) multiplied by $\\chi ^2$ on $\\mathbb {H}_{\\xi ,\\rho }$ and estimate the corresponding terms.", "We first write $\\int _{\\mathbb {H}_{\\xi ,\\rho }} r j_*\\big (\\frac{j(u)}{|u|}-j_*\\big )\\chi ^2 =\\int _{\\mathbb {H}_{\\xi ,\\rho }} r j_*\\big (j(u)-j_*\\big )\\chi ^2 +\\int _{\\mathbb {H}_{\\xi ,\\rho }} r j_*\\big (\\frac{j(u)}{|u|}-j(u)\\big )\\chi ^2$ and we readily estimate $\\Big | \\int _{\\mathbb {H}_{\\xi ,\\rho }} r j_*\\big (\\frac{j(u)}{|u|}-j(u)\\big )\\chi ^2 \\Big |\\le \\frac{C}{\\rho } \\Big ( \\int _{\\mathbb {H}_{\\xi ,\\rho }} r\\frac{j^2(u)}{|u|^2}\\Big )^\\frac{1}{2}\\Big ( \\int _{\\mathbb {H}_{\\xi ,\\rho }} r(1-|u|)^2\\Big )^\\frac{1}{2} \\le C\\frac{\\varepsilon }{\\rho }|\\!\\log \\varepsilon |,$ where we have used the facts that $|j^*| \\le C/\\rho $ on $\\mathbb {H}_{\\xi ,\\rho }$ and that the last two integral factors are dominated by (a constant multiple of) the weighted energy.", "By definition (see Appendix REF ), we have $r j_* = -\\nabla ^\\perp \\big (r\\Psi ^*_\\xi \\big ).$ We modify (truncate) the function $\\Psi _\\xi ^*$ to a function $\\tilde{\\Psi }_\\xi ^*$ in the following way : for each $i=1,\\cdots ,n$ we consider the connected component $\\mathcal {C}_i$ of the superlevel set $\\lbrace \\Psi _\\xi ^* \\ge \\Psi ^*_\\xi (\\xi _i+(\\rho ,0))\\rbrace $ (by convention we include $\\xi _i$ , where $\\Psi ^*_\\xi $ is in principle not defined, in this set) which contains the point $\\xi _i+(\\rho ,0)$ , and we set $\\tilde{\\Psi }^*_\\xi = \\Psi ^*_\\xi (\\xi _i+(\\rho ,0))$ on $\\mathcal {C}_i$ .", "Next, we set $\\tilde{\\Psi }^*_\\xi =\\Psi ^*_\\xi $ on $\\mathbb {H}\\setminus \\cup _{i=1}^n \\mathcal {C}_i.$ By construction, $-\\nabla ^\\perp \\big (r\\tilde{\\Psi }^*_\\xi \\big )=rj_*\\mathbb {1}_{\\mathbb {H}\\setminus \\cup _{i=1}^n\\mathcal {C}_i},$ so that $-\\nabla ^\\perp \\big (r\\tilde{\\Psi }^*_\\xi \\chi ^2\\big )=rj_*\\mathbb {1}_{\\mathbb {H}_{\\xi ,\\rho }} \\chi ^2 +rj_*\\big [\\sum _{i=1}^n(\\mathbb {1}_{B(\\xi _i,\\rho )}-\\mathbb {1}_{\\mathcal {C}_i})\\big ]\\chi ^2 - 2r\\tilde{\\Psi }_\\xi ^* \\chi \\nabla ^\\perp \\chi .$ The latter and integration by parts yields $\\begin{split}\\Big |\\int _{\\mathbb {H}_{\\xi ,\\rho }} r j_*\\big (j(u)-j_*\\big )\\chi ^2 \\Big | &\\le \\sum _{i=1}^n\\int _{B(\\xi _i,\\rho )\\bigtriangleup \\mathcal {C}_i}r|j^*|\\big |j(u)-j_*\\big |+ \\int _\\mathbb {H}2r|\\tilde{\\Psi }_\\xi ^*| |\\nabla ^\\perp \\chi |\\chi |j(u)-j_*|\\\\& \\quad + \\Big |\\int _{\\mathbb {H}}2r\\tilde{\\Psi }^*_\\xi \\big (J(u)-J(u^*_\\xi )\\big )\\chi ^2\\Big | .\\end{split}$ In order to bound the right-hand side of (REF ) we first remark that, from (REF ) and (REF ) in the Appendix, for each $i=1,\\cdots ,n,$ we have $d_{\\mathcal {H}}( \\mathcal {C}_i, B(\\xi _i,\\rho )) \\le \\frac{\\rho ^2}{\\rho _a}\\log (\\frac{\\rho }{\\rho _a}), \\quad \\text{and hence}\\quad \\mathcal {L}^2\\big (\\mathcal {C}_i \\bigtriangleup B(\\xi _i,\\rho )\\big ) \\le C \\frac{\\rho ^3}{\\rho _a}\\log (\\frac{\\rho }{\\rho _a}).$ We write $\\int _{B(\\xi _i,\\rho )\\bigtriangleup \\mathcal {C}_i}r|j^*|\\big |j(u)-j_*\\big | \\le \\int _{B(\\xi _i,\\rho )\\bigtriangleup \\mathcal {C}_i}r|j^*|\\big |\\frac{j(u)}{|u|}-j_*\\big | + \\int _{B(\\xi _i,\\rho )\\bigtriangleup \\mathcal {C}_i}r\\varepsilon |j^*|\\big |\\frac{j(u)}{|u|}\\big |\\frac{||u|-1|}{\\varepsilon },$ and since $|j^*|\\le C/\\rho $ on $\\mathcal {C}_i \\bigtriangleup B(\\xi _i,\\rho )$ , we deduce from (REF ), the Cauchy-Schwarz inequality, and global energy upper bounds, that $\\sum _{i=1}^n\\int _{B(\\xi _i,\\rho )\\bigtriangleup \\mathcal {C}_i}r|j^*|\\big |j(u)-j_*\\big | \\le C \\left( \\frac{\\rho }{\\rho _a}\\log (\\frac{\\rho }{\\rho _a})|\\!\\log \\varepsilon |\\right)^\\frac{1}{2} + C \\frac{\\varepsilon }{\\rho }|\\!\\log \\varepsilon |.$ Concerning the second error term in (REF ), we first decompose it as $\\int _\\mathbb {H}r|\\tilde{\\Psi }_\\xi ^*| |\\nabla ^\\perp \\chi | |j(u)-j_*| \\chi =\\int _\\mathbb {H}r|\\tilde{\\Psi }_\\xi ^*| |\\nabla ^\\perp \\chi | |\\frac{j(u)}{|u|}-j_*|\\chi +\\int _\\mathbb {H}r|\\tilde{\\Psi }_\\xi ^*| |\\nabla ^\\perp \\chi | |\\frac{j(u)}{|u|}||1-|u||\\chi $ and we write by Cauchy-Schwarz inequality on one hand $\\int _\\mathbb {H}r|\\tilde{\\Psi }_\\xi ^*| |\\nabla ^\\perp \\chi ||\\frac{j(u)}{|u|}-j_*|\\chi \\le C\\Vert \\nabla \\chi \\Vert _\\infty \\Big (\\int _{{\\rm spt}(\\nabla \\chi )} r |\\tilde{\\Psi }^*_\\xi |^2\\Big )^\\frac{1}{2}\\Big (\\int _{{\\rm spt}(\\nabla \\chi )} r (e_\\varepsilon (u)+e_\\varepsilon (u^*_\\xi ))\\chi ^2 \\Big )^\\frac{1}{2},$ and by direct comparison with the energy density on the other hand $\\int _\\mathbb {H}r|\\tilde{\\Psi }_\\xi ^*| |\\nabla ^\\perp \\chi | |\\frac{j(u)}{|u|}||1-|u||\\chi \\le C \\varepsilon |\\!\\log \\varepsilon |^2 \\Vert \\nabla \\chi \\Vert _\\infty .$ Coming back to (REF ), and taking into account (REF )-(REF ), we conclude that $\\int _{\\mathbb {H}_{\\xi ,\\rho }} \\!\\!\\!\\!\\!r e_\\varepsilon (u) \\chi ^2 \\ge \\int _{\\mathbb {H}_{\\xi ,\\rho }}\\!\\!\\!\\!\\!", "r \\left[ \\frac{|j_*|^2}{2} + e_\\varepsilon (|u|) +\\big |\\frac{j(u)}{|u|}-j_*\\big |^2\\right] \\chi ^2 - \\left|\\int _{\\mathbb {H}}2r\\tilde{\\Psi }^*_\\xi \\big (J(u)-J(u^*_\\xi )\\big )\\chi ^2\\right|- {\\rm Err}(\\chi ^2),$ where $\\begin{split}{\\rm Err}(\\chi ^2) \\le &C \\Big [ \\Big ( \\frac{\\rho }{\\rho _a}\\log (\\frac{\\rho }{\\rho _a})|\\!\\log \\varepsilon |\\Big )^\\frac{1}{2} + \\frac{\\varepsilon }{\\rho }|\\!\\log \\varepsilon |\\\\&+ \\Vert \\nabla \\chi \\Vert _\\infty \\Big (\\int _{{\\rm spt}(\\nabla \\chi )} r |\\tilde{\\Psi }^*_\\xi |^2\\Big )^\\frac{1}{2}\\Big (\\int _{{\\rm spt}(\\nabla \\chi )} r (e_\\varepsilon (u)+e_\\varepsilon (u^*_\\xi ))\\chi ^2 \\Big )^\\frac{1}{2}\\\\&+ \\Vert \\nabla \\chi \\Vert _\\infty \\varepsilon |\\!\\log \\varepsilon |^2 \\Big ].\\end{split}$ Step 6 : improved lower energy bounds away from the cores.", "The right-hand side of estimate (REF ) contains quantities which we do not yet control: we need good localisation estimates for the jacobian $Ju$ , also outside the cores, and we also have to get rid of the energy term due to the cut-off in (REF ).", "To deal with the localisation, we shall rely on Theorem REF , but in view of the difference between ${\\mathcal {E}}_{\\varepsilon }$ and ${\\mathcal {E}}_\\varepsilon ^{\\rm w}$ (the factor $r$ ), we only expect good localisation estimates when $r$ is not too small.", "To quantify this, we define the set $\\mathcal {S}= \\left\\lbrace s=2^{-k}, k \\in \\mathbb {Z}, \\text{ s.t. }", "{\\mathcal {E}}_\\varepsilon ^{\\rm w}\\Big (u,\\lbrace s\\le r \\le 2s\\rbrace \\setminus \\cup _{i=1}^nB(\\xi _i,\\frac{\\ell _\\epsilon }{2})\\Big ) \\le \\frac{\\pi }{12}s|\\!\\log \\varepsilon |\\right\\rbrace $ and the value $r_\\mathcal {S} := \\min \\left\\lbrace s=2^{-k}, k \\in \\mathbb {Z}, \\text{ s.t. }", "2^{-\\ell }\\in \\mathcal {S} \\ \\forall \\ell \\le k\\right\\rbrace .$ Note that by (REF ) we have $r_\\mathcal {S} \\le C \\left( \\frac{\\Sigma _a^r}{|\\!\\log \\varepsilon |} +\\frac{\\log |\\!\\log \\varepsilon |}{|\\!\\log \\varepsilon |}\\right),$ which we will improve later on in (REF ).", "Also, whenever $\\Omega $ is an open bounded subset contained in $\\lbrace s\\le r \\le 2s\\rbrace \\setminus \\cup _{i=1}^nB(\\xi _i,\\frac{\\ell _\\epsilon }{2})$ for some $s\\ge r_\\mathcal {S}$ , covering it with two of the above slices we obtain ${\\mathcal {E}}_{\\varepsilon }(u,\\Omega )\\le \\frac{1}{s}{\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,\\Omega ) \\le \\frac{\\pi }{4} |\\!\\log \\varepsilon |$ and therefore by Theorem REF $\\Vert Ju\\Vert _{\\dot{W}^{-1,1}(\\Omega )} \\le C{\\mathcal {E}}_{\\varepsilon }(u,\\Omega )\\varepsilon ^\\frac{3}{4}.$ We now take $\\rho :=\\rho _\\epsilon =\\varepsilon ^\\frac{2}{3}$ , and in view of (REF ) we let $ r_\\mathcal {S} \\le \\tilde{r} \\le C \\left( \\Sigma _a^r +\\log |\\!\\log \\varepsilon |\\right)/|\\!\\log \\varepsilon |.$ We choose $0\\le \\chi \\le 1$ a lipschitz function supported in $\\lbrace r\\ge \\tilde{r}\\rbrace $ and such that $\\chi \\equiv 1$ on $\\lbrace r\\ge 2\\tilde{r}\\rbrace $ and $|\\nabla \\chi | \\le C/\\tilde{r}.$ We then invoke estimate (REF ) of Step 5, which we add-up with estimate (REF ) (note that $\\chi \\equiv 1$ on each $B(\\xi _i,\\rho )$ by definition of $\\tilde{r}$ , at least provided $\\varepsilon _1$ and $\\sigma _1$ are chosen small enough) to write ${\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,\\lbrace r\\ge \\tilde{r}\\rbrace ) \\ge \\int _{\\mathbb {H}} re_\\varepsilon (u)\\chi ^2 \\ge T_1-T_2-T_3+T_4$ where $T_1 = \\int _{\\mathbb {H}_{\\xi ,\\rho }}\\!\\!\\!\\!\\!", "r\\frac{|j_*|^2}{2} + \\sum _{i=1}^n r(\\xi _{i})\\Big (\\pi \\log \\frac{\\rho _\\epsilon }{\\varepsilon } + \\gamma \\Big )-\\int _{\\lbrace r\\le \\tilde{r}\\rbrace } r\\frac{|j_*|^2}{2},$ $T_2 = {\\rm Err}(\\chi ^2) \\qquad \\text{,}\\qquad T_3:= \\Big |\\int _{\\mathbb {H}}2r\\tilde{\\Psi }^*_\\xi \\big (J(u)-J(u^*_\\xi )\\big )\\chi ^2\\Big |,$ and $T_4 = \\int _{\\mathbb {H}_{\\xi ,\\rho }}\\!\\!\\!\\!\\!", "r \\left[ e_\\varepsilon (|u|) +\\big |\\frac{j(u)}{|u|}-j_*\\big |^2\\right] \\chi ^2 \\ge 0.$ We invoke Lemma REF (with $a=\\xi $ ) and the definition of $H_\\varepsilon $ to obtain $T_1 \\ge H_\\varepsilon (\\xi ) -C\\big ( \\tilde{r}^2 + \\varepsilon ^\\frac{2}{3}|\\!\\log \\varepsilon |^3\\big ),$ where we have also used $(H_1)$ in order to get rid of $\\rho _a$ wherever it appeared.", "Invoking (REF ) to compute some of the terms in (REF ), we also obtain $T_2 \\le C\\left(\\varepsilon ^\\frac{1}{3}|\\!\\log \\varepsilon |^\\frac{3}{2} + \\tilde{r}\\Big ( {\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,\\lbrace \\tilde{r}\\le r\\le 2\\tilde{r}\\rbrace ) + \\tilde{r}\\Big )^\\frac{1}{2} + \\frac{\\varepsilon }{\\tilde{r}}|\\!\\log \\varepsilon |^2\\right),$ and since $\\tilde{r} \\ge r_\\mathcal {S}$ the definition of the latter yields $T_2 \\le C\\left(\\varepsilon ^\\frac{1}{3}|\\!\\log \\varepsilon |^\\frac{3}{2} + \\tilde{r}^\\frac{3}{2} |\\!\\log \\varepsilon |^\\frac{1}{2} + \\frac{\\varepsilon }{\\tilde{r}}|\\!\\log \\varepsilon |^2\\right).$ It remains to estimate $T_3$ for which we will rely on (REF ) and (REF ).", "To that purpose, we write $\\chi ^2 = \\chi ^2\\left( \\sum _{i=1}^n \\psi ^{\\rm in}_i + \\sum _{j \\in \\mathbb {N}}\\psi _j^{\\rm out}\\right) \\qquad \\text{on } \\mathbb {H}$ for an appropriate partition of unity on $\\mathbb {H}$ verifying the following : Each function of the partition is $\\mathcal {C}^\\infty $ smooth and compactly supported, its support has a smooth boundary.", "Each point of $\\mathbb {H}$ is contained in the support of at most four functions of the partition.", "We have ${\\rm spt}(\\psi ^{\\rm in}_i) \\subset B(\\xi _i,\\ell _\\epsilon ),\\qquad |\\nabla \\psi ^{\\rm in}_i|\\le C/\\ell _\\epsilon , \\qquad \\forall i=1,\\cdots ,n,$ ${\\rm spt}(\\psi ^{\\rm out}_j) \\subset \\mathbb {H}\\setminus \\cup _{i=1}^nB(\\xi _i,\\ell _\\epsilon /2), \\qquad \\forall j \\in \\mathbb {N}.$ For each $j \\in \\mathbb {N}$ , there exists $r_j>\\tilde{r}/2$ such that ${\\rm spt}(\\psi ^{\\rm out}_j) \\subset \\lbrace r_j \\le r \\le 2r_j\\rbrace \\qquad \\text{and}\\qquad |\\nabla \\psi ^{\\rm out}_j|\\le C\\left( \\frac{1}{r_j} + \\frac{1}{\\ell _\\epsilon }\\right).$ The existence of such a partition can be obtained by covering $\\mathbb {H}$ with rectangular tiles with a step size close to being dyadic in the $r$ direction and constant in the $z$ direction and then arranging the round holes corresponding to the $\\xi _i$ 's.", "It may be necessary to shift a little the rectangular tiles so that the balls around the $\\xi _i$ 's do not meet their boundaries (this is the only reason of $r_j$ not being exactly dyadic).", "We use (REF ) for the terms involving $\\psi ^{\\rm in}_i$ and (REF ) for those with $\\psi ^{\\rm out}_j.$ Since $\\chi $ vanishes at $r=0$ we have $|\\chi (r,\\cdot )| \\le r \\Vert \\nabla \\chi \\Vert _\\infty $ and in the dual norm we may crudely estimate $\\Vert r\\tilde{\\Psi }_\\xi ^*\\chi ^2 \\psi _j^{\\rm out}\\Vert _{W^{1,\\infty }} \\le \\frac{C}{\\ell _\\varepsilon }\\le C |\\!\\log \\varepsilon |,\\qquad \\Vert r\\tilde{\\Psi }_\\xi ^*\\chi ^2 \\psi _i^{\\rm in}\\Vert _{W^{1,\\infty }} \\le \\frac{C}{\\rho _\\varepsilon } \\le C \\varepsilon ^{-\\frac{2}{3}},$ so that we finally obtain $T_3 \\le C \\left(\\varepsilon ^\\frac{1}{3}|\\!\\log \\varepsilon |^Ce^{C\\Sigma _a^r}+\\varepsilon ^\\frac{3}{4}|\\!\\log \\varepsilon |^2{\\mathcal {E}}_\\varepsilon ^{\\rm w}\\big ( u,\\mathbb {H}\\setminus \\cup _{i=1}^nB(\\xi _i,\\ell _\\epsilon /2\\big )\\right) \\le C\\varepsilon ^\\frac{1}{3} |\\!\\log \\varepsilon |^Ce^{C\\Sigma _a^r}.$ Combining (REF ),(REF ) and (REF ) in (REF ) we derive ${\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,\\lbrace r\\ge \\tilde{r}\\rbrace ) \\ge H_\\varepsilon (\\xi )+T_4 -C\\Big (\\varepsilon ^\\frac{1}{3}|\\!\\log \\varepsilon |^Ce^{C\\Sigma _a^r}+ \\tilde{r}^\\frac{3}{2} |\\!\\log \\varepsilon |^\\frac{1}{2} + \\frac{\\varepsilon }{\\tilde{r}}|\\!\\log \\varepsilon |^2\\Big ),$ and combining the latter with the definition of $\\Sigma _\\xi $ yields the upper bound ${\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,\\lbrace r\\le \\tilde{r}\\rbrace ) +T_4 \\le \\Sigma _\\xi + C\\Big (\\varepsilon ^\\frac{1}{3}|\\!\\log \\varepsilon |^Ce^{C\\Sigma _a^r}+ \\tilde{r}^\\frac{3}{2} |\\!\\log \\varepsilon |^\\frac{1}{2} + \\frac{\\varepsilon }{\\tilde{r}}|\\!\\log \\varepsilon |^2\\Big ).$ On the other hand, by definition of $r_\\mathcal {S}$ we also have the lower bound ${\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,\\lbrace r\\le r_\\mathcal {S}\\rbrace ) \\ge {\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,\\lbrace r_\\mathcal {S}/2 \\le r \\le r_\\mathcal {S}\\rbrace ) \\ge \\frac{\\pi }{24}r_\\mathcal {S}|\\!\\log \\varepsilon |.$ The comparison of (REF ) specified for $\\tilde{r} = r_\\mathcal {S}$ and (REF ) leads to the conclusion that $r_\\mathcal {S} \\le C \\big ( \\frac{\\Sigma _\\xi }{|\\!\\log \\varepsilon |} + \\varepsilon ^\\frac{1}{3}|\\!\\log \\varepsilon |^Ce^{C\\Sigma _a^r}\\big ).$ Step 7 : improved closeness and upper energy bounds.", "We now choose $\\tilde{r} =C \\big ( \\Sigma _\\xi /|\\!\\log \\varepsilon |+ \\varepsilon ^\\frac{1}{3}|\\!\\log \\varepsilon |^Ce^{C\\Sigma _a^r}\\big )$ in (REF ) to obtain ${\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,\\lbrace r\\le \\tilde{r}\\rbrace )+\\int _{\\mathbb {H}_{\\xi ,\\rho }}\\!\\!\\!\\!\\!", "r \\left[ e_\\varepsilon (|u|) +\\big |\\frac{j(u)}{|u|}-j_*\\big |^2\\right] \\chi ^2 \\le C \\big ( \\Sigma _\\xi + \\varepsilon ^\\frac{1}{3}|\\!\\log \\varepsilon |^Ce^{C\\Sigma _a^r}\\big ).$ The same estimate with $\\tilde{r}$ replaced by half its value, combined with the fact that in the integral of (REF ) the integrand is pointwise dominated by the one of ${\\mathcal {E}}_\\varepsilon ^{\\rm w}$ , allows, in view of the first term of (REF ), to get rid of $\\chi ^2$ in the integrand and conclude that ${\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,\\lbrace r\\le \\tilde{r}\\rbrace )+\\int _{\\mathbb {H}_{\\xi ,\\rho }}\\!\\!\\!\\!\\!", "r \\left[ e_\\varepsilon (|u|) +\\big |\\frac{j(u)}{|u|}-j_*\\big |^2\\right] \\le C \\big ( \\Sigma _\\xi + \\varepsilon ^\\frac{1}{3}|\\!\\log \\varepsilon |^Ce^{C\\Sigma _a^r}\\big ),$ which yields (REF ), for a suitable value of $C_1$ , by taking $\\rho =\\rho _\\epsilon =\\varepsilon ^\\frac{2}{3}.$ Note that combining the lower bound (REF ) (with the error terms now controlled) with the lower bounds (REF ) (used for $\\rho =\\rho _\\epsilon =\\varepsilon ^\\frac{2}{3}$ and for all except one $i$ ) and Lemma REF , we also obtain, in view of the definition of $\\Sigma _a$ , ${\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,B(\\xi _i,\\rho _\\epsilon )) \\le r(\\xi _{i})\\Big (\\pi \\log \\frac{\\rho _\\epsilon }{\\varepsilon } + \\gamma \\Big ) + C\\big (\\Sigma _\\xi +\\varepsilon ^\\frac{1}{3}|\\!\\log \\varepsilon |^Ce^{C\\Sigma _a^r}\\big ),$ so that ${\\mathcal {E}}_{\\varepsilon }(u,B(\\xi _i,\\rho _\\epsilon )) \\le \\pi \\log \\frac{\\rho _\\epsilon }{\\varepsilon } + \\gamma + C\\big (\\Sigma _\\xi +\\varepsilon ^\\frac{1}{3}|\\!\\log \\varepsilon |^Ce^{C\\Sigma _a^r}\\big ),$ for any $1\\le i \\le n.$ Inequality (REF ) is a direct consequence of (REF ) and the explicit form of $H_\\varepsilon .$ Finally, it remains to improve the local estimate (REF ) to the more global one (REF ).", "For that purpose, it suffices to use a (possibly countable) partition of unity, exactly as we did in Step 6, and to rely either on (REF ) or on Theorem REF .", "By the chain rule, the $W^{-1,1}$ norms after the test function is multiplied by the functions of the partition are increased at most by a factor being the sup norm of the gradients of the partition, which in our case is bounded by $C|\\!\\log \\varepsilon |.$ Estimate (REF ) then follows by summation as in (REF ), and adapting $C_1$ if necessary.$\\Box $ Proof of Proposition REF .", "First notice that in view of Remark REF and estimate REF , it suffices to establish an inequality like (REF ) only on each of the balls $B(\\xi _i,\\varepsilon ^\\frac{2}{3})$ .", "The proof is very reminiscent of Step 5 in the proof of Proposition REF .", "We decompose the energy as in (REF ), but with $j_*$ replaced by $j^\\natural $ (here and in the sequel for simplicity we write $j^\\natural $ in place of $j^\\natural (u^*_\\xi )$ ): $e_\\varepsilon (u) = \\frac{1}{2} |j^\\natural |^2 +j^\\natural \\big (\\frac{j(u)}{|u|}-j^\\natural \\big ) +e_\\varepsilon (|u|) + \\frac{1}{2} \\big | \\frac{j(u)}{|u|}-j^\\natural \\big |^2.$ Recall that $\\rho _\\epsilon = \\varepsilon ^\\frac{2}{3}$ and let $\\chi _i$ be a cut-off function with compact support in $B(\\xi _i,2\\rho _\\epsilon )$ and such that $\\chi _i\\equiv 1$ on $B(\\xi _i,\\rho _\\epsilon )$ and $|\\nabla \\chi _i|\\le C/\\rho _\\epsilon .$ On one hand, similar to (REF ) we have the upper bound $\\int r e_\\varepsilon (u) \\chi _i \\le r(\\xi _{i})\\Big (\\pi \\log \\frac{2\\rho _\\epsilon }{\\varepsilon } + \\gamma \\Big ) + C\\big (\\Sigma _a^r +\\varepsilon ^\\frac{1}{3}|\\!\\log \\varepsilon |^Ce^{C\\Sigma _a^r}\\big ).$ On the other hand, by direct computation and the definition (REF ) of $r_\\xi $ we have the lower bound $\\int r\\frac{|j^\\natural |^2}{2}\\chi _i \\ge \\pi r(\\xi _i)\\log \\frac{2\\rho _\\epsilon }{r_\\xi } - C \\ge \\pi r(\\xi _i)\\log \\frac{\\rho _\\epsilon }{\\varepsilon } - C (\\Sigma _a^r+ \\log |\\!\\log \\varepsilon |).$ To conclude, it suffices then to control the cross-term in (REF ).", "We write $\\int r j^\\natural \\big (\\frac{j(u)}{|u|}-j^\\natural \\big ) \\chi _i =\\int r j^\\natural \\big (j(u)-j^\\natural \\big ) \\chi _i +\\int r j^\\natural \\frac{j(u)}{|u|}(|u|-1)\\chi _i$ and then for arbitrary $\\kappa \\in \\mathbb {R},$ $\\begin{split}\\int r j^\\natural \\big (j(u)-j^\\natural \\big ) \\chi _i &= \\int \\nabla ^\\perp (r\\Psi ^\\natural _\\xi -\\kappa )\\big (j(u)-j^\\natural \\big ) \\chi _i\\\\&= -\\int (r\\Psi ^\\natural _\\xi -\\kappa ) {\\rm curl}(j(u)-j^\\natural )\\chi _i -\\int (r\\Psi ^\\natural _\\xi -\\kappa ) (j(u)-j^\\natural )\\nabla ^\\perp \\chi _i.\\end{split}$ Finally, we split $\\int (r\\Psi ^\\natural _\\xi -\\kappa ) (j(u)-j^\\natural )\\nabla ^\\perp \\chi _i =\\int (r\\Psi ^\\natural _\\xi -\\kappa ) (\\frac{j(u)}{|u|}-j^\\natural )\\nabla ^\\perp \\chi _i+\\int (r\\Psi ^\\natural _\\xi -\\kappa ) \\frac{j(u)}{|u|}(|u|-1)\\nabla ^\\perp \\chi _i.$ We choose $\\kappa $ to be the mean value of $r\\Psi ^\\natural _\\xi $ over the support of $\\nabla ^\\perp \\chi _i$ , and therefore in view of the logarithmic nature of $\\Psi ^\\natural _\\xi $ we have the upper bound $|r\\Psi ^\\natural _\\xi -\\kappa |\\le C$ on the support of $\\nabla ^\\perp \\chi _i.$ As in Proposition REF , by Cauchy-Schwarz and the $L^\\infty $ bound on $j^\\natural $ , we estimate $|\\int r j^\\natural \\frac{j(u)}{|u|}(|u|-1)\\chi _i| \\le \\frac{C}{r_\\xi }\\varepsilon {\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,B(\\xi _i,2\\rho _\\epsilon )) \\le C$ and $|\\int (r\\Psi ^\\natural _\\xi -\\kappa ) \\frac{j(u)}{|u|}(|u|-1)\\nabla ^\\perp \\chi _i| \\le C\\frac{\\varepsilon }{\\rho _\\epsilon }{\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,B(\\xi _i,2\\rho _\\epsilon )) \\le C.$ Next, we have $\\begin{split}|\\int (r\\Psi ^\\natural _\\xi -\\kappa )(\\frac{j(u)}{|u|}-j^\\natural )\\nabla ^\\perp \\chi _i| &\\le C\\Vert \\frac{j(u)}{|u|}-j^\\natural \\Vert _{L^2({\\rm supp}(\\nabla \\chi _i))} \\Vert \\nabla \\chi _i\\Vert _{L^2}\\\\&\\le C\\big (\\Sigma _a^r +\\varepsilon ^\\frac{1}{3}|\\!\\log \\varepsilon |^Ce^{C\\Sigma _a^r}\\big )^\\frac{1}{2}\\\\&\\le C(\\Sigma _a^r+1).\\end{split}$ For the last term, we write $\\begin{split}|\\int (r\\Psi ^\\natural _\\xi -\\kappa ) {\\rm curl}(j(u)-j^\\natural )\\chi _i|&\\le C \\Vert 2J(u)-{\\rm curl}j^\\natural \\Vert _{W^{-1,1}(B(\\xi _i,2\\rho _\\epsilon ))} \\Vert (r\\Psi ^\\natural _\\xi -\\kappa )\\chi _i\\Vert _{W^{1,\\infty }}\\\\&\\le C r_\\xi \\frac{1}{r_\\xi } \\le C,\\end{split}$ where we have used (REF ) and the fact that by construction $\\Vert {\\rm curl}j^\\natural -2\\pi \\sum _{i=1}^n \\delta _{\\xi _i}\\Vert _{\\dot{W}^{-1,1}(B(\\xi _i,2\\rho _\\epsilon ))}\\le C r_\\xi .$ The conclusion follows.$\\Box $ Proof of Proposition REF .", "Since $j^\\natural (u_\\xi ^*)$ is not sufficiently regular across the boundaries of the sets $\\mathcal {C}_i$ , defined after (REF ), the computation which follows (REF ) does not hold as is with $X$ replaced by $j^\\natural (u_\\xi ^*)$ and we need instead to divide the integration domain $\\mathbb {H}$ into the union of the pieces $\\mathcal {C}_i$ and of the complement of this union.", "Performing the integration by parts then imply (only) some boundary terms, which actually end up in justifying (REF ) provided ${\\rm curl X}$ is understood in a weak sense according to (REF ) and ${\\rm div}(rX)$ according to (REF ), namely $\\mathcal {F}\\big (j^\\natural (u_\\xi ^*),\\varphi \\big )=-\\sum _{i=1}^n\\int _{\\partial \\mathcal {C}_i} |j(u_\\xi ^*)|j(u_\\xi ^*)\\cdot \\nabla \\varphi .$ For each fixed $i$ , to compute the boundary term on $\\partial \\mathcal {C}_i$ we use a reference polar frame $(\\rho ,\\theta )$ centered at $\\xi _i$ .", "First by construction of $\\mathcal {C}_i$ and (REF )-(REF ) we have $\\rho = r_\\xi +O\\big (r_\\xi ^2\\log (r_\\xi )\\big ),$ so that $\\mathcal {C}_i$ is close to being a circle, and then by (REF ), (REF ) and (REF ), $j^\\natural (u_\\xi ^*) = \\frac{e_\\theta }{r_\\xi } + \\sum _j \\mathbb {J}\\nabla _{a_j}H_\\varepsilon (\\xi _1,\\cdots ,\\xi _n) + O(\\Sigma _a^r + \\log |\\!\\log \\varepsilon |) \\qquad \\text{on } \\partial \\mathcal {C}_i,$ where the main error term, of order $\\Sigma ^r_a + \\log |\\!\\log \\varepsilon |$ , comes from the difference between $|\\!\\log \\varepsilon |$ (as appearing in the definition of $H_\\varepsilon $ ) and $\\log r_\\xi $ (from the value of $\\Psi ^*_\\xi $ on $\\mathcal {C}_i$ ).", "The computation of the right-hand-side of (REF ) is then a direct consequence of (REF ) and (REF ), with a cancellation at main order since $\\frac{e_\\theta }{\\rho }$ integrates to zero on a circle.", "The actual details are left to the reader.", "$\\Box $ Proof of Proposition REF .", "Since estimate (REF ) is only valid for $r$ not too close to zero, we shall split $\\chi _\\varepsilon $ into two pieces.", "More precisely, we write $1 = \\Psi _1 + \\Psi _2$ where $\\Psi _1$ is supported in $r \\le 2\\tau \\equiv 2C_1(\\Sigma _a^r+1)/|\\!\\log \\varepsilon |$ , $\\Psi _2$ is supported in $r\\ge \\tau $ , and $|\\nabla \\Psi _1|+|\\nabla \\Psi _2| \\le 10/\\tau .$ Using (REF ) we immediately obtain $\\left|\\int Ju r^2\\chi _\\varepsilon \\Psi _2 - \\pi \\sum _{i=1}^n r(\\xi _i)^2\\right| \\le C_1 \\varepsilon |\\!\\log \\varepsilon |^{C_1}e^{C_1\\Sigma _a^r}\\Vert r^2\\chi _R\\Psi _2\\Vert _{W^{1,\\infty }} \\le C \\varepsilon |\\!\\log \\varepsilon |^{C_1}e^{C_1\\Sigma _a^r} R_\\varepsilon ^2.$ To estimate the part involving $\\Psi _1$ , and in particular the singularity at $r=0,$ we use Theorem REF (more precisely its higher dimensional extension - see e.g.", "[12]) in the 3D cylinder in cartesian coordinates corresponding to $r\\le 2\\tau $ and $|z| \\le 2R_\\varepsilon .$ Writing back its statement in cylindrical coordinates yields $\\begin{split}\\left|\\int Ju r^2\\chi _\\varepsilon \\Psi _1 \\right| \\le & C \\ \\frac{{\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,\\lbrace r \\le 2\\tau \\rbrace )}{|\\!\\log \\varepsilon |}\\Vert r\\chi _\\varepsilon \\Psi _1\\Vert _\\infty \\\\&\\ + C \\varepsilon ^\\frac{1}{24}(1+ \\frac{{\\mathcal {E}}_\\varepsilon ^{\\rm w}(u,\\lbrace r \\le 2\\tau \\rbrace )}{|\\!\\log \\varepsilon |} )(1+C\\tau ^2R_\\varepsilon )\\Vert r\\chi _\\varepsilon \\Psi _1\\Vert _{\\mathcal {C}^{0,1}}\\\\\\le & \\ C \\frac{1+\\Sigma _a^r}{|\\!\\log \\varepsilon |}\\tau \\\\\\le & \\ C \\frac{(1+\\Sigma _a^r)^2}{|\\!\\log \\varepsilon |^2,}\\end{split}$ provided $\\varepsilon $ is required to be sufficiently small.", "By summation we obain (REF ).", "To obtain (REF ), we notice that in the expansion (REF ) the terms for which the derivatives of $\\varphi $ fall onto $r^2$ exactly cancel (that would correspond without cut-off to the conservation of the momentum) and the remaining ones (where the derivatives fall onto $\\chi _\\varepsilon $ ) are pointwise bounded by $C e_\\varepsilon (u)r|\\nabla \\chi _\\varepsilon |$ , so that the conclusion follows by integration and (REF ).$\\Box $ Proof of Proposition REF .", "In this proof $\\Vert \\cdot \\Vert $ is understood to mean $\\dot{W}^{-1,1}(\\Omega _0)$ and $|\\cdot |$ refers to the Euclidean norm on $\\mathbb {H}.$ We write $\\begin{split}\\Vert Ju_\\varepsilon ^{s_1}-Ju_\\varepsilon ^{s_2}\\Vert &\\le \\Vert Ju_\\varepsilon ^{s_1}-\\pi \\sum _{i=1}^n \\delta _{\\xi _i(s_1)}\\Vert +\\Vert Ju_\\varepsilon ^{s_2}-\\pi \\sum _{i=1}^n \\delta _{\\xi _i(s_2)}\\Vert +\\Vert \\pi \\sum _{i=1}^n (\\delta _{\\xi _i(s_1)}-\\delta _{\\xi _i(s_2)})\\Vert \\\\&\\le r_\\xi ^{s_1}+r_\\xi ^{s_2} + \\pi \\sum _{i=1}^n|\\xi _i(s_1)-\\xi _i(s_2)|.\\end{split}$ If $C_6$ is chosen sufficiently large, it follows from the separation assumption (REF ), the finite speed of propagation of $(\\text{LF})_\\varepsilon $ , and the definition of $S_{\\rm stop}$ , that $\\xi _i(s) \\in B(a_{i,\\varepsilon }(s_1),\\frac{\\rho _{\\rm min}}{4}) \\qquad \\forall \\ s \\in [s_1,s_2].$ Let $\\varphi (x) = \\sum _{i=1}^n\\frac{(x-a_{i,\\varepsilon }(s_1))\\cdot (\\xi _i(s_2)-\\xi _i(s_1))}{|\\xi _i(s_2)-\\xi _i(s_1)|} \\chi \\Big (|x-a_{i,\\varepsilon }(s_1)|\\Big ),$ where $\\chi \\in \\mathcal {C}^\\infty (\\mathbb {R}^+,[0,1])$ is such that $\\chi \\equiv 1$ on $[0,\\rho _{\\rm min}/4],$ $\\chi \\equiv 0$ on $[\\frac{\\rho _{\\rm min}}{2},+\\infty ).$ By construction and the definition of $\\rho _{\\rm min}$ , we have $\\varphi \\in \\mathcal {D}(\\Omega _0)$ and it follows that $\\begin{split}\\pi \\sum _{i=1}^n |\\xi _i(s_1)-\\xi _i(s_2)|&= \\langle \\pi \\sum _{i=1}^n(\\delta _{\\xi _i(s_2)}-\\delta _{\\xi _i(s_1)}), \\varphi \\rangle \\\\&\\le (r_\\xi ^{s_1} + r_\\xi ^{s_2})\\Vert \\varphi \\Vert _{W^{1,\\infty }} +\\langle Ju_\\varepsilon ^{s_2} - J u_\\varepsilon ^{s_1} ,\\varphi \\rangle .\\end{split}$ Combining this with (REF ), we conclude that $\\Vert Ju_\\varepsilon ^{s_2} - Ju_\\varepsilon ^{s_1}\\Vert \\le C( r^{s_1}_\\xi + r^{s_2}_\\xi ) +\\langle Ju_\\varepsilon ^{s_2} - J u_\\varepsilon ^{s_1} ,\\varphi \\rangle .$ By (REF ), $\\left|\\langle Ju_\\varepsilon ^{s_2} - J u_\\varepsilon ^{s_1} ,\\varphi \\rangle \\right| \\le \\frac{1}{|\\!\\log \\varepsilon |}|\\int _{s_1}^{s_2} \\mathcal {F}(\\nabla u_\\varepsilon ^s,\\varphi )\\, ds|.$ Recall that $\\mathcal {F}(\\nabla u_\\varepsilon ^s, \\varphi ):= -\\int _\\mathbb {H}\\varepsilon _{ij} \\frac{\\partial _k r}{r} (\\partial _j u_\\varepsilon ^s ,\\partial _k u_\\varepsilon ^s)\\partial _i\\varphi +\\int _\\mathbb {H}\\varepsilon _{ij}(\\partial _ju_\\varepsilon ^s,\\partial _k u_\\varepsilon ^s)\\partial _{ik}\\varphi ,$ and that by (REF ) we have $\\partial _{ik}\\varphi \\equiv 0 \\qquad \\text{on } \\cup _i B(a_{i,\\varepsilon }(s_1),\\frac{\\rho _{\\rm min}}{4}).$ Since $|\\nabla \\varphi |\\le C$ and $|D^2\\varphi | \\le C/\\rho _{\\rm min}$ , we have $\\left|\\int _\\mathbb {H}\\varepsilon _{ij} \\frac{\\partial _k r}{r} (\\partial _j u_\\varepsilon ^s ,\\partial _k u_\\varepsilon ^s)\\partial _i\\varphi \\right|\\le C{\\mathcal {E}}_\\varepsilon ^{\\rm w}(u_\\varepsilon ^s) \\le C|\\!\\log \\varepsilon |$ and by (REF ), (REF ) and (REF ) $\\left| \\int _\\mathbb {H}\\varepsilon _{ij}(\\partial _ju_\\varepsilon ^s,\\partial _k u_\\varepsilon ^s)\\partial _{ik}\\varphi \\right| \\le \\frac{C}{\\rho _{\\rm min}}\\left( \\Sigma ^0+r_a^s|\\!\\log \\varepsilon |+\\log |\\!\\log \\varepsilon |\\right) \\le C |\\!\\log \\varepsilon |.$ Going back to (REF ) we thus obtain $\\left|\\langle Ju_\\varepsilon ^{s_2} - J u_\\varepsilon ^{s_1} ,\\varphi \\rangle \\right| \\le C|s_1-s_2|$ and therefore $\\Vert Ju_\\varepsilon ^{s_1}-Ju_\\varepsilon ^{s_2}\\Vert \\le C(r^{s_1}_\\xi + r^{s_2}_\\xi + |s_1-s_2|).$ It remains to estimate $r_\\xi ^{s_2}.$ For that purpose, we write $\\begin{split}r_a^{s_2} &= \\Vert J u_\\varepsilon ^{s_2} - \\pi \\sum _{i=1}^n \\delta _{a_{i,\\varepsilon }(s_2)}\\Vert \\\\&\\le \\Vert Ju_\\varepsilon ^{s_2} - J u_\\varepsilon ^{s_1}\\Vert + \\Vert Ju_\\varepsilon ^{s_1} -\\pi \\sum _{i=1}^n\\delta _{a_{i,\\varepsilon }(s_1)}\\Vert + \\Vert \\pi \\sum _{i=1}^n(\\delta _{a_{i,\\varepsilon }(s_1)}-\\delta _{a_{i,\\varepsilon }(s_2)})\\Vert \\\\&\\le r_a^{s_1} + C\\bigl ( |s_1-s_2| + r_\\xi ^{s_1} + r_\\xi ^{s_2} \\bigr ),\\end{split}$ where we have used (REF ) and (REF ).", "By the definition (REF ) of $r_\\xi ^{s_2}$ and (REF ) we obtain $\\begin{split}r_\\xi ^{s_2} &\\le C_1\\varepsilon |\\!\\log \\varepsilon |^{C_1}e^{C_1(\\Sigma ^0+r_a^{s_2}|\\!\\log \\varepsilon |)}\\\\&\\le r_\\xi ^{s_1}e^{C(|s_2-s_1|+2\\varepsilon ^\\frac{5}{6})|\\!\\log \\varepsilon |}\\\\&\\le r_\\xi ^{s_1}(1+C\\left(|s_2-s_1|+\\varepsilon ^\\frac{5}{6}\\right)|\\!\\log \\varepsilon |),\\end{split}$ where we have used the fact that $|s_2-s_1| \\le |\\!\\log \\varepsilon |^{-1}$ by assumption.", "It remains to prove the last assertion of the statement, namely that if $r_a^{s_1} < \\rho _{\\rm min}/16$ then $S_{\\rm stop}\\ge s_1+(C_6\\sqrt{|\\!\\log \\varepsilon |})^{-1}.$ By definition of $S_{\\rm stop},$ the latter follows easily from (REF ) and (REF ), increasing the value of $C_6$ if necessary.", "$\\Box $ Proof of Proposition REF .", "The proof follows very closely the strategy used in [11] Proposition 7.1.", "By () and the definition of $S$ we first remark that $r_\\xi ^\\tau \\le 2 r_\\xi ^s \\qquad \\forall \\tau \\in [s,S].$ Next, note that $\\begin{split}r^S_a - r^s_a &=\\Vert Ju_\\varepsilon ^S - \\pi \\sum _{i=1}^n \\delta _{a_{i,\\varepsilon }(S)}\\Vert -\\Vert Ju_\\varepsilon ^s - \\pi \\sum _{i=1}^n \\delta _{a_{i,\\varepsilon }(s)}\\Vert \\\\&\\le \\pi \\sum _{i=1}^n \\left(|\\xi _i(S)-a_{i,\\varepsilon }(S)| - |\\xi _i(s) - a_{i,\\varepsilon }(s)| \\right)\\ + \\ r^S_\\xi + r^s_\\xi \\\\&\\le \\pi \\sum _{i=1}^n \\nu _i \\cdot \\bigl (\\xi _i(S)- \\xi _i(s) + a_{i,\\varepsilon }(s)- a_{i,\\varepsilon }(S) \\bigr )\\ + \\ r^S_\\xi + r^s_\\xi \\end{split}$ for $\\nu _i = \\frac{\\xi _i(S)-a_{i,\\varepsilon }(S)}{|\\xi _i(S)-a_{i,\\varepsilon }(S)|}$ (unless $\\xi _i(S)- a_{i,\\varepsilon }(S)=0$ , in which case $\\nu _i$ can be any unit vector).", "We let $\\varphi (x) = \\sum _i \\nu _i \\cdot (x - a_{i,\\varepsilon }(s)) \\chi (|x - a_{i,\\varepsilon }(s)|)$ for $\\chi \\in C^\\infty (\\mathbb {R}^+, [0,1])$ such that $\\chi \\equiv 1$ on $[0,\\frac{1}{4} \\rho _{\\rm min}]$ and $\\chi \\equiv 0$ on $(\\frac{1}{2} \\rho _{\\rm min},\\infty )$ .", "It follows from () that $\\pi \\sum _{i=1}^n \\nu _i \\cdot \\bigl (\\xi _i(S)- \\xi _i(s) + a_{i,\\varepsilon }(s)- a_{i,\\varepsilon }(S) \\bigr )\\ =\\pi \\sum _{i=1}^n \\Bigl [ \\varphi (\\xi _i(S))- \\varphi (\\xi _i(s)) - \\varphi (a_{i,\\varepsilon }(S)) + \\varphi (a_{i,\\varepsilon }(s)) \\Bigr ],$ so that (REF ) and the definition of $r^S_\\xi $ imply that $r^S_a-r^s_a \\le \\langle \\varphi , Ju_\\varepsilon ^S - Ju_\\varepsilon ^s \\rangle \\ -\\ \\pi \\sum _{i=1}^n \\Bigl [\\varphi (a_{i,\\varepsilon }(S)) - \\varphi (a_{i,\\varepsilon }(s))\\Bigr ]+\\ C(r^S_\\xi + r^s_\\xi ).$ Our main task in the sequel is therefore to provide an estimate for the quantity $ \\langle \\varphi , Ju_\\varepsilon ^S - Ju_\\varepsilon ^s \\rangle $ .", "By (REF ) (and taking into account the $|\\!\\log \\varepsilon |$ change of scale in time) we have $\\langle \\varphi , Ju_\\varepsilon ^S - Ju_\\varepsilon ^s \\rangle = \\int _s^S \\frac{1}{|\\!\\log \\varepsilon |} \\mathcal {F}(\\nabla u_\\varepsilon ^\\tau ,\\varphi )\\, d\\tau .$ In the sequel for the ease of notation we write $u$ in place of $u_\\varepsilon ^\\tau $ , for $\\tau \\in [s,S].$ Similar to what we did in (REF ), and in view of the definition (REF ) of $\\mathcal {F},$ we decompose here $\\begin{split}\\left( \\partial _j u,\\partial _k u \\right) &= \\partial _j |u|\\partial _k |u| + \\frac{j(u)_j}{|u|}\\frac{j(u)_k}{|u|}\\\\&= \\partial _j |u|\\partial _k |u| + \\left(\\frac{j(u)}{|u|}-j^\\natural \\right)_j\\left(\\frac{j(u)}{|u|}-j^\\natural \\right)_k\\\\&\\quad + \\left(\\frac{j(u)}{|u|}-j^\\natural \\right)_j\\left(j^\\natural \\right)_k +\\left(\\frac{j(u)}{|u|}-j^\\natural \\right)_k\\left(j^\\natural \\right)_j\\\\&\\quad + \\left(j^\\natural \\right)_j\\left(j^\\natural \\right)_k,\\end{split}$ where $j^\\natural \\equiv j^\\natural (u_{\\xi (\\tau )}^*).$ Hence, $\\mathcal {F}(\\nabla u,\\varphi ) = \\mathcal {F}(j^\\natural ,\\varphi )+ \\sum _{p=1}^4 T_p$ where $T_1 := -\\int _\\mathbb {H}\\varepsilon _{ij} \\frac{\\partial _k r}{r} \\left[\\partial _j |u|\\partial _k |u| + \\left(\\frac{j(u)}{|u|}-j^\\natural \\right)_j\\left(\\frac{j(u)}{|u|}-j^\\natural \\right)_k\\right]\\partial _i\\varphi ,$ $T_2 := \\int _\\mathbb {H}\\varepsilon _{ij}\\left[\\partial _j |u|\\partial _k |u| +\\left(\\frac{j(u)}{|u|}-j^\\natural \\right)_j\\left(\\frac{j(u)}{|u|}-j^\\natural \\right)_k\\right]\\partial _{ik}\\varphi ,$ $T_3 := -\\int _\\mathbb {H}\\varepsilon _{ij} \\frac{\\partial _k r}{r} \\left[\\left(\\frac{j(u)}{|u|}-j^\\natural \\right)_j\\left(j^\\natural \\right)_k +\\left(\\frac{j(u)}{|u|}-j^\\natural \\right)_k\\left(j^\\natural \\right)_j \\right]\\partial _i\\varphi ,$ and $T_4 := \\int _\\mathbb {H}\\varepsilon _{ij} \\left[\\left(\\frac{j(u)}{|u|}-j^\\natural \\right)_j\\left(j^\\natural \\right)_k +\\left(\\frac{j(u)}{|u|}-j^\\natural \\right)_k\\left(j^\\natural \\right)_j \\right]\\partial _{ik}\\varphi .$ By Proposition REF we already know that $\\left| \\mathcal {F}(j^\\natural ,\\varphi ) - \\sum _{i=1}^n\\mathbb {J}\\nabla _{a_i}H_\\varepsilon (\\xi _1(\\tau ),\\cdots ,\\xi _n(\\tau ))\\cdot \\nabla \\varphi (\\xi _i(\\tau )) \\right| \\le C_3\\left(\\Sigma ^0+r_a^\\tau |\\!\\log \\varepsilon |+ \\log |\\!\\log \\varepsilon |\\right),$ moreover since $S \\le S_{\\rm stop}$ we have for any $i=1,\\cdots ,n,$ $\\left|\\nabla _{a_i}H_\\varepsilon (\\xi _1(\\tau ),\\cdots ,\\xi _n(\\tau ))-\\nabla _{a_i}H_\\varepsilon (a_{1,\\varepsilon }(\\tau ),\\cdots ,a_{n,\\varepsilon }(\\tau ))\\right| \\le C |\\!\\log \\varepsilon |r_a^\\tau $ and since $\\varphi $ is affine there, $\\nabla \\varphi (\\xi _i(\\tau )) = \\nabla \\varphi (a_{i,\\varepsilon }(\\tau )),$ so that after integration and using the fact that the points $a_{i,\\varepsilon }$ evolve according to the ODE $(\\text{LF})_\\varepsilon $ we obtain $\\left|\\int _s^S \\frac{1}{|\\!\\log \\varepsilon |}\\mathcal {F}(j^\\natural ,\\varphi )\\, d\\tau -\\pi \\sum _{i=1}^n \\Bigl [\\varphi (a_{i,\\varepsilon }(S)) - \\varphi (a_{i,\\varepsilon }(s))\\Bigr ]\\right|\\le C(S-s)\\left(r_a^s + \\frac{\\Sigma ^0}{|\\!\\log \\varepsilon |} +\\frac{\\log |\\!\\log \\varepsilon |}{|\\!\\log \\varepsilon |}\\right).$ Now that we have accounted for the main order, we need to control all the terms $T_p$ (at least integrated in time between $s$ and $S$ ).", "We begin with the terms $T_1$ and $T_2$ for which we already have good estimates (pointwise in time) thanks to Proposition REF and Proposition REF .", "Indeed, by Proposition REF and since $|\\nabla \\varphi | \\le C$ , we have $\\frac{|T_1|}{|\\!\\log \\varepsilon |} \\le C\\left( r_a^s + \\frac{\\Sigma ^0}{|\\!\\log \\varepsilon |} +\\frac{\\log |\\!\\log \\varepsilon |}{|\\!\\log \\varepsilon |}\\right).$ By Proposition REF and (REF ), and since $|D^2\\varphi | \\le C\\sqrt{|\\!\\log \\varepsilon |}$ , we have $\\frac{|T_2|}{|\\!\\log \\varepsilon |} \\le C\\left( r_a^s + \\frac{\\Sigma ^0}{\\sqrt{|\\!\\log \\varepsilon |}}+|\\!\\log \\varepsilon |^{-\\frac{3}{2}} + \\sqrt{|\\!\\log \\varepsilon |}\\Big |P_\\varepsilon (u)-P\\big (a_{1,\\varepsilon }(\\tau )\\cdots ,a_{n,\\varepsilon }(\\tau )\\big )\\Big | \\right).$ In order to deal with the last term involving $P$ and $P_\\varepsilon $ , recall first that $P$ is preserved by the flow $(\\text{LF})_\\varepsilon $ , so that $P\\big (a_{1,\\varepsilon }(\\tau ),\\cdots ,a_{n,\\varepsilon }(\\tau )\\big ) =P\\big (a_{1,\\varepsilon }(0),\\cdots ,a_{n,\\varepsilon }(0)\\big ),$ and that $P_\\varepsilon $ is almost preserved by $(\\text{GP})_\\varepsilon ^{c}$ , as expressed by (REF ), so that $\\left| P_\\varepsilon (u)- P_\\varepsilon (u_\\varepsilon ^0)\\right| \\le C\\frac{S_{\\rm stop}}{|\\!\\log \\varepsilon |^3}\\left(\\Sigma ^0+ \\sqrt{|\\!\\log \\varepsilon |}\\right) \\le C S_{\\rm stop} |\\!\\log \\varepsilon |^{-\\frac{5}{2}}$ where we have used the rough bound (REF ) for $\\Sigma _\\xi $ , the rough estimate $r_a^\\tau \\le C/\\sqrt{|\\!\\log \\varepsilon |}$ which follows from the definition of $S_{\\rm stop}$ , and where we have taken into account the factor $|\\!\\log \\varepsilon |^{-1}$ which arises from the change of time scale which we have here with respect to the one of (REF ).", "On the other hand, at the initial time by (REF ) and the bound (REF ) on $r_a^0$ and $\\Sigma ^0$ we have $\\left|P_\\varepsilon (u_\\varepsilon ^0)- P\\big (\\lbrace a_{i,\\varepsilon }(0)\\rbrace \\big )\\right| \\le \\left|P_\\varepsilon (u_\\varepsilon ^0)- P\\big (\\lbrace \\xi _{i}(0)\\rbrace \\big )\\right| + Cr_a^0 \\le C\\left(r_a^0+\\frac{1}{|\\!\\log \\varepsilon |^2}\\right).$ In total, similar to (REF ) we obtain $\\frac{|T_2|}{|\\!\\log \\varepsilon |} \\le C\\left( r_a^s+\\frac{\\Sigma ^0+r_a^0|\\!\\log \\varepsilon |}{\\sqrt{|\\!\\log \\varepsilon |}} +\\frac{\\log |\\!\\log \\varepsilon |}{|\\!\\log \\varepsilon |}\\right),$ where we have absorbed some of the above error terms by the term $\\log |\\!\\log \\varepsilon |/|\\!\\log \\varepsilon |.$ We decompose $T_3 = T_{3,1}+T_{3,2}+T_{3,3}$ where $T_{3,1}:= -\\int _\\mathbb {H}\\varepsilon _{ij} \\frac{\\partial _k r}{r} \\left[\\left(\\frac{j(u)}{|u|}-j^\\natural \\right)_j\\left(j^\\natural -j^\\natural (u_{\\xi (s)}^*)\\right)_k +\\left(\\frac{j(u)}{|u|}-j^\\natural \\right)_k\\left(j^\\natural -j^\\natural (u_{\\xi (s)}^*)\\right)_j \\right]\\partial _i\\varphi ,$ $T_{3,2}:= -\\int _\\mathbb {H}\\varepsilon _{ij} \\frac{\\partial _k r}{r} \\left[\\left(j(u)-j^\\natural \\right)_j\\left(j^\\natural (u_{\\xi (s)}^*)\\right)_k +\\left(j(u)-j^\\natural \\right)_k\\left(j^\\natural (u_{\\xi (s)}^*)\\right)_j \\right]\\partial _i\\varphi ,$ and $T_{3,3}:= -\\int _\\mathbb {H}\\varepsilon _{ij} \\frac{\\partial _k r}{r} \\left[\\left(\\frac{j(u)}{|u|}\\right)_j\\left(j^\\natural (u_{\\xi (s)}^*)\\right)_k +\\left(\\frac{j(u)}{|u|}\\right)_k\\left(j^\\natural (u_{\\xi (s)}^*)\\right)_j \\right](1-|u|)\\partial _i\\varphi ,$ and accordingly we decompose $T_4 = T_{4,1}+T_{4,2}+T_{4,3}.$ We first deal with $T_{3,3}$ and $T_{4,3}$ , where invoking the inequality $\\frac{|j(u)|}{|u|}(1-|u|) \\le \\frac{\\varepsilon }{2} \\left(\\frac{|j(u)|^2}{|u|^2} + \\frac{(1-|u|)^2}{\\varepsilon ^2}\\right) \\le C\\varepsilon e_\\varepsilon (u)$ combined with the global $|\\!\\log \\varepsilon |$ bound on the energy and the $L^\\infty $ bound $|j^\\natural | \\le Cr_\\xi ^{-1} \\le C\\varepsilon ^{-1}/|\\!\\log \\varepsilon |^{C_1}$ we directly infer (increasing $C_1$ if necessary) that $\\frac{T_{3,3}+T_{4,3}}{|\\!\\log \\varepsilon |} \\le \\frac{C}{|\\!\\log \\varepsilon |}.$ We next turn to the terms $T_{3,1}$ and $T_{4,1}$ , for which we rely on Proposition REF and the definition of $S$ to get the upper bound $\\sum _{i=1}^n |\\xi _i(s)-\\xi _i(\\tau )| \\le C\\left( r_\\xi ^s + \\frac{(r_\\xi ^s)^2}{\\varepsilon }\\right)\\le C \\frac{(r_\\xi ^s)^2}{\\varepsilon }.$ Using the almost explicit form of $j^\\natural $ (more precisely (REF ), (REF ) and the definition of the cut-off at the scale $r_\\xi ^s$ ), we compute that $\\int _{{\\rm supp}(\\varphi )} \\left| j^\\natural -j^\\natural (u_{\\xi (s)}^*)\\right|^2 \\le C(1+\\log \\left(\\frac{\\frac{(r_\\xi ^s)^2}{\\varepsilon }}{r_\\xi ^s}\\right)) \\le C\\left( \\Sigma ^0+r_a^s|\\!\\log \\varepsilon |+ \\log |\\!\\log \\varepsilon |\\right).$ The previous inequality combined with Proposition REF and the Cauchy-Schwarz inequality then yields $\\frac{T_{3,1}}{|\\!\\log \\varepsilon |} \\le C\\left( r_a^s + \\frac{\\Sigma ^0}{|\\!\\log \\varepsilon |} + \\frac{\\log |\\!\\log \\varepsilon |}{|\\!\\log \\varepsilon |}\\right).$ For $T_{4,1}$ , since the integration domain does no longer contain the cores we obtain the stronger estimate $\\int _{{\\rm supp}(D^2\\varphi )} \\left| j^\\natural -j^\\natural (u_{\\xi (s)}^*)\\right|^2 \\le C \\frac{(\\sum _{i=1}^n |\\xi _i(s)-\\xi _i(\\tau )|)^2}{\\rho _{\\rm min}^2} \\le C\\varepsilon ^\\frac{2}{3}|\\!\\log \\varepsilon |,$ where we have used (REF ) and (REF ) for the last inequality.", "Using once more the Cauchy-Schwarz inequality, combined here with Proposition REF (or even simply the crude $|\\!\\log \\varepsilon |$ global energy bound) and the $L^\\infty $ bound $|D^2\\varphi | \\le C\\sqrt{|\\!\\log \\varepsilon |}$ we obtain $\\frac{T_{4,1}}{|\\!\\log \\varepsilon |} \\le C\\varepsilon ^\\frac{1}{3} \\sqrt{|\\!\\log \\varepsilon |}.$ At this stage we are left to estimate $T_{3,2}$ and $T_{4,2}$ , which we will only be able to do after integration in time.", "To underline better the time dependence, it is convenient here to write $j^\\natural _\\tau $ in place of $j^\\natural $ and $j^\\natural _s$ in place of $j^\\natural (u_\\xi ^s).$ The main ingredient in the argument is then to perform a Helmholtz type decomposition of $j(u)-j^\\natural _\\tau .$ More precisely, we first fix a cut-off function $\\chi $ with compact smooth support $B$ in $\\lbrace r \\ge \\frac{r_0}{8}\\rbrace $ , which is identically equal to 1 on the support of $\\varphi $ and which satisfies $|\\nabla \\chi | \\le C$ (its only aim is to get rid of boundary terms, of spatial infinity, and of the singularity at $r=0$ ).", "For every $\\tau \\in [s,S],$ we then set $\\chi (j(u)-j^\\natural _\\tau ) = \\nabla f^\\tau + \\frac{1}{r}\\nabla ^\\perp g^\\tau \\qquad \\text{in } B,$ where $f^\\tau $ and $g^\\tau $ are the unique solutions of the Neumann $\\begin{array}{ll}{\\rm div}\\left( r \\nabla f^\\tau \\right) = {\\rm div}\\left( \\chi r(j(u)-j^\\natural _\\tau )\\right) & \\text{ in } B,\\\\\\partial _n f^\\tau = 0 & \\text{ on } \\partial B,\\\\\\int _B f^\\tau = 0,\\end{array}$ and Dirichlet $\\begin{array}{ll}-{\\rm div}\\left( \\frac{1}{r} \\nabla g^\\tau \\right) = {\\rm curl}\\left( \\chi (j(u)-j^\\natural _\\tau )\\right) & \\text{ in } B,\\\\g^\\tau = 0 & \\text{ on } \\partial B\\end{array}$ boundary value problems.", "By construction, $\\int _s^S \\frac{T_{3,2}}{|\\!\\log \\varepsilon |}\\,d\\tau = -\\frac{1}{|\\!\\log \\varepsilon |}\\int _B \\varepsilon _{ij} \\frac{\\partial _k r}{r}\\left[(\\nabla F + \\frac{\\nabla ^\\perp G}{r})_j(j^\\natural _s)_k+(\\nabla F + \\frac{\\nabla ^\\perp G}{r})_k(j^\\natural _s)_j \\right] \\partial _i\\varphi $ and $\\int _s^S \\frac{T_{4,2}}{|\\!\\log \\varepsilon |}\\,d\\tau = -\\frac{1}{|\\!\\log \\varepsilon |}\\int _B \\varepsilon _{ij}\\left[(\\nabla F + \\frac{\\nabla ^\\perp G}{r})_j(j^\\natural _s)_k+(\\nabla F + \\frac{\\nabla ^\\perp G}{r})_k(j^\\natural _s)_j \\right] \\partial _{ik}\\varphi $ where $F=\\int _s^S f^\\tau \\,d\\tau $ and $G=\\int _s^S g^\\tau \\,d\\tau .$ Integrating (REF ), we split $F=F_1+F_2+F_3$ where ${\\rm div}\\left( r \\nabla F_p\\right) = L_p \\text{ in } B,\\qquad \\partial _n F_p = 0 \\text{ on } \\partial B, \\qquad \\int _B F_p = 0,$ for $p=1,2,3,$ and where $L_1:= \\chi \\int _s^S {\\rm div}(rj(u)),\\qquad L_2:=r\\nabla \\chi \\cdot \\int _s^S \\frac{j(u)}{|u|}(|u|-1),\\qquad L_3:= r\\nabla \\chi \\cdot \\int _s^S (\\frac{j(u)}{|u|}-j^\\natural _\\tau ).$ Similarly, integrating (REF ) we split $G=G_1+G_2+G_3$ where ${\\rm div}\\left( \\frac{1}{r} \\nabla G_p\\right) = M_p \\text{ in } B,\\qquad M_p = 0 \\text{ on } \\partial B,$ for $p=1,2,3,$ and where $M_1:= \\chi \\int _s^S {\\rm curl}(j(u)-j^\\natural _\\tau ),\\qquad M_2:=\\nabla ^\\perp \\chi \\cdot \\int _s^S \\frac{j(u)}{|u|}(|u|-1),\\qquad M_3:= \\nabla ^\\perp \\chi \\cdot \\int _s^S (\\frac{j(u)}{|u|}-j^\\natural _\\tau ).$ Before we state precise bounds for each of them, we note that it should be clear at this stage that all the terms $L_p$ and $M_p$ are small in some sense, except perhaps for the term $L_1$ which requires some more explanation.", "For that last term, we rely on the continuity equation (and this is the main reason for the integration in time) $\\partial _t |u|^2 = \\frac{2}{r}{\\rm div}(rj(u))$ which is a consequence of $(\\text{GP})_\\varepsilon ^{c}$ , and from which we infer that $L_1 = \\varepsilon \\frac{\\chi }{|\\!\\log \\varepsilon |}\\left[ \\frac{(|u|^2-1)}{\\varepsilon } \\right]^S_s,$ so that $\\Vert L_1\\Vert _{L^2} \\le C\\frac{\\varepsilon }{\\sqrt{|\\!\\log \\varepsilon |}}\\le C\\frac{(S-s)}{\\sqrt{|\\!\\log \\varepsilon |}}\\frac{\\varepsilon ^2}{(r_\\xi ^s)^2}\\le C(S-s) |\\!\\log \\varepsilon |^{-2C_1},$ where we have used the definitions of $S$ and $r_\\xi ^s.$ Regarding $L_2$ and $M_2$ , using (REF ) we easily obtain $\\Vert L_2\\Vert _{L^1}+\\Vert M_2\\Vert _{L^1} \\le C\\varepsilon (S-s)|\\!\\log \\varepsilon |.$ For $L_3$ and $M_3$ , we use the fact that $\\nabla \\chi $ lives away from the cores so that Proposition REF and Proposition REF yield (we bound the terms involving $P$ in (REF ) exactly as we did to simplify (REF ) into (REF )) $\\Vert L_3\\Vert _{L^2} +\\Vert M_3\\Vert _{L^2}\\le C(S-s) \\left( r_a^s\\sqrt{|\\!\\log \\varepsilon |}+\\Sigma ^0+r_a^0|\\!\\log \\varepsilon |+ |\\!\\log \\varepsilon |^{-1}\\right)^{\\frac{1}{2}} \\le C(S-s).$ Finally, regarding $M_1$ , we have on one side using (REF ) and (REF ) $\\Vert M_1\\Vert _{W^{-1,1}} \\le C(S-s)r_\\xi ^s,$ and on the other side using the pointwise inequality $Ju \\le Ce_\\varepsilon (u)$ for an arbitrary function $u$ and the global energy bound $\\Vert M_1\\Vert _{L^1} \\le C(S-s)|\\!\\log \\varepsilon |.$ By interpolation, it follows that for any $1<p<2$ $\\Vert M_1\\Vert _{W^{-1,p}} \\le C_p(S-s)(r_\\xi ^s)^\\theta |\\!\\log \\varepsilon |^{1-\\theta },$ where $\\frac{1}{p} = \\theta + \\frac{1-\\theta }{2}.$ These bounds on $L_i$ and $M_i$ turn into bounds on $F_i$ and $G_i$ , since by standard elliptic estimates we have $\\begin{array}{ll}\\Vert \\nabla F_1\\Vert _{L^p} \\le C_p \\Vert L_1\\Vert _{L^2} & \\text{ for all } 1 \\le p<+\\infty ,\\\\\\Vert \\nabla F_2\\Vert _{L^p} + \\Vert \\nabla G_2\\Vert _{L^p} \\le C_p (\\Vert L_2\\Vert _{L^1}+\\Vert M_2\\Vert _{L^1}) &\\text{ for all } 1\\le p<2,\\\\\\Vert \\nabla F_3\\Vert _{L^p} + \\Vert \\nabla G_3\\Vert _{L^p} \\le C_p (\\Vert L_3\\Vert _{L^2}+\\Vert M_3\\Vert _{L^2}) &\\text{ for all } 1\\le p<+\\infty ,\\\\\\Vert \\nabla G_1\\Vert _{L^p} \\le C_p \\Vert M_1\\Vert _{W^{-1,p}} & \\text{ for all } 1<p<2.\\\\\\end{array}$ To estimate (REF ), we then simply input (REF ) into (REF ) where we use the Hölder inequality with $j^\\natural _s$ estimated in $L^{p^{\\prime }}$ (and all the other weights other than $F_i$ or $G_i$ in $L^\\infty $ ).", "The largest contribution arises from $G_1$ since $\\Vert j^\\natural _s\\Vert _{L^{p^{\\prime }}} \\simeq (r^s_\\xi )^{-\\theta }$ when $p<2$ and the final $|\\!\\log \\varepsilon |^{-1^-}$ bound is obtained by choosing $p$ arbitrarilly close to 1 (and hence $\\theta $ arbitrarilly close to 1).", "For the terms involving $p>2$ we use the straightforward bound $\\Vert j^\\natural _s\\Vert _{L^{p^{\\prime }}} \\le C.$ The estimate of (REF ) is at first sight slightly more difficult since $\\partial _{ik}\\varphi $ is diverging like $\\sqrt{|\\!\\log \\varepsilon |}$ whereas in (REF ) $\\partial _{i}\\varphi $ was bounded in absolute value.", "On the other hand, the integrand only lives on the support of $D^2\\varphi $ , which is both away from the cores and of Lebesgues measure of order $|\\!\\log \\varepsilon |^{-1}.$ More precisely, for $F_1$ and $G_1$ we rely exactly on the same estimate as in (REF ), whereas, since the forcing terms $L_2, L_3, M_2$ and $M_3$ have a support disjoint from that of $D^2\\varphi $ , it follows by elliptic regularity that $\\Vert \\nabla F_i\\Vert _{L^\\infty ({\\rm supp}(D^2\\varphi ))} \\le C\\Vert \\nabla F_i\\Vert _{L^1}, \\quad \\Vert \\nabla G_i\\Vert _{L^\\infty ({\\rm supp}(D^2\\varphi ))} \\le C\\Vert \\nabla G_i\\Vert _{L^1}, \\quad i=2,3.$ We then combine (REF ) with our previous estimate (REF ), and therefore in the Hölder estimate of (REF ) for these four terms we can take $p=\\infty $ and hence $p^{\\prime }=1.$ Finally, regarding $j^\\natural _s$ we have for $p=\\infty $ $\\Vert j^\\natural _s\\Vert _{L^{1}({\\rm supp}(D^2\\varphi ))}\\le \\Vert j^\\natural _s\\Vert _{L^{\\infty }({\\rm supp}(D^2\\varphi ))} \\mathcal {L}^2( {\\rm supp}(D^2\\varphi ))) \\le C\\rho _{\\rm min},$ and for $p<2$ $\\Vert j^\\natural _s\\Vert _{L^{p^{\\prime }}({\\rm supp}(D^2\\varphi ))}\\le C (\\rho _{\\rm min})^{-\\theta }$ where $\\frac{1}{p} = \\theta + \\frac{1-\\theta }{2}.$ The conclusion then follows by summation.$\\Box $ Proofs of Theorem REF and Theorem REF Theorem REF follows very directly from Proposition REF .", "Indeed, the iterative use of Proposition REF leads to a discrete Gronwall inequality which is a forward Euler scheme for the corresponding classical (continuous) Gronwall inequality, and the latter has convex solutions which are therefore greater than their discrete equivalent.", "The actual details can be taken almost word for word from the ones used in [11] Proof of Theorem 1.3, and are therefore not repeated here.", "Finally, Theorem REF is also easily deduced from Theorem REF .", "The only point which deserves additional explanation is the fact that in the assumptions of Theorem REF only local norms $\\Vert \\cdot \\Vert _{\\dot{W}^{-1,1}(\\Omega )}$ with $\\Omega $ being of compact closure in the interior of $\\mathbb {H}$ are used whereas the definition of $r_a^0$ for Theorem REF involves the unbounded set $\\Omega _0.$ As the proof of Proposition REF shows (more precisely its Step 6), the closeness estimates expressed in (REF ) (which hold in expanding domains whose union ends up covering the whole of $\\mathbb {H}$ as $\\varepsilon $ tends to zero) only require a first localisation estimate in a neigborhood of size $1/\\sqrt{|\\!\\log \\varepsilon |}$ of the points $a_{i,\\varepsilon }$ , which is of course implied by the assumptions of Theorem REF .", "$\\Box $" ], [ "Vector potential of loop currents", "In the introduction we have considered the inhomogeneous Poisson equation $\\left\\lbrace \\begin{array}{ll}\\displaystyle -{\\rm div} \\left(\\frac{1}{r} \\nabla \\left( r A_a\\right)\\right) = 2\\pi \\delta _{a} &\\qquad \\text{in } \\mathbb {H},\\\\A_a = 0 & \\qquad \\text{on } \\mathbb {H}.\\end{array}\\right.$ Its integration is classical (see e.g.", "[10]) and yields $A_a(r,z) = \\frac{r(a)}{2} \\int _0^{2\\pi } \\frac{\\cos (t)}{\\sqrt{r(a)^2+r^2 +(z-z(a))^2-2r(a)r\\cos (t)}}\\, dt,$ which in turn simplifies to $A_a(r,z) = \\sqrt{\\frac{r(a)}{r}} \\frac{1}{k} \\left[(2-k^2)K(k^2)-2E(k^2)\\right]$ where $k^2 = \\frac{4r(a) r}{ r(a)^2 + r^2 + (z-z(a))^2 + 2 r(a) r}$ and where $E$ and $K$ denote the complete elliptic integrals of first and second kind respectively (see e.g.", "[1]).", "Note that $A_{\\lambda a}(\\lambda r,\\lambda z)= A_a(r,z)$ for any $\\lambda >0$ and that we have the asymptotic expansions [1] of the complete elliptic integrals as $s\\rightarrow 1$ : $K(s) = -\\frac{1}{2}\\log (1-s)\\left(1 + \\frac{1-s}{4}\\right) + \\log (4) + O(1-s),$ $E(s ) = 1 - \\log (1-s)\\frac{1-s}{4} + O(1-s),$ and similarly for their derivatives.", "For $(r,z) \\in \\mathbb {H}\\setminus \\lbrace a\\rbrace ,$ direct computations therefore yield $A_a(r,z) = \\left( \\log (\\tfrac{r(a)}{\\rho }) +3 \\log (2)-2\\right) + O\\left(\\tfrac{\\rho }{r(a)}|\\log (\\tfrac{\\rho }{r(a)})|\\right)\\qquad \\text{as } \\tfrac{\\rho }{r(a)} \\rightarrow 0,$ and $\\partial _{\\rho } A_a = -\\frac{1}{\\rho } + O\\left(\\tfrac{1}{r(a)}\\right)\\qquad \\text{as } \\tfrac{\\rho }{r(a)} \\rightarrow 0,$ where $\\rho := |a-(r,z)|.$ Concerning the asymptotic close to $r=0,$ we have $A_a(r,z) \\simeq \\frac{rr(a)^2}{r(a)^3+|z|^3} \\qquad \\text{as } \\frac{r}{r(a)}\\rightarrow 0.$" ], [ "Singular unimodular maps", "When $a = \\lbrace a_1,\\cdots , a_n\\rbrace $ is a family of $n$ distinct points in $\\mathbb {H}$ , we define the function $\\Psi ^*_a$ on $\\mathbb {H}_{a}:=\\mathbb {H}\\setminus a$ by $\\Psi ^*_a = \\sum _{i=1}^n A_{a_i},$ so that $\\left\\lbrace \\begin{array}{ll}\\displaystyle -{\\rm div} \\left( \\frac{1}{r} \\nabla (r \\Psi ^*_a)\\right) = 2\\pi \\sum _{i=1}^n\\delta _{a_i} &\\qquad \\text{on } \\mathbb {H},\\\\\\displaystyle \\Psi ^*_a = 0 & \\qquad \\text{on } \\partial \\mathbb {H}.\\end{array}\\right.$ Up to a constant phase shift, there exists a unique unimodular map $u^*_a\\in \\mathcal {C}^\\infty (\\mathbb {H}_a,S^1)\\cap W^{1,1}_{\\rm loc}(\\mathbb {H},S^1)$ such that $r (iu^*_a,\\nabla u^*_a) = rj(u^*_a) = -\\nabla ^\\perp (r\\Psi ^*_a).$ In the sense of distributions in $\\mathbb {H}$ , we have $\\left\\lbrace \\begin{array}{ll}\\displaystyle {\\rm div}(rj(u^*_a)) & = 0\\\\\\displaystyle {\\rm curl}(j(u^*_a)) & = 2\\pi \\sum _{i=1}^n \\delta _{a_i}.\\end{array}\\right.$ Let $\\rho _a :=\\frac{1}{4}\\min \\Big ( \\min _{i\\ne j}|a_i-a_j|, \\min _ir(a_i)\\Big ),$ for $\\rho \\le \\rho _a$ we set $\\mathbb {H}_{a,\\rho } :=\\mathbb {H}\\setminus \\cup _{i=1}^n B(a_i,\\rho ).$ Lemma A.1 Under the above assumptions we have $\\int _{\\mathbb {H}_{a,\\rho }} \\frac{|j(u^*_a)|^2}{2} r\\,drdz = \\pi \\sum _{i=1}^n r(a_i)\\Big [\\log \\big (\\tfrac{r(a_i)}{\\rho }\\big )+\\sum _{j\\ne i}A_{a_j}(a_i) +\\big (3\\log (2)-2\\big ) +O\\big (\\tfrac{\\rho }{\\rho _a}\\log ^2(\\tfrac{\\rho }{\\rho _a}) \\big )\\Big ].$ We have the pointwise equality $|j(u^*_a)|^2 r = \\frac{1}{r}|\\nabla ^\\perp (r\\Psi ^*_a)|^2 =\\frac{1}{r}|\\nabla (r\\Psi ^*_a)|^2,$ so that after integration by parts $\\int _{\\mathbb {H}_{a,\\rho }} \\frac{|j(u^*_a)|^2}{2} r\\,drdz = -\\frac{1}{2} \\int _{\\mathbb {H}_{a,\\rho }} {\\rm div}\\left(\\frac{1}{r}\\nabla (r\\Psi ^*_a)\\right)r\\Psi ^*_a \\,drdz +\\frac{1}{2} \\int _{\\partial \\mathbb {H}_{a,\\rho }} \\Psi ^*_a \\nabla (r\\Psi ^*_a)\\cdot \\vec{n},$ and the first integral of the right-hand side in the previous identity vanishes by definition of $\\Psi ^*_a$ and $\\mathbb {H}_{a,\\rho }.$ We next decompose the boundary integral as $\\frac{1}{2} \\int _{\\partial \\mathbb {H}_{a,\\rho }} \\Psi ^*_a \\nabla (r\\Psi ^*_a)\\cdot \\vec{n} =\\frac{1}{2} \\sum _{i,j,k=1}^n \\int _{\\partial B(a_i,\\rho )} A_{a_j}\\nabla (rA_{a_k})\\cdot \\vec{n},$ and for fixed $i,j,k$ we write $A_{a_j} \\nabla (rA_{a_k})\\cdot \\vec{n} = \\left(-A_{a_j}\\partial _\\rho A_{a_k} r + A_{a_j}A_{a_k}n_r\\right).$ Using (REF ), we have $|\\int _{\\partial B(a_i,\\rho )} A_{a_j}A_{a_k}n_r | \\le r(a_i)O\\Big (\\tfrac{\\rho }{\\rho _a}\\log ^2(\\tfrac{\\rho }{\\rho _a})\\Big ).$ When $i=j=k,$ we have by (REF ) and (REF ) $-\\frac{1}{2} \\int _{\\partial B(a_i,\\rho )} A_{a_j}\\partial _\\rho A_{a_k} r = \\pi r(a_i) \\left( \\log (\\frac{r(a_i)}{\\rho }) +3\\log (2) -2 +O\\Big (\\tfrac{\\rho }{\\rho _a}\\log (\\tfrac{\\rho }{\\rho _a})\\Big )\\right)$ while when $i=k\\ne j$ we have $-\\frac{1}{2} \\int _{\\partial B(a_i,\\rho )} A_{a_j}\\partial _\\rho A_{a_k} r = \\pi r(a_i) \\left( A_{a_j}(a_i) +O\\Big (\\tfrac{\\rho }{\\rho _a}\\Big )\\right).$ Finally, when $i\\ne k$ we have $\\big |\\frac{1}{2} \\int _{\\partial B(a_i,\\rho )} A_{a_j}\\partial _\\rho A_{a_k} r \\big | \\le r(a_i)O\\Big ((\\tfrac{\\rho }{\\rho _a})^2\\log (\\tfrac{\\rho }{\\rho _a})\\Big ).$ The conclusion follows by summation.", "If we next fix some constant $K_0>0$ and we assume that the points $a_i$ are of the form $a_{i} := \\Big (r_0+ \\frac{r(b_i)}{\\sqrt{|\\!\\log \\varepsilon |}} ,z_0 + \\frac{z(b_i)}{\\sqrt{|\\!\\log \\varepsilon |}} \\Big ),\\qquad i=1,\\cdots ,n,$ for some $r_0>0$ , $z_0\\in \\mathbb {R}$ and $n$ points $\\lbrace b_1,\\cdots ,b_n\\rbrace \\in \\mathbb {R}^2$ which satisfy $\\max _i |b_i| \\le K_0,\\qquad \\text{and}\\qquad \\min _{i\\ne j} {\\rm dist}(b_i,b_j)\\ge \\frac{1}{K_0},$ we directly deduce from Lemma REF , (REF ) and (REF ) : Lemma A.2 Under the above assumptions we have $\\begin{split}\\int _{\\mathbb {H}_{a,\\rho }} \\frac{|j(u^*_a )|^2}{2} r\\,drdz =&\\ \\pi n r_0 \\Big (|\\log \\rho | + n\\log r_0 + n\\big (3\\log (2)-2\\big )+ \\tfrac{n-1}{2}\\log |\\log \\varepsilon |\\Big ) \\\\&+ \\pi r_0 \\Big ( \\sum _{i} \\frac{r(b_i)}{r_0}\\frac{|\\log \\rho |}{\\sqrt{|\\log \\varepsilon |}} - \\sum _{i\\ne j}\\log |b_i-b_j|\\Big )\\\\&+ O_{K_0,r_0} \\Big ( \\frac{1}{\\sqrt{|\\log \\varepsilon |}}\\Big ).\\end{split}$" ], [ "Jacobian and Excess for 2D Ginzburg-Landau functional", "For the ease of reading, we recall in this appendix a few results from [12], [13] and [14] which we use in our work.", "Theorem B.1 (Thm 1.3 in [13] - Lower energy bound) There exists an absolute constant $C>0$ such that for any $u \\in H^1(B_r,$ satisfying $\\Vert Ju-\\pi \\delta _0\\Vert _{\\dot{W}^{-1,1}(B_r)} < r/4$ we have ${\\mathcal {E}}_{\\varepsilon }(u,B_r) \\ge \\pi \\log \\frac{r}{\\varepsilon } + \\gamma -\\frac{C}{r} \\left(\\varepsilon \\sqrt{\\log \\tfrac{r}{\\varepsilon }} + \\Vert Ju-\\pi \\delta _0\\Vert _{\\dot{W}^{-1,1}(B_r)}\\right).$ Theorem B.2 (from Thm 1.1 in [13] - Jacobian estimate without vortices) There exists an absolute constant $C>0$ with the following property.", "If $\\Omega $ is a bounded domain, $u \\in H^1(\\Omega ,$ , $\\varepsilon \\in (0,1]$ and ${\\mathcal {E}}_{\\varepsilon }(u,\\Omega )< \\pi |\\!\\log \\varepsilon |$ , then $\\Big \\Vert Ju \\Big \\Vert _{\\dot{W}^{-1,1}(\\Omega )} \\le \\varepsilon C {\\mathcal {E}}_{\\varepsilon }(u,\\Omega )\\exp \\Big (\\frac{1}{\\pi }{\\mathcal {E}}_{\\varepsilon }(u,\\Omega )\\Big ).$ Theorem B.3 (Thm 2.1 in [12] - Jacobian estimate with vortices) There exists an absolute constant $C>0$ with the following property.", "If $\\Omega $ is a bounded domain, $u \\in H^1(\\Omega ,$ , and $\\varphi \\in \\mathcal {C}^{0,1}_c(\\Omega )$ , then for any $\\lambda \\in (1,2]$ and any $\\varepsilon \\in (0,1)$ , $\\Big | \\int _\\Omega \\varphi Ju \\, dx \\Big | \\le \\pi d_\\lambda \\Vert \\varphi \\Vert _\\infty + \\Vert \\varphi \\Vert _{\\mathcal {C}^{0,1}}h^\\varepsilon (\\varphi ,u,\\lambda )$ where $d_\\lambda = \\Big \\lfloor \\frac{\\lambda }{\\pi } \\frac{{\\mathcal {E}}_{\\varepsilon }(u,{\\rm spt}(\\varphi ))}{|\\!\\log \\varepsilon |}\\Big \\rfloor ,$ $\\lfloor x \\rfloor $ denotes the greatest integer less than or equal to $x$ , and $h^\\varepsilon (\\varphi ,u,\\lambda ) \\le C\\varepsilon ^\\frac{\\lambda -1}{12\\lambda }\\big (1+\\frac{{\\mathcal {E}}_{\\varepsilon }(u,{\\rm spt}(\\varphi ))}{|\\!\\log \\varepsilon |}\\big )\\big (1+\\mathcal {L}^2({\\rm spt}(\\varphi )\\big ).$ Theorem B.4 (Thm 1.2' in [14] - Jacobian localization for a vortex in a ball) There exists an absolute constant $C>0$ , such that for any $u \\in H^1(B_r,$ satisfying $\\Vert Ju-\\pi \\delta _0\\Vert _{\\dot{W}^{-1,1}(B_r)} < r/4,$ if we write $\\Xi = {\\mathcal {E}}_{\\varepsilon }(u,B_r) - \\pi \\log \\frac{r}{\\varepsilon }$ then there exists a point $\\xi \\in B_{r/2}$ such that $\\Vert Ju - \\pi \\delta _\\xi \\Vert _{\\dot{W}^{-1,1}(B_r)} \\le \\varepsilon C(C+\\Xi )\\left[(C+\\Xi )e^{\\Xi /\\pi } + \\sqrt{\\log \\frac{r}{\\varepsilon }}\\right].$ Theorem B.5 (Thm 3 in [14] - Jacobian localization for many vortices) Let $\\Omega $ be a bounded, open, simply connected subset of $\\mathbb {R}^2$ with $\\mathcal {C}^1$ boundary.", "There exists constants $C$ and $K$ , depending on ${\\rm diam}(\\Omega )$ , with the following property: For any $u \\in H^1(\\Omega ,$ , if there exists $n\\ge 0$ distinct points $a_1,\\cdots ,a_n$ in $\\Omega $ and $d \\in \\lbrace \\pm 1\\rbrace ^{n}$ such that $\\Vert Ju-\\pi \\sum _{i=1}^n d_i \\delta _{a_i}\\Vert _{\\dot{W}^{-1,1}(\\Omega )} \\le \\frac{\\rho _a}{Kn^5},$ where $\\rho _a := \\frac{1}{4}{\\rm min}_i\\left\\lbrace {\\rm min}_{j\\ne i}|a_i-a_j|, {\\rm dist}(a_i,\\partial \\Omega )\\right\\rbrace ,$ and if in addition ${\\mathcal {E}}_{\\varepsilon }(u,\\Omega ) \\ge 1$ and $\\frac{n^5}{\\rho _a}{\\mathcal {E}}_{\\varepsilon }(u,\\Omega ) + \\frac{n^{10}}{\\rho _a^2}\\sqrt{{\\mathcal {E}}_{\\varepsilon }(u,\\Omega )}\\le \\frac{1}{\\varepsilon },$ then there exist $\\xi _1,\\cdots ,\\xi _d$ in $\\Omega $ such that $\\Vert Ju - \\pi \\sum _{i=1}^n d_i \\delta _{\\xi _i} \\Vert _{\\dot{W}^{-1,1}(\\Omega )} \\le C\\varepsilon \\left[n(C+\\Xi _\\Omega ^\\varepsilon )^2e^{\\Xi _\\Omega ^\\varepsilon /\\pi } +(C+\\Xi _\\Omega ^\\varepsilon )\\frac{n^5}{\\rho _a} + {\\mathcal {E}}_{\\varepsilon }(u,\\Omega )\\right],$ where $\\Xi _\\Omega ^\\varepsilon := {\\mathcal {E}}_{\\varepsilon }(u,\\Omega )-n(\\pi \\log \\frac{1}{\\varepsilon } + \\gamma ) +- \\pi \\Big ( \\sum _{i\\ne j} d_id_j \\log |a_i-a_j| + \\sum _{i,j}d_id_j H_\\Omega (a_i,a_j)\\Big )$ and $H_\\Omega $ is the Robin function of $\\Omega .$" ] ]
1606.05103
[ [ "Envelope Words and Return Words Sequences in the Period-doubling\n Sequence" ], [ "Abstract We consider the infinite one-sided sequence generated by the period-doubling substitution $\\sigma(a,b)=(ab,aa)$, denoted by $\\mathbb{D}$.", "Since $\\mathbb{D}$ is uniformly recurrent, each factor $\\omega$ appears infinite many times in the sequence, which is arranged as $\\omega_p$ $(p\\ge 1)$.", "Let $r_p(\\omega)$ be the $p$-th return word over $\\omega$.", "The main result is: for each factor $\\omega$, the sequence $\\{r_p(\\omega)\\}_{p\\geq1}$ is $\\Theta_1$ or $\\Theta_2$, which are substitutive sequences and determined completely in this paper." ], [ "Introduction", "The period-doubling sequence $\\mathbb {D}$ has been heavily studied within mathematics and computer science, etc.", "D.Damanik[3] determined the number of palindromes (resp.", "$k$ -th powers of words) of length $n$ occurring in $\\mathbb {D}$ explicitly.", "Sometimes, the period-doubling sequence is called the first difference of the Thue-Morse sequence.", "Here we use an equivalent substitution $\\hat{\\sigma }(1,0)=(10,11)$ , and the definition of the difference of an integer sequence is natural.", "In 1998, Allouche-Peyri$\\grave{e}$ re-Wen-Wen[1] proved that all the Hankel determinants of the period-doubling sequence are odd integers.", "In 2014, Guo-Wen[5] determined the automaticity of the Hankel determinants of difference sequences of the Thue-Morse sequence, including $\\mathbb {D}$ .", "In 2015, Parreau-Rigo-Rowland-Vandomme[11] proved that $\\mathbb {D}$ have 2-abelian complexity sequences that are 2-regular.", "Fu-Han[4] considered $t$ -extensions of the Hankel determinants of some certain automatic sequences, such as $\\mathbb {D}$ .", "F.Durand[2] introduced the return words and proved that a sequence is primitive substitutive if and only if the set of its return words is finite.", "L.Vuillon[12] proved that an infinite word $\\tau $ is a Sturmian sequence if and only if each nonempty factor $\\omega \\prec \\tau $ has exactly two distinct return words.", "But they have not studied the expressions of the return words and the properties of the sequence composed by the return words.", "Let $\\omega $ be a factor of $\\mathbb {D}$ .", "For $p\\ge 1$ and let $\\omega _p=x_{i+1}\\cdots x_{i+n}$ and $\\omega _{p+1}=x_{j+1}\\cdots x_{j+n}$ .", "The factor $x_{i+1}\\cdots x_{j}$ is called the $p$ -th return word of $\\omega $ and denoted by $r_{p}(\\omega )$ .", "If no confusion happens, we denote by $r_p$ for short.", "The sequence $\\lbrace r_p(\\omega )\\rbrace _{p\\ge 1}$ is called the return word sequence of factor $\\omega $ .", "Denote $r_0(\\omega )$ the prefix of $\\mathbb {D}$ before $\\omega _1$ .", "It is not a return word.", "Huang-Wen [6], [7] determine the structure of the return word sequences of the Fibonacci sequence $\\mathbb {F}$ and the Tribonacci sequence $\\mathbb {T}$ , respectively.", "More precisely, for any factor $\\omega $ of $\\mathbb {F}$ (resp.", "$\\mathbb {T}$ ), the return word sequence $\\lbrace r_p(\\omega )\\rbrace _{p\\ge 1}$ is still $\\mathbb {F}$ (resp.", "$\\mathbb {T}$ ).", "Using these properties, we determined the number of palindromes (resp.", "$k$ -th powers of words) occurring in $\\mathbb {F}[1,n]$ (the prefix of $\\mathbb {F}$ of length $n$ ) and $\\mathbb {T}[1,n]$ for all $n\\ge 1$ , see [8], [9], [10].", "These topics are of great importance in computer science.", "The main tool of the two papers is “kernel word”.", "However, in the studies of the period-doubling sequence $\\mathbb {D}$ , the techniques of kernel words fail.", "In fact, there is no kernel set in $\\mathbb {D}$ , which satisfies the “uniqueness of kernel decomposition”, see [6], [7].", "To overcome this difficulty, we introduce a new notion called “envelope word”.", "The main result of this paper is: for any factor $\\omega $ , the return word sequence $\\lbrace r_p(\\omega )\\rbrace _{p\\ge 1}$ is $\\Theta _1=\\tau _1(\\mathbb {D})$ or $\\Theta _2=\\tau _2(\\mathbb {D})$ , where $\\tau _1(a,b)=(a,bb)$ , $\\tau _2(a,b)=(ab,acac)$ .", "In order to prove this property, we first determine the return word sequences of envelope words in Section 2; then we give two types of “uniqueness of envelope extension” in Section 3; using them, we determine the return word sequences of general factors in Section 4.", "Let $\\mathcal {A}=\\lbrace a,b\\rbrace $ be a binary alphabet.", "We denote the concatenation of $\\nu $ and $\\omega $ by $\\nu \\omega $ or $\\nu \\cdot \\omega $ .", "$\\omega ^k$ means the concatenation of $k$ factors $\\omega $ , called the $k$ -th power factor of $\\omega $ .", "The mirror word of $\\omega $ is defined to be $\\overleftarrow{\\omega }=x_n\\cdots x_2x_1$ .", "A word $\\omega $ is called a palindrome if $\\omega =\\overleftarrow{\\omega }$ .", "We define $\\overline{\\cdot }:\\mathcal {A}\\rightarrow \\mathcal {A}$ by $\\overline{a}=b$ and $\\overline{b}=a$ .", "Let $\\rho =x_1\\cdots x_n$ be a finite word.", "For any $i\\le j\\le n$ , we define $\\rho [i,j]=x_ix_{i+1}\\cdots x_{j-1}x_j$ .", "For convention, we denote $\\rho [i]=\\rho [i,i]=x_i$ , $\\rho [i,i-1]=\\varepsilon $ (empty word).", "We denote by $L(\\omega ,p)$ the position of the first letter of $\\omega _p$ .", "We say that $\\nu $ is a prefix (resp.", "suffix) of a word $\\omega $ if there exists word $u$ such that $\\omega =\\nu u$ (resp.", "$\\omega =u\\nu $ ), $|u|\\ge 0$ , which denoted by $\\nu \\triangleleft \\omega $ (resp.", "$\\nu \\triangleright \\omega $ ).", "In this case, we write $\\nu ^{-1}\\omega =u$ (resp.", "$\\omega \\nu ^{-1}=u$ ), where $\\nu ^{-1}$ is the inverse word of $\\nu $ such that $\\nu \\nu ^{-1}=\\nu ^{-1}\\nu =\\varepsilon $ .", "The period-doubling sequence $\\mathbb {D}$ is the fixed point beginning with $a$ of substitution $\\sigma (a,b)=(ab,aa)$ .", "We denote $A_m=\\sigma ^m(a)$ and $B_m=\\sigma ^m(b)$ for $m\\ge 0$ .", "Then $|A_m|=|B_m|=2^m$ , where $|\\omega |$ is the length of $\\omega $ .", "Let $\\delta _m\\in \\lbrace a,b\\rbrace $ be the last letter of $A_m$ .", "Obviously, $\\delta _m=a$ if and only if $m$ even; $\\delta _{m+1}$ is the last letter of $B_m$ .", "For later use, we list some elementary properties of $\\mathbb {D}$ , which can be proved easily by induction: $A_{m}\\delta _m^{-1}=B_{m}\\delta _{m+1}^{-1}$ and $A_{m}=a\\prod _{j=0}^{m-1} B_{j}$ for $m\\ge 0$ .", "Moreover $\\mathbb {D}=a\\prod _{j=0}^\\infty B_j$ ." ], [ "The return word sequences of envelope words in $\\mathbb {D}$", "As we said in Section 1, there is no “kernel set\" in the period-doubling sequence, which satisfies the “uniqueness of kernel decomposition\".", "To overcome this difficulty, we introduce a new notion called “envelope word\".", "Definition 2.1 (Envelope words) Let $\\mathcal {E}=\\lbrace E_{i,m}:i=1,2,~m\\ge 1\\rbrace $ be a set of factors with $E_{1,m}=A_{m}\\delta _{m}^{-1}\\text{ and }E_{2,m}=B_{m}B_{m-1}\\delta _{m}^{-1}.$ We call $E_{i,m}$ the $m$ -th envelope word of type $i$ .", "Definition 2.2 () Let $\\Theta _1=\\tau _1(\\mathbb {D})$ and $\\Theta _2=\\tau _2(\\mathbb {D})$ , where $\\tau _1(a,b)=(a,bb)$ and $\\tau _2(a,b)=(ab,acac)$ .", "Obviously, $\\Theta _1$ and $\\Theta _2$ are $\\mathbb {D}$ over the alphabets $\\lbrace a,bb\\rbrace $ and $\\lbrace ab,acac\\rbrace $ , respectively.", "In this section, we are going to prove the two theorems below: Theorem 2.3 The return word sequence $\\lbrace r_p(E_{1,m})\\rbrace _{p\\ge 1}$ is $\\Theta _1$ over the alphabet $\\lbrace r_1(E_{1,m}),r_2(E_{1,m})\\rbrace =\\lbrace A_{m},A_{m-1}\\rbrace .$ Theorem 2.4 The return word sequence $\\lbrace r_p(E_{2,m})\\rbrace _{p\\ge 1}$ is $\\Theta _2$ over the alphabet $\\lbrace r_1(E_{2,m}),r_2(E_{2,m}),r_4(E_{2,m})\\rbrace =\\lbrace A_{m-1},A_{m-1}A_{m}B_{m+1},B_{m}B_{m-1}\\rbrace .$" ], [ "Prove of Theorem ", "We first give a criterion to determine all occurrences of a factor, which is useful in our proofs.", "Let $\\omega =u\\nu $ where $|u|,|\\nu |>0$ .", "Obviously, if $\\mathbb {D}[L,L+|\\omega |-1]=\\omega $ , then $\\mathbb {D}[L,L+|u|-1]=u$ .", "Thus in order to determine all occurrences of $\\omega $ , we first find out all occurrences of $u$ , then check whether these $u$ 's can extend to $\\omega $ (i.e.", "be followed by $\\nu $ ) or not.", "Lemma 2.5 For $m\\ge 0$ , $A_m$ occurs exactly twice in $A_mA_m$ (resp.", "$A_mB_mA_m$ ).", "More precisely, $A_m$ occurs as prefix and suffix.", "For $m=0$ , $A_m=a$ occurs exactly twice in $A_mA_m=aa$ (resp.", "$A_mB_mA_m=aba$ ) as prefix and suffix.", "Assume the conclusions hold for $m$ , consider $A_{m+1}=A_{m}B_{m}$ occurs in $A_{m+1}A_{m+1}=\\underbrace{A_{m}}_{[1]}B_{m}\\underbrace{A_{m}}_{[2]}B_{m},~A_{m+1}B_{m+1}A_{m+1}=\\underbrace{A_{m}}_{[3]}B_{m}\\underbrace{A_{m}}_{[4]} \\underbrace{A_{m}}_{[5]}\\underbrace{A_{m}}_{[6]}B_{m}.$ Since $A_m$ occurs exactly twice in $A_mA_m$ and $A_mB_mA_m$ , all possible positions of the prefix $A_{m}$ of $A_{m+1}$ are shown with “underbrace” above.", "(1) The $A_{m}$ locates at [1], [2], [3], [6] followed by $B_m$ , so they can extend to a $A_{m+1}=A_mB_m$ .", "(2) The $A_{m}$ locates at [4], [5] followed by $A_m\\ne B_m$ , so they can't extend to a $A_{m+1}=A_mB_m$ .", "This means, $A_{m+1}$ occurs exactly twice in $A_{m+1}A_{m+1}$ (resp.", "$A_{m+1}B_{m+1}A_{m+1}$ ) as prefix and suffix.", "By induction, the conclusions hold for $m\\ge 0$ .", "Property 2.6 () $r_0(E_{1,m})=\\varepsilon $ , $r_1(E_{1,m})=A_{m}$ and $r_2(E_{1,m})=A_{m-1}$ for $m\\ge 1$ .", "We only need to determine the first three positions of $E_{1,m}$ , denoted by $L(E_{1,m},p)$ , $p=1,2,3$ .", "By the criterion above, we determine $L(A_{m-1},p)$ first, $p=1,2,3$ .", "Since $A_{m+2}=A_{m-1}B_{m-1}A_{m-1}A_{m-1}A_{m-1}B_{m-1}A_{m-1}B_{m-1}\\triangleleft \\mathbb {D},$ By Lemma REF , we have $L(A_{m-1},1)=1$ , $L(A_{m-1},2)=2^m+1$ , $L(A_{m-1},3)=3\\times 2^{m-1}+1$ .", "Since $A_{m}\\delta _m^{-1}=B_{m}\\delta _{m+1}^{-1}$ , all of them are followed by $B_{m-1}\\delta _{m}^{-1}$ , i.e., can extend to $E_{1,m}$ .", "This means, $L(E_{1,m},1)=1$ , $L(E_{1,m},2)=2^m+1$ , $L(E_{1,m},3)=3\\times 2^{m-1}+1$ .", "By the definition of return words $r_p(E_{1,m})$ , we have $r_0(E_{1,m})=\\mathbb {D}[1,L(E_{1,m},1)-1]=\\mathbb {D}[1,0]=\\varepsilon $ ; $r_1(E_{1,m})=\\mathbb {D}[L(E_{1,m},1),L(E_{1,m},2)-1]=\\mathbb {D}[1,2^m]=A_{m}$ ; $r_2(E_{1,m})=\\mathbb {D}[L(E_{1,m},2),L(E_{1,m},3)-1]=\\mathbb {D}[2^m+1,3\\times 2^{m-1}]=A_{m-1}$ .", "Corollary 2.7 $|r_0(E_{1,m})|=0$ , $|r_1(E_{1,m})|=2^{m}$ and $|r_2(E_{1,m})|=2^{m-1}$ for $m\\ge 1$ .", "Let $m\\ge 1$ be fixed.", "Define an alphabet $\\mathcal {A}_{1,m}=\\lbrace \\mathbf {a},\\mathbf {b}\\rbrace =\\lbrace r_1(E_{1,m}),r_2(E_{1,m})r_2(E_{1,m})\\rbrace =\\lbrace A_{m},B_{m}\\rbrace .$ We denote $\\mathbf {A_n}(\\mathcal {A}_{1,m})$ and $\\mathbf {B_n}(\\mathcal {A}_{1,m})$ the $A_n$ and $B_n$ over alphabet $\\mathcal {A}_{1,m}$ for each fixed $m\\ge 1$ .", "If no confusion happens, we simply write $\\mathbf {A_n}$ , $\\mathbf {B_n}$ , $r_1$ and $r_2$ for short.", "Lemma 2.8 Over $\\mathcal {A}_{1,m}=\\lbrace \\mathbf {a},\\mathbf {b}\\rbrace $ , $\\mathbf {A_n}=A_{m+n}$ and $\\mathbf {B_n}=B_{m+n}$ for $n\\ge 0$ .", "Since $\\mathbf {A_0}=\\mathbf {a}=A_{m}$ , $\\mathbf {B_0}=\\mathbf {b}=B_{m}$ , the conclusions hold for $n=0$ .", "Assume they hold for $n$ , then $\\mathbf {A_{n+1}}=\\mathbf {A_nB_n}=A_{m+n}B_{m+n}=A_{m+n+1},~\\mathbf {B_{n+1}}=\\mathbf {A_nA_n}=A_{m+n}A_{m+n}=B_{m+n+1}.$ By induction, the conclusions hold for $n\\ge 0$ .", "By the definition of alphabet $\\mathcal {A}_{1,m}$ and Definition REF , Theorem REF is equivalent to Theorem REF '.", "The return word sequence $\\lbrace r_p(E_{1,m})\\rbrace _{p\\ge 1}$ in $\\mathbb {D}$ is $\\mathbb {D}$ over the alphabet $\\mathcal {A}_{1,m}$ .", "By $A_{m}=a\\prod _{j=0}^{m-1}B_j$ and Lemma REF , we have $\\begin{array}{c}\\mathbb {D}=a\\prod _{j=0}^\\infty B_j=a\\prod _{j=0}^{m-1}B_j\\prod _{j=m}^\\infty B_j=\\mathbf {a}\\prod _{j=0}^\\infty B_{m+j}=\\mathbf {a}\\prod _{j=0}^\\infty \\mathbf {B_{j}}=\\mathbf {D}.\\end{array}$ Here $\\mathbb {D}$ and $\\mathbf {D}$ are the period-doubling sequence over alphabets $\\mathcal {A}$ and $\\mathcal {A}_{1,m}$ , respectively.", "Thus the conclusion holds." ], [ "Prove of Theorem ", "Lemma 2.9 (1) $B_m$ occurs exactly once in $A_mB_m$ , at position $2^{m}+1$ for $m\\ge 0$ ; (2) $B_m$ occurs three times in $B_mA_mB_m$ , at positions 1, $2^{m-1}+1$ and $2^{m+1}+1$ for $m\\ge 1$ ; (3) $B_m$ occurs three times in $B_mA_mA_mA_mB_m$ , at positions 1, $2^{m-1}+1$ and $2^{m+2}+1$ for $m\\ge 1$ .", "Similarly as Property REF and by Lemma REF , the first five positions of $E_{2,m}$ are: $2^m+1,~3\\times 2^{m-1}+1,~5\\times 2^m+1,~11\\times 2^{m-1}+1,~7\\times 2^m+1.$ The expressions of $r_p(E_{2,m})$ , $1\\le p\\le 4$ are Property 2.10 () $r_0(E_{2,m})=A_{m}$ , $r_1(E_{2,m})=r_3(E_{2,m})=A_{m-1}$ , $r_2(E_{2,m})=A_{m-1}A_{m}B_{m+1}$ and $r_4(E_{2,m})=B_{m}B_{m-1}$ for $m\\ge 1$ .", "Corollary 2.11 $|r_0(E_{2,m})|=2^{m}$ , $|r_1(E_{2,m})|=2^{m-1}$ , $|r_2(E_{2,m})|=7\\ast 2^{m-1}$ , $|r_4(E_{2,m})|=3\\ast 2^{m-1}$ .", "Let $m\\ge 1$ be fixed.", "We simply write $r_p(E_{2,m})$ as $r_p$ for short, $p=1,2,4$ .", "Define an alphabet $\\mathcal {A}_{2,m}=\\lbrace \\mathbf {a},\\mathbf {b}\\rbrace =\\lbrace r_1r_2,r_1r_4r_1r_4\\rbrace =\\lbrace A^{-1}_{m}A_{m+2}A_{m},A^{-1}_{m}B_{m+2}A_{m}\\rbrace .$ We denote $\\mathbf {A_n}(\\mathcal {A}_{2,m})$ and $\\mathbf {B_n}(\\mathcal {A}_{2,m})$ the $A_n$ and $B_n$ over alphabet $\\mathcal {A}_{2,m}$ for each fixed $m\\ge 1$ .", "If no confusion happens, we simply write $\\mathbf {A_n}$ and $\\mathbf {B_n}$ for short.", "By induction as Lemma REF , we have Lemma 2.12 Over $\\mathcal {A}_{2,m}=\\lbrace \\mathbf {a},\\mathbf {b}\\rbrace $ , $\\mathbf {A_n}=A^{-1}_{m}A_{m+n+2}A_{m}$ and $\\mathbf {B_n}=A^{-1}_{m}B_{m+n+2}A_{m}$ for $n\\ge 0$ .", "By the definition of alphabet $\\mathcal {A}_{2,m}$ and Definition REF , Theorem REF is equivalent to Theorem REF '.", "The return word sequence $\\lbrace r_p(E_{2,m})\\rbrace _{p\\ge 1}$ in $\\mathbb {D}$ is $\\mathbb {D}$ over the alphabet $\\mathcal {A}_{2,m}$ .", "By $A_{m}=a\\prod _{j=0}^{m-1} B_{j}$ and Lemma REF , we have $\\begin{array}{rl}\\mathbb {D}=&a\\prod _{j=0}^\\infty B_j=a\\prod _{j=0}^{m+1}B_j\\prod _{j=m+2}^\\infty B_j=A_{m+2}\\prod _{j=0}^\\infty B_{m+j+2}\\\\=&A_{m} A^{-1}_{m}A_{m+2}A_{m} \\prod _{j=0}^\\infty (A^{-1}_{m}B_{m+j+2}A_{m})=r_0(E_{2,m})\\mathbf {a}\\prod _{j=0}^\\infty \\mathbf {B_{j}}=r_0(E_{2,m})\\mathbf {D}.\\end{array}$ Here $\\mathbb {D}$ and $\\mathbf {D}$ are the period-doubling sequence over alphabets $\\mathcal {A}$ and $\\mathcal {A}_{2,m}$ , respectively.", "Notice that, by the definition of return word sequence, we omit $r_0(E_{2,m})$ .", "So the conclusion holds." ], [ "Uniqueness of envelope extension", "In this section, we give the two types of uniqueness of envelope extension, which play an important role in our studies.", "Using them, we can extend Theorem REF , REF and other related properties from envelope words to general factors.", "Definition 3.1 (The order of envelope words) $E_{1,m}\\sqsubset E_{2,m}$ , $E_{i,m}\\sqsubset E_{j,m+1}$ for $i,j\\in \\lbrace 1,2\\rbrace $ , $m\\ge 1$ .", "Definition 3.2 (Envelope of factor $\\omega $ , $Env(\\omega )$ ) For any factor $\\omega $ , let $Env(\\omega )=\\min _{\\sqsubset }\\lbrace E_{i,m}:\\omega \\prec E_{i,m},~i=1,2,~m\\ge 1\\rbrace ,$ which is called the envelope of factor $\\omega $ .", "For any $\\omega $ , denote $Env(\\omega )$ by $E_{i,m}$ .", "By Definition REF , we know the envelope of factor $\\omega $ is unique.", "But we don't know: (1) whether $\\omega $ occurs in $Env(\\omega )$ only once or not; (2) the relation between $\\lbrace \\omega _p\\rbrace _{p\\ge 1}$ and $\\lbrace E_{i,m,p}\\rbrace _{p\\ge 1}$ , i.e., the relation between positions $L(\\omega ,p)$ and $L(E_{i,m},p)$ for all $p\\ge 1$ .", "The two types of uniqueness of envelope extension will answer the two questions: $\\bullet $ The weak type (see Definition REF below) shows the relation between $\\omega $ and $Env(\\omega )$ ; $\\bullet $ The strong type (see Definition REF below) shows the relation between $\\omega _p$ and $Env(\\omega )_p$ for each $p\\ge 1$ , i.e.", "$Env(\\omega _p)=Env(\\omega )_p$ ." ], [ "Basic properties of envelope words", "For later use, we give some basic properties of envelope words first.", "By the definition of envelope words, $E_{1,m}=A_m\\delta _{m}^{-1}$ and $E_{2,m}=B_{m}B_{m-1}\\delta _{m}^{-1}$ .", "Thus Property 3.3 $E_{1,m+1}=E_{1,m}\\delta _{m}E_{1,m}$ and $E_{2,m+1}=E_{1,m}\\delta _{m}E_{1,m}\\delta _{m}E_{1,m}$ for $m\\ge 1$ .", "Example.", "When $m=2$ , $E_{1,m}=aba$ , $\\delta _{m}=a$ .", "We have $E_{1,m+1}=aba\\underline{a}aba$ , $E_{2,m+1}=aba\\underline{a}aba\\underline{a}aba$ .", "Here we give all $\\delta _{m}=a$ with underline.", "Corollary 3.4 Both $E_{1,m}$ and $E_{2,m}$ are palindromes.", "By induction, we can give a more general form of Property REF , as Property REF and REF .", "Property 3.5 (Relation between $E_{1,m}$ and $E_{1,n}$ ) For $m>n$ , $E_{1,m}=E_{1,n}x_1E_{1,n}\\cdots E_{1,n}x_hE_{1,n}$ , where $x_1\\cdots x_h=\\overline{E_{1,m-n}}$ for $n$ odd, and $x_1\\cdots x_h=E_{1,m-n}$ for $n$ even.", "Example.", "When $n=1$ (odd) and $m=3$ , we give all $E_{1,1}=a$ with underline: $E_{1,3}=\\underline{a}b\\underline{a}a\\underline{a}b\\underline{a}.$ In this case, $x_1\\cdots x_h=bab=\\overline{aba}=\\overline{E_{1,2}}=\\overline{E_{1,m-n}}$ .", "Property 3.6 (Relation between $E_{2,m}$ and $E_{1,n}$ ) For $m>n$ , $E_{2,m}=E_{1,n}x_1E_{1,n}\\cdots E_{1,n}x_hE_{1,n}$ , where $x_1\\cdots x_h=\\overline{E_{2,m-n}}$ for $n$ odd, and $x_1\\cdots x_h=E_{2,m-n}$ for $n$ even.", "Example.", "When $n=1$ (odd) and $m=3$ , we give all $E_{1,1}=a$ with underline: $E_{2,3}=\\underline{a}b\\underline{a}a\\underline{a}b\\underline{a}a\\underline{a}b\\underline{a}.$ In this case, $x_1\\cdots x_h=babab=\\overline{ababa}=\\overline{E_{2,2}}=\\overline{E_{2,m-n}}$ .", "In Corollary REF , we have all envelope words are palindromes.", "Property REF and REF show stronger relations between envelope words and palindromes.", "We first give some lemmas.", "Since $E_{1,m}=A_m\\delta _m^{-1}$ , it is easy to proof that Lemma 3.7 () For $m\\ge 2$ , we denote $E_{1,m}=x_1x_2\\cdots x_h$ , then $x_1=a$ , $x_2=b$ , $x_h=a$ .", "Lemma 3.8 () The factor $E_{1,n}\\delta _{m}E_{1,k}$ be a palindrome has only two cases: (a) $n=k$ .", "(b) $|n-k|=1$ ; $m$ and $\\min \\lbrace n,k\\rbrace $ have the same parity.", "Obviously, when $n=k$ , $E_{1,n}\\delta _{m}E_{1,k}$ is a palindrome.", "When $n\\ne k$ , since both $E_{1,n}$ and $E_{1,k}$ are palindrome, we assume $n>k$ without loss of generality.", "(1) When $n=k+1$ , by Property REF , $E_{1,n}\\delta _{m}E_{1,k}=E_{1,k}\\underline{\\delta _{k}}E_{1,k}\\underline{\\delta _{m}}E_{1,k}$ .", "Comparing the two letters with underlines, $E_{1,n}\\delta _{m}E_{1,k}$ is a palindrome if and only if $\\delta _{k}=\\delta _{m}$ , i.e., $m$ and $k=\\min \\lbrace n,k\\rbrace $ have the same parity.", "(2) When $n\\ge k+2$ , by Property REF , $E_{1,n}\\delta _{m}E_{1,k}=E_{1,k}x_1E_{1,k}\\underline{x_2}E_{1,k}\\cdots E_{1,k}\\underline{x_h}E_{1,k}\\delta _{m}E_{1,k},$ where $x_1x_2\\cdots x_h=E_{1,n-k}$ for $k$ odd, and $x_1x_2\\cdots x_h=\\overline{E_{1,n-k}}$ for $k$ even.", "No matter $k$ is odd or even, $x_2\\ne x_h$ by Lemma REF .", "Thus $E_{1,n}\\delta _{m}E_{1,k}$ can not be a palindrome in this case.", "In summary, the conclusion holds.", "Lemma 3.9 () For $n,h<m$ , the factor $E_{1,n}\\delta _{m}E_{1,m}\\delta _{m}E_{1,k}$ be a palindrome if and only if $n=k$ .", "Obviously, when $n=k$ , $E_{1,n}\\delta _{m}E_{1,m}\\delta _{m}E_{1,k}$ is palindrome.", "When $n\\ne k$ , since both $E_{1,n}$ and $E_{1,k}$ are palindrome, we assume $n>k$ without loss of generality.", "(1) When $n=k+1$ , by Property REF , $E_{1,n}=E_{1,k}\\delta _{k}E_{1,k}$ and $E_{1,m}=E_{1,k}x_1E_{1,k}x_2E_{1,k}\\ldots E_{1,k}x_hE_{1,k}$ for $h\\ge 3$ , i.e., $\\begin{split}&E_{1,n}\\cdot \\delta _{m}\\cdot E_{1,m}\\cdot \\delta _{m}\\cdot E_{1,k}\\\\=&E_{1,k}\\underbrace{\\delta _{k}}_{[1]}E_{1,k}\\cdot \\underbrace{\\delta _{m}}_{[3]}\\cdot E_{1,k}x_1E_{1,k}x_2E_{1,k}\\ldots E_{1,k}x_{h-1}E_{1,k}x_hE_{1,k}\\cdot \\underbrace{\\delta _{m}}_{[2]}\\cdot E_{1,k}.\\end{split}$ Suppose $E_{1,n}\\delta _{m}E_{1,m}\\delta _{m}E_{1,k}$ is palindrome, then $\\delta _{k}$ locates at [1] is equal to $\\delta _{m}$ locates at [2], i.e.", "$\\delta _{k}=\\delta _{m}$ .", "Moreover $\\delta _{m}$ locates at [3] is equal to $x_h$ , i.e.", "$\\delta _{m}=x_h$ .", "Since $E_{1,m}$ is palindrome, $x_1=x_h$ .", "Since $E_{1,n}\\delta _{m}E_{1,m}\\delta _{m}E_{1,k}$ is palindrome, $x_1=x_{h-1}$ .", "(1) If $m=k+2$ , i.e., $h=3$ , we find a 6-th power factor of $\\alpha \\in \\lbrace a,b\\rbrace $ in $\\Theta _1$ that $\\delta _{k}\\delta _{m}x_1x_2x_h\\delta _{m}$ .", "(2) If $m\\ge k+3$ , $h\\ge 7$ .", "Since $E_{1,m}$ is palindrome, $x_2=x_{h-1}$ .", "We find a 4-th power factor of $\\alpha \\in \\lbrace a,b\\rbrace $ in $\\Theta _1$ that $\\delta _{k}\\delta _{m}x_1x_2$ .", "Both of the two cases above contradict that there is no $k$ -th power factor in $\\Theta _1$ , $k\\ge 4$ .", "Thus $E_{1,n}\\delta _{m}E_{1,m}\\delta _{m}E_{1,k}$ can not be a palindrome for $n\\ne k$ .", "So the conclusion holds.", "Property 3.10 () Let palindrome $\\omega $ be a prefix (resp.", "suffix) of $E_{1,m}$ , then there exists $n$ ($\\le m$ ) such that $\\omega =E_{1,n}$ .", "For $m=1$ , $E_{1,m}=a$ , the palindromical prefix is $a$ .", "For $m=2$ , $E_{1,m}=aba$ , the palindromical prefixes are $a$ and $aba$ .", "The conclusion holds.", "By induction, we assume the conclusions hold for $m$ .", "If $\\omega =E_{1,m+1}$ , the conclusion holds obviously.", "If $\\omega $ is a proper prefix of $E_{1,m+1}=E_{1,m}\\delta _{m}E_{1,m}$ , then Case 1: $\\omega $ is a prefix of $E_{1,m}$ , then there exists $n\\le m$ such that $\\omega =E_{1,n}$ .", "Case 2: $\\omega $ isn't a prefix of $E_{1,m}$ .", "Since $E_{1,m+1}=E_{1,m}\\delta _{m}E_{1,m}$ , $\\omega =E_{1,m}\\delta _{m}\\nu $ , where $\\nu $ is a prefix of $E_{1,m}$ .", "Since $\\omega $ is a palindrome, $\\omega =\\overleftarrow{\\omega }=\\overleftarrow{\\nu }\\delta _{m}E_{1,m}$ .", "This means $\\overleftarrow{\\nu }$ is also the prefix of $E_{1,m}$ .", "So $\\nu =\\overleftarrow{\\nu }$ is a palindrome.", "By assumption and $\\omega \\ne E_{1,m+1}$ , there exists $n<m$ s.t.", "$\\nu =E_{1,n}$ .", "This means $\\omega =E_{1,m}\\delta _{m}E_{1,n}$ , $m\\ne n$ .", "By Lemma REF , $\\omega $ can't be a palindrome, contradicting $\\omega =\\overleftarrow{\\omega }$ .", "By the two cases above and by induction, if palindrome $\\omega $ is a prefix of $E_{1,m}$ , the conclusions hold for all $m$ .", "Similarly, if palindrome $\\omega $ is a suffix of $E_{1,m}$ , there exists $n$ ($\\le m$ ) such that $\\omega =E_{1,n}$ .", "Property 3.11 () Let palindrome $\\omega $ be a proper prefix (resp.", "suffix) of $E_{2,m}$ , then there exists $n$ ($\\le m$ ) such that $\\omega =E_{1,n}$ .", "For $m=1$ , $E_{2,m}=aa$ , the palindromical proper prefix is $a$ .", "For $m=2$ , $E_{2,m}=ababa$ , the palindromical prefixes are $a$ and $aba$ .", "The conclusion holds.", "Assume the conclusions hold for $m$ .", "If palindrome $\\omega $ is a proper prefix of $E_{2,m+1}=E_{1,m+1}\\delta _{m}E_{1,m}$ , then Case 1: $\\omega $ is a prefix of $E_{1,m+1}$ , then there exists $n\\le m+1$ such that $\\omega =E_{1,n}$ .", "Case 2: $\\omega $ isn't a prefix of $E_{1,m+1}$ , then $\\omega =E_{1,m+1}\\delta _{m}\\nu $ , where $\\nu $ is a proper prefix of $E_{1,m}$ .", "Since $\\omega $ is a palindrome, $\\omega =\\overleftarrow{\\omega }=\\overleftarrow{\\nu }\\delta _{m}E_{1,m+1}$ .", "So $\\overleftarrow{\\nu }$ is the prefix of $E_{1,m+2}$ .", "Moreover $|\\nu |<|E_{1,m}|$ , so $\\overleftarrow{\\nu }$ is the prefix of $E_{1,m}$ .", "This means $\\nu =\\overleftarrow{\\nu }$ is a palindrome.", "By Property REF , there exists $n< m$ s.t.", "$\\nu =E_{1,n}$ .", "This means $\\omega =E_{1,m+1}\\delta _{m}E_{1,n}$ , $n\\ne m+1$ .", "By Lemma REF , $\\omega $ can't be a palindrome, contradicting $\\omega =\\overleftarrow{\\omega }$ .", "By the two cases above and by induction, if palindrome $\\omega $ is a prefix of $E_{2,m}$ , the conclusions hold for all $m$ .", "Similarly, if palindrome $\\omega $ is a suffix of $E_{2,m}$ , there exists $n$ ($\\le m$ ) such that $\\omega =E_{1,n}$ ." ], [ "The simplification by palindromic property", "The “weak type of envelope extension” means “each $\\omega $ occurs in $Env(\\omega )$ only once”.", "Though the analysis in this subsection, we only need to prove the last property for $\\omega $ is a palindrome.", "Moreover, if $\\omega $ occurs in $Env(\\omega )$ at least twice, we can pick two of them.", "So we only need to negate the proposition that “there exist a palindrome $\\omega $ occurs in $Env(\\omega )$ twice”.", "Lemma 3.12 () For palindrome $\\Lambda $ , $|\\Lambda |$ is odd, if there exist factor $\\omega $ satisfies: (1) $\\omega $ occurs in $\\Lambda $ twice; (2) Both of the two $\\omega $ 's contain the middle letter of $\\Lambda $ .", "Then there exist a palindrome $\\varpi $ occurs in $\\Lambda $ twice, at symmetric positions.", "Denote the two $\\omega $ in $\\Lambda $ by $\\omega _1$ and $\\omega _2$ , and $\\Lambda =\\alpha \\omega _1\\beta =\\gamma \\omega _2\\eta ,$ where $|\\alpha |<|\\gamma |$ .", "Denote $|\\alpha |:=d_1$ and $|\\eta |:=d_2$ .", "Since both of $\\omega _1$ and $\\omega _2$ contain the middle letter of $\\Lambda $ , they are overlapped.", "(1) $d_1=d_2$$\\overleftarrow{\\Lambda }=\\Lambda $$\\overleftarrow{\\omega _2}$$\\overleftarrow{\\omega _1}$$\\Downarrow $ mirror$\\Lambda $$\\omega _1$$\\omega _2$ $d_1$$d_2$ (2) $d_1<d_2$$\\Lambda $$\\varpi _1$$\\varpi _2$$\\Downarrow $ unite$\\overleftarrow{\\Lambda }=\\Lambda $$\\overleftarrow{\\omega _2}$$\\overleftarrow{\\omega _1}$$\\Downarrow $ mirror$\\Lambda $$\\omega _1$$\\omega _2$ $d_1$$d_2$$\\nu _1$$\\nu _2$$\\nu _3$Fig 3.1: A palindrome $\\varpi $ occurs in $\\Lambda $ twice, at symmetric positions.", "Case 1.", "$d_1=d_2$ , see Fig 3.1(1).", "In this case, $\\omega _1=\\overleftarrow{\\omega _2}$ , i.e., $\\omega =\\overleftarrow{\\omega }$ , $\\omega $ is a palindrome.", "Case 2.", "$d_1\\ne d_2$ , let's just take $d_1<d_2$ , see Fig 3.1(2).", "In this case, let $\\omega _1$ and $\\omega _2$ overlapped at $\\nu _2$ , then $\\omega _1=\\nu _1\\nu _2$ and $\\overleftarrow{\\omega }_2=\\nu _2\\nu _3$ .", "Since $\\omega $ is a palindrome, $\\nu _2$ is a palindrome too, and $\\nu _3=\\overleftarrow{\\nu _1}$ .", "So $\\omega _1$ and $\\overleftarrow{\\omega _2}$ unite to a new word $\\varpi =\\nu _1\\nu _2\\overleftarrow{\\nu _1}$ , which is a palindrome, and occurs in $\\Lambda $ twice at symmetric positions.", "Thus the conclusion holds." ], [ "The weak type of envelope extension", "Theorem 3.13 () The factor $\\omega $ occurs exactly once in $Env(\\omega )$ .", "Suppose the factor $\\omega $ occurs twice in $Env(\\omega )$ , denoted by $\\omega _1$ and $\\omega _2$ .", "Denote $Env(\\omega )$ by $E_{i,m}$ , $i\\in \\lbrace 1,2\\rbrace $ and $m\\ge 1$ .", "Case 1.", "$Env(\\omega )=E_{1,m}$ .", "Since $E_{1,m}=E_{1,m-1}\\underline{\\delta _{m-1}}E_{1,m-1}$ , both $\\omega _1$ and $\\omega _2$ contain the middle letter $\\delta _{m-1}$ .", "Otherwise $\\omega \\prec E_{1,m-1}$ , $Env(\\omega )\\sqsubset E_{1,m-1}$ , contradicting $Env(\\omega )=E_{1,m}$ .", "By Lemma REF , we can assume without loss of generality that: $\\omega $ is palindrome; $\\omega _1$ and $\\omega _2$ occur in $E_{1,m}$ at symmetric positions.", "Denote $\\omega _1=\\mu _1\\delta _{m-1}\\mu _2$ and $\\omega _2=\\eta _1\\delta _{m-1}\\eta _2$ .", "Since $\\omega _1$ and $\\omega _2$ occur at symmetric positions, $|\\mu _1|=|\\eta _2|>|\\mu _2|=|\\eta _1|$ .", "Since $E_{1,m}$ is palindrome, $\\omega _2=\\overleftarrow{\\mu _2}\\delta _{m-1}\\overleftarrow{\\mu _1}$ .", "Since $\\omega $ is palindrome and $|\\mu _1|>|\\mu _2|$ , there exists $\\mu _3$ such that $\\mu _1=\\overleftarrow{\\mu _2}\\delta _{m-1}\\mu _3$ .", "This means $\\omega =\\overleftarrow{\\mu _2}\\delta _{m-1}\\mu _3\\delta _{m-1}\\mu _2$ .", "Moreover $\\mu _3$ is palindrome.", "Now, we consider two subcases.", "Case 1.1.", "$|\\mu _3|>|\\mu _2|$ .", "Since $\\mu _3$ is palindrome, by Property REF , there exists $n$ such that $\\mu _3=E_{1,n}$ .", "Since $\\delta _{m-1}\\mu _3\\delta _{m-1}\\prec E_{1,m}$ , by Property REF , $n\\in \\lbrace m-2k-1,k=1,2,\\ldots ,\\lfloor \\frac{m-2}{2}\\rfloor \\rbrace $ .", "Thus $\\omega =\\overleftarrow{\\mu _2}\\delta _{m-1}\\mu _3\\delta _{m-1}\\mu _2\\prec E_{1,m-3}\\delta _{m-1}E_{1,m-3}\\delta _{m-1}E_{1,m-3}=E_{2,m-2}.$ This contradict the hypotheses of $Env(\\omega )=E_{1,m}$ .", "Case 1.2.", "$|\\mu _3|\\le |\\mu _2|$ .", "(1) If there exists $k\\ge 0$ such that $|\\mu _2|+1=k(|\\mu _3|+1)$ , $\\overleftarrow{\\mu _2}=\\mu _3(\\delta _{m-1}\\mu _3)^k$ .", "Obviously, $\\overleftarrow{\\mu _2}\\delta _{m-1}\\mu _3$ is palindrome.", "By Property REF , there exists $n_1$ such that $\\overleftarrow{\\mu _2}\\delta _{m-1}\\mu _3=E_{1,n_1}$ .", "Similarly, $\\mu _2$ is palindrome, there exists $n_2<n_1$ such that $\\mu _2=E_{1,n_2}$ .", "Thus $\\omega =\\overleftarrow{\\mu _2}\\delta _{m-1}\\mu _3\\underline{\\delta _{m-1}}\\mu _2=E_{1,n_1}\\underline{\\delta _{m-1}}E_{1,n_2}.$ By Lemma REF , $\\omega =E_{1,n_1}\\delta _{m-1}E_{1,n_2}$ does not be a palindrome.", "(2) Otherwise, there exists $k\\ge 1$ and $0<|\\mu _4|<|\\mu _3|$ such that $\\overleftarrow{\\mu _2}=\\overleftarrow{\\mu _4}(\\delta _{m-1}\\mu _3)^k$ .", "Thus $\\omega =\\overleftarrow{\\mu _2}\\delta _{m-1}\\mu _3\\underline{\\delta _{m-1}}\\mu _2=\\overleftarrow{\\mu _4}(\\delta _{m-1}\\mu _3)^k\\delta _{m-1}\\mu _3\\underline{\\delta _{m-1}}(\\mu _3\\delta _{m-1})^k\\mu _4=\\overleftarrow{\\mu _4}\\delta _{m-1}\\mu _5\\delta _{m-1}\\mu _4,$ where $\\mu _5=(\\mu _3\\delta _{m-1})^k\\mu _3(\\delta _{m-1}\\mu _3)^k$ is a palindrome with $|\\mu _5|>|\\mu _4|$ .", "Using the conclusion in Case 1.1, we get a contradiction of $Env(\\omega )=E_{1,m}$ .", "In summary, Case 1 is impossible.", "Case 2.", "$Env(\\omega )=E_{2,m}$ .", "Since $E_{2,m}=E_{1,m-1}\\underline{\\delta _{m-1}E_{1,m-1}\\delta _{m-1}}E_{1,m-1}$ , both $\\omega _1$ and $\\omega _2$ contain the factor $\\delta _{m-1}E_{1,m-1}\\delta _{m-1}$ with underline.", "Otherwise, $\\omega \\prec E_{1,m-1}\\delta _{m-1}E_{1,m-1}=E_{1,m}$ , $Env(\\omega )\\sqsubset E_{1,m}$ , contradicting $Env(\\omega )=E_{2,m}$ .", "Denote $\\omega _1=\\mu _1\\delta _{m-1}\\underline{E_{1,m-1}}\\delta _{m-1}\\mu _2$ and $\\omega _2=\\eta _1\\delta _{m-1}\\underline{E_{1,m-1}}\\delta _{m-1}\\eta _2$ .", "Obviously, the $E_{1,m-1}$ 's in $\\omega _1$ and $\\omega _2$ are overlapped.", "By Theorem REF , $E_{1,m-1}=A_{m-1}\\delta _{m-1}^{-1}$ has only two distinct return words $r_1(E_{1,m-1})=A_{m-1}$ and $r_2(E_{1,m-1})=A_{m-2}$ .", "Moreover, $E_{1,m-1,p}$ and $E_{1,m-1,p+i}$ are separate for $i\\ge 2$ ; $E_{1,m-1,p}$ and $E_{1,m-1,p+1}$ are separate for $\\Theta _1[p]=a$ ; $E_{1,m-1,p}$ and $E_{1,m-1,p+1}$ are overlapped for $\\Theta _1[p]=b$ .", "This means the overlap of the $E_{1,m-1}$ 's in $\\omega _1$ and $\\omega _2$ is $r_2^{-1}(E_{1,m-1})E_{1,m-1}=B_{m-2}\\delta _{m-1}^{-1}$ .", "Thus $\\omega _1=\\mu _1\\delta _{m-1}A_{m-2}\\underline{B_{m-2}\\delta _{m-1}^{-1}}\\delta _{m-1}\\mu _2,~\\omega _2=\\mu _1\\delta _{m-1}\\underline{B_{m-2}\\delta _{m-1}^{-1}}\\delta _{m}B_{m-2}\\mu _2.$ Notice that the first letter followed the overlap $\\underline{B_{m-2}\\delta _{m-1}^{-1}}\\delta _{m}$ in $\\omega _1$ and $\\omega _2$ are $\\delta _{m-1}$ and $\\delta _{m}$ , respectively.", "This contradict that $\\omega _1$ and $\\omega _2$ have the same expression.", "Thus the conclusion holds.", "Let $\\omega $ be a factor, and denote $Env(\\omega )$ by $E_{i,m}$ .", "By Theorem REF , there exist uniquely two words $\\mu _1(\\omega )$ and $\\mu _2(\\omega )$ depending only on $\\omega $ , such that $E_{i,m}=\\mu _1(\\omega )\\omega \\mu _2(\\omega ).$ Definition 3.14 (Weak type of envelope extension) Let $\\omega $ be a factor, the expression (REF ) above is called the weak type of envelope extension of $\\omega $ ." ], [ "The extension of $\\omega _p$", "Property 3.15 (Envelope word inside) For $m>2$ , (1) if $Env(\\omega )=E_{1,m}$ , $E_{1,m-2}\\prec \\omega $ ; (2) if $Env(\\omega )=E_{2,m}$ , $E_{1,m-1}\\prec \\omega $ .", "(1) If $Env(\\omega )=E_{1,m}$ , $\\omega $ can't be the factor of $E_{1,m-1}$ or $E_{2,m-2}$ .", "$E_{1,m-1}$$\\underbrace{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}$$E_{2,m-2}$$\\underbrace{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}$$E_{1,m}=E_{1,m-2}\\delta _{m}E_{1,m-3}\\delta _{m-1}E_{1,m-3}\\delta _{m-1}E_{1,m-3}\\delta _{m-1}E_{1,m-3}\\delta _{m}E_{1,m-2}$$\\overbrace{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}$$E_{2,m-2}$$\\overbrace{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}$$E_{1,m-1}$Fig 3.2: $E_{1,m-1}$ and $E_{2,m-2}$ in $E_{1,m}$ .", "Consider the three overlaps in the figure above, $\\omega $ has three cases: 1) $\\omega \\succ \\delta _{m}E_{1,m-3}\\delta _{m-1}E_{1,m-3}\\delta _{m-1}$ ; 2) $\\omega \\succ \\delta _{m-1}E_{1,m-3}\\delta _{m-1}E_{1,m-3}\\delta _{m-1}$ ; 3) $\\omega \\succ \\delta _{m-1}E_{1,m-3}\\delta _{m-1}E_{1,m-3}\\delta _{m}$ .", "All of them include that $\\omega \\succ E_{1,m-3}\\delta _{m-1}E_{1,m-3}=E_{1,m-2}$ .", "(2) If $Env(\\omega )=E_{2,m}$ , $\\omega $ can't be the factor of $E_{1,m}$ .", "$E_{1,m}$$\\underbrace{~~~~~~~~~~~~~~~~~~~~~~~~}$$E_{2,m}=E_{1,m-1}\\delta _{m-1}E_{1,m-1}\\delta _{m-1}E_{1,m-1}$$\\overbrace{~~~~~~~~~~~~~~~~~~~~~~~~}$$E_{1,m}$Fig 3.3: $E_{1,m}$ in $E_{2,m}$ .", "Consider the overlap in the figure above, $\\omega \\succ \\delta _{m-1}E_{1,m-1}\\delta _{m-1}\\succ E_{1,m-1}$ .", "Denote $Env(\\omega )=E_{i,m}$ .", "Now we turn to prove that, for each $\\omega _p$ there exists a $q$ such that $\\omega \\prec E_{i,m,q}$ .", "More precisely, $L(\\omega ,p)\\ge L(E_{i,m},q)$ and $L(\\omega ,p)+|\\omega |\\le L(E_{i,m},q)+|E_{i,m}|$ .", "In this case, we say that the $\\omega _p$ extend to $E_{i,m,q}$ .", "$\\bullet $    Case 1.", "$\\mathbf {Env(\\omega )=E_{1,m},~m\\ge 1.", "}$ By Theorem REF , we have $\\lbrace r_p(E_{1,m-2})\\rbrace _{p\\ge 1}$ is $\\Theta _1$ over the alphabet $\\lbrace r_1(E_{1,m-2}),r_2(E_{1,m-2})\\rbrace =\\lbrace A_{m-2},A_{m-3}\\rbrace =\\lbrace E_{1,m-2}\\delta _{m},E_{1,m-2}\\delta _{m}B_{m-3}^{-1}\\rbrace ,$ where $E_{1,m-2}=A_{m-2}\\delta _{m}^{-1}$ .", "Since factors with length 3 in $\\Theta _1$ are $\\lbrace abb,bba,baa,aaa,aab,bab\\rbrace $ , $E_{1,m-2}$ occurs in $\\mathbb {D}$ has 6 cases, where $\\mathbf {a}=r_1(E_{1,m-2})$ , $\\mathbf {b}=r_2(E_{1,m-2})$ .", "(1) $\\mathbf {abb}= A_{m-2} E_{1,m-2}\\delta _{m}B_{m-3}^{-1} A_{m-3}=A_{m-2}E_{1,m-2}\\delta _{m-1}\\succ \\delta _{m}E_{1,m-2}\\delta _{m-1}$ ; (2) $\\mathbf {bba}= A_{m-3} E_{1,m-2}\\delta _{m}B_{m-3}^{-1} A_{m-2}=A_{m-3}E_{1,m-2}\\delta _{m-1}B_{m-3}\\succ \\delta _{m-1}E_{1,m-2}\\delta _{m-1}$ ; (3) $\\mathbf {baa}= A_{m-3} E_{1,m-2}\\delta _{m} A_{m-2}\\succ \\delta _{m-1}E_{1,m-2}\\delta _{m}$ ; (4) $\\mathbf {aaa}= A_{m-2} E_{1,m-2}\\delta _{m} A_{m-2}\\succ \\delta _{m}E_{1,m-2}\\delta _{m}$ ; (5) $\\mathbf {aab}= A_{m-2} E_{1,m-2}\\delta _{m} A_{m-3}\\succ \\delta _{m}E_{1,m-2}\\delta _{m}$ ; (6) $\\mathbf {bab}= A_{m-3} E_{1,m-2}\\delta _{m} A_{m-3}\\succ \\delta _{m-1}E_{1,m-2}\\delta _{m}$ .", "Here we always rewrite the middle letter $\\mathbf {a}$ or $\\mathbf {b}$ to be an expression with envelope word $E_{1,m-2}$ .", "Lemma 3.16 The factor $\\delta _{m}E_{1,m-2}\\delta _{m-1}$ occurs in the period-doubling sequence always preceded by the word $E_{1,m-2}=A_{m-2}\\delta _{m}^{-1}$ and followed by the word $E_{1,m-1}=A_{1,m-1}\\delta _{m-1}^{-1}$ .", "By the analysis in the beginning of Case 1, $\\delta _{m}E_{1,m-2}\\delta _{m-1}$ occurs in the period-doubling sequence has only one case: $\\mathbf {abb}$ .", "It extends to $\\mathbf {\\underline{abb}abb}$ or $\\mathbf {\\underline{abb}aa}$ in $\\Theta _1$ .", "$\\mathbf {\\underline{abb}abb}=A_{m-2}E_{1,m-2}\\delta _{m-1}A_{m-2}A_{m-3}A_{m-3}=A_{m-2}E_{1,m-2}\\delta _{m-1}A_{m-1}$ ; $\\mathbf {\\underline{abb}aa}=A_{m-2}E_{1,m-2}\\delta _{m-1}A_{m-2}A_{m-2}=A_{m-2}E_{1,m-2}\\delta _{m-1}B_{m-1}$ .", "Since $B_{m-1}\\delta _{m}^{-1}=A_{m-1}\\delta _{m-1}^{-1}$ , both of them have prefix $A_{m-2}\\delta _{m}^{-1}\\underline{\\delta _{m}E_{1,m-2}\\delta _{m-1}} A_{1,m-1}\\delta _{m-1}^{-1}$ .", "Here we show the factor $\\delta _{m}E_{1,m-2}\\delta _{m-1}$ with underline.", "This means the conclusion holds.", "Lemma 3.17 The factor $\\delta _{m-1}E_{1,m-2}\\delta _{m-1}$ occurs in the period-doubling sequence always preceded by the word $E_{1,m-2}\\delta _{m}E_{1,m-3}=A_{m-2}A_{m-3}\\delta _{m-1}^{-1}$ and followed by the word $E_{1,m-3}\\delta _{m}E_{1,m-2}=B_{m-3}A_{m-2}\\delta _{m}^{-1}$ .", "By the analysis in the beginning of Case 1, $\\delta _{m-1}E_{1,m-2}\\delta _{m-1}$ occurs in the period-doubling sequence has only one case: $\\mathbf {bba}$ .", "It extends to $\\mathbf {a\\underline{bba}a}$ or $\\mathbf {a\\underline{bba}bb}$ in $\\Theta _1$ .", "$\\mathbf {a\\underline{bba}a}=A_{m-2}A_{m-3}E_{1,m-2}\\delta _{m-1}B_{m-3}A_{m-2}$ .", "$\\mathbf {a\\underline{bba}bb}=A_{m-2}A_{m-3}E_{1,m-2}\\delta _{m-1}B_{m-3}A_{m-3}A_{m-3} =A_{m-2}A_{m-3}E_{1,m-2}\\delta _{m-1}B_{m-3}B_{m-2}$ .", "Both of them have prefix $A_{m-2}A_{m-3}\\delta _{m-1}^{-1}\\underline{\\delta _{m-1}E_{1,m-2}\\delta _{m-1}} B_{m-3}A_{m-2}\\delta _{m}^{-1}$ .", "Here we show the factor $\\delta _{m-1}E_{1,m-2}\\delta _{m-1}$ with underline.", "This means the conclusion holds.", "Lemma 3.18 The factor $\\delta _{m-1}E_{1,m-2}\\delta _{m}$ occurs in the period-doubling sequence always preceded by the word $E_{1,m-1}=A_{m-1}\\delta _{m-1}^{-1}$ and followed by the word $E_{1,m-2}=A_{m-2}\\delta _{m}^{-1}$ .", "By the analysis in the beginning of Case 1, $\\delta _{m-1}E_{1,m-2}\\delta _{m-1}$ occurs in the period-doubling sequence has two cases: $\\mathbf {baa}$ and $\\mathbf {bab}$ .", "They extend to $\\mathbf {ab\\underline{baa}}$ and $\\mathbf {ab\\underline{bab}b}$ in $\\Theta _1$ , respectively.", "$\\mathbf {ab\\underline{baa}}=A_{m-2}A_{m-3}A_{m-3}E_{1,m-2}\\delta _{m}A_{m-2}=A_{m-1}E_{1,m-2}\\delta _{m}A_{m-2}$ ; $\\mathbf {ab\\underline{bab}b}=A_{m-2}A_{m-3}A_{m-3}E_{1,m-2}\\delta _{m}A_{m-3}A_{m-3}=A_{m-1}E_{1,m-2}\\delta _{m}B_{m-2}$ .", "Since $A_{m-2}\\delta _{m}^{-1}=B_{m-2}\\delta _{m-1}^{-1}$ , both of them have prefix $A_{m-1}\\delta _{m-1}^{-1}\\underline{\\delta _{m-1}E_{1,m-2}\\delta _{m}} A_{m-2}\\delta _{m}^{-1}$ .", "Here we show the factor $\\delta _{m-1}E_{1,m-2}\\delta _{m}$ with underline.", "This means the conclusion holds.", "Property 3.19 If $Env(\\omega )=E_{1,m}$ , then each $\\omega _p$ can extend to a $E_{1,m}$ in $\\mathbb {D}$ .", "By Lemma REF , $\\delta _{m}E_{1,m-2}\\delta _{m}=\\delta _{m}B_{m-2}\\lnot \\prec E_{1,m}=A_{m-2}B_{m-2}A_{m-2}A_{m-2}\\delta _{m}^{-1}$ .", "By Figure 3.2, for $m>2$ and $Env(\\omega )=E_{1,m}$ , $\\omega $ contains one of the three words below: (1) $\\omega \\succ \\delta _{m}E_{1,m-2}\\delta _{m-1}$ .", "By Lemma REF , $\\omega \\prec A_{m-2}\\delta _{m}^{-1}\\delta _{m}E_{1,m-2}\\delta _{m-1} A_{1,m-1}\\delta _{m-1}^{-1}=E_{1,m}$ .", "(2) $\\omega \\succ \\delta _{m-1}E_{1,m-2}\\delta _{m-1}$ .", "By Lemma REF , $\\omega \\prec A_{m-2}A_{m-3}E_{1,m-2}\\delta _{m-1} B_{m-3}A_{m-2}\\delta _{m}^{-1}=E_{1,m}$ .", "(3) $\\omega \\succ \\delta _{m-1}E_{1,m-2}\\delta _{m}$ .", "By Lemma REF , $\\omega \\prec A_{m-1}\\delta _{m-1}^{-1}\\delta _{m-1}E_{1,m-2}\\delta _{m} A_{m-2}\\delta _{m}^{-1}=E_{1,m}$ .", "In summary, no matter where $\\omega $ occurs, it can extend to a $E_{1,m}$ in $\\mathbb {D}$ .", "$\\bullet $    Case 2.", "$\\mathbf {Env(\\omega )=E_{2,m},~m\\ge 1.", "}$ By Theorem REF , we have $\\lbrace r_p(E_{1,m-1})\\rbrace _{p\\ge 1}$ in $\\mathbb {D}$ is $\\Theta _1$ over the alphabet $\\lbrace r_1(E_{1,m-1}),r_2(E_{1,m-1})\\rbrace =\\lbrace A_{m-1},A_{m-2}\\rbrace =\\lbrace E_{1,m-1}\\delta _{m-1},E_{1,m-1}\\delta _{m-1}B_{m-2}^{-1}\\rbrace ,$ where $E_{1,m-1}=A_{m-1}\\delta _{m-1}^{-1}$ .", "Since factors with length 3 in $\\Theta _1$ are $\\lbrace abb,bba,baa,aaa,aab,bab\\rbrace $ , $\\mathbf {a}=r_1(E_{1,m-1})$ , $\\mathbf {b}=r_2(E_{1,m-1})$ , $E_{1,m-1}$ occurs in $\\mathbb {D}$ has 6 cases as follow: (1) $\\mathbf {abb}= A_{m-1} E_{1,m-1}\\delta _{m-1}B_{m-2}^{-1} A_{m-2}=A_{m-1}E_{1,m-1}\\delta _{m}\\succ \\delta _{m-1}E_{1,m-1}\\delta _{m}$ ; (2) $\\mathbf {bba}= A_{m-2} E_{1,m-1}\\delta _{m-1}B_{m-2}^{-1} A_{m-1}=A_{m-2}E_{1,m-1}\\delta _{m}B_{m-2}\\succ \\delta _{m}E_{1,m-1}\\delta _{m}$ ; (3) $\\mathbf {baa}= A_{m-2} E_{1,m-1}\\delta _{m-1} A_{m-1}\\succ \\delta _{m}E_{1,m-1}\\delta _{m-1}$ ; (4) $\\mathbf {aaa}= A_{m-1} E_{1,m-1}\\delta _{m-1} A_{m-1}\\succ \\delta _{m-1}E_{1,m-1}\\delta _{m-1}$ ; (5) $\\mathbf {aab}= A_{m-1} E_{1,m-1}\\delta _{m-1} A_{m-2}\\succ \\delta _{m-1}E_{1,m-1}\\delta _{m-1}$ ; (6) $\\mathbf {bab}= A_{m-2} E_{1,m-1}\\delta _{m-1} A_{m-2}\\succ \\delta _{m}E_{1,m-1}\\delta _{m-1}$ .", "Here we always rewrite the middle letter $\\mathbf {a}$ or $\\mathbf {b}$ to be an expression with envelope word $E_{1,m-1}$ .", "Lemma 3.20 The word $\\delta _{m-1}E_{1,m-1}\\delta _{m-1}$ occurs in the period-doubling sequence always preceded and followed by the word $E_{1,m-1}=A_{m-1}\\delta _{m-1}^{-1}$ .", "By the analysis above, $\\delta _{m-1}E_{1,m-1}\\delta _{m-1}$ occurs in the period-doubling sequence has two cases: $\\mathbf {aaa}$ and $\\mathbf {aab}$ .", "Here $\\mathbf {\\underline{aaa}}=A_{m-1}E_{1,m-1}\\delta _{m-1}A_{m-1}$ .", "Moreover $\\mathbf {aab}$ extends to $\\mathbf {\\underline{aab}b}$ in $\\Theta _1$ .", "$\\mathbf {\\underline{aab}b}=A_{m-1}E_{1,m-1}\\delta _{m-1}A_{m-2}A_{m-2}=A_{m-1}E_{1,m-1}\\delta _{m-1}B_{m-1}$ .", "Since $B_{m-1}\\delta _{m}^{-1}=A_{m-1}\\delta _{m-1}^{-1}$ , both of them have prefix $A_{m-1}\\delta _{m-1}^{-1}\\underline{\\delta _{m-1}E_{1,m-1}\\delta _{m-1}} A_{m-1}\\delta _{m-1}^{-1}$ .", "This means the conclusion holds.", "Property 3.21 If $Env(\\omega )=E_{2,m}$ , then each $\\omega _p$ can extend to a $E_{2,m}$ in $\\mathbb {D}$ .", "By Lemma REF and REF , the next three words are not the factors of $E_{2,m}=B_mB_{m-1}\\delta _{m}^{-1}$ , $\\delta _{m}E_{1,m-1}\\delta _{m}=\\delta _{m}B_{m-1},~\\delta _{m-1}E_{1,m-1}\\delta _{m}=\\delta _{m-1}B_{m-1},~\\delta _{m}E_{1,m-1}\\delta _{m-1}=\\delta _{m}A_{m-1}.$ By Figure 3.3, $\\delta _{m-1}E_{1,m-1}\\delta _{m-1}\\prec \\omega $ .", "By Lemma REF , $\\omega \\prec A_{m-1}\\delta _{m-1}^{-1}\\delta _{m-1}E_{1,m-1}\\delta _{m-1} A_{m-1}\\delta _{m-1}^{-1}=B_{m}B_{m-1}\\delta _{m}^{-1}=E_{2,m}$ .", "This means no matter where $\\omega $ occurs, it can extend to a $E_{1,m}$ in $\\mathbb {D}$ ." ], [ "The strong type of envelope extension", "Theorem 3.22 $Env(\\omega _p)=Env(\\omega )_p=E_{i,m,p}$ , $i=1,2$ and $p\\ge 1$ .", "By Property REF and REF , we have for all $p$ , there is a $q$ s.t.", "$E_{i,m,q}=Env(\\omega _p)$ .", "By the definition of $Env(\\omega )$ , $\\omega \\prec Env(\\omega )$ .", "So for all $q$ , there is a $p$ s.t.", "$\\omega _p\\prec E_{i,m,q}$ .", "By the weak type of envelope extension, we have (1) if $\\omega _{p_1}\\prec E_{i,m,q}$ and $\\omega _{p_2}\\prec E_{i,m,q}$ , then $p_1=p_2$ ; (2) if $\\omega _p\\prec E_{i,m,q_1}$ and $\\omega _p\\prec E_{i,m,q_2}$ , then $q_1=q_2$ .", "So $Env(\\omega _p)=E_{i,m,q}$ means $p=q$ .", "Thus $Env(\\omega _p)=Env(\\omega )_p=E_{i,m,p}$ .", "Let $\\omega $ be a factor, and denote $Env(\\omega )$ by $E_{i,m}$ .", "By Theorem REF , there exist uniquely two words $\\mu _1(\\omega )$ and $\\mu _2(\\omega )$ depending only on $\\omega $ , such that $E_{i,m,p}=\\mu _1(\\omega )\\omega _p\\mu _2(\\omega ).$ The $\\mu _1(\\omega )$ and $\\mu _2(\\omega )$ have the same expressions with the $\\mu _1(\\omega )$ and $\\mu _2(\\omega )$ in expression (REF ).", "Definition 3.23 (Strong type of envelope extension) Let $\\omega $ be a factor, the expression (REF ) above is called the strong type of envelope extension of $\\omega $ ." ], [ "The return word sequences of general factors in $\\mathbb {D}$", "By the strong type of envelope extension, we can extend Theorem REF and REF to general factors.", "Theorem 4.1 () (1) When $Env(\\omega )=E_{1,m}$ , the return word sequence $\\lbrace r_p(\\omega )\\rbrace _{p\\ge 1}$ in $\\mathbb {D}$ is $\\Theta _1$ over the alphabet $\\lbrace r_1(\\omega ),r_2(\\omega )\\rbrace $ .", "(2) When $Env(\\omega )=E_{2,m}$ , the return word sequence $\\lbrace r_p(\\omega )\\rbrace _{p\\ge 1}$ in $\\mathbb {D}$ is $\\Theta _2$ over the alphabet $\\lbrace r_1(\\omega ),r_2(\\omega ),r_4(\\omega )\\rbrace $ .", "Property 4.2 () Let $Env(\\omega )$ be $E_{i,m}$ .", "$r_0(\\omega )=r_0(E_{i,m})\\mu _1(\\omega )$ , $r_p(\\omega )=\\mu _1(\\omega )^{-1}r_p(E_{i,m})\\mu _1(\\omega )$ for $p\\ge 1$ .", "The proof of the property will be easy by the following figure.", "$\\mu _1$$\\mu _2$$\\omega _1$$E_{i,m,1}$$r_0(E_{i,m})$$r_0(\\omega )$Fig.", "4.1: The relation among $\\omega _1$ , $E_{i,m,1}$ , $r_0(\\omega )$ and $r_0(E_{i,m})$ .", "$\\mu _1$$\\mu _1$$\\mu _2$$\\mu _2$$\\omega _p$$\\omega _{p+1}$$E_{i,m,p}$$E_{i,m,p+1}$$r_p(E_{i,m})$$r_p(\\omega )$Fig.", "4.2: The relation among $\\omega _p$ , $E_{i,m,p}$ , $r_p(\\omega )$ and $r_p(E_{i,m})$ .", "The property above can be proved easily by Figure 4.1 and 4.2.", "By Property REF , REF and REF , we can give the expressions of $r_p(\\omega )$ .", "Since the expressions are complicated, we omit them.", "Corollary 4.3 (The Lengths of $r_p(\\omega )$ ) Denote $Env(\\omega )$ by $E_{i,m}$ , let $E_{i,m}=\\mu _1(\\omega )\\omega \\mu _2(\\omega )$ , then (1) if $i=1$ , $|r_0(\\omega )|=|\\mu _1(\\omega )|$ , $|r_1(\\omega )|=2^m$ , $|r_2(\\omega )|=2^{m-1}$ ; (2) if $i=2$ , $|r_0(\\omega )|=2^m+|\\mu _1(\\omega )|$ , $|r_1(\\omega )|=2^{m-1}$ , $|r_2(\\omega )|=7\\times 2^{m-1}$ , $|r_4(\\omega )|=3\\times 2^{m-1}$ .", "Acknowledgments The research is supported by the Grant NSFC No.11431007, No.11271223 and No.11371210." ] ]
1606.05150
[ [ "Misunderstanding that the Effective Action is Convex under Broken\n Symmetry" ], [ "Abstract The widespread belief that the effective action is convex and has a flat bottom under broken global symmetry is shown to be wrong.", "We show spontaneous symmetry breaking necessarily accompanies non-convexity in the effective action for quantum field theory, or in the free energy for statistical mechanics, and clarify the magnitude of non-convexity.", "For quantum field theory, it is also proved that translational invariance breaks spontaneously when the system is in the non-convex region, and that different vacua of spontaneously broken symmetry cannot be superposed.", "As applications of non-convexity, we study the first-order phase transition which happens at the zero field limit of spontaneously broken symmetry, and we propose a simple model of phase coexistence which obeys the Born rule." ], [ "Introduction", "The effective action $\\Gamma [\\langle \\varphi (x) \\rangle ]$ is one of the most fundamental objects in quantum field theory.", "Much has been said about it under broken global symmetry.", "For simplicity, we here use the language of a field theory with one scalar field $\\varphi $ , symmetric under the operation $\\varphi \\rightarrow -\\varphi $ .", "Let the field has the non-zero vacuum expectation value $|\\langle \\varphi \\rangle | = \\varphi _\\mathrm {sp}$ .", "It is widely believed that: “Because (A) $-\\Gamma $ is downward convex, (B) it has a flat bottom for $|\\langle \\varphi \\rangle | < \\varphi _\\mathrm {sp}.$ (C) States in the flat bottom are realized as a linear superposition of the vacua $|\\pm \\Omega \\rangle $ , where $\\langle \\pm \\Omega | \\varphi (x) | \\pm \\Omega \\rangle = \\pm \\varphi _\\mathrm {sp}$ .” We will show, however, that all the points (A), (B) and (C) are wrong.Almost ubiquitously symanzik1970 is cited, presumably intending the attribution of the point (A).", "However, [24] explicitly excludes the “flat-bottom” region from the discussion.", "See its appendix.", "Based on this misunderstanding, it has been thought that there is contradiction, and the resolution has been attempted.", "Its history is concisely summarized in the recent article by Argyres2009.pathIntegral.", "See also [][sec.", "2.2.5]Delamotte and 1126-6708-2007-04-051 for recent cases where convexity is assumed.", "The origin of this fallacy probably comes from the similarity of the effective action with the free energy in statistical mechanics.", "For example, consider Ising ferromagnet where temperature $T$ is $ < T_c$ , the critical temperature, so there is spontaneous magnetization $M = \\pm M_\\mathrm {sp}$ , where $M$ is the total magnetization.", "Let $F(\\beta , h)$ be the free energy, where $\\beta = 1/T$ and $h$ the external field, and its Legendre transform $C(\\beta , M) := F + hM.$ $C$ plays a role similar to the effective potential $\\mathcal {V}(\\langle \\varphi \\rangle )$ , which is the density of the effective action with uniform vacuum configuration.", "In fact, although $F$ is always convex,Do not confuse it with (exotic) non-convex $F$ which has been of recent interest.", "For examples, see sethna.", "$C$ is not for $|M| < M_\\mathrm {sp}$ .", "It's because it is the region of phase coexistence, and there the cost of domain wall formation arises.", "So the non-convexity of $C$ is of order $\\mathrm {O}(L)$ in the 2-dimensional Ising model, where $L$ is the linear extent of the system.", "Oddly enough, this fact has already been known for long in the communities of mathematics oriented statistical mechanics and Monte Carlo simulation.One of important uses of $C$ in the phase coexistence region is to obtain the interface free energy, for example in Wulff construction.", "For reviews, see Miracle-Sole:2013,Abraham.", "The interface free energy calculation is an old subject also in Monte Carlo simulation.", "[6] Still, the integral understanding of non-convexity is lacking.For example [][sec.", "1]Ioffe1994 says to the effect that the $C/V$ is strictly convex and has a flat bottom, but the decay rate of $\\exp (-\\beta C)$ is of interface order in the 2-dimensional Ising model.", "This statement is correct, but generality and accuracy are not sufficient.", "In a Monte Carlo simulation article, [20] at one hand it is said that there is a maximum in the free energy, while on the other hand the free energy is convex, showing some confusion.", "As a result, arguments on non-convexity is often too much detailed and technical, obscuring the essence.", "In the rest areas of physics, non-convexity is barely known, not to mention field theorists, leading to confusion.", "As we will show, spontaneous symmetry breaking always accompanies non-convexity, which is never zero.", "We will further prove that the order of non-convexity is always $\\mathrm {o}(V)$ , i.e.", "strictly smaller than the volume, including the cases of continuous symmetry, in which domain wall cannot form.", "Non-convexity is therefore indispensable to understand the correct vacuum in quantum field theory, or the correct equilibrium in statistical mechanics.", "We here emphasize that non-convexity itself is so easy that it should be learned even by undergraduate students.", "For example Landau's theory of phase transition is a standard topic in textbooks of statistical physics.", "It is wrongly explained to be a theory with defect by its non-convexity.", "We will be able to see how it can be correctly understood.", "For quantum field theory, two important consequences of non-convexity will be also shown.", "One is the impossibility of superpositions like $\\lambda |+ \\Omega \\rangle + (1-\\lambda ) |- \\Omega \\rangle $ , the point (C) mentioned above.", "This point has remained ambiguous to date.", "The other one is the spontaneous breakdown of the translational invariance inside the region of non-convexity.", "These two are true for general systems.", "Although nothing sure can be said except for the Ising model/the quantum theory with one scalar field, the region of non-convexity seems to introduce topological defects naturally.", "This paper is organized as follows.", "First in section we study the statistical mechanics of lattice systems.", "To be concrete, we first draw on an exact result on Ising model before obtaining general results.", "As an application, in section REF we will examine the first-order phase transition at $|M| = M_\\mathrm {sp}.$ We will find a slight shift of the transition point, and obtain a result more general than already known one.", "Precise degree of non-convexity in general lattice systems is stated in section REF .", "In section we study how the results on lattice models are recast into continuous field theory.", "We will see that the conventional notion of linear superposition can be too naïve in thermodynamic limit.", "In section , as another application of the non-convexity of the free energy, we model the phase coexistence of lattice systems, and obtain a simple system which obeys the Born rule.", "Section 5 gives the conclusion.", "In this section we study spontaneous symmetry breaking of lattice systems.", "We include some basic notes so that it will be readable also to students.", "Rather than stating general results from the beginning, until section REF we take up the 2-dimensional square Ising ferromagnetic model.", "It has its own right, since it is an exceptional case where the explicit dependence of the free energy on the magnetization is known.", "While it is not as complete as Onsager's solution, it is remarkable, since in the latter the dependence on the external field is unknown.", "We think it deserves to be known by every physicist.", "We define our model together with relevant symbols as: $Z(\\beta , h) & := \\sum _{\\lbrace \\sigma _i = \\pm 1\\rbrace } \\exp (-\\beta E(h)), \\\\E(h) & := - J \\sum _\\mathrm {n.n.}", "\\sigma _i \\sigma _j - hM, \\\\M & := \\sum _i \\sigma _i, \\\\m & := M / V, \\\\V & := L^2,\\\\\\beta F(\\beta , h) & := - \\log Z, \\\\\\begin{split}\\beta C(\\beta , m) & := \\beta (F + hM)\\\\& = -\\log \\sum _{\\lbrace \\sum \\sigma _i = M\\rbrace } \\exp (\\beta J \\sum _\\mathrm {n.n.}", "\\sigma _i \\sigma _j).\\end{split}$ Let the interaction be nearest-neighbor, and $m$ the per-spin magnetization.", "When $T < T_c$ , the system exhibits the spontaneous magnetization $m = \\pm m_\\mathrm {sp}(T)$ at zero field limit $h=\\pm 0.$ $C(\\beta , m)$ is the free energy when $(\\beta , m)$ are the variables that specify the state of the system, so $F(h=\\pm 0) = C(m = \\pm m_\\mathrm {sp}).$ We will often present statements and equations which are exactly correct and/or meaningful only in thermodynamic limit $L \\rightarrow \\infty $ , but readers will have no difficulty understanding them.", "The case where $|m| < m_\\mathrm {sp}$ is of our interest.", "Its entire region corresponds to $h=0$ .", "(Consider vapor-liquid coexistence of water.)", "As we will see, it is the region of phase coexistence, but we call it more generally as the “region singular with respect to the external field”, or RSEF.", "We want to write down $C$ .", "To this end, notice $Z(h=0) = \\sum _m \\exp (-\\beta C(m)),$ so $P(m^{\\prime }) := \\exp (-\\beta C(m^{\\prime })) / Z$ is the probability that $m$ takes the value $m^{\\prime }$ when $(T, h = 0)$ are specified.", "This is exactly the inverse of deriving grand canonical ensemble from canonical ensemble.", "The explicit formula of $P$ is obtained for the 2-dimensional Ising model for free and periodic boundaries in thermodynamic limit, at least for low enough temperature.", "[23] To rewrite it for $C$ , first let $\\mathcal {C}_\\text{in}(m) := C(m_\\mathrm {sp}) + \\\\\\quad {\\left\\lbrace \\begin{array}{ll}aLT \\sqrt{m_\\mathrm {sp}- |m|} & \\text{for } m_\\mathrm {f}< |m| < m_\\mathrm {sp}, \\\\aLT \\sqrt{m_\\mathrm {sp}- m_\\mathrm {f}\\phantom{|}} = \\mathrm {const.}", "& \\text{for } |m| < m_\\mathrm {f}\\end{array}\\right.", "}$ where $m_\\mathrm {f}> 0$ and $a > 0$ are finite constants dependent on $\\beta $ and the boundary condition.", "Then for $|m| < m_\\mathrm {sp}$ , $\\begin{split}C(m) =& \\mathcal {C}_\\text{in}(m), \\\\C^+(m) := & C(m) - C(m_\\mathrm {sp}).\\end{split}$ We have expressed $C$ in two steps because $\\mathcal {C}_\\text{in}$ is defined only in the RSEF, while $C$ is defined in the entire $(T, m)$ domain.", "Consider $\\mathcal {C}_\\text{in}$ as a pure mathematical function, while $C$ the free energy, a physical entity.", "It is now obvious that the claims (A), the convexity of $C$ , and (B), the “flat bottom” are illusions.", "$C$ is not always downward convex.", "We here clarify the flaw in the proof of $C$ 's convexity: $F$ is convex inside any finite interval of $h$ , and this fact has to be used to prove $C$ 's convexity.", "But the RSEF corresponds to one point $h=0$ .", "Thus $F$ 's convexity does not imply that $C$ is always convex.", "An intuitive interpretation of $C^+$ is desirable.", "[23] To avoid unnecessary complication, we take the free boundary.", "When $ |m| < m_\\mathrm {f}$ (“flat region” hereafter), the system consists of two domains, the plus-spin rich phase and minus- rich one, separated by a straight wall, running parallel to the two system edges.", "(Although 2-dimensional, we use the word domain wall.)", "As long as $m$ stays in the flat region, the wall can shift but remains straight, and $C$ does not change.", "When $m_\\mathrm {f}< |m| < m_\\mathrm {sp}$ , the smaller domain favors to be a dropletThe word “droplet” is a standard nomenclature.", "Here the droplet is sole and macroscopic, and in equilibrium.", "It is also used in non-equilibrium nucleation in usual first-order phase transitions, where there are many droplets, and the growth of initially microscopic droplets to macroscopic size is studied.", "at a corner to lower the free energy.", "We immediately recognize that (i) the RSEF is a phase coexistence region, (ii) $L\\sqrt{m_\\mathrm {sp}- |m|}$ is of the order of the wall length $l$ and $C^+$ can be written $= sl$ , where $s$ is the interface free energy per unit length, and (iii) the flatness of $C$ means the wall's thickness is finite, or the wall is “sharp”, so does not interact with the boundaries.", "(When the boundary is periodic, there are two walls, and they don't interact with each other.)", "For general boundaries the shape of the domain is determined by “Wulff construction” For reviews, see Miracle-Sole:2013,Abraham.", "There is a phase transition at $|m| = m_\\mathrm {f}$ which is sometimes called droplet-strip transition.", "It is obviously first-order; both domains discontinuously change their geometries, and both sides of the transition point have the metastable branch." ], [ "Normalization—first encounter", "To understand non-convexity correctly, we have to be careful about the normalization of variables.", "This is the key to our entire argument.", "All extensive variables scales as $V$ , or put differently, are $\\mathrm {O}(V).$ So it is tempting to consider “free energy per spin” $c(m) := C(m)/ V.$ Aside from its finiteness, $c$ has another advantage over $C$ that it's differentiable with $m$ : $C$ was originally defined for $M$ , a discrete variable, but we would like to define $\\partial C / \\partial M$ and to expect it to be $=h$ .", "There is no problem by defining it as $(C(M + n) - C(M)) / n$ $(n \\in \\mathbb {Z})$ , since $\\rightarrow \\partial C/ V \\partial m$ as $L\\rightarrow \\infty .$ This means $\\partial c /\\partial m = h$ .", "Then $c(m) = c(m_\\mathrm {sp})$ throughout the RSEF since it corresponds to $h=0$ .", "It is of course consistent with the explicit formula (REF ) of $\\mathcal {C}_\\text{in}$ .", "So $c(m)$ is indeed differentiable and globally convex.", "Notice at the same time $C^+(m) \\rightarrow + \\infty $ when $|m| < m_\\mathrm {sp}$ .", "It has to be so, because $P(m) \\propto \\exp (-\\beta C(m)).$ In words, it is necessary in order for spontaneous symmetry breaking to happen.", "The conclusion of this short section is: $c(m)=c(m_\\mathrm {sp})$ inside the RSEF means $c$ does not contain the information on the RSEF.", "Sticking to $c$ 's convexity is the same as considering $F(\\beta , h)$ and neglecting the RSEF.", "Correct normalization is to consider $C^+/ L.$ Two pedagogical notes are in order: (i) By the “Maxwell construction” the flat-bottomed $C$ can be obtained.", "It is equivalent to divide the system into the domains of different phases, and to ignore the domain wall free energy.", "It will not be much problematic, or rather better e.g.", "when considering bulk equilibrium in chemical reaction.", "(ii) The free energy is equivalent to the partition function, in the sense that in the free energy all information of the system is contained.", "(Prosaically plug $\\exp (-\\beta F)$ in place of $Z$ , and you're done.)", "Unnecessary approximation of the free energy throws away some information.", "In field theory, the counterparts of free energies are generating functionals of connected diagrams." ], [ "Phase transition at $|m| = m_\\mathrm {sp}$", "Next we have a close look at the phase transition at $\\Delta m:= m_\\mathrm {sp}- |m| = 0.$ It is an application and not necessary to understand non-convexity, so uninterested readers can skip this section.", "Because we are using the extensive variable $M$ (in fact $m$ , a normalized one), the transition looks continuous.", "More precisely, it is clearly first-order when approached from the outside of the RSEF, as $|m| \\searrow m_\\mathrm {sp},$ by not showing any sign of transition beforehand, and the transition happens suddenly at $\\Delta m= 0$ .", "After the system passes the transition point, it can go to the metastable branch, too.", "On the other hand, inside the RSEF there is the relation $C^+\\propto \\Delta m^{1/2}$ which looks like scaling laws, just like continuous transitions.", "This is partly expected, because phase coexistence terminates at the point $\\Delta m= 0,$ or put differently cannot extend over that point.", "So there must be a singularity, and by definition, $C^+= 0$ at $\\Delta m= 0.$ There cannot be latent heat and metastability after passing the transition point...", "In reality this transition is not continuous and accompanies discontinuity.", "First let us write: $C(m) = \\mathcal {C}_\\text{out}(m) \\quad \\text{for} \\quad m > m_\\mathrm {sp}.$ Then $\\mathcal {C}_\\text{out}$ is regular at $m = m_\\mathrm {sp}$ : $\\mathcal {C}_\\text{out}(m) := C(m_\\mathrm {sp}) + L^2(c_2 \\Delta m^2 + c_3 \\Delta m^3 + ...),$ where $c_2 = (\\partial m/\\partial h)^{-1}|_{h=0}$ , and $c_2, c_3 = \\mathrm {O}(1).$ Notice that $\\mathcal {C}_\\text{out}$ is defined inside of RSEF too, so compare it with $\\mathcal {C}_\\text{in}$ .", "Then $aLT \\Delta m^{1/2} > c_2 L^2 \\Delta m^2$ , i.e.", "$\\mathcal {C}_\\text{in}> \\mathcal {C}_\\text{out}$ for small enough $\\Delta m>0.$ Therefore the true transition happens at $|\\Delta m| = \\Delta m_\\mathrm {t},$ slightly inside the RSEF, where $\\Delta m_\\mathrm {t}= aTc^{-1}_2 L^{-2/3},$ or at $|M|= M_\\mathrm {t}$ where $M_\\mathrm {sp}- M_\\mathrm {t} \\propto L^{4/3}$ .", "Although $\\Delta m_\\mathrm {t}\\rightarrow 0$ in thermodynamic limit, this shift is observable since $M_\\mathrm {sp}- M_\\mathrm {t} \\rightarrow \\infty .$ So at the transition point the big one droplet appears/disappears all of a sudden, and this discontinuity is inevitable even if the extensive variable $M$ is utilized.", "Latent heat must be there, too.", "In order to avoid discontinuity completely, $E$ or $S$ , the entropy, has to be used instead of $T$ .", "Also notice that $C^+\\propto \\Delta m^{1/2}$ represents a (small) metastable branch for $0 < \\Delta m< \\Delta m_\\mathrm {t}.$ The above shift of the transition point was first proved rigorously by Biskup:EPL:2002,Biskup:CMP:2003.", "They also showed it by a physical argument which is applicable to general vapor-liquid like transitions, which shares much with our argument.", "Although they treat it more deeply, ours can be applicable to more general systems, and in fact we will do so in the next section.", "Similar shift is observed for other systems, for example in a Monte Carlo study of Potts model.", "[20] The shift is an important factor for simulations because it is a finite size effect.", "Another pedagogical remark: If $\\mathcal {C}_\\text{out}$ is regular, why does not it extend into the RSEF, like analytic continuation?", "It's because $\\mathcal {C}_\\text{in}< \\mathcal {C}_\\text{out}$ in the RSEF.Equation () can be rewritten as $-\\beta C(m) = \\log \\sum _\\nu \\exp (-\\beta C_\\nu (m))$ , where $C_\\nu (m) := E_\\nu (m) - TS_\\nu (m)$ , $E_\\nu (m)$ is the energy of the $\\nu $ -th level with $m$ , and $\\exp (S_\\nu (m))$ is the number of states in the $\\nu $ -th level with $m.$ Because $C = \\mathrm {O}(V),$ only the vicinity of the $C_\\nu $ 's minimum can contribute in thermodynamic limit.", "More note on analyticity: In general free energy is a patchwork of regular functions.", "For example, consider the system with a sole ground state at $E = 0$ , and $\\exp (aN)$ -fold excited states at $E = \\epsilon N$ .", "Then $Z = 1 + \\exp ((a-\\epsilon \\beta ) N).$ This $Z$ is analytic for finite $N$ , but in “thermodynamic limit” $N \\rightarrow \\infty $ , $F = {\\left\\lbrace \\begin{array}{ll}F_{-} := 0 & \\text{for} \\quad aT < \\epsilon ,\\\\F_{+} := (\\epsilon - aT)N & \\text{for} \\quad aT > \\epsilon .\\end{array}\\right.", "}$ Notice both $F_{\\pm }$ are analytic for all $T$ , but the smaller one is “chosen”.", "They get cut and sewn together at $aT = \\epsilon $ .", "If a singularity of $F$ has nothing to do with that of $F_{\\pm }$ , then it is the first-order transition point.", "Or $\\mathcal {C}_\\text{in}$ emerges and masks $\\mathcal {C}_\\text{out}$ , so to say.", "Still, $\\mathcal {C}_\\text{out}$ naturally extends into the RSEF, and describes the metastable branch—it is a realm of non-equilibrium theory and not part of the Boltzmann's statistical mechanics.", "We omit this subject." ], [ "Non-convexity and spontaneous symmetry breaking; general lattice systems", "We consider spontaneous symmetry breaking of general lattice theories.", "Although there may be multiple fields and conjugate extensive variables, we suppress labels to distinguish them, and continue to use the same symbols with the former ones; meaning must be clear.", "Let the dimensionality be $d$ , $V=L^d$ , and the symmetry is spontaneously broken at $h=0$ .", "When the symmetry is continuous, domain walls cannot form, and so the RSEF is not a region of phase coexistence.", "The precise magnitude of non-convexity is as follows: Theorem (Magnitude of non-convexity) Suppose there is spontaneous symmetry breaking, being $|m| = m_\\mathrm {sp}$ under zero external field.", "For $|m| < m_\\mathrm {sp}$ , (i) $C(m) - C(m_\\mathrm {sp}) \\stackrel{L\\rightarrow \\infty }{\\rightarrow }+ \\infty $ , because it is necessary for spontaneous symmetry breaking to happen.", "(ii) $C(m) - C(m_\\mathrm {sp}) = \\mathrm {o}(V)$ , because it is equivalent to $\\partial c/\\partial m = h = 0$ .", "The following two points should be common, but other cases are not denied completely: (i) If we write $C = \\mathcal {C}_\\text{out}$ outside of the RSEF, $\\mathcal {C}_\\text{out}$ is regular at $m = m_\\mathrm {sp}$ .", "(ii) Inside the RSEF, $C = \\mathcal {C}_\\text{in}$ and $\\mathcal {C}_\\text{in}$ has a singularity at $m = m_\\mathrm {sp}$ .", "We then study the shift of transition point from $M_\\mathrm {sp}$ , generalizing the previous section.", "Let $\\mathcal {C}_\\text{out}(m) = C(m_\\mathrm {sp}) + V(c_2 \\Delta m^2 + c_3 \\Delta m^3 + ...)$ , and $C(m) = \\mathcal {C}_\\text{out}(m)$ outside the RSEF.", "Let $\\epsilon > 0 $ be such that $0 < C^+/ L^{d-\\epsilon } < \\infty $ inside the RSEF in thermodynamic limit.", "We now assume a scenario where $C^+$ 's dependence on $L^{d - \\epsilon }$ comes exclusively from $M$ 's power inside the RSEF, i.e.", "$C^+\\propto L^{d - \\epsilon } \\propto (M_\\mathrm {sp}- M)^{(d-\\epsilon )/d} = (L^d \\Delta m)^{1-\\epsilon /d}.$ In this case, $\\mathcal {C}_\\text{in}$ is upward convex and $\\mathcal {C}_\\text{in}$ is bigger than $\\mathcal {C}_\\text{out}$ for $1 \\gg |\\Delta m| > 0$ .", "The transition point then lies always inside the RSEF at $|\\Delta m| = \\Delta m_\\mathrm {t}\\propto L^{-d\\epsilon /(d + \\epsilon )} \\stackrel{L\\rightarrow \\infty }{\\rightarrow }0$ or at $|M| = M_t $ where $M_\\mathrm {sp}- M_t \\propto L^{d^2/(d + \\epsilon )} \\stackrel{L\\rightarrow \\infty }{\\rightarrow }\\infty .$ This conclusion is slightly more general than Biskup:EPL:2002, by relating the order of $\\mathcal {C}_\\text{in}$ 's singularity with the shift of the transition point.", "When there is a domain wall, $\\epsilon $ is likely to be $ = 1$ , but can be other values for continuous symmetries." ], [ "General phase equilibria", "We also consider non-lattice systems quickly.", "As we already stated, in general phase equilibria like chemical one, the free energy in extensive variables is not convex by the order of interface.", "The interface free energy has to be positive; otherwise, bubbles would form and phases would mix.", "Pedagogical: When there is no spontaneous symmetry breakdown, there are non-zero external fields, so the free energy per volume under phase coexistence is linear, but not constant." ], [ "Quantum theory of one scalar field", "It should already be clear that in (continuous) quantum field theory under spontaneous broken symmetry, the effective action $\\Gamma [\\langle \\varphi (x) \\rangle ]$ is not convex.", "However conversion from a lattice theory to quantum field theory has some non-trivial points, if straightforward.", "Again, normalization is the point.", "To set up a necessary language, we start this section by studying a theory with one scalar field.", "We first remember how to obtain the $\\varphi ^4$ -theory from the Ising model of $d$ -dimension:This is a standard procedure.", "See for example [][Chap.", "4]Altland-Simons.", "Convert the spin variables $\\lbrace \\sigma _i\\rbrace $ to real-valued variables $\\lbrace \\varphi _i\\rbrace $ by Hubbard-Stratonovich transform and take some more steps to clean up.", "Notice the condition $\\sum \\sigma _i = M$ is mapped to the equation $\\sum \\varphi _i = M,$ since $T \\partial \\log Z / \\partial h_i = \\langle \\sigma _i \\rangle = \\langle \\varphi _i \\rangle $ (with appropriate normalization), where brackets are vacuum expectation value and $h_i$ is the per-site external field.", "Equation (REF ) is an operator identity, because $Z$ is the generating functional.", "Finally Fourier-expand $\\varphi _i$ and retain only low-frequency modes.", "Then the constraint (REF ) becomes $\\tilde{M} := \\int \\mathrm {d}^d x \\, \\varphi $ .", "We have to deal with the obstacle that $m$ was specified in the lattice system.", "This obviously corresponds to non-uniform vacuum.", "We continue to use $L$ to mean the system's linear extent.", "To let the symmetry break spontaneously we have to take $L \\rightarrow \\infty $ .", "Two cases are possible: (i) If you keep the domain wall near the origin, then the wall gets stretched into a straight line, with the boundary conditions say $\\langle \\varphi \\rangle \\rightarrow \\pm \\varphi _\\mathrm {sp}$ as $x^1 \\rightarrow \\pm \\infty ,$ and the original condition of $m$ has vanished.", "Or (ii) if the domain wall is sent infinitely far as $L \\rightarrow \\infty $ , we get the theory $\\langle \\varphi \\rangle = \\pm \\varphi _\\mathrm {sp},$ namely the theory with uniform vacuum, even though we started from the lattice theory inside the RSEF.", "This is not intriguing.", "In both cases, we assumed that we can neglect the constraint about $\\int dx \\varphi (x)$ , which is nonlocal.", "This must be no problem insofar as there are only finite excitations.", "Variant models in which the system edge is there, e.g.", "where $y$ is limited to $\\ge 0,$ are also possible.In this case, renormalization at the system boundary is necessary.", "For reviews on surface/interface renormalization see e.g.", "Diehl:domb-v10,Jasnow:domb-v10 The condition on $M$ in the lattice theory seems to have disappeared in the field theory.", "It is because the purposes of the theories are different—in field theories global quantities like total magnetization are not of much concern, and correlation functions are of prime importance.", "When the non-convexity of the Landau theory of phase transition is concerned for instance, you can focus on $\\Gamma $ with proper normalization.", "The situation is actually the same in lattice theories; $M$ is irrelevant to calculate correlation functions in thermodynamic limit.", "The vacuum with a wall is usually considered to be a topological condition.", "In the present discussion it was naturally introduced as the normalization condition which was originally there in the lattice theory.", "We now explicitly demonstrate the non-convexity.", "We take Minkowski spacetime, but it applies to Euclidean cases too with obvious modification.", "Let the space dimensionality be $d-1$ , $T := \\int dt$ , $V := T \\int \\mathrm {d}^{d-1} x$ , and the Lagrangian density $\\mathcal {L}$ have the potential $U$ : $\\mathcal {L}[J] := \\frac{1}{2} ( \\partial _\\mu \\varphi )^2 - U(\\varphi ) + \\varphi J,$ where $U(\\varphi ) = U(-\\varphi )$ .", "We define $\\Gamma $ as: $\\mathrm {e}^{i \\Gamma [ \\langle \\varphi \\rangle _J ]} &:= \\nonumber \\\\& \\int \\mathcal {D} \\varphi \\exp \\left(i \\int \\mathrm {d}^d\\, x ( \\mathcal {L}[ J ] - \\langle \\varphi (x) \\rangle _J J(x)) \\right),\\\\\\langle \\varphi (x^\\prime ) \\rangle _J & := -i \\frac{\\delta }{\\delta J(x^\\prime )} \\log \\int \\mathcal {D} \\varphi \\exp (i \\int \\mathrm {d}^d\\, x \\mathcal {L}[ J ]),$ so that $- \\Gamma $ takes the minimum for $|\\langle \\varphi (x) \\rangle | = \\varphi _\\mathrm {sp}:= \\langle \\varphi \\rangle _{+ 0}$ .", "$\\Gamma $ is related to the vacuum energy $E_\\mathrm {vac}$ by: $- \\Gamma = TE_\\mathrm {vac}.$ ([24]; for an introduction, see [][Sec 16.3]WeinbergQFT:2) We now calculate the vacuum energy when a wall is present at the classical level.See Bogomolny:1975de.", "For an introduction see also [sec.", "23.1]WeinbergQFT:2.", "Under spontaneous breakdown of spacetime symmetry, counting the number of Goldstone bosons is tricky.", "See PhysRevLett.88.101602.", "Let $U$ take the absolute minimum $U_\\mathrm {min}$ at $\\pm \\varphi _\\mathrm {sp}.$ We impose the boundary condition $\\varphi (x) \\rightarrow \\pm \\varphi _\\mathrm {sp}$ as $x^1 \\rightarrow \\pm \\infty ,$ and uniform in time and other spatial directions.", "(The argument when the boundary condition is spatially uniform and temporarily non-uniform is essentially the same.)", "Then the Hamiltonian $H$ has the minimum value, or the vacuum energy $E_\\mathrm {vac} := L^{d-2} \\int ^{\\varphi _\\mathrm {sp}}_{-\\varphi _\\mathrm {sp}} d\\xi \\sqrt{2U(\\xi )} + L^{d-1} U_\\mathrm {min},$ with the vacuum field configuration which satisfies $\\frac{\\partial }{\\partial x^1} \\varphi = \\sqrt{2U(\\varphi )}.$ The first term of the right hand side of equation (REF ) is the non-convex part.", "Since $U_\\mathrm {min}$ is $\\mathrm {O}(1)$ , notice that $\\Gamma = \\mathrm {O}(V)$ , although $\\Gamma $ itself is dimensionless.", "What is important here is a mere dimensional analysis and quantum correction should not alter the conclusion." ], [ "General field theory; spontaneous breakdown of translational invariance", "As we saw in the previous section, $\\Gamma $ is $\\mathrm {O}(V)$ , but the non-convexity is of order $\\mathrm {o}(V)$ .", "This is necessarily so for any quantum field theory with spontaneous symmetry breakdown, as we saw in the case of lattice theories.", "Now we prove that the translational invariance is spontaneously broken inside the region singular with the external field, namely the region where $\\Gamma $ is non-convex.", "We use the notation of the theory with one scalar field, but it obviously applies to any symmetry.", "If the vacuum configuration $\\langle \\varphi \\rangle $ inside the non-convex region were temporarily and spatially uniform, then $- \\Gamma [\\langle \\varphi \\rangle ] > -\\Gamma [\\varphi _\\mathrm {sp}]$ , by definition of the spontaneously symmetry breakdown.", "But then the left hand side would be bigger than the right hand side by the order $V$ .", "This is a contradiction.", "$\\square $ This can be paraphrased as: By controlling $\\langle \\varphi \\rangle $ , spontaneous symmetry breakdown as $|\\langle \\varphi \\rangle | = \\varphi _\\mathrm {sp}$ is lost, but it has changed its guise as the spontaneous breakdown of the translational invariance.", "We failed to prove this result for statistical mechanics." ], [ "(Seeming) breakdown of linearity in thermodynamic limit", "It is clear (was already from the discussion of lattice systems) that the statement (C), the states in the non-convex region are linear superpositions of the pure-phase vacua, is wrong.", "There is one important corollary: different vacua of spontaneously broken symmetry cannot be superposed.", "In thermodynamic limit, such thing can happen.", "It is because: (i) If it were possible to add the two vacua $|\\pm \\Omega \\rangle ,$ then spontaneous symmetry breaking couldn't happen, by allowing intermediate states.", "(ii) $|\\pm \\Omega \\rangle $ belong to two different theories, described by to two different Lagrangians $\\mathcal {L}[\\pm 0]$ .", "These two explanations would sound somewhat heuristic without our result, but we can now be confident.", "With the last explanation (ii) at hand, a more accurate statement will be: You can not even think of adding $|\\pm \\Omega \\rangle $ , which belong to different theories.", "Of course, in spontaneous symmetry breaking only one vacuum is believed to be chosen.", "Still, this point has not met with a firm, clear consensus to date; [Sec 16.3]WeinbergQFT:2 claims the superposition of different vacua.", "See also Argyres2009.pathIntegral and references therein for the confusion on this point.", "We have finally settled it.", "In fact, denial of superposition of $|\\pm \\Omega \\rangle $ to that effect was already indicated by [][Chap 5]aspects-of-symmetry and by [][Sec 19.1]WeinbergQFT:2 under different contexts, each of which is physically insightful, if not rigorous.", "The former discusses the instability of large systems under small perturbation, and the latter the requirement of cluster decomposition.", "Once it is shown, it may seem a trivial matter, but it will have serious influence over the discussion of quantum measurement.", "There are manifold “interpretations”, but usually linearity is considered to be never broken.", "That's why some people proposed theories which can be categorized as “dynamical reduction models”—those who work in this direction believe that the wave function indeed collapses, and because the Schrödinger equation is strictly linear, they modify it to induce collapses.", "Neither do proponents of decoherence programs think that linearity can be violated.", "For reviews, see RevModPhys.85.471, RevModPhys.76.1267, 1.1356698, Bassi2003257.", "Our result indicates that pure quantum theory intrinsically has a room for non-linearity—to be more accurate, what can be superposed is not self-evident in thermodynamic limit.", "Arguments based on systems with finite degrees of freedom are invalid." ], [ "Definition and result", "As another application of the non-convex free energy of the 2-dimensional Ising model given in equation (REF ), we consider a simple, abstract model which simulates the fluctuation of a rigid domain wall.", "Suppose the “wall” is initially in the flat region (i.e.", "$|M| < M_\\textrm {f}$ ) and then the total magnetization is allowed to variate “quasi-statically”.", "Consider a one dimensional lattice whose points are denoted by $x \\in \\lbrace -M_\\mathrm {f}, -M_\\mathrm {f} + 1, ..., M_\\mathrm {f}\\rbrace , M_\\mathrm {f} \\in \\mathbb {N}$ .", "The wall randomly walks on this lattice.", "Call the regions left and right to the wall as “spin-plus phase” and “-minus” one, respectively.", "At each time step the wall moves to one of the neighbor points with equal probabilities 1/2.", "When the wall reaches an end point $x = \\pm M_\\mathrm {f}$ so getting out of the flat region, then the wall rolls down the slope of the free energy irreversibly, and the system ends up in a uniform state.", "This simplest model is unique in the following point: If the wall is initially at $x$ , or equivalently the plus region shares $v := (x + M_\\mathrm {f}) / 2M_\\mathrm {f}$ part of the system's whole volume $2M_\\mathrm {f}$ , then the probability that the final state is the plus phase is $v$ , which is easy to prove.", "This immediately allows to be made more abstract and more general: Suppose (i) each spin can take $q$ states, (ii) there can be arbitrary number of walls and (iii) when any of two walls meet, or a wall reaches the system edge, they are annihilated.", "The boundary can be periodic, but it does not matter.", "Again, the probability that $i$ -th state domain is the only survivor is proportional to its volume.", "To prove it, notice that the state can be expressed as a point on the lattice on a surface of $q$ -simplex, where $x_i \\in \\lbrace 0, 1, ..., N\\rbrace $ and $ \\sum x_i = N.$ The state hops to one adjacent point with equal probabilities.", "Once one coordinate $x_i$ gets 0, it will be opted out, and the system will be in $(q-1)$ -simplex, and so on.", "The above models satisfy two important conditions of quantum measurement, the irreversible collapse of the state and the Born rule, and we imagined that (i) the time development is deterministic, (ii) randomness is brought about by noise, and (iii) large degrees of freedom (thermodynamic limit) is necessary." ], [ "Discussion", "Probably a pure mathematical model which obeys the Born rule and is equivalent to ours has already been reported; they are the simplest models of random walk, after all.", "(Possibly underneath any theory which obeys the Born rule our model might be hidden.)", "But by the present models we would like to encourage to take the Copenhagen interpretation more seriously: In finite systems, time evolution is unitary, but in thermodynamic limit, it is not always so.", "Admittedly we cannot hint at anything on the model's relation to quantum mechanics." ], [ "Conclusion", "We studied lattice statistical mechanics and quantum field theory under spontaneously broken symmetry, and found that the free energy and the effective action are not convex.", "The order of non-convexity is found to be strictly smaller than the system's volume, but it has to diverge in thermodynamic limit.", "For lattice systems, the first-order transition at $|m| = m_\\mathrm {sp}$ was studied; there is a slight shift of the transition point, and we showed it in a way which may be applicable when the symmetry is continuous.", "For quantum field theory, we proved that the translational invariance breaks in the non-convex region.", "We proved that different vacua of spontaneously broken symmetry can not be added, which has been ambiguous.", "We also modeled the phase coexistence of the Ising model, and obtained a simple model which obeys the Born rule." ] ]
1606.05069
[ [ "Searching the Gamma-ray Sky for Counterparts to Gravitational Wave\n Sources: Fermi GBM and LAT Observations of LVT151012 and GW151226" ], [ "Abstract We present the Fermi Gamma-ray Burst Monitor (GBM) and Large Area Telescope (LAT) observations of the LIGO binary black hole merger event GW151226 and candi- date LVT151012.", "No candidate electromagnetic counterparts were detected by either the GBM or LAT.", "We present a detailed analysis of the GBM and LAT data over a range of timescales from seconds to years, using automated pipelines and new techniques for char- acterizing the upper limits across a large area of the sky.", "Due to the partial GBM and LAT coverage of the large LIGO localization regions at the trigger times for both events, dif- ferences in source distances and masses, as well as the uncertain degree to which emission from these sources could be beamed, these non-detections cannot be used to constrain the variety of theoretical models recently applied to explain the candidate GBM counterpart to GW150914." ], [ "Introduction", "The era of multi-messenger astronomy has fully begun with the regular detections of gravitational waves (GWs) from merging compact objects by the Laser Interferometer Gravitational-wave Observatory (LIGO; [4]), and large multi-wavelength campaigns to pursue electromagnetic (EM) counterparts [3].", "As demonstrated with GW150914, Fermi's GBM and LAT are uniquely capable of providing all-sky observations from hard X-ray to high-energy $\\gamma $ -rays in normal survey operations, including covering the entire localization probability maps of LIGO events [22], [13], [3] within hours of their detections (see also [55]).", "In addition to GW150914 [4], [2], two other candidate compact object merger events were reported by LIGO during the O1 observing run from 2015 September 12 to 2016 January 12.", "GW151226 and the sub-threshold LIGO-Virgo Trigger LVT151012 (if the latter is from a real astrophysical event) are associated with the mergers of two compact objects, likely both stellar-mass black holes (BHs) [1].", "Prior to the watershed discovery of GWs from the binary black hole (BBH) merger GW150914, and the candidate $\\sim 1$ s long $\\gamma $ -ray counterpart GW150914-GBM that was seen 0.4 s later [22], there was little theoretical expectation for EM counterparts to BBH mergers.", "The weak $\\gamma $ -ray signal observed by the GBM is temporally and spatially coincident with the GW trigger, and appears similar to a low-fluence short Gamma-ray Burst (sGRB).", "Note that the candidate GBM counterpart was not detected by the INTEGRAL SPI-ACS (Anti-Coincidence Shield; [54]), and there is debate regarding the nature of the GBM signal [31].", "Since the potential discovery was announced, innovative ideas have emerged to explain an observational signature that possibly resembles a weak sGRB from a BBH (e.g., [37], [28], [33], and [51]); see also [40] for significant constraints on such models.", "Binary Neutron Star (BNS) or Neutron Star - Black Hole (NS-BH) mergers are the most likely progenitors of sGRBs ([25], [47], [36], and [46]), and therefore they are the most similar object class for comparison to Fermi observations of BBH mergers.", "Approximately 68% (for GBM) and 47% (for LAT) of the LVT151012 LIGO localization probability and 83% (for GBM) and 32% (for LAT) of the GW151226 LIGO localization probability were within the Fermi GBM and LAT fields of view (FoV) at the trigger times, respectively.", "The GBM and LAT completed their first post-trigger coverage of the entire localization probability map for LVT151012 within 8 minutes (for GBM) and 113 minutes (for LAT), and for GW151226 within 34 minutes (for GBM) and 140 minutes (for LAT).", "No credible counterpart candidates were detected by either the GBM or the LAT at the trigger times of both events or on the timescales of minutes, hours, days, and months afterwards.", "These non-detections do not constrain models proposed for the candidate GBM counterpart to GW150914, owing to the partial GBM and LAT coverage of the LIGO localization region at the time of trigger for both events, differences in the source distances and system masses, as well the uncertain degree to which emission from these sources could be beamed.", "Therefore, these GBM and LAT non-detections do not provide strong evidence whether $\\gamma $ -ray emission is associated with BBH mergers.", "A statistically-significant sample of BBH mergers, which will be collected over the coming years by the advanced network of GW observatories (including LIGO and Virgo) and wide-field $\\gamma $ -ray instruments, will be required to understand the nature of candidate EM counterparts to BBH merger events, such as GW150914-GBM.", "A summary of the pertinent information regarding the LIGO sources is provided in Section REF , and the custom data analysis and results of specialized searches for $\\gamma $ -ray counterparts are discussed in Section REF (GBM) and Section REF (LAT).", "In Section , we discuss the implications of these non-detections on counterpart searches in general and specifically for GW150914-GBM, placing our GW counterpart limits in the context of sGRB properties.", "We further comment on the relevance of these observations to the recent theoretical developments regarding how a $\\gamma $ -ray counterpart might be produced by a BBH merger.", "Finally, we conclude in Section ." ], [ "Observations and Data Analysis", "This section describes several standard and new extensive searches of the GBM and LAT data within the LIGO localization contours of LVT151012 and GW151226 using a variety of techniques and timescales.", "The timescales referred to throughout this section are summarized in Table REF .", "There were no credible counterpart candidates detected in any of these searches.", "Table: Timescales over which the GBM and LAT data were studied with the various analyses of LVT151012 and GW151226, discussed in Sections & , all referenced to the LIGO trigger times (t LVT t_{\\rm LVT} or t GW t_{\\rm GW}).", "* ^{*}Note that for T adaptive T_{\\rm adaptive} we report the minimum and maximum possible duration.", "# ^{\\#}Note that T EOT T_{\\rm EOT} straddles t LVT t_{\\rm LVT} and t GW t_{\\rm GW} as evenly as possible given limitations of when this analysis was performed relative to the triggers." ], [ "LIGO", "LVT151012 was detected at both the LIGO Hanford and Livingston facilities using the offline data analysis pipelines gstlal [42] and pycbc [58], designed to detect compact binary coalescence (CBC) events, with the candidate source being detected at 09:54:43.4 UTC on 2015 October 12 (hereafter $t_{\\rm LVT}$ ), with $\\sim 2 \\sigma $ significance.", "The LIGO GW analysis of LVT151012 yields a relatively high false alarm rate (FAR) of 1 per 2.3 years, BH masses of $23^{+18}_{-6}$ and $13^{+4}_{-5}$ M$_{\\odot }$ , and a distance of $1100\\pm 500$ Mpc [2].", "GW151226 was detected at both the LIGO Hanford and Livingston facilities, using the gstlal CBC real-time pipeline, at 03:38:53.6 UTC on 2015 December 26 (hereafter $t_{\\rm GW}$ ).", "The GW analysis provides a FAR of less than 1 per 1000 years, and parameter estimation provides BH masses of $14.2^{+8.3}_{-3.7}$ and $7.5\\pm 2.3$ M$_{\\odot }$ , and a distance of $440^{+180}_{-190}$ Mpc [1].", "The LIGO Scientific Collaboration and the Virgo Collaboration reported the discovery and results from Bayesian parameter estimation analyses of LVT151012 and GW151226 under the assumption that the signals arise from a CBC using the latest offline calibration of the GW strain data ([2], [1]).", "The most accurate localization maps for these events (LALInference, [60]) are based on Bayesian Markov-Chain Monte Carlo and nested sampling to forward model the full GW signal including spin precession and regression of systematic calibration errors.", "The analysis of the Fermi observations requires only the trigger times and localization maps as inputs, which were provided via the Gamma-ray Coordinates Network (GCN; [39], [38]) to groups with a memorandum of understanding with LIGO.", "The LIGO localization maps for LVT151012 and GW151226 are shown in Figure REF with the regions occulted by the Earth for Fermi at the times of the GW triggers, indicating the portions of the sky and LIGO localization probability regions visible to both the GBM and LAT.", "All of the GBM and LAT upper limit measurements are calculated for the LIGO localization regions containing 90% of the probability.", "The following sections provide further details on the GBM and LAT observations and analyses.", "Figure: LIGO localization probability maps for LVT151012 (top; ) and GW151226 (bottom; ) indicating the portions of the sky occulted by the Earth for Fermi at the time of the LIGO trigger (blue shaded region).", "The GBM observes the entire unocculted sky.", "The pink shaded region indicates the portions of the sky within the LAT FoV at the GW trigger times." ], [ "GBM", "The GBM is composed of 12 Sodium Iodide (NaI) detectors and two Bismuth Germanate (BGO) detectors [41], with the NaIs providing sensitivity between 8 keV and 1 MeV (NaIs), and the BGOs extending the energy range to 40 MeV.", "The detectors are spaced around the Fermi spacecraft, oriented at different angles to provide approximately uniform sky coverage, resulting in an instantaneous FoV of $\\sim $ 70% of the sky, with the remainder blocked by the Earth.", "The GBM operates continuously except during passages through the South Atlantic Anomaly (SAA), reducing the time-averaged sky exposure to $\\sim $ 60%.", "The data types relevant to the analyses in this paper are CTIME, which is binned at 0.256 s intervals into 8 energy channels, and continuous time-tagged event (CTTE) which is unbinned in time and in 128 energy channels.", "For more detailed explanations of the GBM instrumentation, data types, the triggering algorithms, the sub-threshold searches, and the persistent source searches see [22] and the respective papers for each technique ([41], [17], [61]; Fermi-GBM Collaboration 2016, in preparation).", "The GBM triggers on board in response to impulsive events when the count rates recorded in two or more NaI detectors significantly exceed the background count rate on at least one timescale (from 16 ms to 4.096 s) in at least one of three energy ranges above 50 keV (50–300 keV, $>$ 100 keV, $>$ 300 keV).", "The GBM also triggers on softer events (25–50 keV) on shorter timescales (from 16 to 128 ms).", "Since November 2009 the GBM also triggers on significant increases above the background count rate in the BGOs.", "As described in [22], two new GBM ground pipelines are designed to maximize the chances of detecting counterparts to GW events while carefully accounting for fluctuations common in a background-dominated measurement.", "The GBM offline blind-search pipelinehttp://gammaray.nsstc.nasa.gov/gbm/science/sgrb_search.html (Fermi-GBM Collaboration 2016, in preparation) is sensitive to impulsive transients too weak to trigger on board.", "The pipeline searches CTTE data over 0.1–2.8 s timescales and in four energy bands spanning $\\sim $ 30–1000 keV, approximately doubling the sensitivity of the GBM to sGRBs.", "The GBM seeded-search pipeline [17] uses the GW trigger time and (optionally) the sky location to inform a maximum-likelihood search for modeled burst signals in the GBM data (assuming one of three template source spectra).", "Using an existing catalog of the GBM all-sky instrumental response models [21], the search procedure is to calculate expected source counts for each detector, and compare this predicted signal to any observed excess detector counts over background.", "An overlapping set of short foreground intervals between 0.256 and 8 s long is tested for the contributions from a modeled burst, covering a total search interval of $\\pm $ 30s ($T_{\\rm seeded}$ ) about the GW trigger time.", "The seeded-search pipeline combines NaI and BGO data to provide a sensitive search for short-duration transients.", "This search will be expanded in the future to use the significance of a sub-threshold signal in either the GBM or GWs to strengthen the detection of a signal in the other, provided the false positive rate of the joint search is characterized and the detection levels in both instruments are selected accordingly.", "The ability to validate sub-threshold candidates effectively boosts the LIGO/Virgo horizon by 15-20% and thus the search volume by 50-75% [34], [17].", "In the absence of a detected counterpart signal, we have developed a new technique for setting limits on the strength of impulsive $\\gamma $ -ray emission.", "The LIGO probability map is divided into regions best observed by the same NaI detector.", "A 3$\\sigma $ upper limit on the count rate is defined as three times the standard deviation around a background fit that excludes $\\pm $ 30 s from the GW trigger time.", "This can be converted to a flux upper limit by taking the counts and folding an assumed model through the response.", "We assume a cutoff power-law fit with $E_{peak}$ = 566 keV and a photon index of 0.42, which are the values at peak density for sGRBs best fit by a cutoff power-law from the GBM spectral cataloghttp://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigbrst.html [32], [29] after accounting for parameter correlation.", "With an assumed distance, these upper limits can be converted to luminosity upper limits.", "In addition to searching for impulsive events, the GBM can act as an all-sky monitor for hard X-ray sources over longer timescales using the Earth Occultation Technique (EOT; [61]).", "The EOT stacks the differences in the background count rates as a source sets or rises behind the Earth, and searches the 12–25, 25–50, 50–100, 100–300, and 300–500 keV energy bands.", "We applied the EOT over timescales of 1 day before and after the GW trigger date, 1 month starting at the GW trigger date, and 1 year centered as closely to the GW trigger as possible (given limitations of data collected at the time that this analysis was performed).", "We also now calculate direction-dependent upper limits for persistent emission owing to the extended LIGO localizations." ], [ "GBM Observations of LVT151012", "The GBM collected data continuously, without passing through the SAA, from 24 minutes prior to 50 minutes after the LIGO detection of LVT151012 ($t_{\\rm LVT}$ ).", "Figure REF shows the LIGO sky map from [39] with the blue shaded region indicating the region of sky occulted by the Earth for Fermi at the time of the GW event.", "The GBM was observing 68.2% of the LIGO localization probability at $t_{\\rm LVT}$ , with exposure of the rest of the localization region over the next 8 minutes.", "The only GBM on-board trigger within 12 hours of LVT151012 was misclassified as a GRB by the flight software, and was determined to be caused by a high local particle flux due to an exit from the SAA.", "The offline blind-search pipeline found no credible candidates within 2 days of the LIGO trigger.", "There were also no candidates found by lowering the threshold in a 10-minute time window around $t_{\\rm LVT}$ .", "The seeded-search pipeline was run on the $T_{\\rm seeded}$ interval of $-30<t_{\\rm LVT}<+30$ s, searching for a potential counterpart with duration between 0.256 s and 8 s. The interval was selected a priori roughly guided by the assumption that if GRBs are related to compact binary mergers then the impulsive $\\gamma $ -ray emission should be close in time to the GWs, with a wide enough search window to catch possible precursor emission [57] and possibly unexpected time offsets from $t_{\\rm LVT}$ .", "A light curve showing the summed count rate (ignoring the lowest and highest energy standard CTIME channels) is shown in Figure REF .", "Figure: There is no evidence that the GBM detected any significant emission during LVT151012, demonstrated by the summed count rate light curve over all GBM detectors (NaI from ∼\\sim 10–1000 keV, BGO from 0.4–40 MeV) during the T seeded T_{\\rm seeded} interval: -30<t LVT <+30-30<t_{\\rm LVT}<+30 s. The blue curve shows CTTE data rebinned into 1.024 s bins, the green curve is standard CTIME data with 0.256 s bins, and the red is a sum of non-parametric fits of the background of each detector and CTIME energy channel.", "There are no statistically significant fluctuations within this interval.We find no evidence for the counterpart reported by [15] in their search of the GBM data around LVT151012.", "Our search method combines signals in the 14 GBM detectors in a way that tests for the likelihood of a source from any sky position.", "This is done by weighting both the contribution from each detector and the contribution of each energy channel according to their expected relative contributions for a source at that position.", "By using the detector responses rather than examining just the raw count rates above background, we can find weak sources that are consistent with an astrophysical source while rejecting fluctuations of similar magnitude in counts space.", "That we do not find the candidate counterpart reported in [15] suggests that either the relative rates among detectors or the distribution of counts in energy for their event are not indicative of a physical source from a single sky position.", "Indeed, [15] state they do not use the response information to weight the relative signals when combining detector information, instead weighting the contributions of each detector and energy channel according to signal-to-noise ratio above the background count rates, without consideration as to whether the weighted spectrum is physical or the detector weights are consistent with an arrival direction from a single position.", "Sub-threshold events in background-limited detectors are weak and each detector energy channel is subject to fluctuations.", "The robustness of our technique relies on the combination of 14 individual measurements in a coherent way that uses knowledge of detector responses and typical source energy spectra.", "Given the lack of any significant impulsive $\\gamma $ -ray emission above the background, we set upper limits on the impulsive emission (Figure REF ).", "Using the EOT, we also searched for longer-lasting emission: 1 day before and 1 day after $t_{\\rm LVT}$ , a month starting at $t_{\\rm LVT}$ (2015 October 12 to 2015 November 11), and a year centered around $t_{\\rm LVT}$ (2015 April 12 to 2016 April 12).", "No new sources were detected on any of the searched timescales and energy bands.", "Figure: The area within the LVT151012 LIGO localization contour is shaded to indicate the GBM 10–1000 keV flux upper limits during during the T seeded T_{\\rm seeded} interval: -30<t LVT <+30-30<t_{\\rm LVT}<+30 s. The purple shaded region indicates where the sky was occulted by the Earth for Fermi.", "The Galactic plane is the grey curve, and the Sun is indicated by the yellow disk." ], [ "GBM Observations of GW151226", "The GBM collected data, without passing through the SAA, continuously from nearly 30 minutes before to almost 10 hours after GW151226 ($t_{\\rm GW}$ ).", "Figure REF shows the LIGO sky map from [1], and the regions of the sky accessible to the GBM and LAT at the time of detection of the GW event.", "The GBM observed 83.4% of the LIGO localization probability during the GW emission of GW151226, with exposure of the rest of the localization region over the next 34 minutes.", "There were no GBM on-board triggers within twelve hours of GW151226, and no candidate counterparts found using the blind-search pipeline within 5 days of $t_{\\rm GW}$ .", "There were also no candidates found by lowering the threshold in a 10 minute window around $t_{\\rm GW}$ .", "The seeded-search pipeline also found no credible candidates in the $\\pm $ 30 s $T_{\\rm seeded}$ interval.", "The most significant fluctuation identified has a FAR value of $2.2\\times 10^{-3}$ and occurred 2.0 s before GW151226.", "The post-trials False Alarm Probability (FAP) is 20%; this event is insignificant.", "A summed count rate light curve (ignoring the lowest and highest energy standard CTIME channels) is shown in Figure REF .", "Figure: There is no evidence that the GBM detected any significant emission during GW151226, demonstrated by the summed count rate light curve over all GBM detectors (NaI from ∼\\sim 10–1000 keV, BGO from 0.4–40 MeV) during the T seeded T_{\\rm seeded} interval: -30<t GW <+30-30<t_{\\rm GW}<+30 s. The blue curve shows CTTE data rebinned into 1.024 s bins, the green curve is standard CTIME data with 0.256 s bins, and the red curve is a sum of a non-parametric fit of the background of each detector and CTIME energy channel.", "There are no statistically significant fluctuations within this interval.We use the same method to calculate the upper limits as for LVT151012.", "The resulting upper limit map is shown in Figure REF .", "Using the EOT, we also searched for longer-lasting emission: on timescales of 1 day before and 1 day after $t_{\\rm GW}$ , 1 month starting at $t_{\\rm GW}$ (2015 December 26 to 2016 January 25), and 1 year around $t_{\\rm GW}$ (2015 April 28 to 2016 April 28 - shifted to start at $t_{\\rm GW}$ -242 days and end at $t_{\\rm GW}$ +124 days - given the data available at the time of this analysis).", "No new sources were detected on any of the searched timescales and energy bands.", "Figure: The area within the GW151226 LIGO localization contour is shaded to indicate the GBM 10–1000 keV flux upper limits during the the T seeded T_{\\rm seeded} interval: -30<t GW <+30-30<t_{\\rm GW}<+30 s. The purple shaded region indicates where the sky was occulted by the Earth for Fermi.", "The Galactic plane is the grey curve, and the Sun is indicated by the yellow disk." ], [ "LAT", "The LAT is a pair conversion telescope comprising a $4\\times 4$ array of silicon strip trackers and cesium iodide (CsI) calorimeters covered by a segmented anti-coincidence detector to reject charged-particle background events.", "The LAT covers the energy range from 20 MeV to more than 300 GeV with a FoV of $\\sim 2.4$ sr, observing the entire sky every two orbits ($\\sim $ 3 hours) by rocking north and south about the orbital plane on alternate orbits [14].", "sGRBs at LAT energies are often slightly delayed in their onset, have substantially longer durations and appear to come from a different emission component with respect to their keV-MeV signals [12], [26].", "The late-time $\\gamma $ -ray emission has been shown to be consistent with originating from the same emission component as broadband (radio to X-ray) afterglows [24], [10], [35].", "This warrants the search for a high-energy $\\gamma $ -ray counterpart for GW events on timescales typical of these afterglows (few ks), longer than the prompt emission of an sGRB.", "Thanks to its survey capabilities, the LAT is well suited to look for such signals.", "In addition, given the great uncertainty on the nature of EM signals from BBH mergers, we also search the LAT data over intervals that are much longer than the timescales associated with the afterglow emission of sGRBs, similar to the LAT analysis performed for GW150914 [13].", "We searched the LAT data for evidence of new transient sources.", "Since we did not find any evidence of new sources coincident with the LIGO detections, we set flux upper limits (at 95% c.l.)", "on the $\\gamma $ -ray emission in the energy range 100 MeV – 1 GeV.", "Our analysis is based on the standard unbinned maximum likelihood technique used for LAT data analysishttp://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone.", "We include in our baseline likelihood model all sources (point-like and extended) from the LAT source catalog (3FGL, [7]), as well as the Galactic and isotropic diffuse templates provided by the Fermi-LAT Collaboration [8].", "We employ a likelihood-ratio test [48] to quantify whether the existence of a new source is statistically warranted.", "In doing so, we form a test statistic (TS) that is two times the logarithm of the ratio of the likelihood evaluated at the best-fit model parameters when including a candidate point source at a given position (alternative hypothesis), to the likelihood evaluated at the best-fit parameters under the baseline model (null hypothesis).", "As is standard for LAT analysis, we choose to reject the null hypothesis when the TS is greater than 25, roughly equivalent to a $5 \\sigma $ rejection criterion for a single search.", "In the following, unless stated otherwise, a point in the sky is considered observable by the LAT if it is within 65$^\\circ $ of the LAT boresight (or z-axis) and has an angle with respect to the local zenith smaller than 100$^\\circ $ .", "The latter requirement is used to exclude contamination from terrestrial $\\gamma $ -rays produced by interactions of cosmic rays with the Earth's atmosphere.", "We now describe briefly the different searches we have performed.", "First our Fixed Time Window search is used to search for a possible counterpart and provide a global upper limit on the average flux for a fixed time window.", "This upper limit is relevant if the only information known about the position of a possible counterpart is the LIGO localization map, which is used as a prior.", "Then, our Adaptive time binning search is used to search for counterparts on different time scales, and to provide a map of upper limits which refer to time windows optimized for each location in the map.", "These limits are useful if, after the publication of this paper, a localization of a potential counterpart more accurate than the LIGO localization map become available.", "We refer the reader to Vianello et al.", "(2016, in preparation) for more details about these analyses.", "The results of these analyses for LVT151012 and GW151226 are presented in the following sub-sections.", "Fixed Time Window Search In this analysis we search for high-energy $\\gamma $ -ray emission on a set of fixed time windows ($T_{\\rm fixed}$ ), starting at or slightly before the time of the LIGO triggers.", "For each time window, we start by selecting all pixels [30] that were observable by the LAT within the 90% containment of the LIGO localization maps, down-scaled to a resolution which matches the LAT point-spread function at 100 MeV ($\\sim $$4^\\circ $ ; nside$= 128$ ).", "We then perform an independent likelihood analysis for each pixel, where we test for the presence of a new source at the center of the pixel.", "For all these likelihood analyses we use the Pass 8 P8_TRANSIENTR010E_V6 event class and the corresponding instrument response functionshttp://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone/Cicerone_LAT_IRFs/IRF_overview.html.", "Since we did not detect any new source above our TS threshold in any of the positions, we proceeded with the computation of upper limits with the technique detailed in Vianello et al.", "(2016, in preparation).", "In short, the LIGO probability map is used as a prior on the position of the EM counterpart, and the posterior probability for its flux $F$ is computed by marginalizing the full posterior with respect to the position and all the other free parameters.", "We then compute the upper limit for a given probability $p$ as the upper bound of the credibility interval for $F$ which starts at $F=0$ [50].", "Adaptive time window search In this analysis we optimize the time window for the analysis for each pixel, defining an “adaptive” interval ($T_{\\rm adaptive}$ ) that starts when the pixel becomes observable by the LAT (its angle from the LAT boresight is $<$ 65$^\\circ $ and has a Zenith angle $<$ 92$^\\circ $ –taking into account the 8$^\\circ $ radius Region of Interest, RoI) and ends when it is no longer observable by the LAT.", "We further down-scale the HEALPix map (nside$= 64$ ), and for each pixel we select only the interval that contains the GW trigger time, or the one immediately after (if the center of the pixel was outside the LAT FoV at the GW trigger time).", "This analysis is therefore optimized for the assumption that the source emitted $\\gamma $ rays at the time of the GW event, and the time window of the analysis is designed to only contain a continuous observation.", "As in the fixed-time window analysis, we perform an independent likelihood analysis for each pixel, where we test for the presence of a new source at the center of the pixel.", "We use the Pass 8 P8_TRANSIENTR010E_V6 event class and the corresponding instrument response functions.", "We found no significant excesses, and therefore compute flux upper limits.", "In addition, similar to our analysis for GW150914 [13], we search for a significant excess using adaptive time windows but on longer timescales, from 10 days before to 10 days after the GW event.", "To limit the number of trials, in this analysis we use a coarser spatial resolution (nside$= 8$ ) that roughly matches the size of each RoI, but we compute the TS map (using the ScienceToolhttp://fermi.gsfc.nasa.gov/ssc/data/analysis/ gttsmap) for each RoI.", "As described in [13], we use gtsrcprob to assign, to each event, the probability that the event belongs to each of the sources in the likelihood model.", "We then compute the number of photons that are associated with the source with a probability $>$ 0.9.", "This is useful for filtering the excesses caused by random spatial coincidence of single high-energy events from persistent sources or background.", "Other Standard LAT Searches The Fermi Automatic Science Processing (ASP) pipeline, which is used to search for transient sources (e.g., blazar flares) as regularly reported in Astronomer's Telegrams, was also employed around the GW trigger times.", "The ASP pipeline performs a blind search for sources on all-sky count maps constructed from the event data acquired on 6 hr and 24 hr timescales ($T_{\\rm ASP}$ ).", "Candidate flaring sources are then fit using a standard likelihood analysis modeled along with known sources and the Galactic and isotropic diffuse contributions.", "These candidate sources are then characterized and matched to known sources, allowing for the identification of flaring cataloged sources as well as new unassociated sources.", "The Fermi All-Sky Variability Analysis (FAVA) was also employed to search for excess emission on week-long timescales ($T_{\\rm FAVA}$ ).", "The FAVA weeks are pre-defined: therefore we search the week that includes the corresponding LIGO triggers, and the week afterward.", "FAVA is a blind photometric analysis in which a grid of regions covering the entire sky is searched for deviations from the expected flux based on the observed long-term mission-averaged emission [11].", "This allows the FAVA search to be independent of any model of the $\\gamma $ -ray sky, and therefore complement the standard likelihood analyses." ], [ "LAT Observations of LVT151012", "Fermi was in sky survey mode at the time of the GW signal from LVT151012, $t_{\\rm LVT}$ , rocked 50$^{\\circ }$ South from the orbital plane.", "The LAT was favorably oriented toward LVT151012, covering $\\sim $ 47% of the LIGO localization probability at the time of the trigger.", "Within $\\sim 7$  ks from $t_{\\rm LVT}$ , the LAT had observed $\\sim $ 100% of the LIGO localization probability (Figure REF ).", "The LAT then continued to observe the entire LIGO localization region throughout normal sky-survey operations in the days and months afterward.", "Figure: The LAT observations of LVT151012: cumulative fraction of the LIGO localization probability observed by the LAT as a function of time since t LVT t_{\\rm LVT} (top); Integral of the marginalized posterior for the 0.1–1 GeV energy flux (bottom) during the T fixed 2 T_{\\rm fixed_2} interval.", "The flux at which the blue curve intersects a given probability P(F<x)P(F < x) corresponds to the upper limit at that credibility level.We performed the fixed time window search described in Section REF on two time intervals.", "The first interval $T_{\\rm fixed_1}$ covered from $t_{\\rm LVT} - 10 s$ to $t_{\\rm LVT} + 10$ s, during which the LAT observed $\\sim $ 50% of the LIGO localization map, corresponding to almost the entire Southern region of the LIGO localization contour.", "This search is relevant for finding high-energy emission close in time and of similar duration with respect to the GW signal.", "The second time window $T_{\\rm fixed_2}$ covered from $t_{\\rm LVT}$ to $t_{\\rm LVT} + 8$  ks, which corresponds to the time interval when the LAT fully observed the LIGO localization map (see Figure REF ), plus 1 ks to accrue some exposure of the final regions that became visible to the LAT.", "We found no credible candidate counterparts in $T_{\\rm fixed_1}$ or in $T_{\\rm fixed_2}$ .", "We then performed the upper limit computation described in Section REF for $T_{\\rm fixed_2}$ .", "The integral function of the marginalized posterior for the 0.1–1 GeV energy flux is shown in the bottom panel of Figure REF .", "This function can be used to evaluate the upper limit for different credibility levels.", "In particular, the 95% upper limit is $F_{ul, 95} = 5 \\times 10^{-10}$ erg cm$^{-2}$ s$^{-1}$ .", "Figure: The adaptive time interval analysis for LVT151012 over the first Fermi orbit containing t LVT t_{\\rm LVT}: Flux upper limit map during T adaptive T_{\\rm adaptive} (top), the entry time into the LAT FoV relative to t LVT t_{\\rm LVT} of the RoI for each pixel within the LIGO localization contour (middle), and the upper limit light curves for each RoI (bottom).The horizontal bars in the bottom panel correspond to the values of the LAT upper limits, and their position along the time-axis coincides with the interval of time used in the analysis.The color of each bar indicates the time when the RoI entered the LAT FoV, and matches the color of the pixel in the middle panel.", "The horizontal histogram displays the distribution of upper limits.The adaptive time window ($T_{\\rm adaptive}$ ) analysis did not yield any significant excesses, and no new sources were detected above a likelihood detection threshold of TS=25, neither in the time window containing or just after $t_{\\rm LVT}$ , nor in a scan of 10 days before and 10 days after $t_{\\rm LVT}$ .", "The flux upper limits for the portion of the LIGO localization contour containing 90% of the probability during the adaptive time window are shown in Figure REF .", "The values for the flux upper limits from this analysis range from 2.1 $\\times 10^{-10}$ erg cm$^{-2}$  s$^{-1}$ up to 2.4 $\\times 10^{-8}$ erg cm$^{-2}$  s$^{-1}$ .", "Most of the flux upper limits are below $10^{-9}$ erg cm$^{-2}$  s$^{-1}$ , and the tail extending to higher fluxes is due to a region with poor exposure of the LIGO contour (at RA$\\simeq 30 ^\\circ $ , Dec$\\simeq 0 ^\\circ $ ) entering the LAT FoV at approximately $t_{\\rm LVT}+2$ ks.", "We examined the ASP products during the 6 hour and 24 hour intervals ($T_{\\rm ASP}$ ) containing $t_{\\rm LVT}$ .", "No new unassociated flaring sources were detected within the LIGO localization contour.", "The FAVA search encompassed a pre-defined week before and after $t_{\\rm LVT}$ ($T_{\\rm FAVA}$ ); the weeks from 2015 October 5 to 2015 October 12, and 2015 October 12 to 2015 October 19, respectively.", "The FAVA search over these two periods detected a total of five flaring sources above 5$\\sigma $ within the LIGO localization region.", "A followup likelihood analysis of each seed flare determined their positions to be consistent with known flaring 3FGL sources." ], [ "LAT Observations of GW151226", "Fermi was in sky survey mode at the time of the GW detection of GW151226 (03:38:53.648 UTC on 2015 December 26, $t_{\\rm GW}$ in the following), rocked 50$^{\\circ }$ North from the orbital plane.", "The LAT was favorably oriented toward GW151226, covering $\\sim $ 32% of the LIGO localization probability at the time of the trigger.", "Within $\\sim 1$  ks from $t_{\\rm GW}$ the LAT had observed $\\sim $ 80% of the LIGO localization probability, and $\\sim $ 100% within $\\sim t_{\\rm GW}$ +8.5 ks (Figure REF ).", "The LAT continued to observe the entire LIGO localization region throughout sky-survey operations in the days and months afterwards.", "Figure: The LAT observations of GW151226: cumulative fraction of the LIGO localization probability observed by the LAT as a function of time since t GW t_{\\rm GW} (top); Integral of the marginalized posterior for the 0.1–1 GeV energy flux (bottom) during the T fixed 3 T_{\\rm fixed_3} interval.", "The flux at which the blue curve intersects a given probability P(F<x)P(F < x) corresponds to the upper limit at that credibility level.We performed the fixed time window search described in Section REF on three time intervals.", "The first interval $T_{\\rm fixed_1}$ covered from $t_{\\rm GW} - 10 s$ to $t_{\\rm GW} + 10$ s. During $T_{\\rm fixed_1}$ the LAT observed $\\sim $ 30% of the LIGO localization probability.", "The second interval $T_{\\rm fixed_2}$ covered from $t_{\\rm GW}$ to $t_{\\rm GW} + 1.2$  ks, which corresponds to a shorter time interval when the LAT had an appreciable fractional coverage ($\\sim 80$ %) of the LIGO localization probability (see Figure REF ), with 200 s added to accrue some exposure at the final regions to become visible to LAT.", "The third interval $T_{\\rm fixed_3}$ covered from $t_{\\rm GW}$ to $t_{\\rm GW} + 10$  ks, and corresponds to the time interval during which the LAT had 100% coverage, with $\\sim 1$  ks added to accrue some exposure for the final points to become visible to the LAT.", "We found no candidate counterpart in any of the three time windows.", "We then performed the upper limit computation described in Section REF for $T_{\\rm fixed_3}$ .", "The integral function of the marginalized posterior for the 0.1–1 GeV energy flux is shown in the bottom panel of Figure REF .", "The 95% upper limit is $F_{ul, 95} = 3 \\times 10^{-10}$ erg cm$^{-2}$ s$^{-1}$ .", "Figure: The adaptive time interval analysis for GW151226 over the first Fermi orbit containing t GW t_{\\rm GW}: Flux upper limit map during T adaptive T_{\\rm adaptive} (top), the entry time into the LAT FoV relative to t GW t_{\\rm GW} of the RoI for each pixel within the LIGO localization contour (middle), and the upper limit light curves for each RoI (bottom).The horizontal bars in the bottom panel correspond to the values of the LAT upper limits, and their position along the time-axis coincides with the interval of time used in the analysis.", "The color of each bar indicates the time when the RoI entered the LAT FoV, and matches the color of the pixel in the middle panel.", "The horizontal histogram displays the density of upper limits.The adaptive time window analysis did not lead to the detection of any new $\\gamma $ -ray sources during the first Fermi orbit ($\\sim $ 96 minutes) after $t_{\\rm GW}$ .", "The results of this analysis are shown in Figure REF .", "For this event, the values for the flux upper limits range from 2.6 $\\times 10^{-10}$ erg cm$^{-2}$  s$^{-1}$ up to 7.8 $\\times 10^{-9}$ erg cm$^{-2}$  s$^{-1}$ .", "The tail of upper limits with values $>10^{-9}$ erg cm$^{-2}$  s$^{-1}$ is due to a series of regions entering the FoV at about 3 ks (at R.A.$\\simeq 30^\\circ $ , Dec$\\simeq -15^\\circ $ ) for which the exposure was particularly low.", "As for the previous event, we also searched for excess $\\gamma $ -ray emission on an orbit-by-orbit timescale over $\\pm $ 10 days on either side of $t_{\\rm GW}$ .", "No new sources were detected above a threshold of TS=25.", "The most significant flaring source within the LIGO localization region is the blazar PKS 1424-41, which has flared regularly over the entire Fermi missionhttp://fermi.gsfc.nasa.gov/ssc/data/access/lat/FAVA/SourceReport.php?week=386flare=31$^,$http://fermi.gsfc.nasa.gov/ssc/data/access/lat/msl_lc/source/PKS_1424-41.", "We examined the ASP products during the 6 hour and 24 hour intervals containing $t_{\\rm GW}$ .", "No new unassociated flaring sources were detected within the LIGO localization contour.", "The FAVA search for emission associated with GW151226 encompassed the pre-defined weeks of 2015 December 21 to 2015 December 28 and 2015 December 28 to 2016 January 4, and detected a total of five flaring sources above 5$\\sigma $ within the LIGO localization region.", "Again, a dedicated followup likelihood analysis of each seed flare determined their positions to be consistent with known flaring 3FGL sources, including the highly active blazar PKS 1424-41 (3FGL J1427.9-4206)." ], [ "Implications for Candidate Counterpart GW150914-GBM", "The candidate $\\gamma $ -ray counterpart to GW150914 reported by the GBM [22] that resembles a weak sGRB has surprised the community and also spurred a great deal of theoretical speculations.", "The low significance of the signal, and the lack of corroboration by other experiments has caused the true nature of the GBM signal to remain ambiguous (see also [31], [54]).", "Strong support for the candidate EM counterpart would be achieved if a similar or higher significance counterpart were found associated with other GW BBH merger events.", "The Fermi non-detections of $\\gamma $ -ray counterparts to LVT151012 and GW151226 can neither confirm nor refute the potential association between GW150914 and the GBM candidate counterpart.", "If we assume that all BBH mergers produce sGRB-like signals, the GBM might reasonably not detect them for four reasons: The GBM observed only 68% and 83% of the LIGO localization probability of LVT151012 and GW151226, respectively, at the times of the GW triggers.", "Therefore, there is a significant probability that the LIGO sources could have been simply occulted by the Earth for Fermi at the GW trigger times.", "Without all-sky coverage by the detecting instrument or a set of identical detectors, a non-detection cannot rule out this hypothesis without a sample much larger than the three events from the LIGO O1 observing run.", "The fractional sky coverage alone can account for having a single detection.", "Depending on the source location, orientation, and geomagnetic coordinates of Fermi at the time of the GW trigger, the GBM background rates can vary substantially.", "The background count rates were a few hundred Hz higher (3%) at the time around GW151226 and a few thousand Hz higher (18%) at the time around LVT151012 than around the time of GW150914.", "The reported distance to LVT151012 from GW parameters is a factor of $\\sim 3$ larger than the distance to GW150914.", "If all of these events produced similar $\\gamma $ -ray luminosities, the counterpart to LVT151012 would have been indistinguishable from background.", "If the source producing $\\gamma $ rays in GW150914 is collimated, only a fraction of those objects would be pointed at the Earth.", "This fraction is slightly enhanced by the fact that GW signals from binary mergers, while not truly collimated, have stronger GW emission along the rotation axis of the merger system, which is presumably aligned with the EM jet collimation axis.", "If one assumes that BBH merger counterparts are collimated similarly to sGRBs [27], then only $\\sim $ 15–30% of similar systems would have their $\\gamma $ -ray jets pointed toward Earth.", "The potential detection of a counterpart in one of three objects is entirely consistent with the most conservative assumptions of the degree to which the high-energy emission from such sources is collimated.", "The intrinsic luminosity distribution also limits detectability.", "Even if GW151226 was beamed and on-axis, and the progenitor was not occulted to Fermi, the event still may not be detectable if it was intrinsically dimmer than GW150914-GBM.", "The energy radiated as GWs scales strongly with total progenitor mass.", "If there is also a strong scaling between total progenitor mass and the energy radiated in $\\gamma $ rays, then any $\\gamma $ -ray emission from GW151226 would likely be less luminous than that from GW150914.", "With only three possible GW detections (one with a fairly high FAR), and one candidate counterpart, the statistics are not large, and little can be said of these objects other than that they are broadly consistent.", "As Virgo joins LIGO in upcoming GW observing runs, and they both head toward design sensitivity, the localization regions are anticipated to become smaller and the GW horizon distance increase (with the rate increasing as a cubed factor, [5]).", "Fermi will continue to monitor the sky for potential coincident $\\gamma $ -ray counterparts to all GW source types." ], [ "Comparison to the sGRBs", "Although the potential for EM counterparts to BBH mergers has not been well established in the literature, recent development spurred by the GBM report of a candidate counterpart to GW150914 (e.g., [28], [33], [37]) suggests that mechanisms may exist.", "The connection between BNS (or NS-BH) mergers and sGRBs is much stronger than that of BBH mergers [43], [49], supported by extensive observational evidence (host galaxy observations and offsets, environmental densities inferred from GRB afterglow modeling, observational rates; [56], [27]), and consistency with numerical modeling (jet production, magnetic fields; [53], [52]).", "We do not suggest that GW150914, LVT151012, or GW151226 necessarily produced EM counterparts similar to the population of hundreds of sGRBs observed by BATSE, Konus-Wind, Swift-BAT, GBM and other instruments over the last five decades.", "However, we put our observations (candidate counterpart and upper limits) of these GW detections in the context of the more familiar sGRBs to demonstrate the capability of both the GBM and LAT for these searches in the future.", "In Figure REF , we compare the distribution of sGRB 1-s fluence measurements from the 3rd GBM GRB catalog [16] to our upper limits for LVT151012 and GW151226, as well as the fluence measurement described in [22].", "The fluences from the GBM-detected sGRBs span $2.5\\times 10^{-8}$ to $1.1\\times 10^{-5}$ erg cm$^{-2}$ , with GW150914-GBM around the 40th percentile.", "Compared to sGRBs with known redshifts, GW150914-GBM was unusually close and thus would be very sub-luminous compared to the sGRB population.", "At a more typical sGRB redshift of $z \\sim 0.5$ [23], [27], GW150914-GBM would be undetectable by the GBM.", "The GBM blind-search reveals sGRB candidates that are a factor of two or three weaker than those triggering the GBM on board.", "This opens the possibility of detecting additional fainter sGRBs, and thus testing for the presence of a sub-luminous population that might be associated with BBH mergers (Fermi-GBM Collaboration 2016, in preparation).", "Figure: Integral distribution of GBM fluence of sGRBs from over the duration of the sGRBs, compared to the 1-s fluence measurement for GW150914-GBM, and the upper limits on LVT151012 and GW151226.The LAT has detected far fewer sGRBs than the GBM, only ten to date in the 100 MeV to $>$ 300 GeV band, and only the very luminous GRB 090510 [9] has a measured redshift.", "In [13], we compared the $>$ 100 MeV light curve of GRB 090510 scaled to the distance inferred from the GW measurements for GW150914 to demonstrate the constraining power of the LAT limits.", "We expand that comparison in Figure REF to include additional sGRBs (Fermi-LAT Collaboration 2016, in preparation), and demonstrate that the LAT upper limits from all three GW events are comparable to the measured emission from the LAT-detected sGRBs.", "Therefore, if the GW events had extended high-energy $\\gamma $ -ray emission similar to these sGRBs, it would have been detectable by the LAT within tens-to-hundreds of seconds after the trigger.", "Figure: A comparison between a selection of the longer-lasting LAT-detected sGRBs with the upper limits from the fixed time-intervals for the three GW events.", "The arrows represent the 95% confidence upper limits from the fixed time windows (T 1 T_1 from for GW150914, T 2 T_2 for LVT151012, and T 3 T_3 for GW151226)." ], [ "Theoretical Insights Concerning EM Counterparts for BBH Mergers", "The excitement of the watershed LIGO discovery has precipitated numerous merger models with EM emission components, ranging from sGRBs to optical and radio transients (e.g., [45]) and even luminous neutrino sources (e.g., [33], [44]).", "This discussion is restricted to an incomplete selection of counterpart models, with a view to defining key observational elements that modelers should address in future studies.", "Much of the flurry of very recent activity in GW+EM merger modeling has centered on systems with circumbinary disks or common envelopes that can seed ephemeral accretion onto the resultant BH, perhaps spawning sGRBs.", "The study of [62] explores the evolution of close binaries composed of massive stars, with core collapse in sequence: one companion generates a BH, and the second one facilitates faster precursor inspiral due to the presence of a common envelope.", "After the second BH is formed, the merger takes place amid the ambient shroud that provides fodder for EM emission.", "Such a picture is adopted by [33] as a basis for their neutrino flux predictions.", "A different scenario is that of [37], who discusses a single star progenitor for a BBH merger: the rapid rotation of the massive star yields either a dual helium core or “dumbbell” core configuration that spawns transient BHs that then merge.", "The common envelope again naturally feeds the ergosphere with material for processing into EM form.", "The model of [51] employs an extant BBH system that possesses a residual disk at large radii that is neutral and therefore suppresses the magneto-rotational instability.", "This “fallback” disk remains inert until BBH inspiral revives it through tidal disruption and associated heating.", "The merger then drives belated accretion to generate an sGRB in temporal connection with the GW event.", "Even though the focus in these pictures is on the accretion, there is the suggestion that jet activity will be part of the rapidly evolving system.", "Winds may also be present (e.g., [45]), and the lesser collimation of these can enhance the detectability of energetic EM signals.", "A number of the counterpart models invoke the extraction of energy and angular momentum from the ergospheres of the merging BHs via the Blandford-Znajek mechanism [18], a process that is posited to supply matter and energy to the bases of jets emanating from supermassive BHs.", "Exploring this possibility in detail is beyond the scope of the present suite of incipient models of mergers.", "Yet it should be noted that [40] and [45] indicate that the EM luminosity constraints from such EM induction physics for GW150914 may require TeraGauss magnetic fields, with [40] suggesting that these could be unrealistically large for BH environs.", "The challenge for future theoretical studies of BBH mergers generating EM counterparts is to establish EM templates for observational predictions at a fairly detailed level.", "These must address typical values and ranges for the source luminosity, multi-wavelength spectrum and angular collimation.", "They should also offer clear assessments of the pertinent timescales for the events, including delay relative to the GW event and duration in different wavebands, and also whether or not there is EM precursor activity [20].", "There is also the necessity of establishing a GW merger signal with frequency and frequency derivative character appropriate to the waveforms observed by LIGO and Virgo, i.e.", "matching oscillatory temporal templates calculated assuming a pair of BHs merging in vacuum.", "This array of model discriminants will enable rapid progress should GW+EM mergers become an established astronomical paradigm.", "Depending on the quality of the counterpart data, it may also be possible to constrain elements of fundamental physics.", "Most notable among a plethora of ideas in the literature is using the time separation between the GW and EM signals to limit departures of the light signal speed from $c$ (e.g., [19], [63]).", "This can potentially constrain Lorentz invariance violations that can be attributed to various physics concepts such as quantum gravity.", "Such an enterprise would require significant photon counting statistics and achromatic light curves, as was the case for the GBM+LAT data for the bright burst GRB090510 [6], [59].", "The prospect for probing fundamental physics emphasizes the importance of having $\\gamma $ -ray monitoring capability in place during the era of advanced GW detectors." ], [ "Conclusions", "Fermi GBM and LAT provide the best current wide-field observations of the time-variable $\\gamma $ -ray sky in the keV–GeV band, for comparison to triggers from multi-messenger facilities like LIGO.", "The GBM and LAT observed a substantial fraction of the LIGO localization probabilities at the times of the LIGO triggers for the three potential BBH mergers, and fully observed them within minutes to hours later.", "The GBM candidate counterpart for GW150914 and the non-detections from LVT151012 and GW151226, as well as the LAT non-detections for all three merger candidates, can provide observational constraints for new theoretical models for EM counterparts to BBH mergers.", "Unfortunately, Fermi observations of LVT151012 and GW151226 cannot conclusively resolve the unknown nature of the GBM candidate counterpart to GW150914.", "The partial GBM and LAT coverage of the LIGO localization regions at the time of trigger for both LVT151012 and GW151226 leaves open the possibility that similar EM counterparts occurred outside the GBM and LAT FoVs.", "Ultimately, a statistically large sample of well-observed localization probability maps for BBH mergers will be needed to confidently say whether GW150914-GBM is associated with a BBH merger.", "The era of GW astronomy is an exciting time for facilities like Fermi, that excel at transient source discovery.", "We have developed new pipelines and techniques to search the GBM and LAT data for transient sources, and set constraining upper limits using Fermi data.", "As LIGO and Virgo continue to become more sensitive, and new facilities come online (LIGO India, KAGRA), more BBH mergers will be detected, and BNS mergers (for which expectations for EM counterparts are much more concrete), are also expected to be observed.", "This could finally identify the progenitors of sGRBs.", "The GBM project is supported by NASA.", "Support for the German contribution to GBM was provided by the Bundesministerium für Bildung und Forschung (BMBF) via the Deutsches Zentrum für Luft und Raumfahrt (DLR) under contract number 50 QV 0301.", "AG is funded through the NASA Postdoctoral Fellowship Program.", "The Fermi LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis.", "These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l'Energie Atomique and the Centre National de la Recherche Scientifique / Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden.", "Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d'Études Spatiales in France.", "NC and JB are supported by NSF grant PHY-1505373." ] ]
1606.04901
[ [ "Majorana Neutrino Masses from Neutrinoless Double-Beta Decays and\n Lepton-Number-Violating Meson Decays" ], [ "Abstract The Schechter-Valle theorem states that a positive observation of neutrinoless double-beta ($0\\nu \\beta \\beta$) decays implies a finite Majorana mass term for neutrinos when any unlikely fine-tuning or cancellation is absent.", "In this note, we reexamine the quantitative impact of the Schechter-Valle theorem, and find that current experimental lower limits on the half-lives of $0\\nu \\beta \\beta$-decaying nuclei have placed a restrictive upper bound on the Majorana neutrino mass $|\\delta m^{ee}_\\nu| < 7.43 \\times 10^{-29}~{\\rm eV}$ radiatively generated at the four-loop level.", "Furthermore, we generalize this quantitative analysis of $0\\nu \\beta \\beta$ decays to that of the lepton-number-violating (LNV) meson decays $M^- \\to {M^\\prime}^+ + \\ell^-_\\alpha + \\ell^-_\\beta$ (for $\\alpha$, $\\beta$ = $e$ or $\\mu$).", "Given the present upper limits on these rare LNV decays, we have derived the loop-induced Majorana neutrino masses $|\\delta m^{ee}_\\nu| < 9.7 \\times 10^{-18}~{\\rm eV}$, $|\\delta m^{e\\mu}_\\nu| < 1.6 \\times 10^{-15}~{\\rm eV}$ and $|\\delta m^{\\mu \\mu}_\\nu| < 1.0 \\times 10^{-12}~{\\rm eV}$ from $K^- \\to \\pi^+ + e^- + e^-$, $K^- \\to \\pi^+ + e^- + \\mu^-$ and $K^- \\to \\pi^+ + \\mu^- + \\mu^-$, respectively.", "A partial list of radiative neutrino masses from the LNV decays of $D$, $D_s^{}$ and $B$ mesons is also given." ], [ "Introduction", "It remains an open question whether massive neutrinos are Majorana particles, whose antiparticles are themselves [1].", "The final answer to this fundamental question will tell us whether the lepton number is conserved or not in nature, and help us explore the origin of neutrino masses.", "Currently, the most promising way to answer if massive neutrinos are their own antiparticles is to observe the $0\\nu \\beta \\beta $ decays $N(Z, A) \\rightarrow N(Z+2, A) + e^- + e^-$ , where $Z$ and $A$ stand respectively for the atomic and mass numbers of a nuclear isotope $N(Z, A)$  [2], [3].", "Over the last few decades, a great number of dedicated experiments have been carried out to search for this kind of decays [4], [5].", "So far, we have not observed any positive signals, and a lower bound on the half-life of the implemented nuclear isotope can be drawn from experimental data.", "The GERDA Phase-I experiment [6] has disproved the signals of $0\\nu \\beta \\beta $ decays claimed by the Heidelberg-Moscow experiment [7], and the joint lower bound from all the previous $^{76}{\\rm Ge}$ -based experiments on the half-life turns out to be $T^{1/2}_{0\\nu } > 3.0 \\times 10^{25}~{\\rm yr}$ at the $90\\%$ confidence level [6], [8].", "For $^{136}{\\rm Xe}$ -based experiments, a combined analysis of the EXO-200 [9] and KamLAND-Zen Phase-I data [10] gives rise to a lower bound $T^{1/2}_{0\\nu } > 3.4 \\times 10^{25}~{\\rm yr}$ at the $90\\%$ confidence level.", "More recently, KamLAND-Zen announced their Phase-II result [11], and improved the lower bound to $T^{1/2}_{0\\nu } > 1.07 \\times 10^{26}~{\\rm yr}$ at the $90\\%$ confidence level with both Phase-I and Phase-II data.", "If neutrino mass ordering is inverted (i.e., $m^{}_3 < m^{}_1 < m^{}_2$ ), the next-generation $0\\nu \\beta \\beta $ experiments with a few tons of target mass will be able to discover a remarkable signal in the near future [4].", "The Schechter-Valle theorem [12] states that a clear signal of $0\\nu \\beta \\beta $ decays will unambiguously indicate a finite Majorana mass of neutrinos, if neither a fine-tuning among parameters nor a cancellation among different contributions is assumed.It has been pointed out by Apostolos Pilaftsis that the tree-level parameters can be well chosen to give a vanishing neutrino mass in the type-I seesaw model, while the $0\\nu \\beta \\beta $ decay rate remains nonzero as the nuclear medium effects on quarks may break any intricate cancellation.", "Obviously, this theorem signifies the physical importance of searching for $0\\nu \\beta \\beta $ decays experimentally.", "The quantitative impact of the Schechter-Valle theorem has already been studied by Duerr, Lindner and Merle in Ref.", "[13], where it is found that the Majorana neutrino masses implied by the Schechter-Valle theorem are too small to explain neutrino oscillations.", "Explicitly, assuming one short-range operator to be responsible for $0\\nu \\beta \\beta $ decays, they find that current experimental lower bounds on the half-lives of $0\\nu \\beta \\beta $ -decaying isotopes indicate an upper bound on the Majorana neutrino mass $|\\delta m^{ee}_\\nu | < 5 \\times 10^{-28}~{\\rm eV}$ , where $\\delta m^{\\alpha \\beta }_\\nu $ denotes the effective neutrino mass term associated with $\\overline{\\nu ^{}_{\\alpha {\\rm L}}} \\nu ^{\\rm c}_{\\beta {\\rm L}}$ for $\\alpha , \\beta = e, \\mu , \\tau $ .", "In this paper, we reexamine this problem, and obtain an upper bound $|\\delta m^{ee}_\\nu | < 7.43\\times 10^{-29}~{\\rm eV}$ that agrees with the above result from Ref.", "[13] on the order of magnitude.", "Furthermore, we generalize the analysis of $0\\nu \\beta \\beta $ decays to that of the LNV rare decays of $B$ , $D$ and $K$ mesons.", "For instance, we obtain $|\\delta m^{ee}_\\nu | < 9.7 \\times 10^{-18}~{\\rm eV}$ , $|\\delta m^{e\\mu }_\\nu | < 1.6 \\times 10^{-15}~{\\rm eV}$ and $|\\delta m^{\\mu \\mu }_\\nu | < 1.0 \\times 10^{-12}~{\\rm eV}$ from current upper bounds on the LNV rare decays of $K$ mesons.", "The radiative Majorana neutrino masses related to other LNV decays are also tabulated.", "Therefore, we confirm the conclusion from Ref.", "[13] that although the Schechter-Valle theorem in general implies a tiny Majorana neutrino mass, we have to explore other mechanisms to generate the observed neutrino masses at the sub-eV level.", "The remaining part of this work is organized as follows.", "In Sec.", "2, we recall the calculation of Majorana neutrino masses from the four-loop diagram mediated by the effective operator, which is also responsible for the $0\\nu \\beta \\beta $ decays.", "The generalization to the LNV meson decays is performed in Sec.", "3, where the corresponding Majorana masses are computed.", "Finally, we summarize our main conclusions in Sec.", "4." ], [ "Majorana Masses from $0\\nu \\beta \\beta $ Decays", "In this section, we present a brief review on the calculation of Majorana neutrino masses radiatively generated from the operator that leads to the $0\\nu \\beta \\beta $ decays, following Ref.", "[13] closely.", "Such a calculation can be readily generalized to the case of Majorana neutrino masses induced by the LNV meson decays, as shown in the next section.", "At the elementary-particle level, the $0\\nu \\beta \\beta $ decays can be expressed as $d + d \\rightarrow u + u + e^- + e^-$ , where the up quark $u$ , the down quark $d$ and the electron $e^-_{}$ are all massive fermions.", "If the $0\\nu \\beta \\beta $ decays take place, they can be effectively described by the LNV operator ${\\cal O}^{}_{0\\nu \\beta \\beta } = \\bar{d} \\bar{d} u u e e$ , in which the chiralities of charged fermions have been omitted and will be specified later.", "As already pointed out by Schechter and Valle [12], this operator will unambiguously result in a Majorana neutrino mass term $\\delta m^{ee}_\\nu \\overline{\\nu ^{}_{e\\rm L}} \\nu ^{\\rm c}_{e\\rm L}$ .", "The relevant Feynman diagrams are given in Fig.", "REF .", "It is worthwhile to notice that quark and charged-lepton masses are indispensable for the Schechter-Valle theorem to be valid, as emphasized in Ref. [13].", "In the Standard Model (SM), only left-handed neutrino fields participate in the weak interactions, so the electron masses can be implemented to convert the right-handed electron fields into the left-handed ones, which are then coupled to left-handed neutrino fields via the charged weak gauge boson $W^+$ .", "This does make sense, since the chirality of electrons in the operator ${\\cal O}^{}_{0\\nu \\beta \\beta }$ can in general be either left-handed or right-handed.", "For the same reason, quark masses are also required to realize the hadronic charged-current interactions in the SM.", "In this case, the operator ${\\cal O}^{}_{0\\nu \\beta \\beta }$ in Fig.", "REF (a) can be attached to the left-handed neutrinos through two propagators of $W^+$ , leading to the neutrino self-energy diagram in Fig.", "REF (b).", "Figure: The “ladybird\" diagram (a) for the 0νββ0\\nu \\beta \\beta decays induced by an effective operator 𝒪 0νββ {\\cal O}^{}_{0\\nu \\beta \\beta }, and the “butterfly\" diagram (b) for the corresponding Majorana neutrino mass term δm ν ee ν eL ¯ν eL c \\delta m^{ee}_\\nu \\overline{\\nu ^{}_{e{\\rm L}}} \\nu ^{\\rm c}_{e{\\rm L}} generated at the four-loop level .Assuming that $0\\nu \\beta \\beta $ decays are mediated by short-range interactions, one can write down the most general Lorentz-invariant Lagrangian that contains various point-like operators as follows [14] $ \\mathcal {L}_{0\\nu \\beta \\beta }^{} &=& \\frac{G_{\\rm F}^2}{2 m_p^{}} \\left(\\epsilon _1^{} J J j + \\epsilon _2^{} J^{\\mu \\nu }_{} J_{\\mu \\nu }^{} j + \\epsilon _3^{} J^{\\mu }_{} J_{\\mu }^{} j + \\epsilon _4^{} J^{\\mu }_{} J_{\\mu \\nu }^{} j^{\\nu }_{} + \\epsilon _5^{} J^{\\mu }_{} J j_{\\mu }^{} \\right) \\;,$ where $G_{\\rm F}^{} = 1.166 \\times 10^{-5}~{\\rm GeV}^{-2}$ and $m_p^{} = 938.27~{\\rm MeV}$ denote respectively the Fermi constant and the proton mass, and $\\epsilon ^{}_i$ (for $i = 1, 2, \\cdots , 5$ ) are effective coupling constants.", "In Eq.", "(REF ), the hadronic currents are defined as [14] $J \\equiv \\bar{u} (1 \\pm \\gamma _5^{}) d \\; , \\quad J^{\\mu }_{} \\equiv \\bar{u} \\gamma ^{\\mu }_{} (1 \\pm \\gamma _5^{}) d \\; , \\quad J^{\\mu \\nu }_{} \\equiv \\bar{u} \\frac{{\\rm i}}{2} [\\gamma ^{\\mu }_{}, \\gamma ^{\\nu }_{}] (1 \\pm \\gamma _5^{}) d \\; ,$ while the leptonic currents are given by $j = \\bar{e} (1 \\pm \\gamma _5^{}) e^{\\rm c} \\; , \\quad j^{\\mu }_{} = \\bar{e} \\gamma ^{\\mu }_{} (1 \\pm \\gamma _5^{}) e^{\\rm c} \\; , \\quad j^{\\mu \\nu }_{} = \\bar{e} \\frac{{\\rm i}}{2} [\\gamma ^{\\mu }_{}, \\gamma ^{\\nu }_{}] (1 \\pm \\gamma _5^{}) e^{\\rm c} \\; ,$ where $e^{\\rm c}_{} \\equiv C\\bar{e}^{\\rm T}$ with $C \\equiv i\\gamma ^2 \\gamma ^0$ is the charge-conjugated electron field.", "According to $C \\gamma ^{\\rm T}_\\mu C^{-1} = - \\gamma ^{}_\\mu $ and the fact that fermion fields are Grassmann numbers, one can immediately verify that the tensor leptonic current $j^{}_{\\mu \\nu }$ automatically vanishes.", "Different chiralities of hadronic and leptonic currents in Eqs.", "(REF ) and (REF ) should be distinguished by the left- and right-handed projection operators $P_{{\\rm L},{\\rm R}}^{} = (1 \\mp \\gamma _5^{})/2$ .", "For instance, we have defined $J_{{\\rm L},{\\rm R}}^{} = 2 \\bar{u} P_{{\\rm L}, {\\rm R}}^{} d$ , and similarly for the other types of currents in Eqs.", "(REF ) and (REF ), in which the corresponding subscripts “L\" or “R\" are omitted without causing any confusions.", "In this connection, the effective coupling constants $\\epsilon ^{}_i$ should also be regarded as $\\epsilon ^{\\rm xyz}_i$ (for x, y, z = L, R), which are carrying the superscripts for different chiralities of hadronic and leptonic currents.", "Given one of the five operators in Eq.", "(REF ), one can set an upper limit on their coupling $\\epsilon _i^{}$ by assuming that it is responsible for the $0\\nu \\beta \\beta $ decay and saturates the experimental lower bound on the half-life, as done in Ref. [14].", "Recently, some of those limits have been recalculated in Ref.", "[13], using more recent results for the nuclear matrix elements.", "The effective coupling constants for the operators $J^{}_{\\rm L} J^{}_{\\rm L} j^{}_{\\rm L}$ and $J^\\mu _{\\rm R} J^{}_{\\mu {\\rm R}} j^{}_{\\rm L}$ have been found to be $\\epsilon ^{}_1 < 2\\times 10^{-7}$ and $\\epsilon ^{}_3 < 1.5\\times 10^{-8}$ , respectively.", "Having obtained these couplings, we are then ready to evaluate the induced neutrino mass by inserting the dimension-nine effective operators into the butterfly diagram, as depicted in Fig.", "REF .", "The authors of Ref.", "[13] have demonstrated that the operator $J^{}_{\\rm L} J^{}_{\\rm L} j^{}_{\\rm L}$ leads to a vanishing neutrino mass term via the butterfly diagram, while the other one $J_{\\rm R}^\\mu J_{\\mu {\\rm R}} j_{\\rm L}^{}$ does lead to a tiny Majorana neutrino mass, which will be revisited below.", "Now that the operator $\\epsilon ^{}_3 J_{\\rm R}^\\mu J_{\\mu {\\rm R}} j_{\\rm L}^{}$ is responsible for the $0\\nu \\beta \\beta $ decays, the radiatively induced Majorana mass term for electron neutrinos can be extracted from the self-energy in Fig.", "REF (b) by setting the external momentum to zero as [13], $ \\delta m_{\\nu }^{ee} = \\frac{128 g^4_{} G_{\\rm F}^2 \\epsilon _3^{} m_u^2 m_d^2 m_e^2}{m_p^{}} \\mathcal {I}_{0\\nu \\beta \\beta }^{} \\;,$ where $g$ is the weak gauge coupling, and $m_u^{}, m_d^{}$ and $m_e^{}$ are the up-quark, down-quark and electron masses, respectively.", "In addition, the loop integral is given by $\\mathcal {I}_{0\\nu \\beta \\beta }^{} = \\left[\\mathcal {I}(m_e^2, m_u^2, m_d^2)\\right]^2_{}$ with $\\mathcal {I}(m_e^2, m_u^2, m_d^2) = \\int \\frac{{\\rm d}^4_{} k_1^{}}{(2\\pi )^4} \\frac{{\\rm d}^4_{} q_1^{}}{(2\\pi )^4} \\frac{1}{ (k_1^{2} - m_e^2)[(k_1^{} + q_1^{})^2 - m_u^2] (q_1^2 - m_d^2) (k_1^2-M_{W}^2)} \\; ,$ where $M_W^{} = 80.4~{\\rm GeV}$ is the $W$ -boson mass, $k^{}_1$ and $q^{}_1$ stand for the four-momenta of internal particles running in the loop and can be easily identified via the integrand on the right-hand side of Eq.", "(5) and from Fig.", "REF (b).", "To evaluate this integral, we employ the technique for massive two-loop diagrams in Ref.", "[15] and arrive at $\\mathcal {I}(m_e^2, m_u^2, m_d^2) &=& \\frac{1}{(4\\pi ^2)^{4+\\varepsilon }_{} \\mu ^{2\\varepsilon }_{}} \\int _0^1 {\\rm d}z ~\\mathcal {G} \\left((1-z)M_{W}^{2} + z m_e^2, m_d^2, m_u^2; 0 \\right) \\; ,$ with $\\varepsilon \\equiv 4-n$ as usually introduced in the dimensional regularization and $\\mu $ being the renormalization scale.", "The relevant function reads as [15] $\\mathcal {G} (m^2_i, m^2_j, m^2_k; k^2) &\\equiv & \\int ~ \\frac{{\\rm d}^n p ~ {\\rm d}^n q}{(p^2 + m_i^2)^2 [ (q+k)^2 + m_j^2 ] [ (p+q)^2 + m_k^2 ]} \\\\&=& \\pi ^4 (\\pi m_i^2)^{n-4} \\frac{\\Gamma (2-\\frac{1}{2}n)}{\\Gamma (3-\\frac{1}{2}n)} \\int _0^1 {\\rm d}x \\int _0^1 {\\rm d}y ~ [x (1-x)]^{\\frac{1}{2}n-2} y (1-y)^{2-\\frac{1}{2}n} \\nonumber \\\\& &\\times \\left\\lbrace \\frac{(y^2 \\kappa ^2 + \\eta ^2)\\Gamma (5-n)}{[y(1-y)\\kappa ^2 + y + (1-y)\\eta ^2 ]^{5-n}} + \\frac{n}{2} \\frac{\\Gamma (4-n)}{[y(1-y)\\kappa ^2 + y + (1-y)\\eta ^2 ]^{4-n}} \\right\\rbrace ,$ with $\\eta ^2 = \\frac{ax+b(1-x)}{x(1-x)}\\; , \\quad a = \\frac{m_j^2}{m_i^2} \\; , \\quad b = \\frac{m_k^2}{m_i^2} \\; , \\quad \\kappa ^2 = \\frac{k^2}{m_i^2} \\;.$ As usual, the integral is expanded with respect to $\\varepsilon = 4 - n$ in the limit of $n \\rightarrow 4$ and the ultraviolet divergences can be separated as inverse powers of $\\varepsilon $ .", "Since the loop integral involves the divergent terms proportional to $\\varepsilon ^{-2}_{}$ and $\\varepsilon ^{-1}_{}$ , we need to keep the terms up to $\\varepsilon ^2$ in $\\mathcal {I}(m^2_e, m^2_u, m^2_d)$ , namely, $\\mathcal {I}(m^2_e, m^2_u, m^2_d) \\approx \\frac{8.02\\times 10^{-5}}{\\varepsilon ^{2}} - \\frac{7.96 \\times 10^{-4}}{\\varepsilon } + 0.0041 - 0.0146 \\varepsilon + 0.040 \\varepsilon ^2 + \\mathcal {O}(\\varepsilon ^3) \\; ,$ so as to obtain all the finite parts of ${\\cal I}^{}_{0\\nu \\beta \\beta }$ .", "In our numerical calculations, we have adopted the renormalization scale of $\\mu = 100~{\\rm MeV}$ , which is a characteristic scale of typical energy transfer in nuclear processes.", "The other implemented parameters can be found in Table REF .", "In the scheme of minimal subtraction, we finally get the induced neutrino mass from Eq.", "(REF ) as $|\\delta m_{\\nu }^{ee}| < 7.43 \\times 10^{-29} ~{\\rm eV} \\; ,$ which agrees with the result $|\\delta m^{ee}_\\nu | < 5 \\times 10^{-28}~{\\rm eV}$ from Ref.", "[13] on the order of magnitude.Here we perform a careful treatment on the renormalization scale in the evaluation of the loop integral, and that leads to the above minor discrepancy between two numerical results.", "At this point, we are grateful to Dr. Michael Duerr for kind communications regarding the evaluation of the loop integral.", "Since this mass is extremely small, one has to implement other mechanisms to account for neutrino masses.", "In this sense, the main conclusion in Ref.", "[13] is still valid that the Schechter-Valle theorem is qualitatively correct, but quantitatively irrelevant for the neutrino mass-squared differences required for neutrino oscillation experiments.", "Table: Particle masses, decay constants of mesons, CKM mixing angles and Dirac phase that are used in the evaluation of radiatively-generated neutrino masses ." ], [ "Majorana Masses from LNV Meson Decays", "Then, we consider the LNV meson decays $M_i^- \\rightarrow M_f^+ \\ell _\\alpha ^- \\ell _\\beta ^-$ , where $M_i^-$ and $M_f^+$ are the initial and final charged mesons, while the emitted same-sign charged leptons with flavors $\\alpha $ and $\\beta $ are denoted by $\\ell _{\\alpha }^-$ and $\\ell _{\\beta }^-$ , respectively.", "These processes have been extensively discussed in the presence of heavy Majorana neutrinos or a Higgs triplet [17].", "If the LNV meson decays are observed experimentally, we assume that these processes are caused by some short-range interactions and can be described by a number of Lorentz-invariant operators of dimension-nine.", "However, to carry out an order-of-magnitude estimate of the induced Majorana neutrino masses, one can simply consider just one operator so long as it contributes dominantly.", "The main idea is to generalize the analysis for $0\\nu \\beta \\beta $ decays to the LNV meson decays.", "For instance, we take the operator $\\mathcal {L}_{\\rm MD}^{} = \\varepsilon _{\\alpha \\beta }^{} \\frac{G_{\\rm F}^2}{2 m_{p}^{}} J_{\\rm R}^\\mu J_{\\mu {\\rm R}}^\\prime j_{\\rm L}^{} \\; ,$ where $\\varepsilon _{\\alpha \\beta }^{}$ (for $\\alpha , \\beta $ = $e, \\mu $ ) are real dimensionless couplings, and the hadronic and leptonic currents are defined similarly as before, i.e., $J_{\\mu {\\rm R}}^{(\\prime )} = 2\\overline{U^{(\\prime )}} \\gamma _\\mu ^{} P_{\\rm R}^{} D^{(\\prime )}$ and $j_{\\rm L}^{} = 2\\overline{\\ell _\\alpha ^{}} P_{\\rm L}^{} \\ell _\\beta ^{\\rm c}$ .", "Here $U^{(\\prime )}$ and $D^{(\\prime )}$ are generic up- and down-type quark fields, and we have distinguished two possibly different hadronic currents by a prime symbol.", "For instance, in the decay of $B^-_{} \\rightarrow D^+_{} e^-_{} \\mu ^-_{}$ , the two hadronic currents are $\\bar{c}\\gamma _\\mu ^{}P_{\\rm R}^{} d$ and $\\bar{u}\\gamma _\\mu ^{} P_{\\rm R}^{} b$ , with $b$ being the bottom-quark field, while the leptonic current is $\\overline{e} P_{\\rm L}^{} \\mu ^{\\rm c}_{}$ .", "It should be noticed that the results will also be valid for the CP-conjugated channel $M_i^+ \\rightarrow M_f^- \\ell _\\alpha ^+ \\ell _\\beta ^+$ if the CP violation in these LNV decays is negligible.", "Given the operator in Eq.", "(12), it is straightforward to write down the Feynman amplitude for the LNV meson decay $M_i^-(p^{}_i) \\rightarrow M_f^+(p^{}_f) + \\ell _\\alpha ^-(p^{}_\\alpha ) + \\ell _\\beta ^-(p^{}_\\beta )$ as $ {\\rm i} \\mathcal {M} = {\\rm i} \\varepsilon _{\\alpha \\beta }^{} \\frac{G_{\\rm F}^2}{2 m_{p}^{}}~ 8\\langle M_f^+ (p_f^{}) \\ell _\\alpha ^-(p_\\alpha ^{}) \\ell _\\beta ^- (p_\\beta ^{}) |~ \\overline{U} \\gamma ^\\mu _{} P_{\\rm R}^{} D \\overline{U^\\prime } \\gamma _\\mu ^{} P_{\\rm R}^{} D^\\prime _{} \\overline{\\ell _\\alpha ^{}} P_{\\rm L}^{} \\ell _\\beta ^c ~| M_i^- (p_i^{}) \\rangle \\; ,$ where the four-momenta of initial and final states have been specified explicitly.", "Since the initial and final mesons are bound states, the hadronic processes involving them in general cannot be calculated perturbatively.", "However, in our case, we assume that the hadronic interactions can be factorized out and related to the leptonic meson decay constants, which are defined as follows [16] $\\langle 0 | \\bar{q}\\gamma _\\mu ^{} \\gamma _5^{} q^\\prime | P(p) \\rangle = -ip_\\mu ^{} f_P^{} \\; , \\quad \\langle 0 | \\bar{q} \\gamma _\\mu ^{} q^\\prime |V(p) \\rangle = \\epsilon _\\mu ^{} m_V^{} f_V^{} \\; ,$ where $P$ and $V$ respectively denote pseudoscalar and vector mesons, $\\epsilon _\\mu ^{}$ the polarization vector for $V$ , and $m_V^{}$ the vector-meson mass.", "For the relevant decay constants $f_{P}^{}$ and $f^{}_V$ , we adopt their numerical values from Ref.", "[16] and list them in Table REF .", "For illustration, we first deal with the decay rates of pseudoscalar mesons.", "The results for the vector-meson decays can be similarly obtained, and will be given later in this section.", "With the help of the decay constants, the square of amplitude in Eq.", "(REF ) can be reduced to $|\\mathcal {M}|^2 = \\frac{G_{\\rm F}^4}{4 m_p^4} \\varepsilon _{\\alpha \\beta }^2 f_i^2 f_f^2 (p_i^{} \\cdot p_f^{})^2 (p_\\alpha ^{} \\cdot p_\\beta ^{}) \\; ,$ which will be inserted into the standard formula of the differential rate for three-body decays and lead to $\\frac{d\\Gamma }{ds} = \\frac{1}{2 m_i^{}} \\int \\frac{d^3 p^{}_f}{(2\\pi )^3}\\frac{1}{2 E_f^{}} \\int \\frac{d^3 p_\\alpha ^{}}{(2\\pi )^3} \\frac{1}{2E_\\alpha ^{}} \\int \\frac{d^3 p_{\\beta }^{}}{(2\\pi )^3} \\frac{1}{2E_{\\beta }^{}} |\\mathcal {M}|^2 (2\\pi )^4 \\delta ^4(q-p_\\alpha ^{} - p_\\beta ^{}) \\delta [q^2 - (p_i^{} - p_f^{})^2] \\; ,$ where $E_{f,\\alpha ,\\beta }^{}$ are the energies of final-state particles, and $s \\equiv q^2$ is the invariant momentum square transferred to leptons so that the condition $(m_\\alpha ^{} + m_\\beta ^{})^2 \\le s \\le (m_i^{} - m_f^{})^2$ is satisfied.", "Here $m_{i,f,\\alpha ,\\beta }^{}$ stand for the masses of the initial- and final-state particles.", "After a direct evaluation of the integral, we obtain $\\frac{d\\Gamma }{ds} = \\frac{C_{\\alpha \\beta }^{}}{2}\\frac{\\varepsilon _{\\alpha \\beta }^2 G_{\\rm F}^4 f_i^2 f_f^2}{16 (4\\pi )^3 m_i^3 m_{p}^2} \\lambda ^{1/2}_{} (s, m_\\alpha ^2, m_\\beta ^2) \\lambda ^{1/2}_{} (s, m_i^2, m_f^2) \\frac{(s-m_i^2-m_f^2)^2 (s-m_\\alpha ^2 - m_\\beta ^2)}{s} \\; ,$ where $C_{\\alpha \\beta }^{} = 1 (2)$ for $\\alpha = \\beta $ ($\\alpha \\ne \\beta $ ), and $\\lambda (a,b,c) \\equiv (a-b-c)^2 - 4bc$ is the Källen function.", "Then, the LNV decay rates of vector mesons can be derived in a similar way, and the final results turn out to be $\\frac{d\\Gamma }{ds} = \\frac{C_{\\alpha \\beta }^{}}{2}\\frac{\\varepsilon _{\\alpha \\beta }^2 G_{\\rm F}^4 f_i^2 f_f^2}{16 (4\\pi )^3 m_i^3 m_{p}^2} \\lambda ^{1/2}_{} (s, m_\\alpha ^2, m_\\beta ^2) \\lambda ^{3/2}_{} (s, m_i^2, m_f^2) \\frac{ (s-m_\\alpha ^2 - m_\\beta ^2)}{s} \\; .$ Finally, the partial decay width $\\Gamma $ can be computed by integrating the differential one $d\\Gamma /ds$ over the allowed range of $s$ .", "By comparing between current experimental bounds on the LNV rare decays from Ref.", "[16] and theoretical predictions, one can extract the upper limits on the corresponding coupling constants $\\varepsilon _{\\alpha \\beta }^{}$ .", "In Table REF , we list such upper limits for a number of LNV meson-decay processes, and those numerical values will be used to compute the neutrino masses radiatively generated at the four-loop level, as shown in Fig.", "REF (b).", "Since we have chosen the operator in Eq.", "(12) for the LNV meson decays, which resembles well the one $\\epsilon ^{}_3 J^\\mu _{\\rm R} J^{}_{\\mu {\\rm R}} j^{}_{\\rm L}$ for $0\\nu \\beta \\beta $ decays in the previous section, the calculation of generated Majorana neutrino mass terms from LNV meson decays follows closely that in the case of $0\\nu \\beta \\beta $ decays.", "The only difference is the presence of two possibly different lepton flavors and different hadronic currents, which bring the CKM matrix elements into the calculation.", "In the case where $U \\ne U^\\prime _{}$ and $D \\ne D^\\prime _{}$ do not hold simultaneously, a straightforward evaluation of a similar butterfly diagram leads to an induced neutrino mass $\\delta m^{\\alpha \\beta }_{\\nu }$ for $\\alpha $ and $\\beta $ lepton flavors, namely, $\\delta m^{\\alpha \\beta }_{\\nu } &=& \\frac{64 g^4_{} {\\rm G}_{\\rm F}^2 \\varepsilon _{\\alpha \\beta }^{} m_U^{} m_D^{} m_{U^\\prime }^{} m_{D^\\prime }^{} m_{\\alpha }^{} m_{\\beta }^{}}{C^{}_{U U^\\prime } C^{}_{D D^\\prime } m_{\\rm p}^{}}\\left[ V_{U D}^{*} V_{U^\\prime D^\\prime }^{*} \\cdot \\mathcal {I}(m_\\alpha ^2, m_U^2, m_D^2) \\cdot \\mathcal {I}(m_\\beta ^2, m^2_{U^\\prime }, m^2_{D^\\prime }) + (\\alpha \\leftrightarrow \\beta )\\right] \\; , \\nonumber \\\\$ where $V_{U^{(\\prime )} D^{(\\prime )}}$ is the CKM matrix element, $C^{}_{U U^\\prime }$ and $C^{}_{D D^\\prime }$ follow the same definition of $C^{}_{\\alpha \\beta }$ below Eq.", "(17), and the loop integral $\\mathcal {I}$ is the same as that introduced in Eq. (6).", "On the other hand, when $U \\ne U^\\prime _{}$ and $D \\ne D^\\prime _{}$ are both present, we obtain $\\delta m^{\\alpha \\beta }_{\\nu } &=& 16 g^4_{} {\\rm G}_{\\rm F}^2 \\varepsilon _{\\alpha \\beta }^{} m_U^{} m_D^{} m_{U^\\prime }^{} m_{D^\\prime }^{} m_{\\alpha }^{} m_{\\beta }^{} m_{\\rm p}^{-1}\\left[ V_{U D}^{*} V_{U^\\prime D^\\prime }^{*} \\cdot \\mathcal {I}(m_\\alpha ^2, m_U^2, m_D^2) \\cdot \\mathcal {I}(m_\\beta ^2, m^2_{U^\\prime }, m^2_{D^\\prime }) \\right.", "\\nonumber \\\\& & \\qquad \\qquad \\left.", "+ V_{U^\\prime D}^{*} V_{U D^\\prime }^{*} \\cdot \\mathcal {I}(m_\\alpha ^2, m^2_{U^\\prime }, m_D^2) \\cdot \\mathcal {I}(m_\\beta ^2, m^2_{U}, m^2_{D^\\prime }) + (\\alpha \\leftrightarrow \\beta )\\right] \\; .$ Using numerical values of quark and lepton masses, CKM mixing angles $\\theta _{ij}^{}$ (for $ij=12, 13, 23$ ) and Dirac phase $\\delta $ in Table REF , we have tabulated the Majorana neutrino masses implied by various types of LNV meson decays in Table REF .", "As one can observe from Table REF , depending on the current experimental limits, the values of Majorana neutrino masses from LNV meson decays can be quite different, spanning over many orders of magnitude.", "The LNV meson decays may indicate Majorana neutrino mass terms $\\delta m^{e\\mu }_\\nu $ and $\\delta m^{\\mu \\mu }_\\nu $ , which cannot be obtained from $0\\nu \\beta \\beta $ decays.", "For instance, if the LNV decays $K^- \\rightarrow \\pi ^+ e^- \\mu ^-$ and $K^- \\rightarrow \\pi ^+ \\mu ^- \\mu ^-$ are observed, we arrive at $|\\delta m^{e\\mu }_\\nu | \\sim 1.6 \\times 10^{-15}~{\\rm eV}$ and $|\\delta m^{\\mu \\mu }_\\nu | \\sim 1.0 \\times 10^{-12}~{\\rm eV}$ , which are still far below the required masses from neutrino oscillation experiments." ], [ "Summary", "Whether massive neutrinos are Majorana or Dirac particles remains an unsolved fundamental problem in particle physics.", "According to the Schechter-Valle theorem, if the $0\\nu \\beta \\beta $ decays $N(Z, A) \\rightarrow N(Z+2, A) + e^- + e^-$ are observed in future experiments, one can claim that neutrinos do have Majorana masses.", "In this short note, we have revisited the quantitative impact of the Schechter-Valle theorem and shown that the Majorana neutrino mass radiatively generated at the four-loop level is $|\\delta m^{ee}_\\nu | < 7.43\\times 10^{-29}~{\\rm eV}$ .", "Furthermore, a similar analysis has been performed for the LNV meson decays $M^- \\rightarrow M^{\\prime +} + \\ell ^-_\\alpha + \\ell ^-_\\beta $ , from which the upper bounds $|\\delta m^{ee}_\\nu | < 9.7 \\times 10^{-18}~{\\rm eV}$ , $|\\delta m^{e\\mu }_\\nu | < 1.6 \\times 10^{-15}~{\\rm eV}$ and $|\\delta m^{\\mu \\mu }_\\nu | < 1.0 \\times 10^{-12}~{\\rm eV}$ can be derived.", "A list of radiative neutrino masses from other LNV rare decays of $D$ and $B$ mesons is also given.", "Therefore, even if the $0\\nu \\beta \\beta $ decays or the LNV meson decays are detected and the decay rates are close to current upper bounds, we have to invoke some other mechanisms to produce sub-eV neutrino masses, which can be of either Dirac or Majorana nature.", "In the former case, massive neutrinos should be pseudo-Dirac particles, since a small Majorana mass is implied by the LNV decays.", "In the latter case, compared to the sub-eV neutrino masses at the leading order, the radiative Majorana masses can be neglected." ], [ "Acknowledgments", "One of the authors (S.Z.)", "is grateful to Alexander Merle and Apostolos Pilaftsis for helpful discussions, and to the Mainz Institute for Theoretical Physics (MITP) for its hospitality and its partial support during the completion of this work, which has also been supported in part by the National Recruitment Program for Young Professionals and by the CAS Center for Excellence in Particle Physics (CCEPP).", "Table: A partial list of the LNV decays of KK, DD, D s D^{}_s and BB mesons and current experimental constraints on the branching ratios .", "The upper bounds on the coefficients ε αβ \\varepsilon ^{}_{\\alpha \\beta } and those on radiative neutrino masses |δm ν αβ ||\\delta m^{\\alpha \\beta }_\\nu | (for α,β\\alpha , \\beta = e,μe, \\mu ) are given in the last two columns." ] ]
1606.04886
[ [ "Joint Data Compression and MAC Protocol Design for Smartgrids with\n Renewable Energy" ], [ "Abstract In this paper, we consider the joint design of data compression and 802.15.4-based medium access control (MAC) protocol for smartgrids with renewable energy.", "We study the setting where a number of nodes, each of which comprises electricity load and/or renewable sources, report periodically their injected powers to a data concentrator.", "Our design exploits the correlation of the reported data in both time and space to efficiently design the data compression using the compressed sensing (CS) technique and theMAC protocol so that the reported data can be recovered reliably within minimum reporting time.", "Specifically, we perform the following design tasks: i) we employ the two-dimensional (2D) CS technique to compress the reported data in the distributed manner; ii) we propose to adapt the 802.15.4 MAC protocol frame structure to enable efficient data transmission and reliable data reconstruction; and iii) we develop an analytical model based on which we can obtain efficient MAC parameter configuration to minimize the reporting delay.", "Finally, numerical results are presented to demonstrate the effectiveness of our proposed framework compared to existing solutions." ], [ "Introduction", "The future energy grid is expected to integrate more distributed and renewable energy resources with significantly enhanced communications infrastructure for timely and reliable data exchanges between the control center and various grid control and monitoring points [1].", "Smartgrid is an example of the cyber-physical system (CPS) that integrates different communications, control, and computing technologies [2].", "Smartgrid communications infrastructure is an important component of the future smartgrid that enables to support many critical grid control, monitoring, and management operations and emerging smartgrid applications [2]–[4].", "The smartgrid communications infrastructure is typically hierarchical, i.e., data communications between customer premises (smart meters (SMs)) and local concentrators and between local concentrators and the utility company are supported by field/neighborhood area networks and long-haul wide area networks, respectively [5]–[8].", "The former is usually based on the low bandwidth communications technologies such as Zigbee, WiFi, and power line communications (PLC) while the later is required to have higher capacity, which can be realized by employing LTE, 3G cellular, WiMAX, and fiber optics for example.", "Our current work concerns the design of data compression and MAC protocol for the field/neighborhood area network where PLC is employed to report injected powers from grid reporting points to the local concentrator.", "In fact, several smartgrid projects in Spain [5], and France [6] have chosen PLC for smartgrid deployment since PLC can be realized with low-cost modems and it can utilize available electricity wires for data communications.", "In addition, as reported in Italy's Telegestore project, SMs have been installed at customer premises to send data to a concentrator via PLC [8].", "We focus on reporting injected powers at different grid reporting points once in every reporting interval (RI) [9], [10].", "This is motivated by the fact that information on injected powers can be very useful for various grid applications such as line failure prediction [11]–[13] or congestion management and grid control applications [14], [15].", "Furthermore, the utility control center can utilize the collected data to further estimate the complete phasor data at different nodes [16] which can be then used in the control of voltages and reactive powers or in the active load management [9], [10], [17], outage management [18], [19] as well as illegal electricity usages and data intrusion/anomaly detection [20], [21].", "There have been some existing works that study data compression and MAC protocol design issues in both smartgrids and wireless network contexts.", "There are two popular standards for the PLC technology, namely PRIME and G3-PLC, whose physical- and MAC-layer design aspects are investigated via simulations in [22]–[25].", "In addition, the authors in [9] study the state estimation problem where the voltage phasors at different nodes are recovered based on limited collected data.", "Li et al.", "[26] propose to employ the CS technique for sparse electricity data compression.", "The authors in [27] consider random access exploiting the CS capability for energy efficiency communications in wireless sensor networks.", "However, none of these existing works considers the joint design of communications access and data compression that supports the communication for a large number of nodes (e.g., in smartgrid communication infrastructure).", "We aim to fill this gap in the current work.", "In particular, we make the following contributions.", "We propose the CS-based data compression and PRIME MAC protocol design that supports the communication between a number of grid reporting points and a data concentrator using PLC and random access.", "Specifically, we consider a distributed random reporting (DRR) mechanism where each node decides to report its data in the probabilistic manner by using the 802.15.4-based MAC protocol.", "Then the Kronecker CS technique (2D CS), which can exploit the spatio-temporal correlation of data, is employed for data reconstruction at the control center.", "We develop an analytical model for the MAC protocol, which enables us to achieve efficient and reliable data reconstruction using compressed sensing.", "In addition, we present an algorithm which determines efficient configuration for MAC parameters so that the average reporting time is minimized.", "Also, we analyze the energy consumption and bandwidth usage for our proposed design.", "We present numerical results to demonstrate the significant performance gains of our proposed design compared to the non-compressed solution and the centralized time division multiple access (TDMA) scheme.", "These performance gains are demonstrated for reporting delay, energy consumption, and bandwidth usage.", "The remainder of this paper is organized as follows.", "Section presents the system model and Section describes a CS-based data compression.", "Section describes our design and performance analysis.", "Numerical results are demonstrated in Section followed by the conclusion in Section ." ], [ "System Model", "We consider the communications between $n_S$ grid reporting points and one data concentrator which corresponds to the point-to-multipoint communication in the field area network illustrated in Fig.", "REF .", "There would be a large number of data concentrators collecting various types of monitoring and control data in the distribution network; however, we focus on the design of the point-to-multipoint communication for one such data concentrator without loss of generality.", "For brevity, we refer to the grid reporting points as nodes in the following.", "Each node is assumed to comprise a load and/or a solar/wind generator.", "Deployment of such renewable generators in large scale at customer premises, light poles, or advertising panels has been realized for several urban areas in Europe [5]–[8].", "We further assume that each node is equipped with a SM, which is capable of reporting power data to the data concentrator using the PLC.", "Upon reaching the data concentrator, the collected data is forwarded to the utility control center via the high-speed backhaul network, which is assumed to be error free.This assumption is reasonable since the backhaul network typically has very high capacity and reliability.", "This is the case if the backhaul network is realized by fiber optics for example.", "The data concentrator can be installed at the same place as the transformer to collect power data from its nodes as shown in Fig.", "REF , which enables reliable communications.", "Note, however, that data concentrator can be deployed in the medium voltage (MV) line as well where the power data is transferred over medium-voltage power lines passing coupling devices at transformers [24].", "We assume that each node must report the injected power value once for each fixed-length RI [9], [10].", "Such collected injected powers can be used for various applications as discussed in Section .", "The RI typically ranges from 5 minutes to one day, which is configurable for most available SMs [28] even though we set RI equal to 5 minutes in this work.", "Let $\\mathbf {S}_{l,i}$ , $\\mathbf {S}_{g,i}$ and $\\mathbf {S}_i$ be the load, distributed generation and injected powers at node $i$ , respectively.", "Then, the injected power at node $i$ can be expressed as $\\mathbf {S}_i = \\mathbf {S}_{g,i} - \\mathbf {S}_{l,i}$ .", "We assume that the control center wishes to acquire information about $\\mathbf {S}_i$ for all nodes $i$ $(i \\in \\left[1, n_S\\right])$ in each RI.The proposed framework can be applied to other types of grid data as long as they exhibit sparsity in the time and/or space domains.", "Due to the low bandwidth constraint of PLC [22]–[25], our objectives are to perform joint design of data compression and MAC protocol so that reliable data reporting can be achieved within minimum reporting time (RT).", "This is motivated by the fact that most smartgrid control and monitoring applications have very stringent delay requirements.", "Another important design target is that the MAC protocol is distributed so that it can allow low-cost and large-scale deployment.", "It has been shown in some recent works that power grid data typically exhibits strong correlation over space and time [29]–[31].", "Therefore, the CS technique can be employed for data compression, which can potentially reduce the amount of communication data and thus the delay, communication bandwidth and energy [32], [33], [34].", "To realize such compression benefits, the control center only needs to collect the reported data in $m_T < n_T$ RIs (i.e., compression over time) from a subset of $n_S$ nodes with size $m_S < n_S $ (i.e., compression over space) to reconstruct the complete data for $n_S$ nodes in $n_T$ RIs.", "Note that data reconstruction can be performed locally at the data concentrator or at the control center.", "The later deployment is assumed in this work since it helps save bandwidth usage in the backhaul network.", "Figure: Rolling-based data reconstruction for n S n_S nodes over n T n_T RIsFor practical implementation, the data transmission and reconstruction can be performed in a rolling manner where data reconstruction performed at a particular RI utilizes the reported data over the latest $n_T$ RIs as shown in Fig.", "REF .", "To guarantee the desirable data reconstruction quality, the control center must receive sufficient data which is impacted by the underlying reporting mechanism and MAC protocol.", "Specifically, we must determine the values of $m_T$ and $m_S$ for some given values $n_T$ and $n_S$ to achieve the desirable data reconstruction reliability.", "These design issues are addressed in the following sections." ], [ "CS-Based Data Processing", "Without loss of generality, we consider data reconstruction of one data field for $n_T$ RIs and $n_S$ nodes.", "Let $\\mathbf {Z}$ be an $n_S \\times n_T $ matrix whose $(i,j)$ -th element denotes the injected power at node $i$ and RI $j$ .", "We will refer to the data from one RI (i.e., one column of $\\mathbf {Z}$ ) as one data block.", "From the CS theory, we can compress the data of interest if they possess sparsity properties in a certain domain such as wavelet domain.", "Specifically, we can express the data matrix $\\mathbf {Z}$ as $\\mathbf {Z} = \\mathbf {\\Psi }_S \\mathbf {A} \\mathbf {\\Psi }_T^T$ where $\\mathbf {\\Psi }_S \\in \\mathbb {R}^{n_S \\times n_S} $ and $\\mathbf {\\Psi }_T \\in \\mathbb {R}^{n_T \\times n_T}$ denote wavelet bases in space and time dimensions, respectively [32], [33], [34].", "The sparsity of $\\mathbf {Z}$ can be observed in the wavelet domain if matrix $\\mathbf {A}$ has only $K$ significant (nonzero) coefficients where $K <n_S \\times n_T$ .", "We now proceed to describe the data compression and reconstruction operations.", "Let us denote $\\mathbf {\\Phi }_S \\in \\mathbb {R}^{m_S \\times n_S} $ (for space) and $\\mathbf {\\Phi }_T \\in \\mathbb {R}^{m_T \\times n_T}$ (for time) as the two sparse observation matrices where entries in these two matrices are i.i.d uniform random numbers where $m_S < n_S $ and $m_T < n_T $ .", "Then we can employ $\\mathbf {\\Phi }_S$ and $\\mathbf {\\Phi }_T$ to sample the power data from which we obtain the following observation matrix $\\mathbf {Y} = \\mathbf {\\Phi }_S \\mathbf {Z} \\mathbf {\\Phi }_T^T.$ Let $N_{\\Sigma } = n_T n_S $ and $M = m_T m_S$ be the number of elements of $\\textbf {Z}$ and $\\mathbf {Y}$ , respectively.", "From the CS theory, we can reliably reconstruct the data matrix $\\textbf {Z}$ by using the observation matrix $\\mathbf {Y}$ if $m_T$ and $m_S$ are appropriately chosen.", "For the considered smartgrid communication design, this implies that the control center only needs to collect $M$ injected power elements instead of $N_{\\Sigma }$ values for reliable reconstruction of the underlying data field.", "We now describe the data reconstruction for $\\textbf {Z}$ by using the observation matrix $\\mathbf {Y}$ .", "Toward this end, the control center can determine matrix $\\mathbf {A}$ , which corresponds to the wavelet transform of the original data $\\textbf {Z}$ as described in (REF ), by solving the following optimization problem $\\min _ \\textbf {A} \\left\\Vert \\textbf {A}\\right\\Vert _2 \\,\\,\\,\\text{s.t.}", "\\,\\,\\, \\left\\Vert \\textbf {vec}\\left(\\mathbf {Y}\\right) - \\mathbf {\\bar{Y}}\\right\\Vert _2 \\le \\epsilon $ where $\\mathbf {\\bar{Y}} = \\left(\\mathbf {\\Phi }_S \\otimes \\mathbf {\\Phi }_T \\right) \\left(\\mathbf {\\Psi }_S \\otimes \\mathbf {\\Psi }_T \\right) \\textbf {vec}\\left(\\textbf {A}\\right)$ , $\\otimes $ is the Kronecker product, $\\textbf {vec} \\left(\\mathbf {X}\\right)$ denotes the vectorization of the matrix $\\mathbf {X}$ , which stacks the rows of $\\mathbf {X}$ into a single column vector.", "We can indeed solve problem (REF ) by using the Kronecker CS algorithm [32] to obtain $\\mathbf {A}^* = \\operatornamewithlimits{argmin}_\\textbf {A} \\left\\Vert \\textbf {A}\\right\\Vert _2$ .", "Then, we can obtain the estimation for the underlying data as $\\textbf {vec}\\left(\\textbf {Z}^* \\right) = \\mathbf {\\Psi }_S \\otimes \\mathbf {\\Psi }_T \\mathbf {A}^*$ .", "More detailed discussions of this data reconstruction algorithm can be found in [32].", "Now there are two questions one must answer to complete the design: 1) how can one choose $m_S $ and $m_T$ to guarantee reliable data reconstruction for the underlying data field?", "; and 2) how can one design the data sampling and MAC protocol so that the control center has sufficient information for data reconstruction?", "We will provide the answers for these questions in the remaining of this paper." ], [ "Determination of $m_S$ and {{formula:6a1d0e69-6e4d-49ed-bb34-9172dc375180}}", "We would like to choose $m_S$ and $m_T$ so that $M = m_S m_T$ is minimum.", "Determination of the optimal values of $m_S$ and $m_T$ turns out to be a non-trivial task [32].", "So we propose a practical approach to determine $m_S$ and $m_T$ .", "It is intuitive that $m_S$ and $m_T$ should be chosen according to the compressibility in the space and time dimensions, respectively.", "In addition, these parameters must be chosen so that the reliability of the data reconstruction meets the predetermined requirement, which is quantified by the mean square error (MSE).", "To quantify compressibility in the space and time, we consider two other design options, viz.", "temporal CS and spatial CS alone.", "For the former, the control center reconstructs data for one particular node by using the observations of only that node (i.e., we ignore spatial correlation).", "For the later, the control center reconstructs the data in each RI for all nodes without exploiting the correlation over different RIs.", "For fair quantification, we determine the MSE for one data field (i.e., for data matrix $\\mathbf {Z}$ with $n_S \\times n_T $ elements) for these two design options where $MSE = \\left\\Vert \\mathbf {Z}-\\mathbf {Z}^*\\right\\Vert _2^2/\\left\\Vert \\mathbf {Z}\\right\\Vert _2^2$ .", "For each spatial CS and temporal CS cases, we generate 1000 realizations of the injected powers based on which we perform the data reconstruction using the 1D CS for different values of $m_S$ and $m_T$ , respectively.", "Note that we randomly choose $m_S$ and $m_T$ out of $n_S$ nodes and $n_T$ RIs for spatial CS and temporal CS, respectively.", "Then, we obtain the empirical probability of success for the data reconstruction versus $M_{S} = m_S n_T$ and $M_{T} = n_S m_T$ , respectively where the “success” means that the MSE is less than the target MSE.", "From the obtained empirical probability of success, we can find the required values of $M_S$ and $M_T$ , which are denoted as $M_{S,{\\sf thresh}}$ and $M_{T,{\\sf thresh}}$ , respectively, to achieve the target success probability.", "Having obtained $M_{S,{\\sf thresh}}$ and $M_{T,{\\sf thresh}}$ capturing the compressibility in the space and time as described above, we choose $m_S $ and $m_T$ for the 2D CS so that $m_S/m_T = n_S/n_T \\times M_{S,{\\sf thresh}}/M_{T,{\\sf thresh}}$ .", "Similarly, we obtain the empirical probability of success for the 2D CS based on which we can determine the minimum value of $M_{\\sf thresh} = m_S m_T$ to achieve the target success probability.", "To obtain numerical results in this paper, we choose target $MSE = 0.05$ and target success probability equal 0.95.", "Fig.", "REF shows the empirical probability of success versus $M_{{\\sf thresh}}= m_S m_T$ for $n_S = n_T = 128$ where $M_{{\\sf thresh}}$ in the horizontal axis represents the $M_{S,{\\sf thresh}}$ , $M_{T,{\\sf thresh}}$ , and $M_{{\\sf thresh}}$ for 1D and 2D CS for simplicity.", "The data model for the injected power will be described in Section REF .", "For this particular setting, we can obtain the ratio $m_S/m_T = 0.691 $ from which can obtain the values of $m_S = 47 $ and $m_T = 68 $ .", "Similarly, we determine the $M_{\\sf thresh}$ , $m_S $ and $m_T$ for different scenarios whose corresponding $(n_S, n_T)$ values are given in Table REF .", "For all cases, we can observe that $m_S < n_S$ and $m_T < n_T$ , which demonstrates the benefits of data compression by using the 2D CS.", "Having determined $m_S $ and $m_T$ as described above, the remaining tasks are to design the DDR mechanism and MAC protocol that are presented in the following." ], [ "Distributed Data Reporting", "In any RI, to perform reconstruction for the data field corresponding to the latest $n_T$ RIs, the control center must have data in $m_T$ RIs, each of which comprises $m_S$ injected powers from $m_S$ nodes.", "With the previously-mentioned rolling implementation, the control center can broadcast a message to all nodes to request one more data block (the last column of data matrix $\\textbf {Z}$ ) if it has only $m_T-1$ data blocks to perform reconstruction in the current RI.", "At the beginning, the control center can simply send $m_T$ broadcasts for $m_T$ randomly chosen RIs out of $n_T$ RIs and it performs data reconstruction at RI $n_T$ upon receiving the requested data blocks." ], [ "MAC Protocol Design", "If there is a broadcast message from the control center, we assume that each node participates in the contention process with probability $p_s$ using the slotted CSMA/CA MAC protocol in any RI.", "The slotted CSMA/CA MAC protocol [23] with the proposed frame structure is employed for data transmissions as described in the following.", "We set the optionally contention-free period to zero since we focus on distributed access design in this work.", "Moreover, we assume there are $K_{\\tau } $ superframes (SFs) in the contention period of any RI where $SF_i = SF_0 \\times 2^{BO_i}$ is the length of SF $i$ , $SF_0 $ is the base length of SF, $BO_i \\in \\left[0,BO_{\\sf max}\\right]$ is the beacon order at SF $i$ .", "Therefore, the reporting time (RT) in the underlying RI is $ \\sum _{i=1}^{K_{\\tau }} SF_i$ .", "We will optimize the parameters $K_{\\tau } $ and $\\left\\lbrace BO_i\\right\\rbrace $ to minimize the RT while guaranteeing the desirable data reconstruction quality later.", "The superframe structure in one RI is illustrated in Fig.", "REF .", "In each SF, the nodes that choose to access the channel (with probability $p_s$ ) perform the contention using the standardized slotted CSMA/CA protocol as follows.", "A contending node randomly chooses a number in $\\left[0, W_0\\right]$ as the backoff counter ($W_0 \\!=\\!", "2^{priority}$ ) and starts counting down.", "If the counter reaches zero, the node will perform the clear channel assessment (CCA) for $priority$ times (we set $priority \\!=\\!", "2$ ).", "It will transmit data and wait for ACK if all CCAs are successful.", "The reception of ACK is interpreted as a successful transmission, otherwise this is a collision.", "In the case of failure in any CCA, the node attempts to perform backoff again with doubling backoff window.", "In addition, each node is allowed to access channel up to $NB\\!+\\!1$ times.", "Since the length of the SF is limited, a node may not have enough time to transmit its data and ACK packets at the end of the last SF.", "In this case, we assume that the node will wait for the next SF to access the channel.", "We refer to this as the deference state in the following." ], [ "MAC Parameter Configuration for Delay Minimization", "We consider optimizing the MAC parameters $\\left(K_{\\tau }, p_s, \\left\\lbrace BO_i\\right\\rbrace \\right)$ to minimize the reporting time in each RI by solving the following problem: $\\begin{array}{l}{\\mathop {\\min }\\limits _{K_{\\tau }, p_s, BO_i}} \\quad \\mathcal {D}\\left(K_{\\tau }, p_s, \\left\\lbrace BO_i\\right\\rbrace \\right) \\\\\\mbox{s.t.}", "\\quad \\Pr \\left\\lbrace K_{\\text{succ}} \\ge m_{S} \\right\\rbrace \\ge P_{\\text{suff}}, \\\\\\quad \\quad 0 \\le K_{\\tau } \\le K_{\\tau ,{\\sf max}}, 0 \\le BO_i \\le BO_{\\sf max}, 0 \\le p_s \\le 1, \\\\\\end{array}$ where $\\mathcal {D}(K_{\\tau }, p_s, \\left\\lbrace BO_i\\right\\rbrace ) = \\sum _{i=1}^{K_{\\tau }} SF_0 \\times 2^{BO_i}$ is the reporting time.", "Recall that $K_{\\tau } $ is the number of SFs in one RI where $SF_i$ is the length of SF $i$ and $SF_0 $ is the base length of SF, $BO_i \\in \\left[0,BO_{\\sf max}\\right]$ is the beacon order at SF $i$ , $BO_{\\sf max}$ is the maximum value of $BO_i$ , $p_s$ is the probability that each node performs a contention for a possible channel access.", "In (REF ), the first constraint means that the control center must receive $m_{S}$ packets (i.e., $m_{S}$ injected power values from $m_{S}$ nodes) with probability $P_{\\text{suff}} \\approx 1$ .", "Note that one would not be able to deterministically guarantee the reception of $m_{S}$ packets due to the random access nature of the DDR and MAC protocol.", "The probability in this constraint can be written as $\\Pr \\left\\lbrace K_{\\text{succ}} \\ge m_{S} \\right\\rbrace \\!\\!", "= \\!\\!", "\\sum _{h= m_{S} }^{n_S} \\!\\!", "\\Pr \\left\\lbrace \\widehat{m}_S = h\\right\\rbrace \\!\\!\\!\\!\\!\\!", "\\sum _{K_{\\sf succ} = m_{S} }^h \\sum _{l =1 }^{\\left|\\Xi \\right|} \\prod _{i=1}^{K_{\\tau }} \\\\\\times \\sum _{K_{S,i} = K_{{\\sf succ},i}}^{K_{S,i,{\\sf max}}} \\!\\!\\!\\!\\!\\!\\!\\!", "\\Pr \\!\\left\\lbrace K_i = K_{S,i} \\left|h_i \\right.\\right\\rbrace \\!", "\\Pr \\!\\left\\lbrace {\\bar{K}}_i = K_{\\text{succ},i} \\left|K_{S,i},h_i \\right.\\right\\rbrace $ where $K_{\\text{succ}}$ denotes the number of successfully transmitted packets in the RI and $\\Pr \\!\\left\\lbrace \\widehat{m}_S \\!=\\!", "h\\right\\rbrace $ is the probability of $h$ nodes joining the contention.", "Since each node decides to join contention with probability $p_s$ , $\\Pr \\!\\left\\lbrace \\widehat{m}_S \\!=\\!", "h\\right\\rbrace $ is expressed as $\\Pr \\left\\lbrace \\widehat{m}_S = h\\right\\rbrace = \\left(\\begin{array}{*{20}{c}} {n_S} \\\\ {h} \\\\ \\end{array}\\right) p_s^h \\left(1-p_s\\right)^{n_S - h}.$ In (REF ) and (), we consider all possible scenarios so that the total number of successfully transmitted packets over $K_{\\tau }$ SFs is equal to $K_{\\text{succ}}$ where $K_{\\text{succ}} \\in \\left[m_{S}, h\\right]$ .", "Here, $K_{\\text{succ},i}$ denotes the number of successfully transmitted packets in SF $i$ so that we have $\\sum _{i=1}^{K_{\\tau }} K_{\\text{succ},i} = K_{\\text{succ}}$ .", "In particular, we generate all possible combinations of $\\left\\lbrace K_{\\text{succ},i}\\right\\rbrace $ for $K_{\\tau } $ SFs and $\\Xi $ represents the set of all possible combinations ($\\left|\\Xi \\right|$ is the number of possible combinations).", "For each combination, we calculate the probability that the control center receives $K_{\\text{succ}}$ successful packets.", "Note that a generic frame may experience one of the following events: success, collision, CCA failure and deference.", "Also, there are at most $K_{S,i,{\\sf max}}$ frames in any SF $i$ where $K_{S,i,{\\sf max}} = \\left\\lfloor SF_i/\\min \\left\\lbrace NB+1, L_s+2 \\right\\rbrace \\right\\rfloor $ since the smallest length of a CCA failure frame is $NB +1 $ slots while the minimum length of a successful frame is $L_s+2$ where $L_s$ is the required time for one successful transmission and 2 represents the two CCA slots.", "We only consider the case that $K_{{\\sf succ},i} \\le K_{S,i} \\le K_{S,i,{\\sf max}}, \\forall i \\in \\left[1, K_{\\tau }\\right]$ .", "In (), $\\Pr \\!", "\\left\\lbrace K_i \\!=\\!", "K_{S,i} \\left|h_i \\right.\\right\\rbrace $ is the probability that there are $K_{S,i}$ generic frames in SF $i$ given that $h_i$ nodes join contention where $h_1 \\!=\\!h$ and $h_i \\!=\\!", "h_{i-1}\\!-\\!", "K_{\\text{succ},i-1}$ since successfully transmitting node will not perform contention in the following frames.", "Moreover, $\\Pr \\!\\left\\lbrace {\\bar{K}}_i \\!=\\!", "K_{\\text{succ},i} \\left|K_{S,i},h_i \\right.\\right\\rbrace $ is the probability that $K_{\\text{succ},i}$ nodes transmit successfully in SF $i$ given that there are $h_i$ contending nodes and $K_{S,i}$ generic frames.", "In order to calculate $\\Pr \\left\\lbrace K_i = K_{S,i} \\left|h_i \\right.\\right\\rbrace $ and $\\Pr \\left\\lbrace {\\bar{K}}_i = K_{\\text{succ},i} \\left|K_{S,i},h_i \\right.\\right\\rbrace $ , we have to analyze the Markov chain capturing detailed operations of the MAC protocol.", "For simplicity, we set $NB_i = NB = 5$ , which is the default value.", "We analyze the Markov chain model in Appendix  .", "Then we determine $\\Pr \\left\\lbrace K_i = K_{S,i} \\left|h_i \\right.\\right\\rbrace $ and $\\Pr \\left\\lbrace {\\bar{K}}_i = K_{\\text{succ},i} \\left|K_{S,i},h_i \\right.\\right\\rbrace $ as follows." ], [ "Calculation of $\\Pr \\left\\lbrace K_i = K_{S,i} \\left|h_i \\right.\\right\\rbrace $", "In SF $i $ there are $h_i $ contending nodes and $K_{S,i}$ generic frames where frame $j$ has length $T_{ij}$ .", "We can approximate the distribution of generic frame length $T_{ij}$ as the normal distribution [35].", "So the probability of having $K_{S,i} $ generic frames is written as $\\Pr \\!\\left\\lbrace \\!K_i \\!= \\!K_{S,i} \\left|h_i\\right.", "\\!\\right\\rbrace \\!\\!=\\!\\!", "\\Pr \\lbrace \\sum _{j=1}^{K_{S,i}} \\!", "T_{ij} \\!\\!", "= \\!\\!", "SF_i \\rbrace \\!=\\!\\mathcal {Q} (\\frac{SF_i\\!-\\!K_{S,i} {\\bar{T}}_i}{\\sqrt{K_{S,i} \\sigma ^2_i}})$ where ${\\bar{T}}_i $ and $\\sigma ^2_i $ are the average and variance of the generic frame length, respectively whose calculations are presented in Appendix ." ], [ "Calculation of $\\Pr \\left\\lbrace {\\bar{K}}_i = K_{\\text{succ},i} \\left|K_{S,i},h_i \\right.\\right\\rbrace $", "The second quantity, $\\mathcal {P} = \\Pr \\left\\lbrace {\\bar{K}}_i = K_{\\text{succ},i} \\left|K_{S,i},h_i \\right.\\right\\rbrace $ is equal to $\\mathcal {P} = \\!\\!\\!\\!\\!\\!", "\\sum _{j = 0}^{K_{S,i}- K_{\\text{succ},i}} \\sum _{k= 0}^{1} \\left(\\!\\!\\!\\begin{array}{*{20}{c}} {K_{S,i}} \\\\ { K_{\\text{succ},i},j,k,l} \\\\ \\end{array} \\!\\!\\!\\right) \\mathcal {P}_{\\text{succ},h_i}^{ K_{\\text{succ},i}} \\mathcal {P}_{\\text{coll},h_i}^j \\mathcal {P}_d^k \\mathcal {P}_{\\text{ccas},h_i}^{l} $ where $K_{\\text{succ},i}\\!+\\!j\\!+\\!k\\!+\\!l \\!=\\!", "K_{S,i}$ ; $j$ , $k$ , and $l$ represent the number of frames with collision, deference, and CCA failure, respectively.", "Moreover, $\\mathcal {P}_{\\text{succ},h_i}$ , $\\mathcal {P}_{\\text{coll},h_i}$ , $\\mathcal {P}_{\\text{ccas},h_i}$ , and $\\mathcal {P}_d$ denote the probabilities of success, collision, CCA failure, and deference, respectively, whose calculations are given in Appendix B.", "In (REF ), we generate all possible combinations each of which has different numbers of success, collision, CCA failure, and deference frames.", "Also, the product behind the double summation is the probability of one specific combination." ], [ "MAC Parameter Configuration Algorithm", "[h]Optimization of MAC Parameters [1] each value of $K_{\\tau } \\in [1,K_{\\tau ,{\\sf max}}]$ each possible set $\\left\\lbrace BO_i\\right\\rbrace $ Find optimal ${\\bar{p}}_s $ as ${\\bar{p}}_s = \\mathop {\\operatornamewithlimits{argmin}} \\limits _{0 \\le p_s \\le 1} \\mathcal {D} \\left( K_{\\tau }, \\left\\lbrace BO_i\\right\\rbrace , p_s\\right) $ .", "The best $\\left(\\left\\lbrace \\bar{BO}_i\\right\\rbrace , {\\bar{p}}_s\\right)$ for each $K_{\\tau } $ is $\\left(\\left\\lbrace \\bar{BO}_i\\right\\rbrace , {\\bar{p}}_s\\right) = \\mathop {\\operatornamewithlimits{argmin}} \\limits _{\\left\\lbrace { BO}_i\\right\\rbrace , {\\bar{p}}_s} \\mathcal {D} \\left(K_{\\tau }, \\left\\lbrace BO_i\\right\\rbrace , {\\bar{p}}_s\\right) $ .", "The final solution $\\left( {\\bar{K}}_{\\tau }, \\left\\lbrace \\bar{BO}_i\\right\\rbrace , {\\bar{p}}_s \\right) $ is determined as $\\left( {\\bar{K}}_{\\tau }, \\left\\lbrace \\bar{BO}_i\\right\\rbrace , {\\bar{p}}_s \\right) = \\mathop {\\operatornamewithlimits{argmin}} \\limits _{K_{\\tau }, \\left\\lbrace \\bar{BO}_i\\right\\rbrace , {\\bar{p}}_s} \\mathcal {D} \\left(K_{\\tau }, \\left\\lbrace \\bar{BO}_i\\right\\rbrace , {\\bar{p}}_s\\right) $ .", "The procedure for finding $\\left(K_{\\tau }, p_s, \\left\\lbrace BO_i\\right\\rbrace \\right)$ can be described in Alg.", "REF .", "Since there are only finite number of possible choices for $K_{\\tau } \\in [1,K_{\\tau ,{\\sf max}}]$ and the set $\\left\\lbrace {BO}_i\\right\\rbrace $ , we can search for the optimal value of $p_s$ for given $K_{\\tau }$ and $\\left\\lbrace {BO}_i\\right\\rbrace $ as in step 3.", "Then, we search over all possible choices of $K_{\\tau }$ and the set $\\left\\lbrace {BO}_i\\right\\rbrace $ to determine the optimal configuration of the MAC parameters (in steps 5 and 7)." ], [ "Bandwidth Usage", "To quantify the bandwidth usage, we consider a particular neighborhood with $N>n_S$ nodes, whose simultaneous transmissions can collide with one another.", "In addition, these $N$ nodes must report injected power data to a control center.", "In this case, we would need $N/n_S$ orthogonal channelsWe ignore the fact that this number must be integer for simplicity to support these communications.", "Suppose that the considered smartgrid application has a maximum target delay of $\\mathcal {D}_{\\sf max}$ .", "We should design the group size with $n_S$ nodes as large as possible while respecting this target delay in order to minimize the required bandwidth (i.e., number of channels).", "Let the maximum numbers of nodes for one group under TDMA and CSMA-CS schemes while still respecting the target delay be $n_S^{\\sf TDMA}$ and $n_S^{\\sf CSMA-CS}$ , respectively.", "Note that the TDMA scheme uses all $n_T$ RIs for data transmission while the CSMA-CS scheme only chooses $m_T$ RIs for each $n_T$ RIs to transmit the data.", "Thus, the CSMA-CS scheme allows $n_T/m_T$ groups to share one channel for each interval of $n_T$ RIs.", "As a result, the number of channels needed for $N$ nodes is $N/n_S^{\\sf TDMA}$ for the TDMA scheme and $N/n_S^{\\sf CSMA-CS} \\times m_T/n_T$ for the CSMA-CS scheme." ], [ "Energy Consumption", "We now calculate the average energy consumption for data reporting of one data block.", "In each RI, there are $K_{\\tau } $ SFs where the number of contending nodes decreases over the SFs.", "Let $E_i$ denote the energy consumption per node in SF $i $ with $h_i $ contending nodes.", "The derivation of $E_i$ is given in Appendix  .", "Then, the average energy consumption in one RI can be expressed as follows: $E = \\sum _{h=1}^{n_S} \\Pr \\left\\lbrace \\widehat{m}_S = h\\right\\rbrace \\sum _{K_{\\sf succ} = m_{S}}^h \\sum _{l =1 }^{\\left|\\Xi \\right|} \\prod _{i=1}^{K_{\\tau }} \\sum _{K_{S,i} = K_{{\\sf succ},i}}^{K_{S,i,{\\sf max}}} \\times \\hspace{15.6491pt} \\\\\\Pr \\!\\left\\lbrace K_i = K_{S,i} \\left|h_i \\right.\\right\\rbrace \\!", "\\Pr \\!\\left\\lbrace {\\bar{K}}_i = K_{\\text{succ},i} \\left|K_{S,i},h_i \\right.\\right\\rbrace \\sum _{i=1}^{K_{\\tau }} h_i E_i.", "$ Here, similar to derivation of $\\Pr \\left\\lbrace K_{\\text{succ}} \\ge m_{S}\\right\\rbrace $ in Section REF , we generate all combinations of $\\left\\lbrace h_i\\right\\rbrace $ , which represents the number of contending nodes in SF $i$ .", "For one such combination, we derive the total average energy in $K_{\\tau }$ SFs.", "Note that $\\sum _{i=1}^{K_{\\tau }} h_i E_i$ is the total average energy corresponding to $\\left\\lbrace h_i\\right\\rbrace $ .", "As a result, the average consumed energy for reporting one data field (with size $n_S \\times n_T$ ) is $\\mathcal {E} = m_T \\times E$ ." ], [ "Data Modeling and Simulation Setting", "In order to evaluate the performance of our proposed data compression and MAC protocol, we synthetically generate the data for power loads and distributed generation powers by using available methods [29]–[31], [36]–[41] because real-world data is not available.", "There are various probabilistic and stochastic methods to model power data in the literature.", "While independent and identically distributed (i.i.d.)", "normal and Weibull [41] distributions are commonly used to generate these types of data, the considered power data generated once for every 5-minute RI in this work would be highly dependent.", "Therefore, autoregressive models [41], which belong to the correlated time-series category, are selected to generate the required simulation data.", "To model the load power, we employ the autoregressive moving average (ARMA) [41] as a time series model which consist of two components, namely, deterministic and stochastic parts [36]–[38].", "Hence, the load power can be expressed as $X_t = X_t^d + X_t^s$ where $t$ is the time index, $X$ commonly represents active load power ($P_l$ ) and reactive load power ($Q_l$ ), $X_t^d$ is the deterministic component of $X$ and $X_t^s$ is the stochastic part of $X$ .", "The deterministic component, which captures the trend of data, is represented by the trigonometric functions [36]–[38] as $X^d_t = \\chi _0 + \\sum _{i=1}^{m_h} \\left(\\chi _{\\text{re},i} \\text{sin}\\left(\\frac{2\\pi k_i t}{288}\\right)+ \\chi _{\\text{im},i}\\text{cos}\\left(\\frac{2\\pi k_i t}{288}\\right)\\right)$ where $m_h$ is the number of harmonics (i.e., the number of trigonometric functions), $\\chi _{\\text{re},i}$ and $\\chi _{\\text{im},i}$ are the coefficients of the harmonics, $\\chi _0$ is the constant, and $k_i \\le 288/2$ for $\\forall i \\in \\left\\lbrace 1,\\ldots , m_h\\right\\rbrace $ .", "This form consists of a constant $\\chi _0$ (i.e., the first quantity in (REF )) and $m_h$ trigonometric functions (i.e., the second quantity in (REF )).", "This general model indeed offers flexibility where we can define suitable formulas to present the effects of seasonality and special days which are extensively studied in [36], [37].", "Now, the stochastic component is modeled by the first-order autoregressive process AR(1) which is the simple form of ARMA($n,m$ ) (i.e., $n = 1, m = 0$ ) [41] as follows: $X_{t+1}^s = \\varphi _t X_t^s + U_t$ where $\\varphi _t$ is the AR(1) coefficient and $U_t$ is the white noise process with zero mean and variance of $(1-\\varphi _t)$ .", "Here, we use AR(1) as an appropriate and simple data model; however other more complicated methods can be used as well [36], [37], [41].", "We now present the method to determine the parameters ($\\chi _0$ , $\\chi _{\\text{re},i}$ , $\\chi _{\\text{im},i}$ , $m_h$ , $\\varphi _{t}$ ) in (REF ) and (REF ).", "These parameters are coupled as indicated in (REF ) and (REF ).", "However we can separately estimate the parameters for these deterministic and stochastic components without significantly increasing estimation errors compared to those due to joint estimation [36].", "To estimate these parameters, we use online data set [42] which represents the active and reactive load powers from 2006 to 2010.", "Specifically, we use the ordinary least squares (OLS) [41] to estimate $\\chi _0$ , $\\chi _{\\text{re},i}$ , and $\\chi _{\\text{im},i}$ , $i = 1,\\ldots , m_h$ [36], [38].", "Moreover, we employ the Bayesian information criterion (BIC) [41] to determine $m_h$ significant harmonics which is usually less than 5 for the selected data in [42].", "Then we also use OLS algorithm to estimate the AR(1) coefficient, i.e., $\\varphi _t$ for every 5-minute interval over one day [36], [38].", "We should note that these parameters are updated and stored in every 5-minute interval over one day.", "Then we can generate the required power data in such the way that the output of the current interval is the input of the next interval.", "We can similarly generate the distributed generation power data, which exhibits the temporal and spatial correlation.", "In particular, distributed generation power data model also comprises deterministic and stochastic parts as described in (REF ), (REF ) and (REF ).", "In addition, all parameters in these models are estimated by using online data of wind speeds [43] which represent the 5-minute wind speeds in 2006–2010.", "These data can be transformed to output power data as follows [31], [39]: $S_g = \\left\\lbrace {\\begin{array}{*{20}{c}}0 & {v \\le v_{ci}\\,or\\,v > v_{co}} \\\\{A + Bv + Cv^2} & {v_{ci} < v \\le v_r} \\\\{P_r} & {v_r < v \\le v_{co}}\\end{array}} \\right.$ where $v_{ci}$ , $v_{co}$ , $v_r$ and $P_r$ are the cut-in, cut-out, nominal wind speeds, and the rated power output, respectively.", "These parameters and $(A, B, C)$ are determined as in [40].", "Finally, the spatial correlation of distributed generation powers can be modeled as in [29]–[31] where the correlation coefficient between 2 distributed generators $i$ and $j$ is $\\rho _{i,j} = \\exp \\left(-d_{i,j}/d\\right)$ where $d = 20 km$ is a positive constant and $d_{i,j}$ is the distance between generators $i$ and $j$ , which is randomly chosen in $\\left(0,1 km\\right]$ .", "We assume that wind/solar generators are installed at a half of total nodes for all following experiments.", "We ignore the node index $i$ in these notations for brevity.", "The RI is set as $\\tau _T = 5$ minutes.", "The target probability in the constraint (REF ) is chosen as $P_{\\text{suff}} = 0.9$ .", "The MAC parameters are chosen as $L_s = T_p + t_{ACK} + L_{ACK}$ , $T_p = 5+L_{MAC}$ slots ($L_{MAC} = 2$ is the MAC header), $L_{ACK} = 2$ slots, $t_{ACK} = 1$ slot, $t_{ACK,ti} = 4$ slots where $T_p $ is the length of packet, $t_{ACK} $ is the idle time before the ACK, $L_{ACK} $ is the length of ACK, $t_{ACK,ti}$ is the timeout of the ACK, $BO_{\\sf max} = 8$ , $K_{\\tau ,{\\sf max}} =10$ .", "For all the results presented in this section, we choose $n_T=256$ .", "In Fig.", "REF , we show the variations of sufficient probability, namely $\\Pr \\!\\left\\lbrace K_{\\text{succ}} \\!\\ge \\!", "m_{S}\\right\\rbrace $ , versus $p_s$ for different values of $BO_i \\!=\\!", "BO $ (i.e., all SFs employ the same $BO$ ) where $m_{S} \\!=\\!", "16$ , $K_{\\tau } \\!=\\!", "3$ , and $n_S \\!=\\!", "64$ .", "It can be observed that there exists an optimal $p_s$ that maximizes $\\Pr \\!\\left\\lbrace K_{\\text{succ}} \\!\\ge \\!", "m_{S}\\right\\rbrace $ for each value of $BO$ .", "This optimal value is in the range $\\left[0.3, 0.5\\right]$ .", "Furthermore, $BO$ must be sufficiently large ($BO \\!\\ge \\!", "4$ ) to meet the required target value $P_{\\text{suff}}$ =0.9.", "In addition, larger values of $BO$ lead to longer SF length, which implies that more data packets can be transmitted.", "In Fig.", "REF , we show the probability $\\Pr \\left\\lbrace K_{\\text{succ}} \\ge m_{S}\\right\\rbrace $ versus $p_s $ for different values of $K_{\\tau } $ where we set $n_S = 64 $ and $BO = 3$ .", "This figure confirms that the maximum $\\Pr \\left\\lbrace K_{\\text{succ}} \\ge m_{S}\\right\\rbrace $ becomes larger with increasing $K_{\\tau } $ where we can meet the target probability $P_{\\text{suff}}$ =0.9 if $K_{\\tau } \\ge 6$ .", "Now we fix $BO_i = BO = 4 $ , $K_{\\tau } = 3 $ , then we study $\\Pr \\left\\lbrace K_{\\text{succ}} \\ge m_{S}\\right\\rbrace $ versus $p_s $ for different values of $n_S $ .", "Note that $m_{S}$ can be calculated for each value of $n_S $ as presented in Section REF .", "Specifically, $m_{S}$ is equal to 13, 16, 19, and 22 for $n_S = 48, 64, 80 $ , and 96, respectively.", "Fig.", "REF demonstrates that smaller number of nodes $n_S$ (e.g., $n_S = 48, 64$ ) can achieve the target value of $P_{\\text{suff}} $ at the optimal $p_s$ .", "However, as $n_S$ increases to 80 and 96, there is no $p_s $ that meets the target value of $P_{\\text{suff}}$ .", "This is because the collision probability is lower for the smaller number of nodes, which results in larger values of $\\Pr \\left\\lbrace K_{\\text{succ}} \\ge m_{S}\\right\\rbrace $ ." ], [ "Reporting Delay", "We now show the optimal reporting delay $\\mathcal {D}$ in one RI versus the number of nodes $n_S$ for different schemes, namely TDMA, CSMA, TDMA-CS, and our proposed CSMA-CS schemes in Fig.", "REF .", "Here, X-CS refers to scheme X that integrates the CS-based data compression and X refers to scheme X without data compression.", "Moreover, TDMA is the centralized non-contention time-division-multiple-access MAC, which always achieves better performance that the CSMA scheme.", "For both CSMA and CSMA-CS schemes, their MAC parameters are optimized by using Alg.", "REF .", "It can be seen that our proposed CSMA-CS protocol achieves much smaller delay than the CSMA scheme, which confirms the great benefits of employing the CS.", "In addition, TDMA-CS outperforms our CSMA-CS protocol since TDMA is a centralized MAC while CSMA is a randomized distributed MAC.", "Finally, this figure shows that our CSMA-CS protocol achieves better delay performance than the TDMA scheme.", "We illustrate the variations of the optimal reporting delay with $\\mathcal {P}_{err} $ for different schemes where $\\mathcal {P}_{err} \\!=\\!", "1\\!-\\!\\Pr \\!\\left\\lbrace K_{\\text{succ}} \\!\\ge \\!", "m_{\\text{S}}\\right\\rbrace $ and $n_S \\!=\\!", "64$ in Fig.", "REF .", "This figure shows that as $\\mathcal {P}_{err} $ increases, the reporting delay decreases.", "This indeed presents the tradeoff between the reporting delay and $\\mathcal {P}_{err}$ .", "Note that the delay of the TDMA-CS scheme is the lower bound for all other schemes.", "Interestingly, as $\\mathcal {P}_{err}$ increases the delay gap between the proposed CSMA-CS and the TDMA-CS schemes becomes smaller.", "We compare the delay performance with partial and full optimization of the MAC protocol for our proposed scheme.", "Specifically, we consider two following cases in which we set default value for one of three optimization parameters i) $BO_i = BO = 3$ for all SFs; ii) $p_s = 0.45 $ .", "For these two cases, we optimize the remaining parameters to achieve minimum reporting delay in each RI.", "Fig.", "REF shows that our proposed CSMA-CS protocol with full optimization outperforms the others with the partial optimization.", "Furthermore, the performance for case i) with fixed $BO $ is pretty poor since we only choose the optimal configuration for $p_s $ and $K_{\\tau } $ to minimize the delay performance in this case.", "However, the delay performance degrades more moderately as we fix the access parameter at $p_s = 0.45$ .", "Figure: Reporting delay 𝒟\\mathcal {D} vs. n S n_S.Figure: Reporting delay 𝒟\\mathcal {D} vs. 𝒫 err \\mathcal {P}_{err}.Figure: Optimal 𝒟\\mathcal {D} vs. n S n_S with full and partial optimizations.Figure: Energy consumption EE vs. n S n_S for one RI.Figure: Energy consumption ℰ\\mathcal {E} vs. n S n_S for one data field." ], [ "Energy Consumption", "We present the energy consumption for different schemes in Figs.", "REF , REF under the following parameter setting: $E_{\\sf idle} = 0.228 \\mu J/{\\sf slot}$ , $E_{\\sf tx} = 10.022 \\mu J/{\\sf slot}$ , $E_{\\sf rx} = 11.290 \\mu J/{\\sf slot}$ , and $E_{\\sf sens} =E_{\\sf rx}$ [44].", "Fig.", "REF demonstrates the energy consumption $E$ in one RI for different values of $n_S$ .", "It can be seen that the proposed CSMA-CS scheme results in significant energy saving compared to the TDMA scheme.", "For completeness, we also show the energy consumption for one data field (i.e., for $n_T$ RIs and $n_S$ nodes) in Fig.", "REF .", "Recall that in the proposed framework, only data blocks corresponding to $(m_S, m_T)$ are reported where $(m_S, m_T)$ can be determined for each $(n_S, n_T)$ as described in Section  REF .", "Specifically, we have $(m_S, m_T) = \\left\\lbrace (10, 194), (13, 190), (16, 180), (19, 166), (22, 151)\\right\\rbrace $ for $n_T$ =256 and $n_S = \\left\\lbrace 32, 48, 64, 80, 96\\right\\rbrace $ , respectively.", "Again, this figure confirms that the proposed CSMA-CS scheme outperforms the TDMA scheme.", "Moreover, the consumed energy of the proposed CSMA-CS scheme is slightly higher than that due to the TDMA-CS scheme.", "Figure: Number of channels vs. NN.Figure: Number of channels vs. target delay.Figure: Number of channels vs. n S n_S." ], [ "Bandwidth Usage", "We now examine the bandwidth usage due to different schemes.", "To ease the exposition, we do not show the results of the CSMA and TDMA-CS schemes.", "The group sizes for TDMA and CSMA-CS protocols, namely $n_S^{\\sf TDMA}$ , $n_S^{\\sf CSMA-CS}$ , can be determined for a given target delay $\\mathcal {D}_{\\sf max}$ by using the results shown in Fig.", "REF .", "Specifically, we have $n_S^{\\sf TDMA}\\!\\!", "= \\left\\lbrace 40, 75\\right\\rbrace $ and $n_S^{\\sf CSMA-CS}\\!\\!", "= \\left\\lbrace 55, 128\\right\\rbrace $ for TDMA and CSMA-CS schemes for the target delay values of $\\mathcal {D}_{\\sf max} = \\left\\lbrace 400, 750\\right\\rbrace $ slots, respectively.", "Then we calculate the required bandwidth (i.e., number of channels) for TDMA and CSMA-CS schemes with a given number of nodes $N$ .", "In Fig.", "REF , we show the required number of channels versus $N$ .", "It can be observed that our proposed CSMA-CS scheme requires less than half of the bandwidth demanded by the TDMA scheme.", "Also when the network requires smaller target delay, we need more channels for both schemes as expected.", "Finally, Fig.", "REF illustrates the variations in the number of channels versus the target delay for $N = \\left\\lbrace 2048, 8192\\right\\rbrace $ .", "Again, our proposed CSMA-CS scheme provides excellent bandwidth saving compared to the TDMA scheme.", "In Fig.", "REF , we show the required number of channels versus $n_S$ for $N = \\left\\lbrace 2048, 4096\\right\\rbrace $ .", "These results can be obtained as follows.", "Suppose that the target delay in one RI is 650 slots then the group sizes for TDMA and CSMA-CS protocols can be determined from the results in Fig.", "REF as $\\left\\lbrace n_S^{\\sf TDMA} , n_S^{\\sf CSMA-CS}\\right\\rbrace = \\left\\lbrace 65, 96\\right\\rbrace $ .", "Then using the results in Fig.", "REF , the numbers of required channels for TDMA and CSMA-CS protocols are $\\left\\lbrace 64, 25\\right\\rbrace $ for $N = 4096$ and $\\left\\lbrace 32, 13\\right\\rbrace $ for $N = 2048 $ .", "We can observe that the number of required channels for our proposed scheme is always less than that due to the TDMA protocol, which again demonstrates the efficacy of our proposed design." ], [ "Conclusion", "We have proposed the joint design of data compression using the CS technique and CSMA MAC protocol for smartgrids with renewable energy.", "We have shown how to choose the compression levels in the space and time dimensions to maintain desirable reconstruction performance.", "Then, we have presented the design and optimization of the MAC protocol to minimize the reporting delay.", "Furthermore, we have derived the bandwidth usage and energy consumption for our proposed scheme.", "Numerical results have confirmed the significant performance gains of the proposed design compared to other non-compressed solutions." ], [ "Markov Chain Model for CSMA Protocol", "We study the Markov chain model for the slotted CSMA/CA protocol, which is similar to the one in [23].", "We consider the case with $h_i$ contending nodes and we use the same notations as in [44].", "However, our analysis has the additional “deference state”, which was not considered in [44].", "We consider the 2D Markov chain (MC) for slotted CSMA/CA MAC protocol $\\left(s(t), c(t) \\right)$ where $s(t)=-1$ represents the transmission state, and $s(t)=[0,NB]$ captures the backoff stages (BK); $c(t)=-1,-2$ denote the first and second CCAs; $c(t)=[0,L-1]$ represents transmission slots and $c(t)=[0,W_i-1]$ describes the states of backoff counter.", "The detail of superframe structure in one RI is demonstrated in Fig.", "REF .", "Here, $L$ is used to denote the general transmission time where $L=L_s = T_p + t_{ACK} + L_{ACK}$ for a successful transmission and $L=L_c = T_p + t_{ACK,ti}$ for a collided transmission, $T_p$ is packet transmission time, $L_{ACK}$ is the ACK time, and $t_{ACK,ti}$ is the ACK timeout.", "Fig.", "REF shows the transition diagram of this MC.", "Let $b_{i,k} = {\\sf lim}_{t \\rightarrow \\infty } \\Pr \\left(s(t) = i, c(t) = k\\right)$ denote the stationary probability of the Markov chain.", "In this paper, we asume that $NB = d$ as in [23] where $d = macMaxBE - priority $ , $priority = 2$ is the priority of node and also the number of CCAs.", "In addition, $macMaxBE $ is the maximum backoff exponent and $NB = macMaxattempt-1 $ is the maximum number of backoffs.", "We define the following parameters: $\\lambda = \\alpha + \\beta - \\alpha \\beta $ , $\\omega = \\lambda \\left(1-\\mathcal {P}_d\\right) $ , where $\\alpha $ and $\\beta $ are the probabilities that a node fails to identify an idle channel during $CCA_1 $ and $CCA_2 $ , respectively; $\\mathcal {P}_d = L_s/SF $ is the probability of deference.", "Using the analysis similar to that in [44], we can arrive at the following the relationship for the steady-state probabilities: $b_{i,0} = \\omega ^i b_{0,0} $ , $b_{i,-1} = b_{i,0} (1-\\mathcal {P}_d) $ , $b_{i,-2} = b_{i,0} (1-\\mathcal {P}_d) (1-\\alpha ) $ (for $i \\in \\left[0, NB\\right]$ ), and $b_{-1,k} = (1-\\mathcal {P}_d) (1-\\alpha ) (1-\\beta ) \\sum _{i=0}^{NB} b_{i,0} $ , $b_{i,k} = \\frac{W_i-k}{W_i} b_{i,0} $ (for $i \\in \\left[0, NB\\right], k \\in \\left[-2, {\\sf max} \\left(W_i-1, L_s-1\\right)\\right] $ ).", "Since we have $\\sum _{i,k} b_{i,k} = 1 $ , substitute the above results for all $b_{i,k}$ and perform some manipulations, we can obtain $1=\\frac{b_{0,0}}{2} \\left\\lbrace W_0 \\frac{1-\\left(2\\omega \\right)^{NB+1}}{1-2\\omega }+ \\frac{1-\\omega ^{NB+1}}{1-\\omega } \\times \\right.", "\\nonumber \\\\\\left.", "\\left[3+2\\left(1-\\mathcal {P}_d\\right)\\left(1+\\left(1-\\alpha \\right)\\left(1+\\left(1-\\beta \\right)L_s\\right)\\right)\\right] \\right\\rbrace .", "$ From these results, we can find the relationship among $b_{0,0} $ , $\\alpha $ , $\\beta $ , $\\phi $ as follows [44]: $\\phi = \\sum _{i=0}^{NB} b_{i,0} = \\frac{1-\\omega ^{NB+1}}{1-\\omega } b_{0,0} \\\\\\phi = 1 - \\left(1-\\frac{\\alpha }{L^*\\left(1-\\omega \\right)}\\right)^{\\frac{1}{h-1}} \\\\\\phi = 1 - \\left(1 - \\frac{\\beta _{ACK}}{\\left(1-\\beta _{ACK}\\right)\\left(2-\\mathcal {P}_{\\text{ncol}}\\right)}\\right)^{1/h} $ where $\\phi $ is the probability that a node is at the $CCA_1 $ state after backoff, $L^* = T_p + L_{ACK} \\left(1 - \\mathcal {P}_{\\text{ncol}}\\right)$ , $\\mathcal {P}_{\\text{ncol}} = 1 - \\frac{h\\phi \\left(1-\\phi \\right)^{h-1}}{1-\\left(1-\\phi \\right)^h} $ , $\\beta _{ACK} = \\frac{2-\\mathcal {P}_{\\text{ncol}}}{2-\\mathcal {P}_{\\text{\\text{ncol}}}+\\frac{1}{1-\\left(1-\\phi \\right)^h}} $ .", "From (), (REF ), () and (), we can determine $b_{0,0} $ , $\\alpha $ , $\\beta $ , and $\\phi $ by using the standard numerical method." ], [ "Calculation of ${\\bar{T}}_i $ and {{formula:f6e42ebe-10c1-4c03-bad8-76fc75dc6880}}", "In this appendix, we determine $\\bar{T}_i $ and $\\sigma ^2_i $ for a particular SF $i$ .", "For simplicity, we again omit the index $i$ and $h_i$ in all related parameters when this does not create confusion.", "First, we can express the probability generating function (PGF) of the generic frame, $T(z) $ which includes success, collision, CCA failure and deference, as $T\\left(z\\right) = \\mathcal {P}_{\\sf succ} T_S\\left(z\\right) + \\mathcal {P}_{\\text{coll}} T_C\\left(z\\right) + \\mathcal {P}_{\\text{ccas}} T_F\\left(z\\right) + \\mathcal {P}_d T_D\\left(z\\right)$ Here we denote $\\mathcal {P}_{\\text{ccas}} $ , $\\mathcal {P}_{\\text{coll}} $ and $\\mathcal {P}_{\\text{succ}} $ as the probabilities of CCA failure, collision and success, respectively.", "These probabilities can be calculated as $\\mathcal {P}_{\\text{ccas}} = \\left(1-\\mathcal {P}_d\\right) \\lambda ^{NB+1} $ , $\\mathcal {P}_{\\text{coll}} = p_c \\left(1-\\mathcal {P}_d\\right) \\left(1-\\lambda ^{NB+1}\\right) $ , and $\\mathcal {P}_{\\sf succ} = 1-\\mathcal {P}_{\\text{coll}}- \\mathcal {P}_{\\text{ccas}}- \\mathcal {P}_d $ , where $p_c = 1-\\left(1-\\phi \\right)^{h-1} $ .", "Moreover, we also denote $T_S\\left(z\\right) $ , $T_C\\left(z\\right) $ , $T_F\\left(z\\right)$ and $T_D\\left(z\\right) $ as the PGFs of durations of success, collision, CCA failure and deference, respectively.", "These quantities can be calculated as in [45].", "Finally, we can determine $\\bar{T} $ and $\\sigma ^2$ from the first and second derivation of $T\\left(z\\right) $ at $z = 1 $ , i.e., $\\bar{T} = \\frac{dT}{dz}\\left(1\\right) ;\\sigma ^2 = \\frac{d^2T}{dz^2}\\left(1\\right) + \\bar{T} - \\left(\\bar{T}\\right)^2.", "$ These parameters $\\bar{T} $ and $\\sigma ^2$ will be utilized in (REF )." ], [ "Calculation of ${E}_i $", "The energy consumption per node in SF $i$ with $h_i $ contending nodes can be expressed as $E_i = E_{b,i} + E_{{\\sf ccas},i} + E_{{\\sf suco},i} + E_{d,i}$ where $E_{b,i}$ , $E_{{\\sf ccas},i}$ , $E_{{\\sf suco},i}$ , and $E_{d,i} $ are the average energy consumption due to backoffs, CCAs, transmissions (successful/collided transmissions), and deference, respectively.", "Let us denote $E_{\\sf idle} $ , $E_{\\sf sens} $ , $E_{\\sf tx} $ and $E_{\\sf rx} $ as the energy consumption corresponding to idle, CCA sensing, transmitting and receiving slots, respectively; then, these quantities are determined as follows.", "For brevity, we again omit the index $i$ and $h_i$ in all related parameters if this does not create confusion.", "First, the energy consumption during backoff duration is $E_{b,i} = E_{\\sf idle} \\sum _{j=0}^{NB} \\sum _{k=0}^{W_j -1} b_{j,k} $ .", "After some simple manipulations, we can arrive at [35] $E_{b,i} = E_{\\sf idle} /2 \\left(W_0 b_{0,0} \\frac{1-(2\\omega )^{NB+1}}{1-2\\omega }+3 \\phi \\right).$ Also, $E_{{\\sf ccas},i} $ can be determined as $E_{ccas,i} \\!\\!=\\!\\!", "E_{\\sf sens}\\!\\sum _{j=0}^{NB} (b_{j,-1} \\!+\\!", "b_{j,-2}) \\!\\!=\\!\\!", "E_{\\sf sens} (1-\\mathcal {P}_d)(2-\\alpha ) \\phi .$ Moreover, $E_{{\\sf suco},i} $ is given as [35] $E_{{\\sf suco},i} = E_{\\sf tx} \\sum _{k=0}^{T_p-1} b_{-1,k} + E_{\\sf rx} (1-p_c) \\sum _{k=T_p+t_{ACK}}^{L_s-1} b_{-1,k} + \\nonumber \\\\E_{\\sf idle} \\left((1-p_c)\\!\\!\\!\\sum _{k=T_p}^{T_p+t_{ACK}-1} \\!\\!", "b_{-1,k}+p_c \\!\\!", "\\sum _{k=T_p}^{T_p+t_{ACK,ti}-1} \\!\\!", "b_{-1,k}\\right) \\\\= (1-\\lambda ) (1-\\mathcal {P}_d) \\phi \\left(E_{\\sf tx} T_p + E_{\\sf rx} L_{ACK} (1-p_c) + \\right.", "\\nonumber \\\\\\left.", "E_{\\sf idle} (t_{ACK} (1-p_c) + t_{ACK,ti} p_c)\\right).$ Finally, $E_{d,i} $ can be expressed as $E_{d,i} = E_{\\sf idle} \\mathcal {P}_d L_s$ .", "[Figure: NO_CAPTION [Figure: NO_CAPTION" ] ]
1606.04995
[ [ "Matters of Gravity, The Newsletter of the Division of Gravitational\n Physics of the American Physical Society, Volume 47, June 2016" ], [ "Abstract DGRAV News: DGRAV we hear that.... Research Briefs: GW150914 Obituaries: Remembering Felix Pirani Remembering David Finkelstein Steve, the physicist Remembering Sergio Dain Editorial: Gravitational physics in the modern university" ], [ "Editorial", "The next newsletter is due December 2016.", "This and all subsequent issues will be available on the web at https://files.oakland.edu/users/garfinkl/web/mog/ All issues before number 28 are available at http://www.phys.lsu.edu/mog Any ideas for topics that should be covered by the newsletter should be emailed to me, or Greg Comer, or the relevant correspondent.", "Any comments/questions/complaints about the newsletter should be emailed to me.", "A hardcopy of the newsletter is distributed free of charge to the members of the APS Topical Group on Gravitation upon request (the default distribution form is via the web) to the secretary of the Topical Group.", "It is considered a lack of etiquette to ask me to mail you hard copies of the newsletter unless you have exhausted all your resources to get your copy otherwise.", "David Garfinkle Daniel Holz: Relativistic Astrophysics, Bei-Lok Hu: Quantum Cosmology and Related Topics Veronika Hubeny: String Theory Pedro Marronetti: News from NSF Luis Lehner: Numerical Relativity Jim Isenberg: Mathematical Relativity Katherine Freese: Cosmology Lee Smolin: Quantum Gravity Cliff Will: Confrontation of Theory with Experiment Peter Bender: Space Experiments Jens Gundlach: Laboratory Experiments Warren Johnson: Resonant Mass Gravitational Wave Detectors David Shoemaker: LIGO Project Stan Whitcomb: Gravitational Wave detection Peter Saulson and Jorge Pullin: former editors, correspondents at large.", "Chair: Laura Cadonati; Chair-Elect: Peter Shawhan; Vice-Chair: Emanuele Berti.", "Secretary-Treasurer: Thomas Baumgarte; Past Chair: Deirdre Shoemaker; Members-at-large: Steven Drasco, Tiffany Summerscales, Duncan Brown, Michele Vallisneri, Kelly Holley-Bockelmann, Leo Stein.", "Student Members: Megan Jones, Jessica McIver.", "toc tocDGRAV News: tocsubsubsection DGRAV, by Deirdre Shoemaker Deirdre Shoemaker, Georgia Institute of Technology deirdre.shoemaker-at-physics.gatech.edu We did it!", "After years of trying, the APS Topical group on Gravitation has become the APS Division of Gravitational Physics (DGRAV), as announced at the 2016 GGR Business meeting in Salt Lake City.", "What did it take to make it to division?", "What does it mean to you?", "As a topical group, we needed to achieve 3% of APS’ total membership in two consecutive years.", "We achieved our first 3.07% in January 2015 and then 3.08% in January 2016.", "Once we reached this milestone, we petitioned the APS Council to become a division.", "We were the first group in 17 years to petition for division status.", "What's next?", "We need you, the DGRAV membership, to approve our new Division Bylaws.", "They have mainly changed to reflect the new name of our unit and to add an elected official, the DGRAV Councilor.", "As quoted from the bylaws, “The Division Councilor shall serve as liaison between the Council of the Society and the Executive Committee of the Division.", "Following each Council meeting, the Division Councilor shall report to the Chair and the Secretary-Treasurer regarding Council actions that affect the status and operations of the Division.", "Reports shall be made to the entire Executive Committee during their regularly scheduled meetings.” The term of the Councilor is four years, beginning on January of the year following the election.", "The Councilor may not serve more than two consecutive terms.", "Our new bylaws have one additional change to include the position of webmaster.", "Many of you may not be aware that this Newsletter is our official newsletter for the APS.", "Most of the units in APS have APS handle their newsletters, but Matters of Gravity actually predates the GGR!", "In the same spirit of the position of Editor of the Newsletter standing outside of the executive committee of DGRAV, we have added a webmaster.", "The webmaster has the duty of managing our website, http://dgrav.org, and our social media presence.", "Beverly Berger formed and chaired GGR in 1995.", "Let’s give her and all the others (yes including any of your family and friends that joined GGR!)", "a virtual round of applause for making this possible.", "toc tocsubsubsection we hear that ..., by David Garfinkle David Garfinkle, Oakland University garfinkl-at-oakland.edu Ronald Drever, Kip Thorne, and Rainer Weiss were awarded the Shaw Prize in Astronomy and the Kavli Prize in Astrophysics.", "Emanuele Berti was elected Vice Chair of DGRAV; Kelly Holley-Bockelmann and Leo Stein were elected members at large of the Executive Committee of DGRAV.", "Megan Jones was elected Student Representative of DGRAV.", "Gregory Adkins has been awarded the APS Prize for a Faculty Member for Research in an Undergraduate Institution.", "Raymond Beausoleil has been awarded the APS Distinguished Lectureship Award on the Applications of Physics.", "Douglas Finkbeiner, Shane Larson, Pierre Michel, Dwight Neuenschwander, Scott Ransom, Stephan Schlamminger, and Rodger Thompson have been elected APS Fellows.", "Hearty Congratulations!", "toc tocResearch Briefs: tocsubsubsection GW150914 , by Gabriela González and David Reitze Gabriela González, Louisiana State University gonzalez-at-lsu.edu David Reitze, LIGO Laboratory and Caltech reitze-at-ligo.caltech.edu Those of us who work on LIGO will forever remember exactly where we were on September 14, 2015 when we first learned of a nearly simultaneous `trigger' recorded on the LIGO Hanford and Livingston detectors.", "That trigger would eventually become GW150914, the first gravitational wave ever recorded.", "Perhaps equally momentous, it would also reveal the first ever observation of a binary black hole system in the universe colliding to merge and form a new black hole.", "Almost exactly one hundred years after gravitational waves were first theorized by Einstein, the discovery by the LIGO Scientific Collaboration and Virgo Collaboration marked the culmination of a scientific quest that began over 50 years ago.", "Over that time period, this pursuit has brought together well over a thousand researchers worldwide to develop the new science of gravitational wave physics and astronomy.", "And like most endeavors of this historical magnitude, the path had a few twists and turns along the way.", "Prior to 1960, no one seriously contemplated developing detectors for gravitational waves because, quite simply, no one seriously believed that gravitational waves could ever be detected.", "That changed when Joseph Weber began using large cylindrical aluminum bars to search for gravitational waves.", "While his claims of gravitational wave detections in the 1960s and 70s ultimately proved to be incorrect (resulting in some acrimonious scientific debates), Weber's experimental efforts led Michael Gertsenshtein and Vladislav Pustovoit and, independently Weber himself and also Rainer Weiss to propose using laser interferometers as detectors.", "Gravitational-wave interferometer research programs sprung up in the 1970s at MIT (Weiss), the University of Glasgow (Ron Drever and Jim Hough), and the Max Planck Institute in Garching, Germany (Hans Billing).", "Independently, Kip Thorne (Caltech) began a research group at Caltech focusing on gravitational wave theory, and began collaborating with Vladimir Braginsky (Moscow State University) on some of the more intriguing quantum aspects of suspended-mirror gravitational-wave interferometers.", "This ultimately blossomed into an experimental effort at Caltech, led by Drever (who had moved from Glasgow) and Stan Whitcomb.", "Each of these groups began tackling the challenge of building and understanding the complex subtleties of operating ultrasensitive suspended mirror interferometers.", "The period between 1984 and 1992 saw both innovative advances in interferometer designs and the formulation of a joint Caltech-MIT collaboration to design and build two kilometer-scale gravitational-wave observatories.", "However, funding for LIGO didn't come about quickly or easily.", "When LIGO was first proposed as a large-scale project, it was met with great resistance.", "It was deemed too risky and too expensive; the chance of failure was too high, and the scientific payoff too low relative to more established types of astronomy to justify the expenditure.", "Nonetheless, the US National Science Foundation recognized both the huge scientific potential in gravitational wave physics and astronomy and the cutting edge technology that could result from designing and building a gravitational wave detector.", "The NSF took a huge risk in funding LIGO, but it was a measured risk.", "The technological leap from the prototypes to the advanced detectors was deemed too great to be carried out in a single step.", "A two stage approach was adopted in which an initial set of interferometers (Initial LIGO) would be built with a sensitivity where gravitational waves might be detected (but more likely not), followed by the construction of a second set of interferometers (Advanced LIGO) that would have a high probability of detection.", "In 1991 the US Congress appropriated LIGO's first year of funding.", "In 1992 Hanford, Washington and Livingston, Louisiana were chosen as the sites for LIGO's interferometers, and a cooperative agreement for the management of LIGO was signed between NSF and Caltech.", "In 1994 Barry Barish (Caltech) was appointed LIGO Director and oversaw LIGO's construction phase as well as the installation and commissioning of LIGO's initial interferometers.", "In 1997, the LIGO Scientific Collaboration was created to organize and coordinate LIGO's technical and scientific research and data analysis, and for expanding LIGO to include scientists from institutions beyond Caltech and MIT.", "Initial LIGO was operated from 2002 through 2010, producing over 100 papers and quite a few interesting upper limits on gravitational wave emissions from compact binary systems, pulsars, and even the primordial universe.", "The initial LIGO interferometers were decommissioned in 2010 to make way for the Advanced LIGO interferometers.", "Designed to be ten times more sensitive to gravitational wave strains, Advanced LIGO is completely new - every component and subsystem has been re-designed and rebuilt to achieve the sensitivity goal.", "Following an installation period lasting four years, the Advanced LIGO interferometers were completed in 2014 and commissioned until September 2015 when the inaugural observing run `O1' began.", "The Advanced LIGO interferometers are the most sensitive scientific instruments ever conceived and built.", "Consisting of two identical 4 km arm length interferometers in Hanford, WA and Livingston, LA, they are a technological tour-de-force, bringing together a wide array of technologies that have redefined the state-of-the-art throughout almost every facet of their design - the world’s most stable high power lasers, the most precisely figured and coated 'test mass' mirrors, the most sophisticated low frequency seismic isolation and mirror suspension systems, one of the world’s largest high vacuum systems, and gluing it all together, hundreds of feedback control loops that are capable of sensing and maintaining the 4 km length of the interferometers to almost ${10}^{-19}$ m in their most sensitive bands.", "Owing to its cutting edge nature, the successful construction and commissioning of Advanced LIGO required solving a very large number of problems - related to the handling of high optical power, angstrom-level polishing of massive optical components, development of strong but exquisitely delicate silica fibers for suspending the 40 kg test masses (three suspensions experienced fiber breakage during the installation phase despite stringent protocols), and precision control engineering in a high-vacuum environment.", "In the end, everything came together!", "The superb LIGO commissioning team made rapid progress, and by September 2015, the Livingston and Hanford interferometers were achieving sensitivities four times better than achieved by initial LIGO.", "During the first week of September, the decision was made to officially begin the run on September 18, 2015.", "The interferometers were in an `Engineering Run' phase, and were taking science quality data since mid-August.", "This turned out to be very fortunate.", "During `Engineering Run 8', which began on August 17, operations were conducted 24 hours a day and seven days a week.", "Most of those hours were dedicated to tests for calibrating the detector, tuning the injection methods for simulating gravitational waves in the detector, setting up automated alerts for possible gravitational wave detections, measuring the effects of induced environmental noise in the detector, and many more tasks that needed to be ready before the official first `O1' run.", "During the times that the two LIGO detectors were running unperturbed by these tests, online algorithms were constantly running to search for gravitational waves and test their performance against the new Advanced LIGO data.", "When coincident “triggers” produced by these methods exceeded a (low) significance threshold, automated database entries were produced with a lot of numbers and plots - there had been several entries with injection tests, as well as weak coincidences that were not statistically significant.", "On September 14, at 5:51am US Eastern time, a program looking for short transients registered a very significant coincidence.", "Attentive scientists in Europe and early risers in the US noticed the trigger, and produced a time-frequency plot which showed what is expected from a binary coalescence - but a very short duration one, which would ordinarily correspond to two merging black holes.", "At first glance, this looked much more like an injection than an astrophysical signal - there was no injection label associated with the trigger, but could it still be an injection?", "Perhaps a blind injection?", "Many emails and phone calls later, it was clear that it was not an injection - it was a real coincidence!", "The next question that was asked - Could it be an instrumental artifact?", "- was also eventually ruled out.", "On September 14, many decisions were taken to start the “discovery process”: the instrument configuration was frozen as much as possible to make sure there was not an instrumental source of transients, coincident or not, that looked like binary coalescences.", "After a few days, a minimal set of tests was finished, and the instrument was put into “normal” observation mode.", "Figure: Figure on cover of Physical Review Letters, Feb 12, 2016, showing the data in the LIGO Hanford (blue) and Livingston (red) detectors due to the coalescence of two black holes.We estimated we needed at least 15 days of two-interferometer coincident data with no similar transients to bound the significance at the magical 5-sigma threshold.", "By the end of October, we had collected a sufficient amount of data, and separate `offline' analyses using matched-filtering to look for binary coalescences were ready to `open the box' to answer the question - did the significance hold up?", "It did!", "A few champagne corks popped, but we were not done yet - long periods over the holidays were still needed to finish the review of the methods used to find the event, review the data calibration and its errors, estimate the parameters of the system, write a paper with the observations and other papers with the details, and wait for external peer review of the main result.", "While the LSC and Virgo had approved in early September 2015 (!)", "a “detection procedure” to validate the first detection, publish and announce it, very few if any of us thought that we'd have to apply it so soon.", "But we were very happy to do so!", "Readers of this newsletter have probably already read the articles containing the details: on February 11, the LIGO Scientific and Virgo Collaborations proudly announced the discovery of two black holes, 29 and 36 solar masses respectively, that had merged into a single black hole more than a billion years ago, producing ripples of space time that passed through Earth and are still traveling through the Universe, after leaving a small but very detectable signal first in the LIGO Livingston detector, and seven milliseconds later in the LIGO Hanford detector.", "The signals were consistent with Einstein's theory of General Relativity, and using his theory we could derive bounds for many parameters for the system: the initial and final masses, spins, distance, inclination, and (very rough) sky localization.", "It was a single detection, but contained lots of information!", "All of the papers produced by the LIGO and Virgo collaborations on GW150914 are available at http://papers.ligo.org The news of the first gravitational wave detection made it to first page of most of the major newspapers worldwide, resulting in tremendous interest by both the scientific and general public.", "But this is just the beginning: LIGO and Virgo are still finishing the analysis of the rest of the data - taken until January 12 - when the first observational run finished.", "The analysis of the remaining O1 data should give us more information about the rates of black hole mergers that can be measured.", "The detectors are designed to be about three times more sensitive than the O1 run, and scientists are hard at work on improving the instruments, and making them more robust.", "A second observational run is planned for the fall of 2016, lasting about six months - we expect more detections, and are now prepared to become a real Observatory.", "Soon, the Virgo detector will finish installation and commissioning, and join the network to provide much better sky localization of the sources.", "As the LIGO and Virgo detectors improve both sensitivity and duty cycle, we will see many black hole mergers - and likely other sources, including binary neutron star or neutron star-black hole mergers, a stochastic background of those mergers, possibly close rotating neutron stars in our galaxy - and perhaps more excitingly, gravitational waves of unknown origin.", "Farther into the future, but on the horizon, detectors in Japan and India will join the network of ground-based interferometers.", "Moreover, the success of the LISA Pathfinder mission bodes well for a future gravitational-wave detector in space.", "The future is gravity-bright!", "toc tocObituaries: tocsubsubsection Remembering Felix Pirani, by Stanley Deser Stanley Deser, Brandeis University deser-at-brandeis.edu Felix Pirani, who died at age 87 on the last day of 2015, was one of the leaders in the postwar renaissance of General Relativity.", "When I first met him, at the famous “GR-0” 1955 Bern conference, we constituted half of the younger generation there (and two of those were mere tourists).", "A prodigy, Felix entered UBC at 14, writing his first paper as an undergraduate.", "He learned Relativity at Toronto and Carnegie Tech (where he got his DSc) from Schild.", "Synge and Infeld were also important mentors.", "His thesis was one of the early attempts, and the first serious postwar one, at quantizing GR.", "A subsequent PhD, with Hermann Bondi at Cambridge, led him into cosmology.", "This was followed by a year with Synge at DIAS in Dublin, which he devoted to establishing the reality of gravitational radiation, a controversial topic at the time.", "He then joined Bondi's newly formed group at King's College, London, where he was to remain until his 1983 retirement.", "In 1967, he became head of the group and produced a steady flow of PhDs (many of whom remained close friends) while obtaining major new results.", "Two early examples: a brilliant application of the then new Petrov classification to establish, invariantly, the reality of gravitational waves; with Bondi and Ivor Robinson, a pioneering study of wave solutions in GR.", "During the year he spent as visiting professor at Brandeis in the sixties, I had the pleasure of collaborating with him on several papers, thus seeing firsthand his mastery of our field; his 1964 Brandeis Summer Institute lectures on gravitational radiation are still a classic reference.", "But his contribution to GR is far greater than the standard references to which his name is associated.", "He was one of our giants.", "Felix was also a man of many interests, political and artistic; indeed, he started a successful career as a mosaicist upon retirement. tocsubsubsection Remembering David Finkelstein, by Predrag Cvitanović Predrag Cvitanović, Georgia Institute of Technology predrag.cvitanovic-at-physics.gatech.edu Theoretical physicist David Ritz Finkelstein, Professor Emeritus in the School of Physics of Georgia Tech, died peacefully at home on January 24, 2016.", "He was born in 1929, in New York City.", "He went to Stuyvesant High School in Manhattan, and worked as a page in the NY Public Library, which gave him access to the stacks where he spent much time reading.", "He started out as an engineering major at City College of New York, but graduated in 1949 with honors in both physics and mathematics, and the CCNY Physics Medal.", "In 1953 he received a PhD in Physics from MIT (advisor: Felix Villars), with a thesis entitled “Non-linear meson theory of nuclear forces.” From 1953 to 1960 he worked at Stevens Institute of Technology.", "He was Young Men's Philanthropic League Professor of Physics at Yeshiva University from 1959 to 1976.", "In Sidney Coleman's words, he “was a brilliant scientist with a passion for long shots.", "This meant that nine times out of ten he devoted his talents to ideas that do not pay off, but one time out of ten, they do pay off.", "When this happened, David's work was found to be of a great significance, extraordinary penetration, and ten years ahead of everyone else's, as was the case when topological conservation laws entered the mainstream of quantum field theory.” In a 1955 paper Finkelstein addressed the question of whether “anomalous” spin 1/2 had been overlooked for the gravitational field.", "His discovery of the topological origin of such anomalous spins and a speculation that maybe all physical variables are topological in origin was the thread that led him to kinks and the unidirectional membrane in the 50s and 60s, as well as to anyons in the 80s, antecedents of anomalous quantum numbers in the fractional Hall effect and in high temperature superconductivity.", "Finkelstein was the first to understand the nature of the black hole horizon.", "In 1958 he (age 29) described what is now known as a black hole (“unidirectional membrane”).", "The paper influenced Landau, Penrose and eventually Wheeler, and was instrumental in bringing general relativity into mainstream physics and today's vibrant black-hole research.", "But it took time.", "At the 1962 Relativistic Theories of Gravitation annual conference Finkelstein reported on his Schwarzschild black hole to an audience of three who turned out to be janitorial staff.", "In 1957, following a seminar Finkelstein gave in London, he met Roger Penrose, then a graduate student from Cambridge.", "The seminar, on extending Schwarzschild's metric, both into the past and into the future across null horizons-a basic ingredient of the current understanding of black holes—had been a revelation to Penrose.", "After the seminar Penrose explained to Finkelstein his spin-networks, and the two men exchanged their research subjects, for ever after.", "Finkelstein's extension of the Schwarzschild metric provided Penrose with an opening into general relativity, the subject which has animated his research ever since.", "Finkelstein picked up in return on the combinatorial aspects of quantum spin as a possible route to delving more deeply into the quantum nature of reality and took such ideas to greater lengths than anyone else.", "Another colleague whose career was shaped by Finkelstein's insights is Lenny Susskind.", "In 1967 he told Susskind: “Forget perturbation theory.", "Black holes are the key.” He explained–this was before Bekenstein–that the information in a region of space could not be as rich as the volume because most states would collapse to form a black hole.", "Finkelstein understood the holographic principle long before 't Hooft and Susskind.", "While his “unidirectional membrane” is today considered his key contribution to physics, for Finkelstein that calculation was only an exercise, an illustration for his overarching program to bring topology into quantum physics.", "He was the first to discover “kinks”, topological charges and topological spin-statistics theorems, with Misner (1959) and Rubenstein (1962).", "Until the work of Finkelstein and Rubinstein on topology in quantum field theory, quantum field theory meant Feynman diagrams.", "He was arguably the first to understand the role of quantum vacua, and his papers were among the earliest on solitons in quantum theories, leading to instantons, Higgs particles, etc...", "It took quite a few years for the rest of the world to catch up.", "The 1962-63 papers with Jauch, Schiminovich and Speiser were the first to formulate a unified SU(2) gauge theory of massive vector bosons and light, introducing a type of “Higgs mechanism” before Higgs, and a type of electroweak unification before Glashow, Salam and Weinberg.", "However, by the late 1960s the limitations of the quaternionic formulation led him to shelve the whole quaternionic project.", "In 1976 Finkelstein became the chairman of the Belfer School of Yeshiva University Physics Department, and in 1978 its dean of Natural Sciences and Mathematics.", "On the basis of this administrative experience, he was appointed Georgia Institute of Technology Director of the School of Physics.", "Nobody understood his job colloquium, but all were impressed by recommendation letters the likes of which Georgia Tech had never seen.", "No letter mentioned the fact that his administrative jobs at Yeshiva were ceremonial, to finalize the closing of the already shuttered graduate sciences program.", "The first thing he did upon arrival was to inform the departmental secretary (in those halcyon days the department was run by a single secretary) that he was going to Majorca.", "So for a month the Chair could not be reached.", "But when, by the midyear, he failed to submit a budget, he was deposed by senior faculty, and replaced by an acting chair.", "Finkelstein went into a funk for two weeks, and emerged a changed man, having recognized that his failure as administrator had given him more time for research.", "He was charismatic and involved a number of dedicated students in his efforts to quantize geometry.", "If he did not show up at work, his students would go to his house, where he met them in his bathrobe.", "He was blissfully useless for any faculty committee task.", "In Finkelstein's own words: “My committee services within Georgia Tech have not been onerous; I do not look this gift horse in the mouth, but serve as requested by the School of Physics.” He taught whatever he wanted to teach, so he had to be deposed for the second time, this time by an uprising of undergraduates who found themselves taught quantum logic instead of the introduction to quantum mechanics that they had signed up for.", "In 2003 Finkelstein noticed in a lecture that between glancing at a formula in his notes, and writing it on the blackboard, he had forgotten it.", "So he went to the Chair, and arranged for his retirement.", "Soon enough he found out that it was statins that were making him stupid - once he went off statins, his sharp intellect came back.", "In retrospect, he found this misinformed decision was one of the best of his life - now he could devote himself fully to his research.", "A few days before his death from idiopathic pulmonary fibrosis he had his laptop in his bed, and was still working.", "David was a man who truly loved life.", "He is survived by his wife Shlomit Ritz Finkelstein, his children Daniel Finkelstein, Beth Bosworth, Eve Finkelstein, and Aria Ritz Finkelstein, his five grandchildren and two great-granddaughters.", "tocsubsubsection Steve, the Physicist, by Bernard Whiting and Eric Poisson Bernard Whiting, University of Florida bernard-at-phys.ufl.edu Eric Poisson, University of Guelph epoisson-at-uoguelph.ca As a community, we experienced a most bitter-sweet moment earlier this year when the LIGO announcement of gravitational wave detection came just a few days after we had suffered the sudden and unexpected loss of Steven Detweiler as a friend and colleague on February 8th, 2016.", "Gravitational wave physics was the area of research which drove Steve all his working life.", "Thus, it was with some irony that we found ourselves grieving at his passing while celebrating his life's interest during the announcement of the LIGO detection.", "Steve was not a member of the LIGO Scientific Collaboration that finally made the detection because he wanted the freedom to direct his own research.", "Nevertheless, he was an avid follower of the LIGO experiment and was fortunate enough to have heard, a week or so before his last early morning run, that a detection had been made.", "What would have thrilled Steve more than the announcement itself would have been the news that the waves detected actually came from colliding black holes with an unexpected size, about thirty times the mass of our Sun — not smaller, and not much larger (i.e., not supermassive) either.", "The study of black holes, and the gravitational waves they produce, had been a constant theme permeating Steve's scientific research.", "Steve initially began his work on black holes in conjunction with the Nobel Laureate, Subrahmanyan Chandrasekhar.", "It had been recently realized that when a black hole is disturbed — say, by something falling in — it will actually vibrate, and those vibrations initiate waves which propagate away throughout spacetime.", "The interaction between each black hole and its surrounding spacetime is very stiff, so the vibrations and gravitational waves are rapidly damped down and soon die away to nothing at all.", "But the frequencies and damping times are unique characteristics of the black hole they come from, and Steve was among the first to calculate and tabulate them for later use.", "In fact, that “ring-down” was one of the tell-tale features which the LIGO collaboration was so anxious to identify, since its unique character ensures that a black hole origin is absolutely unambiguous.", "For more than a decade, Steve's most recent area of interest has been in work on the gravitational self-force.", "This refers to the generally tiny effect the motion of a celestial body experiences due to the fact that its movement interacts with its own gravitational field, and it explains why two black holes in orbit around each other will eventually merge.", "The focus of Steve's effort was for a future, space-based, gravitational wave detector, originally called LISA.", "The perturbative methods which have been used in calculations for decades are ideal for this work.", "Steve's valued contribution was in convincing the community of what should actually be compared between different calculations, and in carrying out the computations to high numerical precision.", "The self-force world was now waiting for him to take this work to the next level and produce second order results to compare with post-Newtonian theory.", "One of Steve's very creative ideas, which is barely known among relativists, but is quite well known in the pulsar timing community, was his suggestion to use precise pulsar timing to detect gravitational waves from the collision of supermassive black holes in the centers of other galaxies.", "These very large wavelength waves would cause fluctuations in the arrival times of signals from pulsars — rotating neutron stars in our own galaxy — and the accumulation of the impact of these effects from all over the universe would represent an inescapable, noise-like, signal in the timing precision.", "This was a brilliant idea, far ahead of its time, leading to a technique that is only now coming into fruition.", "Pulsars were not even known when Einstein developed his general relativity theory.", "Though he might have been skeptical about the success of Steve's suggestion, he would certainly have appreciated its inspirational character.", "And when it does work, Steve will be fully vindicated.", "Among his personal traits, Steve was perhaps most admired for his fearless attitude and his fertile creativity.", "It may be strange to speak of a creative mind when describing a scientist.", "After all, science is supposed to be precise and exact, so where can creativity fit in?", "But the best scientists discover fresh insights into the workings of the world, and this does indeed require a creative mind.", "Steve's mind was full of insights, and we have all benefited greatly from his unique ways of thinking.", "This may explain why Steve was admired, but it is not why he was loved by anyone who had the chance to know him and spend time with him.", "It is not just that he would come across as a friendly, nice, and decent guy.", "In addition, Steve had a heightened empathy that allowed him to establish a deep and lasting rapport with new friends and colleagues, often almost immediately.", "Steve's scientific contributions span four decades, with many breakthrough works and deeply original ideas substantiating the impact he has had on the field of general relativity.", "Yet he was nominated for a Fellowship of the American Physical Society only in 2013.", "This was long overdue.", "His citation was “for his many and varied contributions to gravitational physics, which include the computation of black-hole quasi-normal modes, the elucidation of pulsar timing to measure gravitational waves, and foundational contributions to the gravitational self-force.” We should also note Steve's prescient work on Kaluza-Klein theory, which appeared in his most cited paper and yet Steve barely made any reference to it.", "Suffice it to say that Steve had diverse interests, and colleagues were still referring to him for input.", "He is already sorely missed.", "tocsubsubsection Remembering Sergio Dain, by Luis Lehner and Manuel Tiglio Luis Lehner, Perimeter Institute llehner-at-perimeterinstitute.ca Manuel Tiglio, University of California San Diego tiglio-at-ucsd.edu It is with our deepest sadness that we report on the sudden passing away of our friend and colleague Professor Sergio Dain.", "Sergio lost his brave yet short battle to cancer on the 24th of February 2016, at an early age of 46.", "Sergio got his Licenciatura and PhD at the University of Córdoba in Argentina in 1993 and 1999 respectively, spending a significant fraction of his PhD studies at the Albert Einstein Institute (AEI) for Gravitational Physics in Golm, Gemany.", "The research themes for his Licenciatura and PhD were on asymptotically flat spacetimes and topics in gravitational radiation.", "Afterwards, he spent several years at the AEI as a postdoctoral researcher, before returning to Córdoba in 2006 as a faculty member and as an independent investigator at CONICET.", "After his PhD, Sergio's interests transitioned to geometrical analysis, where he made important contributions to the initial data problem in General Relativity and conserved quantities in black hole collisions.", "He then produced a series of seminal papers establishing geometrical inequalities between angular momentum and mass in General Relativity.", "These influential results were recognized in numerous ways, in particular through invitations to give plenary lectures at, for example, the 20th International Conference on General Relativity and Gravitation and the 10th Amaldi meeting, held at Warsaw in 2013, as well as at the Fields Institute at Toronto in 2015.", "Sergio had deep and broad interests, always generously shared his research ideas and was a great mentor and role model.", "About a dozen undergraduate and graduate students received their degrees in Argentina under his supervision, and he also mentored four post-docs.", "Beyond his research activities, Sergio was avidly engaged in the promotion of Science in Argentina at different levels, as well as to the General Relativity community serving, for instance, on the Editorial Board of General Relativity and Gravitation.", "Beyond all of the above, to many of us, Sergio was a dear, exceptional friend and an outstanding person.", "We will always remember him for his sensitivity, fondness for the arts, his great sense of humor, contagious laughter, unconditional friendship, and a deep devotion to science.", "toc tocEditorial: tocsubsubsection Gravitational physics in the modern university, by David Garfinkle David Garfinkle garfinkl-at-oakland.edu (Note: the opinions expressed in this editorial are solely those of the author and do not represent any sort of official pronouncement by either DGRAV or APS).", "Over the past two decades or so, some disturbing trends have taken place in universities in the US: tuition has gone way up (as has student debt).", "State support for public universities has gone way down.", "The percentage of faculty who are tenured or tenure track keeps going down.", "The number of administrators keeps going up.", "And more and more university boards are taking the attitude that universities should be run like businesses.", "This is bad news for everyone.", "However, I will argue that it is even worse news for physics than for other areas of academic endeavor, and even worse news for gravitational physics than for other areas of physics.", "First let's consider what is driving these trends.", "The current trend in state level politics is for state governments to spend less money.", "Since broad support for intellectual pursuits has always been shaky at best in the US, when state governments think about what to cut, support for higher education is always at or near the top of the list.", "In response, state universities make up for lost revenue by raising tuition, and cut costs by hiring more part time faculty and fewer full time faculty.", "In principle, these trends in state universities need not have affected private universities at all.", "However, private universities, especially elite ones, have generally positioned their tuition higher than that of state universities and have marketed themselves as using that higher tuition to provide a higher quality education.", "Thus, when state universities raised tuition, it is not surprising that private universities followed suit.", "However, there is one point missing from the previous analysis: if universities were really motivated to cut costs, wouldn't they hire fewer administrators, not more?", "What then explains the explosion of “administrative bloat”?", "One possible answer is provided by Benjamin Ginsberg in his book The Fall of the Faculty.", "Ginsberg argues that administrative bloat is driven by the need of administrators to feel important.", "Thus, for example, the more people who are working for the Dean, the more important he feels, and since administrative tasks don't actually have to be productive, it is easy for the Dean to invent activities for the administrators in his office to do.", "If Ginsberg's analysis is correct, then from an administrator's point of view, both students and faculty are merely means to generate revenue that can be used to hire more administrators.", "Thus the administration of a university would be motivated to enroll as many students as possible and to charge them as high a tuition as possible; and to hire as few faculty as possible, with the largest teaching load possible, and at the lowest salary possible.", "That is, new faculty hires are likely to be overwhelmingly non-tenure track faculty with such a high teaching load that they have no time at all for scholarship.", "However, the administration cannot hire all part time, non-tenure track faculty, because some tenure track faculty are needed to teach the upper level courses in each department taken by majors and graduate students.", "Thus the administration of a university is likely to target those few tenure track hires to the departments with the largest number of majors.", "This is bad news for physics, since the number of physics majors tends to be rather small compared to those in other departments.", "With a small number of majors, a physics department in a modern university is likely to be regarded by the administration as a service department, delivering its credit hours in introductory courses taken by those students whose major requires an introductory physics course, such as engineering or health science.", "Thus, if the administration hires any tenure track physicists at all, they are likely to be in those areas that most relate to those departments served by the physics department: areas such as materials science or biophysics, but certainly not gravitational physics.", "Thus the rise of the modern, corporate, administrative university is bad news for scholarship in all areas.", "But it is even worse news for physics than for other departments, and even worse news for gravitational physics than for other areas of physics.", "Of course, there will always be elite universities who justify their high tuition in part by the prestige of their faculty, and these universities will continue to hire tenure track physicists (possibly including gravitational physicists) who will continue to do excellent research.", "And from the point of view of the administrations of these elite universities, it is an additional motivating factor that the physicists they hire can bring in grant money whose overhead can be used to hire more administrators.", "However, these bright spots, important as they are, are likely to become ever fewer and farther between if the trend of the modern, corporate, administrative university continues.", "What (if anything) can be done about the problem of administrative bloat in the modern, corporate university?", "Actually, it is somewhat surprising that steps have not already been taken to curtail this problem.", "Recall that the money wasted on hiring superfluous administrators comes from students, their families, state governments, and the federal government.", "Together this could be a powerful constituency that should be highly motivated to do something about the problem.", "However, the modern, corporate university makes use of an extensive modern public relations apparatus to construct and manipulate its image.", "Rather than do something about administrative bloat, this PR apparatus simply denies that the problem exists and blames rising tuition and rising number of administrators on other factors, such as smaller state support, rising government regulation, rising costs of employee health care, and supposed inefficiency of faculty practices.", "This last claim is especially dangerous, as it also provides an excuse for university policies that take ever more decisions out of the hands of faculty and give them to administrators.", "Both individual faculty and the AAUP (American Association of University Professors) have long sounded the alarm about administrative bloat and about the increasing tendency to treat running a university as though it were the same as running a business.", "However, against the professional PR apparatus of the university boards and administrations, the AAUP and individual faculty are losing the war of words.", "Perhaps it is time for professional organizations (such as APS) to weigh in on this issue." ] ]
1606.05232
[ [ "On biadjoint triangles" ], [ "Abstract We prove a biadjoint triangle theorem and its strict version, which are $2$-dimensional analogues of the adjoint triangle theorem of Dubuc.", "Similarly to the $1$-dimensional case, we demonstrate how we can apply our results to get the pseudomonadicity characterization (due to Le Creurer, Marmolejo and Vitale).", "Furthermore, we study applications of our main theorems in the context of the $2$-monadic approach to coherence.", "As a direct consequence of our strict biadjoint triangle theorem, we give the construction (due to Lack) of the left $2$-adjoint to the inclusion of the strict algebras into the pseudoalgebras.", "In the last section, we give two brief applications on lifting biadjunctions and pseudo-Kan extensions." ], [ "Introduction", "Assume that $E:{A}\\rightarrow {C}$ , $J: {A}\\rightarrow {B}$ , $L:{B}\\rightarrow {C}$ are functors such that there is a natural isomorphism ${ {A}[rr]^-{J}[dr]_-{E}&&{B}[dl]^-{L}\\\\&{C}@{}[u]|-\\cong & }$ Dubuc [2] proved that if $L: {B}\\rightarrow {C}$ is precomonadic, $E: {A}\\rightarrow {C}$ has a right adjoint and ${A}$ has some needed equalizers, then $J$ has a right adjoint.", "In this paper, we give a 2-dimensional version of this theorem, called the biadjoint triangle theorem.", "More precisely, let $\\mathfrak {A}$ , $\\mathfrak {B}$ and $\\mathfrak {C}$ be 2-categories and assume that $E:\\mathfrak {A}\\rightarrow \\mathfrak {C}, J: \\mathfrak {A}\\rightarrow \\mathfrak {B}, L:\\mathfrak {B}\\rightarrow \\mathfrak {C}$ are pseudofunctors such that $L$ is pseudoprecomonadic and $E$ has a right biadjoint.", "We prove that, if we have the pseudonatural equivalence below, then $J$ has a right biadjoint $G$ , provided that $\\mathfrak {A}$ has some needed descent objects.", "${ \\mathfrak {A}[rr]^-{J}[dr]_-{E}&&\\mathfrak {B}[dl]^-{L}\\\\&\\mathfrak {C}@{}[u]|-\\simeq & }$ We also give sufficient conditions under which the unit and the counit of the obtained biadjunction are pseudonatural equivalences, provided that $E$ and $L$ induce the same pseudocomonad.", "Moreover, we prove a strict version of our main theorem on biadjoint triangles.", "That is to say, we show that, under suitable conditions, it is possible to construct (strict) right 2-adjoints.", "Similarly to the 1-dimensional case [2], the biadjoint triangle theorem can be applied to get the pseudo(co)monadicity theorem due to Le Creurer, Marmolejo and Vitale [13].", "Also, some of the constructions of biadjunctions related to two-dimensional monad theory given by Blackwell, Kelly and Power [1] are particular cases of the biadjoint triangle theorem.", "Furthermore, Lack [12] proved what may be called a general coherence result: his theorem states that the inclusion of the strict algebras into the pseudoalgebras of a given 2-monad $\\mathcal {T}$ on a 2-category $\\mathfrak {C}$ has a left 2-adjoint and the unit of this 2-adjunction is a pseudonatural equivalence, provided that $\\mathfrak {C}$ has and $\\mathcal {T}$ preserves strict codescent objects.", "This coherence result is also a consequence of the biadjoint triangle theorems proved in Section .", "Actually, although the motivation and ideas of the biadjoint triangle theorems came from the original adjoint triangle theorem [2], [20] and its enriched version stated in Section , Theorem REF may be seen as a generalization of the construction, given in [12], of the right biadjoint to the inclusion of the 2-category of strict coalgebras into the 2-category of pseudocoalgebras.", "In Section , we give a slight generalization of Dubuc's theorem, in its enriched version (Proposition REF ).", "This version gives the 2-adjoint triangle theorem for 2-pre(co)monadicity, but it lacks applicability for biadjoint triangles and pseudopre(co)monadicity.", "Then, in Section we change our setting: we recall some definitions and results of the tricategory 2-${\\rm \\sf CAT}$ of 2-categories, pseudofunctors, pseudonatural transformations and modifications.", "Most of them can be found in Street's articles [18], [19].", "Section gives definitions and results related to descent objects [18], [19], which is a very important type of 2-categorical limit in 2-dimensional universal algebra.", "Within our established setting, in Section we prove our main theorems (Theorem REF and Theorem REF ) on biadjoint triangles, while, in Section , we give consequences of such results in terms of pseudoprecomonadicity (Corollary REF ), using the characterization of pseudoprecomonadic pseudofunctors given by Proposition REF , that is to say, Corollary REF .", "In Section , we give results (Theorem REF and Theorem REF ) on the counit and unit of the obtained biadjunction $J\\dashv G$ in the context of biadjoint triangles, provided that $E$ and $L $ induce the same pseudocomonad.", "Moreover, we demonstrate the pseudoprecomonadicity characterization of [13] as a consequence of our Corollary REF .", "In Section , we show how we can apply our main theorem to get the pseudocomonadicity characterization [8], [13] and we give a corollary of Theorem REF on the counit of the biadjunction $J\\dashv G $ in this context.", "Furthermore, in Section we show that the theorem of [12] on the inclusion $\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}_{{}_{\\mathsf {s}}}\\rightarrow \\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}$ is a direct consequence of the theorems presented herein, giving a brief discussion on consequences of the biadjoint triangle theorems in the context of the 2-(co)monadic approach to coherence.", "Finally, we discuss a straightforward application on lifting biadjunctions in Section .", "Since our main application in Section is about construction of right biadjoints, we prove theorems for pseudoprecomonadic functors instead of proving theorems on pseudopremonadic functors.", "But, for instance, to apply the results of this work in the original setting of [1], or to get the construction of the left biadjoint given in [12], we should, of course, consider the dual version: the Biadjoint Triangle Theorem REF .", "I wish to thank my supervisor Maria Manuel Clementino for her support, attention and useful feedback during the preparation of this work, realized in the course of my PhD program at University of Coimbra." ], [ "Enriched Adjoint Triangles", "Consider a cocomplete, complete and symmetric monoidal closed category $V$ .", "Assume that $L: \\mathbb {B}\\rightarrow \\mathbb {C}$ is a $V$ -functor and $(L\\dashv U, \\eta , \\varepsilon )$ is a $V$ -adjunction.", "We denote by $\\chi : \\mathbb {C}(L-, -)\\cong \\mathbb {B}(-,U-)$ its associated $V$ -natural isomorphism, that is to say, for every object $X$ of $\\mathbb {B}$ and every object $Z$ of $\\mathbb {C}$ , $\\chi _ {{}_{(X,Z)}} = \\mathbb {B}(\\eta _ {{}_X}, UZ )\\circ U_{{}_{LX,Z}} $ .", "Proposition 1.1 (Enriched Adjoint Triangle Theorem) Let $(L\\dashv U, \\eta , \\varepsilon ) $ , $(E\\dashv R, \\rho , \\mu ) $ be $V$ -adjunctions such that ${ \\mathbb {A}[rr]^-{J}[dr]_-{E}&&\\mathbb {B}[dl]^-{L}\\\\&\\mathbb {C}& }$ is a commutative triangle of $V$ -functors.", "Assume that, for each pair of objects $(A\\in \\mathbb {A}, Y\\in \\mathbb {B}) $ , the induced diagram ${ \\mathbb {B}(JA, Y) [r]^-{L_{{}_{JA,Y}}} & \\mathbb {C}(EA, LY)@<-1ex>[rrr]_-{\\mathbb {C}(EA, L(\\eta _{{}_Y} ))}@<1ex>[rrr]^-{L_{{}_{{}_{JA,ULY}}}\\circ \\hspace{1.99997pt} \\chi _ {{}_{{}_{(JA,LY)}}}} &&& \\mathbb {C}(EA, LULY ) }$ is an equalizer in $V$ .", "The $V$ -functor $J$ has a right $V$ -adjoint $G$ if and only if, for each object $Y$ of $\\mathbb {B}$ , the $V$ -equalizer of ${ RLY @<0.9 ex>[rrrr]^-{RL(U(\\mu _{{}_{LY}})\\eta _{{}_{JRLY}})\\rho _ {{}_{RLY}}}@<-0.9ex>[rrrr]_-{RL(\\eta _ {{}_{Y}} )} &&&& RLULY}$ exists in the $V$ -category $\\mathbb {A}$ .", "In this case, this equalizer gives the value of $GY$ .", "For each pair of objects $(A\\in \\mathbb {A}, Y\\in \\mathbb {B}) $ , the $V$ -natural isomorphism $\\mathbb {C}(E-, -)\\cong \\mathbb {A}(-, R-)$ gives the components of the natural isomorphism ${ \\mathbb {B}(JA, Y)@{=}[dd] [r]^-{L_{{}_{JA,Y}}} & \\mathbb {C}(EA, LY)[dd]|-{\\cong }@<-1ex>[rrr]_-{\\mathbb {C}(EA, L(\\eta _{{}_Y} ))}@<1ex>[rrr]^-{L_{{}_{{}_{JA,ULY}}}\\circ \\hspace{1.99997pt} \\chi _ {{}_{{}_{(JA,LY)}}}} &&& \\mathbb {C}(EA, LULY ) [dd]|-{\\cong }\\\\&&&&\\\\\\mathbb {B}(JA, Y) [r] & \\mathbb {A}(A, RLY) @<1ex>[rrr]^{\\mathbb {A}(A, r_ Y)}@<-1ex>[rrr]_{\\mathbb {A}(A, q_Y) } &&& \\mathbb {A}(A, RLULY)}$ in which $q_Y = RL(\\eta _ {{}_{Y}} ) $ and $ r_Y=RL(U(\\mu _{{}_{LY}})\\eta _{{}_{JRLY}})\\rho _ {{}_{RLY}} $ .", "Thereby, since, by hypothesis, the top row is an equalizer, $\\mathbb {B}(JA, Y)$ is the equalizer of $(\\mathbb {A}(A, q_Y), \\mathbb {A}(A, r_Y)) $ .", "Assuming that the pair $(q_ Y,r_ Y)$ has a $V$ -equalizer $GY$ in $\\mathbb {A}$ for every $Y$ of $\\mathbb {B}$ , we have that $\\mathbb {A}(A, GY) $ is also an equalizer of $(\\mathbb {A}(A, q_Y), \\mathbb {A}(A, r_Y) )$ .", "Therefore we get a $V$ -natural isomorphism $\\mathbb {A}( -, GY)\\cong \\mathbb {B}(J-, Y)$ .", "Reciprocally, if $G$ is right $V$ -adjoint to $J$ , since $\\mathbb {A}(-, GY)\\cong \\mathbb {B}(J-, Y) $ is an equalizer of $\\left(\\mathbb {A}(-, q_Y), \\mathbb {A}(-, r_Y)\\right) $ , $GY$ is the $V$ -equalizer of $(q_Y, r_Y) $ .", "This completes the proof that the $V$ -equalizers of $q_Y, r_Y $ are also necessary.", "The results on (co)monadicity in $V$ -${\\rm \\sf CAT}$ are similar to those of the classical context of ${\\rm \\sf CAT}$ (see, for instance, [3], [16]).", "Actually, some of those results of the enriched context can be seen as consequences of the classical theorems because of Street's work [16].", "Our main interest is in Beck's theorem for $V$ -precomonadicity.", "More precisely, it is known that the 2-category $V$ -${\\rm \\sf CAT}$ admits construction of coalgebras [16].", "Therefore every left $V$ -adjoint $L: \\mathbb {B}\\rightarrow \\mathbb {C}$ comes with the corresponding Eilenberg-Moore factorization.", "${\\mathbb {B}[r]^-{\\phi }[rd]_-{L}& {\\rm \\sf CoAlg}[d]\\\\&\\mathbb {C}} $ If $V= {\\rm \\sf Set}$ , Beck's theorem asserts that $ \\phi $ is fully faithful if and only if the diagram below is an equalizer for every object $Y$ of $\\mathbb {B}$ .", "In this case, we say that $L$ is precomonadic.", "${ Y [r]^-{\\eta _ {{}_Y}} & ULY @<-0.9 ex>[rr]_-{UL(\\eta _ {{}_{Y}})}@<0.9ex>[rr]^-{\\eta _ {{}_{ULY}}} && ULULY}$ With due adaptations, this theorem also holds for enriched categories.", "That is to say, $ \\phi $ is $V$ -fully faithful if and only if the diagram above is a $V$ -equalizer for every object $Y$ of $\\mathbb {B}$ .", "This result gives what we need to prove Corollary REF , which is the enriched version for Dubuc's theorem [2].", "Corollary 1.2 Let $(L\\dashv U, \\eta , \\varepsilon ) $ , $(E\\dashv R, \\rho , \\mu ) $ be $V$ -adjunctions and $J$ be a $V$ -functor such that ${ \\mathbb {A}[rr]^-{J}[dr]_-{E}&&\\mathbb {B}[dl]^-{L}\\\\&\\mathbb {C}& }$ commutes and $L$ is $V$ -precomonadic.", "The $V$ -functor $J$ has a right $V$ -adjoint $G$ if and only if, for each object $Y$ of $\\mathbb {B}$ , the $V$ -equalizer of ${ RLY @<0.9 ex>[rrrr]^-{RL(U(\\mu _{{}_{LY}})\\eta _{{}_{JRLY}})\\rho _ {{}_{RLY}}}@<-0.9ex>[rrrr]_-{RL(\\eta _ {{}_{Y}} )} &&&& RLULY}$ exists in the $V$ -category $\\mathbb {A}$ .", "In this case, these equalizers give the value of the right adjoint $G$ .", "The isomorphisms induced by the $V$ -natural isomorphism $\\chi :\\mathbb {C}(L-,-)\\cong \\mathbb {B}(-,U-)$ are the components of the natural isomorphism ${ \\mathbb {B}(JA, Y)@{=}[dd] [rr]^-{L_{{}_{JA,Y}}} && \\mathbb {C}(EA, LY)[dd]|-{\\chi _ {{}_{{}_{(JA,LY)}}}}@<-1ex>[rrr]_-{\\mathbb {C}(EA, L(\\eta _{{}_Y} ))}@<1ex>[rrr]^-{L_{{}_{{}_{JA,ULY}}}\\circ \\hspace{1.99997pt} \\chi _ {{}_{{}_{(JA,LY)}}}} &&& \\mathbb {C}(EA, LULY )[dd]|-{\\chi _ {{}_{{}_{(JA,LULY)}}}}\\\\&& &&&\\\\\\mathbb {B}(JA, Y) [rr]^-{\\mathbb {B}(JA, \\eta _ {{}_{Y}})} && \\mathbb {B}(JA, ULY) @<0.9 ex>[rrr]^-{\\mathbb {B}(JA, \\eta _ {{}_{ULY}})}@<-0.9ex>[rrr]_-{\\mathbb {B}(JA, UL(\\eta _ {{}_{Y}}))} &&& \\mathbb {B}(JA, ULULY)}$ Since $L$ is $V$ -precomonadic, by the previous observations, the top row of the diagram above is an equalizer.", "Thereby, for every object $A$ of $\\mathbb {A}$ and every object $Y$ of $\\mathbb {B}$ , the bottom row, which is the diagram $D^{JA}_Y $ , is an equalizer.", "By Proposition REF , this completes the proof.", "Proposition REF applies to the case of ${\\rm \\sf CAT}$ -enriched category theory.", "But it does not give results about pseudomonad theory.", "For instance, the construction above does not give the right biadjoint constructed in [1], [12] $\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}\\rightarrow \\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}_{{}_{\\mathsf {s}}} .", "$ Thereby, to study pseudomonad theory properly, we study biadjoint triangles, which cannot be dealt with only ${\\rm \\sf CAT}$ -enriched category theory.", "Yet, a 2-dimensional version of the perspective given by Proposition REF is what enables us to give the construction of (strict) right 2-adjoint functors in Subsection REF ." ], [ "Bilimits", "We denote by $2\\textrm {-}{\\rm \\sf CAT}$ the tricategory of 2-categories, pseudofunctors (homomorphisms), pseudonatural transformations (strong transformations) and modifications.", "Since this is our main setting, we recall some results and concepts related to $2\\textrm {-}{\\rm \\sf CAT}$ .", "Most of them can be found in [18], and a few of them are direct consequences of results given there.", "Firstly, to fix notation, we set the tricategory 2-${\\rm \\sf CAT}$ , defining pseudofunctors, pseudonatural transformations and modifications.", "Henceforth, in a given 2-category, we always denote by $\\cdot $ the vertical composition of 2-cells and by $\\ast $ their horizontal composition.", "[Pseudofunctor] Let $\\mathfrak {B}, \\mathfrak {C}$ be 2-categories.", "A pseudofunctor $L:\\mathfrak {B}\\rightarrow \\mathfrak {C}$ is a pair $(L , \\mathfrak {l}) $ with the following data: Function $L : {\\rm obj}(\\mathfrak {B})\\rightarrow {\\rm obj}(\\mathfrak {C}) $ ; For each pair $(X,Y)$ of objects in $\\mathfrak {B}$ , functors $L _{{}_{X,Y}} : \\mathfrak {B}(X,Y)\\rightarrow \\mathfrak {C}(LX, LY) $ ; For each pair $g: X\\rightarrow Y , h: Y\\rightarrow Z $ of 1-cells in $\\mathfrak {B}$ , an invertible 2-cell of $\\mathfrak {C}$ : $\\mathfrak {l}_ {{}_{hg}}: L(h) L(g)\\Rightarrow L(hg); $ For each object $X$ of $\\mathfrak {B}$ , an invertible 2-cell in $\\mathfrak {C}$ : $\\mathfrak {l}_ {{}_{X}}: {\\rm id}_{{}_{LX}}\\Rightarrow L({\\rm id}_ {{}_X} );$ such that, if $\\hat{g}, g: X\\rightarrow Y, \\hat{h}, h:Y\\rightarrow Z, f:W\\rightarrow X $ are 1-cells of $\\mathfrak {B}$ , and $\\mathfrak {x}: g\\Rightarrow \\hat{g}, \\mathfrak {y}: h\\Rightarrow \\hat{h} $ are 2-cells of $\\mathfrak {B}$ , the following equations hold: Associativity: ${LW[rr]^{L(f)}[dd]_{L(hgf)}[ddrr]|{L(gf)}&&LX[dd]^{L(g)}@{}[dl]|{{\\mathfrak {l}_ {{}_{gf}}}}&&LW[rr] ^{L(f) }[dd]_{L(hgf)}@{}[dr]|{{\\mathfrak {l}_ {{}_{(hg)f}}}}&&LX[dd]^{L(g)}[ddll]|{L(hg)}\\\\&&&=&&&\\\\LZ@{}[ru]|{{\\mathfrak {l}_ {{}_{h(gf)}}}}&&LY[ll]^{L(h)}&&LZ&&LY[ll]^{L(h)} @{}[ul]|{{\\mathfrak {l}_ {{}_{hg}}}} }$ Identity: ${ LW [rr]^{L(f)}[dd]_{L({\\rm id}_ {{}_X}f)}&&LX@/_7ex/[dd]|{L({\\rm id}_ {{}_X}) }@{}[dd]|{{\\mathfrak {l}_ {{}_X}} }@/^7ex/[dd]|{{\\rm id}_ {{}_{LX}} }&&LW[dd]_{L(f{\\rm id}_ {{}_W})}&&LW@{=}[ll]@/_7ex/[dd]|{L({\\rm id}_ {{}_W}) }@{}[dd]|{{\\mathfrak {l}_ {{}_W}} }@/^7ex/[dd]|{{\\rm id}_ {{}_{LW}} }&&LW@/_4ex/[dd]|{L(f) }@{}[dd]|{=}@/^4ex/[dd]|{L(f) } \\\\&@{}[l]|{{\\mathfrak {l}_ {{}_{{\\rm id}_ {{}_X} f }}}} &&=&&@{}[l]|{{\\mathfrak {l}_ {{}_{f{\\rm id}_ {{}_W} }}}} &&=& \\\\ LX@{=}[rr]&&LX&&LX &&LW[ll]^{L(f)} && LX }$ Naturality: ${ LX[dddd]_{L(\\hat{h}\\hat{g})}@{=}[r] &LX @{=}[r][dd]_{L(\\hat{g})}&{LX} [dd]^{L(g) } && LX[dddd]_{L(\\hat{h}\\hat{g})}@{=}[r] &LX [rr]^{L(g)}[dddd]^{L(hg)}&&{LY} [dddd]_{L(h)}\\\\&&{}@{}[l]|{{L(\\mathfrak {x})}} && &&&\\\\&{LY}@{}[l]|{{\\mathfrak {l}_ {{}_{\\hat{h}\\hat{g}}}}} [dd]_{L(\\hat{h})} @{=}[r] &{LY}[dd]^{L(h)} &=& &{}@{}[l]|{{L(\\mathfrak {y}\\ast \\mathfrak {x}) }} &{\\hspace{5.0pt}\\mathfrak {l}_ {{}_{hg}} \\hspace{5.0pt}}&\\\\&&{}@{}[l]|{{L(\\mathfrak {y})}} && &&&\\\\LZ@{=}[r] &LZ@{=}[r]& LZ && LZ@{=}[r] &LZ@{=}[rr]&& LZ }$ The composition of pseudofunctors is easily defined.", "Namely, if $(J, \\mathfrak {j}): \\mathfrak {A}\\rightarrow \\mathfrak {B}, (L , \\mathfrak {l}): \\mathfrak {B}\\rightarrow \\mathfrak {C}$ are pseudofunctors, we define the composition by $ L\\circ J := (LJ, (\\mathfrak {l}\\mathfrak {j}) )$ , in which $(\\mathfrak {l}\\mathfrak {j}) _ {{}_{hg}}:= L(\\mathfrak {j}_ {{}_{hg}})\\cdot \\mathfrak {l}_ {{}_{J(h)J(g)}} $ and $(\\mathfrak {l}\\mathfrak {j}) _ {{}_{X}}:= L(\\mathfrak {j}_ {{}_{X}})\\cdot \\mathfrak {l}_ {{}_{JX}} $ .", "This composition is associative and it has trivial identities.", "Furthermore, recall that a 2-functor $L:\\mathfrak {B}\\rightarrow \\mathfrak {C}$ is just a pseudofunctor $(L, \\mathfrak {l}) $ such that its invertible 2-cells $\\mathfrak {l}_ {{}_f}$ (for every morphism $f$ ) and $\\mathfrak {l}_{{}_X} $ (for every object $X$ ) are identities.", "[Pseudonatural transformation] If $ L, E:\\mathfrak {B}\\rightarrow \\mathfrak {C}$ are pseudofunctors, a pseudonatural transformation $\\alpha : L\\longrightarrow E $ is defined by: For each object $X$ of $\\mathfrak {B}$ , a 1-cell $\\alpha _{{}_X}: L X\\rightarrow E X $ of $\\mathfrak {C}$ ; For each 1-cell $g:X\\rightarrow Y $ of $\\mathfrak {B}$ , an invertible 2-cell $\\alpha _{{}_g}: E(g) \\alpha _{{}_X}\\Rightarrow \\alpha _{{}_Y} L(g) $ of $\\mathfrak {C}$ ; such that, if $g, \\hat{g}: X\\rightarrow Y, f:W\\rightarrow X $ are 1-cells of $\\mathfrak {A}$ , and $\\mathfrak {x}: g\\Rightarrow \\hat{g} $ is a 2-cell of $\\mathfrak {A}$ , the following equations hold: Associativity: ${ L W[dddd]_{L (gf)}@{=}[r] &L W [r]^{\\alpha _{{}_W}}[dd]_{L(f)}&{E W } [dd]^{E (f) } && L W[dddd]_{L(gf)}[r]^{\\alpha _{{}_W}} &E W [rr]^{E(f)}[dddd]^{E(gf)}&&{E X} [dddd]^{E (g)}\\\\&&{}@{}[l]|{{\\alpha _{{}_f}}} && &&&\\\\&{L X}@{}[l]|{{\\mathfrak {l}_ {{}_{gf}}}} [dd]_{L(g)} [r]_{\\alpha _{{}_X}} &{E X}[dd]^{E(g)} &=& &{}@{}[l]|{{\\alpha _{{}_{gf}} }} &{\\hspace{5.0pt}\\mathfrak {e}_ {{}_{gf}} \\hspace{5.0pt}}&\\\\&&{}@{}[l]|{{\\alpha _{{}_{g}}}} && &&&\\\\L Y@{=}[r] &L Y[r]_{\\alpha _{{}_Y}}& E Y && L Y[r]_{\\alpha _{{}_Y}} &E Y@{=}[rr]&& E Y }$ Identity: ${ L W@/_7ex/[dd]|-{L ({\\rm id}_ {{}_W}) }@{}[dd]|{{\\mathfrak {l}_ {{}_W}} }@/^7ex/[dd]|-{{\\rm id}_ {{}_{L W }} }[rr]^{\\alpha _{{}_W}}&& E W [dd]^{{\\rm id}_ {{}_ {E W}}}&&L W[rr]^{ \\alpha _ {{}_{W}} }[dd]_{L ({\\rm id}_ {{}_W})}&&E W@/_7ex/[dd]|-{E ({\\rm id}_ {{}_W}) }@{}[dd]|{{\\mathfrak {e}_ {{}_W}} }@/^7ex/[dd]|-{{\\rm id}_ {{}_{E W}} }\\\\&@{}[r]|{\\hspace{3.00003pt}= \\hspace{3.99994pt}} &&=&&@{}[l]|{{ \\alpha _{{}_{{\\rm id}_ {{}_W}}} } } &\\\\L W[rr]_ {\\alpha _ {{}_W}} && E W&&L W[rr]_ {\\alpha _{{}_W}} &&E W }$ Naturality: ${ L X@/_6ex/[dd]_{L (\\hat{g}) }@{}[dd]|{{L (\\mathfrak {x})} }@/^6ex/[dd]^{L (g) }[rr]^{\\alpha _{{}_X}}&& E X[dd]^{E (g)}&&L X[rr]^{ \\alpha _ {{}_{X}} }[dd]_{L (\\hat{g})}&&E X@/_6ex/[dd]_{E (\\hat{g}) }@{}[dd]|{{E (\\mathfrak {x})} }@/^6ex/[dd]^{E (g) }\\\\&@{}[r]|{{\\hspace{1.99997pt}\\alpha _{{}_{g}} \\hspace{1.99997pt}} } &&=&&@{}[l]|{{\\hspace{1.99997pt}\\alpha _{{}_{\\hat{g}}} \\hspace{1.99997pt}} } &\\\\L Y[rr]_ {\\alpha _ {{}_Y}} && E Y&&L Y[rr]_ {\\alpha _{{}_Y}} &&E Y }$ Firstly, we define the vertical composition, denoted by $\\beta \\alpha $ , of two pseudonatural transformations $\\alpha : L\\longrightarrow E, \\beta : E\\longrightarrow U $ by $(\\beta \\alpha ) _ {{}_W} :=\\beta _{{}_W}\\alpha _ {{}_W} $ ${LW[r]^{\\beta _{{}_W}\\alpha _ {{}_W}}[d]_{L(f) }@{}[dr]|{{(\\beta \\alpha ) _{{}_f}} }&U W@{}[drr]|{:=}[d]^{U(f)}&&L W[rr]^{\\alpha _{{}_W}}[d]^{L (f)}@{}[drr]|{{\\alpha _{{}_f}} }&& E W[d]_{E(f) } [r]^{\\beta _ {{}_W}}@{}[dr]|{{\\beta _{{}_f}} }& U W [d]^{U (f)}\\\\L X[r]_{\\beta _ {{}_X}\\alpha _ {{}_X}} & U X&& L X[rr]_ {\\alpha _ {{}_X}} && E X[r]_{\\beta _ {{}_X}} &U X}$ Secondly, assume that $L, E : \\mathfrak {B}\\rightarrow \\mathfrak {C}$ and $ G, J : \\mathfrak {A}\\rightarrow \\mathfrak {B}$ are pseudofunctors.", "We define the horizontal composition of two pseudonatural transformations $\\alpha : L\\longrightarrow E, \\lambda : G\\longrightarrow J $ by $\\left( \\alpha \\ast \\lambda \\right) := (\\alpha J)(L\\lambda )$ , in which $\\alpha J $ is trivially defined and $(L\\lambda ) $ is defined below $\\begin{aligned}(L\\lambda ) _{{}_W} &:=& L(\\lambda _{{}_W})\\end{aligned}\\qquad \\qquad \\qquad \\begin{aligned}(L\\lambda )_ {{}_f} &:=& \\left(\\mathfrak {l}_{{}_{\\lambda _{{}_X}G (f)}}\\right) ^{-1} \\cdot L(\\lambda _{{}_f})\\cdot \\mathfrak {l}_{{}_{J (f)\\lambda _{{}_W}}}\\end{aligned}$ Also, recall that a 2-natural transformation is just a pseudonatural transformation $\\alpha :L\\longrightarrow E $ such that its components $\\alpha _{{}_g}: E(g) \\alpha _{{}_X}\\Rightarrow \\alpha _{{}_Y} L(g) $ are identities (for all morphisms $g$ ).", "[Modification] Let $L, E:\\mathfrak {B}\\rightarrow \\mathfrak {C}$ be pseudofunctors.", "If $\\alpha , \\beta : L\\longrightarrow E $ are pseudonatural transformations, a modification $ \\Gamma : \\alpha \\Longrightarrow \\beta $ is defined by the following data: For each object $X$ of $\\mathfrak {B}$ , a 2-cell $\\Gamma _{{}_X}: \\alpha _ {{}_X}\\Rightarrow \\beta _{{}_X} $ of $\\mathfrak {C}$ ; such that: if $f: W\\rightarrow X $ is a 1-cell of $\\mathfrak {B}$ , the equation below holds.", "${ LW@/_6ex/[dd]_{\\alpha _{{}_W} }@{}[dd]|{{\\Gamma _{{}_W}} }@/^6ex/[dd]^{\\beta _{{}_W} }[rr]^{L (f) } && L X[dd]^{\\beta _ {{}_X} }&&LW[rr]^{ L (f) }[dd]_{\\alpha _ {{}_W}} &&LX@/_6ex/[dd]_{\\alpha _ {{}_X} }@{}[dd]|{{\\Gamma _ {{}_X}} }@/^6ex/[dd]^{\\beta _ {{}_X} }\\\\&@{}[r]|{{\\hspace{1.99997pt}\\beta _{{}_{f}} \\hspace{1.99997pt}} } &&=&&@{}[l]|{{\\hspace{1.99997pt}\\alpha _{{}_{f}} \\hspace{1.99997pt}} } &\\\\E W[rr]_ {E (f)} && E X&&E W[rr]_ {E (f) } &&E X }$ The three types of compositions of modifications are defined in the obvious way.", "Thereby, it is straightforward to verify that, indeed, $2\\textrm {-}{\\rm \\sf CAT}$ is a tricategory, lacking strictness/2-functoriality of the whiskering.", "In particular, we denote by $[\\mathfrak {A}, \\mathfrak {B}]_{PS} $ the 2-category of pseudofunctors $\\mathfrak {A}\\rightarrow \\mathfrak {B}$ , pseudonatural transformations and modifications.", "The bicategorical Yoneda Lemma [18] says that there is a pseudonatural equivalence $[\\mathfrak {S}, {\\rm \\sf CAT}]_ {PS} (\\mathfrak {S}(a, - ), \\mathcal {D})\\simeq \\mathcal {D}a $ given by the evaluation at the identity.", "Lemma 2.1 (Yoneda Embedding [18]) The Yoneda 2-functor $\\mathcal {Y}:\\mathfrak {A}\\rightarrow [\\mathfrak {A}^{{\\rm op}}, {\\rm \\sf CAT}]_ {PS}$ is locally an equivalence (i.e.", "it induces equivalences between the hom-categories).", "Considering pseudofunctors $L: \\mathfrak {B}\\rightarrow \\mathfrak {C}$ and $U:\\mathfrak {C}\\rightarrow \\mathfrak {B}$ , we say that $U$ is right biadjoint to $L$ if we have a pseudonatural equivalence $\\mathfrak {C}(L - , -)\\simeq \\mathfrak {B}(- , U-) $ .", "This concept can be also defined in terms of unit and counit as it is done at Definition .", "Let $U:\\mathfrak {C}\\rightarrow \\mathfrak {B}, L:\\mathfrak {B}\\rightarrow \\mathfrak {C}$ be pseudofunctors.", "$L$ is left biadjoint to $U$ if there exist pseudonatural transformations $\\eta :{\\rm Id}_ {\\mathfrak {B}} \\longrightarrow UL$ and $\\varepsilon :LU \\longrightarrow {\\rm Id}_ { \\mathfrak {C}}$ invertible modifications $s : {\\rm id}_{L} \\Longrightarrow (\\varepsilon L) (L\\eta )$ and $t : (U\\varepsilon ) (\\eta U) \\Longrightarrow {\\rm id}_{U}$ such that the following 2-cells are identities [6]: ${ {\\rm Id}_ {\\mathfrak {B}} [rr]^{\\eta }[dd]_{\\eta }&&{UL} [dd]_{\\eta UL}@{=}@/^6ex/[dddr]@{}[dddr]|{{ tL }} &&{LU} @{=}@/^6ex/[drrr]@{}[drrr]|{{ \\mathfrak {l}_{{}_{U}}^{-1} (Lt) \\mathfrak {l}_ {{}_{(U\\varepsilon ) (\\eta U)}} }}[dr]|{L\\eta U}@{}[dddr]|{{sU}}@{=}@/_6ex/[dddr] &&&\\\\& { \\hspace{5.0pt}\\eta _{(\\eta )}} &&& & {LULU}[rr]^{LU\\varepsilon }[dd]^{\\varepsilon LU} &&{LU} [dd]^{\\varepsilon }\\\\{UL} [rr]_{UL\\eta }@{=}@/_6ex/[drrr]@{}[drrr]|{{\\mathfrak {u}^{-1}_ {{}_{(L\\eta ) (\\varepsilon L)}} (Us) \\mathfrak {u}_{{}_{L}} }} &&{ULUL}[dr]|{U\\varepsilon L} && & &{\\hspace{1.99997pt}\\varepsilon _{(\\varepsilon )}} &\\\\&&& {UL} & & {LU} [rr]_{\\varepsilon } && {\\rm Id}_ {\\mathfrak {C}}}$ By definition, if a pseudofunctor $L$ is left biadjoint to $U$ , there is at least one associated data $(L\\dashv U, \\eta , \\varepsilon , s, t) $ as described above.", "Such associated data is called a biadjunction.", "Also, every biadjunction $(L\\dashv U, \\eta , \\varepsilon , s, t) $ has an associated pseudonatural equivalence $\\chi : \\mathfrak {C}(L-, -)\\simeq \\mathfrak {B}(-,U-)$ , in which $\\chi _ {{}_{(X,Z)}}: &\\mathfrak {C}(LX, Z)&\\rightarrow \\mathfrak {B}(X,UZ)\\\\&f &\\mapsto U(f)\\eta _{{}_X}\\\\&\\mathfrak {m} &\\mapsto U(\\mathfrak {m})\\ast {\\rm id}_ {{}_{\\eta _{{}_X}}}$ $\\left(\\chi _ {{}_{(g,h)}}\\right) _ {{}_f} &:=& \\left(\\mathfrak {u}_{{}_{(hf)Lg}}\\ast {\\rm id}_ {{}_{\\eta _ {{}_{X}} }}\\right) \\cdot \\left(\\mathfrak {u}_ {{}_{hf}}\\ast \\eta _ {{}_{g}}^{-1}\\right)$ Reciprocally, such a pseudonatural equivalence induces a biadjunction $(L \\dashv U, \\eta , \\varepsilon , s, t)$ .", "Similarly to the 1-dimensional case, if $(L\\dashv U, \\eta , \\varepsilon , s, t) $ is a biadjunction, the counit $\\varepsilon : LU\\longrightarrow {\\rm id}_ {\\mathfrak {C}}$ is a pseudonatural equivalence if and only if, for every pair $(X,Y) $ of objects of $\\mathfrak {C}$ , $U_{{}_{X,Y}}: \\mathfrak {C}(X,Y)\\rightarrow \\mathfrak {B}(UX, UY) $ is an equivalence (that is to say, $U$ is locally an equivalence).", "The proof is also analogous to the 1-dimensional case.", "Indeed, given a pair $(X,Y) $ of objects in $\\mathfrak {B}$ , the composition of functors ${ \\mathfrak {B}(X, Y)[r]^-{\\mathfrak {B}(\\varepsilon _ {{}_{X}}, Y) } & \\mathfrak {B}(LUX, Y)[r]^-{\\chi _{{}_{(UX, Y) }} } & \\mathfrak {B}(LX, LY)}$ is obviously isomorphic to $U_{{}_{X,Y}}:\\mathfrak {C}(X,Y)\\rightarrow \\mathfrak {B}(UX, UY)$ .", "Since $\\chi _{{}_{(UX, Y) }}$ is an equivalence, $\\varepsilon _ {{}_{X}}$ is an equivalence for every object $X$ (that is to say, it is a pseudonatural equivalence) if and only if $U $ is locally an equivalence.", "Dually, the unit of this biadjunction is a pseudonatural equivalence if and only if $L$ is locally an equivalence.", "Recall that, if the modifications $s,t$ of a biadjunction $(L\\dashv U, \\eta , \\varepsilon , s, t) $ are identities, $L, U $ are 2-functors and $\\eta , \\varepsilon $ are 2-natural transformations, then $L$ is left 2-adjoint to $U$ and $(L\\dashv U, \\eta , \\varepsilon ) $ is a 2-adjunction.", "If it exists, a birepresentation of a pseudofunctor $\\mathcal {U}:\\mathfrak {C}\\rightarrow {\\rm \\sf CAT}$ is an object $X$ of $\\mathfrak {C}$ endowed with a pseudonatural equivalence $\\mathfrak {C}(X, -)\\simeq \\mathcal {U} $ .", "When $\\mathcal {U}$ has a birepresentation, we say that $\\mathcal {U}$ is birepresentable.", "Moreover, in this case, by Lemma REF , its birepresentation is unique up to equivalence.", "Lemma 2.2 ([18]) Assume that $\\mathcal {U}: \\mathfrak {C}\\rightarrow [\\mathfrak {B}^{{\\rm op}} , {\\rm \\sf CAT}]_ {PS} $ is a pseudofunctor such that, for each object $X$ of $\\mathfrak {C}$ , $\\mathcal {U}X$ has a birepresentation $e_X: \\mathcal {U}X\\simeq \\mathfrak {B}(- , UX ) $ .", "Then there is a pseudofunctor $ U: \\mathfrak {C}\\rightarrow \\mathfrak {B}$ such that the pseudonatural equivalences $e_X $ are the components of a pseudonatural equivalence $\\mathcal {U}\\simeq \\mathfrak {B}(-, U-) $ , in which $\\mathfrak {B}(-,U-) $ denotes the pseudofunctor $\\mathfrak {C}\\rightarrow [\\mathfrak {B}^{{\\rm op}}, {\\rm \\sf CAT}]_ {PS}:\\mbox{ } X \\mapsto \\mathfrak {B}(-, UX)$ As a consequence, a pseudofunctor $L: \\mathfrak {B}\\rightarrow \\mathfrak {C}$ has a right biadjoint if and only if, for each object $X$ of $\\mathfrak {C}$ , the pseudofunctor $\\mathfrak {C}( L-, X) $ is birepresentable.", "Id est, for each object $X$ , there is an object $UX$ of $\\mathfrak {B}$ endowed with a pseudonatural equivalence $\\mathfrak {C}( L-, X) \\simeq \\mathfrak {B}(-,UX)$ .", "The natural notion of limit in our context is that of (weighted) bilimit [18], [19].", "Namely, assuming that $\\mathfrak {S}$ is a small 2-category, if $ \\mathcal {W}: \\mathfrak {S}\\rightarrow {\\rm \\sf CAT}, \\mathcal {D}:\\mathfrak {S}\\rightarrow \\mathfrak {A}$ are pseudofunctors, the (weighted) bilimit, denoted herein by $\\left\\lbrace \\mathcal {W}, \\mathcal {D}\\right\\rbrace _ {{\\rm bi}} $ , when it exists, is a birepresentation of the 2-functor $ \\mathfrak {A}^{{\\rm op}}\\rightarrow {\\rm \\sf CAT}:\\mbox{ } X \\mapsto [\\mathfrak {S}, {\\rm \\sf CAT}]_{PS}(\\mathcal {W}, \\mathfrak {A}(X, \\mathcal {D}-) ).$ Since, by the (bicategorical) Yoneda Lemma, $\\left\\lbrace \\mathcal {W}, \\mathcal {D}\\right\\rbrace _ {{\\rm bi}} $ is unique up to equivalence, we sometimes refer to it as the (weighted) bilimit.", "Finally, if $\\mathcal {W}$ and $\\mathcal {D}$ are 2-functors, recall that the (strict) weighted limit $\\left\\lbrace \\mathcal {W}, \\mathcal {D}\\right\\rbrace $ is, when it exists, a 2-representation of the 2-functor $X \\mapsto [\\mathfrak {S}, {\\rm \\sf CAT}](\\mathcal {W}, \\mathfrak {A}(X, \\mathcal {D}-) )$ , in which $[\\mathfrak {S}, {\\rm \\sf CAT}]$ is the 2-category of 2-functors $\\mathfrak {S}\\rightarrow {\\rm \\sf CAT}$ , 2-natural transformations and modifications [17].", "It is easy to see that ${\\rm \\sf CAT}$ is bicategorically complete.", "More precisely, if $\\mathcal {W}: \\mathfrak {S}\\rightarrow {\\rm \\sf CAT}$ and $ \\mathcal {D}:\\mathfrak {S}\\rightarrow {\\rm \\sf CAT}$ are pseudofunctors, then $\\left\\lbrace \\mathcal {W}, \\mathcal {D}\\right\\rbrace _ {{\\rm bi}} \\simeq [\\mathfrak {S}, {\\rm \\sf CAT}]_{PS}(\\mathcal {W}, \\mathcal {D}) .$ Moreover, from the bicategorical Yoneda Lemma of [18], we get the (strong) bicategorical Yoneda Lemma.", "Lemma 2.3 ((Strong) Yoneda Lemma) Let $\\mathcal {D}: \\mathfrak {S}\\rightarrow \\mathfrak {A}$ be a pseudofunctor between 2-categories.", "There is a pseudonatural equivalence $\\left\\lbrace \\mathfrak {S}(a, -), \\mathcal {D}\\right\\rbrace _ {{\\rm bi}}\\simeq \\mathcal {D}a $ .", "By the bicategorical Yoneda Lemma, we have a pseudonatural equivalence (in $X$ and $a$ ) $[\\mathfrak {S}, {\\rm \\sf CAT}] _ {PS} (\\mathfrak {S}(a, -), \\mathfrak {A}(X, \\mathcal {D}-))\\simeq \\mathfrak {A}(X, \\mathcal {D}a) .$ Therefore $\\mathcal {D}a$ is the bilimit $\\left\\lbrace \\mathfrak {S}(a, -), \\mathcal {D}\\right\\rbrace _{{\\rm bi}} $ .", "Recall that the usual (enriched) Yoneda embedding $\\mathfrak {A}\\rightarrow \\left[ \\mathfrak {A}^{{\\rm op}}, {\\rm \\sf CAT}\\right] $ preserves and reflects weighted limits.", "In the 2-dimensional case, we get a similar result.", "Lemma 2.4 The Yoneda embedding $\\mathcal {Y}: \\mathfrak {A}\\rightarrow \\left[ \\mathfrak {A}^{{\\rm op}}, {\\rm \\sf CAT}\\right] _ {PS} $ preserves and reflects weighted bilimits.", "By definition, a weighted bilimit $\\left\\lbrace \\mathcal {W}, \\mathcal {D}\\right\\rbrace _ {{\\rm bi}} $ exists if and only if, for each object $X$ of $\\mathfrak {A}$ , $\\mathfrak {A}(X, \\left\\lbrace \\mathcal {W}, \\mathcal {D}\\right\\rbrace _ {{\\rm bi}})\\simeq \\left[ \\mathfrak {A}, {\\rm \\sf CAT}\\right] _ {PS}(\\mathcal {W}, \\mathfrak {A}(X, \\mathcal {D}- ))\\simeq \\left\\lbrace \\mathcal {W}, \\mathfrak {A}(X, \\mathcal {D}- ) \\right\\rbrace _ {{\\rm bi}}.$ By the pointwise construction of weighted bilimits, this means that $\\left\\lbrace \\mathcal {W}, \\mathcal {D}\\right\\rbrace _ {{\\rm bi}} $ exists if and only if $\\mathcal {Y}\\left\\lbrace \\mathcal {W}, \\mathcal {D}\\right\\rbrace _{{\\rm bi}}\\simeq \\left\\lbrace \\mathcal {W}, \\mathcal {Y}\\circ \\mathcal {D}\\right\\rbrace _{{\\rm bi}}$ .", "This proves that $\\mathcal {Y}$ reflects and preserves weighted bilimits.", "Let $\\mathfrak {S}$ be a small 2-category and $\\mathcal {D}: \\mathfrak {S}\\rightarrow \\mathfrak {A}$ be a pseudofunctor.", "Consider the pseudofunctor $[\\mathfrak {S}, \\mathfrak {C}]_ {PS} \\rightarrow [ \\mathfrak {A}^{{\\rm op}}, {\\rm \\sf CAT}]_{PS} :\\mbox{ }\\mathcal {W}\\mapsto {D}_\\mathcal {W}$ in which the 2-functor ${D}_\\mathcal {W}$ is given by $X \\mapsto [\\mathfrak {S}, {\\rm \\sf CAT}]_{PS}(\\mathcal {W}, \\mathfrak {A}(X, \\mathcal {D}-) ) $ .", "By Lemma REF , we conclude that it is possible to get a pseudofunctor $\\left\\lbrace -, \\mathcal {D} \\right\\rbrace _ {{\\rm bi}} $ defined in a full sub-2-category of $[\\mathfrak {S}, {\\rm \\sf CAT}]_{PS}$ of weights $\\mathcal {W}:\\mathfrak {S}\\rightarrow {\\rm \\sf CAT}$ such that $\\mathfrak {A}$ has the bilimit $\\left\\lbrace \\mathcal {W},\\mathcal {D}\\right\\rbrace _ {{\\rm bi}}$ ." ], [ "Descent Objects", "In this section, we describe the 2-categorical limits called descent objects.", "We need both constructions, strict descent objects and descent objects [19].", "Our domain 2-category, denoted by $\\Delta $ , is the dual of that defined at Definition 2.1 in [13].", "We denote by $\\dot{\\Delta } $ the 2-category generated by the diagram ${ \\mathsf {0} [rr]^-d && \\mathsf {1}@<1.7 ex>[rrr]^-{d^0 }@<-1.7ex>[rrr]_-{d^1 } &&& \\mathsf {2}[lll]|-{s^0}@<1.7 ex>[rrr]^{\\partial ^0 }[rrr]|-{\\partial ^1}@<-1.7ex>[rrr]_{\\partial ^2} &&& \\mathsf {3} }$ with the invertible 2-cells: $\\sigma _{ik} &:& \\partial ^k d ^i\\cong \\partial ^{i}d^{k-1} ,\\hspace{2.84526pt}\\mbox{if }\\hspace{2.84526pt} i<k \\\\n_0 &:& s^0d^0\\cong {\\rm id}_{{}_\\mathsf {1}} \\\\n_1 &:& {\\rm id}_{{}_\\mathsf {1}}\\cong s^{0}d^{1}\\\\\\vartheta &:& d^1d\\cong d^0d$ satisfying the equations below: Associativity: ${ \\mathsf {0}[r]^-{d}[d]_-{d}@{}[rd]|-{{\\hspace{1.00006pt}\\vartheta \\hspace{1.00006pt}} }&\\mathsf {1}[d]|-{d^0}[r]^-{d^0}@{}[rd]|-{{\\hspace{1.00006pt}\\sigma _{01} \\hspace{1.00006pt}}}&\\mathsf {2}[d]^-{\\partial ^0}@{}[rrdd]|-{=}&&\\mathsf {3}@{}[rd]|{{\\hspace{1.00006pt}\\sigma _{02} \\hspace{1.00006pt}}}&\\mathsf {2}[l]_{\\partial ^0}@{}[rdd]|{{\\hspace{1.00006pt}\\vartheta \\hspace{1.00006pt}}}@{=}[r]&\\mathsf {2}\\\\\\mathsf {1}[r]|-{d^1}[d]_-{d^1}@{}[rrd]|-{{\\hspace{1.00006pt}\\sigma _ {12} \\hspace{1.00006pt}}}&\\mathsf {2}[r]|-{\\partial ^1}&\\mathsf {3}[d]^-{{\\rm id}_ {{}_\\mathsf {3}}}&&\\mathsf {2}@{}[rd]|-{{\\hspace{1.00006pt}\\vartheta \\hspace{1.00006pt}}}[u]^-{\\partial ^2}&\\mathsf {1}[l]|-{d^0}[u]|-{d^1}&\\\\\\mathsf {2}[rr]_{\\partial ^2}&&\\mathsf {3}&&\\mathsf {1}[u]^-{d^1}&\\mathsf {0}[l]^{d}[u]|-{d}[r]_ {d}&\\mathsf {1}[uu]_{d^0}}$ Identity: ${ \\mathsf {0} [rr]^{d}[dd]_{d}&&{\\mathsf {1}} [dd]_{d^1}@{=}@/^4ex/[dddr]@{}[dddr]|{{n_1}} &&&\\mathsf {0} @/_3ex/[ddd]_{d}@{}[ddd]|=@/^3ex/[ddd]^{d}\\\\& {\\hspace{1.00006pt}\\vartheta \\hspace{1.00006pt}} && &=\\\\{\\mathsf {1}} [rr]_{d^0}@{=}@/_4ex/[drrr]@{}[drrr]|{{n_0}} &&{\\mathsf {2}}[dr]|{s^0} \\\\&&& {\\mathsf {1}} && {\\mathsf {1}} }$ The 2-category $\\Delta $ is, herein, the full sub-2-category of $\\dot{\\Delta } $ with objects $\\mathsf {1}, \\mathsf {2}, \\mathsf {3}$ .", "We denote the inclusion by ${\\rm j}: \\Delta \\rightarrow \\dot{\\Delta } $ .", "In fact, the 2-category $\\dot{\\Delta } $ is the locally preordered 2-category freely generated by the diagram and 2-cells described above.", "Moreover, $\\Delta $ is the 2-category freely generated by the corresponding diagram and the 2-cells $\\sigma _{01}, \\sigma _{02}, \\sigma _{12}, n_0, n_1 $ .", "Let $\\mathfrak {A}$ be a 2-category and $\\mathcal {A}: \\Delta \\rightarrow \\mathfrak {A}$ be a 2-functor.", "If the weighted bilimit $\\left\\lbrace \\dot{\\Delta } (\\mathsf {0}, {\\rm j}-), \\mathcal {A}\\right\\rbrace _ {{\\rm bi}} $ exists, we say that $\\left\\lbrace \\dot{\\Delta } (\\mathsf {0}, {\\rm j}-), \\mathcal {A}\\right\\rbrace _ {{\\rm bi}} $ is the descent object of $\\mathcal {A}$ .", "Analogously, when it exists, we call the (strict) weighted 2-limit $\\left\\lbrace \\dot{\\Delta } (\\mathsf {0}, {\\rm j}-), \\mathcal {A}\\right\\rbrace $ the strict descent object of $\\mathcal {A}$ .", "Assuming that $\\mathcal {D}: \\dot{\\Delta }\\rightarrow \\mathfrak {A}$ is a pseudofunctor, we have a pseudonatural transformation $\\dot{\\Delta } (\\mathsf {0}, {\\rm j}- )\\longrightarrow \\mathfrak {A}(\\mathcal {D}\\mathsf {0}, \\mathcal {D}\\circ {\\rm j}- )$ given by the evaluation of $\\mathcal {D}$ .", "By the definition of weighted bilimit, if $\\mathcal {D}\\circ {\\rm j}$ has a descent object, this pseudonatural transformation induces a comparison 1-cell $\\mathcal {D}\\mathsf {0}\\rightarrow \\left\\lbrace \\dot{\\Delta } (\\mathsf {0}, {\\rm j}-), \\mathcal {D}\\circ {\\rm j}\\right\\rbrace _ {{\\rm bi}}.", "$ Analogously, if $\\mathcal {D}$ is a 2-functor, we get a comparison $\\mathcal {D}\\mathsf {0}\\rightarrow \\left\\lbrace \\dot{\\Delta } (\\mathsf {0}, {\\rm j}-), \\mathcal {D}\\circ {\\rm j}\\right\\rbrace $ , provided that the strict descent object of $\\mathcal {D}\\circ {\\rm j}$ exists.", "[Effective Descent Diagrams] We say that a 2-functor $\\mathcal {D}: \\dot{\\Delta }\\rightarrow \\mathfrak {A}$ is of effective descent if $\\mathfrak {A}$ has the descent object of $\\mathcal {D}\\circ {\\rm j}$ and the comparison $\\displaystyle \\mathcal {D}\\mathsf {0}\\rightarrow \\left\\lbrace \\dot{\\Delta } (\\mathsf {0}, {\\rm j}-), \\mathcal {D}\\circ {\\rm j}\\right\\rbrace _ {{\\rm bi}} $ is an equivalence.", "We say that $\\mathcal {D}$ is of strict descent if $\\mathfrak {A}$ has the strict descent object of $\\mathcal {D}\\circ {\\rm j}$ and the comparison $\\mathcal {D}\\mathsf {0}\\rightarrow \\left\\lbrace \\dot{\\Delta } (\\mathsf {0}, {\\rm j}-), \\mathcal {D}\\circ {\\rm j}\\right\\rbrace $ is an isomorphism.", "Lemma 3.1 Strict descent objects are descent objects.", "Thereby, strict descent diagrams are of effective descent as well.", "Also, if $\\mathfrak {A}$ has strict descent objects, a 2-functor $\\mathcal {D}: \\dot{\\Delta }\\rightarrow \\mathfrak {A}$ is of effective descent if and only if the comparison $\\mathcal {D}\\mathsf {0}\\rightarrow \\left\\lbrace \\dot{\\Delta } (\\mathsf {0}, {\\rm j}-), \\mathcal {D}\\circ {\\rm j}\\right\\rbrace $ is an equivalence.", "Lemma 3.2 Assume that $\\mathcal {A}, \\mathcal {B}, \\mathcal {D}: \\dot{\\Delta } \\rightarrow \\mathfrak {A}$ are 2-functors.", "If there are a 2-natural isomorphism $\\mathcal {A}\\longrightarrow \\mathcal {B}$ and a pseudonatural equivalence $\\mathcal {B}\\longrightarrow \\mathcal {D}$ , then $\\mathcal {A}$ is of strict descent if and only if $\\mathcal {B}$ is of strict descent; $\\mathcal {B}$ is of effective descent if and only if $\\mathcal {D}$ is of effective descent.", "We say that an effective descent diagram $\\mathcal {D}:\\dot{\\Delta }\\rightarrow \\mathfrak {B}$ is preserved by a pseudofunctor $L:\\mathfrak {B}\\rightarrow \\mathfrak {C}$ if $L\\circ \\mathcal {D}$ is of effective descent.", "Also, $\\mathcal {D}:\\dot{\\Delta }\\rightarrow \\mathfrak {B}$ is said to be an absolute effective descent diagram if $ L\\circ \\mathcal {D}$ is of effective descent for any pseudofunctor $L$ .", "In this setting, a pseudofunctor $L:\\mathfrak {B}\\rightarrow \\mathfrak {C}$ is said to reflect absolute effective descent diagrams if, whenever a 2-functor $\\mathcal {D}:\\dot{\\Delta }\\rightarrow \\mathfrak {B}$ is such that $L\\circ \\mathcal {D}$ is an absolute effective descent diagram, $\\mathcal {D}$ is of effective descent.", "Moreover, we say herein that a pseudofunctor $L:\\mathfrak {B}\\rightarrow \\mathfrak {C}$ creates absolute effective descent diagrams if $L$ reflects absolute effective descent diagrams and, whenever a diagram $\\mathcal {A}: \\Delta \\rightarrow \\mathfrak {B}$ is such that $L\\circ \\mathcal {A}\\simeq \\mathcal {D}\\circ {\\rm j}$ for some absolute effective descent diagram $\\mathcal {D}: \\dot{\\Delta } \\rightarrow \\mathfrak {C}$ , there is a diagram $\\mathcal {B}: \\dot{\\Delta }\\rightarrow \\mathfrak {B}$ such that $L\\circ \\mathcal {B}\\simeq \\mathcal {D}$ and $\\mathcal {B}\\circ {\\rm j}= \\mathcal {A}$ .", "Recall that right 2-adjoints preserve strict descent diagrams and right biadjoints preserve effective descent diagrams.", "Also, the usual (enriched) Yoneda embedding $\\mathfrak {A}\\rightarrow \\left[ \\mathfrak {A}^{{\\rm op}}, {\\rm \\sf CAT}\\right]$ preserves and reflects strict descent diagrams, and, from Lemma REF , we get: Lemma 3.3 The Yoneda embedding $\\mathcal {Y}: \\mathfrak {A}\\rightarrow \\left[ \\mathfrak {A}^{{\\rm op}}, {\\rm \\sf CAT}\\right] _ {PS} $ preserves and reflects effective descent diagrams.", "The dual notion of descent object is that of codescent object, described by Lack [12] and Le Creurer, Marmolejo, Vitale [13].", "It is, of course, the descent object in the opposite 2-category.", "The 2-category ${\\rm \\sf CAT}$ is ${\\rm \\sf CAT}$ -complete.", "In particular, ${\\rm \\sf CAT}$ has strict descent objects.", "More precisely, if $\\mathcal {A}: \\Delta \\rightarrow {\\rm \\sf CAT}$ is a 2-functor, then $\\left\\lbrace \\dot{\\Delta }(\\mathsf {0}, {\\rm j}-), \\mathcal {A}\\right\\rbrace \\cong \\left[ \\Delta , {\\rm \\sf CAT}\\right] \\left( \\dot{\\Delta }(\\mathsf {0}, {\\rm j}-), \\mathcal {A}\\right) .$ Thereby, we can describe the strict descent object of $\\mathcal {A}:\\Delta \\rightarrow {\\rm \\sf CAT}$ explicitly as follows: Objects are 2-natural transformations $\\mathsf {f}: \\dot{\\Delta }(\\mathsf {0}, {\\rm j}-)\\longrightarrow \\mathcal {A}$ .", "We have a bijective correspondence between such 2-natural transformations and pairs $(f, \\varrho _ {{}_{\\mathsf {f}}})$ in which $f$ is an object of $ \\mathcal {A}\\mathsf {1} $ and $\\varrho _ {{}_{\\mathsf {f}}}: \\mathcal {A}(d^1)f\\rightarrow \\mathcal {A}(d^0)f $ is an isomorphism in $ \\mathcal {A}\\mathsf {2} $ satisfying the following equations: Associativity: $\\left(\\mathcal {A}(\\partial ^0)(\\varrho _ {{}_{\\mathsf {f} }} )\\right) \\left( \\mathcal {A}(\\sigma _ {{}_{02}}) _ {{}_{f}}\\right)\\left(\\mathcal {A}(\\partial ^2)(\\varrho _ {{}_{\\mathsf {f}}} )\\right)\\left(\\mathcal {A}(\\sigma _ {{}_{12}} ) ^{-1}_ {{}_f}\\right) = \\left(\\mathcal {A}(\\sigma _ {{}_{01}}) _ {{}_{f}}\\right)\\left(\\mathcal {A}(\\partial ^1)(\\varrho _ {{}_{\\mathsf {f}}})\\right) $ Identity: $\\left(\\mathcal {A}(n_0) _ {{}_f}\\right)\\left(\\mathcal {A}(s^0) (\\varrho _ {{}_{\\mathsf {f}}}) \\right)\\left(\\mathcal {A}(n_1) _ {{}_f}\\right) = {\\rm id}_ {{}_{f}} $ If $\\mathsf {f}: \\dot{\\Delta }(\\mathsf {0}, {\\rm j}-)\\longrightarrow \\mathcal {A}$ is a 2-natural transformation, we get such pair by the correspondence $\\mathsf {f}\\mapsto (\\mathsf {f} _ {{}_{\\mathsf {1} }}(d), \\mathsf {f} _ {{}_{\\mathsf {2} }}(\\vartheta )) $ .", "The morphisms are modifications.", "In other words, a morphism $\\mathsf {m} : \\mathsf {f}\\rightarrow \\mathsf {h} $ is determined by a morphism $\\mathfrak {m}: f\\rightarrow h $ such that $\\mathcal {A}(d^0)(\\mathfrak {m} )\\varrho _ {{}_\\mathsf {f}} = \\varrho _ {{}_h}\\mathcal {A}(d^1)(\\mathfrak {m} ) $ ." ], [ "Biadjoint Triangles", "In this section, we give our main theorem on biadjoint triangles, Theorem REF , and its strict version, Theorem REF .", "Let $L: \\mathfrak {B}\\rightarrow \\mathfrak {C}$ and $U: \\mathfrak {C}\\rightarrow \\mathfrak {B}$ be pseudofunctors, and $(L\\dashv U, \\eta , \\varepsilon , s, t)$ be a biadjunction.", "We denote by $\\chi : \\mathfrak {C}(L-, -)\\simeq \\mathfrak {B}(-,U-)$ its associated pseudonatural equivalence as described in Remark .", "In this setting, for every pair $(X,Y)$ of objects of $\\mathfrak {B}$ , we have an induced diagram $\\mathcal {D}_Y^X :\\dot{\\Delta }\\rightarrow {\\rm \\sf CAT}$ ${1.7pc}{ \\mathfrak {B}(X, Y) [d]|-{L_{{}_{X,Y}}}\\\\ \\mathfrak {C}(LX, LY)@<-2.5ex>[rrr]_-{\\mathfrak {C}(LX, L(\\eta _{{}_Y} ))}@<2.5ex>[rrr]^-{L_{{}_{{}_{X,ULY}}}\\circ \\hspace{1.99997pt} \\chi _ {{}_{{}_{(X,LY)}}}} &&& \\mathfrak {C}(LX, LULY )[lll]|-{\\mathfrak {C}(LX,\\varepsilon _ {{}_{LY}})}@<-2.5 ex>[rrrr]_-{\\mathfrak {C}(LX, LUL(\\eta _{{}_Y} ))}[rrrr]|-{\\mathfrak {C}(LX, L(\\eta _ {{}_{ULY}}))}@<2.5ex>[rrrr]^-{L_{{}_{{}_{X,(UL)^2Y}}}\\circ \\hspace{1.99997pt}\\chi _ {{}_{{}_{(X,LULY)}}}} &&&& \\mathfrak {C}(LX, L(UL)^2Y) }\\qquad \\mathrm {(\\mathcal {D}_Y^X)}$ in which the images of the 2-cells of $\\dot{\\Delta } $ by $\\mathcal {D}_Y^X: \\dot{\\Delta }\\rightarrow {\\rm \\sf CAT}$ are defined as: $\\begin{aligned}&\\mathcal {D}_Y^X(\\vartheta ) _{{}_g} & : = & L\\left(\\eta _ {{}_g}^{-1}\\right)\\cdot \\mathfrak {l}_{{}_{\\eta _ {{}_Y} g}} \\\\&\\mathcal {D}_Y^X (\\sigma _{12}) _{{}_f} & : =& \\left(L\\eta \\right) _ {{}_{\\eta _ {{}_Y} }}\\ast {\\rm id}_ {{}_f}\\\\&\\mathcal {D}_Y ^X(n_1)_{{}_f} &: = &s_{{}_Y}\\ast {\\rm id}_ {{}_f}\\end{aligned}\\qquad \\begin{aligned}&\\mathcal {D}_Y^X (\\sigma _{01})_{{}_f} & : = & \\mathfrak {l}_ {{}_{UL(U(f)\\eta _ {{}_{X}} )\\eta _ {{}_{X}}}}\\cdot \\left( L\\eta \\right) _ {{}_{U(f)\\eta _ {{}_{X}} }}^{-1}\\\\&\\mathcal {D}_Y^X(\\sigma _{02}) _{{}_f} &: = & L\\left(\\mathfrak {u}_ {{}_{L(\\eta _ {{}_Y}) f }}\\ast {\\rm id}_ {{}_{\\eta _ {{}_X}}}\\right)\\cdot \\mathfrak {l}_ {{}_{UL(\\eta _ {{}_Y})L(U(f)\\eta _ {{}_X})}}\\\\&\\mathcal {D}_Y^X(n_0)_{{}_f} &: = &\\left( {\\rm id}_ {{}_f}\\ast s_ {{}_X} ^{-1} \\right)\\cdot \\left( \\varepsilon _ {{}_f} ^{-1}\\ast {\\rm id}_ {{}_{\\eta _ {{}_X}}}\\right)\\cdot \\left( {\\rm id}_ {{}_{\\varepsilon _ {{}_{LY}}}}\\ast \\mathfrak {l}_ {{}_ {U(f)\\eta _ {{}_X}}}^{-1}\\right)\\end{aligned}$ We claim that $\\mathcal {D}^X_Y $ is well defined.", "In fact, by the axioms of naturality and associativity of Definition  (of pseudonatural transformation), for every morphism $g\\in \\mathfrak {B}(X,Y) $ , we have the equality ${ LX[d]_{L(\\eta _ {{}_{X}})}[r]^{L(g)}@{}[rd]|{{\\gamma }} &LY [d]^{L(\\eta _ {{}_{Y}})}[rd]^{L(\\eta _ {{}_{Y}})} &&LX[d]_{L(\\eta _ {{}_{X}}) }[r]^{L(g)}[rd]|{L(\\eta _ {{}_{X}} )}@{}[rrd]|{{\\gamma } }@{}[rdd]|{{\\left( L\\eta \\right)_ {{}_{\\eta _ {{}_X}}}^{-1}}}&LY[rd]^{L(\\eta _ {{}_Y})}&\\\\LULX[r]_{LUL(g)}[rd]_{LUL(\\eta _ {{}_{X}} )}@{}[rrd]|{{\\widehat{LU(\\gamma )} }}&LULY[rd]|{LUL(\\eta _ {{}_{Y}}) }@{}[r]|{{\\left( L\\eta \\right)_ {{}_{\\eta _ {{}_Y}}}^{-1} }}&LULY[d]^{L(\\eta _{{}_{ULY}})}@{}[r]|{=}&LULX[rd]_{LUL(\\eta _ {{}_X}) } &LULX[d]|{L(\\eta _ {{} _ {ULX}})}[r]^{LUL(g)}@{}[rd]|{{ (L\\eta )_ {{}_ {UL(g)}}^{-1} } }&LULY[d] ^{L(\\eta _ {{}_ {ULY}})}\\\\&LULULX[r]_{LULUL(g)}&LULULY &&LULULX [r]_{LULUL(g)}&LULULY }$ in which $\\begin{aligned}\\gamma &: = \\mathfrak {l}_{{}_{UL(g)\\eta _ {{}_X}}}^{-1}\\cdot \\mathcal {D}_Y^X(\\vartheta ) _{{}_g} = (L\\eta ) _ {{}_g} ^{-1}\\end{aligned}\\qquad \\qquad \\begin{aligned}\\widehat{LU(\\gamma )} &: = (\\mathfrak {l}\\mathfrak {u}) _ {{}_{LUL(g)L(\\eta _{{}_{X}})}}^{-1}\\cdot LU(\\gamma )\\cdot (\\mathfrak {l}\\mathfrak {u}) _ {{}_{L(\\eta _{{}_{X}})L(g)}}\\\\\\end{aligned}$ By the definition of $\\mathcal {D}_Y^X $ given above, this is the same as saying that the equation ${\\mathcal {D}_Y^X \\mathsf {3}@{=}[rr]@{}[rrdd]|{{\\mathcal {D}_Y^X(\\sigma _ {12})^{-1}}}&&\\mathcal {D}_Y^X \\mathsf {3}@{}[rd]|{{\\mathcal {D}_Y^X(\\sigma _ {02})}}&\\mathcal {D}_Y^X \\mathsf {2}@{=}[r]@{}[rdd]|{{\\mathcal {D}_Y^X(\\vartheta )}}[l]_{\\mathcal {D}_Y^X(\\partial ^0)}&\\mathcal {D}_Y^X \\mathsf {2}@{}[rdd]|{=}&\\mathcal {D}_Y^X \\mathsf {0}[r]^{\\mathcal {D}^X_Y (d) }[dd]|{\\mathcal {D}_Y^X (d) }@{}[rdd]|{{\\mathcal {D}^X_Y(\\vartheta ) }}&\\mathcal {D}_Y^X \\mathsf {1}[dd]|{\\mathcal {D}_Y^X(d^0)}[rr]^{\\mathcal {D}_Y^X(d^0)}@{}[rrdd]|{{\\mathcal {D}_Y^X(\\sigma _ {01})}}&&\\mathcal {D}_Y^X \\mathsf {2}[dd]|{\\mathcal {D}_Y^X(\\partial ^0 )}\\\\&&\\mathcal {D}_Y^X \\mathsf {2}@{}[rd]|{{\\mathcal {D}_Y^X(\\vartheta )}}[u]^{\\mathcal {D}_Y^X(\\partial ^2)}&\\mathcal {D}_Y^X \\mathsf {1}[l]^{\\mathcal {D}_Y^X(d^0)}[u]_{\\mathcal {D}_Y^X(d^1)}&&&&&\\\\\\mathcal {D}_Y^X \\mathsf {2}[uu]|{\\mathcal {D}_Y^X(\\partial ^1) }&&\\mathcal {D}_Y^X \\mathsf {1} [ll]^{\\mathcal {D}_Y^X(d^1) }[u]^{\\mathcal {D}_Y^X(d^1)}&\\mathcal {D}_Y^X \\mathsf {0} [l]^{\\mathcal {D}_Y^X(d)}[u]_{\\mathcal {D}_Y^X(d)}[r]_{\\mathcal {D}_Y^X(d)}&\\mathcal {D}_Y^X \\mathsf {1}[uu]|{\\mathcal {D}_Y^X (d^0) }&\\mathcal {D}_Y^X \\mathsf {1} [r]_{\\mathcal {D}_Y^X(d^1)}&\\mathcal {D}_Y^X \\mathsf {2} [rr]_{\\mathcal {D}_Y^X(\\partial ^1) }&&\\mathcal {D}_Y^X \\mathsf {3}}$ holds, which is equivalent to the usual equation of associativity given in Definition .", "Also, by the naturality of the modification $s : {\\rm id}_{L} \\Longrightarrow (\\varepsilon L) (L\\eta )$ (see Definition ), for every morphism $g\\in \\mathfrak {B}(X,Y)$ , the pasting of 2-cells ${LX[rrrr]^{L(g)}[rd]|{L(\\eta _ {{}_X})}@{=}[dd]&&&&LY[ld]|{L(\\eta _ {{}_Y}) }@{=}[dd]\\\\&LULX@{}[l]|{{s_{{}_X} ^{-1}}}[rr]|{LUL(g)}[ld]|{\\varepsilon _ {{}_{LX}} }&@{}[u]|{{(L\\eta )_{{}_g}^{-1}}}@{}[d]|{{(\\varepsilon L) _ {{}_g}^{-1}}}&LULY@{}[r]|{{s_{{}_Y}} }[rd]|{\\varepsilon _ {{}_{LY}}}&\\\\LX[rrrr]_{L(g)}&&&&LY}$ is equal to the identity $L(g)\\Rightarrow L(g) $ in $\\mathfrak {C}$ .", "This is equivalent to say that ${ \\mathcal {D}_Y^X\\mathsf {0} [rr]^-{\\mathcal {D}_Y^X(d)}[dd]|-{\\mathcal {D}_Y^X(d)}&&{\\mathcal {D}_Y^X\\mathsf {1}} [dd]|-{\\mathcal {D}_Y^X(d^1)}@{=}@/^7ex/[dddr]@{}[dddr]|-{{\\mathcal {D}_Y^X(n_1)}} &&&\\mathcal {D}_Y^X\\mathsf {0} @/_3ex/[ddd]_-{\\mathcal {D}_Y^X(d)}@{}[ddd]|=@/^3ex/[ddd]^-{\\mathcal {D}_Y^X(d)}\\\\& {\\hspace{1.00006pt}\\mathcal {D}_Y^X(\\vartheta )} && &=\\\\{\\mathcal {D}_Y^X\\mathsf {1}} [rr]|{\\mathcal {D}_Y^X(d^0)}@{=}@/_7ex/[drrr]@{}[drrr]|{{\\mathcal {D}_Y^X(n_0)}} &&{\\mathcal {D}_Y^X\\mathsf {2}}[dr]|-{\\mathcal {D}_Y^X(s^0)} \\\\&&& {\\mathcal {D}_Y^X\\mathsf {1}} && {\\mathcal {D}_Y^X\\mathsf {1}} }$ holds, which is the usual identity equation of Definition .", "Thereby it completes the proof that indeed $\\mathcal {D}_ Y^X $ is well defined.", "As in the enriched case, we also need to consider another special 2-functor induced by a biadjoint triangle.", "Let $(E\\dashv R, \\rho , \\mu , v,w)$ and $(L\\dashv U, \\eta , \\varepsilon , s,t)$ be biadjunctions such that the triangle ${ \\mathfrak {A}[rr]^-{J}[dr]_-{E}&&\\mathfrak {B}[dl]^-{L}\\\\&\\mathfrak {C}& }$ is commutative.", "In this setting, for each object $Y$ of $\\mathfrak {B}$ , we define the 2-functor $\\mathcal {A}_ Y : \\Delta \\rightarrow \\mathfrak {A}$ ${ RLY@<-2.5ex>[rrrr]_-{RL(\\eta _{{}_Y})}@<2.5ex>[rrrr]^-{RL(U(\\mu _{{}_{LY}})\\eta _{{}_{JRLY}})\\rho _ {{}_{RLY}}} &&&& RLULY [llll]|-{R(\\varepsilon _{{}_{LY}})}@<-2.5 ex>[rrrrr]_-{RLUL(\\eta _{{}_{Y}})}[rrrrr]|-{RL(\\eta _{{}_{ULY}})}@<2.5ex>[rrrrr]^-{ RL(U(\\mu _{{}_{LULY}})\\eta _ {{}_{JRLULY}})\\rho _{{}_{RLULY}}} &&&&& RLULULY }\\qquad \\mathrm {(\\mathcal {A}_Y)}$ in which $\\mathcal {A}_Y (\\sigma _{12}) &: =& (RL\\eta )_{{}_{\\eta _ {{}_{Y}} }}\\\\\\mathcal {A}_Y (n_1) &: =& \\mathfrak {r}_ {{}_{\\varepsilon _ {{}_{LY}}\\cdot L(\\eta _ {{}_{Y}}) }}^{-1} R(s_ {{}_{Y}})\\cdot \\mathfrak {r}_ {{}_{LY}}\\\\\\mathcal {A}_Y (n_0) &: =&\\left(w_{{}_{LY}}\\right)\\\\&&\\cdot \\left({\\rm id}_ {{}_{R(\\mu _ {{}_{LY}} )}}\\ast \\left( \\mathfrak {r}_ {{}_{ERLY}}^{-1}\\cdot R(s _ {{}_{JRLY}}^{-1})\\cdot \\mathfrak {r}_ {{}_{\\varepsilon _ {{}_{ERLY}}L(\\eta _ {{}_{JRLY}}) }}\\right)\\cdot {\\rm id}_ {{}_{\\rho _ {{}_{RLY}} }}\\right)\\\\&&\\cdot \\left( (R\\varepsilon )^{-1}_ {{}_{\\mu _ {{}_{LY}} }}\\ast {\\rm id}_ {{}_{RL(\\eta _ {{}_{JRLY}})\\rho _ {{}_{RLY}} }}\\right)\\\\&&\\cdot \\left( {\\rm id}_ {{}_{R(\\varepsilon _ {{}_{LY}}) }}\\ast (\\mathfrak {r}\\mathfrak {l}) ^{-1}_ {{}_{U(\\mu _ {{}_{LY}})\\eta _ {{}_{JRLY}} }}\\ast {\\rm id}_ {{}_{\\rho _ {{}_{RLY}} }}\\right)$ $\\mathcal {A}_Y (\\sigma _{02}) &: =&\\left( (\\mathfrak {r}\\mathfrak {l})_{{}_{ U(\\mu _ {{}_{RLULY}})\\eta _ {{}_{JRLULY}} }}\\ast {\\rm id}_ {{}_{\\rho _ {{}_{RLULY}} RL(\\eta _ {{}_{Y}}) }}\\right)\\cdot \\left( \\left(RLU\\mu L\\right)\\left(RL\\eta J RL\\right)\\left( \\rho RL\\right)\\right) _ {{}_{\\eta _ {{}_{Y}} }}\\\\&&\\cdot \\left( {\\rm id}_ {{}_{RLUL(\\eta _ {{}_{Y}}) }}\\ast (\\mathfrak {r}\\mathfrak {l}) ^{-1} _ {{}_{U(\\mu _ {{}_{LY}})\\eta _ {{}_{JRLY}} }}\\ast {\\rm id}_ {{}_{\\rho _ {{}_{RLY}} }}\\right)\\\\\\mathcal {A}_Y (\\sigma _{01}) &: =&\\left( (\\mathfrak {r}\\mathfrak {l})_{{}_{U(\\mu _{{}_{LULY}})\\eta _ {{}_{JRLULY}} }}\\ast \\rho _{{}_{RL(U(\\mu _{{}_{LY}})\\eta _ {{}_{JRLY}}) }}\\ast {\\rm id}_ {{}_{\\rho _{{}_{RLY}} }}\\right)\\\\&&\\cdot \\left(\\left( (RLU\\mu L)(RL\\eta JRL)\\right) _ {{}_{U(\\mu _{{}_{LY}})\\eta _ {{}_{JRLY}} }}\\ast \\rho _ {{}_{\\rho _{{}_{RLY}} }}\\right)\\\\&&\\cdot \\left( {\\rm id}_ {{}_{RLUL(U(\\mu _{{}_{LY}})\\eta _ {{}_{JRLY}})RLU(\\mu _{{}_{ERLY}}) }}\\ast \\left( RL\\eta J\\right) _{{}_{\\rho _{{}_{RLY}}}}\\ast {\\rm id}_ {{}_{\\rho _{{}_{RLY}}}}\\right)\\\\&&\\cdot \\left( {\\rm id}_ {{}_{RLUL(U(\\mu _{{}_{LY}})\\eta _ {{}_{JRLY}}) }}\\ast \\left((\\mathfrak {r}\\mathfrak {l}\\mathfrak {u})^{-1}_{{}_{\\mu _ {{}_{ERLY}}E(\\rho _ {{}_{RLY}})}}\\cdot RLU(v_{{}_{RLY}})\\cdot (\\mathfrak {r}\\mathfrak {l}\\mathfrak {u})_ {{}_{ERLY}}\\right)\\ast {\\rm id}_ {{}_{RL(\\eta _ {{}_{JRLY}})\\rho _ {{}_{RLY}} }}\\right)\\\\&&\\cdot \\left(\\left( RL\\eta \\right) ^{-1}_{{}_{U(\\mu _ {{}_{LY}})\\eta _ {{}_{JRLY}} }}\\ast {\\rm id}_ {{}_{\\rho _ {{}_{RLY}} }}\\right)$ Theorem 4.1 (Biadjoint Triangle) Let $(E\\dashv R, \\rho , \\mu , v,w)$ and $(L\\dashv U, \\eta , \\varepsilon , s,t)$ be biadjunctions such that ${ \\mathfrak {A}[rr]^-{J}[dr]_-{E}&&\\mathfrak {B}[dl]^-{L}\\\\&\\mathfrak {C}& }$ is a commutative triangle of pseudofunctors.", "Assume that, for each pair of objects $(Y\\in \\mathfrak {B}, A\\in \\mathfrak {A})$ , the 2-functor ${ \\mathfrak {B}(JA, Y) [d]|-{L_{{}_{JA,Y}}}\\\\ \\mathfrak {C}(LJA, LY)@<-2.5ex>[rrr]_-{\\mathfrak {C}(LJA, L(\\eta _{{}_Y} ))}@<2.5ex>[rrr]^-{L_{{}_{{}_{JA,ULY}}}\\circ \\hspace{1.99997pt} \\chi _ {{}_{{}_{(JA,LY)}}}} &&& \\mathfrak {C}(LJA, LULY )[lll]|-{\\mathfrak {C}(LJA,\\varepsilon _ {{}_{LY}})}@<-2.5 ex>[rrr]_-{\\mathfrak {C}(LJA, LUL(\\eta _{{}_Y} ))}[rrr]|-{\\mathfrak {C}(LJA, L(\\eta _ {{}_{ULY}}))}@<2.5ex>[rrr]^-{L_{{}_{{}_{JA,(UL)^2Y}}}\\circ \\hspace{1.99997pt}\\chi _ {{}_{{}_{(JA,LULY)}}}} &&& \\mathfrak {C}(LJA, L(UL)^2Y) }\\qquad \\mathrm {(\\mathcal {D}_Y^{JA})}$ is of effective descent.", "The pseudofunctor $J$ has a right biadjoint if and only if, for every object $Y$ of $\\mathfrak {B}$ , the descent object of the diagram $\\mathcal {A}_Y: \\Delta \\rightarrow \\mathfrak {A}$ exists in $\\mathfrak {A}$ .", "In this case, $J$ is left biadjoint to $G$ , defined by $GY:= \\left\\lbrace \\dot{\\Delta } (\\mathsf {0}, {\\rm j}- ), \\mathcal {A}_Y\\right\\rbrace _{{\\rm bi}}.$ We denote by $\\xi : \\mathfrak {C}(E-, -)\\simeq \\mathfrak {A}(-,R-)$ the pseudonatural equivalence associated to the biadjunction $(E\\dashv R, \\rho , \\mu , v,w)$ (see Remark ).", "For each object $A$ of $\\mathfrak {A}$ and each object $Y$ of $\\mathfrak {B}$ , the components of $\\xi $ induce a pseudonatural equivalence $\\psi : \\mathcal {D}_Y^{JA}\\circ {\\rm j}\\longrightarrow \\mathfrak {A}(A, \\mathcal {A}_Y -) $ in which $\\psi _ {{}_{\\mathsf {1}}} &:=& \\xi _ {{}_{(A, LY)}}: \\mathfrak {C}(EA, LY)\\rightarrow \\mathfrak {A}(A, RLY) \\\\\\psi _ {{}_{\\mathsf {2}}} &:=& \\xi _ {{}_{(A, LULY)}}: \\mathfrak {C}(EA, LULY)\\rightarrow \\mathfrak {A}(A, RLULY) \\\\\\psi _ {{}_{\\mathsf {3}}} &:=& \\xi _ {{}_{(A, LULULY)}}: \\mathfrak {C}(EA, LULULY)\\rightarrow \\mathfrak {A}(A, RLULULY)$ $\\begin{aligned}\\left(\\psi _ {{}_{s^0}}\\right)_{{}_f} &:=& \\mathfrak {r}_{{}_{\\varepsilon _ {{}_{LY}}f}}\\ast {\\rm id}_{{}_{\\rho _ {{}_A}}}\\\\\\left(\\psi _ {{}_{d^1}}\\right)_{{}_f} &:=& \\mathfrak {r}_{{}_{L(\\eta _ {{}_{Y}})f}}\\ast {\\rm id}_{{}_{\\rho _ {{}_A}}}\\\\\\end{aligned}\\qquad \\qquad \\qquad \\begin{aligned}\\left(\\psi _ {{}_{\\partial ^1}}\\right) _ {{}_f} &:=& \\mathfrak {r}_{{}_{L(\\eta _ {{}_{ULY}})f}}\\ast {\\rm id}_{{}_{\\rho _ {{}_A}}} \\\\\\left( \\psi _ {{}_{\\partial ^2}}\\right) _ {{}_f} &:=& \\mathfrak {r}_{{}_{LUL(\\eta _ {{}_{Y}})f}}\\ast {\\rm id}_{{}_{\\rho _ {{}_A}}}\\end{aligned}$ $\\left(\\psi _ {{}_{d^0}}\\right)_{{}_f} &:=& \\left( (\\mathfrak {r}\\mathfrak {l})_ {{}_{U(f)\\eta _ {{}_{JA}}}}\\ast {\\rm id}_ {{}_{\\rho _ {{}_{A}}}}\\right)\\\\&&\\cdot \\left( {\\rm id}_ {{}_{RLU(f)}}\\ast \\left((\\mathfrak {r}\\mathfrak {l}\\mathfrak {u})_ {{}_{EA}}^{-1}\\cdot RLU(v_{{}_A}^{-1})\\cdot (\\mathfrak {r}\\mathfrak {l}\\mathfrak {u})_{{}_{\\mu _ {{}_{EA}}E(\\rho _ {{}_{A}})}}\\right)\\ast {\\rm id}_ {{}_{RLU(\\eta _ {{}_{JA}})\\rho _ {{}_{A}}}}\\right)\\\\&&\\cdot \\left( {\\rm id}_ {{}_{RLU(f)RL(\\mu _ {{}_{EA})}}}\\ast \\left((RL\\eta J) \\rho \\right)_ {{}_{\\rho _ {{}_{A}}}}^{-1}\\right)\\cdot \\left(\\left( (RLU\\mu )(RL\\eta JR)(\\rho R)\\right) ^{-1}_{{}_f}\\ast {\\rm id}_ {{}_{\\rho _ {{}_{A}}}}\\right)\\\\&&\\cdot \\left((\\mathfrak {r}\\mathfrak {l})^{-1}_{{}_{U(\\mu _ {{}_{LY})\\eta _ {{}_{JRLY}} }}}\\ast {\\rm id}_ {{}_{\\rho _ {{}_{RLY}}R(f)\\rho _ {{}_{A}}}}\\right)\\\\\\left(\\psi _ {{}_{\\partial ^0}}\\right)_{{}_f} &:=&\\left( (\\mathfrak {r}\\mathfrak {l})_ {{}_{U(f)\\eta _ {{}_{JA}}}}\\ast {\\rm id}_ {{}_{\\rho _ {{}_{A}}}}\\right)\\\\&&\\cdot \\left( {\\rm id}_ {{}_{RLU(f)}}\\ast \\left((\\mathfrak {r}\\mathfrak {l}\\mathfrak {u})_ {{}_{EA}}^{-1}\\cdot RLU(v_{{}_A}^{-1})\\cdot (\\mathfrak {r}\\mathfrak {l}\\mathfrak {u})_{{}_{\\mu _ {{}_{EA}}E(\\rho _ {{}_{A}})}}\\right)\\ast {\\rm id}_ {{}_{RLU(\\eta _ {{}_{JA}})\\rho _ {{}_{A}}}}\\right)\\\\&&\\cdot \\left( {\\rm id}_ {{}_{RLU(f)RL(\\mu _ {{}_{EA})}}}\\ast \\left((RL\\eta J) \\rho \\right)_ {{}_{\\rho _ {{}_{A}}}}^{-1}\\right)\\cdot \\left(\\left( (RLU\\mu )(RL\\eta JR)(\\rho R)\\right) ^{-1}_{{}_f}\\ast {\\rm id}_ {{}_{\\rho _ {{}_{A}}}}\\right)\\\\&&\\cdot \\left((\\mathfrak {r}\\mathfrak {l})^{-1}_{{}_{U(\\mu _ {{}_{LULY})\\eta _ {{}_{JRLULY}} }}}\\ast {\\rm id}_ {{}_{\\rho _ {{}_{RLULY}}R(f)\\rho _ {{}_{A}}}}\\right)$ First of all, we assume that $\\mathcal {D}_Y^{JA}$ is of effective descent for every object $A$ of $\\mathfrak {A}$ and every object $Y$ of $\\mathfrak {B}$ .", "Then the descent object of $\\mathcal {D}_Y^{JA}\\circ {\\rm j}\\simeq \\mathfrak {A}(A, \\mathcal {A}_Y -) $ is $\\mathcal {D}_Y^{JA}\\mathsf {0}$ .", "Moreover, since this is true for all objects $A$ of $\\mathfrak {A}$ , we conclude that the descent object of $ \\mathcal {Y}\\circ \\mathcal {A}_Y $ is $\\mathfrak {C}(J-,Y): \\mathfrak {A}^{{\\rm op}}\\rightarrow {\\rm \\sf CAT}$ .", "If, furthermore, $\\mathfrak {A}$ has the descent object of $\\mathcal {A}_ Y $ , we get that $\\mathcal {Y}\\left\\lbrace \\dot{\\Delta } (\\mathsf {0}, {\\rm j}- ), \\mathcal {A}_Y\\right\\rbrace _{{\\rm bi}} $ is also a descent object of $\\mathcal {Y}\\circ \\mathcal {A}_Y $ .", "Therefore we get a pseudonatural equivalence $\\mathfrak {C}\\left(J-, Y\\right)\\simeq \\mathfrak {A}\\left( -, \\left\\lbrace \\dot{\\Delta } (\\mathsf {0}, {\\rm j}- ), \\mathcal {A}_Y\\right\\rbrace _{{\\rm bi}}\\right) .$ This proves that $J$ is left biadjoint to $G$ , provided that the descent object of $\\mathcal {A}_ Y $ exists for every object $Y$ of $\\mathfrak {B}$ .", "Reciprocally, if $J$ is left biadjoint to a pseudofunctor $G$ , since $\\mathfrak {C}(-,GY)\\simeq \\mathfrak {C}(J-, Y) $ is the descent object of $\\mathfrak {A}(-, \\mathcal {A}_Y -) $ , we conclude that $GY$ is the descent object of $\\mathcal {A}_ Y $ .", "We establish below the obvious dual version of Theorem REF , which is the relevant theorem to the usual context of pseudopremonadicity [13].", "For being able to give such dual version, we have to employ the observations given in Remark on codescent objects.", "Also, if $(L\\dashv U, \\eta , \\varepsilon , s,t)$ is a biadjunction, we need to consider its associated pseudonatural equivalence $\\tau : \\mathfrak {C}(-, U-)\\rightarrow \\mathfrak {B}(L-,-)$ .", "In particular, $\\tau _ {{}_{(X,Z)}}: \\mathfrak {C}(X, UZ)\\rightarrow \\mathfrak {B}(LX,Z):\\qquad \\qquad f \\mapsto \\varepsilon _{{}_Z}L(f);\\qquad \\mathfrak {m}\\mapsto {\\rm id}_ {{}_{\\varepsilon _{{}_Z} }}\\ast L(\\mathfrak {m}) $ Theorem 4.2 (Biadjoint Triangle) Let $(E\\dashv R, \\rho , \\mu , v,w)$ and $(L\\dashv U, \\eta , \\varepsilon , s,t)$ be biadjunctions such that ${ \\mathfrak {A}[rr]^-{J}[dr]_-{R}&&\\mathfrak {B}[dl]^-{U}\\\\&\\mathfrak {C}& }$ is a commutative triangle of pseudofunctors.", "Assume that, for each pair of objects $(Y\\in \\mathfrak {B}, A\\in \\mathfrak {A})$ , the 2-functor $\\dot{\\Delta }\\rightarrow {\\rm \\sf CAT}$ ${ \\mathfrak {B}(Y, JA) [d]|-{U_{{}_{Y,JA}}}\\\\ \\mathfrak {C}(UY, UJA)@<2.5ex>[rrr]^-{\\mathfrak {C}(U(\\varepsilon _{{}_A} ), UJA)}@<-2.5ex>[rrr]_-{U_{{}_{{}_{LUY,JA}}}\\circ \\hspace{1.99997pt} \\tau _ {{}_{{}_{(UY,JA)}}}} &&& \\mathfrak {C}(ULUY, UJA )[lll]|-{\\mathfrak {C}(\\eta _ {{}_{UY}}, UJA)}@<2.5 ex>[rrrr]^-{\\mathfrak {C}(ULU(\\varepsilon _{{}_Y}),UJA)}[rrrr]|-{\\mathfrak {C}(U(\\varepsilon _{{}_{LUY}}),UJA)}@<-2.5ex>[rrrr]_-{U_{{}_{{}_{(LU)^2Y, JA}}}\\circ \\hspace{1.99997pt}\\tau _ {{}_{{}_{(ULUY, JA)}}}} &&&& \\mathfrak {C}(U(LU)^2Y, UJA) }$ (with omitted 2-cells) is of effective descent.", "We have that $J$ has a left biadjoint if and only if, for every object $Y$ of $\\mathfrak {B}$ , $\\mathfrak {A}$ has the codescent object of the diagram (with the obvious 2-cells) $\\Delta ^{{\\rm op}}\\rightarrow \\mathfrak {A}$ ${ EUY[rrrrr]|-{E(\\eta _{{}_{UY}})} &&&&& EULUY @<-2.5ex>[lllll]_-{EU(\\varepsilon _{{}_Y})}@<2.5ex>[lllll]^-{\\mu _ {{}_{EUY}} EU(\\varepsilon _{{}_{JEUY}}L(\\rho _{{}_{UY}}))}&&&&& EULULUY@<-2.5 ex>[lllll]_-{EULU(\\varepsilon _{{}_{Y}})}[lllll]|-{EU(\\varepsilon _{{}_{LUY}})}@<2.5ex>[lllll]^-{ \\mu _{{}_{EULUY}} EU(\\varepsilon _ {{}_{JEULUY}} L(\\rho _{{}_{ULUY}}))} }$" ], [ "Strict Version", "The techniques employed to prove strict versions of Theorem REF are virtually the same.", "We just need to repeat the same constructions, but, now, by means of strict descent objects and 2-adjoints.", "For instance, we have: Theorem 4.3 (Strict Biadjoint Triangle) Let $(L\\dashv U, \\eta , \\varepsilon , s, t)$ be a biadjunction between 2-functors and $(E\\dashv R, \\rho , \\mu ) $ be a 2-adjunction such that the triangle of 2-functors ${ \\mathfrak {A}[rr]^-{J}[dr]_-{E}&&\\mathfrak {B}[dl]^-{L}\\\\&\\mathfrak {C}& }$ commutes and $\\left(\\eta J\\right): J\\longrightarrow UE $ is a 2-natural transformation.", "We assume that, for every pair of objects $(A\\in \\mathfrak {A}, Y\\in \\mathfrak {B})$ , the diagram $\\mathcal {D}_Y^{JA}:\\dot{\\Delta }\\rightarrow {\\rm \\sf CAT}$ induced by $(L\\dashv U, \\eta , \\varepsilon , s, t)$ is of strict descent.", "The 2-functor $J$ has a right 2-adjoint if and only if, for every object $Y$ of $\\mathfrak {B}$ , the strict descent object of $\\mathcal {A}_Y: \\Delta \\rightarrow \\mathfrak {A}$ exists in $\\mathfrak {A}$ .", "In particular, we have the setting of Theorem REF .", "Therefore, again, we can define $\\psi : \\mathcal {D}_Y^{JA}\\circ {\\rm j}\\longrightarrow \\mathfrak {A}(A, \\mathcal {A}_Y -) $ as it was done in the proof of Theorem REF .", "However, since $(E\\dashv R, \\rho , \\mu ) $ is a 2-adjunction, $J, E, R, L, U $ are 2-functors and $(\\eta J) $ is a 2-natural transformation, the components $\\psi _ {{}_{d^0}}$ , $\\psi _ {{}_{d^1}}$ , $\\psi _ {{}_{s^0}}$ , $\\psi _ {{}_{\\partial ^0}}$ , $\\psi _ {{}_{\\partial ^1}}$ , $\\psi _ {{}_{\\partial ^2}} $ are identities.", "Thereby $\\psi $ is a 2-natural transformation.", "Moreover, since $(E\\dashv R, \\rho , \\mu ) $ is a 2-adjunction, $\\psi $ is a pointwise isomorphism.", "Thus it is a 2-natural isomorphism.", "Firstly, we assume that $\\mathcal {D}_Y^{JA}$ is of strict descent for every object $A$ of $\\mathfrak {A}$ and every object $Y$ of $\\mathfrak {B}$ .", "Then the strict descent object of $\\mathfrak {A}(A, \\mathcal {A}_Y -) $ is $\\mathcal {D}_Y^{JA}\\mathsf {0}$ .", "If, furthermore, $\\mathfrak {A}$ has the strict descent object of $\\mathcal {A}_ Y$ , we get a 2-natural isomorphism $\\mathfrak {C}\\left(J-, Y\\right)\\cong \\mathfrak {A}\\left( -, \\left\\lbrace \\dot{\\Delta } (\\mathsf {0}, {\\rm j}- ), \\mathcal {A}_Y\\right\\rbrace \\right) .$ This proves that $J$ is left 2-adjoint, provided that the strict descent object of $\\mathcal {A}_ Y $ exists for every object $Y$ of $\\mathfrak {B}$ .", "Reciprocally, if $J$ is left 2-adjoint to a 2-functor $G$ , since $\\mathfrak {C}(-,GY)\\cong \\mathfrak {C}(J-, Y) $ is the strict descent object of $\\mathfrak {A}(-, \\mathcal {A}_Y -) $ , we conclude that $GY$ is the strict descent object of $\\mathcal {A}_ Y $ ." ], [ "Pseudoprecomonadicity", "A pseudomonad [11], [14] is the same as a doctrine, whose definition can be found in page 123 of [18], while a pseudocomonad is the dual notion.", "Similarly to the 1-dimensional case, for each pseudocomonad $\\mathcal {T}$ on a 2-category $\\mathfrak {C}$ , there is an associated right biadjoint to the forgetful 2-functor $\\mathsf {L}: \\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}\\rightarrow \\mathfrak {C}$ , in which $\\mathsf {Ps}$ -$\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}$ is the 2-category of pseudocoalgebras [11] of Definition .", "Also, every biadjunction $(L\\dashv U, \\eta , \\varepsilon , s, t) $ induces a comparison pseudofunctor and an Eilenberg-Moore factorization [13] ${\\mathfrak {B}[r]^-{\\mathcal {K}}[rd]_-{L}& \\mathsf {Ps} \\textrm {-} \\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}[d]\\\\&\\mathfrak {C}}$ in which $\\mathcal {T}$ denotes the induced pseudocomonad.", "Before proving Corollary REF which is a consequence of Theorem REF in the context of pseudocomonads, we sketch some basic definitions and known results needed to fix notation and show Lemma REF .", "Some of them are related to the formal theory of pseudo(co)monads developed by Lack [11].", "There, it is employed the coherence result of tricategories [4] (and, hence, with due adaptations, the formal theory developed therein works for any tricategory).", "[Pseudocomonad] A pseudocomonad $\\mathcal {T}= (\\mathcal {T}, \\varpi , \\varepsilon , \\Lambda , \\delta , \\mathsf {s} ) $ on a 2-category $\\mathfrak {C}$ is a pseudofunctor $(\\mathcal {T}, \\mathfrak {t}) : \\mathfrak {C}\\rightarrow \\mathfrak {C}$ with Pseudonatural transformations: $\\begin{aligned}\\varpi &:& \\mathcal {T}\\longrightarrow \\mathcal {T}^2\\end{aligned}\\qquad \\qquad \\begin{aligned}\\varepsilon &:&\\mathcal {T}\\longrightarrow {\\rm id}_ {{}_{\\mathfrak {C}}}\\end{aligned}$ Invertible modifications: $\\Lambda &:& (\\varpi \\mathcal {T})(\\varpi )\\Longrightarrow (\\mathcal {T}\\varpi )(\\varpi )\\\\\\mathsf {s} &:&(\\varepsilon \\mathcal {T})(\\varpi )\\Longrightarrow {\\rm id}_ {{}_{\\mathcal {T}}}\\\\\\delta &:&{\\rm id}_ {{}_{\\mathcal {T}}}\\Longrightarrow (\\mathcal {T}\\varepsilon )(\\varpi )$ such that the following equations hold: Associativity: ${ \\mathcal {T}[d]_{\\varpi }[r]^{\\varpi }@{}[rd]|{{\\Lambda }} &\\mathcal {T}^2 [d]^{\\varpi \\mathcal {T}}[rd]^{\\varpi \\mathcal {T}} &&\\mathcal {T}[d]_{\\varpi }[r]^{\\varpi }[rd]|{\\varpi }@{}[rrd]|{{\\Lambda } }@{}[rdd]|{{\\Lambda }}&\\mathcal {T}^2[rd]^{\\varpi \\mathcal {T}}&\\\\\\mathcal {T}^2[r]_{\\mathcal {T}\\varpi }[rd]_{\\mathcal {T}\\varpi }@{}[rrd]|{{\\widehat{\\mathcal {T}\\Lambda }} }&\\mathcal {T}^3[rd]|{\\mathcal {T}\\varpi \\mathcal {T}}@{}[r]|{{\\Lambda \\mathcal {T}}}&\\mathcal {T}^3[d]^{\\varpi \\mathcal {T}^2}@{}[r]|{=}&\\mathcal {T}^2[rd]_{\\mathcal {T}\\varpi } &\\mathcal {T}^2[d]|{\\varpi \\mathcal {T}}[r]^{\\mathcal {T}\\varpi }@{}[rd]|{{\\varpi _ {{}_{\\varpi }}^{-1} } }&\\mathcal {T}^3[d] ^{\\varpi \\mathcal {T}^2}\\\\&\\mathcal {T}^3[r]_{\\mathcal {T}^2\\varpi }&\\mathcal {T}^4 &&\\mathcal {T}^3 [r]_{\\mathcal {T}^2\\varpi }&\\mathcal {T}^4}$ Identity: ${ &\\mathcal {T}[dl]_-{\\varpi }[dr]^-{\\varpi }@{}[dd]|{{\\Lambda }}&&&&\\mathcal {T}[d]|-{\\varpi }&\\\\\\mathcal {T}^2[dr]_-{\\mathcal {T}\\varpi }&&\\mathcal {T}^2[dl]^-{\\varpi \\mathcal {T}}&&&\\mathcal {T}^2[dl]|-{\\mathcal {T}\\varpi }[dr]|-{\\varpi \\mathcal {T}}@{=}[dd]&\\\\&\\mathcal {T}^3[d]|-{\\mathcal {T}\\varepsilon \\mathcal {T}}&&=&\\mathcal {T}^3@{}[r]|-{{\\delta \\mathcal {T}}}[dr]|-{\\mathcal {T}\\varepsilon \\mathcal {T}}&&\\mathcal {T}^3[dl]|-{\\mathcal {T}\\varepsilon \\mathcal {T}}@{}[l]|-{{{\\widehat{\\mathcal {T}\\mathsf {s}}} } }\\\\&\\mathcal {T}^2&&&&\\mathcal {T}^2&}$ in which $\\widehat{\\mathcal {T}\\mathsf {s}}, \\widehat{\\mathcal {T}\\Lambda } $ denote “corrections” of domain and codomain given by the isomorphisms induced by the pseudofunctor $\\mathcal {T}$ .", "That is to say, $\\begin{aligned}\\widehat{\\mathcal {T}\\mathsf {s}} &: =& \\mathfrak {t}^{-1} _ {{}_{(\\varepsilon \\mathcal {T}) (\\varpi )}}(\\mathcal {T}\\mathsf {s} )\\mathfrak {t}_ {{}_{\\mathcal {T}^2 }}\\end{aligned}\\qquad \\qquad \\begin{aligned}\\widehat{\\mathcal {T}\\Lambda } &: =& \\mathfrak {t}^{-1}_{{}_{(\\mathcal {T}\\varpi ) (\\varpi )}}(\\mathcal {T}\\Lambda ) \\mathfrak {t}_ {{}_{(\\varpi \\mathcal {T})(\\varpi )}}\\end{aligned}$ [Pseudocoalgebras] Let $\\mathcal {T}= (\\mathcal {T}, \\varpi , \\varepsilon , \\Lambda , \\delta , \\mathsf {s} ) $ be a pseudocomonad in $\\mathfrak {C}$ .", "We define the objects, 1-cells and 2-cells of the 2-category $\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}$ as follows: Objects: pseudocoalgebras are defined by $\\mathsf {z}= (Z, \\varrho _{{}_{\\mathsf {z}}}, \\varsigma _ {{}_{\\mathsf {z}}}, \\Omega _{{}_\\mathsf {z}} )$ in which $\\varrho _{{}_{\\mathsf {z}}}: Z\\rightarrow \\mathcal {T}Z $ is a morphism in $\\mathfrak {C}$ and $\\begin{aligned}\\varsigma _ {{}_{\\mathsf {z}}}&:& {\\rm id}_ {{}_{Z}} \\Rightarrow \\varepsilon _ {{}_Z}\\varrho _ {{}_{\\mathsf {z}}}\\end{aligned}\\qquad \\qquad \\begin{aligned}\\Omega _ {{}_{\\mathsf {z}}} &:& \\varpi _ {{}_{Z}} \\varrho _{{}_{\\mathsf {z}}}\\Rightarrow \\mathcal {T}(\\varrho _{{}_{\\mathsf {z}}}) \\varrho _{{}_{\\mathsf {z}}}\\end{aligned}$ are invertible 2-cells of $\\mathfrak {C}$ such that the equations ${ Z[d]_{\\varrho _ {{}_\\mathsf {z}} }[r]^{\\varrho _ {{}_\\mathsf {z}} }@{}[rd]|{{\\Omega _ {{}_{\\mathsf {z}}} }} &\\mathcal {T}Z [d]^{\\varpi _{{}_{Z}} }[rd]^{\\varpi _{{}_Z} } &&Z[d]_{\\varrho _ {{}_{\\mathsf {z}}} }[r]^{\\varrho _ {{}_{\\mathsf {z}}} }[rd]|{\\varrho _ {{}_{\\mathsf {z}}} }@{}[rrd]|{{\\Omega _ {{}_{\\mathsf {z}}} } }@{}[rdd]|{{\\Omega _ {{}_{\\mathsf {z}}}}}&\\mathcal {T}Z[rd]^{\\varpi _{{}_Z}}&\\\\\\mathcal {T}Z [r]_{\\mathcal {T}(\\varrho _ {{}_{\\mathsf {z}}} ) }[rd]_{\\mathcal {T}(\\varrho _ {{}_{\\mathsf {z}}}) }@{}[rrd]|{{\\widehat{\\mathcal {T}(\\Omega _ {{}_{\\mathsf {z}}}) }} }&\\mathcal {T}^2 Z[rd]|{(\\mathcal {T}\\varpi ) _ {{}_{Z}} }@{}[r]|{{\\Lambda _ {{}_Z} }}&\\mathcal {T}^2 Z [d]^{\\varpi _ {{}_{ \\mathcal {T}Z}} }@{}[r]|{=}&\\mathcal {T}Z[rd]_{\\mathcal {T}(\\varrho _ {{}_{\\mathsf {z}}}) } &\\mathcal {T}Z [d]|{\\varpi _ {{}_{Z}}}[r]^{\\mathcal {T}(\\varrho _ {{}_{\\mathsf {z}}} ) }@{}[rd]|{{\\varpi _ {{}_{\\varrho _ {{}_{\\mathsf {z}}} }}^{-1} } }&\\mathcal {T}^2 Z[d] ^{\\varpi _{{}_{\\mathcal {T}Z}}}\\\\&\\mathcal {T}^2 Z [r]_{\\mathcal {T}^2(\\varrho _ {{}_{\\mathsf {z}}} ) }&\\mathcal {T}^3 Z &&\\mathcal {T}^2 Z [r]_{\\mathcal {T}^2(\\varrho _ {{}_{\\mathsf {z}}})}&\\mathcal {T}^3 Z}$ ${ &Z [dl]_-{\\varrho _ {{}_{\\mathsf {z}}} }[dr]^-{\\varrho _ {{}_{\\mathsf {z}}} }@{}[dd]|{{\\Omega _ {{}_{\\mathsf {z}}} }}&&&&Z [d]|-{\\varrho _ {{}_{\\mathsf {z}}} }&\\\\\\mathcal {T}Z[dr]_-{\\mathcal {T}(\\varrho _ {{}_{\\mathsf {z}}}) }&&\\mathcal {T}Z [dl]^-{\\varpi _{{}_{Z}} }&&&\\mathcal {T}Z[dl]|-{\\mathcal {T}(\\varrho _ {{}_{\\mathsf {z}}} ) }[dr]|-{\\varpi _ {{}_Z} }@{=}[dd]&\\\\&\\mathcal {T}^2 Z [d]|-{(\\mathcal {T}\\varepsilon ) _ {{}_{Z}} }&&=&\\mathcal {T}^2 Z @{}[r]|-{{\\widehat{\\mathcal {T}(\\varsigma _ {{}_{\\mathsf {z}}})}} }[dr]|-{(\\mathcal {T}\\varepsilon ) _ {{}_{Z}} }&&\\mathcal {T}^2 Z [dl]|-{(\\mathcal {T}\\varepsilon ) _{{}_{Z}} }@{}[l]|-{{\\delta _ {{}_{Z}} } }\\\\&\\mathcal {T}Z&&&&\\mathcal {T}Z&}$ are satisfied, in which $\\begin{aligned}\\widehat{\\mathcal {T}(\\varsigma _ {{}_{\\mathsf {z}}})} &:= & \\mathfrak {t}^{-1} _ {{}_{\\varepsilon _ {{}_{Z}} \\varrho _ {{}_{\\mathsf {z} }} }} \\mathcal {T}(\\varsigma _ {{}_{\\mathsf {z}}})\\mathfrak {t}_ {{}_{Z }}\\end{aligned}\\qquad \\qquad \\begin{aligned}\\widehat{\\mathcal {T}(\\Omega _ {{}_{\\mathsf {z}}}) } &:= &\\mathfrak {t}^{-1} _ {{}_{\\varpi _ {{}_{Z}} \\varrho _ {{}_{\\mathsf {z} }} }} \\mathcal {T}(\\Omega _ {{}_{\\mathsf {z}}})\\mathfrak {t}_ {{}_{\\mathcal {T}(\\varrho _ {{}_{\\mathsf {z}}} )\\varrho _ {{}_{\\mathsf {z} }} }}\\end{aligned}$ Morphisms: $\\mathcal {T}$ -pseudomorphisms $\\mathsf {f}:\\mathsf {x}\\rightarrow \\mathsf {z} $ are pairs $\\mathsf {f} = (f, \\varrho _ {{}_{f}} ^{-1} ) $ in which $f: X\\rightarrow Z $ is a morphism in $\\mathfrak {C}$ and $\\varrho _ {{}_{\\mathsf {f}}}: \\mathcal {T}(f) \\varrho _{{}_{\\mathsf {x} }} \\Rightarrow \\varrho _{{}_{\\mathsf {z} }}f $ is an invertible 2-cell of $\\mathfrak {C}$ such that, defining $\\widehat{\\mathcal {T}(\\varrho _ {{}_{\\mathsf {f}}} ^{-1})} : = \\mathfrak {t}^{-1} _ {{}_{\\mathcal {T}(f) \\varrho _ {{}_{\\mathsf {x} }} }} \\mathcal {T}(\\varrho _ {{}_{\\mathsf {f}}} ^{-1})\\mathfrak {t}_ {{}_{\\varrho _ {{}_{\\mathsf {z}}}f }}$ , ${ X[d]_{\\varrho _ {{}_{\\mathsf {x}}}}[r]^{f}@{}[rd]|{{\\varrho _ {{}_{\\mathsf {f}}} ^{-1} }} &Z [d]^{\\varrho _ {{}_{\\mathsf {z}}}}[rd]^{\\varrho _ {{}_{\\mathsf {z}}}} &&X[d]_{\\varrho _ {{}_{\\mathsf {x}}} }[r]^{f}[rd]|{\\varrho _ {{}_{\\mathsf {x}}} }@{}[rrd]|{{\\varrho _ {{}_{\\mathsf {f} }} ^{-1} } }@{}[rdd]|{{\\Omega _ {{}_{\\mathsf {x}}} }}&Z[rd]^{\\varrho _{{}_{\\mathsf {z}}} }&\\\\\\mathcal {T}X[r]_{\\mathcal {T}(f)}[rd]_{\\mathcal {T}(\\varrho _ {{}_{\\mathsf {x}}} )}@{}[rrd]|{{\\widehat{\\mathcal {T}(\\varrho _ {{}_{\\mathsf {f}}} ^{-1})}}}&\\mathcal {T}Z[rd]|{\\mathcal {T}(\\varrho _ {{}_{\\mathsf {z}}} ) }@{}[r]|{{\\Omega _ {{}_{\\mathsf {z}}} } }&\\mathcal {T}Z [d]^{\\varpi _ {{}_{Z}} }@{}[r]|{=}&\\mathcal {T}X [rd]_{\\mathcal {T}(\\varrho _ {{}_{\\mathsf {x} }} ) } &\\mathcal {T}X[d]|{\\varpi _{{}_{X}} }[r]^{\\mathcal {T}(f)}@{}[rd]|{{ \\varpi _ {{}_{f}}^{-1} } }&\\mathcal {T}Z [d] ^{\\varpi _ {{}_{Z}} }\\\\&\\mathcal {T}^2 X[r]_{\\mathcal {T}^2 (f)}&\\mathcal {T}^2 Z &&\\mathcal {T}^2 X [r]_{\\mathcal {T}^2 (f) }&\\mathcal {T}^2 Z}$ holds and the 2-cell below is the identity.", "${X[rrrr]|{f}[rd]|{\\varrho _ {{}_{\\mathsf {x}}}}@{=}[dd]&&&&Z[ld]|{\\varrho _ {{}_{\\mathsf {z}}} }@{=}[dd]\\\\&\\mathcal {T}X@{}[l]|{{\\varsigma _ {{}_{\\mathsf {x}}} }}[rr]|{\\mathcal {T}(f)}[ld]|{\\varepsilon _ {{}_{X}} }&@{}[u]|{{\\varrho _ {{}_{\\mathsf {f}}} }}@{}[d]|{{(\\varepsilon ) _ {{}_f}^{-1}}}&\\mathcal {T}Z @{}[r]|{{\\varsigma _ {{}_{\\mathsf {z}}}^{-1}}}[rd]|{\\varepsilon _ {{}_{Z}}}&\\\\X[rrrr]|{f}&&&&Y}$ 2-cells: a $\\mathcal {T}$ -transformation between $\\mathcal {T}$ -pseudomorphisms $\\mathsf {m} : \\mathsf {f}\\Rightarrow \\mathsf {h} $ is a 2-cell $\\mathfrak {m} : f\\Rightarrow h $ in $\\mathfrak {C}$ such that the equation below holds.", "${ X@/_6ex/[dd]_{f }@{}[dd]|{{\\mathfrak {m} } }@/^6ex/[dd]^{h }[rr]^{\\varrho _ {{}_{\\mathsf {x}}} } && \\mathcal {T}X[dd]^{\\mathcal {T}(h) }&&X[rr]^{ \\varrho _ {{}_{\\mathsf {x}}} }[dd]_{f } &&\\mathcal {T}X@/_6ex/[dd]_{\\mathcal {T}(f) }@{}[dd]|{{\\mathcal {T}(\\mathfrak {m} ) } }@/^6ex/[dd]^{\\mathcal {T}(h) }\\\\&@{}[r]|{{\\hspace{1.99997pt}\\varrho _{{}_{h}} \\hspace{1.99997pt}} } &&=&&@{}[l]|{{\\hspace{1.99997pt}\\varrho _{{}_{f}} \\hspace{1.99997pt}} } &\\\\Z[rr]_ {\\varrho _ {{}_{\\mathsf {z}}} } && \\mathcal {T}Z&&Z[rr]_ {\\varrho _ {{}_{\\mathsf {z}}} } &&\\mathcal {T}Z }$ If $\\mathcal {T}= (\\mathcal {T}, \\varpi , \\varepsilon , \\Lambda , \\delta , \\mathsf {s} ) $ is a pseudocomonad on $\\mathfrak {C}$ , then $\\mathcal {T}$ induces a biadjunction $(\\mathsf {L}\\dashv \\mathsf {U}, \\varrho , \\varepsilon , \\underline{\\mathsf {s}}, \\underline{\\mathsf {t} })$ in which $\\mathsf {L}, \\mathsf {U} $ are defined by $\\begin{aligned}\\mathsf {L}: \\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}& \\rightarrow \\mathfrak {C}\\\\\\mathsf {z}= (Z, \\varrho _{{}_{\\mathsf {z}}}, \\varsigma _ {{}_{\\mathsf {z}}}, \\Omega _{{}_\\mathsf {z}} )&\\mapsto Z\\\\\\mathsf {f} = (f, \\varrho _ {{}_{f}} ^{-1} )&\\mapsto f\\\\\\mathsf {m} &\\mapsto \\mathfrak {m}\\end{aligned}\\qquad \\qquad \\begin{aligned}\\mathsf {U} : \\mathfrak {C}&\\rightarrow \\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}\\\\Z &\\mapsto \\left( \\mathcal {T}(Z), \\varpi _ {{}_{Z}}, \\mathsf {s} _ {{}_{Z}}, \\Lambda _ {{}_{Z}}\\right)\\\\f &\\mapsto \\left(\\mathcal {T}(f), \\varpi _ {{}_{f}}^{-1}\\right)\\\\\\mathfrak {m} &\\mapsto \\mathcal {T}(\\mathfrak {m}) \\end{aligned}$ Reciprocally, we know that each biadjunction $(L\\dashv U, \\eta , \\varepsilon , s, t) $ induces a pseudocomonad $\\mathcal {T}= (LU, L\\eta U, \\varepsilon , (L\\eta ) _ {{}_{\\eta _ {{}_{U}} }}^{-1} , \\widehat{\\left(Lt\\right)}, sU ) $ Lemma REF gives some further aspects of these constructions (which follows from calculations on the formal theory of pseudocomonads in 2-${\\rm \\sf CAT}$ ).", "Lemma 5.1 Let $L: \\mathfrak {B}\\rightarrow \\mathfrak {C}$ be a pseudofunctor.", "A biadjunction $(L\\dashv U, \\eta , \\varepsilon , s, t)$ induces commutative triangles ${\\mathfrak {B}[r]^-{\\mathcal {K}}[rd]_-{L}& \\mathsf {Ps} \\textrm {-} \\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}[d]^{\\mathsf {L} } &&\\mathfrak {C}[r]^-{U}[rd]_-{\\mathsf {U}}& \\mathfrak {B}[d]^{\\mathcal {K}}\\\\&\\mathfrak {C}&&& \\mathsf {Ps} \\textrm {-} \\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}}$ in which $\\mathcal {T}= (\\mathcal {T}, \\varpi , \\varepsilon , \\Lambda , \\delta , \\mathsf {s} ) $ is the pseudocomonad induced by $(L\\dashv U, \\eta , \\varepsilon , s, t)$ , $(\\mathsf {L}\\dashv \\mathsf {U}, \\varrho , \\varepsilon , \\underline{\\mathsf {s}}, \\underline{\\mathsf {t} })$ is the biadjunction induced by $\\mathcal {T}$ and $\\mathcal {K}: \\mathfrak {B}\\rightarrow \\mathsf {Ps} \\textrm {-} \\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}$ is the unique (up to pseudonatural isomorphism) comparison pseudofunctor making the triangles above commutative.", "Namely, $\\mathcal {K}: &\\mathfrak {B}&\\rightarrow \\mathsf {Ps} \\textrm {-} \\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}\\\\&Y &\\mapsto \\left(LY, L(\\eta _ {{}_{Y}}), s_ {{}_{Y}}^{-1}, \\left(L\\eta \\right) _ {{}_{\\eta _ {{}_{Y}} } }^{-1} \\right)\\\\&g &\\mapsto \\left(L(g), \\left( L\\eta \\right) _{{}_{g}}^{-1}\\right)\\\\&\\mathfrak {m}&\\mapsto L(\\mathfrak {m})$ Furthermore, we have the obvious equalities $\\begin{aligned}L(\\eta _ {{}_{Y}}) &=&\\varrho _ {{}_{\\mathcal {K}Y}}\\end{aligned}\\qquad \\qquad \\begin{aligned}\\varpi _ {{}_{ LY }} &=& (L\\eta U) _ {{}_{LY}}.\\end{aligned}$ Proposition 5.2 Let $\\mathcal {T}= (\\mathcal {T}, \\varpi , \\varepsilon , \\Lambda , \\delta , \\mathsf {s} ) $ be a pseudocomonad on $\\mathfrak {C}$ .", "Given $\\mathcal {T}$ -pseudocoalgebras $\\mathsf {x}= (X, \\varrho _{{}_{\\mathsf {x}}}, \\varsigma _ {{}_{\\mathsf {x}}}, \\Omega _{{}_\\mathsf {x}} ), \\mathsf {z}= (Z, \\varrho _{{}_{\\mathsf {z}}}, \\varsigma _ {{}_{\\mathsf {z}}}, \\Omega _{{}_\\mathsf {z}} ),$ the category $\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}(\\mathsf {x}, \\mathsf {z}) $ is the strict descent object of the diagram $\\mathbb {T}_{\\mathsf {z}}^{\\mathsf {x}}: \\Delta \\rightarrow {\\rm \\sf CAT}$ ${ \\mathfrak {C}(\\mathsf {L}\\mathsf {x}, \\mathsf {L}\\mathsf {z})@<-2.5ex>[rrr]_-{\\mathfrak {C}(\\mathsf {L}\\mathsf {x}, \\varrho _ {{}_{\\mathsf {z}}} )}@<2.5ex>[rrr]^-{\\mathfrak {C}(\\varrho _ {{}_\\mathsf {x}}, \\mathcal {T}\\mathsf {L} \\mathsf {z} )\\circ \\hspace{1.99997pt}\\mathcal {T}_ {{}_{(\\mathsf {L}\\mathsf {x},\\mathsf {L}\\mathsf {z})}} } &&& \\mathfrak {C}(\\mathsf {L}\\mathsf {x}, \\mathcal {T}\\mathsf {L}\\mathsf {z} )[lll]|-{\\mathfrak {C}(\\mathsf {L}\\mathsf {x},\\varepsilon _ {{}_{\\mathsf {L}\\mathsf {z} }})}@<-2.5 ex>[rrr]_-{\\mathfrak {C}(\\mathsf {L}\\mathsf {x}, \\mathcal {T}(\\varrho _{{}_\\mathsf {z}}) )}[rrr]|-{\\mathfrak {C}(\\mathsf {L}\\mathsf {x}, \\varpi _ {{}_{\\mathsf {L}\\mathsf {z}}})}@<2.5ex>[rrr]^-{\\mathfrak {C}(\\varrho _ {{}_\\mathsf {x}}, \\mathcal {T}\\mathsf {L} \\mathsf {z} )\\circ \\hspace{1.99997pt}\\mathcal {T}_ {{}_{(\\mathsf {L}\\mathsf {x},\\mathcal {T}\\mathsf {L}\\mathsf {z})}}} &&& \\mathfrak {C}(\\mathsf {L}\\mathsf {x}, \\mathcal {T}^2 \\mathsf {L}\\mathsf {z} ) }\\qquad \\mathrm {(\\mathbb {T}_{\\mathsf {z}}^{\\mathsf {x}})}$ such that $\\begin{aligned}&\\mathbb {T}_{\\mathsf {z}}^{\\mathsf {x}}(\\sigma _ {02}) _{{}_{f}} &:=& \\left( \\mathfrak {t}_ {{}_{\\varrho _ {{}_{\\mathsf {z}}}f }}\\ast {\\rm id}_ {{}_{ \\varrho _ {{}_{ \\mathsf {x} }} }}\\right) \\\\&\\mathbb {T}_{\\mathsf {z}}^{\\mathsf {x}}(\\sigma _ {12}) _{{}_{f}} &:=& \\left( \\Omega _ {{}_{\\mathsf {z}}}^{-1}\\ast {\\rm id}_ {{}_{f}}\\right)\\\\&\\mathbb {T}_{\\mathsf {z}}^{\\mathsf {x}}(n _ {1}) _{{}_{f}} &:=& \\left(\\varsigma _ {{}_{\\mathsf {z}}}^{-1}\\ast {\\rm id}_ {{}_{f}}\\right)\\end{aligned}\\qquad \\qquad \\begin{aligned}&\\mathbb {T}_{\\mathsf {z}}^{\\mathsf {x}}(\\sigma _ {01}) _{{}_{f}} &:=& \\left(\\mathfrak {t}_ {{}_{\\mathcal {T}(f) \\varrho _ {{}_{\\mathsf {x} }} }}\\ast {\\rm id}_ {{}_{\\varrho _ {{}_{\\mathsf {x}}} }} \\right)\\cdot \\left( {\\rm id}_ {{}_{\\mathcal {T}^2(f) }}\\ast \\Omega _{{}_{\\mathsf {x} }}\\right)\\cdot \\left( \\varpi _ {{}_{f}}^{-1}\\ast {\\rm id}_ {{}_{ \\varrho _ {{}_{\\mathsf {x} }} }}\\right) \\\\&\\mathbb {T}_{\\mathsf {z}}^{\\mathsf {x}}(n _ {0}) _{{}_{f}} &:=& \\left({\\rm id}_ {{}_{f}}\\ast \\varsigma _ {{}_{\\mathsf {x}}}\\right)\\cdot \\left(\\varepsilon ^{-1}_{{}_{f}}\\ast {\\rm id}_ {{}_{\\varrho _ {{}_{\\mathsf {x} }} }}\\right)\\end{aligned}$ It follows from Definition and Remark .", "Recall that every biadjunction induces diagrams $\\mathcal {D}_ Y^X: \\dot{\\Delta } \\rightarrow {\\rm \\sf CAT}$ (Definition ).", "Also, for every pseudocomonad $\\mathcal {T}$ and objects $\\mathsf {x}, \\mathsf {z} $ of $\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}$ , we defined in Proposition REF a diagram $\\mathbb {T}_{\\mathsf {z}}^{\\mathsf {x}}: \\Delta \\rightarrow {\\rm \\sf CAT}$ whose strict descent object is $\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}(\\mathsf {x}, \\mathsf {z}) $ .", "Now, we give the relation between these two diagrams.", "Lemma 5.3 Let $L: \\mathfrak {B}\\rightarrow \\mathfrak {C}$ be a pseudofunctor, $(L\\dashv U, \\eta , \\varepsilon , s, t) $ be a biadjunction and $\\mathcal {T}= (\\mathcal {T}, \\varpi , \\varepsilon , \\Lambda , \\delta , \\mathsf {s} ) $ be the induced pseudocomonad.", "For each pair $(X,Y)$ of objects in $\\mathfrak {B}$ , $(L\\dashv U, \\eta , \\varepsilon , s, t) $ induces the diagram $\\mathcal {D}^X_Y : \\dot{\\Delta } \\rightarrow {\\rm \\sf CAT}$ and $\\mathcal {T}$ induces the diagram $\\mathbb {T}^{\\mathcal {K}X}_{\\mathcal {K}Y}: \\Delta \\rightarrow {\\rm \\sf CAT}$ defined in Proposition REF , in which $\\mathcal {K}: \\mathfrak {B}\\rightarrow \\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}$ is the comparison pseudofunctor.", "In this setting, there is a pseudonatural isomorphism $\\beta : \\mathcal {D}^X_Y\\circ {\\rm j}\\longrightarrow \\mathbb {T}^{\\mathcal {K}X}_{\\mathcal {K}Y} $ for every such pair $(X,Y)$ of objects in $\\mathfrak {B}$ .", "Moreover, if $L$ is a 2-functor, $\\beta $ is actually a 2-natural isomorphism.", "We can write $\\mathbb {T}^{\\mathcal {K}X}_{\\mathcal {K}Y}: \\Delta \\rightarrow {\\rm \\sf CAT}$ as follows ${ \\mathfrak {C}(LX, LY)@<-2.5ex>[rrr]_-{\\mathfrak {C}(LX, L(\\eta _ {{}_{Y}}) )}@<2.5ex>[rrr]^-{\\mathfrak {C}(L(\\eta _ {{}_{X}}), LUL Y )\\circ \\hspace{1.99997pt}(LU) _ {{}_{L X, L Y}} } &&& \\mathfrak {C}(L X, LUL Y )[lll]|-{\\mathfrak {C}( LX,\\varepsilon _ {{}_{LY }})}@<-2.5 ex>[rrrr]_-{\\mathfrak {C}( LX, LUL(\\eta _ {{}_{Y}}) )}[rrrr]|-{\\mathfrak {C}(LX, L(\\eta _ {{}_{LY}}))}@<2.5ex>[rrrr]^-{\\mathfrak {C}(L(\\eta _ {{}_{X}}), LUL Y )\\circ \\hspace{1.99997pt}(LU) _ {{}_{LX, LUL Y}}} &&&& \\mathfrak {C}(L X, LULUL Y ) }$ Furthermore, by Lemma REF and the observations given in this section, we can define a pseudonatural isomorphism $\\beta : \\mathcal {D}^X_Y\\circ {\\rm j}\\longrightarrow \\mathbb {T}^{\\mathcal {K}X}_{\\mathcal {K}Y} $ such that $\\beta _ {{}_{\\mathsf {1} }},\\beta _ {{}_{\\mathsf {2} }},\\beta _ {{}_{\\mathsf {3} }}$ are identity functors, $\\beta _ {{}_{d^1 }},\\beta _ {{}_{\\partial ^1 }},\\beta _ {{}_{\\partial ^2 }}, \\beta _ {{}_{s^0}} $ are identity natural transformations, $\\left(\\beta _ {{}_{d^0 }}\\right) _ {{}_{f}}:= \\mathfrak {l}_ {{}_{U(f)\\eta _ {{}_{X}} }}$ and $\\left( \\beta _ {{}_{\\partial ^0 }}\\right) _ {{}_{f}} := \\mathfrak {l}_ {{}_{U(f)\\eta _ {{}_{X}} }}$ .", "This completes the proof.", "Let $(L\\dashv U, \\eta , \\varepsilon , s, t) $ be a biadjunction and $\\mathcal {T}$ be the induced pseudocomonad.", "By Lemma REF , Proposition REF and Lemma REF , $\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}(\\mathcal {K}X, \\mathcal {K}Y) $ is a descent object of $\\mathcal {D}^X_Y\\circ {\\rm j}$ for every pair of objects $(X,Y)$ of $\\mathfrak {B}$ .", "Moreover, $\\mathcal {K}_{{}_{X,Y}}:\\mathfrak {B}(X,Y)\\rightarrow \\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}(\\mathcal {K}X, \\mathcal {K}Y) $ is the comparison $\\mathcal {D}^X_Y\\mathsf {0}\\rightarrow \\left\\lbrace \\dot{\\Delta }(\\mathsf {0}, {\\rm j}- ), \\mathcal {D}^X_Y\\circ {\\rm j}\\right\\rbrace $ .", "Thereby we get: Proposition 5.4 Let $(L\\dashv U, \\eta , \\varepsilon , s, t) $ be a biadjunction, $\\mathcal {T}$ be the induced pseudocomonad and $\\mathcal {K}: \\mathfrak {B}\\rightarrow \\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}$ be the comparison pseudofunctor.", "For each pair of objects $(X,Y) $ in $\\mathfrak {B}$ , $\\mathcal {D}^X_Y : \\dot{\\Delta } \\rightarrow {\\rm \\sf CAT}$ is of effective descent if and only if $\\mathcal {K}_{{}_{X,Y}}:\\mathfrak {B}(X,Y)\\rightarrow \\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}(\\mathcal {K}X, \\mathcal {K}Y) $ is an equivalence.", "Furthermore, if $L$ is a 2-functor, $\\mathcal {D}^X_Y$ is of strict descent if and only if $\\mathcal {K}_{{}_{X,Y}}$ is an isomorphism." ], [ "Biadjoint Triangles", "In this subsection, we reexamine the results of Section in the context of pseudocomonad theory.", "More precisely, we prove Corollary REF of our main theorems in Section , Theorem REF and Theorem REF .", "Let $(L,U, \\eta , \\varepsilon , s, t) $ be a biadjunction and $\\mathcal {T}$ be its induced pseudocomonad.", "We say that $L: \\mathfrak {B}\\rightarrow \\mathfrak {C}$ is pseudoprecomonadic, if its induced comparison pseudofunctor $\\mathcal {K}:\\mathfrak {B}\\rightarrow \\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}$ is locally an equivalence.", "As a consequence of Proposition REF , we get a characterization of pseudoprecomonadic pseudofunctors.", "Corollary 5.5 (Pseudoprecomonadic) Let $(L\\dashv U, \\eta , \\varepsilon , s, t) $ be a biadjunction.", "The pseudofunctor $L: \\mathfrak {B}\\rightarrow \\mathfrak {C}$ is pseudoprecomonadic if and only if $\\mathcal {D}^X_Y: \\dot{\\Delta } \\rightarrow {\\rm \\sf CAT}$ is of effective descent for every pair of objects $(X,Y) $ of $\\mathfrak {B}$ .", "By Corollary REF , assuming that $(L\\dashv U, \\eta , \\varepsilon , s, t) $ is a biadjunction and $J: \\mathfrak {A}\\rightarrow \\mathfrak {B}$ is a pseudofunctor, if $L: \\mathfrak {B}\\rightarrow \\mathfrak {C}$ is pseudoprecomonadic, then, in particular, $\\mathcal {D}_Y^{JA}:\\dot{\\Delta }\\rightarrow {\\rm \\sf CAT}$ is of effective descent for every object $A$ of $\\mathfrak {A}$ and every object $Y$ of $\\mathfrak {B}$ .", "Thereby, as a consequence of Theorem REF , Theorem REF and Propostion REF , we get: Corollary 5.6 (Biadjoint Triangle Theorem) Assume that $(E\\dashv R, \\rho , \\mu , v,w), (L\\dashv U, \\eta , \\varepsilon , s,t)$ are biadjunctions such that the triangle of pseudofunctors ${ \\mathfrak {A}[rr]^-{J}[dr]_-{E}&&\\mathfrak {B}[dl]^-{L}\\\\&\\mathfrak {C}& }$ is commutative and $L$ is pseudoprecomonadic.", "Then $J$ has a right biadjoint if and only if, for every object $Y$ of $\\mathfrak {B}$ , $\\mathfrak {A}$ has the descent object of the diagram $\\mathcal {A}_Y : \\Delta \\rightarrow \\mathfrak {A}$ .", "In this case, $J$ is left biadjoint to $GY:= \\left\\lbrace \\dot{\\Delta } (\\mathsf {0}, {\\rm j}- ), \\mathcal {A}_Y\\right\\rbrace _{{\\rm bi}}$ .", "If, furthermore, $E, R, J, L, U$ are 2-functors, $(E\\dashv R, \\rho , \\mu ) $ is a 2-adjunction, $\\left(\\eta J\\right)$ is a 2-natural transformation and the comparison 2-functor $\\mathcal {K}: \\mathfrak {B}\\rightarrow \\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}$ induced by the biadjunction $L\\dashv U $ is locally an isomorphism, then $J$ is left 2-adjoint if and only if the strict descent object of $\\mathcal {A}_Y $ exists for every object $Y$ of $\\mathfrak {B}$ .", "In this case, $GY:= \\left\\lbrace \\dot{\\Delta } (\\mathsf {0}, {\\rm j}- ), \\mathcal {A}_Y\\right\\rbrace $ defines the right 2-adjoint to $J$ ." ], [ "Unit and Counit", "In this section, we show that the pseudoprecomonadicity characterization given in Theorem 3.5 of [13] is a consequence of Corollary REF .", "Secondly, we study again biadjoint triangles.", "Namely, in the context of Corollary REF , we give necessary and sufficient conditions under which the unit and the counit of the obtained biadjunction $J\\dashv G $ are pseudonatural equivalences, provided that $E$ and $L$ induce the same pseudocomonad.", "In other words, we prove the appropriate analogous versions of Corollary 1 and Corollary 2 of page 76 in [2] within our context of biadjoint triangles.", "Again, we need to consider another type of 2-functors induced by biadjunctions.", "The definition below is given in Theorem 3.5 of [13].", "Assume that $L:\\mathfrak {B}\\rightarrow \\mathfrak {C}$ is a pseudofunctor and $(L\\dashv U, \\eta , \\varepsilon , s, t)$ is a biadjunction.", "For each object $Y$ of $\\mathfrak {B}$ , we get the 2-functor $\\mathcal {V}_Y : \\dot{\\Delta }\\rightarrow \\mathfrak {B}$ ${ Y [r]^-{\\eta _ {{}_{Y}}} & ULY@<-2.5ex>[rrr]_-{UL(\\eta _{{}_Y})}@<2.5ex>[rrr]^-{\\eta _ {{}_{ULY}}} &&& ULULY[lll]|-{U(\\varepsilon _ {{}_{LY}})}@<-2.5 ex>[rrr]_-{ULUL(\\eta _ {{}_{Y}})}[rrr]|-{UL(\\eta _ {{}_{ULY}})}@<2.5ex>[rrr]^-{\\eta _ {{}_{ULULY}}} &&& ULULULY }\\qquad \\mathrm {(\\mathcal {V}_ Y)}$ in which $\\displaystyle \\mathcal {V}_Y (\\vartheta ) := \\left( \\eta _ {{}_{\\eta _{{}_Y}}}:UL(\\eta _{{}_Y})\\eta _ {{}_Y}\\cong \\eta _ {{}_{ULY}}\\eta _{{}_Y}\\right) $ is the invertible 2-cell component of the unit $\\eta $ at the morphism $\\eta _ {{}_Y} $ .", "Analogously, the images of the 2-cells $\\sigma _ {ik}, n_0, n_1$ are defined below.", "$\\begin{aligned}\\mathcal {V}_Y (\\sigma _{01}) &:= \\eta _ {{}_{\\eta _{{}_{ULY}}}}\\\\\\mathcal {V}_Y (\\sigma _{02}) &:= \\eta _ {{}_{UL(\\eta _{{}_Y})}}\\\\\\mathcal {V}_Y (\\sigma _{12}) &:= (UL\\eta ) _ {{}_{\\eta _{{}_Y}}}\\end{aligned}\\qquad \\qquad \\begin{aligned}\\mathcal {V}_Y (n_0) & := t_{{}_{LY}}\\\\\\mathcal {V}_Y (n_1) & := \\left(\\mathfrak {u}_{{}_{\\varepsilon _ {{}_{LY}},L(\\eta _{{}_Y})}}\\right)^{-1}\\cdot U(s_{{}_Y})\\cdot \\mathfrak {u}_ {{}_{ULY}}\\end{aligned}$ We verify below that $ \\mathcal {V}_Y $ is well defined.", "That is to say, we have to prove that $ \\mathcal {V}_Y $ satisfies the equations given in Definition .", "Firstly, the associativity and naturality equations of Definition give the following equality ${ &&\\mathcal {V}_Y\\mathsf {0}[rr]|-{\\mathcal {V}_Y(d)}[d]|-{\\mathcal {V}_Y(d)}[lld]|-{\\mathcal {V}_Y(d)}@{}[rrd]|-{{\\mathcal {V}_Y(\\vartheta ) } }&&\\mathcal {V}_Y\\mathsf {1}[d]|-{\\mathcal {V}_Y(d^1)}@{}[rrdd]|-{=}&&\\mathcal {V}_Y\\mathsf {0}[rr]|-{\\mathcal {V}_Y(d)}@{}[rrd]|-{{\\mathcal {V}_Y(\\vartheta )}}[d]|-{\\mathcal {V}_Y(d)}&&\\mathcal {V}_Y\\mathsf {1}[d]|-{\\mathcal {V}_Y (d^1)}[rrd]|-{\\mathcal {V}_Y(d^1)}&&\\\\\\mathcal {V}_Y\\mathsf {1}@{}[rr]|-{{\\mathcal {V}_Y(\\vartheta ) } }[rrd]|-{\\mathcal {V}_Y(d^0)}&&\\mathcal {V}_Y\\mathsf {1}[rr]|-{\\mathcal {V}_Y(d^0)}[d]|-{\\mathcal {V}_Y(d^1)}@{}[rrd]|-{{\\mathcal {V}_Y(\\sigma _ {02}) }}&&\\mathcal {V}_Y\\mathsf {2}[d]|-{\\mathcal {V}_Y(\\partial ^2) }&&\\mathcal {V}_Y\\mathsf {1}[rr]|-{\\mathcal {V}_Y(d^0)}[d]|-{\\mathcal {V}_Y(d^0)}@{}[rrd]|-{{\\mathcal {V}_Y(\\sigma _ {01}) } }&&\\mathcal {V}_Y\\mathsf {2}@{}[rr]|-{{\\mathcal {V}_Y(\\sigma _ {12}) } }[d]|-{\\mathcal {V}_Y(\\partial ^1)}&&\\mathcal {V}_Y\\mathsf {2}[lld]|-{\\mathcal {V}_Y(\\partial ^2)}\\\\&&\\mathcal {V}_Y\\mathsf {2}[rr]|-{\\mathcal {V}_Y(\\partial ^0)}&&\\mathcal {V}_Y\\mathsf {3}&&\\mathcal {V}_Y\\mathsf {2}[rr]|- {\\mathcal {V}_Y(\\partial ^0)}&&\\mathcal {V}_Y\\mathsf {3}}$ which is the associativity equation of Definition .", "Furthermore, by Definition of biadjunction, we have that ${ \\mathcal {V}_Y\\mathsf {0} [rr]^-{\\mathcal {V}_Y(d)=\\eta _ {{}_Y}}[dd]|-{\\mathcal {V}_Y(d)=\\eta _ {{}_Y}}&&{\\mathcal {V}_Y\\mathsf {1}} [dd]|-{\\mathcal {V}_Y(d^1)=UL(\\eta _ {{}_Y})}@{=}@/^5ex/[dddr]@{}[dddr]|-{{\\mathcal {V}_Y(n_1)}} &&&\\mathcal {V}_Y\\mathsf {0} @/_4ex/[ddd]_-{\\mathcal {V}_Y(d)}@{}[ddd]|=@/^4ex/[ddd]^-{\\mathcal {V}_Y(d)}\\\\& {\\hspace{1.00006pt}\\mathcal {V}_Y(\\vartheta )} && &=\\\\{\\mathcal {V}_Y\\mathsf {1})} [rr]_{\\mathcal {V}_Y(d^0)=\\eta _ {{}_{ULY}}}@{=}@/_5ex/[drrr]@{}[drrr]|{{\\mathcal {V}_Y(n_0)}} &&{\\mathcal {V}_Y\\mathsf {2}}[dr]|-{\\mathcal {V}_Y(s^0)} \\\\&&& {\\mathcal {V}_Y\\mathsf {1}} && {\\mathcal {V}_Y\\mathsf {1}} }$ which proves that $\\mathcal {V}_Y$ satisfies the identity equation of Definition .", "As mentioned before, Corollary REF is Theorem 3.5 of [13].", "Below, it is proved as a consequence of Corollary REF .", "Corollary 6.1 ([13]) Let $(L\\dashv U, \\eta , \\varepsilon , s, t)$ be a biadjunction.", "The pseudofunctor $L$ is pseudoprecomonadic if and only if, for every object $Y$ of $\\mathfrak {B}$ , the 2-functor $\\mathcal {V}_Y : \\dot{\\Delta }\\rightarrow \\mathfrak {B}$ is of effective descent.", "On one hand, by Corollary REF , $L$ is pseudoprecomonadic if and only if $\\mathcal {D}^X_Y:\\dot{\\Delta }\\rightarrow {\\rm \\sf CAT}$ is of effective descent for every pair $(X,Y)$ of objects in $\\mathfrak {B}$ .", "On the other hand, by Lemma REF , $\\mathcal {V}_Y $ is of effective descent if and only if $\\mathfrak {B}(X, \\mathcal {V}_Y-): \\dot{\\Delta }\\rightarrow {\\rm \\sf CAT}$ is of effective descent for every object $X$ in $\\mathfrak {B}$ .", "Therefore, by Lemma REF , to complete our proof, we just need to verify that $ \\mathcal {D}^X_Y\\simeq \\mathfrak {B}(X, \\mathcal {V}_Y-)$ .", "Indeed, there is a pseudonatural equivalence $\\iota : \\mathcal {D}^X_Y\\longrightarrow \\mathfrak {B}(X, \\mathcal {V}_Y-) $ induced by $\\chi : \\mathfrak {C}(L-,-)\\simeq \\mathfrak {B}(-, U-)$ such that $\\iota _ {{}_{\\mathsf {0}}} &:=& {\\rm Id}_ {{}_{\\mathfrak {B}(X,Y) }} \\\\\\iota _ {{}_{\\mathsf {1}}} &:=& \\chi _ {{}_{(X, LY)}}: \\mathfrak {C}(LX, LY)\\rightarrow \\mathfrak {B}(X, ULY) \\\\\\psi _ {{}_{\\mathsf {2}}} &:=& \\chi _ {{}_{(X, LULY)}}: \\mathfrak {C}(LX, LULY)\\rightarrow \\mathfrak {B}(X, ULULY) \\\\\\psi _ {{}_{\\mathsf {3}}} &:=& \\chi _ {{}_{(X, LULULY)}}: \\mathfrak {C}(LX, LULULY)\\rightarrow \\mathfrak {B}(X, ULULULY)$ $\\begin{aligned}\\left(\\iota _ {{}_{d}}\\right)_{{}_f} &:= \\eta _{{}_{f}}^{-1}\\\\\\left(\\iota _ {{}_{d^0}}\\right)_{{}_f} &:= \\eta _ {{}_{U(f)\\eta _ {{}_{X}}}}^{-1}\\\\\\left(\\iota _ {{}_{d^1}}\\right)_{{}_f} &:= \\mathfrak {u}_ {{}_{L(\\eta _ {{}_{Y}})f }}\\ast {\\rm id}_ {{}_{\\eta _ {{}_{X}} }}\\end{aligned}\\qquad \\begin{aligned}\\left(\\iota _ {{}_{\\partial ^0}}\\right)_{{}_f} &:= \\eta _ {{}_{U(f)\\eta _ {{}_{X}}}}^{-1}\\\\\\left(\\iota _ {{}_{\\partial ^1}}\\right) _ {{}_f} &:= \\mathfrak {u}_ {{}_{L(\\eta _ {{}_{ULY}})f }}\\ast {\\rm id}_ {{}_{\\eta _ {{}_{X}} }}\\end{aligned}\\qquad \\begin{aligned}\\left( \\iota _ {{}_{\\partial ^2}}\\right) _ {{}_f} &:= \\mathfrak {u}_ {{}_{LUL(\\eta _ {{}_{Y}})f }}\\ast {\\rm id}_ {{}_{\\eta _ {{}_{X}} }} \\\\\\left(\\iota _ {{}_{s^0}}\\right)_{{}_f} &:= \\mathfrak {u}_ {{}_{\\varepsilon _ {{}_{LY}} f }}\\ast {\\rm id}_ {{}_{\\eta _ {{}_{X}} }}\\end{aligned}$ We assume the existence of a biadjunction $J\\dashv G $ in the commutative triangles below and study its counit and unit, provided that the biadjunctions $(E\\dashv R, \\rho , \\mu , v,w), (L\\dashv U, \\eta , \\varepsilon , s,t)$ induce the same pseudocomonad.", "We start with the unit.", "Theorem 6.2 (Unit) Assume that $(E\\dashv R, \\rho , \\mu , v,w), (L\\dashv U, \\eta , \\varepsilon , s,t), (J\\dashv G, \\bar{\\eta }, \\bar{\\varepsilon }, \\bar{s}, \\bar{t})$ are biadjunctions such that the triangles ${ \\mathfrak {A}[rr]^-{J}[dr]_-{E}&&\\mathfrak {B}[dl]^-{L} & \\mathfrak {A}[rr]^-{J}&&\\mathfrak {B}\\\\&\\mathfrak {C}& & &\\mathfrak {C}[ul]^-{R}[ur]_-{U} & }$ are commutative.", "If $(E\\dashv R, \\rho , \\mu , v,w)$ , $(L\\dashv U, \\eta , \\varepsilon , s,t)$ induce the same pseudocomonad $\\mathcal {T}$ , then the following statements are equivalent: The unit $\\overline{\\eta }: {\\rm Id}_ {{}_\\mathfrak {A}}\\longrightarrow GJ $ is a pseudonatural equivalence; $E$ is pseudoprecomonadic; The following 2-functor is of effective descent for every pair of objects $(A,B)$ in $\\mathfrak {A}$ $\\widehat{\\mathcal {D}} ^A_B:\\dot{\\Delta }\\rightarrow {\\rm \\sf CAT}$ ${\\mathfrak {A}(A, B) [d]|-{E_{{}_{A,B}}}\\\\ \\mathfrak {C}(EA, EB)@<-2.5ex>[rrr]_-{\\mathfrak {C}(EA, E(\\rho _{{}_B} ))}@<2.5ex>[rrr]^-{E_{{}_{{}_{A,REB}}}\\circ \\hspace{1.99997pt} \\xi _ {{}_{{}_{(A,EB)}}}} &&& \\mathfrak {C}(EA, EREB )[lll]|-{\\mathfrak {C}(EA,\\mu _ {{}_{EB}})}@<-2.5 ex>[rrrr]_-{\\mathfrak {C}(EA, ERE(\\rho _{{}_B} ))}[rrrr]|-{\\mathfrak {C}(EA, E(\\rho _ {{}_{REB}}))}@<2.5ex>[rrrr]^-{E_{{}_{{}_{A,(RE)^2B}}}\\circ \\hspace{1.99997pt}\\xi _ {{}_{{}_{(A,EREB)}}}} &&&& \\mathfrak {C}(EA, E(RE)^2B) }$ in which $\\xi : \\mathfrak {C}(E-,-)\\simeq \\mathfrak {A}(-,R-) $ is the pseudonatural equivalence induced by the biadjunction $(E\\dashv R, \\rho , \\mu , v,w)$ described in Remark .", "For each object $B$ of $\\mathfrak {A}$ , the following diagram is of effective descent.", "$\\widehat{\\mathcal {V}} _ {{}_{B}}: \\dot{\\Delta }\\rightarrow \\mathfrak {A}$ ${ B [r]^-{\\rho _ {{}_{B}}} & REB@<-2.5ex>[rrr]_-{RE(\\rho _{{}_B})}@<2.5ex>[rrr]^-{\\rho _ {{}_{REB}}} &&& REREY[lll]|-{R(\\mu _ {{}_{EB}})}@<-2.5 ex>[rrr]_-{RERE(\\rho _ {{}_{B}})}[rrr]|-{RE(\\rho _ {{}_{REB}})}@<2.5ex>[rrr]^-{\\rho _ {{}_{REREB}}} &&& REREREB }$ By Remark , the unit $\\bar{\\eta } $ is a pseudonatural equivalence if and only if $J$ is locally an equivalence.", "Moreover, by the hypothesis and the universal property of the 2-category of pseudocolagebras, we have the following diagram ${ \\mathfrak {A}[r]|{J} @/^5ex/[rr]|{\\widetilde{\\mathcal {K}} }@{ { }}@/^3ex/[rr]|{\\cong }[rrd]|{E}&\\mathfrak {B}[r]|-{\\mathcal {K}}[rd]|{L}&\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}[d]|{\\mathsf {L}}\\\\&& \\mathfrak {C}}$ such that $\\widetilde{\\mathcal {K}}, \\mathcal {K}$ are the comparison pseudofunctors.", "Since, by hypothesis, we know that $\\mathcal {K}$ is locally an equivalence, we conclude that $J $ is locally an equivalence if and only if $\\widetilde{\\mathcal {K}}$ is locally an equivalence.", "Thereby, to conclude, we just need to apply the characterizations of pseudoprecomonadic pseudofunctors: that is to say, Corollary REF and Corollary REF .", "Before studying the counit, for future references, we need the following result about the diagram $\\mathcal {V}_ Y: \\dot{\\Delta }\\rightarrow \\mathfrak {B}$ in the context of biadjoint triangles.", "Lemma 6.3 Let ${ \\mathfrak {A}[rr]^-{J}[dr]_-{E}&&\\mathfrak {B}[dl]^-{L} & \\mathfrak {A}[rr]^-{J}&&\\mathfrak {B}\\\\&\\mathfrak {C}& & &\\mathfrak {C}[ul]^-{R}[ur]_-{U} & }$ be commutative triangles of pseudofunctors such that we have biadjunctions $(E\\dashv R, \\rho , \\varepsilon , v,w)$ and $(L\\dashv U, \\eta , \\varepsilon , s,t)$ inducing the same pseudocomonad $\\mathcal {T}= (\\mathcal {T}, \\varpi , \\varepsilon , \\Lambda , \\delta , \\mathsf {s} ) $ .", "We consider the diagram $\\mathcal {A}_Y : \\Delta \\rightarrow \\mathfrak {A}$ .", "Then, for each object $Y$ of $\\mathfrak {B}$ , there is a pseudonatural isomorphism $\\zeta ^{{}^Y}: J\\circ \\mathcal {A}_ Y\\longrightarrow \\mathcal {V}_ Y\\circ {\\rm j}.$ Again, we have the same diagram of the proof of Theorem REF .", "In particular, for each object $Y$ of $\\mathfrak {B}$ , there is an invertible 2-cell $\\mathfrak {y}_{{}_{Y}}: J(\\rho _ {{}_{RLY}})\\Rightarrow \\eta _ {{}_{ULY}} $ .", "Thereby, we can define $\\zeta ^{{}^Y} : J\\circ \\mathcal {A}_ Y\\longrightarrow \\mathcal {V}_ Y\\circ {\\rm j}$ such that the components $\\zeta _ {{}_{\\mathsf {1}}}, \\zeta _ {{}_{\\mathsf {2}}}, \\zeta _ {{}_{\\mathsf {3}}}$ are identity 1-cells, the components $\\zeta ^{{}^Y} _ {{}_{d^1}} $ , $\\zeta ^{{}^Y} _ {{}_{s^0}}$ , $\\zeta ^{{}^Y}_ {{}_{\\partial ^1}}$ , $\\zeta ^{{}^Y} _ {{}_{\\partial ^2}} $ are identity 2-cells and $\\begin{aligned}\\left(\\zeta ^{{}^Y}_ {{}_{d^0}}\\right) &:= \\mathfrak {y}_{{}_{Y}}\\cdot J\\left(\\left((\\mathfrak {r}\\mathfrak {l})_{{}_{LY}}^{-1}\\cdot RL(t_{{}_{LY}})\\right)\\ast {\\rm id}_ {{}_{\\rho _ {{}_{RLY}}}}\\right);\\end{aligned}\\begin{aligned}\\left(\\zeta ^{{}^Y}_ {{}_{\\partial ^0}}\\right) &:= \\mathfrak {y}_{{}_{ULY}}\\cdot J\\left(\\left((\\mathfrak {r}\\mathfrak {l})_{{}_{LULY}}^{-1}\\cdot RL(t_{{}_{LY}})\\right)\\ast {\\rm id}_ {{}_{\\rho _ {{}_{RLULY}}}}\\right).\\end{aligned}$ Theorem 6.4 (Counit) Let $(E\\dashv R, \\rho , \\varepsilon , v,w)$ and $(L\\dashv U, \\eta , \\varepsilon , s,t)$ be biadjunctions inducing the same pseudocomonad $\\mathcal {T}= (\\mathcal {T}, \\varpi , \\varepsilon , \\Lambda , \\delta , \\mathsf {s} )$ such that the triangles of pseudofunctors ${ \\mathfrak {A}[rr]^-{J}[dr]_-{E}&&\\mathfrak {B}[dl]^-{L} & \\mathfrak {A}[rr]^-{J}&&\\mathfrak {B}\\\\&\\mathfrak {C}& & &\\mathfrak {C}[ul]^-{R}[ur]_-{U} & }$ commute.", "We assume that $(J\\dashv G, \\bar{\\eta }, \\bar{\\varepsilon }, \\bar{s}, \\bar{t})$ is a biadjunction and $L$ is pseudoprecomonadic.", "We consider the diagram $\\mathcal {A}_Y : \\Delta \\rightarrow \\mathfrak {A}$ .", "Then $J \\left\\lbrace \\dot{\\Delta } (\\mathsf {0}, {\\rm j}-), \\mathcal {A}_ Y\\right\\rbrace _ {{\\rm bi}} $ is the descent object of $J\\circ \\mathcal {A}_ Y $ for every object $Y$ of $\\mathfrak {B}$ if and only if the counit $\\overline{\\varepsilon } : JG\\longrightarrow {\\rm Id}_ {{}_{\\mathfrak {B}}} $ is a pseudonatural equivalence.", "Actually, this is a corollary of Lemma REF , Corollary REF and Corollary REF .", "More precisely, by Lemma REF , $ J\\circ \\mathcal {A}_ Y\\simeq \\mathcal {V}_ Y\\circ {\\rm j}$ .", "By Corollary REF , since $L$ is pseudoprecomonadic, $\\mathcal {V}_ Y $ is of effective descent.", "Moreover, by the constructions of Theorem REF (which proves Corollary REF ), the counit is pointwise defined by the comparison 1-cells $J\\left\\lbrace \\dot{\\Delta }(\\mathsf {0}, {\\rm j}-), \\mathcal {A}_ Y \\right\\rbrace _{{\\rm bi}}\\rightarrow Y= \\mathcal {V}_ Y\\mathsf {0}\\simeq \\left\\lbrace \\dot{\\Delta }(\\mathsf {0}, {\\rm j}-), \\mathcal {V}_ Y\\circ {\\rm j}\\right\\rbrace .$ This completes the proof." ], [ "Pseudocomonadicity", "Similarly to the 1-dimensional case, to prove the characterization of pseudocomonadic pseudofunctors employing the biadjoint triangle theorems, we need two results: Lemma REF and Proposition REF , which are proved in [13] in Lemma 2.3 and Proposition 3.2 respectively.", "We start with Lemma REF , which is a basic and known property of the diagram $\\mathcal {V}_ Y $ .", "It follows from explicit calculations using the definition of descent objects: we give a sketch of the proof below.", "Lemma 7.1 ([13]) Let $(L\\dashv U, \\eta , \\varepsilon , s, t)$ be a biadjunction.", "For each object $Y$ of $\\mathfrak {B}$ , the diagram $L\\circ \\mathcal {V}_Y$ is of absolute effective descent.", "Trivially, given a pseudofunctor $\\mathcal {F}: \\mathfrak {C}\\rightarrow \\mathfrak {Z} $ , we can see $\\mathcal {F}\\circ L\\circ \\mathcal {V}_Y$ as a 2-functor, taking, if necessary, the obvious pseudonaturally equivalent version of $\\mathcal {F}\\circ L\\circ \\mathcal {V}_ Y $ .", "Then, for each objects $Z$ of $\\mathfrak {Z} $ , by Remark , we can consider the strict descent object of the 2-functor explicitly $\\mathfrak {Z}( Z, \\mathcal {F}\\circ L\\circ \\mathcal {V}_Y\\circ {\\rm j}-): \\Delta \\rightarrow {\\rm \\sf CAT}$ ${ \\mathfrak {Z}(Z, \\mathcal {F}LULY)@<-2.5ex>[rrr]_-{\\mathfrak {Z}(Z,\\mathcal {F}LUL(\\eta _{{}_Y}))}@<2.5ex>[rrr]^-{\\mathfrak {Z}(Z, \\mathcal {F}L(\\eta _ {{}_{ULY}}))} &&& \\mathfrak {Z}(Z, \\mathcal {F}LULULY)[lll]|-{\\mathfrak {Z}(Z, \\mathcal {F}LU(\\varepsilon _ {{}_{LY}}))}@<-2.5 ex>[rrr]_-{\\mathfrak {Z}(Z, \\mathcal {F}LULUL(\\eta _ {{}_{Y}}))}[rrr]|-{\\mathfrak {Z}(Z,\\mathcal {F}LUL(\\eta _ {{}_{ULY}}))}@<2.5ex>[rrr]^-{\\mathfrak {Z}(Z, \\mathcal {F}L(\\eta _ {{}_{ULULY}}))} &&& \\mathfrak {Z}(Z, \\mathcal {F}LULULULY) }$ Thereby, by straightforward calculations, taking Remark into account, we conclude that $\\mathfrak {Z}(Z, \\mathcal {F}LY)&\\rightarrow & \\left\\lbrace \\dot{\\Delta }(\\mathsf {0}, {\\rm j}-),\\mathfrak {Z}( Z, \\mathcal {F}\\circ L\\circ \\mathcal {V}_Y\\circ {\\rm j}-)\\right\\rbrace \\\\f&\\mapsto & (\\mathcal {F}L(\\eta _ {{}_{Y}}) f, (\\mathcal {F}L\\eta ) _ {{}_{\\eta _ {{}_{Y}} }}\\ast {\\rm id}_ {{}_{f}} )\\\\\\mathfrak {m} &\\mapsto & {\\rm id}_ {{}_{{}_{\\mathcal {F}L(\\eta _ {{}_{Y}})}}}\\ast \\mathfrak {m}$ gives an equivalence of categories (and it is the comparison functor).", "This completes the proof.", "Proposition 7.2 ([13]) Let $\\mathcal {T}= (\\mathcal {T}, \\varpi , \\varepsilon , \\Lambda , \\delta , \\mathsf {s} ) $ be a pseudocomonad on $\\mathfrak {C}$ .", "The forgetful pseudofunctor $L: \\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}\\mathsf {CoAlg}\\rightarrow \\mathfrak {C}$ creates absolute effective descent diagrams.", "In this section, henceforth we work within the following setting (and notation): given a biadjunction $(E\\dashv R, \\rho , \\mu , v,w)$ , recall that, by Lemma REF , it induces a biadjunction, herein denoted by $(L\\dashv U, \\eta , \\varepsilon , s, t)$ .", "We also get commutative triangles ${\\mathfrak {A}[r]^-{\\mathcal {K}}[rd]_-{E}& \\mathsf {Ps} \\textrm {-} \\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}[d]^{L } &&\\mathfrak {C}[r]^-{R}[rd]_-{U}& \\mathfrak {A}[d]^{\\mathcal {K}}\\\\&\\mathfrak {C}&&& \\mathsf {Ps} \\textrm {-} \\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}}$ in which, clearly, the biadjunctions $E\\dashv R$ , $L\\dashv U $ induce the same pseudocomonad $\\mathcal {T}$ .", "In this context, if the comparison pseudofunctor $\\mathcal {K}$ is a biequivalence, we say that $E$ is pseudocomonadic.", "In other words, we say that $E $ is pseudocomonadic if there is a pseudofunctor $ G:\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}\\rightarrow \\mathfrak {A}$ such that $G\\circ \\mathcal {K}\\simeq {\\rm Id}_ {{}_{\\mathfrak {A}}} $ and $\\mathcal {K}\\circ G\\simeq {\\rm Id}_ {{}_{ \\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}}}$ .", "Of course, in the triangle above, the forgetful pseudofunctor $L$ is always pseudocomonadic.", "In particular, $L$ is always pseudoprecomonadic.", "Therefore the triangle satisfies the basic hypothesis of Corollary REF .", "Observe that, to verify the pseudocomonadicity of a left biadjoint pseudofunctor $L$ , we can do it in three steps: Verify whether $\\mathcal {K}$ has a right biadjoint via Corollary REF ; If it does, the next step would be to verify whether the counit of the biadjunction $\\mathcal {K}\\dashv G $ is a pseudonatural equivalence via Theorem REF ; The final step would be to verify whether the unit of the biadjunction $\\mathcal {K}\\dashv G $ is a pseudonatural equivalences via Theorem REF .", "These are precisely the steps used below.", "Theorem 7.3 (Pseudocomonadicity [13]) A left biadjoint pseudofunctor $E: \\mathfrak {A}\\rightarrow \\mathfrak {C}$ is pseudocomonadic if and only if it creates absolute effective descent diagrams.", "By Proposition REF , pseudocomonadic pseudofunctors create absolute effective descent diagrams.", "Reciprocally, assume that $E$ creates absolute effective descent diagrams.", "$\\mathcal {K}$ has a right biadjoint $G$ : In this proof, we take a biadjunction $(E\\dashv R, \\rho , \\varepsilon , v,w)$ and assume that $\\mathcal {T}$ is its induced pseudocomonad.", "Also, we denote by $(L\\dashv U, \\eta , \\varepsilon , s,t)$ the biadjunction induced by $\\mathcal {T}$ (as described above).", "On one hand, by Lemma REF and Lemma REF , for each object $\\mathsf {z} $ of $\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}$ , the diagram $\\mathcal {A}_\\mathsf {z} : \\Delta \\rightarrow \\mathfrak {A}$ is such that $E\\circ \\mathcal {A}_\\mathsf {z}\\simeq L\\circ \\mathcal {V}_ {\\mathsf {z}}\\circ {\\rm j}$ is an absolute effective descent diagram, in which $\\mathcal {V}_ {\\mathsf {z}}:\\dot{\\Delta } \\rightarrow \\mathsf {Ps} \\textrm {-} \\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}$ is induced by the biadjunction $L\\dashv U$ .", "Therefore, since $E$ creates absolute effective diagrams, we conclude that there is an effective descent diagram $\\mathcal {B}_ {\\mathsf {z}}$ such that $\\mathcal {A}_\\mathsf {z} = \\mathcal {B}_ {\\mathsf {z}}\\circ {\\rm j}$ and $E\\circ \\mathcal {B}_ {\\mathsf {z}}\\simeq L\\circ \\mathcal {V}_ {\\mathsf {z}} $ .", "Thus, by Corollary REF , we conclude that $\\mathcal {K}$ has a right biadjoint $G$ .", "The counit of the biadjunction $\\mathcal {K}\\dashv G $ is a pseudonatural equivalence: Since $L\\circ \\mathcal {K}\\circ \\mathcal {B}_ {\\mathsf {z}}= E\\circ \\mathcal {B}_ {\\mathsf {z}}\\simeq L\\circ \\mathcal {V}_ {\\mathsf {z}} $ is of absolute effective descent and $L$ creates absolute effective descent diagrams, we conclude that $\\mathcal {K}\\circ \\mathcal {B}_\\mathsf {z} $ is of effective descent.", "By Theorem REF , it completes this second step.", "The unit of the biadjunction $\\mathcal {K}\\dashv G $ is a pseudonatural equivalence: By Lemma REF , for every object $A$ of $\\mathfrak {A}$ , $E\\circ \\widehat{\\mathcal {V}} _A : \\dot{\\Delta }\\rightarrow \\mathfrak {C}$ is of absolute effective descent, in which $\\widehat{\\mathcal {V}} _A$ is induced by the biadjunction $E\\dashv R $ .", "Since $E$ creates absolute effective descent diagrams, we get that $\\widehat{\\mathcal {V}} _A $ is of effective descent.", "Therefore, by Corollary REF , $E$ is pseudoprecomonadic.", "By Theorem REF , it completes the proof of the final step.", "As a consequence of Theorem REF , within the setting of Theorem REF , if $J$ has a right biadjoint and $ E$ is pseudocomonadic, then $J$ is pseudocomonadic as well.", "Furthermore, it is worth to point out that the second step of the proof of Theorem REF follows directly from the fact that $E$ preserves the effective descent diagrams $\\mathcal {B}_ {{}_{\\mathsf {z}}}$ and from the pseudocomonadicity of $L$ .", "More precisely, as direct consequence of Lemma REF , Theorem REF and Proposition REF , we get: Corollary 7.4 (Counit) Let $(E\\dashv R, \\rho , \\mu , v,w), (L\\dashv U, \\eta , \\varepsilon , s,t)$ be biadjunctions inducing the same pseudocomonad $\\mathcal {T}= (\\mathcal {T}, \\varpi , \\varepsilon , \\Lambda , \\delta , \\mathsf {s} ) $ such that ${ \\mathfrak {A}[rr]^-{J}[dr]_-{E}&&\\mathfrak {B}[dl]^-{L}\\\\&\\mathfrak {C}& }$ commutes.", "Assume that $L$ is pseudocomonadic, $J\\circ R=U$ and $(J, G, \\overline{\\varepsilon }, \\overline{\\eta }, \\overline{s}, \\overline{t})$ is a biadjunction.", "The counit $\\overline{\\varepsilon }: JG\\longrightarrow {\\rm Id}_ {{}_{\\mathfrak {B}}}$ is a pseudonatural equivalence if and only if, for every object $Y$ of $\\mathfrak {B}$ , $E$ preserves the descent object of $\\mathcal {A}_Y : \\Delta \\rightarrow \\mathfrak {A}$ .", "By Corollary REF , since $J$ is left biadjoint, for each object $Y$ of $\\mathfrak {B}$ , there is an effective descent diagram $\\mathcal {B}_ Y : \\dot{\\Delta }\\rightarrow \\mathfrak {A}$ such that $\\mathcal {B}_ Y\\circ {\\rm j}\\simeq \\mathcal {A}_ Y $ .", "By the commutativity of the triangles $L\\circ J=E$ and $J\\circ R=U $ , since $(E\\dashv R, \\rho , \\mu ) $ and $(L\\dashv U, \\eta , \\varepsilon , s, t)$ induce the same pseudocomonad, our setting satisfies the hypotheses of Lemma REF .", "Thus, for each object $Y$ of $\\mathfrak {B}$ , there is a pseudonatural equivalence $J\\circ \\mathcal {B}_ Y\\circ {\\rm j}\\simeq J\\circ \\mathcal {A}_ Y\\simeq \\mathcal {V}_Y \\circ {\\rm j}.$ By Theorem REF , to complete this proof, it is enough to show that $J\\circ \\mathcal {B}_ Y $ is of effective descent if and only if $E\\circ \\mathcal {B}_ Y $ is of effective descent.", "Firstly, we assume that $J\\circ \\mathcal {B}_ Y $ is of effective descent.", "In this case, since $J\\circ \\mathcal {B}_ Y\\circ {\\rm j}\\simeq \\mathcal {V}_Y \\circ {\\rm j}$ and $\\mathcal {V}_ Y $ is of effective descent, we conclude that $\\mathcal {V}_ Y\\simeq J\\circ \\mathcal {B}_ Y$ .", "Thus, by Lemma REF , $L\\circ \\mathcal {V}_ Y\\simeq L\\circ J\\circ \\mathcal {B}_ Y = E\\circ \\mathcal {B}_ Y $ is, in particular, of effective descent.", "Reciprocally, we assume that $E\\circ \\mathcal {B}_ Y $ is of effective descent.", "Again, since $E\\circ \\mathcal {B}_ Y\\circ {\\rm j}\\simeq L\\circ \\mathcal {V}_Y \\circ {\\rm j}$ and $L\\circ \\mathcal {V}_Y$ is of absolute effective descent, we conclude that $E\\circ \\mathcal {B}_ Y\\simeq L\\circ \\mathcal {V}_ Y $ is of absolute effective descent.", "Therefore, since $L$ is pseudocomonadic, by Proposition REF , we conclude that $J\\circ \\mathcal {B}_ Y $ is of effective descent." ], [ "Coherence", "A 2-(co)monadic approach to coherence consists of studying the inclusion of the 2-category of strict (co)algebras into the 2-category of pseudo(co)algebras of a given 2-(co)monad to get general coherence results [1], [12], [15].", "More precisely, one is interested, firstly, to understand whether the inclusion of the 2-category of strict coalgebras into the 2-category of pseudocoalgebras has a right 2-adjoint $G$ (what is called a “coherence theorem of the first type” in [12]).", "Secondly, if there is such a right 2-adjoint, one is interested in investigating whether every pseudocoalgebra $\\mathsf {z} $ is equivalent to the strict replacement $G(\\mathsf {z})$ (what is called a “coherence theorem of the second type” in [12]).", "We fix the notation of this section as follows: we have a 2-comonad $\\mathcal {T}= (\\mathcal {T}, \\varpi , \\varepsilon ) $ on a 2-category $\\mathfrak {C}$ .", "We denote by $\\mathcal {T}$ -${\\rm \\sf CoAlg}_{{}_{\\mathsf {s}}} $ the 2-category of strict coalgebras, strict morphisms and $\\mathcal {T}$ -transformations, that is to say, the usual ${\\rm \\sf CAT}$ -enriched category of coalgebras of the ${\\rm \\sf CAT}$ -comonad $\\mathcal {T}$ .", "The 2-adjunction $E\\dashv R :\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}_{{}_{\\mathsf {s}}}\\rightarrow \\mathfrak {C}$ induces the Eilenberg-Moore factorization w.r.t.", "the pseudocoalgebras: ${\\mathcal {T}\\textrm {-} {\\rm \\sf CoAlg}_{{}_{\\mathsf {s}}}[r]^-{J}[rd]_-{E}& \\mathsf {Ps} \\textrm {-} \\mathcal {T}\\textrm {-} {\\rm \\sf CoAlg}[d]^-{L}\\\\&\\mathfrak {C}}$ in which $J: \\mathcal {T}\\textrm {-} {\\rm \\sf CoAlg}_{{}_{\\mathsf {s}}}\\rightarrow \\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}$ is the usual inclusion.", "Firstly, Corollary REF gives, in particular, necessary and sufficient conditions for which a 2-comonad satisfies the “coherence theorem of the first type” and a weaker version of it, that is to say, it also studies when $J$ has a right biadjoint $G$ .", "Secondly, Corollary REF gives necessary and sufficient conditions for getting a stronger version of the “coherence theorem of the second type”, that is to say, it studies when the counit of the obtained biadjunction/2-adjunction is a pseudonatural equivalence.", "Corollary 8.1 (Coherence Theorem) Let $\\mathcal {T}= (\\mathcal {T}, \\varpi , \\varepsilon ) $ be a 2-comonad on a 2-category $\\mathfrak {C}$ .", "It induces a 2-adjunction $(E\\dashv R, \\rho , \\varepsilon )$ and a biadjunction $(L\\dashv U, \\eta , \\varepsilon , s,t)$ such that ${\\mathcal {T}\\textrm {-} {\\rm \\sf CoAlg}_{{}_{\\mathsf {s}}} [r]^-{J}[rd]_-{E}& \\mathsf {Ps} \\textrm {-} \\mathcal {T}\\textrm {-} {\\rm \\sf CoAlg}[d]^-{L}\\\\&\\mathfrak {C}}$ commutes.", "The inclusion $J: \\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}_{{}_{\\mathsf {s}}}\\rightarrow \\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}$ has a right biadjoint if and only if $\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}_{{}_{\\mathsf {s}}}$ has the descent object of ${ RL\\mathsf {z}@<-2.5ex>[rrrr]_-{RL(\\eta _{{}_\\mathsf {z}})}@<2.5ex>[rrrr]^-{\\rho _ {{}_{RL\\mathsf {z}}}} &&&& R\\mathcal {T}L\\mathsf {z} [llll]|-{R(\\varepsilon _{{}_{L\\mathsf {z}}})}@<-2.5 ex>[rrrr]_-{R\\mathcal {T}L(\\eta _{{}_{\\mathsf {z}}})}[rrrr]|-{RL(\\eta _{{}_{UL\\mathsf {z} }})}@<2.5ex>[rrrr]^-{ \\rho _{{}_{R\\mathcal {T}L\\mathsf {z}}}} &&&& R\\mathcal {T}^2 L\\mathsf {z} }\\qquad \\mathrm {(\\mathcal {A}_\\mathsf {z})}$ for every pseudocoalgebra $\\mathsf {z}$ of $\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}$ .", "In this case, $J$ is left biadjoint to $G$ , given by $G\\mathsf {z}:= \\left\\lbrace \\dot{\\Delta } (\\mathsf {0}, {\\rm j}- ), \\mathcal {A}_\\mathsf {z}\\right\\rbrace _{{\\rm bi}}$ .", "Moreover, assuming the existence of the biadjunction $(J\\dashv G, \\overline{\\varepsilon }, \\overline{\\eta }, \\overline{s}, \\overline{t})$ , the counit $\\overline{\\varepsilon }: JG\\longrightarrow {\\rm id}_ {{}_{\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}}}$ is a pseudonatural equivalence if and only if $E$ preserves the descent object of $\\mathcal {A}_ \\mathsf {z} $ for every pseudocoalgebra $\\mathsf {z}$ .", "Furthermore, $J$ has a genuine right 2-adjoint $G$ if and only if $\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}_{{}_{\\mathsf {s}}}$ admits the strict descent object of $\\mathcal {A}_\\mathsf {z}$ for every $\\mathcal {T}$ -pseudocoalgebra $\\mathsf {z}$ .", "In this case, the right 2-adjoint is given by $\\displaystyle G\\mathsf {z}:= \\left\\lbrace \\dot{\\Delta } (\\mathsf {0}, {\\rm j}- ), \\mathcal {A}_\\mathsf {z}\\right\\rbrace $ .", "Since $(E\\dashv R, \\rho , \\varepsilon )$ and $(L\\dashv U, \\eta , \\varepsilon , s,t)$ induce the same pseudocomonad and $(\\eta J ) = (J\\rho ) $ is a 2-natural transformation, it is enough to apply Corollary REF and Corollary REF to the triangle $L\\circ J = E$ .", "We say that a 2-comonad $\\mathcal {T}$ satisfies the main coherence theorem if there is a right 2-adjoint $\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}\\rightarrow \\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}_{{}_{\\mathsf {s}}} $ to the inclusion and the counit of such 2-adjunction is a pseudonatural equivalence.", "To get the original statement of [12], we have to employ the following well known result (which is a consequence of a more general result on enriched comonads): Let $\\mathcal {T}$ be a 2-comonad on $\\mathfrak {C}$ .", "The forgetful 2-functor $\\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}_{{}_{\\mathsf {s}}}\\rightarrow \\mathfrak {C}$ creates all those strict descent objects which exist in $\\mathfrak {C}$ and are preserved by $\\mathcal {T}$ and $\\mathcal {T}^2 $ .", "Employing this result and Corollary REF , we prove Theorem 3.2 and Theorem 4.4 of [12].", "For instance, we get: Corollary 8.2 ([12]) Let $\\mathcal {T}$ be a 2-comonad on a 2-category $\\mathfrak {C}$ .", "If $\\mathfrak {C}$ has and $\\mathcal {T}$ preserves strict descent objects, then $\\mathcal {T}$ satisfies the main coherence theorem." ], [ "On lifting biadjunctions", "One of the most elementary corollaries of the adjoint triangle theorem [2] is about lifting adjunctions to adjunctions between the Eilenberg-Moore categories.", "In our case, let $\\mathcal {T}:\\mathfrak {A}\\rightarrow \\mathfrak {A}$ and $\\mathcal {S}: \\mathfrak {C}\\rightarrow \\mathfrak {C}$ be 2-comonads (with omitted comultiplications and counits), if ${ \\mathcal {T}\\textrm {-}{\\rm \\sf CoAlg}_{{}_{\\mathsf {s}}}[d]_{\\widehat{L}}[r]^J & \\mathcal {S}\\textrm {-}{\\rm \\sf CoAlg}_{{}_{\\mathsf {s}}}[d]^{L}\\\\\\mathfrak {A}[r]_{E}& \\mathfrak {C}}$ is a commutative diagram, such that $E$ has a right 2-adjoint $R$ , then Proposition REF gives necessary and sufficient conditions to construct a right 2-adjoint to $J$ .", "Also, of course, as a consequence of Corollary REF , we have the analogous version for pseudocomonads.", "Corollary 9.1 Let $\\mathcal {T}:\\mathfrak {A}\\rightarrow \\mathfrak {A}$ and $\\mathcal {S}:\\mathfrak {C}\\rightarrow \\mathfrak {C}$ be pseudocomonads.", "If the diagram ${ \\mathsf {Ps} \\textrm {-}\\mathcal {T}\\textrm {-} {\\rm \\sf CoAlg}[d]_{\\widehat{L}}[r]^J & \\mathsf {Ps} \\textrm {-} \\mathcal {S}\\textrm {-} {\\rm \\sf CoAlg}[d]^{L}\\\\\\mathfrak {A}[r]_{E}& \\mathfrak {C}}$ commutes and $E$ has a right biadjoint, then $J$ has a right biadjoint provided that $\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-} {\\rm \\sf CoAlg}$ has descent objects.", "Recall that $\\mathsf {Ps}\\textrm {-}\\mathcal {T}\\textrm {-} {\\rm \\sf CoAlg}$ has descent objects if $\\mathfrak {A}$ has and $\\mathcal {T}$ preserves descent objects.", "Therefore the pseudofunctor $J$ of the last result has a right biadjoint in this case." ], [ "On pseudo-Kan extensions", "One simple application of Corollary REF is about pseudo-Kan extensions.", "In the tricategory 2-${\\rm \\sf CAT}$ , the natural notion of Kan extension is that of pseudo-Kan extension.", "More precisely, a right pseudo-Kan extension of a pseudofunctor $\\mathcal {D}: \\mathfrak {S}\\rightarrow \\mathfrak {A}$ along a pseudofunctor ${\\rm h}: \\mathfrak {S}\\rightarrow \\dot{\\mathfrak {S}} $ , denoted by ${\\rm Ps}\\textrm {-}{\\mathcal {R}an}_ {{\\rm h}} \\mathcal {D} $ , is (if it exists) a birepresentation of the pseudofunctor $\\mathcal {W}\\mapsto [\\mathfrak {S}, \\mathfrak {A}]_ {PS}(\\mathcal {W}\\circ {\\rm h}, \\mathcal {D})$ .", "Recall that birepresentations are unique up to equivalence and, therefore, right pseudo-Kan extensions are unique up to pseudonatural equivalence.", "Assuming that ${\\rm h}:\\mathfrak {S}\\rightarrow \\dot{\\mathfrak {S}} $ is a pseudofunctor between small 2-categories, in the setting described above, the following are natural problems on pseudo-Kan extensions: (1) investigating the left biadjointness of the pseudofunctor $\\mathcal {W}\\rightarrow \\mathcal {W}\\circ {\\rm h}$ , namely, investigating whether all right pseudo-Kan extensions along ${\\rm h}$ exist; (2) understanding pointwise pseudo-Kan extensions (that is to say, proving the existence of right pseudo-Kan extensions provided that $\\mathfrak {A}$ has all bilimits).", "It is shown in [1] that, if $\\mathfrak {S}_ 0 $ denotes the discrete 2-category of the objects of $\\mathfrak {S}$ , the restriction $[\\mathfrak {S}, \\mathfrak {A}]\\rightarrow [\\mathfrak {S}_ 0 , \\mathfrak {A}] $ is 2-comonadic, provided that $[\\mathfrak {S}, \\mathfrak {A}]\\rightarrow [\\mathfrak {S}_ 0 , \\mathfrak {A}] $ has a right 2-adjoint ${\\mathcal {R}an}_ {\\mathfrak {S}\\rightarrow \\mathfrak {S}_ 0 } $ .", "It is also shown there that the 2-category of pseudocoalgebras of the induced 2-comonad is $ [\\mathfrak {S}, \\mathfrak {A}] _ {PS} $ .", "It actually works more generally: $[\\mathfrak {S}, \\mathfrak {A}] _{PS}\\rightarrow [\\mathfrak {S}_ 0 , \\mathfrak {A}] _ {PS} = [\\mathfrak {S}_ 0 , \\mathfrak {A}]$ is pseudocomonadic whenever there is a right biadjoint ${\\rm Ps}\\textrm {-}{\\mathcal {R}an}_ {\\mathfrak {S}_ 0\\rightarrow \\mathfrak {S}} : [\\mathfrak {S}_ 0, \\mathfrak {A}] _{PS}\\rightarrow [\\mathfrak {S}, \\mathfrak {A}] _{PS} $ because existing bilimits of $\\mathfrak {A}$ are constructed objectwise in $[\\mathfrak {S}, \\mathfrak {A}] _{PS}$ (and, therefore, the hypotheses of the pseudocomonadicity theorem [13] are satisfied).", "Thus, we get the following commutative square: ${ [\\dot{\\mathfrak {S}}, \\mathfrak {A}] _{PS}[d][r]^{[{\\rm h}, \\mathfrak {A}] _{PS} } & [\\mathfrak {S}, \\mathfrak {A}] _{PS}[d]\\\\[\\dot{\\mathfrak {S}_ 0 }, \\mathfrak {A}] [r]_{[{\\rm h}, \\mathfrak {A}] _{PS}}& [\\mathfrak {S}_ 0, \\mathfrak {A}] }$ Thereby, Corollary REF gives a way to study pseudo-Kan extensions, even in the absence of strict 2-limits.", "That is to say, on one hand, if the 2-category $\\mathfrak {A}$ is complete, our results give pseudo-Kan extensions as descent objects of strict 2-limits.", "On the other hand, in the absence of strict 2-limits and, in particular, assuming that $\\mathfrak {A}$ is bicategorically complete, we can construct the following pseudo-Kan extensions: ${\\rm Ps}\\textrm {-}{\\mathcal {R}an}_ {{}_{\\mathfrak {S}_ 0 \\rightarrow \\dot{\\mathfrak {S}}_0 } } : & [\\mathfrak {S}_ 0, \\mathfrak {A}] &\\rightarrow [\\dot{\\mathfrak {S}}_0, \\mathfrak {A}] _{PS}\\\\& \\mathcal {D} & \\mapsto {\\rm Ps}\\textrm {-}{\\mathcal {R}an}_ {{}_{\\mathfrak {S}_ 0 \\rightarrow \\dot{\\mathfrak {S}}_0 } } \\mathcal {D}: \\left( x\\mapsto \\prod _{{\\rm h}(a)=x } \\mathcal {D}a\\right)\\\\&&\\\\{\\rm Ps}\\textrm {-}{\\mathcal {R}an}_ {{}_{\\dot{\\mathfrak {S}} _ 0 \\rightarrow \\dot{\\mathfrak {S}}} } : & [\\dot{\\mathfrak {S}} _ 0, \\mathfrak {A}] &\\rightarrow [\\dot{\\mathfrak {S}}, \\mathfrak {A}] _{PS}\\\\& \\mathcal {D} & \\mapsto {\\rm Ps}\\textrm {-}{\\mathcal {R}an}_ {{}_{\\dot{\\mathfrak {S}} _ 0 \\rightarrow \\dot{\\mathfrak {S}}} } \\mathcal {D} : \\left( x\\mapsto \\prod _{y\\in \\dot{\\mathfrak {S}} _ 0 } \\dot{\\mathfrak {S}} (x, y)\\pitchfork \\mathcal {D} y\\right)\\\\&&\\\\{\\rm Ps}\\textrm {-}{\\mathcal {R}an}_ {{}_{\\mathfrak {S}_ 0 \\rightarrow \\mathfrak {S}} } : & [\\mathfrak {S}_ 0, \\mathfrak {A}] &\\rightarrow [\\mathfrak {S}, \\mathfrak {A}] _{PS}\\\\& \\mathcal {D} & \\mapsto {\\rm Ps}\\textrm {-}{\\mathcal {R}an}_ {{}_{\\mathfrak {S}_ 0 \\rightarrow \\mathfrak {S}} } \\mathcal {D}: \\left( a\\mapsto \\prod _{b\\in \\mathfrak {S}_ 0 } \\mathfrak {S}(a, b)\\pitchfork \\mathcal {D} b\\right)$ in which $\\prod $ and $\\pitchfork $ denote the bilimit versions of the product and cotensor product, respectively.", "Thereby, by Corollary REF , the pseudo-Kan extension ${\\rm Ps}\\textrm {-}{\\mathcal {R}an}_{{\\rm h}} $ can be constructed pointwise as descent objects of a diagram obtained from the pseudo-Kan extensions above.", "Namely, ${\\rm Ps}\\textrm {-}{\\mathcal {R}an}_ {{\\rm h}}\\mathcal {D}x $ is the descent object of a diagram ${ \\mathfrak {a}_ 0@<1.5ex>[rr]@<-1.5ex>[rr]&& \\mathfrak {a}_1[ll]@<1.5 ex>[rr][rr]@<-1.5ex>[rr]&& \\mathfrak {a}_2 }$ in which, by Theorem REF and the last observations, $\\mathfrak {a}_ 0 &=& \\displaystyle \\prod _{y\\in \\dot{\\mathfrak {S}_ 0 }} \\left(\\dot{\\mathfrak {S}} (x, y) \\pitchfork \\prod _{{\\rm h}(a)=y } \\mathcal {D}a \\right)\\simeq \\displaystyle \\prod _{a\\in \\mathfrak {S}_ 0 } \\left( \\dot{\\mathfrak {S}}(x,{\\rm h}(a))\\pitchfork \\mathcal {D}a\\right) \\\\\\\\\\mathfrak {a}_ 1 & = & \\left(\\dot{\\mathfrak {S}} (x, y) \\pitchfork \\prod _{{\\rm h}(a)=y } \\left( \\prod _{b\\in \\mathfrak {S}_ 0 } \\mathfrak {S}(a, b)\\pitchfork \\mathcal {D} b\\right) \\right)\\\\&\\simeq & \\displaystyle \\prod _{a\\in \\mathfrak {S}_ 0 } \\left( \\dot{ \\mathfrak {S}}(x,{\\rm h}(a))\\pitchfork \\left( \\prod _{b\\in \\mathfrak {S}_ 0 } \\mathfrak {S}(a, b)\\pitchfork \\mathcal {D} b\\right)\\right)\\\\&\\simeq & \\displaystyle \\prod _ {(a,b)\\in \\mathfrak {S}_ 0\\times \\mathfrak {S}_ 0 } \\left(\\left(\\mathfrak {S}(a,b)\\times \\dot{\\mathfrak {S}}(x,{\\rm h}(a))\\right) \\pitchfork \\mathcal {D}b\\right)\\\\\\\\\\mathfrak {a}_ 2 &\\simeq & \\displaystyle \\prod _ {(a, b, c)\\in \\mathfrak {S}_ 0\\times \\mathfrak {S}_ 0\\times \\mathfrak {S}_ 0} \\left( \\left( \\mathfrak {S}(b, c)\\times \\mathfrak {S}(a, b)\\times \\dot{\\mathfrak {S}} (x, {\\rm h}(a)) \\right)\\pitchfork \\mathcal {D}c\\right)$ This implies that, indeed, if $\\mathfrak {A}$ is bicategorically complete, then ${\\rm Ps}\\textrm {-} {\\mathcal {R}an}_ {{\\rm h}} \\mathcal {D} $ exists and, once we assume the results of [19] related to the construction of weighted bilimits via descent objects, we conclude that: Proposition 9.2 (Pointwise pseudo-Kan extension) Let $\\mathfrak {S}, \\dot{\\mathfrak {S}} $ be small 2-categories and $\\mathfrak {A}$ be a bicategorically complete 2-category.", "If ${\\rm h}: \\mathfrak {S}\\rightarrow \\dot{\\mathfrak {S}} $ is a pseudofunctor, then ${\\rm Ps}\\textrm {-} {\\mathcal {R}an}_ {{\\rm h}} \\mathcal {D} x = \\left\\lbrace \\dot{\\mathfrak {S}} (x, {\\rm h}-), \\mathcal {D}\\right\\rbrace _ {{\\rm bi}} $ Corollary 9.3 If $\\mathcal {A}:\\Delta \\rightarrow \\mathfrak {A}$ is a pseudofunctor and $\\mathfrak {A}$ has the descent object of $\\mathcal {A}$ , then ${\\rm Ps}\\textrm {-} {\\mathcal {R}an}_ {{\\rm j}} \\mathcal {A}\\mathsf {0}$ is the descent object of $\\mathcal {A} $ .", "Moreover, by the bicategorical Yoneda Lemma, we get: Corollary 9.4 If ${\\rm h}: \\mathfrak {S}\\rightarrow \\dot{\\mathfrak {S}} $ is locally an equivalence and there is a biadjunction $[{\\rm h}, \\mathfrak {A}]\\dashv {\\rm Ps}\\textrm {-}{\\mathcal {R}an}_ {{\\rm h}} $ , its counit is a pseudonatural equivalence.", "Finally, let $\\mathfrak {A}$ be a 2-category with all descent objects and $\\mathcal {T}$ be a pseudocomonad on $\\mathfrak {A}$ .", "Recall that, if $\\mathcal {T}$ preserves all effective descent diagrams, $\\mathsf {Ps} \\textrm {-}\\mathcal {T}\\textrm {-} {\\rm \\sf CoAlg}$ has all descent objects.", "Therefore, if ${\\rm h}: \\mathfrak {S}\\rightarrow \\dot{\\mathfrak {S}} $ is a pseudofunctor, in this setting, the commutative diagram below satisfies the hypotheses of Corollary REF (and, thereby, it can be used to lift pseudo-Kan extensions to pseudocoalgebras).", "${ [\\dot{\\mathfrak {S}} , \\mathsf {Ps} \\textrm {-}\\mathcal {T}\\textrm {-} {\\rm \\sf CoAlg}] _{PS}[d][r] & [\\mathfrak {S}, \\mathsf {Ps} \\textrm {-}\\mathcal {T}\\textrm {-} {\\rm \\sf CoAlg}] _{PS}[d]\\\\[\\dot{\\mathfrak {S}}, \\mathfrak {A}] _{PS}[r]& [\\mathfrak {S}, \\mathfrak {A}] _{PS} }$ Assume that ${\\rm h}: \\mathfrak {S}\\rightarrow \\dot{\\mathfrak {S}} $ is a pseudofunctor, in which $\\mathfrak {S}, \\dot{\\mathfrak {S}} $ are small 2-categories.", "There is another way of proving Proposition REF .", "Firstly, we define the bilimit version of end.", "That is to say, if $T: \\mathfrak {S}\\times \\mathfrak {S}^{{\\rm op}}\\rightarrow {\\rm \\sf CAT}$ is a pseudofunctor, we define $ \\displaystyle \\int _ {\\mathfrak {S}} T : = \\left[ \\mathfrak {A}\\times \\mathfrak {A}^{{\\rm op}}, {\\rm \\sf CAT}\\right] _{PS} \\left( \\mathfrak {A}(-,-), T\\right) $ From this definition, it follows Fubini's theorem (up to equivalence).", "And, if $\\mathcal {B}, \\mathcal {D}: \\mathfrak {S}\\rightarrow \\mathfrak {A}$ are pseudofunctors, the following equivalence holds: $ \\displaystyle \\int _ {\\mathfrak {S}} \\mathfrak {A}(\\mathcal {B}a, \\mathcal {D}a) \\simeq \\left[ \\mathfrak {S}, \\mathfrak {A}\\right] _{PS} \\left( \\mathcal {B}, \\mathcal {D} \\right) $ Therefore, if ${\\rm h}: \\mathfrak {S}\\rightarrow \\dot{\\mathfrak {S}} $ is a pseudofunctor and we define ${\\rm Ps}{\\mathcal {R}an}_ {{\\rm h}} \\mathcal {D}x = \\left\\lbrace \\dot{\\mathfrak {S}} (x, {\\rm h}-), \\mathcal {D}\\right\\rbrace _{{\\rm bi}} $ , we have the pseudonatural equivalences (analogous to the enriched case [9]) $\\left[\\dot{\\mathfrak {S}}, \\mathfrak {A}\\right] _ {PS}(\\mathcal {W}, {\\rm Ps}{\\mathcal {R}an}_ {{\\rm h}} \\mathcal {D}) &\\simeq & \\int _ {\\dot{\\mathfrak {S}}}\\mathfrak {A}(\\mathcal {W}x, {\\rm Ps}{\\mathcal {R}an}_ {\\rm h}\\mathcal {D}x)\\\\&\\simeq & \\int _{\\dot{\\mathfrak {S}}}\\mathfrak {A}(\\mathcal {W}x, \\left\\lbrace \\dot{\\mathfrak {S}} (x, {\\rm h}-), \\mathcal {D}\\right\\rbrace _{{\\rm bi}}) \\\\& \\simeq & \\int _ {\\dot{\\mathfrak {S}} }\\left[ \\mathfrak {S}, {\\rm \\sf CAT}\\right]_ {PS}(\\dot{\\mathfrak {S}} (x, {\\rm h}-), \\mathfrak {A}(\\mathcal {W}x, \\mathcal {D}-))\\\\& \\simeq & \\int _ {\\dot{\\mathfrak {S}}} \\int _ {\\mathfrak {S}} {\\rm \\sf CAT}(\\dot{\\mathfrak {S}} (x, {\\rm h}(a)), \\mathfrak {A}(\\mathcal {W}x, \\mathcal {D}a))\\\\& \\simeq & \\int _ {\\mathfrak {S}} \\int _ {\\dot{\\mathfrak {S}}} {\\rm \\sf CAT}(\\dot{\\mathfrak {S}} (x, {\\rm h}(a)), \\mathfrak {A}(\\mathcal {W}x, \\mathcal {D}a))\\\\&\\simeq & \\int _ {\\mathfrak {S}} \\left[\\dot{\\mathfrak {S}}^{{\\rm op}}, {\\rm \\sf CAT}\\right] _ {PS} (\\dot{\\mathfrak {S}} (-, {\\rm h}(a)), \\mathfrak {A}(\\mathcal {W}-, \\mathcal {D}a))\\\\&\\simeq & \\int _ {\\mathfrak {S}} \\mathfrak {A}(\\mathcal {W}\\circ {\\rm h}(a), \\mathcal {D}a)\\\\&\\simeq & \\left[ \\mathfrak {S}, \\mathfrak {A}\\right]_ {PS} (\\mathcal {W}\\circ {\\rm h}, \\mathcal {D})$ This completes the proof that if the pointwise right pseudo-Kan extension ${\\rm Ps}{\\mathcal {R}an}_{{\\rm h}} $ exists, it is a right pseudo-Kan extension.", "Within this setting and assuming this result, the original argument used to prove Proposition REF using biadjoint triangles gets the construction via descent objects of weighted bilimits originally given in [19]." ] ]
1606.05009
[ [ "Linear radial Regge trajectories for mesons with any quark flavor" ], [ "Abstract In the Regge phenomenology, the radial spectrum of light mesons is given by a linear relation $M_n^2=a(n+b)$, where $a$ is a universal slope, the dimensionless intercept $b$ depends on quantum numbers, and $n$ enumerates the excited states in radial recurrences.", "The usual extensions of this relation to heavy quarkonia in the framework of hadron string models typically lead to strong nonlinearities which seem to be at variance with the available experimental data.", "Introducing a radially static string picture of mesons, we put forward a linear generalization $(M_n-m_1-m_2)^2=a(n+b)$, where $m_{1,2}$ are quark masses.", "The vector channel contains enough experimental states to check this new relation and a good agreement is observed.", "It is shown that this generalization leads to a simple estimate of current quark masses from the radial spectra." ], [ "Linear radial Regge trajectories for mesons with any quark flavorA talk presented at QUARKS-2016.", "S. S. Afonin and I. V. Pusenkov Saint Petersburg State University, 7/9 Universitetskaya nab., St.Petersburg, 199034, Russia In the Regge phenomenology, the radial spectrum of light mesons is given by a linear relation $M_n^2=a(n+b)$ , where $a$ is a universal slope, the dimensionless intercept $b$ depends on quantum numbers, and $n$ enumerates the excited states in radial recurrences.", "The usual extensions of this relation to heavy quarkonia in the framework of hadron string models typically lead to strong nonlinearities which seem to be at variance with the available experimental data.", "Introducing a radially static string picture of mesons, we put forward a linear generalization $(M_n-m_1-m_2)^2=a(n+b)$ , where $m_{1,2}$ are quark masses.", "The vector channel contains enough experimental states to check this new relation and a good agreement is observed.", "It is shown that this generalization leads to a simple estimate of current quark masses from the radial spectra.", "The Regge phenomenology grew out of the dual amplitudes [1] and gave rise to various hadron string models.", "This approach continues to play an important role in the study of hadron spectroscopy.", "The dual amplitudes and some related string approaches predict the following behavior of meson masses, $M_n^2=a(J+n+c), \\qquad J,n=0,1,2,\\dots ,$ where $J$ is the spin, $n$ enumerates the radially excited states, $a$ represents a universal slope and $c$ is a constant.", "The relation (REF ) reproduces the classical Regge behavior $M^2\\sim J$ observed in the light baryons and mesons.", "Also this relation predicts the equidistant daughter Regge trajectories.", "The number $n$ enumerates the states on these \"radial\" Regge trajectories.", "The available experimental data on light non-strange mesons seem to confirm the linear Regge behavior (REF ) [2], [3], [4], [5].", "Various attempts to generalise the relation (REF ) to the sector of heavy quarks resulted in the emergence of strong non-linearities with respect to $J$ and $n$ .", "Let us look, however, at the experimental data.", "A relatively rich set of data on radial recurrences in this sector exists only for the unflavored vector heavy quarkonia.", "Using the relevant data from the Particle Data [6] which are displayed in Table 1 (see Ref.", "[7] for detailed discussions on the data choice) we plot the masses squared of known $\\omega $ , $\\varphi $ , $\\psi $ , and $\\Upsilon $ -mesons as a function of consecutive number $n$ in Figs.", "1 – 4.", "Table: The masses of known ω\\omega , φ\\phi , ψ\\psi andΥ\\Upsilon mesons (in MeV) which are used in ouranalysis.", "The experimental error is not displayed if it is lessthan 1 MeV.Figure: The spectrum of ω\\omega -mesons.The experimental points (for this and subsequent figures) are taken from Table 1.Figure: The spectrum of φ\\phi -mesons.Figure: The spectrum of ψ\\psi -mesons.Figure: The spectrum of Υ\\Upsilon -mesons.Aside from the ground state, the masses approximately lie on the linear radial trajectories $M_n^2=a(n+b),$ where we re-denoted the vector intercept $b=1+c$ .", "The slope and intercept in (REF ) depend strongly on the quark flavor, see Table 2.", "Table: The radial Regge trajectories () (inGeV 2 ^2) for the data from Table 1 (see text).The universal linearity seen in Figs.", "1 – 4 suggests that some universal gluodynamics lies behind the observed behavior.", "We propose a simple string scheme explaining this universality.", "The central idea is that at the conditions when a quark-antiquark pair form a resonance in the QCD vacuum, the pair can be viewed as a radially static system.", "The binding is provided by the exchange of some massless particle (the pion or gluon in concrete realizations of the scheme).", "And one must quantize the motion of this particle (not the radial motion of quarks as in the standard hadron string approaches!).", "The total energy (mass) of the system is $M=m_1+m_2+p+\\sigma r.$ Here $m_1$ and $m_2$ are the masses of quark and antiquark separated by the distance $r$ , $p$ is the momentum of exchanged particle, and $\\sigma $ represents the standard string tension.", "Let us apply the semiclassical quantization to the momentum $p$ $\\int _0^l p\\,dr=\\pi (n+b),\\qquad n=0,1,2,\\dots .$ Here $l$ is the maximal quark separation and the constant $b$ depends on the boundary conditions.", "Substituting $p$ from (REF ) to (REF ) and making use of the definition $\\sigma =\\frac{M}{l}$ we obtain the linear radial trajectory $(M_n-m_1-m_2)^2=2\\pi \\sigma (n+b).$ In our unflavored case $m_1=m_2\\equiv m$ and the relation (REF ) can be simplified to $(M_n-2m)^2=a(n+b),$ which is our generalization of the linear spectrum (REF ) to non-zero quark masses.", "In this relation, the universal gluodynamics (the slope $a$ ) and dependence on quantum numbers (the dimensionless intercept $b$ ) are clearly separated from the contribution of quark masses.", "For this reason, the parameters $a$ and $b$ in the relation (REF ) should be flavor-independent.", "Let us test our expectations.", "We will consider two cases — with the light quark mass set to zero (Fit I) and with all quark masses unfixed (Fit II).", "The results of interpolation of the data in Table 1 by the ansatz (REF ) are given in Table 3.", "The ensuing two variants for the spectrum are depicted in Fig.", "5 and Fig. 6.", "Taking into account a simplicity of the relation (REF ), the agreement is remarkable.", "Table: The quark masses (in GeV), the slope aa (inGeV 2 ^2) and the dimensionless intercept parameter bb in therelation ().Figure: The spectrum () for m u,d m_{u,d} fixed (Fit I).Figure: The spectrum () for m u,d m_{u,d} unfixed (Fit II).The results in Table 3 demonstrate that the radial meson trajectories are able to \"measure\" the current quark masses with surprisingly good accuracy.", "If we set $m_{u,d}=0$ , the current masses of other quarks turn out to be very close to their phenomenological values [6].", "If we keep all quark masses unfixed, they acquire an additional contribution about 360 MeV.", "This contribution may be interpreted as an averaged value of constituent quark mass emerging due to the chiral symmetry breaking in QCD.", "A more detailed analysis (see Ref.", "[7]) shows that the Fit I works better and the interpretation with the current quark masses looks preferable.", "Many related discussions and further fits can be found in Ref. [7].", "An independent fit of the relation (REF ) was also performed by the authors of Ref. [8].", "In summary, we proposed a new generalization of linear radial Regge trajectories to the case of massive quarks.", "Within this generalization, the form of contribution to the meson masses due to confinement is universal for any quarkonia.", "Our generalization is well consistent with the experimental data on the unflavored vector mesons.", "Although our considerations were simple and did not take into account various effects which may cause some mass shifts, the quality of final fits is comparable with typical results of semirelativistic potential models, hadron strings and other technically nontrivial approaches.", "A natural question emerging after our analysis is whether the relation (REF ) can be extended to other types of mesons?", "Unfortunately, the available experimental data are too scarce for making any definite conclusion.", "A possible extensions of (REF ) is $(M_n-m_1-m_2)^2=a(n+x+b),$ where we may have $x=\\beta L$ or $x=\\beta J$ (here $L$ and $J$ mean the orbital momentum of valent quarks and total spin).", "The constant $\\beta $ should be fixed from the phenomenology.", "An intriguing possibility $\\beta =1$ would lead to a large degeneracy observed in the light non-strange mesons [3], [4], [5].", "The fact that the slope $a=2\\pi \\sigma $ in (REF ) coincides with the slope of rotating open string may give a theoretical explanation for this degeneracy.", "Acknowledgments.", "The work was supported by the Saint Petersburg State University research grant 11.38.189.2014 and by the RFBR grant 16-02-00348-a." ] ]
1606.05218
[ [ "Wronskians of Fourier and Laplace Transforms" ], [ "Abstract Associated with a given suitable function, or a measure, on $\\mathbb{R}$, we introduce a correlation function, so that the Wronskian of the Fourier transform of the function is the Fourier transform of the corresponding correlation function, and the same holds for the Laplace transform.", "We obtain two types of results.", "First, we show that Wronskians of the Fourier transform of a nonnegative function on $\\mathbb{R}$ are positive definite functions and the Wronskians of the Laplace transform of a nonnegative function on $\\mathbb{R}_+$ are completely monotone functions.", "Then we establish necessary and sufficient conditions in order that a real entire function, defined as a Fourier transform of a positive kernel $K$, belongs to the Laguerre-P\\'olya class, which answers an old question of P\\'olya.", "The characterization is given in terns of a density property of the correlation kernel related to $K$, via classical results of Laguerre and Jensen and employing Wiener's $L^1$ Tauberian theorem.", "As a consequence we provide a necessary and sufficient condition for the Riemann hypothesis in terms of a density of the translations of the correlation function related to the Riemann $\\xi$-function." ], [ "Introduction", "A real entire function $\\varphi $ is in the Laguerre-Pólya class, written $\\varphi \\in \\mathcal {LP}$ , if $\\varphi (z) = c z^m e^{-{\\alpha }z^2+{\\beta }z} \\prod _{k=1}^\\infty (1+ z/x_k) e^{-z/x_k}$ for some $c,{\\beta }\\in {\\mathbb {R}}$ , ${\\alpha }>0$ , $m \\in {\\mathbb {N}}_0$ and $x_k \\in {\\mathbb {R}}\\setminus \\lbrace 0\\rbrace $ , such that $\\sum _k x_k^{-2} < \\infty $ .", "The class $\\mathcal {LP}$ consists of entire functions that are uniform limits on the compact sets of the complex plane of polynomials with only real zeros.", "This class of functions was studied first by Laguerre in the ninetieth century and then more extensively by Jensen, Pólya, Schur, Obrechkoff and others in the beginning of the twentieth century because of the efforts towards the Riemann hypothesis.", "The latter connection is straightforward and we recall it very briefly.", "The Riemann $\\xi $ -function is defined in terms of the $\\zeta $ -function by (see [19]) $\\xi (s) = \\frac{1}{2} s(s-1) \\pi ^{-s/2} \\Gamma (s/2) \\zeta (s).$ Define also $\\Xi (z) = \\xi (1/2+iz)$ .", "The Riemann hypothesis states that $\\Xi $ , represented also as $\\Xi (z) = \\int _{-\\infty }^{\\infty } \\Phi (u) e^{-izu} du,$ with $\\Phi (t) = 2 \\sum _{n=1}^{\\infty } (2 n^{4} \\pi ^{2} e^{9t/2} - 3 n^{2}\\pi e^{5t/2}) \\exp (- n^{2} \\pi e^{2t}),$ has only real zeros.", "Since $\\Xi (z)$ is an entire function of order one, the Riemann hypothesis is equivalent to the fact that it belongs to $\\mathcal {LP}$ .", "Attempts to provide general tractable necessary and sufficient conditions for an entire function to be in $\\mathcal {LP}$ had failed, so that in 1926 Pólya [14] raised the question of characterizing the kernels $K$ whose Fourier transforms $\\int _{-\\infty }^{\\infty } K(u) e^{-izu} du$ belong to $\\mathcal {LP}$ .", "We establish necessary and sufficient condition for Pólya's problem for a subclass of Fourier transforms that contains $\\Xi (z)$ , which is given in terms of a density of a family of functions in $L^1(\\mathbb {R})$ .", "In order to formulate it, for a given real function $f$ , we denote by ${\\mathcal {T}}(f)$ the span of its translations (or translates), that is, ${\\mathcal {T}}(f) := \\left\\lbrace \\sum _{k=1}^n c_k f(x+a_k),\\ a_k \\in \\mathbb {R}, n \\in {\\mathbb {N}}\\right\\rbrace .$ Our result implies the following necessary and sufficient condition for the Riemann hypothesis: Theorem 1.1 The Riemann hypothesis is true if and only if, for each $y \\in (-1/2,1/2)\\setminus \\lbrace 0\\rbrace $ , the translates ${\\mathcal {T}}(\\Phi _{2,y})$ of the kernel $ \\Phi _{2,y} (t) := \\cosh (ty) \\int _{-\\infty }^{\\infty } (t-2s)^2\\, \\Phi (t-s)\\, \\Phi (s)\\, ds$ are dense in $L^1(\\mathbb {R})$ .", "Furthermore, the translates ${\\mathcal {T}}(\\Phi _{2,y})$ of $\\Phi _{2,y} (t)$ are dense in $L^1(\\mathbb {R})$ for every fixed $y \\in (-1/2,1/2)$ if and only if the zeros of $\\Xi (z)$ are real and simple.", "It is worth mentioning that there are other density criteria for the Riemann hypothesis.", "We mention the classical Nyeman-Beurling criterion [3], [13] and its various generalizations and refinements due to Báez-Duarte and his collaborators [1], [2].", "Other equivalent sufficient conditions for the Riemann hypothesis in terms of properties of the correlation kernel $\\Phi _{2,y} (t)$ will be stated in the end of Section 3.", "The basic ingredients in the proof of Theorem REF are classical results of Jensen and Laguere about entire functions in the Laguerre-Pólya class, Wiener's $L^1$ Tauberian theorem, known also as The Wiener Approximation Theorem, and a tool that we develop in this paper, called the correlation function associated to a given function or a measure, which we now describe.", "Let ${\\mathcal {M}}(E)$ be the set of Borel measures on $E \\in {\\mathbb {R}}$ .", "For $\\mu \\in {\\mathcal {M}}(E)$ and $m=0,1,\\ldots $ , let $\\mu _m: = \\int _{\\mathbb {R}}t^m d\\mu $ be the moment of $\\mu $ .", "Let ${\\mathcal {M}}_N(E)$ denote the set that consists of $\\mu \\in {\\mathcal {M}}(E)$ for which $\\mu _n$ is finite for $n =0,1,\\ldots N$ .", "Furthermore, we denote by ${\\mathcal {M}}^+(E)$ and ${\\mathcal {M}}_N^+$ the subset of non-negative Borel measures, respectively.", "Definition 1.2 Let $\\mu \\in {\\mathcal {M}}_{2n-2}({\\mathbb {R}})$ be an absolutely continuous measure.", "For $n=2,3,\\ldots $ we define a correlation function $\\nu _n(t):= \\nu _n(d\\mu ;t) = \\int _{T^n(t)} \\prod _{1 \\le i< j \\le n} (s_i-s_j)^2 \\mu ^{\\prime }(s_1)\\ldots \\mu ^{\\prime }(s_n) d{\\mathbf {s}},$ where $T^n(t)$ is the simplex in ${\\mathbb {R}}^n$ defined by $T^n(t): = \\lbrace (s_1,\\ldots ,s_n) \\in {\\mathbb {R}}^n: s_1+\\cdots + s_n =t\\rbrace $ and $d{\\mathbf {s}}$ is the Lebesgue measure on $T^n(t)$ .", "When $d\\mu = w(t) dt$ , we also write $\\nu _n(w; t)$ and define $\\nu _1(t): = w(t)$ .", "The correlation function is well–defined and is closely related to the Wronskian determinants of integral transforms.", "We study this function in view of the Fourier and Laplace transforms below.", "For $\\mu \\in {\\mathcal {M}}({\\mathbb {R}})$ , let ${\\mathcal {F}}$ be the Fourier transform of $\\mu $ defined by, with ${\\mathtt {i}}=\\sqrt{-1}$ , ${\\mathcal {F}}\\mu (x):= \\widehat{\\mu }(x) = \\int _{\\mathbb {R}}e^{- {\\mathtt {i}}x t }d\\mu (t), \\qquad x \\in {\\mathbb {R}}.$ A measure $\\mu \\in {\\mathcal {M}}({\\mathbb {R}})$ is called even if $d\\mu (t) = d\\mu (-t)$ .", "If $\\mu $ is even, then ${\\mathcal {F}}\\mu $ is a real valued function.", "Let ${\\mathbb {R}}_+ = [0, \\infty )$ .", "For $\\mu \\in {\\mathcal {M}}({\\mathbb {R}}_+)$ , let ${\\mathcal {L}}\\mu $ be the Laplace transform of $\\mu $ defined by ${\\mathcal {L}}\\mu (x):= \\int _{\\mathbb {R}}e^{- x t }d\\mu (t), \\qquad x \\in {\\mathbb {R}}.$ Let $f$ be a function in $C^{2n-2}({\\mathbb {R}})$ , the class of functions that have continuous derivatives of order $2n-2$ .", "The $n$ -th Wronskian determinant of the function $f$ is defined by $W_n (f;x) := \\det \\left[f^{(i+j)}(x) \\right]_{i,j =0}^{n-1}=\\det \\left[ \\begin{matrix} f(x) & f^{\\prime }(x) & \\ldots & f^{(n-1)}(x) \\\\f^{\\prime }(x) & f^{\\prime \\prime }(x) & \\ldots & f^{(n)}(x) \\\\\\cdots & \\cdots & \\cdots & \\cdots \\\\f^{(n-1)}(x) & f^{(n)}(x) & \\ldots & f^{(2n-2)}(x)\\end{matrix} \\right] .$ The key ingredient, and our starting point, in this study is the observation that the Wronskian of the Fourier (or Laplace) transform of a function is the Fourier (respectively, Laplace) transform of the corresponding correlation function.", "More precisely, we have the following theorem: Theorem 1.3 For $n =2,3,\\ldots $ and $f \\in L^1(E)$ such that $\\int _E |t^{2n-2} f(t)| dt < \\infty $ , $W_n({\\mathcal {F}}f; \\cdot ) = (-1)^{n(n+1)/2}{\\mathcal {F}}(\\nu _n(f))$ if $E = {\\mathbb {R}}$ ; $W_n({\\mathcal {L}}f; \\cdot ) = {\\mathcal {L}}( \\nu _n(f) )$ if $E={\\mathbb {R}}_+$ .", "This result allows us to prove, for example, that if $f$ is a nonnegative even function, then $(-1)^{n(n-1)/2} W_n({\\mathcal {F}}f;x)$ is a strictly positive definite function on the real line, and $W_n({\\mathcal {L}}f;x)$ is a completely monotone function on ${\\mathbb {R}}_+$ .", "The paper is organised as follows.", "The next section is devoted to the study of the correlation functions, where we establish the latter results.", "Examples that illustrate our results are also given in the section.", "The functions in the Laguerre-Pólya class, represented as a Fourier transform, are studied in Section 3, where we give the proof of Theorem 1.1.", "In Section 4 we provide some comments and results concerning Wiener's Tauberian theorem related to the main result established in Section 3." ], [ "Correlation functions and Wronskians of integral transforms", "We discuss general properties of the correlation functions and its immediate applications in the first subsection.", "Several examples are given in the second subsection." ], [ "Correlation functions and Wronskians", "The integral in the definition (REF ) has $n-1$ folds and it can be written explicitly so as an integral over the simplex $T^n(t) := \\lbrace s \\in {\\mathbb {R}}^{n-1}: {\\alpha }(s):= s_1+ \\cdots +s_{n-1} \\le t\\rbrace $ of ${\\mathbb {R}}^{n-1}$ .", "Indeed, we can write the correlation function $\\nu _n$ as $ \\nu _n(w;t) = \\int _{T^n(t)} \\prod _{1 \\le i < j \\le {n-1}}& (s_i-s_j)^2 \\prod _{i=1}^{n-1} (t- {\\alpha }(s) -s_i)^2 \\\\& \\times \\mu ^{\\prime }(s_1)\\cdots \\mu ^{\\prime }(s_{n-1}) \\mu ^{\\prime }(t-{\\alpha }(s))ds_1\\cdots ds_{n-1} $ by setting $s_n = t - {\\alpha }(s) = t- s_1-\\cdots - s_{n-1}$ in (REF ).", "We need a classical identity that is the cornerstone of our results.", "The identity can be found in [16].", "Lemma 2.1 Let $f_i, g_i$ , $1 \\le i \\le n$ , be integrable functions defined on ${\\mathbb {R}}$ such that $f_i g_j \\in L^1({\\mathbb {R}})$ for $1 \\le i \\ne j \\le n$ .", "Then $ \\det \\left[ \\int _{{\\mathbb {R}}} f_i(t) g_j(t) d\\mu (t) \\right]_{i,j=1}^n = \\int _{{\\mathbb {R}}^n} \\det \\left[ f_j(t_i) \\right]_{i,j=1}^n\\det \\left[ g_j(t_i) \\right]_{i,j=1}^n \\prod _{i=1}^n d\\mu (t_i).$ For $\\mu \\in {\\mathcal {M}}(E)$ and $n=0,1,\\ldots $ , let $M_n(d\\mu )$ be the moment matrix defined by $M_n(d\\mu ): = \\det \\left[ \\mu _{i+j} \\right]_{i,j =0}^n.$ If $d\\mu (t) =w(t) dt$ , we write this determinant as $M_n(w)$ .", "It is known that, if $\\mu \\in {\\mathcal {M}}_{2n}^+(E)$ , then $M_n(d\\mu )$ is positive definite and, in particular, $\\det M_n(d\\mu ) > 0$ .", "Lemma 2.2 Let $\\mu \\in {\\mathcal {M}}_{2n-2} ({\\mathbb {R}})$ be absolutely continuous.", "Then the correlation function $\\nu _n$ is integrable and $\\int _{{\\mathbb {R}}} \\nu _n(d\\mu ; x) dx = \\det M_{n-1}(d\\mu ).$ Furthermore, if $d\\mu (t) = w(t)dt$ and $w \\in L^1({\\mathbb {R}})$ , then $\\nu _n \\in L^1({\\mathbb {R}})$ .", "Applying the previous lemma and using the Vandermond determinant $V(t_1,\\ldots ,t_n) := \\det \\left[ t_i^{j-1} \\right]_{i,j = 0}^{n-1} = \\prod _{1 \\le i< j \\le n} (t_j - t_i),$ we obtain $\\int _{{\\mathbb {R}}} \\nu _n(t) dt & = \\int _{\\mathbb {R}}\\int _{{\\mathcal {S}}^n(t)} \\prod _{1 \\le i< j \\le n} (s_i-s_j)^2 \\mu ^{\\prime }(s_1)\\ldots \\mu ^{\\prime }(s_n)d {\\mathbf {s}}dt \\\\& = \\int _{{\\mathbb {R}}^n} V(s_1,\\ldots ,s_n)^2 d\\mu (s_1)\\ldots d\\mu (s_n) \\\\& = \\det \\left[ \\int _{\\mathbb {R}}t^{i+j} d\\mu (t) \\right]_{i,j =0}^{n-1} = \\det M_{n-1}(d\\mu ),$ where we have used (REF ) with $f_j(t) = g_j(t) = t^j$ .", "If $d\\mu (t) =w(t) dt$ and $w\\in L^1({\\mathbb {R}})$ , then the same proof shows that $\\int _{{\\mathbb {R}}} |\\nu _n(t)| dt \\le \\int _{{\\mathbb {R}}^n} V(s_1,\\ldots ,s_n)^2 w(s_1)\\ldots w(s_n) ds_1\\cdots ds_n = \\det M_{n-1}(|w|),$ which is finite and positive, so that $\\nu _n \\in L^1({\\mathbb {R}})$ .", "Lemma 2.3 If $d\\mu $ is an even measure, then $\\nu _n$ is an even function for $n =2,3,\\ldots $ .", "Changing variables $s_i \\rightarrow - s_i$ in the definition of $\\nu _n$ shows that $\\nu _n(t) = \\nu _n(-t)$ if $d\\mu $ is symmetric with respect to the origin.", "Lemma 2.4 If $d\\mu (t) =f(t)dt $ and $f$ is supported on the interval $(a,b)$ , then $\\nu _n(f)$ is supported on $(na, nb)$ .", "If $f$ is supported on $(a,b)$ , then $a \\le s_i \\le b$ in (REF ).", "As a result, if $t \\le na$ , then $t - s_1-\\cdots -s_{n-1} \\le a$ , so that $f(t- {\\alpha }(s)) =0$ .", "Similarly, if $t \\ge nb$ , then $f(t-{\\alpha }(s)) =0$ .", "We are now ready to prove Theorem REF , which we restate below: Theorem 2.5 For $n =2,3,\\ldots $ , $f \\in L^1(E)$ such that $t^{2n-2} f \\in L^1(E)$ , $W_n({\\mathcal {F}}f; \\cdot ) = (-1)^{n(n+1)/2}{\\mathcal {F}}(\\nu _n(f))$ if $E= {\\mathbb {R}}$ ; $W_n({\\mathcal {L}}f; \\cdot ) = {\\mathcal {L}}( \\nu _n(f) )$ if $E ={\\mathbb {R}}_+$ .", "We prove (2) first.", "Let $d\\mu = f (t) dt$ .", "The condition $t^{2n-2} f \\in L^1({\\mathbb {R}}_+)$ implies that the derivatives of the Laplace transform ${\\mathcal {L}}f$ , up to $2n-2$ order, are well defined continuous functions, so is $W_n({\\mathcal {L}}f;\\cdot )$ .", "Applying (REF ) with $f_j(t) = g_j(t) = t^j$ , we obtain $W_n({\\mathcal {L}}f;x) & = \\det \\left[ \\int _{{\\mathbb {R}}_+} (-t)^{i+j} e^{-t x} d \\mu (t) \\right]_{i,j =0 }^{n-1} \\\\& = \\int _{{\\mathbb {R}}_+^n} \\prod _{1 \\le i < j \\le n} (s_i- s_j)^2 e^{- x(s_1+\\cdots + s_n)} d\\mu (s_1)\\cdots d\\mu (s_n) \\\\& = \\int _{\\mathbb {R}}\\nu _n(f;t) e^{-t x} dt = {\\mathcal {L}}\\nu _n(f;x),$ which proves (2).", "The proof of (1) is similar.", "Taking the derivatives of ${\\mathcal {F}}\\mu $ introduces powers of ${\\mathtt {i}}$ .", "Using $\\prod _{1 \\le i < j \\le n} (- {\\mathtt {i}}s_j + {\\mathtt {i}}s_i)^2 = {\\mathtt {i}}^{n(n-1)/2} \\prod _{1 \\le i < j \\le n} (s_i-s_j)^2= (-1)^{n(n-1)/2} \\prod _{1 \\le i < j \\le n} (s_i-s_j)^2,$ the proof follows as in the case (2).", "Although these are simple relations, we give two nontrivial applications below to show that they are not as obvious as they appear to be.", "Recall that a function $\\psi : {\\mathbb {R}}\\mapsto {\\mathbb {R}}$ is called positive definite if $\\sum _{i =1}^N \\sum _{j=1}^N c_i c_j \\psi (x_i - x_j) \\ge 0$ for all $x_1\\ldots , x_N \\in {\\mathbb {R}}$ and $N =1,2,\\ldots $ , and it is called strictly positive definite if $\\ge 0$ is replaced by $>0$ whenever $(c_1,\\ldots ,c_N)$ is not identically zero.", "The positive definite functions are characterized by Bochner's theorem: a continuous function $\\psi $ is positive definite if and only if it is the Fourier transform of a finite non-negative Borel measure.", "Together with Theorem REF , we can then state the following theorem: Theorem 2.6 For each $n \\in {\\mathbb {N}}$ , let $\\mu $ be an even measure in ${\\mathcal {M}}_{2n-1}^+({\\mathbb {R}})$ .", "Then the multiple of the Wronskian determinant $W_n^{\\mathcal {F}}(\\mu ; x):= (-1)^{n(n-1)/2} W_n({\\mathcal {F}}\\mu ;x)$ is a strictly positive definite function on the real line.", "We only have to comment on the strictly positiveness of the statement, which is not covered by Bochner's theorem on the positive definite functions.", "However, it is known that if $\\psi $ is the Fourier transform of a finite non-negative Borel measure and the measure is not discrete, then $\\psi $ is strictly positive.", "A function $\\phi : {\\mathbb {R}}_+ \\mapsto {\\mathbb {R}}$ in $C({\\mathbb {R}}_+) \\cap C^\\infty ({\\mathbb {R}}_+)$ is called a completely monotone function if it satisfies $(-1)^k \\phi ^{(k)} (x) \\ge 0, \\qquad x > 0, \\quad k =0,1,2,\\ldots .$ The completely monotone functions are characterized by Bernstein's theorem: a function is completely monotone if and only if it is the Laplace transform of a finite non-negative Borel measure $\\mu $ on ${\\mathbb {R}}_+$ .", "Together with Theorem REF , we can then state the following theorem, which appeared first in [8] and motivated our study in the present paper: Theorem 2.7 Let $n \\in {\\mathbb {N}}$ and $\\mu \\in {\\mathcal {M}}_{2n-2}^+({\\mathbb {R}}_+)$ .", "For each $n \\in {\\mathbb {N}}$ , $W_n({\\mathcal {L}}\\mu ; \\cdot )$ is a completely monotone function on ${\\mathbb {R}}_+$ .", "We state a corollary of Theorem REF and Theorem REF .", "Let $({\\mathcal {F}}\\mu )^{(k)}$ and $({\\mathcal {L}}\\mu )^{(k)}$ denote the $k$ -th order derivative of ${\\mathcal {F}}\\mu $ and ${\\mathcal {L}}\\mu $ , respectively.", "Theorem 2.8 Let $k =1,2,\\ldots $ and $n=2,3,\\ldots $ .", "Then If $\\mu \\in {\\mathcal {M}}_{k+n-2}^+{\\mathbb {R}})$ is even, then $(-1)^{n k/2} W_n( ({\\mathcal {F}}\\mu )^{(k)}; \\cdot )$ is a strictly positive definite function on the real line provided either $k$ or $n$ is even.", "If $\\mu \\in {\\mathcal {M}}_{k+n-2}^+({\\mathbb {R}}_+)$ , then $(-1)^{n k} W_n(({\\mathcal {L}}\\mu )^{(k)}; \\cdot )$ is a completely monotone function on the real line.", "For $k=1,2,\\ldots $ , let $\\lbrace \\cdot \\rbrace ^k \\mu $ be the measure defined by $d (\\lbrace \\cdot \\rbrace ^k \\mu )(x) := x^k d\\mu (x)$ .", "Since $({\\mathcal {F}}\\mu )^{(k)}(x) = (- {\\mathtt {i}})^k {\\mathcal {F}}(\\lbrace \\cdot \\rbrace ^k \\mu ; x)$ , it is easy to see that $W_n \\left(({\\mathcal {F}}\\mu )^{(k)}; x \\right) = (-{\\mathtt {i}})^{n(n+k+1)} W_n \\left( {\\mathcal {F}}(\\lbrace \\cdot \\rbrace ^k \\mu ); x \\right),$ which is real valued and $(-{\\mathtt {i}})^{n(n+k+1)} = (-1)^{n(n+k+1)/2}$ if $n k $ is even.", "Since $\\lbrace \\cdot \\rbrace ^k \\mu \\in {\\mathcal {M}}_{n-2}^+({\\mathbb {R}})$ , (1) follows from Theorem REF .", "Similarly, (2) follows as a corollary of Theorem REF .", "In particular, in the case of $n =2$ , this shows that $W_2 \\left(({\\mathcal {F}}\\mu )^{(k)}; x \\right) =(-1)^k \\left( [({\\mathcal {F}}\\mu )^{(k+1)}(x)]^2 - ({\\mathcal {F}}\\mu )^{(k)}(x)({\\mathcal {F}}\\mu )^{(k+2)}(x) \\right)$ is a strictly positive definite function.", "For a nontrivial example of such results, we refer to Corollary REF in the next subsection." ], [ "Examples", "We give several examples to illustrate our results.", "First we point out that, by Theorem REF and Theorem REF , $\\nu _n(d\\mu ; x) = {\\mathcal {L}}^{-1} [W_n ({\\mathcal {L}}(d\\mu ); \\cdot )](x) \\quad \\hbox{and}\\quad \\nu _n(d\\mu ; x) = {\\mathcal {F}}^{-1} [W_n^{\\mathcal {F}}(d\\mu ; \\cdot )](x),$ where we assume that $d\\mu $ is supported on ${\\mathbb {R}}_+$ in the first identity.", "In general, taking the Fourier or Laplace transform of $W_n^{\\mathcal {F}}(d\\mu ; \\cdot )$ or $W_n ({\\mathcal {L}}(d\\mu ); \\cdot )$ , respectively, is difficult and the above identity may not be useful for determining the explicit formula of $\\nu _n$ .", "In some cases, however, it can be used as shown in our first two examples.", "Proposition 2.9 If $d\\mu (t) = e^{-t^2/2} dt $ , then $\\nu _n(d\\mu ; t) = a_n e^{-t^2/{2n}} \\quad \\hbox{with}\\quad a_n = \\frac{1}{\\sqrt{n}} (2\\pi )^{(n-1)/2} \\prod _{k=1}^{n-1}k!.$ In particular, the span of $\\lbrace e^{-(t-a)^2/{2n}}: a \\in {\\mathbb {R}}\\rbrace $ is dense in $L^1({\\mathbb {R}})$ .", "Let $h(t) = e^{-t^2/2}$ .", "It is well-known that $\\widehat{h} (t) = \\sqrt{2 \\pi } e^{-t^2}$ .", "Furthermore, by the Rodrigues formula of the Hermite polynomials, it is easy to see that $\\frac{d^n}{dx^n} \\widehat{h}(x) = \\sqrt{2\\pi } 2^{-n/2} (-1)^n H_n(x/2).$ The closed formula of the Wronskian of the Hermite polynomials is known; see, for example, the identity [11], from which we deduce that $W_n({\\mathcal {F}}h; x) = \\left( (-1)^{n(n-1)/2} (2 \\pi )^{n/2} \\prod _{k=1}^{n-1} k!", "\\right) e^{-n x^2/2}.$ Taking the inverse Fourier transform of this identity, the formula for $\\nu _n(t)$ follows from (1) in Theorem REF .", "Proposition 2.10 Let $d\\mu _{\\alpha }(t) = t^{\\alpha }e^{-t} dt$ , ${\\alpha }> -1$ , be supported on ${\\mathbb {R}}_+$ .", "Then $\\nu _n(d\\mu _{\\alpha };t) = a_n t^{ n(n+{\\alpha })-1} e^{-t} \\quad \\hbox{with} \\quad a_n =\\frac{\\Gamma ({\\alpha }+1)^n}{\\Gamma (n(n+{\\alpha }))} \\prod _{k=1}^{n-1} k!", "({\\alpha }+1)_k.$ Let $h_{\\alpha }(t) = t^{\\alpha }e^{-t}$ on ${\\mathbb {R}}_+$ .", "The Laplace transform of $h_{\\alpha }$ is given by ${\\mathcal {L}}h_{\\alpha }(t) = \\Gamma ({\\alpha }+1) (1+t)^{-{\\alpha }-1}$ , so that $({\\mathcal {L}}h_{\\alpha }(t))^{(k)} = (-1)^k\\Gamma ({\\alpha }+k) (1+t)^{-{\\alpha }- k}$ .", "It follows that $W_n( {\\mathcal {L}}h_{\\alpha }; x) = \\left(\\prod _{k=1}^{n-1} k!", "({\\alpha }+1)_k \\right) \\frac{\\Gamma ({\\alpha }+1)^n}{(1+x)^{n (n+{\\alpha })}}.$ Indeed, the power of $1+x$ can be completely factored out from the determinant, so that $W_n({\\mathcal {L}}h_{\\alpha };x) = c (1+x)^{-n (n+{\\alpha })}$ and the constant determinant can be evaluated to the value given.", "Taking the inverse Laplace transform of this identity, the formula of $\\nu _n(t)$ follows from (2) in Theorem REF .", "Our next example uses the definition of the correlation function, or rather (REF ), to derive an explicit formula for $\\nu _2$ for a family of functions.", "Let ${}_2F_1$ be the standard Gauss hypergeometric function.", "Proposition 2.11 For ${\\alpha }, {\\beta }> -1$ , let $w_{{\\alpha },{\\beta }}(t) = (1-t)^{\\alpha }(1+t)^{\\beta }\\chi _{[-1,1]}(t)$ .", "Then $\\nu _2(w_{{\\alpha },{\\beta }}; t) =0$ if $|t| > 2$ and $\\nu _2(w_{{\\alpha },{\\beta }} ;t) = \\frac{\\Gamma (\\frac{3}{2}) \\Gamma ({\\alpha }+1)}{2^{2 {\\alpha }+1} \\Gamma ({\\alpha }+\\frac{5}{2})}(2-t)^{2{\\alpha }+3} (2 t)^{{\\beta }} {}_2F_1\\left( \\begin{matrix} - {\\beta }, {\\alpha }+1 \\\\ {\\alpha }+\\frac{5}{2} \\end{matrix}; - \\frac{(2-t)^2}{8t}\\right)$ if $0 \\le t \\le 2$ , and $\\nu _2(w_{{\\alpha },{\\beta }};- t) = \\nu _2(w_{{\\beta },{\\alpha }};t)$ .", "Directly from the expression of $\\nu _2$ at (REF ), we obtain that $\\nu _2(t) = \\int _{-\\infty }^{\\infty } (2s-t)^2 w(s) w(t-s) ds,$ from which it follows immediately that $\\nu _2(w_{{\\alpha },{\\beta }};t) = 0$ if $|t| > 2$ and $\\nu _2(w_{{\\alpha },{\\beta }};- t) = \\nu _2(w_{{\\beta },{\\alpha }};t)$ .", "Furthermore, for $0 \\le t \\le 2$ , changing variable $u = -1+2 (1-s)/(2-t)$ gives $\\nu _2(w_{{\\alpha },{\\beta }};t) & = \\int _{-1+t}^{1} (2s-t)^2 w_{{\\alpha },{\\beta }}(s) w_{{\\alpha },{\\beta }}(t-s) ds \\\\& = \\frac{(2-t)^{2 {\\alpha }+3}}{2^{ 2 {\\alpha }+1}} \\int _{-1}^1 \\left( 2 t + \\frac{(2-t)^2}{4} (1-u^2)\\right)^{\\beta }u^2 (1-u^2)^{\\alpha }du \\\\& =\\frac{(2-t)^{2 {\\alpha }+3}}{2^{ 2 {\\alpha }+1}} (2 t)^{\\beta }\\int _{-1}^1 \\left( 1 + \\frac{(2-t)^2}{8 t}(1-u^2)\\right)^{{\\beta }}u^2 (1-u^2)^{{\\alpha }} du.$ Let $s = - (2-t^2)/(8t)$ .", "Expanding $(1 + s (1-u^2))^{{\\beta }}$ in infinite series, we obtain $\\int _{-1}^1 \\left( 1 - s (1-u^2)\\right)^{{\\beta }} u^2 (1-u^2)^{{\\alpha }} du& = \\sum _{n=0}^\\infty \\frac{(-{\\beta })_n}{n!", "}\\int _{-1}^1 u^2 (1-u^2)^{n+{\\alpha }} du s^n \\\\& = \\frac{\\Gamma (3/2) \\Gamma ({\\alpha }+1)}{\\Gamma ({\\alpha }+\\frac{5}{2})}\\sum _{n=0}^\\infty \\frac{(-{\\beta })_n ({\\alpha }+1)_n }{({\\alpha }+\\frac{5}{2})_n n!}", "s^n,$ the last summation is the ${}_2F_1$ function.", "This completes the proof.", "Corollary 2.12 For ${\\lambda }> -1/2$ , let $w_{\\lambda }(t) := (1-t^2)^{{\\lambda }-1/2} \\chi _{[-1,1]}(t)$ .", "Then $\\nu _2(w_{\\lambda };t) = \\frac{\\Gamma (\\frac{3}{2}) \\Gamma ({\\lambda }+\\frac{1}{2})}{2^{{\\lambda }+\\frac{1}{2}} \\Gamma ({\\lambda }+2)}(2-|t|)^{2{\\lambda }+2} |t|^{{\\lambda }-\\frac{1}{2}} {}_2F_1\\left( \\begin{matrix} - {\\lambda }+ \\frac{1}{2}, {\\lambda }+\\frac{1}{2} \\\\ {\\lambda }+2 \\end{matrix}; \\frac{(2-|t|)^2}{8|t|}\\right)\\chi _{[-2,2]}(t).$ In particular, the above ${}_2F_1$ function is nonnegative on the interval $[-2,2]$ .", "The Fourier transform of $w_{{\\alpha },{\\beta }}$ is given by ${\\mathcal {F}}w_{{\\alpha },{\\beta }} (x) = \\frac{2^{{\\alpha }+{\\beta }+1}\\Gamma ({\\alpha }+1)\\Gamma ({\\beta }+1)}{\\Gamma ({\\alpha }+{\\beta }+2)}e^{ {\\mathtt {i}}t} {}_1F_1\\left( \\begin{matrix} {\\beta }+1 \\\\ {\\alpha }+{\\beta }+2 \\end{matrix}; -2 {\\mathtt {i}}t\\right).$ More interestingly, the Fourier transform of $w_{\\lambda }$ can be expressed in terms of the Bessel function $J_{\\lambda }(x)$ .", "Indeed, by the integral representation of the Bessel function, ${\\mathcal {F}}w_{\\lambda }(x) = \\int _{-1}^1 e^{-{\\mathtt {i}}tx} (1-t^2)^{{\\lambda }-1/2} dt = \\sqrt{\\pi }\\Gamma ({\\lambda }+1/2) \\left(\\frac{2}{x}\\right)^{\\lambda }J_{\\lambda }(x).$ Using the identity $J_{\\lambda }^{\\prime }(x) = (J_{{\\lambda }-1}(x) - J_{{\\lambda }+1}(x))/2$ , it follows that $- W_2({\\mathcal {F}}w_{\\lambda };x) =\\, & 4^{\\lambda }\\pi \\Gamma ({\\lambda }+\\tfrac{1}{2})^2 x^{-2(1+{\\lambda })} \\\\& \\times \\left[ x^2 J_{{\\lambda }-1}(x)^2 - (2 {\\lambda }-1) x J_{{\\lambda }+1}(x)J_{\\lambda }(x)+ (x^2 -2{\\lambda })J_{\\lambda }(x)^2\\right].$ By Theorem REF , this function is equal to the Fourier transform of $\\nu _2(w_{\\lambda };t)$ , which is non-trivial since a direct verification is not immediate and, in fact, looks formidable for a generic parameter ${\\lambda }$ .", "In the case of ${\\lambda }= 1/2$ , $w_{1/2}$ is the characteristic function $w_{1/2}(t) = \\chi _{(-1,1)}(t)$ .", "In this case, ${\\mathcal {F}}w_{1/2}(t) = 2 \\frac{\\sin t}{t} \\quad \\hbox{and} \\quad \\nu _2(w_{1/2};t) = \\frac{1}{3} (2-|t|)^3.$ It is known that if $\\phi $ is a positive definite function, then $\\phi (0) > 0 \\quad \\hbox{and} \\quad |\\phi (x)| \\le \\phi (0), \\quad \\forall x\\in {\\mathbb {R}}.$ However, a positive definite function is not necessarily positive everywhere.", "Corollary 2.13 For ${\\lambda }> -1/2$ , let ${\\mathcal {J}}_{\\lambda }(x):= \\left(\\frac{1}{t}\\right)^{\\lambda }J_{\\lambda }(t)$ .", "For $n = 2, 3,\\ldots $ , the Wronskian determinant $W_n^{({\\lambda })} (x): =(-1)^{n(n-1)/2} W_n({\\mathcal {J}}_{\\lambda }; x)$ is a strictly positive definite function on ${\\mathbb {R}}$ and $W_n^{({\\lambda })} (x) \\le W_n^{({\\lambda })} (0)$ for $x \\in {\\mathbb {R}}$ .", "Even in the case of $w_0(t) = \\chi _{[-1,1]}(t)$ , that the Wronskian $W_n^{\\mathcal {F}}(x)$ leads to a positive definite function appears to be nontrivial.", "For example, the case $n =2$ shows that $W_2^{\\mathcal {F}}(x)= 2 \\frac{-1 + 2 x^2 + \\cos (2 x)}{x^4}$ is a strictly positive function on the real line and $W_2^{\\mathcal {F}}(x) \\le 4/3$ for all $x\\in {\\mathbb {R}}$ .", "As an application of Proposition REF in Section 4, $W_2^{\\lambda }(x) > 0$ on ${\\mathbb {R}}$ .", "In this case, the inequality in Corollary REF gives: Corollary 2.14 For ${\\lambda }> -1/2$ and $x \\in {\\mathbb {R}}$ , $0 < ({\\mathcal {J}}_{\\lambda }^{\\prime }(x))^2 - {\\mathcal {J}}_{\\lambda }(x) {\\mathcal {J}}^{\\prime \\prime }_{\\lambda }(x) \\le \\frac{1}{2} \\left(\\frac{1}{\\Gamma ({\\lambda }+1)^2} -\\frac{1}{\\Gamma ({\\lambda })\\Gamma ({\\lambda }+2)} \\right).$ In particular, the right hand side becomes $1/2$ when ${\\lambda }= 0$ .", "We give another example, where the “tent function” $w_\\Lambda $ is such that its Fourier transform is the function $(\\sin t/t)^2$ .", "Proposition 2.15 Let $w_\\Lambda (t) = \\frac{1}{4} (2-|t|)_+$ , where $x_+ = x$ if $x \\ge 0$ and $x_+ = 0$ otherwise.", "Then the correlation function $\\nu _2(w_\\Lambda ;\\cdot )$ is given by $\\nu _2(w_\\Lambda ; t) = \\frac{1}{480} \\chi _{[-4,4]}(t){\\left\\lbrace \\begin{array}{ll} (4-|t|)^5 -2 (2-|t|)^5 - 40 (2-|t|)^3, & 0 \\le t \\le 2, \\\\(4-|t|)^5, & t \\ge 2.\\end{array}\\right.", "}$ As in the proof of Proposition REF , it is easy to see that $\\nu _2(w_\\Lambda )$ is supported on $[-4,4]$ and it is an even function.", "Assume $0 \\le t \\le 4$ .", "It follows then that $\\nu _2(w_\\Lambda ;t) = \\frac{1}{16}\\int _{-2+t}^2 (2s -t)^2 (2 - s) (2 - |t - s|) ds.$ Evaluating the integral gives the stated result.", "For the Laplace transform, we need $d\\mu $ supported on ${\\mathbb {R}}_+$ .", "We give one example.", "Proposition 2.16 For ${\\alpha }, {\\beta }> -1$ , let $u_{{\\alpha },{\\beta }}(t) = t^{\\alpha }(1-t)^{\\beta }\\chi _{[0,1]}(t)$ .", "Then $\\nu _2(u_{{\\alpha },{\\beta }}; t) =0$ if $t > 2$ , $\\nu _2(u_{{\\alpha },{\\beta }} ;t) = \\frac{\\Gamma (\\frac{3}{2}) \\Gamma ({\\alpha }+1)}{2^{2 {\\alpha }+1} \\Gamma ({\\alpha }+\\frac{5}{2})}t^{2{\\alpha }+3} (1-t)^{{\\beta }} {}_2F_1\\left( \\begin{matrix} - {\\beta }, {\\alpha }+1 \\\\ {\\alpha }+\\frac{5}{2} \\end{matrix}; \\frac{t^2}{4(1-t)}\\right)$ if $0 \\le t \\le 1$ , and $\\nu _2(u_{{\\alpha },{\\beta }} ;t) = \\frac{\\Gamma (\\frac{3}{2}) \\Gamma ({\\beta }+1)}{2^{2 {\\beta }+1} \\Gamma ({\\beta }+\\frac{5}{2})}(2-t)^{2{\\beta }+3} (t-1)^{{\\alpha }} {}_2F_1\\left( \\begin{matrix} - {\\alpha }, {\\beta }+1 \\\\ {\\beta }+\\frac{5}{2} \\end{matrix}; \\frac{(2-t)^2}{4(1-t)}\\right)$ if $1 \\le t \\le 2$ .", "Since $u_{{\\alpha },{\\beta }} (\\frac{1-t}{2}) = w_{{\\alpha },{\\beta }}(t) /2^{{\\alpha }+{\\beta }}$ , a simple change variable shows that $\\nu _2(u_{{\\alpha },{\\beta }};t) & = 2^{-2{\\alpha }-2{\\beta }-1} \\int _{u_1+u_2 = 2-2t} w_{{\\alpha },{\\beta }}(u_1)w_{{\\alpha },{\\beta }}(u_2)du_1du_2 \\\\& = 2^{-2{\\alpha }-2{\\beta }-1} \\nu _2(w_{{\\alpha },{\\beta }}; 2(1-t)),$ from which the stated formula follows from the previous proposition.", "For ${\\alpha },{\\beta }> -1$ , the Laplace transform of $u_{{\\alpha },{\\beta }}$ is given by ${\\mathcal {L}}u_{{\\alpha },{\\beta }} (x) = \\frac{\\Gamma ({\\alpha }+1)\\Gamma ({\\beta }+1)}{\\Gamma ({\\alpha }+{\\beta }+2)}{}_1F_1\\left( \\begin{matrix} {\\alpha }+1 \\\\ {\\alpha }+{\\beta }+2 \\end{matrix}; -t\\right).$ According to Theorem REF , the Wronskian of this function is a completely monotone function.", "In the case of ${\\alpha }= {\\beta }= {\\lambda }-1/2$ , we write $u_{\\lambda }(t) = (t (1-t))^{{\\lambda }-1/2}$ and the Laplace transform is given in terms of the modified Bessel function $I_n(t)$ of the first kind, ${\\mathcal {L}}u_{{\\lambda }} (t) = \\sqrt{\\pi }\\Gamma (a+\\tfrac{1}{2}) t^{-a} e^{-t} I_n\\left(\\frac{t}{2} \\right) e^{-t},$ where $I_n(t) = {\\mathtt {i}}^n J_n({\\mathtt {i}}t)$ is real valued.", "In particular, in the case of ${\\alpha }= {\\beta }=0$ , we have ${\\mathcal {L}}\\chi _{[0,1]} (t) = \\frac{1 - e^{-t}}{t},$ which is a completely monotone function.", "The Wronskin of this function is also completely monotone.", "The simplest case $n=2$ shows that $W_2^{\\mathcal {L}}(t) = \\frac{2 \\cosh (t) - t^2 -2}{t^4} e^{-t}$ is completely monotone." ], [ "Fourier transforms in the Laguerre-Polya class", "In this section we establish the necessary and sufficient conditions for an entire function, represented as a Fourier transform of an absolutely continuous Borel measure, to belong to the Laguerre-Pólya class in terms of density of the corresponding correlation functions.", "In fact, we consider an additional assumption on the entire function that is satisfied by the Riemann $\\Xi $ -function.", "As it has been mentioned, our result provides an answer of the problem of Pólya [14] for this specific subclass.", "Recall that a real entire function $\\varphi $ is in the Laguerre-Pólya class, $\\varphi \\in \\mathcal {LP}$ , if $\\varphi (z) = c z^m e^{-{\\alpha }z^2+{\\beta }z} \\prod _{k=1}^\\infty (1+ z/x_k) e^{-z/x_k}$ for some $c,{\\beta }\\in {\\mathbb {R}}$ , ${\\alpha }>0$ , $m \\in {\\mathbb {N}}_0$ and $x_k \\in {\\mathbb {R}}\\setminus \\lbrace 0\\rbrace $ such that $\\sum _k x_k^{-2} < \\infty $ .", "Already Laguerre observed that if $\\varphi \\in \\mathcal {LP}$ , then $[\\varphi ^{(j)}(x)]^2 - \\varphi ^{(j-1)}(x) \\varphi ^{(j+1)}(x) \\ge 0,\\ \\ x\\in \\mathbb {R},\\ \\ j\\in \\mathbb {N}.$ Although the Laguerre inequalities (REF ) consist of an infinite set of conditions, they are only necessary for an entire function $\\varphi $ to belong to the Laguerre-Pólya class.", "In fact, (REF ) follow from the most simple Laguerre inequalities $[\\varphi ^{\\prime }(x)]^2 - \\varphi (x) \\varphi ^{\\prime \\prime }(x) \\ge 0,\\ \\ x\\in \\mathbb {R}$ and the fact that $\\mathcal {LP}$ is closed under differentiation.", "In 1913 Jensen [9] made the ingenious observation that $\\varphi $ should have only real zeros if $|\\varphi (z)|^2$ is either an increasing function along all rays perpendicular to the real line or convex along all such lines.", "Let $\\varphi $ be a real entire function $ \\varphi (z) = \\phi (x,y) + {\\mathtt {i}}\\psi (x,y),\\qquad z=x+{\\mathtt {i}}y,$ whose real and imaginary parts are $\\phi (x,y): = \\Re \\varphi (z) = \\frac{1}{2} \\left(\\varphi (z) + \\varphi ({\\bar{z}}) \\right) \\quad \\hbox{and} \\quad \\psi (x,y): = \\Im \\varphi (z) = \\frac{1}{2 {\\mathtt {i}}} \\left(\\varphi (z) - \\varphi ({\\bar{z}}) \\right).$ Jensen's criteria state that, under a mild additional restriction, the zeros of $\\varphi $ should be real if and only if $ | \\varphi (z)|^2=[\\phi (x,y)]^2 + [\\psi (x,y)]^2$ is either an increasing function of $y\\in [0,\\infty )$ or it is a convex function of $y\\in (-\\infty ,\\infty )$ .", "He observed that $\\frac{1}{2} \\frac{\\partial ^2}{\\partial y^2} | \\varphi (z)|^2 = | \\varphi ^\\prime (z)|^2 - \\Re (\\varphi (z) \\overline{\\varphi ^{\\prime \\prime }(z)}).$ Jensen's convexity criterion states: Theorem A Let $\\varphi (z)=e^{-az^2} \\varphi _1(z)$ , $a\\ge 0$ , $\\varphi \\lnot \\equiv 0$ , where $\\varphi _1$ is a real entire function of genus 0 or 1.", "Then $\\varphi \\in \\mathcal {LP}$ if and only if $| \\varphi ^\\prime (z)|^2 - \\Re (\\varphi (z) \\overline{\\varphi ^{\\prime \\prime }(z)}) \\ge 0\\ \\ \\mathrm {for\\ all}\\ z\\in \\mathbb {C}.$ Jensen's result was refined recently by Csordas and Vishnyakova [6] who proved that if $\\varphi (z)$ is a real entire function, $\\varphi \\lnot \\equiv 0$ , and Jensen's inequalities (REF ) hold then $\\varphi \\in \\mathcal {LP}$ .", "However, if one differentiates the right-hand side of (REF ) twice with respect to $y$ , uses the fact that both $\\phi (x,y)$ and $\\psi (x,y)$ are harmonic functions and applies the Cauchy-Riemann equations, obtains (see also [4], [6]) $|\\varphi ^\\prime (z)|^2 - \\Re (\\varphi (z) \\overline{\\varphi ^{\\prime \\prime }(z)}) = & \\ [\\phi _x (x,y)]^2 - \\phi (x,y) \\phi _{xx}(x,y)\\\\\\ & + [\\psi _x (x,y)]^2 - \\psi (x,y) \\psi _{xx}(x,y)\\\\= & \\ - \\left[ W_2 (\\phi (\\cdot ,y);x) + W_2 (\\psi (\\cdot ,y);x) \\right].$ Therefore, Jensen's result can be rewritten in the following form: Corollary 3.1 Let $\\varphi (z)$ , defined by (REF ), be a real entire function that obeys the requirements in Theorem REF .", "Then $\\varphi \\in \\mathcal {LP}$ if and only if $W_2 (\\phi (\\cdot ,y);x) + W_2 (\\psi (\\cdot ,y);x) \\le 0 \\ \\ \\ \\mathrm {for\\ all}\\ z=x+{\\mathtt {i}}y\\in \\mathbb {C}.$ Let $K$ be an even function that decreases rapidly enough at infinity, so that $\\varphi (z) := \\int _{-\\infty }^\\infty K(u) e^{-i u z} du$ is an entire function of the form described in Theorem REF .", "Then its real and imaginary part are given by $\\phi (x, y) = \\int _{\\mathbb {R}}\\cosh (s y) K(s) \\cos (s x) ds, \\quad \\psi (x, y) = \\int _{\\mathbb {R}}\\sinh (s y) K(s) \\sin (s x) ds.$ The observations presented in this section up to now are classical and they are due to Jensen [9], Pólya [15] and developed further by Csordas and Varga [5].", "Roughly speaking, they say that $\\varphi \\in \\mathcal {LP}$ if and only if the sum of the Laguerre quantities (REF ) for the real and imaginary part of $\\varphi $ , considered as functions of $x$ , are nonnegative for any fixed $y$ .", "In what follows we restrict our study to a narrower class of entire function which obey specific properties that are verified for the Riemann $\\Xi (z)$ .", "The main features we employ are the facts that the zeros of $\\Xi $ lie in the horizontal strip $S_{1/2}= \\lbrace z\\, :\\, |\\Im z|<1/2\\rbrace $ because the nontrivial zeros of $\\zeta $ lie in the critical strip and it possesses the Hadamard factorization $\\Xi (z) = \\Xi (0) \\prod _{k=1}^\\infty \\left( 1 - \\frac{z}{z_k} \\right),\\ \\ z_k \\in S_{1/2}.$ It is worth mentioning that Riemann stated the latter representation without proof in [17] and it was established rigorously by Hadamard.", "Therefore, in what follows, we consider real entire functions $\\varphi $ with the following properties: (i) $\\varphi (z)$ is represented as a Fourier trasform (REF ) of an even kernel $K$ in the Schwartz space $\\mathcal {S}({\\mathbb {R}})$ (see [18]); (ii) The zeros of $\\varphi $ belong to the horizontal strip $S_{\\alpha }= \\lbrace z=x+iy: - {\\alpha }< y < {\\alpha }\\rbrace $ with width $2{\\alpha }> 0$ ; (iii) The function $\\varphi $ possesses the Hadamard factorization $\\varphi (z) = \\varphi (0) \\prod _{k=1}^\\infty \\left( 1 - \\frac{z}{z_k} \\right),\\ \\ z_k \\in S_{\\alpha }.$ We shall prove that an entire function which obeys these requirements belongs to the Laguerre-Pólya class if and only if it satisfies the strict inequalities (REF ), or equivalently the strict inequalities (REF ), for every $y\\ne 0$ .", "We shall employ the results from the previous sections and the celebrated Wiener's $L^1$ Tauberian Theorem [20]: Theorem B If $K\\in L^1({\\mathbb {R}})$ , a necessary and sufficient condition for the set of all its translations to be dense in $L^1({\\mathbb {R}})$ is that its Fourier transform ${\\mathcal {F}}(K)$ should have no real zeros.", "It is worth mentioning that Wiener's $L^2$ Tauberian Theorem [20] states that the translates of $K\\in L^1({\\mathbb {R}}) \\cap L^2({\\mathbb {R}})$ are dense in $L^2({\\mathbb {R}})$ if and and only if its Fourier transform, extended to $L^2({\\mathbb {R}})$ via Plancherel's theorem, is such that its real zeros should form a set of zero measure.", "Thus, the main result in this section reads as follows: Theorem 3.2 Let the real entire function $\\varphi $ satisfy the above properties (i), (ii) and (iii).", "Then $\\varphi \\in \\mathcal {LP}$ if and only if, for every fixed $y\\in (-\\alpha ,\\alpha )\\setminus \\lbrace 0\\rbrace $ , the translates ${\\mathcal {T}}(K_{2,y})$ of the kernel $K_{2,y} (t) = \\cosh (ty) \\int _{-\\infty }^{\\infty } (t-2s)^2 K(t-s) K(s)\\, ds$ are dense in $L^1(\\mathbb {R})$ .", "Furthermore, the translates ${\\mathcal {T}}(K_{2,y})$ of $K_{2,y} (t)$ are dense in $L^1(\\mathbb {R})$ for every fixed $y\\in (-\\alpha ,\\alpha )$ if and only if the zeros of $\\varphi $ are real and simple.", "We shall need the following simple technical result: Lemma 3.3 Let $\\varphi $ be defined by (REF ).", "For a fixed $y$ , let $\\mu _y^\\phi (s):= \\cosh (s y) K(s)$ and $\\mu _y^\\psi (s):= \\sinh (s y) K(s)$ .", "Then $\\phi (x,y) = ({\\mathcal {F}}\\mu _y^\\phi )(x) \\quad \\hbox{and} \\quad \\psi (x,y) = ({\\mathcal {F}}\\mu _y^\\psi )(x).$ Since $K$ is even, changing variable $s \\mapsto -s$ shows that $({\\mathcal {F}}\\mu _y^\\phi ) (x) = \\int _{{\\mathbb {R}}} \\cosh (s y) K(s) e^{-i s x} ds = \\int _{{\\mathbb {R}}} \\cosh (s y) K(s) \\cos (s x) ds= \\phi (x,y)$ The case of ${\\mathcal {F}}\\mu _y^\\psi $ follows similarly.", "We shall need also some results concerning entire functions in the Hermite–Biehler class (see [12]).", "The entire function $\\omega $ is said to be in this class, denoted $\\omega \\in HB$ , if it has no zeros in the closed lower half-plane $\\Im z \\le 0$ and $|\\omega (z)/\\bar{\\omega }(z)| < 1$ for every $z$ with $\\Im z >0$ .", "Here $\\bar{\\omega }(z)=\\overline{\\omega (\\bar{z})}$ .", "Let $\\omega (z) = P(z) + {\\mathtt {i}}Q(z)$ , where the real and imaginary parts $P$ and $Q$ are real entire functions.", "An important fact we shall need is the following (see [12]): Theorem C If $\\omega \\in HB$ then the zeros of $P(z)$ and $Q(z)$ are real and strictly interlace.", "Similarly, the Hemite-Biehler class $\\overline{HB}$ consists of entire functions $\\omega $ with no zeros in the open lower half-plane $\\Im z < 0$ and $|\\omega (z)/\\bar{\\omega }(z)| \\le 1$ for every $z$ with $\\Im z >0$ .", "An essential difference between $HB$ and $\\overline{HB}$ is that the real and imaginary parts $P$ and $Q$ of a function from $\\overline{HB}$ may have common real zeros.", "In general, if $R(z)$ is the canonical product corresponding to those common zeros of $\\omega \\in \\overline{HB}$ , then $\\omega (z)=R(z) \\omega _1(z)$ with $\\omega _1 \\in HB$ .", "Now we are in a position to prove Theorem REF ." ], [ "Proof of the sufficiency", "In this subsection we prove that the condition about the density of the correlation functions implies that $\\varphi \\in \\mathcal {LP}$ under milder conditions that do not require the Hadmard factorization (REF ).", "By Corollary REF the entire function $\\varphi $ certainly belongs to $\\mathcal {LP}$ provided that, for any fixed $y \\in {\\mathbb {R}}$ , the inequality $W_2 (\\phi (\\cdot ,y);x) + W_2 (\\psi (\\cdot ,y);x) \\le 0$ holds for every $x \\in {\\mathbb {R}}$ .", "By Lemma REF , $\\phi (x,y) = ({\\mathcal {F}}\\mu _y^\\phi )(x) \\quad \\hbox{and} \\quad \\psi (x,y) = ({\\mathcal {F}}\\mu _y^\\psi )(x).$ It follows immediately from Proposition 3.1 in [5] that when $y$ , with $|y|\\ge {\\alpha }$ , is fixed, $\\phi (x,y)$ and $\\psi (x,y)$ are entire functions of the variable $x$ and that they belong to $\\mathcal {LP}$ .", "Hence, the Laguerre inequalities (REF ) hold for them with $j=1$ .", "In other words $\\left[ \\phi _x (x,y) \\right]^2 - \\phi (x,y) \\phi _{xx}(x,y) \\ge 0, & \\\\\\left[\\psi _x (x,y)\\right]^2 - \\psi (x,y) \\psi _{xx}(x,y) \\ge 0, &$ for every $z=x+i y \\notin S_{\\alpha }$ .", "It remains to establish (REF ) for $z \\in S_{\\alpha }$ .", "By Theorem REF , the Wrosnkians of $\\phi (x,y)$ and $\\psi (x,y)$ , considered as functions of $x$ , are the Fourier transforms of the corresponding correlations functions, $W_2 (\\phi (\\cdot ,y);x) = W_2 ({\\mathcal {F}}\\mu _y^\\phi ;x) = - {\\mathcal {F}}(\\nu _2(\\mu _y^\\phi ) )(x) = -{\\mathcal {F}}(\\nu _2(\\cosh (\\cdot \\, y) K(\\cdot )))(x),$ and $W_2 (\\psi (\\cdot ,y);x) = W_2 ({\\mathcal {F}}\\mu _y^\\psi ;x) = - {\\mathcal {F}}(\\nu _2(\\mu _y^\\psi ) )(x) = - {\\mathcal {F}}(\\nu _2(\\sinh (\\cdot \\, y) K(\\cdot )))(x).$ By the definition of the correlation function, it is easy to see that $\\nu _2(\\cosh (\\cdot \\, y) K(\\cdot ); t) + \\nu _2(\\sinh (\\cdot \\, y) K(\\cdot );t) = \\cosh (t y ) \\nu _2(K; t).", "$ Therefore, it follows that $W_2 (\\phi (\\cdot ,y);x) + W_2 (\\psi (\\cdot ,y);x) = - {\\mathcal {F}}(\\cosh (\\cdot \\, y)\\nu _2(K; \\cdot ) )\\, (x).$ We shall prove now that for every $y \\in (-{\\alpha },{\\alpha })\\setminus \\lbrace 0\\rbrace $ , $ \\ {\\mathcal {F}}(\\cosh (\\cdot \\, y)\\nu _2(K;\\cdot ) )(x) \\ge 0 \\ \\ \\mathrm {for\\ every}\\ x \\in {\\mathbb {R}}.$ It is clear that the inequalities (REF ) hold for every such $y$ and for $x=0$ .", "Indeed, $[\\phi _x (0,y)]^2 - \\phi (0,y) \\phi _{xx}(0,y) & = \\int _{\\mathbb {R}}\\cosh (sy) K(s) ds\\int _{\\mathbb {R}}s^2 \\cosh (sy) K(s) ds > 0, \\\\[\\psi _x (0,y)]^2 - \\psi (0,y) \\psi _{xx}(0,y) & = \\left( \\int _{\\mathbb {R}}s \\sinh (sy) K(s) ds\\right)^2 > 0,$ which shows that both $W_2 (\\phi (\\cdot ,y);0)$ and $W_2 (\\psi (\\cdot ,y);0)$ are negative.", "Since the translates of $\\cosh (\\cdot \\, y)\\nu _2(K; \\cdot )$ are dense in $L^1({\\mathbb {R}})$ , then its Fourier transform does not change sign for $x\\in {\\mathbb {R}}$ , so that the strict inequalities (REF ) hold for $y \\in (-{\\alpha },{\\alpha })\\setminus \\lbrace 0\\rbrace $ .", "It is clear then that (REF ) also hold for $y=0$ by continuity.", "Finally, observe that, if the $L^1$ -density holds for $y=0$ too, then by Wiener's theorem, since $\\varphi (x) = \\phi (x,0)$ , $[\\varphi ^\\prime (x)]^2 - \\varphi (x) \\varphi ^{\\prime \\prime }(x) > 0,\\ \\ x\\in \\mathbb {R}.$ Therefore, $\\varphi $ does not have multiple zeros." ], [ "Proof of the necessity", "We shall prove that if $\\varphi \\in \\mathcal {LP}$ is represented as a Fourier transform and has an Hadamard factorization (REF ), then the translations of the correlation function are dense in $L^1(\\mathbb {R})$ not only for every $y \\in (-{\\alpha },{\\alpha })\\setminus \\lbrace 0\\rbrace $ but for every $y\\in \\mathbb {R}\\setminus \\lbrace 0\\rbrace $ .", "Because of Wiener's Tauberian Theorem, we need to show that the strict inequalities (REF ) hold.", "In fact, we shall see that $\\left[ \\phi _x (x,y) \\right]^2 - \\phi (x,y) \\phi _{xx}(x,y) > 0$ and $\\left[\\psi _x (x,y)\\right]^2 - \\psi (x,y) \\psi _{xx}(x,y) > 0$ for all $z=x+{\\mathtt {i}}y$ , $y\\ne 0$ , which, in Jensen's terminology means that $\\varphi $ is strictly convex with respect to $y$ .", "However, we shall employ the Hadamard factorization (REF ).", "It is worth mentioning that Levin [12] pointed out that the requirement $|\\omega (z)/\\bar{\\omega }(z)| < 1$ , $\\Im z >0$ is secured by the fact that the zeros lie in upper half plane when $\\omega $ is a polynomial and the same holds for entire functions of order zero because of the Phragmén-Lindelöf theorem.", "In fact, the same holds for functions with Hadamard factorization (REF ).", "Indeed, if $\\varphi $ is such an entire function in the Laguerre-Pólya class, and $a >0$ , then both $\\varphi (z-{\\mathtt {i}}a)$ and $\\varphi ({\\mathtt {i}}a-z)$ belong to $HB$ .", "We sketch the proof that the “horizontal translation” $\\varphi _a(z):=\\varphi (z-{\\mathtt {i}}a)\\in HB$ .", "Since $\\varphi (z) = \\varphi (0) \\prod (1-z/x_k)$ , $x_k \\in \\mathbb {R}$ , then $\\varphi _a(z) = \\varphi (0) \\varphi (-{\\mathtt {i}}a) \\prod _{k=1}^\\infty \\left( 1 - \\frac{z}{x_k +{\\mathtt {i}}a} \\right).$ Hence, the zeros of $\\varphi _a$ belong to the upper half-plane.", "Moreover, if $\\Im z>0$ , $z=x+ {\\mathtt {i}}y$ with $y >0$ , then $\\left| \\frac{\\varphi _a(z)}{\\overline{\\varphi _a(\\bar{z})}} \\right|= \\left| \\frac{\\prod _{k=1}^\\infty (1-(x+ {\\mathtt {i}}y)/(x_k +{\\mathtt {i}}a))}{\\prod _{k=1}^\\infty (1-(x- {\\mathtt {i}}y)/(x_k +{\\mathtt {i}}a))} \\right|.$ So the quotient of the corresponding terms is $|x_k-x +{\\mathtt {i}}(a-y)|/|x_k-x +{\\mathtt {i}}(a+y)|<1$ .", "This implies immediately that there is $q\\in (0,1)$ , such that the quotients of the finte products above are limited from above by $q$ .", "Thus, $| \\varphi _a(z)/\\overline{\\varphi _a(\\bar{z})}| <1$ when $\\Im z>0$ and $\\varphi (z-{\\mathtt {i}}a) \\in HB$ .", "Then, if $\\varphi (z-{\\mathtt {i}}a) = P_a(z) + {\\mathtt {i}}Q_a(z)$ , the real and imaginary parts $P_a(z)$ and $Q_a(z)$ are (see [5]) $P_a (x) = \\int _\\mathbb {R} K(u) \\cosh (au) \\cos (xu) du = \\phi (x,a),\\\\Q_a (x) = \\int _\\mathbb {R} K(u) \\sinh (au) \\sin (xu) du = \\psi (x,a).$ Since, by Theorem REF , the zeros of $P_a (x)$ and $Q_a (x)$ are real and strictly interlace, then each has only simple zeros and, consequently, satisfies the strict Laguerre inequalities $\\left[ \\phi _x (x,a) \\right]^2 - \\phi (x,a) \\phi _{xx}(x,a) > 0, & \\\\\\left[\\psi _x (x,a)\\right]^2 - \\psi (x,a) \\psi _{xx}(x,a) > 0.", "&$ Although the reasonings up to now concern the case $a>0$ , they hold analogously for $a<0$ , so that the latter inequalities hold for all $a\\ne 0$ .", "The above relations between the Laguerre inequalities and the Wronskians yield that, for every fixed $y \\ne 0$ , $ \\ {\\mathcal {F}}(\\cosh (\\cdot \\, y)\\nu _2(K;\\cdot ) )(x) > 0 \\ \\ \\mathrm {for\\ every}\\ x \\in {\\mathbb {R}}.$ Finally, Theorem REF implies that the translates of the the correlation kernel $K_{2,y} (t)$ must be dense in $L^1(\\mathbb {R})$ for every fixed $y\\ne 0$ .", "It is worth mentioning that without the restriction (iii), one may prove only that $\\varphi (z-{\\mathtt {i}}a)$ and $\\varphi ({\\mathtt {i}}a-z)$ belong to $\\overline{HB}$ , as was done in [5].", "In that case one guarantees that the Laguerre inequalities do hold but not the strict ones and in order to prove the necessity, the strict ones are fundamental." ], [ "Theorem ", "It is clear now that Theorem REF follows from Theorem REF by setting $K(t)=\\Phi (t)$ and having in mind that $\\Xi (z)$ obeys all the requirements imposed on $\\varphi (z)$ .", "Wiener's $L^1$ Tauberian theorem has several equivalent formulations, as given in [10], which can be used to derive other equivalent forms of Theorem REF .", "We state one such result given in terms of the convolution $f*g$ .", "Theorem 3.4 Under the assumption in Theorem REF , $\\varphi \\in \\mathcal {LP}$ if and only if, for every fixed $y\\in (-{\\alpha },{\\alpha })\\setminus \\lbrace 0\\rbrace $ , the testing equation $K_{2,y} * g = 0$ for bounded $g$ implies $g = 0$ .", "Recall that $\\Phi _{2,y}$ is defined in (REF ).", "Applying the above theorem with $K = \\Phi $ gives the following corollary.", "Corollary 3.5 The Riemann hypothesis is true if and only if, for each $y \\in (-1/2,1/2)\\setminus \\lbrace 0\\rbrace $ , the testing equation $\\Phi _{2,y} * g =0$ for bounded $g$ implies $g =0$ .", "There are two additional equivalent statements of Wiener's theorem in [10] that provide further equivalent forms of Theorems REF and REF .", "We omit the details." ], [ "Further results concerning Wiener's theorems", "It is quite clear that Wiener proved his density theorems aiming at a proof of the Prime Number Theorem.", "At the first glance, these theorems are peculiar.", "They guarantee that the translates of the the Gauss kernels $\\exp (-a x^2)$ , $a>0$ , are dense in both $L^1(\\mathbb {R})$ and $L^2(\\mathbb {R})$ .", "However, according to Wiener's results, the translates of the characteristic functions of a compact interval and of the “tent function” in Example REF are dense in $L^2(\\mathbb {R})$ but not in $L^1(\\mathbb {R})$ .", "Indeed, their Fourier transforms are, up to a normalization or scaling, the sinc function $\\mathrm {sinc}(t)=\\sin t/t$ and its square.", "The same holds for the whole family of the Gegenbauer weights $\\omega _\\lambda $ in Corollary REF whose Fourier transforms are $\\mathcal {J}_\\lambda $ defined in Corollary REF .", "This is so because translations of the argument are permitted, but not scaling.", "However, Theorem REF allows us to build a vast class of functions whose translates are dense in $L^1(\\mathbb {R})$ .", "This class is generated by the correlation functions that correspond to those $\\mathcal {LP}$ -functions that obey the requirements in Theorem REF .", "Let us first state the following general consequence of Wiener's theorems and Theorem REF : Theorem 4.1 Let $f \\in L^1({\\mathbb {R}})$ be a nonnegative function, and $t^{2n-2} f \\in L^1({\\mathbb {R}})$ for $n \\in {\\mathbb {N}}$ , $n \\ge 2$ .", "$W_n({\\mathcal {F}}f; x)$ has no real zeros if, and only if, the span of the translates of the functions $\\nu _{n,a}(x):= \\nu _n(f;x+a)$ , $a\\in {\\mathbb {R}}$ , are dense in $L^1({\\mathbb {R}})$ .", "If $\\nu _n(f) \\in L^2({\\mathbb {R}})$ , then the real zeros of $W_n({\\mathcal {F}}f; x)$ form a set of measure zero set if, and only if, the span of the translates of the functions $\\nu _{n,a}(x):= \\nu _n(f;x+a)$ , $a\\in {\\mathbb {R}}$ , are dense in $L^2({\\mathbb {R}})$ .", "By Theorem REF , $W_n({\\mathcal {F}}f; x)$ is equal to, up to a sign, ${\\mathcal {F}}\\nu _n(f)$ .", "By Lemma REF , $\\nu _n(f) \\in L^1({\\mathbb {R}})$ .", "Hence, (1) follows immediately from Wiener's $L^1$ Tauberian theorem (see [20] or Theorem REF in the next section).", "Similarly, $(2)$ is a consequence of Wiener's $L^2$ Tauberian theorem [20].", "More specifically, Theorem REF immediately yields: Corollary 4.2 Let $\\varphi \\in \\mathcal {LP}$ satisfy the properties (i), (ii) and (iii) in Section 3.", "Then the translates ${\\mathcal {T}}(K_{2,y})$ of the kernel $K_{2,y} (t) = \\cosh (ty) \\int _{-\\infty }^{\\infty } (t-2s)^2 K(t-s) K(s)\\, ds$ are dense in $L^1(\\mathbb {R})$ for every fixed $y\\in \\mathbb {R} \\setminus \\lbrace 0\\rbrace $ .", "Furthermore, if zeros of $\\varphi $ are real and simple, then the translates ${\\mathcal {T}}(K_{2,y})$ of $K_{2,y} (t)$ are dense in $L^1(\\mathbb {R})$ for every fixed $y\\in \\mathbb {R}$ .", "It is known that both $\\mathrm {sinc}^2(t)$ and ${\\mathcal {J}}_{\\lambda }(t)$ (see [7]) belong to the Laguerre-Pólya class and they obey the requirements of Theorem REF ; moreover, the zeros of ${\\mathcal {J}}_{\\lambda }(t)$ are simple while the zeros of $\\mathrm {sinc}^2(t)$ are obviously double.", "Then the Corollaries REF and REF imply the following: Corollary 4.3 The translates of the following functions are dense in $L^1({\\mathbb {R}})$ : a) $\\cosh (yt)\\, \\nu _2(w_{\\lambda };t),\\ \\ y\\in \\mathbb {R}$ , where $ \\nu _2(w_{\\lambda };t)$ is defined in (REF ); b) $\\cosh (yt)\\, (2-|t|)^3,\\ \\ y\\in \\mathbb {R}$ ; (${\\lambda }=1/2$ of (a)) c) $\\cosh (yt)\\, \\nu _2(w_\\Lambda ; t) ,\\ \\ y\\in \\mathbb {R}\\setminus \\lbrace 0\\rbrace $ , where $\\nu _2(w_\\Lambda ; t)$ is defined in (REF ).", "Finally, we prove that the translates of the correlation functions $\\nu _2(w_{{\\alpha },{\\beta }};x)$ in Proposition REF are also dense in $L^1({\\mathbb {R}})$ .", "Proposition 4.4 For ${\\alpha },{\\beta }> 0$ , the family of functions $\\lbrace \\nu _2(w_{{\\alpha },{\\beta }};x+a): a \\in {\\mathbb {R}}\\rbrace $ is dense in $L^1({\\mathbb {R}})$ .", "Equivalently, the Wronskian of the Fourier transform of $w_{{\\alpha },{\\beta }}$ , $W_n({\\mathcal {F}}w_{a,{\\beta }};x)$ does not change sign on ${\\mathbb {R}}$ .", "As shown in [10], one of the equivalent statements for the density of $\\lbrace \\nu _2(w_{{\\alpha },{\\beta }}; x+a): a \\in {\\mathbb {R}}\\rbrace $ in $L^1({\\mathbb {R}})$ is that the test equation $\\int _{{\\mathbb {R}}} \\nu _2(w_{{\\alpha },{\\beta }};t) \\phi (x-t) dt =0 \\qquad \\forall x \\in {\\mathbb {R}},$ where $\\phi $ is a continuous and bounded function, has only trivial solution $\\phi (x) =0$ .", "Since $\\nu _2(w_{{\\alpha },{\\beta }})$ is nonnegative and has compact support on $[-2,2]$ , the measure $d m(t):= \\nu _2(w_{{\\alpha },{\\beta }};t) dt$ is a finite nonnegative measure.", "If $\\int _{\\mathbb {R}}\\phi (x-t) d\\mu (t) =0$ for all $x \\in {\\mathbb {R}}$ then, by Fatou's lemma, $\\int _{{\\mathbb {R}}} \\lim _{n\\rightarrow \\infty } \\frac{\\phi (t+n^{-1}) - \\phi (t)}{n^{-1}} d m(t) \\le \\liminf _{n\\rightarrow \\infty } \\int _{\\mathbb {R}}\\frac{\\phi (t+n^{-1}) - \\phi (t)}{n^{-1}} d m(t) =0$ and the same inequality holds if the left hand side contains a negative sign, from which it follows that $\\int _{\\mathbb {R}}\\phi ^{\\prime }(t)dm(t) =0$ almost everywhere.", "Hence, $\\phi (t) =0$ .", "This proves the density.", "By Theorem REF and Wiener's $L^1$ Tauberian theorem, the Wronskian of the Fourier transform of $w_{{\\alpha },{\\beta }}$ does not change sign." ] ]
1606.05011
[ [ "Contact matrix in dilute quantum systems" ], [ "Abstract Contact has been well established as an important quantity to govern dilute quantum systems, in which the pairwise correlation at short distance traces a broad range of thermodynamic properties.", "So far, studies have been focusing on contact in individual angular momentum channels.", "Here, we point out that, to have a complete description of the pairwise correlation in a general dilute quantum systems, contact should be defined as a matrix.", "Whereas the diagonal terms of such matrix include contact of all partial wave scatterings, the off-diagonal terms, which elude previous studies in the literature, characterise the coherence of the asymptotic pairwise wavefunction in the angular momentum space and determine important thermodynamic quantities including the momentum distribution.", "Contact matrix allows physicists to access unexplored connections between short-range correlations and macroscopic quantum phenomena.", "As an example, we show the direct connection between contact matrix and order parameters of a superfluid with mixed partial waves." ], [ "Contact matrix in dilute quantum systems Shao-Liang Zhang$^*$ , Mingyuan He$^*$ , and Qi Zhou Department of Physics, The Chinese University of Hong Kong, Shatin, New Territories, HK Contact has been well established as an important quantity to govern dilute quantum systems, in which the pairwise correlation at short distance traces a broad range of thermodynamic properties.", "So far, studies have been focusing on contact in individual angular momentum channels.", "Here, we point out that, to have a complete description of the pairwise correlation in a general dilute quantum systems, contact should be defined as a matrix.", "Whereas the diagonal terms of such matrix include contact of all partial wave scatterings, the off-diagonal terms, which elude previous studies in the literature, characterise the coherence of the asymptotic pairwise wavefunction in the angular momentum space and determine important thermodynamic quantities including the momentum distribution.", "Contact matrix allows physicists to access unexplored connections between short-range correlations and macroscopic quantum phenomena.", "As an example, we show the direct connection between contact matrix and order parameters of a superfluid with mixed partial waves.", "Since S. Tan first invented the concept of contact in quantum dilute systems in 2005, the study of contact and the universal thermodynamic relations have become a fundamentally important topic in ultracold atom physics, and have also influenced considerably related fields [1], [2], [3].", "Due to the length scale separation that the average interparticle distance $k_F^{-1}$ , where $k_F$ is the Fermi momentum, is much larger than the range of interaction $r_0$ in a dilute quantum system, the correlation between a pair of particles at short distance determines a wide range of thermodynamic quantities, and allows physicists to establish deep connections among very different physical quantities.", "Within a decade, both theoretical and experimental efforts have made significant progresses towards unveiling thermodynamic relations that are universal regardless of the details of microscopic physics [4], [5], [9], [10], [11], [12], [13], [6], [7], [8], [14], [15], [16], [17].", "Contact was originally invented for systems with a delta-function interaction $U({\\bf r}_i-{\\bf r}_j)\\sim \\delta ({\\bf r}_i-{\\bf r}_j)$ , where ${\\bf r}_i$ is the real space coordinate of the $i$ th particle in the system [1], [2], [3].", "For such modelling potential, only $s$ -wave scattering exists, and the $s$ -wave contact alone is sufficient to characterise thermodynamics of the many-body system.", "Recently, it was realised that contact also exist for $p$ -wave scattering[18], [19], [20], and an experiment has probed two $p$ -wave contact [21].", "In particular, in reference [20], we pointed out that contact can be defined for a general short-range interaction including any partial wave scatterings.", "Putting all these contact together, one gets contact spectrum, which enables universal relations beyond thermodynamics and has powerful applications in atomic quantum Hall states.", "Later, a work obtained consistent result for the momentum distribution determined by $d$ -wave contact [22].", "Another work also found out in a system with both strong $s$ and $p$ -wave interactions, the thermodynamics replies on multiple contact[23].", "In this Letter, we point out that, to completely describe the pairwise correlation at short distance in a generic dilute many-body system, contact should be defined as a matrix $C_{\\alpha \\beta }$ , where $\\alpha $ is the short-hand notation of the quantum numbers of angular momentum.", "For instance, $\\alpha =(l,m)$ and $\\alpha =l$ in three and two dimensions, respectively.", "The diagonal terms, $C_{\\alpha \\alpha }$ , is the contact studied in the literature for an individual partial wave scattering.", "The importance of the off-diagonal terms, $C_{\\alpha \\ne \\beta }$ , are summarised as follows.", "First, $C_{\\alpha \\ne \\beta }$ have the same origin as diagonal terms.", "When the distance between two particles is much smaller than $k_F^{-1}$ , the many-body wave function takes a universal asymptotic form as a pairwise wavefunction describing the relative motion of two particles.", "$C_{\\alpha \\ne \\beta }$ must be required to characterise the coherence of such pairwise wavefunction in the angular momentum space as analogous to the textbook example of a spin, whose transverse magnetisation needs to be measured for probing the spin coherence, regardless of the choice of quantisation axis.", "In particular, $C_{\\alpha \\ne \\beta }$ is crucial for a generic system at low temperatures whose total angular momentum is not conserved, due to either anisotropic external potentials or anisotropic interactions.", "Second, any physical quantity determined by the asymptotic pairwise wavefunction depends on both $C_{\\alpha \\alpha }$ and $C_{\\alpha \\ne \\beta }$ , if it is not angular momentum selective.", "A prototypical example is the momentum distribution $n(\\bf k)$ at large $k=|{\\bf k}|$ , which crucially replies on $C_{\\alpha \\ne \\beta }$ .", "Third, $C_{\\alpha \\ne \\beta }$ allows one to access many-body physics beyond the scope of diagonal terms $C_{\\alpha \\alpha }$ .", "For instance, $C_{\\alpha \\ne \\beta }$ directly reflects the phase coherence between different order parameters of a superfluid with mixed partial waves, and thus allows one to trace macroscopic quantum phenomena from short-range correlations.", "Figure: (a) small dark blue spheres represent two particles picked up from a dilute quantum system, which is represented by the big light blue cloud.", "The relative motion of such pair in general contains multiple partial waves and is entangled with the rest of the system.", "(b) the solid red curve represents the dependence of n(𝐤)n({\\bf k}) of an anisotropic pp-wave superfluid Y 10 (𝐤 ^)+(λ/2)(Y 11 (𝐤 ^)+Y 1,-1 (𝐤 ^))Y_{10}({\\hat{\\bf k}})+(\\lambda /\\sqrt{2})(Y_{11}({\\hat{\\bf k}})+Y_{1,-1}({\\hat{\\bf k}})) on the azimuthal angle ϕ 𝐤 ^ \\varphi _{\\hat{\\bf k}} in the momentum space.", "λ=0.4\\lambda =0.4, k/k F ≫1k/k_F\\gg 1, and the polar angle θ 𝐤 ^ =π/4\\theta _{\\hat{\\bf k}}=\\pi /4.", "The blue dotted(green dashed) curve is the contribution from C 1010 C_{1010}(C 1111 C_{1111} or C 1-11-1 C_{1-11-1}), respectively.", "The black dash dotted curve includes the contribution from all diagonal contact.", "Inset is the three-dimensional plot of n(𝐤)n({\\bf k}).", "The off-diagonal contact C α≠β C_{\\alpha \\ne \\beta }, which is proportional to Δ α * Δ β \\Delta _{\\alpha }^*\\Delta _{\\beta }, gives rise to the difference between the solid and dash dotted curve.To concretise discussions, we consider a single component system with $N$ particles, whose total angular momentum does not need to be a good quantum number.", "When the $i$ th and the $j$ th particles are close to each other, the asymptotic form of the many body wavefunction in three dimensions $\\Psi ({\\bf r}_1,{\\bf r }_2,...,{\\bf r}_N)$ can be written as $\\Psi ({\\bf r}_1,{\\bf r }_2,...,{\\bf r}_N) \\stackrel{|{\\bf r}_{ij}|\\ll k_F^{-1}}{\\xrightarrow{} } \\sum _{\\alpha } \\psi _{\\alpha }({\\bf r}_{ij})G_{\\alpha }({\\bf R}_{ij}) $ where $\\alpha =(l, m)$ is the quantum number for the angular momentum in three dimensions, ${\\bf r}_{ij}={\\bf r}_i-{\\bf r}_j$ is the real space coordinate of the relative motion of the $i$ th and $j$ th particle, ${\\bf R}_{ij}=\\lbrace \\frac{{\\bf r}_i+{\\bf r}_j}{2},{\\bf r}_{k\\ne i,j}\\rbrace $ is a short-hand notation, including the center of mass coordinate of $i$ th and $j$ th particles and the coordinates of all other ones.", "$\\psi _{\\alpha }({\\bf r}_{ij})$ is an unnormalized zero-energy solution of the Hamiltonian, $\\hat{h}=-\\frac{\\hbar ^2}{M}\\nabla ^2+\\hat{U}({\\bf r}_{ij})$ .", "To simplify notations, we have considered the zero energy expansion of the two-body wave function $\\psi _{\\alpha }({\\bf r}_{ij})\\equiv \\psi _{\\alpha }({\\bf r}_{ij};0) \\approx \\psi _{\\alpha }({\\bf r}_{ij};\\epsilon )$ , where $\\epsilon $ is the energy of the relative motion of the $i$ th and $j$ th particles.", "Though the energy dependence of $\\psi _{\\alpha }({\\bf r}_{ij};\\epsilon )$ gives rise to interesting structures of contact of each partial wave scattering, it does not affect discussions of the main results regarding the contact matrix.", "Thus, in the main text, we focus on such zero energy expansion.", "Finite energy corrections are given in the supplementary material.", "We now consider a class of operators $\\hat{O}$ , which relies on the short-range behaviour of the relative motion of a pair of particles.", "The interaction energy $\\hat{U}_{int}=\\sum _{i<j} U({\\bf r}_i-{\\bf r}_j)$ is such example.", "Recall that we consider short-range interaction, the range $r_0$ of which is much smaller than the interparticle spacing, one sees that the expectation value of $\\hat{O}$ is indeed determined by the asymptotic form of the many-body wave function, as shown in Eq.", "(REF ).", "Other well known examples in this category of operators include momentum distribution $n({\\bf k})$ at large momentum $k$ , photoassociation rate, rf-spectroscopy, and etc [1], [2], [3], [4], [5], [6], [7], [8], [24], [25], [26], [27].", "Using Eq.", "(REF ), we obtain the expectation value of $\\hat{O}$ , $\\langle \\hat{O}\\rangle =\\sum _{\\alpha \\beta } \\frac{1}{32\\pi ^2} \\langle \\psi _{\\alpha }|\\hat{O}|\\psi _{\\beta }\\rangle C_{\\alpha \\beta }, $ where $\\langle \\psi _{\\alpha }|\\hat{O}|\\psi _{\\beta }\\rangle =\\int d{\\bf r}_{ij} \\psi _{\\alpha }^*({\\bf r}_{ij})\\hat{O} \\psi _{\\beta }({\\bf r}_{ij})\\equiv O_{\\alpha \\beta }$ is a quantity purely determined by two-body physics, and $C_{\\alpha \\beta }=32\\pi ^2 \\frac{N(N-1)}{2} \\int d{\\bf R}_{ij} G^*_{\\alpha }({\\bf R}_{ij})G_{\\beta }({\\bf R}_{ij})$ encodes all many-body physics.", "The prefactor $\\frac{N(N-1)}{2}$ comes from the number of pairs of identical particles and $32\\pi ^2$ is introduced to simplify later expressions of $n({\\bf k})$ .", "When $\\alpha =\\beta $ , Eq.", "(REF ) recovers the contact we defined for an arbitrary partial wave scattering [20].", "If the total angular momentum $L$ is conserved, $\\alpha $ uniquely fixes the angular momenta of both the pair of particle and the rest of the system, which is represented by ${\\bf R}_{ij}$ , so that $G_{\\alpha }({\\bf R}_{ij})$ must be orthogonal to each other.", "However, in a generic system with broken rotational symmetry, due to either anisotropic external trapping potential or anisotropic interaction, different $G_{\\alpha }({\\bf R}_{ij})$ may not be orthogonal to each other, i.e., $\\int d{\\bf R}_{ij} G^*_{\\alpha }({\\bf R}_{ij})G_{\\beta }({\\bf R}_{ij})\\ne 0$ when $\\alpha \\ne \\beta $ .", "Thus the off-diagonal contact $C_{\\alpha \\ne \\beta }$ becomes finite.", "Eq.", "(REF ) and Eq.", "(REF ) allow one to fully unveils the structure of the pairwise correlations at short distance.", "Formally, it is equivalent to a bipartite decomposition of an arbitrary many-body system into two parts.", "The relative motion of an arbitrarily picked up pair of particles is regarded as one (small) subsystem $A$ , and the rest of the many-body system, including the center of mass of such pair and all other $N-2$ particles, is regarded as the other (big) subsystem $B$ , as shown in Fig.", "1(a).", "Since $\\hat{O}$ only acts on the subsystem $A$ , its expectation value also depends on the overlap integral of the wavefunctions of subsystem $B$ .", "The off-diagonal terms $ C_{\\alpha \\ne \\beta }$ thus characterises the coherence of the subsystem $A$ in the angular momentum space and how much it is entangled with the subsystem $B$ , which can be viewed as the environment of $A$ .", "Two extreme cases can be used to illuminate the physics.", "$\\int d{\\bf R}_{ij} G^*_{\\alpha }({\\bf R}_{ij})G_{\\beta }({\\bf R}_{ij})=0\\,\\,\\,(\\alpha \\ne \\beta ), & \\mathrm {case (I)} \\\\ \\forall \\alpha , G_{\\alpha }({\\bf R}_{ij})=G({\\bf R}_{ij}), \\,\\,\\,\\,\\,\\,\\,\\,\\, & \\mathrm {case (II)} $ In case (I), the off-diagonal contact $C_{\\alpha \\ne \\beta }$ vanishes, and each $\\psi _{\\alpha }({\\bf r}_{ij})$ is coupled to a unique one in an orthogonal set of wavefunctions $G_{\\alpha }({\\bf R}_{ij})$ .", "In other words, the relative motion of the pair is highly entangled with the rest of the system, and thus losses its own coherence.", "In case (II), all $G_{\\alpha }({\\bf R}_{ij})$ are identical, and Eq.", "() is satisfied for any $\\alpha $ .", "The right hand side of Eq.", "(REF ) becomes $[\\sum _{\\alpha } \\psi _{\\alpha }({\\bf r}_{ij})]G({\\bf R}_{ij})$ , which is a product state.", "In this case, the relative motion of the pair is not entangled with the rest of the system at all, and its own coherence retains.", "Later, we will discuss examples of many-body states in both cases, i.e., quantum Hall states in (I) and BCS-superfluids in (II), respectively.", "It is worth mentioning that Eq.", "(REF ) is analogous to the central spin problem, in which the coherence of the electronic spin is controlled by the entanglement with a bath of nuclear spins, i.e., $\\langle \\hat{\\sigma }_x \\rangle _e\\sim \\langle I_1|I_2\\rangle $ for an entangled state $|\\uparrow \\rangle _e|I_1\\rangle +|\\downarrow \\rangle _e|I_2\\rangle $ , where $|I_1\\rangle $ ($|\\uparrow \\rangle _e$ ) and $|I_2\\rangle $ ($|\\downarrow \\rangle _e$ ) are nuclear(electronic) spin states[28], [29], [30], [31].", "Spin coherence allows one to detect a wide range of many-body physics in the bath[32], [33], [34], [35], [36].", "Here, $\\alpha $ may be regarded as a pseudospin index.", "Moreover, a unique feature is that the $A$ subsystem is actually a pair of particles within the many-body system of interest.", "Due to the length scale separation $k_Fr_0\\ll 1$ , and the resultant Eq.", "(REF ) and Eq.", "(REF ), many thermodynamic quantities of the many-body system, such as the momentum distribution $n({\\bf k})$ , depends on the off-diagonal contact $C_{\\alpha \\ne \\beta }$ .", "More importantly, Eq.", "(REF ) allows one to trace many-body physics using observable that are dependent on the pairwise correlations at short distance.", "For instance, the symmetry and the coherence of the order parameters in a superfluid can be traced from the momentum distributions, as shown later.", "The momentum distribution, a typical measurable in ultracold atoms, can be computed using $n_{\\bf k}=\\sum _{i}\\int \\prod _{j\\ne i}d{\\bf r}_{j}\\Big |\\int d{\\bf r}_i\\Psi ({\\bf r}_1,{\\bf r}_2,... {\\bf r}_N)e^{ -i {{\\bf k}\\cdot {\\bf r}_i}}\\Big |^2$ .", "It has been shown that for large $k\\gg k_F$ , the form of $n({\\bf k})$ is determined by the Fourier transform of the asymptotic form in Eq.", "(REF ), i.e., $\\psi _{\\bf k}({\\bf R}_{ij})\\equiv \\int d{\\bf r}_{ij} e^{ -i {\\bf k}\\cdot {\\bf r}_{ij}}[\\sum _{\\alpha } \\psi _{\\alpha }({\\bf r}_{ij})G_{\\alpha }({\\bf R}_{ij}) ] $ .", "Recall that $\\psi _\\alpha ({\\bf r}_{ij})=\\varphi _{\\alpha }(r_{ij})Y_{\\alpha }(\\hat{\\bf r}_{ij})$ , where $\\varphi _{\\alpha }(r_{ij})$ is the radial part of the wavefunction, $r_{ij}=|{\\bf r}_{ij}|$ , $\\hat{\\bf r}_{ij}={\\bf r}_{ij}/| {\\bf r}_{ij}|$ , and $Y_{\\alpha }(\\hat{\\bf r}_{ij})$ is the spherical Harmonics with $\\alpha =(l,m)$ , and that $e^{i{\\bf k}\\cdot {\\bf r}}=4\\pi \\sum _{l=0}^{\\infty }\\sum _{m=-l}^{l}i^lj_l(kr)Y^*_{lm}(\\hat{\\bf k})Y_{lm}(\\hat{\\bf r})$ , where $j_l(kr)$ is the first kind spherical Bessel function, one sees that $\\psi _{\\bf k}({\\bf R}_{ij})$ is a superposition of many partial waves in the momentum space.", "$|\\psi _{\\bf k}({\\bf R}_{ij})|^2$ thus naturally has the cross terms $G^*_\\alpha ({\\bf R}_{ij})G_\\beta ({\\bf R}_{ij})$ .", "Consider a short-range interaction, in the regime $k_F\\ll k\\ll r_0^{-1}$ , $\\varphi _\\alpha (r_{ij})$ has the asymptotic form $r^{-l-1}_{ij}$ .", "A straightforward calculation shows that $\\begin{split}n_{\\bf k} &\\stackrel{k_F\\ll k\\ll r_0^{-1}}{\\xrightarrow{} } \\sum \\limits _{lm} { \\frac{ C_{lmlm}}{k^{4-2l}} |Y_{lm}(\\hat{\\bf k})|^2 } \\\\& + \\sum \\limits _{(l,m) \\ne (l^{\\prime },m^{\\prime })} {i^{l-l^{\\prime }} \\frac{C_{lml^{\\prime }m^{\\prime }}}{k^{4-l-l^{\\prime }} } Y^*_{lm}(\\hat{\\bf k}) Y_{l^{\\prime }m^{\\prime }}(\\hat{\\bf k}) }\\end{split}$ Details of the calculation are presented in the supplementary materials.", "Eq.", "(REF ) readily allows one to measure contact matrix in experiments by fitting the angular part of the momentum distribution based on the partial wave expansion in this equation.", "The first line in Eq.", "(REF ) is the contribution from each individual contact, which has been discussed before.", "The second line comes from the off-diagonal contact.", "It inevitably leads to not only new power-law dependence $k^{l+l^{\\prime }-4}$ , but also interference pattern in the momentum space.", "Such interference pattern is a direct probe of the coherence of the pairwise wave function in Eq.", "(REF ) in the angular momentum space.", "For many-body states in case (I), such interference pattern vanishes.", "In contrast, the amplitude of the interference pattern of many-body states in case (II) directly reflects the strength of the off-diagonal contact $C_{\\alpha \\ne \\beta }$ .", "Whereas we have been focusing on three dimensions, it is rather clear that the above discussions can be directly generalised to two dimensions.", "One defines contact matrix in two dimensions, $C_{\\alpha \\beta }=8\\pi ^2 \\frac{N(N-1)}{2} \\int d{\\bf R}_{ij} G^*_{\\alpha }({\\bf R}_{ij})G_{\\beta }({\\bf R}_{ij}).$ The only difference from Eq.", "(REF ) is the prefactor.", "Here $\\alpha =l$ represents the angular momentum quantum number.", "We do not use a different symbol to denote contact in two dimensions, since it is rather apparent in later discussions whether it means the one in three dimensions or two dimensions.", "The expression for the momentum distribution in Eq.", "(REF ) remains unchanged.", "We now discuss how to use contact matrix to trace many-body physics and macroscopic quantum phenomena.", "In a recent work of us, we have discussed contact of quantum Hall states, which belong to case (I).", "In such states, the off-diagonal contact vanishes, and each pair of particles is highly entangled with the rest of the system.", "We have used all the diagonal contact $C_{\\alpha }$ to define contact spectrum $\\lbrace C_{\\alpha }\\rbrace $ , which serves as a unique tool to probe the quantum Hall states.", "Here, we focus on BCS-superfluids, which belong to case (II).", "It is well known that the first quantisation form of a BCS wavefunction of a (spinless) superfluid is written as $\\Psi _{BCS}=\\mathcal {A}[ \\phi ({\\bf r}_1-{\\bf r}_{2})\\phi ({\\bf r}_3-{\\bf r}_{4})\\cdots \\phi ({\\bf r}_{N-1}-{\\bf r}_{N})]$ where $\\mathcal {A}$ is the antisymmetrizing operator, $\\phi ({\\bf r}_i-{\\bf r}_j)$ is the pair wavefunction.", "Considering the asymptotic behaviour when ${\\bf r}_i\\rightarrow {\\bf r}_j$ , one sees that it is indeed described by Eq.().", "This is not surprising, since a BCS wave function can be viewed as a condensate of pairs, in which each pair is not entangled with others.", "Sometimes, $\\phi ({\\bf r}_i-{\\bf r}_j)$ contains only a single partial wave $Y_{\\alpha }(\\hat{\\bf r}_{ij})$ .", "Nevertheless, many important superfluids are mixtures of multiple partial waves.", "One example is the cyclic state of $l=2$ superfluid.", "The order parameter $\\Delta _{\\bf k}$ can be written as $\\Delta _{\\bf k}=\\Delta (k) \\tilde{\\Delta }(\\hat{\\bf k})$ , $\\tilde{\\Delta }(\\hat{\\bf k})=e^{-\\pi i/6}/2(Y_{22}(\\hat{\\bf k}) +Y_{2-2}(\\hat{\\bf k}))+e^{4\\pi i/3}/\\sqrt{2} Y_{20}(\\hat{\\bf k}) \\propto \\hat{k}_x^2+e^{2\\pi i/3}\\hat{k}_y^2+e^{4\\pi i/3}\\hat{k}_z^2$ [37].", "It is a superposition of different magnetic quantum numbers $m$ .", "The other example is the anisotropic $p$ -wave superfluid, when the an anisotropic interaction, such as dipole-dipole interaction, breaks the rotation symmetry [38].", "In two dimensions, $s+d$ superfluid is an important example for discussing topological phase transitions.", "It is a mixture of $l=0$ and $l=2$ .", "These superfluids have a unique feature that the order parameter is a coherent superposition of different partial wave components, distinct from incoherent mixtures of multiple order parameters.", "However, it remains challenging to directly probe this phase coherence.", "We discuss how to use contact matrix and the momentum distributions to access such phase coherence.", "To compute the contact matrix of a superfluid, it is simple to make use of the second quantisation form to obtain the momentum distribution.", "Consider a single component Fermi gases with short range interaction $V({\\bf r})$ , the Hamiltonian $\\hat{H}=\\hat{H}_0 +\\hat{V}$ , $\\hat{H}_0=\\sum \\nolimits _{\\bf k} \\epsilon _{\\bf k} \\hat{a}^\\dag _{{\\bf k}}\\hat{a}_{{\\bf k}}$ , $\\epsilon _{\\bf k}=\\hbar ^2 k^2/(2M)$ , $M$ is the mass of each particle, $\\hat{V}=\\Omega ^{-1} \\sum \\nolimits _{{\\bf k},{\\bf k^{\\prime }}} V_{{\\bf k^{\\prime }}-{\\bf k}} \\hat{a}^\\dag _{{\\bf k^{\\prime }}}\\hat{a}^\\dag _{-{\\bf k^{\\prime }}}\\hat{a}_{-{\\bf k}}\\hat{a}_{{\\bf k}},$ $\\Omega $ is the volume of the system, $V_{{\\bf k^{\\prime }}-{\\bf k}}=\\int d {\\bf r} e^{-i({\\bf k^{\\prime }}-{\\bf k})\\cdot {\\bf r}} V({\\bf r})$ is the Fourier transform of the interaction of $V({\\bf r})$ .", "In standard BCS theory, $\\left| {G} \\right\\rangle =\\prod \\nolimits _{\\bf k} (u_{\\bf k}+v_{\\bf k} \\hat{a}_{{\\bf k}}^\\dag \\hat{a}_{-{\\bf k}}^\\dag )\\left| {0} \\right\\rangle $ , $|u_{\\bf k}|^2+|v_{\\bf k}|^2=1$ , and the momentum distribution is written as $n({\\bf k})= \\left( 1-(\\epsilon _{\\bf k}-\\mu )/{E_{\\bf k}} \\right)/2$ , where $E_{\\bf k}=\\sqrt{(\\epsilon _{\\bf k}-\\mu )^2+|\\Delta _{\\bf k}|^2}$ and $\\mu $ is the chemical potential.", "Note that $\\Delta _{\\bf k}$ in general may contain multiple partial waves, $\\Delta _{\\bf k}=\\sum \\limits _{lm} (-i)^l\\Delta _{lm} k^l Y_{lm}(\\hat{\\bf k}),$ where $\\Delta _{lm}$ is the strength of the order parameter in a given partial wave channel.", "Eq.", "(REF ) is valid for $k\\ll k_{l}$ , where $k_{l}\\sim 1/r_0$ is a momentum cutoff that reproduces the realistic two-body scattering phase shift [39], [40].", "Under the condition $\\epsilon _{\\bf k} \\gg \\mu $ and $\\epsilon _{\\bf k} \\gg |\\Delta _{\\bf k}|^2$ , one obtains the momentum distribution at large $k$ , $n({\\bf k}) \\stackrel{k_F\\ll k\\ll r_0^{-1}}{\\xrightarrow{} }\\sum \\limits _{lml^{\\prime }m^{\\prime }} i^{l-l^{\\prime }}\\frac{M^2}{ \\hbar ^4}\\frac{ \\Delta _{lm}^* \\Delta _{l^{\\prime }m^{\\prime }}}{k^{4-l-l^{\\prime }}} Y^*_{lm}(\\hat{\\bf k}) Y_{l^{\\prime }m^{\\prime }}(\\hat{\\bf k})$ Compare it with Eq.", "(REF ), one sees that in such BCS superfluid, contact matrix is directly related to superfluid order parameters, $C_{lml^{\\prime }m^{\\prime }}= \\frac{M^2}{ \\hbar ^4} \\Delta _{lm}^* \\Delta _{l^{\\prime }m^{\\prime }}$ Eq.", "(REF ) thus establish a direct relation between contact matrix and the superfluid order parameters.", "In particular, the phase coherence between different order parameters is revealed by the off diagonal contact.", "As a demonstration, Fig.1(b) shows the momentum distribution of an anisotropic $p$ -wave superfluid $\\hat{k}_z-i\\lambda \\hat{k}_y\\propto Y_{10}({\\hat{\\bf k}})+(\\lambda /\\sqrt{2})(Y_{11}({\\hat{\\bf k}})+Y_{1,-1}({\\hat{\\bf k}}))$ .", "For such superfluid, the contact matrix is a $3\\times 3$ one.", "The off-diagonal contact $C_{\\alpha \\ne \\beta }$ , such as $C_{101-1}$ , $C_{1011}$ , and $C_{111-1}$ , which is proportional $\\Delta _{\\alpha }^*\\Delta _{\\beta }$ , gives rise to the dependence of $n({\\bf k})$ on the azimuthal angle $\\varphi _{\\hat{\\bf k}}$ in the momentum space, i.e., the difference between the red solid curve and the black dash dotted one in Fig.1(b).", "Besides momentum distributions, it is useful to comment on the relations between contact matrix and other universal relations.", "S. Tan first showed that the internal energy of a dilute system with $s$ -wave scattering can be written as a functional of the momentum distribution[1].", "In a single component system, such integral is written as $ E=\\int \\frac{d{\\bf k}}{(2\\pi )^3} \\frac{\\hbar ^2 k^2}{2M}(n({\\bf k}) -C_{0000}/k^4 |Y_{00}(\\hat{\\bf k})|^2) +\\frac{\\hbar ^2C_{0000}}{32\\pi ^2 Ma_0}$ [10].", "Such a functional fixes the divergent problem in the zero-range interaction limit.", "In a recent work, we generalise such functional to a generic short range interaction, which include contact of all partial wave scatterings, i.e., the diagonal terms $C_{\\alpha \\alpha }$ in contact matrix [20].", "Here, we have verified that such energy functional is not affected by the off-diagonal terms.", "Apparently, the cross terms $C_{\\alpha \\ne \\beta }$ in Eq.", "(REF ) do not contribute to the integral $\\int d{\\bf k} k^2n({\\bf k})$ , due to the orthogonal condition $\\int d\\Omega _{\\bf k} Y^*_{\\alpha }(\\hat{\\bf k})Y_{\\beta }(\\hat{\\bf k})=\\delta _{\\alpha \\beta }$ , where $\\int d\\Omega _{\\bf k} $ denotes the angular part of the integral in the momentum space.", "S. Tan also found out the adiabatic relation, in which the derivative of the energy with respect to the inverse of the $s$ -wave scattering length is given by $s$ -wave contact, i.e., $\\frac{dE}{d(-1/a_0)}\\sim C_{0000}$[2].", "Such relation were recently generalised to high partial wave scatterings[18], [19], [21], [22].", "Here, the off-diagonal contact $C_{\\alpha \\beta }$ is apparently not associated with any scattering length and thus is beyond the scope of the conventional adiabatic relations.", "Results of photoassociation and rf-spectroscopy are presented in the supplementary material.", "It is useful to highlight a few potential applications of contact matrix.", "Anisotropic interactions, such as the magnetic dipole-dipole interaction, are important in many cases, for instance, the $p$ -wave Feshbach resonance [41], [42], [43].", "Recent experiments on Er and Dr has also shown the importance of anisotropic interaction in magnetic lanthanide atoms [44], [45].", "For those anisotropic interactions, different partial waves naturally mix with each other in two-body scattering.", "A full description of contact matrix is thus required and our general results apply.", "We have shown that contact matrix provides one a complete description of the pairwise correlations at short distance in a dilute quantum system.", "The off-diagonal contact, which elude previous studies, determines thermodynamic quantities and allows one to trace macroscopic quantum phenomena.", "Whereas we focus on single component systems here, it is straightforward to generalise our results to multi-component systems.", "We hope that our work will inspire more interests in applying contact matrix in many-body physics.", "This work is supported by RGC/GRF(14306714).", "*SZ and MH contribute equally to this work.", "Supplementary Materials In this supplementary material, we present the results on the large momentum distribution including the finite energy corrections, photoassociation and rf-spectroscopy.", "Large momentum distribution and contact matrix For a $N$ -particle dilute quantum systems with short range interaction, the many-body wave function ${\\Psi } = \\Psi \\left( {{{\\bf r}_1},{{\\bf r}_2}, \\cdots ,{{\\bf r}_N}} \\right)$ , asymptotically, can be written as $\\Psi \\stackrel{r_{ij}\\ll k_F^{-1}}{\\xrightarrow{} } \\sum _{lm} \\int d\\epsilon \\psi _{lm}({\\bf r}_{ij};\\epsilon )G_{lm}({\\bf R}_{ij};E-\\epsilon )$ where ${\\bf r}_{ij} = {\\bf r}_i -{\\bf r}_j$ , $ r_{ij}=|{\\bf r}_{ij}|$ , ${\\bf R}_{ij}=\\lbrace ({\\bf r}_i + {\\bf r}_j)/2, {\\bf r}_{k \\ne i,j}\\rbrace $ , $\\epsilon =q^2_{\\epsilon }/M$ is the energy of the the relative motion of the $i$ th and $j$ th particle pair, and ${\\psi _{lm}}\\left( {{{\\bf r}_{ij}};\\epsilon } \\right) = \\frac{q_{\\epsilon }^{l+1}}{\\tan {[\\eta _l]}}\\left\\lbrace {{j_l}\\left( {{q_\\epsilon }{r_{ij}}} \\right) - \\tan \\left[ {{\\eta _l}} \\right]{n_l}\\left( {{q_\\epsilon }{r_{ij}}} \\right)} \\right\\rbrace {Y_{lm}}(\\hat{\\bf r}_{ij})$ is the solution of the relative motion of an isolated two-body system.", "The momentum distribution is computed using the expression in the main text ${n_{\\bf k}} = \\sum \\limits _i {\\int {\\prod \\nolimits _{j \\ne i} {d{{\\bf r}_j}} {{\\left| {\\int {d{{\\bf r}_i}\\Psi \\left( {{{\\bf r}_1},{{\\bf r}_2}, \\cdots ,{{\\bf r}_N}} \\right){e^{ - i {\\bf k} \\cdot {{\\bf r}_i}}}} } \\right|}^2}} }$ .", "By defining $ {\\Psi _i}\\left( {\\bf k} \\right) = \\int {d{{\\bf r}_i}\\Psi \\left( {{{\\bf r}_1},{{\\bf r}_2}, \\cdots ,{{\\bf r}_N}} \\right){e^{ - i {\\bf k} \\cdot {{\\bf r}_i}}}} $ , we obtain ${n_{\\bf k}} = \\sum \\limits _i {\\int {\\prod \\nolimits _{j \\ne i} {d{{\\bf r}_j}} {{\\left| {{\\Psi _i}\\left( {\\bf k} \\right)} \\right|}^2}} }.$ By defining a pure two-body quantity, ${F_{lm}}\\left( {{\\bf k};\\epsilon } \\right) = \\int {d{{\\bf r}_{ij}}{\\psi _{lm}}\\left( {{{\\bf r}_{ij}};\\epsilon } \\right){e^{ - i {\\bf k} \\cdot \\left( {{{\\bf r}_i} - {{\\bf r}_j}} \\right)}}}$ so that, in the regime $|{\\bf k}| \\gg k_F$ , one has ${\\Psi _i}\\left( {\\bf k} \\right) \\rightarrow \\sum \\limits _{j \\ne i} {{e^{ - i{\\bf k} \\cdot {{\\bf r}_j}}}} \\sum \\limits _{lm} {\\int {d\\epsilon {G_{lm}}\\left( {{{\\bf R}_{ij}};E - \\epsilon } \\right){F_{lm}}\\left( {{\\bf k};\\epsilon } \\right)} }.$ If one extends the wave function $\\psi _{lm}({\\bf r}_{ij};\\epsilon )$ in the regime $[r_0, \\infty ]$ to $[0,\\infty ]$ , $F_{lm}({\\bf k}; \\epsilon )$ in the regime $k_F \\ll k$ , $k=|{\\bf k}|$ , can be given by $\\begin{split}{F_{lm}}\\left( {{\\bf k};\\epsilon } \\right) &= \\int {d{{\\bf r}_{ij}}\\left( {\\frac{{{\\beta _{l0}}}}{{r_{ij}^{l + 1}}} + \\frac{{{\\beta _{l1}}q_\\epsilon ^2}}{{r_{ij}^{l - 1}}} + \\cdots } \\right){Y_{lm}}\\left( \\hat{\\bf r}_{ij} \\right){e^{ - i{\\bf k} \\cdot {{\\bf r}_{ij}}}}} \\\\&= 4\\pi {\\left( { - i} \\right)^l}\\left( {{k^{l - 2}} + q_\\epsilon ^2{k^{l - 4}} + \\cdots } \\right){Y_{lm}}( {\\hat{\\bf k}} )\\end{split}$ where one has used $\\beta _{ls}=(2l-2s-1)!!/(2s)!", "!$ , ${e^{ - i{\\bf k} \\cdot {{\\bf r}_{ij}}}} = 4\\pi \\sum \\nolimits _{l = 0}^\\infty {\\sum \\nolimits _{m = - l}^l {{{\\left( { - i} \\right)}^l}{j_l}\\left( {k{r_{ij}}} \\right){Y_{lm}}( \\hat{\\bf k} )Y_{lm}^ * ( \\hat{\\bf r}_{ij} )} }$ and $f_{ls}(k \\Lambda )=\\int _0^{k\\Lambda } {\\left[ {{\\beta _{ls}}/{x^{l - 2s - 1}}} \\right]{j_l}\\left( x \\right)dx} = 1 - {\\beta _{ls}}\\sum \\nolimits _{i = 0}^s {\\left[ {\\left( {2s} \\right)!!", "{j_{l - i - 1}}\\left( {k\\Lambda } \\right)} \\right]/[(2s-2i)!!", "{{\\left( {k\\Lambda } \\right)}^{l - 2s + i - 1}}]} \\stackrel{k \\Lambda \\rightarrow \\infty }{\\xrightarrow{} } 1$ .", "One can then obtain ${\\Psi _i}\\left( {\\bf k} \\right) \\rightarrow \\sum \\limits _{j \\ne i} {{e^{ - i{\\bf k} \\cdot {{\\bf r}_j}}}} \\sum \\limits _{lm} {4\\pi {{\\left( { - i} \\right)}^l}\\left( {g_{lm}^0{k^{l - 2}} + g_{lm}^2{k^{l - 4}} + \\cdots } \\right){Y_{lm}}( \\hat{\\bf k} )},$ where $g_{lm}^{s} = \\int {d\\epsilon q_\\epsilon ^{s}{G_{lm}}\\left( {{{\\bf R}_{ij}};E - \\epsilon } \\right)}$ .", "And since the cross term $e^{i{\\bf k} \\cdot ({\\bf r}_{j} - {\\bf r}_{j^{\\prime }} )}$ vanishes in the large $k$ limit, and $d{{\\bf R}_{ij}} = \\prod \\nolimits _{k \\ne i,j} {d{{\\bf r}_k}} d\\left( {{{\\bf r}_i} + {{\\bf r}_j}} \\right)/2$ , one has ${n_{\\bf k}} \\stackrel{k_F\\ll k\\ll r_0^{-1}}{\\xrightarrow{} } \\sum \\limits _{lml^{\\prime }m^{\\prime }} {{i^{l - l^{\\prime }}}\\left( {{k^{l+l^{\\prime } - 4}}{C_{lml^{\\prime }m^{\\prime }}} + {k^{l+l^{\\prime } - 6}}C_{lml^{\\prime }m^{\\prime }}^1 + {k^{l+l^{\\prime } - 8}}C_{lml^{\\prime }m^{\\prime }}^2 + \\cdots } \\right)Y_{lm}^ * ( \\hat{\\bf k}){Y_{l^{\\prime }m^{\\prime }}}( \\hat{\\bf k} )}$ which mainly includes all the terms that can cause the divergence in the energy functional discussed in reference[17], i.e., $k^{s \\ge -4}$ , where we define a contact matrix $\\lbrace C_{lml^{\\prime }m^{\\prime }} \\rbrace $ as $C_{lml^{\\prime }m^{\\prime }}^{ij} = {\\left( {4\\pi } \\right)^2}N\\left( {N - 1} \\right)\\int {d{{\\bf R}_{ij}}g_{lm}^{2i * }g_{l^{\\prime }m^{\\prime }}^{2j}} ,\\quad C_{lml^{\\prime }m^{\\prime }} =C_{lml^{\\prime }m^{\\prime }}^{00} , \\quad C_{lml^{\\prime }m^{\\prime }}^{s>0} = \\sum \\limits _{j = 0}^s {C_{lml^{\\prime }m^{\\prime }}^{\\left( j \\right)\\left( {s - j} \\right)}}.$ Note that for the 2D dilute quantum systems, one also has ${\\psi _{l}}\\left( {{{\\bf r}_{ij}};\\epsilon } \\right) =\\frac{\\pi }{2} \\frac{q_{\\epsilon }^{l}}{\\tan {[\\eta _l]}}\\left\\lbrace {{J_l}\\left( {{q_\\epsilon }{r_{ij}}} \\right) - \\tan \\left[ {{\\eta _l}} \\right]{N_l}\\left( {{q_\\epsilon }{r_{ij}}} \\right)} \\right\\rbrace {Y_{l}}(\\hat{\\bf r}_{ij})$ where $Y_l( \\hat{\\bf r}_{ij})=e^{il \\theta _{ij}}/\\sqrt{2\\pi }$ , $\\theta _{ij}=\\mathrm {arg}\\lbrace {\\bf r}_{ij}\\rbrace $ .", "By repeating the above steps, one end up with the same results.", "Photoassociation Photoassocation couples two atoms to an electronically excited molecular state.", "If the photoassociation is not angluar momentum selective, multiple partial waves can all be coupled to the excited state.", "For a two-body problem, the coupling matrix element of each partial wave to the excited state is written as $V^{[2]}_\\alpha =\\sqrt{\\frac{2\\pi I}{c}}\\langle \\varphi _e|{\\bf d}_M\\cdot \\hat{\\bf e}|\\psi _\\alpha \\rangle =\\sqrt{\\frac{2\\pi I}{c}}\\Omega _\\alpha \\big |{\\bf d}_M\\big |$ where $c$ is the speed of light, $I$ is the laser intensity, $\\varphi _e$ is the electronically excited molecule wave function, $\\psi _\\alpha $ is the two-body wave function of $\\alpha $ -th partial wave, ${\\bf d}_M$ is the electric dipole moment, $\\hat{\\bf e}$ is the polarization vector of the laser and $\\Omega _\\alpha =\\langle \\varphi _e|\\hat{\\bf d}_M\\cdot \\hat{\\bf e}|\\psi _\\alpha \\rangle $ where $\\hat{\\bf d}_M={\\bf d}_M/\\big |{\\bf d}_M\\big |$[46], [47], [48].", "For a many-body problem, since the molecule size is much smaller than the average interparticle distance, $G_\\alpha ({\\bf R}_{ij})$ can be considered as a normalization factor for $\\psi _\\alpha ({\\bf r}_{ij})$ , the coupling strength is written as $\\begin{split}\\Gamma &=\\frac{2\\pi I}{c}\\frac{N(N-1)}{2}\\int d{\\bf R}_{ij}\\big |\\sum _\\alpha \\langle \\varphi _e|{\\bf d}_M\\cdot \\hat{\\bf e}|\\psi _\\alpha \\rangle G_\\alpha ({\\bf R}_{ij})\\big |^2=\\frac{I}{16\\pi c}\\big |{\\bf d}_M\\big |^2\\sum _{\\alpha ,\\beta }\\Omega ^*_\\alpha \\Omega _\\beta C_{\\alpha \\beta }\\end{split}$ The off-diagonal contact thus determines $\\Gamma $ .", "Radiofrequency spectroscopy A rf field transfers atoms to an different hyperfine spin state.", "When the rf frequency between two hyperfine spin states $\\omega $ is large compared with relevant frequencies in many-body physics, the transition rate has the asymptotic form[49] $\\Gamma _\\mathrm {rf}=2\\pi \\Omega ^2_\\mathrm {rf}\\hbar \\sum _{\\bf k}{n}_{\\bf k}\\delta (\\hbar \\omega -\\hbar ^2k^2/M)$ where $\\Omega _\\mathrm {rf}$ is the rf Rabi frequency.", "Using the result of momentum distribution in equation (6) of the supplementary materials, one obtain $\\begin{split}\\Gamma _\\mathrm {rf}&=2\\pi \\Omega ^2_\\mathrm {rf}\\hbar \\frac{\\Omega }{(2\\pi )^3}\\int d{\\bf k}{n}_{\\bf k}\\delta (\\hbar \\omega -\\hbar ^2k^2/M) \\\\&\\approx \\frac{\\Omega ^2_\\mathrm {rf}M}{8\\pi ^2\\hbar }\\Omega \\sum _{lm}\\Big \\lbrace C_{lmlm}\\Big (\\frac{M\\omega }{\\hbar }\\Big )^{l-3/2}+C^1_{lmlm}\\Big (\\frac{M\\omega }{\\hbar }\\Big )^{l-5/2}+\\cdots \\Big \\rbrace \\end{split}$ where $\\Omega $ is the volume of the system, $C_{lmlm}^{s\\ge 1}$ are contact we defined for a high partial wave scattering associated with the subleading divergent terms and beyond in energy functional[17].", "The off-diagonal contact does not contribute to rf spectroscopy, due to the orthogonality $\\int d\\Omega _{\\bf k} Y^*_{\\alpha }(\\hat{\\bf k})Y_{\\beta }(\\hat{\\bf k})=\\delta _{\\alpha \\beta }$ , where $\\int d\\Omega _{\\bf k} $ denotes the angular part of the integral in the momentum space." ] ]
1606.05176
[ [ "A Goldilocks principle for modeling radial velocity noise" ], [ "Abstract The doppler measurements of stars are diluted and distorted by stellar activity noise.", "Different choices of noise models and statistical methods have led to much controversy in the confirmation of exoplanet candidates obtained through analysing radial velocity data.", "To quantify the limitation of various models and methods, we compare different noise models and signal detection criteria for various simulated and real data sets in the Bayesian framework.", "According to our analyses, the white noise model tend to interpret noise as signal, leading to false positives.", "On the other hand, the red noise models are likely to interprete signal as noise, resulting in false negatives.", "We find that the Bayesian information criterion combined with a Bayes factor threshold of 150 can efficiently rule out false positives and confirm true detections.", "We further propose a Goldilocks principle aimed at modeling radial velocity noise to avoid too many false positives and too many false negatives.", "We propose that the noise model with RHK-dependent jitter is used in combination with the moving average model to detect planetary signals for M dwarfs.", "Our work may also shed light on the noise modeling for hotter stars, and provide a valid approach for finding similar principles in other disciplines." ], [ "Introduction", "Almost all natural phenomena are studied by collecting and modeling data, comparing models and inferring model parameters.", "Since the data collection and reduction are usually standardised to remove bias and systematics, the results of data analyses are more influenced by the choice of models and inference methods than by data reductionActually, data reduction is also a kind of modeling that converts the primary observations into secondary data such as catalogues.", "But the process is well understood in a theoretical sense.. Due to the limited explanation power of theories and models, natural phenomena are always left with inadequate or incomplete modeling.", "Thus our understanding of these unexplained variations (so-called “noise”) significantly influences how well we can explain data particularly when noise levels are similar to those of variations.", "For example, the glacial-interglacial cycles over the Pleistocene era were probably caused by the orbital variations of the Earth through modulating the incoming solar radiation [35], [27].", "However, this theory is challenged by various authors [26], [39], [65] using stochastic processes to model the climate change.", "In the case of detections of exoplanets in the doppler measurements of stars, the activity induced radial velocity (RV) variations are called “excess noise” or “jitter”, compared with the RV variations caused by Keplerian motions of planets (called “signals”).", "Jitter is typically correlated (or red) over various time scales [5], and is caused by various mechanisms such as instability of instruments, magnetic cycles, oscillation, rotation and granulation of stars [12].", "Jitter actually consists of unmodeled variations as well as pure noise.", "This jitter is poorly understood and modeled, leading to the problem of model incompleteness [17].", "To separate jitter from planetary signals, many noise models are proposed based either on statistical properties of the RV time series (e.g., [5]) or on astrophysical studies of stellar variability (e.g.", "[42]).", "The number of planetary candidates are sometimes greatly influenced by the choice of these noise models.", "For example, six planets have been claimed to orbit around GJ 581 [63] based on data analysis applying the white noise model.", "However, [5] could not confirm all of them using Gaussian process models.", "Similar controversies exist in the confirmation of exoplanets around GJ 667C using the white noise, moving average and Gaussian process models [3], [16].", "These controversies show that the more flexible the noise model is, the less planetary signals it can find.", "We will see this effect in the comparison of noise models.", "Another factor that causes uncertainties in data analysis is the usage of different statistical methods.", "For example, many studies claimed periodicities in the data of mass extinctions and terrestrial impact craters based on the periodogram or other frequentist approach [2], [43], [34].", "But there seems to be no evidence for periodicities in the data based on the Bayesian inference [4], [14], [15].", "The tension caused by statistics also exists in the confirmation of planetary candidates even if the same noise model is used.", "For example, [52] only found four planets using Bayesian methods while [63] found six using the same radial velocity data set of GJ 581 and the same noise model but based on the periodogram.", "The solution to such controversies relies on the exploration of appropriate modeling and inference methods based on a better understanding of the mechanisms underlying certain phenomena and a proper choice of statistical tools.", "Most data analysis of radial velocity data was seen based on frequentist methods, in particular, the Lomb-Scargle periodogram and adaptations of it, e.g.", "two dimensional Keplerian Lomb-Scargle periodogram [38].", "However, the periodogram assumes that the noise in the time series is not correlated and that there is only one periodic signal in the data.", "Despite this, it is misused to search for multiple periodic signals.", "For example, only one Keplerian component is used to model a superposition of several Keplerian signals.", "Both of these assumptions are problematic if the strength of signals is comparable with the noise level [55], [17].", "Furthermore, the periodogram assumes periodicity in the data rather than testing it by comparing periodic models with other models.", "Thus periodogram, by definition, is biased in terms of model comparison.", "This is particularly true when the mechanisms responsible for certain phenomena are complex and poorly modeled (e. g. aperiodic and/or quasi-periodic phenomena).", "Despite these problems, various periodograms are broadly employed to identify planetary signals in RV observations because they are easy to calculate.", "To test the significance of a signal, a selected false alarm probability (FAP) of a periodogram is commonly used as a detection threshold.", "This metric is equivalent to the p-value which is used to reject null hypotheses such as the white noise model.", "However, the choice of null hypothesis is always arbitrary, and thus makes FAP unable to properly estimate the significance of a signal.", "Considering these drawbacks, periodograms should be used cautiously particularly in cases when the signal to noise ratio is not high and the host star is perturbed by multiple planets [9], [18].", "To avoid the above problems of the periodogram, we need a statistical tool to compare models on the same footing rather than rejecting simple null hypothesis.", "If we know exactly the underlying physics of certain phenomena, there would be no need to compare different models.", "But this is often not the case for natural phenomena.", "Hence a proper way to account for model incompleteness, model complexity and uncertainties of models and data is crucial for robust data analyses.", "Fortunately, such inference problems can be properly dealt with in the Bayesian framework (e.g., [30], [47], [23], [64]).", "For example, Bayesian inference methods assess the overall plausibility of a model by calculating its likelihood averaged over its prior distribution.", "This approach naturally accounts for the model complexity and thus models can be compared properly.", "In addition to an appropriate inference method, a modeling principle should be established through quantifying the limitations of stochastic and deterministic models.", "We do this for the RV data of M dwarfs by comparing various noise models in the following steps.", "First, we generate artificial data sets using noise models and the Keplerian model.", "For these data sets, we compare noise models using various signal detection criteria.", "Then we select the best criterion which confirms most true detections and rejects most false positives.", "We further apply the criterion to compare models for the data sets with injected Keplerian signals.", "Based on the results, we quantify the limitations of various noise models and devise a framework of noise models to detect planetary signals.", "Our aim is to provide a quantitative comparison between noise models used in the literature.", "We quantify the disadvantages and advantages of each noise model within the Bayesian framework.", "Various inference criteria are investigated for representative RV data sets.", "We also present a new principle to model stellar jitter and identify planetary signals.", "This paper is structured as follows.", "We describe the Bayesian inference method and signal detection criteria in section .", "In section , we introduce the models of RV variations, and define their prior distributions.", "Then we compare various noise models and signal detection criteria for artificial data sets in section .", "In section , we introduce three RV data sets and inject planetary signals into them for model comparison.", "Finally, we discuss the results and conclude in section .", "The Bayesian model comparison relies on the Bayes theorem which is $P(M_i|D)= \\frac{P(D|M_i)P(M_i)}{\\sum \\limits _{j} P(D|M_j)},$ where $P(M_i|D)$ is the posterior of model $M_i$ for data $D$ , $P(D|M_i)$ and $P(M_i)$ are the evidence (also called the integrated likelihood) and the prior of model $M_i$ , and the denominator is a normalisation factor.", "Then the ratio of the posteriors of two models is $\\frac{P(M_i|D)}{P(M_j|D)} =\\frac{P(D|M_i)}{P(D|M_j)}\\frac{P(M_i)}{P(M_j)}.$ If no model is favoured a priori, i.e.", "$P(M_i)/P(M_j)=1$ , the posterior ratio becomes $\\frac{P(M_i|D)}{P(M_j|D)} =\\frac{P(D|M_i)}{P(D|M_j)}\\equiv \\textrm {BF}_{ij}~,$ where BF$_{ij}$ is the odds of evidences of model $M_i$ and $M_j$ , and is called Bayes factor.", "Following [30], we interpret $\\textrm {BF}_{ij}>150$ as a strong evidence for $M_{i}$ and against $M_j$ .", "For model $M$ with parameters $\\theta $ , the evidence is $P( D| M)= \\int _{\\theta } P( D|\\theta , M) P(\\theta |M)\\textrm {d}\\theta ~,$ where $\\theta $ is the parameter vector of model $M$ , $P(\\theta |M)$ is the prior distribution of parameters, and $\\mathcal {L}(\\theta )\\equiv P( D|\\theta ,M)$ is the likelihood.", "The evidence is actually the normalisation factor of the posterior distribution of model parameters, $P(\\theta |D,M) = \\frac{P(D|\\theta ,M)P(\\theta |M)}{P(D|M)}~.$ In most cases, the evidence cannot be calculated analytically due to the complexity of the likelihood.", "Thus a Monte Carlo approach is required either to sample the prior density $P(\\theta |M)$ (prior sampling) or to sample the posterior density $P(\\theta |D,M)$ (posterior sampling) or to sample both.", "The prior sampling is not appropriate for RV models because the posterior density always contains multiple modes in the period space related to planets and activity-induced variations.", "The modes are typically narrow that the prior samples may not resolve the posterior distribution properly.", "Considering these difficulties in prior sampling, we sample the posterior and calculate the Bayes factors using various estimators which will be introduced in section REF ." ], [ "Posterior sampling", "To sample the posterior density, we use an adaptive Metropolis-Hastings algorithm of Markov Chain Monte Carlo (MCMC) developed by [24].", "This algorithm adjusts the step of the sampler to explore the posterior efficiently.", "Considering possible nonlinear correlation between parameters and a non-Gaussian posterior, we run adaptive Metropolis-Hastings algorithms to obtain posterior samples of $10^6-10^7$ for inference.", "We use the Gelman-Rubin criteria to judge whether a chain approximately converges to a stationary distribution [21].", "Specifically, we conduct the following steps to produce posterior samples.", "First, we run four chains in parallel, and drop out one half of each chain as “burn-in” part.", "Second, we estimate the so-called “potential scale reduction factor” ($\\hat{R}$ ) by calculating the variance between and within the chains according to the Gelman-Rubin criteria.", "If $\\hat{R}$ is less than 1.1, we combine these chains to provide a statistically representative posterior sample for inference.", "Third, we repeat the above two steps to generate chains with different tempering parameters.", "A chain is tempered if it is generated with a probability of move that is proportional to a power of the posterior ratio of proposed parameters and current parameters.", "Tempering is used to improve the dynamic properties of a chain to explore the whole parameter space efficiently.", "The chain without tempering is called “cold chain” while the tempered chain is called “hot chain”.", "Considering that the optimal acceptance rate of a Metropolis-Hastings algorithm is around 0.234 under general conditions [44], we select the hot chains with acceptance rate between 10% and 35%.", "Then we identify the potential signal based on the maximum a posteriori estimation, and use the corresponding parameters as the initial conditions of a cold chain.", "Finally, the cold chain provides a sample drawn from the posterior density of the model.", "For models without a Keplerian component, we run cold chains directly to obtain their unimodal posterior densities.", "Because we aim at comparing noise models rather than models with multiple Keplerian components, we only obtain samples for models with at most one planetary component." ], [ "Signal detection criteria", "Given a statistically representative sample drawn from the posterior density, we move on to calculate the evidence using various methods.", "The integral in Eqn.", "(REF ) can be calculated by the “importance sampling” method [30], which generates samples from a density.", "For example, the harmonic mean (HM) estimator of the evidence is calculated by averaging the likelihood over samples approximately drawn from the posterior density.", "However, this estimator cannot converge efficiently due to the occasional occurrence of samples with very low likelihoods.", "To solve the convergence problem of HM, [56] introduce the truncated posterior mixture (TPM) by drawing samples from different sections of a MCMC chain to avoid the divergence caused by low likelihood values.", "This method is easy to implement because it only uses the output of Metropolis-Hastings algorithms.", "But this method is biased if its free parameter $\\lambda $ is large [56], [10].", "In addition to importance sampling methods, we introduce the one-block Metropolis-Hastings method developed by [8].", "The Chib's estimator (CHIB) is based on the calculation of the posterior of a single point using samples drawn from the posterior density and the proposal density of a Metropolis-Hastings sampler.", "Although the evidence can be approximately calculated by the above methods, they have limitations in applications to complex problems due to unrealistic assumptions or computation inefficiency (see [20] and [25] for a review).", "Considering these, we also introduce various information criteria which are easy to calculate and thus are frequently used by practitioners.", "We introduce three of them: the Akaike Information Criterion (AIC; [1]), the Bayesian Information Criterion (BIC; [46]) and the Deviance Information Criterion (DIC; [47]).", "The AIC and DIC are criteria motivated from information theory while the BIC is derived in the Bayesian frameworkAlthough the BIC is derived using the Laplace approximation of a Gaussian likelihood distribution and the likelihood distribution in our case is always multimodal, we use it because the likelihood is always dominated by the Keplerian signal if there is, and the local distribution around the maximum is always Gaussian..", "Considering that the sample size of RV data sets may be small, we use a revised version of AIC introduced by [28].", "We write the three criteria as follows.", "$AIC&\\equiv &-2\\ln {\\mathcal {L}_{\\textrm {max}}}+\\frac{2k(k+1)}{N-k-1}\\\\BIC&\\equiv & -2\\ln {\\mathcal {L}_{\\textrm {max}}}+k\\ln {N}\\\\DIC&\\equiv &D(\\bar{\\theta })+2p_{D}=\\bar{D}(\\theta ) +p_{D}~,$ where $\\mathcal {L}_{\\textrm {max}}$ is the maximum likelihood, $k$ is the number of free parametersWe assume that a free parameter could be any variable in a model as in the case of linear models.", "Although a more complex definition of the parameter number could be helpful for nonlinear models, this is equivalent to changing the Bayes factor threshold which we will do in section REF  ., $N$ is the number of data points, the deviance $D(\\theta )=-2\\ln \\mathcal {L}(\\theta )$ and the effective number of parameters $p_{D}=\\bar{D}-D(\\bar{\\theta })$ .", "To compare the above information criteria with the Bayes factor estimators, we transform these information criteria into a Bayes factor like quantities To make the notation simple, we still use BF to name this quantity..", "It is straightforward to convert the BIC into a Bayes factor because [30] argued that $e^{-\\Delta BIC_{10}/2} \\rightarrow BF_{10}$ when the sample size is large.", "Here we define $\\Delta \\textrm {BIC}_{10}=\\textrm {BIC}_1-\\textrm {BIC}_0$ .", "We also define Bayes factor using the relative likelihood derived from AIC, i.e.", "$BF_{10}=e^{-\\Delta \\textrm {AIC}_{10}/2}$ , where $\\Delta \\textrm {AIC}_{10}=\\textrm {AIC}_1-\\textrm {AIC}_0$ .", "We then derive Bayes factor from DIC in the same fashion, since the DIC probably approaches the AIC when parameters are well constrained [32].", "Note that the transformations from AIC and DIC to Bayes factor are without theoretical foundation.", "Rather, it is used to transform the threshold of AIC or DIC to the threshold of Bayes factor, making AIC or DIC approximately suitable for Bayesian inference.", "With the above evidence estimators and information criteria, we adopt the following diagnostics for the presence of a Keplerian signal.", "The period $P$ of the signal can be constrained from above and below in the posterior density.", "In other words, it converges to a stationary distribution.", "The amplitude $K$ of the signal is significantly greater than zero.", "Specifically, the posterior of $K=0$ , i.e.", "$P(K=0|D,M)$ , is less than 1% In reality, we fit a normal distribution to the posterior sample, and from the best fitted posterior density we determine $P(K=0|D,M)$ ..", "The evidence of a model with one Keplerian component should be at least 150 times higher than the evidence for the model without any Keplerian component, i.e.", "BF$_{10}>150$ [30].", "The above procedure is also used by [53] in combination with the moving average model which we will introduce in the following section.", "The measured doppler shifts of a star are generated by gravitational force from star-planet(s) interactions and stellar activity.", "The spectroscopic measurements of these doppler shifts yield RV data with instrument uncertainties and various activity indexes.", "To account for these factors, we model the data by combining Keplerian components and various noise components.", "In the following sections, we introduce the basic model which includes the white noise model and the Keplerian component.", "Then we add various noise components onto the basic model to build other models in such a way that the basic model is nested in the full model given all noise components." ], [ "White noise model", "There is good evidence in the architectures of the Solar System and exoplanetary systems for orbital resonances playing some role (e.g., semi-major axes of resonant trans-Neptunian objects).", "However, the importance is limited over the typically time span of RV data (e.g., [6]), and so we make the simplifying assumption that planetary orbits are indenpendent of each other in a planetary system.", "Although we only consider at most one Keplerian signal in this work, we introduce a model of multiple Keplerian signals for general cases.", "We adopt the following basic model of RV variations, $\\hat{v}_b(t_i, \\theta ) &=& \\sum _{j=1}^{n}f_j(t_i)+a\\,t_i+b+\\sum _{k}c_kI_k\\nonumber \\\\f_j(t_i)&=& K_j [\\cos (\\omega _j + \\nu _j(t_i))+e_j\\cos (\\omega _j)]~,$ where $K_j$ , $\\omega $ , $\\nu _j$ , $e_j$ are the amplitude, the longitude of periastron, the true anomaly and the eccentricity for the $j^{\\text{th}}$ planetary signal.", "The true anomaly $\\nu $ is an implicit function of time, and depends on the orbital period $P$ and the orbital phase at the reference time $M_0$ .", "It can be calculated by solving the Kepler's equation.", "Thus the Keplerian component for each planet contains five free parameters: $K$ , $P$ , $e$ , $\\omega $ and $M_0$ .", "In addition to the above parameters, we use two parameters, $a$ and $b$ , to model the acceleration caused either by a companion or by the long period activity cycles of the star and the reference velocity.", "We also use $c_k$ to model the linear dependence of radial velocity on the activity index $I_k$ .", "Specifically, we use $I_F$ , $I_B$ and $I_R$ to denote the width of the spectral lines (FWHM), the bisector span (BIS) and the $log(R^{\\prime }hk)$ ($R_{\\rm HK}$ ), respectively.", "Note that these indexes are included into the model in a deterministic way.", "But they will be used in section REF and REF to model the jitter in a stochastic way .", "The white noise model accounts for the excess noise through the likelihood function: $\\mathcal {L}(\\theta )\\equiv P(D|\\theta ,M_w)=\\prod _i\\frac{1}{\\sqrt{2\\pi (\\sigma _i^2+s_w^2)}}\\exp \\left[-\\frac{(\\hat{v}_b (t_i,{\\theta } )-v_i)^2}{2(\\sigma _i^2+s_w^2)}\\right],$ where $\\sigma _i$ is the observational noise at time $t_i$ , $s_w$ is the constant amplitude of the white noise, and $v_i$ is the observed RV at time $t_i$ .", "The jitter depends on stellar activity levels which are partly measured by various shape indexes of spectrum such as FWHM and BIS and activity proxies such as the $R_{\\rm HK}$ index.", "The noises caused by activity and instruments are typically correlated [5], and are too complex to be modeled deterministically.", "Thus a range of red noise models are proposed to remove correlated noises which may mimic Keplerian signals.", "In the following sections, we introduce two of them: the moving average and Gaussian process models." ], [ "Moving average", "The moving average (MA) model is used to model the dependence of current noise on previous noise.", "The moving average of order $q$ or MA$(q)$ is $\\hat{v}(t_i)=\\hat{v}_b(t_i)+\\epsilon _{t_i}+\\sum _{j=1}^{q}k(t_i,t_{i-j})\\epsilon _{t_{i-j}},$ where $k(t_i,t_{i-j})$ is the kernel used to weight the white noise at time $t_{i-j}$ .", "We introduce two kernel functions, the Laplacian kernel $k_L(t_i,t_{i-j}) = w_j \\exp (-\\beta |t_i-t_{i-j}|)$ and the squared exponential kernel $k_{\\textrm {se}}(t_i,t_{i-j}) = w_j\\exp (-\\beta (t_i-t_{i-j})^2/2) ~,$ where $w_j$ is a positive number for white noise at $t_{i-j}$ , and $\\beta $ is a free parameter.", "According to [57], MA(1) is the best moving average model which enable the detection of weak signals.", "In addition, this model seems to outperform other red noise models in recent RV Challenge [13] conducted by Xavier Dumusque The details of RV Challenge data sets and results can be found online at https://rv-challenge.wikispaces.com.. Hereafter we will use moving average to denote MA(1) and use the Laplacian kernel if not mentioned otherwise." ], [ "Gaussian process ", "The Gaussian process (GP) is included in the RV model by adding non-diagonal parts to the covariance matrix of the likelihood function (see Eqn.", "REF ) which is $\\mathcal {L}(\\theta )\\equiv P(D|\\theta ,M_{\\text{g}p})=\\frac{1}{\\sqrt{|2\\pi \\mathcal {C}|}}\\exp \\left[-\\frac{1}{2}(v-\\hat{v}_b)\\mathcal {C}(v-\\hat{v}_b)\\right]~,$ where $v$ is the observed RV sequence (i.e.", "$\\lbrace v_i\\rbrace $ ), $\\hat{v}_b$ is the RV model expressed by Eqn.", "(REF ) (i.e.", "$\\lbrace \\hat{v}_b(t_i,\\theta )\\rbrace $ ), and $\\mathcal {C}$ is the covariance matrix.", "The covariance matrix is composed of diagonal and non-diagonal components.", "The former is related to the Gaussian measurement noise $\\lbrace \\sigma _i\\rbrace $ and the excess white noise $s_w$ , while the latter is determined by a kernel.", "To calculate the covariance matrix, we introduce three types of kernels: Laplacian (L), squared exponential (se) and quasi-periodic (qp) kernels, which are formulated as follows.", "$k_{\\text{L}}(t,t^{\\prime })&=&s_r\\exp (-|t-t^{\\prime }|/l)~,\\nonumber \\\\k_{\\text{s}e}(t,t^{\\prime })&=&s_r\\exp \\left[-\\frac{(t-t^{\\prime })^2}{2l^2}\\right]~,\\nonumber \\\\k_{\\text{q}p}(t,t^{\\prime })&=&s_r\\exp \\left[-\\frac{\\sin ^2(\\pi (t-t^{\\prime })/P_q)}{2l_p^2}-\\frac{(t-t^{\\prime })^2}{2l^2}\\right]~,$ where $s_r$ is the red noise amplitude, $l$ , $l_p$ and $l_e$ are correlation time scales, and $P_q$ is the period of the quasi-periodic kernel.", "The L kernel is used by [5] and [16], while the se and qp kernels are advocated by [42] in their Gaussian process framework used to disentangle activity-induced variations from planetary signals.", "In the comparison of noise models, we will focus on the Laplacian kernel, and use the other kenels for sensitivity tests." ], [ "$R_{\\rm HK}$ -dependent jitter", "It is well known that the solar activity is determined by the magnetohydrodynamic turbulence in the atmosphere, and thus is difficult to be accurately predicted due to the chaotic nature of turbulence [40].", "Like the Sun, stars also show complex activity-induced variations [51] which are partly recorded by activity indexes in doppler measurements.", "Therefore the relationship between RV variations and activity indexes are probably complex [60], and a deterministic relationship may not be appropriate to model the activity-induced RV counterpart.", "As a result, the dependence of RV on indexes should be modeled statistically.", "For example, [10] have proposed a linear dependence of jitter on the $R_{\\rm HK}$ index.", "We call this model “$R_{\\rm HK}$ -dependent jitter” (RJ) which replace the $s_w$ in Eqn.", "(REF ) with $s(t_i)=s_w+\\alpha I_R(t_i)~.$ Although [10] used a truncated version of the linear function, we don't explore more versions because the $R_{\\rm HK}$ -dependent jitter model is flexible and representative, and does not require a fine-tuned threshold to truncate the $R_{\\rm HK}$ index.", "For RV data sets that do not have $R_{\\rm HK}$ index, $I_R$ denotes the activity index which is determined by measuring the flux at the Ca II H&K lines with respect to continuum." ], [ "Lagged $R_{\\rm HK}$ -dependent jitter", "The activity of a star is determined by the underlying complex and nonlinear dynamics.", "Thus the time series generated by stellar activity typically have long correlation time scale [7].", "In a nonlinear dynamical system, the time series of state variables are actually projected from the motion on the manifold of a set of states.", "According to [50] the non-linear state space of the dynamics could be reconstructed by using only the lagged time series of one variable, e.g.", "the $R_{\\rm HK}$ index in our case.", "Thus the jitter in the RV variations can be modeled as a function of the lagged activity indexes.", "Considering that the $R_{\\rm HK}$ index is more sensitive to stellar activity [11], we let the jitter $s(t_i)$ depend on the previous and subsequent $R_{\\rm HK}$ indexes.", "Then the jitter noise becomes $s(t_i) &=& s_w+\\alpha I_R(t_i)+f(t_i,t_{i-1}) I_R( t_{i-1})+f(t_i,t_{i+1}) I_R( t_{ i+1})) \\nonumber \\\\f(t, t^{\\prime }) & =&\\eta \\exp (-\\kappa |t-t^{\\prime }|),$ where $\\eta $ is the amplitude of the correlation between jitter, and $\\kappa $ is the inverse of the correlation time scale.", "Replacing the white noise jitter $s_w$ in Eqn.", "(REF ) by the above $R_{\\rm HK}$ -dependent jitter, we define the lagged RJ model or LRJ.", "Considering that the lagged $R_{\\rm HK}$ may induce the RV counterpart in a deterministic way, we propose another version of lagged RJ which is $\\hat{v}(t_i)=\\hat{v}_b(t_i)+f(t_i,t_{i-1}) I_R( t_{i-1})+f(t_i,t_{i+1}) I_R( t_{ i+1}).$ We call the stochastic version in Eqn.", "(REF ) LRJ(S) and the deterministic version in Eqn.", "(REF ) LRJ(R).", "They are more alike than different in terms of accounting for time lagged $R_{\\rm HK}$ , and thus share the same name.", "In other words, the lagged $R_{\\rm HK}$ of LRJ(S) is put into the denominator of the exponential term of the likelihood (Eqn.", "REF ) while the LRJ(R) model contains the lagged $R_{\\rm HK}$ in the numerator.", "For a simulated data set, only one LRJ version will be chosen according to the ability of each LRJ model in finding signals.", "In addition to the above mentioned noise models, we combine them to build compound models to make the comparison more comprehensive.", "We combine moving average with $R_{\\rm HK}$ -dependent jitter to make the MARJ model, and combine moving average with lagged RJ to make the MALRJ model.", "For models with and without one Keplerian component, we add 1 and 0 after the model name, respectively.", "For example, MA1 means the moving average model with one Keplerian component while MA0 means the moving average model without Keplerian componentThis is not to be mixed with $q^{\\mathrm {th}}$ order moving average models denoted as MA(q), we only apply MA(1) model." ], [ "Prior distributions", "As a necessary part in Bayesian inference, the prior distributions of parameters are explicitly given for all models in table REF .", "For the Keplerian component in the white noise model, we adopt a Jeffreys prior for the period with a range from one day to the time span of the RV data sets.", "Following [54] we adopt a Gaussian prior distribution over eccentricity, i.e.", "$P(e)\\propto \\mathcal {N}(0,\\sigma _e)$ with $\\sigma _e=0.2$ , to account for observed eccentricity distribution [31] Although [31] recommend beta distribution, we adopt the Gaussian distribution which is simpler and also flexible enough to describe eccentricities.. We adopt a uniform prior for the amplitude $K$ with a upper limit of twice the maximum absolute value of RV.", "This prior is set not only to speed up the convergence of the chain but also to account for the fact that the long period signal (longer than the RV time span) is already modeled as a linear trend in Eqn.", "(REF ).", "The jitter amplitude $s$ follows a uniform distribution from 0 to the upper limit of the prior of $K$ .", "This is also the range of $R_{\\rm HK}$ -dependent jitter $s( t_i)$ .", "To make the chain converge quickly, we scale the $R_{\\rm HK}$ index such that it has zero mean and unit variance.", "For the same reason, we define the ratio of the upper boundary of the prior of $K$ , $K_{\\text{max}}$ , and the difference between the maximum and minimum of the index variation as the upper limit of the parameters for the dependence of RV on activity indexes, $c_R$ , $c_F$ and $c_B$ .", "For the Gaussian process model, we find that the likelihood of the Gaussian process model is not sensitive to the time scale $l$ , and thus set a narrow boundary for its prior.", "For the quasi-periodic kernel, we adopt the priors used by [36].", "Table: The prior distributions of model parameters.", "Theunit of c 1 c_1, c 2 c_2, c 3 c_3, α\\alpha and η\\eta is m/s because theactivity indexes are scaled before included in a model.", "The maximumand minimum time of the RV data are denoted by t max t_{\\textrm {max}} andt min t_{\\textrm {min}}, respectively." ], [ "Model comparison for artificial data sets", "To ensure that a model is suitable for the data, it is important to test them using artificial data sets with known noise.", "Otherwise, the model or its prior may not be appropriate for certain applications [17].", "We will do this with two types of simulated data sets, artificial (with artificial signals and noise) and injection (with artificial signals and real noise) data sets.", "Comparing all models for these data sets, we quantify the limitations of these models and form a signal detection strategy.", "Fitting different models to the artificial data sets, we report the signals and models recovered by testing models according to the diagnostics in section REF .", "We generate artificial data sets using three time samples and corresponding measurement errors: the sample from the Keck measurements of GJ 515A which contains 282 data points and two subsamples which are generated by randomly drawing 100 time samples from the full sample of GJ 515A.", "We denote these three samples as “full”, “sub1” and “sub2”, respectively.", "Then we generate the RV values using three models: white noise with one Keplerian component (W1), moving average with one Keplerian component (MA1) and Gaussian process with one Keplerian component (GP1).", "We vary the period $P\\in \\lbrace 20,40,80\\rbrace $  days and the amplitude of constant jitter $s\\in \\lbrace 0.5,1,2\\rbrace $  m/s.", "The other parameters are fixed, such that $K=1$  m/s, $e=0.1$ , $\\omega =\\pi /2$ , $M_0=\\pi /2$ , $a=-0.1$  m s$^{-1}$ yr$^{-1}$ , $b=1$  m/s, $(w,\\beta )=(0.9,0.25~\\textrm {day}^{-1})$ and $(s_r,l)=(1~\\textrm {m/s},3~\\textrm {day})$ .", "The total number of these artificial data sets is 54.", "Then we analyze each data set with the W0, W1, MA0, MA1, GP0 and GP1 models, and apply the signal detection criteria (see section REF ) to confirm/reject potential signals.", "The results are shown in table REF .", "Table: Comparison of noise models for artificial data sets.", "TheBayes factor of the one planet models (e.g.", "MA1) and correspondingzero planet models (e.g.", "MA0) are calculated for each dataset.", "Without applying the threshold of BF 10 >150BF_{10}>150, the results ofthe test are denoted as A, B and C, which means that the signal iscorrectly recovered, not recovered (false negative) and falselyidentified (false positive).", "The results obtained without and with applying the BIC-based BF threshold are written in normal and bold fonts, and corresponding detection numbers (denoted by NN and nn with subscripts) are reported without and with brackets, respectively.", "The flag A is underlined if the true model is recovered based on any estimator of the Bayes factor.", "The numbers of the data sets that have true models selected by any estimator and by the BIC is denoted by N T N_T and n T n_T, respectively." ], [ "Comparing models", "In table REF , we observe that the white noise model recovers signals better than the red noise models.", "However, the white noise model also detects more false positives because it interprets correlated noise as signal.", "It is a surprise that the moving average model does not recover itself for the full and sub2 data sets with $(P,K)=$ (40 day, 1 m/s).", "That means the moving average model interprets both the moving average noise and part of the signal as correlated noise, leading to an underestimation of the significance of the true signal.", "This indicates that a red noise model may not successfully separate correlated noise from planetary signals.", "We also find that the white noise and moving average models could be recovered through applying at least one Bayes factor estimator for 10 and 7 data sets, respectively.", "However, the Gaussian process model is recovered for only 3 data sets.", "Due to the flexibility of the Gaussian process model in fitting a data set, it overfits the noise and thus underfits the signal, leading to a lower evidence.", "This problem is also evident from the fact that the Gaussian process model does not recover any true signals while the white noise and moving average models recover 4 and 3, respectively, for the Gaussian process generated data sets.", "We can see the above differences between the white noise and moving average and Gaussian process models from their posterior densities in figure REF .", "We calculate the posterior densities by binning the posterior samples over the period and choosing the maximum posterior in each bin as an approximation of the marginalised posterior.", "The significance of the signal can be inferred from the posterior difference between thresholds and the difference between the signal and noise.", "We find that the signal detected by the white noise model is more significant than those detected by red noise models, and the signal detected by the Gaussian process model is the weakest.", "We also observe that the broad peaks around 70 days are probably false positives.", "The red noise models reduce the significance of the false positives together with that of the signal.", "In other words, the cost of decreasing the false positive rate is increasing the false negative rate.", "We also notice that the peak around 1.05 day is probably an alias of the signal.", "But it is not as significant as the signal for all noise models.", "In particular, the red noise models seem to remove the alias efficiently due to their ability of modeling correlated noise over short time scales.", "Figure: The unnormalised logarithm posterior density from thetempered chains for the white noise (upper), moving average (middle)and Gaussian process (lower) models for the full data setsimulated using the Gaussian process model with P=20P=20day ands=1s=1m/s.", "To show the significance of signals in each panel, the 10%, 1% and 0.1% of the maximum a posterior is shown by dashed lines.", "The true period of the artificial signal is denoted by a vertical dotted line.The performance of the models is also revealed by their maximum likelihoods shown in figure REF .", "We see that the planetary component can improve the maximum likelihood of white noise model more significantly than those of other models.", "This property of the white noise model is generic for all artificial data sets.", "Figure: The maximum likelihood ratios (MLR) of various models and theW0 model for the full data set simulated by the Gaussian process model withP=20P=20day and s=1s=1m/s (the same data used in figure )." ], [ "Choosing the optimum Bayes factor estimators", "To further confirm the signal detections in table REF , we compare models with and without Keplerian component using the Bayes factor threshold BF $>150$ .", "We cannot calculate the Bayes factors of models which do not have any chains converged to the target signal (denoted by flag “A” in table REF ) due to a lack of statistically representative posterior samples.", "For data sets with recovered signals, we calculate the Bayes factor using the AIC, BIC, CHIB, DIC, HM and TPM estimators.", "To ensure the convergence of each method, we increase the size of the posterior sample gradually and calculate the Bayes factor for each sample size.", "Since the DIC cannot converge properly due to the asymmetry and multimodes of the posterior density, we only show the results for other estimators in figure REF .", "Figure: The convergence of Bayes factor estimators for the artificial data setgenerated by the moving average model with P=80P=80day and s=1s=1m/s.", "The upper, middle and bottom panels show the logarithm Bayes factors of W1, MA1, GP1, with respect to W0, respectively.", "The TPM estimator of Bayes factor is calculated with λ=10 -4 \\lambda =10^{-4} as recommended by .We find that the HM and TPM estimators give similar results since the HM is just a special case of the TPM [56].", "However, neither of them converge very well due to the occasional occurrence of samples with very low likelihoods.", "The Bayes factor estimated by AIC is always higher than that estimated by the BIC and CHIB.", "We also see these differences from the Bayes factors calculated using the full posterior sample in figure REF .", "For models with one Keplerian component, we find that the DIC always estimate much higher Bayes factors while the BIC and CHIB estimate the lowest Bayes factors.", "However, for models without any Keplerian component, all methods give simliar Bayes factors, which justifies the usage of each method in the cases of unimodal posterior densities.", "We further investigate all data sets, and find that the Bayes factors estimated by AIC, HM and TPM are comparable, and the BIC and CHIB estimators always give similar results.", "Figure: The Bayes factors estimated by various methods.", "The heights of thelines within each gray bar represent the Bayes factors of a certain modelwith respect to W0 estimated by AIC, BIC, CHIB, DIC, HM and TPM from left to right.We then test the significance of signals using the Bayes factor threshold of 150.", "For each estimator of the Bayes factor, we apply the threshold to confirm recovered signals (denoted by “ A”), false positives (denoted by “ C”) and recovered models (denoted by “ T”).", "The ratios of confirmed and total number of them for all data setsTo keep the notation simple, we continue to use $N$ and $n$ with subscripts to denote them (see table REF ).", "are used to characterize the ability of estimators combined with the Bayes factor threshold in confirming true signals and noise properties.", "The results for all estimators are reported in table REF .", "Table: The ratios used to characterise estimators.", "The notations are similar to those in table .In this table, we find that the DIC is not appropriate for confirming detections because it cannot rule out any false positive.", "On the contrary, the CHIB estimator is able to get rid of almost all false positives, but can only confirm 47% true detections.", "An appropriate choice is the AIC which could rule out about one quarter of the false positives, and confirm 96% of the true detections.", "Although the AIC is not a Bayesian criterion, we regard the AIC combined with a threshold as a practical tool to confirm detections.", "In the case of exoplanet detection, avoiding false positives is more important than avoiding false negatives.", "Such requirements can be satisfied by the BIC which rules out 75% false positives and confirms 58% true detections.", "In addition, the BIC recovers 75% true models while the CHIB method recovers 60%, justifying our choice of the BIC rather than the CHIB estimator to rule out false positives.", "Although the HM and TPM estimators confirm most true detections, they have convergence problems as we have mentioned (see figure REF ).", "In summary, the white noise model is able to detect weak signals efficiently while the Gaussian process are so flexible that signals are interpreted as correlated noise.", "The performance of the moving average model is somewhere between the performance of Gaussian process and white noise models.", "Moreover, most false positives could be ruled out by the BIC-estimated Bayes factor threshold of 150." ], [ "Model comparison for injection data sets", "In the above section, we have analysed the artificial data sets with known noises and signals.", "To make our analysis more general, we apply the same analysis method to data sets with known signals but with noises from real data.", "We adopt three data sets, the HARPS measurements of GJ 1 (44 epochs) and GJ 361 (101 epochs), and the Keck measurements of GJ 445 (64 epochs).", "For each data set, we use the RVs, measurement errors and activity indexes of $R_{\\rm HK}$ .", "We also consider the RV dependence on the FWHM and BIS index (see Eqn.", "REF ) if they are available.", "Considering that the data of GJ 445 is not published before, we show it in the appendix.", "We introduce the three targets in the following section." ], [ "Radial velocity data", "Most nearby stars now have precision radial velocities recorded for them.", "As of 2016 April the ESO archive for the HARPS instrument finds over 7000 different targets although most only have a few epochs some objects have large numbers, e.g., nearly 20,000 for alpha Centauri B.", "We focus our attention on radial velocity data for nearby M dwarfs.", "We choose these targets because they appear to have relatively lower activity noise.", "We have attempted to focus on targets which have reasonable sampling, good precision and for which there is enough radial velocity data to make detections but where there is not a strong known signal.", "We have drawn these from the sample of [58] and specifically choose to study data from both HARPS [33] and HIRES [62].", "These are two of the pre-eminent radial velocity instruments whose design, calibration and processing can be considered both reliable and independent.", "GJ 1 is a metal-poor (e.g., [Fe/H]=-0.45; [37]) M2 dwarf at a distance of 4.5 pc [59] without any reported planetary companions (e.g.", "[66]).", "[48] find that it has a rotation period of $60.1\\pm 5.7$  d based on spectral activity indexes.", "Analysis of the ASAS photometric data [41] does not confirm this rotation period [58] though there are relatively modest 42 photometric data points spanning less than a year.", "We consider the 44 HARPS epochs which we have extracted from the ESO archive.", "GJ 445 is a metal-poor (e.g., [Fe/H]=-0.30; [37]) M4 dwarf at a distance of 5.4 pc [59] without any reported planetary companions or rotational signals.", "Here we consider 64 epochs of Keck data.", "GJ 361 is a slightly metal-poor (e.g., [Fe/H]=-0.11; [37]) M1.5 dwarf at 11 pc [59].", "[58] find a low amplitude signal (3.82m/s) with a period of 28.9 d which is removed from the data.", "[58] do not find any evidence for significant periodicities in 241 ASAS V-band photometric observations of the star spanning 2,298 days.", "We utilise 101 epochs of HARPS radial velocity data.", "To extract the noise from the GJ 361 data set, we analyze the data of GJ 361 with the moving model and identify the signal, and subtract the signal from the data set.", "This subtracted version is called “GJ361 subtracted”.", "To ensure that no significant signal exist in the GJ 1, GJ 445 and GJ 361 subtracted data sets, we fit all models in section to them and do not find any significant signal which satisfies the signal detection criteria.", "Fitting the W0 model to all data sets, we obtain the posterior of jitter-induced white noise $s_w$ , and use the mean value as a reference point for choosing the amplitudes of injected Keplerian signals." ], [ "Recovering signals", "To inject signals, we vary the amplitude and period of the Keplerian component and keep other parameters fixed.", "The amplitude of a signal is varied in such a way that the signal strength is lower, comparable or higher than the jitter level.", "We adopt $P\\in \\lbrace 20,40,80\\rbrace $  day for all data sets, $K\\in \\lbrace 1,2,4\\rbrace $  m/s for GJ 1 and GJ 361, and $K\\in \\lbrace 4,6,8\\rbrace $  m/s for GJ 445.", "The other Keplerian parameters are set $e=0.1$ , $\\omega =\\pi /2$ , $M_0=\\pi /2$ .", "Finally, we fit all noise models with and without planet to the injection data sets, and report the detections confirmed by the signal detection criteria in table REF .", "Table: Model comparison for RV data sets without and with injected signals.", "We use the LRJ(S) and MALRJ(S) models to fit the GJ1 data set, andapply the LRJ(R) and MALRJ(R) models to fit the other data sets.", "The meanings of A,B and C are described in table .As seen from table REF , no strong signals are found by any noise model, although the W1, RJ1 and LRJ1 models seem to identify weak signals which fail to pass the Bayes factor threshold.", "The table shows that the LRJ1 model detects the most signals without applying the BIC-estimated Bayes factor threshold (BF$_{10}>150$ ) while the W1 and RJ1 models find the most signals once applying the threshold.", "On the contrary, red noise models only recover less than half of the injected signals and even less if applying the Bayes factor threshold, implying that they are much more conservative and prone to false negatives.", "However, if without applying the Bayes factor threshold, the moving average model can recover 13 signals.", "As a red noise model, moving average is not as flexible as Gaussian process, and thus is able to identify more signals.", "In addition, most true signals recovered by the W1, RJ1 and LRJ1 models are also strong in the posterior densities of the moving average model, although they may not satisfy the detection criteria.", "On the contrary, the false positives are never strong in the posterior distributions of moving average.", "Hence the moving average model can be used to confirm true detections and reject false ones.", "To ensure that the results are not sensitive to the choice of kernels, we adopt the squared exponential kernel for the moving average models (see Eqn.", "REF ), the squared exponential and quasi-periodic kernels for the Gaussian process models (see Eqn.", "REF ).", "We fit the red noise models with these new kernels to the GJ1 data set with $(P,K)=$ (20 day, 2 m/s), the GJ445 data set with $(P,K)=$ (80 day, 8 m/s), and the GJ361 subtracted data set with $(P,K)=$ (80 day, 2 m/s).", "For all these three data sets, we don't find any statistically significant improvement by changing kernels.", "As we have mentioned in section , the red noise models interprets signals as noise, and thus adding a Keplerian component to a red noise model would not improve the likelihood as much as the W1 model does.", "This is evident from the comparison of the maximum likelihoods of all models in figure REF .", "We observe that the likelihoods of W1, RJ1 and LRJ1 are much higher than those of W0, RJ0 and LRJ0, indicating the necessity of adding one Keplerian component into the noise model.", "On the contrary, the Keplerian component does not significantly improve the likelihoods of the red noise models, in particular the Gaussian process model.", "Figure: The maximum likelihood ratio (MLR) of noise models and the W0model for the GJ445 data set with P=40P=40 day and K=8K=8 m/s andthe GJ361 subtracted data sets with P=80P=80 day andK=2K=2 m/s.", "Each grey bar encloses the MLRs for one noise modelwith and without planetary component.Among the W1, RJ1 and LRJ1 models, the W1 model give similar results as the RJ1 and LRJ1 models, although RJ1 and LRJ1 can model a few data sets slightly better due to adjusting extra free parameters.", "The LRJ1 model is favoured by the GJ445 data sets with $(P,K)=$ (20 day, 6 m/s) and (40 day, 4 m/s).", "For the latter one, we show the posterior distribution of parameters $\\alpha $ , $\\eta $ and $\\kappa $ of LRJ1 in figure REF .", "We observe that the white noise $s_w$ dominates the total noise while the index dependent noises is consistent with zero.", "For all the other data sets, we don't find strong dependence of noise on the RHS index either.", "Despite this, the RJ1 and LRJ1 models give fewer false positives than the white noise model does, indicating a weak dependence of jitter on the RHS index.", "Furthermore, the false positives detected by RJ1 and LRJ1 failed to pass the Bayes factor threshold.", "This is not caused by the Bayesian penalization of model complexity because the Bayes factor of RJ1 (or LRJ1) and RJ0 (or LRJ0) do not depend on the number of parameters of the RJ (or LRJ) model.", "To test this further, we vary the Bayes factor threshold to see whether there is an optimal threshold which can reject more false positives and keep all true detections.", "But we failed to find such a value.", "For example, we increase the Bayes factor threshold to be 200, and apply this new threshold to test the signals detected by the white noise model.", "We find that the false positives for the GJ445 data set with $(P,K)=$ (40 day, 4 m/s) cannot be ruled out, and the true signal in the GJ1 data set with $(P,K)=$ (20 day, 2 m/s) is rejected.", "It means that we cannot confirm all true signals recovered by the white noise model and reject false positives simultaneously by adjusting the Bayes factor threshold.", "Considering these problems of the white noise model and the complexity of the LRJ models, we recommend the $R_{\\rm HK}$ -dependent jitter to model the excess noise in RV observations.", "Figure: The unnormalised posterior distribution of s w s_w, α\\alpha ,η\\eta and κ\\kappa of the LRJ1 model.", "The histograms are made with90,5004 posterior samples of a cold chain.", "For each panel, the redcurve shows the fit of Gaussian distribution to the unnormalisedposterior density.", "The mode, mean (μ\\mu ), standard deviation(σ\\sigma ), skewness (μ 3 \\mu _3) and kurtosis (μ 4 \\mu _4) are shown foreach posterior distribution.Considering the limitations of different noise models, we set up a rule for modeling RV noise and selecting signals in order to avoid as many false positives and negatives as possible.", "We call this rule “Goldilocks principle”.", "Specifically, we suggest combining the white noise model and the $R_{\\rm HK}$ -dependent jitter with the moving average model in the following way to confirm detections.", "First, we apply the three criteria introduced in section REF to confirm a signal detected by the model of $R_{\\rm HK}$ -dependent jitter.", "Then the signal is further confirmed if it is also strong and unique (without local maxima exceeding the 10% threshold, see figure REF ) in the posterior distribution of the moving average model.", "Finally, the signal is confirmed as a planet if it is not found to be strong in the posterior distributions of the white noise model (with zero eccentricity) for the activity indexes to avoid detecting activity-induced false positives that have a different phase in the RVs and activity indexes." ], [ "Discussions and conclusions", "This work aims at comparing various noise models and inference criteria for detecting weak signals in radial velocity data sets.", "We define different noise models and introduce estimators of Bayes factor to analyse artificial data sets.", "We find that the white noise model is better than red noise models in detecting true signals.", "However, the white noise model tends to interpret correlated noise as a signal, and thus detect false positives.", "On the contrary, the red noise models, particularly the Gaussian process, usually interprets the signal as noise, at least partially, leading to false negatives.", "This is also the reason why the Gaussian process model is not favored even by the Gaussian process generated data sets (see table REF ).", "This challenges the view that a simultaneous modeling of noise and signal components in data would not result in overfitting or underfitting problems (e.g.", "[19]).", "The solution of the problem is not only to perform modeling in the Bayesian framework but also to properly model noise and signal according to a Goldilocks principle which could be obtained for each specific scientific question.", "Comparing various Bayes factor estimators, we find that the BIC estimation of Bayes factor combined with a Bayes factor threshold of 150 can reject most false positives while other criteria either confirm more false positives or reject a large proportion of true detections.", "In addition, the truncated posterior mixture, harmonic mean and Deviance Information Criterion estimators do not converge properly.", "Meanwhile the Akaike Information Criterion and Chib's estimators penalize the one planet models too little and too much, leading to false positives and negatives, respectively.", "Given that all estimators of Bayes factors have short-comings [18], we adopt the BIC for practical reasons.", "We have applied the BIC-based signal detection threshold to analyze data sets with injected signals.", "We have simulated 27 data sets by injecting signals with various periods and amplitudes into the HARPS measurements of GJ1 and GJ361 and the Keck observations of GJ445.", "We find that the white noise model and the (lagged) $R_{\\rm HK}$ -dependent jitter models recover most injected signals.", "However, the Bayes factor threshold cannot reject all false positives found by the while noise model.", "Increasing the threshold cannot rule out all false positives and confirm all true detections simultaneously.", "On the contrary, the Bayes factor threshold successfully reject all false positives detected by the $R_{\\rm HK}$ -dependent noise models, although the dependence of jitter on the $R_{\\rm HK}$ is weak probably due to the low activity level of our targets.", "To make the planet detection conservative, we suggest to form a noise model framework by combining the $R_{\\rm HK}$ -dependent noise model (RJ) and the moving average model.", "Since most planet hosts are M dwarfs, our conclusions on modeling the RV noise of M dwarfs are probably generic for exoplanet detections and so this work may also shed light on the noise modeling for hotter stars.", "We also test the sensitivity of the evidences for red noise models to their kernels, and don't find any significant improvement for the test cases analysed here.", "Since the injection data sets are from different instruments and with different sizes, our quantification of the limitations of various noise models are probably generic for detecting planets in RV observations.", "Our results indicate that flexible noise models such as Gaussian processes may underestimate the number of Keplerian signals.", "This is supported by [57]'s choice of first order moving average model to reduce jitter rather than higher order moving average models.", "In addition, the Tuomi et al.", "group “won” the RV Challenge using the moving average model while other groups failed to recover as many signals using more flexible models.", "This is consistent with our findings that the usage of flexible noise models tend to result in false negatives when the models do not correctly reflect the underlying physics of stellar activity.", "This difference between noise models is also evident from the controversy over the validation of the number of planets, where red noise models find less signals than other models, e.g.", "GJ 581 and GJ 667C discussed in the introduction section.", "These controversies are consistent with our conclusion that red noise models lead to false negatives while the white (or $R_{\\rm HK}$ -dependent) noise model lead to false positives.", "To avoid both, we define a Goldilocks principle by combining the $R_{\\rm HK}$ -dependent noise model with the moving average model and a BIC-based signal detection criterion.", "This principle also provides a clue for noise modeling in other fields.", "For example, stochastic models may not be appropriate for modeling the glacial-interglacial cycles over the Pleistocene because they tend to give false negatives.", "This can be investigated through injecting Earth's orbital variations into noisy climate data and recovering them using stochastic noise models in combination with orbital models.", "Another example is the detection of periodic signals in quasar light curves.", "The optical variability of quasars could be caused by random processes, rotations of binary black holes, uneven sampling and/or correlated noise.", "Since RV variations have similar characteristics, our work may also provide insights for disentangling periodic signals from stochastic variability in quasar light curves (e.g.", "[22], [61]).", "In summary, the Goldilocks principle provides an approach to balance between overfitting and underfitting of noise by statistical models, which may poorly reflect the underlying physics and thus are unable to disentangle noise from signals.", "Although a Gaussian process framework has been proposed to partly account for the underlying physics [42], the jitter may not be properly modeled due to the flexibility of Gaussian process models and the simplification of the complex relationship between RV variations and stellar activity indexes.", "Further studies on the statistical property of stellar activity proxies and their connection with RV variations are essential steps towards an astrophysically motivated modeling of stellar jitter.", "A probable method is the nonlinear time series analysis which connects the nonlinear dynamical system with the time series of some system outputs [29], [49].", "This idea has inspired us to build the lagged $R_{\\rm HK}$ -dependent jitter model which performs well in our analyses.", "Moreover, correlated noise and deterministic signals can be well distinguished using surrogate time series, a concept developed in the community of nonlinear time series [45].", "These facts justify further investigations into the nonlinear approach of modeling RV noise." ], [ "Acknowledgements", "FF, MT and HJ are supported by the Leverhulme Trust (RPG-2014-281) and the Science and Technology Facilities Council (ST/M001008/1).", "We used the ESO Science Archive Facility to collect radial velocity data sets.", "We also thank the referee, David Kipping, for valuable comments." ] ]
1606.05196
[ [ "Models of low-mass helium white dwarfs including gravitational settling,\n thermal and chemical diffusion, and rotational mixing" ], [ "Abstract A large number of extremely low-mass helium white dwarfs (ELM WDs) have been discovered in recent years.", "The majority of them are found in close binary systems suggesting they are formed either through a common-envelope phase or via stable mass transfer in a low-mass X-ray binary (LMXB) or a cataclysmic variable (CV) system.", "Here, we investigate the formation of these objects through the LMXB channel with emphasis on the proto-WD evolution in environments with different metallicities.", "We study, for the first time, the combined effects of rotational mixing and element diffusion (e.g.", "gravitational settling, thermal and chemical diffusion) on the evolution of proto-WDs and on the cooling properties of the resulting WDs.", "We present state-of-the-art binary stellar evolution models computed with MESA for metallicities between Z=0.0002 and Z=0.02, producing WDs with masses between 0.16-0.45 M$_{\\odot}$.", "Our results confirm that element diffusion plays a significant role in the evolution of proto-WDs that experience hydrogen shell flashes.", "The occurrence of these flashes produces a clear dichotomy in the cooling timescales of ELM WDs, which has important consequences e.g.", "for the age determination of binary millisecond pulsars.", "Rotational mixing is found to counteract the effect of gravitational settling in the surface layers of young, bloated ELM proto-WDs and therefore plays a key role in determining their surface chemical abundances.", "We predict that these proto-WDs have helium-rich envelopes through a significant part of their lifetime, a crucial ingredient for understanding the newly observed ELM proto-WD pulsators.", "The hydrogen envelope at detachment, although small compared to the total mass of the WD, contains enough angular momentum such that the spin frequency of the resulting WD on the cooling track is well above the orbital frequency." ], [ "Introduction", "Extremely low-mass white dwarfs (ELM WDs) are low-mass helium-core WDs with masses below $0.2-0.3\\;M_{\\odot }$ and with surface gravities of $5<\\log g<7$ [14].", "A large number of such objects have been discovered in recent years through dedicated or general surveys such as ELM, SPY, WASP, SDSS and the Kepler mission [15], [66], [16], [67], [14], [73], [78], [65], [13].", "Soon after the discovery of the first ELM WDs, it was recognised that they have to be a product of binary evolution [77].", "From an evolutionary point of view, these ELM WDs cannot be formed from single-star progenitors as the nuclear evolution timescale of such low-mass objects would exceed the Hubble time – unless they have an extremely high metallicity [71] or the star lost its envelope from an inspiralling giant planet [83].", "Indeed, the vast majority of ELM WDs are found in binary systems with a companion star such as a neutron star in millisecond pulsar (MSP) systems [108], an A-type star in EL CVn-type systems [79] or another (typically a carbon-oxygen) WD.", "ELM WDs have been discovered in various environments, from the Galactic disk to open and globular clusters [93], [17], and thus they can be formed from progenitors with different metallicities.", "The revived interest in ELM WDs was fostered by the discovery of pulsations in several of these objects [49], [47], [48], [69] as well as ELM proto-WDs [81], [80], [25], [35].", "The ELM WD pulsators extend the ZZ Ceti instability strip to lower effective temperatures and higher luminosities.", "This instability strip contains stars with a convective driving mechanism for pulsations acting at the base of the convective zone associated with hydrogen recombination [107].", "In the newly discovered ELM proto-WD pulsators, the excitation mechanism is instead the usual $\\kappa -$ mechanism for which the presence of He in the envelope is thought to play a key role [59], [24].", "The pulsational behaviour of ELM WDs and ELM proto-WDs provide an unique insight into their interior properties, such as the hydrogen envelope mass and their total mass and rotation rate, which will place stronger constraints on the theoretical models [22], [23], [24].", "Another interesting and not completely understood feature of ELM WDs is the observed presence of metals in their atmospheres.", "[36] provided for the first time systematic measurements of the atmospheric abundances of He, Ca and Mg for this type of stars and examined their distribution as a function of effective temperature and mass.", "In the observed sample, all the WDs with $\\log ~g<5.9$ show Ca II K lines, suggesting that the presence of metals in these objects is a ubiquitous phenomenon, possibly linked to their evolution.", "Detailed abundance analyses exist for only a handful of objects [63], [37], [45], [74] but already suggest a diversity of metallicities, as in the case of sdB stars.", "Gravitational settling depletes the metals in the atmospheres of WDs on a very short timescale compared to their evolutionary timescale [109], [86], [73], indicating that a process should be at work that counteracts it or replenishes the depleted metals.", "In addition to the formation and evolutionary history of these objects, their future outcome is also of theoretical interest.", "Short-period double WD binaries are candidate progenitors for transient explosive phenomena such as Type Ia, underluminous .Ia and Ca-rich supernovae [11], [53], [90], [32], as well as exotic systems such as AM CVn stars, R Coronae Borealis (R CrB), and single subdwarf B/O stars [70], [100], [21], [40].", "Moreover, they are expected to be excellent sources of gravitational waves [46], [68] and verification sources for gravitational detectors such as eLISA [7]." ], [ "Formation and evolution of ELM WDs", "From a theoretical point of view, an ELM WD can be formed either through common-envelope evolution or stable Roche-lobe overflow (RLO) mass transfer in a low-mass X-ray binary (LMXB) or a cataclysmic variable (CV) system.", "The formation and evolution of low-mass WDs through a stable mass-transfer phase (or by artificially removing envelope mass from its progenitor star) has been studied intensively over the years [29], [95], [84], [4], [85], [2], [56], [57].", "In comparison, the common-envelope channel is less studied and far more uncertain [82].", "Although the majority of ELM WDs are found in double WD systems [8], almost all evolutionary calculations that involve stable mass transfer producing an ELM WD consider a neutron star companion (i.e.", "an LMXB progenitor system).", "For the structure of the final ELM WDs, the results of these LMXB calculations can also be applied to CV systems producing ELM WDs in double WD binaries, as the stellar properties of the produced ELM WDs do not depend on the mass of their accreting companion, but instead on the initial orbital period and mass of the donor (progenitor) star [84], [26], [56].", "Only the orbital periods of the produced ELM WDs will be different." ], [ "Hydrogen shell flashes and proto-WDs", "After the RLO mass-transfer phase ends, the remaining donor star goes through a so-called (bloated) proto-WD phase in which a significant part of the hydrogen left in the envelope is burned through stable hydrogen shell burning.", "In addition to this, depending on the mass of the proto-WD, its metallicity and the physics included in the modelling, hydrogen may be burned through short-lived phases of unstable burning through CNO hydrogen shell flashes [29], [6], [84].", "Figure: Hertzsprung-Russel (HR) diagram showing the formation and cooling of a 0.28 M ⊙ _{\\odot } helium WD (produced in an LMXB) that undergoes a hydrogen shell flash.", "The initial progenitor mass is 1.4M ⊙ 1.4\\;M_{\\odot } (Z=0.02Z=0.02), the neutron star mass is 1.2M ⊙ 1.2\\;M_{\\odot }, and the initial orbital period is 5.0 days.", "See Table  for ages at each stage.Figure REF shows an example of the formation of an ELM WD through the LMXB channel, including the evolution as a proto-WD as well as its further cooling.", "The stellar track is computed from the zero-age main sequence (ZAMS) until the donor star reaches an age of 14 Gyr, points 0 and 12, respectively, in Fig.", "REF .", "In this case, the star experiences one hydrogen shell flash.", "After the Roche-lobe detachment (point 2), the proto-WD goes through a phase of contraction at almost constant luminosity and increasing effective temperature (between points 2 and 3).", "The total luminosity is dominated by CNO burning while the contribution due to release of gravitational binding energy from contraction is negligible.", "When the proto-WD reaches point 3, which is at the beginning of the cooling branch, the temperature in the burning shell is too low to sustain CNO burning, therefore the main contribution to the total luminosity is for a while given by contraction, until the star switches to pp-burning.", "The unstable burning starts around point 4 and CNO burning becomes dominant again.", "The increasing energy release during the flash development creates a steep temperature gradient close to the location of maximum energy production.", "This will give rise to a pulse-driven convection zone within the hydrogen burning shell.", "After the convection zone is fully developed, the evolution becomes faster (between points 5 and 6).", "Around point 7, the convection zone reaches the stellar surface, and consequently, its surface chemical composition is altered.", "The maximum hydrogen luminosity reached during the flash is supplied by the pp-burning, although the onset of the instability is triggered by the CNO cycling.", "Between points 7 and 8, the lower boundary of the pulse-driven convection zone moves upwards, and at point 8 it completely vanishes.", "Beyond point 8, the contraction of the inner shells resumes, while the surface layers react by expansion, resulting in a redward motion in the HR-diagram that almost brings the proto-WD back to the red-giant branch.", "At point 9, the star fills its Roche lobe again and a short episode of mass transfer is initiated (between points 9 and 10) with a high mass-transfer rate that approaches $\\sim \\!10^{-7}\\;M_{\\odot }\\,{\\rm yr}^{-1}$ .", "After point 10, the star again evolves towards a high surface temperature at almost constant luminosity, and before reaching the final cooling track, it develops a so-called subflash (near log $T_{\\mathrm {eff}}=4.4$ ).", "The time intervals for each of the above described phases are shown in Table REF .", "Table: Evolution as a function of time for a 0.28M ⊙ 0.28\\;M_{\\odot } proto-WD evolving through a hydrogen shell flash, as plotted in Fig. .", "The ZAMS is at point 0, and the onset of RLO (the LMXB phase) is at point 1.", "The relative age is in comparison to the previous point of evolution and the WD age is with respect to the Roche-lobe detachment (point 2).", "See text for more details.The proto-WD phase has an associated timescale, $\\Delta t_{\\rm proto}$ , which is the time it takes the star to evolve (and contract) from the Roche-lobe detachment until it reaches its maximum effective temperature on the (final) cooling track.", "In Fig.", "REF this corresponds to the time interval during the evolution from point 2 to point 11.", "The duration of this contraction phase is mainly given by the burning rate of the residual hydrogen in the envelope.", "For evolved low-mass stars there is a well-known correlation between the degenerate core mass and its luminosity [92].", "Therefore, after Roche-lobe detachment, the rate at which the residual hydrogen in the envelope is consumed is directly proportional to the luminosity and thus increases strongly with $M_{\\mathrm {WD}}$ .", "A detailed analysis of the dependence of $\\Delta t_{\\rm proto}$ on the mass of the WD is given in [57].", "This timescale is especially important in MSP systems and should be added to the optically determined cooling age of the WD to yield the true age of the recycled radio pulsar.", "Unfortunately, the true age of a recycled pulsar cannot be determined from its characteristic spin-down age as this method has proved unreliable by a factor of 10 or more [18], [76], [102], [103].", "Hydrogen shell flashes occur in a range of proto-WD masses that is dependent on the metallicity and whether or not element diffusion is included in the modelling.", "The lower mass limit for flashes, $M_{\\mathrm {flash, min}}$ , is determined by the size of the burning shell, such that if $M_{\\mathrm {proto-WD}}$ < M$_{\\mathrm {flash, min}}$ , then the shell is too thick to trigger unstable hydrogen burning.", "The upper mass limit, $M_{\\mathrm {flash, max}}$ , is determined by the cooling time of the burning shell, which needs to be long enough to avoid an extinction of the shell before the instability is fully established [29].", "These conditions are altered when element diffusion is included [6], and this issue is investigated more carefully in Sect.", "." ], [ "Age dichotomy in helium WD cooling?", "The occurrence of hydrogen shell flashes, when element diffusion is taken into account, has been found to be responsible for a dichotomy in the cooling ages of helium WDs [4], [108], [2], [10].", "The occurrence of flashes in relatively massive helium WDs ($>0.2\\;M_{\\odot }$ ), with initially thin hydrogen envelopes, leaves behind an even thinner envelope, giving rise to relatively fast cooling.", "On the other hand, less massive (proto) helium WDs ($<0.2\\;M_{\\odot }$ ) have thicker hydrogen envelopes after RLO, resulting in stable shell hydrogen burning, and will therefore continue residual hydrogen burning on the cooling track on a long timescale.", "Recently, [57] found no evidence for such a dichotomy in the case of thermal evolution of proto-WDs but rather a smooth transition with the mass of the WD.", "The authors showed that the thermal evolution timescale mainly depends on the proto-helium WD luminosity, which in turn depends on the mass of the proto-WD and not on the occurrence of hydrogen shell flashes.", "These new findings questioned whether a dichotomy exists in the cooling ages of ELM WDs and if the responsible process might be the occurrence of hydrogen shell flashes." ], [ "Aims of this investigation of ELM WDs", "The focus of this paper is on the proto-WD phase of ELM WDs, which are investigated through a series of binary stellar evolution calculations of LMXBs.", "The following aspects are addressed: (i) the hydrogen envelope mass as a result of binary evolution, (ii) the role played by rotational mixing in the evolution of ELM proto-WDs, (iii) the influence of element diffusion and rotation on $\\Delta t_{\\rm proto}$ as well as on the cooling timescale, (iv) the existence of a dichotomy in the cooling ages of ELM WDs as a result of the occurrence of hydrogen shell flashes, (v) the presence of metals in the atmospheres of proto-WDs, and (vi) the relation between the mass of a proto-WD and its orbital period at the end of the mass-transfer phase.", "All these aspects are addressed not only as a function of the proto-WD mass, but also as a function of metallicity.", "Answering these open questions is essential for understanding the formation of ELM WDs and their age determination, for providing accurate models for astroseismology calculations, and for determining the correct age of MSP binaries.", "This work extends the previous work by [57] by including element diffusion and rotational mixing in the evolution of the donor star and during the proto-WD and the WD cooling phase.", "Moreover, the study is extended to include the effect of metallicity as well, for which we investigate four metallicities: $Z=0.02$ , $0.01$ , $0.001$ , and $0.0002$ .", "The evolutionary tracks presented in this paper are calculated using the publicly available binary stellar evolution code MESA, version 7624 [87], [88], [89].", "The nuclear network used is cno$\\_$ extras.net and accounts for the CNO burning with the following isotopes: $^{1}$ H, $^{3}$ He, $^4$ He, $^{12}$ C, $^{13}$ C, $^{13}$ N, $^{14}$ N, $^{15}$ N, $^{14}$ O, $^{15}$ O, $^{16}$ O, $^{17}$ O, $^{18}$ O, $^{17}$ F, $^{18}$ F, $^{19}$ F,$^{18}$ Ne, $^{19}$ Ne, $^{20}$ Ne, $^{22}$ Mg and $^{24}$ Mg. Radiative opacities are taken from [31] for $2.7\\le \\log T \\le 4.5$ and OPAL [54], [55] for $3.75\\le \\log T \\le 8.7$ and conductive opacities are adopted from [19].", "Convective regions are treated using the mixing-length theory (MLT) in the [43] formulation with $\\alpha _{\\rm {MLT}}=2.0$ .", "Transport of angular momentum is treated as a diffusive process which results in rigid rotation in convective zones.", "The boundaries of convective regions are determined using the Schwarzschild criterion.", "A step function overshooting extends the mixing region for 0.2 pressure scale heights beyond the convective boundary during core H-burning.", "We here refer to element diffusion as the physical mechanism for mixing of chemical elements that is due to pressure gradients (or gravity, i.e.", "gravitational settling), temperature (thermal diffusion) and composition gradients (chemical diffusion).", "Gravitational settling tends to concentrate heavier elements towards the centre of the star.", "Thermal diffusion generally acts in the same direction, although to a lesser degree, by bringing highly charged and more massive species towards the hottest region of the star (its centre).", "Chemical diffusion, on the other hand, has the opposite effect [52], [106].", "MESA includes the treatment of element diffusion through gravitational settling, chemical and thermal diffusion [106], and radiative accelerations [50].", "Radiative forces are proportional to the reciprocal of the temperature and are thus negligible in hot regions where nuclear burning is of importance.", "In addition, calculating these forces is computationally demanding.", "We therefore here neglected the effects of radiative levitation (which is important for determining photospheric composition of hot WDs [33].", "The detailed description of how element diffusion is implemented in MESA can be found in [89].", "We take into account the effects of element diffusion due to gravitational settling and chemical and thermal diffusion for the following elements $^1$ H, $^{3}$ He, $^{4}$ He, $^{12}$ C, $^{13}$ C, $^{14}$ N, $^{16}$ O, $^{20}$ N, $^{24}$ Mg, and $^{40}$ Ca.", "MESA includes the effects of the centrifugal force on stellar structure, chemical mixing, and transport of angular momentum that is due to rotationally induced hydrodynamic and secular instabilities as described in [41].", "Here, we take into account the mixing due to dynamical shear instability, secular shear instability, Eddington-sweet circulation, and Goldreich-Schubert-Fricke instability with a mixing efficiency factor of $f_{c}=1/30$ [41].", "The mixing of angular momentum that is due to dynamo-generated magnetic fields in radiative zones is also included [101], [42] as is the angular momentum transport due to electron viscosity [58].", "A decrease of the mean molecular weight with radius has a damping effect on mixing processes driven by rotation or even prevents these from occurring.", "The strength of this effect is regulated by the parameter f$_{\\mu }$ , for which we follow [41] and set f$_{\\mu }$ =0.05.", "The initial metallicity was set to $Z=0.02$ (Y=0.28), with initial abundances from [39].", "The lower metallicities were obtained by scaling both X and Y by the same factor such that $X+Y+Z=1$ .", "For the WD evolution and for $T_{\\rm {eff}}<10\\;000\\;{\\rm K}$ , the outer boundary conditions were derived using non-grey model atmospheres [94].", "To calculate the rate of change of orbital angular momentum, we took into account contributions from gravitational wave radiation, mass loss, magnetic braking, and spin orbit couplings: $\\dot{J}_{\\rm orb} = \\dot{J}_{\\rm gwr} + \\dot{J}_{\\rm ml} +\\dot{J}_{ \\rm mb} + \\dot{J}_{\\rm ls}\\,,$ as described in [89].", "The contribution of spin-orbit couplings to $\\dot{J}_{\\mathrm {orb}}$ was computed by demanding conservation of total angular momentum (except for losses due to gravitational wave radiation, magnetic braking, and mass loss), that is, changes in spin angular momentum were compensated for by changing the orbital angular momentum.", "The initial rotation velocity of the donor star was set by requiring that its spin period be synchronized with the initial orbital period.", "The time evolution of the angular velocity of the donor star is given by $\\frac{d\\Omega _{i}}{dt}=\\frac{\\Omega _{\\rm {orb}}-\\Omega _{i}}{\\tau _{\\rm {sync}}},$ where $\\Omega _{i}$ is the angular velocity of cell $i$ [28].", "The synchronization time, $\\tau _{\\rm sync}$ was calculated using the formalism of tidal effects from [51] and depends on whether the envelope is convective or radiative." ], [ "Grid of models", "To produce our grid of models, we followed the detailed binary evolution of the donor star from the ZAMS until it reached an age of 14 Gyr.", "The neutron star was treated as a point mass.", "The final outcome of these LMXB systems is very sensitive to the initial orbital period and to the treatment of orbital angular momentum loss [56].", "We calculated binary tracks for four metallicities: $Z=0.02$ , 0.01, 0.001, and 0.0002.", "For each metallicity, the models were divided into three categories: (i) basic models (with no diffusion nor rotation), (ii) diffusion models (with element diffusion) and, (iii) diffusion+rotation models (with element diffusion plus rotation).", "In both the diffusion and diffusion+rotation models, we included the effects of centrifugal forces and angular momentum transport, which means that these two models only differ by the presence of rotational mixing in the rotation models.", "For $Z=0.02$ , the initial binary configuration has a $1.4\\;M_{\\odot }$ donor star and a $1.2\\;M_{\\odot }$ neutron star accretor.", "For all the other metallicities, our models were calculated with a $1.0\\;M_{\\odot }$ donor star and a $1.4\\;M_{\\odot }$ neutron star (to facilitate direct comparison with previous work in the literature, see Sect.", "REF ).", "All the models were computed using a magnetic braking index of $\\gamma =4$ , and we assumed that 30 per cent of the transferred mass is ejected from the neutron star as a fast wind carrying its specific orbital angular momentum.", "We note that the structure of the ELM WDs is not sensitive to the above choices of mass-transfer parameters which only affect their final orbital periods.", "A comprehensive study of the influence of the magnetic braking index and the accretion efficiency on LMXB evolution can be found in [56].", "We point out again that we here refer to the mass of the proto-WD as being the (bloated) donor star mass at the end of the RLO mass-transfer phase (before the occurrence of flashes, which can lead to additional mass-transfer episodes), and the mass of the WD as being the mass at the beginning of the cooling track.", "We calculated models just above the bifurcation period, which is defined as the shortest initial orbital period that produces a WD [56].", "In the context of low-mass helium WDs, element diffusion was investigated in detail over the past few years by the La Plata group [1], [97], [4], [6], [5], [85], [3], [2] using the stellar evolution code LPCODE for various ranges of helium WD masses and metallicity.", "To the best of our knowledge, there is only one other study that used MESA for low-mass helium WDs [34].", "Our models include element diffusion from the ZAMS and not only from the proto-WD phase, as in the previous works.", "Moreover, for the first time, we investigate in detail the role played by rotational mixing in addition to element diffusion in the evolution of ELM WDs.", "Figure: Post-RLO evolution of surface gravity versus effective temperature for a ∼0.23M ⊙ \\sim \\!0.23\\;M_{\\odot } proto-WD produced from an LMXB donor star with Z=0.01Z=0.01.Three different models are shown: basic configuration (top panel), diffusion configuration (middle panel) and diffusion+rotation configuration (bottom panel).Note that the computations are stopped when the age of the model star reaches 14 Gyr (since ZAMS).Element diffusion has a strong effect on the surface composition of a proto-WD and on the chemical profile deep inside the star close to the helium core.", "At the surface, gravitational settling increases the hydrogen abundance given that hydrogen is the lightest element.", "Close to the helium core boundary, chemical diffusion tends to smooth it out by mixing the hydrogen downwards into hotter layers because a large hydrogen abundance gradient exists.", "It has been shown that this hydrogen tail promotes the occurrence of hydrogen shell flashes [4].", "Moreover, when element diffusion is included, a proto-WD experiences more flashes than when diffusion is neglected [4].", "The WD mass interval in which they occur is also changed compared to the case when element diffusion is ignored.", "The number of flashes and other information for all the models studied in this work are given in Appendix .", "Figure: Evolution of hydrogen surface abundance (top panel) and log gg (bottom panel) for the three proto-WDs shown in Fig.", "illustrating the effect of gravitational settling, rotational mixing and the mixing due to convection zones developed during the hydrogen shell flashes on the surface composition of these objects.Figure REF shows the evolution of surface gravity versus effective temperature for a proto-WD of $\\sim \\!0.23\\;M_{\\odot }$ obtained from the following three model configurations: basic, diffusion, and diffusion+rotation.", "The basic model experiences three hydrogen shell flashes, while the diffusion and the diffusion+rotation models experience one additional flash.", "The radial expansion following the CNO burning is more pronounced when element diffusion is included, in some cases leading to additional episodes of RLO.", "In general, the models with diffusion and diffusion+rotation behave in a very similar way.", "Figure: Kippenhahn diagrams showing the proto-WD phase for the same systems as inFigure .", "The plots show cross sections of the outer ∼\\sim 0.01M ⊙ 0.01\\;M_{\\odot } envelope of the proto-WD in mass coordinates, along the yy-axis, as a function of stellar age on the xx-axis (relative to the ZAMS age).", "The green areas denote zones with convection; the dotted white lines define lines of constant hydrogen abundance, 10 -2 ^{-2} to 10 -5 ^{-5} (from top to bottom).The intensity of the blue and red colour indicates the hydrogen abundance by mass fraction, as shown on the colour scale to the right.", "As a result of different input physics, the proto-WDs have slightly different masses and ages.", "See text for details.Figure REF shows the evolution of hydrogen surface abundance (top panel) and log $g$ (bottom panel) for the same (proto)WDs as in Fig.", "REF .", "As already mentioned, gravitational settling changes the surface abundances.", "All the elements heavier than hydrogen sink below the surface, leaving a pure hydrogen envelope behind.", "When rotational mixing is included, gravitational settling and rotational mixing compete with each other to determine the chemical composition of the surface.", "At the beginning of the proto-WD phase, rotational mixing dominates.", "However, the surface gravity of the proto-WD increases with time, while the efficiency of rotational mixing decreases, as described in Sect.", "REF .", "Thus, in later phases of the evolution, the gravitational settling overcomes the mixing induced by rotation.", "By the beginning of the last flash, the surface structure of the model that only includes element diffusion is nearly identical to the structure of the model that includes both element diffusion and rotational mixing.", "Helium in the envelopes of ELM proto-WDs is a crucial ingredient for exciting pulsation modes through the $\\kappa -$ mechanism, as shown by [59] and [24] for radial and nonradial modes.", "[35] recently provided the first empirical evidence that pulsations in ELM proto-WDs can only occur when a significant amount of helium is present in their atmospheres.", "In contrast with evolutionary models that only include element diffusion, our new evolutionary models including rotational mixing produce proto-WDs that have mixed He/H envelopes during most of their evolution before settling on the cooling track.", "Another effect of element diffusion, resulting mainly from the competition between chemical and thermal diffusion, is the development of a hydrogen tail that reaches down into the hot helium-rich layers, as shown in Fig.", "REF .", "This effect is responsible for the larger number of flashes compared to the case where element diffusion is ignored (basic model).", "Rotational mixing is seen not to change the chemical structure of the deep layers.", "This can also be concluded from the very similar behaviour in terms of the number of flashes and the structure of the flashes in the case that includes both diffusion and rotation compared to the case that only includes element diffusion, cf.", "Figs.", "REF and REF .", "Figure: Hydrogen burning luminosity versus hydrogen envelope mass, for the same systems as in Fig. .", "The grey stars represent the moment of Roche-lobe detachmentwhile the grey squares denote the point of maximum effective surface temperature.", "The evolution is from the right to the left.In Fig.", "REF we plot the luminosity produced by hydrogen burning versus the hydrogen envelope mass.", "For all three models, around 70 per cent of the hydrogen remaining from the end of the LMXB phase (Roche-lobe detachment) is processed before the occurrence of flashes while the bloated proto-WD crosses the HR–diagram.", "The occurrence of additional flashes, which applies to the cases where element diffusion is included, reduces the hydrogen envelope mass available on the cooling track (i.e.", "after reaching maximum $T_{\\rm eff}$ , marked by squares in Fig.", "REF ) by a factor of $\\sim \\!3$ compared with the basic model.", "The basic model still experiences significant residual hydrogen burning on the cooling track.", "As a result, the basic model only cools down to a temperature of $T_{\\rm eff}\\approx 8400\\;K$ within 14 Gyr (since the ZAMS), while the two models that include diffusion will cool down to roughly $T_{\\mathrm {eff}}\\approx 4000\\;K$ (cf.", "Fig.", "REF ).", "The cooling properties of the ELM WDs are discussed in more detail in Sect.", "REF .", "Figure: Orbital period evolution for the same models as in Fig. .", "The grey circles represent the onset of the mass transfer, the grey stars represent Roche-lobe detachment and the grey squares mark the maximum T eff T_{\\mathrm {eff}}.Figure: Kippenhahn diagram for the same proto-WD as in Fig.", ", including both element diffusion and rotational mixing.", "The intensity of the orange and indigo colour indicates the ratio of the spin angular velocity to the orbital angular velocity, Ω/Ω orb \\Omega /\\Omega _{\\mathrm {orb}}, as shown on the colour scale to the right.", "The green areas and the dotted white lines have the same meaning as in Fig. .", "The black arrows point to the position of the profiles in Fig.", ".The orbital evolution of the models described above is shown in Fig.", "REF .", "One difference between the three models is that those with element diffusion (and rotation) require a longer initial orbital period to form approximately the same proto-WD.", "Figure: Angular momentum diffusion coefficient, ν\\nu as function of mass coordinate from the centre (left) to the surface of the star (right) for the same proto-WD as in Fig. .", "Processes included are Spruit-Tayler dynamo (st), Eddington-Sweet circulation (es), and electron viscosity (visc).", "Each panel (from top to bottom) corresponds to a profile as marked by the black arrows in Fig.", "(from left to right).", "The grey dotted lines define lines of constant hydrogen abundance, from 10 -5 ^{-5} to 10 -2 ^{-2}, while the black dashed line represents the hydrogen mass abundance.The orbital period at the Roche-lobe detachment is $\\sim \\!7.05\\;{\\rm days}$ for the model that includes diffusion and rotation, $\\sim \\!6.73\\;{\\rm days}$ for the model with diffusion only, and $\\sim \\!5.19\\;{\\rm days}$ for the basic model.", "As the diffusion-induced flashes are stronger and because almost every flash causes the star to expand and fill its Roche lobe again, the mass-transfer episodes widen the orbit during each flash.", "In the end, this effect accounts for an increase of a few per cent in the orbital period.", "As shown in Fig.", "REF , the tidal coupling is strong enough to completely synchronize the donor with the orbit up until the end of the LMXB phase.", "This changes dramatically after detachment from the Roche lobe; while the helium core barely contracts and spins up only slightly above the orbital frequency, the extended hydrogen envelope spins up significantly during contraction, resulting in strong shear at the core-envelope boundary.", "As hydrogen flashes develop and the star expands and then contracts, the envelope successively spins down and up, with the rotational period at the surface of the proto-WD at its maximum being up to 20 times shorter than the orbital period.", "Even though the proto-WD expands back and fills its Roche lobe during flashes, these phases are very short.", "The convective layers developed during the flashes disappear well before the next phase of Roche–lobe overflow such that tidal synchronization past the LMXB phase is negligible.", "Although a strong shear is developed, the composition gradients that help stabilize the instabilities driven by rotation prevent the mixing of elements and angular momentum from the envelope to the core.", "This is shown in Fig.", "REF , where the different processes contributing to the angular momentum diffusion coefficient, $\\nu $ are shown at four different times.", "As depicted in the first panel, there is a very steep H-gradient immediately after a flash (and also after detachment from the LMXB phase) that completely prevents mixing to the core.", "As burning proceeds between flashes (second and third panels in Fig.", "REF ), the H-gradient is softened and starts to move outwards in mass, which allows some angular momentum to be transported to the core mainly through magnetic torques from the Spruit-Tayler dynamo.", "Finally, after settling on the cooling track (final panels panel in Figs.", "REF and REF ), most of the remaining hydrogen has been burnt, and angular momentum has mixed efficiently between the envelope and the core.", "Despite the small mass of the envelope relative to the total WD mass, this results in the WD having a spin period more than four times shorter (i.e.", "faster) than its orbital period.", "Because we have assumed that magnetic torques do not contribute to the mixing of elements, rotational mixing in our models barely affects the formation of the hydrogen tail due to element diffusion, and thus has a weak effect on the strength and the occurrence of flashes.", "At the surface, however, the fast rotation induces element mixing through Eddington-Sweet circulation (see Fig.", "REF ), which counteracts the rapid settling of elements heavier than hydrogen.", "Figure: HR-diagram showing the formation and evolution of a ∼0.28M ⊙ \\sim \\!0.28\\;M_{\\odot } WD from progenitors with different metallicities.", "The grey symbols represent the beginning of RLO (circles), the end of RLO (stars) and the maximum T eff T_{\\mathrm {eff}} (squares).", "The grey dashed lines represent lines of constant radius." ], [ "Effect of metallicity ", "For a given stellar mass, decreasing the metallicity produces a decrease in the radiative opacity which has an impact on the stellar evolution.", "The ZAMS and the RGB-phase are shifted towards the blue region of the HR-diagram, with the luminosity and effective temperature being higher during these phases than for models at solar metallicity.", "In other words, at low metallicity stars tend to be hotter, have smaller radii, and evolve more quickly than their high-metallicity counterparts.", "An immediate consequence of lower metallicities is therefore a higher mass of the helium WD formed through the LMXB phase at a given initial orbital period because the Roche lobe is filled at a more advanced stage in the evolution as a result of the smaller radius.", "As previously demonstrated by [98] and [84], the threshold mass for the occurrence of hydrogen shell flashes increases with lower metallicity.", "This is confirmed by our calculations, as shown in Table REF .", "For example, for the basic models (without diffusion and rotation), the minimum mass above which the flashes occur is $\\sim \\!0.21\\;M_{\\odot }$ for $Z=0.02$ , $\\sim \\!0.22\\;M_{\\odot }$ for $Z=0.01$ , $\\sim \\!0.25\\;M_{\\odot }$ for $Z=0.001$ , and $\\sim \\!0.28\\;M_{\\odot }$ for $Z=0.0002$ .", "Models with element diffusion have lower threshold values for flashes to occur, with the following dependence on metallicity: all the models studied for $Z=0.02$ experience flashes, the lowest mass proto-WD produced being $0.167\\;M_{\\odot }$ .", "For $Z=0.01$ flashes occur above $0.169\\;M_{\\odot }$ , for $Z=0.001$ the limit is $\\sim \\!0.22\\;M_{\\odot }$ , while for $Z=0.0002$ the lower threshold value is $\\sim \\!0.26\\;M_{\\odot }$ .", "When rotational mixing is included, all the threshold values are slightly higher than only diffusion is included, cf.", "Table REF .", "The upper limit for the occurrence of flashes is not as well constrained because fewer models are calculated models in this mass range, given that the focus of this work is towards the lowest masses of helium WDs, which are the ELM WDs.", "The obtained limits for hydrogen shell flashes agree well with those found in the literature for low metallicity [97], but at solar metallicity we obtain somewhat lower values than [2].", "Table: Proto-WD mass ranges for hydrogen shell flashes.", "For a given model category and a given metallicity, the threshold mass for flashes also depends on the initial mass of the donor star.The change in metallicity not only affects the threshold for flashes, but also the extent of the loops in the HR diagram.", "In Fig.", "REF the formation and evolution of a proto-WD with a mass of $\\sim 0.28\\;M_{\\odot }$ is shown in the HR-diagram for all the investigated metallicities.", "The lower the metal content, the weaker the CNO burning, and thus the loops during the CNO flashes are markedly less extended than in models with higher metallicity.", "Moreover, the number of flashes increases with decreasing metallicity: while the models for $Z=0.02$ and $Z=0.01$ experience just one hydrogen shell flash, the model at $Z=0.001$ goes through two flashes, and at $Z=0.0002$ the star experiences three hydrogen shell flashes.", "The interval of masses for which flashes occur is also affected by metallicity.", "For $Z=0.02$ and $Z=0.01$ , a $0.28\\;M_{\\odot }$ helium WD is close to the upper mass limit where hydrogen shell flashes occur, while for $Z=0.001$ and $Z=0.0002$ , a $0.28\\;M_{\\odot }$ helium WD is located close to the lower mass limit of the hydrogen shell flash interval.", "We stress that the number of flashes decreases with increasing mass of the WD and varies between 0 and 7 flashes for our computed models (see Appendix )." ], [ "Inheritance of proto-WDs: the hydrogen envelope mass", "Figure REF shows the hydrogen envelope mass at the end of the mass-transfer phase (Roche-lobe detachment), $M_{\\rm H, det}$ , as a function of the proto-WD mass for all the computed models.", "For a given metallicity, the models with diffusion and with diffusion+rotation have very similar values of $M_{\\rm H, det}$ as the basic models.", "The general trend is that the lower the mass of the proto-WD, the higher $M_{\\rm H, det}$ .", "The features in $M_{\\rm H, det}$ are given by the evolutionary history of the progenitor (donor) star and depend on the point in its evolution at which mass transfer is initiated.", "We note a jump in the hydrogen envelope mass at $\\sim \\!0.21\\;M_{\\odot }$ , $\\sim \\!0.23\\;M_{\\odot }$ , $\\sim \\!0.29\\;M_{\\odot }$ , and $\\sim \\!0.34\\;M_{\\odot }$ for $Z=0.02$ , $Z=0.01$ , $Z=0.001$ , and $Z=0.0002$ , respectively.", "This can be understood as discussed below.", "The shell hydrogen burning produces a convective envelope.", "When the convective envelope reaches its deepest extent, a hydrogen abundance gradient is produced between the region of the star mixed by the convective envelope and the layers below (which are rich in helium).", "When the hydrogen burning shell passes through this chemical discontinuity, the hydrogen burning rate drops, the radius contracts on a Kelvin-Helmholtz timescale and, as a result, the mass transfer will cease.", "The same phenomenon is responsible for the occurrence of the luminosity bump in red-giant stars [105], [20], first discussed in the context of temporary Roche-lobe detachment in LMXBs in [104].", "The interruption of the mass transfer can be a temporary effect if the envelope is massive enough, such that when the burning shell has passed through the discontinuity, the star still has enough material to burn and can therefore resume its mass transfer.", "If its envelope has been stripped to a greater extent, then the donor star is unable to resume mass transfer and a proto-WD is formed.", "This discontinuity in $M_{\\rm H, det}$ , observed at all the metallicities studied, distinguishes the systems that undergo this type of temporary detachment (the systems on the right-hand or upper side of the discontinuity) from the systems in which the hydrogen shell burning passes through the hydrogen abundance discontinuity without being able to resume mass transfer afterwards (the systems on the left-hand or lower side of the discontinuity).", "This explains the increasing values of $M_{\\rm H, det}$ with $M_{\\mathrm {proto-WD}}$ just below the discontinuity.", "Figure: Hydrogen envelope mass at the end of the mass-transfer phase (Roche-lobe detachment) for the basic stars (purple circles), for the stars with diffusion only (orange stars) and for the stars with diffusion+rotation (blue squares) for Z=0.02Z=0.02 (top panel), Z=0.01Z=0.01 (second panel), Z=0.001Z=0.001 (third panel) and Z=0.0002Z=0.0002 (bottom panel) as a function of proto-WD mass.", "The grey shaded area denotes the stars that undergo a temporary Roche-lobe detachment (see text for details)." ], [ "$\\Delta t_{\\rm proto}$ : the contraction timescale for proto-WDs", "As has been discussed earlier in this work, after the end of the LMXB mass-transfer phase, a certain amount of time, $\\Delta t_{\\rm proto}$ , is required by the newly formed object, the proto-WD, to contract and reach its cooling track.", "This timescale, from Roche-lobe detachment to the beginning of the cooling track (defined as when $T_{\\rm eff}$ reaches its maximum value), depends on the mass of the proto-WD and can reach up to 2 Gyr for the lowest mass proto-WDs [57] down to $10-100\\;{\\rm Myr}$ for the highest mass helium WDs.", "More importantly, [57] have shown that $\\Delta t_{\\rm proto}$ is not influenced by the occurrence of hydrogen shell flashes, and consequently, the suggested dichotomy in WD cooling times produced by hydrogen flashes was called into question.", "The determination of $\\Delta t_{\\rm proto}$ is important, especially for determining the age of MSPs with helium WD companions independently of the spin-down of the MSP [108], [9], [10].", "During this phase, the proto-WD appears to be bloated, meaning that its radius is significantly larger than the radius of a cold WD of similar mass.", "As the timescale for this contraction phase ($\\Delta t_{\\rm proto}$ ) is predicted to be relatively long, a number of ELM WDs should be observed in this bloated stage.", "One example suggested by [57] is PSR J1816+4510, based on observations by [64], [63].", "Figure REF shows $\\Delta t_{\\rm proto}$ for all our computed models.", "One feature is the occurrence of clustering in the data that groups the proto-WDs that undergo the same number of flashes (see Appendix ).", "As discussed before, the models with diffusion only and diffusion+rotation behave in a very similar way.", "In general, $\\Delta t_{\\rm proto}$ is larger than in the basic models when diffusion is included because of the additional flashes.", "For $Z=0.02$ and $Z=0.01$ there is a smooth transition of $\\Delta t_{\\rm proto}$ around the limit of the occurrence of flashes (models that experienced hydrogen shell flashes are plotted with open symbols).", "We recall that for these high metallicities all the models with diffusion only and diffusion+rotation (except for the model with the lowest mass at $Z=0.01$ ) undergo unstable burning through CNO hydrogen shell flashes.", "However, for $Z=0.001$ and $Z=0.0002$ we note a slight increase in $\\Delta t_{\\rm proto}$ around the lowest threshold for flashes for all models where diffusion is included.", "The maximum value of $T_{\\rm eff}$ reached during the proto-WD phase, however, is very sensitive to both the time and the spatial resolution with which the stellar structure is computed.", "With this in mind, and taking into account that $\\Delta t_{\\rm proto}$ is relatively small around the lowest threshold for flashes at these low metallicities the results shown in Fig.", "REF do not present evidence for a dichotomy in $\\Delta t_{\\rm proto}$ that is due to hydrogen flashes.", "However, for the long-term evolution on the WD cooling track the situation is different, as we discuss below.", "Figure: Hydrogen envelope mass at the beginning of the cooling track (maximum T eff T_{\\rm eff}) for the basic models (purple circles), for the models with diffusion only (orange stars) and for the models with diffusion+rotation (blue squares) as a function of proto-WD mass.", "See Fig.", "for further explanations.Figure: Timescale t cool ,L -2 t_{\\mathrm {cool, L_{-2}}} (see text) for the basic models (purple circles), for the models with diffusion only (orange stars), and for the models with diffusion+rotation (blue squares) as a function of proto-WD mass.", "See Fig.", "for further explanations." ], [ "Dichotomy on ELM WD cooling tracks", "The hydrogen envelope mass is an important parameter that determines the long-term cooling timescale for WDs.", "Following the work of [57], we consider the beginning of the cooling track as the moment at which the proto-WD reaches its maximum value of $T_{\\mathrm {eff}}$ .", "Figure REF shows the remaining hydrogen envelope mass when the proto-WD reaches the maximum $T_{\\rm eff}$ ($M_{\\mathrm {H,T_{\\mathrm {eff, max}}}}$ ) and finally settles on the cooling track, as a function of the mass of the proto-WD.", "Again, the large scatter is related to the number of flashes (between $0-7$ ) that the proto-WD experiences.", "At all metallicities we note a jump in $M_{\\mathrm {H,T_{\\mathrm {eff, max}}}}$ that occurs at the lowest threshold for flashes.", "At low metallicities, the effect is more pronounced in models with diffusion and diffusion+rotation, whereas at Z=0.02 the discontinuity is only seen in the basic models (all the systems for which element diffusion is considered experience hydrogen flashes).", "When element diffusion is included, the hydrogen envelope mass at the beginning of the cooling track is typically twice as small as when element diffusion is neglected (basic models), except for models at low metallicities and masses below the flash threshold.", "This significantly affects the cooling times of these objects.", "For example, consider an ELM WD with a mass of $\\sim 0.20\\;M_\\odot $ .", "Figure REF shows that flashes at high metallicities ($Z=0.02$ and $Z=0.01$ ) cause the amount of remaining hydrogen envelope mass at $T_{\\rm eff}^{\\rm max}$ ($M_{\\mathrm {H,T_{\\mathrm {eff, max}}}}\\!\\sim \\!1.0\\!\\times 10^{-3}\\;M_{\\odot }$ ) to be up to five times smaller than for the cases with lower metallicities ($Z=0.001$ and $Z=0.0002$ ) where flashes do not develop and a thick $\\sim \\!5.0\\times 10^{-3}\\;M_{\\odot }$ residual hydrogen envelope remains at the onset of the cooling track.", "Hence, it is clear that we do see a dichotomy in the long-term cooling ages of ELM WDs, such that those proto-WDs that experience flashes will have thin hydrogen envelopes and therefore shorter cooling timescales, whereas proto-WDs that avoid flashes will have relatively thick hydrogen envelopes and cool on a much longer timescale.", "It is important to stress that the threshold mass at which this transition occurs is dependent on metallicity.", "We now analyse the dichotomy in long-term cooling in more detail.", "Figure REF shows the time from the end of the LMXB mass transfer until the WD luminosity reaches $\\log (L/L_{\\odot })=-2.0$ , $t_{\\mathrm {cool, L_{-2}}}$ (including $\\Delta t_{\\rm proto}$ ).", "Some of our computed models are not plotted because they reached an age of 14 Gyr before $\\log (L/L_{\\odot })=-2.0$ .", "For the basic models, independent of metallicity, there is a small difference in cooling times between the WDs that experience flashes and those for which the hydrogen burning in the shell is stable.", "However, for the WDs computed with diffusion and diffusion+rotation, the difference in cooling times between systems with and without flashes can be as large as 3 Gyr, see Fig.REF .", "We conclude that when element diffusion is included, the occurrence of hydrogen shell flashes does indeed produce a (metallicity-dependent) dichotomy in the cooling times of helium WDs.", "Only when element diffusion is neglected in the modelling there is no or only a very small difference between the WDs that experience unstable hydrogen burning compared to those for which the residual hydrogen burning is stable, independent of metallicity.", "This can explain the findings of [57], who evolved their stellar models without element diffusion and thus questioned the dichotomy idea.", "Figure: Different cooling curves illustrating the differences between basic models (without diffusion and rotation) and models that include element diffusion and rotation.", "All models have a metallicity of Z=0.001Z=0.001.", "The value of M WD /M ⊙ M_{\\rm WD}/M_{\\odot } is shown for each track, and the difference in masses can partly explain the different cooling rates.", "However, more important in this respect is the occurrence of hydrogen shell flashes (blue and green curves), which accelerates the cooling and creates a dichotomy in WD cooling ages (see text).Figure: Evolution of log( Ca /H)\\log {\\rm (Ca/H)} at the stellar surface (top panel) and log gg (bottom panel) for a 0.185M ⊙ 0.185\\;M_{\\odot } proto-WD computed with diffusion only (orange) and diffusion+rotation (blue) for Z=0.02.", "The starting point (t=0t=0) is defined at the moment of Roche-lobe detachment." ], [ "Rotational mixing: source of surface metals?", "In the past few years, metals, especially calcium, were detected in the spectra of ELM WDs with a surface gravity lower than $\\sim $ 5.9.", "Metals sink below the atmosphere on a timescale much shorter than the evolutionary timescale of the proto-WD, which means that another process is required to either counteract the gravitational settling or replenish the depleted metals.", "There are several possible processes that can be responsible for the observed surface composition of ELM WDs.", "For a detailed discussion, we refer to [36] and [45].", "For higher mass (carbon-oxygen) WDs, the presence of metals in their atmosphere is explained by accretion from circumstellar debris discs formed by tidal disruption of planetary bodies [27], [62], which are detectable through excess flux in the IR [30], [72].", "This scenario seems unlikely for ELM WDs given that their compact orbits make the existence of a debris disk dynamically difficult to explain.", "[63] suggested that the observed metals are brought to the surface by the pulse-driven convection developed during a hydrogen shell flash.", "However, shortly after the convection zone vanishes, the metals will sink below the stellar surface as a result of gravitational settling.", "A mechanism that can counteract diffusion would be radiative levitation.", "However, as [44] showed, radiative levitation alone cannot explain the observed abundances, especially in the case of calcium.", "They suggested that in addition to radiative levitation, another support mechanism such as rotational mixing is likely required to explain the observed pattern in the metal abundances of ELM WDs.", "Here, we discuss the effect of rotational mixing in determining the surface composition of ELM WDs.", "Figure: Comparison of evolutionary tracks of and this work (including diffusion only) for a ∼0.18M ⊙ \\sim \\!0.18\\;M_{\\odot } WD (a) and ∼0.27M ⊙ \\sim \\!0.27\\;M_{\\odot } WD (b).The post-RLO evolution of luminosity, given by CNO burning, is plotted as a function of the hydrogen envelope mass.", "Time goes from right to left.", "The inset shows an artefact of hydrogen production during the shell flashes in the models, see text.Figure: Evolutionary tracks in the (T eff ,loggT_{\\rm eff},\\log g)–diagram for the same models as in Fig. .", "The circles, stars, and squares indicate cooling ages of 1, 4, and 8 Gyr, respectively.", "For the 0.18 M ⊙ _{\\odot } WD, the 1 Gyr symbols for the two models almost coincide.", "Moreover, we note that the models from this work are only calculated to an age of 14 Gyr, which prevents cooling ages exceeding 3 Gyr for the 0.18 M ⊙ _{\\odot } model and 6 Gyr for the 0.27 M ⊙ _{\\odot } model.Figure REF shows the evolution of $\\log {\\rm (Ca/H)}$ at the stellar surface from the beginning of the proto-WD phase (Roche-lobe detachment) to several hundred Myr onto the cooling track.", "In the model that only includes diffusion calcium sinks much faster beneath the surface than the proto-WD evolutionary timescale after the mass transfer ends.", "It is brought back to the surface through to the pulse-driven convection zone that is developed by the occurrence of a hydrogen shell flash, only to quickly sink again as a result of the gravitational settling.", "In the diffusion+rotation model, rotational mixing at the surface acts against the gravitational settling (see Sect.", "REF ).", "As the proto-WD advances towards higher surface gravity, rotational mixing becomes less efficient than gravitational settling.", "During the proto-WD phase, the star may experience several episodes of radial expansion followed by contraction and may also develop zones of convection through the hydrogen flashes.", "This interplay between convection, expansion (low surface gravity), contraction (high surface gravity), and rotational mixing explains the pattern shown in Fig.", "REF .", "When the proto-WD enters the cooling track, the surface gravity steadily increases and gravitational settling finally overcomes the mixing that is due to rotation.", "As a long-term result, the metals will sink below the surface, leaving behind a pure hydrogen envelope.", "We plott in Fig.", "REF all the models with $Z=0.02$ computed with diffusion only (left panel) and diffusion+rotation (right panel).", "The points are spaced at intervals of 0.5 Myr and colour-coded according to the value of $\\log {\\rm (Ca/H)}$ .", "Over-plotted are the data points from [36].", "The left panel clearly shows that the flash scenario discussed above cannot explain the observations.", "On the other hand, when rotational mixing is included, we can qualitatively explain the presence of calcium in the spectra of proto-WDs as a natural result of their evolution.", "We recall that the observational data most likely belong to populations with different metallicities, while in Fig.", "REF we only plot our models with $Z=0.02$ .", "The lack of observations of proto-WDs at high T$_{\\mathrm {eff}}$ arises because the detection limit of Ca lines depends on the effective temperature (see Fig.", "9 in [36])." ], [ "Comparison with previous work", "As discussed in [38] and [12], the models of [57] (from here on I14) and [2](A13) show a relatively large discrepancy in their cooling ages.", "Although the initial binary parameters, and to some extent the metallicities, are different in the two sets of models, the main difference is that the models of A13 include element diffusion, which has an important role in reducing the hydrogen envelope mass through flashes (cf.", "Sect.", "REF ), and thus consequently leads to accelerated cooling and therefore younger cooling ages than in I14 models.", "Here, we compare our new models including element diffusion (but without rotational mixing to enable comparison) (I16) with the A13 models.", "We also adopted the same initial binary parameters, namely a $1.0\\;M_{\\odot }$ donor star and a $1.4\\;M_{\\odot }$ neutron star accretor, and also applied the same metallicity of $Z=0.01$ .", "A13 found that the lower mass limit for which hydrogen shell flashes occur is somewhere in the interval $0.176-0.182\\;M_{\\odot }$ (i.e.", "between the last model with stable shell burning and the first model that experiences flashes), while we find a lower mass limit of $0.165-0.169\\;M_{\\odot }$ .", "Figure REF compares the I16 models with those of A13 by showing the evolution of luminosity produced by CNO burning as a function of hydrogen envelope mass for a $\\sim \\!0.18\\;M_{\\odot }$ and a $\\sim \\!0.27\\;M_{\\odot }$ (proto) helium WD.", "One important difference is the hydrogen envelope mass left at Roche-lobe detachment.", "For the $0.18\\;M_{\\odot }$ WD, the model of I16 initially has a more massive hydrogen envelope, while for the $0.27\\;M_{\\odot }$ WD it is the opposite.", "We mention again that in our models diffusion acts from the ZAMS, in contrast with the A13 models, where diffusion is turned on during the proto-WD evolution.", "Figure REF also shows that the A13 models contain a numerical artefact by which hydrogen is created during CNO burning (see inset).", "In Fig.", "REF we present a ($T_{\\rm eff},\\log g$ )–diagram and compare the evolutionary tracks from Fig.", "REF .", "The main difference are additional mass-transfer episodes in the I16 models as a result of a few vigorous flashes.", "This effect will in the end leave a slightly lower WD mass at the beginning of the cooling track.", "These differences in proto-WD evolution, combined with the artificial creation of hydrogen in the A13 models, result in slight differences on the cooling track.", "However, the difference between the A13 models and the I16 models are significantly smaller than when comparing the A13 models with the I14 models [38], [12]." ], [ "Relation of mass to orbital period in WDs", "When low-mass stars ($< 2.3\\;M_{\\odot }$ ) reach the red-giant branch, the radius of the star mainly depends on the mass of the degenerate helium core and is almost entirely independent of the mass of the envelope [92], [110].", "For the formation of binary MSPs, this relation proves to be very important because it provides a correlation between the mass of the newly formed WD and $P_{\\mathrm {orb}}$ following the mass transfer episode [96], [61], [91], [104], [84], [26], [75], [99], [60], [56].", "Figure REF shows the ($M_{\\rm WD},P_{\\rm orb}$ )-relation for all the models computed in this work.", "Our results are in fine agreement with [104] for systems with $P_{\\rm orb}>1-2\\;{\\rm days}$ .", "For close-orbit systems with $P_{\\rm orb}<1\\;{\\rm day}$ our results agree well with [75] and [56].", "We note a slight discontinuity in the relation, which is dependent on metallicity as discussed in Sect.", "REF .", "This weak break in the ($M_{\\rm WD},P_{\\rm orb}$ )-relation was previously reported by other authors [84], [60].", "Figure: (M WD ,P orb M_{\\rm WD},P_{\\rm orb})-relation for Z=0.02Z=0.02 (top panel), Z=0.01Z=0.01 (second panel), Z=0.001Z=0.001 (third panel), and Z=0.0002Z=0.0002 (bottom panel).", "The grey shaded area denotes the LMXB systems that undergo a temporary detachment (see text).", "The purple circles represent the basic models, the orange stars the models with diffusion only and the blue squares the models with diffusion+rotation.", "Over-plotted are the theoretical relations by and for the respective metallicity." ], [ "Conclusions", "We computed a grid of models for ELM WDs with different metallicities.", "For each metallicity, we computed three types of models with different physics included: i) basic models (with no element diffusion nor rotation), ii) diffusion (including element diffusion), and iii) diffusion+rotation.", "For the first time, we took into account the combined effects of rotational mixing and element diffusion in the evolution of WD progenitors and during the proto-WD phase and WD cooling.", "The main results obtained are summarized as follows: We confirm that element diffusion plays a significant role in the evolution of proto-WDs that experience hydrogen shell flashes.", "We also confirm that the unstable burning is triggered by the diffusive hydrogen tail reaching the hot deep layers inside the star [4].", "The formation of the hydrogen tail is a cyclic process and depends on the available hydrogen in the envelope.", "Consequently, the number of flashes experienced by a proto-WD of a given mass is increased, leading to reduced hydrogen envelope mass and subsequent accelerated cooling, compared with the models without element diffusion.", "Rotational mixing counteracts the effect of gravitational settling in the surface layers of young bloated ELM proto-WDs, but its efficiency is reduced towards the end of the proto-WD phase, when the star contracts and its surface gravity increases.", "As a consequence, our new evolutionary models including rotational mixing predict that ELM proto-WDs have mixed H/He envelopes during a significant part of their evolution before settling on the cooling track, in accordance with recent observational evidence from pulsations in ELM proto-WDs [35].", "Except for this, the general properties, such as the number of flashes, are not strongly influenced by the presence of rotational mixing.", "Although the hydrogen envelope left after detachment from the LMXB phase is a small fraction of the total WD mass, it has a very high angular momentum content compared to the core.", "Because the proto-WD contracts while this hydrogen is burnt and angular momentum mixes inwards following each flash, the resulting WD on the cooling track is spun up significantly to a rotation period well below the orbital period.", "The hydrogen envelope mass in newborn proto-WDs is influenced by the evolutionary stage of the donor star at the moment when LMXB mass transfer is initiated.", "In particular, we found that LMXB donor stars that experience a temporary contraction will produce proto-WDs with a significantly reduced hydrogen envelope mass at the moment of the final Roche-lobe detachment.", "In general, the shorter the orbital period at the onset of the LMXB phase, the lower the mass of the proto-WD and the higher the final envelope mass.", "The hydrogen envelope mass is also metallicity dependent, such that (for the same proto-WD mass) the lower the metallicity, the higher the envelope mass.", "In general, our resulting mass range for the occurrence of flashes is similar to those found in the literature.", "For $Z=0.02$ , all our models with diffusion experience flashes, and for $Z=0.01$ we obtain a lower limit of $\\sim \\!0.16\\;M_{\\odot }$ , compared to $\\sim \\!0.18\\;M_{\\odot }$ found by [2].", "We identified two timescales relevant for understanding the evolution of (proto) WDs.", "The evolutionary timescale of the contraction of proto-WDs, $\\Delta t_{\\rm proto}$ , can reach up to 2.5 Gyr for the lowest proto-WD masses, while for the higher WD masses it is $<100\\;{\\rm Myr}$ .", "When element diffusion is included, this timescale is slightly increased by more numerous flashes than for the case when element diffusion is neglected.", "As concluded in [57], we did not find a dichotomy in the $\\Delta t_{\\rm proto}$ distribution with respect to the occurrence of flashes, but rather a smooth transition.", "However, we note a small increase in $\\Delta t_{\\rm proto}$ close to the flash limit for $Z=0.001$ and $Z=0.002$ .", "The cooling timescale $t_{\\rm cool,L_{-2}}$ describes the evolution of the WD from LMXB detachment until the the WD has evolved well down the cooling track and reaches log ($L/L_{\\odot })=-2$ .", "This timescale is mostly determined by the hydrogen envelope mass that remains after the proto-WD phase.", "In the basic models (without element diffusion or rotation), independent of metallicity, there is a very small difference between the models that experience flashes and those that do not.", "However, we confirm, as first stated by [4], that the situation is different when element diffusion is included, which leads to a dichotomy in the cooling timescale of ELM WDs.", "We investigated whether the observed metal features in ELM proto-WDs might be linked to the internal evolution of these objects.", "In particular, we analysed the evolution of the surface abundance of calcium and concluded that rotational mixing is a key component for producing the observed pattern in the ($\\log g,T_{\\rm eff}$ )–diagram.", "The systematic investigation presented here for the effects of thermal and chemical diffusion, gravitational settling, and rotational mixing on a wide range of LMXBs with different initial masses, orbital periods, and metallicity, leads us to conclude that theoretical models must include these aspects when they are compared to observational data of ELM WDs.", "Hence, with these improved models at hand, the implications are better constraints on the true ages and masses of these WDs and therefore also on the ages and masses of their companion stars, such as millisecond radio pulsars.", "Moreover, the grid of models we presented might be further used for astroseismology calculations to determine the pulsational behaviour of ELM proto-WDs and ELM WDs." ], [ "Acknowledgements", "We thank the anonymous referee for helpful advice.", "A.G.I.", "thanks Alejandra Romero and Elvijs Matrozis for very useful discussions and BWin Technologies Ltd. for the computing cores on which the simulations in this work were performed." ] ]
1606.04947
[ [ "Collecting and Analyzing Data from Smart Device Users with Local\n Differential Privacy" ], [ "Abstract Organizations with a large user base, such as Samsung and Google, can potentially benefit from collecting and mining users' data.", "However, doing so raises privacy concerns, and risks accidental privacy breaches with serious consequences.", "Local differential privacy (LDP) techniques address this problem by only collecting randomized answers from each user, with guarantees of plausible deniability; meanwhile, the aggregator can still build accurate models and predictors by analyzing large amounts of such randomized data.", "So far, existing LDP solutions either have severely restricted functionality, or focus mainly on theoretical aspects such as asymptotical bounds rather than practical usability and performance.", "Motivated by this, we propose Harmony, a practical, accurate and efficient system for collecting and analyzing data from smart device users, while satisfying LDP.", "Harmony applies to multi-dimensional data containing both numerical and categorical attributes, and supports both basic statistics (e.g., mean and frequency estimates), and complex machine learning tasks (e.g., linear regression, logistic regression and SVM classification).", "Experiments using real data confirm Harmony's effectiveness." ], [ "Introduction", "Smart devices connected to the Internet, including mobile phones, wearables, home appliances, sensors and vehicles, have become a part of everyday life in many parts of the world.", "The data collected by these devices could be an invaluable asset to hardware designers and application developers.", "For instance, a smartphone maker with its own customized UI such as Samsung TouchWiz could learn about the usage patterns of the various UI features such as Multi Window and One-Handed Mode, and focus on improving the popular ones.", "However, privacy concerns remain a major hurdle in collecting users' data.", "For example, the user may not want others to know her web browsing history, apps installed and locations visited.", "Even if the user (reluctantly) allows trusted organizations to collect her data, the possession of large amounts of sensitive personal data poses a major security risk.", "Accidental leakage of such personal data, which happened to AOLhttps://en.wikipedia.org/wiki/AOL_search_data_leak, Netflixhttp://www.wired.com/2009/12/netflix-privacy-lawsuit/ and Ashley Madisonhttp://edition.cnn.com/2015/08/27/opinions/yang-ashley-madison-hack/, led to serious consequences and substantial damage.", "So, organizations with a large user base commonly face a dilemma: either they collect users' personal data and become exposed to the risk of privacy breaches, or they do not collect such data and lose the opportunity of mining them.", "Local differential privacy (LDP), which has been used in Google's Chrome browser [11], addresses the above dilemma.", "The idea is to compute aggregates of users' data without collecting individuals precise personal information.", "Unlike other models of differential privacy [8], [9], which publish randomized aggregates but still collect the exact sensitive data, LDP avoids collecting exact personal information in the first place, thus providing a stronger assurance to the users and to the aggregator.", "Meanwhile, LDP satisfies the strong and rigorous privacy guarantees of differential privacy, i.e., the adversary cannot infer sensitive information of an individual with high confidence, regardless of the adversary's background knowledge.", "Google's LDP solution in its Chrome browser, namely Rappor [11], has rather limited functionalities.", "The core of Rappor is a randomized response mechanism [20] for a user to answer a yes/no question to the aggregator.", "A classic example is to collect statistics about a sensitive group (e.g., communists in the US), in which the aggregator asks each individual: “Are you a communist?” To answer this question, each individual tosses a coin, gives the true answer if it is a head, and a random yes/or answer otherwise.", "Clearly, this randomized approach provides plausible deniability to the individuals.", "Meanwhile, it is shown to satisfy $\\epsilon $ -differential privacy, and the strength of privacy protection (i.e., $\\epsilon $ ) can be controlled by using an unfair coin [11].", "Based on the collected randomized answers, the aggregator estimates the percentage of users whose true answer is “yes” (resp.", "“no”).", "Besides simple counting, a follow-up paper [12] shows that Rappor can also compute other types of statistics such as joint-distribution estimation and association testing.", "However, three major limitations remain: first, Rappor cannot compute aggregates on numeric attributes, e.g., the average running time of an app.", "Second, the accuracy of Rappor deteriorates quickly with increasing number of attributes; as we show later in Section REF , to compute aggregates on $d$ independent attributes, the error of Rappor grows linearly with $d$ , which is sub-optimal [5], [7].", "Third, it is unclear whether/how Rappor can handle complex and yet commonly used machine learning tasks, such as logistic regression and SVM classification.", "LDP has also drawn considerable interest from the theory community.", "However, as we review in Section , their focus lies mostly in analyzing the asymptotical performance of basic building blocks of LDP, rather than practical systems and performance.", "In other words, the problem they study is “what is the best that LDP can do”, rather than “how to design a practical LDP solution”.", "Perhaps due to this reason, as we show in Section REF , the state-of-the-art solution (to our knowledge) for estimating mean values of multiple attributes under LDP is buggy and requires a fix.", "Meanwhile, we are not aware of any systematic approach that can perform common machine learning tasks under LDP on a large data domain containing both numeric and categorical attributes.", "Motivated by this, we have been building Harmony, an advanced data analytics tool that conforms to LDP requirements.", "Harmony supports a multitude of common data analysis tasks over an arbitrary number of numerical or categorical attributes.", "Further, Harmony achieves both non-trivial asymptotical error bounds and improved accuracy in practice, compared to the current state of the art.", "As a case study, we show how Harmony can improve diagnostic information reporting in Samsung smartphones with strong privacy guarantees, as follows.", "Samsung currently collects mobile phone usage information from users through a diagnostic tool bundled with the Samsung Android OSThe tool is available in the system under “Settings > About device > Report diagnostic info”..", "The information collected include the mobile phone's settings (e.g., display settings, whether or not the location functionality is turned on), memory and battery usage, as well as other log data.", "The tool transmits the data in an unperturbed format to Samsung, and the transmission explicitly requires users' consent.", "However, as the collected information could potentially reveal sensitive information, it is beneficial to enhance the tool with privacy protection mechanisms, so as to provide rigorous privacy assurance to users.", "Towards this end, local differential privacy is an attractive approach due to its strong privacy guarantee, and its practicability that has been demonstrated in Rappor [11].", "However, Rappor's LDP mechanism focuses on collecting a single categorical attribute, whereas the data collected by the Samsung diagnostic tool include both numeric ones (e.g., battery usage) and categorical ones (e.g., display settings).", "Furthermore, Rappor does not support complex learning tasks (e.g., regressions, SVM), whereas such tasks are important for Samsung to build analytical models from the data, e.g., building an early symptom model to predict system errors based on other measurements such as battery usage, memory usage, active applications, etc.", "This necessitates the development of new data collection technique based on local differential privacy.", "In the following, Section provides the necessary background on LDP.", "Sections presents the fundamental LDP mechanisms in Harmony.", "Section applies Harmony to common data analytics tasks based on empirical risk minimization, including linear regression, logistic regression and SVM classification.", "Section contains an extensive set of experiments.", "Section reviews related work.", "Finally, Section concludes with directions for future work." ], [ "Preliminaries", "In our setting, an aggregator (e.g., Samsung) collects data from a set of users (e.g., smart device owners), and computes statistical models of the collected data.", "The goal is to maximize the accuracy of these statistical models, while preserving the privacy of the users.", "Following the local differential privacy model [11], [2], [7], we assume that the aggregator already knows the identities (e.g., IP addresses) of the users, but not their private data.", "Formally, let $n$ be the total number of users, and $u_i$ ($1 \\le i \\le n$ ) denote the $i$ -th user.", "Each user $u_i$ 's private data is represented by a tuple $t_i$ , which contains $d$ attributes $A_1, A_2, \\ldots , A_d$ .", "These attributes can be either numerical or categorical.", "Without loss of generality, we assume that each numeric attribute has a domain $[-1, 1]$ , and each categorical attribute with $k$ distinct values has a discrete domain $\\lbrace 1, 2, \\ldots , k\\rbrace $ .", "To protect privacy, each user $u_i$ first perturbs her tuple $t_i$ using a randomized perturbation function $f$ .", "Then, she sends the perturbed data $f(t_i)$ to the aggregator instead of her true data record $t_i$ .", "The perturbation function determines the privacy / utility tradeoff.", "As an extreme case, if $f(t_i) = t_i$ (i.e., no perturbation), the aggregator obtains perfect utility since it computes the statistical models based on the exact data; however, the privacy of the users is completely lost as the aggregator receives their sensitive data.", "The other extreme is that $f$ simply outputs a random tuple regardless of $t_i$ , which leads to the highest level of privacy and zero utility, i.e., the aggregator learns nothing about the users' data.", "Given a privacy parameter $\\epsilon > 0$ that controls the privacy-utility tradeoff, we require that $f$ satisfies $\\epsilon $ -local differential privacy ($\\epsilon $ -LDP) [11], defined as follows: Definition 1 ($\\epsilon $ -local differential privacy) A randomized function $f$ satisfies $\\epsilon $ -local differential privacy if and only if for any two input tuples $t, t^\\prime \\in \\text{Dom(f)}$ and for any possible output $t^*$ of $f$ , we have: $\\Pr [f(t) = t^*] \\le e^\\epsilon \\times \\Pr [f(t^\\prime ) = t^*].$ Basically, local differential privacy is a special case of differential privacy [9] where the random perturbation is performed by the users, not by the aggregator.", "In other words, the aggregator never possesses the exact private data of any user.", "According to the above definition, the aggregator, who receives the perturbed tuple $t^*$ , cannot distinguish whether the true tuple is $t$ or another tuple $t^{\\prime }$ with high confidence (controlled by parameter $\\epsilon $ ), regardless of the background information of the aggregator.", "This provides plausible deniability to the user.", "Note that in $\\epsilon $ -LDP, since random perturbation is done at each user, it is possible to achieve personalized privacy protection by using different values of the privacy parameter $\\epsilon $ at different users, depending on their respective privacy requirements.", "In this paper, we assume a universal $\\epsilon $ for the ease of presentation and analysis.", "We aim to support the following types of analytics tasks under $\\epsilon $ -LDP: Mean value and frequency estimation.", "These are two basic types of statistics.", "For each numeric attribute $A_j$ , we aim to estimate the mean value of $A_j$ over all $n$ users, $\\frac{1}{n} \\sum _{i=1}^n {t_i[A_j]}$ .", "For each categorical attribute $A_j^{\\prime }$ , we aim to estimate the frequency of each possible value of $A_j^{\\prime }$ , i.e., a histogram.", "Note that it is also possible to build such a histogram for a numeric attribute with a finite number of possible values.", "Empirical Risk Minimization.", "These are advanced statistics commonly used in machine learning.", "Examples include linear regression, logistic regression, and support vector machines (SVM) [3].", "Unless otherwise specified, all expectations in this paper are taken over the random choices made by the algorithms considered.", "Remark.", "In practice, the tuple $t_i$ of a user may change overtime (e.g., the phone usage information of a user would change day by day); accordingly, the aggregator may want to re-collect information from users after a certain time period (e.g., one week).", "In that case, we aim to ensure that the collection of each individual snapshot of $t_i$ satisfies $\\epsilon $ -differential privacy.", "One may argue that it is more desirable to ensure that all collected snapshots jointly achieve $\\epsilon $ -differential privacy, but to our knowledge, this is an open problem when the number of snapshots to be collected can be arbitrarily large." ], [ "Estimating Means and Frequencies", "This section investigates the design of the perturbation function $f$ to support accurate estimation of mean values (resp.", "frequencies) of numeric (resp.", "categorical) attributes.", "For ease of exposition, Section REF considers the case when all attributes $A_1, A_2, \\ldots , A_d$ in the users' data have a numeric domain $[-1, 1]$ ; after that, Section REF extends our discussions to the case when both numeric and categorical attributes are present." ], [ "Estimating Mean Values for Numeric Attributes", "Given a tuple $t_i$ containing $d$ numeric attributes, a naive design of the perturbation function $f$ is to apply the Laplace Mechanism [8].", "In particular, let $t^*_i = f(t_i)$ be the perturbed tuple of user $u_i$ , we have: $\\forall j \\in [d], t^*_i[A_j] = t_i[A_j] + \\operatorname{Lap}\\left(\\frac{2d}{\\epsilon }\\right),$ where $\\operatorname{Lap}(\\lambda )$ denotes a random variable that follows a Laplace distribution of scale $\\lambda $ , with the following probability density function: $pdf(x) = \\frac{1}{2\\lambda } \\exp \\left(-\\frac{|x|}{\\lambda }\\right).$ Once the aggregator receives all perturbed tuples, it simply computes their average $\\frac{1}{n} \\sum _{i = 1}^n t^*_i[A_j]$ as an estimate of the mean of $A_j$ .", "Clearly, this estimate is unbiased, since the injected Laplace noise $\\operatorname{Lap}\\left(\\frac{2d}{\\epsilon }\\right)$ in each $t^*_i[A_j]$ has zero mean.", "Meanwhile, it is easy to calculate that the expected error incurred by this estimator is $O\\left(\\frac{d}{\\epsilon \\sqrt{n}}\\right)$ , which is linear to the number of attributes $d$ and, thus, could be excessively large when there are many attributes.", "Note that this is a fundamental problem that also exists in the traditional differential privacy setting, when publishing statistics for multiple independent attributes.", "Perhaps rather surprisingly, this problem has not received much attention in the differential privacy literature; the first and only solution we are aware of is proposed by Duchi et al.", "[7], [6] under the local differential privacy setting, presented below.", "Duchi et al.", "'s method.", "Algorithm REF shows the pseudo-code of Duchi et al.", "'s method.", "The authors claim that this method satisfies $\\epsilon $ -local differential privacy, yields unbiased estimates for the mean value of each attribute, and incurs $O\\big (\\sqrt{d \\log d} / (\\epsilon \\sqrt{n})\\big )$ expected error for each attribute, which is proven to be asymptotically optimal.", "As we explain later, all three claims are incorrect, i.e., their method can violate differential privacy, lead to a biased estimate and incur a much higher amount of error.", "It takes as input the exact tuple $t_i \\in [-1, 1]^d$ of user $u_i$ and a privacy parameter $\\epsilon $ , and outputs a perturbed vector $t^*_i \\in \\lbrace -B, B\\rbrace ^d$ , where $B$ is a constant decided by $d$ and $\\epsilon $ .", "Note that the output is binary for each attribute, i.e., it is either $B$ or $-B$ .", "Therefore, it suffices for each user to transmit only one bit for each attribute to the aggregator.", "Upon receiving the perturbed tuples, the aggregator simply computes the average value for each attribute over all users, and outputs these averages as the estimates of the mean values for their corresponding attributes.", "Next we focus on the calculation of $B$ , which is rather complicated.", "Essentially, $B$ is a scaling factor to ensure that the expected value of a perturbed attribute is the same as that of the exact attribute value.", "First, we calculate: $ C_d = {\\left\\lbrace \\begin{array}{ll}2^{d-1}, &\\text{if $d$ is odd}\\\\2^{d-1} - \\frac{1}{2} \\binom{d}{d/2}, &\\text{otherwise}\\end{array}\\right.", "}$ Then, $B$ is calculated by: $ B = {\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{2^d + C_d \\cdot (e^\\epsilon -1)}{ \\binom{d-1}{(d-1)/2} \\cdot (e^\\epsilon - 1)}, &\\text{if $d$ is odd} \\vspace{8.53581pt} \\\\\\displaystyle \\frac{2^d + C_d \\cdot (e^\\epsilon -1)}{ \\binom{d-1}{d/2} \\cdot (e^\\epsilon - 1)}, &\\text{otherwise}\\end{array}\\right.", "}$ Duchi et al.", "show that $\\frac{1}{n} \\sum _{i = 1}^n t^*_i[A_j]$ is an unbiased estimator of the mean of $A_j$ , and $ \\mathbb {E}\\left[\\max _{j \\in [d]} \\left|\\frac{1}{n} \\sum _{i = 1}^n t^*_i[A_j] - \\frac{1}{n} \\sum _{i = 1}^n t_i[A_j]\\right|\\right] = O\\left(\\frac{\\sqrt{d \\log d}}{\\epsilon \\sqrt{n}}\\right),$ which is asymptotically optimal [7].", "[t] Duchi et al.", "'s Method [7], [6] Inputinput Outputoutput tuple $t^*_i \\in \\lbrace -B, B\\rbrace ^d.$ Generate a random tuple $v \\in \\lbrace -1, 1\\rbrace ^d$ by sampling each $v[A_j]$ independently from the following distribution: $\\Pr [v[A_j] = x] = {\\left\\lbrace \\begin{array}{ll}\\frac{1}{2} + \\frac{1}{2} t_i[A_j], & \\text{if $x = 1$} \\vspace{5.69054pt} \\\\\\frac{1}{2} - \\frac{1}{2} t_i[A_j], & \\text{if $x = -1$}\\\\\\end{array}\\right.", "}$ Let $T^+$ (resp.", "$T^-$ ) be the set of all tuples $t^* \\in \\lbrace -B, B\\rbrace ^d$ such that $t^* \\cdot v > 0$ (resp.", "$t^* \\cdot v \\le 0$ ) Sample a Bernoulli variable $u$ that equals 1 with ${e^\\epsilon / (e^\\epsilon + 1)}$ probability $u = 1$ a tuple uniformly at random from $T^+$ a tuple uniformly at random from $T^-$ Problems in Duchi et al.", "'s method and a possible fix.", "We implemented and evaluated Duchi et al.", "'s method, but found that whenever the number $d$ of attribute is even, the method yields a biased estimation of the mean of each attribute and incurs significant error.", "Then, we also found that it violates differential privacy when $d$ is even.", "To illustrate, consider that $d=2$ and we have an input tuple $t_i = \\langle 1, 1 \\rangle $ , i.e., $t_i[A_1] = t_i[A_2] = 1$ .", "Then, Line 1 in Algorithm REF would generates a tuple $v = \\langle 1, 1\\rangle $ .", "Let $B$ be as defined in Equation (REF ), and $T^+$ and $T^-$ be as defined in Line 2 in Algorithm REF .", "It can be verified that $T^+$ and $T^-$ contain 1 and 3 tuples, respectively, with T+ = { B, B }, and T- = { -B, -B , -B, B , B, -B }.", "Then, by Lines 3-8 in Algorithm REF , the method outputs $\\langle B, B \\rangle $ with $\\frac{e^\\epsilon }{e^\\epsilon + 1}$ probability.", "In contrast, each tuple in $T^-$ has only $\\frac{1}{3e^\\epsilon + 3}$ probability to be output.", "Now consider another input tuple $t^\\prime _i = \\langle -1, -1 \\rangle $ .", "It follows that, for $t^\\prime _i$ , the algorithm outputs $\\langle B, B \\rangle $ with only $\\frac{1}{3e^\\epsilon + 3}$ probability.", "As a consequence, [f(ti) = B, B ] = 3 e[f(ti) = B, B ] > e[f(ti) = B, B ], which indicates that the algorithm does not satisfy $\\epsilon $ -differential privacy.", "We find that the above problem is caused by Line 3 in Algorithm REF , in that the Bernoulli variable $u$ is incorrectly defined for the case for $d$ is even.", "To address the problem, one possible fix we found is to re-define $u$ as a Bernoulli variable such that $\\Pr [u = 1] = \\frac{e^\\epsilon \\cdot C_d}{(e^\\epsilon - 1)C_d + 2^d}.$ It can be shown that, with this revised choice of $u$ , Algorithm REF achieves $\\epsilon $ -differential privacy and ensures the error bound in Equation REF .", "We omit the proofs for brevity.", "Proposed method.", "In what follows, we present an algorithm used in Harmony for perturbing a tuple that is conceptually simpler than Duchi et al.", "'s method, but achieves the same privacy assurance and asymptotic error bound.", "Furthermore, our experiments (in Section ) show that the algorithm slightly outperforms Duchi et al.", "'s method in terms of the empirical accuracy of the estimated means of numeric attributes.", "Additionally, our method is more efficient: in particular, each user only needs to transmit one bit to the aggregator, which is clearly optimal.", "Our algorithm is inspired by an existing approach [2] for publishing categorical data, which we will discuss in Section REF .", "[t] Proposed Method for Handling Numeric Attributes Inputinput Outputoutput tuple $t^*_i \\in \\left\\lbrace -\\frac{e^\\epsilon +1}{e^\\epsilon -1}d, \\:\\: 0, \\:\\: \\frac{e^\\epsilon +1}{e^\\epsilon -1} d\\right\\rbrace ^d.$ Let $t^*_i = \\langle 0, 0, \\ldots , 0\\rangle $ Sample $j$ uniformly at random from $[d]$ Sample a Bernoulli variable $u$ such that $\\Pr [u = 1] = \\frac{t_i[A_j]\\cdot (e^\\epsilon -1) + e^\\epsilon + 1}{2e^\\epsilon + 2}$ $u = 1$ $t^*[A_j] = \\frac{e^\\epsilon +1}{e^\\epsilon -1} \\cdot d$ $t^*[A_j] = -\\frac{e^\\epsilon +1}{e^\\epsilon -1} \\cdot d$ $t^*_i$ Algorithm REF shows the pseudo-code of our method.", "Given a tuple $t_i \\in [-1, 1]^d$ , the algorithm returns a perturbed tuple $t^*_i$ that has non-zero value on only one attribute $A_j$ ($j \\in [d]$ ).", "Specifically, $A_j$ is selected uniformly at random from all $d$ attributes of $t_i$ , and $t^*_i[A_j]$ is sampled from the following distribution: $ \\Pr \\big [t^*_i[A_j] = x\\big ] ={\\left\\lbrace \\begin{array}{ll}\\frac{t_i[A_j]\\cdot (e^\\epsilon -1) + e^\\epsilon + 1}{2e^\\epsilon + 2}, & \\textrm {if x = \\frac{e^\\epsilon +1}{e^\\epsilon -1} \\cdot d} \\vspace{5.69054pt} \\\\\\end{array}\\frac{-t_i[A_j]\\cdot (e^\\epsilon -1) + e^\\epsilon + 1}{2e^\\epsilon + 2}, & \\textrm {if \\right.x = -\\frac{e^\\epsilon +1}{e^\\epsilon -1} \\cdot d}}$ Observe that in the above method, the output $t^*_i$ contains only one non-zero value, for the randomly chosen attribute $A_j$ .", "This value is binary; hence, the user $u_i$ only needs to transmit 1 bit to the aggregator indicating its sign, and the latter can re-scale it using parameters $\\epsilon $ and $d$ .", "Further, as we show below the correctness of this method does not depend on the choice of $A_j$ as long as it is chosen uniformly at random.", "Therefore, the value of $j$ can be obtained, e.g., using a public source of random numbers such as a hash value of the user's ID.", "Therefore, the communication overhead between each user and the aggregator is exactly 1 bit.", "The following lemmas establish the theoretical guarantees of Algorithm REF .", "Lemma 1 Algorithm REF satisfies $\\epsilon $ -local differential privacy.", "Let $t^*$ be an output of Algorithm REF , and $A_j$ be the only attribute such that $t^*[A_j] \\ne 0$ .", "Let $t$ and $t^\\prime $ be any two tuples, and $u$ (resp.", "$u^\\prime $ ) be the Bernoulli variable generated in Line 3 of Algorithm REF given $t$ (resp.", "$t^\\prime )$ as the input.", "In the following, we focus on the case when $t^*[A_j] = \\frac{e^\\epsilon + 1}{e^\\epsilon - 1}d$ ; the case when $t^*[A_j] = -\\frac{e^\\epsilon + 1}{e^\\epsilon - 1}d$ can be analyzed in a similar manner.", "By Algorithm REF , we have [t* t][t* t] = 1/d [u = 1 t]1/d [u= 1 t] t [u = 1 t] t [u= 1 t] = t[Aj][-1,1] (t[Aj] (e-1) + e+ 1)t[Aj][-1,1] (t[Aj](e-1) + e+ 1) = e. This completes the proof.", "Lemma 2 Let $t^*_i$ be the output of Algorithm REF given an input tuple $t_i$ .", "Then, for any $j \\in [d]$ , $\\mathbb {E}[t^*[A_j]] = t[A_j]$ .", "By Equation (REF ), E[t*[Aj]] = [t*[Aj] = e+1e-1d] e+1e-1 d + [t*[Aj] = -e+1e-1 d] (- e+1e-1 d ) + [t*[Aj] = 0] 0 = 2 t[Aj] (e- 1)2e- 2 = t[Aj].", "By Lemma REF , the server can use $\\frac{1}{n} \\sum _{i = 1}^n t^*[A_j]$ as an unbiased estimator of the mean of $A_j$ .", "The following lemma shows the accuracy guarantee of this estimator.", "Lemma 3 For any $j \\in [d]$ , let $Z[A_j] = \\frac{1}{n} \\sum _{i=1}^n t^*_i[A_j]$ and $X[A_j] = \\frac{1}{n} \\sum _{i=1}^n t_i[A_j]$ .", "With at least $1 - \\beta $ probability, $\\max _{j \\in [d]} \\big |Z[A_j] - X[A_j]\\big | = O\\left(\\frac{\\sqrt{d \\log (d/\\beta )}}{\\epsilon \\sqrt{n}}\\right).$ First, observe that for any $i \\in [d]$ and any $j \\in [d]$ , the variance of $t^*_i[a_j] - t_i[a_j]$ equals: Var[t*i[aj] - ti[aj]] = Var[t*i[aj]] = E[(t*i[aj])2] - (E[t*i[aj])2 = 1d (e+1e-1 d)2 - (ti[aj])2 (e+1e-1)2 d. By Bernstein's inequality, [|Z[aj] - X[aj]| ] 2 (-n22 n i=1n Var[t*i[aj] - ti[aj]] + 23e+ 1e-1 2d) = 2(-n 22d (O(1/2) + O(1/) ) ).", "By the union bound, there exists $\\lambda = O\\left(\\frac{\\sqrt{d \\log (d/\\beta )}}{\\epsilon \\sqrt{n}}\\right)$ such that $\\max _{j \\in [d]} |Z[a_j] - X[a_j]| < \\lambda $ holds with at least $1-\\beta $ probability." ], [ "Estimating Frequencies for Categorical Attributes", "We now focus on the case where each user's data record contains not only numeric attributes but also categorical ones.", "For each categorical attribute, the aggregator aims to build an accurate histogram containing the frequency estimate for each possible value in the attribute's domain.", "For example, Samsung may want to know the percentage of users who enable a specific setting, through the diagnostic information report app described in Section REF .", "Note that we can convert a numeric attribute a categorical one (e.g., display brightness can be discretized to three levels: low, medium and high) and build a histogram accordingly.", "Randomized response for binary attributes.", "For a single binary attribute (e.g., WiFi on/off), it suffices to use the classic randomized response method [20] (also used in Rappor [11]) to estimate the distribution of users.", "Specifically, suppose that the domain of the binary attribute (let $A_j$ ) contains two possible values, $-1$ and $+1$ .", "Each user $u_i$ reports her true answer $t_i[A_j]$ with probability $p$ , and a random answer with probability $1-p$ .", "The latter has the same probability to be $-1$ and $+1$ ; hence, its expected value is zero.", "Therefore, the expected value for $u_i$ 's reported value is $p \\cdot t_i[A_j]$ ; thus, we can obtain an unbiased estimate by multiplying the reported value by a scaling factor $c_\\epsilon = 1/p$ .", "Meanwhile, comparing $u_i$ 's true attribute value and her reported one, the two are the same with probability ${p+(1-p)/2}$ , and they are different with probability $(1-p)/2$ .", "According to Definition REF , $\\epsilon $ -local differential privacy requires that $\\frac{p+(1-p)/2}{(1-p)/2} \\le e^{\\epsilon }$ .", "The equality holds when $p = \\frac{e^\\epsilon - 1}{e^\\epsilon + 1}$ .", "We thus arrive at the following unbiased mechanism that satisfies $\\epsilon $ -LDP: each user $u_i$ reports $c_\\epsilon \\cdot t_i[A_j] = \\frac{e^\\epsilon + 1}{e^\\epsilon - 1}\\cdot t_i[A_j]$ with probability $p+(1-p)/2 = \\frac{e^\\epsilon }{e^\\epsilon + 1}$ , and $-c_\\epsilon \\cdot t_i[A_j]$ otherwise (i.e., with probability $\\frac{1}{e^\\epsilon + 1}$ ).", "Once the aggregator receives all reported values for attribute $A_j$ , it computes the average over all users, which is an estimate of the mean value $\\mathbb {E}[A_j]$ for $A_j$ .", "Since $A_j$ can be either $+1$ or $-1$ , the percentage of users with $+1$ (resp.", "$-1$ ) is $\\frac{1+\\mathbb {E}[A_j]}{2}$ (resp.", "$\\frac{1-\\mathbb {E}[A_j]}{2}$ ).", "[t] Bassily and Smith's method [2] Inputinput Outputoutput Frequency estimate for each of the $k$ values in attribute $A_j$ Compute $\\gamma = \\sqrt{\\frac{\\log (2k/\\beta )}{\\epsilon ^2 n}}$   and $m = \\frac{\\log (k+1) \\log (2/\\beta )}{\\gamma ^2}$ Generate random matrix $\\Phi \\in \\lbrace \\pm \\frac{1}{\\sqrt{m}}\\rbrace ^{m \\times k}$ $i$ = 1 to $n$ User $u_i$ : draw $s ~\\sim \\text{Uniform}(\\lbrace 1,2\\dots , m\\rbrace )$ User $u_i$ : draw $t \\sim \\text{Bern}(\\frac{e^\\epsilon }{e^\\epsilon + 1})$ User $u_i$ : if $t=1$ then $\\alpha = c_\\epsilon m \\Phi [s, t_i[A_j]]$ else $\\alpha = -c_\\epsilon m \\Phi [s, t_i[A_j]]$ , where $c_\\epsilon = \\frac{e^\\epsilon +1}{e^\\epsilon - 1}$ User $u_i$ : submit $\\langle s, \\alpha \\rangle $ , which represents a $k$ -dimensional vector $z_i$ where the $s$ -th entry is $\\alpha $ and the other entries are 0 Compute $\\bar{z} = \\frac{1}{n} \\sum _{i=1}^n z_i$ $l$ = 1 to $k$ Estimate the frequency of the $l$ -th value by the inner product of the $l$ -th column of $\\Phi $ and $\\bar{z}$ ; $k$ frequency estimates obtained above; Bassily and Smith's method.", "The problem is more complicated when the categorical attribute $A_j$ contains $k>2$ possible values.", "In this situation, the aggregator aims to build a histogram that contains the estimated frequency for each of the $k$ possible values.", "The current state-of-the-art to our knowledge is by Bassily and Smith [2], shown in Algorithm REF , which is proven to satisfy $\\epsilon $ -LDP and achieve an optimal asymptotical error bound.", "There are two main ideas in this method.", "First, the authors assume that the number of possible values $k$ in the categorical attribute is far larger than the number of users $n$ ; hence, the method applies random projection to reduce the dimensionality from $k$ to $m$ .", "The value of $m$ is chosen carefully so as to obtain the asymptotically optimal error bound.", "This step essentially transforms the categorical attribute into $m$ binary ones.", "Specifically, the random projection is done with a $m \\times k$ matrix $\\Phi $ in which each element is randomly set to either $+\\frac{1}{\\sqrt{m}}$ or $-\\frac{1}{\\sqrt{m}}$ with equal probability.", "This ensures that (i) the inner product of any column in $\\Phi $ with itself is 1 (which is where the absolute value of each element $\\frac{1}{\\sqrt{m}}$ comes from) and (ii) the inner product of two different columns in $\\Phi $ has zero expected value, since the signs are randomly generated.", "Each user $u_i$ 's attribute value $t_i[A_j]$ is then transformed to $m$ binary values by taking the $t_i[A_j]$ -th column in $\\Phi $ .", "The second idea is for each user to randomly pick one of the converted $m$ binary attributes, and report a randomized response using the method described earlier.", "Note that the randomized response needs to be scaled by a factor of $m$ , since each of the $m$ binary attributes has probability $1/m$ to be chosen.", "The aggregator collects the average for each of the $m$ binary attributes, which is stored as a vector $\\bar{z}$ .", "To obtain the frequency estimate of a particular attribute value $l$ , the method takes the inner product of $\\bar{z}$ and the $l$ -th column of $\\Phi $ , which can be proven to yield an unbiased frequency estimate for the $l$ -th value in $A_j$ .", "Meanwhile, the frequency estimate is proven to be within $O\\left(\\frac{\\sqrt{\\log (k/\\beta )}}{\\epsilon \\sqrt{n}}\\right)$ error with probability $1-\\beta $ [2], where $\\beta $ is an input to the algorithm.", "Proposed method for a single categorical attribute.", "Harmony generally follows Bassily and Smith's method to estimate value frequencies for a categorical attribute.", "However, we found that although Bassily and Smith's method achieves optimal asymptotic accuracy, in practice its accuracy tends to be unstable, especially for relatively small categorical domains.", "The reason is that the random projection matrix $\\Phi $ introduces considerable noise; in particular, the inner product of two different columns is often non-zero unless $k$ is very large, which is magnified by a large number of users $n$ .", "Hence, we propose an alternative solution that obtains higher accuracy when $k = o(n)$ .", "In particular, instead of generating random matrix $\\Phi $ (size $m \\times k$ where $m = O(n)$ ), we construct a binary matrix of size $k \\times k$ satisfying that any two column vectors are always orthogonal.", "The construction algorithm of this matrix can be found in the appendix.", "Proposed method for multiple numeric and categorical attributes.", "Bassily and Smith's method is limited to a single categorical attribute.", "To extend it to multiple categorical attributes, a straightforward approach is to apply the method once for each attribute separately.", "In that case, however, the privacy budget $\\epsilon $ needs to be divided among all attributes, so as to ensure $\\epsilon $ -LDP as a whole.", "Without loss of generality, assume that we have $d$ categorical attributes, and we assign $\\epsilon /d$ budget to each of them.", "Then, the amount of noise incurred by Bassily and Smith's method on each attribute is increased $d$ times to $O\\left(\\frac{d\\sqrt{\\log (d/\\beta )}}{\\epsilon \\sqrt{n}}\\right)$ , which is unsatisfactory when $d$ is large.", "Another approach for extension is to (i) convert the $d$ categorical attributes into a “composite” attribute whose domain equals the Cartesian product of the individual attribute domains, and then (ii) apply Bassily and Smith's method on the composite attribute.", "This, however, only allows us to (accurately) derive the frequency each composite value (i.e., combination of values from all $d$ individual attributes), but does not provide quality estimation of the frequency of each individual value.", "Note that these limitations are not specific to Bassily and Smith's method; to our knowledge, there is no existing work (including Rappor) that are designed to handle multiple categorical attributes, let alone a mixture of numeric and categorical ones.", "In Harmony, we use a simple and elegant solution to handle multiple attributes: for each numerical attribute, the aggregator estimates its mean value; for each categorical attribute, the aggregator estimates its value frequencies.", "In particular, given $d$ attributes $A_1, A_2, \\ldots , A_d$ , the solution asks each user to perform the following: Draw $j$ uniformly at random from set $\\lbrace 1, 2, \\ldots , d\\rbrace $ ; If $A_j$ is a numeric attribute, then submit a noisy version of $t[A_j]$ computed as in Lines 3-7 of Algorithm REF ; Otherwise (i.e., $A_j$ is a categorical attribute), compute ${\\langle s, \\alpha \\rangle }$ as in Lines 4-8 of Algorithm REF , then submits ${\\langle s, d \\cdot \\alpha \\rangle }$ to represent a $d$ -dimensional vector where the $s$ -th entry is $d \\cdot \\alpha $ and all other entries are zero.", "The above solution satisfies $\\epsilon $ -LDP, which follows from the fact that (i) each user randomly selects one attribute to submit, and (ii) the algorithm used for submitting the selected attribute is $\\epsilon $ -differentially private.", "For each numeric attribute, it is easy to see that the solution provides the same accuracy guarantee as Algorithm REF , since both methods handle numeric attributes in exactly the same way.", "The following lemma states the accuracy guarantee of our solution for categorical attributes.", "Lemma 4 For each categorical attribute $A_j$ with a domain $\\lbrace 1, 2, \\ldots , k\\rbrace $ , let $x_l$ be the frequency of the $l$ -th value of $A_j$ , and $y_l$ be the estimation of $x_l$ returned by our solution.", "With at least $1 - \\beta $ probability, $\\max _{l \\in [k]} \\big |y_l - x_l\\big | = O\\left(\\frac{\\sqrt{d \\log (k/\\beta )}}{\\epsilon \\sqrt{n}}\\right).$ [(Sketch)] Consider any user with a tuple $t$ .", "With respect to $A_j$ , our solution can be regarded as a method that (i) outputs nothing with ${d-1}/d$ probability, and (ii) with the remaining $1/d$ probability, applies Bassily and Smith's algorithm on $A_j$ and scale its output up by $d$ times.", "It follows that the variance of each of our frequency estimators for $A_j$ is $d$ times that of Bassily and Smith's estimator.", "Based on the analysis in [2], it can be shown that the variance of Bassily and Smith's estimator is $O\\left(\\frac{1}{n \\epsilon ^2}\\right)$ .", "Therefore, the variance of each of our frequency estimators is $O\\left(\\frac{d}{n \\epsilon ^2}\\right)$ .", "Combining this with Bernstein's inequality and the union bound, it can be proven that with at least $1-\\beta $ probability, the error in any $k$ of our estimators is $O\\left(\\frac{\\sqrt{d \\log (k/\\beta )}}{\\epsilon \\sqrt{n}}\\right).$ By Lemma REF , when there exist multiple categorical attributes, the error incurred by our approach is a factor of $O(\\sqrt{d})$ smaller than that of a solution that repeatedly apply Bassily and Smith's method on each categorical attribute." ], [ "Building Machine Learning Models using Stochastic Gradient Descent", "This section investigates building a large class of machine learning models that can be expressed as empirical risk minimization under $\\epsilon $ -local differential privacy.", "In particular, we focus on three common types of learning tasks: linear regression, logistic regression, and SVM classification.", "Section REF introduces the basic approaches for building these models, while Section REF discusses optimizations that lead to improved results in practice." ], [ "Basic Methods", "Suppose that each user $u_i$ has a pair $\\langle x_i, y_i\\rangle $ , where $x_i \\in [-1, 1]^d$ and $y_i \\in [-1, 1]$ (for linear regression) or $y_i \\in \\lbrace -1, 1\\rbrace $ (for logistic regression and SVM classification).", "Let $\\ell (\\cdot )$ be a loss function that (i) maps a $d$ -dimensional parameter vector $\\beta $ into a real number and (ii) is parameterized by $x_i$ and $y_i$ .", "We aim to identify a parameter vector $\\beta ^*$ such that $\\beta ^* = \\arg \\min _{\\beta } \\frac{1}{n}\\left(\\sum _{i=1}^n \\ell (\\beta ; x_i, y_i)\\right) + \\frac{\\lambda }{2} \\Vert \\beta \\Vert ^2_2,$ where $\\lambda > 0$ is a regularization parameter.", "We consider three specific loss functions: Linear regression: $\\ell (\\beta ; x_i, y_i) = (x_i^T \\beta - y_i)^2$ ; Logistic regression: $\\ell (\\beta ; x_i, y_i) = \\log \\left(1 + e^{-y_ix_i^T\\beta }\\right)$ ; SVM (hinge loss): $\\ell (\\beta ; x_i, y_i) = \\max \\lbrace 0, 1 -y_i x_i^T\\beta \\rbrace $ .", "For convenience, we define $\\ell ^\\prime (\\beta ; x_i, y_i) = \\ell (\\beta ; x_i, y_i) + \\frac{\\lambda }{2} \\Vert \\beta \\Vert ^2_2.$ One of the most common solutions to compute $\\beta ^*$ is stochastic gradient descent (SGD).", "It starts from an initial parameter vector $\\beta _0$ , and iteratively updates it into $\\beta _1, \\beta _2, \\ldots $ based on the following equation: $\\beta _{k+1} = \\beta _k - \\gamma _k \\cdot \\nabla \\ell ^\\prime (\\beta _k; x, y),$ where $\\langle x, y \\rangle $ is the tuple of a randomly selected user, $\\nabla \\ell ^\\prime (\\beta _k; x, y)$ is the gradient of $\\ell ^\\prime $ at $\\beta _k$ , and $\\gamma _k$ is a constant typically set to $O(1/\\sqrt{k})$ .", "It terminates when the difference between $\\beta _{k+1}$ and $\\beta _k$ is sufficiently small.", "Under our problem setting, however, $\\nabla \\ell ^\\prime $ is not directly available to the aggregator, and needs to be collected in a private manner.", "Towards this end, existing work [13], [7] has suggested that the aggregator may ask the selected user in each iteration to submit a noisy version of $\\nabla \\ell ^\\prime $ , by using the Laplace mechanism or Duchi et al.", "'s method (i.e., Algorithm REF ).", "We can straightforwardly improve these existing approaches by perturbing $\\nabla \\ell ^\\prime $ using Algorithm REF instead; however, we observe that such a solution is insufficient for our target application, as we explain in Section REF ." ], [ "Improvements", "Mini-batching.", "We observe in our experiments that the aforementioned SGD approach yields rather inaccurate results, due to the noise injected in the gradient $\\nabla \\ell ^\\prime $ returned by each user.", "In particular, if each user applies Algorithm REF to compute $\\nabla \\ell ^\\prime (\\beta _k; x, y)$ , the amount of noise in $\\nabla \\ell ^\\prime $ is $O\\left(\\frac{\\sqrt{d \\log d}}{\\epsilon }\\right)$ , which is excessively large given that $x \\in [-1, 1]^d$ .", "To address this issue, we adopt mini-batch gradient descent instead of SGD.", "That is, each iteration of the algorithm, we involve a group $G$ of users, and ask each of them to submit a noisy version of the gradient; after that, we update the parameter vector $\\beta _k$ with the mean of the noisy gradients, i.e., $\\textstyle \\beta _{k+1} = \\beta _k - \\gamma _k \\cdot \\frac{1}{|G|}\\sum _{i=1}^{|G|} \\nabla \\ell ^*_i,$ where $\\nabla \\ell ^*_i$ is the noisy gradient submitted by the $i$ -th user in $G$ .", "This helps because the amount of noise in the average gradient is $O\\left(\\frac{\\sqrt{d \\log d}}{\\epsilon \\sqrt{|G|}}\\right)$ , which could be acceptable if $|G| = \\Omega \\left(d \\log d/\\epsilon ^2\\right)$ .", "However, when $d$ is sizable, $|G| = \\Omega \\left(d \\log d/\\epsilon ^2\\right)$ is large.", "As a consequence, when we allow each user to participate in at most one iteration of the algorithm, the maximum number of iterations (i.e., $n/|G|$ ) is small.", "In that case, the algorithm may terminate prematurely and return an inferior parameter vector.", "One may attempt to mitigate this problem by allowing each user to be involved in $m > 1$ iterations, but it would further increase the amount of noise in each gradient returned.", "To explain this, suppose that the $i$ -th ($i \\in [1, m]$ ) gradient returned by the user satisfies $\\epsilon _i$ -differential privacy.", "By the composition property of differential privacy [17], if we are to enforce $\\epsilon $ -differential privacy for the user's data, we should have $\\sum _{i=1}^m \\epsilon _i \\le \\epsilon $ .", "Consider that we set $\\epsilon _i = \\epsilon /m$ .", "Then, the amount of noise in each gradient becomes $O\\left(\\frac{m \\sqrt{d \\log d}}{\\epsilon }\\right)$ ; accordingly, the acceptable mini-batch size becomes $|G| = \\Omega \\left(m^2 d \\log d/\\epsilon ^2\\right)$ , which is $m^2$ times the acceptable size when we allow each user to participate in at most one iteration.", "It then follows that the total number of iterations in the algorithm is inversely proportional to $1/m$ , i.e., setting $m > 1$ only degrades the performance of the algorithm.", "Dimension reduction.", "For linear regression, instead of increasing $m$ , we propose to apply dimensionality reduction on each user's data, so as to reach an acceptable size of mini-batches.", "Specifically, the curator first generates a random $r \\times d$ matrix $P$ where $r < d$ and each entry has an equal probability to be assigned $1/d$ or $-1/d$ .", "Then, the curator shares $P$ with all users, and asks each user to convert her tuple $\\langle x_i, y_i\\rangle $ into a reduced tuple $\\langle x^\\prime _i, y_i\\rangle $ , where $x^\\prime _i = P x$ .", "In other words, we project $\\lbrace x_i\\rbrace $ into a random $r$ -dimensional sub-space, and such a projection is known to preserve several important characteristics of the original data [1].", "It can be verified that $x^\\prime _i \\in [-1, 1]^r$ .", "Subsequently, each user uses the reduced tuple $\\langle x^\\prime _i, y_i\\rangle $ to participate in the mini-batch gradient descent algorithm.", "In other words, each noisy gradient $\\nabla \\ell ^*_i$ returned by the user is $r$ -dimensional instead of $d$ -dimensional.", "As such, the average noisy gradient obtained from a mini-batch $G$ of users has an error of $O\\left(\\frac{\\sqrt{r \\log r}}{\\epsilon \\sqrt{|G|}}\\right)$ instead of $O\\left(\\frac{\\sqrt{d \\log d}}{\\epsilon \\sqrt{|G|}}\\right)$ .", "Accordingly, the acceptable mini-batch size is reduced to $|G| = \\Omega \\left(r \\log r/\\epsilon ^2\\right)$ .", "Algorithm REF shows the pseudo-code of our mini-batch gradient descent method with dimension reduction, in the context of the Samsung diagnostic tool.", "The aggregator first generates a random $r \\times d$ matrix $P$ , and maintains a $r$ -dimensional parameter vector $\\alpha $ (Lines 1-3).", "(We use $\\alpha $ instead of $\\beta $ to denote the parameter vector to avoid confusion on its dimensionality.)", "After that, whenever a user with a tuple $\\langle x_i, y_i\\rangle $ comes online, she obtains $P$ and the current $\\alpha $ from the aggregator (Line 6).", "Then, the user computes a reduced tuple $\\langle x^\\prime _i, y_i\\rangle $ , as well as the gradient $\\nabla _i = \\nabla \\ell ^\\prime (\\alpha ; x^\\prime _i, y_j)$ (Line 7).", "If any entry of $\\nabla _i$ is larger than 1 (resp.", "smaller than $-1$ ), then the user resets the entry to 1 (resp.", "$-1$ ) (Lines 8-9).", "This ensures that $\\nabla _i \\in [-1, 1]^d$ , so that it can be a valid input to Algorithm REF .", "After that, the user computes a noisy gradient $\\nabla ^*_i$ using Algorithm REF , submits it to the aggregator, and then logs off (Line 10).", "The aggregator computes the average noisy gradient from every $g$ users (where $g$ is an input parameter), and updates the parameter vector $\\alpha $ accordingly (Lines 11-14).", "When the update to $\\alpha $ is sufficiently small or when a sufficiently large number of users have participated, the aggregator terminates the algorithm (Lines 15-16).", "[t] An Iterative Method for Empirical Risk Minimization Inputinput Outputoutput parameter vector $\\alpha \\in \\mathbb {R}^r$ generates a random $r \\times d$ matrix $P$ each entry has an equal probability to $1/d$ or $-1/d$ initialize a counter $k = 0$ , and learning rate $\\gamma $ initialize a $r$ -dimensional vector $\\nabla = \\langle 0, 0, \\ldots , 0\\rangle $ true a user with a tuple $\\langle x_i, y_i\\rangle $ comes online send $P$ to the user the user computes $x^\\prime _i = P x_i$ , and derives $\\nabla _i = \\nabla \\ell ^\\prime (\\alpha ; x^\\prime _i, y_j)$ $\\nabla _i \\notin [-1, 1]^r$ the user projects $\\nabla _i$ onto $[-1, 1]^r$ the user applies Algorithm REF on $\\nabla _i$ , and submits a noisy gradient $\\nabla ^*_i$ $k = k + 1$ , and $\\nabla = \\nabla + \\nabla ^*_i$ $k \\mod {g} = 0$ $\\nabla = \\frac{\\nabla }{g}$ , and $\\gamma = 1/\\sqrt{k/g}$ $\\alpha = \\alpha - \\gamma \\cdot \\nabla $ $k$ is sufficiently large or $\\alpha $ changes sufficiently small in the last update break $\\alpha $ Figure: Experiments on estimating means and frequencies." ], [ "Experimental Settings", "For experimental repeatability, we use two public datasets extracted from the Integrated Public Use Microdata Series [14], US and BR, which contains census records from the United States and Brazil, respectively.", "US contains 9M tuples and 23 attributes, among which 6 are numeric (e.g., age) and 17 are categorical (e.g., gender); BR has 4M records and 18 attributes, among which 6 are numeric and 12 are categorical.", "Both datasets contain a numeric attribute “total income”, which we use as the dependent attribute in linear regression, logistic regression, and SVM.", "We normalize the domain of each numeric attribute into $[-1, 1]$ ." ], [ "Estimating Means and Frequencies", "In the first set of experiments, we consider the task of collecting a noisy tuple from each user to estimate the mean of each numeric attribute and the frequency of each categorical value.", "As mentioned in Section , none of the existing solutions can directly support this task, since they are designed for either numeric or categorical attributes, but not both.", "To address this issue, we construct a method (referred to as Hybrid) by combining the best existing solutions as follows.", "Let $t$ be a tuple with $d_n$ numeric attributes and $d_c$ categorical attributes.", "Given $t$ and a privacy budget $\\epsilon $ , Hybrid first constructs a $d_n$ -dimensional tuple $t^\\prime $ that contains all numeric values in $t$ , and then release a noisy version of $t^\\prime $ by invoking Duchi et al.", "'s method (see Section REF ) on $t^\\prime $ , using a privacy budget of $d_n\\epsilon /d$ .", "After that, for each categorical value in $t$ , Hybrid applies Bassily and Smith's method (see Section REF ) to release a noisy version with a privacy budget of $\\epsilon /d$ .", "By the composition property of differential privacy [17], Hybrid ensures $\\epsilon $ -LDP.", "Intuitively, Hybrid is a best-effort approach to incorporate two states of the art that are designed only for numeric attributes (i.e., Duchi et al.", "'s method) and a single categorical attribute (i.e., Bassily and Smith's method), respectively.", "We apply our solution in Section  and Hybrid on both US and BR to generate noisy tuples, and then use the noisy tuples to estimate the frequency of each value in each categorical domain in US and BR.", "For each method, we measure the $L_\\infty $ error in the estimated value frequencies, and we take the average measurement from 100 runs.", "Figures REF a and REF b illustrate the results as $\\epsilon $ varies.", "Observe that our method is significantly more accurate than Hybrid in all cases, and its error is only around $1/4$ of the error incurred by Hybrid.", "This is consistent with the analysis in Section REF that (i) our method has $O\\left(\\frac{\\sqrt{d\\log k}}{\\epsilon \\sqrt{n}}\\right)$ estimation error, and (ii) repeatedly applying Bassily and Smith's method on each categorical leads to $O\\left(\\frac{d\\sqrt{\\log k}}{\\epsilon \\sqrt{n}}\\right)$ error.", "We also use the noisy tuples to estimate the mean of each numeric attribute, and we measure the $L_\\infty $ error of each method, averaged over 100 runs.", "Figure REF c (resp.", "REF d) shows the results on US (resp.", "BR) as a function of the privacy budget $\\epsilon $ .", "Observe that our solution slightly outperforms Hybrid, regardless of the dataset used and the value of $\\epsilon $ .", "We also note that, compared with Hybrid (which applies Duchi et al.", "'s method to handle numeric attributes), our solution has a much lower communication cost on each user (since it only requires each user to transfer 1 bit), and is much simpler." ], [ "Empirical Risk Minimization", "In the second set of experiments, we consider linear regression, logistic regression, and SVM classification on US and BR.", "For both datasets, we use the numeric attribute “total income” as the dependent variable, and all other attributes as independent variables.", "Following the standard practice, we transform each categorical attribute $A_j$ with $k$ values into $k-1$ binary attributes with a domain $\\lbrace -1, 1\\rbrace $ , such that (i) the $l$ -th ($l < k$ ) value in $A_j$ is represented by a 1 on the $l$ -th binary attribute and a $-1$ on each of the remaining $k-2$ attributes, and (ii) the $k$ -th value in $A_j$ is represented by $-1$ on all binary attributes.", "After this transformation, the dimensionality of US (resp.", "BR) becomes 85 (resp.", "95).", "For logistic regression and SVM (which requires the dependent variable to be binary), we convert “total income” into a binary attribute by mapping all values which are greater than or equal to the mean to 1, and all other values to $-1$ .", "We evaluate four methods: (i) a private version of SGD that involves one user in each iteration, and asks the user to submit a noisy gradient using Duchi et al.", "'s method; (ii) mini-batch gradient descent (MGD), which involves $g$ users in each iteration, and uses the average noisy gradients of those users (generated with Algorithm REF ) to update the parameter vector; (iii) MGD with dimension reduction (DR), which is an improved version of MGD that projects the users' data onto an $r$ -dimensional sub-space before the learning task (this method is applied for linear regression only); (iv) a non-private version of SGD.", "Based on the analysis in Section , we set the mini-batch size for MGD and MGD-DR to $g = \\max \\lbrace 2d\\log d /\\epsilon ^2, n/1000\\rbrace $ and $g = \\max \\lbrace 2r\\log r / \\epsilon ^2, n/1000\\rbrace $ , respectively.", "The term $n/1000$ is to guarantee that our mini-batch size is not too small when $\\epsilon $ is large.", "In addition, we set $r = 20$ .", "For all methods, we set the regularization factor $\\lambda = 10^{-4}$ .", "On each dataset, we use 10-fold cross validation to assess the performance of each method.", "Figure REF a (resp.", "REF d) shows the mean squared error (MSE) of the linear regression model generated by each method, under various values of $\\epsilon $ .", "Private SGD incurs prohibitive errors in all cases, due to the large amount of noise in the gradient that it obtains in each iteration.", "MGD alleviates this issue with mini-batches, but still provides unsatisfactory accuracy for linear regression.", "The reason, as we mentioned in Section REF , is that MGD requires using a large mini-batch size $g$ when $d$ is large, which in turn leads to a small total number of iterations and degrades its performance.", "MGD-DR overcomes the drawback of MGD by incorporating dimension reduction to reduce the required mini-batch size, and hence, it is able to achieve an accuracy that is close to the non-private SGD.", "Figures REF b and REF e (REF c and REF f) illustrate the misclassification rate of each method for logistic regression (resp.", "SVM).", "Overall, our experimental results demonstrate the effectiveness of mini-batches and dimension reduction for empirical risk minimization under local differential privacy." ], [ "Related Work", "Differential privacy [4], [9], [8] is a strong, mathematically rigorous framework for privacy protection.", "Unlike earlier privacy-preserving data publication methods which are largely syntactic, differential privacy provides semantic, information-theoretic guarantees on individuals' privacy.", "Hence, since its proposal in 2003 it had attracted much attention from various research communities in computer science, including theory [10], machine learning [18], data management [21], and systems [19].", "Earlier models of differential privacy assume a trusted data curator, who collects and manages the exact private information of individuals, and releases statistics derived from the data under differential privacy requirements.", "In practice, however, users may not want to share private information with anyone, including the central data curator.", "Recently, much attention has been shifted to the local differential privacy model, which eliminates the data curator and the collection of exact private information.", "Specifically, Duchi et al.", "[5] systematically investigate the concept of local differential privacy, propose the minimax framework for LDP based on the information theory, prove upper and lower error bounds of LDP-compliant methods, and analyze the trade-off between privacy and accuracy.", "Kairouz et al.", "[15] show that a version of randomized response is an optimal mechanism for frequency estimation on a single binary attribute.", "Kairouz et al.", "[16] study the problem with a categorical attribute with an arbitrary number of possible values, propose two mechanisms: binary and randomized response mechanisms, and prove their optimality when the privacy budget is low and high, respectively.", "Bassily and Smith [2] propose an asymptotically optimal solution for building succinct histograms over a large categorical domain under LDP.", "Erlingsson et al.", "[11] propose the RAPPOR framework, which is based on the randomized response mechanism for publishing a vector of binary values under LDP.", "They use this mechanism with a Bloom filter, which intuitively adds another level of protection and increases the difficulty for the adversary to infer private information.", "As a result, it also becomes more difficult derive statistics from the collected data, and they propose a sophisticated solution for this purpose.", "A follow-up paper [12] extends Rappor to more complex statistics such as joint distributions and association testing, as well as categorical attributes that contain a large number of potential values, such as a user's home page." ], [ "Conclusion", "This work systematically investigates the problem of collecting and analyzing users' personal data under $\\epsilon $ -local differential privacy, in which the aggregator only collects randomized data from the users, and computes statistics based on such data.", "The proposed solution Harmony is able to collect data records that contain multiple numeric and categorical attributes, and compute accurate statistics from simple ones such as mean and frequency to complex machine learning models such as linear regression, logistic regression and SVM classification.", "Harmony achieves both optimal asymptotic error bound and high accuracy in practice.", "Meanwhile, it is highly efficient in terms of communication and computational overhead.", "Extensive experiments demonstrate its effectiveness on real data.", "In the next step, we plan to investigate the application of Harmony in a real use case such as Samsung's diagnostic info report app.", "[t] Generation of orthogonal set Inputinput Outputoutput Set $S$ of $d$ vectors.", "$S = \\lbrace [1, -1], [1, 1] \\rbrace $ $|S| < d$ $S^\\prime = \\emptyset $ $v \\in S$ $S^\\prime \\leftarrow S^\\prime \\cup \\lbrace v|| v, v || (-v)\\rbrace $ $S \\leftarrow S^\\prime $ $S$ Lemma 5 The set $S$ returned from Algorithm  is an orthogonal set.", "We prove this lemma by induction.", "For the base case, observe that the initial value of $S = \\lbrace [1, -1], [1, 1] \\rbrace $ is a orthogonal set.", "Now assume that $S= \\lbrace v_i\\rbrace $ is a orthogonal set.", "We will prove that $S^\\prime = \\lbrace v_i || v_i, v_i|| (-v_i)\\rbrace $ is also orthogonal set.", "For any $v^{\\prime \\prime }, v^\\prime \\in S^\\prime $ , consider the dot product $\\langle v^{\\prime \\prime }, v^\\prime \\rangle $ , there are two general cases: (i) $\\langle v^{\\prime \\prime }, v^\\prime \\rangle = \\langle v_i || \\pm v_i, v_j || \\pm v_j\\rangle $ and (ii) $\\langle v^{\\prime \\prime }, v^\\prime \\rangle = \\langle v_i ||v_i ,v_i || - v_i\\rangle $ .", "In both cases, the inner product equals zero.", "Thus, lemma is proved." ] ]
1606.05053
[ [ "Large Antenna Analysis of Multi-Cell Full-Duplex Networks" ], [ "Abstract We study a multi-cell multi-user MIMO full-duplex network, where each base station (BS) has multiple antennas with full-duplex capability supporting single-antenna users with either full-duplex or half-duplex radios.", "We characterize the up- and downlink ergodic achievable rates for the case of linear precoders and receivers.", "The rate analysis includes practical constraints such as imperfect self- interference cancellation, channel estimation error, training overhead and pilot contamination.", "We show that the 2X gain of full-duplex over half-duplex system remains in the asymptotic regime where the number of BS antennas grows infinitely large.", "We numerically evaluate the finite SNR and antenna performance, which reveals that full-duplex networks can use significantly fewer antennas to achieve spectral efficiency gain over the half-duplex counterparts.", "In addition, the overall full-duplex gains can be achieved under realistic 3GPP multi-cell network settings despite the increased interference introduced in the full-duplex networks." ], [ "Introduction", "One of the emerging techniques to significantly improve the spectral efficiency in wireless networks is full-duplex wireless communication [1].", "In-band full-duplex wireless allows simultaneous transmission and reception using the same frequency band, and thus opens up new design opportunities to increase the spectral efficiency of wireless systems.", "The feasibility of a (near-)full-duplex radio has been demonstrated by many groups, see e.g.", "[1], [2], [3], [4], [5], [6], [7] and references therein.", "A side-effect of the full-duplex operation is that additional interference is introduced because there are more simultaneous active links and hence there is a possibility that the full-duplex gain can be offset by the loss due to additional interference.", "In the example shown in Fig.", "REF , the uplink rate will be affected by the new interference from neighboring full-duplex BSs, and downlink rate will be affected by the new interference from uplink users (UE).", "Figure: The full-duplex BS in each cell has MM antennas, and the UE has single antenna with either full-duplex or half-duplex radio.", "Besides the conventional interference, there will be new BS-BS and UE-UE interference highlighted by the red dash lines.In this paper, we study if and how large antenna arrays at BSs can be used to manage the increased intra- and inter-cell interference in full-duplex enabled networks.", "Recently, the use of a very large antenna array at the BS has become very attractive [8], [9], [10], known as massive MIMO, where a BS has orders of magnitude more antennas compared with the current use.", "The large antenna array at the BS not only can increase the network capacity many-fold, but also enable a new network architecture to simplify baseband signal processing [8], eliminate inter-cell interference [11], and reduce node transmit power for energy saving [12].", "The experimental evidence on the benefits of massive MIMO has already sparked strong industry interest and 64-antenna configuration [13] is now being considered for 5G systems.", "Our contributions are three-fold.", "First, we provide a general analysis to characterize the uplink and downlink ergodic achievable rates in multi-cell multi-user MIMO (MU-MIMO) full-duplex networks.", "Focusing on computationally efficient linear receivers and precoders, we consider the case where each BS has multiple antennas with full-duplex capability, while each UE has a single antenna with either full-duplex or half-duplex radio.", "Practical constraints such as imperfect self-interference cancellation, channel estimation error, training overhead and pilot contamination are considered in our analysis.", "Second, we analyze the system performance in the asymptotic regime where the number of BS antennas grows infinitely large.", "We show that the transmit power of BSs and UEs can be scaled down with an increasing number of BS antennas to maintain a fixed asymptotic rate.", "The impact of imperfect self-interference cancellation at full-duplex BSs and full-duplex UEs, intra-cell and inter-cell interference in the multi-cell MU-MIMO full-duplex networks disappears as the number of BS antennas becomes infinitely large.", "Under the assumption of perfect channel knowledge, full-duplex system asymptotically achieves 2$\\times $ spectral efficiency gain over the half-duplex system.", "When channel estimation error and channel training overhead are considered, the 2$\\times $ asymptotic full-duplex gain is achieved when serving only full-duplex UEs.", "Lastly, we numerically evaluate the system performance in finite SNR and finite antenna regimes.", "Our numerical results reveal that full-duplex networks can use significantly fewer antennas to achieve spectral efficiency gain over the half-duplex counterparts.", "In addition, under realistic 3GPP multi-cell network scenarios [14], the overall full-duplex gains can be achieved despite the increased interference introduced in the full-duplex networks.", "Related work: There are two lines of work in the area of MU-MIMO full-duplex cellular networks.", "The first line of work focuses on information theoretic limits while the other line of work focuses on practical network design.", "Towards that end, we discuss [15], [16], [17], [18], [19], [20], [21], [22], [23]; note that not all papers are multi-cell studies.", "In [15], [16], [17], [18], [19], only a single-cell MIMO full-duplex case is studied.", "The authors in [15], [16], [17] study the information theoretic limits of a single-cell MU-MIMO full-duplex network, where the high signal-to-noise (SNR) approximation, i.e., degrees of freedom or multiplexing gains have been characterized.", "In [18], [19], the authors propose opportunistic scheduling and a distributed power control method to mitigate UE-UE interference in a single-cell MIMO full-duplex network.", "Authors in [20] present several schemes to manage UE-UE interference via out-of-band wireless side-channels in the full-duplex network.", "All the papers [21], [22], [23] study multi-cell full-duplex networks.", "In [21], scheduling and power control algorithms with BS cooperation in the multi-cell SISO full-duplex networks are investigated and [22] studies the MIMO case with 4 BS antennas.", "The performance analyses are based on extensive simulations, largely because of the challenge of analyzing complex scheduling and power control methods.", "In [23], the authors study the degrees of freedom region when the BSs have full coordination.", "This converts the problem into a network MIMO problem, and essentially allows one to treat the multi-cell problem as one giant MIMO cell.", "This, in turn, allows the use of interference alignment to achieve the highest possible degrees of freedom.", "While this approach provides insights into the maximum possible degrees of freedom, it relies on full coordination and proposes a very complex transmission method - both of them are extremely challenging to implement in practice.", "Compared to the above works, we focus on the case when there is no BS coordination and hence we cannot convert the problem to the more tractable single-cell problem.", "In addition, we allow only simple linear processing at the BSs, namely conjugate beamforming, and hence complex schemes like zero-forcing or interference-alignment cannot be employed.", "The rest of paper is organized as follows.", "We first describe the system model in Section .", "In Section , we characterize the ergodic achievable rates of uplink and downlink under both perfect and imperfect CSI assumptions.", "The large-scale system performance is studied in Section .", "The numerical results with realistic network evaluation are presented in Section .", "Section  concludes the paper." ], [ "Multi-cell Full-Duplex System Model", "We consider a multi-cell MU-MIMO full-duplex system with $L~(\\ge 1)$ cells, where each cell has one in-band full-duplex BS with $M$ antennas.", "In each cell, single-antenna UEs with either full-duplex or half-duplex radios are supported.", "Each BS can serve $K_f$ full-duplex (FD) users, $K_{h}^{u}$ half-duplex (HD) uplink users and $K_{h}^d$ half-duplex downlink users.", "We denote $\\mathcal {K}_{u}=\\lbrace \\underbrace{1,\\cdots ,K_{f}}_{\\mathrm {FD~UE}},\\underbrace{K_{f}+1,\\cdots ,K_f+K_h^u}_{\\mathrm {HD~UE}}\\rbrace $ as the set of all uplink users, where the first $K_f$ elements represent $K_f$ full-duplex users; and the last $K_{h}^{u}$ elements represent half-duplex uplink users.", "The set of all downlink users are denoted as $\\mathcal {K}_d=\\lbrace \\underbrace{1,\\cdots ,K_{f}}_{\\mathrm {FD~UE}},\\underbrace{K_{f}+1,\\cdots ,K_f+K_h^d}_{\\mathrm {HD~UE}}\\rbrace $ , where the first $K_f$ elements represent full-duplex users, and the last $K_{h}^{d}$ elements represent half-duplex downlink users.", "We further denote the set of full-duplex users as $\\mathcal {K}_f\\triangleq \\lbrace 1,\\cdots ,K_f\\rbrace $ and set of half-duplex downlink users as $\\mathcal {K}_h^d\\triangleq \\lbrace K_f+1,\\cdots ,K_f+K_h^d\\rbrace $ .", "The total number of uplink users is $|\\mathcal {K}_{u}|\\triangleq K_u$ , where $K_u=K_{f}+K_h^u$ and the total number of downlink users is $|\\mathcal {K}_{d}|\\triangleq K_d$ , where $K_d=K_{f}+K_h^d$ .", "The uplink and downlink data are transmitted over the same time-frequency slot with block fading.", "The analysis in this paper can be applied to wide-band channels like OFDM system.", "In this work, we will consider practical constraints on the full-duplex radio chains such as non-ideal power amplifier, oscillator phase noise, non-ideal digital-to-analog converter and analog-to-digital converter, which can be captured by the dynamic range model [24], [25].", "The dynamic range model approximates the imperfect full-duplex transmit radio chain as an additive white Gaussian “transmitter noise\" added to each transmit antenna.", "The variance of the transmitter noise is $\\kappa $  ($\\kappa \\ll 1$ ) times the power of the transmit signals, where $\\kappa $ is the dynamic range parameter.", "The full-duplex transmitter noise will propagate over the self-interference (SI) channel and become nontrivial compared to the receiver thermal noise.", "However, the effect of transmitter noise that propagates over the uplink/downlink channels can be ignored compared to the receiver thermal noise [25]." ], [ "Uplink", "The $j$ -th full-duplex BS will receive an $M\\times 1$ uplink signal vector $\\mathbf {y}_{u,j}^\\prime $ : $\\begin{aligned}\\mathbf {y}_{u,j}^\\prime =\\sum _{l=1}^L\\mathbf {G}_{u,jl} \\mathbf {x}_{u,l}+\\sum _{l=1}^{L}\\mathbf {V}_{jl}\\mathbf {x}_{d,l}+\\mathbf {V}_{jj}\\mathbf {e}_{bs,j}+\\mathbf {n}_{u,j},\\end{aligned}$ where $\\mathbf {x}_{u,l} = \\sqrt{P_u}\\mathbf {u}_l$ denotes the uplink transmit signal vector, $\\mathbf {u}_l\\triangleq [u_{l,1},\\cdots ,u_{l,K_u}]^T$ is a $K_u\\times 1$ vector consisting of uplink messages of the $K_u$ uplink users in the $l$ -th cell.", "Each user has an average power constraint $P_u$ , and $\\mathbb {E}(|{u}_{l,k}|^2)=1$ , for $k\\in \\mathcal {K}_u$ .", "$\\mathbf {x}_{d,l}$ is an $M\\times 1$ vector denoting downlink transmit signal in the $l$ -th cell.", "Each BS has an average power constraint $P_d$ .", "We assume $\\mathbf {G}_{u,jl}$ is an $M\\times K_u$ matrix denoting the channel between the uplink users in the $l$ -th cell and the $j$ -th BS.", "The propagation channel model in our system considers both small-scale fading due to mobility and multipath, and large-scale fading due to geometric attenuation and shadowing effect, thus allowing arbitrary cell layout.", "The uplink channel $\\mathbf {G}_{u,jl}$ encompasses independent small-scale fading and large-scale fading $(\\mathbf {G}_{u,jl})_{m,n}\\triangleq g_{u,jmln}=h_{u,jmln}\\sqrt{\\beta _{u,jln}},$ where $j,l\\in \\lbrace 1,\\cdots ,L\\rbrace ,m\\in \\lbrace 1,\\cdots ,M\\rbrace ,n\\in \\mathcal {K}_{u}$ .", "$h_{u,jmln}$ is the small-scale fading value between the $n$ -th uplink user in the $l$ -th cell and the $m$ -th antenna at the $j$ -th BS, following independent and identically distributed (i.i.d.)", "circularly-symmetric complex Gaussian distribution $\\mathcal {CN}(0,1)$ .", "We use ${\\beta _{u,jln}}$ to model the path loss and shadow fading between the $n$ -th uplink user in the $l$ -th cell and the $j$ -th BS, which is independent of $m$ .", "Such long-term parameters can be measured at the BS.", "The BS-BS channel $\\mathbf {V}_{j,l}$ is an $M\\times M$ matrix denoting the channel between the $l$ -th BS and the $j$ -th BS, and $\\mathbf {V}_{j,j}$ denotes the SI channel for the $j$ -th full-duplex BS.", "We assume $\\mathbf {V}_{j,l}$ contains i.i.d $\\mathcal {CN}(0,\\beta _{b,jl})$ elements.", "The imperfect full-duplex transmit front-end chain is modeled as transmit noise $\\mathbf {e}_{bs,j}\\in \\mathbb {C}^{M\\times 1}$ added to each transmit antenna at the $j$ -th full-duplex BS.", "And $\\mathbf {e}_{bs,j}$ will propagate through the SI channel and follow $\\mathcal {CN}\\left(0,\\frac{\\kappa P_d}{M}\\mathbf {I}_M\\right)$ assuming equal power allocation among downlink users, where $\\mathbf {I}_M$ denotes an $M\\times M$ identity matrix.", "The receiver thermal noise is denoted as $\\mathbf {n}_{u,j}$ which contains i.i.d $\\mathcal {CN}(0,\\sigma ^2)$ entries.", "The $j$ -th BS then performs SI cancellation by knowing its SI channel and its downlink signal.", "Hence from (REF ), we have $\\begin{aligned}\\mathbf {y}_{u,j}=\\sum _{l=1}^L\\mathbf {G}_{u,jl} \\mathbf {x}_{u,l}+\\sum _{l\\ne j}^{L}\\mathbf {V}_{j,l}\\mathbf {x}_{d,l}+\\mathbf {z}_{u,j},\\end{aligned}$ where $\\mathbf {z}_{u,j}\\sim \\mathcal {CN}\\left(0,\\left(\\sigma ^2+\\kappa P_d\\beta _{b,jj}\\right)\\mathbf {I}_M\\right)$ ." ], [ "Downlink", "The received signals for the $K_d$ downlink users in the $l$ -th cell can be expressed as a $K_d\\times 1$ vector $\\mathbf {y}_{d,l}$ , given by $\\begin{aligned}\\mathbf {y}_{d,l}=\\sum _{j=1}^L\\mathbf {G}^T_{d,jl}\\mathbf {x}_{d,j}+\\sum _{j=1}^L\\mathbf {F}_{lj}\\mathbf {x}_{u,j}+\\mathbf {J}_{l}+\\mathbf {n}_{d,l}, \\end{aligned}$ where $\\mathbf {G}_{d,jl}$ is an $M\\times K_d$ matrix denoting the channel between the downlink users in the $l$ -th cell and the $j$ -th BS, the downlink channel can be represented as $\\mathbf {G}_{d,jl}^T$ and we have $(\\mathbf {G}_{d,jl})_{m,k}^T\\triangleq g^T_{d,jmlk}=h^T_{d,jmlk}\\sqrt{\\beta _{d,jlk}},$ where $k\\in \\mathcal {K}_{d}$ , the superscript $``T\"$ denotes transpose.", "$h_{d,jmlk}$ is the small-scale fading value between the $k$ -th downlink user in the $l$ -th cell and the $m$ -th antenna at the $j$ -th BS, following $\\mathcal {CN}(0,1)$ ; ${\\beta _{d,jlk}}$ incorporates path loss and shadow fading between the $k$ -th downlink user in the $l$ -th cell and the $j$ -th BS .", "The UE-UE interference channel $\\mathbf {F}_{lj}$ is a $K_d\\times K_u$ matrix denoting the interference channel from $K_u$ uplink users in the $j$ -th to $K_d$ downlink users in the $l$ -th cell; $\\mathbf {J}_{l}=\\begin{pmatrix}\\mathsf {diag}(\\mathbf {S}_{ll})\\mathbf {e}_{ue,l} \\\\\\mathbf {0}\\end{pmatrix}$ is a $K_d\\times 1$ vector, where $\\mathbf {S}_{ll}$ is a $K_f\\times K_f$ matrix denoting the interference channels between full-duplex users in the $l$ -th cell.", "The diagonal elements of $\\mathbf {S}_{ll}$ constitute SI channels for each full-duplex user in the $l$ -th cell.", "And $\\mathbf {S}_{ll}$ is a sub-matrix of $\\mathbf {F}_{ll}$ , where $ \\left(\\mathsf {diag}(\\mathbf {S}_{ll})\\right)_k=(\\mathbf {F}_{ll})_{k,k}\\triangleq f_{lklk},~k\\in \\mathcal {K}_f$ .", "We assume each entry in $(\\mathbf {F}_{lj})_{k,n}\\triangleq f_{lkjn}$ models both the large-scale and fast fading channel coefficient between the $n$ -th uplink user in the $j$ -th cell and the $k$ -th downlink user in the $l$ -th cell, and follows i.i.d.", "$\\sim \\mathcal {CN}(0,\\beta _{I,lkjn})$ , where $j,l\\in \\lbrace 1,\\cdots ,L\\rbrace $ and $n\\in \\mathcal {K}_u,k\\in \\mathcal {K}_d$ .", "The transmit noise at each full-duplex user in the $l$ -th cell $\\mathbf {e}_{ue,l}\\in \\mathbb {C}^{K_f\\times 1}$ will propagate through the SI channel and follow $\\mathcal {CN}\\left(0,\\kappa P_u\\mathbf {I}_{K_f}\\right)$ .", "The receiver thermal noise $\\mathbf {n}_{d,l}$ follows $\\sim \\mathcal {CN}(0,\\sigma ^2\\mathbf {I}_{K_d})$ .", "Next, the $k$ -th full-duplex user in the $l$ -th cell where $k\\in \\mathcal {K}_f$ , will perform SI cancellation by subtracting out its own interference from the received signal in (REF ).", "Thus we have $\\begin{aligned}{r}_{d,lk}=\\sum _{j=1}^L\\mathbf {g}^T_{d,jlk}\\mathbf {x}_{d,j}+\\sum _{j=1}^L\\sum _{n=1}^{K_u}\\sqrt{P_u}f_{lkjn}{u}_{j,n}-\\sqrt{P_u}f_{lklk}{u}_{l,k}+{z}_{d,lk}, \\end{aligned}$ where $\\mathbf {g}_{d,jlk}$ is the $k$ -th column of matrix $\\mathbf {G}_{d,jl}$ and ${z}_{d,lk}\\sim \\mathcal {CN}\\left(0,\\left(\\sigma ^2+\\kappa P_u\\beta _{I,lklk}\\right)\\right).$ The expressions of the received downlink signals for full-duplex users and half-duplex users will differ, because for a full-duplex user, after SI cancellation there is no self UE-UE interference but an additional transmit noise is added to the received signal, while for a half-duplex downlink user, it will suffer the interference from all uplink users.", "Hence we can rewrite the received downlink signal for the $k^\\prime $ -th half-duplex user in the $l$ -th cell where $k^\\prime \\in \\mathcal {K}_h^d$ as $\\begin{aligned}{r}_{d,lk^\\prime }=\\sum _{j=1}^L\\mathbf {g}^T_{d,jlk^\\prime }\\mathbf {x}_{d,j}+\\sum _{{\\begin{array}{c}j=1\\end{array}}}^L\\sum _{{n=1}}^{K_u} \\sqrt{P_u}f_{lk^\\prime jn}u_{j,n}+{n}_{d,lk^\\prime }, \\end{aligned}$ where ${n}_{d,lk^\\prime }$ is the $k^\\prime $ -th element in $\\mathbf {n}_{d,l}$ ." ], [ "Achievable rates in full-duplex networks", "In this section, we will derive the up- and downlink ergodic achievable rates in multi-cell MU-MIMO full-duplex networks.", "We first study the case of perfect channel state information (CSI) where the channel information is obtained perfectly at no cost.", "Next, we consider the channel estimation error, where CSI is estimated from uplink pilot sequences.", "We assume synchronized reception from all cells, which will reflect the worst possible scenario of pilot contamination [11]." ], [ "Uplink with maximum ratio combining", "The $j$ -th BS will receive data transmitted by its associated $K_u$ uplink users, together with the interference from other cells.", "We apply a low-complexity linear receiver, i.e., maximum ratio combing for uplink signal detection.", "The $j$ -th BS will multiply the received signal after SI cancellation by the conjugate-transpose of its uplink channel $\\mathbf {G}_{u,jj}^H$ to obtain a $K_u\\times 1$ signal vector $\\begin{aligned}\\mathbf {r}_{u,j}&=\\mathbf {G}_{u,jj}^H\\mathbf {y}_{u,j}\\\\&=\\mathbf {G}_{u,jj}^H\\mathbf {G}_{u,jj}\\mathbf {x}_{u,j}+\\sum _{{l\\ne j}}\\mathbf {G}_{u,jj}^H\\mathbf {G}_{u,jl}\\mathbf {x}_{u,l}+\\sum _{l\\ne j}\\mathbf {G}_{u,jj}^H\\mathbf {V}_{j,l}\\mathbf {x}_{d,l}+\\mathbf {G}_{u,jj}^H\\mathbf {z}_{u,j}, \\end{aligned}$ where superscript “$H$ \" denotes conjugate-transpose, $\\mathbf {y}_{u,j}$ is given in (REF )." ], [ "Downlink with conjugate beamforming", "The $l$ -th BS will transmit an $M\\times 1$ downlink signal vector $\\mathbf {x}_{d,j}$ by precoding the downlink messages using a conjugate beamforming linear precoder $\\begin{aligned}\\mathbf {x}_{d,l}=\\frac{\\mathbf {G}^*_{d,ll}}{\\sqrt{\\gamma _l}}\\mathbf {s}_{d,l},\\end{aligned}$ where superscript “*\" denotes conjugate.", "$\\mathbf {s}_{d,l}=\\sqrt{\\frac{P_d}{K_d}}\\mathbf {d}_l$ , $\\mathbf {d}_l\\triangleq [d_{l,1},\\cdots ,d_{l,K_d}]^T$ is a $K_d\\times 1$ vector consisting of downlink messages of the $K_d$ downlink users in the $l$ -th cell with $\\mathbb {E}(|d_{l,k}|^2)=1,k\\in \\mathcal {K}_d$ ; $\\gamma _l$ is the $l$ -th cell power normalization factor to meet the average power constraint such that $\\mathbb {E}(\\mathbf {x}_{d,l}^H\\mathbf {x}_{d,l})=P_d$ , hence $\\gamma _l=\\frac{\\mathbb {E}(\\mathbf {d}_l^H\\mathbf {G}^T_{d,ll}\\mathbf {G}^*_{d,ll}\\mathbf {d}_l)}{K_d}=\\frac{{M\\sum _{k=1}^{K_d} \\beta _{d,llk}}}{K_d}$ .", "Substituting (REF ) into (REF ), we can first obtain the downlink signal at the $k$ -th full-duplex user in the $l$ -th cell where $k\\in \\mathcal {K}_f$ as $\\begin{aligned}{r}_{d,lk}=\\sum _{j=1}^L\\sum _{i=1}^{K_d}\\sqrt{\\frac{P_d}{K_d\\gamma _j}}\\mathbf {g}^T_{d,jlk}\\mathbf {g}^*_{d,jji}{d}_{j,i}+\\sum _{j=1}^L\\sum _{n=1}^{K_u}\\sqrt{P_u}f_{lkjn}{u}_{j,n}-\\sqrt{P_u}f_{lklk}{u}_{l,k}+{z}_{d,lk}.", "\\end{aligned}$ Next, we substitute (REF ) into (REF ) to obtain the received downlink signal at the $k^\\prime $ -th half-duplex user in the $l$ -th cell where $k^\\prime \\in \\mathcal {K}_h^d$ as $\\begin{aligned}{r}_{d,lk^\\prime }=\\sum _{j=1}^L\\sum _{i=1}^{K_d}\\sqrt{\\frac{P_d}{K_d\\gamma _j}}\\mathbf {g}^T_{d,jlk^\\prime }\\mathbf {g}^*_{d,jji}{d}_{j,i}+\\sum _{{\\begin{array}{c}j=1\\end{array}}}^L\\sum _{{n=1}}^{K_u} \\sqrt{P_u}f_{lk^\\prime jn}u_{j,n}+{n}_{d,lk^\\prime }.", "\\end{aligned}$" ], [ "Ergodic achievable rates", "We will treat the interference terms in (REF ), (REF ) and (REF ) as noise in our rate analysis.", "By coding over infinitely large number of the realizations of the fast ading channels, we can obtain the ergodic achievable rate of the $n$ -th uplink user in the $j$ -th cell (bits/s/Hz) as $\\begin{aligned}\\tilde{R}^{fd,p}_{u,jn}&= \\mathbb {E}\\left\\lbrace \\mathrm {log}_2\\left(1+\\frac{P_u\\left\\Vert \\mathbf {g}_{u,jjn}\\right\\Vert ^4}{P_u\\sum _{(l,m)\\ne (j,n)}|\\mathbf {g}^H_{u,jjn}\\mathbf {g}_{u,jlm}|^2+I_{bs-bs}+\\left\\Vert \\mathbf {g}_{u,jjn}\\right\\Vert ^2(\\sigma ^2+\\kappa P_d\\beta _{b,jj})}\\right)\\right\\rbrace , \\end{aligned}$ where $n\\in \\mathcal {K}_u$ , $I_{bs-bs}=\\sum _{l\\ne j}\\sum _{k\\in \\mathcal {K}_d}\\frac{P_d}{K_d \\gamma _l}|\\mathbf {g}^H_{u,jjn}\\mathbf {V}_{jl}\\mathbf {g}^*_{d,llk}|^2$ .", "The notation of $\\sum _{(l,m)\\ne (j,n)}$ denotes the summation over all tuples $(l,m)\\in \\lbrace 1,\\cdots ,L\\rbrace \\times \\mathcal {K}_u\\backslash \\lbrace (l=j,m=n)\\rbrace $ .", "Similarly, the downlink ergodic achievable rates of the $k$ -th full-duplex user and the $k^\\prime $ -th half-duplex user in the $l$ -th cell are respectively given as $\\begin{aligned}\\tilde{R}^{fd,p}_{d,lk}&= \\mathbb {E}\\left\\lbrace \\mathrm {log}_2\\left(1+\\frac{\\frac{P_d}{K_d\\gamma _l}\\left\\Vert \\mathbf {g}_{d,llk}\\right\\Vert ^4}{\\sum _{(j,i)\\ne (l,k)}\\frac{P_d}{K_d\\gamma _j}|\\mathbf {g}^T_{d,jlk}\\mathbf {g}^*_{d,jji}|^2+I_{ue-ue}(k)- P_u|f_{lklk}|^2+\\sigma ^2+\\kappa P_u\\beta _{I,lklk}}\\right)\\right\\rbrace ,\\\\\\tilde{R}^{fd,p}_{d,lk^\\prime }&= \\mathbb {E}\\left\\lbrace \\mathrm {log}_2\\left(1+\\frac{\\frac{P_d}{K_d\\gamma _l}\\left\\Vert \\mathbf {g}_{d,llk^\\prime }\\right\\Vert ^4}{\\sum _{(j,i)\\ne (l,k^\\prime )}\\frac{P_d}{K_d\\gamma _j}|\\mathbf {g}^T_{d,jlk^\\prime }\\mathbf {g}^*_{d,jji}|^2+I_{ue-ue}(k^\\prime )+\\sigma ^2}\\right)\\right\\rbrace ,\\end{aligned}$ where $k\\in \\mathcal {K}_f,k^\\prime \\in \\mathcal {K}_h^d$ .", "$I_{ue-ue}(k)=\\sum _{j=1}^L\\sum _{n\\in \\mathcal {K}_u}|P_uf_{lkjn}|^2$ .", "The notation of $\\sum _{(j,i)\\ne (l,k)}$ denotes the summation over all tuples $(j,i)\\in \\lbrace 1,\\cdots ,L\\rbrace \\times \\mathcal {K}_d\\backslash \\lbrace (j=l,i=k)\\rbrace $ .", "Proposition 1 For perfect CSI, lower bounds on the ergodic achievable rates of multi-cell MU-MIMO full-duplex networks when $M\\ge 3$ are given as $\\begin{aligned}\\text{Uplink user:}~R^{fd,p}_{u,jn}&= \\mathrm {log}_2\\bigg (1+\\frac{P_u(M-1)\\beta _{u,jjn}}{I_{up}+\\sigma ^2+\\kappa P_d\\beta _{b,jj}}\\bigg ),\\\\\\text{Downlink, FD user:}~R^{fd,p}_{d,lk}&= \\mathrm {log}_2\\Bigg (1+\\frac{\\eta _l P_d(M-1)(M-2)\\beta ^2_{d,llk}}{I_{down}(k)-P_u\\beta _{I,lklk}+\\sigma ^2+\\kappa P_u\\beta _{I,lklk}}\\Bigg ),\\\\\\text{Downlink, HD user:}~R^{fd,p}_{d,lk^\\prime }&= \\mathrm {log}_2\\Bigg (1+\\frac{\\eta _l P_d(M-1)(M-2)\\beta ^2_{d,llk^\\prime }}{I_{down}(k^\\prime )+\\sigma ^2}\\Bigg ),\\end{aligned}$ where $n\\in \\mathcal {K}_u,~k\\in \\mathcal {K}_f,~k^\\prime \\in \\mathcal {K}_h^d$ , $I_{up}=P_u\\sum _{(l,m)\\ne (j,n)}\\beta _{u,jlm}+P_d\\sum _{l\\ne j} \\beta _{b,jl}$ , $I_{down}(k) = \\sum _{i\\ne k,i\\in \\mathcal {K}_d}\\eta _l P_d\\beta _{d,llk}\\beta _{d,lli}(M-2)+P_d\\sum _{j\\ne l}\\beta _{d,jlk}+\\sum _{j=1}^L\\sum _{n\\in \\mathcal {K}_u}P_u\\beta _{I,lkjn}$ , and $\\eta _l=\\frac{1}{M\\sum _{i\\in \\mathcal {K}_d}\\beta _{d,lli}}$ .", "See Appendix REF ." ], [ "Imperfect channel state information", "In order to perform up- and downlink beamforming in MU-MIMO networks, the BS needs to know the up- and downlink channels for uplink coherent detection and downlink precoding.", "In this section, we will derive the ergodic achievable rates in the presence of channel estimation error.", "In the full-duplex system, the up- and downlink channels are estimated through uplink training sequences, and thus the pilot overhead is only proportional to the number of users.", "Simultaneous up-and downlink data transmission starts after uplink training, as shown in Fig.", "REF ." ], [ "Pilot-aided channel estimation", "During the uplink training period, within a coherence interval of $T$ , $K_{tot}\\triangleq K_u+K_d-K_f$ mutually orthogonal pilot sequences of length $\\tau $  ($\\tau \\ge K_{tot}$ ) symbols are used to estimate the channels between each BS and its associated UEs.", "The same set of pilot sequences will be reused by $L$ cells.", "The channel estimate will be corrupted by pilot contamination [11] due to the non-orthogonality of the reused pilots.", "We assume that each user has an average channel training power of $P_{tr}$ , which is a parameter that depends on the length of the pilot sequences.", "The $j$ -th BS will correlate the received signal from uplink training with the pilot sequences assigned for the $k$ -th user to obtain an $M$ -dimensional vector $\\mathbf {y}_{tr,jk}$ $\\mathbf {y}_{tr,jk}=\\mathbf {g}_{\\Phi ,jjk}+\\sum _{l\\ne j}^L\\mathbf {g}_{\\Phi ,jlk}+\\frac{\\mathbf {n}_{jk}}{\\sqrt{P_{tr}}},$ where $\\Phi \\in \\lbrace u,d\\rbrace $ , $k\\in \\mathcal {K}_u\\cup \\mathcal {K}_h^d$ , $\\mathbf {g}_{\\Phi , jjk}$ is the $k$ -th column of the channel matrix $\\mathbf {G}_{\\Phi , jj}$ , and $\\mathbf {n}_{jk}\\sim \\mathcal {CN}(\\mathbf {0},\\sigma ^2\\mathbf {I}_M)$ .", "For the full-duplex users, due to the channel reciprocity, we have $\\mathbf {g}_{u,jji}=\\mathbf {g}_{d,jji}$ for $i\\in \\mathcal {K}_f$ .", "Hence the estimated uplink channels for full-duplex users can also be used for downlink precoding.", "Figure: Uplink channel training in full-duplex networks.The MMSE channel estimate of the $k$ -th user in the $j$ -th cell $\\mathbf {g}_{\\Phi ,jjk}$ can be obtained as [26] $\\begin{aligned}\\hat{\\mathbf {g}}_{\\Phi ,jjk}=\\frac{P_{tr}\\beta _{\\Phi ,jjk}}{\\lambda _{\\Phi ,jk}}\\left(\\sum _{l=1}^L \\mathbf {g}_{\\Phi ,jlk}+\\frac{\\mathbf {n}_{jk}}{\\sqrt{P_{tr}}}\\right),\\end{aligned}$ where $\\lambda _{\\Phi ,jk}=\\sigma ^2+P_{tr}\\sum _{l=1}^L\\beta _{\\Phi ,jlk}$ , and $\\hat{\\mathbf {g}}_{\\Phi ,jjk}\\sim \\mathcal {CN}\\left(\\mathbf {0},\\frac{P_{tr}\\beta ^2_{\\Phi ,jjk}}{\\lambda _{\\Phi ,jk}}\\mathbf {I}_M\\right)$ Due to the orthogonality principle of the MMSE estimator, the true channel can be decomposed as the estimated channel and channel estimation error.", "Hence we have $\\begin{aligned}\\mathbf {g}_{\\Phi ,jjk}=\\hat{\\mathbf {g}}_{\\Phi ,jjk}+\\mathbf {\\epsilon }_{\\Phi ,jjk},\\end{aligned}$ where $\\mathbf {\\epsilon }_{\\Phi ,jjk}\\sim \\mathcal {CN}\\left(\\mathbf {0},\\frac{\\beta _{\\Phi ,jjk}\\left(\\sigma ^2+P_{tr}\\sum _{l\\ne j}\\beta _{\\Phi ,jlk}\\right)}{\\lambda _{\\Phi ,jk}}\\mathbf {I}_M\\right)$ , and the error $\\mathbf {\\epsilon }_{\\Phi ,jjk}$ is independent of the estimate $\\hat{\\mathbf {g}}_{\\Phi ,jjk}$ ." ], [ "Uplink and downlink data transmission", "In the uplink, the $j$ -th BS will apply maximum ratio combining detector by multiplying the conjugate-transpose of channel estimate $\\hat{\\mathbf {G}}_{u,jj}^H$ with the received signal $\\mathbf {r}_{u,j}&=&\\hat{\\mathbf {G}}_{u,jj}^H\\mathbf {y}_{u,j}\\\\&=&\\hat{\\mathbf {G}}_{u,jj}^H\\mathbf {G}_{u,jj}\\mathbf {x}_{u,j}+\\sum _{{l\\ne j}}\\hat{\\mathbf {G}}_{u,jj}^H\\mathbf {G}_{u,jl}\\mathbf {x}_{u,l}+\\sum _{l\\ne j}\\hat{\\mathbf {G}}_{u,jj}^H\\mathbf {V}_{j,l}\\mathbf {x}_{d,l}+\\hat{\\mathbf {G}}_{u,jj}^H\\mathbf {z}_{u,j}\\\\&=&\\hat{\\mathbf {G}}_{u,jj}^H\\hat{\\mathbf {G}}_{u,jj}\\mathbf {x}_{u,j}+\\hat{\\mathbf {G}}_{u,jj}^H\\mathbf {E}_{u,jj}\\mathbf {x}_{u,j}+\\sum _{{l\\ne j}}\\hat{\\mathbf {G}}_{u,jj}^H\\mathbf {G}_{u,jl}\\mathbf {x}_{u,l}+\\sum _{l\\ne j}\\hat{\\mathbf {G}}_{u,jj}^H\\mathbf {V}_{j,l}\\mathbf {x}_{d,l} \\\\&&+~\\hat{\\mathbf {G}}_{u,jj}^H\\mathbf {z}_{u,j}\\nonumber $ where the $k$ -th column of $\\mathbf {E}_{u,jj}$ is $\\mathbf {\\epsilon }_{u,jjk}$ , and (REF ) follows from $\\mathbf {G}_{u,jj}=\\hat{\\mathbf {G}}_{u,jj}+\\mathbf {E}_{u,jj}$ .", "In the downlink, the $l$ -th BS will employ conjugate beamforming to precode the downlink messages using the channel estimate $\\hat{\\mathbf {G}}_{d,ll}$ , and transmit an $M\\times 1$ signal vector $\\mathbf {x}_{d,l}$ , $\\begin{aligned}\\mathbf {x}_{d,l}=\\frac{\\hat{\\mathbf {G}}^*_{d,ll}}{\\sqrt{\\tilde{\\gamma }_l}}\\mathbf {s}_{d,l}, \\end{aligned}$ where $\\tilde{\\gamma }_l=\\frac{\\mathbb {E}(\\mathbf {d}_l^H\\hat{\\mathbf {G}}^T_{d,ll}\\hat{\\mathbf {G}}^*_{d,ll}\\mathbf {d}_l)}{K_d}=\\frac{M}{K_d}\\sum _{i=1}^{K_d}\\frac{P_{tr}\\beta _{d,lli}^2}{\\sigma ^2+P_{tr}\\sum _{j=1}^L\\beta _{d,lji}}$ .", "We assume that the downlink users do not have the channel estimate for reception (otherwise, additional pilot overhead needs to be considered), but are aware of the channel statistics.", "Each downlink user can perfectly track the average effective channel gain.", "Thus the received downlink signal can be decomposed as an average effective channel gain times the desired signal symbol, plus a composite term to denote effective noise as in [27].", "Substituting (REF ) into (REF ), the received downlink signal at the $k$ -th full-duplex user in the $l$ -th cell where $k\\in \\mathcal {K}_f$ can be written as $\\begin{aligned}{r}_{d,lk}&=\\sqrt{\\frac{P_d}{K_d\\tilde{\\gamma }_l}}\\mathbb {E}[\\mathbf {g}^T_{d,llk}\\mathbf {v}_{lk}]d_{l,k}+\\sqrt{\\frac{P_d}{K_d\\tilde{\\gamma }_l}}\\left(\\mathbf {g}^T_{d,llk}\\mathbf {v}_{lk}-\\mathbb {E}[\\mathbf {g}^T_{d,llk}\\mathbf {v}_{lk}]\\right)d_{l,k}\\\\&+\\sum _{(j,i)\\ne (l,k)}\\sqrt{\\frac{P_d}{K_d\\tilde{\\gamma }_j}}\\mathbf {g}^T_{d,jlk}{\\mathbf {v}}_{ji}d_{j,i}+\\sum _{j=1}^L\\sum _{n=1}^{K_u}\\sqrt{P_u}f_{lkjn}{u}_{j,n}-\\sqrt{P_u}f_{lklk}{u}_{l,k}+z_{d,lk}.\\end{aligned}$ where $\\mathbf {v}_{lk}=\\hat{\\mathbf {g}}^*_{d,llk}$ for conjugate beamforming precoder.", "Similarly, we substitute (REF ) into (REF ) to obtain the downlink signal at the $k^\\prime $ -th half-duplex user in the $l$ -th cell where $k^\\prime \\in \\mathcal {K}_h^d$ $\\begin{aligned}{r}_{d,lk^\\prime }&=\\sqrt{\\frac{P_d}{K_d\\tilde{\\gamma }_l}}\\mathbb {E}[\\mathbf {g}^T_{d,llk^\\prime }\\mathbf {v}_{lk^\\prime }]d_{l,k^\\prime }+\\sqrt{\\frac{P_d}{K_d\\tilde{\\gamma }_l}}\\left(\\mathbf {g}^T_{d,llk^\\prime }\\mathbf {v}_{lk^\\prime }-\\mathbb {E}[\\mathbf {g}^T_{d,llk^\\prime }\\mathbf {v}_{lk^\\prime }]\\right)d_{l,k^\\prime }\\\\&+\\sum _{(j,i)\\ne (l,k^\\prime )}\\sqrt{\\frac{P_d}{K_d\\tilde{\\gamma }_j}}\\mathbf {g}^T_{d,jlk^\\prime }{\\mathbf {v}}_{ji}d_{j,i}+\\sum _{{\\begin{array}{c}j=1\\end{array}}}^L\\sum _{{n=1}}^{K_u} \\sqrt{P_u}f_{lk^\\prime jn}u_{j,n}+{n}_{d,lk^\\prime }.\\end{aligned}$" ], [ "Ergodic achievable rates", "For the uplink, the channel estimate will be treated as the true channel.", "Hence we can obtain the ergodic achievable rate for the $n$ -th uplink user in the $j$ -th cell as follows: $\\begin{aligned}\\tilde{R}^{fd,ip}_{u,jn}&= \\mathbb {E}\\left\\lbrace \\mathrm {log}_2\\left(1+\\frac{P_u\\left\\Vert \\hat{\\mathbf {g}}_{u,jjn}\\right\\Vert ^4}{P_u|\\hat{\\mathbf {g}}_{u,jjn}^H\\mathbf {\\epsilon }_{u,jjn}|^2+P_u\\sum _{(l,m)\\ne (j,n)}|\\hat{\\mathbf {g}}^H_{u,jjn}\\mathbf {g}_{u,jlm}|^2+I^{ip}_{bs-bs}+N}\\right)\\right\\rbrace , \\end{aligned}$ where $n\\in \\mathcal {K}_u$ , $I^{ip}_{bs-bs}=\\sum _{l\\ne j}\\sum _{k\\in \\mathcal {K}_d}\\frac{P_d}{K_d \\tilde{\\gamma }_l}|\\hat{\\mathbf {g}}^H_{u,jjn}\\mathbf {V}_{jl}\\hat{\\mathbf {g}}^*_{d,llk}|^2,~N=\\left\\Vert \\hat{\\mathbf {g}}_{u,jjn}\\right\\Vert ^2(\\sigma ^2+\\kappa P_d\\beta _{b,jj})$ .", "For the downlink, the effective noise is uncorrelated with the signal, using worst-case independent Gaussian noise results in [27], the downlink ergodic achievable rates of the $k$ -th full-duplex user and the $k^\\prime $ -th half-duplex user in the $l$ -th cell are respectively given as $\\begin{aligned}{R}_{d,lk}^{fd,ip}&= \\mathrm {log}_2\\left(1+\\frac{\\frac{P_d}{K_d\\tilde{\\gamma }_l}\\left|\\mathbb {E}\\left[\\mathbf {g}^T_{d,llk}\\hat{\\mathbf {g}}^*_{d,llk}\\right]\\right|^2}{\\frac{P_d}{K_d\\tilde{\\gamma }_l}\\mathsf {var}\\left[\\mathbf {g}^T_{d,llk}\\hat{\\mathbf {g}}^*_{d,llk}\\right]+I_{other}(k)- P_u\\mathbb {E}\\left[{f_{lklk}}^2\\right]+\\sigma ^2+\\kappa P_u\\beta _{I,lklk}}\\right),\\\\{R}_{d,lk^\\prime }^{fd,ip}&= \\mathrm {log}_2\\left(1+\\frac{\\frac{P_d}{K_d\\tilde{\\gamma }_l}\\left|\\mathbb {E}\\left[\\mathbf {g}^T_{d,llk^\\prime }\\hat{\\mathbf {g}}^*_{d,llk^\\prime }\\right]\\right|^2}{\\frac{P_d}{K_d\\tilde{\\gamma }_l}\\mathsf {var}\\left[\\mathbf {g}^T_{d,llk^\\prime }\\hat{\\mathbf {g}}^*_{d,llk^\\prime }\\right]+I_{other}(k^\\prime )+\\sigma ^2}\\right),\\end{aligned}$ where $k\\in \\mathcal {K}_f,k^\\prime \\in \\mathcal {K}_h^d$ .", "$I_{other}(k)=\\sum _{(j,i)\\ne (l,k)}\\frac{P_d}{K_d\\tilde{\\gamma }_j}\\mathbb {E}\\left[{\\mathbf {g}^T_{d,jlk}\\hat{\\mathbf {g}}^*_{d,jji}}^2\\right]+\\sum _{j=1}^L\\sum _{n\\in \\mathcal {K}_u}P_u\\mathbb {E}\\left[{f_{lkjn}}^2\\right]$ .", "The notation $\\mathsf {var}[x]\\triangleq \\mathbb {E}[(x -\\mu )(x -\\mu )^H],~\\mathbb {E}[x]=\\mu $ .", "Proposition 2 For imperfect CSI with MMSE estimation, the following sets of rates are achievable in multi-cell MU-MIMO full-duplex networks when $M\\ge 3$ $\\begin{aligned}\\text{Uplink user:}~R^{fd,ip}_{u,jn}&= \\mathrm {log}_2\\bigg (1+\\frac{P_{tr}P_u(M-1)\\beta ^2_{u,jjn}}{P_u\\beta _{u,jjn}(\\sigma ^2+P_{tr}\\sum _{l\\ne j}\\beta _{u,jln})+\\tilde{I}_{up}+\\tilde{N}}\\bigg ),\\\\\\text{Downlink, FD user:}~R^{fd,ip}_{d,lk}&= \\mathrm {log}_2\\Bigg (1+\\frac{\\tilde{\\eta }_lP_{tr} P_dM\\beta ^4_{d,llk}}{\\lambda ^2_{d,lk}(\\tilde{I}_{down}(k)-P_u\\beta _{I,lklk}+\\sigma ^2+\\kappa P_u\\beta _{I,lklk})}\\Bigg ),\\\\\\text{Downlink, HD user:}~R^{fd,ip}_{d,lk^\\prime }&= \\mathrm {log}_2\\Bigg (1+\\frac{\\tilde{\\eta }_lP_{tr} P_dM\\beta ^4_{d,llk^\\prime }}{\\lambda ^2_{d,lk^\\prime }(\\tilde{I}_{down}(k^\\prime )+\\sigma ^2)}\\Bigg ),\\end{aligned}$ where $\\tilde{I}_{up}=P_{tr}P_u\\sum _{l\\ne j}\\left[(M+1)\\beta ^2_{u,jln}+\\sum _{l_1\\ne l}\\beta _{u,jl_1 n}\\beta _{u,jln}+\\frac{\\beta _{u,jln}\\sigma ^2}{P_{tr}}\\right]$ , $\\tilde{N}=\\lambda _{u,jn}(P_d\\sum _{l\\ne j} \\beta _{b,jl}+P_u\\sum _{l=1}^L\\sum _{m\\in \\mathcal {K}_u,m\\ne n}\\beta _{u,jlm}+\\sigma ^2+\\kappa P_d\\beta _{b,jj})$ , $\\tilde{I}_{down}(k) = \\frac{\\tilde{\\eta }_l P_d\\beta ^3_{d,llk}}{\\lambda _{d,lk}}+\\sum _{j\\ne l}\\frac{\\tilde{\\eta }_jP_{tr}P_d(M+1)\\beta ^2_{d,jlk}\\beta ^2_{d,jjk}}{\\lambda ^2_{d,jk}}+\\sum _{j\\ne l}\\frac{\\tilde{\\eta }_j P_{d}\\beta _{d,jjk}^2(\\sigma ^2+P_{tr}\\sum _{l_1\\ne l}\\beta _{d,jl_1k})\\beta _{d,jlk}}{\\lambda ^2_{d,jk}}+\\sum _{j=1}^L\\sum _{i\\ne k,i\\in \\mathcal {K}_d}\\frac{\\tilde{\\eta }_jP_d\\beta ^2_{d,jji}\\beta _{d,jlk}}{\\lambda _{d,ji}}+\\sum _{j=1}^L\\sum _{n\\in \\mathcal {K}_u}P_u\\beta _{I,lk jn}$ .", "$\\tilde{\\eta }_l=\\left(\\sum _{i\\in \\mathcal {K}_d}\\frac{\\beta ^2_{d,lli}}{\\lambda _{d,li}}\\right)^{-1}$ , $n\\in \\mathcal {K}_u,~k\\in \\mathcal {K}_f,~k^\\prime \\in \\mathcal {K}_h^d.$ See Appendix REF .", "We use TDD system as a baseline half-duplex system for comparison since both full-duplex and TDD systems use uplink training for channel estimation.FDD system requires downlink training with channel feedback which incurs a much higher pilot overhead as the overhead is not only proportional to the number of users but also the number of BS antennas.", "Compared with a full-duplex system, in a TDD system, the uplink does not have BS-BS interference and transmit noise which accounts for imperfect FD radios, and the downlink does not have UE-UE interference and transmit noise.", "We assume the corresponding TDD system has an uplink and downlink user sets of $\\mathcal {K}_u$ and $\\mathcal {K}_d$ , respectively, with the same total number of uplink and downlink users as in the full-duplex system, i.e., $|\\mathcal {K}_u|=K_u$ and $|\\mathcal {K}_d|=K_d$ .", "For the TDD baseline system, the up- and downlink transmissions are in two different time slots, and we assume an equal time sharing between up- and downlink transmission.", "By treating interference as noise, the up- and downlink ergodic achievable rates in a TDD system under perfect CSI assumption are given as $\\begin{aligned}\\text{Uplink user:}~{R}^{tdd,p}_{u,jn}&=\\frac{1}{2}\\mathbb {E}\\left\\lbrace \\mathrm {log}_2\\left(1+\\frac{P_u\\left\\Vert \\mathbf {g}_{u,jjn}\\right\\Vert ^4}{P_u\\sum _{(l,m)\\ne (j,n)}|\\mathbf {g}^H_{u,jjn}\\mathbf {g}_{u,jlm}|^2+\\left\\Vert \\mathbf {g}_{u,jjn}\\right\\Vert ^2\\sigma ^2}\\right)\\right\\rbrace ,\\\\\\text{Downlink user:}{R}^{tdd,p}_{d,lk}&= \\frac{1}{2} \\mathbb {E}\\left\\lbrace \\mathrm {log}_2\\left(1+\\frac{\\frac{P_d}{K_d\\gamma _l}\\left\\Vert \\mathbf {g}_{d,llk}\\right\\Vert ^4}{\\sum _{(j,i)\\ne (l,k)}\\frac{P_d}{K_d\\gamma _j}|\\mathbf {g}^T_{d,jlk}\\mathbf {g}^*_{d,jji}|^2+\\sigma ^2}\\right)\\right\\rbrace ,\\end{aligned}$ where $\\gamma _l=\\frac{{M\\sum _{k=1}^{K_d} \\beta _{d,llk}}}{K_d}$ , $n\\in \\mathcal {K}_u,~k\\in \\mathcal {K}_d$ .", "When training sequences are used for channel estimation, within a coherence interval $T$ , $K_u$ mutually orthogonal pilot sequences of length $\\tau _u$ ($\\tau _u\\ge K_u$ ) are used for the uplink users and $K_d$ mutually orthogonal pilot sequences of length $\\tau _d$ ($\\tau _d\\ge K_d$ ) are used for the downlink users.", "The same sets of training sequences are also reused by all cells.", "The corresponding up- and downlink ergodic achievable rates with MMSE estimation in TDD system are given below, $\\begin{aligned}\\text{Uplink user:}~ {R}^{tdd,ip}_{u,jn}&=\\frac{1}{2} \\mathbb {E}\\left\\lbrace \\mathrm {log}_2\\left(1+\\frac{P_u\\left\\Vert \\hat{\\mathbf {g}}_{u,jjn}\\right\\Vert ^4}{P_u|\\hat{\\mathbf {g}}_{u,jjn}^H\\mathbf {\\epsilon }_{u,jjn}|^2+P_u\\sum _{(l,m)\\ne (j,n)}|\\hat{\\mathbf {g}}^H_{u,jjn}\\mathbf {g}_{u,jlm}|^2+N^\\prime }\\right)\\right\\rbrace ,\\\\\\text{Downlink user:}~{R}^{tdd,ip}_{d,lk}&= \\frac{1}{2} \\mathbb {E}\\left\\lbrace \\mathrm {log}_2\\left(1+\\frac{\\frac{P_d}{K_d\\tilde{\\gamma }_l}\\left|\\mathbb {E}\\left[\\mathbf {g}^T_{d,llk}\\hat{\\mathbf {g}}^*_{d,llk}\\right]\\right|^2}{Z^\\prime +\\sigma ^2}\\right)\\right\\rbrace ,\\end{aligned}$ where $N^\\prime =\\left\\Vert \\hat{\\mathbf {g}}_{u,jjn}\\right\\Vert ^2\\sigma ^2$ , $n\\in \\mathcal {K}_u,~k\\in \\mathcal {K}_d$ , and $Z^\\prime =\\frac{P_d}{K_d\\tilde{\\gamma }_l}|\\mathbf {g}^T_{d,llk}\\hat{\\mathbf {g}}^*_{d,llk}-\\mathbb {E}[\\mathbf {g}^T_{d,llk}\\hat{\\mathbf {g}}^*_{d,llk}]|^2+\\sum _{(j,i)\\ne (l,k)}\\frac{P_d}{K_d\\tilde{\\gamma }_j}|\\mathbf {g}^T_{d,jlk}{\\mathbf {v}}_{ji}d_{j,i}|^2$ .", "Note that the uplink and downlink ergodic achievable rates in the baseline TDD system follow [12] and [27] but without lower bounding the achievable rates, as we will use them to compute the full-duplex versus half-duplex rate ratios in the next section." ], [ "Large-Scale full-duplex System Performance", "While the general ergodic achievable rates in the full-duplex system are given in the previous section, it is of interest to study the impact of large antenna arrays at BSs as the next generation of wireless systems will employ significantly more antennas at the infrastructure nodes [13].", "In this section, we will investigate large-scale system performance as the number of full-duplex BS antennas, $M$ , becomes arbitrarily large." ], [ "Leveraging large antenna arrays for multi-cell interference mitigation", "In this section, we will show that using large BS antenna arrays can mitigate multi-cell interference in the full-duplex system.", "With increasing number of BS antennas, the signal strength will become stronger due to beamforming, and the transmit power can be scaled down proportionally to maintain the same quality-of-service.", "In what follows, we will present two theorems which characterize the asymptotic full-duplex spectral efficiency gain over TDD system under both perfect and imperfect CSI assumptions.", "Theorem 1 (Asymptotic FD spectral efficiency gain with perfect CSI) For perfect CSI, we scale the transmit power of each node proportional to $1/M$ as $P_u=E_u/M$ and $P_d=E_d/M$ , where $E_u$ and $E_d$ are fixed.", "As $M\\rightarrow \\infty $ , the full-duplex spectral efficiency gains over the TDD system in the uplink and downlink, denoted by $\\mathsf {Gain}^p_{u}$ and $\\mathsf {Gain}^p_{d}$ , respectively, are given below, where fixed asymptotic up- and downlink rates are maintained.", "$\\begin{aligned}\\mathsf {Gain}^{p}_{u}&\\triangleq \\lim _{M\\rightarrow \\infty }\\frac{\\sum _{j=1}^L\\sum _{n\\in \\mathcal {K}_u}R^{fd,p}_{u,jn}}{\\sum _{j=1}^L\\sum _{n\\in \\mathcal {K}_u}R_{u,jn}^{tdd,p}}=2,\\\\\\mathsf {Gain}^p_{d}&\\triangleq \\lim _{M\\rightarrow \\infty }\\frac{\\sum _{l=1}^L\\sum _{k\\in \\mathcal {K}_d}R^{fd,p}_{d,lk}}{\\sum _{l=1}^L\\sum _{k\\in \\mathcal {K}_d}R_{d,lk}^{tdd,p}}=2.\\end{aligned}$ The asymptotic ergodic achievable rate of the $n$ -th uplink user in the $j$ -th cell and the achievable rate of the $k$ -th downlink user in the $l$ -th cell are given below, $\\begin{aligned}R^{fd,p}_{u,jn}&\\rightarrow \\mathrm {log}_2\\left(1+\\frac{\\beta _{u,jjn}E_u}{\\sigma ^2}\\right), ~n\\in \\mathcal {K}_u\\\\R^{fd,p}_{d,lk}&\\rightarrow \\mathrm {log}_2\\left(1+\\frac{\\beta ^2_{d,llk}E_d}{\\sum _{i\\in \\mathcal {K}_d}\\beta _{d,lli}\\sigma ^2}\\right), ~k\\in \\mathcal {K}_d.\\end{aligned}$ For two mutually independent $M\\times 1$ vectors $a\\triangleq [a_1,\\cdots ,a_M]^T$ and $b\\triangleq [b_1,\\cdots ,b_M]^T$ whose entries are i.i.d.", "zero-mean random variables with $\\mathbb {E}(|a_i|^2=\\sigma ^2_a)$ and $\\mathbb {E}(|b_i|^2=\\sigma ^2_b),~\\forall i\\in \\lbrace 1,\\cdots ,M\\rbrace $ , by law of large numbers, we have the following almost sure convergence according to [28], $\\begin{aligned}\\frac{1}{M}\\mathbf {a}^H\\mathbf {a}\\xrightarrow{} \\sigma ^2_a,~\\frac{1}{M}\\mathbf {a}^H\\mathbf {b}\\xrightarrow{} 0,~\\text{as}~M\\rightarrow \\infty .\\end{aligned}$ Under the favorable propagation condition in [11] where the fast fading channels are i.i.d.", "with zero mean and unit variance, from (REF ), in the limit of $M$ , we have $\\lim _{M\\rightarrow \\infty }\\frac{\\mathbf {G}_{u,jl}^H\\mathbf {G}_{u,jl}}{M}=\\mathbf {D}_{u,jl}\\delta _{jl}$ , $\\lim _{M\\rightarrow \\infty }\\frac{\\mathbf {G}_{d,jl}^H\\mathbf {G}_{d,jl}}{M}=\\mathbf {D}_{d,jl}\\delta _{jl}$ , where $\\mathbf {D}_{u,jl}$ is a $K_u\\times K_u$ diagonal matrix, and each diagonal element is $(\\mathbf {D}_{u,jl})_{n}=\\beta _{u,jln}$ ; $\\mathbf {D}_{d,jl}$ is a $K_d\\times K_d$ diagonal matrix, and each diagonal element is $(\\mathbf {D}_{d,jl})_{k}=\\beta _{d,jlk}$ ; $\\delta _{jl}=1$  for $j=l$ , $\\delta _{jl}=0$  for $j\\ne l$ .", "By substituting $P_u=E_u/M$ and $P_d=E_d/M$ into (REF ), (REF ) and (REF ), we can obtain the desired results as $M\\rightarrow \\infty $ .", "Remark 1 When a full-duplex BS employs a large antenna array, the transmit power of each node can possibly be scaled down proportionally to $1/M^{C}$ to achieve the same rate.", "As $M\\rightarrow \\infty $ , $1/M$ is the fastest rate at which we can scale down the transmit power to maintain fixed asymptotic up- and downlink rates.", "Otherwise, if $C>1$ , the rates will go to zero and if $C<1$ , the rates will go to infinity.", "The full-duplex system preserves the same power scaling law as in the half-duplex system [12] despite increased interference in the multi-cell full-duplex networks.", "Full-duplex system asymptotically doubles the spectral efficiency over TDD system since all $K_u$ uplink streams and $K_d$ downlink streams can be supported in the same time-frequency slot.", "Theorem 2 (Asymptotic FD spectral efficiency gain with imperfect CSI) For imperfect CSI with MMSE estimation, we scale the power of each node for channel training and data transmission proportional to $1/\\sqrt{M}$ as $P_{tr}=E_{tr}/{\\sqrt{M}}$ , $P_u=E_u/{\\sqrt{M}}$ and $P_d={E_d}/{\\sqrt{M}}$ , where $E_{tr}$ , $E_u$ and $E_d$ are fixed.", "Within a coherence interval $T$ , let $\\tau _u=K_u$ , $\\tau _d=K_d$ and $\\tau =K_u+K_d-K_f$ .", "As $M\\rightarrow \\infty $ , the full-duplex spectral efficiency gains over the TDD system in the uplink and downlink, denoted by $\\mathsf {Gain}^{ip}_{u}$ and $\\mathsf {Gain}^{ip}_{d}$ , respectively, are given below, where fixed asymptotic up- and downlink rates are maintained.", "$\\begin{aligned}\\mathsf {Gain}^{ip}_{u}&\\triangleq \\lim _{M\\rightarrow \\infty }\\frac{\\frac{T-\\tau }{T}\\sum _{j=1}^L\\sum _{n\\in \\mathcal {K}_u}R^{fd,ip}_{u,jn}}{\\frac{T-\\tau _u}{T}\\sum _{j=1}^L\\sum _{n\\in \\mathcal {K}_u}R_{u,jn}^{tdd,ip}}=2\\Big (1-\\frac{K_h^d}{T-K_u}\\Big ),\\\\\\mathsf {Gain}^{ip}_{d}&\\triangleq \\lim _{M\\rightarrow \\infty }\\frac{\\frac{T-\\tau }{T}\\sum _{l=1}^L\\sum _{k\\in \\mathcal {K}_d}R^{fd,ip}_{d,lk}}{\\frac{T-\\tau _d}{T}\\sum _{l=1}^L\\sum _{k\\in \\mathcal {K}_d}R_{d,lk}^{tdd,ip}}=2\\Big (1-\\frac{K_h^u}{T-K_d}\\Big ).\\end{aligned}$ The asymptotic ergodic achievable rate of the $n$ -th uplink user in the $j$ -th cell and the achievable rate of the $k$ -th downlink user in the $l$ -th cell are given below, $\\begin{aligned}R_{u,jn}^{fd,ip}&\\rightarrow \\mathrm {log}_2\\left(1+\\frac{E_{tr}E_u\\beta ^2_{u,jjn}}{E_{tr} E_u\\sum _{l\\ne j} \\beta ^2_{u,jln}+\\sigma ^4}\\right),\\\\R_{d,lk}^{fd,ip}&\\rightarrow \\mathrm {log}_2\\left(1+\\frac{E_{tr}E_d \\beta ^4_{d,llk}}{Z_l\\left(\\sum _{j\\ne l} \\frac{E_{tr}E_d\\beta ^2_{d,jjk}\\beta ^2_{d,jlk}}{Z_j}+\\sigma ^6\\right)}\\right),\\end{aligned}$ where $Z_l=\\sum _{i\\in \\mathcal {K}_d}\\frac{\\beta ^2_{d,lli}}{\\sigma ^2},~n\\in \\mathcal {K}_u,~k\\in \\mathcal {K}_d$ .", "When channel estimation overhead is taken into account, the up- and downlink spectral efficiency in the full-duplex system are $\\frac{T-\\tau }{T}\\sum _{j=1}^L\\sum _{n\\in \\mathcal {K}_u}R^{ip}_{u,jn}$ and $\\frac{T-\\tau }{T}\\sum _{l=1}^L\\sum _{k\\in \\mathcal {K}_d}R^{ip}_{d,lk}$ , respectively.", "While for the TDD system, since uplink and downlink transmission are in two different time slots, the corresponding up- and downlink spectral efficiency are $\\frac{T-\\tau _u}{T}\\sum _{j=1}^L\\sum _{n\\in \\mathcal {K}_u}R_{u,jn}^{tdd,ip}$ and $\\frac{T-\\tau _d}{T}\\sum _{l=1}^L\\sum _{k\\in \\mathcal {K}_d}R_{d,lk}^{tdd,ip}$ , respectively.", "Following similar steps used in the proof of Theorem REF , substituting $P_{tr}=E_{tr}/{\\sqrt{M}}$ , $P_u=E_u/{\\sqrt{M}}$ , $P_d={E_d}/{\\sqrt{M}}$ , $\\tau _u=K_u$ , $\\tau _d=K_d$ and $\\tau =K_u+K_d-K_f$ into (REF ), (REF ) and (REF ), and using the fact that $K_u=K_h^u+K_f$ and $K_d=K_h^d+K_f$ , the desired results can be obtained as $M\\rightarrow \\infty $ .", "Corollary 1 When $K_u=K_d=K_f$ , i.e., when only full-duplex users are served, full-duplex system asymptotically doubles the spectral efficiency over the TDD system under the imperfect CSI assumption, where $\\mathsf {Gain}^{ip}_{u}=\\mathsf {Gain}^{ip}_{d}=2$ .", "However, if there exists half-duplex users (i.e., $K_h^d>0$ or $K_h^u>0$ ), then $\\mathsf {Gain}^{ip}_{u/d}<2$ , and the spectral efficiency gains decrease as the number of half-duplex users increases due to channel training overhead.", "Remark 2 In case of imperfect CSI, the fastest rate at which we can cut down the transmit power to maintain fixed asymptotic up- and downlink rates is $1/\\sqrt{M}$ .", "As the transmit power scale down with increasing $M$ , the impact of imperfect SI cancellation, intra-cell and inter-cell interference in the multi-cell MU-MIMO full-duplex networks will vanish as $M\\rightarrow \\infty $ ." ], [ "System with only full-duplex users", "For tractability, we consider a homogeneous network by assuming all links in the same cell have the same channel statistic and all the cross-cell links have the same channel statistics, i.e., let $\\beta _{u,lln}=\\beta _{d,llk}=\\beta _{I,lkln}=\\beta _{b,ll}=1$ , $\\beta _{u,jln}=\\beta _{d,jlk}=\\beta _{I,lkjn}=\\beta _{b,jl}=\\beta $ , where $\\beta \\in [0,1]$ for $j\\ne l\\in [1,\\cdots ,L],~n\\in \\mathcal {K}_u$ , $k\\in \\mathcal {K}_d$ , and the noise variance is set as $\\sigma ^2=1$ .", "We consider a simple scenario where all users are full-duplex and the number of full-duplex users in each cell is $K_f\\triangleq K$ .", "Similar results can be derived in case of mixed half-duplex UEs or only half-duplex UEs.", "With such simplification, we can compute the up- and downlink spectral efficiency per cell (bits/s/Hz/cell) when CSI is perfect as $\\begin{aligned}R^{fd,p}_{u}&=K\\mathrm {log}_2\\left(1+\\frac{P_u(M-1)}{P_u(K-1)+(L-1)\\beta (P_uK+P_d)+\\kappa P_d+1}\\right),\\\\R^{fd,p}_{d}&=K\\mathrm {log}_2\\left(1+\\frac{P_d(M-1)(M-2)}{P_d(K-1)(M-2)+MK(L-1)\\beta P_d+V+MK(\\kappa P_u+1)}\\right),\\end{aligned}$ where $V=MK\\left(K-1+(L-1)K\\beta \\right)P_u$ .", "When CSI is imperfect, the up- and downlink spectral efficiency per cell (bits/s/Hz/cell) are $\\begin{aligned}R^{fd,ip}_{u}&\\!=\\frac{K(T-\\tau )}{T}\\mathrm {log}_2\\left(\\!1+\\frac{P_{tr} P_u(M-1)}{P_{tr} P_u\\left(K\\bar{L}^2-1+\\beta (\\bar{L}-1)M\\right)+J}\\right),\\\\R^{fd,ip}_{d}&\\!=\\frac{K(T-\\tau )}{T}\\mathrm {log}_2\\left(\\!1+\\frac{P_{tr}P_dM}{\\left(1+P_{tr}\\bar{L}\\right)U_1+\\left(\\bar{L}-1\\right)U_2}\\right),\\end{aligned}$ where $J=P_d(1+P_{tr}\\bar{L})(\\bar{L}-1)+P_uK\\bar{L}+P_{tr}(1+ \\kappa P_d)\\bar{L}+\\kappa P_d+1$ , $U_1=P_d\\left(1+(K-1)\\bar{L}\\right)+K\\left(P_u(K-1)+P_u(\\bar{L}-1)K+1+\\kappa P_u\\right),$ $U_2=P_{tr}P_d\\left(M\\beta +\\bar{L}\\right)+P_d,~\\bar{L}=1+(L-1)\\beta $ .", "We first evaluate the tightness of our derived bounds on achievable rates in the homogenous networks.", "We consider a scenario with $L=7$ cells and the inter-cell interference level $\\beta =0.3$ , each cell has $K=5$ full-duplex UEs.", "The up- and downlink transmit power are assumed as $P_{tr}=P_u=10$  dB and $P_d=20$  dB, respectively.", "The dynamic range parameter is $\\kappa =-50$  dB.", "In case of imperfect CSI, considering an OFDM system, we assume the coherence time is 1 ms (one subframe in LTE standard where there are 14 OFDM symbols in each subframe), and the “frequency smoothness interval\" is 14 as given in [11].", "Hence the coherence interval which is a time-frequency product is equal to $T=196$ .", "The pilot length is assumed to be the same as the number of users, i.e., $\\tau =K$ .", "From Figure REF , we can see that all bounds are very tight in both perfect and imperfect CSI cases, particularly with increasing $M$ .", "Next, we use these bounds for the full-duplex system to compute the uplink and downlink spectral efficiency ratios between full-duplex system and half-duplex system.", "Note that for the half-duplex system, we still numerically evaluate the ergodic achievable rates with no simplification.", "Figure: Comparisons between lower bounds in Proposition  and and numerically evaluated values of the ergodic achievable rates (bits/s/Hz/cell) under both perfect and imperfect CSI assumptions, where P tr =P u =10P_{tr}=P_u=10 dB and P d =20P_d=20 dB.We compute the spectral efficiency ratios between full-duplex and half-duplex by comparing the full-duplex rates given in (REF ) and (REF ) with the half-duplex achievable rates in (REF ) and (REF ) which can be evaluated numerically.", "We consider the same setting as in Fig.", "REF under both perfect and imperfect CSI assumptions.", "In Figure REF , in the case of perfect CSI, we can see that as $M$ increases, the up- and downlink spectral efficiency gain between full-duplex system versus TDD system will converge to 2 as we scale down the transmit power according to Theorem REF .", "The convergence rate is fast at the beginning, and becomes slower for large $M$ .", "To reach the asymptotic $2\\times $ gain, it will require remarkably large number of BS antennas.", "However, full-duplex spectral efficiency gains in both uplink and downlink are achievable for finite $M$ .", "For example, when $M=64$ , full-duplex achieves about 1.7$\\times $ downlink gain and 1.3$\\times $ uplink gain.", "We numerically show that even without scaling down the transmit power, similar full-duplex gains are achievable.", "Figure REF shows the full-duplex over half-duplex spectral efficiency gains in the case of imperfect CSI.", "We observe that even with channel estimation error and pilot contamination, full-duplex uplink and downlink gains exist for finite $M$ with and without scaling down the transmit power.", "Figure: Spectral efficiency gains of using full-duplex for finite MM with and without power scaling.We also investigate the spectral efficiency gain and antenna reduction tradeoff between full-duplex system and TDD system.", "The antenna reduction is the reduction of BS antennas due to full-duplex operation which can be characterized by the ratio between the number of BS antennas in TDD system ($M_{\\rm {TDD}}$ ) and the number of BS antennas in FD system.", "The spectral efficiency gain and antenna reduction tradeoff is essentially the tradeoff between full-duplex multiplexing gain due to simultaneous transmission and reception and beamforming gain due to large $M$ .", "In Fig.", "REF , we consider the same setting as in Fig.", "REF under imperfect CSI assumption.", "We illustrate the spectral efficiency gain and antenna reduction tradeoff for both uplink and downlink with different $M_{\\rm {TDD}}$ in the TDD system.", "Larger full-duplex antenna reduction can be achieved at the cost of reducing the spectral efficiency gain.", "In the regimes above the dashed arrows as shown in Fig.", "REF , full-duplex system achieves both spectral efficiency gain and antenna reduction over TDD system.", "We can see that full-duplex system can require an order of magnitude fewer BS antennas compared with TDD to obtain the same performance in some cases.", "In addition, as $M_{\\rm {TDD}}$ increases, full-duplex system can achieve higher spectral efficiency gain and antenna reduction.", "Figure: Spectral efficiency gain and antenna reduction tradeoff for various numbers of BS antennas employed in the TDD system under imperfect CSI assumption." ], [ "A small cell scenario", "Based on the state-of-art self-interference cancellation capability [1], the coverage of a full-duplex system is more likely to be within a small-cell communication ranges.", "Hence in this section, we present the numerical simulation in realistic small-cell network settings used in 3GPP [14] to evaluate the system performance.", "Twelve small cell BSs are uniformly and randomly distributed within a hexagonal region with a radius of 300 meters All the small-cell BSs have full-duplex capability with multiple antennas.", "Each small-cell BS is associated with five single-antenna half-duplex uplink UEs and downlink UEs, respectively, which are uniformly and randomly dropped within a radius of 40 meters of the BS.", "The numerical results are shown for the case of imperfect CSI with channel estimation error.", "We consider an OFDM system and the coherence interval is $T=196$ .", "The channel bandwidth is assumed as 20 MHz for both TDD and full-duplex systems.", "The large-scale fading models for BS-UE, BS-BS and UE-UE channels which include path loss and shadowing effect follow 3GPP model in [14].", "The SI channel model is based on the existing experiment data [1], where the propagation loss of SI channel in a separate-antenna system includes path loss, isolation, cross-polarization and antenna directionality [3], and in a shared-antenna system includes isolation using a circulator.", "We assume the SI channel has a propagation loss of 40 dB.", "We run hundreds of random drops of BSs and UEs in the simulation, and parameter details are given in Tabel REF .", "Table: Simulation parametersThe average full-duplex spectral efficiency gain is illustrated in Fig.", "REF with varying numbers of BS antennas, where the dynamic range parameter $\\kappa $ is $-60$  dB.", "We verify that the full-duplex gains in realistic network scenarios exist for a range of finite number of BS antennas and the gains will scale with BS antennas.", "Fig REF depicts the average spectral efficiency gain of the full-duplex system for different dynamic range parameters $\\kappa $ when $M=100$ , which demonstrates the impact of imperfect SI cancellation.", "Since all users are half-duplex, the downlink gains are not affected by $\\kappa $ .", "However, since all the BSs are full-duplex, the uplink gains are severely affected by the dynamic range values especially when the dynamic range value is low.", "We can see that larger dynamic range $\\kappa ^{-1}$ , (i.e., smaller $\\kappa $ ) will result in less residual SI, thus increasing the uplink gain.", "Once the dynamic range $\\kappa ^{-1}$ exceeds certain threshold, there is not much impact of the residual SI on the uplink performance.", "Figure: Average full-duplex spectral efficiency gain with BS antenna arrays when L=12L=12 and κ=-60\\kappa =-60.Figure: Average full-duplex spectral efficiency gain with different dynamic ranges when M=100M=100 and L=12L=12." ], [ "Conclusion", "In this paper, we have investigated the multi-cell MU-MIMO full-duplex networks where single-antenna full-duplex and half-duplex UEs are served by the full-duplex BSs with multiple antennas.", "Using low complexity linear receivers and precoders, the ergodic achievable rates of uplink and downlink are characterized for the full-duplex system.", "Several practical constraints are modeled in the analysis such as imperfect full-duplex radio chains, channel estimation error, pilot overhead and pilot contamination.", "The large scale system performance is analyzed when each BS has a large antenna array.", "When the number of BS antennas grows infinitely large, the 2$\\times $ asymptotic full-duplex spectral efficiency gain can be achieved over the TDD system with perfect CSI.", "When channel estimation error and channel training overhead are considered, the 2$\\times $ asymptotic full-duplex spectral efficiency gain is achieved when serving only full-duplex UEs.", "Furthermore, we show by numerical simulation that full-duplex system can achieve both spectral efficiency gain and antenna reduction as the full-duplex system can require one order of magnitude fewer antennas than TDD to achieve the same or better performance in certain cases." ], [ "Proof of Proposition ", "Based on Jensen's inequality and the convexity of $\\mathrm {log}_2\\left(1+x^{-1}\\right)$ , we have $\\mathbb {E}\\left[\\mathrm {log}_2\\left(1+x^{-1}\\right)\\right]\\ge \\mathrm {log}_2\\left(1+\\mathbb {E}^{-1}(x)\\right)$ .", "Hence we can lower bound the ergodic achievable rate of the $n$ -th uplink UE in the $j$ -th cell in (REF ) as $\\begin{aligned}&\\tilde{R}^{fd,p}_{u,jn}\\ge R^{fd,p}_{u,jn}\\triangleq \\mathrm {log}_2\\Bigg (1+\\\\&\\mathbb {E}^{-1}\\left[\\frac{P_u\\sum _{(l,m)\\ne (j,n)}|\\mathbf {g}^H_{u,jjn}\\mathbf {g}_{u,jlm}|^2+I_{bs-bs}+\\left\\Vert \\mathbf {g}_{u,jjn}\\right\\Vert ^2(\\sigma ^2+\\kappa P_d\\beta _{b,jj})}{P_u\\left\\Vert \\mathbf {g}_{u,jjn}\\right\\Vert ^4}\\right]\\Bigg )\\\\&=\\mathrm {log}_2\\left(1+\\mathbb {E}^{-1}\\left[\\frac{P_u\\sum _{(l,m)\\ne (j,n)}|\\tilde{g}_{u,jlm}|^2+\\sum _{l\\ne j}\\sum _{k\\in \\mathcal {K}_d}\\frac{P_d}{K_d \\gamma _l}|\\tilde{v}_{jlk}|^2+\\sigma ^2+\\kappa P_d\\beta _{b,jj}}{P_u\\left\\Vert \\mathbf {g}_{u,jjn}\\right\\Vert ^2}\\right]\\right), \\end{aligned}$ where $\\tilde{g}_{u,jlm}\\triangleq \\frac{\\mathbf {g}^H_{u,jjn}\\mathbf {g}_{u,jlm}}{\\left\\Vert \\mathbf {g}_{u,jjn}\\right\\Vert }$ , $\\tilde{v}_{jlk}\\triangleq \\frac{\\mathbf {g}^H_{u,jjn}\\mathbf {V}_{jl}\\mathbf {g}^*_{d,llk}}{\\left\\Vert \\mathbf {g}_{u,jjn}\\right\\Vert }$ .", "Conditioned on $\\mathbf {g}_{u,jjn}$ , $\\tilde{g}_{u,jlm}\\sim \\mathcal {CN}(0,\\beta _{u,jlm})$ and $\\tilde{v}_{jlk}\\sim \\mathcal {CN}(0,M\\beta _{b,jl}\\beta _{d,llk})$ are independent of $\\mathbf {g}_{u,jjn}$ .", "Thus we have $\\begin{aligned}&\\mathbb {E}\\left[\\frac{P_u\\sum _{(l,m)\\ne (j,n)}|\\tilde{g}_{u,jlm}|^2+\\sum _{l\\ne j}\\sum _{k\\in \\mathcal {K}_d}\\frac{P_d}{K_d \\gamma _l}|\\tilde{v}_{jlk}|^2+\\sigma ^2+\\kappa P_d\\beta _{b,jj}}{P_u\\left\\Vert \\mathbf {g}_{u,jjn}\\right\\Vert ^2}\\right]\\\\&=\\left(\\sum _{(l,m)\\ne (j,n)}P_u\\mathbb {E}[|\\tilde{g}_{u,jlm}|^2]+\\sum _{l\\ne j}\\sum _{k\\in \\mathcal {K}_d}\\frac{P_d}{K_d \\gamma _l}\\mathbb {E}[|\\tilde{v}_{jlk}|^2]+\\sigma ^2+\\kappa P_d\\beta _{b,jj}\\right)\\mathbb {E}\\left[\\frac{1}{P_u\\left\\Vert \\mathbf {g}_{u,jjn}\\right\\Vert ^2}\\right].", "\\end{aligned}$ From Lemma 2.10 in [29], for a central Wishart matrix $\\mathbf {W}\\sim \\mathcal {W}_m(n,\\mathbf {I})$ with $n\\ge m$ , we have $\\mathbb {E}[\\mathsf {tr}\\lbrace \\mathbf {W}^{-1}\\rbrace ]=\\frac{m}{n-m}$ for $n>m$ .", "Hence $\\begin{aligned}\\mathbb {E}\\left[\\frac{1}{P_u\\left\\Vert \\mathbf {g}_{u,jjn}\\right\\Vert ^2}\\right]=\\frac{1}{(M-1)P_u\\beta _{u,jjn}}~\\text{for}~M\\ge 2.", "\\end{aligned}$ Combing (REF ), (REF ) and (REF ), we can obtain the uplink achievable rate in (REF ).", "To derive the lower bound on the downlink achievable rate, we need to evoke the Lemma 2.10 in [29] again where $\\mathbb {E}[\\mathsf {tr}\\lbrace \\mathbf {W}^{-2}\\rbrace ]=\\frac{mn}{(n-m)^3-(n-m)}$ for $n>m+1$ .", "Thus we have $\\begin{aligned}\\mathbb {E}\\left[\\frac{1}{\\left\\Vert \\mathbf {g}_{d,llk}\\right\\Vert ^4}\\right]=\\frac{1}{(M-1)(M-2)\\beta _{d,llk}^2}~\\text{for}~M\\ge 3.\\end{aligned}$ Following the same approach used in the derivation of uplink rate, we can obtain the achievable rate of each downlink user given in proposition REF .", "The details are omitted to avoid redundancy." ], [ "Proof of Proposition ", "Similar to the proof of Proposition REF , applying Jensen's inequality, we can lower bound the ergodic achievable rate of the $n$ -th uplink user in the $j$ -th cell in (REF ) as $\\begin{aligned}&\\tilde{R}^{fd,ip}_{u,jn}\\ge R^{fd,ip}_{u,jn}\\triangleq \\mathrm {log}_2\\Bigg (1+\\\\&\\mathbb {E}^{-1}\\left[\\frac{P_u{\\tilde{\\epsilon }_{jjn}}^2+P_u\\sum _{(l,m)\\ne (j,n)}|\\tilde{\\tilde{g}}_{u,jlm}|^2+\\sum _{l\\ne j}\\sum _{k\\in \\mathcal {K}_d}\\frac{P_d}{K_d \\gamma _l}|\\tilde{\\tilde{v}}_{jlk}|^2+\\sigma ^2+\\kappa P_d\\beta _{b,jj}}{P_u\\left\\Vert \\hat{\\mathbf {g}}_{u,jjn}\\right\\Vert ^2}\\right]\\Bigg )\\end{aligned}$ where $\\tilde{\\epsilon }_{jjn}\\triangleq \\frac{\\hat{\\mathbf {g}}^H_{u,jjn}\\mathbf {\\epsilon }_{u,jjn}}{\\left\\Vert \\hat{\\mathbf {g}}_{u,jjn}\\right\\Vert }$ , $\\tilde{\\tilde{g}}_{u,jlm}\\triangleq \\frac{\\hat{\\mathbf {g}}^H_{u,jjn}\\mathbf {g}_{u,jlm}}{\\left\\Vert \\hat{\\mathbf {g}}_{u,jjn}\\right\\Vert }$ , $\\tilde{\\tilde{v}}_{jlk}\\triangleq \\frac{\\hat{\\mathbf {g}}^H_{u,jjn}\\mathbf {V}_{jl}\\hat{\\mathbf {g}}^*_{d,llk}}{\\left\\Vert \\mathbf {g}_{u,jjn}\\right\\Vert }$ .", "Conditioned on $\\hat{\\mathbf {g}}_{u,jjn}$ , $\\tilde{\\epsilon }_{jjn}\\sim \\mathcal {CN}\\left(0,\\frac{\\beta _{u,jjk}\\left(\\sigma ^2+P_{tr}\\sum _{l\\ne j}\\beta _{u,jlk}\\right)}{\\lambda _{u,jk}}\\right)$ , $\\tilde{\\tilde{v}}_{jlk}\\sim \\mathcal {CN}\\left(0,M\\beta _{b,jl}\\frac{P_{tr}\\beta ^2_{d,llk}}{\\lambda _{d,lk}}\\right)$ and $\\tilde{\\tilde{g}}_{u,jlm}\\sim (0,\\mathrm {var}(\\tilde{\\tilde{g}}_{u,jlm}))$ are independent of $\\hat{\\mathbf {g}}_{u,jjn}$ .", "Using (REF ), we have $\\begin{aligned}\\tilde{\\tilde{g}}_{u,jlm}=\\frac{P_{tr}\\beta _{u,jjn}}{\\lambda _{u,jn}}\\left(\\sum _{l_1=1}^L \\mathbf {g}^H_{u,jl_1n}+\\frac{\\mathbf {n}^H_{jn}}{\\sqrt{P_{tr}}}\\right)\\frac{\\mathbf {g}_{u,jlm}}{\\left\\Vert \\hat{\\mathbf {g}}_{u,jjn}\\right\\Vert }.\\end{aligned}$ Evoking Lemma 2.9 in [29], where for a central Wishart matrix $\\mathbf {W}\\sim \\mathcal {W}_m(n,\\mathbf {I})$ with $n\\ge m$ , $\\mathbb {E}[\\mathsf {tr}\\lbrace \\mathbf {W}^{2}\\rbrace ]=mn(m+n)$ , we can calculate the variance of $\\tilde{\\tilde{g}}_{u,jlm}$ as $\\begin{aligned}\\mathbb {E}[{\\tilde{\\tilde{g}}_{u,jlm}}^2]={\\left\\lbrace \\begin{array}{ll}\\frac{P_{tr}}{\\lambda _{u,jn}}\\left[(M+1)\\beta ^2_{u,jln}+\\sum _{l_1\\ne l}\\beta _{u,jl_1 n}\\beta _{u,jln}+\\frac{\\beta _{u,jln}\\sigma ^2}{P_{tr}}\\right] & ~~m=n,l\\ne j\\\\\\beta _{u,jlm} &~~ m\\ne n.\\end{array}\\right.", "}\\end{aligned}$ Now we can rewrite the expectation in (REF ) as $\\begin{aligned}&\\mathbb {E}\\left[\\frac{P_u{\\tilde{\\epsilon }_{jjn}}^2+P_u\\sum _{(l,m)\\ne (j,n)}|\\tilde{\\tilde{g}}_{u,jlm}|^2+\\sum _{l\\ne j}\\sum _{k\\in \\mathcal {K}_d}\\frac{P_d}{K_d \\gamma _l}|\\tilde{\\tilde{v}}_{jlk}|^2+\\sigma ^2+\\kappa P_d\\beta _{b,jj}}{P_u\\left\\Vert \\hat{\\mathbf {g}}_{u,jjn}\\right\\Vert ^2}\\right]\\\\&=\\left(P_u\\mathbb {E}[|\\tilde{\\epsilon }_{jjn}|^2]+P_u\\sum _{{(l,m)\\ne (j,n)}}\\mathbb {E}[|\\tilde{\\tilde{g}}_{u,jlm}|^2]+\\sum _{l\\ne j}\\sum _{k\\in \\mathcal {K}_d}\\frac{P_d}{K_d \\gamma _l}\\mathbb {E}[|\\tilde{\\tilde{v}}_{jlk}|^2]+\\sigma ^2+\\kappa P_d\\beta _{b,jj}\\right)\\mathbb {E}\\left[\\frac{1}{P_u\\left\\Vert \\hat{\\mathbf {g}}_{u,jjn}\\right\\Vert ^2}\\right].", "\\end{aligned}$ Combing (REF ), (REF ) and (REF ), we can obtain the uplink achievable rate in (REF ).", "For the downlink achievable rate in (REF ), we first compute $\\mathbb {E}\\left[\\mathbf {g}^T_{d,llk}\\hat{\\mathbf {g}}^*_{d,llk}\\right]$ .", "Let $\\mu =\\mathbf {g}^T_{d,llk}\\hat{\\mathbf {g}}^*_{d,llk}$ , since $\\mathbf {g}_{d,llk}=\\hat{\\mathbf {g}}_{d,llk}+\\mathbf {\\epsilon }_{d,llk}$ and $\\hat{g}_{d,llk}$ is independently of $\\epsilon _{d,llk}$ , we have $\\begin{aligned}\\mathbb {E}[\\mu ]&=\\mathbb {E}\\left[(\\hat{\\mathbf {g}}^T_{d,llk}+\\mathbf {\\epsilon }^T_{d,llk})\\hat{\\mathbf {g}}^*_{d,llk}\\right]\\\\&=\\mathbb {E}\\left[\\left\\Vert \\hat{\\mathbf {g}}_{d,llk}\\right\\Vert ^2\\right]=\\frac{MP_{tr}\\beta ^2_{d,llk}}{\\lambda _{d,lk}}.\\end{aligned}$ Again invoking Lemma 2.9 in [29], we have $\\begin{aligned}\\mathbb {E}\\left[\\mu ^2\\right]&=\\mathbb {E}\\left[\\left\\Vert \\hat{\\mathbf {g}}_{d,llk}\\right\\Vert ^4\\right]+\\mathbb {E}\\left[\\mathbf {\\epsilon }^T_{d,llk}\\hat{\\mathbf {g}}^*_{d,llk}\\hat{\\mathbf {g}}^T_{d,llk}\\mathbf {\\epsilon }^*_{d,llk}\\right]\\\\&=\\frac{M(M+1)P^2_{tr}\\beta ^4_{d,llk}+MP_{tr}\\beta ^3_{d,llk}(\\sigma ^2+P_{tr}\\sum _{j\\ne l}\\beta _{d,ljk})}{\\lambda _{d,lk}^2}.\\end{aligned}$ Since $\\mathsf {var}\\left[\\mathbf {g}^T_{d,llk}\\hat{\\mathbf {g}}^*_{d,llk}\\right]\\triangleq \\mathsf {var}(\\mu )=\\mathbb {E}(\\mu ^2)-\\mathbb {E}^2(\\mu )$ , we can obtain that $\\begin{aligned}\\mathsf {var}(\\mu )&=\\frac{MP_{tr}\\beta ^3_{d,llk}}{\\lambda _{d,lk}}.\\end{aligned}$ Next, we compute $\\mathbb {E}\\left[{\\mathbf {g}^T_{d,jlk}\\hat{\\mathbf {g}}^*_{d,jji}}^2\\right]$ as $\\begin{aligned}\\mathbb {E}\\left[{\\mathbf {g}^T_{d,jlk}\\hat{\\mathbf {g}}^*_{d,jji}}^2\\right]={\\left\\lbrace \\begin{array}{ll}\\frac{MP^2_{tr}\\beta ^2_{d,jjk}}{\\lambda ^2_{d,jk}}\\!\\!\\left[(M+1)\\beta ^2_{d,jlk}+\\sum _{l_1\\ne l}\\beta _{d,jl_1 k}\\beta _{d,jlk}+\\frac{\\beta _{d,jlk}\\sigma ^2}{P_{tr}}\\right] & ~~i=k,j\\ne l\\\\\\frac{MP_{tr}\\beta ^2_{d,jji}\\beta _{d,jlk}}{\\lambda _{d,ji}} &~~ i\\ne k.\\end{array}\\right.", "}\\end{aligned}$ The rest terms in (REF ) can be calculated easily, and thus the details are omitted.", "Combing all the results above, we can obtain downlink achievable rate given in proposition REF ." ] ]
1606.05025
[ [ "QSWalk: a Mathematica package for quantum stochastic walks on arbitrary\n graphs" ], [ "Abstract We present a Mathematica package, QSWalk, to simulate the time evaluation of Quantum Stochastic Walks (QSWs) on arbitrary directed and weighted graphs.", "QSWs are a generalization of continuous time quantum walks that incorporate both coherent and incoherent dynamics and as such, include both quantum walks and classical random walks as special cases.", "The incoherent component allows for quantum walks along directed graph edges.", "The dynamics of QSWs are expressed using the Lindblad formalism, originally developed for open quantum systems, which frames the problem in the language of density matrices.", "For a QSW on a graph of $N$ vertices, we have a sparse superoperator in an $N^2$-dimensional space, which can be solved efficiently using the built-in MatrixExp function in Mathematica.", "We illustrate the use of the QSWalk package through several example case studies." ], [ "Introduction", "Over the past two decades, quantum computing and information theory have emerged as one of the most exciting and promising frontiers of research, at the forefront of both quantum physics and computer science (see, e.g.", "[1] and references therein).", "One particularly interesting avenue of research is the quantum walk (QW) [2], [3], [4], [5], a quantum mechanical analogue of the classical random walk (CRW).", "CRWs model the random motion of a “walker” over the vertices of some graph, with each step chosen at random from the edges connecting to adjacent vertices [6], [7].", "The CRW is ubiquitous throughout physics and applied mathematics, being associated with phenomena as diverse as Brownian motion in liquids [8] and price fluctuations in financial markets [9].", "QWs were first described by Aharanov in 1993 et al.", "[10], who introduced a discrete-time formulation in which random steps are effected by means of a unitary coin operator [11], [12].", "A continuous-time model, with no need of a coin operator, was subsequently introduced by Farhi and Gutmann in 1998 [13].", "The key difference of both discrete- and continuous-time QWs with respect to the equivalent CRW is that the QW evolves according to quantum amplitudes instead of probabilities, and so the diffusion process of the CRW is replaced by a phase-coherent unitary evolution in which different paths can interfere.", "This phase-coherence leads to drastically different behaviour, which has been exploited to develop algorithms on QWs achieving an exponential speedup vis-à-vis the corresponding CRW [14].", "A significant limitation of QWs is that the time evolution must be unitary in order to conserve probability.", "This presents a particular challenge when attempting to apply the QW to directed graphs, i.e.", "graphs containing edges which can only be traversed in one direction.", "While there have been several approaches to extend discrete-time QWs to directed graphs [15], [16], these do not carry over to the continuous-time case.", "One approach that does yield a version of continuous-time QWs applicable to directed graphs is the so-called Quantum Stochastic Walk (QSW), a generalization of both QWs and CRWs developed by Whitfield et al.", "[17].", "The object of study in a QSW is the density matrix, which combines the coherent dynamics of quantum mechanical states with the incoherent evolution of a probability vector.", "The QSW posits a master equation comprising a combination of the QW Hamiltonian, representing unitary evolution, and a term describing incoherent scattering.", "The latter is based on the Lindblad formalism from open quantum systems theory [18], [19].", "By tuning the relative weighting of these components, the QSW can describe a continuum of behavior with the QW and CRW at either end.", "Importantly, the incoherent scattering component of the QSW is not subject to the same constraints as the Hamiltonian of the QW and is thus able to represent irreversible scattering, e.g.", "along one-way edges in directed graphs.", "QSWs have recently been applied to several problems at the forefront of current quantum information theory research, including the quantum PageRank algorithm [20], quantum neural networks [21] and, most recently, decision-making networks [22].", "In this paper we present a Mathematica package, QSWalk, for the evaluation of QSWs on arbitrary directed graphs.", "The numerical approach used in our package is based on vectorization of the density matrix and master equation, allowing a solution in terms of a matrix exponential for which efficient sparse matrix methods can be used.", "By writing our package in Mathematica we are able to make use of its extensive built-in functionality for graphs and linear algebra, making our implementation straightforward and efficient.", "Furthermore, Mathematica offers the advantage of an intuitive notebook environment and visualization functions, and is thus an ideal tool for exploring QSWs for either research or educational purposes.", "Although a python library for quantum walks has recently been made available [23], to our knowledge there is currently no software in the public domain for the computation of QSWs.", "A python package, QuTiP, has been written for the study of open quantum systems [24], [25], which could potentially provide similar functionality to our package.", "However, it is not specifically tailored to the context of quantum walks on graphs, and it requires working in Python, so may not be as accessible as our Mathematica package.", "The remainder of this paper is structured as follows.", "In the next section we give an overview of the basic theory underlying QSWs.", "In Section we describe the QSWalk package and some of the computational methods used therein.", "We then present several illustrative applications, including a study of FMO complex in photosynthesis and the quantum Page Rank algorithm, in Section ." ], [ "Theory", "Our goal in this section is to present the background theory relevant to QSWs in a concise and self-contained way.", "We also aim to establish a clear and consistent notation, which is particularly important owing to the many different (and sometimes mutually contradictory) notations in the literature (ours is most similar to that of Refs.", "[17] and [26]).", "We start by briefly defining graphs and the associated CRWs, followed by QWs, and finish with a more detailed description of QSWs.", "For the sake of brevity, and because they are irrelevant to QSWs, we omit the discrete-time CRW and QW models and discuss only the continuous-time case.", "Throughout this section, and the rest of the paper, we work in units with $\\hbar =1$ ." ], [ "Classical random walks on graphs", "For our purposes, a graph $G$ is defined as a set of vertices (or nodes) $\\lbrace 1,\\ldots ,N\\rbrace $ together with an $N\\times N$ -dimensional adjacency matrix $A$ describing the connections (or edges) between them.", "In the simplest case of an unweighted graph, $A_{ij}=1$ if there is an edge from $j$ to $i$ and 0 otherwise.", "For a $weighted$ graph, on the other hand, edges have arbitrary weights $A_{ij}>0$ representing the strength of the connection from $j$ to $i$ .", "If $A$ is symmetric, so that $A_{ij}=A_{ji}$ for all $i,j$ , $G$ is said to be undirected; otherwise it is a directed graph (or digraph).", "For each vertex, the sum of weights for all edges leaving it is called its out-degree: $\\mathrm {outDeg}(j) = \\sum _{i\\ne j}A_{ij}$ .", "We note that the diagonal entries $A_{ii}$ represent self-loops, edges which start and finish at the same vertex; if $G$ has no self-loops it is called simple.", "In the literature on QWs and QSWs simple graphs are usually considered, however, non-simple graphs can in principle be used.", "A (continuous-time) CRW on $G$ may be defined as an $N$ -component probability vector $\\mathbf {p}(t)$ whose components $p_i(t)$ sum to unity and represent the probability for the hypothetical walker being found at vertex $i$ at time $t$ .", "The initial condition typically places the walker on a particular vertex at time zero, $p_i(0) = \\delta _{i,q}$ for some $q$ .", "The time evolution of $\\mathbf {p}(t)$ is governed by a master equation: $\\frac{\\mathrm {d}\\mathbf {p}}{\\mathrm {d}t} = -M \\cdot \\mathbf {p}(t),$ with generator matrix $M$ : $M_{ij} = {\\left\\lbrace \\begin{array}{ll}-\\gamma A_{ij}, & i\\ne j, \\\\\\gamma \\, \\mathrm {outDeg}(j), & i = j.\\end{array}\\right.", "}$ The parameter $\\gamma >0$ determines the CRW transition rate between vertices.", "The off-diagonal elements of $M_{ij}$ represent the individual probability flows $j\\rightarrow i$ along each edge from vertex $j$ , while the diagonal elements $M_{jj}$ account for the total outflow from vertex $j$ per unit time.", "The solution of Eq.", "(REF ) can be expressed as a matrix exponential: $\\mathbf {p}(t) = \\mathrm {e}^{-M t} \\cdot \\mathbf {p}(0).$" ], [ "Quantum walks", "A QW may be defined in a natural way by analogy to the CRW.", "The graph structure of $G$ is mapped onto a quantum mechanical Hilbert space in which the set of vertices forms an orthonormal basis $\\lbrace \\left|1\\right>,\\ldots ,\\left|N\\right>\\rbrace $ .", "The probability vector $\\mathbf {p}(t)$ from the CRW is replaced by a quantum state vector of probability amplitudes: $\\left|\\psi (t)\\right> = \\sum _{i=1}^{N}\\left|i\\right>\\left<i|\\psi (t)\\right>.$ The amplitudes $\\left<i|\\psi (t)\\right>$ represent a coherent superposition over all vertices, such that the probability associated with vertex $i$ at time $t$ is $|\\left<i|\\psi (t)\\right>|^2$ .", "The CRW master equation (Eq.", "REF ) is replaced by a Schrödinger equation: $\\frac{\\mathrm {d}\\left|\\psi (t)\\right>}{\\mathrm {d}t} = -\\mathrm {i}\\hat{H} \\left|\\psi (t)\\right>,$ where $\\hat{H}$ is the Hamiltonian, whose matrix elements come from the CRW generator matrix: $\\left<i\\right|\\hat{H}\\left|j\\right> = M_{ij}.$ In the last two equations we have introduced the $\\hat{}$ notation to denote that we regard $\\hat{H}$ as an operator, independent of any basis.", "However, in the remainder of this paper we will switch freely between the operator ($\\hat{H}$ ) and matrix ($H$ ) viewpoints depending on the context; the distinction is generally not significant.", "The imaginary $\\mathrm {i}$ factor on the right hand side of Eq.", "(REF ) is responsible for the uniquely “quantum” properties of the QW.", "It should be noted that, in order for the time evolution of the QW to be unitary (and hence probability-conserving), $\\hat{H}$ must be Hermitian.", "From Eqs.", "() and (REF ), it follows that $M$ must be symmetric, and hence the graph $G$ must be undirected for a QW.", "The solution to Eq.", "(REF ) can once again be expressed as a matrix exponential: $\\left|\\psi (t)\\right> = \\mathrm {e}^{-\\mathrm {i}\\hat{H} t} \\left|\\psi (0)\\right>,$ with the initial condition $\\left|\\psi (0)\\right> = \\left|q\\right>$ for some $q$ ." ], [ "Quantum stochastic walks", "The QW just described is restricted to undirected graphs on which the Hamiltonian operator is Hermitian.", "It is also, by definition, limited to pure quantum states undergoing phase-coherent unitary evolution.", "By contrast, the CRW represents the opposite limit, in which scattering is non-unitary and evolution is purely phase-incoherent.", "The QSW was proposed by Whitfield et al.", "[17] as a generalization that includes both QWs and CRWs as limiting special cases, and allows random walks with a combination of coherent and incoherent dynamics.", "The incoherent part of the QSW can be used to incorporate scattering along directed edges and thus allows directed graphs to be treated.", "In place of the probability vector $\\mathbf {p}(t)$ of the CRW or wavefunction $\\left|\\psi (t)\\right>$ of the QW, the QSW is framed in terms of the density matrix $\\hat{\\rho }(t)$ [1], [27], [28].", "In the general case, this operator describes a statistical ensemble of pure quantum states (a so-called “mixed” state): $\\hat{\\rho }(t) = \\sum _{k} p_k \\left|\\psi _k(t)\\right>\\left<\\psi _k(t)\\right|.$ The weights $p_k\\ge 0$ satisfy $\\sum _k p_k=1$ and represent the probabilities for the system to be in each of the quantum states $\\left|\\psi _k(t)\\right>$ .", "This probability reflects a lack of knowledge of the true system state, which is distinct from the quantum mechanical uncertainty associated with the measurement a single quantum state that happens to be a superposition of observable basis states.", "The special case of a known quantum state is recovered when only one of the $p_k$ is non-zero, and is known as a “pure” state.", "We note several relevant properties of $\\hat{\\rho }(t)$ which follow from Eq.", "(REF ) and are relevant here: $\\hat{\\rho }(t)$ is Hermitian; $\\mathrm {tr}(\\hat{\\rho }(t))=1$ and $\\mathrm {tr}(\\hat{\\rho }(t)^2)\\le 1$ , with equality holding only for pure states; the expectation value of an operator $\\hat{A}$ is given by $\\langle \\hat{A}\\rangle =\\mathrm {tr}(\\hat{\\rho }(t)\\hat{A})$ .", "For a QSW on graph $G$ , the natural representation for $\\hat{\\rho }(t)$ is a $N\\times N$ matrix in the basis of vertex states $\\lbrace \\left|1\\right>,\\ldots ,\\left|N\\right>\\rbrace $ , with elements $\\rho _{ij}(t)=\\left<i\\right|\\hat{\\rho }(t)\\left|j\\right>$ .", "As for the QW, the usual initial condition for a QSW will have the walker start in the state $\\left|q\\right>$ at $t=0$ , so that $\\hat{\\rho }(0) = \\left|q\\right>\\left<q\\right|$ , or equivalently, $\\rho _{ij}(0)=\\delta _{iq}\\delta _{jq}$ .", "For $t\\ge 0$ , the diagonal elements $\\rho _{ii}(t)$ represent the probability density at vertex $i$ (and are therefore referred to as “populations”), while the off-diagonal elements $\\rho _{ij}(t)$ ($i\\ne j$ ) describe the phase coherence between distinct vertices $i$ and $j$ (and are known as “coherences”).", "Following Ref.", "[17], we can now define the QSW by the following master equation: $ \\frac{\\mathrm {d}\\hat{\\rho }}{\\mathrm {d}t}= -(1-\\omega ) \\mathrm {i}[\\hat{H},\\hat{\\rho }(t)] + \\\\\\omega \\sum _{k=1}^K \\left(\\hat{L}_k \\hat{\\rho }(t) \\hat{L}_k^\\dagger - \\frac{1}{2}\\left(\\hat{L}_k^\\dagger \\hat{L}_k\\hat{\\rho }(t) +\\hat{\\rho }(t)\\hat{L}_k^\\dagger \\hat{L}_k\\right)\\right),$ where $\\hat{H}$ is the Hamiltonian operator, describing coherent evolution; $\\hat{L}_k$ are the Lindblad operators, which describe phase-incoherent scattering (see below); and $0\\le \\omega \\le 1$ is a weighting factor that interpolates between the coherent and incoherent terms.", "Eq.", "(REF ) is an example of the Kossakowski-Lindblad master equation [29], [30], widely used to model interactions with an external environment in the study of open quantum systems [18], [19].", "In its usual context the $1-\\omega $ and $\\omega $ factors are not present, but it is convenient to include them in the QSW as they permit an explicit variation between completely phase coherent ($\\omega =0$ ) and incoherent ($\\omega =1$ ) regimes.", "The first term on the right hand side of Eq.", "(REF ), representing the phase coherent component of the evolution, uses the same Hamiltonian $\\hat{H}$ that we introduced in the QW.", "Indeed, when $\\omega =0$ , Eq.", "(REF ) reduces to the Liouville-von Neumann equation, which is the density matrix equivalent of the Schrödinger equation Eq.", "(REF ).", "As for the QW, $\\hat{H}$ must be Hermitian and hence based on an undirected graph.", "Unlike the QW case, however, in a QSW the graph $G$ may be undirected in general.", "We therefore define $\\hat{H}$ in terms of a symmetrized version of $G$ , as follows: $ H_{ij} = {\\left\\lbrace \\begin{array}{ll}-\\gamma \\, \\mathrm {max}(A_{ij},A_{ji}), & i\\ne j, \\\\-\\sum _{k\\ne j}H_{kj}, & i = j.\\end{array}\\right.", "}$ The directedness of $G$ is manifest in the second term of Eq.", "(REF ), which represents the incoherent component of the evolution.", "It comprises a sum over the Lindblad operators $\\hat{L}_k$ , each of which corresponds to a particular scattering channel.", "The range of the sum, $K$ , depends on the particular choice of $\\lbrace \\hat{L}_k\\rbrace $ .", "For a QSW, a natural choice is the set of operators representing scattering between each pair of vertices.", "Identifying $k$ with the pair $i,j$ (say, $k \\equiv N(j-1)+i$ ) we define $ \\hat{L}_k = \\sqrt{|M_{ij}|}\\left|i\\right>\\left<j\\right|,$ representing incoherent scattering from vertex $j$ to $i$ .", "The sum extends over all $i,j$ so that $K = N^2$ , but only pairs with non-zero $M_{ij}$ contribute.", "In this way, the scattering represented by the $\\hat{L}_k$ operators incorporates the full directed structure of $G$ ; this is what is meant by saying that a QSW can be applied to a directed graph.", "It is straightforward to show that with this set of $\\hat{L}_k$ , the QSW reduces to the CRW when $\\omega =1$ , if we make the obvious identification $\\rho _{ii}(t)\\equiv p_i(t)$ .", "Other choices for $\\hat{L}_k$ are possible and may lead to QSWs with different physical interpretations.", "However, the definition in Eq.", "() is of particular interest, since it allows us to recover the well-known and widely studied QW ($\\omega =0$ ) and CRW ($\\omega =1$ ) as special cases.", "By varying $\\omega $ between 0 and 1 one can then explore intermediate regimes with both quantum and classical behaviour.", "The QSWalk package is written in the Mathematica system (also known since 2013 as the Wolfram Language) [31].", "One of the key strengths of Mathematica is its vast number of built-in functions (approximately 5000 as of version 10.3), providing efficient and sophisticated numerical and symbolic algorithms.", "Of particular value to our application is Mathematica's built-in graph theory functionality: graphs are “first-class citizens” [32] and many standard graph types are available as built-in functions.", "This makes it easy to apply our package to immediately study a wide range of known graph types, in addition to readily defining new ones.", "Mathematica also offers powerful linear algebra routines, which are essential for the efficient evaluation of QSWs.", "In particular, we make use of the built-in sparse array methods for the matrix exponential calculation.", "The use of these methods is seamless and mostly transparent to the user, which greatly streamlines the development of efficient code." ], [ "Package overview", "The main purpose of the QSWalk package is the implementation of QSWs on graphs.", "In this section we illustrate this generic use case with a simple example that nevertheless incorporates all the necessary steps.", "For ease of presentation the following input and output is taken from a command-line terminal session (although we recommend working within the Mathematica notebook interface in general).", "Mathematica 10.3.0 for Mac OS X x86 (64-bit) Copyright 1988-2015 Wolfram Research, Inc. We begin by loading the package.", "For this to work, the QSWalk.m file must be in the current path (as specified by the $Path variable).", "Alternatively, the full path to the file can be specified.", "In[1]:= <<QSWalk.m Our calculation begins by defining a graph object; here we use a built-in function to create an Erdös-Renyi random graph with 10 vertices and 30 (directed) edges: In[2]:= G = RandomGraph[{10,30}, DirectedEdges->True]; Assuming a transition rate $\\gamma =1$ , we define the Hamiltonian as the generator matrix of the symmetrized graph (Eq.", "REF ): In[3]:= gamma = 1.; In[4]:= H = GeneratorMatrix[UndirectedGraph[G], gamma]; We then define the Lindblad operators using the generator of the original directed graph (Eq.", "REF ): In[5]:= LkSet = LindbladSet[GeneratorMatrix[G,gamma]]; Next, we define the remaining inputs.", "The initial density matrix is $\\hat{\\rho }(0) = \\left|1\\right>\\left<1\\right|$ , which is conveniently implemented in matrix form using the SparseArray function: In[6]:= omega = 0.5; t = 10.; In[7]:= rho0 = SparseArray[{{1,1}->1.", "}, {n,n}]; The final step is to carry out the QSW calculation: In[8]:= rho1 = QuantumStochasticWalk[H,LkSet,omega,rho0,t]; The output is a density matrix; to get the final vertex probabilities we take the diagonal elements: In[8]:= p = Diagonal[rho1] // Chop Out[8]= {0.311295, 0.0266081, 0.0938145, 0.0797414, 0.0884782, >  0.0687126, 0.0747984, 0.121446, 0.0900074, 0.0450989} The above eight short lines of code can, with only slight modifications, be used to calculate QSWs on arbitrary graphs.", "It is often of interest to evaluate a walk at a sequence of times, with the result of each time step forming the input to the next step.", "In this case, it is advantageous to avoid re-generating the master equation matrix for each time step.", "The following approach shows how this can be done in a straightforward way.", "First, define a time increment dt.", "We then call QuantumStochasticWalk as before, but with a symbolic (i.e.", "undefined) initial state rho.", "The result is a partially evaluated expression in which the master equation has been generated, but the matrix exponential has remained unevaluated because its second argument is symbolic.", "We save this as a function qsw, which can subsequently be called with specific numerical values for rho: In[12]:= dt = 1.; In[13]:= qsw[rho_]=QuantumStochasticWalk[H,LkSet,omega,rho,dt] Out[13]= Matricize[MatrixExp[SparseArray[<1170>, {100, 100}], >  Vectorize[rho]]] We use the NestList function to call qsw recursively, starting with rho = rho0 to build up a list of density matrices for each time: In[14]:= pt = Diagonal /@ NestList[qsw,rho0,Round[t/dt]]; As a check, we can verify that the final element in this list agrees with our previous calculation: In[20]:= Chop[Last[pt] - p] Out[20]= {0, 0, 0, 0, 0, 0, 0, 0, 0, 0} The Chop function is used here to zero out the small differences arising from the limitations of machine precision arithmetic." ], [ "Functions of the QSWalk package", "In this section we provide a brief summary of the key functions in the QSWalk package." ], [ "Propagator functions", "The function QuantumStochasticWalk implements QSWs on graphs and is the main function in the QSWalk package.", "We also provide functions to implement QWs and CRWs, which are much faster to evaluate (as they work with vectors of length $N$ rather than $N^2$ ) and provide a useful comparison.", "QuantumStochasticWalk[H,LkSet,omega,rho0,t]: computes a QSW with Hamiltonian matrix H, Lindblad operators LkSet, and weighting factor omega; starting from density matrix rho0 at time 0 and returning the density matrix at time t. H should be an $N\\times N$ Hermitian matrix, and LkSet should be a list of $N\\times N$ matrices; rho0 should be an $N\\times N$ Hermitian matrix with a trace of unity.", "QuantumWalk[H,psi0,t]: computes a QW with Hamiltonian matrix H, starting from state vector psi0 at time 0 and returning the state vector at time t. H should be an $N\\times N$ Hermitian matrix; psi0 should be a length-$N$ list of complex-valued probability amplitudes with squared magnitudes summing to unity.", "ClassicalRandomWalk[M,p0,t]: computes a CRW with generator matrix M, starting from probability vector p0 at time 0 and returning the probability vector at time t. M should be a real-valued $N\\times N$ matrix; p0 should be a length-N list of non-negative probabilities summing to unity.", "These functions are used to construct the inputs to the functions from the previous section (i.e.", "H, M, and LkSet).", "GeneratorMatrix[G,gamma]: returns the generator matrix $M$ for the graph G, with transition rate gamma.", "LindbladSet[mat]: returns of set of Lindblad matrices corresponding to each non-zero element of mat.", "GoogleMatrix[G,alpha]: returns the “Google matrix” with damping factor alpha for the graph G. The following two functions convert between an $N\\times N$ matrix and its $N\\times 1$ vectorized form (see Section REF ).", "Vectorize[mat]: returns the vectorization of matrix mat.", "Matricize[vec]: returns the matrix formed by applying the inverse action of the Vectorize function on vec.", "The length of vec must be an integer squared.", "Here we provide definitions for several graph types that have been used in studies on quantum walks, but are not already available in Mathematica.", "CayleyTree[d,n]: returns a graph representing an $n$ -th generation Cayley tree of order $d$ [33].", "QWs have been applied to such graphs in the study of quantum search algorithms [34], [35].", "GluedBinaryTree[n]: returns a graph comprising two complete, $(n+1)$ -level binary trees glued together in order along their leaf nodes.", "This graph type was used by Childs et al.", "as an example for which QWs provide an exponential speedup over a CRW [26].", "RandomGluedBinaryTree[n] returns a graph comprising two complete, $(n+1)$ -level binary trees glued together in random order along their leaf nodes.", "These were important in demonstrating the existence of graphs for which no classical algorithm could compete with a quantum walk [14]." ], [ "Vectorizing the master equation", "We now describe our approach for solving the QSW master equation Eq.", "(REF ) in the QSWalk package.", "For clarity, in this section we adopt the matrix viewpoint and drop the $\\hat{}$ notation throughout.", "The master equation is a superoperator acting on the density matrix $\\rho (t)$ .", "In this form, a solution to Eq.", "(REF ) using the matrix exponential, by analogy to Eqs.", "(REF , ), cannot be found.", "We can, however, recast $\\rho (t)$ as the $N^2$ -element column vector $\\tilde{\\rho }(t)$ , formed by concatenating its columns in order: $\\tilde{\\rho }(t) =(\\rho _{11}(t),\\ldots ,\\rho _{N1}(t),\\rho _{12}(t),\\ldots ,\\rho _{N2}(t),\\ldots ,\\rho _{1N}(t),\\ldots ,\\rho _{NN}(t))^T.$ This operation is known in linear algebra as vectorization [36].", "In vectorized form, Eq.", "(REF ) reduces to a linear operator (albeit in an expanded $N^2\\times N^2$ space), whose solution can be expressed as a matrix exponential.", "To vectorize Eq.", "(REF ) we make use of a standard identity ([36], Theorem 14.14) for arbitrary $N\\times N$ matrices $X,Y,Z$ : $\\mathrm {vec}(X\\cdot Y \\cdot Z) = (Z^T \\otimes X) \\cdot \\mathrm {vec}(Y),$ where $\\otimes $ denotes the Kronecker (direct) product, ${}^T$ is the matrix transpose, and $\\cdot $ is standard matrix multiplication.", "The resulting vectorized QSW master equation is: $\\frac{\\mathrm {d}\\tilde{\\rho }}{\\mathrm {d}t} = \\mathcal {L}\\cdot \\tilde{\\rho }(t)$ where $ \\mathcal {L} =-(1-\\omega )\\mathrm {i}\\left(I_N\\otimes H - H^T\\otimes I_N\\right) + \\\\\\omega \\sum _{k=1}^{K}\\left(L_k^{\\ast }\\otimes L_k - \\frac{1}{2}\\left(I_N\\otimes L_k^\\dagger L_k + L_k^TL_k^{\\ast }\\otimes I_N\\right)\\right).$ The value of this representation is that each term is a direct product involving one or more sparse matrices ($I_N$ and $L_k$ ), and therefore $\\mathcal {L}$ is a sparse matrix.", "Fig.", "REF illustrates this point for a complete graph, for which $H$ is dense.", "Figure: (a) Hamiltonian matrix of a complete undirected graph with N=8N=8 andrandom edge weights between 0 and 1; (b) the corresponding sparse matrixℒ\\mathcal {L} (Eq. ).", "The plots are shaded withdarkness proportional to the magnitude of matrix entries, so that whiteareas correspond to zero entries.The matrix exponential solution of Eq.", "(REF ) is then $\\tilde{\\rho }(t) = \\mathrm {e}^{\\mathcal {L} t}\\cdot \\tilde{\\rho }(0).$ The density matrix $\\rho (t)$ is trivially recovered from the vector $\\tilde{\\rho }(t)$ .", "Eq.", "(REF ) can be implemented using Mathematica's built-in functions; in particular, KroneckerProduct, IdentityMatrix, Conjugate, Transpose, ConjugateTranspose, as well as matrix multiplication (the “.” operator).", "The vectorization approach outlined above leads us to work with matrices of size $N^2\\times N^2$ , which quickly become rather large for even moderately-sized graphs.", "It therefore becomes essential to exploit sparse matrix representations.", "In Mathematica, sparse arrays are implemented via the SparseArray function [37], and all of the built-in matrix functionality includes handling to work with SparseArray objects efficiently.", "Therefore, provided we correctly initialize all the relevant matrices to be in SparseArray form, Mathematica will automatically carry out all matrix operations using sparse matrix methods if possible.", "The performance of our approach depends on both the size (number of vertices) and density (number of edges) of the graph being studied.", "Fig.", "REF gives some indicative estimates of running time and memory requirements for representative dense and sparse graphs.", "It can be seen that the primary bottleneck on a typical machine is memory: typically, memory requirements for a QSW can be as high as 4GB with run times of only several minutes.", "This corresponds to $N\\approx 150$ for dense graphs, and $N\\approx 350$ for sparse graphs.", "For QWs, unsurprisingly, the scaling is much better: for comparable run time and memory we can calculate QWs on graphs with $N\\approx 10^4$ (dense) and $N\\approx 10^6$ (sparse).", "Figure: Scaling behaviour for QSWs and QWs as a function of graph size NN, forrepresentative examples of dense (D) and sparse (S) undirected graphs.", "Thedense graphs have N(N-1)/2N(N-1)/2 edges (defined with CompleteGraph), whilethe sparse graphs are of Erdös-Renyi type with ∼N log N\\sim N \\mathrm {log}N edges(defined with RandomGraph); in both cases the edges are assigned randomweights uniformly between 0 and 1.", "In these figures, the slope of the linesgives the scaling exponent with respect to NN.", "In (a), these areapproximately (in the same order as the legend): 3.3, 2.7, 2.2, and 1.1; in(b), they are 3.7, 3.0, 2.0, and 1.1.", "The parameters used are γ=1\\gamma =1,ω=0.5\\omega =0.5, t=10t=10, with initial conditions ρ ij (0)=δ ij /n\\rho _{ij}(0)=\\delta _{ij}/n(QSW) and i|ψ(0)=1/n\\left<i|\\psi (0)\\right>=1/\\sqrt{n} (QW).", "Calculations were performedon a Late-2013 iMac with 8GB RAM, running Mathematica 10.3 on Mac OS 10.11." ], [ "Applications", "In this section we describe several applications of the QSWalk package to systems similar to those studied recently in the literature.", "Our emphasis is on demonstrating the functionality of the package rather than describing any new physics." ], [ "Quantum-classical transition on a line", "One of the key properties of the QSW is that it permits an interpolation between coherent (QW) and incoherent (CRW) dynamics, via the parameter $\\omega $ .", "To illustrate this transition, we consider a QSW on one of the simplest graphs possible: an undirected $N\\times 1$ line graph (which is equivalent to the one-dimensional tight-binding model studied in electron transport theory [38]).", "The adjacency matrix for such a graph has entries $A_{ij}=\\delta _{i,j+1}+\\delta _{i,j-1}$ .", "The matrix elements of the Hamiltonian and Lindblad operators can be found using Eqs.", "(REF ) and (REF ): $\\left<i\\right|\\hat{H}\\left|j\\right> = \\gamma \\left(\\delta _{ij}\\left(2-\\delta _{i,1}-\\delta _{iN}\\right)- \\delta _{i,j+1} - \\delta _{i,j-1}\\right),$ $\\left<i\\right|\\hat{L}_{k^{\\prime }}\\left|j\\right> = \\delta _{i^{\\prime }i}\\delta _{j^{\\prime }j}\\sqrt{\\gamma }\\left(\\delta _{i^{\\prime }j^{\\prime }} \\left(2-\\delta _{i^{\\prime },1}-\\delta _{i^{\\prime }N}\\right)+ \\delta _{i^{\\prime },j^{\\prime }+1} + \\delta _{i^{\\prime },j^{\\prime }-1}\\right),$ where $k^{\\prime }\\equiv (i^{\\prime },j^{\\prime })$ and $\\hat{L}_{k^{\\prime }}$ is the Lindblad operator corresponding to the scattering process $\\left|i^{\\prime }\\right>\\left<j^{\\prime }\\right|$ .", "As initial condition we assume the state is localized in the middle of the line at vertex $q=(N+1)/2$ (assuming odd $N$ ), so that $\\hat{\\rho }(0) = \\left|q\\right>\\left<q\\right|$ .", "The code to implement this QSW in QSWalk is straightforward, and uses the built-in GridGraph function for the line graph: n = 51; gamma = 1.; t = 10.; omega = 0.5; G = GridGraph[{n}]; H = GeneratorMatrix[G, gamma]; LkSet = LindbladSet[H]; rho0 = SparseArray[{{(n+1)/2,(n+1)/2}->1.", "},{n,n}]; rho1 = QuantumStochasticWalk[H,LkSet,omega,rho0,t]; Fig.", "REF shows the dynamics of the resulting QSW.", "In Fig.", "REF a, the transition from an oscillatory distribution (characteristic of coherent dynamics) at $\\omega =0$ , to a purely diffusive distribution at $\\omega =1$ , is clearly evident.", "Fig.", "REF b shows the time evolution of a particular vertex population ($i\\approx 0.6N$ ) for several values of $\\omega $ , illustrating the damping which occurs when $\\omega >0$ .", "This is actually a useful property, as it causes the QSW populations to converge to a stationary state rather than oscillate indefinitely, while still retaining some (transitory) coherence.", "Figure: (a) A sequence of QSWs showing the transition from pure QW (ω=0\\omega =0)to pure CRW (ω=1\\omega =1), on a line graph with N=51N=51 vertices and propagationtime t=5t=5 (in atomic units).", "(b) Time evolution of QSWs on the same graph,for a fixed vertex i=31i=31 (≈0.6N\\approx 0.6N) for several values of ω\\omega ." ], [ "Pure dephasing scattering", "As noted in Ref.", "[17], an alternative approach to incoherent scattering is via pure dephasing scattering processes [3].", "In this model, the Lindblad operators are defined by $ \\hat{L}_k = \\left|k\\right>\\left<k\\right|,$ for $k = 1,\\ldots ,N$ .", "The implementation is the same as the previous section, with the exception of the definition of $\\hat{L}_k$ , which now reads: LkSet = Table[SparseArray[{{i,i}->1}, {n,n}], {i,n}]; In Fig.", "REF we consider a QSW using the pure dephasing model of Eq.", "(REF ) on the line graph from the previous section.", "There are several notable differences: firstly, from Fig.", "REF a it is clear that $\\omega \\rightarrow 1$ no longer produces the CRW.", "In fact, in this limit there are no dynamics at all since the $\\hat{L}_k$ operators do not scatter between vertices.", "In this case, for $\\omega =1$ we have $\\hat{\\rho }(t) = \\hat{\\rho }(0)$ for all $t$ .", "Secondly, comparing Fig.", "REF to Fig.", "REF we see that pure dephasing scattering causes less dephasing than the model of Eq.", "(REF ), in the sense that the wave-like oscillations persist for larger $\\omega $ .", "This is actually not surprising since for the line graph the model of Eq.", "(REF ) has dephasing terms with weighting $\\sqrt{2}$ (in contrast to the weighting of 1 in Eq.", "), in addition to the off-diagonal scattering elements which also contribute to the dephasing.", "Figure: The equivalent of the plots from Fig.", "butfor a QSW with pure dephasing scattering (Eq.", ")." ], [ "Photosynthetic light-harvesting in the FMO complex", "QSWs have found application to an important problem in biophysics, concerning the capture of energy (in the form of photons from sunlight) by photosynthetic protein complexes [39].", "A prototypical example which has been extensively studied is the so-called Fenna-Matthews-Olson (FMO) complex from green sulphur bacteria.", "This molecule is comprised of seven regions called chromophores, through which electronic excitations (excitons) are propagated by a hopping process in which phase coherence plays an important role [40].", "An excitation generated by photon absorption on a particular chromophore is transmitted through the complex to a reaction centre where it is converted to stored chemical energy.", "Theoretical studies aimed at understanding the phase coherence properties in the FMO have modeled the chromophores as vertices in an undirected graph on which excitons follow a quantum walk-like process [41], [42], [43] which, like the QSW, combines coherent and incoherent evolution through a master equation.", "These models are not strictly equivalent to QSWs as they incorporate a loss term, representing absorption of the exciton into the reaction centre, and therefore are not probability-conserving.", "Nevertheless, it is straightforward to develop a QSW model which describes the same essential physics, as we now show.", "As a concrete example, we consider the model of Hoyer et al.", "[42], which uses a Hamiltonian obtained by Adolphs and Renger [44]: $ H = \\left(\\begin{array}{ccccccc}\\mathbf {200} & \\mathbf {-96} & 5 & -4.4 & 4.7 & -12.6 & -6.2 \\\\\\mathbf {-96} & \\mathbf {320} & \\mathbf {33.1} & 6.8 & 4.5 & 7.4 & -0.3 \\\\5 & \\mathbf {33.1} & \\mathbf {0} & \\mathbf {-51.1} & 0.8 & -8.4 & 7.6 \\\\-4.4 & 6.8 & \\mathbf {-51.1} & \\mathbf {110} & \\mathbf {-76.6} & -14.2 & \\mathbf {-67} \\\\4.7 & 4.5 & 0.8 & \\mathbf {-76.6} & \\mathbf {270} & \\mathbf {78.3} & -0.1 \\\\-12.6 & 7.4 & -8.4 & -14.2 & \\mathbf {78.3} & \\mathbf {420} & \\mathbf {38.3} \\\\-6.2 & -0.3 & 7.6 & \\mathbf {-67} & -0.1 & \\mathbf {38.3} & \\mathbf {230}\\end{array}\\right).$ The bold entries represent the chromophores with the largest hopping probabilities, and correspond to vertices 1 to 7 of the graph shown in Fig.", "REF a.", "The units of energy are $\\mathrm {cm}^{-1}$ (using the spectroscopy convention of expressing energy in terms of the wavelength of a photon with that energy, i.e.", "$1\\mathrm {cm}^{-1} \\equiv 1.23984 \\times 10^{-4}\\mathrm {eV}$ ).", "It is also customary to work in units with $\\hbar =1$ , so that the unit of time is $5.309\\mathrm {ps}$ .", "In the FMO model, the exciton is created at chromophore 6, so for our QSW we use initial condition $\\hat{\\rho }(0) = \\left|6\\right>\\left<6\\right|$ .", "Chromophore 3 is the site where the exciton is absorbed into the reaction centre, which has been modeled in previous studies by including non-probability conserving “loss” terms in the master equation.", "Here, we take a simpler approach and add an extra vertex (labeled 8 in Fig.", "REF a).", "This “sink” vertex is connected via a directed edge to vertex 3, but it is not included in the Hamiltonian; it therefore contributes incoherent scattering via a Lindblad operator but does not participate in the coherent evolution.", "Our QSW model then takes $H$ from Eq.", "(REF ) (padded with zeros to make an $8\\times 8$ matrix) to describe the coherent evolution.", "For the incoherent evolution we use the usual set of Lindblad operators from Eq.", "(REF ) (using $H_{ij}$ in place of $M_{ij}$ ), representing dephasing scattering as well as incoherent scattering between chromophores; this is a slight departure from Hoyer et al.", ", who only included the dephasing scattering.", "The exciton absorption at the sink vertex is represented by an extra Lindblad operator, $\\hat{L}_{k}=\\sqrt{\\alpha \\gamma } \\left|8\\right> \\left<3\\right|$ , where the factor $\\alpha $ determines the rate of absorption.", "In Fig.", "REF b we show the vertex populations as a function of time for a particular set of parameters, which demonstrates excellent qualitative agreement with Fig.", "3d of Ref.", "[42] (whose calculation used dephasing scattering rates based on a temperature of 77K).", "A key feature is the damping of oscillations due to the combination of coherent and incoherent dynamics.", "Also evident is the accumulation of probability at the sink vertex ($i=8$ ), at the expense of the remaining vertices, representing the absorption of the exciton into the reaction centre.", "Although temperature does not appear explicitly in our model, its effect is approximated by the factor $\\omega $ : $\\omega \\rightarrow 0$ is equivalent to the low-temperature limit of a phase coherent QW, while $\\omega \\rightarrow 1$ corresponds to the high-temperature limit of a CRW with no phase coherence.", "Figure: (a) Simplified graph structure of the FMO complex.", "(b) Vertexpopulations as a function of time for a QSW on the FMO model (with parametervalues γ=1\\gamma = 1, ω=0.1\\omega =0.1, α=100\\alpha =100)." ], [ "Quantum Page Rank", "As an interesting application of QSWs to directed graphs, we now briefly discuss the recent continuous-time quantum page rank algorithm (QPR) introduced by Sánchez-Burillo et al.", "[20], [45], based on the well-known (classical) Page Rank algorithm (CPR) [46] which is a key component of the Google search engine.", "The basic idea of CPR is to represent the world wide web as a directed graph, with each webpage representing a vertex and the links on each page representing outward edges to the pages linked to; the algorithm then carries out a CRW on this graph to determine the centrality, or “rank”, of each page.", "In the QPR algorithm, the CRW is replaced by a QSW which respects the directed nature of the graph.", "A detailed discussion of the QPR and CPR algorithms can be found in Refs.", "[20], [45]; here we just describe the main features.", "We begin with a directed graph $G$ having $N$ vertices and adjacency matrix $A$ .", "The Hamiltonian is defined as usual in terms of a symmetrized form of $G$ (Eq.", "REF ).", "The Lindblad operators are defined in terms of the so-called “Google matrix”, $\\mathcal {G}$ : $ \\hat{L}_k = \\sqrt{\\gamma \\mathcal {G}_{ij}} \\left|i\\right>\\left<j\\right|,$ where $ \\mathcal {G}_{ij} = {\\left\\lbrace \\begin{array}{ll}\\alpha A_{ij}/\\mathrm {outDeg}(j) + (1-\\alpha )\\frac{1}{N}, &\\mathrm {outDeg}(j) > 0, \\\\\\frac{1}{N}, & \\mathrm {outDeg}(j) = 0.\\end{array}\\right.", "}$ The effect of $\\mathcal {G}$ is to avoid “dead ends” (nodes with out-degree 0 are scattered equally to all other vertices) and to introduce random jumps between all vertices.", "The “damping factor” $\\alpha $ controls the weighting of the random jumps relative to the edges in the original graph ($\\alpha =0.85$ is a common choice in the literature).", "As with all QSWs, the parameter $\\omega $ determines the relative weighting of coherent and incoherent components.", "For $\\omega =0$ we get a QW on a symmetrized graph; this does not in general provide a useful ranking algorithm, as it does not incorporate the graph directionality.", "For $\\omega =1$ , on the other hand, we recover the original CPR algorithm.", "For intermedate $\\omega $ , the QPR algorithm proceeds by evaluating the QSW (Eq.", "REF ) from arbitrary initial conditions with $t$ large enough for $\\hat{\\rho }(t)$ to reach a stationary state.", "The vertex populations, $\\rho _{ii}(t)$ , then represent their rank.", "The implementation using QSWalk is once again straightforward.", "The only new step is the definition of $\\hat{L}_k$ (Eq.", "REF ), for which we now use the GoogleMatrix function:   LkSet = LindbladSet[Sqrt[gamma] GoogleMatrix[G, alpha]]; The remainder of the calculation proceeds as in Section REF .", "It is worth noting the built-in Mathematica function PageRankCentrality, which computes the CPR and is a useful reference point.", "As a simple illustration, in Fig.", "REF we show an example from Ref. [20].", "The QPR and CPR values are broadly similar, although the QPR calculation “lifts” the degeneracy of two vertices whose ranking under CPR is equal.", "In Ref.", "[20], the authors show that this effect also holds for larger real-world networks, suggesting that QPR could become an important application for QSWs.", "Figure: (a) The graph used by Sánchez-Burillo et al.", "to illustrate degeneracy-breaking by the quantumpage rank algorithm.", "(b) Quantum and classical Page Ranks of the vertices ofthis graph; the highlighted vertices have equal CPR but differentQPR.", "(The parameters used in these calculations are t=100t=100, γ=1\\gamma =1,α=0.85\\alpha =0.85, ω=0.8\\omega =0.8.)" ], [ "Conclusion", "The intensive research in quantum computing and information theory over recent decades has spurred interest in a range of quantum walk models such as the recently-developed quantum stochastic walks.", "As well as the exciting potential for practical applications down the line, these models are uncovering a wealth of interesting physics.", "In practice, for all but the simplest examples much of the insight is gained with the help of efficient numerical computing tools.", "There is thus a clear need for software packages which abstract away the computational details and allow the focus to remain on the physical application at hand.", "It is with this perspective in mind that we have developed the QSWalk package; we hope that it will be a useful tool, both in the hands of researchers active in this field as well as instructors wishing to bring cutting-edge models into a computer lab teaching environment.", "On a typical modern desktop computer, the implementation provided by QSWalk can be applied to graphs with a few hundred vertices with calculation times on order of the tens of seconds, which should make it suitable for a wide range of applications.", "However, there will be specialized applications involving larger graphs where it becomes necessary to turn to high performance computing (HPC) resources (supercomputers), which would be beyond the scope of our package in its current form.", "One possibility for a future version of QSWalk, therefore, would be to incorporate calls to a program using parallelized linear algebra libraries running in an HPC environment; an approach along these lines was used successfully in Ref.", "[23] to simulate phase-coherent quantum walks." ], [ "Acknowledgments", "The authors would like to acknowledge helpful discussions with Jingwei Tang, on the Lindblad formalism, and Tania Loke, on the Quantum Page Rank algorithm.", "PEF gratefully acknowledges the hospitality of the UWA School of Physics." ] ]
1606.04974
[ [ "Tricritical behaviour of the frustrated Ising antiferromagnet on the\n honeycomb lattice" ], [ "Abstract We use the effective-field theory with correlations based on different cluster sizes to investigate phase diagrams of the frustrated Ising antiferromagnet on the honeycomb lattice with isotropic interactions of the strength $J_1 < 0$ between nearest-neighbour pairs and $J_2 < 0$ between next-nearest neighbour pairs of spins.", "We present results for the ground-state energy as a function of the frustration parameter $R = J_2 /|J_1|$.", "We find that the cluster-size has a considerable effect on the existence and location of a tricritical point in the phase diagram at which the phase transition changes from the second order to the first one." ], [ "[12pt]article Tricritical behaviour of the frustrated Ising antiferromagnet on the honeycomb lattice A. Bobák$\\,^a$ Corresponding author.", "Fax: +421-55-6222124.", "E-mail address: [email protected] (A. Bobák).", ", T. Lučivjanský$\\,^{a,b}$ , M. Žukovič$\\,^a$ , M. Borovský$\\,^a$ , Corresponding author.", "Fax: +421-55-6222124.", "E-mail address: [email protected] (A. Bobák).", "T. Balcerzak$\\,^c$ , $^a\\,$ Department of Theoretical Physics and Astrophysics, Faculty of Science, P. J. Šafárik University, Park Angelinum 9, 041 54 Košice, Slovak Republic $^b\\,$ Fakultät für Physik, Universität Duisburg-Essen, D-47048 Duisburg, Germany $^c\\,$ Department of Solid State Physics, University of Łódź, Pomorska 149/153, 90-236 Łódź, Poland Abstract     We use the effective-field theory with correlations based on different cluster sizes to investigate phase diagrams of the frustrated Ising antiferromagnet on the honeycomb lattice with isotropic interactions of the strength $J_1 < 0$ between nearest-neighbour pairs and $J_2 < 0$ between next-nearest neighbour pairs of spins.", "We present results for the ground-state energy as a function of the frustration parameter $R = J_2/|J_1|$ .", "We find that the cluster-size has a considerable effect on the existence and location of a tricritical point in the phase diagram at which the phase transition changes from the second order to the first one.", "Keywords: Ising antiferromagnet; Frustrated honeycomb lattice; Phase diagrams; Tricritical point 1.", "Introduction     Since a honeycomb lattice antiferromagnet with only nearest-neighbour exchange interactions ($J_1$ ) is considered as a bipartite lattice, the ground state exhibits long-range ordering.", "The system becomes frustrated like the square lattice, if the next-nearest-neighbour exchange interactions ($J_2$ ) are considered.", "However, spin fluctuations are expected to be larger for the honeycomb lattice than the square lattice because the coordination number $z=3$ in the honeycomb lattice is smaller than that of $z=4$ in the square lattice.", "Hence, it is interesting to study the magnetic ordering on the honeycomb lattice under frustrating interactions.", "We note that investigations of the frustrated two-dimensional Ising antiferromagnet (AF) with spin-$\\frac{1}{2}$ on a square lattice have a long history (see, e.g.", "$[1-8]$ ).", "In particular, it has been found that the introduction of competing interactions is accompanied by the appearance of new ground states at the critical point $R\\equiv J_2/|J_1| = -0.5$ and due to the ground-state degeneracy there is no long-range order at finite temperatures $[4, 9-11]$ .", "Despite the simplicity of the model, it has been proved difficult to precisely determine the order of the phase transition.", "Now, it is well established by using different approximate studies $[10-12]$ and the Monte Carlo method $[13-18]$ that in the region of $R<-0.5$ , the phase transition changes at a tricritical temperature from the second order to the first order.", "However, a very recent cluster mean-field calculation $[18]$ with a cluster of the size $4\\times 4$ and the effective-field theory with correlations based on the different cluster sizes $[19]$ give change in the order of the phase transition not only for $R < -0.5$ but also in the region of $R > -0.5$ .", "Interestingly, a similar attention has not been paid so far to the frustrated Ising AF with spin-$\\frac{1}{2}$ on the honeycomb lattice.", "A special feature of this lattice is that it is not a Bravais lattice, i.e., a translation invariance of the full lattice is broken for any type of state $[20]$ .", "This non-Bravais lattice can be viewed as a composition of two interlacing triangular sublattices and the lattice is constructed by two vectors of the triangular Bravais lattice (see Fig.", "1 in $[21]$ ).", "Hence, for a transition from a paramagnetic state to a magnetically ordered phase, the spatial symmetry is not reduced as for the square lattice.", "We expect that the non-Bravais character of this bipartite lattice results in a behaviour that cannot be observed in the square lattice or other Bravais lattices $[22]$ .", "Moreover, in view of recent experimental activities $[23-28]$ , materials regarded as various types of spin systems on honeycomb lattices are expected to be synthesized.", "Motivated by the above considerations, in this paper we investigate the phase diagram and critical properties of the frustrated $J_1-J_2$ Ising AF on the honeycomb lattice.", "As far as we know, this model has not been analyzed in the literature.", "An interest in the honeycomb lattice is also promoted in recent years because of its relevance to graphen $[29]$ .", "However, when second-neighbor interactions are taken into account or when a magnetic field is applied to the honeycomb lattice, the Hamiltonian is no longer exactly solvable and only approximate analytical studies or numerical approaches are possible to attack this more general problem.", "In this paper we employ the effective-field theory with correlations (EFT) based on different cluster sizes which has been used for an investigation of frustration in the square case $[19]$ .", "Therefore, it will be interesting to compare effects of frustration on the phase diagram of these bipartite lattices.", "This approach is based on the differential operator technique introduced into exact Ising spin identities and has been successfully applied to a variety of spin-$\\frac{1}{2}$ and higher spin problems (for a review see, e.g., Ref.", "$[30, 31]$ ) including a geometrically frustrated triangular lattice Ising AF $[32-34]$ .", "Namely, here we will study the frustrated $J_1-J_2$ Ising AF on the honeycomb lattice in its parameter space using EFT based on one-, two-, four-, and six-spin clusters.", "It is important that the present EFT allows us to treat large clusters in a simpler and more efficient computational manner.", "2.", "Theory     We consider the frustrated honeycomb Ising AF with competing nearest-neighbour $(J_1 < 0)$ and next-nearest-neighbour $(J_2 < 0)$ interactions.", "The Hamiltonian of the model is given by $H = -J_1\\sum _{\\langle i,j\\rangle }s_i s_j - J_2 \\sum _{\\langle i,i_{2}\\rangle }s_i s_{i_2},$ with $s_i = \\pm 1$ , where the first and second sums are taken over all pairs of nearest-neighbours (nn) and next-nearest-neighbours (nnn) of spins, respectively.", "Before calculation of the transition line between ordered and paramagnetic phases, it is appropriate to first consider the ground state of this model.", "For $J_2 = 0$ the ground state of the Hamiltonian (REF ) is the known AF solution with the energy per site $E_{AF}/N = -3/2 |J_1|$ .", "However, adding the nnn AF interactions yields an increase of the ground state energy per site for the AF state: $\\frac{E_{AF}}{N} = -\\frac{3}{2} (|J_1| + 2 J_2).$ In this case each site has its three nn on the other sublattice and six nnn on its own sublattice.", "For a large negative $J_2$ the system orders in the collinear striped states (CS) described either by alternate single ferromagnetic columns of antiparallel spins (Fig.", "1(a)) or alternate pairs of columns consisting of AF coupled spins (Fig.", "1(b)) (see Ref.", "$[35, 36]$ ).", "In such case the ground state is degenerate and its energy per site is given by $\\frac{E_{CS}}{N} = -\\frac{1}{2}(|J_1| - 2 J_2).$ A critical point separating these ordered phases is located at $R_c = -1/4$ , where the transition temperature is suppressed to $T = 0$ K. This value may be compared to that of the frustrated $J_1-J_2$ Ising model on the square lattice $R_c = -1/2$ , where the energy of the collinear (or superantiferromagnetic) state depends only on the value of $J_2$ coupling $[14]$ .", "Due to the degeneracy of the ground state the system remains disordered at all finite temperatures for $R < -1/4$ .", "Therefore, we focus only on the AF phase which exists for $R > -1/4$ .", "A starting point of the EFT for our Ising spin system is generalized Callen-Suzuki $[37, 38]$ exact identity ${\\langle O_{\\lbrace {n}\\rbrace }\\rangle } = {\\Bigg \\langle \\frac{\\rm {Tr_{\\lbrace n\\rbrace }}[\\it O_{\\lbrace {n}\\rbrace } \\exp (-\\beta {\\it H_{\\rm \\it \\lbrace n\\rbrace }})]}{\\rm {Tr_{\\lbrace n\\rbrace }}[\\exp (-\\beta {\\it H_{\\rm \\it \\lbrace n\\rbrace }})]}\\Bigg \\rangle },$ where the partial trace $\\rm {Tr_{\\lbrace n\\rbrace }}$ is to be taken over the set $\\lbrace {n\\rbrace }$ of spin variables specified by the cluster spin Hamiltonian $\\it H_{\\rm \\it \\lbrace n\\rbrace }$ .", "Here, $\\it O_{\\lbrace n\\rbrace }$ denotes any arbitrary spin function including the set of all $\\lbrace {n\\rbrace }$ spin variables (finite cluster) and $\\langle \\cdots \\rangle $ denotes the usual thermal average.", "2.1.", "Single-spin cluster approach     Let us consider first the cluster containing only one spin on site $i$ and $A$ sublattice which interacts with other nn and nnn spins from the neighbourhood.", "In this approach the multispin Hamiltonian $\\it H_{\\rm \\it \\lbrace n\\rbrace }$ for the AF single-spin cluster ($n = 1$ ) on the honeycomb lattice is given by $H_{\\lbrace {1}\\rbrace }^{AF} = -s_i^A h_i^{AF},$ with $h_i^{AF}= J_1\\sum _{i_1=1}^{3}s_{i_1}^B + J_2 \\sum _{i_2=1}^{6}s_{i_2}^A,$ where $s_i^A$ and $s_j^B$ are spin variables on sublattices $A$ and $B$ , respectively, and the superscript $\\rm {AF}$ denotes the antiferromagnetic system.", "After performing the trace over the selected spin $s_i^A$ on the right-hand side of the relation (REF ), applying the differential operator technique, and using the van der Waerden identity for the two-state Ising spin system, one finds $m_A \\equiv \\langle s_i^A \\rangle = {\\Bigg \\langle \\prod _{i_1=1}^{3}(A_1 + B_1 s_{i_1}^B) \\prod _{i_2=1}^{6}(A_2 + B_2 s_{i_2}^A) \\Bigg \\rangle }\\tanh (\\beta x)\\Big |_{x=0},$ where $A_\\nu = \\cosh (J_\\nu D_x)$ , $B_\\nu = \\sinh (J_\\nu D_x)$ $(\\nu = 1, 2)$ , and $D_x = \\partial /\\partial x$ is the differential operator.", "To proceed further, one has to approximate the thermal multiple correlation functions occurring on the right-hand side of Eq.", "(REF ) as follows: ${\\langle s_{i_1}^B s_{i^{\\prime }_1}^B \\cdots s_{i_2}^A\\rangle } \\approx {\\langle s_{i_1}^B\\rangle }{\\langle s_{i^{\\prime }_1}^B \\rangle } \\cdots {\\langle s_{i_2}^A \\rangle },$ which means that nn and nnn of site $i$ are assumed to be completely independent of each other.", "It should be noted here that the approximation (REF ) is quite superior to the standard mean-field theory since even though it neglects correlations between different spins but takes the single-site kinematic relations exactly into account through the van der Waerden identity.", "Based on this approximation, Eq.", "(REF ) reduces to $m_A = (A_1 + B_1 m_B)^3(A_2 + B_2 m_A)^6\\tanh (\\beta x)\\Big |_{x=0},$ where $m_\\alpha $ ($\\alpha = A, B$ ) are the sublattice magnetizations per site.", "At this place, in order to solve the problem generally, we need to evaluate the sublattice magnetization $m_B$ .", "It can be derived in the same way as $m_A$ by the use of (REF ) for the selected spin $s_j$ on $B$ sublattice.", "However, at zero magnetic field we have $m_{AF} \\equiv m_A = -m_B$ and the equation for $m_B$ is the same as Eq.", "(REF ).", "Therefore, in what follows we use only Eq.", "(REF ), which in this case takes the final form $m_{AF} = \\sum _{{n}=0}^{4} K_{2n+1}^{AF} m_{AF}^{2n+1},$ where the coefficients $K_{2n+1}^{AF}$ , which depend on $T$ and $R$ , can be easily calculated within the symbolic programming by using the mathematical relation $\\exp ( \\lambda D_x) f(x) = f(x+\\lambda )$ .", "Because the final expressions for these coefficients are lengthy, their explicit form is omitted.", "We are now interested in studying the transition temperature (or the phase diagram) and the tricritical point of the model where the transition changes from the second order to the first order.", "In the neigbourhood of a second-order transition line where the order parameter $m_{AF}$ is small, Eq.", "(REF ) can be rewritten as $m_{AF} = K_1^{AF} m_{AF} + K_3^{AF} m_{AF}^3 + \\cdots .$ The second-order phase transition line is then determined by the conditions $K_1^{AF} = 1 \\quad \\quad {\\mathrm {a}nd} \\quad \\quad K_3^{AF} < 0.$ Note that we have verified that the coefficient $K_3^{AF}$ is negative in the entire $(T, R)$ plane for $R > -1/4$ .", "Thus, within the present EFT based on the single-spin cluster we have only a second-order transition line between the AF and paramagnetic (P) phases.", "2.2.", "Multi-spin cluster approach     In order to take into account effects of frustration within the present EFT more precisely, it is necessary to consider at least a two-spin cluster.", "In this approach, we select two nn spins, labeled $i$ and $j$ , which interact with other nn and nnn spins from the neighborhood $[39]$ .", "Hence, the multi-spin Hamiltonian $\\it H_{\\rm \\it \\lbrace n\\rbrace }$ for the AF two-spin cluster ($n=2$ ) on the honeycomb lattice (Fig.", "2) is given by ${\\it H_{\\rm \\it \\lbrace ij\\rbrace }}^{AF} = -J_1 s_i^As_j^B - s_i^A h_i^{AF} - s_j^B h_j^{AF},$ with $h_i^{AF} = J_1\\sum _{i_1=1}^{2}s_{i_1}^B + J_2 \\sum _{i_2=1}^{6}s_{i_2}^A, \\quad \\quad h_j^{AF} = J_1\\sum _{j_1=1}^{2}s_{j_1}^A + J_2 \\sum _{j_2=1}^{6}s_{j_2}^B,$ where the terms $i_1=j$ and $j_1=i$ are excluded from summations over the indices $i_1$ and $j_1$ , respectively.", "At this point one should notice that the neighbourhood of the sites $i$ and $j$ of the two-spin cluster for the $J_1-J_2$ model on a honeycomb lattice contains a set of common spins, namely the spins at the sites labeled by $(i_1, j_2)$ or $(j_1, i_2)$ in Fig. 2.", "These spins interact with spins of the cluster and are frustrated directly within the two-spin cluster theory, which is not the case of the one-spin cluster approximation.", "Now, taking this into account and using the same procedure as for the single-spin cluster, one derives the equation analogous to Eq.", "(REF ), which now reads $m_{AF} &=& \\Big [A_x(1) A_y(2) + B_x(1) B_y(2) + m_B \\Big (A_x(1)B_y(2)+A_y(2)B_x(1)\\Big )\\Big ]^2 \\nonumber \\\\& & \\times \\Big [A_y(1) A_x(2) + B_y(1) B_x(2) + m_A \\Big (A_y(1)B_x(2)+A_x(2)B_y(1)\\Big )\\Big ]^2 \\nonumber \\\\& & \\times \\Big [\\Big (A_x(2) + m_A B_x(2)\\Big )\\Big (A_y(2) + m_B B_y(2)\\Big )\\Big ]^4f_{AF} (x,y)\\Big |_{x=0,y=0},$ where $A_\\mu (\\nu ) = \\cosh (J_\\nu D_\\mu )$ , $B_\\mu (\\nu ) = \\sinh (J_\\nu D_\\mu )$ ($\\nu = 1, 2$ ), $D_\\mu = \\partial /\\partial \\mu $ ($\\mu = x, y$ ) are the differential operators and function $f_{AF}(x, y)$ is defined by $f_{AF} (x, y) = \\frac{\\sinh \\beta (x-y)}{\\cosh \\beta (x-y) + e^{2\\beta J_1}\\cosh \\beta (x+y)}.$ Now, by using the condition $m_{AF} \\equiv \\langle (s_i^A-s_j^B)/2\\rangle = m_A = -m_B$ , Eq.", "(REF ) can be finally recast in the form $m_{AF} = \\sum _{{n}=0}^{5} L_{2n+1}^{AF} m_{AF}^{2n+1},$ where the coefficients $L_{2n+1}^{AF}$ , which depend on $T$ and $R$ , can be again easily calculated within the symbolic programming by using the mathematical relation $\\exp (\\lambda D_x + \\gamma D_y) f_{AF}(x, y) = f_{AF}(x+\\lambda , y+\\gamma )$ .", "We also note that in obtaining Eq.", "(REF ) we have made use of the fact that $f_{AF}(x, y) = - f_{AF}(-x, -y)$ and therefore only odd differential operator functions give nonzero contributions.", "The second-order phase transition line is then determined by $L_1^{AF} = 1 \\quad \\quad {\\mathrm {a}nd} \\quad \\quad L_3^{AF} < 0.$ In the vicinity of the second-order phase transition line, the order parameter $m_{AF}$ is given by $m_{AF}^2 = \\frac{1-L_1^{AF}}{L_3^{AF}}.$ The right-hand side of Eq.", "(REF ) must be positive.", "If this not the case, the transition is of the first order, and hence the point at which $L_1^{AF} = 1 \\quad \\quad {\\mathrm {a}nd} \\quad \\quad L_3^{AF} = 0$ is the tricritical point (TCP) $[40]$ .", "To get a more convincing evidence for the existence a TCP in the phase diagram, we have also considered four- and six-spin clusters.", "However, analytical calculations for such large clusters would have been very lengthy and tedious, therefore, the results were obtained in a completely numerical way within the symbolic programming by using Mathematica software package $[41]$ .", "It should be noted here that the calculation times for the large clusters become rather long even using the symbolic programming.", "Therefore, the highest approximation used to study the frustrated Ising honeycomb lattice is the one based on the six-spin cluster.", "3.", "Results and discussion     Numerical results for the critical temperature $k_BT_N/|J_1|$ versus $R$ for various cluster sizes are shown in Fig. 3.", "In this figure the solid lines indicate the second-order phase transitions and the black circles denote the positions of TCPs at which the phase transitions change from the second to the first order.", "First, by solving Eq.", "(REF ) numerically, we obtain a phase diagram between the AF and P phases in the $(R, T)$ plane for the single-spin cluster.", "In this case, as seen from Fig.", "3, the corresponding AF-P transition ($n =1$ ) is always of the second order and the critical temperature gradually reduces from the value $k_BT_N/|J_1| =2.1038$ at $R = 0$ to $k_BT_N/|J_1| =0$ at $R_c =-1/4$ , as expected from the ground-state arguments.", "However, when a larger cluster than the single-spin one is used, the second-order transition line for the frustrated $J_1-J_2$ Ising honeycomb lattice terminates at the TCP.", "Thus, for the larger clusters there are second-order as well as first-order transitions.", "An example of such a phase diagram, obtained by solving Eqs.", "(REF ) and (REF ) numerically for the two-spin cluster, is shown in Fig.", "3 ($n = 2$ ).", "In this figure we show also a phase diagram for the four-spin cluster ($n = 4$ ) obtained within the present EFT in a completely numerical way.", "The scheme of this cluster, which consists of the spins $s_i^A, s_j^B, s_k^B$ , and $s_l^B$ , is illustrated in Fig. 4.", "It is seen from the figure that, similar to the two-spin cluster approximation, the corresponding 'fields' $h_i^{AF}$ , $h_j^{AF}$ , $h_k^{AF} $ , and $h_l^{AF}$ of the four-spin cluster contain a set of common spins, namely the spins at the sites labeled by two indices in Fig. 4.", "(These 'fields', for brevity, are not presented explicitly here.)", "Further, it is seen from Fig.", "4 that within the four-spin cluster approximation we take into account exactly three nn interactions $J_1$ and three nnn interactions $J_2$ .", "For the nonfrustrated model ($R = 0$ ), we find that values of $k_BT_N/|J_1|$ are $1.9869$ and $1.9261$ for the two- and four-spin clusters, respectively, which indicates a relatively slow convergence to the exact value of $k_BT_N/|J_1| = 1.5186$ with the increasing cluster size.", "On the other hand, our estimates for the coordinates of the TCP ($k_BT_t/|J_1|; R_t$ ) are $(1.1352; -0.0907)$ and $(1.0820; -0.9125)$ for the two- and four-spin cluster approximations, respectively.", "Thus, the cluster-size has a considerable effect on the existence and location of the TCP at which the phase transition between the AF and P phases changes from the second order to the first one.", "We note here that the first-order transition line is not possible to calculate on the basis of Eq.", "(REF ) since then we are not allowed to linearize Eq.", "(REF ) in the vicinity of the transition point.", "To solve this problem, one needs to calculate the free energy for the AF and P phases and to find a point of intersection.", "Since only an approximate expression exists for the free energy at finite temperature in the frame of the EFT based on any spin cluster (see, e.g.", "$[12, 19]$ ), we have confined our calculations only to the second-order phase transitions, including the TCP.", "To further investigate this tricritical behaviour, we determine the phase diagram for the six-spin cluster.", "We note that the choice of a six-spin cluster is not unambiguous.", "Indeed, one can choose a cluster with six spins in the form of a 'dumbbell' or hexagon (see Fig. 5).", "In this case the neighbourhood of the sites $i, j, k, l, m$ , and $n$ of the six-spin cluster contains a set of common spins for both the 'dumbbell' spin- and the hexagon spin-clusters.", "In Fig.", "5 these spins are labeled at the sites by two or three indices.", "It is worth noticing that in the hexagon-spin cluster there are only sites labeled by three indices, contrary to the 'dumbbell'-spin cluster where sites with two or three indices exist.", "Since the phase diagrams for these clusters are qualitatively the same, in Fig.", "3 we show only the corresponding phase diagram for the hexagon-spin cluster.", "In particular, we have found that the critical temperature for $R =0$ is $k_BT_N/|J_1|=1.9064$ ('dumbbell'-spin cluster) and $k_BT_N/|J_1|=1.8673$ (hexagon-spin cluster).", "By comparing these values of $k_BT_N/|J_1|$ to the exact value ($k_BT_N/|J_1| = 1.5186$ ), it can be seen that the EFT based on the spin cluster in the form of the hexagon produces a larger improvement in the $k_BT_N/|J_1|$ than that for the 'dumbbell' cluster.", "This is not surprising because the present treatment based on the hexagon cluster approximation takes into account exactly six nn interactions while the EFT based on the 'dumbbell' cluster takes into account exactly only five nn interactions between the pair of spins defining the cluster (see Fig. 5).", "Therefore, a further improvement to the theory is possible if clusters with a larger number of nn interactions are considered.", "This is, as mentioned above, a difficult task due to the fact that calculation times for large clusters become rather long even using the symbolic programming.", "Finally, we estimate coordinates of the TCP at the $(R, T)$ plane, which are $(1.0528; -0.0969)$ and $(0.9189; -0.1133)$ for the 'dumbbell'- and hexagon-spin cluster approximations, respectively.", "Generally it is seen that within the present approach the TCP occurs at a fractionally higher negative value of the frustration parameter with an increasing cluster size, but at temperature considerably lower.", "4.", "Conclusions     We have studied the phase diagram in the (R, T) plane of the frustrated $J_1-J_2$ Ising model with spin-$\\frac{1}{2}$ on a honeycomb lattice using the EFT based on different cluster sizes.", "We have determined that the ground-state is the AF phase for $R >-1/4$ , while the system orders in the CS phase for $R < -1/4$ .", "However, for $R < -1/4$ , we have not found a long-range order at $T \\ne 0$ K due to the degeneracy of the ground state.", "This behaviour has been also confirmed by our preliminary Monte Carlo calculations.", "Further, in the AF region ($R > -1/4$ ), we have found the phase transition line between the AF and P phases.", "However, the present EFT predicts the TCP in the phase diagram only for clusters $n > 1$ , but not for the single-spin $(n=1)$ cluster where only the second-order phase transition was observed.", "Since by using larger and larger clusters, better results are expected, we are forced to conclude that the frustrated $J_1-J_2$ Ising system on a honeycomb lattice exhibits the TCP in the phase diagram between the AF and P phases.", "Therefore, we believe that our effective-field results are qualitatively correct and the tricritical behaviour is due to stronger effects of frustration for the clusters $n > 1$ than for the single-spin $(n=1)$ cluster.", "A thorough Monte Carlo study or more reliable calculations for this frustrated $J_1-J_2$ model would be desirable.", "To our knowledge, no such studies have been attempted yet.", "Acknowledgment     This work was supported by the Scientific Grant Agency of Ministry of Education of Slovak Republic (Grant VEGA No.", "1/0331/15).", "Figure captions Figure 1: Ground-state configurations of the $J_1-J_2$ Ising model on the honeycomb lattice showing two, (a) and (b), degenerate collinear striped states.", "Two sublattices are marked by black and white circles.", "Figure 2: Ground-state configurations of the $J_1-J_2$ Ising model on the honeycomb lattice showing aniferromagnetic states for the two-spin cluster approximation defined by the Hamiltonian (REF ) with spins $s_i$ and $s_j$ (thick line).", "The sites occupied by spins that interact with one or two spins of the cluster are labeled by one or two corresponding indices, respectively.", "Two sublattices are marked by black and white circles.", "Figure 3: Phase diagram in the coupling-temperature plane for the $J_1-J_2$ Ising model on the honeycomb lattice based on the one- ($n=1$ ), two- ($n=2$ ), four- ($n=4$ ), and six-spin ($n=6$ ) clusters.", "The latter cluster corresponds to the hexagon-spin one (see also text).", "The solid lines indicate second-order transitions and the black circles denote the position of a tricritical point.", "AF and $P$ are the antiferromagnetic and paramagnetic phases.", "Figure 4: Ground-state configurations of the $J_1-J_2$ Ising model on the honeycomb lattice showing aniferromagnetic states for the four-spin cluster approximation with spins $s_i, s_j, s_k$ , and $s_l$ (thick lines).", "The sites occupied by spins which interact with one or two spins of the cluster are labeled by one or two corresponding indices, respectively.", "Two sublattices are marked by black and white circles.", "Figure 5: Two options of the six-spin cluster with spins $s_i, s_j, s_k, s_l, s_m$ , and $s_n $ for the antiferromagnetic arrangement on the honeycomb lattice (thick lines): (a) for the 'dumbbell'-spin cluster and (b) for the hexagon-spin cluster.", "The sites occupied by spins which interact with one, two or three spins of the cluster are labeled by one, two or three corresponding indices, respectively.", "Two sublattices are marked by black and white circles." ] ]
1606.05076
[ [ "The Mondrian Kernel" ], [ "Abstract We introduce the Mondrian kernel, a fast random feature approximation to the Laplace kernel.", "It is suitable for both batch and online learning, and admits a fast kernel-width-selection procedure as the random features can be re-used efficiently for all kernel widths.", "The features are constructed by sampling trees via a Mondrian process [Roy and Teh, 2009], and we highlight the connection to Mondrian forests [Lakshminarayanan et al., 2014], where trees are also sampled via a Mondrian process, but fit independently.", "This link provides a new insight into the relationship between kernel methods and random forests." ], [ "INTRODUCTION", "Kernel methods such as support vector machines and Gaussian processes are very popular in machine learning.", "While early work relied on dual optimization, recent large-scale kernel methods focus on the primal optimization problem where the input data are mapped to a finite-dimensional feature space and the weights are learned using fast linear optimization techniques, e.g., stochastic gradient descent.", "[10] proposed to approximate shift-invariant kernels by mapping the inputs to so-called random features, constructed so that the inner product of two mapped data points approximates the kernel evaluated at those two points (which is the inner product in the feature space corresponding to the kernel).", "[10] proposed two random feature construction schemes: random Fourier features, where data points are projected onto random vectors drawn from the Fourier transform of the kernel and then passed through suitable non-linearities; and random binning, where the input space is partitioned by a random regular grid into bins and data points are mapped to indicator vectors identifying which bins they end up in.", "Both of these approaches require specifying the kernel hyperparameters in advance, so that the appropriate distribution is used for sampling the random vectors or random grids, respectively.", "However, a suitable kernel width (length-scale) is often not known a priori and is found by cross-validation, or, where available, marginal likelihood optimization.", "In practice, this entails constructing a new feature space and training a linear learner from scratch for each kernel width, which is computationally expensive.", "Using a suitable kernel width is often more important than the choice of kernel type [14], so a fast kernel width selection method is desirable.", "We describe a connection between the Laplace kernel and the Mondrian process [13], and leverage it to develop a random feature approximation to the Laplace kernel that addresses the kernel width selection problem.", "This approximation, which we call the Mondrian kernel, involves random partitioning of data points using a Mondrian process, which can be efficiently reused for all kernel widths.", "The method preserves the nonparametric nature of kernel learning and is also suitable for online learning.", "The Mondrian kernel reveals an interesting link between kernel methods and decision forests [2], [3], another popular class of nonparametric methods for black-box prediction tasks.", "The Mondrian kernel resembles Mondrian forests, a decision-forest variant introduced by [7], where a Mondrian process is used as the randomization mechanism.", "The efficiently trainable Mondrian forests excel in the online setting, where their distribution is identical to the corresponding batch Mondrian forest, and have been successfully applied to both classification and regression [7], [8].", "Mondrian forests and the Mondrian kernel both lead to randomized, non-linear learning algorithms whose randomness stems from a Mondrian process.", "The former fits parameters corresponding to different Mondrian trees independently, while the latter fits them jointly.", "We compare these methods theoretically and thus establish a novel connection between the Laplace kernel and Mondrian forests via the Mondrian kernel.", "The contributions of this paper are: a review of the Mondrian process using the simple notion of competing exponential clocks (Section ); a novel connection between the Mondrian process and the Laplace kernel (Section ), yielding a fast approximation to learning with the Laplace kernel; an efficient procedure for learning the kernel width from data (Section ); and a comparison between Mondrian kernel and Mondrian forest that provides another connection between kernel learning and random forests (Section )." ], [ "MONDRIAN PROCESS", "For completeness, we review the Mondrian process [13], [12].", "Although simple and perhaps well known to experts, our exposition through competing exponential clocks has not explicitly appeared in this form in the literature.", "Readers familiar with the Mondrian process may skip this section on first reading." ], [ "TERMINOLOGY", "An axis-aligned box $\\mathcal {X}= \\mathcal {X}_1 \\times \\cdots \\times \\mathcal {X}_D \\subseteq \\mathbb {R}^D$ is a Cartesian product of $D$ bounded intervals $\\mathcal {X}_d \\subseteq \\mathbb {R}$ .", "Their total length $|\\mathcal {X}_1| + \\cdots + |\\mathcal {X}_D|$ is the linear dimension of $\\mathcal {X}$ .", "A guillotine partition of $\\mathcal {X}$ is a hierarchical partitioning of $\\mathcal {X}$ using axis-aligned cuts.", "Such a partition can be naturally represented using a strictly binary tree.", "An exponential clock with rate $r$ takes a random time $T \\sim \\operatorname{Exp}(r)$ to ring after being started, where $\\operatorname{Exp}(r)$ is the exponential distribution with rate (inverse mean) $r$ .", "The notion of competing exponential clocks refers to $D$ independent exponential clocks with rates $r_1, \\ldots , r_D$ , started at the same time.", "It can be shown that (1) the time until some clock rings has $\\operatorname{Exp}( \\sum r_d)$ distribution, (2) it is the $d$ -th clock with probability proportional to $r_d$ , and (3) once a clock rings, the remaining $D - 1$ clocks continue to run independently with their original distributions." ], [ "GENERATIVE PROCESS", "The Mondrian process on an axis-aligned box $\\mathcal {X}\\subseteq \\mathbb {R}^D$ is a time-indexed stochastic process taking values in guillotine-partitions of $\\mathcal {X}$ .", "It starts at time 0 with the trivial partition of $\\mathcal {X}$ (no cuts) and as time progresses, new axis-aligned cuts randomly appear, hierarchically splitting $\\mathcal {X}$ into more and more refined partitions.", "The process can be stopped at a lifetime $\\lambda \\in [0, \\infty )$ , which amounts to ignoring any cuts that would appear after time $\\lambda $ .", "To describe the distribution of times and locations of new cuts as time progresses, we associate an independent exponential clock with rate $|\\mathcal {X}_d|$ to each dimension $d$ of $\\mathcal {X}$ .", "Let $T$ be the first time when a clock rings and let $d$ be the dimension of that clock.", "If $T > \\lambda $ then this process terminates.", "Otherwise, a point $a$ is chosen uniformly at random from $\\mathcal {X}_d$ and $\\mathcal {X}$ is split into $\\mathcal {X}^{<} = \\lbrace \\mathbf {x}\\in \\mathcal {X}\\mid x_d < a \\rbrace $ and $\\mathcal {X}^{>} = \\lbrace \\mathbf {x}\\in \\mathcal {X}\\mid x_d > a \\rbrace $ by a hyperplane in dimension $d$ that is perpendicular to $\\mathcal {X}_d$ at point $a$ .", "After making this first cut, the remaining $D - 1$ clocks are discarded and the generative process restarts recursively and independently on $\\mathcal {X}^{<}$ and $\\mathcal {X}^{>}$ .", "However, those processes start at time $T$ rather than 0 and thus have less time left until the lifetime $\\lambda $ is reached.", "The specification of the generative process on $\\mathcal {X}$ is now complete.", "Due to the properties of competing exponential clocks, the time until the first cut appears in $\\mathcal {X}$ has exponential distribution with rate equal to the linear dimension of $\\mathcal {X}$ and the dimension $d$ in which the cut is made is chosen proportional to $|\\mathcal {X}_d|$ .", "This confirms equivalence of our generative process to the one proposed by [13].", "Finally, we note that a.s. the Mondrian process does not explode, i.e., for every lifetime $\\lambda \\in [0, \\infty )$ , the process generates finitely many cuts with probability 1 [12].", "Figure: (a) Sample of a Mondrian process on the axis-aligned box 𝒳=[0,1]×[0,1]⊆ℝ 2 \\mathcal {X}= [0, 1] \\times [0, 1] \\subseteq \\mathbb {R}^2 with lifetime λ=1.0\\lambda = 1.0.", "Numbers on the cuts (shown in green) indicate the times when they appeared.", "The first cut appeared at time T=0.23T = 0.23, in dimension d=1d = 1, at location a=0.66∈𝒳 1 a = 0.66 \\in \\mathcal {X}_1.", "(b) Representing the Mondrian sample as a strictly binary tree, with new nodes (shown as circles) appearing as time (y-axis) progresses.", "The two numbers below each node show the rates of the two exponential clocks competing to split that node, with the winning clock's rate shown in green." ], [ "PROJECTIVITY", "If a Mondrian process runs on $\\mathcal {X}$ , what distribution of random partitions does it induce on an axis-aligned subbox $\\mathcal {A}\\subseteq \\mathcal {X}$ ?", "(See Figure REF for an illustration in $D = 2$ dimensions.)", "The Mondrian process was constructed so that the answer is the Mondrian process itself [12].", "Here we explain this projectivity property using the notion of competing exponential clocks.", "To argue that the resulting process on $\\mathcal {A}$ is indeed a Mondrian process, we show that the process running on $\\mathcal {X}$ generates cuts in $\\mathcal {A}$ in the same way as a Mondrian process running directly on $\\mathcal {A}$ would.", "Recall that each dimension $d$ of $\\mathcal {X}$ is associated with an exponential clock with rate $|\\mathcal {X}_d|$ and if it rings first, the cut location is chosen uniformly at random from $\\mathcal {X}_d$ .", "This procedure can be equivalently represented using two competing clocks for each dimension (rather than just one): Clock $\\mathcal {C}^d_{\\mathcal {A}}$ with rate $|\\mathcal {A}_d|$ .", "If this clock rings first, the cut location is chosen uniformly at random from $\\mathcal {A}_d$ .", "Clock $\\mathcal {C}^d_{\\lnot \\mathcal {A}}$ with rate $|\\mathcal {X}_d| - |\\mathcal {A}_d|$ .", "If it rings first, the cut location is sampled uniformly from $\\mathcal {X}_d \\setminus \\mathcal {A}_d$ .", "(See Figure REF .)", "Note that the clocks $\\mathcal {C}^1_{\\mathcal {A}}, \\ldots , \\mathcal {C}^D_{\\mathcal {A}}$ represent the same cut distribution as a Mondrian process running on $\\mathcal {A}$ would.", "If a clock $\\mathcal {C}^d_{\\lnot \\mathcal {A}}$ rings first, a cut is made outside of $\\mathcal {A}$ and all of $\\mathcal {A}$ remains on one side of this cut.", "None of the clocks $\\mathcal {C}^d_{\\mathcal {A}}$ have rung in that case and would usually be discarded and replaced with fresh clocks of identical rates, but by property (3) of competing exponential clocks, we can equivalently reuse these clocks (let them run) on the side of the cut containing $\\mathcal {A}$ .", "(Figure REF shows a cut in dimension $d=1$ that misses $\\mathcal {A}$ and the reused clocks $\\mathcal {C}^d_{\\mathcal {A}}$ ).", "Hence, cuts outside $\\mathcal {A}$ do not affect the distribution of the first cut crossing $\\mathcal {A}$ , and this distribution is the same as if a Mondrian process were running just on $\\mathcal {A}$ .", "When a cut is made within $\\mathcal {A}$ (see Figure REF ), the process continues on both sides recursively and our argument proceeds inductively, confirming that the Mondrian process on $\\mathcal {X}$ generates cuts in $\\mathcal {A}$ in the same way as a Mondrian process on $\\mathcal {A}$ would.", "Figure: (a) A Mondrian process running on 𝒳=𝒳 1 ×𝒳 2 \\mathcal {X}= \\mathcal {X}_1 \\times \\mathcal {X}_2 generates cuts (dashed lines), some of which intersect 𝒜=𝒜 1 ×𝒜 2 \\mathcal {A}= \\mathcal {A}_1 \\times \\mathcal {A}_2 (green lines) and thus induces a random partition of 𝒜\\mathcal {A}.", "(b) Representing the first cut distribution using 2D=42D = 4 competing exponential clocks: in each dimension dd, clock 𝒞 𝒜 d \\mathcal {C}^d_{\\mathcal {A}} corresponds to the region where making a cut splits 𝒜\\mathcal {A} (shown in green) and clock 𝒞 ¬𝒜 d \\mathcal {C}^d_{\\lnot \\mathcal {A}} to the (disconnected) region where making a cut misses 𝒜\\mathcal {A} (shown in red).", "(c) Cut outside 𝒜\\mathcal {A}: reusing the clocks 𝒞 𝒜 1 \\mathcal {C}^1_{\\mathcal {A}}, 𝒞 𝒜 2 \\mathcal {C}^2_{\\mathcal {A}} on the side of the cut containing 𝒜\\mathcal {A}.", "(d) Cut inside 𝒜\\mathcal {A} (shown in black): the argument proceeds by induction on both sides." ], [ "MONDRIAN PROCESS ON $\\mathbb {R}^D$", "The Mondrian process on $\\mathbb {R}^D$ is defined implicitly as a time-indexed stochastic process such that its restriction to any axis-aligned box $\\mathcal {X}\\subseteq \\mathbb {R}^D$ is a Mondrian process as defined in section REF .", "Fortunately, this infinite-dimensional object can be compactly represented by instantiating the Mondrian process only in regions where we have observed data.", "As we observe new data points, the Mondrian sample can be extended using the conditional Mondrian algorithm [13], a simple and fast sampling procedure for extending a Mondrian sample in an axis-aligned box $\\mathcal {A}$ to a larger axis-aligned box $\\mathcal {X}\\supseteq \\mathcal {A}$ .", "The conditional Mondrian is useful for online learning and prediction, as it can be used to extend Mondrian samples to (yet) unobserved parts of the input space [7]." ], [ "MONDRIAN KERNEL", "For concreteness, our running example will be regression: the problem of learning a function $f : \\mathbb {R}^D \\rightarrow \\mathbb {R}$ from a set of $N$ training examples $(\\mathbf {x}_1, y_1), \\ldots , (\\mathbf {x}_N, y_N)$ .", "However, the Mondrian kernel applies equally well to classification, or any other learning task.", "Learning with kernels involves choosing a kernel function $k : \\mathbb {R}^D \\times \\mathbb {R}^D \\rightarrow \\mathbb {R}$ to act as a similarity measure between input data points.", "Evaluating $k(\\cdot , \\cdot )$ on all pairs of $N$ data points takes $\\Omega (N^2)$ operations, with some models also requiring a $\\Theta (N^3)$ operation on an $N \\times N$ kernel matrix.", "This generally makes exact kernel methods unsuitable for large-scale learning.", "[10] proposed a fast approximation through a randomized construction of a low-dimensional feature map $\\phi : \\mathbb {R}^D \\rightarrow \\mathbb {R}^C$ such that $\\forall {\\mathbf {x}, \\mathbf {x}^{\\prime } \\in \\mathbb {R}^D}\\hspace{20.0pt}k(\\mathbf {x}, \\mathbf {x}^{\\prime })\\approx \\phi (\\mathbf {x})^T \\phi (\\mathbf {x}^{\\prime })$ and then using a linear learning method in the feature space $\\mathbb {R}^C$ implied by $\\phi $ .", "For example, linear regression $\\mathbf {y}\\approx \\mathbf {\\Phi }\\mathbf {w}$ , where $\\mathbf {\\Phi }\\in \\mathbb {R}^{N \\times C}$ is the feature matrix with $n$ -th row $\\phi (\\mathbf {x}_n)^T$ , is solvable exactly in time linear in $N$ .", "In general, the primal problem also lends itself naturally to stochastic gradient descent approaches for learning $\\mathbf {w}$ .", "We use the Mondrian process to construct a randomized feature map for the (isotropic) Laplace kernel: $k(\\mathbf {x}, \\mathbf {x}^{\\prime })= \\exp (- \\lambda \\Vert \\mathbf {x}- \\mathbf {x}^{\\prime } \\Vert _1)= \\exp (- \\lambda \\sum _{d = 1}^D |x_d - x^{\\prime }_d|).$ Here $\\lambda \\ge 0$ is the inverse kernel width (inverse length-scale), which we call the lifetime parameter of the kernel.", "We use a non-standard parametrization as this lifetime parameter will be linked to the Mondrian process lifetime." ], [ "MONDRIAN KERNEL", "Consider the following randomized construction of a feature map $\\phi : \\mathbb {R}^D \\rightarrow \\mathbb {R}^C$ : Sample a partition of $\\mathbb {R}^D$ via a Mondrian process on $\\mathbb {R}^D$ with lifetime $\\lambda $ .", "Label the cells of the generated partition by $1, 2, \\ldots $ in arbitrary order.", "To encode a data point $\\mathbf {x}\\in \\mathbb {R}^D$ , look up the label $c$ of the partion cell $\\mathbf {x}$ falls into and set $\\phi (\\mathbf {x})$ to be the (column) indicator vector that has a single non-zero entry at position $c$ , equal to 1.", "The Mondrian process on $\\mathbb {R}^D$ generates infinitely many partition cells and cannot be stored in memory, but projectivity comes to the rescue.", "As we only ever need to evaluate $\\phi $ on finitely many data points, it suffices to run the Mondrian on the smallest axis-aligned box containing all these points.", "Also, we only label partition cells containing at least one data point, in effect removing features that would be 0 for all our data points.", "Then, the dimensionality $C$ of $\\phi $ equals the number of non-empty partiton cells and each data point has a single non-zero feature, equal to 1.", "Figure: Feature expansions of 4 data points in ℝ 2 \\mathbb {R}^2.However, note that the set of points on which the feature map $\\phi $ is evaluated need not be known in advance and can even grow in an online fashion.", "Indeed, the conditional Mondrian algorithm discussed in section REF allows us to extend Mondrian samples to larger boxes as necessary, and we can increase the dimensionality of $\\phi $ whenever a data point is added to a previously empty partition cell.", "This feature map $\\phi $ induces a kernel $k_1(\\mathbf {x}, \\mathbf {x}^{\\prime }) &:=\\phi (\\mathbf {x})^T \\phi (\\mathbf {x}^{\\prime }) \\nonumber \\\\ &=\\begin{dcases}1 & \\text{ if } \\mathbf {x}, \\mathbf {x}^{\\prime } \\text{ in same partition cell } \\\\0 & \\text{ otherwise }\\end{dcases}$ which we call a Mondrian kernel of order 1.", "Instead of using a single Mondrian sample (partition), we can use $M$ independent samples and construct a feature map $\\phi $ by concatenating and normalizing the feature maps $\\phi ^{(1)}, \\ldots , \\phi ^{(M)}$ obtained from each individual sample as above: $\\phi (\\mathbf {x}):= \\frac{1}{\\sqrt{M}} \\left[ \\phi ^{(1)}(\\mathbf {x})^T \\;\\;\\cdots \\;\\; \\phi ^{(M)}(\\mathbf {x})^T \\right]^T.$ This feature expansion is sparse: every data point has exactly $M$ non-zero features.", "The corresponding kernel, which we call a Mondrian kernel of order $M$ , is $k_M(\\mathbf {x}, \\mathbf {x}^{\\prime }) &:=\\phi (\\mathbf {x})^T \\phi (\\mathbf {x}^{\\prime })\\\\ &=\\frac{1}{M} \\sum _{m = 1}^M \\phi ^{(m)}(\\mathbf {x})^T \\phi ^{(m)}(\\mathbf {x}^{\\prime }).$ This is the empirical frequency with which points $\\mathbf {x}$ and $\\mathbf {x}^{\\prime }$ end up in the same partition cell of a Mondrian sample.", "[H] Mondrian kernel [1] $m = 1$ to $M$ construct feature map $\\phi ^{(m)}$ section  REF join and rescale $\\phi ^{(1)}, \\ldots , \\phi ^{(M)}$ into $\\phi $ equation (REF ) map data $\\mathbf {X}$ to feature representations $\\mathbf {\\Phi }$ using $\\phi $ use linear learning method on $\\mathbf {\\Phi }$" ], [ "MONDRIAN–LAPLACE LINK", "By independence of the $M$ Mondrian samples, a.s. $\\lim _{M \\rightarrow \\infty } k_M(\\mathbf {x}, \\mathbf {x}^{\\prime })= \\mathbb {E}\\left[ \\phi ^{(1)}(\\mathbf {x})^T \\phi ^{(1)}(\\mathbf {x}^{\\prime }) \\right]= \\mathbb {E}\\left[ k_1(\\mathbf {x}, \\mathbf {x}^{\\prime }) \\right]$ with convergence at the standard rate $\\mathcal {O}_p(M^{-1/2})$ .", "We thus define the Mondrian kernel of order $\\infty $ as $k_{\\infty }(\\mathbf {x}, \\mathbf {x}^{\\prime }) := \\mathbb {E}[ k_1(\\mathbf {x}, \\mathbf {x}^{\\prime })].$ Proposition 1 (Mondrian-Laplace link) The Mondrian kernel of order $\\infty $ coincides with the Laplace kernel.", "As $k_1(\\mathbf {x}, \\mathbf {x}^{\\prime })$ (defined in (REF )) is a binary random variable, $k_{\\infty }(\\mathbf {x}, \\mathbf {x}^{\\prime })$ equals the probability that $\\mathbf {x}$ and $\\mathbf {x}^{\\prime }$ fall into the same partition cell of a Mondrian sample, which is equivalent to the sample having no cut in the minimal axis-aligned box spanned by $\\mathbf {x}$ and $\\mathbf {x}^{\\prime }$ .", "Figure: NO_CAPTIONBy projectivity, this probability is the same as the probability of not observing any cuts in a Mondrian process with lifetime $\\lambda $ running on just this minimal box.", "Noting that the linear dimension of this box is $\\Vert \\mathbf {x}- \\mathbf {x}^{\\prime } \\Vert _1$ , we obtain $k_{\\infty }(\\mathbf {x}, \\mathbf {x}^{\\prime }) & =\\mathbb {P}(\\text{no cut between } \\mathbf {x}, \\mathbf {x}^{\\prime } \\text{ until time } \\lambda )\\\\ & =\\mathbb {P}\\left( T > \\lambda \\right) \\text{ where } T \\sim \\text{Exp}\\left( \\Vert \\mathbf {x}- \\mathbf {x}^{\\prime } \\Vert _1 \\right)\\\\ & =e^{- \\lambda \\Vert \\mathbf {x}- \\mathbf {x}^{\\prime } \\Vert _1}.$ Note that the lifetime (inverse width) $\\lambda $ of the Laplace kernel corresponds to the lifetime of the Mondrian process used in the construction of the Mondrian kernel.", "This link allows us to approximate the Laplace kernel with a Mondrian kernel $k_M$ , which, unlike the Laplace kernel, admits a finite-dimensional feature expansion.", "The finite order $M$ trades off kernel approximation error and computational costs (indirectly through the complexity of $\\phi $ ).", "The following result confirms that the convergence of the Mondrian kernel approximation is exponentially fast in $M$ uniformly on any fixed bounded input domain $\\mathcal {X}$ .", "Proposition 2 For any bounded input domain $\\mathcal {X}\\subseteq \\mathbb {R}^D$ and $\\delta > 0$ , as $M \\rightarrow \\infty $ , $&\\;\\;\\;\\;\\;\\mathbb {P}\\left[ \\sup _{\\mathbf {x}, \\mathbf {x}^{\\prime } \\in \\mathcal {X}} \\left| k_M(\\mathbf {x}, \\mathbf {x}^{\\prime }) - k_{\\infty }(\\mathbf {x}, \\mathbf {x}^{\\prime }) \\right| > \\delta \\right]\\\\ &=\\mathcal {O}\\left( M^{2/3} e^{- M \\delta ^2 / (12D + 2)} \\right).$ Given in Supplement ." ], [ "FAST KERNEL WIDTH LEARNING", "This section discusses the main advantage of our Mondrian approximation to the Laplace kernel: the efficient learning of kernel width from data.", "In particular, the approximation allows for efficient evaluation of all kernel lifetimes (inverse widths) $\\lambda \\in [0, \\Lambda ]$ , where the terminal lifetime $\\Lambda > 0$ need not be fixed a priori." ], [ "FEATURE SPACE REUSAL", "We make the following recollections from earlier sections: the Mondrian process runs through time, starting at time 0 and only refining the generated partition as time progresses (cuts are never removed) the Mondrian process with lifetime $\\lambda $ is obtained by ignoring any cuts that would occur after time $\\lambda $ the lifetime $\\lambda $ of the Mondrian process used in constructing an explicit feature map $\\phi $ for a Mondrian kernel corresponds to the lifetime (inverse width) of the Laplace kernel that it approximates Running the Mondrian process from time 0 to some terminal lifetime $\\Lambda $ thus sweeps through feature spaces approximating all Laplace kernels with lifetimes $\\lambda \\in [0, \\Lambda ]$ .", "More concretely, we start with $\\lambda = 0$ and $\\phi $ the feature map corresponding to $M$ trivial partitions, i.e., for any data point $\\mathbf {x}$ , the vector $\\phi (\\mathbf {x})$ has length $M$ and all entries set to the normalizer $M^{-1/2}$ .", "As we increase $\\lambda $ , at discrete time points new cuts appear in the $M$ Mondrian samples used in constructing $\\phi $ .", "Suppose that at some time $\\lambda $ , the partition cell corresponding to the $c$ -th feature in $\\phi $ is split into two by a new cut that first appeared at this time $\\lambda $ .", "We update the feature map $\\phi $ by removing the $c$ -th feature and appending two new features, one for each partition cell created by the split.", "See Figure REF for an example with $M = 1$ .", "Figure: A new cut (shown in thick blue) appeared, splitting cell c=2c = 2 (cf.", "Figure ) into two new cells c=4c = 4 and c=5c = 5.", "The table shows the update to φ\\phi , with the removed feature in gray italics and the two new features in bold blue.This procedure allows us to approximate all Laplace kernels with lifetimes $\\lambda \\in [0, \\Lambda ]$ without having to resample new feature spaces for each lifetime.", "The total computational cost is the same (up to a multiplicative constant) as of constructing a single feature space just for the terminal lifetime $\\Lambda $ .", "This is because a strictly binary tree with $C^{(m)}$ leaves (partition cells in the $m$ -th Mondrian sample at time $\\Lambda $ ) contains at most $C^{(m)} - 1$ internal nodes (features that had to be removed at some time point $\\lambda < \\Lambda $ )." ], [ "LINEAR LEARNER RETRAINING", "Evaluating suitability of a lifetime (inverse kernel width) $\\lambda $ requires training and evaluating a linear model in the feature space implied by $\\phi $ .", "This can also be done more efficiently than retraining a new model from scratch every time a new cut is added and $\\phi $ updated.", "We discuss the example of ridge regression with exact solutions, and a general case of models trainable using gradient descent methods." ], [ "Ridge regression", "The MAP weights of the primal ridge regression problem are $\\hat{\\mathbf {w}} = \\mathbf {A}^{-1} \\mathbf {\\Phi }^T \\mathbf {y}$ , where $\\mathbf {A} := (\\mathbf {\\Phi }^T \\mathbf {\\Phi }+ \\delta ^2 \\mathbf {I}_C)$ is the regularized feature covariance matrix and $\\delta ^2$ is the regularization hyperparameter.", "Instead of inverting $\\mathbf {A}$ , it is numerically more stable to work with its Cholesky factor $\\operatorname{chol}(\\mathbf {A})$ [16].", "Phrasing the problem as Bayesian linear regression with, say, observation noise variance $\\sigma _{y}^2 = \\delta ^2$ and prior weights variance $\\sigma _{w}^2 = 1$ , we can also obtain the log marginal likelihood $\\mathcal {L}(\\lambda )$ of the form $\\mathcal {L}(\\lambda )= - \\frac{\\Vert \\mathbf {y}- \\mathbf {\\Phi }\\hat{\\mathbf {w}} \\Vert _2^2}{2 \\delta ^2}- \\frac{\\Vert \\hat{\\mathbf {w}} \\Vert _2^2}{2}- \\frac{1}{2} \\ln \\det \\mathbf {A}+ \\text{const},$ where the dependence on $\\lambda $ is implicit through $\\phi $ .", "When a new cut appears in one of the $M$ Mondrian samples and $\\phi $ is updated by deleting the $c$ -th feature and appending two new ones, the corresponding update to the regularized feature covariance matrix $\\mathbf {A}$ is to delete its $c$ -th row and $c$ -th column, and append two new rows and columns.", "Then both $\\mathbf {A}^{-1}$ and $\\operatorname{chol}(\\mathbf {A})$ can be appropriately updated in $\\mathcal {O}(C^2)$ time, faster than $\\mathcal {O}(C^3)$ recomputation from scratch.", "Updating the Cholesky factor when the $c$ -th row and column are removed is slightly involved but can be achieved by first permuting the rows and columns so that the ones to be removed are the last ones [17], after which the Cholesky factor is updated by deleting its last row and column.", "If $C$ is the number of features at the terminal lifetime $\\Lambda $ , this $\\mathcal {O}(C^2)$ update is performed $\\mathcal {O}(C)$ times, for a total computational cost $\\mathcal {O}(C^3)$ .", "Note that performing the inversion or Cholesky factorization at just the terminal lifetime $\\Lambda $ would have the same time complexity.", "After updating $\\mathbf {A}^{-1}$ or $\\operatorname{chol}(\\mathbf {A})$ , the optimal weights $\\hat{\\mathbf {w}}$ can be updated in $\\mathcal {O}(C^2 + N)$ time and the determinant of $\\mathbf {A}$ required for the marginal likelihood $\\mathcal {L}(\\lambda )$ can be obtained from $\\operatorname{chol}(\\mathbf {A})$ as the squared product of its diagonal elements in $\\mathcal {O}(C)$ time.", "Exploiting sparsity of $\\phi $ , evaluating the model on $N_{\\text{test}}$ data points takes $\\mathcal {O}(N_{\\text{test}} M)$ time.", "Finally, we note that computing the marginal likelihood $\\mathcal {L}(\\lambda )$ for all $\\lambda \\in [0, \\Lambda ]$ and combining it with a prior $p(\\lambda )$ supported on $[0, \\Lambda ]$ allows Bayesian inference over the kernel width $\\lambda ^{-1}$ .", "We refer to Supplement  for more details." ], [ "Models trainable using gradient descent", "Consider a linear model trained using a gradient descent method.", "If (an approximation to) the optimal weight vector $\\mathbf {w}$ is available and then $\\phi $ is updated by removing the $c$ -th feature and appending two new features, a natural way of reinitializing the weights for subsequent gradient descent iterations is to remove the $c$ -th entry of $\\mathbf {w}$ and append two new entries, both set to the removed value (as points in the split cell are partitioned into the two new cells, this preserves all model predictions).", "Note that we have the freedom of choosing the number of gradient descent iterations after each cut is added, and we can opt to only evaluate the model (on a validation set, say) at several $\\lambda $ values on the first pass through $[0, \\Lambda ]$ .", "One iteration of stochastic gradient descent takes $O(M)$ time thanks to sparsity of $\\phi $ .", "This efficient kernel width selection procedure can be especially useful with models where hyperparameters cannot be tweaked by marginal likelihood optimization (e.g., SVM)." ], [ "ONLINE LEARNING", "In this section, we describe how the Mondrian kernel can be used for online learning.", "When a new data point $\\mathbf {x}_{N + 1} \\in \\mathbb {R}^D$ arrives, incorporating it into $M$ existing Mondrian samples (using the conditional Mondrian algorithm discussed in section REF ) can create $0 \\le k \\le M$ new non-empty partition cells, increasing the dimensionality of the feature map $\\phi $ .", "We set the new features to 0 for all previous data points $\\mathbf {x}_1, \\ldots , \\mathbf {x}_N$ .", "In our running example of ridge regression, exact primal updates can again be carried out efficiently.", "The inverse $\\mathbf {A}^{-1}$ or Cholesky factor $\\operatorname{chol}(\\mathbf {A})$ of the regularized feature covariance matrix $\\mathbf {A}$ can be updated in two steps: extend $\\mathbf {A}^{-1}$ or $\\operatorname{chol}(\\mathbf {A})$ to incorporate the $k$ new features (set to 0 for all existing data points) in $\\mathcal {O}(C^2)$ incorporate the new data point $\\mathbf {x}_{N + 1}$ , which is now a simple rank-1 update on $\\mathbf {A}$ , so $\\mathbf {A}^{-1}$ or $\\operatorname{chol}(\\mathbf {A})$ can again be updated efficiently in $\\mathcal {O}(C^2)$ time We refer to Supplement  for more details.", "With gradient descent trainable models, we maintain (an approximation to) the optimal weights $\\mathbf {w}$ directly.", "When a new data point arrives, we expand the dimensionality of $\\phi $ as described above.", "The previously optimal weights can be padded with 0's in any newly added dimensions, and then passed to the gradient descent method as initialization." ], [ "LINK TO MONDRIAN FOREST", "We contrast Mondrian kernel with Mondrian forest [7], [8], another non-linear learning method based on the Mondrian process.", "They both start by sampling $M$ independent Mondrians on $\\mathbb {R}^D$ to provide $M$ independent partitions of the data.", "However, these partitions are then used differently in the two models: In a Mondrian forest, parameters of predictive distributions in each tree are fitted independently of all other trees.", "The prediction of the forest is the average prediction among the $M$ trees.", "With Mondrian kernel, the weights of all random features are fitted jointly by a linear learning method.", "Let $C^{(m)}$ count the leaves (non-empty partition cells) in the $m$ -th Mondrian sample and let $C = \\sum _{m = 1}^M C^{(m)}$ be the total number of leaves.", "Let $\\mathbf {\\phi }^{(m)}_n := \\phi ^{(m)}(\\mathbf {x}_n) \\in \\mathbb {R}^{C^{(m)}}$ be the indicator of the partition cell in the $m$ -th sample into which the $n$ -th data point falls (as in section REF ).", "Also, as in equation (REF ), let $\\mathbf {\\phi }_n := \\phi (\\mathbf {x}_n) \\in \\mathbb {R}^C$ be the normalized concatenated feature encoding of the $n$ -th data point.", "Recall that each vector $\\mathbf {\\phi }_n \\in \\mathbb {R}^C$ contains exactly $M$ non-zero entries, all of which equal the normalizer $M^{-1/2}$ .", "For simplicity, we restrict our attention to ridge regression in this section and compare the learning objective functions of Mondrian kernel and Mondrian forest." ], [ "MONDRIAN KERNEL OBJECTIVE", "The primal ridge regression problem in the feature space implied by $\\phi $ is $\\min _{\\mathbf {w}\\in \\mathbb {R}^C}\\;\\sum _{n = 1}^N (y_n - \\mathbf {w}^T \\mathbf {\\phi }_n)^2+ \\delta ^2 \\Vert \\mathbf {w}\\Vert _2^2.$ Decomposing $\\mathbf {w}= M^{-1/2} [ \\mathbf {w}^{(1)T} \\cdots \\mathbf {w}^{(M)T}]^T$ , so that each (rescaled) subvector $\\mathbf {w}^{(m)}$ corresponds to features from the $m$ -th Mondrian, denoting by $\\hat{y}_n^{(m)} := \\mathbf {w}^{(m)T} \\mathbf {\\phi }_n^{(m)}$ the “contribution\" of the $m$ -th Mondrian to the prediction at the $n$ -th data point, and writing $\\text{loss}(y, \\hat{y}) := (y - \\hat{y})^2$ , the Mondrian kernel objective function can be restated as $\\min _{\\mathbf {w}\\in \\mathbb {R}^C}\\sum _{n = 1}^N \\text{loss}\\left(y_n, \\frac{1}{M} \\sum _{m = 1}^M \\hat{y}_n^{(m)} \\right)+ \\delta ^2 \\Vert \\mathbf {w}\\Vert _2^2.$" ], [ "MONDRIAN FOREST OBJECTIVE", "Assuming a factorizing Gaussian prior over the leaves in each Mondrian tree (i.e., without the hierarchical smoothing used by [8]), the predictive mean parameters $\\mathbf {w}^{(m)}$ in the leaves of the $m$ -th Mondrian tree are fitted by minimizing $\\min _{\\mathbf {w}^{(m)} \\in \\mathbb {R}^{C^{(m)}}}\\sum _{n = 1}^N (y_n - \\mathbf {w}^{(m)T} \\mathbf {\\phi }^{(m)}_n)^2+ \\gamma ^2 \\Vert \\mathbf {w}^{(m)} \\Vert _2^2$ where $\\gamma ^2$ is the ratio of noise and prior variance in the predictive model.", "The parameters $\\mathbf {w}^{(m)}$ are disjoint for different trees, so these $M$ independent optimization problems are equivalent to minimizing the average of the $M$ individual objectives.", "Writing $\\hat{y}_n^{(m)} := \\mathbf {w}^{(m)T} \\mathbf {\\phi }_n^{(m)}$ for the $m$ -th tree's prediction at the $n$ -th data point and concatenating the parameters $\\mathbf {w}:= M^{-1/2} [\\mathbf {w}^{(1)T} \\cdots \\mathbf {w}^{(M)T}]^T$ , the Mondrian forest objective can be stated as $\\min _{\\mathbf {w}\\in \\mathbb {R}^C}\\sum _{n = 1}^N \\frac{1}{M} \\sum _{m = 1}^M \\text{loss}(y_n, \\hat{y}_n^{(m)})+ \\gamma ^2 \\Vert \\mathbf {w}\\Vert _2^2.$" ], [ "DISCUSSION", "Comparing (REF ) and (REF ), we see that subject to regularization parameters (priors) chosen compatibly, the two objectives only differ in the contribution of an individual data point $n$ to the total loss: $\\text{Mondrian kernel:} \\hspace{10.0pt}& \\text{loss}\\left(y_n, \\frac{1}{M} \\sum _{m = 1}^M \\hat{y}_n^{(m)} \\right)\\\\\\text{Mondrian forest:} \\hspace{10.0pt}& \\frac{1}{M} \\sum _{m = 1}^M \\text{loss}(y_n, \\hat{y}_n^{(m)})$ Specifically, the difference is in the order in which the averaging $\\frac{1}{M} \\sum _{m = 1}^M$ over Mondrian samples/trees and the non-linear $\\text{loss}$ function are applied.", "In both models predictions are given by $\\hat{y} = \\frac{1}{M} \\sum _{m = 1}^M \\hat{y}^{(m)}$ , so the Mondrian kernel objective is consistent with the aim of minimizing empirical loss on the training data, while the forest objective minimizes average loss across trees, not the loss of the actual prediction (when $M > 1$ ) [11].", "[11] address this inconsistency between learning and prediction by proposing to extend random forests with a global refinement step that optimizes all tree parameters jointly, minimizing the empirical training loss.", "Our approximation of the Laplace kernel via the Mondrian kernel can be interpreted as implementing this joint parameter fitting step on top of Mondrian forest, revealing a new connection between random forests and kernel methods.", "Figure: NO_CAPTION" ], [ "RELATED WORK", "The idea of [10] to approximate shift-invariant kernels by constructing random features has been further developed by [9] and [20], providing a faster method of constructing the random features when the input dimension $D$ is high.", "The fast method of [4] can adapt the number of random features, making it better-suited for streaming data.", "To the best of our knowledge, these methods require random features to be reconstructed from scratch for each new kernel width value; however, our solution allows us to efficiently learn this hyperparameter for the Laplace kernel.", "Decision forests are popular for black-box classification and regression thanks to their competitive accuracy and computational efficiency.", "The most popular variants are Breiman's Random Forest [2] and Extremely Randomized Trees [6].", "[1] established a link between the Laplace kernel and random forests with an infinite number of trees, but unlike our work, made two additional strong assumptions, namely infinite data and a uniform distribution of features.", "From a computational perspective, [19] approximated evaluation of an isotropic kernel using $kd$ -trees, reducing computational complexity as well as memory requirements.", "[5] constructed `supervised' kernels using random forests and demonstrated that this can lead to linear-time inference.", "We refer to [15] for a recent discussion on the connection between decision forests and kernel methods.", "A key difference between decision forests and kernel methods is whether parameters are fit independently or jointly.", "In decision forests, the leaf node parameters for each tree are fit independently, whereas the weights of random features are fit jointly.", "[15] shows that random forests can be interpreted as adaptive kernel estimates and discusses the theoretical properties of fitting parameters jointly.", "[11] propose to extend random forests with a global refinement step, optimizing all tree parameters jointly to minimize empirical training loss.", "The proposed Mondrian kernel establishes a link between Mondrian trees and Laplace kernel for finite data, without any assumptions on the distribution of the features.", "Unlike prior work, we exploit this connection to construct an adaptive random feature approximation and efficiently learn the kernel width." ], [ "EXPERIMENTS", "We conducted three sets of experiments, with these goals: verify that Mondrian kernel approximates the Laplace kernel, and compare to other random feature generation schemes (Section REF ); demonstrate usefulness of our efficient kernel width selection procedure, showing that it can quickly learn a suitable kernel width from data (Section REF ); and empirically compare the Mondrian kernel and Mondrian forests, supporting the insight into their relationship from Section  (Section REF ).", "With the exception of two experiments on synthetic data, we carried out our evaluation on the CPU dataset from [10], containing $N = 6554$ training and $N_{\\text{test}} = 819$ test points with $D = 21$ attributes.", "Note that the CPU dataset is an adversarial choice here, as [10] report that random Fourier features perform better than binning schemes on this task.", "In all experiments, the ridge regularization constant was set to $\\delta ^2 = 10^{-4}$ , the value used by [10], and the primal optimization problems were solved using stochastic gradient descent." ], [ "LAPLACE KERNEL APPROXIMATION", "First we examined the absolute kernel approximation error $|k_{\\infty }(\\cdot , \\cdot ) - k_M(\\cdot , \\cdot )|$ directly.", "To this end, we sampled $N = 100$ data points uniformly at random in the unit square $[0, 1]^2$ and computed the maximum absolute error over all $N^2$ pairs of points.", "The Laplace kernel $k_{\\infty }$ and Mondrian kernels $k_M$ had a common lifetime (inverse width) $\\lambda = 10$ , so that several widths fit into the input domain $[0, 1]^2$ .", "We repeated the experiment 5 times for each value of $M$ , showing the results in Figure REF .", "We plot the maximum error against the number $M$ of non-zero features per data point, which is relevant for solvers such as Pegasos SVM [18], whose running time scales with the number of non-zero features per data point.", "Under this metric, the Mondrian kernel and Random binning converged to the Laplace kernel faster than random Fourier features, showing that in some cases they can be a useful option.", "(The error of Random Fourier features would decrease faster when measured against the total number of features, as Mondrian kernel and Random binning generate sparse feature expansions.)", "Figure: Maximum absolute kernel approximation error on all pairs of N=100N = 100 data points in [0,1] 2 [0, 1]^2.Second, we examined the approximation error indirectly via test set error on the CPU dataset.", "We repeated the experiment 5 times for each value of $M$ and show the results in Figure REF .", "Even though Fourier features are better suited to this task, for a fast approximation with few ($M < 15$ ) non-zero features per data point, random binning and Mondrian kernel are still able to outperform the Fourier features.", "Figure: Test set error on the CPU dataset.", "The horizontal line at 3.1%3.1\\% indicates the error achieved with an exact, but expensive computation using the Laplace kernel." ], [ "FAST KERNEL WIDTH LEARNING", "First, using a synthetic regression dataset generated from a Laplace kernel with known ground truth lifetime $\\lambda _0 = 10$ , we verified that the lifetime could be recovered using our kernel width selection procedure from Section .", "To this end, we let the procedure run until a terminal lifetime $\\Lambda = 100$ and plotted the error on a held-out validation set as a function of the lifetime $\\lambda $ .", "The result in Figure REF shows that the ground truth kernel lifetime $\\lambda _0 = 10$ was recovered within an order of magnitude by selecting the lifetime $\\hat{\\lambda }$ minimizing validation set error.", "Moreover, this value of $\\hat{\\lambda }$ led to excellent performance on an independent test set.", "Figure: Recovering the ground truth lifetime λ 0 =10\\lambda _0 = 10 by selecting the value λ ^≈19\\hat{\\lambda } \\approx 19 minimizing validation set error.Second, we evaluated our kernel width selection procedure on the CPU dataset in order to demonstrate its practical usefulness.", "While the Mondrian kernel allows to efficiently sweep through lifetimes $\\lambda $ , Fourier features and random binning need to be reconstructed and retrained for each attempted lifetime value.", "We started the Fourier features and random binning at $\\lambda = 1$ , and in each step, we either doubled the maximum lifetime or halved the minimum lifetime considered so far, based on which direction seemed more promising.", "Once a good performing lifetime was found, we further optimized using a binary search procedure.", "All schemes were set to generate $M = 350$ non-zero features per datapoint.", "Figure REF shows the performance of each scheme on a held-out validation set as a function of computation time.", "The result suggests that our kernel width learning procedure can be used to discover suitable lifetimes (inverse kernel widths) at least an order of magnitude faster than random Fourier features or random binning." ], [ "MONDRIAN KERNEL VS FOREST", "We compared the performance of Mondrian kernel and “Mondrian forest\" (quotes due to omission of hierarchical smoothing) based on the same $M = 50$ Mondrian samples, using the CPU dataset and varying the lifetime $\\lambda $ .", "Recall that higher values of $\\lambda $ lead to more refined Mondrian partitions, allowing more structure in the data to be modeled, but also increasing the risk of overfitting.", "Figure REF shows that Mondrian kernel exploits the joint fitting of parameters corresponding to different trees and achieves a lower test error at lower lifetime values, thus producing a more compact solution based on simpler partitions.", "Figure REF shows the parameter values learned by Mondrian kernel and Mondrian forest at the lifetime $\\lambda = 2 \\times 10^{-6}$ .", "The distribution of weights learned by Mondrian kernel is more peaked around 0, as the joint fiting allows achieving more extreme predictions by adding together several smaller weights.", "Figure: Comparison of Mondrian kernel and Mondrian forest models based on the same set of Mondrian samples.Figure: Weights learned by Mondrian forest and Mondrian kernel at the lifetime λ=2×10 -6 \\lambda = 2 \\times 10^{-6} in Figure ." ], [ "CONCLUSION", "We presented the Mondrian kernel, a fast approximation to the Laplace kernel that admits efficient kernel width selection.", "When a different kernel or a different approximation is used, our procedure can provide a fast and simple way of initializing the kernel width for further optimization.", "While a Gaussian kernel is often considered a default choice, in many situations it imposes an inappropriately strong smoothness assumption on the modelled function and the Laplace kernel may in fact be a preferable option.", "Our approach revealed a novel link between the Mondrian process and the Laplace kernel.", "We leave the discovery of similar links involving other kernels for future work." ], [ "Acknowledgements", "We would like to thank Nilesh Tripuraneni for useful discussions.", "Part of this research was carried out while MB was at the University of Oxford.", "BL gratefully acknowledges generous funding from the Gatsby Charitable Foundation.", "ZG acknowledges funding from the Alan Turing Institute, Google, Microsoft Research and EPSRC Grant EP/N014162/1.", "DMR is supported by an NSERC Discovery Grant.", "YWT's research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) ERC grant agreement no.", "617071.", "The Mondrian Kernel Supplementary material" ], [ "Proofs", "Definition 1 The linear dimension of an axis-aligned box $\\mathcal {X}= \\mathcal {X}_1 \\times \\cdots \\times \\mathcal {X}_D \\subseteq \\mathbb {R}^D$ is $|\\mathcal {X}| := |\\mathcal {X}_1| + \\cdots + |\\mathcal {X}_D|$ .", "Our first result is a tail bound on the number of partition cells generated by a Mondrian process.", "We will use it as a Lemma in Proposition REF , but it also confirms that with probability 1, the Mondrian process does not explode (does not generate infinitely many partition cells in finite time).", "Proposition 3 Let $\\mathcal {M}$ be a Mondrian process on an axis-aligned box $\\mathcal {X}$ .", "For $t \\ge 0$ , let $N_t$ be the number of partition cells generated by $\\mathcal {M}$ until time $t$ .", "Then $\\forall {n \\in \\mathbb {R}_{+}}\\hspace{20.0pt}\\mathbb {P}[N_t > n]\\le \\frac{e^{|\\mathcal {X}| t}}{n}.$ In particular, the Mondrian process does not explode.", "At any time $s$ , by lack of memory of the exponential distribution, the residual time until a partition cell $c$ splits into two has $\\operatorname{Exp}(|c|)$ distribution and is independent of all other cells by construction of the Mondrian process.", "As $|c| \\le |\\mathcal {X}|$ , this cell splitting process is dominated by a Yule process with birth rate $|\\mathcal {X}|$ .", "The number $\\tilde{N}_t$ of individuals at time $t$ of a Yule process with birth rate $|\\mathcal {X}|$ has geometric distribution with mean $e^{|\\mathcal {X}| t}$ and Markov's inequality yields $\\mathbb {P}[N_t > n]\\le \\mathbb {P}[\\tilde{N}_t > n]\\le \\frac{e^{|\\mathcal {X}| t}}{n}.$ as claimed.", "Hence $\\mathbb {P}[ N_t = \\infty ] = \\lim _{n \\rightarrow \\infty } \\mathbb {P}[ N_t > n ] = 0$ for any $t$ .", "We define an $\\varepsilon $ -grid covering a (closed) interval as a set of points at most $\\varepsilon $ distance apart, including the boundary points, and with minimal possible cardinality: Definition 2 Let $\\mathcal {X}_1 = [a_1, b_1]$ be an interval of length $|\\mathcal {X}_1| = b_1 - a_1$ and let $0 < \\varepsilon < |\\mathcal {X}_1|$ .", "Define $K := \\lceil \\frac{|\\mathcal {X}_1|}{\\varepsilon } \\rceil $ .", "An $\\varepsilon $ -grid covering $\\mathcal {X}_1$ is a set $\\mathcal {U}_1$ of $K + 1$ points $u_0 < u_1 < \\cdots < u_K$ in $\\mathcal {X}_1$ such that $u_0 = a_1$ , $u_K = b_1$ and $|u_i - u_{i - 1}| \\le \\varepsilon $ for all $1 \\le i \\le K$ .", "Note that such an $\\varepsilon $ -grid exists by our choice of $K$ , as we can take, e.g., $u_i = i \\varepsilon $ for $1 \\le i < K$ .", "The next lemma bounds the probability that two arrivals of a Poisson process running on a bounded interval occur between two consecutive points of an $\\varepsilon $ -grid covering that interval.", "Lemma 1 Consider a Poisson process with rate $\\lambda $ running on a bounded interval $[0, L]$ .", "Let $\\mathcal {U}$ be an $\\varepsilon $ -grid covering of $[0, L]$ .", "Then the probability that two or more arrivals of the process occur between any two consecutive points of $\\mathcal {U}$ is at most $2 \\lambda ^2 L \\varepsilon $ .", "As the distance between any two consecutive points of the $\\varepsilon $ -grid is at most $\\varepsilon $ by definition, the number of arrivals in a line segment between such two points is dominated by a Poisson random variable with mean $\\lambda \\varepsilon $ .", "As there are $\\lceil \\frac{L}{\\varepsilon } \\rceil $ such segments, the sought probability $p$ can be upper bounded using a union bound as $p \\le \\left\\lceil \\frac{L}{\\varepsilon } \\right\\rceil \\left( 1 - e^{- \\lambda \\varepsilon } - e^{- \\lambda \\varepsilon } \\lambda \\varepsilon \\right)$ and using $1 - e^{-x} \\le x$ twice, we obtain as claimed $p \\le \\left\\lceil \\frac{L}{\\varepsilon } \\right\\rceil \\left( \\lambda \\varepsilon - e^{- \\lambda \\varepsilon } \\lambda \\varepsilon \\right)\\le \\left\\lceil \\frac{L}{\\varepsilon } \\right\\rceil \\left( \\lambda \\varepsilon \\right)^2\\le 2 L \\lambda ^2 \\varepsilon .$ Definition REF also set us up for defining the concept of an $\\varepsilon $ -grid on higher-dimensional axis-aligned boxes: Definition 3 Let $\\mathcal {X}= \\mathcal {X}_1 \\times \\cdots \\times \\mathcal {X}_D \\subseteq \\mathbb {R}^D$ be an axis-aligned box and let $\\varepsilon > 0$ .", "An $\\varepsilon $ -grid covering $\\mathcal {X}$ is a cartesian product $\\mathcal {U} = \\mathcal {U}_1 \\times \\cdots \\times \\mathcal {U}_D$ , where each $\\mathcal {U}_d$ is an $\\varepsilon $ -grid covering of $\\mathcal {X}_d$ in the sense of Definition REF .", "Proposition 4 For any bounded input domain $\\mathcal {X}\\subseteq \\mathbb {R}^D$ and $\\delta > 0$ , as $M \\rightarrow \\infty $ , $&\\;\\;\\;\\;\\;\\mathbb {P}\\left[ \\sup _{\\mathbf {x}, \\mathbf {x}^{\\prime } \\in \\mathcal {X}} \\left| k_M(\\mathbf {x}, \\mathbf {x}^{\\prime }) - k_{\\infty }(\\mathbf {x}, \\mathbf {x}^{\\prime }) \\right| > \\delta \\right]\\\\ &=\\mathcal {O}\\left( M^{2/3} e^{- M \\delta ^2 / (12D + 2)} \\right).$ By extending $\\mathcal {X}$ if necessary, we may assume without loss of generality that $\\mathcal {X}$ is an axis-aligned box with linear dimension $|\\mathcal {X}|$ .", "Recall that a Mondrian kernel of order $M$ corresponds to a random features obtained from $M$ independent Mondrians with lifetime $\\lambda $ .", "Let $\\mathcal {U}$ be an $\\varepsilon $ -grid covering $\\mathcal {X}$ , where $\\varepsilon > 0$ will be specified later.", "The proof will upper bound the probability of the following three “bad\" events: [leftmargin=*,label=] $A_1$ := { any of the $M$ Mondrian samples contains more than $n$ partition cells } $A_2$ := { the common refinement of the $M$ Mondrian partitions, disregarding any potential cuts after generating $n$ cells in one Mondrian, has a partition cell that does not contain an element of $\\mathcal {U}$ } $A_3$ := { $\\frac{\\delta }{2}$ -approximation fails on $\\mathcal {U}$ , i.e., for some $\\mathbf {u}_1$ , $\\mathbf {u}_2 \\in \\mathcal {U}$ , $|k_M(\\mathbf {u}_1, \\mathbf {u}_2) - k_{\\infty }(\\mathbf {u}_1, \\mathbf {u}_2)| > \\frac{\\delta }{2}$ } The constant $n \\in \\mathbb {R}{+}$ will be specified (optimized) later.", "Note that $A_1 \\cap A_2$ implies that all partition cells in the common refinement of all $M$ Mondrian partitions contain a grid point from $\\mathcal {U}$ .", "Since $k_M$ is constant in each such cell, making $\\varepsilon $ small enough, smoothness of the Laplace kernel $k_{\\infty }$ will ensure that if $A_3^c$ holds then $\\delta $ -approximation holds throughout $\\mathcal {X}$ .", "Proposition REF and a union bound over the $M$ Mondrian samples give immediately that $\\mathbb {P}(A_1)\\le M \\frac{e^{|\\mathcal {X}| \\lambda }}{n}.$ Note that the $\\varepsilon $ -grid $\\mathcal {U}$ contains at most $( 2 | \\mathcal {X}| / \\varepsilon )^D$ grid points.", "Hoeffding's inequality and a union bound over all pairs of grid points gives for any $\\varepsilon > 0$ : $\\mathbb {P}(A_3)\\le \\left[ \\left( 2 \\frac{| \\mathcal {X}|}{\\varepsilon } \\right)^D \\right]^2 \\left[ 2 \\exp \\left(- M \\delta ^2 / 2 \\right) \\right].$ To upper bound the probability of $A_2$ , note that at any time $t < \\lambda $ , in each partition cell generated so far by any of the $M$ Mondrian processes, an exponential clock is associated to each dimension $d$ of the cell, and if that clock rings, the cell is split at a random location $a$ by a hyperplane lying in dimension $d$ .", "Consider the point process obtained by projecting the cut points from all partition cells onto their respective coordinate axes.", "If each Mondrian process generates no more than than $n$ partition cells until its lifetime $\\lambda $ is exhausted, the cut points on the $d$ -th coordinate axis come from at most $M n$ partition cells, each having width at most $|\\mathcal {X}_d|$ in dimension $d$ .", "Therefore this point process on the $d$ -th coordinate axis can be thought of as taking a suitable subset of points generated by a Poisson point process with intensity $M n |\\mathcal {X}_d| \\lambda $ .", "Thus by Lemma REF , the probability that two cut points in dimension $d$ fall between two adjacent coordinates of the $\\varepsilon $ -grid $\\mathcal {U}$ is upper bounded by $2 (M n \\lambda )^2 |\\mathcal {X}_d| \\varepsilon $ .", "Observe that if this does not happen in any of the $D$ dimensions then all partition cells in the common refinement must contain a grid point from $\\mathcal {U}$ .", "Hence, taking the union bound over all $D$ dimensions, $\\mathbb {P}(A_2)\\le \\sum _{d = 1}^D 2 (M n \\lambda )^2 |\\mathcal {X}_d| \\varepsilon = 2 (M n \\lambda )^2 |\\mathcal {X}| \\varepsilon .$ Thus the probability of a “bad\" event occuring is at most $& \\mathbb {P}(A_1 \\cup A_2 \\cup A_3)\\\\ & \\le \\mathbb {P}(A_1) + \\mathbb {P}(A_2) + \\mathbb {P}(A_3)\\\\ & \\le M \\frac{e^{|\\mathcal {X}| \\lambda }}{n}+ 2 (M n \\lambda )^2 |\\mathcal {X}| \\varepsilon + 2 \\left(2 \\frac{| \\mathcal {X}|}{\\varepsilon } \\right)^{2D} e^{- M \\delta ^2 / 2}.$ and minimizing over $n \\in \\mathbb {R}_{+}$ gives $&\\mathbb {P}(A_1 \\cup A_2 \\cup A_3)\\\\ &\\le \\left( 4 \\lambda ^2 M^2 |\\mathcal {X}| \\varepsilon e^{2 \\lambda |\\mathcal {X}|} \\right)^{1/3}+ 2 \\left( \\frac{|\\mathcal {X}|}{\\varepsilon } \\right)^{2D} e^{- M \\delta ^2 / 2}.$ If $A_1 \\cap A_2$ holds then each cell in the common refinement of the $M$ Mondrian partitions contains an element of the $\\varepsilon $ -grid $\\mathcal {U}$ , and the Laplace kernel of lifetime $\\lambda $ changes by at most $1 - e^{-D \\lambda \\varepsilon }$ when moving from any point in $\\mathcal {X}$ to the nearest grid point in its partition cell (in the common refinement).", "Therefore, as long as $2(1 - e^{-D \\lambda \\varepsilon }) < \\frac{\\delta }{2}$ (i.e., $\\varepsilon \\le \\frac{1}{\\lambda D} \\ln (1 - \\frac{\\delta }{4})$ ), the event $(A_1 \\cup A_2 \\cup A_3)^c$ implies that $\\delta $ -approximation holds throughout $\\mathcal {X}$ .", "The upper bound on $\\mathbb {P}(A_1 \\cup A_2 \\cup A_3)$ above is minimized for $\\varepsilon _0= \\left( \\frac{12 D |\\mathcal {X}|^{2D} e^{-M\\delta ^2/2}}{(4 \\lambda |\\mathcal {X}|)^{1/3} e^{2 \\lambda |\\mathcal {X}| / 3}} \\right)$ which tends to 0 as $M \\rightarrow \\infty $ and so for large enough $M$ , we do have $\\varepsilon _0 \\le \\frac{1}{\\lambda D} \\ln (1 - \\frac{\\delta }{4})$ .", "For these large enough $M$ it then holds that $&\\mathbb {P}\\left[ \\sup _{\\mathbf {x}, \\mathbf {x}^{\\prime } \\in \\mathcal {X}} \\left|\\phi (\\mathbf {x})^T \\phi (\\mathbf {x}^{\\prime }) - k(\\mathbf {x}, \\mathbf {x}^{\\prime }) \\right| > \\delta \\right]\\\\ &\\le \\mathbb {P}(A_1 \\cup A_2 \\cup A_3)\\\\ &\\le \\left( 4 \\lambda ^2 M^2 |\\mathcal {X}| \\varepsilon _0 e^{2 \\lambda |\\mathcal {X}|} \\right)^{1/3}+ 2 \\left( \\frac{|\\mathcal {X}|}{\\varepsilon _0} \\right)^{2D} e^{- M \\delta ^2 / 2}\\\\ &=\\left( 2^{1/(2D)} 4 \\lambda ^2 M^2 |\\mathcal {X}|^2 e^{2 \\lambda L} / D \\right)^{1/(3+1/2D)}e^{- \\frac{M \\delta ^2}{12 D + 2}}\\\\ &\\in \\mathcal {O}\\left( M^{2/3} e^{- \\frac{M \\delta ^2}{12 D + 2}} \\right).$ Proposition 5 In a Mondrian regression forest with a factorizing Gaussian prior over leaf predictions, the learning objective function can be stated as $\\min _{\\mathbf {w}\\in \\mathbb {R}^C}\\sum _{n = 1}^N \\frac{1}{M} \\sum _{m = 1}^M \\text{loss}(y_n, \\hat{y}_n^{(m)})+ \\gamma ^2 \\Vert \\mathbf {w}\\Vert _2^2.$ The predictive mean parameters $\\mathbf {w}^{(m)}$ in the leaves of the $m$ -th tree are fitted by solving $\\min _{\\mathbf {w}^{(m)} \\in \\mathbb {R}^{C^{(m)}}}\\sum _{n = 1}^N (y_n - \\mathbf {w}^{(m)T} \\mathbf {\\phi }^{(m)}_n)^2+ \\gamma ^2 \\Vert \\mathbf {w}^{(m)} \\Vert _2^2$ where $\\gamma ^2$ is the ratio of noise and prior variance in the predictive model.", "The parameters $\\mathbf {w}^{(m)}$ are disjoint for different trees, so these $M$ independent optimization problems are equivalent to minimizing the average $\\min _{\\mathbf {w}^{(1)}, \\ldots , \\mathbf {w}^{(M)}}\\frac{1}{M} \\sum _{m = 1}^M \\left( \\sum _{n = 1}^N (y_n - \\hat{y}_n^{(m)})^2+ \\gamma ^2 \\Vert \\mathbf {w}^{(m)} \\Vert _2^2 \\right)$ where $\\hat{y}_n^{(m)} := \\mathbf {w}^{(m)T} \\mathbf {\\phi }_n^{(m)}$ is the $m$ -th tree's prediction at data point $n$ .", "Rewriting in terms of the squared loss $\\text{loss}(y, \\hat{y}) := (y - \\hat{y})^2$ and the normalized concatenated weights $\\mathbf {w}:= M^{-1/2} [\\mathbf {w}^{(1)T} \\cdots \\mathbf {w}^{(M)T}]^T$ , the learning objective function becomes $\\min _{\\mathbf {w}\\in \\mathbb {R}^C}\\sum _{n = 1}^N \\frac{1}{M} \\sum _{m = 1}^M \\text{loss}(y_n, \\hat{y}_n^{(m)})+ \\gamma ^2 \\Vert \\mathbf {w}\\Vert _2^2.$" ], [ "Bayesian kernel width learning", "Section REF described how in a ridge regression setting, the marginal likelihood $\\mathcal {L}(\\lambda ) = p( \\mathbf {y}| \\mathbf {X}, \\lambda )$ can be efficiently computed for all $\\lambda \\in [0, \\Lambda ]$ .", "With a prior $p(\\lambda )$ over the lifetime (inverse kernel width) $\\lambda $ whose support is included in $[0, \\Lambda ]$ , the posterior distribution over $\\lambda $ is $p(\\lambda | \\mathbf {y}, \\mathbf {X})\\propto p(\\lambda ) p( \\mathbf {y}| \\mathbf {X}, \\lambda )$ with normalizing constant $p(\\mathbf {y}| \\mathbf {X})= \\sum _{c = 0}^{C-M} p( \\mathbf {y}| \\mathbf {X}, \\lambda = \\tau _c) \\int _{\\tau _c}^{\\tau _{c + 1}} p(\\lambda ) \\,\\mathrm {d}\\lambda $ where $0 = \\tau _0 < \\tau _1 < \\cdots < \\tau _{C - M}$ is the sequence of times when new cuts appeared in any of the $M$ Mondrian samples.", "The predictive distribution at a new test point $\\mathbf {x}_{*}$ is obtained by marginalizing out $\\lambda $ : $& p(y_{*} | \\mathbf {x}_{*}, \\mathbf {X}, \\mathbf {y}) \\\\&=\\int p(y_{*} | \\mathbf {x}_{*}, \\mathbf {X}, \\mathbf {y}, \\lambda ) p(\\lambda | \\mathbf {y}, \\mathbf {x}) \\,\\mathrm {d}\\lambda \\\\&=\\sum _{c = 0}^{C - M} p(y_{*} | \\mathbf {x}_{*}, \\lambda = \\tau _c) p(\\tau _c \\le \\lambda < \\tau _{c + 1} | \\mathbf {y}, \\mathbf {X})\\\\&=\\sum _{c = 0}^{C - M} p(y_{*} | \\mathbf {x}_{*}, \\lambda = \\tau _c) \\int _{\\tau _c}^{\\tau _{c + 1}} p(\\lambda | \\mathbf {y}, \\mathbf {X}) \\,\\mathrm {d}\\lambda \\\\&=\\sum _{c = 0}^{C - M} p(y_{*} | \\mathbf {x}_{*}, \\lambda = \\tau _c) \\int _{\\tau _c}^{\\tau _{c + 1}} \\frac{p(\\lambda ) p( \\mathbf {y}| \\mathbf {X}, \\lambda )}{p(\\mathbf {y}| \\mathbf {X})} \\,\\mathrm {d}\\lambda \\\\&=\\sum _{c = 0}^{C - M} p(y_{*} | \\mathbf {x}_{*}, \\lambda = \\tau _c) p( \\mathbf {y}| \\mathbf {X}, \\lambda = \\tau _c) \\frac{\\int _{\\tau _c}^{\\tau _{c + 1}} p(\\lambda ) \\,\\mathrm {d}\\lambda }{p(\\mathbf {y}| \\mathbf {X})}\\\\&=\\frac{\\sum _{c = 0}^{C - M} p(y_{*} | \\mathbf {x}_{*}, \\lambda = \\tau _c) p( \\mathbf {y}| \\mathbf {X}, \\lambda = \\tau _c) \\int _{\\tau _c}^{\\tau _{c + 1}} p(\\lambda ) \\,\\mathrm {d}\\lambda }{\\sum _{c = 0}^{C-M} p( \\mathbf {y}| \\mathbf {X}, \\lambda = \\tau _c) \\int _{\\tau _c}^{\\tau _{c + 1}} p(\\lambda ) \\,\\mathrm {d}\\lambda }\\\\ &=\\sum _{c = 0}^{C - M} k_c p(y_{*} | \\mathbf {x}_{*}, \\lambda = \\tau _c)$ where the mixing coefficients $k_c:= \\frac{p( \\mathbf {y}| \\mathbf {X}, \\lambda = \\tau _c) \\int _{\\tau _c}^{\\tau _{c + 1}} p(\\lambda ) \\,\\mathrm {d}\\lambda }{\\sum _{c = 0}^{C-M} p( \\mathbf {y}| \\mathbf {X}, \\lambda = \\tau _c) \\int _{\\tau _c}^{\\tau _{c + 1}} p(\\lambda ) \\,\\mathrm {d}\\lambda }$ can be precomputed and cached for faster predictions.", "The integrals $\\int _{\\tau _c}^{\\tau _{c + 1}} p(\\lambda ) \\,\\mathrm {d}\\lambda $ can be readily evaluated if we have access to the cumulative distribution function of our prior $p(\\lambda )$ , which we assume." ], [ "Online learning", "Mirroring Section REF , we discuss the example of ridge regression where exact online updates can be carried out.", "Assume we have access to the regularized feature covariance matrix $\\mathbf {A} = \\mathbf {\\Phi }^T \\mathbf {\\Phi }+ \\delta ^2 \\mathbf {I}_C$ and its inverse $\\mathbf {A}^{-1}$ or Cholesky decomposition $\\operatorname{chol}(\\mathbf {A})$ before a new data point $\\mathbf {x}\\in \\mathbb {R}^D$ arrives, and we wish to update these efficiently.", "If the dimensionality of $\\phi $ increases by $k$ due to $\\mathbf {x}$ creating $k$ new non-empty partition cells, we first append $k$ rows and columns to $\\mathbf {A}$ , containing $0s$ only, except on the main diagonal we put $\\delta ^2$ .", "Correspondingly, $\\mathbf {A}^{-1}$ or $\\operatorname{chol}(A)$ are updated by appending $k$ rows and columns, with non-zero entries only on the main diagonal.", "(These entries would equal $\\delta ^{-2}$ in $\\mathbf {A}^{-1}$ and $\\delta $ in $\\operatorname{chol}(\\mathbf {A})$ ).", "This ensures the feature map $\\phi $ now incorporates all necessary features.", "Noting that the $(i, j)$ -entry of $\\mathbf {A} - \\delta ^2 \\mathbf {I}_C$ counts data points belonging to partition cells $i$ and $j$ at the same time (this can be non-zero only if $i$ , $j$ correspond to different Mondrian samples), normalized by $1/M$ , and that the $(i, j)$ -entry of the outer product $\\phi (\\mathbf {x}) \\phi (\\mathbf {x})^T$ is $1/M$ if the new data point $\\mathbf {x}$ falls into both cells $i$ and $j$ , and 0 otherwise, we see that $\\mathbf {A}_{\\text{new}}\\leftarrow \\mathbf {A}_{\\text{old}} + \\phi (\\mathbf {x}) \\phi (\\mathbf {x})^T$ is a rank-1 update.", "Therefore both $\\mathbf {A}^{-1}$ and $\\operatorname{chol}(\\mathbf {A})$ can be updated efficiently in $\\mathcal {O}(C^2)$ time and the new MAP weights $\\hat{\\mathbf {w}}_{\\text{new}} = \\mathbf {A}_{\\text{new}}^{-1} (\\mathbf {\\Phi }^T \\mathbf {y})$ in $\\mathcal {O}(M C)$ by exploiting sparsity of $\\phi (\\mathbf {x})$ .", "The determinant of the rank-1 updated matrix $\\mathbf {A}_{\\text{new}}$ can also be updated in $\\mathcal {O}(C^2)$ time using the Matrix determinant lemma, or obtained directly from the Cholesky decomposition (as the squared product of its diagonal entries) in $\\mathcal {O}(C)$ time, allowing the training marginal likelihood to be updated in $\\mathcal {O}(N M + C^2)$ ." ] ]
1606.05241
[ [ "Swift follow-up of gravitational wave triggers: results from the first\n aLIGO run and optimisation for the future" ], [ "Abstract During its first observing run, in late 2015, the advanced LIGO facility announced 3 gravitational wave (GW) triggers to electromagnetic follow-up partners.", "Two of these have since been confirmed as being of astrophysical origin: both are binary black hole mergers at ~500 Mpc; the other trigger was later found not to be astrophysical.", "In this paper we report on the Swift follow up observations of the second and third triggers, including details of 21 X-ray sources detected; none of which can be associated with the GW event.", "We also consider the challenges that the next GW observing run will bring as the sensitivity and hence typical distance of GW events will increase.", "We discuss how to effectively use galaxy catalogues to prioritise areas for follow up, especially in the presence of distance estimates from the GW data.", "We also consider two galaxy catalogues and suggest that the high completeness at larger distances of the 2MASS Photometric Redshift Catalogue (2MPZ) makes it very well suited to optimise Swift follow-up observations." ], [ "Introduction", "In the last quarter of 2015, the Advanced Laser Interferometer Gravitational-wave Observatory observatory (aLIGO; , [6]) performed its first observing run (`O1') searching for gravitational waves (GW).", "Each potential GW event was assigned a false alarm rate (FAR) indicating the frequency with which a noise event with a signal of the observed strength is expected to arise.", "Partner electromagnetic (EM) facilities, including Swift were notified of GW signals with an FAR of less than one per month [1].", "O1 yielded the detection of two GW events, which have been confidently identified as binary black hole (BBH) mergers: GW150914 [4] and GW151226 [5], and there was a further trigger (G194575) from the online analysis which was later determined to be a noise event.", "Another possible merger event was detected in offline analysis of the O1 data [LVT151012; ].", "Details of that event were not provided to EM partners until 2016 April, so no Swift follow up was performed.", "The full results of O1 were reported by [2].", "Whilst the direct detection of GW was a significant achievement which marked the beginning of a new era of astronomy, in order to maximise the scientific potential of such discoveries, complementary EM data are needed.", "The three events reported so far are all believed to be stellar-mass BBH mergers, which were not expected to produce significant EM emission .", "However, Fermi-GBM reported a possible low-significance event 0.4 s after the GW trigger for GW150914, which may be associated with the GW event .", "In the days following the announcement of this, many authors suggested that EM emission from stellar-mass BBH is possible given the correct binary parameters, or a charged black hole , , , , although others suggested that a physical association between the GW and GBM events was unlikely .", "Further, INTEGRAL reported no detection and suggest that this casts doubt over whether the object detected by GBM was astrophysical in origin.", "This issue will likely only be resolved by future GW detections of BBH with both contemporaneous and follow-up EM observations.", "Regardless of whether BBH mergers give rise to EM emission, aLIGO is also expected to detect GW from the coalescence of binary neutron star systems or neutron star-black hole systems.", "These are both expected to produce multi-wavelength EM radiation, for example in the form of a short gamma-ray burst (sGRB, e.g.", "[10]) if the binary is viewed close to face-on, or a kilonova regardless of the jet orientation; see e.g.", ", , for a discussion of possible EM counterparts to such events.", "In an earlier paper (; hereafter `paper I') we presented the Swift observations of GW150914.", "In this work we present the results of the Swift observations of the other two triggers reported to the EM teams during O1, and consider how the Swift follow-up strategy may best evolve for the second run (O2) expected in the second half of 2016.", "Throughout this paper, all errors are given at the 1-$\\sigma $ level, unless stated otherwise." ], [ "The Swift satellite contains three complementary instruments.", "The Burst Alert Telescope (BAT; [9]) is a 15–350 keV coded-mask instrument with a field of view $\\sim \\,$ 2 sr. Its primary role is to trigger on new transient events such as GRBs.", "The other two instruments are narrow-field instruments, used for example to follow up GRBs detected by BAT.", "The X-ray telescope (XRT; ) is a 0.3–10 keV focusing instrument with a peak effective area of 110 cm$^2$ at 1.5 keV and a roughly circular field of view with radius 12.3$^\\prime $ .", "The ultra-violet/optical telescope (UVOT; ) has 6 optical filters covering 1600–6240 Å and a white filter covering 1600–8000 Å), with a peak effective area of 50 cm$^2$ in the $u$ band.", "The field of view is square, $\\sim \\,$ 17$^\\prime $ to a side.", "The ideal scenario for Swift to observe a GW event would be for BAT to detect EM emission (e.g.", "an sGRB) independently of the GW trigger on the same event.", "Swift would then automatically slew and gather prompt XRT and UVOT data.", "An sGRB is only seen if the coalescing binary is inclined such that the jet is oriented towards Earth; the opening angles of sGRB jets are not well known, however the observational limits are in the range $\\sim \\,$ 5–25$^{\\circ }$, , , , , therefore from purely geometrical constraints, we expect only a minority of binary neutron star/neutron star-black hole mergers detected in GW to be accompanied by an sGRB (e.g.", "for a jet angle of 10$^{\\circ }$ , only 1.5% will be viewed on-axis), whereas the GW signal is only modestly affeted by binary inclination.", "Some authors (e.g.", ", ) have suggested that the neutron star crust can be disrupted prior to the merger and that this could give rise to an isotropic precursor, i.e.", "BAT could in principle detect such emission from an off-axis GRB.", "However, these could well be too faint to trigger Swift-BAT.", "Also, while an excellent GRB-detection machine, Swift-BAT can only observe $\\sim \\,$ 1/6 of the sky at any given time.", "The combination of these factors means that, while a simultaneous ALIGO-BAT detection would be scientifically optimal, it is not a particularly likely occurrance.", "In addition, Swift can respond to the GW trigger, and observe a portion of the GW error region (which typically hundreds of square degrees in size) rapidly with its narrow-field telescopes.", "discussed optimal ways to do this, focusing primarily on the XRT, since it has a larger field of view than the UVOT, and the expected rate of unrelated transient events in the X-ray range, while not well constrained, is expected to be lower than in the optical bands (see , for a discussion of X-ray transient rates).", "Their suggested approach was to modify the GW error region by means of a galaxy catalogue (they used the Gravitational Wave Galaxy Catalogue (GWGC): ), weighting each pixel in the GW skymap by the luminosity of the catalogued galaxies in that pixel, and then to observe in a succession of short observations, in decreasing order of probability in this combined map.", "A more detailed Bayesian approach to this was discussed by .", "As we reported in paper I, the ability to observe a large number of fields with short exposures required operational changes for Swift which were not completed in time for O1, therefore we were only able to observe a relatively small part of the GW error regions for the triggers during that run.", "As described in paper I for GW150914, we combined the GW error region for each trigger in O1 with GWGC, weighting the galaxies according to their $B$ -band luminosities, and selecting XRT fields based on the resultant probability map.", "The data analysis approach was described in paper I so here we offer only a précis.", "For XRT, the source detection system was based on that of , slightly modified to support shorter exposures.", "Every source detected was automatically assigned a Rank of 1–4 describing how likely it was to be the counterpart to the GW event, with 1 being the most likely.", "This was based on whether the source was previously catalogued, its flux compared to previous detection or upper limits and its proximity to a known galaxy (full definitions of the ranks are in paper I).", "The detection system also produces warning flags for sources which it believes may be spurious due to effects, such as diffuse X-ray emission (the detection system is designed for point sources), or instrumental artefacts, such as stray light or optical loading (see section 3.4 and fig. 5).", "For each source detected, a GCN `Counterpart' notice was automatically produced as soon as the source was detected; this contained standard details (position, time of detection, flux) and also the rank and any warning flagsFor most of O1 these extra fields were only included in the email form of the GCN notice.", "Towards the end of O1 these were also added to the binary-format notice.. All sources were checked by humans, and any which were spurious were removed, and the verified sources reported in LVC/GCN circulars (, , , , , , , ).", "UVOT data were analysed using standard heasoft tools, and an automated pipeline was used to search for transients.", "Visual screening was applied to UVOT images, using the Digitized Sky Survey as a comparison.", "Although no rank 1 or 2 X-ray sources were found during O1, the UVOT data around any such sources would also have been closely inspected by eye." ], [ "GW150914", "GW150914 was detected at the end of the aLIGO engineering run immediately prior to O1, and was the first ever direct detection of GW.", "The Swift results for this event were reported in full in paper I.", "Recently (2016 April) we reobserved the GW error region of GW150914, as the final step to commission the ability to perform large-scale rapid tiling with Swift.", "Swift observed 426 fields during the UT day 2016 April 21 with 60 s of exposure per fieldThe GW error region is not observable for the entire Swift orbit, which is why the total exposure was $426\\times 60$ s $\\ll 1$ day., covering a total of 53 sq deg.", "Only one X-ray source was detected in these observations, the known X-ray emitter 1RXS J082709.9$-$ 650447, which was detected with a flux consistent with that from the Rosat observations .", "The scientific `result' from these observations is that, as expected, the only background X-ray source found was a known (rank 4) object; therefore we are optimistic that a transient should be easy to distinguish.", "However, these observations also demonstrate that Swift is now capable of performing large-scale tiling in response to a GW trigger." ], [ "Trigger G194575", "The aLIGO `compact binary coalescence' (CBC) pipeline, which uses a template library of expected GW waveforms from merging compact binaries, triggered on 2015 October 22 at 13:33:19.942 UT.", "The detected signal had an FAR of 9.65$\\times 10^{-8}$ Hz, equivalent to one per four months .", "Unfortunately, most of the higher probability areas of the error region were too close to the Sun for observations with Swift (Fig.", "REF ), therefore only the low probability regions were observable.", "Additionally, offline analysis of the GW signal reduced the FAR to 8.19$\\times 10^{-6}$ Hz (one per 1.41 days), and it was therefore determined not to be a real GW event .", "Although this trigger is therefore not of astrophysical significance, one point of procedural interest for future triggers is worth noting.", "Before the significance of the trigger had been downgraded, two sources identified by ground-based observatories were reported as being of potential interest: LSQ15bjb and iPTF15dld , which were detected by the La Silla QUEST and iPTF ground-based facilities respectively.", "Swift observed both of these sources and, finding no X-ray counterpart, we reported upper limits , .", "LSQ15bjb was originally reported as an uncatalogued and rapidly brightening optical source , which was subsequently classified as a type Ia supernova .", "iPTF15dld was one of several optical transients reported by that were consistent in position with a known galaxy at $z<0.1$ ; it later transpired that this source had been detected by by La Silla QUEST 19 days before the GW event .", "Details of the Swift observations and results for these two sources are given in Table REF .", "This demonstrates the ability to be flexible when performing Swift GW follow up, and perform targetted observation of point sources detected by other facilities, as well the blind searches." ], [ "GW 151226", "The CBC pipeline triggered on 2015 December 26 at 03:38:53.648 UT, with a signal with FAR lower than one per month ; this was later refined to an FAR lower than one per hundred years .", "The GW waveform indicated that this was a high-mass event, most likely a BH-BH merger .", "As with both previous triggers, a portion of the error region was too close to the Sun to observe with Swift (Fig.", "REF ).", "The trigger was announced to the follow-up community on 2015 December 27 at 16:28 UT, and Swift observations began at 18:35 UT on the same day.", "We followed the same procedure as for the earlier triggers, selecting the most probable XRT fields after combining with the GWGC galaxy catalogue.", "However, after the first field had been observed, we modified this approach and instead of selecting single XRT fields, we selected regions covered by a set of 19 tiled pointings (Fig.", "REF ).", "We uploaded four such observation sets, as detailed in Table REF .", "We also performed additional sets of observations, observing the locations of PS15dqa and PS15dpn : optical sources highlighted as potentially interesting.", "PS15dpn was observed repeatedly for several days in order to track the UV evolution of its light curve.", "This source is not believed to be related to the GW trigger, and the PS15dpn data are not presented in this work.", "The GW localisation of GW151226 is shown in Fig.", "REF .", "At the time of the Swift observations, only the `BAYESTAR' map (produced by the low-latency pipeline; ) was available (top two panels).", "A revised skymap produced by the offline `LALInference' pipeline [; bottom panel of Fig.", "REF ] was made available on 2016 January 16 , well after our observations had been completed.", "The BAT field of view overlaps the GW localisations, covering 14% of the probability in the `BAYESTAR' map and 15% from the revised map [these probabilities are higher, $\\sim \\,$ 29%  and 33% after weighting by the GWGC, however since the distance to this merger is large ($440^{+180}_{-190}$ Mpc; [5]) and GWGC only contains galaxies to 100 Mpc, the galaxy-weighted map is not appropriateThe distance estimate from the GW data was not available at the time of the observations.].", "We created a 15–350 keV BAT light curve from $T_0-100$ to $T_0+100$ s ($T_0$ is the GW trigger time) with bins of 1.024 s. No signal is seen, at the 4-$\\sigma $ level with an upper limit (also 4 $\\sigma $ ) of 303.6 counts in a single bin.", "We used 4 $\\sigma $ as the limit rather than 3 $\\sigma $ because, for a 1.024-s binned light curve we expect a 3-$\\sigma $ noise fluctuation every $\\sim \\,$ 6 minutes, therefore the chance of a spurious signal in our data is high.", "At 4 $\\sigma $ noise fluctuations are expected only every 4.4 hours.", "To convert this to a flux limit we assumed a typical short GRB BAT spectrum: a power-law with a photon index of 1.32.", "The counts-to-flux conversion depends on the angle of the source to the BAT boresight which, since the source position is poorly contsrained, is not known.", "If the GW source was close to the BAT boresight (`fully coded' by the BAT mask) the upper limit is 4.3$\\times 10^{-8}$ erg $$ cm$^{-2}$ s$^{-1}$ .", "For a source which is half coded by the mask (45$^{\\circ }$ off axis) the limit is 1.7$\\times 10^{-7}$ erg $$ cm$^{-2}$ s$^{-1}$ , and if the source was only 10% coded by the BAT mask (56$^{\\circ }$ off axis) the limit is 9.0$\\times 10^{-7}$ erg $$ cm$^{-2}$ s$^{-1}$ .", "The initial XRT observations (i.e.", "not including the observations of PS15dqa or PS15dpn) covered 8.5 square degrees and enclosed 0.9% of the probability in both the original and revised sky mapsThis rises to 12% after galaxy convolution – which was performed as we did not know at that time that the source was a BBH.. 55 sources were detected in these observations, however 39 of these were artefacts of an area of extended emission (all but 2 of which were correctly flagged as such by the automated system and in the counterpart notices, the final two were removed by visual inspection).", "Details of the 16 genuine sources, none of which is believed to be the counterpart, are given in Table REF ; eight of these were rank 3 sources (uncatalogued, but below previous catalogue deteciton limits), and eight rank 4 (catalogued sources at fluxes consistent with their catalogued values).", "No uncatalogued sources were found in the UVOT data." ], [ "Late observations", "On 2016 January 13 we performed a new set of observations of the error region of GW151226.", "This consisted of 201 short ($\\sim \\,60$ s) exposures, and was primarily performed as part of commissioning the ability to rapidly tile GW error regions with Swift.", "These observations precede those reported in Section REF (i.e.", "the 2016 April observations of GW150914), and the test was only allowed to run for a few hours.", "This test revealed a bug in our software affecting low resolution GW localisations (healpixhttp://healpix.sourceforge.net-format maps with nside$<512$ ), as a result of which the 201 fields selected did not lie within the GW error region (this bug is now fixed).", "As with the 2016 April 24-hour test (Section REF ), the only X-ray sources found were two rank 4 sources.", "Fortuitously, these both lay in an area of the sky previously observed by Swift-XRT, and the two sources were in the 1SXPS catalogue , which allows us to compare their fluxes with no spectral assumptions.", "The sources were 1SXPS J090436.8+553600 (catalogued XRT count-rate: $0.168\\pm 0.004$ s$^{-1}$ , rate in the GW observations: $0.15\\pm 0.06$ s$^{-1}$ ) and 1SXPS J101504.1+492559 (catalogued at $1.300\\pm 0.008$ s$^{-1}$ , observed at $2.0\\pm 0.4$ s$^{-1}$ , i.e.", "both were consistent with the observed rate at the 1.5 $\\sigma $ level)." ], [ "Optimisations for future GW runs", "The second aLIGO observing run (`O2') is expected to take place in the second half of 2016, with Advanced VIRGO (AdVIRGO; [7]) also anticipated to be collecting data during the latter part of this run (the anticipated timeline for the aLIGO/AdVIRGO commissioning is given by [3]).", "As noted earlier, a new observing mode for Swift has now been commissioned, so it will be able to cover $\\sim \\,$ 50 sq deg per day, representing a significant improvement over the O1 response.", "The core approach to Swift observations during O2 is expected to be as recommended by : combining the GW error region with an appropriate galaxy catalogue, and performing 60 s suggested 50-s exposures, but for technical reasons we cannot have observations shorter than 60 s. observations of as many of the most probable fields as possible, as soon as possible, for the first 48 hours (when afterglow emission from an on-axis sGRB will be brightest).", "Thereafter we will re-observe these fields for longer (500 s) exposures, as argued that more than 48 hours after the GW trigger the population of detectable sGRBs will be dominated by off-axis objects (which require longer observations to detect).", "However, this broad plan hides a key detail: what galaxy catalogue should we use, and indeed, how should we use it?" ], [ "Selecting a galaxy catalogue", "For O1 we used the GWGC, since this extends to 100 Mpc, which the predictions of and the simulations of suggested was an appropriate horizon for binary neutron star mergers detectable by aLIGO during O1.", "These same authors predict the horizon distance will be higher (up to $\\sim \\,$ 250 Mpc) during O2.", "The two GW sources detected so far were both at much larger distances, $\\sim \\,$ 500 Mpc.", "As discussed in Section , while these sources are BBH mergers which were not believed to be strong EM emitters, the possible detection of a sGRB by Fermi coincident with GW150914 renders this uncertain, and it would be preferable to be able to observe the error regions from such triggers.", "If we still wish to reduce the sky area searched by using galaxy catalogues, we therefore need a catalogue with a reasonable degree of completeness out to at least 500 Mpc.", "However, when extending to such a distance the number of galaxies becomes so large that the benefits of targetting galaxies are diminished, therefore some means of selecting which galaxies to target preferentially is needed.", "One method was proposed by , who noted that by selecting only the brightest galaxies (those that produce 50% of galactic light) the probability of selecting the GW host galaxy is immediately reduced by 50%, but the number of galaxies one has to search is reduced by more than 50% (around 68% according to our analysis).", "Our approach is similar to this.", "We do not reject the fainter galaxies, but our galaxy map is weighted by the luminosity of the galaxy, and each possible XRT field of view (over the whole sky) is assigned a probability: $\\mathcal {P}\\propto \\sum _i{\\frac{L_i}{L_{\\rm tot}} \\mathcal {P}_{\\rm GW}}$ where $\\mathcal {P}_{\\rm GW}$ is the probability that the GW event lies within the XRT field of view according to the GW skymaps, $L_{\\rm tot}$ is the total luminosity contained in the galaxy catalogue, and the summation is over all galaxies within the XRT field of view.", "Swift pointings are performed approximately in decreasing order of probability (modulo observing constraints and some optimisations to keep the time observing to time slewing ratio high), and the number of fields observed is not based on the probability enclosed, but is limited by the amount of observing time committed to the follow up.", "This process could be further optimised if the distance to the GW event is available promptly, so that only galaxies at an appropriate distance are selected.", "showed that 3D skymaps could be rapidly produced by aLIGO during O2.", "For these, each pixel in the skymap would contain not just a probability, but also the parameters for how that probability is distributed in distance.", "This would allow equation (1) to be modified to include the distance to the galaxy (which may itself be a probability distribution) and the distance dependence of $P_{\\rm GW}$ , as we demonstrate shortly.", "Figure: The effect of extinction on the colour and redshift accuracy of the 2MPZ V1.1 catalogue.Top: The extinction-corrected K-BK-B colour as a function of extinction.", "For valuesof E(B-V)≥0.5E(B-V)\\ge 0.5 a clear trend starts to emerge, indicating a selection effect due to reddening.The contours indicate the density of the points, and the red lines mark E(B-V)=0.5E(B-V)=0.5 and 1.Bottom: Histograms of z photo -z spec z_{\\rm photo}-z{\\rm spec} for objects with spectroscopic redshift measurements,for E(B-V)<0.5E(B-V)<0.5 (black), 0.5≤E(B-V)<10.5\\le E(B-V)<1 (red) and E(B-V)≥1E(B-V)\\ge 1 (blue).", "The red histogram showsa small systematic shift, such that the mean photometric redshift is 0.004 too low compared to spectroscopicvalues; the widths are comparible.", "At high extinction, the photometric redshift calibration becomes very poor.Figure: The completeness of the CLU (black) and 2MPZ V1.1 (red).", "In each plot the upper pane shows thetheoretical total L/L * L/L^* predicted by a Schechter function (dotted line) as a function of distance, and the observed valuefrom the catalogues (solid line); the lower pane shows the ratio of the theoretical to observed,which we interpret as the completeness of the catalogue.The CLU data and theoretical valuesrelate to BB-band magnitudes.", "The 2MPZ data use KK-band magnitudes.", "For 2MPZ, only sourcesin regions with E(B-V)<1E(B-V)<1 are included; the magnitudeshave been corrected from isophotal to total values; see text for details.Left: Luminosity is integrated out to the distance on the x-axis, hence completeness refers to howcomplete the catalogue is out to distance D.Right: Luminosity is calculated in distance bins, hence completeness refers to how complete the catalogueis at distance D.First, we must select an appropriate galaxy catalogue.", "Ideally this will be highly complete out to at least 500 Mpc, have uniform sky coverage, and reliable luminosity (in a single band) and distance measurements for every galaxy.", "introduced the `Census of the Local Universe' (CLU) catalogue, and suggested the 2MASS Photometric Redshift catalogue [2MPZ; , [8] also suggested using 2MPZ].", "The CLU is a meta catalogue created from several existing catalogues (Kasliwal, in preparation), whereas 2MPZ is based on a cross-correlation of the 2MASS extended source catalogue with the WISE and SuperCOSMOS all-sky catalogues.", "Following and earlier works (e.g. )", "we estimate the completeness of the catalogues by comparing the integrated luminosity observed out to a given distance, with that predicted by a Schechter function .", "Using the terminology of we can define this function in terms of $x=L/L^*$ and the integrated luminosity per unit volume is therefore given by: $L dV = \\int _0^\\infty \\phi ^* L^* x^{\\alpha +1} e^{-x} dx$ where $L^*$ , $\\phi ^*$ and $\\alpha $ are measured from observations.", "For the $B$ -band data used in the CLUThe CLU is a meta catalogue, so for component catalogues where $B$ is not available, a pseudo $B$ magnitude is inferred and supplied., give $M_B^*=-19.7 + 5\\log h$ , $\\alpha _B=-1.07$ and $\\phi _B^*=0.016 h^3$ ($M^*$ is the absolute magnitude of a galaxy with luminosity $L^*$ ); for 2MPZ we use the $K$ -band magnitudes (as the catalogue is IR-selected) and the parameters from : $M_K^*=-23.39 + 5\\log h$ , $\\alpha _B=-1.09$ and $\\phi _B^*=0.0116 h^3$ .", "We assumed $h=0.7(=H_0/100)$ .", "To avoid questions of photometric zeropoints, rather than comparing the observed luminosity with that predicted by the Schechter function, we compare $x=L/L^*$ , thus the zeropoints cancel out.", "From equation REF , the theoretical value for $x$ within volume $V$ is $x = \\int _0^\\infty \\phi ^* x^{\\alpha +1} e^{-x} dx V = \\phi ^*\\Gamma (\\alpha +2, 0) V$ where $\\Gamma $ is the incomplete gamma function.", "The 2MPZ catalogue contains not the total infra-red magnitude of each galaxy, but instead those measured out to the 20mag/sq arcsec isophote (for the $K$ band, labelled as `k_m_k20fe' in the 2MASS database).", "Such magnitudes will systematically miss some flux and need to be corrected for the total light.", "We follow and use the mean correction advocated by Kochanek et al.", "(2001), subtracting 0.2 mag from the $K$ -band magnitudes of every entry in 2MPZ.", "We used a pre-release version 1.1 of 2MPZNow publicly available for download from the Wide Field Astronomy Unit at the Institute for Astronomy, Edinburgh: http://surveys.roe.ac.uk/ssa/TWOMPZ.", "which contains $\\sim \\,$ 6,000 extra galaxies compared to 2MPZ (added after the correction of WISE instrumental artefacts), and the version we used contains more data that the public one as it had no cuts made for Galactic extinction or stellar density.", "noted that such cuts are important to preserve uniformity.", "At high extinction the dust maps of may saturate (i.e.", "become inaccurate); more significantly, at high extinction the intrinsically fainter galaxies become undetectable, and this is a function of wavelength, so intrinsically redder galaxies tend to be retained while bluer ones are lost, biasing the sample.", "In areas of high stellar density, where galaxies and stars may be blended, the colours can also become unreliable.", "For these reasons the publicly released catalogues (both v1 and v1.1) do not include sources for which $E(B-V)>1.5$ mag or log(stellar density, sources per square degree)$>$ 4.0; no such filters were applied to the dataset we began with (providing an extra 6,700 sources compared to the public dataset): we wish for a sample that is as homogeneous as possible, yet also accurate and complete, therefore we explored what cuts were necessary to achieve this.", "Fig.", "REF shows the $K-B$ colour (both values are corrected for Galactic extinction)In 2MPZ the magnitudes are corrected for Galactic extinction, but not for cosmological effects (i.e.", "the $k$ -correction).", "The 2MASS and WISE magnitudes are in the Vega system, whereas the WISE and SuperCOSMOS magnitudes are AB-like magnitudes (Peacock et al.", "in prep).", "as a function of extinction, $E(B-V)$ .", "A clear bias begins to emerge when the extinction exceeds 0.5 mag, demonstrating that the extinction correction, and therefore colours and hence photometric redshifts ($z_{\\rm photo}$ ), are unreliable in this regime.", "The bottom panel of Fig.", "REF shows the difference between the photometric redshift and spectroscopic redshift ($z_{\\rm spec}$ ) for those objects with both, for three samples: $E(B-V)<0.5$ , $0.5\\le E(B-V)<1$ and $E(B-V)\\ge 1$ .", "The $0.5\\le E(B-V)<1$ sample is slightly off-centre, suggesting that at these extinctions, $z_{\\rm photo}$ is systematically underestimated by 0.004, however this shift is small compared to the uncertainty (discussed below) and can be ignored.", "We therefore applied a cut of $E(B-V)<1$ to 2MPZ resulting in 9,638 (1.0%) of the sources being discarded; all future discussion of 2MPZ in this work refers to this sample.", "We also investigated the effect that stellar density has on the accuracy of the catalogue.", "As with extinction, a clear bias in colour is visible at high stellar densities, however the extinction filtering just described removes all the sources with colours affected by stellar density, so an independent density filter is not needed.", "In Fig.", "REF we show the completeness of CLU and 2MPZ.", "At $D<40$ Mpc, CLU is over-complete (i.e.", "contains more than the expected luminosity): this was also true of GWGC which attributed to the effect of the Virgo cluster.", "2MPZ does not show this overcompleteness, this is likely the result both of the (comparatively) low sensitivity of 2MASS to low-surface-brightness galaxies (which dominate the nearby sample), and the inaccuracy of the photometric redshift: we return to the latter point below.", "Beyond $\\sim \\,$ 60 Mpc the 2MPZ survey is significantly more complete than CLU.", "2MPZ also has the advantage of being more uniform across the sky than CLU; compare fig.", "13 of with fig.", "1 of .", "In particular, completeness of CLU will depend on sky position, as this dataset is constructed from spectroscopic surveys, which at present do not cover the full sky beyond $\\sim \\,$ 130 Mpc [$z=0.03$ : completeness of the 2MASS Redshift Survey, ].", "Such a limitation does not apply to photometric all-sky surveys, such as 2MASS or WISE, which are only limited by their respective flux limits and Galactic plane nuisances.", "These considerations suggest that 2MPZ represents the better catalogue to use, although if the GW information shows that the object is $<60$ Mpc away it may be better to instead use CLU.", "However, the accuracy of the redshift information must also be considered.", "While CLU employs only spectroscopic redshift measurements, roughly 2/3 of the sources in 2MPZ have only photometric redshifts, which will have a larger uncertainty.", "To determine the accuracy of the photometric redshifts, we selected from 2MPZ only those objects with $E(B-V)<1$ and with both spectroscopic and photometric redshifts (the latter had the systematic correction above applied for $E(B-V)\\ge 0.5$ ).", "We then created histograms of $z_{\\rm photo}-z_{\\rm spec}$ in several $z_{\\rm photo}$ bins; two examples are shown in Fig.", "REF .", "These are approximately Gaussian, but the width ($\\sigma _{z_p}$ ) varies with redshift.", "We fit this variation (Fig.", "REF , lower panel) with a broken powerlaw: $\\sigma _{z_p} = \\left\\lbrace \\begin{array}{ll}0.043\\ z_{\\rm photo}^{0.402} & z_{\\rm photo} < 0.10 \\\\0.023\\ z_{\\rm photo}^{0.14} & z_{\\rm photo} \\ge 0.10 \\\\\\end{array}\\right.$ Figure: The accuracy of the photometric redshifts in 2MPZ.", "Top: Histogramsof z photo -z spec z_{\\rm photo}-z_{\\rm spec} for galaxies with z photo <0.05z_{\\rm photo}<0.05 (black) and0.2≤z photo <0.250.2\\le z_{\\rm photo} < 0.25 (red).", "Bottom: The variation of the Gaussian σ\\sigma of the histograms as a function of photometric redshift; the model is given in equation .as shown in the lower panel of Fig.", "REF .", "This uncertainty must be taken into account when comparing the distance of a galaxy with the distance inferred from the GW data, as will be described in Section REF .", "This relatively large uncertainty in photometric redshift will also distort the completeness curve slightly.", "The probability of a galaxy with true redshift $z_1$ being assigned $z_{\\rm photo}=z_1+\\Delta z=z_2$ is slightly lower than the inverse: the probability of a galaxy with a true redshift $z_2$ being assigned $z_{\\rm photo}=z_2-\\Delta z=z_1$ , because there is more volume (and so more galaxies) at $z_2$ than at $z_1$This is analogous to the Malmquist bias present for redshift-independent distance indicators and peculiar velocities derived from them..", "This is clearly a bigger effect at lower redshift, where $\\Delta z/z$ is higher.", "Similarly towards the limit of the catalogue's redshift range (i.e.", "$z\\rightarrow 0$ and $z\\rightarrow z_{\\rm lim}$ ) the completeness will be underestimated because some galaxies with true redshifts inside the catalogue's redshift range will receive photometric values outside of it, but there are no (or few) galaxies outside of the limit to `compensate'.", "The overall effect of this is that the completeness shown in Fig.", "REF is underestimated until the catalogue limit ($z\\sim \\,0.3$ corresponding to $D\\sim \\,1.6$ Gpc) is approached, i.e.", "2MPZ is more complete within the distance range we are interested in than implied from Fig.", "REF , which strengthens the argument that 2MPZ is the better choice of the two catalogues we have studied.", "Looking further in the future as the horizon distance of aLIGO/AVIRGO rises further, we may wish to consider the forthcoming WISExSuperCOSMOS photometric redshift catalogue (Bilicki et al.", "2016 in press), which covers less of sky than 2MPZ (70%), but reaches much deeper in redshift (median $z\\sim \\,0.2, D\\sim \\,850$ Mpc), however we defer study on the cost/benefit of this to a future work." ], [ "Using the distance and completeness", "In our GW response to date, we have used galaxy catalogues in a simplistic way: we ignored the (in)completeness of the catalogue (i.e.", "regions of the sky without known galaxies were given zero probability of hosting the GW event) and did not weight galaxies by their distance compared to the expected distance to the GW event – the latter was not possible due to the lack of GW distance estimate available at trigger-time.", "In O2 and beyond, the horizon distance is such that the incompleteness of galaxy catalogues is significant (Fig.", "REF ), and the aLIGO/VIRGO teams are likely to produce rapid distance estimates ; therefore, we describe now a new method of galaxy convolution to produce higher-fidelity skymaps than we have used to date.", "Considering first completeness: if the distance $D$ to the GW source is known perfectly, then we can estimate the completeness of the galaxy catalogue at this distance from the lower panel of Fig.", "REF , we will call this $C$ .", "The probability that the GW event occurred in a known galaxy is thus $C$ , and the probability that it occurred in an unknown one is $1-C$ .", "demonstrated that the distance $D$ deduced from the GW data is a function of direction on the sky.", "The GW error regions are distributed as healpix-format skymaps, and each pixel in this map in the approach has its own $D$ distribution and hence $C$ value, which we calculate thusNote that we are implicitly assuming that the completeness of the catalogue is not a function of direction.", "This is a reasonable assumption for 2MPZ, but for current spectroscopic surveys the non-uniformity on the sky would need to be factored into $C_p$ .", ": $C_p = \\frac{\\int P_p(D) C(D) dD }{\\int P_p(D) dD}$ where $P_p(D)$ is the probability distribution of the distance, defined for the pixel $p$ .", "Therefore, for each given pixel in the skymap, the probability of the GW event occurring in an uncatalogued galaxy within that pixel is: $\\mathcal {P}_{{\\rm nogal},p}=\\mathcal {P}_{{\\rm GW},p} \\left( 1-C_p \\right)$ where $\\mathcal {P}_{{\\rm GW},p}$ is the probability in the original skymap from the aLIGO/AVIRGO team for pixel $p$ .", "For pixels containing galaxies, an extra factor $\\mathcal {P}_{{\\rm gal},p}$ must be included.", "Previously (paper I) we defined this as in equation REF : the GW probability multiplied by the ratio of galaxy luminosity in the pixel to the total catalogued galaxy luminosity.", "This now needs to refer only to the luminosity within the distance indicated by the GW dataset, and needs a correction for completeness.", "We therefore redefine the probability of the GW event occurring within a known galaxy in pixel $p$ thus: $\\mathcal {P}_{{\\rm gal},p}=\\mathcal {P}_{{\\rm GW},p} C_p \\sum _g \\left( \\mathcal {P}(g|P_p[D]) \\frac{L_gN_P}{L_{\\rm tot}} \\right)$ The summation is over all galaxies $g$ in pixel $p$ .", "$L_g$ is the luminosity of galaxy $g$ divided by the number of pixels it covers.", "$N_p$ is the number of pixels in the map, and $L_{\\rm tot}$ is the total catalogued galaxy luminosity within the GW volume, so $\\frac{L_gN_P}{L_{\\rm tot}}$ gives the ratio of the actual luminosity in pixel $p$ compared to that expected if the galaxies were homogeniously distributed on the sky, i.e.", "the relative probability of this pixel hosting a merger event compared to any other pixel.", "$L_{\\rm tot}$ is given by: $L_{\\rm tot} = \\sum _p \\sum _g \\left[ \\mathcal {P}(g|P_p[D]) L_g \\right]$ where $\\mathcal {P}(g|P(D)_p)$ is the probability that the galaxy $g$ is at the correct distance to host the GW event.", "This is simply: $\\mathcal {P}(g|P_p[D]) = \\int P_p(D) P_g(D) dD$ where $P(D)_p$ is the probability as a function of distance for pixel P, determined from the GW data.", "For the low-latency analysis this is a Gaussian multiplied by distance squared .", "$P_g(D)$ is the probability distribution of the distance $D$ of galaxy $g$ .", "2MPZ does not contain uncertainties on the photometric redshift measurements, therefore we need to decide on the form of $P_g(D)$ .", "For galaxies in 2MPZ with spectroscopic redshift we assume that the dominant source of error is the peculiar velocity of the galaxy, and we take 500 km s$^{-1}$ as representative of this.", "This corresponds to a distance error of $500/H_0 = 7.4$ Mpc (assuming $H_0=70$ km s$^{-1}$ Mpc$^{-1}$ ), so for galaxies with spectroscopic redshift we treat $P_g(D)$ as a Gaussian with $\\sigma =7.4$ Mpc.", "For photometric redshifts we use the prescription given in Section REF (the peculiar velocity correction is insignificant compared to this and can be ignored).", "Having now calculated the probability of the GW event occurring in an unknown galaxy, or in any specific known galaxy, the probability that the GW event is in pixel $p$ of the skymap is simply: $\\mathcal {P}_p=\\mathcal {P}_{{\\rm nogal},p}+\\mathcal {P}_{{\\rm gal},p}$ Finally, the map must be renormalised such that it sums to unityRenormalisation is not essential for planning observations, since it is the relative probability in each pixel that matters; however in order to calculate the probability that one has observed the true GW location, the map must be normalised to 1..", "The result of this is a modified probability map on the sky which accounts for both the GW localisation and our prior knowledge of the structure of the local universe.", "This can then be used in a manner similar to that proposed by , i.e.", "by selecting fields in (descending) probability order until some threshold probability has been selected (50% in the method of ).", "Alternatively, as suggested in Section , we can observe as many fields as possible in a given time interval, but again observing in order of priority.", "Based on local structure in the universe it could be argued that the unknown galaxies are not homogeneously distributed on the sky, but are instead more likely near known galaxies.", "In this case, more exotic definitions of $P_{{\\rm nogal},p}$ could be created to account for the distance to nearby galaxies.", "For the present, we will limit ourselves to the simple prescription above.", "Similarly since the values of binary inclination and distance determined from the GW are degenerate, the probability distribution of binary inclination for each pixel could in principle be produced and then, from a template library of GRB light curves for different inclinations, one could determine the probability of detecting a GRB from a given pixel as a function of time.", "While this is under investigation it is not likely to be possible on the timescale of O2, and is beyond the scope of this paper." ], [ "Which luminosity to use", "The above calculations weigh each galaxy by its luminosity.", "However, which band one uses is also pertinentTechnically one should consider the rest-frame band rather than the observer frame, however since we are considering only the relatively nearby universe, we neglect this issue..", "The $K$ -band provides a reasonable proxy for stellar mass in the galaxy, which we may take as being a proxy for the number of binary neutron-star systems in the galaxy and hence the probability of hosting a merger of such a system.", "However, recent observations , suggest that short GRBs are more common in late-type galaxies (suggesting that the probability of a compact binary coalescing is influenced by recent star formation), which suggest that it is more appropriate to weight galaxies by their $B$ band luminosity.", "2MPZ contains both infrarad magnitudes (from 2MASS and WISE) and the optical $R$ and $B$ magnitudes (from SuperCOSMOS), which gives us the flexibility to select which band we wish to use, if the theoretical (or observation) priors change.", "To investigate this, and the impact of the galaxy convolution, we performed a series of simulations.", "We started with the GW simulations of https://dcc.ligo.org/public/0119/P1500071/005/index.html, which provide 3-D probability maps for 250 simulated binary neutron star mergers in the 2-detector configurationLabelled as `O1' online..", "These simulations assume that the mergers are simply distributed homogeneously in space, whereas we wish to seed them in galaxies.", "Each GW simulation has the position and distance to the simulated event.", "We calculated the completeness $C$ of 2MPZ at this distance, and then generated a random number $0\\le \\mathcal {R}<1$ .", "If $\\mathcal {R}\\ge C$ the GW event was treated as occuring in an uncatalogued galaxy, so the data needed no changes.", "Otherwise, a host galaxy for the event was selected at random from the 2MPZ catalogue, with each galaxy having a probabilty of being the host, proportional to $L\\mathcal {P}(D)$ (where $L$ is the galaxy luminosity and $\\mathcal {P}(D)$ is the probability that this galaxy is at the distance of the simulated merger).", "Since the LIGO probability maps are strongly dependent on the geocentric direction to the merger, rather than rotate these maps such that the GW events occured in galaxies, instead we rotated the galaxy catalogue such that the selected host was at the position of the simulated merger.", "We then created a series of XRT fields tiled on the sky, arranged them in decreasing order of probability and determined which field contained the merger event.", "We did this five times with different models.", "In the first instance, we performed no galaxy convolution at all, i.e.", "we simulated tiling the original GW error region.", "In the other four simulations, we selected host galaxies based on either their $B$ - or $K$ -band luminosities, and then convolved the GW map with 2MPZ using the $B$ - or $K$ -band luminosities.", "That is, we simulated the cases where our assumption about which galaxies are more likely to host GW events are correct (the same band was used in selecting the host and convolving the GW region), and when they are incorrect (one band was used to select the hosts, and the other used in convolution).", "In Fig.", "REF we show the results of this.", "Plotted is the cumulative distribution of which field contained the GRB in the 250 simulated mergers, for the different simulations runs.", "This confirms that the galaxy convolution significantly reduces the typical number of XRT fields we have to observe before we reach the correct location.", "With no convolution, 50% of the time at least 1,200 XRT fields are needed to reach the correct location.", "With convolution this falls to $\\sim \\,$ 170 (a factor of $\\sim \\,$ 7 decrease) if the same band is used in the simulations and search, or $\\sim \\,$ 300 fields (a factor of $\\sim \\,$ 4 improvement over the no-convolution approach) if different bands are used.", "Therefore, while the choice of which band to use when convolving galaxies does have a significant effect, choosing the wrong band is still much better than not using galaxy convolution.", "Figure: The cumulative distribution of which XRT field (in probability order) contains the GW event,from 250 simulated GW events (per colour), showing the benefit of galaxy convolution and the impact ofusing an incorrect assumption as to which luminosity band the galaxies should be weighted by.Black: Simulations where the XRT fields are generatedbased on the original GW map with no galaxy convolution.", "Red: The GW events are seeded in hostsweighted by BB-band luminosity, and the GW error region is convolved with a BB-band weighted galaxy catalogue.Cyan: Seeding is weighted by the BB-band, but convolution uses the KK-band.", "Blue: Seeding and convolutionboth use the KK-band.", "Magenta: Seeding is weighted by the KK-band and convolution by the BB-band." ], [ "Conclusions", "Swift performed rapid-response follow up to all three GW triggers released to the EM partners by the aLIGO team during the O1 operating run of aLIGO.", "No compelling X-ray, optical or gamma-ray counterpart was found, however this is not surprising, since only a small fraction of the GW error region was covered.", "Additionally, one of the GW triggers was spurious and the other two are believed to be BBH mergers, which may not be expected to give rise to EM emission.", "For the second trigger, we can place a limit on the hard X-ray emission of (4.3–90)$\\times 10^{-8}$ erg $$ cm$^{-2}$ s$^{-1}$ (15–350 keV) for a region enclosing 15% of the GW probability.", "In the future Swift will be able to observe a much larger fraction of the GW error region as a new observing capability has been commissioned, which will enable large-scale, short-exposure tiling.", "Given both the increased horizon distance expected during O2, and the fact that both real GW events in O1 were at large distances ($\\sim \\,$ 500 Mpc), targeting galaxies in the GWGC, which is limited to 100 Mpc, is not a good approach.", "The 2MPZ catalogue, which uses a mixture of spectroscopic and photometric redshifts, offers a better prospect (unless the GW localisation identifies the object as being $<60$ Mpc from Earth), and we have shown how we can use the completeness measurements for this catalogue, and the GW distance estimates expected to be rapidly available in O2, to optimise the skymap produced by the aLIGO/AVIRGO teams.", "In the future, as new catalogues become available (such as the WISExSuperCOSMOS, GLADE in prep/press) or when photometric redshifts are added to Wise-2MASS complilation , and the sensitivity and localisation characteristics of aLIGO/AVIRGO increase it may be valuable to reassess the benefits of galaxy targeting and choice of catalogue." ], [ "Acknowledgements", "We thank András Kovács for helpful discussion on galaxy catalogues.", "This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester.", "This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.", "This research has made use of the XRT Data Analysis Software (XRTDAS) developed under the responsibility of the ASI Science Data Center (ASDC), Italy.", "This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.", "The GW probability maps and our related galaxy maps are in healpix format .", "PAE, JPO and KLP acknowledge UK Space Agency support.", "SC and GT acknowledge ASI for support (contract I/004/11/1).", "MB is supported by the Netherlands Organization for Scientific Research, NWO, through grant number 614.001.451; through FP7 grant number 279396 from the European Research Council; and by the Polish National Science Centre under contract #UMO-2012/07/D/ST9/02785." ] ]
1606.05001
[ [ "SZ/X-ray scaling relations using X-ray data and Planck Nominal maps" ], [ "Abstract We determine the relation between the Comptonization parameter predicted using X-ray data $Y_{C,Xray}$ and the X-ray luminosity $L_X$, both magnitudes derived from ROSAT data, with the Comptonization parameter $Y_{C,SZ}$ measured on {\\it Planck} 2013 foreground cleaned Nominal maps.", "The 560 clusters of our sample includes clusters with masses $M\\ge 10^{13}M_\\odot$, one order of magnitude smaller than those used by the Planck Collaboration in a similar analysis.", "It also contains eight times more clusters in the redshift interval $z\\le 0.3$.", "The prediction of the $\\beta=2/3$ model convolved with the Planck antenna beam agrees with the anisotropies measured in foreground cleaned Planck Nominal maps within the X-ray emitting region, confirming the results of an earlier analysis (Atrio-Barandela et al.", "2008).", "The universal pressure profile overestimates the signal by a 15-21\\% depending on the angular aperture.", "We show that the discrepancy is not due to the presence of {\\it cool-core} systems but it is an indication of a brake in the $L_X-M$ relation towards low mass systems.", "We show that relation of the Comptonization parameter averaged over the region that emits 99\\% of the X-ray flux and and the X-ray luminosity is consistent with the predictions of the self-similar model.", "We confirm previous findings that the scaling relations studied here do not evolve with redshift within the range probed by our catalog." ], [ "Introduction.", "Clusters of galaxies are the largest virialized structures in the Universe first observed as concentrations of optical galaxies.", "Compression and shock-heating processes raise the temperature of the Intra-Cluster Medium (ICM) to $T_X\\sim 1-10$ keV and clusters can be observed through their X-ray emission and their thermal Sunyaev-Zeldovich (TSZ, [72]) distortion of the Cosmic Microwave Background (CMB).", "The self-similar model predicts simple scaling relations between cluster observables and their mass [30].", "More specifically, hydrodynamical N-body simulations have shown that the TSZ signal integrated over the cluster volume scales with the cluster mass [79], [19], [44], [45], [80], [1].", "X-ray properties are also related to the cluster mass and gas temperature [42].", "Scaling relations have been determined observationally but their form does not necessarily coincide with the prediction of the self-similar model [78], [4], [5], [65], [76].", "Therefore, non-gravitational processes such as mergers, cooling and energy injection from Active Galactic Nuclei (AGN) have a significant contribution to the equilibrium state of clusters and scaling relations can be used to test the physics of clusters of galaxies [12], [39], [6], [42], [3], [17].", "The redshift evolution of the scaling relations seems to follow the self-similar prediction [13], [40], suggesting that cluster properties evolve with the density of the Universe.", "Large-scale surveys of the TSZ effect such as those carried out by the Atacama Cosmology Telescope (ACT, [33]) the South Pole Telescope (SPT, [14]) and the Planck satellite [55] have helped to constrain cosmological parameters using SZ clusters [75], [70], [60], [63] and to establish scaling relations between X-ray magnitudes and TSZ measurements [64], [41], [65], [6], [17], [3], [34], [68], [18].", "The effectiveness of clusters as cosmological probes depends on obtaining reliable mass estimates.", "In this respect, the scatter on the scaling relations needs to be understood in order to account for observational biases.", "The scatter includes statistical errors, systematic biases and the intrinsic differences between clusters.", "The intrinsic scatter is dominated by the cluster cores [50], [16] since cool-core (CC) clusters deviate from self-similarity of the no cool-core (NCC) population.", "The difference in the cluster population have been thought to be responsible for discrepancies between numerical predictions and observations.", "[34] found that the universal pressure profile [6] overestimated the TSZ amplitude by a 30%.", "Since the universal pressure profile agreed with TSZ data for CC but disagreed for the NCC clusters, it was thought that the excess was related to the dynamical state of the ICM.", "Nevertheless, subsequent analysis did not find any evidence of this discrepancy [42], [3], [68].", "This was further confirmed by the Planck Collaboration using a sample of $\\sim 1600$ clusters [52], [53].", "In this study, the TSZ anisotropy was individually measured for each object but, in order to maximize the statistical significance of the result, the signal was averaged in bins of X-ray luminosity.", "The Planck Collaboration also carried out an analysis of scaling relations using a sample of 62 clusters (later extended to 78) without binning their properties [53], [62].", "In this paper we extent this later analyses using a sample of 560 clusters with redshifts $z\\le 0.3$ that includes systems with masses one order of magnitude smaller than the latter Planck sample.", "Briefly, in Sec.", "2 we review cluster pressure profiles models, in Sec.", "3 we describe our cluster catalog and the CMB data used in this study and in Sec.", "4 we discuss the scaling relations to be determined, the regression routines used and the associated errors.", "Finally, in Sec.", "5 we present our results and in Sec.", "6 we summarize our conclusions.", "Table: The universal pressure profile parameters determinedby and ." ], [ "The Integrated Comptonization Parameter.", "The TSZ effect is a distortion of the CMB black-body spectrum produced when CMB photons are scattered off by the free electrons of the ICM.", "The TSZ effect is independent of redshift and it is usually expressed in terms of the Comptonization parameter $y_c=(k_B\\sigma _T/m_ec^2)\\int _l T_Xn_e dl$ , where $n_e, T_X$ are the electron density and temperature along the line of sight $l$ , $\\sigma _T$ the Thomson cross section, $m_e$ the electron rest mass and $c$ the speed of light.", "The temperature anisotropy $\\Delta T_{TSZ}(\\hat{n})= T_0G(\\nu )y_c$ , with $T_0$ is the CMB black-body temperature and $G(\\nu )$ is the spectral dependence of the TSZ effect, that is different from that of any other known astrophysical foreground.", "In the non-relativistic limit $G(\\nu )= \\tilde{\\nu }{\\rm coth}(\\tilde{\\nu }/2)-4$ , where $\\tilde{\\nu }=h\\nu /k_BT_X$ is the reduced frequency and $h,\\; k_B$ are the Planck and Boltzmann constants, respectively.", "In this work we will include relativistic corrections up to fourth order in the electron temperature ([28], [48], [49]).", "The SZ effect integrated over the solid angle subtended by the cluster is $&Y_C=\\int y_cd\\Omega =D_A^{-2}(z)\\int y_cdA\\nonumber \\\\[0.3cm]& =\\frac{k_B\\sigma _{\\rm T}}{mc^2}D_A^{-2}(z)\\int _VT_{\\rm e}n_{\\rm e}dV=\\frac{\\sigma _{\\rm T}}{mc^2}D_A^{-2}(z)\\int _VP_{\\rm e}dV,$ where $D_A(z)$ is the angular diameter distance, $dA$ the projected area and $dV=dldA$ the volume element.", "The integrated Comptonization parameter is dimensionless; i.e., it is measured in units of solid angle, generally in $(arcmin)^2$ .", "An alternative convention is to use $D^2_A~Y_C$ and express this magnitude in units of Mpc$^2$ .", "The pressure profile of the hot ICM was initially described by an spherically symmetric isothermal gas with the electron density profile given by the $\\beta $ model [15] $n_e(r)=n_{e,0}\\left[1+\\left(\\frac{r}{r_c}\\right)^2 \\right]^{-3\\beta /2},$ where the cluster core radius $r_c$ , central electron density $n_{e,0}$ and slope $\\beta $ are determined from observations.", "By fitting the cluster X-ray surface brightness [29] estimated $\\beta =0.6-0.8$ .", "The TSZ emission predicted for a cluster sample using the fiducial value $\\beta =2/3$ , convolved with the WMAP antenna beam has been shown by [7] to agree with the measured anisotropy in WMAP 3yr maps within the region emitting 99% of the X-ray flux.", "Based on the Navarro-Frenk-White (NFW) dark matter profile [47] and using the the results of N-body simulations, [46] proposed the dimensionless pressure profile $p(x)=\\frac{P_0}{(c_{500}x)^\\gamma [1+(c_{500}x)^\\alpha ]^{(\\beta -\\gamma )/\\alpha }},$ where $x=r/r_{500}$ is the distance from the cluster center in units of the radius at which the mean overdensity of the cluster is 500 times the critical density, $c_{500}$ is the gas concentration parameter at $r_{500}$ , $(\\gamma ,\\alpha ,\\beta )$ are the central, intermediate and outer slopes and $P_0$ is given in Table REF .", "[6] assumed the profile of eq.", "(REF ) to be universal and used X-ray data from a sample of 33 clusters at redshift $z<0.2$ to constrain the model parameters.", "The dimensional electron pressure profile derived from that data was $P_e(x)&=3.36~h^2E^{8/3}(z)\\times \\nonumber \\\\[0.3cm]&\\times \\left[\\frac{M_{500}}{2.1\\times 10^{14}h^{-1}M_\\odot }\\right]^{2/3+\\alpha _p+\\alpha ^{\\prime }_p} p(x)~[{\\rm eV~cm^{-3}}],$ with $\\alpha _p=0.12$ , and $\\alpha ^{\\prime }_p(x)=0.1−(\\alpha _p+0.10)(x/0.5)^3(1.0+(x/0.5)^3)^{-1}$ .", "In this expression $h=H_0/[100{\\rm km s^{-1}Mpc^{-1}}]$ is the reduced Hubble constant with $H_0$ its current value.", "The parameters of eq.", "(REF ) have also been determined using the TSZ anisotropy of 62 nearby clusters, measured in the 2013 Planck CMB data [54].", "To test how well the model predictions agree with the measured anisotropy, in this article we will analyze if the amplitude of the TSZ effect predicted from the X-ray data convolved with the beam at each Planck frequency agrees with the measured anisotropy using the isothermal $\\beta $ model of eq.", "(REF ) and the pressure profile of eq.", "(REF ) with the two sets of parameters given in Table REF ." ], [ "Data.", "We use a sample of 560 X-ray clusters and the Planck Nominal data released in 2013Data downloaded from http://www.cosmos.esa.int/web/planck to determine the X-ray/SZ scaling relation.", "Our work differs from previous studies in that all magnitudes used in our analyses have been derived from observations, except the X-ray temperature and $r_{500}$ scale that were themselves obtained from scaling relations." ], [ "X-ray Cluster Catalog.", "The cluster catalog was compiled from three ROSAT X-ray flux limited surveys: the ROSAT-ESO Flux Limited X-ray catalog (REFLEX, [10]), the extended Brightest Cluster Sample (eBCS, [23], [24]) and the Clusters in the Zone of Avoidance (CIZA, [25]).", "These three samples differ in selection techniques, flux measuring algorithms and are affected by different systematic effects.", "To construct a homogeneous all-sky sample, the different selection technique and the flux determination method employed have to be taken into account to guarantee that all three samples are complete to the same depth.", "A full discussion of the method used to combine the individual catalogs into a homogeneous all-sky sample is given in [32] and is briefly summarized here.", "First, the flux is recomputed using ROSAT All Sky Survey (RASS) data.", "The centroid of the cluster X-ray emission is determined and the X-ray count rate is computed taking into account the local RASS exposure time.", "The X-ray background is determined from an annulus of radius 1 and 1.5$h^{-1}$ Mpc around the cluster centroid and subtracted from the measured counts.", "The resulting count rates are deconvolved from the telescope Point Spread Function (PSF) and converted to unabsorbed fluxes in the [0.1-2.4]keV band.", "For the RASS, the PSF is the weighted average of the PSF's at all off-axis angles [23].", "Clusters whose emission is dominated by a point source were removed and a cut of $F_x[0.1-2.4{\\rm keV}]\\ge 3\\times 10^{-12}$ ergs cm$^{-2}$ s$^{-1}$ was applied.", "The merged catalog contains 782 clusters with well measured positions, spectroscopic redshifts, X-ray fluxes in the [0.1-2.4]keV band and angular extents of the region emitting 99% of the X-ray flux, hereafter $\\theta _X$ .", "Of those, only 623 clusters survive the point-source and the Planck galactic masks.", "Foreground contamination reduced the total number of clusters used in this study to $N_{cl}=560$ .", "All the clusters in our sample were fitted to an isothermal $\\beta $ model [15].", "If $S(r)$ is the projected surface brightness distribution, then $S(r) = S_0 \\left[1+(r/r_c)^2\\right]^{-3\\beta +1/2}$ where $S_0$ , $r_c$ , and $\\beta $ are the central surface brightness, the core radius, and the $\\beta $ parameter characterizing the profile, respectively.", "Due to the low angular resolution of the RASS the surface brightness of our clusters is poorly sampled except for nearby clusters.", "The correlation between $r_c$ and $\\beta $ introduces further uncertainties and makes the results for both parameters sensitive to the radius of the cluster chosen to fit the model.", "Due to these limitations, we take $\\beta =2/3$ , the canonical value [29].", "Reassuringly, the values of $r_c$ derived from the data agree with the values derived from the $L_X-r_c$ relation determined by [66].", "Cluster luminosities and electron temperatures are used to determine central electron densities.", "The ICM temperature is derived from the bias corrected $L_{\\rm X}[0.1-2.4keV]-T_{\\rm X}$ relation of [37].", "From the RASS data, X-ray luminosities are measured within a radius of angular size $\\theta _X$ , and are k-corrected to rest frame $[0.1-2.4]$ keV from the REFLEX/CIZA/eBCS surveys.", "Conversions between angular extents and physical dimensions are made using the $\\Lambda $ CDM model with Planck measured parameters [59].", "Errors are due to Poisson noise in the number of the photons detected for each cluster and are, at most, 20%H.", "Ebeling, private communication.", "By fitting a $\\beta =2/3$ model to the X-ray surface density, the core radii ($r_c$ ) and central electron density ($n_{e,0}$ ) have been determined.", "To compare with previous analyses, we also evaluate our scaling relations at $r_{500}$ .", "We derive this scale from the $r_{500}-L_X$ relation of [11].", "From the latter magnitude we define the angular size $\\theta _{500}=r_{500}/D_A(z)$ and mass scale $M_{500}=({4\\pi }/{3})500 \\rho _c(z)r_{500}^3$ , where $\\rho _c(z)$ is the critical density at the cluster redshift.", "These clusters are located within the redshift interval $z=[0,0.3]$ , have luminosities $L_X=[0.3 - 22.5]\\times 10^{44}$ erg/s that correspond to $T_X=[0.87 - 11.5]$ keV, and $M_{500}=[0.2-14.7]\\times 10^{14}$ M$_\\odot $ .", "By comparison, the Planck Collaboration study used 78 individual clusters with $z=[0.0, 0.5]$ , $T_X=[3-14]$ keV and $M_{500}=[2-20]\\times 10^{14}M_\\odot $ .", "The larger number of clusters and the wider mass range will allow us to test the accuracy with which the cluster pressure profiles of eqs.", "(REF ) and (REF ) fit the data and how well the average properties of the cluster population are described by the self-similar model." ], [ "Foreground cleaned Planck Nominal Maps.", "Planck data were originally released in 2013 in a Healpix format with resolution $N_{side} = 2048$ [26].", "The Nominal maps contained foreground emissions from galactic dust, CO lines, synchrotron, point sources and extended infrared sources.", "The angular resolution of the Low Frequency Instrument channels is $\\theta _{FWHM}>13^{\\prime }$ , larger than the angular extent of the clusters in our sample, so we will restrict our analysis to the High Frequency Instrument data.", "Prior to compute the TSZ anisotropies we clean the data from foreground and cosmological contributions as described in [20].", "First, we subtract the intrinsic CMB and kinematic SZ anisotropies using the LGMCA CMB template [8], [9].", "Thermal dust emission is subtracted using the 857 GHz channel as a dust-template [56], [57].", "We clean this contribution on sky patches $\\mathcal {P}(\\nu , i)$ centered on each cluster $i$ at each frequency $\\nu $ following [22].", "The CO emission at 100 and 217 GHz is removed using the Type 2 maps described in [58].", "We used the PCCS-SZ-Union mask to excise point sources and mask the residual Galactic Plane emission [61], [62].", "Temperature fluctuations ($\\delta \\bar{T}$ ) are measured by averaging the anisotropy over a disc of size $\\theta $ on the foreground cleaned patches.", "The method is not fully effective and we rejected those clusters for which $Y_{C,500}<0$ , corresponding to high redshift and low luminosity systems with high levels of foreground residuals.", "In total, $N_{cl}=560$ clusters were used in this study.", "In addition to the TSZ signal of interest, $Y_C(x)G(\\nu )$ , these foreground cleaned patches $\\mathcal {P}$ contain instrumental noise $N(\\nu ,x)$ and some degree of CMB and foreground residuals.", "To estimate error bars, we placed discs the size of each cluster on 1,000 random positions.", "The patch covered by each random disc is cleaned using the procedure described above and then the mean temperature anisotropy on the disc is computed.", "The process is repeated for all clusters.", "To avoid overlapping these random clusters with the real population, we mask an area of one degree radius around all known clusters.", "The error bar associated with the measured anisotropy of a cluster is $\\sigma ^2(\\theta ,\\nu _i)=\\langle [\\delta \\bar{T}(\\theta ,\\nu _i)-\\mu (\\theta ,\\nu _i)]^2\\rangle ^{1/2},$ where $\\mu (\\theta ,\\nu _i)=\\langle \\delta \\bar{T}(\\theta ,\\nu _i)\\rangle $ and averages are taken over the 1,000 random positions.", "Our cleaning method performs better for small apertures.", "The average relative error was $\\sim 13\\%$ at $\\theta _{500}$ and it grows to $\\sim 40\\%$ at $2\\theta _{500}$ .", "Therefore, we will perform our analysis on apertures equal or smaller than $\\theta _{500}$ .", "The integrated Comptonization parameter $Y_C(\\theta )$ is measured using two different methods: (A) At a cluster location and for all frequencies we define the signal as $Y_{C}(\\theta ,\\nu )=\\delta \\bar{T}/(T_0 G(\\nu ))$ and the associated error as $\\sigma _{Y_C}(\\theta ,\\nu )=\\sigma (\\theta ,\\nu )/(T_0G(\\nu ))$ .", "Since $G(217{\\rm GHz})\\approx 0$ , $Y_{C}(\\theta ,{\\rm 217 GHz})$ will be dominated by the errors (see below), the Comptonization parameter will be computed as the weighted average over all frequencies except 217 GHz.", "It is given by $&\\bar{Y}_{C,\\nu }(\\theta )=\\sigma _{\\bar{Y}_{C,\\nu }}^2(\\theta )\\Sigma _{\\nu }\\left[\\frac{Y_C(\\theta ,\\nu )}{\\sigma ^2_{Y_C}(\\theta ,\\nu )}\\right]; \\nonumber \\\\&\\sigma _{\\bar{Y}_{C,\\nu }}^{-2}(\\theta )=\\Sigma _{\\nu }\\sigma ^{-2}_{Y_C}(\\theta ,\\nu ).$ This expression does not include the negligible error on the CMB blackbody temperature $T_0$ .", "(B) The foreground cleaned patches at all frequencies are combined using the Internal Linear Combination (ILC) method described in [21] and the Comptonization parameter is measured directly on the combined map.", "To illustrate how the ILC technique is applied to estimate the TSZ emission, we assume that the residual cosmological signal $B(\\nu )T(x)$ dominates over the foreground residuals.", "Then, the anisotropy in each patch will have the following components $\\mathcal {P}(\\nu ,x)=Y_C(x)G(\\nu )+B(\\nu )T(x)+N(\\nu ,x).$ Following [67], we can estimate the TSZ emission in each patch as $\\hat{Y}_C(x)=w(\\nu _i)\\mathcal {P}(\\nu _i, x)$ .", "The weights $w(\\nu )$ are obtained by minimizing $\\chi ^2=N_{\\rm pix}^{-1}\\sum _x\\left(\\hat{Y}_C(x)-\\langle \\hat{Y}_C\\rangle \\right)^2$ where $N_{\\rm pix}$ is the number of pixels of each patch.", "The weights satisfy $\\Sigma w(\\nu _i)G(\\nu _i)=1$ , $\\Sigma w(\\nu _i)B(\\nu _i)=0$ and are given by $w(\\nu _i)=\\frac{\\left(B_k\\hat{R}^{-1}_{kl}B_l\\right)G_j\\hat{R}^{-1}_{ij}-\\left(G_k\\hat{R}^{-1}_{kl}B_l\\right)B_j\\hat{R}^{-1}_{ij}}{\\left(G_k\\hat{R}^{-1}_{kl}G_l\\right)\\left(B_m\\hat{R}^{-1}_{mn}B_n\\right)-\\left(G_k \\hat{R}^{-1}_{kl} B_l \\right)^2} .$ Here $\\hat{R}_{ij}=N_{\\rm pix}^{-1}\\sum _p\\left(T_i(p)-\\langle T_i\\rangle \\right)\\left( T_j(p)-\\langle T_j \\rangle \\right)$ is the empirical covariance matrix computed on the foreground cleaned random patches and the indices $i,j,k$ run over the frequencies $\\nu =[100,143,217,353]$ GHz.", "Like before, the Comptonization parameter $\\bar{Y}_{C,ILC,\\theta }$ is obtained by averaging on a disc of the radius $\\theta $ ; the associated error is estimated using the ILC weights from 1,000 foreground cleaned patches placed randomly outside the known cluster positions.", "Figure: Planck Nominal and foreground cleaned patchescentered on the position of A1656, PSZ1 G067.19+67.44, and PSZ1 G081.01-50.92.at 100-545 GHz, and in the ILC foreground cleaned patch.The angular size of the patches is 1°×\\times 1°.In each clusters, the black circle corresponds to a disc of radiusθ 500 \\theta _{500}.To demonstrate how efficiently our two pipelines remove foregrounds and the different effect on the low redshift and extended, the intermediate and the high redshift and compact clusters, in Fig.", "REF we show the original Planck Nominal maps and the foreground cleaned patches in units of $T_0G(\\nu )$ for each frequency.", "In these units, the TSZ anisotropy does not change sign but the CMB residuals do.", "Patches subtend a solid angle of 1°$\\times $ 1°.", "We selected three Planck clusters: A1656 (Coma) with redshift $z = 0.023$ and an angular extent of $\\theta _{500} = 48.1^\\prime $ , PSZ1 G067.19+67.44 with $z = 0.1712$ and $\\theta _{500} = 9.34^\\prime $ and PSZ1 G081.01-50.92 $z = 0.2998$ and $\\theta _{500} = 5.45^\\prime $ .", "For each cluster, we also present the ILC reconstruction of the TSZ signal of the same data.", "While the TSZ emission is zero at 217 GHz, it dominates over the foregrounds residuals at all other frequencies except at 545 GHz, where dust residuals are still the dominant contribution.", "Therefore, this channel will not be used to avoid biasing the results.", "Figure: (a) Histograms of the Comptonization parameters measured on thecleaned patches centered at the cluster positions.", "Y ¯ 500 \\bar{Y}_{500} was computed ondiscs of size θ 500 \\theta _{500} from 100 to 353 GHz usingeq.", "(); (b) Comparison of the Comptonization parametermeasured in the ILC map and measured by combining frequencies; the red solid linerepresents the best linear fit: Y ¯ ν,500 =A+BY ¯ ILC,500 \\bar{Y}_{\\nu , 500}=A+B\\bar{Y}_{ILC,500},with parameters A=0.001±0.007A=0.001\\pm 0.007 and B=0.97±0.03B=0.97\\pm 0.03.", "(c) Histogram of the errors on the measured Comptonization parametersusing the ILC (red line) and the combined frequencies method (black line).Fig.", "REF illustrates that the estimated Comptonization parameter is independent of our foreground cleaning technique and estimation method.", "In Fig.", "REF a, we plot the distribution of the Comptonization parameter measured on discs of angular radius $\\theta _{500}$ at difference frequencies, $Y_{C,500}$ , and its average over all channels.", "All measured values are very similar except those at 217 GHz (long-dashed green line) since dividing by $G({\\rm 217GHz})\\simeq 0$ boosts the errors.", "Fig.", "REF b demonstrates that the Comptonization parameters derived using weighted frequency averages (method A) and the ILC map (method B) are fully compatible.", "To avoid overcrowding the plot, error bars are not shown.", "The red line represents $\\ln Y_{C,\\nu , 500}=A+B\\ln Y_{C,ILC,500}$ whose best fit parameters are $A=0.001\\pm 0.007$ and $B=0.97\\pm 0.03$ , consistent with the expected values of $A=0$ and $B=1$ at the $1\\sigma $ confidence level (CL).", "In Fig.", "REF c we compare the error using each estimator to show that while the values of $Y_{C,500}$ measured by both methods are comparable, the errors given by the ILC method (red histogram) are slightly smaller.", "This is logical since the 217 GHz channel is used to construct the ILC map while it is not used in the weighted average of eq.", "(REF ).", "Therefore, in our subsequent analysis we will quote the results using the ILC method." ], [ "Scaling Relations.", "To determine the scaling relation between the Comptonization parameter measured from the ILC map as described above $Y_{C,SZ}$ and the value predicted using X-ray data $Y_{C,Xray}$ we will use the isothermal $\\beta =2/3$ profile and the universal pressure profile with the two set of parameters given in Table REF .", "We will compare the measured and the predicted values at two angular scales, $\\theta _X$ and $\\theta _{500}$ .", "We subdivide our sample in five bins to study if the scaling relations evolve with redshift.", "All bins have width $\\Delta z=0.05$ except the last one where $\\Delta z=0.1$ since only 19 clusters have $z\\ge 0.25$ .", "The average cluster properties of the full sample and the different subsample are given in Table REF ." ], [ "Self-Similar Scaling Relations.", "While the dynamical evolution of clusters is dominated by the collapse of the Dark Matter (DM) component, their observational properties are determined by the physical processes undergone by the baryon component.", "In the self-similar model all cluster observables scale with the cluster mass; in particular, the mass of the gas $M_g$ , the luminosity $L_X$ , the gas temperature $T_X$ and the Comptonization parameter $Y_C$ scale as: $M_g\\sim M$ , $L_X\\sim E(z)^{7/3}M^{4/3}$ , $T_X\\sim E(z)^{2/3}M^{2/3}$ and $Y_C\\sim E(z)^{2/3}M^{5/3}$ [30].", "In this expression, $E(z)$ is the Hubble function in units of the Hubble constant today.", "In the Kaiser model, $Y_C\\propto E(z)^{9/4}L_X^{5/4}$ [41].", "To facilitate the comparison with earlier results we will fit scaling relations of the form $E(z)^\\gamma [D_A^2Y_{C,SZ}]=10^A[E(z)^\\kappa X/X_0]^B$ .", "Specifically, $& E(z)^{\\gamma _Y}\\frac{D_A^2 Y_{C,SZ}}{{\\rm Mpc}^2}=10^A\\left(\\frac{D_A^2Y_{C,Xray}}{10^{-4}{\\rm Mpc}^2}E(z)^{\\kappa _Y} \\right)^B,\\\\[0.3cm]& E(z)^{\\gamma _L}\\frac{D_A^2 Y_{C,SZ}}{\\rm Mpc^2}=10^A\\left(\\dfrac{L_X}{7\\times 10^{44}\\, {\\rm erg/s}}E(z)^{\\kappa _L}\\right)^B ,$ where the luminosity is measured in the $[0.1-2.4\\, {\\rm keV}]$ band.", "The chosen normalizations are those of [53].", "In eq.", "(REF ), $\\gamma _Y$ and $\\kappa _Y$ parametrize any possible redshift dependence due to observational biases on the measured cluster X-ray and TSZ magnitudes.", "If there were no such systematics, $(\\kappa _Y,\\gamma _Y)=(0,0)$ and $(A,B)=(-4,1)$ .", "In eq.", "(), X-ray luminosity and Comptonization parameter depend on electron density and temperature differently.", "In the self-similar model $(\\kappa _L,\\gamma _L)=(-7/3,-2/3)$ and $B=5/4$ while $A$ is determined observationally.", "In this equation, the X-ray luminosity normalization roughly corresponds to the mean X-ray luminosity of our sample.", "To simplify the analysis we will fix $(\\kappa ,\\gamma )$ to their self-similar values.", "To test the redshift evolution we will subdivide the sample in the redshift bins described above and we will compute $(A,B)$ for each subsample.", "As mentioned in the introduction, the predictions of the self-similar model do not coincide exactly with the observations.", "Deviations from the self-similar predictions are not unexpected.", "Since the concentration parameter that characterize the DM profile depends on mass [47], if the gas is in hydrostatic equilibrium within the DM potential well, one can expect that also the gas density and temperature profiles will deviate from the self-similarity assumed in the Kaiser model.", "If we parametrize $M_g\\sim M^{1+\\alpha _g}$ and $T\\sim M^{2/3+\\alpha _T}$ , then the scaling relations become: $L\\sim E(z)^{7/3}M^{4/3+2\\alpha _g+\\alpha _T/2}$ , $Y_C\\sim E(z)^{2/3}M^{5/3+\\alpha _g+\\alpha _T}$ [35].", "This would correspond to $B=(5/3+\\alpha _g+\\alpha _T)/(4/3+2\\alpha _g+\\alpha _T/2)$ in eq. ().", "Then, a deviation from the value $B=5/4$ would indicate to what extent the gas is better described by this extension of the self-similar model.", "Table: Average properties of full cluster sample and subsamples." ], [ "Fit methods.", "Linear fits are the most used regression algorithms; different statistical estimators can be used to determine the intercept $A$ , the slope $B$ and their respective uncertainties.", "If $(X_i, Y_i)$ are the data points and $(\\sigma _{X_i},\\sigma _{Y_i})$ their respective uncertainties, the commonly used least squares method is a biased estimator.", "As an alternative, [2] introduced the Bivariate Correlated Errors and intrinsic Scatter (BCES) that accounts for errors in both variables and their intrinsic scatter with respect to the regression line.", "If there are low-precision measurements that dominate over the other data points the method could be useless [74].", "For comparison we use a Maximum Likelihood Estimator (MLE) introduced by [31]Code downloaded from http://idlastro.gsfc.nasa.gov/ that also accounts for correlated errors in both variables and their intrinsic scatter.", "We computed the regression coefficients using both methods and found the results to be in excellent agreement.", "The MLE performs marginally better than BCES when the measurement errors and the intrinsic scatter are large.", "Then, we will quote only the results estimated using the MLE technique.", "When fitting a scaling relation, it is important to distinguish between the raw scatter, $\\sigma _{raw}$ , the dispersion around the best fit model, and the intrinsic scatter, $\\sigma _{int}$ , due to the differences on physical properties of clusters.", "The former is computed as the error weighted residual $\\sigma _{raw}^2= \\frac{1}{N_{cl}-2}\\Sigma _{i=1}^{N_{cl}} \\lambda _i(Y_i-BX_i-A)^2 ,$ where $N_{cl}$ is the number of clusters and the weights are given by $\\lambda _i= \\frac{N_{cl}\\sigma _i^{-2}}{\\Sigma _{i=1}^{N_{cl}} \\sigma _i^{-2}},\\qquad \\sigma _i^{2}=\\sigma _{Y_i}^{2}+A\\sigma _{X_i}^{2},$ while $\\sigma _{int}^2=\\sigma _{raw}^2-\\sigma _{stat}^2$ is the difference between the `raw' scatter and the statistical uncertainty ($\\sigma _{stat}^2=N_{cl}^{-1}\\Sigma _{i=0}^{N_{cl}} \\sigma _i^2$ ) obtained by propagating the error on the measured quantities.", "A simple estimator of the intrinsic error is given by adding in quadrature uniform values to the measured uncertainties of each individual cluster and finding the value for which $\\chi ^2$ per degree of freedom is equal to unity, i.e., $\\chi ^2=\\sum _i^{N_{cl}}(d_i-Y_i-BX_i-A)^2/(\\sigma _{raw,i}^2+\\sigma _{int}^2)\\equiv N_{cl}-2$ ([41], [17]).", "An alternative estimator has been used in [52] that differs from the one described previously on the fifth decimal place." ], [ "Error bars.", "For the $\\beta $ model, the errors on $Y_{C,Xray}$ are dominated by the uncertainties in $T_X$ and $n_{e,0}$ , while for the universal pressure profile the dominant uncertainty is that of $r_{500}$ and, consequently, of $M_{500}$ .", "The error on $n_{e,0}$ is negligible compared with the error on $T_X$ , a magnitude derived from a scaling relation.", "In the case of the universal pressure profile, we propagated the error on $M_{500}$ and added in quadrature the uncertainty $\\Delta _{par}$ due to the difference between the parameters of [6] and [54].", "We estimated $Y_{C,Xray}$ using both sets of parameters, denoted by subindices A and P, respectively, and we take this uncertainty to be the absolute value of their difference: $\\Delta _{par}=|Y_{C,Xray}^{P}-Y_{C,Xray}^{A}|$ .", "To estimate the errors on magnitudes derived from scaling relations we generated 10,000 realizations of $T_X$ and $r_{500}$ for each cluster, assuming that the errors on the parameters of their respective scaling relations were Gaussian distributed.", "On average, the relative error on $T_X$ was found to be $\\sim 8\\%$ , while in $r_{500}$ was $ \\sim 12\\%$ .", "The corresponding error on $M_{500}$ is $\\sim 35\\%$ .", "Since $\\Delta _{par}$ contributes to the total relative error budget with less than $10\\%$ , the final error on the universal profiles was dominated by the uncertainty on the mass estimates." ], [ "Effect of selection biases.", "Selection biases affect X-ray flux limited samples in two ways: Malmquist bias, due to higher luminosity clusters being preferentially selected out to higher redshifts and Eddington bias, due objects above the flux limit having above average luminosities for their mass as a result of the intrinsic or statistical scatter in their luminosity for any given mass.", "Scaling relations need to be corrected from these effects [27], [71], [51], [65], [76], [38], [43].", "To determine the selection biases we follow the method outlined in [18].", "We generated a sample of $5\\times 10^4$ halo objects out to $z\\le 0.3$ by sampling the number density of halos of a given mass given by [73].", "All relevant cosmological parameters were fixed to the Planck measured values [59].", "To each halo mass we assign two magnitudes $X=[Y_{C,X-ray},L_X]$ using scaling relations; the Comptonization parameter comes from the relation $Y_{C,X-ray,500}-M_{500}$ derived by [6] and the X-ray luminosity from the $L_X-M_{500}$ relation given in [37].", "Both these relations are corrected from statistical biases.", "The properties of individual clusters differ from one another according to the measured intrinsic dispersion.", "Then, we imposed the same flux cut than in the data and selected a sample of 560 clusters with the same redshift distribution than in the actual catalog.", "We repeat the process till three hundred samples have been selected.", "The functional form of the scalings given in eqs.", "(REF , ) is $\\log (Y_{C,SZ}/Y_{C,SZ,0})= B\\dot{\\log }(X/X_0) + A$ .", "We assign a value of $\\log (X)$ from the scaling relations taking into account the intrinsic scatter.", "The coefficients $(A,B)$ are varied within predefined intervals.", "For the scaling relation of eq.", "(REF ), the normalization and slope were varied in the range $A=[-4.50, -3.50]$ and $B=[0.85, 1.45]$ while for the scaling of eq.", "() the range of variation was $A=[-4.50, -3.50]$ and $B=[1.00, 1.40]$ , respectively.", "We fit the scaling relation to the 300 subsamples of simulated clusters for each pair of grid points $(A,B)$ and we vary these coefficients in steps of 0.01.", "We use the same scaling relations to correct the selection biases determined at the $\\theta _X$ aperture.", "This aperture is slightly larger than $\\theta _{2500}$ .", "The scalings measured at the latter aperture are not very different from those at $\\theta _{500}$ [12] and are well within the range defined above.", "In each realization, coefficients are measured using the MLE estimator.", "Like in [53] slopes and amplitudes are adjusted until the mock observed samples match those recovered from the actual data.", "Then, the unbiased scaling relation is that of the parent population." ], [ "Results and Discussion.", "We have used a sample of 560 X-ray selected clusters and the 2013 foreground cleaned Planck Nominal maps to determine two scaling relations, $Y_{C,SZ}-Y_{C,Xray}$ and $Y_{C,SZ}-L_X$ , using the BCES(Y$|$ X) and the MLE regression methods but the differences are below 1% and only the results from the latter method will be quoted.", "The uncertainties were determined by 10,000 bootstrap re-samplings [2], [31].", "Table: Scaling relations with MLE regression coefficients (eqs.", ",) corrected and not corrected from statistical biases." ], [ "The SZ-Xray scalings from pressure profiles.", "In Figs.", "REF and REF we represent the data and scaling parameters of the $Y_{C,SZ}-Y_{C,Xray}$ relation.", "The Comptonization parameters were obtained by averaging the temperature anisotropies on discs on angular size $\\theta _X$ (Fig.", "REF ) and $\\theta _{500}$ (Fig.", "REF ).", "$Y_{C,SZ}$ was measured directly on the foreground cleaned Planck Nominal maps while $Y_{C,Xray}$ was computed using the X-ray profiles described in Sec.", "2: $\\beta =2/3$ model and universal profile with [6] and [54] parameters.", "In the top three panels we plot the full data and its error bars.", "The solid red line corresponds to the best fit and the dashed lines to scaling relation with the parameters modified by $1\\sigma $ .", "The dotted line represents the scaling relation corrected of the statistical biases as described in Sec.", "REF .", "The cyan square shows the region of the parameter space occupied by the clusters used in [53].", "In Table REF we give the best fit parameters and the raw and intrinsic errors for the full catalog.", "The intrinsic scatter is always smaller but similar to the raw scatter, demonstrating that the uncertainties on the scaling relations are due to the physical differences within the cluster population and not to the statistical uncertainties on the measured magnitudes.", "For comparison we give the parameters of the scaling relation derived from the data and corrected from the statistical biases as indicated in Sec REF .", "The raw and intrinsic errors are identical (differences are seen only at the fourth decimal place) since, by construction, the mock catalogs of simulated clusters used to correct for selection biases had the same dispersion than the data.", "At the $\\theta _X$ aperture, the bias corrected relation shows that the intercept and slope of the $\\beta =2/3$ model are $A=-3.99\\pm 0.04$ and $B=1.11\\pm 0.04$ , deviating by about $2.7\\sigma $ from the expected values of $A=-4$ and $B=1$ .", "The deviations are larger when the bias corrected relations of the universal profile are used.", "Evaluating magnitudes at the $\\theta _{500}$ aperture, the intercept from the $\\beta =2/3$ model deviates by a $2.5\\sigma $ but the slope is compatible with $B=-1$ while the universal profile predict slopes that deviate by more than $4\\sigma $ but the intercepts are closer to the expected value.", "To quantify these deviations, we compute the mean Comptonization parameter weighted by the angular extent of each cluster, $\\bar{Y}_C=\\sum _i(Y_{C,i}\\theta _i^2)/\\sum _i\\theta _i^2$ .", "At $\\theta _X$ and $\\theta _{500}$ , this average is $\\bar{Y}_{C,Xray}=(1.04,1.17)\\bar{Y}_{C,SZ}$ for the $\\beta $ model and is $\\bar{Y}_{C,Xray}=(1.15,1.21)\\bar{Y}_{C,SZ}$ for the universal profile, respectively.", "Then, on average the $\\beta =2/3$ model correctly predicts the TSZ amplitude at $\\theta _X$ (4% excess) but overpredicts it beyond this radius (17% excess) as already demonstrated by [7].", "The universal profile overpredicts the signal by a 21% at $\\theta _{500}$ , contrary to earlier findings by the Planck Collaboration [53].", "Similar results were obtained using a universal profile with the [54] parameters.", "We can verify if the discrepancy between the prediction of the universal pressure profile and the measured anisotropy is related to the cooling process of the ICM by dividing clusters into cool-core and non cool-core systems.", "We adopt the [50] classification: CC clusters are those with central cooling times below the Hubble time ($t_H$ ).", "We used the central electron densities, X-ray temperatures listed in our catalog and the definition of cooling time $t_{cool}=8.5\\times 10^{10}\\rm {yr}\\left(n_e/10^{-3}\\rm {cm}^{-3}\\right)^{-1}\\left(T_X/10^8\\rm {K}\\right)^{1/2}$ by [69] to distinguish between CC and NCC clusters.", "The error on this cooling time was estimated by propagating the uncertainty of the X-ray temperature.", "In total, only 63 clusters in our sample were cool-core clusters.", "Reproducing the analysis, we found no significant difference with the results obtained using the full sample.", "When analyzing both subsamples separately, the results were similar in each subset; the only noticeable effect was the expected increment on the statistical uncertainty due to the smaller number of clusters, but the intrinsic scatter was still the largest component of the total scatter.", "Therefore, we can not ascribe the discrepancy to the presence of CC systems in our catalog.", "At $\\theta _{500}$ our result using the universal profile is $(A,B)=(-3.96\\pm 0.03, 1.20\\pm 0.05)$ , rather different from $(A,B)=(-3.91\\pm 0.01, 0.96\\pm 0.04)$ found by [53].", "To understand the source of this discrepancy, we repeat the analysis using only the 358 in the same mass and temperature range than the ones used by the Planck Collaboration, $M_{500}=[2 - 20]\\cdot 10^{14}\\rm {M}_\\odot $ and $T_X=[3-14]$ keV.", "For this subsample we obtain $(A,B)=(-3.99\\pm 0.03, 1.08\\pm 0.05)$ and the discrepancy is reduced to less than $3\\sigma $ .", "When averaging over the clusters angular extent, we obtain $\\bar{Y}_{C,Xray}=1.035\\bar{Y}_{C,SZ}$ at $\\theta _{500}$ , an excess of less than 4%.", "The intrinsic scatter in this subsample is also reduced from $\\sigma _{log_{int}}=0.476$ to $\\sigma _{log_{int}}=0.17$ , i.e.", "the dispersion around the best fit is much smaller, but it is still greater than the one in the Planck analysis, $\\sigma _{log_{int}}=0.10$ .", "The small difference with respect to the Planck result suggests that the universal pressure profile with a unique set of parameters describes well clusters with masses $\\mathrel {\\hbox{$>$}\\hbox{$\\sim $}}$ 1014M$ and X-ray temperatures $ 3$keV,comparable to the range analyzed by \\cite {arnaud2010}) whilefor less massive clusters the profile is not as accurate.This confirms the trend found by \\cite {lovisari_15} with a muchsmaller sample: the more massive systems had a shallower slopeand could be an indication that the $ LX-M$ relation is graduallysteepening when moving toward the low-mass objects suggesting thata simple power law cannot be used to convert the measured luminositiesinto masses.$ The diamonds and error bars in the middle and bottom panels of Figs.", "REF , REF show the value of intercept and slope $(A,B)$ in the five redshift bins described in Sec.", "(not corrected from statistical biases).", "The solid dotted and dashed line show the mean, 68% and 99% CL of the full sample.", "There is no clear trend in either parameter for the three pressure profiles considered.", "The value of the coefficients A and B at each redshift is always compatible with the fit of the full sample, indicating that the scaling does not evolve with time and confirming the results of [53].", "Nevertheless, we can not disregard the importance of statistical biasing effects since in the high redshift bins massive clusters are better represented compared with the low mass ones.", "Unfortunately, our redshift bins contain few clusters and the error bars on the measured values are consequently large so no statistical analysis could be made." ], [ "The $Y_{C,SZ}$ -Xray luminosity relation.", "Equally important is the scaling relation of the Comptonization parameter with X-ray luminosity since we can test the extensions of the self-similar model described in Sec.", "REF .", "In this case, we limit our study to $\\theta _X$ , the aperture at which the X-ray magnitudes have been measured.", "In Fig.", "REF we present the data, the best fit (solid line), the best corrected from statistical biases (dotted line) and the $1\\sigma $ deviations from the best fit.", "We quote results assuming a 10% uncertainty in the X-ray luminosity of all clusters, but the results remain unchanged if we conservatively increase the relative errors to 20%.", "The slope and intercept are given in Table REF .", "The measured correlation parameter corrected from statistical biases is $B=1.25\\pm 0.04$ , fully compatible with the self-similar value $B=5/4$ .", "The deviations from self-similarity are compatible with zero: if we assume $\\alpha _T=0$ then $\\alpha _g\\simeq 0.0\\pm 0.03$ while if $\\alpha _g=0$ then $\\alpha _T\\simeq 0.0\\pm 0.15$ .", "Our value is not directly comparable to $\\alpha _g=0.13\\pm 0.03$ found by [36] using 94 clusters in the range $z=0-0.6$ since magnitudes were evaluated at $\\theta _{500}$ .", "In [52] a similar analysis was carried out using a catalog of $\\sim 1600$ clusters up to redshift $z\\sim 1$ .", "For this sample, the measured value was $B\\simeq 1.095\\pm 0.025$ .", "Again, the value is not directly comparable to ours since X-ray luminosities are measured at $\\theta _{500}$ and the data was binned in luminosity to reduce errors.", "In Fig.", "REF also we present the parameters measured at different redshift using the same notation as in Figs.", "REF and REF .", "Like before, we found no indication of redshift evolution, confirming the results of [42] and [53].", "However, we have to consider that our redshift range could be too small to be sensitive to any hypothetical evolution.", "For instance, [41] used a sample of 115 clusters at redshifts $z=0.1-1.3$ to carry out his analysis.", "Computing the cluster mass from a $M-T_X$ relation and using a universal gas to dark matter fraction, he estimated the Comptonization parameter for each cluster and determined the $Y_C-L_X$ relation for different apertures.", "Since $Y_C$ was not measured from CMB temperature anisotropy maps, his results are not directly comparable with ours.", "Nevertheless, he systematically finds the exponent to be $B\\le 1.1$ .", "Since his cluster sample extends to a greater redshift range than the one used here, it would be interesting to test if his results, when compared to ours, are an indication of the time evolution of the scaling parameters." ], [ "Conclusions.", "We have fitted scaling relations between the Comptonization parameter predicted using X-ray data and the measured X-ray luminosity with the one measured from foreground cleaned Planck 2013 Nominal maps using a sample of 560 X-ray selected clusters.", "Prediction and measurement are compared at two angular scales, $\\theta _X$ that correspond to the region that emits 99% of the X-ray flux and $\\theta _{500}$ , the scale at which the cluster density is 500 times the critical density.", "Our catalog contains cluster and rich groups with masses $M_{500}\\mathrel {\\hbox{$>$}\\hbox{$\\sim $}}$ 1013M$, one order of magnitude below the massrange explored by the Planck Collaboration.", "We found that the Comptonizationparameter $ YC,Xray$ predicted using the $ =2/3$ model agrees withthe measured value within $ X$ but overestimates it at $ 500$indicating that clusters are not isothermal.", "We also show that theUniversal profile with either \\cite {arnaud2010} or \\cite {planck_int_05} parametersoverestimates the SZ anisotropy.", "Averaged over our cluster sample, wefind an excess ranging from 15\\% to 21\\% depending on the aperture andthe set of parameters used.", "This is slightly lower than the 30\\%required to explain the TSZ power determined by WMAP \\cite {komatsu2011}.We verified that the discrepancy is not due to the presence ofCC clusters.", "The discrepancy is greatly reduced ifthe analysis is restricted to clusters with $ M5001014M$,suggesting that the low and high mass systems are not well described by auniversal pressure profile with the same set of parameters, an indicationthat the dynamical evolution of baryons in low and high mass systemswas different.", "This conclusion supports the \\cite {lovisari_15} suggestionof a brake in the $ LX-M$ relation after correcting for selectionbiases.", "Then a simple power law can not be used to describe both clustersand groups and to translate X-ray luminosities into masses.Finally, we have also shown that the relation of the Comptonization parameteraveraged over the region that emits 99\\% of the X-ray fluxand the X-ray luminosity are consistent with the prediction of the self-similarmodel.$ We found that the scaling relation $Y_{C,SZ}-L_X$ is consistent with the self-similar model described in Sec.", "REF .", "If the temperature scales with mass as in the self-similar model ($\\alpha _T=0$ ) then deviations of the scaling of the gas mass with the total mass are compatible with zero ($\\alpha _g=0.0\\pm 0.03$ ) and is in tension with the value of $\\alpha _g=0.13\\pm 0.03$ measured by [36], although the two results are not directly comparable since $\\alpha _g$ was determined from scaling relations measured at different apertures.", "We tested the redshift evolution by dividing the catalog in five redshift subsamples.", "We found no evidence of evolution within $z<0.3$ in the two scaling relations analyzed.", "A comparison with the earlier results of [41] is not straightforward since this author did not obtain $Y_{C,SZ}$ from CMB data but used scaling relations.", "His sample of 115 clusters includes systems with $z=0.1-1.3$ and he finds the exponent to be $B\\le 1.1$ , depending on the angular scale used.", "The difference is significant enough to suggest that the scaling relations could be evolving in time but our sample is not deep enough to be sensitive to the effect.", "This question will require a separate study." ], [ "Acknowledgments", "We warmly thank H. Ebeling and D. Kocevski for sharing with us their cluster X-ray data and the referee for helping us to improve the paper.", "IDM acknowledges financial support from the University of the Basque Country UPV/EHU under the program \"Convocatoria de contratación para la especialización de personal investigador doctor en la UPV/EHU 2015\", and from the Spanish Ministry of Economy and Competitiveness through research project FIS2014-57956-P (comprising FEDER funds); FAB acknowledges financial support from the Spanish Ministerio de Educación y Ciencia (grant FIS2015-65140-P) and to the “Programa de Profesores Visitantes Severo Ochoa” of the Instituto de Astrofísica de Canarias." ] ]
1606.04983
[ [ "Incident-energy-dependent spectral weight of resonant inelastic x-ray\n scattering in doped cuprates" ], [ "Abstract We theoretically investigate the incident-photon energy omega_i dependence of resonant inelastic x-ray scattering (RIXS) tuned for the Cu L edge in cuprate superconductors by using the exact diagonalization technique for a single-band Hubbard model.", "Depending on the value of core-hole Coulomb interaction in the intermediate state, RIXS for non-spin-flip channel shows either a omega_i-dependent fluorescencelike or omega_i-independent Raman-like behavior for hole doping.", "An analysis of x-ray absorption suggests that the core-hole Coulomb interaction is larger than on-site Coulomb interaction in the Hubbard model, resulting in a fluorescencelike behavior in RIXS consistent with recent RIXS experiments.", "A shift on the high-energy side of the center of spectral distribution is also predicted for electron-doped systems though spectral weight is small.", "Main structures in spin-flip channel exhibit a Raman-like behavior as expected, accompanied with a fluorescencelike behavior with small intensity." ], [ "Introduction", "Resonant inelastic x-ray scattering (RIXS) experiments tuned for the Cu $L$ edge have provided a lot of new insights about spin dynamics of the spin-flip channel in cuprate superconductors when incident photon has the $\\pi $ polarization [1], [2].", "The non-spin-flip channel involving charge dynamics as well as two-magnon excitations can be predominantly detected by $\\sigma $ polarization for the incident photon.", "Not only polarization dependence but also incident-photon energy $\\omega _\\mathrm {i}$ dependence of the RIXS spectrum gives us useful information on the electronic states of cuprates.", "Experimentally it has been reported that the $\\omega _\\mathrm {i}$ dependence for the $\\pi $ -polarized incident photon gives a Raman-like behavior independent of $\\omega _\\mathrm {i}$ , while the $\\sigma $ polarization induces a fluorescencelike shift of spectral weight with increasing $\\omega _\\mathrm {i}$  [3], [4].", "Theoretically it has been pointed out that the $\\omega _\\mathrm {i}$ dependence is sensitive to particle-hole excitations of quasiparticles [5], [6] depending on band structures of materials [7].", "On the other hand, a model calculation for the doped-Mott insulator, i.e., a calculation of the single-band Hubbard model, has also captured Raman-like and fluorescencelike behaviors [4]: The Raman-like excitation comes from collective spin excitations, while the fluorescencelike shift is due to the continuum of particle-hole excitations.", "In the fluorescence of the normal x-ray emission spectroscopy (XES), there is no correlation between a photoexcited electron in the vacuum continuum and a valence electron decaying into the core hole so that the emission spectrum is almost independent of $\\omega _\\mathrm {i}$ .", "Then the energy loss of the x ray increases as $\\omega _\\mathrm {i}$ increases.", "In the present RIXS, however, the fluorescencelike behavior would come from a different origin [5], [6] since these electrons are rather correlated.", "The intermediate state of the RIXS process contains a core hole created by an incident photon.", "Therefore, the Coulomb interaction between the core hole and valence electron ($3d_{x^2-y2}$ electron in cuprates) acting in the intermediate state influences the RIXS spectrum.", "It is thus expected that the core-hole Coulomb interaction also affects the $\\omega _\\mathrm {i}$ dependence of the RIXS spectrum.", "In this paper, we perform the Lanczos-type exact diagonalization calculation of the RIXS spectrum for the single-band Hubbard model describing hole- and electron-doped cuprates.", "Examining the $\\omega _\\mathrm {i}$ dependence of non-spin-flip RIXS spectra, we find that the dependence is strongly affected by the value of core-hole Coulomb interaction in hole doping: A $\\omega _\\mathrm {i}$ -dependent fluorescencelike behavior appears when the core-hole Coulomb interaction is larger than the on-site Coulomb interaction in the Hubbard model, while a $\\omega _\\mathrm {i}$ -independent Raman-like behavior appears for smaller core-hole Coulomb interaction although the distribution of spectral weight depends on $\\omega _\\mathrm {i}$ .", "Analyzing main and satellite structures in x-ray absorption (XAS) for hole doping, we find that it is reasonable to take the core-hole Coulomb interaction larger than the on-site Coulomb interaction in modeling RIXS by a single-band model of cuprates.", "This suggests a fluorescence-like behavior in RIXS for the $\\sigma $ -polarized geometry detecting non-spin-flip excitations, being consistent with recent experiments [3], [4].", "In such a case, the dynamical charge structure factor is observed through $\\sigma $ -polarized RIXS by tuning $\\omega _\\mathrm {i}$ to a satellite structure in XAS for hole doping.", "Using the larger core-hole Coulomb interaction, we predict a shift on the high-energy side of the center of spectral distribution in electron doping, which has not yet been observed experimentally.", "In the spin-flip channel, main structures exhibit a Raman-like behavior as expected.", "In addition, there is a fluorescencelike behavior similar to the non-spin-flip channel, although spectral weight is very small.", "The behavior is enhanced with increasing hole carriers.", "A detailed experimental work to detect these behaviors is desired in the future.", "This paper is organized as follows.", "The Hubbard model and RIXS spectra decomposed into spin-flip and non-spin-flip channels are introduced in Sec. .", "In Sec.", ", we calculate the dependence of XAS on the core-hole Coulomb interaction for both hole and electron doping.", "The $\\omega _\\mathrm {i}$ dependence of RIXS spectra are shown in Sec.", "and the origin of fluorescencelike and Raman-like behaviors is discussed.", "Finally, a summary is given in Sec.", "." ], [ "Model and Method", "In order to describe $3d$ electrons in the CuO$_2$ plane, we take a single-band Hubbard model given by $H_\\mathrm {3d}=-t\\sum _{i\\delta \\sigma } c^\\dagger _{i\\sigma } c_{i+\\delta \\sigma } -t^{\\prime }\\sum _{i\\delta ^{\\prime }\\sigma } c^\\dagger _{i\\sigma } c_{i+\\delta ^{\\prime }\\sigma } + U\\sum _i n_{i\\uparrow }n_{i\\downarrow },$ where $c^\\dagger _{i\\sigma }$ is the creation operator of an electron with spin $\\sigma $ at site $i$ , number operator $n_{i\\sigma }=c^\\dagger _{i\\sigma }c_{i\\sigma }$ , $i+\\delta $ ($i+\\delta ^{\\prime }$ ) represents the four first (second) nearest-neighbor sites around site $i$ , and $t$ , $t^{\\prime }$ , and $U$ are the nearest-neighbor hopping, the next-nearest-neighbor hopping, and on-site Coulomb interaction, respectively.", "We take $U/t=10$ and $t^{\\prime }/t=-0.25$ , which are typical values appropriate for cuprates.", "Based on the Hubbard model, the XAS for the $L$ edge accompanied by the excitation of an electron from the core Cu$2p$ to Cu$3d$ orbital can be described by $I^\\mathrm {XAS}\\left(\\omega \\right) = -\\frac{1}{\\pi }\\mathrm {Im} \\left\\langle 0 \\right| \\sum _{l\\sigma } c_{l\\sigma }\\frac{1}{\\omega -H_l^j+E_0+i\\Gamma } c^\\dagger _{l\\sigma }\\left| 0 \\right\\rangle ,$ where $\\left|0 \\right\\rangle $ represents the ground state with energy $E_0$ ; $j$ is the total angular momentum of Cu$2p$ with either $j=1/2$ or $j=3/2$ ; $\\Gamma $ is the relaxation time of the core hole; and $H_l^j=H_{3d}-U_\\mathrm {c} \\sum _\\sigma n_{l\\sigma } + \\varepsilon _j$ with $U_\\mathrm {c}$ and $\\varepsilon _j$ being the Cu $2p$ -$3d$ Coulomb interaction and energy level of Cu $2p$ , respectively.", "Here, we assume the presence of a Cu$2p$ core hole at site $l$ .", "In RIXS, tuning polarization of incident and outgoing photons, we can separate excitation with the change of total spin by one ($\\Delta S=1$ ) and excitation with no change of total spin ($\\Delta S=0$ ) [8], [9], [10], [11].", "We call the former (the latter) spin-flip (non-spin-flip) process hereafter.", "The two excitations can be defined as $I^{\\Delta S=0}_\\mathbf {q} \\left(\\Delta \\omega \\right) &=& \\sum \\limits _f \\left| \\left\\langle f \\right|N^j_\\mathbf {q} \\left| 0 \\right\\rangle \\right|^2 \\delta \\left( \\Delta \\omega - E_f + E_0 \\right),\\\\I^{\\Delta S=1}_\\mathbf {q} \\left(\\Delta \\omega \\right) &=& \\sum \\limits _f \\left| \\left\\langle f \\right|S^j_\\mathbf {q} \\left| 0 \\right\\rangle \\right|^2 \\delta \\left( \\Delta \\omega - E_f + E_0 \\right),$ with $S^j_\\mathbf {q}=(B^j_{\\mathbf {q}\\uparrow \\uparrow }-B^j_{\\mathbf {q}\\downarrow \\downarrow })/2$ , $N^j_\\mathbf {q}=B^j_{\\mathbf {q}\\uparrow \\uparrow }+B^j_{\\mathbf {q}\\downarrow \\downarrow }$ , and $B^j_{\\mathbf {q}\\sigma ^{\\prime }\\sigma }=\\sum _l e^{-i\\mathbf {q}\\cdot \\mathbf {R}_l} c_{l\\sigma ^{\\prime }}\\frac{1}{\\omega _\\mathrm {i}-H_l^j+E_0+i\\Gamma } c^\\dagger _{l\\sigma },$ where $\\left|f \\right\\rangle $ represents the final state with energy $E_f$ ; and $\\mathbf {R}_l$ is the position vector at site $l$ .", "When $\\Gamma $ is much larger than the remaining terms in the denominator of (REF ), $S^j_\\mathbf {q}$ and $N^j_\\mathbf {q}$ reduce to $S^z_\\mathbf {q}=\\sum _l e^{-i\\mathbf {q}\\cdot \\mathbf {R}_l} S^z_l$ and $N_\\mathbf {q}=\\sum _l e^{-i\\mathbf {q}\\cdot \\mathbf {R}_l} N_l$ , respectively, with the $z$ component of the spin operator $S^z_l$ and electron-number operator $N_l$ (the first-collision approximation).", "In this approximation, (REF ) and () read the dynamical charge structure factor, $N(\\mathbf {q},\\omega )=\\sum \\limits _f \\left| \\left\\langle f \\right|N_\\mathbf {q} \\left| 0 \\right\\rangle \\right|^2 \\delta \\left( \\omega - E_f + E_0 \\right), $ and the dynamical spin structure factor, $S(\\mathbf {q},\\omega )=\\sum \\limits _f \\left| \\left\\langle f \\right|S^z_\\mathbf {q} \\left| 0 \\right\\rangle \\right|^2 \\delta \\left( \\omega - E_f + E_0 \\right), $ respectively.", "When $\\Gamma $ is comparable with the remaining terms in (REF ), intersite operators emerge as effective operators of (REF ) in addition to the on-site charge $N_i$ and spin $S_i^z$  [12].", "In the non-spin-flip process, a two-magnon-type operator is easily expected to contribute to $N^j_\\mathbf {q}$ .", "We thus define the $\\mathbf {q}$ -dependent dynamical two-magnon correlation function, $M(\\mathbf {q},\\omega )=\\sum \\limits _f \\left| \\left\\langle f \\right|M_\\mathbf {q}^\\pm \\left| 0 \\right\\rangle \\right|^2 \\delta \\left( \\omega - E_f + E_0 \\right), $ with $M_\\mathbf {q}^\\pm =\\sum _\\mathbf {k} \\left(\\cos k_x \\pm \\cos k_y \\right) \\mathbf {S}_{\\mathbf {k}+\\mathbf {q}}\\cdot \\mathbf {S}_\\mathbf {-k}$ , where $+$ ($-$ ) corresponds to A$_{1\\mathrm {g}}$ (B$_{1\\mathrm {g}}$ ) representation.", "In order to calculate Eqs.", "(REF ), (), (REF ), (REF ), and (REF ), we use a Lanczos-type exact diagonalization technique on a $\\sqrt{18}\\times \\sqrt{18}$ cluster under periodic boundary conditions.", "We consider hole and electron doping with the carrier concentration $x=n/18$ , corresponding to $18-n$ electrons and $18+n$ electrons in the cluster for hole and electron doping, respectively.", "We set $\\Gamma =t$ for both XAS and RIXS." ], [ "x-ray absorption spectrum (XAS)", "The value of $U_\\mathrm {c}$ used in the literature is ranged from $\\sim U/2$  [12] to $\\sim 3U/2$  [11] in describing $L$ -edge RIXS for cuprates.", "In Fig.", "REF , the $U_\\mathrm {c}$ dependence of XAS spectrum $I^\\mathrm {XAS}\\left(\\omega \\right)$ is shown for $x=0.11$ .", "In electron doping, the XAS spectra exhibit a single peak independent of $U_\\mathrm {c}$ , located near $\\omega \\sim \\varepsilon _j+U-2U_\\mathrm {c}$ .", "This is reasonable since kicking an electron from Cu$2p$ to Cu$3d$ creates a doubly occupied state at the core-hole site, similar to half filling as long as $U/2<U_\\mathrm {c}$ .", "This process is denoted as $\\tilde{d}^1\\rightarrow \\underline{\\tilde{c}}\\tilde{d}^2$ , where $\\tilde{d}^{1(2)}$ and $\\underline{\\tilde{c}}$ represent a singly (doubly) occupied state and a core hole, respectively.", "We note that an electron carrier cannot contribute to XAS because the carrier already creates the $\\tilde{d}^2$ state.", "Figure: The core-hole potential U c U_\\mathrm {c} dependence of XAS spectrum I XAS ωI^\\mathrm {XAS}\\left(\\omega \\right) for the 18×18\\sqrt{18}\\times \\sqrt{18} periodic cluster of the Hubbard model.", "(a) Electron doping and (b) hole doping with x=2/18∼0.11x=2/18\\sim 0.11.", "U=10tU=10t and t ' =-0.25tt^{\\prime }=-0.25t.", "The value of U c U_\\mathrm {c} is changed in the range of U/2≥U c ≥3U/2U/2\\ge U_\\mathrm {c} \\ge 3U/2.", "The solid lines are obtained by a Lorenzian broadening with Γ=t\\Gamma =t for δ\\delta functions shown by bars.On the other hand, a clear $U_\\mathrm {c}$ dependence appears in hole doping accompanied with a two-peak structure as shown in Fig.", "REF (b).", "For $U_\\mathrm {c}=15t$ there are two peaks at $\\omega -\\varepsilon _j-U+2U_\\mathrm {c}=0.5t$ and $8t$ .", "One peak comes from the process $\\tilde{d}^1\\rightarrow \\underline{\\tilde{c}}\\tilde{d}^2$ same as the case of electron doping and its energy is given by $\\varepsilon _j+U-2U_\\mathrm {c}$ neglecting the hopping terms.", "The other peak is related to a singly occupied state at the core-hole site, where a hole carrier is annihilated by an electron from core Cu$2p$ as denoted by $\\tilde{d}^0\\rightarrow \\underline{\\tilde{c}}\\tilde{d}^1$ and its peak position is higher in energy by $U_\\mathrm {c}-U$ (neglecting the hopping) as compared with the $\\underline{\\tilde{c}}\\tilde{d}^2$ peak.", "With decreasing $U_\\mathrm {c}$ in Fig.", "REF (b), the $\\underline{\\tilde{c}}\\tilde{d}^1$ peak shifts to lower energy and overlaps with the $\\underline{\\tilde{c}}\\tilde{d}^2$ peak near $U_\\mathrm {c}\\sim U=10t$ .", "With further decrease, the $\\underline{\\tilde{c}}\\tilde{d}^1$ peak is lower in energy than the $\\underline{\\tilde{c}}\\tilde{d}^2$ peak: The former peak is at $\\omega -\\varepsilon _j-U+2U_\\mathrm {c}=-2.5t$ and the latter one is at $0.5t$ for $U_\\mathrm {c}=5$ .", "These results suggest that the $U_\\mathrm {c}$ dependence of RIXS spectra is strong in hole doping.", "The Cu $L_3$ -edge XAS in hole-doped cuprates has detected two structures near 930 eV [13], [14]: One is a large peak coming from the process of $3d^9\\rightarrow \\underline{2p}3d^{10}$ ($\\underline{2p}=$ Cu$2p$ core hole) and the other is a weak structure related to carrier concentration [15] given by the process of $3d^9\\underline{L}\\rightarrow \\underline{2p}3d^{10}\\underline{L}$ ($\\underline{L}=$ oxygen ligand hole).", "The latter structure is higher by roughly 1.5 eV in energy than the former one [13], [14], [15].", "By using Cu-O clusters with appropriate parameters, the two structures have been obtained in hole doping [16].", "A two-band CuO model used in RIXS calculations [11] also reproduced a two-structure behavior similar to the experiments (not shown).", "In comparing the experimental data with the XAS in the single-band Hubbard model, it would be reasonable to assign $3d^9\\rightarrow \\underline{2p}3d^{10}$ ($3d^9\\underline{L}\\rightarrow \\underline{2p}3d^{10}\\underline{L}$ ) to $\\tilde{d}^1\\rightarrow \\underline{\\tilde{c}}\\tilde{d}^2$ ($\\tilde{d}^0\\rightarrow \\underline{\\tilde{c}}\\tilde{d}^1$ ) by taking into account the correspondence between the Zhang-Rice singlet state ($3d^9\\underline{L}$ ) and the lower-Hubbard-band state ($\\tilde{d}^0$ ).", "Taking a standard value of $t=0.35$  eV, we can find that the energy separation of the two structures in hole-doped XAS shown in Fig.", "REF (b) becomes an experimental value (1.5 eV) when $U_\\mathrm {c}\\sim 12t$ .", "This means that $U_\\mathrm {c}>U=10t$ .", "However, a smaller value of $U_\\mathrm {c}$ satisfying $U_\\mathrm {c}<U$ has been used in the literature [4], [12].", "Therefore, it is interesting to examine the effect of $U_\\mathrm {c}$ on the $\\omega _\\mathrm {i}$ dependence of the RIXS spectrum.", "In the following, keeping $U=10t$ , we use $U_\\mathrm {c}=12t$ and $U_\\mathrm {c}=8t$ as a representative parameter for $U_\\mathrm {c}>U$ and $U_\\mathrm {c}<U$ , respectively." ], [ "Hole doping", "The incident-photon energy $\\omega _\\mathrm {i}$ dependence of non-spin-flip intensity $I^{\\Delta S=0}_\\mathbf {q}$ at $\\mathbf {q}=(2\\pi /3,0)$ in hole doping with $x=0.11$ is shown for the case of $U_\\mathrm {c}=12t>U$ in Fig.", "REF (a).", "The bottom spectrum exhibits the spectrum obtained by tuning $\\omega _\\mathrm {i}$ at the edge of the lowest-energy peak in the XAS spectrum shown in Fig.", "REF (b).", "Here, the edge is defined as the lowest-energy eigenstate represented by the lowest-energy vertical bar in Fig.", "REF (b).", "There are two peaks at $\\omega =1.8t$ and $\\omega =0.2t$ together with a broad tail around $\\omega =3t$ .", "The peak at $\\omega =1.8t$ is similar to that at half filling, which is originated from two-magnon excitation.", "We note that in this energy region there are also charge excitations coming from hole carrier as seen from the dynamical charge structure factor $N(\\mathbf {q},\\omega )$ in Fig.", "REF (c).", "On the other hand, the peak at $\\omega =0.2t$ does not have a corresponding structure in $N(\\mathbf {q},\\omega )$ , indicating that the peak is related to an excitation beyond a simple two-particle response [12].", "Actually, we find that the peak at $\\omega =0.2t$ appears in $M(\\mathbf {q},\\omega )$ with B$_{1\\mathrm {g}}$ , as shown in Fig.", "REF (d).", "The broad tail around $\\omega =3t$ may partly come from charge excitations, but spectral shape is different from $N(\\mathbf {q},\\omega )$ .", "This is reasonable since the edge of XAS is mainly composed of the $\\underline{\\tilde{c}}\\tilde{d}^2$ state to which hole carriers do not contribute.", "Figure: (Color online) The incident-phonon energy ω i \\omega _\\mathrm {i} dependence of the non-spin-flip spectrum I 𝐪 ΔS=0 I^{\\Delta S=0}_\\mathbf {q} at 𝐪=(2π/3,0)\\mathbf {q}=(2\\pi /3,0) for the hole-doped 18×18\\sqrt{18}\\times \\sqrt{18} Hubbard cluster with x=2/18∼0.11x=2/18\\sim 0.11.", "(a) U c =12tU_\\mathrm {c}=12t and (b) U c =8tU_\\mathrm {c}=8t.Parameters are U=10tU=10t, t ' =-0.25tt^{\\prime }=-0.25t, and Γ=t\\Gamma =t.ω i \\omega _\\mathrm {i} at the bottom panel is set to the edge of the XAS spectrum and increases by tt from bottom to top.", "(c) The dynamical charge structure factor N(𝐪,ω)N(\\mathbf {q},\\omega ) at 𝐪=(2π/3,0)\\mathbf {q}=(2\\pi /3,0).", "(d) The 𝐪\\mathbf {q}-dependent dynamical two-magnon correlation function M(𝐪,ω)M(\\mathbf {q},\\omega ) at 𝐪=(2π/3,0)\\mathbf {q}=(2\\pi /3,0).", "Black (red) line, B 1g _{1\\mathrm {g}} (A 1g _{1\\mathrm {g}}) mode.Figure: (Color online) Same as Fig.", "but x=4/18∼0.22x=4/18\\sim 0.22.When $\\omega _\\mathrm {i}$ is increased by $t$ from the edge, $I^{\\Delta S=0}_\\mathbf {q}$ increases as shown in Fig.", "REF (a).", "This is simply due to the fact that the peak of the XAS is located slightly above the edge position and thus resonance becomes maximum when $\\omega _\\mathrm {i}$ is larger than the edge energy.", "With further increasing $\\omega _\\mathrm {i}$ , spectral weight gradually decreases as the resonance condition becomes weaker.", "Furthermore, the distribution of spectral weight becomes wider, accompanied by the reduction of peak heights.", "When $\\omega _\\mathrm {i}$ is higher than the edge position by $5t$ , where the incident photon resonates to a satellite structure in XAS, spectral distribution becomes similar to $N(\\mathbf {q},\\omega )$ .", "This is again reasonable since the incident photon resonates mainly to the $\\underline{\\tilde{c}}\\tilde{d}^1$ state [see Fig.", "REF (b)] and thus hole carriers contribute to the RIXS process through the intermediate state.", "As a consequence, a characteristic structure in the spectral weight gradually changes from the low-energy peak near $\\Delta \\omega =2t$ to the high-energy broad peak near $\\Delta \\omega =4t$ with increasing $\\omega _\\mathrm {i}$ .", "This evolution of $I^{\\Delta S=0}_\\mathbf {q}$ with $\\omega _\\mathrm {i}$ exhibits a fluorescencelike $\\omega _\\mathrm {i}$ dependence: The energy position of main spectral weight increases with $\\omega _\\mathrm {i}$ .", "Such dependence is due to the fact that particle-hole intraband excitations can couple to the $\\underline{\\tilde{c}}\\tilde{d}^1$ state extended above the absorption edge in XAS.", "Note that the amount of the energy shift is not the same as the increase of $\\omega _\\mathrm {i}$ , since a photoexcited electron is no longer independent of other electrons unlike the case of the normal XES.", "In contrast to the $U_\\mathrm {c}=12t$ case, the $\\omega _\\mathrm {i}$ dependence for $U_\\mathrm {c}=8t<U$ shows less dispersive spectral distribution as seen in Fig.", "REF (b).", "When $\\omega _\\mathrm {i}$ is tuned to the edge of XAS, there is a peak structure at $\\omega =4t$ , which is originated from the charge excitation shown in Fig.", "REF (c).", "This structure remains with increasing $\\omega _\\mathrm {i}$ , resulting in a Raman-like $\\omega _\\mathrm {i}$ dependence.", "This behavior in sharp contrast with the $U_\\mathrm {c}=12t$ case can be attributed to the difference of intermediate states.", "The main peak in XAS [Fig.", "REF (b)] shows a similar single-peak structure for both the $U_\\mathrm {c}=8t$ and $U_\\mathrm {c}=12t$ cases.", "However, in contrast to the $U_\\mathrm {c}=12t$ case, not only a doubly occupied state $\\underline{\\tilde{c}}\\tilde{d}^2$ but also a singly occupied state $\\underline{\\tilde{c}}\\tilde{d}^1$ contributes to the main peak for $U_\\mathrm {c}=8t$ as discussed above.", "The singly occupied state can couple to particle-hole excitations in the final state of RIXS, resulting in a wide energy spectral distribution independent of the value of $\\omega _\\mathrm {i}$ .", "The fluorescencelike $\\omega _\\mathrm {i}$ dependence at $U_\\mathrm {c}=12t$ becomes more clear when hole carriers are increased.", "Figure REF shows $I^{\\Delta S=0}_\\mathbf {q}$ , $N(\\mathbf {q},\\omega )$ , and $M(\\mathbf {q},\\omega )$ at $\\mathbf {q}=(2\\pi /3,0)$ for $x=0.22$ .", "As for $x=0.11$ , low-energy excitations below $\\Delta \\omega \\sim 2t$ in $I_\\mathbf {q}^{\\Delta S=0}$ at the absorption edge [see Figs.", "REF (a) and REF (b)] are partly contributed by two-magnon excitations as expected from $M(\\mathbf {q},\\omega )$ shown in Fig.", "REF (d).", "It is clear in Fig.", "REF (a) that, with increasing $\\omega _\\mathrm {i}$ from the edge position, the spectral distribution smoothly shifts to the high-energy side accompanied by the reduction of low-energy weight similar to the $x=0.11$ case and eventually resembles $N(\\mathbf {q},\\omega )$ shown in Fig.", "REF (c).", "This smooth shift indicates that the fluorescence-like behavior can be seen more clearly in the overdoped region.", "On the other hand, $\\omega _\\mathrm {i}$ dependence at $U_\\mathrm {c}=8t$ exhibits a Raman-like behavior consistent with the case of $x=0.11$ .", "We note that these $\\omega _\\mathrm {i}$ dependencies mentioned above are seen at not only $\\mathbf {q}=(2\\pi /3,0)$ but also $\\mathbf {q}=(\\pi /3,\\pi /3)$ defined in the $\\sqrt{18}\\times \\sqrt{18}$ cluster (not shown).", "The $\\omega _\\mathrm {i}$ dependence of the RIXS spectrum has been measured in YBa$_2$ Cu$_3$ O$_{6+x}$  [3].", "In the experiment, the $\\sigma $ polarization of the incident photon approximately correspond to non-spin-flip spectrum $I^{\\Delta S=0}_\\mathbf {q}$ .", "Comparing the calculated $I^{\\Delta S=0}_{\\mathbf {q}=(2\\pi /3,0)}$ in Figs.", "REF and REF with the experimental data at $\\mathbf {q}=(0.74\\pi ,0)$ for the overdoped sample [3], we find that the $U_\\mathrm {c}=12t$ case corresponds to the experiment, since there is a fluorescencelike behavior.", "Encouraged by the correspondence, we use $U_\\mathrm {c}=12t$ in the following as a representative value for cuprates.", "Figure: (Color online) (a) The incident-phonon energy ω i \\omega _\\mathrm {i} dependence of the non-spin-flip spectrum I 𝐪 ΔS=0 I^{\\Delta S=0}_\\mathbf {q} at 𝐪=(2π/3,0)\\mathbf {q}=(2\\pi /3,0) for the electron-doped 18×18\\sqrt{18}\\times \\sqrt{18} Hubbard cluster with x=2/18∼0.11x=2/18\\sim 0.11.Parameters are U=10tU=10t, t ' =-0.25tt^{\\prime }=-0.25t, U c =12tU_\\mathrm {c}=12t, and Γ=t\\Gamma =t.ω i \\omega _\\mathrm {i} at the bottom panel is set to the edge of the XAS spectrum and increases by tt from bottom to top.", "(b) The dynamical charge structure factor N(𝐪,ω)N(\\mathbf {q},\\omega ) at 𝐪=(2π/3,0)\\mathbf {q}=(2\\pi /3,0).", "(c) The 𝐪\\mathbf {q}-dependent dynamical two-magnon correlation function M(𝐪,ω)M(\\mathbf {q},\\omega ) at 𝐪=(2π/3,0)\\mathbf {q}=(2\\pi /3,0).Black (red) line, B 1g _{1\\mathrm {g}} (A 1g _{1\\mathrm {g}}) mode." ], [ "Electron doping", "Figure REF shows the $\\omega _\\mathrm {i}$ dependence of the non-spin-flip spectrum $I^{\\Delta S=0}_\\mathbf {q}$ at $\\mathbf {q}=(2\\pi /3,0)$ for electron doping.", "When $\\omega _\\mathrm {i}$ is tuned to the edge, there are two main structures at $\\Delta \\omega =1.2t$ and $2.5t$ .", "The latter is partly contributed from charge excitations due to electron carriers as expected from $N(\\mathbf {q},\\omega )$ shown in Fig.", "REF (b).", "In $N(\\mathbf {q},\\omega )$ , however, there is no corresponding structure for the former structure, indicating a possible two-magnon origin.", "In fact, $M(\\mathbf {q},\\omega )$ shown in Fig.", "REF (c) exhibits an enhancement at the former energy.", "We further notice that there is no prominent structure at $\\omega =6t$ where $N(\\mathbf {q},\\omega )$ shows a peak structure.", "With increasing $\\omega _\\mathrm {i}$ , the two main structures gradually lose their weight and a broad structure near $\\Delta \\omega =3t$ remains accompanied with an small enhancement around $\\Delta \\omega =6t$ .", "The spectral distribution, for example, for $\\omega _\\mathrm {i}$ higher by $\\sim 3t$ from the edge, resembles $N(\\mathbf {q},\\omega )$ .", "With further increasing $\\omega _\\mathrm {i}$ , the center of spectral distribution shifts to the high-energy side, although the spectral weight is strongly reduced as compared with the case of hole doping.", "We note that the origin of such a fluorescencelike behavior is different from the case of hole doping.", "In hole doping, the spectral weight similar to $N(\\mathbf {q},\\omega )$ at high $\\omega _\\mathrm {i}$ is constructed by hole carriers through the $\\underline{\\tilde{c}}\\tilde{d}^1$ state.", "On the other hand, in electron doping there is no direct contribution from electron carriers in the intermediate state in RIXS, since the carriers already make an inactive $\\tilde{d}^2$ states for XAS.", "Therefore, the spectral weight similar to $N(\\mathbf {q},\\omega )$ in electron doping may be due to an indirect effect of electron carriers appearing in the XAS spectra as a broad structure with small spectral weight above the main peak as seen in Fig.", "REF (a).", "This is the reason for weak fluorescencelike intensity in electron doping.", "Detailed examinations of $\\omega _\\mathrm {i}$ dependence in the $\\sigma $ -polarized RIXS is desired to confirm theoretical predictions for electron doping.", "Figure: The incident-phonon energy ω i \\omega _\\mathrm {i} dependence of the spin-flip spectrum I 𝐪 ΔS=1 I^{\\Delta S=1}_\\mathbf {q} at 𝐪=(2π/3,0)\\mathbf {q}=(2\\pi /3,0) for the 18×18\\sqrt{18}\\times \\sqrt{18} Hubbard cluster with x=2/18∼0.11x=2/18\\sim 0.11 for (a) electron doping and (c) hole doping.Parameters are U=10tU=10t, t ' =-0.25tt^{\\prime }=-0.25t, U c =15tU_\\mathrm {c}=15t, and Γ=t\\Gamma =t.ω i \\omega _\\mathrm {i} at the bottom panel is set to the edge of the XAS spectrum and increases by tt from bottom to top.The dynamical spin structure factor S(𝐪,ω)S(\\mathbf {q},\\omega ) at 𝐪=(2π/3,0)\\mathbf {q}=(2\\pi /3,0) for (b) electron doping and (d) hole doping.Figure: (Color online) The contour plot of incident-phonon energy ω i \\omega _\\mathrm {i} dependence of spin-flip spectra I 𝐪 ΔS=1 I^{\\Delta S=1}_\\mathbf {q} and non-spin-flip spectra I 𝐪 ΔS=0 I^{\\Delta S=0}_\\mathbf {q} at 𝐪=(2π/3,0)\\mathbf {q}=(2\\pi /3,0) for the hole-doped 18×18\\sqrt{18}\\times \\sqrt{18} Hubbard cluster with parameters U=10tU=10t, t ' =-0.25tt^{\\prime }=-0.25t, U c =12tU_\\mathrm {c}=12t and Γ=t\\Gamma =t.", "(a) x=0x=0, (b) x=2/18∼0.11x=2/18\\sim 0.11, and (c) x=4/18∼0.22x=4/18\\sim 0.22 for I 𝐪 ΔS=1 I^{\\Delta S=1}_\\mathbf {q}.", "(d) x=0x=0, (e) x∼0.11x\\sim 0.11, and (f) x∼0.22x\\sim 0.22 for I 𝐪 ΔS=0 I^{\\Delta S=0}_\\mathbf {q}.Since particle-hole excitations are involved in not only the non-spin-flip channel but also the spin-flip one, it is interesting to examine how the spin-flip spectrum $I^{\\Delta S=1}_\\mathbf {q}$ is dependent on $\\omega _\\mathrm {i}$ .", "Figure REF shows $I^{\\Delta S=1}_\\mathbf {q}$ at $\\mathbf {q}=(2\\pi /3,0)$ for both electron and hole dopings.", "Comparing Figs.", "REF (a) and REF (b) for electron doping and Figs.", "REF (c) and REF (d) for hole doping, we find that $I^{\\Delta S=1}_\\mathbf {q}$ for $\\omega _\\mathrm {i}$ tuned to the absorption edge resembles $S(\\mathbf {q},\\omega )$ .", "This is in contrast to the case of $I^{\\Delta S=0}_\\mathbf {q}$ as discussed in Sec.", "REF  [12].", "With increasing $\\omega _\\mathrm {i}$ from the edge, low-energy spin-flip excitations below $\\omega =2t$ in both dopings remain showing a Raman-like behavior.", "In addition, there is a dispersive fluorescencelike structure, which is similar to the case of the non-spin-flip spectrum $I^{\\Delta S=0}_\\mathbf {q}$ , although the intensity is very small above $\\Delta \\omega =2t$ .", "This indicates the presence of particle-hole excitations in the spin-flip channel.", "In order to make clear the effect of hole carriers on the $\\omega _\\mathrm {i}$ dependence, we show the contour map of $I^{\\Delta S=1}_\\mathbf {q}$ at $\\mathbf {q}=(2\\pi /3,0)$ together with the non-spin flip $I^{\\Delta S=0}_\\mathbf {q}$ for the realistic value of $U_\\mathrm {c}=12t$ in Fig.", "REF .", "At half filling $x=0$ [Figs.", "REF (a) and REF (d)], both spectra exhibit Raman-like $\\omega _\\mathrm {i}$ dependence.", "$I^{\\Delta S=0}_\\mathbf {q}$ clearly shows fluorescencelike $\\omega _\\mathrm {i}$ dependence with increasing $x$ as expected.", "The spin-flip spectra in Figs.", "REF (b) and  REF (c) also show fluorescencelike $\\omega _\\mathrm {i}$ dependence in the same $\\Delta \\omega $ region as $I^{\\Delta S=0}_\\mathbf {q}$ , though its intensity is weak as compared with low-energy excitations.", "We can find that the fluorescencelike intensity increases with increasing $x$ , being consistent with the view that particle-hole excitations contribute to RIXS spectra in the spin-flip channel when carriers become more itinerant.", "A detailed examination of $\\omega _\\mathrm {i}$ dependence of RIXS for the $\\pi $ -polarized incident photon and different carrier concentrations is desired in the near future." ], [ "Summary", "Examining the $\\omega _\\mathrm {i}$ dependence of non-spin-flip RIXS spectra by the exact diagonalization calculation for the single-band Hubbard model, we have found that the dependence is strongly affected by the value of the core-hole Coulomb interaction in hole doping: A fluorescencelike behavior appears when the core-hole Coulomb interaction $U_\\mathrm {c}$ is larger than the on-site Coulomb interaction $U$ in the Hubbard model ($U_\\mathrm {c}>U$ ), while a Raman-like behavior appears for $U_\\mathrm {c}<U$ , although the distribution of spectral weight depends on $\\omega _\\mathrm {i}$ .", "Comparing the calculated energy separation between a main peak and a satellite structure in XAS with the corresponding experimental data, we have confirmed that $U_\\mathrm {c}>U$ for the single-band Hubbard model, suggesting a fluorescencelike behavior in RIXS for the $\\sigma $ -polarized geometry detecting non-spin-flip excitations.", "This is consistent with recent experimental observations for overdoped YBa$_2$ Cu$_3$ O$_{6+x}$  [3].", "We predict that the dynamical charge structure factor is observed through RIXS by tuning $\\omega _\\mathrm {i}$ to the satellite structure.", "Using the same $U_\\mathrm {c}$ value, we predict a shift on the high-energy side of the center of spectral distribution in electron doping, whose intensity is reduced as compared with the case of hole doping.", "In the spin-flip channel, main structures exhibit a Raman-like behavior as expected but there is a fluorescencelike behavior similar to the non-spin-flip case, although the spectral weight is very small.", "A detailed experimental work to detect these behaviors is highly desired in the near future.", "This work was supported by the Japan Society for the Promotion of Science, KAKENHI (Grants No.", "26287079, 15H03553 and 16H04004) and by HPCI Strategic Programs for Innovative Research (Grants No.", "hp140078) and Computational Materials Science Initiative from Ministry of Education, Culture, Sports, Science, and Technology." ] ]
1606.05048
[ [ "Kremer-Grest models for universal properties of specific common polymer\n species" ], [ "Abstract The Kremer-Grest (KG) bead-spring model is a near standard in Molecular Dynamic simulations of generic polymer properties.", "It owes its popularity to its computational efficiency, rather than its ability to represent specific polymer species and conditions.", "Here we investigate how to adapt the model to match the universal properties of a wide range of chemical polymers species.", "For this purpose we vary a single parameter originally introduced by Faller and M\\\"uller-Plathe, the chain stiffness.", "Examples include polystyrene, polyethylene, polypropylene, cis-polyisoprene, polydimethylsiloxane, polyethyleneoxide and styrene-butadiene rubber.", "We do this by matching the number of Kuhn segments per chain and the number of Kuhn segments per cubic Kuhn volume for the polymer species and for the Kremer-Grest model.", "We also derive mapping relations for converting KG model units back to physical units, in particular we obtain the entanglement time for the KG model as function of stiffness allowing for a time mapping.", "To test these relations, we generate large equilibrated well entangled polymer melts, and measure the entanglement moduli using a static primitive-path analysis of the entangled melt structure as well as by simulations of step-strain deformation of the model melts.", "The obtained moduli for our model polymer melts are in good agreement with the experimentally expected moduli." ], [ "Introduction", "Polymers are long chain molecules built by covalent linkage of a large numbers of identical monomers [1], [2].", "Some properties of polymeric materials such as their density or glass transition temperature depend on chemical details at the monomer scale.", "Others, like the variation of the melt viscosity with the molecular weight of the chains, are controlled by the large scale chain statistics [3].", "These properties, which are characteristic of polymeric systems, are universal [4], [5] in that they are shared by large numbers of chemically different systems.", "The character of the target properties is crucial for the choice of a model in theoretical or computational investigations.", "Universal properties can be studied using simple, numerically convenient lattice and off-lattice models, see e.g.", "refs.", "[6], [7], [8] for reviews.", "In contrast, predicting specific materials properties for a given chemical species requires atom-scale modeling [9].", "A growing body of work aims at developing coarse-grained models [10], [11], [12], [13], [14] designed for specific polymer chemistries, for example in the case of polyethylene [15], [16], [17], polyisoprene [18], [19], [20], [21], polystyrene [22], [23], polyamide [24], [25], polydimethylsiloxane [26], [27], polymethacrylate [28], [29], bisphenol-A polycarbonate [30], [31], [32], [33], polybutadiene [34], [35], polyvinyl [36], polyterephthalate [37], and polyimide [38].", "Common to these approaches is the selected inclusion of specific chemical details.", "These models offer insights into which atomistic details of the chemical structure are relevant for particular non-universal polymer properties and perhaps also allow for transferability to a larger domain of state space.", "[39], [13] In the present paper, we use a more theory-inspired route to including specificity, which requires experimental input on the local equilibrium chain structure and dynamics.", "From universality, we expect that by matching a very limited number of microscopic length and time scales [40], we can 1) design generic polymer models that match the properties of real chemical polymers, and 2) map the predictions of such simulations back to experiment.", "In contrast to theoretical approaches and without needing to be accurate on the atomic scale, a properly mapped generic polymer model should automatically reproduce many non-trivial static and dynamic features such as correlation holes, as well as stress relaxation via constraint release and contour length fluctuations [3], [41].", "Generic models allow for the investigation of effects due to polymer branching, chain polydispersity, and/or chemical cross-linking, see [42], [43], [44], [45] for examples.", "Generic models not only reproduce the properties of bulk polymer materials, but also films, tethered polymer chains, spatially confined polymers materials, welding of polymer interfaces or composite materials formed by adding filler particles to a polymer melt or solid, see [46], [47], [48], [49], [50], [51], [52], [53], [54], [55] for examples.", "The KG model is used in a vast number of publications as a basis for studying generic polymer and materials physics, see e.g.", "[7], [8], [13] for reviews.", "In this model approximately hard sphere beads are connected by strong non-linear springs.", "The spring potential is chosen to energetically prevent two polymer chains from passing through each other, allowing to sample entangled dynamics [56].", "It can be mapped to experimental systems [57], [40] by adjusting the chain stiffness via a local bending potential [58], [59], [60].", "Compared to lattice models, the deformation response of polymer materials can be straightforwardly studied by deforming the simulation domain e.g.", "via shear [61], [62], uniaxial [63], [64] or biaxial elongation without worrying about artifacts introduced by a underlying lattice.", "We apply this philosophy to the most popular Molecular Dynamics polymer model introduced by Kremer and Grest (KG) [65], [56].", "Here we use KG models of varying stiffness to model real polymers species.", "We generate huge well equilibrated melt states using a new equilibration process.", "[66] We predict the plateau modulus from the model melts using the Primitive-path analysis [67] and stress-relaxation after step-strain and compare results to experimental values.", "The paper is structured as follows; We present the relevant theory in Sec.", ", the Kremer-Grest model and its mappings to real polymers is introduced and characterized in Sec.", ".", "We present our results for the plateau moduli for KG models of real polymers in Sec.", ", and conclude with our conclusions in Sec.", "." ], [ "Theory", "Before introducing the Kremer-Grest model and how to map it to a given chemical polymer species, we introduce the basic quantities used to characterize static and dynamic properties of polymer melts and in particular their dependence on the number of Kuhn segments per cubic Kuhn length.", "This will be the central quantity that we want to reproduce by our KG model systems." ], [ "Chain and melt structure", "At a given state point, a long monodisperse polymer melt is characterized by just a few experimental observables: the molecular mass of a polymer chain $M_{c},$ the mass density $\\rho _{bulk}$ , the average chain end-to-end distance $\\langle R^{2}\\rangle $ , the contour length of the chains, $L$ , and the maximal intra-chain relaxation time, $\\tau _{max}$ .", "From these experimental observables we can derive a set of microscopic parameters characterizing the melt, such as the chain number density, $\\rho _{c}=\\rho _{bulk}/M_{c}$ , its inverse, the volume per chain, $V_{c}=1/\\rho _{c}$ , and the Kuhn length, $l_{K}=\\frac{\\langle R^{2}\\rangle }{L}$ which is the fundamental length scale characterizing chain configurations beyond the monomer scale.", "The number of Kuhn segments per chain is $N_{K}=\\frac{\\langle R^{2}\\rangle }{l_{K}^{2}}=\\frac{L^{2}}{\\langle R^{2}\\rangle }$ and the mass and density of Kuhn segments are given by $M_{K}=M_{c}/N_{K}$ and $\\rho _{K}=V_{K}^{-1}=N_{K}/V_{c}$ respectively.", "There are various manners to characterize the mutual chain interpenetration in polymer melts.", "The Flory number, $n_{F}=\\rho _{c}\\langle R^{2}\\rangle ^{3/2}=\\left(\\rho _{K}l_{K}^{3}\\right)N_{K}^{1/2},$ is defined as the number of chains populating, on average, the volume spanned by one chain.", "For a given polymer material, the Flory number increases with the square root of the chain length, since $\\rho _c\\propto N_K^{-1}$ and $R\\propto N_K^{1/2}$ .", "The chemistry-specific prefactor in this relation is the reduced Kuhn density, $n_{K}=\\rho _{K}l_{K}^{3},$ defined as the number of Kuhn segments within a cubic Kuhn length.", "In most melts of flexible polymers we have $1\\le n_{K}\\le 10$ , whereas in gels of tightly entangled filamentous proteins such as f-actin we have $n_{K}\\gg 10$ .", "[68] To define a length scale characterizing the chain packing in a polymeric material, one can consider a spherical region centered on a chosen monomer.", "If the region is small, then most monomers found inside will belong to the same chain as the chosen monomer.", "On the other hand, if the region is large, then most monomers inside the region will belong to other chains.", "The packing length[69], [70], $p=\\frac{V_{c}}{\\langle R^{2}\\rangle }=\\left(l_{K}^{2}\\rho _{K}\\right)^{-1}=\\frac{l_{K}}{n_{K}},$ defines the crossover between these two regimes.", "It corresponds to the root-mean square end-to-end distance of polymers with Flory number $n_{F}\\equiv 1$ and $N_{K}=1/n_{K}^{2}$ ." ], [ "Local dynamics", "The dynamics of short unentangled polymers is described by the Rouse model.", "[71] In this model a single Gaussian polymer is modeled by a chain of Kuhn segments, and the dynamics is modeled by Brownian motion with a friction term describing the mean field friction due to the surrounding chains.", "In polymer melts, hydrodynamic interactions are strongly screened and can be neglected.", "From the Rouse model, the fundamental Kuhn time is given by $\\tau _{K}=\\frac{\\zeta _{K}l_{K}^{2}}{3\\pi ^{2}k_{B}T}.$ This is time that it takes a single Kuhn segment to diffuse ($D_{K}=k_{B}T/\\zeta _{K}$ ) its own size.", "The Kuhn friction or equivalently the Kuhn time sets the fundamental time scale of all dynamical polymer properties.", "The maximal internal relaxation time of a chain with $N_{k}$ Kuhn units is given by the Rouse time $\\tau _{R}=\\tau _{K}N_{K}^{2}$ and the melt viscosity is predicted to be $\\eta =\\zeta _{K}N_{K}/(36l_{K})$ .", "[71]" ], [ "Predicting emergent large scale dynamics and rheological behaviour", "While chains undergoing Brownian motion can slide past each other, but their backbones cannot cross[3].", "As a consequence, the motion of long chains is subject to transient topological constraints[72], an effect which is familiar from the manipulation of knotted strings.", "These constraints become relevant at scales beyond the entanglement (contour) length[73], [70], $L_{e}$ , or the equivalent the number of Kuhn units between entanglements, $N_{eK}=L_e/l_K$ .", "For loosely entangled polymers, entanglements contribute to the linear elastic response on the level of the entanglement modulus $\\frac{G_{e}l_{K}^{3}}{k_{B}T}=\\frac{\\rho _{K}l_{K}^{3}}{N_{eK}}=\\frac{n_{K}^{3}}{\\alpha ^{2}}.$ In this limit [68], there are $N_{eK}=\\left(\\frac{\\alpha }{n_{K}}\\right)^{2}$ Kuhn segment per entanglement length.", "The (spatial) tube diameter is given by $\\frac{d_{T}}{l_{K}}=\\sqrt{\\frac{\\langle R^{2}(N_{eK})\\rangle }{l_{K}^{2}}}=\\sqrt{N_{eK}}=\\frac{\\alpha }{n_{K}}$ while $\\tau _{e}=\\tau _{K}N_{eK}^{2}=\\tau _{K}\\left(\\frac{\\alpha }{n_{K}}\\right)^{4}.$ defines the corresponding entanglement time.", "The relevance of the packing length, $d_{T} = \\alpha p\\ ,$ to the microscopic topological state can be understood through the primitive path analysis (PPA) [67], [40], [68].", "The numerical prefactor (see also fig.", "REF ), describes the number of entanglement strands per entanglement volume [74], [75] $\\alpha = \\frac{\\rho _K}{N_{eK}} d_T^3$ and appears to be a universal constant for all flexible polymers.", "A geometric argument [76] suggests $\\alpha =20.49$ for the local pairwise entanglement of Gaussian chains [77].", "The analysis of large experimental data sets for polymers at $T=413K$ and $T=298K$ yielded $\\alpha =19.36$ and $\\alpha =17.68$ respectively [78].", "Modern theories of polymer dynamics and rheology [3] describe the universal aspects of the viscoelastic behavior based on the idea that molecular entanglements confine individual filaments to a one-dimensional, diffusive motion (reptation [79]) in tube-like regions in space [80].", "In the long chain limit, the maximal relaxation time is [79] $\\tau _{max}=3\\left(\\frac{N_{K}}{N_{eK}}\\right)^{3}\\tau _{e}.$ In slowing down the chain equilibration after a deformation, entanglements dominate the viscoelastic behavior of high molecular weight polymeric liquids.", "For $\\tau _{e}<t<\\tau _{max}$ the shear relaxation modulus, $G(t)$ exhibits a rubber-elastic plateau, $G_{N}=\\frac{4}{5}G_{e}$ , of the order of the entanglement modulus, increasing the melt viscosity to $\\eta =G_{e}\\tau _{max}\\sim \\frac{N_{K}^{3}}{N_{eK}^{2}}.$ As illustrated by this brief outline of (linear) melt viscoelasticity, two polymeric systems characterized by the same number of Kuhn segments, $N_{K}$ , and the same dimensionless Kuhn density, $n_{K}$ , (and hence identical numbers of entanglements $Z=N_{K}/N_{eK}$ ) are expected to show the same universal large scale behavior.", "By matching the Kuhn length, $l_{K}$ , and the Kuhn time, $\\tau _{K}$ , and by measuring energy in units of the thermal excitation energy, $k_{B}T$ , results obtained for one system can be converted into predictions for the other.", "This analogy is not restricted to experimental systems, but also extends to computational models provided they preserve the key features of polymer melts: chain connectivity, local liquid-like monomer packing, and the impossibility of chain backbones to dynamically cross through each other.", "Nor is the analogy restricted to the linear viscoelasticity of monodisperse melts of linear chains, but should hold more generally provided the two systems are characterized by comparable amounts of branching, polydispersity, or cross-linking.", "Even without being accurate on the atomic scale, a properly mapped simulated model should automatically reproduce many non-trivial static and dynamic features (e.g.", "correlation holes[41], contour-length fluctuations, constraint release, and dynamic dilution effects[3], [81]) which are difficult to preserve in theoretical approaches." ], [ "The choice of the contour length", "Before proceeding, it is useful to briefly reflect on the importance of the choice of the contour length, $L$ , for the emergent properties of the coarse-grain model.", "In other words, what changes, if we choose a different contour length, $L^{\\prime }=\\lambda L$ , for the mapping?", "By construction, this choice modifies the Kuhn length and all quantities associated with it: $l_K^{\\prime } &=& \\frac{\\langle R^{2}\\rangle }{L^{\\prime }} = \\frac{l_K}{\\lambda }\\\\N_K^{\\prime } &=& \\frac{L^{\\prime }}{l_K^{\\prime }} = \\lambda ^2 N_K\\\\\\rho _K^{\\prime } &=& \\rho _c N_K^{\\prime } = \\lambda ^2 \\rho _K\\\\n_K^{\\prime } &=& \\rho _K^{\\prime } l_K^{\\prime 3} = \\frac{n_K }{\\lambda }$ However, on first sight, this seems to have no consequences for the quantities discussed in the preceding section: the Flory number, $n_F^{\\prime } = n_K^{\\prime } N_K^{\\prime 1/2} = n_F\\ ,$ the packing length, $p^{\\prime } = \\frac{l_K^{\\prime }}{n_K^{\\prime }} = p \\ ,$ the tube diameter $d_T^{\\prime } = \\left(\\frac{l_K^{\\prime }}{n_K^{\\prime }}\\right) \\alpha = d_T$ and the entanglement modulus, $\\frac{G_e^{\\prime }}{k_BT} = \\left(\\frac{n_K^{\\prime }}{l_K^{\\prime }}\\right)^3 \\alpha ^{-2} = \\frac{G_e}{k_BT}$ appear to remain unaffected.", "What does change, however, are ratios like $N_{eK}^{\\prime }=\\frac{L_e^{\\prime }}{l_K^{\\prime }} = \\left(\\frac{d_T^{\\prime }}{l_K^{\\prime }} \\right)^2 = \\left(\\lambda \\frac{d_T}{l_K} \\right)^2= \\lambda ^2 \\frac{L_e}{l_K} = \\lambda ^2 N_{eK}\\ .$ The number of Kuhn segments per entanglement strand, $N_{eK}^{\\prime }$ , defines the maximal elongation of entanglement strands and is relevant for the non-linear viscoelastic response of strongly elongated melts and networks [3] as well as craze formation in glassy polymers [82], [83].", "But even for modelling linear viscoelasticity, $L^{\\prime }$ and hence $N_{eK}^{\\prime }$ , cannot be chosen freely, at least if the model is meant remain in the loosely entangled regime.", "Flexible chain behaviour on the entanglement scale requires, that the Kuhn length be smaller than the tube diameter: $1\\ll d_T^{\\prime }/l_K^{\\prime }$ or $N_{eK}^{\\prime }\\gg 1$ .", "If this is the only constraint, then the computationally most efficient approach is to choose a (relatively stiff) model, where $n_K^{\\prime } \\approx \\alpha /2$ close to the crossover to the tightly entangled regime [84], [85].", "In the remainder of the article we employ a physically realistic choice of the contour length to parameterize coarse-grain models, which can be used to explore larger deformations." ], [ "Kremer-Grest model", "The Kremer-Grest (KG) model[65], [56] is a quasi-standard in Molecular Dynamics investigations of generic polymer properties.", "The KG model is a bead-spring model, where the mutual interactions between all beads are given by the Weeks-Chandler-Anderson (WCA) potential (the truncated and shifted repulsive part of the 12-6 Lennard-Jones potential), $U_{WCA}(r)=4\\epsilon \\left[\\left(\\frac{\\sigma }{r}\\right)^{-12}-\\left(\\frac{\\sigma }{r}\\right)^{-6}+\\frac{1}{4}\\right]\\quad \\mbox{for}\\quad r<2^{1/6}\\sigma \\ ,$ while bonded beads interact through the finite-extensible-non-linear spring (FENE) potential given by $U_{FENE}(r)=-\\frac{kR^{2}}{2}\\ln \\left[1-\\left(\\frac{r}{R}\\right)^{2}\\right]\\ .$ We choose $\\epsilon =k_{B}T$ and $\\sigma $ as the simulation units of energy and distance, respectively.", "The standard choices for the Kremer-Grest model is to set $R=1.5\\sigma $ and $k=30\\epsilon \\sigma ^{-2}$ .", "We also choose the standard number of beads per unit volume $\\rho _{b}=0.85\\sigma ^{-3}$ .", "Here and below we use subscript “b” to denote bead properties to distinguish these from Kuhn units used above.", "With these choices, the bond length becomes $l_{b}=0.965\\sigma $ .", "We add an additional bending interaction defined by $U_{bend}(\\Theta )=\\kappa \\left(1-\\cos \\Theta \\right),$ where $\\Theta $ denotes the angle between subsequent bonds.", "The stiffness parameter $\\kappa $ controls the Kuhn length $l_{K}$ and the reduced Kuhn density $n_{K}$ of the resulting polymer model.", "The same potential has previously been studied by Faller and Müller-Plathe.", "[58], [59], [60] For other choices, see Ref. [86].", "For integrating the dynamics of the KG model we use Langevin dynamics $m_{b}\\frac{\\partial ^{2}{\\bf R}_{i}}{\\partial t^{2}}=-\\nabla _{{\\bf R_{i}}}U-\\Gamma \\frac{\\partial {\\bf R}_{i}}{\\partial t}+{\\bf \\xi }_{i}(t)$ where ${\\bf R}_{i}$ denotes the position of bead $i$ and $U$ the total potential energy.", "${\\bf \\xi }_{i}(t)$ is a Gaussian distributed random vector with $\\langle {\\bf \\xi }_{i}(t)\\rangle =0$ and $\\langle {\\bf \\xi }_{i}(t)\\cdot {\\bf \\xi }{}_{j}(t^{\\prime })\\rangle =\\frac{6k_{B}T}{\\Gamma \\Delta t}\\delta (t-t^{\\prime })\\delta _{ij}$ .", "The mass of a bead is denoted $m_{b}$ , and we choose this as our mass scale for the simulations.", "From the currently defined units we can derive a simulation unit of time $\\tau =\\sigma \\sqrt{m_{b}/\\epsilon }$ .", "We use the standard Kremer-Grest choice of $\\Gamma =0.5m_{b}\\tau ^{-1}$ for the friction of the Langevin thermostat together with integration time steps $\\Delta t=0.01\\tau $ .", "For integrating the dynamics of our systems, we use the Grønbech-Jensen/Farago Langevin integration algorithm as implemented in the Large Atomic Molecular Massively Parallel Simulator (LAMMPS).", "[87], [88], [89] Before we can generate KG models of specific polymers, we need to characterize the Kremer-Grest model in terms of the dependency of the Kuhn length and the Kuhn friction on the chain stiffness $\\kappa $ ." ], [ "Kuhn length", "It is difficult to predict the Kuhn length of a polymer model directly from the microscopic force field.", "While excluded volume interactions are approximately screened in melts (the Flory ideality hypothesis[90]), the incompressibility constraint of melts leads to deviations from polymer $\\Theta $ -solutions via a correlation hole, which causes the chain stiffness to increase due to a long range effective repulsive interaction between polymer blobs.", "[41], [91], [92], [93], [94], [95] Here we brute force equilibrate medium length entangled melts with $M=2000$ chains of length $N_{b}=400$ for varying chain stiffness.", "Each initial melt conformation was simulated for at least $2\\times 10^{5}\\tau $ while performing double-bridging hybrid Monte Carlo/Molecular Dynamics simulations[96], [15], [97], [16].", "Configurations from the last $5\\times 10^{4}\\tau $ of the trajectory were used for analysis.", "We chose $N=400$ , since the chains are long enough to estimate the Kuhn length (see below) without compromising the efficiency of the double-bridging algorithm.", "The Kuhn length is estimated using $l_{K}=\\frac{\\langle R^{2}(N_{b})\\rangle }{N_{b}\\sqrt{\\langle l_{b}^{2}\\rangle }}=2\\sqrt{\\langle l_{b}^{2}\\rangle }\\int _{0}^{N_{b}}\\left(1-\\frac{n}{N_{b}}\\right)C(n)\\mbox{d}n,$ where $C(n)=\\left\\langle {\\bf b}(m)\\cdot {\\bf b}(m+n)\\right\\rangle _{m}$ is the bond correlation function averaged along the contour of the chain.", "The result is shown in Fig.", "REF .", "As expected, the Kuhn length increases with the bending stiffness.", "The simulation data are well fitted by a cubic polynomial, which allows us to obtain the chain stiffness parameter $k$ required to reproduce a desired Kuhn length $l_{K}(\\kappa )$ .", "For $l_{K}>7\\sigma $ , KG polymer melts undergo a isotropic-nematic phase transition,[58] however the present systems are in the isotropic state.", "Figure: Kuhn length l K l_{K} vs stiffness parameterκ\\kappa for Kremer-Grest polymer model (green symbols), also shownis an interpolation given by l K (κ)/σ=1.795+0.358ϵ -1 κ+0.172ϵ -2 κ 2 +0.019ϵ -3 κ 3 l_{K}(\\kappa )/\\sigma =1.795+0.358\\epsilon ^{-1}\\kappa +0.172\\epsilon ^{-2}\\kappa ^{2}+0.019\\epsilon ^{-3}\\kappa ^{3}(hashed black line)." ], [ "Primitive-path analysis", "To estimate the entanglement length as function of chain stiffness, we have generated highly entangled melt states with $M=500$ chains of length $N_{b}=10000$ for stiffness parameter $\\kappa =-1,-0.5,0,0.5,1.0,1.5,2.0,2.5$ .", "The details of the equilibration procedure can be found in Ref.", "[66].", "We have performed primitive-path analysis (PPA) of the melt states.", "During the primitive path analysis a melt conformation is converted into the topologically equivalent primitive-path mesh work characterizing the microscopic topological melt state.", "[67] During the analysis the chain ends are fixed, hence the mean-square end-to-end distance is constant.", "The Kuhn length of the tube is given by $a_{pp}=\\langle R^{2}\\rangle /L_{pp}$ where $L_{pp}$ is the average contour length obtained from the primitive-path mesh.", "Here we distinguish between the tube diameter $d_{T}$ predicted by rheology and $a_{pp}$ predicted from the static PPA analysis, even though these two parameters describes the same physics, and are expected to be the same on a scaling level.", "Hence we can obtain the number of Kuhn segments between entanglements as $N_{eK}(\\kappa )=a_{pp}(\\kappa )/l_{K}^{2}(\\kappa )$ .", "We have performed a version of the PPA analysis that preserves self-entanglements by only disabling pair interactions between beads within a chemical distance of $2N_{eK}$ .", "Subsequently the potential energy was minimized.", "The minimization was performed using the steepest descent algorithm implemented in LAMMPS followed by dampened Langevin dynamics.", "We have used the method of Hoy et al.", "[98] to check for finite chain length effects, however, this can be completely neglected for all the the chain lengths used here.", "Figure: Entanglement length N eK N_{eK} vs stiffnessparameter for Kremer-Grest polymer model (green symbols).", "The insetshows the same data but as function of the reduced Kuhn density.", "Shownis an interpolation given by N eK (κ)=41.774-29.605ϵ -1 κ+5.842ϵ -2 κ 2 N_{eK}(\\kappa )=41.774-29.605\\epsilon ^{-1}\\kappa +5.842\\epsilon ^{-2}\\kappa ^{2}(hashed black line) and the theoretical relation N eK (n K )=α 2 n K -2 N_{eK}(n_{K})=\\alpha ^{2}n_{K}^{-2}with α=18.0\\alpha =18.0 (solid black line in the inset with hashed linesindicating a ±10%\\pm 10\\% error).Fig.", "REF shows the chain stiffness dependence of the entanglement length.", "As expected as the chain become stiffer, their spatial size increases, and hence also the number of entanglements.", "This leads to the observed reduction in the distance between entanglements.", "The inset shows the theoretical prediction based on the universal number of $\\alpha =18.0$ chains per entanglement volume, which is in very good agreement with the simulation results." ], [ "Time mapping", "To define a relevant time scale for the polymer dynamics, we measure the Kuhn friction $\\zeta _{K}$ .", "For short chains $N_{K}<N_{eK}(\\kappa )$ , Rouse theory applies, and the Kuhn friction can be obtained by $\\zeta _{K}(\\kappa )=\\frac{k_{B}T}{D_{cm}(\\kappa ,N_{K})N_{K}}.$ We have performed a series of simulations of melts of 2000 chains of length $N_{K}=3,4,5,8,10,15,20,30$ , which were equilibrated for a period of $10^{4}\\tau $ using double bridging hybrid MC/MD[96], [15], [97], [16] as above.", "The resulting equilibrium melt states were run for up to $2-10\\times 10^{5}\\tau $ and the center of mass diffusion coefficient $D_{cm}(\\kappa ,N_{K})$ was obtained from the plateau of $\\langle [R_{cm}(t)-R_{cm}(0)]^{2}\\rangle /[6t]$ for $t>10^{5}\\tau $ by sampling plateau values for log-equidistant times, and discarding simulations where the standard deviation of the samples exceeded $2\\%$ of their average value.", "Figure: Kuhn friction ζ K \\zeta _{K} (top) and renormalizedfriction ζ K R \\zeta _{K}^{R} (bottom) vs stiffness parameter.", "The frictionwas estimated from simulations with N K =3,4,5,8,10,15,20,30N_{K}=3,4,5,8,10,15,20,30 (black circle,red box, green diamond, blue triangle up, yellow triangle left,magenta triangle down, orange triangle right, respectively).Simulations where the Rouse model is expected to be valid (N K <N eK (κ)/2N_{K}<N_{eK}(\\kappa )/2)are shown with large open symbols, the rest are shown with small closed symbols.Shown is also is a trend line givenby ζ K (κ)/[m b τ -1 ]=34.85+10.31(κ/ϵ)+5.659(κ/ϵ) 2 \\zeta {}_{K}(\\kappa )/[m_{b}\\tau ^{-1}]=34.85+10.31(\\kappa /\\epsilon )+5.659(\\kappa /\\epsilon )^{2}(solid black line).Fig.", "REF shows the Kuhn friction obtained from the analysis of the simulations using eq.", "(REF ).", "We observe that the friction increases with stiffness for all chain lengths, but most strongly for the long chains.", "This is to be expected since spatial size of the chains increase, and hence more strongly confine the diffusive dynamics.", "The Rouse model should apply for chains that are shorter than the entanglement length (open symbols in the figure), and we would expect these to collapse to a single line, however we see a systematic and significant spread in the simulation data.", "This is due to larger mobility of beads towards the chain ends compared to beads far away from the chain ends.", "We have corrected for this by defining a renormalized Kuhn Friction $\\zeta _{K}^{R}(\\kappa )=\\zeta _{K}(\\kappa )g(N_{K}),$ where the $g$ function describes the chain-length dependence of the reduction of friction due to chain ends.", "Opting for simplicity, we use a linear interpolation where $N_{end}$ Kuhn segments have friction reduced by a factor $f_{end}$ while the friction due to the rest of the Kuhn segments is unaffected, hence $g(N_{k})={\\left\\lbrace \\begin{array}{ll}f_{end} & \\quad \\mbox{for}\\quad N_{K}<N_{end}\\\\N_{K}^{-1}((N_{K}-N_{end})+f_{end}N_{end}) & \\quad \\mbox{for}\\quad N_{K}\\ge N_{end}\\end{array}\\right.", "}.$ The parameters were determined by minimizing the variance between Kuhn friction estimates where 1) the Rouse model applies and where 2) we have at least three values estimates for the friction.", "We obtained $f_{end}=0.70$ and $N_{end}=7$ , which corresponds to completely neglecting the friction due to the outermost Kuhn segment at each end.", "This choice reduced the total variance of the renormalized frictions by more than an order of magnitude.", "The result is shown in Fig.", "REF , where we observe an significantly improved collapse of the data allowing us to estimate the effective Kuhn friction required for the time mapping of long entangled chains.", "Note that since the entanglement length drops rapidly for large stiffnesses ($N_{eK}(2.5\\epsilon )<5$ ), it is not possible to perform simulations where chains are long and the Rouse model applies at the same time.", "This necessitates a renormalization procedure that resolves the end effect.", "Figure: Entanglement time τ e (κ)\\tau _{e}(\\kappa ) and Kuhn timeτ K (κ)\\tau _{K}(\\kappa ) in KG units as function of stiffness parameterpredicted using the interpolations given in figs.", ",, and .Experimentally observable is the entanglement time $\\tau _{e}(\\kappa )=\\tau _{K}(\\kappa )N_{eK}^{2}(\\kappa )$ .", "Empirical relations for the chain stiffness dependent entanglement and Kuhn times are shown in Fig.", "REF .", "The Kuhn time rises steeply with increased chain stiffness, since the friction itself rises steeply with stiffness.", "This is most likely due to the fact that as chains become stiffer they interact with more chains.", "This slows down the dynamics, which we rationalize as an apparent increase in the friction coefficient felt by the center-of-mass.", "On the other hand, since the entanglement length drops with stiffness the net effect is that the entanglement time drops with increasing chain stiffness.", "Time mappings have previously been studied for the standard KG model ($\\kappa =0$ ) Likhtman[99] obtained $\\tau _{e}(\\kappa =0)\\approx 5800\\tau $ by fitting shear-relaxtion moduli produced by KG simulations, while Kremer and Grest[56] obtained $\\tau _{e}(\\kappa =0)=1.5N_{eb}^{2}$ from Rouse mode analysis of chain dynamics.", "Here $N_{eb}=c_{b}N_{eK}$ denotes the entanglement length expressed in beads.", "Using the most recent value for the entanglement length $N_{eb}=85\\pm 7$ [98] results in $\\tau _{e}(\\kappa =0)=10800\\pm 1800\\tau $ .", "However, looking into the data of Kremer and Grest more carefully, they also give an effective bead rouse friction of $\\zeta _{b}=25\\pm 2m_{b}\\tau ^{-1}$ , and using the relations in the present paper to derive an estimate for the entanglement time results in a somewhat reduced entanglement time of $\\tau _{e}(\\kappa =0)=8800\\pm 700\\tau $ .", "Our present result $\\tau _{e}(0)=6780\\tau $ is in reasonable good agreement with both Likhtman and our refined Kremer and Grest value." ], [ "Mapping", "Given a specific chemical polymer species characterized by its Kuhn length $l_{K}$ , and reduced density of Kuhn segments $n_{K}$ .", "We would like to design a Kremer-Grest model and a mapping that reproduces these properties.", "This gives rise to the following equations $c_{b}(\\kappa )l_{b}\\equiv l_{K},$ and $c_{b}^{2}(\\kappa )l_{b}^{3}\\rho _{b}=l_{K}^{3}(\\kappa )\\rho _{K}(\\kappa )\\equiv n_{K}.$ Here we have the physical quantities on the right hand side, and the corresponding KG expressions on the left hand side.", "Hence from the second equation we learn the numerical value of $c_{b}(\\kappa )$ which is dimensionless, while from the first equation we learn the mapping the KG length unit to the real physical unit of the Kuhn length.", "The physical interpretation of these two equations is that they provide the number of beads per Kuhn segment as well as the width of the beads required to fulfill the constraints.", "Given a concrete number of molecules with $N_{K}$ Kuhn segments per molecule, we can generate a corresponding initial state with $N_{b}=c_{b}(\\kappa )N_{K}$ beads per chain and $M$ such chains in the volume required to produce the target density $\\rho _{b}$ .", "What remains is to map the simulation results back to real physical units specific for the chosen polymer species.", "With the energy scale set to the (target) thermal excitation energy, $\\epsilon =k_{B}T$ , all that remains to be done is to fix the bead mass $m$ .", "Equating the masses of the Kuhn segments, $c_{b}(\\kappa )m_{b}\\equiv M_{K}.$ This is a one parameter family of KG polymer models, since we keep the density fixed and adjust only a single parameter, the chain stiffness $\\kappa $ , to the specific polymer species.", "We expect this to work well for dense polymer melts, however, modeling dilute or semi-dilute solutions would require KG models where we also adapt the density of beads to match the volume fraction of monomers, and perhaps also modify the bead interactions to take solvent induced effects into account.", "These additional complexities are outside the scope of the present paper.", "With this present approach different polymer species with matching pairs of $l_{K}$ and $n_{K}$ are mapped unto the same KG model.", "Recently Zhang et al.", "presented an similar approach[100], where a given polymer species was represented by KG models varying both the stiffness and density, where the mapping was based on keeping the Flory number $n_{F}=N_{K}^{3/2}M^{-1}\\times n_{K}$ constant.", "Equilibrated low-resolution precursor melts was rescaled and successively fine grained to reach the resolution of a target model as determined.", "Here take as a starting point a real specific polymer chemistry, and derive a corresponding KG polymer model varying only the stiffness while retaining the standard density.", "We also produce a set of mapping relations for units.", "We subsequently generate an equilibrated melt for this model using a fast multiscale equilibration procedure that we recently developed[66].", "In table REF we summarize the relevant experimental and chemical properties of eight chosen polymers.", "These are the numbers that are required to define equivalent KG models.", "In Tab.", "REF we characterize the same polymers in terms of Kuhn and entanglement properties required.", "As expected the reduced Kuhn densities varies within $1\\le n_{K}\\le 10$ .", "Note that the average ratio of the tube diameter to the packing parameter is approximately a universal constant $\\alpha =\\langle d_{T}/p\\rangle =18.0\\pm 1.3$ for all the polymers.", "Table: Polymer properties: Reference temperatureT ref ,T_{ref}, end-to-end distance per molecular weight 〈R 2 〉/M c \\langle R^{2}\\rangle /M_{c},bulk mass density ρ bulk \\rho _{bulk}, monomer mass, and back bone contourlength per monomer.", "The physical properties are from Ref.", ".", "1 ^{1}SBR rubber has 25% styrene content.Table: Descriptions of polymers interms of Kuhn length l K l_{K}, reduced Kuhn density n K n_{K}, monomersper Kuhn length c ∞ ,c_{\\infty }, packing length pp, and the dimensionlessratio of tube diameter d T d_{T} and the packing length.", "The tube diameterwas derived using the experimental plateau moduli in Tab.", "." ], [ "Kremer-Grest models of common polymers", "Using the relations presented above, we have generated Kremer-Grest model parameters and mappings for the eight chosen polymer species shown in Tab.", "REF , the mapping relations are shown in Tab.", "REF .", "From the mapping relations, we note that the only free parameter, the stiffness parameter varies from $\\kappa =-0.36$ up to $\\kappa =2.0$ , which falls into the range where our empirical relations for Kuhn length, entanglement length, and Kuhn friction are valid.", "The number of beads per monomer produced by the mapping is about $1\\pm 0.5$ .", "The length scale of the beads is around $5\\pm 2\\mathring{A}$ , the energy scale is by construction always $kT$ .", "Despite fairly small variations in the length and energy scales, the unit of stress $\\epsilon \\sigma ^{-3}$ is seen to be quite sensitive to the specific polymer species.", "Based on our time mappings, we can estimate the Kuhn and entanglement times for the polymers, which can be compared directly to results obtained from rheology or neutron spin-echo scattering to provide a time mapping.", "For example, for PDMS the entanglement time is $\\tau _{e}(288K)=1.0\\times 10{}^{-6}s$[102], and for cis-PI $\\tau _{e}(298K)=13.21\\times 10^{-6}s$[103], based on these values we obtain time mappings $1\\tau =0.1-2.4ns$ .", "Hence the simulation time steps $\\Delta t=0.01\\tau $ corresponds to $\\approx 10^{4}fs$ , secondly due to the number of atoms represented by a single bead, the coarse-graining alone gives factor of ten speed increase.", "Hence simulations with the KG models are about $10^{5}$ times faster than atomistic simulations of the same polymer.", "However, the time mapping should by no means be assumed valid for other polymer species or state points, especially if the reference temperature is close to the glass transition temperature.", "In such cases the speedup of the present approach for representing the large scale behaviour is exponentially larger, since the standard KG model without attractive interactions does not exhibit a glass transition [104].", "Table: Kremer-Grest models andunit mappings for the polymers shown in Tab.", "and .We have generated a single equilibrated melt sample for each of the polymer models in Table REF .", "We chose the chain length such that the number of entanglements per chain $Z=N_{k}/N_{e}\\approx 100$ , and number of molecules such that the total number of beads in the system was approximately $5\\times 10^{6}$ .", "This ensures that the resulting systems are in the strongly entangled regime.", "For comparison the largest systems that were equilibrated with the recent method of Zhang et al.", "[100], [105] contained $2\\times 10^{6}$ beads and $Z=24$ entanglements.", "The resulting melt states are listed in Tab.", "REF .", "Also shown in the table are estimates of their Rouse and maximal relaxation times.", "Each melt state was equilibrated using a new computationally very fast method, that we have developed.", "[66] To summarize the process; lattice melts states were initially generated using a lattice chain model where blobs belonging to different polymers can occupy the same lattice site.", "The lattice Hamiltonian contains a term that penalizes density fluctuations[106], and the large-scale chain statistics was equilibrated using Monte Carlo simulated annealing.", "The resulting lattice melt states were fine grained to a bead-spring polymer model, which was then further equilibrated at short and intermediate using a KG force field but with a force capped pair-interaction.", "This models allows some overlap between beads, and the resulting Rouse dynamics allows the lattice artifacts to be equilibrated while further reducing density fluctuations.", "Finally, the melt states were energy minimized with the KG force field and reheated to the standard temperature $T=1\\epsilon $ .", "We choose the number of molecules to produce a target systems with $5\\times 10^{6}$ beads, but self-consistently adjusted the number of molecules to minimize a round off introduced due to the lattice used in the first stage of the equilibration process.", "As a result the actual number of beads vary from $2.1\\times 10^{6}$ to $8.0\\times 10^{6}$ for aPS and cis-PI respectively.", "The reason for the variation of $Z$ is that we used preliminary simulation results from shorter melts to provide an $N_{e}(\\kappa )$ estimate.", "We have subsequently recalculated $Z$ with the more accurate estimate obtained from the long entangled melts presented above.", "Table: Prepared Kremer-Grest melt states.MM number of molecules, ZZ number of entanglements per chain,N K N_{K} number of Kuhn units per chain, N b N_{b} number of beadsper chain, τ R \\tau _{R} Rouse time, and τ max \\tau _{max} maximal relaxationtime." ], [ "Results and discussion", "Having generated well-equilibrated melt configurations for our KG models shown in tab.", "REF , we now proceed to calculate their emergent rheological properties and to compare the results to experiment.", "We have performed primitive-path analysis (PPA) of the melt states.", "During the primitive path analysis a melt conformation is converted into the topologically equivalent primitive-path mesh work characterizing the tube structure.", "[67] From the mesh work we can estimate the plateau modulus as $G_{N}^{est}=\\frac{4\\rho _{K}k_{B}T}{5N_{eK}}=\\frac{4\\rho _{K}k_{B}Tl_{K}^{2}}{5a_{pp}^{2}}.$ The numerator is specified by the polymer model and the mapping of units, while PPA provides the required Kuhn length of the primitive path.", "During the analysis the chain ends are fixed.", "The mean-square end-to-end distance is constant, and hence the Kuhn length of the tube is given by $a_{pp}=\\langle R^{2}\\rangle /L_{pp}$ where $L_{pp}$ is the average contour length obtained from the primitive-path mesh.", "The primitive-path analysis was performed as described above.", "Table REF shows the comparison between plateau moduli from experiments and those predicted by the primitive path analysis applied to the model polymer melts.", "Note that no rheological information was build into the KG models nor the mappings of units, hence this is a prediction of macroscopic elastic properties based solely on matching mesoscale conformational properties of the polymer molecules.", "We observe quite good agreement between most of the models and experimental values with five of the eight polymers being within 10% of the experimental value, and all polymers are within 30% of the experimental plateau value.", "From this we can conclude that our melt states provide good descriptions of the true polymer melts properties, and secondly that our mapping of simulation to SI units is also correct.", "Table: Comparison between predictions of experimental plateaumoduli and moduli predicted from the primitive path mesh of the equilibratedpolymer melt states.Figure: Reduced entanglement moduli for thechosen polymers (symbols), the one parameter family of KG models forthe range of stiffness parameters κ=-1,⋯,2.5\\kappa =-1,\\dots ,2.5 (solid blackline) using the N eK (κ)N_{eK}(\\kappa ) interpolation from Fig.", "including ten and twenty percent error intervals (red hashed and greenhashed lines, respectively).", "The insert shows a comparison betweenKG reduced moduli, flexible chain relation eq.", "()with α=18.0\\alpha =18.0 (red dotted line), and an extrapolation from Ref., see the text for details.", "(green hashed line).Fig.", "REF shows a comparison between the reduced experimental entanglement moduli of our polymers and the reduced plateau moduli produced by the KG models we can generate by varying the stiffness.", "The deviations we saw in Tab.", "REF are completely consistent with the deviation between the experimental entanglement moduli and the KG model line.", "Almost all of the experimental values are within the $20\\%$ error interval around the KG models.", "The scatter observed between the experimental plateau moduli and the predicted plateau modulus line must be attributed either to chemical details causing some small degree of non-universal behaviour[107], or to experimental uncertainties in accurately estimating the plateau modulus which can be quite difficult.", "[108] Nonetheless, the KG models are clearly able to predict the entanglement modulus of the chemical polymer species with good accuracy.", "Also shown in Fig.", "REF is a comparison between reduced plateau moduli of the KG models compared to the theoretical prediction eq.", "(REF ) using $\\alpha =18.0$ consistent with Fig.", "REF .", "The agreement is good for smaller values of $n_{K}$ i.e.", "for flexible polymers, but increasingly deviate for stiffer polymers.", "A universal description for entanglement moduli of dense melts of flexible chains and tightly entangled gels of rigid rods was developed by Uchida et al.[68].", "The result was an extrapolation describing the reduced entanglement modulus using two free parameters, $c_{\\xi }=0.06$ and $c_{G}=0.6$ , that fixes the the modulus in the flexible chain and rigid rod regimes, respectively.", "We retain the latter parameter, but replace the former parameter by $c_{\\xi }=2/[\\alpha \\sqrt{5c_{G}}]=0.064$ consistent with $\\alpha =18.0$ in the flexible chain limit.", "With this choice, eqs.", "(5-7) of Ref.", "[68] reduce to the simple extrapolation formula $\\frac{G_{e}l_{K}^{3}}{k_{B}T}(n_{K})=\\frac{n_{K}^{7/5}}{4\\left(243n_{K}^{-2}+3n_{K}^{-2/5}+1\\right)^{4/5}}.$ This expression is also shown in the Fig.", "REF and is in excellent agreement with the reduced entanglement moduli produced by the KG models for all values of the reduced Kuhn density or equivalently chain stiffness.", "To estimate the plateau modulus from a purely rheological approach, we have also performed rapid uniaxial step-strain simulations of some of the model melts.", "The deformation is followed by a long stress relaxation simulation at constant strain.", "We obtained the parallel and perpendicular stress components $\\sigma _{\\parallel }$ and $\\sigma _{\\perp }$ from the instantaneous microscopic virial tensor.", "The melts were strained by an elongation factor $\\lambda =1.5,2.0,3.0$ during a fast simulation of just $\\Delta t=10^{2}\\tau =3-29\\tau _{K}$ .", "These deformations are far inside the non-linear response regime for the polymer melts.", "The systems were then run $2\\times 10^{5}\\tau $ corresponding to $22-191\\tau _{e}$ or to $0.002-0.023\\tau _{R}$ at constant strain while the stress relaxed.", "The simulations should be long enough to see the onset of a stress plateau, but not long enough for the chains to contract to their equilibrium length, and certainly far too short to start seeing the terminal stress relaxation at $\\tau _{max}$ .", "We thus expect to see stress relaxation to the level of the entanglement modulus rather than to the level of the plateau modulus.", "The shear relaxation modulus $G(t)$ can be obtained from $\\sigma _{T}(\\lambda ;t)=h(\\lambda )G(t),$ where $\\sigma _{T}(\\lambda ;t)=\\sigma _{\\parallel }-\\sigma _{\\perp }$ is the normal tension, and $h(\\lambda )$ is a damping function.", "The damping function describes the tube reorganization due to the applied strain as well as initial fast stress relaxation processes due to chain length redistribution inside the tube, which we do not expect to be relevant for the present simulations.. Our ability to estimate the plateau modulus from the simulation data depends on having expressions for the damping function that takes into account the non-linear large-strain response of the polymer melts.", "Several expressions have been proposed based on different assumptions for the strain response of polymer melts.", "The classical damping function $h^{CL}(\\lambda )=\\lambda ^{2}-\\lambda ^{-1}$ is based on the assumption of affine deformations.", "Doi and Edwards (see Ref.", "[3] eqs.", "7.137-7.141) derived a time-dependent damping function, which includes a uniform contour length contraction dynamics of the chain inside an affine deforming strain independent tube mesh work.", "In the early time limit, where contour length contraction does not occur, this damping function reduce to $\\nonumber h^{DE}(\\lambda )=\\left(\\frac{15\\lambda ^{2}\\left(\\lambda ^{3}+\\frac{1}{2}\\right)}{16(\\lambda ^{3}-1)}\\right)\\left(\\frac{\\sinh ^{-1}\\left(\\sqrt{\\lambda ^{3}-1}\\right)}{\\sqrt{\\lambda ^{3}\\left(\\lambda ^{3}-1\\right)}}+1\\right)$ $\\times \\left(1-\\frac{\\left(4\\lambda ^{3}-1\\right)\\sinh ^{-1}\\left(\\sqrt{\\lambda ^{3}-1}\\right)}{\\sqrt{\\lambda ^{3}\\left(\\lambda ^{3}-1\\right)}\\left(2\\lambda ^{3}+1\\right)}\\right)$ Rubinstein and Panyukov[109] derived a non-affine tube model, where the tube diameter is assumed to change with deformation as as $d_{T}\\sim \\lambda ^{1/2}$ .", "This is believed to model the weaker tube localization effect of entanglements due to chain slipping compared to localization due to chemical cross-links, which is strain independent.", "[110], [111] This assumption gives rise to the following damping function for a melt $h^{RPNA}(\\lambda )=\\frac{\\lambda ^{2}-\\lambda ^{-1}}{\\lambda -\\lambda ^{1/2}+1}$ Later, Rubinstein-Panyukov additionally allowed for chain length redistribution inside the tube.", "Their their slip-tube model[112] predicts $h^{RPST}(\\lambda )=\\frac{\\lambda ^{2}-\\lambda ^{-1}}{0.74\\lambda +0.61\\lambda ^{-1/2}-0.35}$ Previously, we concluded that the Rubinstein-Panyukov slip-tube damping function gave the best agreement to simulation results for step-strained KG melts by comparing to shear relaxation moduli estimated using a Green-Kubo approach.", "[113] However, these results were obtained for KG melts that were significantly shorter than the present systems, furthermore the deformations were up to two orders of magnitude slower, and hence we would expect complete or at least partial relaxation of chain length inside the tubes after the deformation for these results.", "Figure: Shear relaxation modulus using the classicaldamping function (top left), the Doi-Edwards damping function (topright), and for the Rubinstein-Panyukov non-affine tube model (bottomleft) and slip-tube (bottom right) for step-strains λ=1.5\\lambda =1.5(circle), 2.02.0 (square), and 3.03.0 (diamond) for PDMS (cyan), aPP (violet),PI (black), PEO (blue), PE413 (green), PE298 (red).", "The experimentaland PPA entanglement moduli are illustrated as horizontal solid anddashed lines, respectively.Fig.", "REF shows the shear relaxation modulus obtained using different damping functions.", "Also shown in the figure are the experimental entanglement moduli as well as moduli estimates from the primitive-path analysis.", "Since chain contraction occurs on much longer time scales than our simulations, we expect to measure the entanglement modulus rather than the plateau modulus.", "After an entanglement time $\\tau _{e}$ , we observe that the modulus for each melt has dropped down and the start of a plateau can be seen.", "Overall for all the damping functions we see a reasonable collapse of the data obtain from different strains.", "The classical and Doi-Edwards damping functions give rise to very similar shear relaxation moduli, that generally undershoot the expected entanglement moduli.", "Comparing the two theories of Rubinstein and Panyukov, we observe that they also give rise to very similar relaxation moduli, that are in better agreement with the experimental moduli than the classical and Doi-Edwards damping functions.", "This supports the validity of the assumption that the entanglement tube strain response is indeed sub-affine [114].", "Most importantly, the results suggests that we can indeed extract consistent moduli of our model materials from either the stress relaxation after a step-strain deformation or from the computationally much less expensive primitive path analysis." ], [ "Conclusion", "We have shown how to model specific chemical polymer species with an extension of the Kremer-Grest polymer model introduced by Faller et al.[58].", "The force field has just a single adjustable parameter, the chain stiffness.", "By matching the Kuhn length and the reduced Kuhn density of the KG polymer model to the specific polymer, we can build KG melts of any flexible polymer species.", "Secondly we produce mapping relations converting simulation units to SI units, such that we can compare our predict macroscopic viscoelastic properties to the experimental values of the real polymers.", "We have illustrated this approach by designing KG polymer models for eight standard polymer species of both commercial and academic interest.", "We have build well equilibrated massively entangled model polymer melts for these species, and measured the plateau moduli of the model melts using both primitive-path analysis (PPA) of the static conformations and extensive stress relaxation simulations after a rapid step-strain deformation.", "PPA seems to be the method of choice.", "The stress relaxation simulations are not only computationally much more expensive, but the evaluation of the results suffers from the uncertainty in the choice of damping function.", "For most of the polymers, we observed excellent agreement between the PPA predicted plateau moduli and the experimental values.", "Five out of eight polymer models were within 5% and the remaining three being within 30% of the experimental plateau moduli.", "We observed the largest deviations for PE298.", "They are comparable to those reported for another bead-spring model of PE [12], while evaluating the proper PPA measures [77] for atomistic models appears to reproduce experimental entanglement moduli [115].", "Drawing on the analogy to entanglements in $\\theta $ -solutions [68], [116], we can only speculate that details in the local packing or the different bending angle distributions modify the entanglement density in the present KG model relative to the experimental target system.", "The problems might be particularly pronounced for PE298, which has with $n_K=9.15$ the highest reduced Kuhn density of all investigated systems (Table REF ) and is thus closest to the crossover to the tightly entangled regime expected around $n_K\\approx \\alpha =20$  [68].", "Computationally the KG models are very cheap compared to atomistic detailed simulations.", "Based on our time mapping of polymer dynamics, we have estimated that the KG models are $10^{5}$ times faster than atomistic models for PI and PDMS.", "For systems close to the glass transition, the speedup in modelling the large scale behaviour along the present lines would be exponentially larger.", "Compared to an atomistic model, this obviously comes at the price of loosing the ability to predict any of the glassy behaviour.", "In terms of computational effort, we have studied melts of $2-8\\times 10^{6}$ beads with about 100 entanglements per chain.", "For each of the systems studied, the equilibration procedure took less than 1000 core hours.", "The primitive path analysis required less than 20 core hours.", "Finally the deformation and subsequent stress relaxation required about 30000 node hours.", "The present coarse-graining level is about one bead per chemical monomer, or two to three beads per Kuhn segment.", "It results from matching the Kuhn length and reduced Kuhn segment density, and is not directly adjustable except by changing the equilibrium bond length of the KG model or the density of beads.", "Here we have chosen to keep these at their standard KG values.", "Hence there are a number of future avenues for improving and accelerating the dynamics.", "Alternatively, the number of beads per Kuhn unit could be reduced by using rods rather than beads as the fundamental build blocks of coarse-grained models.", "Computation/simulation for the work described in this paper was supported by the DeiC National HPC Center, SDU." ] ]
1606.05008
[ [ "A linear time algorithm for a variant of the max cut problem in series\n parallel graphs" ], [ "Abstract Given a graph $G=(V, E)$, a connected sides cut $(U, V\\backslash U)$ or $\\delta (U)$ is the set of edges of E linking all vertices of U to all vertices of $V\\backslash U$ such that the induced subgraphs $G[U]$ and $G[V\\backslash U]$ are connected.", "Given a positive weight function $w$ defined on $E$, the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut $\\Omega$ such that $w(\\Omega)$ is maximum.", "MAX CS CUT is NP-hard.", "In this paper, we give a linear time algorithm to solve MAX CS CUT for series parallel graphs.", "We deduce a linear time algorithm for the minimum cut problem in the same class of graphs without computing the maximum flow." ], [ "Introduction", "Sets and their characterisitic vectors will not be distinguished.", "We refer to Bondy and Murty [5] about graph theory terminolgy and facts.", "Given an undirected graph $G = (V, E)$ and positive weights $w_{ij} = w_{ji}$ on the edges $(i, j)\\in E$ , the maximum cut problem (MAX CUT) is that of finding the set of vertices $S$ that maximizes the weight of the edges in the cut $(S, V\\backslash S)$ or $\\delta (S)$ or $\\delta (V\\backslash S)$ ; that is, the weight of the edges with one endpoint in $S$ and the other in $V\\backslash S$ .", "The (decision variant of the) MAX CUT is one of the Karp’s original NP-complete problems [9], and has long been known to be NP-complete even if the problem is unweighted; that is, if $w_{ij} = 1$ for all $(i, j)\\in E$ [6].", "This motivates the research to solve the MAX CUT problem in special classes of graphs.", "The MAX CUT problem is solvable in polynomial time for the following special classes of graphs: planar graphs [2], [8], [10], line graphs [7], graphs with bounded treewidth, or cographs [4].", "But the problem remains NP-complete for chordal graphs, undirected path graphs, split graphs, tripartite graphs, graphs that are the complement of a bipartite graph [4] and planar graphs if the weights are of arbitrary sign [13].", "Besides its theoretical importance, the MAX CUT problem has applications in circuit layout design and statistical physics [1].", "For a comprehensive survey of the MAX CUT problem, the reader is referred to Poljak and Tuza [11] and Ben-Ameur et al.", "[3].", "The best known algorithm for MAX CUT in planar graphs has running time complexity $O(n^{3/2} log n)$ , where $n$ is the number of vertices of the graph [12].", "The main result of this paper is to exhibit a linear time algorithm for a special variant of MAX CUT in series parallel graphs.", "Let us give some definitions.", "Given an undirected graph $G = (V, E)$ and a subset of vertices $U$ , a connected sides cut $\\delta (U)$ is a cut where both induced subgraphs $G[U]$ and $G[V\\backslash U]$ are connected.", "Special connected sides cuts are trivial cuts, i.e.", "cuts with one single vertex in one side.", "The corresponding weighted variant of MAX CUT for connected sides cuts is called MAX CONNECTED SIDES CUT problem (MAX CS CUT).", "It is clear that MAX CUT and MAX CS CUT are the same problem for complete graphs.", "Since MAX CUT is NP-hard for complete graphs (see [9]) then MAX CS CUT is NP-hard in the general case.", "Another motivation is that MAX CS CUT gives a lower bound for MAX CUT.", "A parallel closure of a graph is an induced subgraph on two vertices.", "A series extension of the graph $G = (V, E)$ based on the edge $e\\in E$ is adding a vertex $v$ of degree 2 in the middle of $e$ in order to have two edges instead of $e$ .", "A parallel extension of $G$ based on the edge $e$ is adding an edge $f$ having the same incident vertices as $e$ .", "Series parallel graphs are graphs obtained by applying recursively series and/or parallel extensions starting from one edge.", "A series degree of a vertex $v$ in a graph $G$ is the degree of $v$ after replacing every parallel closure of $G$ by one single edge.", "A series labeling of the vertices of a series parallel graph is a labeling of the vertices from 0 to $n-1 = |V|-1$ starting from the first two vertices $v_0$ and $v_1$ and so on to the last added vertex.", "Any series parallel graph contains at least one vertex of series degree 2.", "So, given a vertex $v$ of series degree 2 with the two parallel closures $P_0$ and $P_1$ incident to $v$ , and the two adjacent vertices $u_0$ and $u_1$ to $v$ , we can contract all edges of $P_0$ (or $P_1$ ) and replace $v$ by $u_0$ (or $u_1$ ), and we obtain a new series parallel graph with a new vertex of series degree 2.", "Each involved graph in any step of this process is labeled $G_j, 0 \\le j\\le n-1$ , with $G_{n-1} = G$ and $G_1$ is the induced subgraph on the two vertices $v_0$ and $v_1$ .", "Let $G_1$ and $G_2$ be two graphs with $e_j$ an edge of $G_j, j = 1, 2$ .", "The 2-sum of $G_1$ and $G_2$ , denoted $G_1\\oplus _e G_2$ , based on the edges $e_1$ and $e_2$ is the graph obtained by identifying $e_1$ and $e_2$ on an edge $e$ , and keeping $G_j/e_j, j = 1, 2$ , as it is.", "We say that MAX CS CUT is linear for a class of graphs if there is a linear time algorithm to solve it in such class.", "The remaining of the paper is organized as follows: in section 2, we give a linear time algorithm for MAX CS CUT in series parallel graphs, in section 3, we prove that 2-sums preserve the linearity of MAX CS CUT.", "We deduce a linear time algorithm for MIN CUT in series parallel graphs in section 4, and we conclude in section 5." ], [ "MAX CS CUT is linear for series parallel graphs", "MAXCSCUTSP Algorithm: Input: A series parallel graph $G = (V, E)$ with a series labeling of $V$ , a positive weight function $w$ defined on $E$ .", "Output: A $w$ -maximum connected sides cut $\\Omega $ in $G$ .", "0) Begin 1) $j := n-1$ ; 2) While $j > 1$ do 3) Begin 4) Let $P_0$ and $P_1$ be the two parallel closures incident to $v_j$ in $G_j$ : 5) If $w(P_0) > w(P_1)$ then contract $P_1$ ; 6) Else: contract $P_0$ ; 7) $j := j-1$ ; 6) End of While 7) $j := 2$ ; 8) $\\Omega := E(G_1)$ ; 9) While $j\\le n-1$ do 10) Begin 11) Let $P_0$ and $P_1$ the two parallel closures incident to $v_j$ in $G_j$ : 12) If $w(P_0)+w(P_1)>w(\\Omega )$ then $\\Omega := P_0\\cup P_1$ ; 13) $j := j+1$ ; 14) End of While 15) End of MAXCSCUTSP algorithm.", "This algorithm has two phases: Phase I (steps 1-6) and Phase II (steps 7-14).", "In each step, we do roughly $n$ operations, so the complexity of MAXCSCUTSP is $O(n)$ , where $n=|V|$ .", "Theorem 2.1 MAXCSCUTSP algorithm solves MAX CS CUT in series parallel graphs.", "The summary of the algorithm is as follows: MAXCSCUT chooses a vertex v with series degree 2 (step 4) and contract the less weighted parallel closure incident to v (steps 5 and 6).", "And so on the resulted graph until it reaches $G_1$ , the starting single parallel closure (Phase I).", "In $G_1$ , the $w$ -maximum connected sides cut is $E(G_1)$ (step 8).", "After that, it goes in the reverse path (Phase II): the $w$ -maximum connected sides cut is either the trivial cut based on the current vertex $v_j$ with series degree 2 or the current computed connected sides cut (step 12).", "Let $v_j$ be the chosen vertex with series degree 2 in $G_j$ , $P_0$ and $P_1$ the two parallel closures incident to $v_j$ .", "Without loss of generality, we can suppose that $w(P_0)<w(P_1)$ and $G_{j-1}=G_j/P_0$ .", "Let $\\Omega _j$ be the w-maximum connected sides cut in $G_j, 1\\le j\\le n-1$ .", "It suffices to prove that $w(\\Omega _j) = Max \\lbrace w(\\Omega _{j-1}), w(P_0\\cup P_1)\\rbrace $ .", "Let $\\Omega $ be a connected sides cut in $G_j$ distinct from $P_0\\cup P_1$ .", "Since $w(P_0)<w(P_1)$ , we have only two cases: Case 1: $P_1\\subseteq \\Omega $ then $\\Omega $ is a connected sides cut in $G_{j-1} = G_j/P_0$ containing $P_1$ .", "And vice versa, any connected sides cut in $G_{j-1} = G_j/P_0$ containing $P_1$ is a connected sides cut in $G_j$ containing $P_1$ .", "Case 2: $P_1\\nsubseteq \\Omega $ then $\\Omega $ is a connected sides cut in $G_{j-1}=G_j/P_0$ not containing $P_1$ .", "And vice versa, any connected sides cut in $G_{j-1}=G_j/P_0$ not containing $P_1$ is a connected sides cut in $G_j$ not containing $P_1$ .", "So the connected sides cuts candidates for the $w$ -maximum connected sides cut in $G_j$ and $G_{j-1}$ are the same, except $P_0\\cup P_1$ .", "Note that MAXCSCUT algorithm solves MAX CS CUT in series parallel graphs even for arbitrary sign weight functions." ], [ "2-sums preserve linearity of MAX CS CUT", "Let $\\mathcal {C}(G)$ be the class of connected sides cuts of $G$ .", "We need the following lemma.", "Lemma 3.1 $\\mathcal {C}(G_1\\oplus _e G_2)=\\lbrace \\Omega _j\\in \\mathcal {C}(G_j) : e_j\\notin \\Omega _j, j=1, 2\\rbrace \\cup \\lbrace \\Omega _1\\oplus _e \\Omega _2 : \\Omega _j\\in \\mathcal {C}(G_j)$ and $e_j\\in \\Omega _j, j = 1, 2\\rbrace $ .", "It follows that a $w$ -maximum connected sides cut in $G_1\\oplus _e G_2$ is one of the three following connected sides cuts: (cases 1-2) one of the two $w$ -maximum connected sides cuts in $G_j$ which does not contain $e_j, j = 1, 2$ , (case 3) or the 2-sum of the $w$ -maximum connected sides cuts containing $e_j, j = 1, 2$ .", "To find a $w$ -maximum connected sides cut in $G_j$ which does not contain $e_j, j = 1, 2$ (case 2), we have to contract $e_j$ .", "We need then to perform at most $c(n_j-1)$ operations, where $c$ is the linearity coefficient and $n_j, j=1, 2$ is the number of vertices of $G_j$ (by induction).", "To find $\\Omega _1\\oplus _e \\Omega _2$ (case 3), we have to put $w(e_j), j = 1, 2$ , as big as possible, e.g.", "sum of the positive weights of all edges, and find $\\Omega _j, j = 1, 2$ .", "In this case, we need to perform at most $c(n_1+n_2)$ operations (by induction).", "So we have to compute MAX CS CUT twice in each graph and compare three cuts.", "The total number of operations is bounded then by $2c(n_1+n_2-1)=2c(n-1)$ , where $n$ is the number of vertices of $G_1\\oplus _e G_2$ .", "So linearity of the problem is preserved." ], [ "MIN CUT is linear for series parallel graphs", "MINCUTSP Algorithm: Input: A series parallel graph $G = (V, E)$ with a series labeling of $V$ , a positive weight function $w$ defined on $E$ .", "Output: A $w$ -minimum connected sides cut $\\Omega $ in $G$ .", "We keep the same steps as MAXCSCUTSP algorithm except the following changes in two steps: 5) If $w(P_0) < w(P_1)$ then contract $P_1$ ; 12) If $w(P_0)+w(P_1)<w(\\Omega )$ then $\\Omega := P_0\\cup P_1$ ; Since this algorithm is similar to MAXCSCUTSP, then its complexity is $O(n)$ , where $n=|V|$ .", "And it is not difficult to see, similarly to MAXCSCUTSP, that MINCUTSP gives the minimum weighted connected sides cut in a series parallel graph without computing the maximum flow.", "We can conclude with the following result.", "Theorem 4.1 Given a connected graph $G=(V, E)$ and a positive weight function $w$ defined on $E$ .", "Then any $w$ -minimum cut is a connected sides cut of $G$ .", "Let $\\delta (U)$ be a cut with $G[U]$ disconnected.", "It suffices to prove that $\\delta (U)$ is not a $w$ -minimum cut.", "Let $G[U_1]$ be one connected component of $G[U]$ .", "Since $G$ is connected, then $w(V\\backslash U, U_1)>0$ (i.e.", "there are edges between $V\\backslash U$ and $U_1$ ).", "It follows that $w(\\delta (U\\backslash U_1))=w(\\delta (U))-w(V\\backslash U, U_1)<w(\\delta (U))$ .", "Another consequence of Lemma 3.1 and Theorem 4.1 is the following corollary.", "Corollary 4.2 2-sums preserves the linearity of MIN CUT." ], [ "Conclusion", "We have introduced a new variant of MAX CUT: MAX CS CUT, which is also NP-hard.", "We have provided two linear time algorithms for MAX CS CUT and MIN CUT, respectively, in series parallel graphs.", "We have proved that 2-sums preserve the linearity of MAX CS CUT and MIN CUT.", "Further directions are to study MAX CS CUT in larger classes of graphs than series parallel graphs.", "Acknowledgements The author is grateful to the deanship of Scientific Research at Al Imam Mohammad Ibn Saud Islamic University (IMSIU) for supporting financially this research under the grant No 331203." ] ]
1606.05240
[ [ "The dustier early-type galaxies deviate from late-type galaxies' scaling\n relations" ], [ "Abstract Several dedicated surveys focusing on early-type galaxies (ETGs) reveal that significant fractions of them are detectable in all interstellar medium phases studied to date.", "We select ETGs from the Herschel Reference Survey that have both far-infrared Herschel and either HI or CO detection (or both).", "We derive their star formation rates (SFR), stellar masses and dust masses via modelling their spectral energy distributions.", "We combine these with literature information on their atomic and molecular gas properties, in order to relate their star formation, total gas mass and dust mass on global scales.", "The ETGs deviate from the dust mass-SFR relation and the Schmidt-Kennicutt relation that SDSS star forming galaxies define: compared to SDSS galaxies, ETGs have more dust at the same SFR, or less SFR at the same dust mass.", "When placing them in the M*-SFR plane, ETGs show a much lower specific SFR as compared to normal star-forming galaxies.", "ETGs show a large scatter compared to the Schmidt-Kennicutt relation found locally within our Galaxy, extending to lower SFRs and gas mass surface densities.", "Using an ETG's SFR and the Schmidt-Kennicutt law to predict its gas mass leads to an underestimate.", "ETGs have similar observed-gas-to-modelled-dust mass ratios to star forming-galaxies of the same stellar mass, as well as they exhibit a similar scatter." ], [ "Introduction", "Cosmological simulations predict a two-phase galaxy formation scenario: an early epoch where stars formed within the galaxy, and a later epoch where merging and satellite accretion become important in building the mass of the parent galaxy, often forming an early–type galaxy , , .", "analyze a large sample of ETGs with SDSS and find the early epoch of star formation to be governed by internal processes, while later phases of star formation activity are influenced by the environment, supporting the two-phase galaxy formation scenario.", "Additional evidence for a two-phase galaxy formation process comes from observations of globular clusters and metallicity gradients in ETGs , .", "The occurrence of gravitational interactions and satellite accretion have thus changed the long-standing picture of ETGs being simple stellar systems, passively evolving and devoid of interstellar medium (ISM) with only traces of dust and gas.", "The importance of minor merging and satellite accretion in driving a significant amount of star formation activity in massive galaxies has been established in , .", "The star formation and satellite accretion histories in ETGs lead to complex ISM histories.", "Multiwavelength observational studies have unveiled these complex ISM histories in ETGs, with the detection of significant amounts of gas and/or dust.", "The detection of dust , , , , , and gas [3], , , , , as well as several stellar tidal features , , lend credence to both an external and an internal origin of the ISM in ETGs.", "Internal sources are primarily the evolved mass-losing stars that dominate the stellar populations of ETGs polluting the ISM with enriched gas raising their dust abundance , .", "External sources are material accreted through gravitational interactions , , leading to a contribution of stars, gas and dust to the parent galaxy, probably driving star formation in these systems as studied in .", "An interplay between an internal origin of the dust found in ETGs from evolved mass-losing stars that formed during an external event, i.e.", "a past gravitational interaction event, has been suggested for NGC 4125 .", "NGC 5128 is the most prominent nearby example of an ETG with a rich ISM .", "What is prominent in NGC 5128 is the gaseous dusty disk, which has been associated with the remnant of an accreted gas–rich galaxy.", "Moreover, star formation is ongoing in NGC 5128 in the form of Hii regions .", "Going beyond NGC 5128, recent or current star formation activity has been detected directly in nearby ETGs with Hubble Space Telescope and GALEX observations , , as well as in the form of Hii regions , while Spitzer IRS spectroscopy has unveiled PAH emission in the dustier ETGs , , .", "The ongoing star formation activity clearly observed in many ETGs together with the richness of the ISM compel us to investigate their relation.", "An important physical parameter to constrain chemical evolution models of galaxies is the gas–to–dust (G/D) mass ratio , which shows the amount of metals locked up in dust.", "Even though several phases of the ISM of ETGs have been detected, their G/D mass ratio is still an uncertain physical parameter .", "Yet, while the bulk of the stars in ETGs were formed in the early Universe, their assembly continues throughout cosmic time , and understanding how their gas and dust relate provides an important constraint to their evolution.", "With the availability of Herschel observations necessary to constrain the cold dust properties, complemented with surveys focusing on other ISM tracers, this paper aims at exploring the relation among dust, gas and star formation activity in ETGs." ], [ "Sample & analysis", "The present galaxy sample is drawn from the Herschel Reference Survey .", "This is a volume-limited and K$_{S}$ –band flux-limited survey of 323 galaxies in the local universe observed initially with the Herschel Space Observatory with all three SPIRE bands.", "The SPIRE photometry for the whole HRS galaxy sample is presented by and the PACS observations of the HRS were followed up by .", "The sample includes galaxies of all morphological types that lie within a distance between 15 Mpc to 25 Mpc, and have a 2MASS K$_{S}$ –band total magnitude of less than or equal to 12 mag for the late–type galaxies and 8.7 mag for the ETGs.", "The selection criteria have been optimized to study the ISM content of more quiescent galaxies that have a lower limit in the dust mass of $\\sim $ 10$^{4}$  M$_{\\odot }$ .", "Many additional observations across the electromagnetic spectrum complement the HRS galaxy sample.", "For example H$\\alpha $ imaging is presented by , and SDSS and GALEX imaging by .", "Moreover, HRS galaxies observed in atomic hydrogen and in CO emission have been homogenized and presented by .", "Thus, the HRS galaxy sample is unique and complete in terms of wavelength coverage, making it well-suited to understand the ISM properties of local galaxies.", ", study the molecular and total gas scaling relations, as well as the effect of the environment on the HRS data set.", "The present study focuses on the ETG–sample in the HRS with a cold dust component present.", "There is a total of 62 ETGs in the HRS, first studied by .", "From these, we select those ETGs that have, first, a Herschel detection in all three SPIRE bands and, second, either an Hi detection or a CO detection or both.", "The SPIRE detection in all three bands is required to examine the cold dust component and derive accurate dust masses.", "There are 18 ETGs detected in the far-infrared (FIR)/sub-millimeter (submm) with all three SPIRE bands, which also have Hi or CO: 14 ETGs have Hi detection, and 6 ETGs are detected in CO, of which 2 have both Hi and CO detection.", "These galaxies are listed in Table REF , along with their global properties.", "We use all available imaging for these 18 ETGs in order to construct their global multiwavelength spectral energy distributions (SEDs).", "apply the model of to the IR SEDs (from 8$\\mu $ m to 500$\\mu $ m) of HRS galaxies, focusing on the sub-sample of gas-rich late-type galaxies.", "also study the dust properties of HRS galaxies and derive dust masses from only three SPIRE bands and employing empirical relations.", "analyze the dust properties of the 62 ETGs in the HRS using single-temperature modified blackbody fits to their dust SED.", "use a wavelength coverage ranging from 60$\\mu $ m to 500$\\mu $ m, using images for their sample wherever available from IRAS (60 and 100$\\mu $ m), ISO (60-200$\\mu $ m; ), Spitzer (70-160$\\mu $ m; ), PACS (100 and 160$\\mu $ m; Herschel Virgo Cluster Survey ), and SPIRE (250-500$\\mu $ m; ).", "In our study, we model the full SEDs using two SED models and the same apertures across all wavelengths, in order to derive the star formation and dust properties of the 18 ETGs.", "Modelling the panchromatic SED using the same extraction aperture throughout the spectrum can provide a consistent constraint to interpret the physical properties of a galaxy .", "The available data set to construct the SED covers from the ultraviolet (UV) to the FIR/submm regime, and consists of imaging in the GALEX , SDSS , 2MASS , WISE , Spitzer IRAC and MIPS , and Herschel PACS and SPIRE bands.", "The available imaging is drawn from several sources/archiveshttp://irsa.ipac.caltech.edu/images.html and http://hedam.lam.fr/HRS/: from for GALEX; the IRAC/WISE/2MASS imaging is accessed through the NASA/ IPAC Infrared Science Archive (IRSA); the MIPS three-band imaging from ; the Herschel PACS 100 $\\mu $ m and 160 $\\mu $ m photometry are drawn from , while the Herschel SPIRE images from .", "We process all the available multiwavelength imaging using IMAGECUBE and performing image registration, convolution to the same point spread function (PSF), and regridding to a common pixel size.", "In brief, the available imaging is initially converted to a common unit (Jy/pixel) and registered to the same world coordinate system.", "Then, we background subtract all images and correct the UV–to–MIR imaging for Galactic foreground extinction: $F_{\\rm i} = F_{\\rm obs} 10^ {0.4 A_{\\rm \\lambda }},$ where F$_{\\rm i}$ is the flux density corrected for foreground extinction, F$_{\\rm obs}$ is the observed flux density, and A$_{\\rm \\lambda }$ is the extinction at the wavelength ${\\rm \\lambda }$ , estimated using the extinction law by improved by in the infrared.", "In the case of the GALEX photometry, we correct for the Galactic foreground extinction using the selective extinction A$_{FUV}/$ E(B-V)=8.29 and A$_{NUV}/$ E(B-V)=8.18 .", "Table: Wavelength coverage and adopted flux calibration uncertainties.Subsequently, we convolve all imaging to a common PSF, that of SPIRE 500$\\mu $ m, using Gaussian kernels with appropriate full width at half-maximum (FWHM) to match that of SPIRE 500$\\mu $ m [1].", "Finally, we re–sample the images to the same pixel size of 15$$ .", "The uncertainties are estimated from the convolved images, in the same way as described by [2].", "No colour corrections are applied to the WISE, Spitzer, and Herschel images (or any other imaging data set used here), as these are inserted as an additional source of uncertainty.", "In estimating the photometric uncertainties, we also take into account the calibration uncertainties listed in Table REF .", "We add all uncertainty components (calibration, background variation, colour correction) in quadrature.", "The processed imaging is used to construct the global observed SEDs of the sample ETGs, from which the star formation and dust emission properties are derived via SED modelling.", "We perform photometry in apertures larger than the common PSF used (a FWHM of 43$$ for the SPIRE 500 PSF; [1]), in order to derive statistically independent properties.", "We consistently apply the same aperture to all available imaging, in order to extract the flux densities used to model the ETGs SEDs.", "Table: Semimajor and semiminor axes, and position angles used to extract photometry in this work, denoted with a, b, and PA, respectively, and in , denoted with a C12 _{C12}, b C12 _{C12}, and PA C12 _{C12}, respectively.The apertures used are listed in Table 3.", "Figure: Flux densities from our image processing compared to those of .", "The x–axis shows our flux density, while the y–axis shows the difference between Ciesla's flux density and ours over our flux densities.", "Blue asterisks correspond to SPIRE 250 flux densities, green diamonds to SPIRE 350 flux densities, and red squares to SPIRE 500 flux densities.Fig.", "REF compares the SPIRE flux densities obtained from our image processing with that listed by ; the latter authors have used the beam areas listed in their Table 1.", "Hence, for the comparison shown in Fig.", "REF , we use the same beam areas as the ones used by , while anywhere else in this text we use beam areas 465 arcsec$^{2}$ , 823 arcsec$^{2}$ , and 1769 arcsec$^{2}$ for SPIRE 250$\\mu $ m, 350$\\mu $ m and 500$\\mu $ m, respectivelyhttp://herschel.esac.esa.int/Docs/SPIRE/spire_handbook.pdf.", "For the sake of this comparison, we also use the same photometry apertures as those used by , with the exception of HRS 93, HRS 123, HRS 162, HRS 200, HRS 209, HRS 243, HRS 260, HRS 286 for which the multiwavelength coverage yields smaller field-of-view, as well as HRS 161 and HRS 174 which classify as point sources; we have adjusted our apertures to take this into account, and the apertures used are those listed in Table REF .", "Fig.", "REF shows that the fluxes agree within a mean value of 4%, 7%, and 13%, for the SPIRE 250$\\mu $ m, SPIRE 350$\\mu $ m, and SPIRE 500$\\mu $ m, respectively, while the standard deviation is 15%, 18%, and 25%, respectively for each band.", "The larger standard deviation is contributed by the galaxies which have different apertures than those used by .", "The choice of apertures takes into account the dust emission seen at the SPIRE bands, while in the same time consistently using the same elliptical apertures throughout the whole bands available, which drives our final choice of apertures independently of what is presented by .", "The molecular hydrogen masses for the sample are adopted from and listed in their Table 1.", "These authors collect and homogenize for the whole HRS sample all available observations in CO emission, which are adapted either from the literature or from their own observations.", "To convert the CO emission to molecular hydrogen masses, a conversion factor $X_{\\rm CO}$ is applied.", "use both a constant $X_{\\rm CO}$ and a luminosity–dependent $X_{\\rm CO}$ , the latter adapted from .", "Here, we choose to show results based on the luminosity–dependent $X_{\\rm CO}$ ; using a constant $X_{\\rm CO}$ yields similar trends.", "A luminosity–dependent $X_{\\rm CO}$ is preferred due to the dependence of the $X_{\\rm CO}$ on metallicity , , and taking into account the relation between luminosity and metallicity.", "As most CO surveys target the central part of a galaxy, while CO emission may be found extended beyond thisNGC 5128 is the most prominent nearby ETG with molecular gas detected beyond the central parts, in which detect CO at a distance of 15 kpc from its centre, corresponding to a radius of about 2.7$\\times $ R$_{eff}$ ., have applied a correction for such aperture effects.", "More specifically, they have assumed a three-dimensional distribution for the CO emission, as has been done in the case of the dust distribution in the z-direction in edge-on galaxies in radiative transfer modelling , .", "Finally, compile from the literature the atomic hydrogen masses for the HRS sample with available measurements, and we use their values here.", "For those galaxies which are undetected in CO, the M$_{ \\hbox{H\\,{\\sc i}} }$ dominates over their M$_{H_{2}}$ , as indicated by the latter's upper limits; hence, when we compute the total gas mass, we include those M$_{H_{2}}$ upper limits.", "We compute the total gas mass, M$_{gas}$ , for each galaxy taking into account the atomic gas mass, M$_{ \\hbox{H\\,{\\sc i}} }$ , and the molecular gas mass, M$_{H_{2}}$ : $M_{\\rm gas} = 1.3 (M_{\\hbox{H\\,{\\sc i}}} + M_{H_{2}}).$ The adopted atomic and molecular hydrogen masses are listed in Table REF .", "To obtain the surface densities of the gas mass, dust mass and star formation rate, we divide these quantities by each galaxy's projected area within the adopted flux extraction aperture.", "The surface density of a physical property is preferred over its integrated value so as to remove any dependence on galaxy size." ], [ "SED modelling", "We derive the star formation and dust emission properties of our ETGs sample via modelling of their SEDs using two models: PCIGALE (v0.9.0)http://cigale.lam.fr , , and MAGPHYShttp://www.iap.fr/magphys/magphys/MAGPHYS.html .", "Both these SED models use the energy balance between the absorption by dust and the re-radiation in the IR to fit the multiwavelength SEDs, from UV to FIR/submm part of the spectrum.", "The basic components of MAGPHYS are described by , while PCIGALE is described by .", "One important aspect with modelling the SED with PCIGALE is the inclusion of an active galactic nucleus (AGN) component in the SED, while this implementation is currently lacking in MAGPHYS.", "MAGPHYS uses the stellar population synthesis of to model the star formation histories, assuming both a continuous star formation rate and additional star formation bursts.", "The attenuation law of describes the attenuation of the stellar emission by dust in the diffuse ISM and in the stellar birth clouds.", "No contribution from an AGN to the heating of the ISM is assumed.", "No detailed modelling of the physical properties of the dust grains is performed in MAGPHYS.", "While in MAGPHYS pre-computed libraries of SED models already exist, in PCIGALE the user builds the library of the SED models based on input parameters; for a detailed description on input parameters we refer to , .", "The assumptions entering the modelling with PCIGALE are as follows.", "A double exponentially declining star formation history (SFH), with the functional form as in eq.", "3 in , i.e.", "$SFR(t) = exp(-t/\\tau _{1})$ , if $t < t_{1} - t_{2}$ , and $SFR(t) = exp(-t/\\tau _{1}) + k\\ exp(-t/\\tau _{2})$ , if $t \\ge t_{1} - t_{2}$ , using the stellar population synthesis models of .", "The optical emission is attenuated in the IR following a modified curve, including a UV bump to take into account emission from younger stars as in .", "The dust emission is modelled with the model.", "An AGN component is assumed for a type–I and a type–II AGN, adopting the model of which assumes a smooth distribution for the dust around the AGN.", "Another class of AGN models assumes a clumpy dust distribution, and these two assumptions on the dust distribution can describe several features in the SED of an AGN; a comparison between the two most popular AGN models is given in .", "Table: Input parameter choice for PCIGALE.We list in Table REF the non–default input parameters we have adjusted while running PCIGALE.", "The population synthesis model and the initial mass function are chosen in PCIGALE so as to be common with MAGPHYS.", "Figure: Modelled (best-fitting) and observed SED for NGC 4203 (HRS 93) with PCIGALE shown with the blue line, and with MAGPHYS shown with the red line.", "Black asterisks correspond to observed flux densities, a total of 20 data points used to constrain the SED of NGC 4203.Fig.", "REF shows an example of a best–fit model SED to an ETG of our sample derived with PCIGALE and MAGPHYS.", "An important aspect in both SED models is the adoption of a Bayesian approach where probability density functions are constructed for the physical parameters of interest; the median likelihood is adopted in MAGPHYS and the mean value in PCIGALE as the physical parameter value.", "The 16th–84th percentile range is then used to reflect the associated confidence interval in MAGPHYS , and the standard deviation in PCIGALE .", "We adopt in this study the Bayesian approach built in the SED models, rather than the physical parameters inherent to the best–fit SED model.", "We consider the parameters of the mass of the dust (M$_{D}$ ), the star formation rate (SFR) averaged over the last 10$^{8}$ yr, and the stellar mass (M$_{\\star }$ ).", "We adopt the following naming convention throughout the remaining text: we use a subscript “P” to denote results derived with PCIGALE, while a subscript “M” is used to denote results with MAGPHYS.", "As the SED model libraries in either MAGPHYS or PCIGALE are built based on different assumptions for the input parameters, e.g.", "SFH or dust model assumption or attenuation law, the derived properties with the two modelling techniques are not directly comparable.", "Each of the two SED models defines a unique ”scale” for the derived properties.", "The use of “property scale” is the same in concept as that of metallicity scale: the value of the metallicity (or property) is tied to the method used to derive it; for example, the or the metallicity scale for stellar metallicities derived using the Ca ii triplet, as well as the metallicity scale for gas–phase metallicities.", "Hence, any comparison of the derived properties between the two SED models performed in this study is only illustrative and has the purpose of understanding the intrinsic difference between the two scales, rather than to derive a calibration between them.", "The reason for this is that a calibration between the two SED model scales is not unique, as it depends on the input parameters chosen to build each SED library used in the fitting process.", "Hence, deriving a calibration requires a complete search of the available parameter space of the input parameters in the SED models; this goes beyond the scope of this paper.", "The same reasoning holds for the comparison of the derived properties between each of the two SED models and those assuming a modified blackbody.", "Figure: Comparison between M D _{D} derived in this work to that derived by (upper panel), along with their relative residual, i.e.", "M D,S12 _{D,S12} minus M D _{D} over M D,S12 _{D,S12} (lower panel).MAGPHYS results are shown in grey squares, while PCIGALE results in blue circles.", "Dark-coloured filled symbols indicate ETGs with both H i and CO; light-coloured filled symbols are ETGs with only H i; open symbols are ETGs with only CO.", "The solid line in the upper panel is the one-to-one relation.Fig.", "REF illustrates the difference in M$_{D}$ derived here, grey squares for MAGPHYS and blue circles for PCIGALE, with that derived by using modified blackbody fits to the IR/submm SED (60–to–500 $\\mu $ m), denoted as M$_{D,S12}$ and shown in the upper panel.", "The lower panel shows the relative residuals of the comparison.", "use the SPIRE photometry from with beam areas as listed in Table 1 of the latter.", "Hence, for the comparison between Smith's and our dust masses (both with MAGPHYS and PCIGALE) we have used the same apertures to extract fluxes, as well as the same beam areas as by (see also explanation for Fig.", "REF in previous section).", "In this comparison shown in Fig.", "REF , the mean dust mass from is log$_{10}$ (M$_{D,S12})$ =6.5$\\pm $ 0.5, while the mean dust mass with MAGPHYS is log$_{10}$ (M$_{D,M})$ =6.0$\\pm $ 0.6 and with PCIGALE is log$_{10}$ (M$_{D,P})$ =6.3$\\pm $ 0.5.", "The mean relative residual of the comparison shown in the lower panel of Fig.", "REF is 51%, with a standard deviation of 28% for MAGPHYS, while in the case of PCIGALE the mean relative residual is 18% with a standard deviation of 38%.", "Fig.", "REF shows a systematic tendency of dust masses derived with MAGPHYS to be underestimated as compared to dust masses derived with modified blackbody fits; this seems to be random in the case of PCIGALE.", "study the star formation and dust properties in NGC 3226; they find that dust masses derived with MAGPHYS are underestimated by 30% as compared to dust masses derived with modified blackbody fits.", "Their dust mass estimate with MAGPHYS on NGC 3226 (HRS 3), equal to 6.09 in logarithmic space, is consistent with our finding of 5.8$\\pm $ 0.1 in logarithmic space, considering that we probe a slightly different area.", "Figure: Comparison of the physical property derived with the two SED models: ratio of MAGPHYS property over PCIGALE property as a function of the PCIGALE property.", "The physical properties compared are: M D _{D} (in M ⊙ _{\\odot }) shown with grey circles, SFR (in M ⊙ _{\\odot }yr -1 ^{-1}) shown with red squares, M ☆ _{\\star } (in M ⊙ _{\\odot }) shown with blue diamonds.Fig.", "REF illustrates the difference in the derived properties between PCIGALE and MAGPHYS: M$_{D}$ (in M$_{\\odot }$ ) shown with grey circles, SFR (in M$_{\\odot }$ yr$^{-1}$ ) shown with red squares, M$_{\\star }$ (in M$_{\\odot }$ ) shown with blue diamonds.", "Note that this comparison is performed on beam areas and apertures used in this paper (see Table REF ).", "The dust masses derived with MAGPHYS are underestimated as compared to those derived with PCIGALE, by a factor of 4$\\pm $ 6 on average.", "The comparison of the SFRs derived with the two models shows a large scatter with a variation of the MAGPHYS SFR between 0.02 to 120 times that of PCIGALE.", "The stellar masses derived with MAGPHYS are underestimated as compared to those derived with PCIGALE by a factor of 4.4$\\pm $ 8.6 on average.", "Overall, as stated earlier, the two SED models can be regarded each as defining a separate scale for each physical parameter, with model SED libraries built based on different assumptions, hence they are expected to have differences in the derived parameters.", "use CIGALE on mock galaxies, in order to investigate the influence of an AGN component in modelling their SEDs; they find that including an AGN component in the SED modelling is important to constrain the derived physical properties.", "While PCIGALE includes an AGN component in the SED, this is not currently the case in MAGPHYS.", "There are two ETGs in our sample hosting weak AGN activity , .", "suggest the likely presence of a low-luminosity AGN contribution to the IR for NGC 3226, after modelling its SED using two different methods each including an AGN component; they constrain the AGN contribution to the IR luminosity to be between 20$\\pm $ 5% to 18$\\pm $ 4%.", "Table 5 by suggests that the dust mass of NGC 3226 derived when using MAGPHYS can be underestimated by a factor of 3 as compared to the dust mass derived using one of the SED modelling method that includes an AGN component.", "As stated earlier, in addition to the inclusion or not of an AGN component, different modelling techniques can lead to different dust mass estimates.", "This is the conclusion reached in studies where detailed dust grain modelling versus modified blackbody fits are compared , .", "To this end, discuss, as also points out, that adopting the same mass absorption coefficient of dust ($\\kappa _{\\nu }$ ) and the same dust emissivity index ($\\beta $ ) is important in deriving consistent dust masses when comparing different dust modelling techniques.", "In this aspect, absolute dust masses, and direct comparison between dust mass estimates derived from different modelling techniques, are not as meaningful as is the examination of trends in scaling relations within each of the SED scales, as these are defined earlier in this section.", "Table: Derived properties obtained with MAGPHYS, indicated with an ”M” subscript, and with PCIGALE, indicated with a ”P” subscript, for the ETG sample.Table REF lists the results obtained with PCIGALE and MAGPHYS, along with their associated uncertainties, following the Bayesian approach.", "For these 18 ETGs, the mean and standard deviation of the derived properties are: log$_{10}$ (M$_{D,M})$  = 5.9$\\pm $ 0.6 and log$_{10}$ (M$_{D,P})$  = 6.3$\\pm $ 0.5, log$_{10}$ (M$_{\\star ,M}$ ) = 10.3$\\pm $ 0.4 and log$_{10}$ (M$_{\\star ,P}$ ) = 10.6$\\pm $ 0.2 (both M$_{D}$ and M$_{\\star }$ in M$_{\\odot }$ ), log$_{10}$ (SFR$_{M}$ ) = $-$ 1.7$\\pm $ 0.6 and log$_{10}$ (SFR$_{P}$ ) = $-$ 1.7$\\pm $ 0.6 (M$_{\\odot }$ yr$^{-1}$ ).", "Overall, while taking the mean value and standard deviation of each property indicates consistency between values derived based on each of the two SED models, nevertheless, the comparison shown in Fig.", "REF shows large one-to-one differences among the properties, as discussed earlier." ], [ "Dust, gas, and star formation", "If we attribute the cold dust emission of ETGs as due to recent star formation activity, then we expect the dust mass and SFR to be tracing each other.", "Fig.", "REF shows the dust mass as a function of the SFR, with dust masses derived with MAGPHYS shown in grey squares and with PCIGALE in blue circles.", "study a sample of low–redshift galaxies drawn from the SDSS DR6 main spectroscopic sample with additional GALEX, 2MASS and IRAS wavelength coverage.", "These authors find that the galaxies in their sample follow a tight correlation between dust mass and SFR averaged over the last 10$^{8}$  yr; they suggest this relation as a calibration to derive dust masses of galaxies given their SFRs.", "The relation of is shown in Fig.", "REF with the black solid line, while the intrinsic scatter is shown with the grey solid lines.", "In the case of the ETGs, Fig.", "REF shows that these deviate from this tight relation: the majority of the ETGs are scattered beyond the intrinsic scatter of the relation introduced by .", "In addition, ETGs fall into the lower left regime of Fig.", "5 in , i.e.", "towards lower dust masses and lower SFRs.", "There may be several factors contributing to this scatter seen for the ETGs in Fig.", "REF .", "First, the dust emission is not solely due to recent star formation, but can also be attributed to more evolved stars.", "Such dust emission, though, is not adequate to justify the amount of dust mass detected in these ETGs .", "Then, the interstellar medium in these galaxies may have been externally acquired, via mergers and satellite accretion.", "This latter suggestion is in line with the study of , who find short grain lifetimes in passive elliptical galaxies, assuming their dust mass is solely due to evolved stars.", "suggest that in the case of FIR/submm-detected ETGs the dust mass should be externally acquired.", "By selection, our ETGs fall into this category.", "Figure: Star formation rate, SFR, as a function of stellar mass, M ☆ _{\\star }.", "The solid black line indicates the relation of local galaxies studied in scaled down by a factor of 400, while the grey lines indicate the 0.26 dex rms of the relation.MAGPHYS results are shown in grey squares, while PCIGALE results in blue circles.", "Dark-coloured filled symbols indicate ETGs with both H i and CO; light-coloured filled symbols are ETGs with only H i; open symbols are ETGs with only CO.The error bars represent the 16th–84th percentile range for MAGPHYS and the standard deviation of the probability distribution function for PCIGALE.Fig.", "REF places the ETGs in the SFR versus M$_{\\star }$ plane.", "Normal star-forming galaxies are found to follow a tight correlation in this plane , , .", "The slope of this relation is consistent with $\\sim $ one independent of redshift, but the normalization changes such that at higher redshift the star formation is higher for a given mass .", "The solid line in Fig.", "REF represents a scaled-down (by 400) version of the relation defined for local star forming galaxies studied by , i.e.", "for the scaled down relation SFR $\\propto $  M$_{\\star }$ /[400$\\times $ 4$\\times $ 10$^{9}$ M$_{\\odot }$ ]; the scaling-down has been performed to facilitate the presentation of the figure.", "The grey lines in Fig.", "REF show the 0.26 dex rms scatter of the original relation presented in .", "It is remarkable that this scatter in SFR remains the same across all redshift, suggesting that stellar feedback processes likely play a role to its constancy .", "Fig.", "REF shows that ETGs scatter around the scaled-down version of the relation of , i.e.", "ETGs fall systematically below the original relation found for local star-forming galaxies by , having lower SFRs for a given mass as compared to normal star-forming galaxies.", "In other words, while normal star-forming galaxies in the SFR-M$_{\\star }$ plane have a constant specific SFR, defined as the ratio of SFR with stellar mass, of 0.25Gyr$^{-1}$ , the ETGs in this study have specific SFRs lower by a factor of 30 to 4000 than that of normal galaxies.", "Figure: Upper panel: D/G mass ratio, M D _{D}/ M H _{H} as a function of the stellar mass, M ☆ _{\\star }, assuming a Schmidt-Kennicutt law to derive total gas masses from the SFR.", "Lower panel: G/D mass ratio, M Gas _{Gas}/M D _{D}, using observed total gas masses.", "The solid red lines indicate the range of values found for the G/D mass ratio as a function of M ☆ _{\\star } by .In both panels, MAGPHYS results are shown in grey squares, while PCIGALE results in blue circles.", "Dark-coloured filled symbols indicate ETGs with both H i and CO; light-coloured filled symbols are ETGs with only H i; open symbols are ETGs with only CO.The error bars represent the 16th–84th percentile range for MAGPHYS and the standard deviation of the probability density function for PCIGALE.", "investigated the dust–to–gas (D/G) mass ratio, M$_{D}$ / M$_{H}$ , using the ratio of M$_{D}$ to SFR.", "The assumption is that the SFR can be used to infer the gas mass, using a Schmidt-Kennicutt law.", "We show the D/G mass ratio as a function of stellar mass in the upper panel of Fig.", "REF , using the SFR from the SED modelling and the Schmidt-Kennicutt law to obtain the gas mass.", "We have assumed the same star formation law as that used by , i.e.", "$\\Sigma _{SFR}= (2.5\\pm 0.7)\\times 10^{-4}\\Sigma _{H}^{1.4\\pm 0.15}$ , where $\\Sigma _{SFR}$ is the surface density of SFR, and $\\Sigma _{H}$ is the surface density of the (HI+H$_{2}$ ) gas mass.", "We have applied the same factor as by to correct from a to a initial mass function.", "The stellar mass can also be used to infer the metallicity, given their relation , hence the D/G mass ratio as a function of stellar mass can indirectly provide information regarding the D/G mass ratio as a function of metallicity.", "This may be useful given the difficulty of obtaining gas-phase metallicities for a large sample of ETGs, including the current sample, while SED modelling can provide reliable information regarding stellar masses.", "Fig.", "REF shows that there is no clear correlation between stellar mass and D/G mass ratio in the case of ETGs; there is a higher scatter among the ETGs than that observed for the star-forming galaxies in Fig.", "8 of .", "This is the case for both results obtained with either MAGPHYS or PCIGALE.", "Fig.", "B.3 in shows the G/D mass ratio as a function of stellar mass for 126 local galaxies.", "Their dust masses are modelled as in , their gas masses are compiled from the literature, and their stellar masses are derived using the formula of .", "Their sample includes galaxies from the Dwarf Galaxy Survey , and the KINGFISH survey , .", "We derive a mean G/D mass ratio, $\\langle $ log$_{10}$ (M$_{D,M}$ /M$_{H})\\rangle $  = -1.1 $\\pm $  0.5, and $\\langle $ log$_{10}$ (M$_{D,P}$ /M$_{H})\\rangle $  = -0.8 $\\pm $  0.6, where the quoted uncertainties reflect the standard deviation.", "With a mean stellar mass of $\\langle $ log$_{10}$ (M$_{\\star ,M})\\rangle $  = 10.3 $\\pm $  0.4 M$_{\\odot }$ and $\\langle $ log$_{10}$ (M$_{\\star ,P})\\rangle $  = 10.6 $\\pm $  0.2 M$_{\\odot }$ , the ETGs fall in the lower right part of Fig.", "B.3 by , below their galaxies at that location, reflecting the lower gas reservoirs inferred from the SFR as compared to their dust masses.", "The lower panel of Fig.", "REF shows the observed-gas-to-modelled-dust mass ratio as a function of the stellar mass.", "In this panel, we choose to present the G/D mass ratio for a direct comparison with .", "The mean observed G/D mass ratio is $\\langle $ log$_{10}$ (M$_{Gas}$ /M$_{D,M})\\rangle $  = 2.8 $\\pm $  0.5, and $\\langle $ log$_{10}$ (M$_{Gas}$ /M$_{D,P})\\rangle $  = 2.4 $\\pm $  0.6, now placing the ETGs at a location well within the galaxies studied in Fig.", "B.3 by .", "The lower panel of Fig.", "REF shows a large scatter, as well.", "This large scatter is consistent with the idea that the ISM in ETGs is acquired from an external source.", "If the ISM is acquired from a satellite, then the G/D mass ratio would indicate the properties of the ISM in the accreted satellite, showing no correlation with the properties of the ETG itself, as is the case in both panels in Fig.", "REF .", "Even though the scatter of the G/D mass ratio as a function of M$_{\\star }$ seen in Fig.", "REF for the ETGs is large, nevertheless when they are placed in Fig.", "B.3 of the trend of the G/D mass ratio decreasing with increasing M$_{\\star }$ emerges, making the latter a useful tool to approximate the G/D mass ratio of ETGs knowing their M$_{\\star }$ .", "On the other hand, using a Schmidt-Kennicutt law similar to that describing disk galaxies together with an estimate of SFR through SED modelling, for example, underestimates the total gas mass and the G/D mass ratio of ETGs.", "We now explore the Schmidt-Kennicutt law for the ETGs.", "Figure: Surface density of star formation rate, ∑ SFR \\sum _{SFR}, versus surface density of the total gas mass, ∑ M Gas \\sum _{M_{Gas}}.", "MAGPHYS results are shown in grey squares, while PCIGALE results in blue circles.", "Crosses correspond to galaxies taken from .", "Dark-coloured filled symbols indicate ETGs with both H i and CO; light-coloured filled symbols are ETGs with only H i; open symbols are ETGs with only CO.The dark–grey thick solid line is the Schmidt-Kennicutt law used here and by .", "The light–grey thick solid line is the internal Milky Way star formation law from .", "The thin light-grey lines indicate the star formation efficiency per 10 8 ^{8}yr for values 0.1%, 1%, 10%, and 100%.The grey error bars in all the data points represent the 16th–84th percentile range for MAGPHYS and the standard deviation of the probability density function for PCIGALE.In Fig.", "REF we show the surface density of the star formation rate, $\\sum _{SFR}$ as a function of the surface density of the total gas mass, $\\sum _{M_{Gas}}$ , for the 18 galaxies in our sample.", "The thick dark–grey solid line is the Schmidt-Kennicutt law used here and by ; the thick light–grey solid line is the internal Milky Way star formation law from ; The thin light-grey lines indicate the star formation efficiency per 10$^{8}$ yr for values 0.1%, 1%, 10%, and 100%.", "obtained the Schmidt–Kennicutt law locally within the Milky Way and derived a power–law index equal to 2.18$\\pm $ 0.20 (see eq.", "16 of ).", "called this star formation law for the Milky Way an internal Schmidt law.", "The ETGs of our sample extend towards lower gas masses and SFRs and show a larger scatter as compared to the internal Milky Way star formation law of , or the galaxies studied by shown in Fig.", "REF .", "The majority of the ETGs have star formation efficiency per 0.1 Gyr between 1% to 0.1%, i.e.", "lower than that of normal galaxies.", "In addition, the ETGs are offset from the Schmidt-Kennicutt law, independent of the SED model used to derive SFRs.", "investigate the Schmidt-Kennicutt law via simulations of gas disks embedded in ETGs, and reach similar conclusions regarding the star formation efficiency and scatter, signifying the process of morphological quenching .", "The high scatter and the deviation from both the internal Milky Way star formation law and the Schmidt-Kennicutt law followed by disk galaxies indicate the complex star formation and ISM histories of the ETGs.", "Indeed, the majority of the galaxies studied here show many morphological signs of tidal interactions in FIR/submm imaging.", "Past studies investigating the star formation law of ETGs include the work of , , and .", "The latter authors study molecular gas-rich ETGs, and investigate their star formation law using their total gas mass and SFR tracer based on FUV+22$\\mu $ m. These authors find overall higher SFR surface densities for their ETGs, comparable to those of normal spiral galaxies (see their Fig.", "5), but they lie systematically lower than a Schmidt-Kennicutt law, indicating lower star formation efficiency.", "Lower star formation efficiency is also found for molecular gas-rich dust-lane ETGs by , who suggest that the lower star formation efficiency is due to a suppression of star formation by minor mergers.", "They assume that the gas and dust in these dust-lane ETGs comes from the accreted galaxy through minor merging, and they use the observed gas–to–dust mass ratio as a function of stellar mass to infer the stellar mass of the accreted galaxy .", "For the dustier galaxies studied here, given the scatter of the cold dust–rich ETGs when placing them in the Schmidt-Kennicutt law, their SFR is not a good tracer of their gas mass.", "The SFR underestimates ETGs' gas mass in the case of a power–law index appropriate for disk galaxies, as used by and here for the upper panel of Fig.", "REF .", "Figure: Upper panel: M D _{D}/SFR as a function of the total gas mass.", "Lower panel: M D _{D} as a function of the total gas mass.In all panels, MAGPHYS results are shown in grey squares, while PCIGALE results in blue circles.", "Dark-coloured filled symbols indicate ETGs with both H i and CO; light-coloured filled symbols are ETGs with only H i; open symbols are ETGs with only CO.The grey error bars represent the 16th–84th percentile range for MAGPHYS or the standard deviation for PCIGALE.Indeed, the upper panel of Fig.", "REF shows that there is no correlation between the ratio M$_{D}$ /SFR and the total gas mass, M$_{Gas}$ .", "For a given value of the total gas mass, there is between 1 to 3 dex difference in the ratio M$_{D}$ /SFR.", "This latter ratio, as a property derived from the SED modelling, may serve as a practical proxy to infer indirectly the D/G mass ratio, as discussed earlier (by assuming that the gas mass is derived through the SFR assuming a Schmidt–Kennicutt law).", "In the case of ETGs, though, this ratio provides a poor proxy to estimate their D/G mass ratio.", "In the lower panel of Fig.", "REF we explore how the observed total gas mass M$_{Gas}$ relates to the dust mass, M$_{D}$ .", "This figure reveals that there is no trend between the dust mass and the gas mass, as for a given M$_{Gas}$ there is between 1 to 4 dex scatter in M$_{D}$ .", "These results suggest that assuming a dust mass for ETGs will not provide a good estimate of their M$_{Gas}$ ." ], [ "Summary & conclusions", "We analyze UV–to–submm imaging for a sample of 18 early-type galaxies, drawn from the Herschel Reference Survey , in order to derive their star formation and dust properties via SED modelling.", "The sample is selected to be detected both in FIR/submm with all three Herschel/SPIRE bands, and in either H i or CO or both.", "We combine the dust and star formation results with literature information on their atomic and molecular gas masses , in order to relate the dust and star formation properties with the total gas mass in these ETGs.", "Our major findings are summarized as follows: When the ETGs are compared to the M$_{D}$ –SFR relation for disk galaxies studied by , they do not follow the disk galaxies' correlation and show a large scatter beyond the intrinsic scatter of the relation (Fig.", "REF ).", "When placing the ETGs in the stellar mass versus SFR plane, they fall below the relation defined by normal star forming galaxies, showing much smaller specific SFRs, by a factor between 30 to 4000 smaller than that of the normal star-forming galaxies (Fig.", "REF ).", "The ETGs studied here do not follow the same Schmidt–Kennicutt law as the disk galaxies studied by , neither do they follow the internal Schmidt law found locally within Milky Way by , showing a large scatter and extending to lower SFRs and gas masses (Fig.", "REF ).", "The Schmidt–Kennicutt law valid for disk galaxies , can not be used along with the SFR to infer the total gas mass of ETGs, as the total gas masses inferred are under-predicted (upper panel of Fig.", "REF ).", "Contrary to the case of disk galaxies, this renders the ratio M$_{D}$ /SFR a poor proxy for inferring the D/G mass ratio of ETGs (Fig.", "REF ).", "The total gas masses derived observationally (i.e., using eq.", "2) provide G/D mass ratios consistent, on average, with the ones found for the galaxy sample studied by , and fall in the right place for their M$_{\\star }$ , on average, as those galaxies studied by , making the latter a useful relation to approximate the G/D mass ratio of ETGs (lower panel of Fig.", "REF ).", "The large scatter seen in individual ETGs in Fig.", "REF , though, indicates an externally acquired origin for their ISM.", "ETGs have complex ISM and star formation histories.", "This complexity is a result of their merging and satellite accretion history.", "The majority of the ETGs studied here show morphology of recent interactions.", "This is not uncommon among ETGs , and their ISM history coupled to the satellite accretion history has been uncovered with deep optical imaging from the ATLAS$^{3D}$ survey , as well as imprinted in their colour properties .", "Such interaction events may be the cause of star formation responsible for significant dust production , hence their star formation history coupled to their ISM history leads to deviation or significant scatter from a Schmidt–Kennicutt law relation." ], [ "Acknowledgements", "We are grateful to an anonymous referee for useful suggestions, which helped us to improve the manuscript.", "Support for the work of SL and PB is provided by an NSERC Discovery Grant and by the Academic Development Fund of the University of Western Ontario.", "SL thanks Médéric Boquien for support with PCIGALE, and Pierre-Alain Duc for useful discussions.", "SL, EX, and SM acknowledge support from DustPedia, a collaborative focused research project supported by the European Union under the Seventh Framework Programme (2007-2013) call (proposal no.", "606847), with participating institutions: Cardiff University, UK; National Observatory of Athens, Greece; Ghent University, Belgium; Université Paris Sud, France; National Institute for Astrophysics, Italy and CEA (Paris), France.", "This research has made use of data from HRS project.", "HRS is a Herschel Key Programme utilizing Guaranteed Time from the SPIRE instrument team, ESAC scientists and a mission scientist.", "The HRS data was accessed through the Herschel Database in Marseilles (HeDaM; http://hedam.lam.fr) operated by CeSAM and hosted by the Laboratoire d'Astrophysique de Marseilles.", "This research made use of several facilities and open source software: Astropy , a community-developed core Python package for Astronomy; APLpy (http://aplpy.github.com), an open-source plotting package for Python; IRAF, distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation; STSDAS and PyRAF, products of the Space Telescope Science Institute, which is operated by AURA for NASA; NASA/IPAC Extragalactic Database (NED), operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration; NASA/IPAC Infrared Science Archive (IRSA), operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration; Aladin; NASA's Astrophysics Data System Bibliographic Services; SAOImage DS9, developed by Smithsonian Astrophysical Observatory; the Spanish Virtual Observatory (http://svo.cab.inta-csic.es) supported from the Spanish MICINN / MINECO through grants AyA2008-02156, AyA2011-24052.", "This work made extensive use of the free software GNU Octave and the authors are grateful to the Octave development community for their support." ] ]
1606.04867
[ [ "Process-theoretic characterisation of the Hermitian adjoint" ], [ "Abstract We show that the physical principle \"the adjoint associates to each state a `test' for that state\" fully characterises the Hermitian adjoint for pure quantum theory, therefore providing the adjoint with operational meaning beyond its standard mathematical definition.", "Also, we show that for general process theories, which all admit a diagrammatic representation, this physical principle induces a reflection operation." ], [ "Introduction", "The process theoretic approach [1], [17], [18] provides a novel perspective on the structure of quantum theory by focusing on compositionality.", "Moreover, it leads to a diagrammatic representation of quantum processes, simplifying calculations and providing an intuitive way to reason about quantum theory [11], [13].", "Furthermore, process-theoretic ideas have been: adopted within other frameworks and corresponding reconstruction theorems [21], [9], [22], [10]; provided a natural setting for theories other than quantum theory [15] which has led to, for example, the study of connections between physical principles and computation [25], [26]; helped solve open problems in quantum computing e.g.", "[20], [7], [23]; and provided a general framework for resource theories [16].", "However, one particular diagrammatic ingredient that plays a central role in quantum theory lacks a clear physical interpretation, namely, the diagrammatic counterpart to the Hermitian adjoint.", "Earlier work has introduced dagger process theories [2], [27] as a generalisation of the adjoint to other theories.", "However, this is not entirely satisfactory, raising three questions which we answer in this paper.", "Firstly, why should a theory have have a dagger at all?", "Section 4 introduces the concept of a test structure, associating to each state a test for that state.", "Section 5 shows that this leads to a dagger.", "Secondly, why is it always assumed that the dagger should be involutive?", "Section 5 demonstrates how this is a consequence of the definition of the test structure.", "Finally, why is the particular dagger of interest in quantum theory the Hermitian adjoint, rather than some other dagger such as the transpose?", "Section 6 shows that for quantum theory the test structure forces the dagger to be the Hermitian adjoint.", "To finish in section 7 we consider process theories with a notion of probabilistic mixing, and show that such theories typically fail to have a test structure.", "The reason for this is that `testability' is tightly intertwined with that of `purity'.", "This provides a new perspective on what it means for states to be pure – they are pure if they are `testable'." ], [ "Related Work", "The physical principle used in this work is closely related to the `logical sharpness' axiom used by Hardy in the GPT setting [22].", "A similar approach to ours was recently suggested by Chiribella [8] who relates the adjoint to time-reversal symmetry.", "By taking this as an axiom he has shown that the adjoint must be a process theoretic dagger.", "This however does not uniquely pick out the Hermitian adjoint as a dagger; for example, it also allows for the transpose.", "Vicary [28] also characterised the adjoint in process-theoretic terms.", "However, he required extremely strong assumptions, namely that all dagger-limits exist, which isn't ever the case if one eliminates redundant global phases from quantum theory." ], [ "Process theories", "A process theory consists of: a collection of `systems', these could represent a physical system such as a photon, but also a mathematical object such as a vector space, or some computational notion such as a data type; a collection of `processes', each of which has an input consisting of a set of systems and another set of systems as an output, these could represent a physical process such as a parametric down converter (this would have a single photon as its input, and a pair of photons as its output), but again, these processes can also be mathematical processes such as a linear transformation, or computational processes such as a sorting algorithm; a composition operation for systems and processes.", "This composition operation can be best understood in terms of a diagrammatic notation for systems and processes.", "In this notation, systems are represented by labelled wires, for example: $\\text{single system}:\\quad \\ ,\\qquad \\quad \\text{composite system}:\\quad \\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (0, -1) {};\\node [style=none] (1) at (0, 1) {};\\node [style=none] (2) at (0.5, -0.5) {A_1};\\node [style=none] (3) at (1.5, -1) {};\\node [style=none] (4) at (1.5, 1) {};\\node [style=none] (5) at (3.25, -0) {...};\\node [style=none] (6) at (4.5, -1) {};\\node [style=none] (7) at (4.5, 1) {};\\node [style=none] (8) at (2, -0.5) {A_2};\\node [style=none] (9) at (5, -0.5) {A_n};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(0.center) to (1.center);(3.center) to (4.center);(6.center) to (7.center);\\end{pgfonlayer}\\end{tikzpicture} \\ .$ Processes are represented as boxes which have a set of input wires at the bottom and a set of output wires at the top.", "This gives a sense of a `direction of time' as flowing up the page.", "Dropping system labels for convenience we can denote a process $f$ as, for example: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (-1.5, -0.5) {};\\node [style=none] (1) at (1.5, -0.5) {};\\node [style=none] (2) at (2, 0.5) {};\\node [style=none] (3) at (-1.5, 0.5) {};\\node [style=none] (4) at (-1, 0.5) {};\\node [style=none] (5) at (-0.25, 0.5) {};\\node [style=none] (6) at (1.25, 0.5) {};\\node [style=none] (7) at (-0.75, -0.5) {};\\node [style=none] (8) at (0.75, -0.5) {};\\node [style=none] (9) at (-1, 1.75) {};\\node [style=none] (10) at (-0.25, 1.75) {};\\node [style=none] (11) at (1.25, 1.75) {};\\node [style=none] (12) at (0.5, 1.25) {...};\\node [style=none] (13) at (0, -1.25) {...};\\node [style=none] (14) at (-0.75, -1.75) {};\\node [style=none] (15) at (0.75, -1.75) {};\\node [style=none] (16) at (0, 0) {f};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(3.center) to (2.center);(2.center) to (1.center);(1.center) to (0.center);(0.center) to (3.center);(4.center) to (9.center);(5.center) to (10.center);(6.center) to (11.center);(14.center) to (7.center);(15.center) to (8.center);\\end{pgfonlayer}\\end{tikzpicture} .$ The shape of this box is irrelevant at this point but the asymmetry will be useful later on.", "There are three special types of processes: those with no input, called state preparations (or states for short); those with no outputs, called measurement outcomes (or effects for short); and those with neither inputs nor outputs, called scalars, which are often taken to be probabilities.", "Each of these has its own special notation: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=point] (0) at (0, -0.25) {s};\\node [style=none] (1) at (0, 0.7500001) {};\\node [style=none] (2) at (1, -0) {,};\\node [style=none] (3) at (3, -0.75) {};\\node [style=copoint] (4) at (3, 0.25) {e};\\node [style=none] (5) at (4, -0) {,};\\node [style=scalar] (6) at (6, -0) {p};\\node [style=none] (7) at (7, -0) {,};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(0) to (1.center);(3.center) to (4);\\end{pgfonlayer}\\end{tikzpicture} $ inspired by Dirac-notation [11].", "The box around a scalar is often omitted for clarity.", "We can now easily define the composition operation of a process theory: given some collection of processes then these can be wired together to form diagrams, for example: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (0, -1.5) {};\\node [style=none] (1) at (2, -1.5) {};\\node [style=none] (2) at (0.9999999, -2) {};\\node [style=none] (3) at (0.4999999, -1.5) {};\\node [style=none] (4) at (0.4999999, -0.4999999) {};\\node [style=none] (5) at (1.5, -1.5) {};\\node [style=none] (6) at (1.5, -0.4999999) {};\\node [style=none] (7) at (0.9999999, -0.4999999) {};\\node [style=none] (8) at (3, -0.4999999) {};\\node [style=none] (9) at (3.5, 0.4999999) {};\\node [style=none] (10) at (0.9999999, 0.4999999) {};\\node [style=none] (11) at (1.5, 0.4999999) {};\\node [style=none] (12) at (3, 0.4999999) {};\\node [style=none] (13) at (2.5, -0.4999999) {};\\node [style=none] (14) at (2.5, -2.5) {};\\node [style=none] (15) at (0.4999999, 1.5) {};\\node [style=none] (16) at (0.4999999, 1.5) {};\\node [style=none] (17) at (1.5, 1.5) {};\\node [style=none] (18) at (2.5, 1.5) {};\\node [style=none] (19) at (3, 1.5) {};\\node [style=none] (20) at (3.5, 1.5) {};\\node [style=none] (21) at (3, 2) {};\\node [style=none] (22) at (1.75, 1.5) {};\\node [style=none] (23) at (2.25, 2.5) {};\\node [style=none] (24) at (0, 2.5) {};\\node [style=none] (25) at (0, 1.5) {};\\node [style=none] (26) at (0.4999999, 3.5) {};\\node [style=none] (27) at (1.5, 3.5) {};\\node [style=none] (28) at (0.4999999, 2.5) {};\\node [style=none] (29) at (1.5, 2.5) {};\\node [style=none] (30) at (0, 0.25) {A};\\node [style=none] (31) at (2, 0.9999999) {B};\\node [style=none] (32) at (2, -0.9999999) {C};\\node [style=none] (33) at (3.5, 0.9999999) {D};\\node [style=none] (34) at (3, -1.5) {E};\\node [style=none] (35) at (0.9999999, 3.5) {F};\\node [style=none] (36) at (2, 3.5) {G};\\node [style=none] (37) at (4, -0) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(0.center) to (2.center);(2.center) to (1.center);(1.center) to (0.center);(3.center) to (4.center);(5.center) to (6.center);(7.center) to (8.center);(8.center) to (9.center);(9.center) to (10.center);(10.center) to (7.center);(4.center) to (16.center);(11.center) to (17.center);(12.center) to (19.center);(19.center) to (18.center);(18.center) to (21.center);(21.center) to (20.center);(20.center) to (19.center);(14.center) to (13.center);(25.center) to (22.center);(22.center) to (23.center);(23.center) to (24.center);(24.center) to (25.center);(28.center) to (26.center);(29.center) to (27.center);\\end{pgfonlayer}\\end{tikzpicture} ,$ which are also processes in the theory.", "This composition is subject to the condition that when forming diagrams, any two systems wired together have to be (of) the same (type).", "Essentially, processes in a process theory are closed under forming diagrams.", "In order to describe `wiring processes together' explicitly, in particular when the processes are represented by means of standard mathematical models, it is often convenient to consider the following two primitive forms of composition, sequential composition and parallel composition: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (0.5, -2) {};\\node [style=map] (1) at (0, -1.25) {f};\\node [style=map] (2) at (0, 1.25) {g};\\node [style=none] (3) at (0, 2.25) {};\\node [style=none] (4) at (0, -2.25) {};\\node [style=none] (5) at (0.5, -0) {B};\\node [style=none] (6) at (0.5, 2) {};\\node [style=none] (7) at (2, -0) {,};\\node [style=map] (8) at (4, -0) {h};\\node [style=map] (9) at (6, -0) {i};\\node [style=none] (10) at (4, 1.25) {};\\node [style=none] (11) at (6, 1.25) {};\\node [style=none] (12) at (4, -1.25) {};\\node [style=none] (13) at (6, -1.25) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(4.center) to (1);(1) to (2);(2) to (3.center);(12.center) to (8);(8) to (10.center);(13.center) to (9);(9) to (11.center);\\end{pgfonlayer}\\end{tikzpicture} ,$ which we symbolically denote by $g\\circ f$ and $h\\otimes i$ respectively.", "There are also some special scalars for a process theory that are fully characterised by their behaviour under composition.", "One special scalar is `certain', which is either written as 1 or by the empty diagram, which, of course, when composed with any other process leaves that process invariant: $\\,{\\node [style=empty diagram] (x) {};}\\,\\ \\ \\ \\ = \\ \\ \\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (-1.5, -0.5) {};\\node [style=none] (1) at (1.5, -0.5) {};\\node [style=none] (2) at (2, 0.5) {};\\node [style=none] (3) at (-1.5, 0.5) {};\\node [style=none] (4) at (-1, 0.5) {};\\node [style=none] (5) at (-0.25, 0.5) {};\\node [style=none] (6) at (1.25, 0.5) {};\\node [style=none] (7) at (-0.75, -0.5) {};\\node [style=none] (8) at (0.75, -0.5) {};\\node [style=none] (9) at (-1, 1.75) {};\\node [style=none] (10) at (-0.25, 1.75) {};\\node [style=none] (11) at (1.25, 1.75) {};\\node [style=none] (12) at (0.5, 1.25) {...};\\node [style=none] (13) at (0, -1.25) {...};\\node [style=none] (14) at (-0.75, -1.75) {};\\node [style=none] (15) at (0.75, -1.75) {};\\node [style=none] (16) at (0, 0) {f};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(3.center) to (2.center);(2.center) to (1.center);(1.center) to (0.center);(0.center) to (3.center);(4.center) to (9.center);(5.center) to (10.center);(6.center) to (11.center);(14.center) to (7.center);(15.center) to (8.center);\\end{pgfonlayer}\\end{tikzpicture} \\ \\ \\ .$ Another one is `impossible', which is written as 0, and `eats' all other diagrams, in the sense that for each set of input and output wires there is a 0-process, again simply denoted by 0, and when composing any process with the 0-scalar we obtain the corresponding 0-process: $0\\ \\ \\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (-1.5, -0.5) {};\\node [style=none] (1) at (1.5, -0.5) {};\\node [style=none] (2) at (2, 0.5) {};\\node [style=none] (3) at (-1.5, 0.5) {};\\node [style=none] (4) at (-1, 0.5) {};\\node [style=none] (5) at (-0.25, 0.5) {};\\node [style=none] (6) at (1.25, 0.5) {};\\node [style=none] (7) at (-0.75, -0.5) {};\\node [style=none] (8) at (0.75, -0.5) {};\\node [style=none] (9) at (-1, 1.75) {};\\node [style=none] (10) at (-0.25, 1.75) {};\\node [style=none] (11) at (1.25, 1.75) {};\\node [style=none] (12) at (0.5, 1.25) {...};\\node [style=none] (13) at (0, -1.25) {...};\\node [style=none] (14) at (-0.75, -1.75) {};\\node [style=none] (15) at (0.75, -1.75) {};\\node [style=none] (16) at (0, 0) {f};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(3.center) to (2.center);(2.center) to (1.center);(1.center) to (0.center);(0.center) to (3.center);(4.center) to (9.center);(5.center) to (10.center);(6.center) to (11.center);(14.center) to (7.center);(15.center) to (8.center);\\end{pgfonlayer}\\end{tikzpicture} \\ \\ = \\ \\ 0 \\ \\ \\ .$ More details on this process-theoretic framework can be found in [17], [18].", "Example 1 In the process theory ${C}LM$ the systems are finite dimensional complex vector spaces and the processes are linear maps between these vector spaces.", "Sequential composition is composition of linear maps and parallel composition is the tensor product.", "State preparations can be identified with complex vectors, effects with covectors, and scalars with the complex numbers.", "This `mathematical' process theory will now be used to construct quantum theory as a process theory.", "Example 2 We can construct the process theory of pure post-selected quantum processes directly from the process theory ${C}LM$ .", "One way to do so is to consider equivalence classes of linear maps, by ignoring global phases.", "A more elegant manner of doing the same is by means of doubling, that is, every process is replaced by its double: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=map] (0) at (0, 0) {f};\\node [style=none] (1) at (0, -1.5) {};\\node [style=none] (2) at (0, 1.5) {};\\node [style=none] (3) at (-2, 0) {{D}};\\node [style=none] (4) at (2, 0) {\\mapsto };\\node [style=none] (5) at (4, -1.5) {};\\node [style=mapconj] (6) at (4, 0) {f};\\node [style=none] (7) at (4, 1.5) {};\\node [style=none] (8) at (5.75, -1.5) {};\\node [style=map] (9) at (5.75, -0) {f};\\node [style=none] (10) at (5.75, 1.5) {};\\node [style=none] (11) at (-1.25, -0) {\\emph {::}};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(1.center) to (0);(0) to (2.center);(5.center) to (6);(6) to (7.center);(8.center) to (9);(9) to (10.center);\\end{pgfonlayer}\\end{tikzpicture} ,$ where, $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (0, -1.5) {};\\node [style=mapconj] (1) at (0, 0) {f};\\node [style=none] (2) at (0, 1.5) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(0.center) to (1);(1) to (2.center);\\end{pgfonlayer}\\end{tikzpicture} \\ ,$ represents the conjugate of $f$ , hence obtaining a new process theory ${D}[{C}LM]$ [12].", "Composing a state and an effect now gives a positive real number corresponding to the probability of, for example, measuring the effect $\\phi $ given state $\\psi $ according to the Born-rule: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=kpointtrans] (0) at (0, 0.75) {\\phi };\\node [style=kpointconj] (1) at (0, -0.75) {\\psi };\\node [style=kpoint] (2) at (1.25, -0.75) {\\psi };\\node [style=kpointadj] (3) at (1.25, 0.75) {\\phi };\\node [style=none] (5) at (4.75, -0) {=\\ c^* c \\in {R}^+};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(1) to (0);(2) to (3);\\end{pgfonlayer}\\end{tikzpicture} \\ .$ Clearly, processes differing by only a global phase become identical, ${D}(e^{i\\theta }f)=e^{i\\theta }e^{-i\\theta }f^*\\otimes f = {D}(f).$ Perhaps the most obvious description of ${D}[{C}LM]$ is as a generalisation of `Dirac' notation, when representing states by ket-bras [11].", "Remark 1 Symbolically, in category-theoretic terms, process theories can be defined as strict symmetric monoidal categories (SMCs).", "The systems of the process theory are the objects in the category, the processes are morphisms, and the states for some object are the morphisms from the tensor unit into that object [19], [18]." ], [ "Types of process theories", "For physical process theories we will always assume that the scalars are the positive real numbers, where the interval $[0,1]$ corresponds to `probabilistic weights'.", "Inspired by the physical notion of process tomography, and given the interpretations of states and effects, we can define the following special kind of process theories: Definition 1 Tomographic process theories are process theories where all processes can be characterised in terms of scalars: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (-0.7500001, -2) {};\\node [style=none] (1) at (-0.7500001, 2) {};\\node [style=none] (2) at (1.5, -0) {=};\\node [style=none] (3) at (0.25, 0.5) {};\\node [style=none] (4) at (-0.25, -0.5) {};\\node [style=none] (5) at (-2.75, -0.5) {};\\node [style=none] (6) at (-2.75, 0.5) {};\\node [style=none] (7) at (-0.75, 0.5) {};\\node [style=none] (8) at (-2.25, 0.5) {};\\node [style=none] (9) at (-2.25, -0.5) {};\\node [style=none] (10) at (-2.25, -2) {};\\node [style=none] (11) at (-2.25, 2) {};\\node [style=none] (12) at (-0.75, -0.5) {};\\node [style=none] (13) at (-1.5, -0) {f};\\node [style=none] (14) at (3.5, -0.5) {};\\node [style=none] (15) at (5, 2) {};\\node [style=none] (16) at (5, -0.5) {};\\node [style=none] (17) at (3, -0.5) {};\\node [style=none] (18) at (3.5, -2) {};\\node [style=none] (19) at (5.5, -0.5) {};\\node [style=none] (20) at (6, 0.5) {};\\node [style=none] (21) at (4.25, -0) {g};\\node [style=none] (22) at (5, -2) {};\\node [style=none] (23) at (3.5, 2) {};\\node [style=none] (24) at (5, 0.5) {};\\node [style=none] (25) at (3, 0.5) {};\\node [style=none] (26) at (3.5, 0.5) {};\\node [style=none] (27) at (-1.5, 1.25) {\\ldots };\\node [style=none] (28) at (-1.5, -1.25) {\\ldots };\\node [style=none] (29) at (4.25, 1.25) {\\ldots };\\node [style=none] (30) at (4.25, -1.25) {\\ldots };\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(6.center) to (3.center);(3.center) to (4.center);(4.center) to (5.center);(5.center) to (6.center);(8.center) to (11.center);(1.center) to (7.center);(9.center) to (10.center);(0.center) to (12.center);(25.center) to (20.center);(20.center) to (19.center);(19.center) to (17.center);(17.center) to (25.center);(26.center) to (23.center);(15.center) to (24.center);(14.center) to (18.center);(22.center) to (16.center);\\end{pgfonlayer}\\end{tikzpicture} \\quad \\iff \\quad \\forall \\phi ,\\psi , C \\quad \\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (-0.5, -0) {=};\\node [style=none] (1) at (-3.5, 0.5) {};\\node [style=none] (2) at (-3.75, -0.5) {};\\node [style=none] (3) at (-6.25, -0.5) {};\\node [style=none] (4) at (-6.25, 0.5) {};\\node [style=none] (5) at (-4.25, 0.5) {};\\node [style=none] (6) at (-5.75, 0.5) {};\\node [style=none] (7) at (-5.75, -0.5) {};\\node [style=none] (8) at (-4.25, -0.5) {};\\node [style=none] (9) at (-5, -0) {f};\\node [style=none] (10) at (-5.75, 1.5) {};\\node [style=none] (11) at (-4.25, 1.5) {};\\node [style=none] (12) at (-4.25, -1.5) {};\\node [style=none] (13) at (-5.75, -1.5) {};\\node [style=none] (14) at (-2.75, -1.5) {};\\node [style=none] (15) at (-2.75, 1.5) {};\\node [style=none] (16) at (-2, 1.5) {};\\node [style=none] (17) at (-6.5, 1.5) {};\\node [style=none] (18) at (-6.5, -1.5) {};\\node [style=none] (19) at (-2, -1.5) {};\\node [style=none] (20) at (-4.25, -2.75) {};\\node [style=none] (21) at (-4.25, 2.75) {};\\node [style=none] (22) at (-4.25, 2) {\\psi };\\node [style=none] (23) at (-4.25, -2) {\\phi };\\node [style=none] (24) at (-2.25, -0) {C};\\node [style=none] (25) at (2, -0.5) {};\\node [style=none] (26) at (5.75, 1.5) {};\\node [style=none] (27) at (3.5, -2) {\\phi };\\node [style=none] (28) at (2, -1.5) {};\\node [style=none] (29) at (3.5, -2.75) {};\\node [style=none] (30) at (5.5, -0) {C};\\node [style=none] (31) at (2.75, -0) {g};\\node [style=none] (32) at (1.25, 1.5) {};\\node [style=none] (33) at (3.5, 0.5) {};\\node [style=none] (34) at (3.5, 2.75) {};\\node [style=none] (35) at (5, -1.5) {};\\node [style=none] (36) at (4.25, 0.5) {};\\node [style=none] (37) at (1.5, 0.5) {};\\node [style=none] (38) at (2, 1.5) {};\\node [style=none] (39) at (3.5, -0.5) {};\\node [style=none] (40) at (1.25, -1.5) {};\\node [style=none] (41) at (4, -0.5) {};\\node [style=none] (42) at (3.5, 2) {\\psi };\\node [style=none] (43) at (2, 0.5) {};\\node [style=none] (44) at (5, 1.5) {};\\node [style=none] (45) at (3.5, -1.5) {};\\node [style=none] (46) at (5.75, -1.5) {};\\node [style=none] (47) at (1.5, -0.5) {};\\node [style=none] (48) at (3.5, 1.5) {};\\node [style=none] (49) at (-5, 0.9999998) {\\ldots };\\node [style=none] (50) at (-5, -0.9999998) {\\ldots };\\node [style=none] (51) at (2.75, 0.9999998) {\\ldots };\\node [style=none] (52) at (2.75, -0.9999998) {\\ldots };\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(4.center) to (1.center);(1.center) to (2.center);(2.center) to (3.center);(3.center) to (4.center);(17.center) to (16.center);(16.center) to (21.center);(21.center) to (17.center);(18.center) to (19.center);(19.center) to (20.center);(20.center) to (18.center);(13.center) to (7.center);(6.center) to (10.center);(11.center) to (5.center);(8.center) to (12.center);(14.center) to (15.center);(37.center) to (36.center);(36.center) to (41.center);(41.center) to (47.center);(47.center) to (37.center);(32.center) to (26.center);(26.center) to (34.center);(34.center) to (32.center);(40.center) to (46.center);(46.center) to (29.center);(29.center) to (40.center);(28.center) to (25.center);(43.center) to (38.center);(48.center) to (33.center);(39.center) to (45.center);(35.center) to (44.center);\\end{pgfonlayer}\\end{tikzpicture}.$ Locally tomographic process theories are tomographic process theories which satisfy the stronger condition of `local process tomography' [3], [6], [29], [10]: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (-0.7500001, -2) {};\\node [style=none] (1) at (-0.7500001, 2) {};\\node [style=none] (2) at (1.5, -0) {=};\\node [style=none] (3) at (0.25, 0.5) {};\\node [style=none] (4) at (-0.25, -0.5) {};\\node [style=none] (5) at (-2.75, -0.5) {};\\node [style=none] (6) at (-2.75, 0.5) {};\\node [style=none] (7) at (-0.75, 0.5) {};\\node [style=none] (8) at (-2.25, 0.5) {};\\node [style=none] (9) at (-2.25, -0.5) {};\\node [style=none] (10) at (-2.25, -2) {};\\node [style=none] (11) at (-2.25, 2) {};\\node [style=none] (12) at (-0.75, -0.5) {};\\node [style=none] (13) at (-1.5, -0) {f};\\node [style=none] (14) at (3.5, -0.5) {};\\node [style=none] (15) at (5, 2) {};\\node [style=none] (16) at (5, -0.5) {};\\node [style=none] (17) at (3, -0.5) {};\\node [style=none] (18) at (3.5, -2) {};\\node [style=none] (19) at (5.5, -0.5) {};\\node [style=none] (20) at (6, 0.5) {};\\node [style=none] (21) at (4.25, -0) {g};\\node [style=none] (22) at (5, -2) {};\\node [style=none] (23) at (3.5, 2) {};\\node [style=none] (24) at (5, 0.5) {};\\node [style=none] (25) at (3, 0.5) {};\\node [style=none] (26) at (3.5, 0.5) {};\\node [style=none] (27) at (-1.5, 1.25) {\\ldots };\\node [style=none] (28) at (-1.5, -1.25) {\\ldots };\\node [style=none] (29) at (4.25, 1.25) {\\ldots };\\node [style=none] (30) at (4.25, -1.25) {\\ldots };\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(6.center) to (3.center);(3.center) to (4.center);(4.center) to (5.center);(5.center) to (6.center);(8.center) to (11.center);(1.center) to (7.center);(9.center) to (10.center);(0.center) to (12.center);(25.center) to (20.center);(20.center) to (19.center);(19.center) to (17.center);(17.center) to (25.center);(26.center) to (23.center);(15.center) to (24.center);(14.center) to (18.center);(22.center) to (16.center);\\end{pgfonlayer}\\end{tikzpicture}\\quad \\iff \\quad \\forall \\lbrace \\phi _i\\rbrace ,\\lbrace \\psi _j\\rbrace \\quad \\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=point] (0) at (-0.4999996, -2) {\\psi _n};\\node [style=copoint] (1) at (-0.4999996, 2) {\\phi _m};\\node [style=none] (2) at (1.75, -0) {=};\\node [style=none] (3) at (0.4999996, 0.5000003) {};\\node [style=none] (4) at (0, -0.5000003) {};\\node [style=none] (5) at (-3, -0.5000003) {};\\node [style=none] (6) at (-3, 0.5000003) {};\\node [style=none] (7) at (-0.4999996, 0.5000003) {};\\node [style=none] (8) at (-2.500001, 0.5000003) {};\\node [style=none] (9) at (-2.500001, -0.5000003) {};\\node [style=point] (10) at (-2.500001, -2) {\\psi _1};\\node [style=copoint] (11) at (-2.500001, 2) {\\phi _1};\\node [style=none] (12) at (-0.4999996, -0.5000003) {};\\node [style=none] (13) at (-1.5, -0) {f};\\node [style=none] (14) at (3.25, -0.5) {};\\node [style=none] (15) at (3.25, 0.5) {};\\node [style=none] (16) at (5.75, 0.5000003) {};\\node [style=none] (17) at (3.75, 0.5) {};\\node [style=point] (18) at (3.75, -2) {\\psi _1};\\node [style=none] (19) at (4.75, -0) {g};\\node [style=none] (20) at (5.75, -0.5000003) {};\\node [style=none] (21) at (6.25, -0.5000003) {};\\node [style=point] (22) at (5.75, -2) {\\psi _n};\\node [style=none] (23) at (6.749999, 0.5000003) {};\\node [style=copoint] (24) at (5.75, 2) {\\phi _m};\\node [style=copoint] (25) at (3.75, 2) {\\phi _1};\\node [style=none] (26) at (3.75, -0.5) {};\\node [style=none] (27) at (-1.5, 0.9999998) {\\ldots };\\node [style=none] (28) at (-1.5, -0.9999998) {\\ldots };\\node [style=none] (29) at (4.75, 0.9999998) {\\ldots };\\node [style=none] (30) at (4.75, -0.9999998) {\\ldots };\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(6.center) to (3.center);(3.center) to (4.center);(4.center) to (5.center);(5.center) to (6.center);(8.center) to (11);(1) to (7.center);(9.center) to (10);(0) to (12.center);(15.center) to (23.center);(23.center) to (21.center);(21.center) to (14.center);(14.center) to (15.center);(17.center) to (25);(24) to (16.center);(26.center) to (18);(22) to (20.center);\\end{pgfonlayer}\\end{tikzpicture}.$ Example 3 Quantum theory, in the form of ${D}[{C}LM]$ is a locally tomographic process theory [22], [18].", "Previous work on generalising the Hermitian adjoint to the process theoretic setting led to the notion of dagger process theories.", "Definition 2 Dagger process theories are process theories which come with a `dagger', ${\\dagger _{pt}}$ , an operation that assigns to each processes another process as its reflection: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=map] (0) at (-4, 0) {f};\\node [style=mapdag] (1) at (3.75, 0) {f};\\node [style=none] (2) at (-4, 2) {};\\node [style=none] (3) at (3.75, 2) {};\\node [style=none] (4) at (-4, -2) {};\\node [style=none] (5) at (3.75, -2) {};\\node [style=none] (6) at (1.5, -0.5) {};\\node [style=none] (7) at (1.5, 0.5) {};\\node [style=none] (8) at (0, 1.25) {{\\dagger _{pt}}};\\node [style=none] (9) at (-1.5, -0.5) {};\\node [style=none] (10) at (0, -1.25) {{\\dagger _{pt}}};\\node [style=none] (11) at (-1.5, 0.5) {};\\node [style=none] (12) at (-3.5, -1.5) {A};\\node [style=none] (13) at (4.25, 1.5) {A};\\node [style=none] (14) at (-3.5, 1.5) {B};\\node [style=none] (15) at (4.25, -1.5) {B};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(4.center) to (0);(0) to (2.center);(5.center) to (1);(1) to (3.center);[style=arrow plain, bend left=15, looseness=1.00] (11.center) to (7.center);[style=arrow plain, bend left=15, looseness=1.00] (6.center) to (9.center);\\end{pgfonlayer}\\end{tikzpicture} \\ ,$ which moreover preserves diagrams, that is, diagrams are reflected `as a whole': $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (-6.5, -2.25) {};\\node [style=none] (1) at (-4.5, -2.25) {};\\node [style=none] (2) at (-5.5, -2.75) {};\\node [style=none] (3) at (-6, -2.25) {};\\node [style=none] (4) at (-6, -1.25) {};\\node [style=none] (5) at (-5, -2.25) {};\\node [style=none] (6) at (-5, -1.25) {};\\node [style=none] (7) at (-5.5, -1.25) {};\\node [style=none] (8) at (-3.5, -1.25) {};\\node [style=none] (9) at (-3, -0.25) {};\\node [style=none] (10) at (-5.5, -0.25) {};\\node [style=none] (11) at (-5, -0.25) {};\\node [style=none] (12) at (-3.5, -0.25) {};\\node [style=none] (13) at (-4, -1.25) {};\\node [style=none] (14) at (-4, -3.25) {};\\node [style=none] (15) at (-6, 0.75) {};\\node [style=none] (16) at (-6, 0.75) {};\\node [style=none] (17) at (-5, 0.75) {};\\node [style=none] (18) at (-4, 0.75) {};\\node [style=none] (19) at (-3.5, 0.75) {};\\node [style=none] (20) at (-3, 0.75) {};\\node [style=none] (21) at (-3.5, 1.25) {};\\node [style=none] (22) at (-4.75, 0.75) {};\\node [style=none] (23) at (-4.25, 1.75) {};\\node [style=none] (24) at (-6.5, 1.75) {};\\node [style=none] (25) at (-6.5, 0.75) {};\\node [style=none] (26) at (-6, 2.75) {};\\node [style=none] (27) at (-5, 2.75) {};\\node [style=none] (28) at (-6, 1.75) {};\\node [style=none] (29) at (-5, 1.75) {};\\node [style=none] (30) at (3.5, -1.25) {};\\node [style=none] (31) at (4, 2.25) {};\\node [style=none] (32) at (3.5, 0.75) {};\\node [style=none] (33) at (6.5, -0.25) {};\\node [style=none] (34) at (3.5, -2.25) {};\\node [style=none] (35) at (4.5, -2.25) {};\\node [style=none] (36) at (5, 1.75) {};\\node [style=none] (37) at (5.5, 2.75) {};\\node [style=none] (38) at (6, 0.75) {};\\node [style=none] (39) at (4.5, -0.25) {};\\node [style=none] (40) at (4.75, -1.25) {};\\node [style=none] (41) at (3.5, -3.25) {};\\node [style=none] (42) at (3, 1.75) {};\\node [style=none] (43) at (5.5, 0.75) {};\\node [style=none] (44) at (4.5, 1.75) {};\\node [style=none] (45) at (5.25, -2.25) {};\\node [style=none] (46) at (6, -0.25) {};\\node [style=none] (47) at (5.5, -1.25) {};\\node [style=none] (48) at (4.5, 0.75) {};\\node [style=none] (49) at (6, -1.75) {};\\node [style=none] (50) at (3, -2.25) {};\\node [style=none] (51) at (4, -0.25) {};\\node [style=none] (52) at (3, -1.25) {};\\node [style=none] (53) at (3.5, -1.25) {};\\node [style=none] (54) at (6, -1.25) {};\\node [style=none] (55) at (3.5, 1.75) {};\\node [style=none] (56) at (4.5, -1.25) {};\\node [style=none] (57) at (4, 0.75) {};\\node [style=none] (58) at (6.5, -1.25) {};\\node [style=none] (59) at (4.5, -3.25) {};\\node [style=none] (60) at (-1.5, 0.25) {};\\node [style=none] (61) at (1.5, 0.25) {};\\node [style=none] (62) at (1.5, -0.75) {};\\node [style=none] (63) at (-1.5, -0.75) {};\\node [style=none] (64) at (0, 1) {{\\dagger _{pt}}};\\node [style=none] (65) at (0, -1.5) {{\\dagger _{pt}}};\\node [style=none] (66) at (7, -0) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(0.center) to (2.center);(2.center) to (1.center);(1.center) to (0.center);(3.center) to (4.center);(5.center) to (6.center);(7.center) to (8.center);(8.center) to (9.center);(9.center) to (10.center);(10.center) to (7.center);(4.center) to (16.center);(11.center) to (17.center);(12.center) to (19.center);(19.center) to (18.center);(18.center) to (21.center);(21.center) to (20.center);(20.center) to (19.center);(14.center) to (13.center);(25.center) to (22.center);(22.center) to (23.center);(23.center) to (24.center);(24.center) to (25.center);(28.center) to (26.center);(29.center) to (27.center);(42.center) to (31.center);(31.center) to (36.center);(36.center) to (42.center);(55.center) to (32.center);(44.center) to (48.center);(57.center) to (38.center);(38.center) to (33.center);(33.center) to (51.center);(51.center) to (57.center);(32.center) to (53.center);(39.center) to (56.center);(46.center) to (54.center);(54.center) to (47.center);(47.center) to (49.center);(49.center) to (58.center);(58.center) to (54.center);(37.center) to (43.center);(52.center) to (40.center);(40.center) to (45.center);(45.center) to (50.center);(50.center) to (52.center);(34.center) to (41.center);(35.center) to (59.center);[style={arrow plain}, bend left=15, looseness=1.00] (60.center) to (61.center);[style={arrow plain}, bend left=15, looseness=1.00] (62.center) to (63.center);\\end{pgfonlayer}\\end{tikzpicture} ,$ and which, as the notation already suggests, is involutive.", "Remark 2 In category-theoretic terms, a dagger process theory is an SMC equipped with a identity-on-objects involutive contravariant monoidal endofunctor [27].", "Example 4 Pure quantum theory ${D}[{C}LM]$ involves a dagger structure, namely the Hermitian adjoint.", "More specifically, for the process theory ${C}LM$ underpinning quantum theory, for a process $f:X\\rightarrow Y$ it is defined by: $\\langle f^{\\dagger }\\psi ,\\phi \\rangle _X = \\langle \\psi ,f\\phi \\rangle _Y \\quad \\forall \\psi ,\\phi .$ by doubling this lifts to ${D}[{C}LM]$ .", "This raises an important question: Why does it `mean' for quantum theory to have a dagger structure?", "and in particular the following two sub-questions: Why does it have to be involutive?", "Given that there are other candidate dagger structures for quantum theory, most notably the transpose, why is that central role played specifically by the Hermitian adjoint?" ], [ "The new physical principle", "In what sense can something be considered a state if there is no way to check that it is indeed that state?", "Therefore we state the following physical principle: for each state there exists a corresponding `test'.", "In the remainder of this paper, after giving formal substance to this principle for general process theories, we will show that the assignment of tests for pure quantum theory is given by the Hermitian adjoint, after showing that for general process theories this principle gives rise to a dagger.", "Definition 3 A test structure ${\\sharp }$ is a mapping from states to effects on the same type subject to the following conditions: Composability of tests: $\\left( \\right)^{\\sharp }\\ =\\ \\ \\ \\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (0, -1) {};\\node [style=copoint] (1) at (0, 0) {\\psi ^{\\sharp }};\\node [style=none] (2) at (1.5, -0.9999999) {};\\node [style=copoint] (3) at (1.5, -0) {\\phi ^{\\sharp }};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(0.center) to (1);(2.center) to (3);\\end{pgfonlayer}\\end{tikzpicture} \\ \\ .$ Transformability of tests: $\\forall f\\ \\exists f^{\\sharp }\\quad \\text{ s.t.", "}\\quad \\left(\\right)^{\\sharp }\\ =\\ \\ \\ \\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (0, -2.25) {};\\node [style=map] (1) at (0, -0.75) {f^{\\sharp }};\\node [style=copoint] (2) at (0, 1) {\\psi ^{\\sharp }};\\node [style=none] (3) at (0.5, -1.75) {B};\\node [style=none] (4) at (0.5, 0.25) {A};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}[style=none] (0.center) to (1);[style=none] (1) to (2);\\end{pgfonlayer}\\end{tikzpicture} \\ \\ .$ Tests produce probabilities: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (-11.25, 0) {If};\\node [style=copoint] (1) at (-8.75, 0.4999999) {\\psi ^{\\sharp }};\\node [style=point] (2) at (-8.75, -0.4999999) {\\psi };\\node [style=none] (3) at (-7.25, -0) {=};\\node [style=none] (4) at (-6.5, 0) {1};\\node [style=none] (5) at (-5.75, -0) {=};\\node [style=point] (6) at (-4.25, -0.4999999) {\\phi };\\node [style=copoint] (7) at (-4.25, 0.4999999) {\\phi ^{\\sharp }};\\node [style=none] (8) at (-1.25, 0) {then,};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(2) to (1);(6) to (7);\\end{pgfonlayer}\\end{tikzpicture} \\ \\quad \\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=point] (0) at (-0.9999999, -0.4999999) {\\phi };\\node [style=copoint] (1) at (-0.9999999, 0.4999999) {\\psi ^{\\sharp }};\\node [style=none] (2) at (0.5, 0) {\\le };\\node [style=none] (3) at (1.5, 0) {1};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(0) to (1);\\end{pgfonlayer}\\end{tikzpicture} \\ .$ Testability of all states: $\\forall \\ \\chi \\ne 0 \\quad \\exists r\\quad \\text{s.t.", "}\\quad \\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=point] (0) at (0.25, -0.75) {\\chi };\\node [style=copoint] (1) at (0.2499997, 0.7499999) {\\chi ^{\\sharp }};\\node [style=scalar] (2) at (-1, -1) {r};\\node [style=scalar] (3) at (-1, 1) {r^{\\sharp }\\!", "};\\node [style=none] (4) at (-2.5, 0.2500001) {};\\node [style=none] (5) at (-0.5000002, 2.75) {};\\node [style=none] (6) at (1.5, 0.2500001) {};\\node [style=none] (7) at (1.5, -0.2500001) {};\\node [style=none] (8) at (-0.5000002, -2.75) {};\\node [style=none] (9) at (-2.5, -0.2500001) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(0) to (1);[dashed] (4.center) to (5.center);[dashed] (5.center) to (6.center);[dashed] (6.center) to (4.center);[dashed] (9.center) to (8.center);[dashed] (8.center) to (7.center);[dashed] (7.center) to (9.center);\\end{pgfonlayer}\\end{tikzpicture} \\ \\ =1\\ .$ Sharpness of tests: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (-11.25, 0) {If};\\node [style=copoint] (1) at (-8.75, 0.4999999) {\\psi ^{\\sharp }};\\node [style=point] (2) at (-8.75, -0.4999999) {\\psi };\\node [style=none] (3) at (-7.25, -0) {=};\\node [style=none] (4) at (-6.5, 0) {1};\\node [style=none] (5) at (-5.75, -0) {=};\\node [style=point] (6) at (-4.25, -0.4999999) {\\phi };\\node [style=copoint] (7) at (-4.25, 0.4999999) {\\phi ^{\\sharp }};\\node [style=none] (8) at (-1.25, 0) {then,};\\node [style=copoint] (9) at (1.5, 0.4999999) {\\psi ^{\\sharp }};\\node [style=point] (10) at (1.5, -0.4999999) {\\phi };\\node [style=none] (11) at (3, 0) {=};\\node [style=none] (12) at (3.75, -0) {1};\\node [style=none] (13) at (5.499999, -0) {\\iff };\\node [style=point] (14) at (7.75, -0.25) {\\phi };\\node [style=none] (15) at (9.000001, -0) {=};\\node [style=point] (16) at (10.25, -0.25) {\\psi };\\node [style=none] (17) at (7.75, 0.7499999) {};\\node [style=none] (18) at (10.25, 0.7499999) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(2) to (1);(6) to (7);(10) to (9);(14) to (17.center);(16) to (18.center);\\end{pgfonlayer}\\end{tikzpicture} \\ \\ .$ Clearly, sharpness constitutes the core of these axioms, the others mainly guaranteeing that test structures are `well-behaved' with respect the interpretation of $[0,1]$ as probabilities and the structure of process theories.", "Of course, for pure quantum theory (up to a global phase), there is a unique normalised effect which gives unit probability for a particular normalised state.", "So it may seem at first like sharpness trivially suffices to single out the Hermitian adjoint (or at least, it's action on states and effects), however this is not the case, as in quantum theory normalisation of a state itself is defined in terms of the Hermitian adjoint.", "Note also that in transformability the use of the notation ${\\sharp }$ for arbitrary processes that transform effects is justified, since when we take $f$ to be a state, it can be taken to be the test structure ${\\sharp }$ ." ], [ "Test structures provide daggers", "Lemma 11 Test structures preserves certainty, i.e.", "${\\sharp }(1)=1$ .", "For scalars $r_1, r_2$ composability implies that ${\\sharp }(r_1r_2)={\\sharp }(r_1){\\sharp }(r_2)$ .", "Let $r_1=1$ then ${\\sharp }(r_2)={\\sharp }(1){\\sharp }(r_2)$ .", "There are two solutions for this equation, i) ${\\sharp }(r_2)=0\\ \\forall r_2$ which violates testability leaving just ii) ${\\sharp }(1)=1$ .", "Lemma 22 Test structures, extended to effects via transformability, are involutive on normalised states i.e.", "$\\psi ^{{\\sharp }^{\\sharp }}=\\psi $ .", "Using transformability and lemma REF , for normalised states we have, $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (-2, 0) {{\\sharp }};\\node [style=point] (1) at (-0.25, -0.4999999) {\\psi \\ };\\node [style=copoint] (2) at (-0.25, 0.4999999) {\\psi ^{{\\sharp }}};\\node [style=none] (3) at (2.25, 0) {=};\\node [style=copoint] (4) at (-5.75, 0.7499999) {\\psi ^{{\\sharp }\\ }};\\node [style=point] (5) at (-5.75, -0.7499999) {\\psi ^{{\\sharp }^{\\sharp }}};\\node [style=none] (6) at (-1.25, 1.5) {};\\node [style=none] (7) at (-1.25, -1.25) {};\\node [style=none] (8) at (0.7499999, -1.25) {};\\node [style=none] (9) at (0.7499999, 1.5) {};\\node [style=none] (10) at (3.75, 0) {{\\sharp }(1)};\\node [style=none] (11) at (5.25, 0) {=};\\node [style=none] (12) at (6, 0) {1};\\node [style=none] (13) at (-3.5, 0) {=};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(1) to (2);(5) to (4);[bend left=15, looseness=1.00] (7.center) to (6.center);[bend right=15, looseness=1.00] (8.center) to (9.center);\\end{pgfonlayer}\\end{tikzpicture} \\ \\ ,$ then, by sharpness, $\\psi ^{{\\sharp }^{\\sharp }}=\\psi $ .", "Lemma 33 For tomographic process theories, a test structure is involutive on tests for normalised states, i.e.", "$\\psi ^{{\\sharp }^{{\\sharp }^{\\sharp }}}=\\psi ^{\\sharp }$ .", "For $\\psi ^{\\sharp }$ , a test for a normalised state $\\psi $ , we have, $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (3.75, -0) {{\\sharp }};\\node [style=point] (1) at (5.499999, -0.4999999) {\\psi \\ };\\node [style=copoint] (2) at (5.499999, 0.4999999) {\\psi ^{{\\sharp }}};\\node [style=none] (3) at (7.999999, -0) {=};\\node [style=copoint] (4) at (-5.75, 0.7499999) {\\psi ^{{\\sharp }^{{\\sharp }^{\\sharp }}}};\\node [style=point] (5) at (-5.75, -0.7499999) {\\hspace{4.2679pt} \\psi \\hspace{4.2679pt} };\\node [style=none] (6) at (4.5, 1.5) {};\\node [style=none] (7) at (4.5, -1.25) {};\\node [style=none] (8) at (6.499999, -1.25) {};\\node [style=none] (9) at (6.499999, 1.5) {};\\node [style=none] (10) at (9.5, -0) {{\\sharp }({\\sharp }(1))};\\node [style=none] (11) at (11, -0) {=};\\node [style=none] (12) at (11.75, -0) {1};\\node [style=none] (13) at (-3.5, 0) {=};\\node [style=point] (14) at (-0.7499999, -0.7499999) {\\psi ^{{\\sharp }^{{\\sharp }}}};\\node [style=copoint] (15) at (-0.7499999, 0.7499999) {\\psi ^{{\\sharp }^{{\\sharp }^{\\sharp }}}};\\node [style=none] (16) at (1.75, -0) {=};\\node [style=none] (17) at (2.75, -0) {{\\sharp }};\\node [style=none] (18) at (3.5, -1.25) {};\\node [style=none] (19) at (3.5, 1.5) {};\\node [style=none] (20) at (6.999999, -1.25) {};\\node [style=none] (21) at (6.999999, 1.5) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(1) to (2);(5) to (4);[bend left=15, looseness=1.00] (7.center) to (6.center);[bend right=15, looseness=1.00] (8.center) to (9.center);(14) to (15);[bend left=15, looseness=1.00] (18.center) to (19.center);[bend right=15, looseness=1.00] (20.center) to (21.center);\\end{pgfonlayer}\\end{tikzpicture} \\ \\ ,$ using lemma REF for the first equality and transformability twice for the second.", "Then, by sharpness, $\\psi ^{{\\sharp }^{{\\sharp }^{\\sharp }}}=\\psi ^{\\sharp }$ .", "Lemma 44 Test structure preserves probabilities, i.e.", "${\\sharp }(r)=r$ .", "Consider, $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=point] (0) at (12, -0.4999999) {\\phi };\\node [style=none] (1) at (0.9999999, -0) {=};\\node [style=none] (2) at (10, -0) {=1=};\\node [style=copoint] (3) at (12, 0.4999999) {\\phi ^{\\sharp }};\\node [style=point] (4) at (2.5, -0.4999999) {\\psi };\\node [style=none] (5) at (0, -0) {p};\\node [style=none] (6) at (5, -0) {where};\\node [style=copoint] (7) at (2.5, 0.4999999) {\\phi ^{\\sharp }};\\node [style=point] (8) at (8.000001, -0.4999999) {\\psi };\\node [style=copoint] (9) at (8.000001, 0.4999999) {\\psi ^{\\sharp }};\\node [style=none] (10) at (13, -0) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(4) to (7);(8) to (9);(0) to (3);\\end{pgfonlayer}\\end{tikzpicture} ,$ then tests-produce-probabilities implies that $0\\le p\\le 1$ .", "Now consider, $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=copoint] (0) at (4, 0.625) {\\psi ^{{\\sharp }\\ }};\\node [style=copoint] (1) at (8.500001, 0.4999999) {\\psi ^{\\sharp }};\\node [style=point] (2) at (4, -0.625) {\\phi ^{{\\sharp }^{\\sharp }}};\\node [style=none] (3) at (1.75, -0) {=};\\node [style=none] (4) at (6.499999, -0) {=};\\node [style=point] (5) at (8.500001, -0.4999999) {\\phi };\\node [style=none] (6) at (0, -0) {{\\sharp }(p)};\\node [style=none] (7) at (9.5, -0) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(0) to (2);(1) to (5);\\end{pgfonlayer}\\end{tikzpicture} ,$ and again tests-produce-probabilities implies that $0\\le {\\sharp }(p) \\le 1$ , and so, ${\\sharp }$ preserves $[0,1]$ .", "Now note that lemmas REF and REF imply that, $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (1, -0) {{\\sharp }({\\sharp }(p))=};\\node [style=none] (1) at (2.75, -0) {{\\sharp }};\\node [style=none] (2) at (3.5, -0) {{\\sharp }};\\node [style=copoint] (3) at (4.75, 0.4999999) {\\psi ^{\\sharp }};\\node [style=point] (4) at (4.75, -0.4999999) {\\phi };\\node [style=none] (5) at (6.75, -0) {=};\\node [style=copoint] (6) at (8, 0.5000001) {\\psi ^{\\sharp }};\\node [style=point] (7) at (8, -0.5000001) {\\phi };\\node [style=none] (8) at (9.5, -0) {=p};\\node [style=none] (9) at (4, 1.25) {};\\node [style=none] (10) at (4, -1.25) {};\\node [style=none] (11) at (5.5, -1.25) {};\\node [style=none] (12) at (5.5, 1.25) {};\\node [style=none] (13) at (3.25, 1.25) {};\\node [style=none] (14) at (3.25, -1.25) {};\\node [style=none] (15) at (5.75, 1.25) {};\\node [style=none] (16) at (5.75, -1.25) {};\\node [style=none] (17) at (10.25, -0) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(3) to (4);(6) to (7);[bend right=15, looseness=1.00] (9.center) to (10.center);[bend right=15, looseness=1.00] (13.center) to (14.center);[bend left=15, looseness=1.00] (12.center) to (11.center);[in=75, out=-75, looseness=1.00] (15.center) to (16.center);\\end{pgfonlayer}\\end{tikzpicture} ,$ so the test structure is involutive for $[0,1]$ .", "Moreover, composability implies that ${\\sharp }$ is multiplicative on all scalars, ${\\sharp }(r_1r_2)={\\sharp }(r_1){\\sharp }(r_2).$ The result that ${\\sharp }(r)=r$ is an application of standard results regarding functional equations [24].", "Multiplicativity and preservation of $[0,1]$ imply that ${\\sharp }(r)=r^a$ where $a \\in {R}^+$ .", "Involutivity on $[0,1]$ demands that $a=1$ and so ${\\sharp }(r)=r$ .", "Theorem 55 For tomographic process theories, the test structure is involutive.", "Firstly we extend to all states using testability; write an arbitrary state $\\chi = r \\psi $ where $r$ is a scalar and $\\psi $ normalised.", "Then lemmas REF and REF imply that, $\\chi ^{{\\sharp }^{\\sharp }} = r^{{\\sharp }^{\\sharp }}\\psi ^{{\\sharp }^{\\sharp }}=r\\psi = \\chi .$ Tomography then allows us to extend this to all effects, $e$ , $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=copoint] (0) at (1.5, 0.4999999) {e^{{\\sharp }^{\\sharp }}};\\node [style=point] (1) at (1.5, -0.4999999) {\\chi };\\node [style=none] (2) at (-0.4999999, 0.2500001) {\\forall \\chi };\\node [style=none] (3) at (2.75, -0) {=};\\node [style=none] (4) at (3.5, -0) {{\\sharp }};\\node [style=none] (5) at (4.25, -0) {{\\sharp }};\\node [style=copoint] (6) at (5.5, 0.4999999) {e};\\node [style=point] (7) at (5.5, -0.4999999) {\\chi };\\node [style=none] (8) at (4.75, 1.25) {};\\node [style=none] (9) at (4.75, -1.25) {};\\node [style=none] (10) at (4, -1.25) {};\\node [style=none] (11) at (4, 1.25) {};\\node [style=none] (12) at (6.25, 1.25) {};\\node [style=none] (13) at (6.25, -1.25) {};\\node [style=none] (14) at (6.5, -1.25) {};\\node [style=none] (15) at (6.5, 1.25) {};\\node [style=point] (16) at (8.75, -0.4999999) {\\chi };\\node [style=none] (17) at (7.5, -0) {=};\\node [style=copoint] (18) at (8.75, 0.4999999) {e};\\node [style=none] (19) at (10.75, -0) {\\iff };\\node [style=copoint] (20) at (13, 0.4999999) {e^{{\\sharp }^{\\sharp }}};\\node [style=none] (21) at (13, -0.4999999) {};\\node [style=none] (22) at (14.25, -0) {=};\\node [style=copoint] (23) at (15.25, 0.4999999) {e};\\node [style=none] (24) at (15.25, -0.4999999) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(0) to (1);[bend right=15, looseness=1.00] (11.center) to (10.center);[in=105, out=-105, looseness=1.00] (8.center) to (9.center);[bend left=15, looseness=1.00] (12.center) to (13.center);[bend left=15, looseness=1.00] (15.center) to (14.center);(6) to (7);(18) to (16);(20) to (21.center);(23) to (24.center);\\end{pgfonlayer}\\end{tikzpicture} \\ \\ .$ And finally, using tomography again we can extend this to all processes, $f$ , $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (0, 0.5) {\\forall \\chi , e, C};\\node [style=none] (1) at (5.75, -0) {=};\\node [style=none] (2) at (6.5, -0) {{\\sharp }};\\node [style=none] (3) at (7.25, -0) {{\\sharp }};\\node [style=none] (4) at (8, 2.25) {};\\node [style=none] (5) at (8, -2.25) {};\\node [style=none] (6) at (7.25, -2.25) {};\\node [style=none] (7) at (7.25, 2.25) {};\\node [style=none] (8) at (11, 2.25) {};\\node [style=none] (9) at (11, -2.25) {};\\node [style=none] (10) at (11.25, -2.25) {};\\node [style=none] (11) at (11.25, 2.25) {};\\node [style=none] (12) at (12.75, -0) {=};\\node [style=none] (13) at (18.5, -0) {\\iff };\\node [style=none] (14) at (22.75, -0) {=};\\node [style=map] (15) at (2.75, -0) {f^{{\\sharp }^{\\sharp }}};\\node [style=none] (16) at (2.75, 1) {};\\node [style=none] (17) at (2, 1) {};\\node [style=none] (18) at (3.5, 2) {};\\node [style=none] (19) at (5, 0.9999999) {};\\node [style=none] (20) at (4.25, 0.9999999) {};\\node [style=none] (21) at (4.25, -0.9999999) {};\\node [style=none] (22) at (2.75, -0.9999998) {};\\node [style=none] (23) at (2, -0.9999998) {};\\node [style=none] (24) at (3.5, -2) {};\\node [style=none] (25) at (5, -0.9999999) {};\\node [style=none] (26) at (3.5, 1.5) {e};\\node [style=none] (27) at (4.75, -0) {C};\\node [style=none] (28) at (3.5, -1.5) {\\chi };\\node [style=none] (29) at (5, 0.9999999) {};\\node [style=map] (30) at (8.75, -0) {f};\\node [style=none] (31) at (8.75, 0.9999999) {};\\node [style=none] (32) at (10.75, -0) {C};\\node [style=none] (33) at (11, -0.9999999) {};\\node [style=none] (34) at (8, -0.9999999) {};\\node [style=none] (35) at (9.5, -2) {};\\node [style=none] (36) at (9.5, 1.5) {e};\\node [style=none] (37) at (11, 0.9999999) {};\\node [style=none] (38) at (9.5, -1.5) {\\chi };\\node [style=none] (39) at (8, 0.9999999) {};\\node [style=none] (40) at (9.5, 2) {};\\node [style=none] (41) at (10.25, 0.9999999) {};\\node [style=none] (42) at (10.25, -0.9999999) {};\\node [style=none] (43) at (8.75, -0.9999999) {};\\node [style=none] (44) at (11, 0.9999999) {};\\node [style=map] (45) at (14.5, -0) {f};\\node [style=none] (46) at (14.5, 0.9999999) {};\\node [style=none] (47) at (16.5, -0) {C};\\node [style=none] (48) at (16.75, -0.9999999) {};\\node [style=none] (49) at (13.75, -0.9999999) {};\\node [style=none] (50) at (15.25, -2) {};\\node [style=none] (51) at (15.25, 1.5) {e};\\node [style=none] (52) at (16.75, 0.9999999) {};\\node [style=none] (53) at (15.25, -1.5) {\\chi };\\node [style=none] (54) at (13.75, 0.9999999) {};\\node [style=none] (55) at (15.25, 2) {};\\node [style=none] (56) at (16, 0.9999999) {};\\node [style=none] (57) at (16, -0.9999999) {};\\node [style=none] (58) at (14.5, -0.9999999) {};\\node [style=none] (59) at (16.75, 0.9999999) {};\\node [style=map] (60) at (21, -0) {f^{{\\sharp }^{\\sharp }}};\\node [style=none] (61) at (21, -0.9999999) {};\\node [style=none] (62) at (21, 0.9999999) {};\\node [style=none] (63) at (24, 1) {};\\node [style=none] (64) at (24, -1) {};\\node [style=map] (65) at (24, -0) {f};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}[bend right=15, looseness=1.00] (7.center) to (6.center);[in=105, out=-105, looseness=1.00] (4.center) to (5.center);[bend left=15, looseness=1.00] (8.center) to (9.center);[bend left=15, looseness=1.00] (11.center) to (10.center);(17.center) to (19.center);(19.center) to (18.center);(18.center) to (17.center);(16.center) to (15);(15) to (22.center);(23.center) to (25.center);(25.center) to (24.center);(24.center) to (23.center);(20.center) to (21.center);(39.center) to (37.center);(37.center) to (40.center);(40.center) to (39.center);(31.center) to (30);(30) to (43.center);(34.center) to (33.center);(33.center) to (35.center);(35.center) to (34.center);(41.center) to (42.center);(54.center) to (52.center);(52.center) to (55.center);(55.center) to (54.center);(46.center) to (45);(45) to (58.center);(49.center) to (48.center);(48.center) to (50.center);(50.center) to (49.center);(56.center) to (57.center);(62.center) to (60);(60) to (61.center);(63.center) to (65);(65) to (64.center);\\end{pgfonlayer}\\end{tikzpicture} \\ \\ .$ Theorem 66 For test structures of tomographic process theories, the operation $\\sharp $ in transformability can always be chosen in a manner such that it is a dagger.", "Theorem REF demonstrates that ${\\sharp }$ is involutive, and transformability imposes that if $f:A\\rightarrow B$ then $f^{\\sharp }: B\\rightarrow A$ , therefore for a test structure to provide a dagger all we need to check is that it preserves diagrams.", "This is easiest to prove by checking that ${\\sharp }$ preserves the two primitive forms of composition, $\\otimes $ and $\\circ $ .", "Consider the action of ${\\sharp }$ on the diagram, $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=point] (0) at (2, -1.5) {\\psi };\\node [style=map] (1) at (2, -0) {f};\\node [style=none] (2) at (2, 1.25) {};\\node [style=point] (3) at (4, -1.5) {\\phi };\\node [style=map] (4) at (4, -0) {g};\\node [style=none] (5) at (4, 1.25) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(0) to (1);(1) to (2.center);(3) to (4);(4) to (5.center);\\end{pgfonlayer}\\end{tikzpicture} ,$ there are two ways to apply the composability and transformability axioms to this giving the following constraint, $\\forall \\psi , \\phi \\qquad \\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (-0.7499999, -0) {{\\sharp }};\\node [style=point] (1) at (0.9999998, -1.5) {\\psi };\\node [style=map] (2) at (0.9999998, -0) {f};\\node [style=none] (3) at (0.9999998, 1.5) {};\\node [style=point] (4) at (3, -1.5) {\\phi };\\node [style=map] (5) at (3, -0) {g};\\node [style=none] (6) at (3, 1.5) {};\\node [style=none] (7) at (5.25, -0) {=};\\node [style=copoint] (8) at (7, 1.5) {\\psi ^{\\sharp }};\\node [style=copoint] (9) at (9, 1.5) {\\phi ^{\\sharp }};\\node [style=none] (10) at (7, 0.5) {};\\node [style=none] (11) at (9, 0.5) {};\\node [style=none] (12) at (10, 0.5) {};\\node [style=none] (13) at (9.5, -1) {};\\node [style=none] (14) at (6.5, -1) {};\\node [style=none] (15) at (6.5, 0.5) {};\\node [style=none] (16) at (7, -1) {};\\node [style=none] (17) at (9, -1) {};\\node [style=none] (18) at (7, -1.75) {};\\node [style=none] (19) at (9, -1.75) {};\\node [style=none] (20) at (8, -0.25) {{\\sharp }(f\\otimes g)};\\node [style=none] (21) at (0.25, 1.75) {};\\node [style=none] (22) at (0.25, -2.25) {};\\node [style=none] (23) at (4, -2.25) {};\\node [style=none] (24) at (4, 1.75) {};\\node [style=map] (25) at (-7.25, -0.25) {{\\sharp }(f)};\\node [style=none] (26) at (-4.25, -1.75) {};\\node [style=copoint] (27) at (-4.25, 1.25) {\\phi ^{\\sharp }};\\node [style=map] (28) at (-4.25, -0.25) {{\\sharp }(g)};\\node [style=none] (29) at (-7.25, -1.75) {};\\node [style=copoint] (30) at (-7.25, 1.25) {\\psi ^{\\sharp }};\\node [style=none] (31) at (-1.999999, -0) {=};\\node [style=none] (32) at (10.5, -0) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(1) to (2);(2) to (3.center);(4) to (5);(5) to (6.center);(8) to (10.center);(9) to (11.center);(15.center) to (12.center);(12.center) to (13.center);(13.center) to (14.center);(14.center) to (15.center);(16.center) to (18.center);(17.center) to (19.center);[bend left=15, looseness=1.00] (22.center) to (21.center);[bend right=15, looseness=1.00] (23.center) to (24.center);(30) to (25);(25) to (29.center);(27) to (28);(28) to (26.center);\\end{pgfonlayer}\\end{tikzpicture} .$ Next consider applying ${\\sharp }$ to the diagram, $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=point] (0) at (2, -2.25) {\\psi };\\node [style=map] (1) at (2, -0.7499999) {f};\\node [style=map] (2) at (2, 0.7499999) {g};\\node [style=none] (3) at (2, 2.25) {};\\node [style=none] (4) at (3.5, -0) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(0) to (1);(1) to (2);(2) to (3.center);\\end{pgfonlayer}\\end{tikzpicture} ,$ here there are again two different ways to apply the transformability axiom, and so we obtain, $ \\forall \\psi \\qquad \\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (0, -0) {{\\sharp }};\\node [style=point] (1) at (1.75, -1.75) {\\psi };\\node [style=map] (2) at (1.75, -0.25) {f};\\node [style=map] (3) at (1.75, 1.25) {g};\\node [style=none] (4) at (1.75, 2.25) {};\\node [style=none] (5) at (4.25, -0) {=};\\node [style=copoint] (6) at (7, 1) {\\psi ^{\\sharp }};\\node [style=map] (7) at (7, -0.5) {{\\sharp }(f\\circ g)};\\node [style=none] (8) at (7, -1.5) {};\\node [style=map] (9) at (-4, -1.5) {{\\sharp }(g)};\\node [style=copoint] (10) at (-4, 1.5) {\\psi ^{\\sharp }};\\node [style=none] (11) at (-4, -2.5) {};\\node [style=map] (12) at (-4, -0) {{\\sharp }(f)};\\node [style=none] (13) at (1, 2.5) {};\\node [style=none] (14) at (2.75, 2.5) {};\\node [style=none] (15) at (1, -2.5) {};\\node [style=none] (16) at (2.75, -2.5) {};\\node [style=none] (17) at (-1.5, -0) {=};\\node [style=none] (18) at (8.75, -0) {};\\node [style=none] (19) at (8.75, -0.25) {};\\node [style=none] (20) at (9.25, -0) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(1) to (2);(2) to (3);(3) to (4.center);(7) to (6);(8.center) to (7);(10) to (12);(12) to (9);(9) to (11.center);[bend left=15, looseness=1.00] (15.center) to (13.center);[bend right=15, looseness=1.00] (16.center) to (14.center);\\end{pgfonlayer}\\end{tikzpicture} .$ The above two conditions (eq.", "REF & REF ) are direct consequences of the test structure axioms and so must be satisfied for the axioms to hold.", "There is an obvious solution to these: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (0, 0.5) {};\\node [style=none] (1) at (3.25, 0.5) {};\\node [style=none] (2) at (3, -0.5) {};\\node [style=none] (3) at (0, -0.5) {};\\node [style=none] (4) at (0.5, 0.5) {};\\node [style=none] (5) at (2.5, 0.5) {};\\node [style=none] (6) at (2.5, -0.5) {};\\node [style=none] (7) at (0.5, -0.5) {};\\node [style=none] (8) at (0.4999999, 1.5) {};\\node [style=none] (9) at (2.5, 1.5) {};\\node [style=none] (10) at (0.4999999, -1.5) {};\\node [style=none] (11) at (2.5, -1.5) {};\\node [style=none] (12) at (1.5, -0) {{\\sharp }(f\\otimes g)};\\node [style=none] (13) at (4, -0) {=};\\node [style=none] (14) at (6, -1.5) {};\\node [style=map] (15) at (6, -0) {{\\sharp }(f)};\\node [style=none] (16) at (6, 1.5) {};\\node [style=none] (17) at (9.000001, -1.5) {};\\node [style=map] (18) at (9, -0) {{\\sharp }(g)};\\node [style=none] (19) at (9.000001, 1.5) {};\\node [style=none] (20) at (12, -0) {\\&};\\node [style=none] (21) at (15, -1.5) {};\\node [style=map] (22) at (15, -0) {{\\sharp }(g\\circ f)};\\node [style=none] (23) at (15, 1.5) {};\\node [style=none] (24) at (18, -0) {=};\\node [style=map] (25) at (20, -0.7499999) {{\\sharp }(g)};\\node [style=map] (26) at (20, 0.7499999) {{\\sharp }(f)};\\node [style=none] (27) at (20, 2.25) {};\\node [style=none] (28) at (20, -2.25) {};\\node [style=none] (29) at (21.75, -0) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(0.center) to (1.center);(1.center) to (2.center);(2.center) to (3.center);(3.center) to (0.center);(4.center) to (8.center);(5.center) to (9.center);(7.center) to (10.center);(6.center) to (11.center);(14.center) to (15);(15) to (16.center);(17.center) to (18);(18) to (19.center);(21.center) to (22);(22) to (23.center);(28.center) to (25);(25) to (26);(26) to (27.center);\\end{pgfonlayer}\\end{tikzpicture} ,$ and so there will always exist an operation satisfying the axioms as well as eq.", "REF , and so there is always a ${\\sharp }$ which is a process theoretic dagger, ${\\sharp }$ .", "Theorem 77 All test structures of local tomographic process theories induce daggers.", "The only solution to equations REF & REF for a tomographically local theory are eq.", "REF and so ${\\sharp }$ must be a dagger.", "In the process-theoretic context, daggers have always been taken to be involutive.", "Now, finally, we have shown that there is a good justification for doing so by demonstrating how this important feature of the Hermitian adjoint follows from the notion of a test structure." ], [ "Deriving the Hermitian adjoint for quantum theory", "In this section we demonstrate that for quantum theory any test structure must be a Hermitian adjoint.", "In order to do so, we will need to extend the process theory representing quantum theory so that it also includes mixed states, rather than just the pure theory described in the previous sections.", "Example 5 We construct the process theory of mixed post-selected quantum processes from the process theories ${D}[{C}LM]$ and ${C}LM$ .", "First note that we have an embedding ${D}[{C}LM]\\hookrightarrow {C}LM$ .", "This embedding allows us to take sums of arbitrary processes of ${D}[{C}LM]$ within ${C}LM$ .", "These sums, together with the availability of the scalars as processes, allows one to form all linear combinations: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=kpointconj] (0) at (0.7500001, -0.5) {\\psi _i};\\node [style=kpoint] (1) at (2.25, -0.5) {\\psi _i};\\node [style=none] (2) at (0.7500001, 0.75) {};\\node [style=none] (3) at (2.25, 0.75) {};\\node [style=none] (4) at (-2.25, -0) {\\sum _i};\\node [style=scalar] (5) at (-1, -0) {r_i};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(0) to (2.center);(1) to (3.center);\\end{pgfonlayer}\\end{tikzpicture} \\ \\ .$ which, in particular, includes all mixtures: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=kpointconj] (0) at (0.7500001, -0.5) {\\psi _i};\\node [style=kpoint] (1) at (2.25, -0.5) {\\psi _i};\\node [style=none] (2) at (0.7500001, 0.75) {};\\node [style=none] (3) at (2.25, 0.75) {};\\node [style=none] (4) at (-2.25, -0) {\\sum _i};\\node [style=scalar] (5) at (-1, -0) {p_i};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(0) to (2.center);(1) to (3.center);\\end{pgfonlayer}\\end{tikzpicture} \\quad \\text{where}\\quad \\sum _ip_i=1\\ .$ We call the resulting process theory ${M}[{C}LM]$ .", "For notational convenience we denote processes in this theory with bold lines, for example: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=dmap] (0) at (0, -0) {f};\\node [style=none] (1) at (0, 1) {};\\node [style=none] (2) at (0, -1) {};\\node [style=none] (3) at (2, -0) {=};\\node [style=none] (4) at (3.25, -0) {\\sum _i};\\node [style=scalar] (5) at (4.25, -0) {r_i};\\node [style=mapconj] (6) at (6, -0) {f_i};\\node [style=map] (7) at (7.75, -0) {f_i};\\node [style=none] (8) at (6, 1) {};\\node [style=none] (9) at (6, -1) {};\\node [style=none] (10) at (7.75, -1) {};\\node [style=none] (11) at (7.75, 1) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}[style=doubled] (1.center) to (0);[style=doubled] (0) to (2.center);(8.center) to (6);(6) to (9.center);(11.center) to (7);(7) to (10.center);\\end{pgfonlayer}\\end{tikzpicture} \\ .$ We will now assume that transformability applies to ${M}[{C}LM]$ , that is: $\\forall f\\ \\exists f^{\\sharp }\\quad \\text{ s.t. }", "\\quad \\left(\\right)^{\\sharp }\\ =\\ \\ \\ \\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (0, -2.25) {};\\node [style=dmap] (1) at (0, -0.75) {f^{\\sharp }};\\node [style=dcopoint] (2) at (0, 1.25) {\\psi ^{\\sharp }};\\node [style=none] (3) at (0.5, -1.75) {B};\\node [style=none] (4) at (0.5, 0.25) {A};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}[style=doubled](0.center) to (1);[style=doubled](1) to (2);\\end{pgfonlayer}\\end{tikzpicture} \\ \\ \\ .$ where now $f$ can now be an arbitrary sum of processes, $f=\\sum _i f_i$ .", "Conceptually, this is a very natural assumption, which states that uncertainty about how a state transforms translates into uncertainty about how the test for that state transforms (see also section  below).", "Before showing how the test structure provides a Hermitian adjoint we show how another candidate dagger, the transpose, fails to be a test-structure.", "Theorem 88 The transpose, $T$ , does not provide a test structure for ${M}[{C}LM]$ .", "Consider any states $\\psi $ and $\\phi $ in ${C}LM$ such that ${D}(\\psi )^T\\circ {D}(\\psi )= 1= {D}(\\phi )^T \\circ {D}(\\phi )$ and, ${D}(\\psi )^T\\circ {D}(\\phi )=0$ (e.g.", "the computational basis states).", "Then ${D}(\\psi + i \\phi )^T\\circ {D}(\\psi +i \\phi ) = 1+0+0+i^2= 0$ violating testability.", "Theorem 99 The test structure provides a Hermitian adjoint for ${M}[{C}LM]$ .", "First note that quantum theory is a locally tomographic additive process theory, and so any test structure provides a dagger.", "This provides an inner product on the states defined as: $\\langle \\psi ,\\phi \\rangle \\ :=\\ \\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=dcopoint] (0) at (0, .5) {\\psi ^{\\sharp }};\\node [style=dpoint] (1) at (0, -.5) {\\phi };\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}[style=doubled](1) to (0);\\end{pgfonlayer}\\end{tikzpicture} \\ \\ .$ It is simple to check that the inner product axioms are satisfied: Symmetry: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (1, -0) {\\left<\\psi ,\\phi \\right>=};\\node [style=dcopoint] (1) at (3.75, 0.5000001) {\\psi ^{\\sharp }};\\node [style=dpoint] (2) at (3.75, -0.5000001) {\\phi };\\node [style=none] (3) at (5.5, -0) {=};\\node [style=none] (4) at (6.5, -0) {{\\sharp }};\\node [style=dcopoint] (5) at (8, 0.5000001) {\\psi ^{\\sharp }};\\node [style=dpoint] (6) at (8, -0.5000001) {\\phi };\\node [style=none] (7) at (7.25, 1.5) {};\\node [style=none] (8) at (7.25, -1.25) {};\\node [style=none] (9) at (8.75, -1.25) {};\\node [style=none] (10) at (8.75, 1.5) {};\\node [style=none] (11) at (10, -0) {=};\\node [style=dcopoint] (12) at (11.5, 0.5000001) {\\phi ^{\\sharp }};\\node [style=dpoint] (13) at (11.5, -0.5000001) {\\psi };\\node [style=none] (14) at (14, -0) {=\\left<\\phi ,\\psi \\right>};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}[style=doubled] (2) to (1);[style=doubled] (6) to (5);[bend left=15, looseness=1.00] (8.center) to (7.center);[bend right=15, looseness=1.00] (9.center) to (10.center);[style=doubled] (13) to (12);\\end{pgfonlayer}\\end{tikzpicture} \\ \\ ,$ where the second equality follows by lemma REF and the third from the extended form of transformability.", "Linearity: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (1.25, -0) {\\left<\\phi ,a\\psi +b\\chi \\right>\\ =};\\node [style=scalar] (1) at (4.75, -0) {a};\\node [style=dcopoint] (2) at (6.25, 0.5000001) {\\phi ^{\\sharp }};\\node [style=dpoint] (3) at (6.25, -0.5000001) {\\psi };\\node [style=scalar] (4) at (8.25, -0) {b};\\node [style=dcopoint] (5) at (9.75, 0.5000001) {\\phi ^{\\sharp }};\\node [style=dpoint] (6) at (9.75, -0.5000001) {\\chi };\\node [style=none] (7) at (7.25, -0) {+};\\node [style=none] (8) at (14.25, -0) {=\\ a\\left<\\phi ,\\psi \\right>+b\\left<\\phi ,\\chi \\right>};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}[style=doubled] (3) to (2);[style=doubled] (6) to (5);\\end{pgfonlayer}\\end{tikzpicture} \\ \\ .$ Positivity: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=none] (0) at (-6.25, 0) {\\left<\\chi ,\\chi \\right>\\ =};\\node [style=dcopoint] (1) at (-3.5, 0.4999999) {\\chi ^{\\sharp }};\\node [style=dpoint] (2) at (-3.5, -0.4999999) {\\chi };\\node [style=none] (3) at (-1.5, 0) {=};\\node [style=dcopoint] (4) at (2, 0.4999999) {\\psi ^{\\sharp }};\\node [style=dpoint] (5) at (2, -0.4999999) {\\psi };\\node [style=scalar] (6) at (0.25, 0.5) {r};\\node [style=scalar] (7) at (0.25, -0.5) {r};\\node [style=none] (8) at (4, 0) {=};\\node [style=scalar] (9) at (5.25, -0.5) {r};\\node [style=scalar] (10) at (5.25, 0.5) {r};\\node [style=scalar] (11) at (6, 0) {1};\\node [style=none] (12) at (7.5, 0) {\\ge \\ 0};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}[style=doubled] (2) to (1);[style=doubled] (5) to (4);\\end{pgfonlayer}\\end{tikzpicture} \\ \\ ,$ where the second equation uses testability, and similarly positive definiteness follows: $\\left<\\chi ,\\chi \\right>=0\\quad \\iff \\quad r^2=0 \\quad \\iff \\quad \\psi = 0 \\ .$ The test structure is therefore the Hermitian adjoint associated to this inner product, defined as $\\langle \\cdot ,A\\cdot \\rangle =\\langle A^\\dagger \\cdot ,\\cdot \\rangle $ .", "This explains why the Hermitian adjoint – rather than any other dagger such as the transpose – plays such a prominent role in quantum theory: it has an operational interpretation in terms of a test structure." ], [ "Test structures for additive theories", "Mixed quantum theory ${M}[{C}LM]$ is an example of an additive process theory.", "Definition 4 Additive process theories are process theories that come with a notion of `sum of diagrams' that distributes over diagrams: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=map] (0) at (-5.25, 0) {f_i};\\node [style=none] (1) at (-5.25, -1.25) {};\\node [style=none] (2) at (-5.25, 1.25) {};\\node [style=none] (3) at (-7, 0) {\\sum _i};\\node [style=none] (4) at (-7.75, 0.75) {};\\node [style=none] (5) at (-7.75, -0.75) {};\\node [style=none] (6) at (-3.75, 0.75) {};\\node [style=none] (7) at (-3.75, -0.75) {};\\node [style=none] (8) at (-3, 1.25) {};\\node [style=none] (9) at (-3, -1.25) {};\\node [style=none] (10) at (-6.25, 1.25) {};\\node [style=none] (11) at (-2.25, 1.25) {};\\node [style=none] (12) at (-1.75, 2.25) {};\\node [style=none] (13) at (-6.25, 2.25) {};\\node [style=none] (14) at (-4.25, 2.25) {};\\node [style=none] (15) at (-4.25, 3) {};\\node [style=none] (16) at (-2.25, -2.25) {};\\node [style=none] (17) at (-3, -1.25) {};\\node [style=none] (18) at (-5.25, -1.25) {};\\node [style=none] (19) at (-6.25, -2.25) {};\\node [style=none] (20) at (-3, -1.25) {};\\node [style=none] (21) at (-6.25, -1.25) {};\\node [style=none] (22) at (-5.25, -1.25) {};\\node [style=none] (23) at (-4.25, -2.25) {};\\node [style=none] (24) at (-1.75, -1.25) {};\\node [style=none] (25) at (-4.25, -3) {};\\node [style=none] (26) at (-4.25, -2.25) {};\\node [style=none] (27) at (5.25, -1.25) {};\\node [style=none] (28) at (7.5, 1.25) {};\\node [style=none] (29) at (4.25, 2.25) {};\\node [style=none] (30) at (5.25, -1.25) {};\\node [style=none] (31) at (3.75, -3.25) {};\\node [style=none] (32) at (6.5, -2.25) {};\\node [style=none] (33) at (7.5, -1.25) {};\\node [style=none] (34) at (9, 3) {};\\node [style=none] (35) at (3.75, 3) {};\\node [style=none] (36) at (5.25, -1.25) {};\\node [style=none] (37) at (4.25, -2.25) {};\\node [style=none] (38) at (6.5, 2.25) {};\\node [style=none] (39) at (6.5, -3) {};\\node [style=none] (40) at (8.25, -2.25) {};\\node [style=none] (41) at (7.5, -1.25) {};\\node [style=none] (42) at (5.25, 1.25) {};\\node [style=none] (43) at (4.25, -1.25) {};\\node [style=none] (44) at (8.75, 2.25) {};\\node [style=none] (45) at (2.25, 0) {\\sum _i};\\node [style=none] (46) at (6.5, 3) {};\\node [style=none] (47) at (4.25, 1.25) {};\\node [style=map] (48) at (5.25, 0) {f_i};\\node [style=none] (49) at (8.25, 1.25) {};\\node [style=none] (50) at (7.5, -1.25) {};\\node [style=none] (51) at (8.75, -1.25) {};\\node [style=none] (52) at (9, -3.25) {};\\node [style=none] (53) at (6.5, -2.25) {};\\node [style=none] (54) at (0, 0) {=};\\node [style=none] (55) at (-4.25, -1.75) {f};\\node [style=none] (56) at (-4.25, 1.75) {g};\\node [style=none] (57) at (6.5, -1.75) {f};\\node [style=none] (58) at (6.5, 1.75) {g};\\node [style=none] (59) at (12.5, 0) {\\forall \\ f,\\ g,\\ C};\\node [style=none] (60) at (-2.5, 0) {C};\\node [style=none] (61) at (8, 0) {C};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(1.center) to (0);(0) to (2.center);[bend right=15, looseness=1.00] (4.center) to (5.center);[bend left=15, looseness=1.00] (6.center) to (7.center);(13.center) to (12.center);(12.center) to (11.center);(11.center) to (10.center);(10.center) to (13.center);(14.center) to (15.center);(8.center) to (9.center);(21.center) to (24.center);(24.center) to (16.center);(16.center) to (19.center);(19.center) to (21.center);(18.center) to (22.center);(20.center) to (17.center);(25.center) to (26.center);(36.center) to (48);(48) to (42.center);[bend right=15, looseness=1.00] (35.center) to (31.center);[bend left=15, looseness=1.00] (34.center) to (52.center);(29.center) to (44.center);(44.center) to (49.center);(49.center) to (47.center);(47.center) to (29.center);(46.center) to (38.center);(28.center) to (50.center);(43.center) to (51.center);(51.center) to (40.center);(40.center) to (37.center);(37.center) to (43.center);(27.center) to (30.center);(41.center) to (33.center);(32.center) to (39.center);\\end{pgfonlayer}\\end{tikzpicture} .$ Remark 3 In category-theoretic terms, additivity of process theories means enrichment in commutative monoids [18].", "That the numbers are positive reals means that the morphisms from the tensor unit to itself are the positive reals.", "Having both sums and numbers, together with the fact that since numbers have no outputs they can freely move around in diagrams, it also follows that: Proposition 1010 In additive process theories convex combinations (or equivalent, probabilistic mixtures) distribute over diagrams: $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=map] (0) at (-5.25, 0) {f_i};\\node [style=none] (1) at (-5.25, -1.25) {};\\node [style=none] (2) at (-5.25, 1.25) {};\\node [style=none] (3) at (-7.5, 0) {\\sum _i p_i};\\node [style=none] (4) at (-8.5, 0.75) {};\\node [style=none] (5) at (-8.5, -0.75) {};\\node [style=none] (6) at (-3.75, 0.75) {};\\node [style=none] (7) at (-3.75, -0.75) {};\\node [style=none] (8) at (-3, 1.25) {};\\node [style=none] (9) at (-3, -1.25) {};\\node [style=none] (10) at (-6.25, 1.25) {};\\node [style=none] (11) at (-2, 1.25) {};\\node [style=none] (12) at (-2, 2.25) {};\\node [style=none] (13) at (-6.25, 2.25) {};\\node [style=none] (14) at (-4.25, 2.25) {};\\node [style=none] (15) at (-4.25, 3) {};\\node [style=none] (16) at (-2, -2.25) {};\\node [style=none] (17) at (-3, -1.25) {};\\node [style=none] (18) at (-5.25, -1.25) {};\\node [style=none] (19) at (-6.25, -2.25) {};\\node [style=none] (20) at (-3, -1.25) {};\\node [style=none] (21) at (-6.25, -1.25) {};\\node [style=none] (22) at (-5.25, -1.25) {};\\node [style=none] (23) at (-4.25, -2.25) {};\\node [style=none] (24) at (-2, -1.25) {};\\node [style=none] (25) at (-4.25, -3) {};\\node [style=none] (26) at (-4.25, -2.25) {};\\node [style=none] (27) at (5.75, -1.25) {};\\node [style=none] (28) at (8, 1.25) {};\\node [style=none] (29) at (4.75, 2.25) {};\\node [style=none] (30) at (5.75, -1.25) {};\\node [style=none] (31) at (4.25, -3.25) {};\\node [style=none] (32) at (7, -2.25) {};\\node [style=none] (33) at (8, -1.25) {};\\node [style=none] (34) at (9.5, 3) {};\\node [style=none] (35) at (4.25, 3) {};\\node [style=none] (36) at (5.75, -1.25) {};\\node [style=none] (37) at (4.75, -2.25) {};\\node [style=none] (38) at (7, 2.25) {};\\node [style=none] (39) at (7, -3) {};\\node [style=none] (40) at (9, -2.25) {};\\node [style=none] (41) at (8, -1.25) {};\\node [style=none] (42) at (5.75, 1.25) {};\\node [style=none] (43) at (4.75, -1.25) {};\\node [style=none] (44) at (9, 2.25) {};\\node [style=none] (45) at (2.25, 0) {\\sum _i p_i};\\node [style=none] (46) at (7, 3) {};\\node [style=none] (47) at (4.75, 1.25) {};\\node [style=map] (48) at (5.75, 0) {f_i};\\node [style=none] (49) at (9, 1.25) {};\\node [style=none] (50) at (8, -1.25) {};\\node [style=none] (51) at (9, -1.25) {};\\node [style=none] (52) at (9.5, -3.25) {};\\node [style=none] (53) at (7, -2.25) {};\\node [style=none] (54) at (0, 0) {=};\\node [style=none] (55) at (-4.25, -1.75) {f};\\node [style=none] (56) at (-4.25, 1.75) {g};\\node [style=none] (57) at (7, -1.75) {f};\\node [style=none] (58) at (7, 1.75) {g};\\node [style=none] (59) at (12.5, 0) {\\forall \\ f,\\ g,\\ C};\\node [style=none] (60) at (-2.5, 0) {C};\\node [style=none] (61) at (8.5, 0) {C};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(1.center) to (0);(0) to (2.center);[bend right=15, looseness=1.00] (4.center) to (5.center);[bend left=15, looseness=1.00] (6.center) to (7.center);(13.center) to (12.center);(12.center) to (11.center);(11.center) to (10.center);(10.center) to (13.center);(14.center) to (15.center);(8.center) to (9.center);(21.center) to (24.center);(24.center) to (16.center);(16.center) to (19.center);(19.center) to (21.center);(18.center) to (22.center);(20.center) to (17.center);(25.center) to (26.center);(36.center) to (48);(48) to (42.center);[bend right=15, looseness=1.00] (35.center) to (31.center);[bend left=15, looseness=1.00] (34.center) to (52.center);(29.center) to (44.center);(44.center) to (49.center);(49.center) to (47.center);(47.center) to (29.center);(46.center) to (38.center);(28.center) to (50.center);(43.center) to (51.center);(51.center) to (40.center);(40.center) to (37.center);(37.center) to (43.center);(27.center) to (30.center);(41.center) to (33.center);(32.center) to (39.center);\\end{pgfonlayer}\\end{tikzpicture} ,$ where $\\sum _ip_i=1$ .", "The notion of an additive process theory is similar to the well-studied framework of generalised probabilistic theories [5], [4].", "There are many presentations of this framework which are subtly different but in their most recent incarnation they have the structure of an additive process theory [10].", "While the usual GPTs have no unique parallel composition operation, the underlying structure of the processes allows one nonetheless to talk about composite systems.", "Theorem 1111 Additive process theories do not have test structures.", "Consider a state that is a convex mixture of normalised states and assume the existence of a test structure, $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=point] (0) at (-0.4999999, -0.2500001) {\\psi };\\node [style=none] (1) at (-0.4999999, 0.9999999) {};\\node [style=none] (2) at (1, -0) {=};\\node [style=none] (3) at (2.5, -0) {\\sum _i p_i};\\node [style=point] (4) at (4.25, -0.2500001) {\\psi _i};\\node [style=none] (5) at (4.25, 0.9999999) {};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(1.center) to (0);(5.center) to (4);\\end{pgfonlayer}\\end{tikzpicture} \\ \\ ,$ where $\\sum _ip_i=1$ .", "Then if we demand that we have a test for that state, $\\psi ^{\\sharp }$ , then by definition this test must produce probabilities, $\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=point] (0) at (-0.25, -0) {\\psi };\\node [style=copoint] (1) at (-0.25, 0.9999999) {e};\\node [style=none] (2) at (8.5, 0.4999999) {=};\\node [style=none] (3) at (5.5, 0.5) {\\sum _i p_i};\\node [style=point] (4) at (7.25, -0) {\\psi _i};\\node [style=copoint] (5) at (7.25, 0.9999999) {e};\\node [style=none] (6) at (1, 0.5) {=};\\node [style=none] (7) at (2, 0.5) {1};\\node [style=none] (8) at (3.5, 0.5) {\\Rightarrow };\\node [style=none] (9) at (9.5, 0.4999999) {1};\\node [style=point] (10) at (5.5, -2.5) {\\psi _i};\\node [style=none] (11) at (3.5, -2) {\\Rightarrow };\\node [style=none] (12) at (6.75, -2) {=};\\node [style=copoint] (13) at (5.5, -1.5) {e};\\node [style=none] (14) at (7.75, -2) {1\\ ,};\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}(1) to (0);(5) to (4);(13) to (10);\\end{pgfonlayer}\\end{tikzpicture} $ which violates sharpness giving a contradiction.", "This is not surprising; one cannot deterministically test for a probabilistic mixture of states.", "However in the previous section we showed that this wasn't a problem for quantum theory, similarly, theorem REF can be extended to additive process theories.", "Theorem 1212 A process theory with a test structure, embedded within an additive theory, has an inner product defined as: $\\langle \\psi ,\\phi \\rangle :=\\begin{tikzpicture}\\begin{pgfonlayer}{nodelayer}\\node [style=copoint] (0) at (0, .5) {\\psi ^{\\sharp }};\\node [style=point] (1) at (0, -.5) {\\phi };\\end{pgfonlayer}\\begin{pgfonlayer}{edgelayer}[style=none](1) to (0);\\end{pgfonlayer}\\end{tikzpicture} \\ \\ ,$ if we demand that transformability extends to all processes in the additive theory.", "Identical to the quantum case in theorem REF ." ], [ "Summary and outlook", "In this paper we considered the physical principle of: for each state there exists a corresponding `test'.", "We show that for tomographic process theories this principle leads to a `test structure' which we show provides a process-theoretic dagger.", "This explains why the dagger should have various properties, involutivity being a particularly surprising consequence given the test structure's definition.", "Moreover, for pure quantum theory, we show that the particular dagger provided is the Hermitian adjoint, explaining why this plays such an important role both in the process-theoretic description of quantum theory, and of course, in quantum theory itself in the form of the Hilbert space inner-product.", "In the final section we begin to explore the connections between test structures and purity of a theory, showing that theories with mixed states cannot have a test structure.", "In fact, if we consider process theories that come with a discarding operation – for which we require causality [9], [14] – then we can turn this around and show that a test structure characterises when a process is pure by its testability." ] ]
1606.05086
[ [ "Invariant measures for continued fraction algorithms with finitely many\n digits" ], [ "Abstract In this paper we consider continued fraction (CF) expansions on intervals different from $[0,1]$.", "For every $x$ in such interval we find a CF expansion with a finite number of possible digits.", "Using the natural extension, the density of the invariant measure is obtained in a number of examples.", "In case this method does not work, a Gauss-Kuzmin-L\\'evy based approximation method is used.", "Finally, a subfamily of the $N$-expansions is studied.", "In particular, the entropy as a function of a parameter $\\alpha$ is estimated for $N=2$ and $N=36$.", "Interesting behavior can be observed from numerical results." ], [ "Introduction", "In general, studies on continued fraction expansions focus on expansions for which almost all $x$All `almost all $x$ ' statements are wrt.", "Lebesgue measure.", "have an expansion with digits from an infinite alphabet.", "A classical example is the regular continued fraction [2], [3], [4].", "An example of continued fraction expansions with only finitely many digits has been introduced in [5] by Joe Lehner, where the only possible digits are 1 and 2; see also [6].", "More recently, continued fractions have been investigated for which all $x$ in a certain interval have finitely many possible digits.", "In [7] the following 4-expansion has been (briefly) studied.", "Let $T:[1,2]\\rightarrow [1,2]$ be defined as $T(x) = \\left\\lbrace \\begin{array}{l l l}\\dfrac{4}{x}-1 & \\text{for} &x\\in (\\frac{4}{3},2] \\\\[0.75em]\\dfrac{4}{x}-2 & \\text{for} &x\\in [1,\\frac{4}{3}] \\ .", "\\\\\\end{array} \\right.$ Figure: The CF-map TT from ().By repeatedly using this map we find that every $x\\in [1,2]$ has an infinite continued fraction expansion of the form $x= \\frac{4}{\\displaystyle d_1+\\frac{4}{\\displaystyle d_2+ \\ddots }}\\\\$ with $d_n\\in \\lbrace 1,2\\rbrace $ for all $n\\ge 1$ .", "The class of continued fractions algorithms that give rise to digits from a finite alphabet is very large.", "In this paper we will give examples of such expansions and in Section  we will take a closer look at an interesting sub-family.", "Most of the examples will be a particular case of $N$ -expansions (see [8], [9], [7]).", "Other examples are closely related and can be found by combining the $N$ -expansions with flipped expansions (cf.", "[10] for 2-expansions; see also [11] for flipped expansions).", "For all these examples we refer to [11] for ergodicity (which can be obtained in all these cases in a similar way) and existence of an invariant measure.", "In a number of cases however, it is difficult to find the invariant measure explicitly, while in seemingly closely related cases it is very easy.", "In case we cannot give an analytic expression for the invariant measure we will give an approximation using a method that is very suitable (from a computational point of view) for expansions with finitely many different digits.", "This method is based on a Gauss-Kuzmin-Lévy theorem.", "For greedy $N$ -expansions this theorem is proved by Dan Lascu in [13].", "The method yields smoother results than by simulating in the classical way (looking at the histogram of the orbit of a typical point as described in Choe's book [14], and used in his papers [12], [15]).", "We also give an example in which we do know the density and where we use this method to show its strength.", "In Section  we will give the general form of the continued fraction maps we study in this paper.", "After that we give several examples of such maps and a way of finding the density of the invariant measure by using the natural extension.", "In Section REF we will see how we simulated the densities in case we were not able to find them explicitly.", "In the last section we will consider a subfamily of the $N$ -expansions which can be parameterized by $\\alpha \\in (0,\\sqrt{N}-1]$ .", "We study the entropy as function of $\\alpha $ .", "We will do so mainly on a numerical basis.", "In the past decades, it turned out to be very interesting to look at entropy of a family of continued fractions as a function of a parameter.", "In [16], [17], [18], [19], [20], [21], [22] the entropy of Nakada's $\\alpha $ -expansions is studied.", "For example in [22] it is shown that in any neighborhood of 0 you can always find an interval on which the entropy function is increasing, an interval on which the entropy function is decreasing and an interval on which this function is constant.", "In [18] it is shown that there is a countable set of open intervals on which the entropy function is monotonic.", "The union of these intervals has Lebesgue measure 1.", "As matching plays an important role in these papers we will take a glimpse at how matching works for our subfamily (see Section  for the definition and use of matching)." ], [ "The general form of our maps", "Throughout this paper we will look at continued fraction algorithms of the following form.", "Fix an integer $N\\ge 2$ and let $[a,b]$ be a subinterval of $[0,N]$ with $b-a\\ge 1$ .", "Let $T:[a,b]\\rightarrow [a,b]$ be defined as $T(x)=\\frac{\\varepsilon (x)N}{x}-\\varepsilon (x)d(x)$ where $\\varepsilon (x)$ is either $-1$ or 1 depending on $x$ and $d(x)$ is a positive integer such that $T(x)\\in [a,b]$ .", "Note that if $b-a=1$ then there is exactly one positive integer such that $T(x)\\in [a,b)$ if $\\varepsilon (x)$ is fixed.", "For $N=2$ we find the family that is studied in [10] and for $\\varepsilon (x)=1$ for all $x$ we find the $N$ -expansions from [7].", "Whenever $a>0$ this map can only have finitely many different digits.", "This family is closely related to the $(a,b)$ -continued fractions introduced and studied by Svetlana Katok and Ilie Ugarcovici in [23], [24], [25].", "For $(a,b)$ -continued fraction $\\varepsilon (x)=-1$ for all $x\\in [a,b]$ and $N=1$ .", "Also there are restrictions on $a,b$ .", "These are chosen such that $a\\le 0\\le b, \\ b-a\\ge 1, \\ -ab\\le 1$ .", "The case $b-a=1$ was further studied by Carlo Carminati, Stefano Isola and Giulio Tiozzo in [26], where they study the entropy as a function of $\\alpha $ .", "Note that this family is rather `large'.", "For the examples in the next section $\\varepsilon (x)$ will be plus or minus one on fixed interval(s).", "In Section  other restrictions are imposed." ], [ "Two seemingly closely related examples and their natural extension", "In [7], using the natural extension the invariant measure of the 4-expansion map $T$ given in (REF ) was easily obtained.", "To (briefly) illustrate the method and the kind of continued fraction algorithms we are interested in we consider a slight variation of this continued fraction.", "Let $\\tilde{T}:[1,2]\\rightarrow [1,2]$ be defined as $\\tilde{T}(x) = \\left\\lbrace \\begin{array}{l l l}\\dfrac{4}{x}-1 & \\text{for} & x\\in (\\frac{4}{3},2] \\\\[0.75em]5-\\dfrac{4}{x} & \\text{for} & x\\in [1,\\frac{4}{3}] \\ , \\\\\\end{array} \\right.$ i.e.", "we `flipped' the map $T$ on the interval $[1,\\frac{4}{3}]$ ; see also Figure REF .", "Figure: The CF map T ˜\\tilde{T} from ().Setting $\\varepsilon _1(x) = \\left\\lbrace \\begin{array}{l l}1 & \\text{for $x\\in (\\frac{4}{3},2]$}\\\\-1 & \\text{for $x\\in [1,\\frac{4}{3}]$}\\\\\\end{array} \\right.\\quad \\text{and } \\quad d_1(x) = \\left\\lbrace \\begin{array}{l l}1 & \\quad \\text{for $x\\in (\\frac{4}{3},2]$}\\\\5 & \\quad \\text{for $x\\in [1,\\frac{4}{3}]$}\\ ,\\\\\\end{array} \\right.$ we define $\\varepsilon _n(x)=\\varepsilon _1\\left(T^{n-1}(x)\\right)$ and $d_n(x)=d_1\\left(T^{n-1}(x)\\right)$ .", "From $T(x)=\\varepsilon _1\\cdot \\left(\\frac{4}{x}-d_1\\right)$ , it follows that $x = \\frac{4}{d_1+\\varepsilon _1 T(x)} =\\ldots = \\frac{4}{\\displaystyle d_1+\\frac{4 \\,\\varepsilon _1}{\\displaystyle d_2+ \\ddots + \\frac{4 \\,\\varepsilon _{n-1}}{\\displaystyle d_n +\\varepsilon _n T^n(x)}}} \\ .$ Taking finite truncations, we find the so called convergents $c_n=\\frac{p_n}{q_n}=\\frac{4}{\\displaystyle d_1+\\frac{4 \\,\\varepsilon _1}{\\displaystyle d_2+ \\ddots + \\frac{4 \\,\\varepsilon _{n-1}}{\\displaystyle d_n}}}$ of $x$ .", "One can show that $\\lim _{n\\rightarrow \\infty }c_n=x$ ; see [10] for further details.", "Therefore we write $x= \\frac{4}{\\displaystyle d_1+\\frac{4 \\,\\varepsilon _1}{\\displaystyle d_2+ \\ddots }}\\\\ \\ ,$ or in short hand notation $x=[4/d_1,4\\varepsilon _1/d_2,\\ldots ]$ or $x=[d_1,\\varepsilon _1/d_2,\\ldots ]_4$ .", "To demonstrate the method we will use $\\tilde{T}$ from (REF ).", "The idea now is to build a two-dimensional system (the natural extension) $\\left(\\Omega =[1,2]\\times [A,B],\\mathcal {T}\\right)$ which is almost surely invertible and contains $([1,2],\\tilde{T})$ as a factor.", "In [7] it was shown that a suitable candidate for the natural extension map $\\mathcal {T}$ is given by $\\mathcal {T}(x,y)=\\left(\\tilde{T}(x),\\frac{4\\varepsilon _1(x)}{d_1(x)+y}\\right) \\ .$ Figure: The suitable domain for 𝒯\\mathcal {T}.Now we choose $A$ and $B$ in such a way the system is indeed (almost surely) invertible.", "We define fundamental intervals $\\Delta _{n}=\\lbrace (x,y)\\in \\Omega :d_1(x)=n\\rbrace $ if $\\varepsilon =1$ and $\\Delta _{-n}=\\lbrace (x,y)\\in \\Omega :d_1(x)=n\\rbrace $ if $\\varepsilon =-1$ .", "When the fundamental intervals fit exactly under the action of $\\mathcal {T}$ , the system is almost surely invertible; see Figure REF .", "An easy calculation shows that $A=-1$ and $B=\\infty $ is the right choice here.", "It is shown in [7], that the density of the invariant measure (for the 2-dimensional system) is given by $f(x,y)=C\\frac{4}{(4+xy)^2} , \\quad \\text{for } (x,y)\\in \\Omega \\ ,$ where $C$ is a normalizing constant (which is $\\frac{1}{\\log (3)}$ in this example).", "Projecting on the first coordinate yields the invariant measure for the 1-dimensional system $([1,2],\\tilde{T})$ , with density $\\frac{1}{\\log (3)}\\left(\\frac{1}{x}+\\frac{1}{4-x}\\right) , \\quad \\text{for } x\\in [1,2] \\ .$ Note that if we would consider the map $\\hat{T}(x) = \\left\\lbrace \\begin{array}{l l l}4-\\dfrac{4}{x} & \\text{for } x\\in (\\frac{4}{3},2] \\\\[0.75em]\\dfrac{4}{x}-2 & \\text{for } x\\in [1,\\frac{4}{3}] \\ , \\\\\\end{array} \\right.$ Figure: The CF map T ^\\hat{T} from ().which is a `flipped version' of the map $T$ from (REF ) where we flipped the branch on the interval $(\\frac{4}{3},2]$ , we get another continued fraction of the form (REF ) but now with digits $d_n\\in \\lbrace 2,4\\rbrace $ ; see Figure REF .", "Our approach would give $A=-2$ and $B=\\infty $ which shows that the underlying dynamical system has a $\\sigma $ -finite infinite measure with `density' $f(x)$ , given by $f(x)=\\frac{1}{x}+\\frac{1}{2-x}, \\quad \\text{for } x\\in [1,2].$ The method from [7] we just used does not always `work'.", "As an example we will use an expansion given in [10].", "Let $\\bar{T}(x)$ be defined as $\\bar{T}(x) = \\left\\lbrace \\begin{array}{l l l}\\dfrac{2}{x}-3 & \\text{for} & x\\in (\\frac{1}{2},\\frac{4}{7}] \\\\[0.75em]4-\\dfrac{2}{x} & \\text{for} & x\\in (\\frac{4}{7},\\frac{2}{3}] \\\\[0.75em]\\dfrac{2}{x}-2 & \\text{for} & x\\in (\\frac{2}{3},\\frac{4}{5}] \\\\[0.75em]3-\\dfrac{2}{x} & \\text{for} & x\\in [\\frac{4}{5},1] \\ , \\\\\\end{array} \\right.$ see Figure .", "Figure: NO_CAPTIONWhen trying to construct the domain of the natural extension one quickly notices that `holes' appear.", "This is not an entirely new phenomena in continued fractions, it also appears in constructing the natural extension of Nakada's $\\alpha $ -expansions when $\\alpha \\in (0,\\sqrt{2}-1)$ ; see [21].", "One might hope that there are finitely many holes, but a simulation of the domain indicates otherwise; see Figure REF .", "Figure: A simulation of the domain of the natural extension for the map T ¯\\bar{T} from ().Although the method might still work in this case, it does not really seem to help us to find a description of the invariant density.", "In order to get an idea of the density we will use two different approaches.", "One will be based on the Gauss-Kuzmin-Lévy Theorem.", "The other will be a more classical approach based on Choe's book [14]." ], [ "Two different ways of approximating the density", "The first way is based on the Gauss-Kuzmin-Lévy theorem.", "This theorem states that for the regular continued fraction the Lebesgue measure of the pre-images of a measurable set $A$ will converge to the Gauss measure.", "$\\lambda \\left(T^{-n}(A)\\right)\\rightarrow \\mu (A) \\quad \\text{as } n\\rightarrow \\infty .$ This was stated as an hypothesis by Gauss in his mathematical diary in 1800 and proved by Kuzmin in 1928 who also obtained a bound on the speed of convergence.", "Independently, Lévy proved the same theorem in 1929 but found a sharper bound for the speed of convergence namely $|\\lambda \\left(T^{-n}(A)\\right)-\\mu (A)|=\\mathcal {O}(q^n)$ with $0<q<1$ instead of $\\mathcal {O}(q^{\\sqrt{n}})$ which is the bound Kuzmin found.", "There are many proofs of this theorem and refinements on the speed of convergence; see e.g.", "Khinchine's book [3], or [27] for various refinements.", "The idea for our method is to look at the pre-images of $[\\frac{1}{2},z]$ for our map $\\bar{T}$ from (REF ) and take the Lebesgue measure of the intervals found.", "Note that in the first iteration you will find 2 intervals and after the first iteration the number of intervals found is multiplied by 4.", "Also the size of the intervals shrink relatively fast.", "Fortunately it seems that a low number of iterates (around 10) is already enough to give a good approximation; see Figure REF where the theoretical density and its approximation are displayed and figure REF , where both methods of approximating are compared for a density we do not know the theoretical density.", "The other way of finding an approximation is by iterating points and look at the histogram of the orbits.", "The way we iterated is that we used a lot of points and iterated them just a few times.", "To be more precise we iterated 2500 uniformly random points 20 times, repeated this process 400 times and took the average density of all points.", "Then we redid the process but instead of sampling uniformly we sampled from the previously found density (see also [28]) In Figure REF we see both methods applied to our example.", "Figure: Approximations of the density of the invariant measure of T ¯(x)\\bar{T}(x) using the Gauss-Kuzmin-Lévy method (red) and the classical way (blue).The two methods give results that are relatively close but the approximation found with the Gauss-Kuzmin-Lévy method is far more smooth.", "Since we do not know the density we cannot compare the theoretical density with the approximation.", "Since the Gauss-Kuzmin-Lévy method is the new method we will look how well this method performs in an example in which we know the invariant density explicitly.", "For the map $T$ from (REF ) we know the density which was given in [7].", "In Figure REF we see a plot of both the theoretical density and the approximation found by the Gauss-Kuzmin-Lévy method.", "Figure: An approximation of the true density for the CF map TT from () and the true TT-invariant density.The difference can barely be seen by the naked eye.", "If we look at the difference in 2-norm we get $\\left(\\int _1^2 (f(x)-\\hat{f}(x))^2 \\, dx\\right)^{\\frac{1}{2}}=1.1235*10^{-5}$ where $f(x)$ is the true density and $\\hat{f}(x)$ the approximation." ], [ "A sub-family of the $N$ -expansions", "In this section we study a subfamily of the $N$ -expansions (so $\\varepsilon (x)=1$ for all $x$ in the domain) with digits from a finite alphabet and an interval $[\\alpha ,\\beta ]$ as domain.", "For our subfamily we want that it has finitely many digits.", "Furthermore we would like that there is a unique digit such that $T(x)\\in [\\alpha ,\\beta )$ .", "This results in $\\alpha >0$ and $\\beta -\\alpha =1$ .", "In this way the map is uniquely determined by the domain.", "With these restrictions we define for $\\alpha \\in (0,\\sqrt{N}-1]$ the map $T_{\\alpha ,N}:[\\alpha ,\\alpha +1]\\rightarrow [\\alpha ,\\alpha +1]$ as $T_{\\alpha ,N}=\\frac{N}{x}-\\left\\lfloor \\frac{N}{x}-\\alpha \\right\\rfloor \\ .$ Note that for all these expansions we have a finite number of digits since $\\alpha >0$ .", "Also note that this is the largest range in which we can choose $\\alpha $ because for $\\alpha >\\sqrt{N}-1$ the digit would be 0 or less (see Section REF for a calculation).", "Simulations show that a lot of maps have an attractor smaller than $[\\alpha ,\\beta ]$ .", "When $N\\ge 9$ we find that if $\\alpha =\\sqrt{N}-1$ there is always an interval $[c,d]\\subsetneq [\\alpha ,\\alpha +1]$ for which the $T_{\\alpha ,N}$ -invariant measure of $[c,d]$ is zero.", "Whenever $N>4$ we have that $T_{\\alpha ,N}$ always has 2 branches for $\\alpha =\\sqrt{N}-1$ .", "Calculations of these observations are given in Section REF in which we take a closer look on which sequences are admissible for a given $N$ and $\\alpha $ .", "In Section REF we study the behavior of the entropy as a function of $\\alpha $ for a fixed $N$ .", "The examples in [7] with `fixed range' are all member of this kind of sub-family of the $N$ -expansions.", "Though, these examples are cases for which all the branches of the mapping are full.", "In such case the natural extension can be easily build using the method described previously.", "If not all branches are full we can still make the natural extension in some cases.", "We will start this section with such case." ], [ "A 2-expansion with $\\alpha =\\sqrt{2}-1$", "Let $T(x):[\\sqrt{2}-1,\\sqrt{2}]\\rightarrow [\\sqrt{2}-1,\\sqrt{2}]$ be defined by $T(x) = \\left\\lbrace \\begin{array}{l l l }\\dfrac{2}{x}-1 & \\text{for} & 2(\\sqrt{2}-1) < x \\le \\sqrt{2} \\\\[0.75em]\\dfrac{2}{x}-2 & \\text{for} & 2-\\sqrt{2} < x \\le 2(\\sqrt{2}-1) \\\\[0.75em]\\dfrac{2}{x}-3 & \\text{for} & \\frac{1}{7}(6-2\\sqrt{2})< x \\le 2-\\sqrt{2} \\\\[0.75em]\\dfrac{2}{x}-4 & \\text{for} & \\sqrt{2}-1 \\le x \\le \\frac{1}{7}(6-2\\sqrt{2}) \\ .", "\\\\\\end{array} \\right.$ Figure: A 2-expansion on the interval [2-1,2][\\sqrt{2}-1,\\sqrt{2}].A graph of this map is shown in Figure REF .", "We can find the invariant measure for this map by using the method as in Section REF though we now need to determine 3 `heights' in order to make the mapping of the natural extension almost surely bijective on the domain (see Figure REF ).", "We get the following equations for the heights $A, B$ and $C$ : $A=\\frac{2}{4+C} \\ , \\quad B=\\frac{2}{3+C} \\quad \\text{and } C=\\frac{2}{1+B} \\ .$ Figure: Ω\\Omega and 𝒯(Ω)\\mathcal {T}(\\Omega ).This results in $A=\\frac{1}{2}(\\sqrt{33}-5)$ , $B=\\frac{1}{6}(\\sqrt{33}-3)$ and $C=\\frac{1}{2}(\\sqrt{33}-3)$ .", "We find the following invariant density up to a normalizing constant (which is given in Theorem REF ) $f(x)= \\left\\lbrace \\begin{array}{l l}\\frac{\\sqrt{33}-3}{4+(\\sqrt{33}-3)x}-\\frac{\\sqrt{33}-5}{4+(\\sqrt{33}-5)x} & \\text{for } \\sqrt{2}-1 < x \\le 2(\\sqrt{2}-1) \\\\[0.75em]\\frac{\\sqrt{33}-3}{4+(\\sqrt{33}-3)x}-\\frac{\\sqrt{33}-3}{12+(\\sqrt{33}-3)x} & \\text{for } 2(\\sqrt{2}-1) < x \\le \\sqrt{2} \\ .", "\\\\\\end{array}\\right.$ The graph of the density is given in Figure REF .", "Figure: The density of the invariant measure for the 2-expansion on [2-1,2][\\sqrt{2}-1,\\sqrt{2}].In this case we were lucky.", "But in general it seems to be very hard to construct the natural extension explicitly.", "Still we can simulate the densities and calculate the entropy for a given $\\alpha $ .", "Also for the 2-expansions we can extend the above result to all $\\alpha \\in [\\frac{\\sqrt{33}-5}{2},\\sqrt{2}-1]$ ; see Theorem  REF" ], [ "Admissibility", "In this section we look at how the alphabet is determined by $\\alpha $ for a fixed $N$ .", "It turns out that not all different sequences of such alphabet will occur in a continued fraction expansion (or only finitely many times).", "This is a consequence of some cylinders having zero mass.", "The range of the first digits of a continued fraction for given $\\alpha $ and $N$ are easily described since the smallest digit will be attained by the right end point of the domain largest digits will be attained by the left end point of the domain.", "Let $n_{min}=\\left\\lfloor \\frac{N}{\\alpha +1}-\\alpha \\right\\rfloor \\quad \\text{and} \\quad n_{max}=\\left\\lfloor \\frac{N}{\\alpha }-\\alpha \\right\\rfloor \\ .$ Note that $n_{min}\\le 0$ when $\\alpha >\\sqrt{N}-1$ and therefore $\\alpha =\\sqrt{N}-1$ is the largest value for which we have positive digits.", "Furthermore the alphabet is given by $\\lbrace n_{min},\\ldots ,n_{max}\\rbrace $ .", "Now to see for which $N$ we have that for $\\alpha =\\sqrt{N}-1$ there are two branches we must check that $n_{max}=2$ .", "This happens when $\\frac{N}{\\sqrt{N}-1}-2\\in [\\sqrt{N}-1,\\sqrt{N}]$ .", "We find 2 inequalities $\\frac{N}{\\sqrt{N}-1}-2\\le \\sqrt{N}$ and $\\sqrt{N}-1\\le \\frac{N}{\\sqrt{N}-1}-2 \\ .$ Inequality (REF ) gives $4\\le N$ and inequality (REF ) gives $N-1\\le N$ .", "This yields that for all $N\\ge 4$ we have two branches for $\\alpha =\\sqrt{N}-1$ .", "If $N\\ge 9$ we also have an attractor which is strictly smaller than the entire interval for $\\alpha =\\sqrt{N}-1$ as the following calculation shows $T_{\\alpha ,N}\\left([\\alpha ,\\frac{N}{\\alpha +2}]\\right)&=&\\left[\\alpha ,\\frac{N}{\\alpha }-2\\right]\\\\T_{\\alpha ,N}\\left([\\alpha ,\\frac{N}{\\alpha }-2]\\right)&=&\\left[\\alpha ,\\frac{N}{\\alpha }-2]\\cup [\\frac{N\\alpha }{N-2\\alpha }-1,\\alpha +1\\right]\\\\T_{\\alpha ,N}\\left([\\frac{N\\alpha }{N-2\\alpha }-1,\\alpha +1]\\right)&=&\\left[\\alpha ,\\frac{N^2-(1-3\\alpha )N-2\\alpha }{(\\alpha -1)N+2\\alpha }\\right] \\ .\\\\$ If we substitute $\\alpha $ with $\\sqrt{N}-1$ and the following two inequalities hold we find an attractor strict smaller than the interval $[\\alpha ,\\alpha +1]$ ; $\\frac{N}{\\sqrt{N}-1}-2<\\frac{N(\\sqrt{N}-1)}{N-2(\\sqrt{N}-1)}-1$ $\\frac{N^2-(4-3\\sqrt{N})N-2(\\sqrt{N}-1)}{(\\sqrt{N}-2)N+2(\\sqrt{N}-1)} <\\frac{N}{\\sqrt{N-1}}-2 \\ ,$ yielding that $N\\ge 9$ .", "We take a closer look at $N=9$ and $\\alpha =\\sqrt{9}-1=2$ .", "This example is briefly discussed in [7] where it is stated that computer experiments suggest that the orbit of 2 never becomes periodic and therefore it is hard to find the natural extension explicitly.", "However, when simulating the natural extension, it seems that there are finitely many discontinuities; see Figure REF .", "Figure: A simulation of the natural extension for N=9N=9 and α=2\\alpha =2.We can also simulate the density of the invariant measure; see Figure REF .", "Figure: A Simulation of the density for N=9N=9 and α=2\\alpha =2 using the Gauss-Kuzmin-Lévy method.Remark that cylinders with zero mass tells us which sequences are not apparent in any continued fraction of numbers outside the attractor and for those numbers not in the attractor these sequences only appear in the start of the continued fraction.", "We can describe which cylinders these are.", "The hole is given by $[2.5,2.6]$ .", "Now $2.5=[1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,2,2,1,1,1,\\ldots ]_9$ and $2.6=[1,1,1,1,1,1,1,2,1,1,1,1,1,1,2,2,1,1,1,1\\ldots ]_9$ .", "The boundary of a cylinder $\\Delta (a_1,\\ldots ,a_n)$ is given by $[a_1,\\ldots ,a_n,1,r]_9$ and $[a_1,\\ldots ,a_n,2,r]_9$ where $r$ is the expansion of 2.", "Now a cylinder is contained in $[2.5,2.6]$ if $2.5<[a_1,\\ldots ,a_n,1,r]_9<2.6$ and $2.5<[a_1,\\ldots ,a_n,2,r]_9<2.6$ .", "Note that here we can have a clear description of the attractor and therefore for the admissible sequences.", "Simulation shows us that there are a lot of different settings in which you find an attractor strictly smaller than the interval.", "In Figure REF simulations for several values of $N$ are shown.", "On the $y$ -axes $\\alpha $ is given and on the $x$ -axes the attractor is plotted.", "For example, for $N=9$ we see that for $\\alpha =1$ there is no attractor strictly smaller than the interval.", "There is an attractor for $\\alpha =1.8$ and also for example for $\\alpha =2$ .", "The pattern seems to be rather regular.", "Moreover, more `holes' seem to appear for large $N$ and large $\\alpha $ .", "Figure: Attractors plotted for several values of NN." ], [ "Entropy", "Entropy for a dynamical system has been introduced by Kolmogorov and Sinai in 1959.", "For a dynamical system related to continued fraction expansions it gives information about the speed of convergence for a typical point in that system.", "Entropy as a function of a parameter $\\alpha $ is widely studied for Nakada's $\\alpha $ -expansions (see [17], [18], [19], [21], [22]).", "Also for the $(a,b)$ -continued fractions the entropy is studied [26].", "A special interest goes to monotonicity which can be proven (for both cases) by using matching.", "It is shown that matching implies monotonicity and that matching holds almost everywhere.", "In this section we look at entropy as a function of $\\alpha \\in (0,\\sqrt{N}-1]$ for a fixed $N$ .", "For our family it we do not know whether matching holds almost everywhere.", "In fact, it is not even clear whether matching occurs for all $N$ .", "Also, if matching implies monotonicity is not clear.", "Certain conditions used to prove it in the case of Nakada $\\alpha $ -expansions do not hold for our family and therefore we cannot mimic the proof of [18].", "Let us first give the definition of matching.", "Definition 1 (Matching) We say matching holds for $\\alpha $ if there is an $K,M$ such that $T_{N,\\alpha }^{K}(\\alpha +1)=T_{N,\\alpha }^M(\\alpha )$ .", "The numbers $K,M$ are called the matching exponents, $K-M$ is called the matching index and an interval $(c,d)$ such that for all $\\alpha \\in (c,d)$ we have the same matching exponents is called a matching interval.", "Note that for Nakada's $\\alpha $ -expansions all rationals have a finite expansion and thus match in 0 or before.", "For any rational in the domain a matching interval is found.", "For our family all rationals will have an infinite expansion.", "Since for all $\\alpha \\in (0,\\sqrt{N}-1)$ the expansion is infinite there are no values for which we find matching `trivially'.", "Even if we find an $\\alpha $ for which matching holds it is hard to conclude it must hold in a neighborhood of $\\alpha $ as well since we cannot find an algebraic relation.", "The presence of the holes for some $\\alpha $ might also lead to problems.", "The only reason to believe matching can help us to prove monotonicity is the fact that it works for related families and for specific choices of $N$ and $\\alpha $ we can find matching where we can see the function is monotonic.", "We will now discuss the $2_\\alpha $ -expansions in detail.", "Moreover, we proof that the entropy is constant for $\\alpha \\in \\left( \\frac{\\sqrt{33}-5}{2}, \\sqrt{2}-1\\right)$ when $N=2$ ." ], [ "The entropy of $2_\\alpha $ -expansions", "We start with an example for which there is no $\\alpha $ such that we have an attractor strictly smaller than the interval.", "Also, simulation indicates that there seems to be a plateau in the neighborhood of $\\sqrt{2}-1$ .", "For this value we can calculate the entropy since we have the density for this specific case of $\\alpha $ ; see Section REF also see Figure REF for a plot of the entropy function.", "Figure: Entropy as function of α\\alpha for N=2N=2.When taking a closer look at this plateau we found that on the interval $[\\frac{\\sqrt{33}-5}{2},\\sqrt{2}-1]$ the entropy is constant on this interval.", "The point $\\frac{\\sqrt{33}-5}{2}$ is the point for which all smaller $\\alpha $ have that there are always 5 or more branches and for all larger $\\alpha $ there are always 4 branches.", "If we look at a simulation of the natural extension it seems that for these values of $\\alpha $ we can construct a natural extension.", "Indeed this turned out to be the case (see Theorem REF ).", "For $\\frac{\\sqrt{33}-5}{2}$ we find matching exponents $(0,2)$ and for $\\sqrt{2}-1$ we find matching exponents $(0,1)$ .", "Inside the interval itself we find $(3,3)$ .", "As the value of the entropy, These values were first found by simulation, in Theorem REF we give a proof of this.", "Theorem 3.1 Let $N=2$ , and let $\\alpha \\in \\left( \\frac{\\sqrt{33}-5}{2}, \\sqrt{2}-1\\right)$ .", "Then $T^3(\\alpha ) = T^3(\\alpha +1)$ .", "Note that the interval $(\\alpha , \\alpha +1)$ has as natural partition $\\lbrace I_1,I_2,I_3,I_4\\rbrace $ , where $I_1= \\Big ( \\frac{2}{\\alpha +2},\\alpha +1\\Big ) , \\quad I_2= \\Big ( \\frac{2}{\\alpha +3},\\frac{2}{\\alpha +2}\\Big ] ,\\quad I_3= \\Big ( \\frac{2}{\\alpha +4},\\frac{2}{\\alpha +3}\\Big ] ,$ and $I_4 = \\Big [ \\alpha , \\frac{2}{\\alpha +4}\\Big ] ,$ where $T(x) = \\frac{2}{x} - d,\\quad \\text{if $x\\in I_d$\\,\\, for $d=1,2,3,4$}.$ An easy calculation shows that $T(\\alpha ) = \\frac{2-4\\alpha }{\\alpha }\\in I_1,$ (and $T(\\alpha ) = \\frac{2}{\\alpha +2}$ when $\\alpha = \\sqrt{2}-1$ , and $T(\\alpha ) = \\alpha +1$ when $\\alpha =\\frac{\\sqrt{33}-5}{2}$ ), so that $T^2(\\alpha ) = \\frac{3\\alpha -1}{1-2\\alpha }.$ Furthermore, we have that $T(\\alpha +1) = \\frac{1-\\alpha }{\\alpha +1} \\in I_4,$ (and $T(\\alpha +1) = \\sqrt{2}-1$ when $\\alpha = \\sqrt{2}-1$ ; $T(\\alpha +1) = \\frac{2}{\\alpha +4}$ when $\\alpha = \\frac{\\sqrt{33}-5}{2}$ ), so that $T^2(\\alpha +1) = \\frac{6\\alpha -2}{1-\\alpha }.$ Now let $K_1= \\left(\\frac{\\sqrt{33}-5}{2},\\frac{\\sqrt{51}-6}{3}\\right], \\quad K_2= \\left(\\frac{\\sqrt{51}-6}{3},\\frac{\\sqrt{129}-9}{6}\\right] \\\\$ and $K_3= \\left(\\frac{\\sqrt{129}-9}{6},\\sqrt{2}-1\\right) \\ .$ For $\\alpha \\in K_1$ we have that $T^2(\\alpha )\\in I_3$ and so $T^3(\\alpha ) = \\frac{5-13\\alpha }{3\\alpha -1}$ and $T^2(\\alpha +1 )\\in I_4$ which results in $T^3(\\alpha +1) = \\frac{5-13\\alpha }{3\\alpha -1} = T^3(\\alpha ) \\ .$ For $\\alpha \\in K_2$ we have that $T^2(\\alpha )\\in I_2$ and so $T^3(\\alpha ) = \\frac{4-10\\alpha }{3\\alpha -1}$ and $T^2(\\alpha +1)\\in I_3$ which results in $T^3(\\alpha +1) = \\frac{4-10\\alpha }{3\\alpha -1} = T^3(\\alpha ) \\ .$ For $\\alpha \\in K_3$ we have that $T^2(\\alpha )\\in I_1$ and so $T^3(\\alpha ) = \\frac{3-7\\alpha }{3\\alpha -1}$ and $T^2(\\alpha +1)\\in I_2$ which results in $T^3(\\alpha +1) = \\frac{3-7\\alpha }{3\\alpha -1} = T^3(\\alpha ) \\ .$ Earlier we thought we were just lucky finding the natural extension in case $N=2$ and $\\alpha = \\sqrt{2}-1$ .", "Note that from this natural extension we immediately also have the case $N=2$ , $\\alpha = \\frac{\\sqrt{33}-5}{2}$ ; just “invert” the time and exchange the two coordinates in the natural extension we found for $N=2$ and $\\alpha = \\sqrt{2}-1$ .", "However, from Theorem REF it immediately follows that we can also `build' the natural extension for every $\\alpha \\in \\left( \\frac{\\sqrt{33}-5}{2},\\sqrt{2}-1\\right)$ .", "Clearly, from Theorem REF we see that we have three different cases.", "Theorem 3.2 For $\\alpha \\in \\left( \\frac{\\sqrt{33}-5}{2}, \\sqrt{2}-1\\right)$ the natural extension can be build as in Figure REF .", "Moreover the invariant density is given by $f(x)&=&H \\large (\\frac{D}{2+Dx}\\textbf {1}_{(\\alpha ,T(\\alpha +1))} +\\frac{E}{2+Ex}\\textbf {1}_{(T(\\alpha +1),T^2(\\alpha ))}+\\frac{F}{2+Fx}\\textbf {1}_{(T^2(\\alpha ),\\alpha +1)}\\\\&-&\\frac{A}{2+Ax}\\textbf {1}_{(\\alpha ,T^2(\\alpha +1))}-\\frac{B}{2+Bx}\\textbf {1}_{(T^2(\\alpha +1),T(\\alpha ))}-\\frac{C}{2+Cx}\\textbf {1}_{(T(\\alpha ),\\alpha +1)} \\large ) \\\\$ with $A=\\frac{\\sqrt{33}-5}{2}, B=\\sqrt{2}-1, C=\\frac{\\sqrt{33}-3}{6}, D=2\\sqrt{2}-2, E=\\frac{\\sqrt{33}-3}{2}, F=\\sqrt{2}$ and $H^{-1}=\\log \\left(\\frac{1}{32}(3+2\\sqrt{2})(7+\\sqrt{33})(\\sqrt{33}-5)^2\\right)\\approx 0.25$ the normalizing constant.", "We guessed the shape of the domain of natural extension by studying a simulation.", "For the map on this domain we used $\\mathcal {T}(x,y)=\\left(T(x),\\frac{2}{d_1(x)+y}\\right)$ .", "Figure: Ω\\Omega and 𝒯(Ω)\\mathcal {T}(\\Omega ) with α∈K 1 \\alpha \\in K_1.For $\\alpha \\in K_1$ , we find the following equations: $\\begin{array}{l c l}A=\\frac{2}{4+E} & \\quad & A=\\frac{\\sqrt{33}-5}{2}\\\\B=\\frac{2}{4+D} & \\quad & B=\\sqrt{2}-1\\\\C=\\frac{2}{3+E} & \\quad & C=\\frac{\\sqrt{33}-3}{6}\\\\D=\\frac{2}{2+A} & \\text{implying that} & D=2\\sqrt{2}-2\\\\E=\\frac{2}{1+C} & \\quad & E=\\frac{\\sqrt{33}-3}{2}\\\\F=\\frac{2}{1+B} & \\quad & F=\\sqrt{2} \\ .\\\\\\end{array}$ A similar picture emerges for $\\alpha \\in K_2$ and $\\alpha \\in K_3$ .", "Moreover, you will find the same set of equations and thus the same heights!", "Note that for $\\alpha <\\frac{2}{5}$ we have $T^2(\\alpha )<T(\\alpha )$ for $\\alpha =\\frac{2}{5}$ we have $T^2(\\alpha )=T(\\alpha )$ and for $\\alpha >\\frac{2}{5}$ we have $T^2(\\alpha )>T(\\alpha )$ .", "When you integrate over the second coordinate you find the density given in the statement of the theorem.", "For the normalizing constant we have the following integral $H&=&\\int _{\\alpha }^{\\alpha +1} \\frac{D}{2+Dx}\\textbf {1}_{(\\alpha ,T(\\alpha +1))} \\ldots -\\frac{C}{2+Cx}\\textbf {1}_{(T(\\alpha ),\\alpha +1)} \\, dx\\\\&=& \\log \\left(\\frac{2+D T(\\alpha +1)}{2+D\\alpha }\\right)+\\ldots +\\log \\left(\\frac{2+CT(\\alpha )}{2+C(\\alpha +1)}\\right)\\\\&=& \\log \\left(\\frac{2+D T(\\alpha +1)}{2+D\\alpha }\\cdots \\frac{2+CT(\\alpha )}{2+C(\\alpha +1)}\\right)\\\\$ It seems that $H$ depends on $\\alpha $ but this is not the case.", "We will calculate one term since all others have a similar calculation.", "$\\log \\left(\\frac{2+D T(\\alpha +1)}{2+D\\alpha }\\right)&=&\\log \\left(\\frac{2+(2\\sqrt{2}-2)T(\\alpha +1)}{2+(2\\sqrt{2}-2)\\alpha }\\right)\\\\&=&\\log \\left(\\frac{2+(2\\sqrt{2}-2)\\frac{1-\\alpha }{1+\\alpha }}{2+(2\\sqrt{2}-2)\\alpha }\\right)\\\\&=&\\log \\left(\\frac{2(1+\\alpha )+(2\\sqrt{2}-2)(1-\\alpha )}{2+(2\\sqrt{2}-2)\\alpha }\\right)+\\log (1+\\alpha )\\\\&=&\\log \\left(\\sqrt{2}\\frac{\\sqrt{2}(1+\\alpha )+\\sqrt{2}(\\sqrt{2}-1)(1-\\alpha )}{2+(2\\sqrt{2}-2)\\alpha }\\right)+\\log (1+\\alpha )\\\\&=&\\log \\left(\\sqrt{2}\\frac{2+(2\\sqrt{2}-2)\\alpha }{2+(2\\sqrt{2}-2)\\alpha }\\right)+\\log (1+\\alpha )\\\\&=&\\log (\\sqrt{2})+\\log (1+\\alpha )\\\\$ When calculating the second term one finds $\\frac{1}{4}(\\sqrt{33}-5)-\\log (1+\\alpha )$ and so this cancels the $\\log (1+\\alpha )$ term.", "One might hope that when calculating the entropy using Rohlin's formula, terms will cancel as well.", "These integrals result in $Li_2$ functions depending on $\\alpha $ and things are not so easy anymore.", "We provide a more elegant proof to show that the entropy is constant on $\\left( \\frac{\\sqrt{33}-5}{2}, \\sqrt{2}-1\\right)$ and then calculate the entropy for $\\alpha =\\sqrt{2}-1$ .", "We will use quilting introduced in [29].", "Proposition 1 in this article can be formulated (specific to our case) in the following way: Proposition 3.3 Let $(\\mathcal {T}_{\\alpha },\\Omega _{\\alpha },\\mathcal {B}_{\\alpha },\\mu )$ and $(\\mathcal {T}_{\\beta },\\Omega _{\\beta },\\mathcal {B}_{\\beta },\\mu )$ be two dynamical systems as in our setting.", "Furthermore let $D_1=\\Omega _{\\alpha }\\backslash \\Omega _{\\beta }$ and $A_1=\\Omega _{\\alpha }\\backslash \\Omega _{\\beta }$ .", "If there is a $k\\in \\mathbb {N}$ such that $\\mathcal {T}_{\\alpha }^k(D_1)= \\mathcal {T}_{\\beta }^k(A_1)$ then the dynamical systems are isomorphic.", "Since isomorphic systems have the same entropy it will give us the following corollary.", "Corollary 3.4 For $N=2$ the entropy function is constant on $\\left( \\frac{\\sqrt{33}-5}{2}, \\sqrt{2}-1\\right)$ and the value is approximately $1.14$ .", "We show that for $k=3$ we satisfy the condition in Proposition REF .", "Define $D_i=\\mathcal {T}_{\\alpha }^{i-1}(D_1)$ and $A_i=\\mathcal {T}_{\\beta }^{i-1}(A_1)$ for $k=1,2,3,4$ .", "We find the following regions (see Figure REF ): $D_1 &=& [\\alpha ,\\beta ]\\times [A,D],\\\\D_2 &=& [T(\\beta ),T(\\alpha )]\\times [B,C],\\\\D_3 &=& [T^2(\\beta ),T^2(\\alpha )]\\times [E,F],\\\\D_4 &=& [T^3(\\beta ),T^3(\\alpha )]\\times \\left[\\frac{2}{3+F},\\frac{2}{3+E}\\right],\\\\A_1 &=& [\\alpha +1,\\beta +1]\\times [C,F],\\\\A_2 &=& [T(\\beta +1),T(\\alpha +1)]\\times [D,E],\\\\A_3 &=& [T^2(\\beta +1),T^2(\\alpha +1)]\\times [A,B],\\\\A_4 &=& [T^3(\\beta +1),T^3(\\alpha +1)]\\times \\left[\\frac{2}{4+B},\\frac{2}{4+A}\\right].\\\\ $ Note that since we have matching $[T^3(\\beta ),T^3(\\alpha )]=[T^3(\\beta +1),T^3(\\alpha +1)]$ .", "Now $\\frac{2}{3+F}=\\frac{2}{3+\\sqrt{2}}= \\frac{2}{4+B}=\\frac{2}{4+\\sqrt{2}-1} \\ ,$ $\\frac{2}{3+E}=\\frac{2}{3+\\frac{\\sqrt{33}-3}{2}}= \\frac{2}{4+A}=\\frac{2}{4+\\frac{\\sqrt{33}-5}{2}}$ and so we find $D_4=A_4$ .", "Figure: Illustration of the quilting.For the value of the entropy we use Rohlin's formula for $\\alpha =\\sqrt{2}-1$ (see [2], [4]); $h(T_{\\sqrt{2}-1})&=&\\int _{\\sqrt{2}-1}^{\\sqrt{2}}\\log |T^{\\prime }(x)|f(x)\\, dx\\\\&=&H\\int _{\\sqrt{2}-1}^{\\sqrt{2}}\\left(\\log (2)-2\\log (x)\\right)f(x)\\, dx\\\\&=&\\log (2)-2H\\int _{\\sqrt{2}-1}^{\\sqrt{2}}\\log (x)f(x)\\, dx\\\\&=&\\log (2)-2H\\int _{\\sqrt{2}-1}^{\\sqrt{2}}\\log (x)\\large ( (\\frac{\\sqrt{33}-3}{4+(\\sqrt{33}-3)x}-\\frac{\\sqrt{33}-5}{4+(\\sqrt{33}-5)x})1_{\\sqrt{2}-1,2(\\sqrt{2}-1)}\\\\& & \\, +(\\frac{\\sqrt{33}-3}{4+(\\sqrt{33}-3)x}-\\frac{\\sqrt{33}-3}{12+(\\sqrt{33}-3)x})1_{2(\\sqrt{2}-1),\\sqrt{2}}\\large )\\, dx\\\\&=&\\log (2)-2H\\large ( (Li_2(-\\frac{x(\\sqrt{33}-3)}{4})+\\log (\\frac{x(\\sqrt{33}-3)}{4}+1)\\\\& &\\, -Li_2(-\\frac{x(\\sqrt{33}-5)}{4})+\\log (\\frac{x(\\sqrt{33}-5)}{4}+1))\\large |_{\\sqrt{2}-1}^{2(\\sqrt{2}-1)}\\\\& & \\, + (Li_2(-\\frac{x(\\sqrt{33}-3)}{4})+\\log (\\frac{x(\\sqrt{33}-3)}{4}+1)\\\\& &\\, -Li_2(-\\frac{x(\\sqrt{33}-5)}{12})+\\log (\\frac{x(\\sqrt{33}-5)}{12}+1))\\large |_{2(\\sqrt{2}-1)}^{\\sqrt{2}}\\large )\\\\&\\approx &1.14 \\ .$ By looking at the graph displayed in Figure REF we can't really find other matching exponents easily.", "To check for other matching exponents we can do the following.", "Suppose we are interested in finding a matching interval with exponents $(n_1,n_2)$ .", "We select a large number random points (say $10\\,000$ ) from $(0,\\sqrt{N}-1)$ .", "Then we looked at $T_{\\alpha }^{n_1}(\\alpha )-T_{\\alpha }^{n_2}(\\alpha +1)$ for these random points and we checked to see whether it is very close to 0.", "Note that if an interval was found this way with matching exponents $(n_1,n_2)$ then we also find that interval for $(n_1+1,n_2+1)$ .", "Table REF shows which matching exponents we found.", "Table: observed matching exponents for N=2N=2: 1 if seen, 0 if not.This is very different from Nakada's $\\alpha $ -continued fractions where you can find all possible matching exponents.", "The fact that we did not observe them does not mean they are not there.", "Maybe they are too small to observe using this method." ], [ "The entropy of $36_\\alpha $ -expansions", "For $N\\ge 9$ we expect different behavior due to the fact that for some $\\alpha $ there is at least one subinterval on which the invariant measure is zero.", "If we pick $N=36$ we have a map with only full branches for $\\alpha =1,2,3$ .", "Figure REF shows the entropy as function of $\\alpha $ .", "The stars indicate those values which we could calculate theoretically.", "Figure: Entropy as function of aa for N=36N=36.Clearly, we can observe plateaus but if we look at the matching exponents we can observe that for all $M,K\\le 10$ the only matching exponents we find are $(3,3),(4,4),\\ldots (10,10)$ ." ], [ "Conclusion", "We have seen that the general form of the examples given yields a rather large family.", "In some examples we were able to construct the natural extension and therefore to find the invariant measure.", "In other examples this was not the case.", "There does not seem to be an easy rule which tells you when the method will work and when it does not.", "The subfamily of the $N$ -expansions we studied is not new, but it has not been studied in this detail with finitely many digits.", "Note that having the Gauss-Kuzmin-Lévy method for approximating the densities allowed us to study the entropy much easier due to computation time.", "We have seen that matching is helpful to prove monotonicity even though we did not mimic the proof for $\\alpha $ -expansions.", "Motivated by similar results in the case of Nakada's $\\alpha $ -expansions the following questions about entropy arise: [noitemsep] For every $N\\in \\mathbb {N}_{\\ge 2}$ is there an interval in $(0,\\sqrt{N}-1)$ for which the entropy function is constant?", "For a fixed $N \\in \\mathbb {N}_{\\ge 2}$ for which $\\alpha \\in (0,\\sqrt{N}-1)$ do we have matching?", "Does matching holds on an open dense set?", "Does matching hold almost everywhere?", "What is the influence of an attractor strictly smaller than the interval $[\\alpha , \\alpha +1]$ on the entropy?" ] ]
1606.05099
[ [ "Tropical Geometry and Mechanism Design" ], [ "Abstract We develop a novel framework to construct and analyze finite valued, multidimensional mechanisms using tropical convex geometry.", "We geometrically characterize incentive compatibility using cells in the tropical convex hull of the type set.", "These cells are the sets of incentive compatible payments and form tropical simplices, spanned by generating payments whose number equals the dimension of the simplex.", "The analysis of the collection of incentive compatible mechanisms via tropical simplices and their generating payments facilitates the use of geometric techniques.", "We use this view to derive a new geometric characterization of revenue equivalence but also show how to handle multidimensional mechanisms in the absence of revenue equivalence." ], [ "Introduction", "Mechanisms are engineered games, devised to implement outcomes that depend on the private information of individuals in the economy.", "They provide a theoretical model to study which economic allocations an institution can achieve as equilibrium of a game in which individuals' preferences, called their type is not known by the institution.", "When this information is relevant for taking a decision, and individuals can misreport their preferences in order to manipulate the outcome in their favor, they need to be incentivized to reveal it truthfully.", "A mechanism that elicits agents' types truthfully is said to be incentive compatible (IC).", "In this paper we focus on single agent mechanisms, which form a fundamental building block of dominant strategy mechanism design, cf.", "[27], [11].", "The analysis of such mechanisms is predominantly analytic or algebraic [20], [22], [27].", "We propose an entirely different approach via tropical convex geometry and tropical combinatorics.", "Within this framework, the study of incentive compatibility becomes a question about tropical eigenspaces and point configurations in the tropical affine space $\\mathbb {TP}$ .", "To handle these economic problems geometrically, we put forward the definition of the basic region $\\operatornamewithlimits{\\mathsf {basic}}(T)$ and the set of basic cells $\\operatornamewithlimits{\\mathsf {cells}}(T)$ for an arbitrary multiset of types $T\\subset \\mathbb {TP}^{m-1}$ .", "Our first main result states that any IC mechanism on $T$ and its payments can be constructed from these objects.", "Theorem 1.1 Let $T\\subset \\mathbb {TP}^{m-1}$ be a multiset of types, and $(g,p)$ a single agent mechanism with outcome function $g: T \\rightarrow [m]$ and payment vector $p \\in \\mathbb {TP}^{m-1}$ .", "A mechanism $(g,p)$ is IC if and only if $p$ lies in a basic cell $P \\in \\operatornamewithlimits{\\mathsf {cells}}(T)$ , and $g$ is contained in $\\underline{\\mathsf {coVec}}_T(P)$ as a subgraph.", "Then $P$ is the set of all IC payments of $g$ .", "A vector $p \\in \\mathbb {TP}^{m-1}$ is the payment function of some IC mechanism $(g,p)$ if and only if $p$ lies in the basic region $\\operatornamewithlimits{\\mathsf {basic}}(T)$ .", "If $T$ is finite and generic, then the basic cells $\\operatornamewithlimits{\\mathsf {cells}}(T)$ are full-dimensional cells of the max-plus tropical polytope generated by $T$ .", "Let $(T^n, n \\ge 1)$ be a sequence of finite sets which approximate $T$ .", "Then the basic cells in $\\operatornamewithlimits{\\mathsf {cells}}(T)$ are limits in the Hausdorff metric of the basic cells of $T^n$ as $n \\rightarrow \\infty $ .", "Theorem REF supplies a novel framework to study IC mechanisms, and more importantly, the set of possible IC payments, on a given type space.", "In particular, parts (1) and (2) state that the basic cells $\\operatornamewithlimits{\\mathsf {basic}}(T)$ and their covectors determine the set of all IC mechanisms and IC payments on $T$ , while parts (3) and (4) say that they are easy to discern from the geometry of $T$ .", "Our second main result, Theorem REF , gives a geometric construction and meaning to important classes of mechanisms, such as weakly monotone and revenue equivalent mechanisms on an arbitrary type space $T$ .", "This theorem relates geometry and linear algebra, giving a clear understanding of how the outcome function interacts with the geometry of the type space in affecting these properties.", "The geometric approach developed here is conceptually distinct from large parts of the existing literature on mechanism design.", "Our theorems provide new tools that make checking and visualizing incentive compatibility easy, both theoretically and computationally.", "Furthermore, Theorem REF and REF provide powerful techniques to construct all possible IC outcome functions together with their payments for arbitrary type spaces.", "To our knowledge such a joint characterization has not been obtained previously and appears to be beyond the scope of the traditional mathematical machinery of mechanism design.", "Let us demonstrate the insights that can be gained using these theorems.", "An IC mechanism is revenue equivalent (RE) if there exists a unique IC payment for its outcome function, up to an additive constant.", "We shall say that a type space $T$ is revenue equivalent if all IC mechanisms on $T$ are revenue equivalent.", "Prior to our work, the characterization of RE type spaces, due to Chung and Olszewski [6], reads as follows: $T$ is RE if and only if there do not exist disjoint subsets $B_1, B_2$ , a function $r: B_1 \\cup B_2\\rightarrow \\operatorname{\\mathbb {R}}$ , and an $\\epsilon >0$ , such that $T$ equals the union of two non-empty sets $V_+(B_1,\\epsilon ,r)$ and $V_-(B_2,\\epsilon ,r)$ , whose definitions depend on the parameters given.", "Given a specific type space $T$ , it is not immediate how to apply this theorem to check whether $T$ is RE.", "In contrast, we obtain the following characterization as a consequence of Theorem REF and REF .", "Theorem 1.2 A type space $T \\subset \\mathbb {TP}^{m-1}$ is RE if and only if for each $p~\\in ~\\operatornamewithlimits{\\mathsf {basic}}(T)$ the graph of $p$ is strongly connected.", "The graph of $p$ is the directed graph on $n$ nodes, with edge $(i,j)$ if and only if the distance between $T\\cap \\overline{\\mathcal {H}}_i(-p)$ and $\\overline{\\mathcal {H}}_{ij}(-p)$ is zero, where $\\overline{\\mathcal {H}}_{i}(-p)$ and $\\overline{\\mathcal {H}}_{ij}(-p)$ denote max-plus half-spaces of type $\\lbrace i\\rbrace $ and $\\lbrace i,j\\rbrace $ respectively.", "The max-plus hyperplane $\\overline{\\mathcal {H}}(-p)$ at $p$ equals the max-plus hyperplane at 0 translated by $p$ , so that checking whether a type space $T$ is RE using Theorem REF only requires considering translations of a fixed set by points in $\\operatornamewithlimits{\\mathsf {basic}}(T)$ and recording which sectors have positive distances.", "This makes checking revenue equivalence for a given type space surprisingly simple, while such an algorithm is not apparent from Chung and Olszewski's characterization.", "In addition, Theorem REF and REF allow for more detailed conclusions.", "For example, on an arbitrary type space $T$ , one can precisely identify the set of all RE and non-RE mechanisms.", "We illustrate these techniques in Examples REF and REF .", "To demonstrate the insights gained from the combinatorial view, consider the following problem.", "Suppose $T \\subset \\mathbb {TP}^{m-1}$ is a multiset consisting of $r$ points.", "Let $d(T) \\in \\mathbb {N}$ be the number of IC outcome functions on $T$ .", "We are not aware of ways to compute or bound $d(T)$ using traditional techniques.", "On the other hand, this easily follows from Theorem REF .", "Corollary 1.3 Suppose $T \\subset \\mathbb {TP}^{m-1}$ consists of $r$ points.", "Then $1 \\le d(T) \\le \\binom{r-1}{m-1}$ .", "If $T$ is generic, then $d(T) = \\binom{r-1}{m-1}$ .", "Our theorems were obtained by applying results from tropical convexity and tropical combinatorics to problems in mechanism design.", "However, we go further, by adapting and developing tropical convex geometry to handle economic problems.", "The basic region $\\operatornamewithlimits{\\mathsf {basic}}(T)$ and basic cells $\\operatornamewithlimits{\\mathsf {cells}}(T)$ of a multiset $T$ in tropical affine space are entirely new objects in tropical convex geometry, motivated by incentive compatibility problems.", "Thus concepts in mechanism design lend useful interpretations to objects in tropical convex geometry.", "Conversely the mathematical apparatus we advocate in this paper allows for the analysis of mechanisms in a unified way.", "It applies equally to mechanism design with and without money, and to multidimensional mechanism design.", "At the same time it supplies a complementary geometric view to the existing edifice of mechanism design theory, allowing for additional insights, as demonstrated in the preceding paragraphs.", "Mechanism design is a cornerstone of economic theory with many prominent applications, ranging from the analysis and design of voting procedures [19], [13], trading and auctioning mechanisms [20], [21], [26], to contract design and regulatory policy [17].", "The plenitude of open problems, both theoretical and applied, make it a very active research area.", "Tropical mathematics offers a new tool box to study and interpret mechanisms, with a novel perspective on open problems.", "This promises further developments between tropical geometry and mechanism design, though we certainly do not claim that all problems in mechanism design can be solved tropically.", "More broadly, there have been applications of tropical geometry to other economic problems, such as mean-payoff games [1], trade theory [24], [14], and auction theory [3], [25].", "These papers attest to a fruitful interaction between tropical geometry and economics." ], [ "Further Literature", "This paper is written for a general math audience with minimal background in either mechanism design or tropical geometry.", "While the paper is self-contained, it is not intended as an introduction to either field.", "For streamlined exposition with minimal notation, we only consider the simplest case of dominant strategy mechanism design, namely, direct mechanisms with a finite set of outcomes and one agent in a quasi-linear setting.", "These can be embedded into richer models.", "For an in-depth treatment of mechanism design, with economic interpretations and further generalizations, we refer to the excellent monograph of Vohra [27].", "Similarly, for a fuller picture of tropical algebra and geometry, there exists the comprehensive texts of Butkovivc [5] and Bacelli et al.", "[2] on solving tropical linear equations, or Joswig [15] and Maclagan and Sturmfels [18] on tropical geometry.", "We introduce the essential background in mechanism design and tropical convex geometry in Section .", "In Section , we prove Theorem REF by developing auxiliary results along the way.", "In particular we formalize the concept of basic cells and the basic set.", "In Section , we state and prove Theorem REF , providing a series of examples to demonstrate its power, and including a proof of Theorem REF stated above.", "For an integer $n$ , define $[n] := \\lbrace 1, 2, \\ldots , n\\rbrace $ .", "For sets $A, B \\subset \\operatorname{\\mathbb {R}}^m$ , write $\\mathsf {d}(A,B)$ for the infimum of the distance between pairs of points in these sets, and $\\mathsf {d_H}(A,B)$ for their Hausdorff distance, with respect to the Euclidean norm.", "We shall use the underline notation, such as $\\underline{\\oplus }, \\underline{\\odot }, \\underline{\\mathcal {H}},\\ldots $ to indicate objects defined with arithmetic done in the min-plus algebra, and the overline notation $\\overline{\\oplus }, \\overline{\\odot }, \\overline{\\mathcal {H}},\\ldots $ to indicate the same objects defined with arithmetic in the max-plus tropical algebra.", "When we write a multiset, we will use the same notation as for sets.", "When listing or enumerating the elements, we will list copies.", "The cardinalities of multisets are understood to include copies.", "Identify a graph with its incidence matrix." ], [ "Mechanism Design in Tropical Algebra", "Consider a game with one agent and $m~\\in ~\\mathbb {N}$ possible outcomes.", "Fix $T \\subset \\operatorname{\\mathbb {R}}^m$ , called the type space.", "At the beginning of the game, Nature chooses a true type $t^*\\in T$ which only the agent knows.", "The $i$ -th coordinate $t^\\ast _i$ measures how much the agent values outcome $i$ .", "A mechanism is a pair $(g, p)$ , consisting of an outcome function $g: T \\rightarrow [m]$ , which is always assumed to be onto, and a payment function $p: T \\rightarrow \\operatorname{\\mathbb {R}}$ .", "The agent's action is to declare to the mechanism a type $s \\in T$ , which may be different from the true type $t^\\ast $ .", "If $s$ is declared, the game's outcome is $g(s)$ , and the agent needs to pay $p(s)$ .", "In this case, the agent's utility is $u([g(s),p(s)], t^\\ast ) = t^\\ast _{ g(s)}- p(s).$ The agent, knowing $(g,p)$ , will declare a type $s \\in T$ that maximizes utility.", "A central goal of mechanism design is to identify incentive compatible mechanisms, these are mechanisms under which the agent will always tell the truth.", "Definition 2.1 Say that a mechanism $(g,p)$ is incentive compatible (IC) if regardless of $t^\\ast $ , the agent always maximizes utility by declaring the true type.", "That is, $t^\\ast _{g(t^*)}- p(t^*)\\ge t^\\ast _{ g(s)}- p(s)\\quad \\mbox{ for all } \\quad s,t^\\ast \\in T.\\qquad \\mathrm {(IC)}$ Say that an outcome function $g: T \\rightarrow [m]$ is incentive compatible if there exists $p:T \\rightarrow \\operatorname{\\mathbb {R}}$ such that $(g,p)$ is IC.", "We now proceed to state the essential terminology of mechanism design using tropical mathematics.", "Tropical linear algebra and its geometric version, tropical convex geometry, is the study of matrices, linear spaces and convex sets with arithmetic done in the tropical semi-ring.", "The min-plus semi-ring $(\\mathbb {R}\\cup \\lbrace +\\infty \\rbrace , \\underline{\\oplus },\\odot )$ has addition and multiplication defined by $a \\underline{\\oplus } b := \\min (a,b), \\quad a \\odot b := a + b\\quad \\mbox{ for } a,b \\in \\operatorname{\\mathbb {R}}\\cup \\lbrace +\\infty \\rbrace .$ Analogously, the max-plus semi-ring $(\\mathbb {R}\\cup \\lbrace -\\infty \\rbrace , \\overline{\\oplus },\\odot )$ is defined by $a \\overline{\\oplus } b := \\max (a,b), \\quad a \\odot b := a + b\\quad \\mbox{ for } a,b \\in \\operatorname{\\mathbb {R}}\\cup \\lbrace -\\infty \\rbrace .$ A matrix $L \\in \\operatorname{\\mathbb {R}}^{m \\times m}$ is said to have min-plus eigenvalue-eigenvector pair $(\\lambda ,p) \\in \\operatorname{\\mathbb {R}}\\times \\operatorname{\\mathbb {R}}^m$ if $ L \\,\\,\\underline{\\odot }\\,\\, p = \\lambda \\,\\,\\underline{\\odot }\\,\\,p, $ where the matrix-vector and scalar-vector multiplications take place in the min-plus semiring.", "Explicitly, for all $i =1,\\ldots ,m$ , $ \\min _{j=1,\\ldots ,m} L_{ij} + p_j = \\lambda + p_i.", "$ By a theorem of Cuninghame-Green [8], a $m \\times m$ matrix with finite entries $L \\in \\operatorname{\\mathbb {R}}^{m \\times m}$ has a unique min-plus eigenvalue, interpreted as the smallest normalized cycle length on a graph with edge weights given by $L$ .", "Thus one can speak of the min-plus eigenvalue of a matrix $L$ , denoted $\\underline{\\lambda }(L)$ .", "The min-plus eigenspace $\\underline{\\mathsf {Eig}}(L)$ of $L \\in \\operatorname{\\mathbb {R}}^{m \\times m}$ is its set of eigenvectors $ \\underline{\\mathsf {Eig}}(L) = \\lbrace x \\in \\operatorname{\\mathbb {R}}^m: L \\underline{\\odot } x = \\underline{\\lambda }(L) \\underline{\\odot } x\\rbrace .", "$ Returning to mechanism design, the allocation matrix $L^g$ of a given outcome function $g: T \\rightarrow [m]$ is an $m \\times m$ matrix, with $jk$ -th entry $L^g_{jk} = \\inf _{t \\in g^{-1}(j)}\\lbrace t_j - t_k\\rbrace .$ As is common in the literature [27], we shall always assume $T$ and $g$ to be such that the entries of $L^g$ are finite.", "It follows from Definition REF that $p(t) = p(s)$ whenever $g(t) = g(s)$ , for any IC mechanism $(g,p)$ .", "Thus we may assume that $p \\in \\operatorname{\\mathbb {R}}^m$ .", "By algebraic manipulations, one can check that equation (REF ) holds if and only if the matrix $L^g \\in \\operatorname{\\mathbb {R}}^{m \\times m}$ has a tropical eigenvector $p \\in \\operatorname{\\mathbb {R}}^m$ with tropical eigenvalue zero.", "That is, $(g,p)$ is an IC mechanism if and only if $L^g \\,\\,\\underline{\\odot }\\,\\, p = 0\\,\\, \\underline{\\odot }\\,\\, p = p.$ Hence, deciding IC for a given $g$ involves solving the tropical linear equation (REF ).", "Cuninghame-Green's theorem applied to equation (REF ) gives the classical characterization of IC outcome functions obtained by Rochet [22].", "Theorem 2.2 ([8], [22]) An outcome function $g$ is IC if and only if the matrix $L^g$ has min-plus tropical eigenvalue zero, that is, $\\underline{\\lambda }(L^g) = 0$ .", "Definition 2.3 If an outcome function $g: T \\rightarrow [m]$ is IC, call $\\underline{\\mathsf {Eig}}(L^g)$ the set of incentive compatible payments of $g$ .", "Definition 2.4 Say that $g$ is weakly monotone if the matrix $L^g + (L^g)^\\top $ is element-wise nonnegative.", "Definition 2.5 Say that $g$ is revenue equivalent (RE) if it is IC, and its set of incentive compatible payments $\\underline{\\mathsf {Eig}}(L^g)$ consists of exactly one point up to tropical scalar multiplication.", "That is, $g$ is RE if for any pair $p,q \\in \\underline{\\mathsf {Eig}}(L^g)$ , there exists a constant $c$ such that $p_i = q_i + c$ for all $i = 1, \\ldots , m$ .", "Say that a type space is revenue equivalent if any incentive compatible mechanism defined on it is revenue equivalent." ], [ "Mechanism design and tropical convex geometry", "Our key departure from the network flow approach to mechanism design, developed in [27], will be the use of tropical convex geometry.", "This subsection shall make Theorem REF precise.", "In a first step we replace type sets in $\\operatorname{\\mathbb {R}}^m$ by multisets in tropical affine space $\\mathbb {TP}^{m-1}$ .", "In a second step we define fundamental concepts in tropical convex geometry including tropical hyperplanes, polytopes, covectors, basic cells and basic sets.", "The $(m-1)$ -dimensional tropical affine space $\\mathbb {TP}^{m-1}$ is $\\operatorname{\\mathbb {R}}^m$ modulo the line spanned by the all-one vector $\\mathbb {TP}^{m-1} \\equiv \\operatorname{\\mathbb {R}}^m / \\operatorname{\\mathbb {R}}\\cdot (1, \\ldots , 1).$ We denote by $\\pi : \\operatorname{\\mathbb {R}}^m \\rightarrow \\mathbb {TP}^{m-1}$ the canonical projection from $\\operatorname{\\mathbb {R}}^m$ to $\\mathbb {TP}^{m-1}$ .", "The space $\\mathbb {TP}^{m-1}$ arises when we want to regard a tropical vector $t \\in \\operatorname{\\mathbb {R}}^m$ and its scalar multiple $a \\odot t = (a + t_1, \\ldots , a + t_m)$ as equivalent.", "Whenever convenient, such as visualizing examples, we shall identify $\\mathbb {TP}^{m-1}$ with $\\operatorname{\\mathbb {R}}^{m-1}$ via the homeomorphism $\\lbrace a \\odot (x_1, \\ldots , x_m): a \\in \\operatorname{\\mathbb {R}}\\rbrace \\in \\mathbb {TP}^{m-1} \\mapsto (x_2-x_1, \\ldots , x_m-x_1) \\in \\operatorname{\\mathbb {R}}^{m-1}.$ A multiset in $\\mathbb {TP}^{m-1}$ is a set of unique elements in $\\mathbb {TP}^{m-1}$ together with a function counting the number of copies of the underlying set's elements.", "A multiset in which each element has only one copy is a set.", "As a convention, we shall write a multiset by listing copies of its unique elements.", "Functions defined on a multiset may assign distinct values to the copies of the unique elements.", "To translate incentive compatibility problems to $\\mathbb {TP}^{m-1}$ , we proceed as follows.", "Given a type set $T^{\\prime }\\subset \\operatorname{\\mathbb {R}}^m$ , define the multiset $T\\subset \\mathbb {TP}^{m-1}$ with underlying set $T= \\pi (T^{\\prime })$ , by assigning to each $t \\in T$ , $m$ copies whenever $\\pi ^{-1}(t)\\cap T^{\\prime }$ contains at least $m$ points, and $k$ copies whenever $\\pi ^{-1}(t)\\cap T^{\\prime }$ contains $k<m$ points.", "Note that this definition depends only upon the type set $T^{\\prime }\\subset \\operatorname{\\mathbb {R}}^m$ .", "Suppose now we were given an outcome function $g^{\\prime }:T^{\\prime }\\rightarrow [m]$ .", "We can then define an induced outcome function $g:T \\rightarrow [m]$ which maps the copies of $t \\in T$ to the values of $g^{\\prime }$ on $\\pi ^{-1}(t)\\cap T^{\\prime }$ .", "Note that $L^g = L^{g^{\\prime }}$ , so that $g$ is IC if and only if $g^{\\prime }$ is.", "Furthermore, if $p$ is an IC payment of $g$ , then so is $a \\odot p$ for all $a\\in \\operatorname{\\mathbb {R}}$ , so that IC payments naturally are subsets of $\\mathbb {TP}^{m-1}$ .", "Hence, for incentive compatibility problems, there is no loss in passing from $g^{\\prime }$ , an outcome function defined on a subset of $\\operatorname{\\mathbb {R}}^m$ , to $g$ , an outcome function defined on a sufficiently rich multiset in $\\mathbb {TP}^{m-1}$ .", "In economic terms, this means only the relative valuation of the agent matter for truthfulness.", "In the following we recall some fundamental concepts in tropical convex geometry.", "Most definitions we build on appeared in the seminal work of Develin and Sturmfels [9], and were developed further in subsequent works by the authors of [10], [16], [12].", "However, tropical convex geometry itself has a much longer history, for a comprehensive introduction consult [15], [18] and references therein.", "For a point $t \\in \\mathbb {TP}^{m-1}$ , the min-plus hyperplane with apex $t$, denoted $\\underline{\\mathcal {H}}(-t)$ , is the set of $z \\in \\mathbb {TP}^{m-1}$ such that the minimum in the tropical inner product $(-t)^\\top \\underline{\\odot } z = \\min \\lbrace z_1 - t_1, \\ldots , z_m - t_m\\rbrace $ is achieved at least twice.", "For a subset $I \\subseteq [m]$ , denote by $\\underline{\\mathcal {H}}_I(-t)$ the min-plus half-space $I$ of $\\underline{\\mathcal {H}}(-t)$ , which is the set of $z \\in \\mathbb {TP}^{m-1}$ such that the minimum in the tropical inner product is achieved at all indices $i \\in I$ (and possibly more).", "That is, $ \\underline{\\mathcal {H}}_I(-t) = \\lbrace z \\in \\mathbb {TP}^{m-1}: z_i - t_i \\le z_j - t_j \\mbox{ for all } i \\in I, j \\in [m]\\rbrace .", "$ When $I = \\lbrace i\\rbrace $ or $I = \\lbrace i,j\\rbrace $ , we write $\\underline{\\mathcal {H}}_i$ and $\\underline{\\mathcal {H}}_{ij}$ , respectively, instead of $\\underline{\\mathcal {H}}_I$ .", "We shall call $\\underline{\\mathcal {H}}_i$ the $i$ -th sector of the hyperplane.", "For a multiset $T \\subset \\mathbb {TP}^{m-1}$ , the union of min-plus hyperplanes $\\bigcup _{t \\in T}\\underline{\\mathcal {H}}(-t)$ partitions $\\mathbb {TP}^{m-1}$ into a polyhedral complex, called the min-plus hyperplane arrangement based on $T$ , denoted $\\underline{\\mathcal {H}}(-T)$ .", "For a finite set $T$ , the set of bounded cells of $\\underline{\\mathcal {H}}(-T)$ is the max-plus polytope generated by $T$[9].", "Say that a multiset $T \\subset \\mathbb {TP}^{m-1}$ is generic if there is no subset of $2 \\le k \\le m$ points in $T$ whose projection onto $k$ coordinates lie on a tropical hyperplane in $\\mathbb {TP}^{k-1}$ .", "Note that a generic multiset $T$ necessarily has no copies, i.e.", "it is a set.", "The min-plus covector of a point $p \\in \\mathbb {TP}^{m-1}$ with respect to a multiset $T ~\\subseteq ~\\mathbb {TP}^{m-1}$ , denoted $\\mathsf {\\underline{coVec}}_T(p)$ , is the bipartite graph with nodes $[m] \\times T$ and edge $(i,t)\\in [m]\\times T$ if and only if $p\\in \\mathcal {\\underline{H}}_i(-t)$ .", "Definition 2.6 Fix a multiset $T \\subset \\mathbb {TP}^{m-1}$ .", "Let $g$ be a bipartite graph on $[m] \\times T$ .", "We will say that $g$ is an outcome graph if it defines an outcome function $g: T \\rightarrow [m]$ via $g(t) = i$ if and only if $(i,t)$ is an edge of the graph $g$ .", "Definition 2.7 To a bipartite graph $g$ on $[m] \\times T$ , the possibly empty polyhedron of $g$ is $\\mathcal {P}(g) = \\lbrace q \\in \\mathbb {TP}^{m-1}: q \\in \\mathcal {\\underline{H}}_i(-t) \\mbox{ for all } (i,t) \\in g\\rbrace .$ By min-max duality, $\\mathcal {P}(g)$ can be equivalently written as $\\mathcal {P}(g) = \\lbrace q \\in \\mathbb {TP}^{m-1}: t \\in \\mathcal {\\overline{H}}_i(-q) \\mbox{ for all } (i,t) \\in g\\rbrace .$ It is immediate that if a polyhedron $\\mathcal {P}(g)$ is non-empty, then all points in its relative interior have the same covector with respect to $T$ .", "Call this the basic covector of $g$ , denoted $\\nu (g)$ .", "Definition 2.8 Say that a covector $\\nu $ is basic, and that $\\mathcal {P}(\\nu )$ is a basic cell, if $\\nu = \\nu (g)$ for some outcome function $g$ .", "The set of basic cells of $T$ is $\\operatornamewithlimits{\\mathsf {cells}}(T) = \\lbrace \\mathcal {P}(\\nu ): \\nu \\mbox{ is a basic covector}\\rbrace .$ The set union of the basic cells, denoted $\\operatornamewithlimits{\\mathsf {basic}}(T) = \\bigcup _{P \\in \\operatornamewithlimits{\\mathsf {cells}}(T)}P.$ will be called the basic set of $T$.", "Figure: Hyperplanes and arrangements.", "Figures accompany Examples and .", "For all figures in this paper, axis orientation and sector labels follow the convention set in Panel (a).Example 2.9 (Hyperplanes, half-spaces and basic cells) Panel (a) in Figure REF depicts the min-plus hyperplane $\\underline{\\mathcal {H}}(-t_0)$ and its sectors (top) and the max-plus hyperplane $\\overline{\\mathcal {H}}(-t_0)$ and its sectors (bottom).", "Both are in $\\mathbb {TP}^2$ with apex $t_0$ .", "As usual, we identify $\\mathbb {TP}^2$ with $\\operatorname{\\mathbb {R}}^2$ via the map given in (REF ).", "Example 2.10 (Covectors and basic cells) Panel (b) depicts a set $T=\\lbrace t_1, t_2, t_3\\rbrace $ , the min-plus arrangement $\\underline{\\mathcal {H}}(-T)$ , and the covectors of a selection of cells.", "There is only one basic cell, shaded gray.", "Its covector defines the outcome function $g: T \\rightarrow [3]$ with $g(t_1)~=~1, g(t_2)~=~2, g(t_3)~=3$ .", "By Theorem REF , $g$ is the only IC outcome function amongst all possible outcome functions on $T$ , and its set of IC payments is the basic cell shaded gray.", "One can also apply Corollary REF : $T$ consists of three generic points, so the number of IC outcome functions on $T$ is $\\binom{m-1}{m-1} = 1$ ." ], [ "Auxiliary Results and Proof of the Main Theorem", "In this section, we prove Theorem REF by establishing a series of lemmata and propositions.", "For readability, we split the Theorem into three propositions followed by their proofs.", "Proposition REF covers parts (1) and (2) of Theorem REF .", "Parts (3) and (4) are covered by Propositions REF and REF , respectively.", "In the process we clarify all subtleties and discuss some examples to aide understanding." ], [ "Covectors and incentive compatibility", "An outcome function $g: T \\rightarrow [m]$ is onto, and each $t \\in T$ must be assigned to an outcome.", "Thus, a bipartite graph $g$ is an outcome graph if and only if each node $t \\in T$ has degree one, and each node $i \\in [m]$ has degree at least one.", "The goal is to discern which outcome functions $g$ are IC.", "Proposition REF gives a characterization in terms of the covector of the polyhedron $\\mathcal {P}(g)$ .", "Lemma 3.1 Let $g: T \\rightarrow [m]$ be an outcome function.", "Then $\\mathcal {P}(g)$ is the set of incentive compatible payments of $g$ .", "In particular, $\\mathcal {P}(g) \\ne \\emptyset $ if and only if $g$ is IC.", "Note that $p \\in \\mathcal {P}(g)$ if and only if for all $t \\in g^{-1}(i) \\subset T$ and all $j \\in [m]$ , $p_i - t_i \\ge p_j - t_j.$ Thus $p$ is an incentive compatible payment of $g$ .", "Lemma 3.2 Let $h$ be a bipartite graph on $[m] \\times T$ .", "The following are equivalent.", "$\\mathcal {P}(h)$ is not empty, $h$ is contained as a subgraph in the covector of some cell of $\\underline{\\mathcal {H}}(-T)$ , $\\mathcal {P}(h) = \\mathcal {P}(\\nu (h))$ for the covector $\\nu (h)$ of a point in the relative interior of $\\mathcal {P}(h)$ .", "Suppose (1).", "Let $p$ be a point in the relative interior.", "By definition of covector, all points in the relative interior of $\\mathcal {P}(h)$ have the same covector.", "Let us denote this covector by $\\nu (h)$ .", "If $(i,t) \\in h$ , then $p \\in \\underline{\\mathcal {H}}_i(-t)$ , so $(i,t) \\in \\nu (h)$ .", "Thus $h$ is contained in $\\nu (h)$ .", "This implies (2).", "In addition, it also implies $\\mathcal {P}(\\nu (h)) \\subseteq \\mathcal {P}(h)$ , by definition of the polyhedra $\\mathcal {P}(h)$ and $\\mathcal {P}(\\nu (h))$ .", "Now, $\\mathcal {P}(h)$ and $\\mathcal {P}(\\nu (h))$ are closed polyhedra.", "So if $\\mathcal {P}(\\nu (h))$ is a strict subset of $\\mathcal {P}(h)$ , then there must exists points in the relative interior of $\\mathcal {P}(h)$ which do not belong to $\\mathcal {P}(\\nu (h))$ , and in particular, cannot have covector $\\nu (h)$ .", "This is a contradiction, so $\\mathcal {P}(\\nu (h)) = \\mathcal {P}(h)$ .", "This establishes (3).", "Finally, (3) trivially implies (1).", "Proposition 3.3 (Covector characterization) Let $g: T \\rightarrow [m]$ be an outcome function on a multiset $T$ .", "Then $g$ is IC with payment $p \\in \\mathbb {TP}^{m-1}$ if and only if $\\mathsf {\\underline{coVec}}_T(p)$ contains $g$ as a subgraph.", "In this case, $\\mathcal {P}(\\nu (g))$ is the set of incentive compatible payments of $g$ .", "By Lemma REF the mechanism $(g, p)$ is IC if and only if $p\\in \\mathcal {P}(g)$ .", "By Lemma REF it is without loss to assume that $p$ is in the relative interior, with covector $\\underline{\\mathsf {coVec}}_T(p)$ .", "It follows from the same lemma that $p\\in \\mathcal {P}(g)$ if and only if $g$ is contained in $\\underline{\\mathsf {coVec}}_T(p)$ and $\\mathcal {P}(\\underline{\\mathsf {coVec}}_T(p)) = \\mathcal {P}(g)$ .", "In other words, to each IC outcome function there corresponds a basic cell containing its payments.", "Conversely, each basic cell can be associated with a set of IC outcome functions whose payments it contains.", "This set consists of precisely those outcome functions contained in the cell's covector.", "Example 3.4 (Algebraic construction of IC mechanisms) Let $\\nu _1$ (resp.", "$\\nu _2$ ) be the cells containing $p$ (resp.", "$q$ ) in its relative interior.", "The covectors of $p$ and $q$ are given by $\\begin{matrix}\\underline{\\mathsf {coVec}}_T(p)=\\begin{pmatrix}1&0&0\\\\ 0&1&0\\\\0 &0&1 \\end{pmatrix}, &\\mathsf {\\underline{coVec}}_T(q)=\\begin{pmatrix}0&1&0\\\\ 1&0&1\\\\0&1&0 \\end{pmatrix}\\end{matrix}.$ There is a unique outcome function $g: T \\rightarrow [3]$ whose set of IC payments equals $\\nu _1$ , namely, $g(t_1)=1, g(t_2)=2, g(t_3)=3$ .", "For the cell $\\nu _2$ , there is no outcome function contained in $\\underline{\\mathsf {coVec}}_T(q)$ , so there exists no onto IC mechanism with payment set $\\nu _2$ .", "Example 3.5 (Geometric construction of IC mechanisms) In Figure REF , consider the max-plus hyperplane with apex $p$ (dotted).", "The sets $\\overline{\\mathcal {H}}_i(-p)\\cap T$ for $i\\in [m]$ partition $T$ into non-empty, disjoint subsets.", "By Lemma REF and (REF ), any outcome function $g$ such that $(g,p)$ is IC must map types in the interior of $\\overline{\\mathcal {H}}_i(-p)\\cap T$ to $i$ , for $i\\in [3]$ .", "Thus $g(t_1)=1, g(t_2)=2, g(t_3)=3$ is the unique such outcome function, agreeing with Example REF .", "Example 3.6 (Non-basic cells cannot contain IC payments) In Figure REF (b), consider the max-plus hyperplane with apex $q$ (dotted).", "Suppose for contradiction that there exists an outcome function $g$ such that $(g,q)$ is IC.", "By (REF ) and Lemma REF , $g(t_1) = g(t_3) = 2$ , while $t_2 \\in \\overline{\\mathcal {H}}_{13}(-q) \\cap T$ , $g(t_2)$ must be either 1 or 3.", "But in either assignment of $t_2$ , $g$ is not onto, so it cannot be an outcome function.", "Thus, $q$ cannot be an IC payment of some IC mechanism.", "Figure: Here T={t 1 ,t 2 ,t 3 }T = \\lbrace t_1, t_2, t_3\\rbrace with no repeated points.", "The min-plus arrangement ℋ ̲(-T)\\underline{\\mathcal {H}}(-T) is drawn using solid lines.", "Axis orientation and sector labels follow the convention set in Figure .The unique basic cell is shaded gray.", "Figure accompanies Examples to ." ], [ "Basic cells via generic approximations", "By Proposition REF , the set of basic cells $\\operatornamewithlimits{\\mathsf {cells}}(T)$ and its union, the basic set $\\operatornamewithlimits{\\mathsf {basic}}(T)$ , encode the IC payments of any possible mechanism.", "We now study some properties of basic cells, starting with the case where $T$ is a multiset of generic points.", "Recall that this means there does not exist a subset of $2 \\le k \\le m$ points in $T$ whose projection onto $k$ coordinates lie on a tropical hyperplane in $\\mathbb {TP}^{k-1}$ .", "Thus, in particular, $T$ is a set.", "Proposition 3.7 Suppose $T \\subset \\mathbb {TP}^{m-1}$ contains $r$ generic points.", "Then $\\operatornamewithlimits{\\mathsf {cells}}(T)$ is precisely the set of full-dimensional cells of $\\underline{\\mathcal {H}}(-T)$ .", "In particular, the cardinality of $\\operatornamewithlimits{\\mathsf {cells}}(T)$ is $\\binom{r-1}{m-1}$ .", "By [9], $P = \\mathcal {P}(\\nu )$ is a full-dimensional cell of $\\underline{\\mathcal {H}}(-T)$ if and only if $\\nu $ is an outcome graph whose polyhedron $\\mathcal {P}(\\nu )$ is non-empty.", "Thus, the set of full-dimensional cells of $\\underline{\\mathcal {H}}(-T)$ is a subset of $\\operatornamewithlimits{\\mathsf {cells}}(T)$ .", "On the other hand, let $P = \\mathcal {P}(\\nu ) \\in \\operatornamewithlimits{\\mathsf {basic}}(T)$ be a basic cell with respect to some outcome graph $g$ .", "Since $\\nu $ is basic, $\\nu = \\nu (g) = g$ , so $P$ is a full-dimensional cell of $\\underline{\\mathcal {H}}(-T)$ .", "Finally, the cardinality follows from [9].", "To handle the case of a general multiset $T$ we use approximations by generic perturbations, which are common technique in tropical geometry, see [18].", "Definition 3.8 Suppose the multiset $T=\\lbrace t^1, \\ldots t^r\\rbrace $ in $\\mathbb {TP}^{m-1}$ consists of $r < \\infty $ points, counting copies.", "A sequence $(T^k, k \\ge 1)$ is called a generic perturbation of $T$ if each set $T^k= \\lbrace t^{k,1}, \\ldots , t^{k,r}\\rbrace \\subset \\mathbb {TP}^{m-1}$ consists of generic points and $\\lim _{k\\rightarrow \\infty }t^{k,i}=t^i$ for all $i\\in [r]$ .", "Lemma 3.9 Suppose the multiset $T \\subset \\mathbb {TP}^{m-1}$ is finite, counting copies.", "Let $(T^k, k \\ge 1)$ be a generic perturbation of $T$ .", "Then $\\sigma $ is a basic cell of $T$ if and only if there exists a sequence $(\\sigma ^k,k\\ge 1)$ , where $\\sigma ^k$ is a full-dimensional cell in $\\underline{\\mathcal {H}}(-T^k)$ , such that $\\lim _{k\\rightarrow \\infty }\\mathsf {d_H}(\\sigma ^k,\\sigma )=0$ .", "In other words, the basic cells of $\\underline{\\mathcal {H}}(-T)$ are limits as $k \\rightarrow \\infty $ of full-dimensional cells in $\\underline{\\mathcal {H}}(-T^k)$ with respect to the Hausdorff distance.", "For the following proof, we shall write $\\nu _T(g)$ instead of the usual shorthand $\\nu (g)$ to mean the basic covector of $g$ with respect to $T$ .", "Let $r$ be the number of points in the multiset $T$ , counting copies.", "Enumerate the points $T = \\lbrace t^1, \\ldots , t^r\\rbrace $ .", "Let $T^k = \\lbrace t^{k,j}, j=1,\\ldots ,r\\rbrace $ be a generic perturbation from Definition REF .", "Suppose $\\sigma ^k\\in \\underline{\\mathcal {H}}(-T^k)$ is a sequence of full-dimensional cells such that $\\lim _{k\\rightarrow \\infty } \\mathsf {d_H}(\\sigma ^k, \\sigma )=0$ .", "Let $\\nu ^k:=\\underline{\\mathsf {coVec}}_{T^k}(\\sigma ^k)$ be their covector.", "Each $\\nu ^k$ is a bipartite graph on $[m] \\times [r]$ .", "Since there are only finitely many such graphs, the sequence $(\\nu ^k,k\\ge 1)$ contains a constant subsequence $\\nu ^{k^{\\prime }}=\\nu ^{\\prime }$ to which is associated the sequence $(\\sigma ^{k^{\\prime }}, k^{\\prime }\\ge 1)$ .", "Since each $\\sigma ^{k^{\\prime }}$ is full-dimensional, $\\nu ^{\\prime } = \\nu _T(g)$ for some outcome function $g$ .", "Evidently $\\lim _{k^{\\prime }\\rightarrow \\infty } \\mathsf {d_H}(\\sigma ^{k^{\\prime }}, \\sigma ) = 0$ , so that $\\sigma $ is a basic cell of $T$ .", "Conversely, suppose $\\sigma $ is a basic cell of $T$ .", "A cell $\\sigma $ of $\\underline{\\mathcal {H}}(-T)$ has the form $\\sigma = \\bigcap _{j=1}^r \\underline{\\mathcal {H}}_{I^j}(-t^j)$ for some subsets $I^j \\subseteq [m]$ , $j=1,\\ldots ,r$ .", "Let $g^1, \\ldots , g^n$ be all the IC outcome functions such that $\\sigma = \\mathcal {P}(\\nu _T(g))$ , for $g \\in \\lbrace g^1, \\ldots , g^n\\rbrace $ .", "By (REF ) at least one such function exists.", "For each $k$ and each $i \\in [n]$ , consider the cell $\\mathcal {P}(\\nu _{T^k}(g^i))$ in $\\underline{\\mathcal {H}}(-T^k)$ .", "Since $T^k$ is generic and $g^i$ is an outcome function, by Proposition REF , either this cell is empty or that it is full-dimensional.", "Let $\\mathcal {Q}^k$ be the set of non-empty such cells, $ \\mathcal {Q}^k = \\lbrace \\mathcal {P}(\\nu _{T^k}(g^i)): \\mathcal {P}(\\nu _{T^k}(g^i)) \\ne \\emptyset , i =1,\\ldots ,n\\rbrace .", "$ Since $T^k$ is a generic perturbation of $T$ , $\\mathcal {Q}^k \\ne \\emptyset $ .", "For each $k$ , choose an arbitrary cell $\\sigma ^k \\in \\mathcal {Q}^k$ .", "But $\\lim _{k\\rightarrow \\infty }t^{k,j} = t^j$ for all $j\\in [r]$ , implies $\\lim _{k\\rightarrow \\infty }\\mathsf {d_H}(\\sigma ^k, \\sigma )=0$ , so we are done.", "The approximation result below allows us to identify the basic cells of an infinite multiset $T$ as limit of finite approximations.", "For its proof it will be convenient to have another characterization of cells in $\\underline{\\mathcal {H}}(-T)$ using a generalization of [9].", "Lemma 3.10 Introduce variables $(z_t: t \\in T)$ .", "A polyhedron $\\sigma $ is a cell of the tropical hyperplane arrangement $\\underline{\\mathcal {H}}(-T)$ with covector $\\nu $ if and only if it is the projection onto the $y$ coordinate of the set $B := \\lbrace (y, z): y_i + z_t \\ge t_i \\mbox{ for all } t \\in T, i \\in [m], y_i + z_t = t_i \\mbox{ if } (i,t) \\in \\nu \\rbrace .$ For each $y \\in \\sigma $ , define $z(y)$ coordinate-wise by $z_t(y) = \\max _{k\\in [m]}\\lbrace t_k-y_k\\rbrace , \\hspace{10.0pt} t \\in T.$ By definition, $y \\in \\sigma $ if and only if $y_i - t_i \\le y_j - t_j$ for all $(i,t) \\in \\nu $ .", "For each $y \\in \\sigma $ , the pair $(y,z(y))$ belongs to $B$ .", "Conversely, if $(y,z) \\in B$ for some $z$ , then $y_i - t_i \\le y_j - t_j$ , so $y \\in \\sigma $ .", "Proposition 3.11 Let $T \\subset \\mathbb {TP}^{m-1}$ be a multiset and $\\sigma \\in \\underline{\\mathcal {H}}(-T)$ .", "Let $(T^k, k \\ge 1)$ be a sequence of finite multisets, counting copies, such that $\\lim _{k\\rightarrow \\infty }\\mathsf {d}_H(T^k, T)=0 $ .", "Then $\\sigma \\in \\operatornamewithlimits{\\mathsf {cells}}(T)$ if and only if there is a sequence of basic cells $\\sigma ^k \\in \\operatornamewithlimits{\\mathsf {cells}}(T^k)$ such that $\\lim _{k\\rightarrow \\infty }\\mathsf {d_H}(\\sigma ^k,\\sigma )=0$ .", "Without loss of generality, one can assume that $T^k$ is an increasing sequence, i.e.", "$T^k \\subseteq T$ , and $T^k \\subseteq T^{k+1}$ for all $k\\ge 1$ , and that the limit is dense in $T$ .", "Let $\\sigma $ be a cell of $T$ with covector $\\nu $ .", "Employing Lemma REF , we see that $\\sigma $ is a polytope in $\\mathbb {TP}^{m-1}$ , where each $t \\in T$ contributes a constraint of the form $ y_i + z_t > t_i \\mbox{ if } (i,t) \\notin \\nu , \\hspace{10.0pt} y_i + z_t = t_i \\mbox{ if } (i,t) \\in \\nu .", "$ In particular, if $t,t^{\\prime }$ are arbitrarily close, then $(i,t) \\in \\nu $ if and only if $(i,t^{\\prime }) \\in \\nu $ .", "This implies that the covector of $\\sigma $ with respect to $T$ is completely determined by its covector with respect to a dense subset of $T$ .", "This implies the result.", "We remark that by combining Proposition REF and REF , one can choose $T^k$ to be a finite, generic sequence that approximates $T$ .", "This way, each basic cell of $T$ is the limit of a sequence of full-dimensional cells of an approximating sequence of max-plus tropical polytopes.", "This is particularly useful in specific examples, as full-dimensional cells of a generic approximation of $T$ are easy to identify.", "We conclude this section with some examples.", "Figure: A non-generic arrangement on three types and possible generic perturbations thereof.", "The cells shaded gray are basic.", "Figure accompanies Example .Example 3.12 (Generic perturbation of finite point-configurations) Figure REF depicts arrangements on three points, with no repeated points.", "The points in Panel (a) are not generic: the projections of $t_1$ and $t_3$ onto the first coordinate is a point.", "Panels (b) and (c) show possible generic perturbations of these points.", "The covectors of the cells shaded gray are $\\begin{matrix}\\underline{\\mathsf {coVec}}_T(S)=\\begin{pmatrix}1&0&1\\\\ 1&0&1\\\\0 &1&0 \\end{pmatrix}, &\\mathsf {\\underline{coVec}}_L(S^{\\prime })=\\begin{pmatrix}0&0&1\\\\ 1&0&0\\\\0&1&0 \\end{pmatrix}, &\\mathsf {\\underline{coVec}}_L(S^{\\prime \\prime })=\\begin{pmatrix}1&0&0\\\\ 0&0&1\\\\0&1&0 \\end{pmatrix}\\end{matrix} $ The cells $S^{\\prime }$ and $S^{\\prime \\prime }$ are full-dimensional and thus basic by Proposition REF .", "Indeed, their covectors are outcome graphs $g$ and $h$ respectively, with $g(t^{\\prime }_1)=2, g(t_2)=3, g(t_3)=1$ and $h(t_1)=1, h(t_2)=3, h(t_3^{\\prime \\prime })=2$ .", "The cell $S$ in Panel (a) is basic, as $P(\\nu (g)) = P(\\nu (h)) = P(\\underline{\\mathsf {coVec}}_T(S))$ .", "One can also verify via Proposition REF , as $S$ can be obtained as the limit of $S^{\\prime }$ as $t_1^{\\prime }$ approaches $t_1$ , or as the limit of $S^{\\prime \\prime }$ as $t_3^{\\prime \\prime }$ approaches $t_3$ .", "Example 3.13 (A multiset with copies) We could also view the point configurations in Figure REF (b) and (c) as arising from a single point with three copies that have been perturbed to become generic.", "In an arrangement arising from a point with three copies, the basic cell is the point itself and the covector is the $3\\times 3$ all ones matrix.", "Such a multiset admits $3!$ different IC mechanisms that all have the same price, namely the point we started out with.", "Figure: The set TT, drawn in black, consists of the circle and an isolated point.", "Panel (A) depicts the basic cells in gray for a finite approximation of TT consisting of five points.", "In Panel (B), the basic region is obtained by applying Proposition .Figure accompanies Example and .Example 3.14 (Limiting approximation) The set $T \\subset \\mathbb {TP}^2$ in Figure REF consists of the black dot and the circle drawn using opaque lines.", "This set $T$ can be approximated as the closed limit of a finite sequence $(T^k)$ .", "Each $T^k$ consists of the isolated point and $k-1$ points on the circle.", "Panel (a) depicts $T^5$ and the corresponding min-plus arrangement $\\underline{\\mathcal {H}}(-T^5)$ .", "The basic cells of $\\underline{\\mathcal {H}}(-T^5)$ are shaded gray.", "In Panel (b), the basic region $\\operatornamewithlimits{\\mathsf {basic}}(T)$ is shaded gray.", "By Proposition REF , it is computed by taking limits of the basic cells of $T^k$ .", "Basic cells in the dotted triangular region to the left of the circle consist of parallel lines of slope $(1,1)$ .", "The basic cells elsewhere consist of a single point.", "In particular, $T$ is not RE.", "In ExampleREF , we give another way to verify that $T$ is not RE given its basic region." ], [ "Mechanisms as Allocation Matrices", "We now revisit Rochet's theorem using tropical geometry.", "Identify an outcome function $g$ with its allocation matrix $L^g$ defined in (REF ).", "The main result of this section, Theorem REF , specifies the set of matrices that parametrizes all outcome functions on a given multiset $T\\subseteq \\mathbb {TP}^{m-1}$ .", "It allows one to infer properties of $g$ from $L^g$ , such as incentive compatibility, the set of IC payments, and the dimension of this set, without computing cycle weights in$L^g$ .", "This geometric insight allows us to easily construct examples of outcome functions, as illustrated in the examples below.", "In particular, Theorem REF follows as an easy consequence.", "We introduce some notational shortcuts and definitions.", "For a matrix $L \\in \\operatorname{\\mathbb {R}}^{m \\times m}$ with zero diagonal, let $L_1, \\ldots , L_m \\in \\mathbb {TP}^{m-1}$ be the $m$ rows of $L$ , viewed as vectors in $\\mathbb {TP}^{m-1}$ .", "For $j,k \\in [m], j \\ne k$ , write $\\overline{\\mathcal {L}}_j$ for $\\overline{\\mathcal {H}}_j(-L_j)$ .", "Let $\\overline{\\mathcal {L}}_j^\\circ $ denote the interior of this cone.", "Write $\\overline{\\mathcal {L}}_{jk}$ for $\\overline{\\mathcal {H}}_{jk}(-L_j)$ .", "As before, we use the underline notation to mean the analogous quantity in min-plus.", "For a matrix $L$ , define the zero eigenspace of $L$ to be the possibly empty set $\\mathsf {Eig}_0(L) := \\bigcap _{j=1}^m\\underline{\\mathcal {L}}_j.$ For a multiset $T \\subseteq \\mathbb {TP}^{m-1}$ , say that $L$ is realizable with respect to $T$ if there exists some outcome function $g: T \\rightarrow [m]$ such that $L = L^g$ .", "In that case, say that $L$ is realized by $g$.", "Definition 4.1 For $j,k \\in [m], j \\ne k$ , define $\\mathcal {I}_{jk} = \\mathcal {\\overline{L}}_{jk}\\cap T$ .", "A $(j,k)$ -witness is a sequence $\\lbrace s^{j,r}: r \\ge 1\\rbrace \\subseteq T \\cap \\overline{\\mathcal {L}}_j^\\circ $ such that $ \\lim _{r \\rightarrow \\infty } \\mathsf {d}(s^{j,r}, \\mathcal {\\overline{L}}_{jk} )=0.$ We say that $L$ separates $T$ at $(j,k)$ if $\\mathsf {d}(T \\cap \\mathcal {\\overline{L}}_j, \\mathcal {\\overline{L}}_{jk})=0,$ and in addition, whenever $\\mathcal {I}_{jk} = \\mathcal {I}_{kj} = \\lbrace s\\rbrace $ for some $s \\in \\mathbb {TP}^{m-1}$ , then there exists a $(j,k)$ -witness or a $(k,j)$ -witness.", "Say that $L$ separates $T$ if $L$ separates $T$ for all $j,k \\in [m], j \\ne k$ .", "Definition 4.2 For $p \\in \\mathbb {TP}^{m-1}$ , the graph of $p$ (with respect to $T$ ) is the directed graph on $m$ nodes, with edge $(i,j)$ if and only if $ \\mathsf {d}(T\\cap \\overline{\\mathcal {L}}_i,\\overline{\\mathcal {H}}_{ij}(-p)) = 0.", "$ Theorem 4.3 Let $L \\in \\operatorname{\\mathbb {R}}^{m \\times m}$ be a matrix with zero diagonal, $T \\subseteq \\mathbb {TP}^{m-1}$ be a multiset.", "$L$ is realizable if and only if $L$ separates $T$ , and $T \\subseteq \\bigcup _{k=1}^m \\overline{\\mathcal {L}}_k.$ Suppose $L$ is realized by $g$ .", "Then $g$ is weakly monotone if and only if the open sets $\\overline{\\mathcal {L}}^\\circ _1, \\ldots , \\overline{\\mathcal {L}}^\\circ _m$ are pairwise disjoint.", "Suppose $L$ is realized by $g$ .", "The set of incentive compatible payments of $g$ is $\\mathsf {Eig}_0(L)$ .", "In particular, $g$ is IC if and only if $\\mathsf {Eig}_0(L) \\ne \\emptyset $ .", "Suppose $L$ is realized by $g$ and $\\mathsf {Eig}_0(L)\\ne \\emptyset $ .", "Let $p$ be a point in the relative interior of $\\mathsf {Eig}_0(L)$ .", "Then the dimension of $\\mathsf {Eig}_0(L)$ is the number of strongly connected components in the graph of $p$ with respect to $T$ minus 1.", "We defer the proof to Section REF .", "Instead, we illustrate the witnessing condition and the implications of the theorem with examples.", "They show that it is simple to construct mechanisms and verify realizability, incentive compatibility and revenue equivalence.", "Figure: Geometric construction of allocation matrices.", "The type set consists of the black dots that are apices of the min-plus hyperplanes drawn using dashed lines.", "In Panel (a), types have been labeled according to their outcome under an outcome function gg.", "In Panel (b), the gray dots are the rows of the matrix L g L^g, viewed as points in 𝕋ℙ 2 \\mathbb {TP}^2.", "The set of IC payments of gg is shaded gray in both panels.", "Figure accompanies Example .Example 4.4 (Geometric construction of allocation matrices) Figure REF (a) shows how to construct the apices of the sectors $\\overline{\\mathcal {L}}_i$ geometrically.", "The set $T$ consists of the black generic points.", "The dashed lines are hyperplanes of $\\underline{\\mathcal {H}}(-T)$ .", "The labels next to the points define an outcome function $g: T \\rightarrow [3]$ .", "For $i \\in [3]$ , the heavy black lines define the boundary of the max-plus sectors $\\overline{\\mathcal {L}}_i$ , whose apex is the $i$ -th row of $L^g$ .", "For example, for $i=1$ , the first row of $L^g$ can be written as $ L^g_1 = (0, ~ \\sup _{t\\in g^{-1}(1)}\\lbrace t_2-t_1 \\rbrace ,~ \\sup _{t\\in g^{-1}(1)}\\lbrace t_3-t_1 \\rbrace )$ .", "The set of IC payments of $g$ equals the basic cell shaded gray, computed using Theorem REF .", "Panel(b) verifies that this basic cell equals $\\mathsf {Eig}_0(L)$ , as stipulated by Theorem REF , part (2).", "Here $\\mathsf {Eig}_0(L)$ is a polytope in $\\mathbb {TP}^2$ of dimension 2.", "To verify with Theorem REF part (4), let $p$ be a point in the interior of $\\mathsf {Eig}_0(L)$ as shown in Panel (a), with $\\overline{\\mathcal {H}}(-p)$ shown in dotted lines.", "One can readily verify that the graph of $p$ consists of three points with no edges, thus it has three strongly connected components.", "So $\\mathsf {Eig}_0(L)$ has dimension 2, as expected.", "Figure: Realizable and non-realizable matrices, defined by the apices of ℒ ¯ i \\overline{\\mathcal {L}}_i for i∈[3]i \\in [3].Figure accompanies Example .Example 4.5 (Realizable and non-realizable matrices) In Figure REF , $T = \\lbrace t_1,t_2,t_3,t_4\\rbrace $ is the set of four black points.", "Each panel defines a matrix $L$ whose rows give rise to the sectors $\\overline{\\mathcal {L}}_1, \\overline{\\mathcal {L}}_2$ and $\\overline{\\mathcal {L}}_3$ .", "The matrix in Panel (a) is realizable: it equals $L^g$ for the mechanism $g$ is defined by $g(t_2) = g(t_3) = 3$ , $g(t_1) = 1$ and $g(t_4) = 2$ .", "The matrix $L$ in Panel (b) is not realizable.", "Suppose for contradiction that it is realized by some outcome function $g$ .", "By Theorem REF , we must have $g(t_4) = 2$ , $g(t_1) = 1$ , $g(t_3) = 3$ , and $g(t_2)$ is either 1 or 3.", "If $g(t_2) = 3$ , then $L^g$ must equal that shown in Figure REF  (a) and thus in not equal to $L$ .", "So $g(t_2) = 1$ , but computations show that for this $g$ , $L^g \\ne L$ .", "Thus $L$ is not realizable.", "Realizability fails since $L$ does not separate $T$ at $\\lbrace 1,3\\rbrace $ .", "Here, one has $\\mathcal {I}_{13} = \\overline{\\mathcal {L}}_{13} \\cap \\overline{\\mathcal {L}}_{31} \\cap T = \\lbrace t_2\\rbrace , $ but there is no $(1,3)$ - nor a $(3,1)$ -witness.", "Example 4.6 (A type space without RE) Consider Figure REF (b), where the basic region of a certain type set $T \\subset \\mathbb {TP}^2$ is shaded in gray.", "To each point $p$ in this basic region, put a max-plus hyperplane and compute its graph.", "We find that for all $p$ in the circle, the graph of $p$ is the complete graph on 3 nodes, and for $p$ in the top right region outside the circle, the graph has edges $(1,3), (3,1), (2,3)$ and $(3,2)$ .", "These graphs are all strongly connected, thus for each such $p$ , outcome functions with IC payment $p$ is RE.", "On the other hand, for $p$ in the dotted triangular region to the left of the circle, the graph has edges $(2,3)$ and $(3,2)$ .", "In Figure REF (b), we show two such $p$ as black dots, their max-plus hyperplanes in dotted lines.", "By Theorem REF part (1) and REF part (4), the set of IC payments for outcome functions with IC payment $p$ has dimension 1.", "In particular, such outcome functions are not RE.", "So $T$ is not RE.", "One can also verify the dimension of the basic cells via finite approximation, as done in Example REF .", "Figure: Demonstration of weak monotonicity, lack of incentive compatibility, on a revenue equivalent domain.", "Figure accompanies Example .Example 4.7 (Weak monotonicity, IC and an RE type space) In Figure REF , the set $T$ consists of all points in the region shaded gray.", "This type space $T$ is RE, but not all weakly monotone outcome functions on $T$ are IC, on convex type spaces this is cannot happen cf. [23].", "Panel (a) defines a weakly monotone outcome function via $L^g$ .", "Weak monotonicity follows from part (2) of Theorem REF , since the open sectors $\\overline{\\mathcal {L}}_1^\\circ , \\overline{\\mathcal {L}}_2^\\circ , \\overline{\\mathcal {L}}_3^\\circ $ are pairwise disjoint.", "Panel (b) verifies that $\\mathsf {Eig}_0(L^g) = \\emptyset $ , so $g$ is not IC, by Theorem REF part (3).", "The basic region of $T$ is all of $\\mathbb {TP}^2$ .", "By considerations of various max-plus hyperplanes, such as the one depicted in Figure REF (c), one can confirm that $T$ is RE.", "Note that the closure of this type space is not path-connected, nor `boundedly grid-wise connected', which are other known and easily verified sufficient conditions for a type space to be RE, cf.", "[4]." ], [ "Proofs of Theorem ", "Suppose $L$ is realized by $g$ .", "For each $j = 1, 2, \\ldots , m$ , $g^{-1}(j) \\subseteq \\overline{\\mathcal {L}}_j$ , since $\\max _{k \\in [m]} (L_{jk} + t_k) = L_j \\overline{\\odot } t = t_j$ .", "The sets $g^{-1}(1), \\ldots , g^{-1}(m)$ partition $T$ , so $T \\subseteq \\bigcup _{j=1}^m \\overline{\\mathcal {L}}_j$ .", "It remains to show that $L$ separates $T$ at an arbitrary pair $\\lbrace j,k\\rbrace $ , where $j, k \\in [m], j\\ne k$ .", "We have that $\\mathsf {d}(T \\cap \\mathcal {\\overline{L}}_j, \\mathcal {\\overline{L}}_{jk})=0$ since $L_{jk} = \\inf _{t \\in g^{-1}(j)}\\lbrace t_j - t_k\\rbrace $ .", "If $\\mathcal {I}_{jk} = \\mathcal {I}_{kj} = \\lbrace s\\rbrace $ , then if $g(s) = j$ , the infimum in the definition of $L_{kj}$ must be achieved by a $(k,j)$ -witness, so there must exist a $(k,j)$ -witness.", "Conversely, if $g(s) = k$ , then there must exist a $(j,k)$ -witness.", "This shows that $L$ separates $T$ at $\\lbrace j,k\\rbrace $ , as desired.", "For the converse direction, suppose $L$ is a matrix with zero diagonal such that $T \\subseteq \\bigcup _{j=1}^m \\overline{\\mathcal {L}}_j$ and $L$ separates $T$ .", "Define $g: T \\rightarrow [m]$ as follows.", "For a point $t \\in T \\cap \\overline{\\mathcal {L}}_j^\\circ $ , let $g(t) = j.$ The remaining points must lie on $\\bigcup _{j,k \\in [m], j \\ne k}\\mathcal {I}_{jk}$ by definition of $\\mathcal {I}_{jk}$ 's.", "Assign these points such that points on $\\mathcal {I}_{jk}$ have either outcome $j$ or $k$ , and such that on every non-empty boundary $\\mathcal {I}_{jk}$ there exists a point with outcome $j$ .", "The only case where this cannot be done is if $\\mathcal {I}_{jk} = \\mathcal {I}_{kj} = \\lbrace s\\rbrace $ .", "In this case, if there is a $(j,k)$ -witness, set $g(s) = k$ , else, since $L$ separates $T$ , there must exists a $(k,j)$ -witness, so set $g(s) = j$ .", "We claim that $L$ is realized by $g$ .", "Fix $j, k \\in [m], j\\ne k$ .", "By definition of $g^{-1}(j)$ , $\\inf _{s \\in g^{-1}(j)} (s_j - s_k) \\ge L_{jk}.$ Furthermore, there must be a point in $\\mathcal {I}_{jk}$ or a $(j,k)$ -witness in $g^{-1}(j)$ .", "So (REF ) holds with an equality, thus $L = L^g$ , as claimed.", "Fix a pair of indices $j,k \\in [m]$ , $j \\ne k$ .", "We claim that $L_{jk} + L_{kj} > 0 &\\Leftrightarrow \\overline{\\mathcal {L}}_j \\cap \\overline{\\mathcal {L}}_k = \\emptyset , \\\\L_{jk} + L_{kj} = 0 &\\Leftrightarrow \\overline{\\mathcal {L}}_j \\cap \\overline{\\mathcal {L}}_k \\mbox{ on their boundaries, i.e. }", "\\overline{\\mathcal {L}}_j \\cap \\overline{\\mathcal {L}}_k = \\partial \\overline{\\mathcal {L}}_j \\cap \\partial \\overline{\\mathcal {L}}_k \\\\L_{jk} + L_{kj} < 0 & \\Leftrightarrow \\overline{\\mathcal {L}}_j \\cap \\overline{\\mathcal {L}}_k \\mbox{ in their interiors, i.e.", "}\\overline{\\mathcal {L}}^\\circ _j \\cap \\overline{\\mathcal {L}}^\\circ _k \\ne \\emptyset .$ This claim implies statement (4) by definition of weakly monotone.", "Now let us prove the claim.", "Note that $\\overline{\\mathcal {L}}_j$ and $\\overline{\\mathcal {L}}_k$ are closed polyhedra, so they either do not intersect, intersect on their boundaries, or intersect in their interiors.", "So it is sufficient to prove the last two equivalences in the statements.", "Suppose there exists $t \\in \\overline{\\mathcal {L}}_j \\cap \\overline{\\mathcal {L}}_k$ .", "Then by definition of the sectors, $ L_{jk} + L_{kj} \\le (t_j - t_k) + (t_k - t_j) = 0.", "$ In particular, strict equality holds if and only if $t$ lies on the boundary of $\\overline{\\mathcal {L}}_j \\cap \\overline{\\mathcal {L}}_k$ , while strict inequality holds if and only if $t$ lies in either of their interiors.", "In that case, one can pick a $t^{\\prime } \\in \\overline{\\mathcal {L}}^\\circ _j \\cap \\overline{\\mathcal {L}}^\\circ _k$ , then $L_{jk} + L_{kj} < (t^{\\prime }_j - t^{\\prime }_k) + (t^{\\prime }_k - t^{\\prime }_j) = 0$ .", "This proves the desired statement.", "By definition, $\\mathsf {Eig}_0(L) = \\lbrace p \\in \\mathbb {TP}^{m-1}: L \\odot p = p\\rbrace .$ Part (3) of the theorem follows from uniqueness of the tropical eigenvalue [8].", "For part (4), by Proposition REF , $\\mathsf {Eig}_0(L^g) = \\mathcal {P}(\\nu (g))$ , where $\\nu (g) = \\underline{\\mathsf {coVec}}_T(p)$ .", "Note that $\\mathsf {d}(\\overline{\\mathcal {H}}_{ij}(-p), T\\cap \\overline{\\mathcal {L}}_i) = \\left(\\inf _{t \\in g^{-1}(i)} (t_i - t_j)\\right) - (p_i-p_j) = L^g_{ij} - (p_i - p_j).$ Therefore, the graph of $p$ has edge $(i,j)$ if and only if $p_i - p_j = L^g_{ij}$ .", "Form the matrix $L^{\\prime }$ via $L^{\\prime }_{ij} = L^g_{ij} - (p_i - p_j)$ .", "Then $L^{\\prime }$ has eigenvalue 0, and any cycle in the graph of $p$ is a zero-length cycle.", "In tropical linear systems terminology, the graph of $p$ is the critical graph of $L^{\\prime }$ .", "It follows from [5] that the dimension of the eigenspace of $L^{\\prime }$ equals the number of connected components of $L^{\\prime }$ .", "But $\\mathsf {Eig}(L^{\\prime }) = -p + \\mathsf {Eig}(L^g)$ , so $\\mathsf {Eig}(L^g)$ has the same dimension.", "This proves the theorem.", "This follows immediately from Theorem REF , part (4).", "If $T$ is RE, then for every $p \\in \\operatornamewithlimits{\\mathsf {basic}}(T)$ , its graph equals the graph of $L^g$ for some RE mechanism $(g,p)$ , which is connected.", "Conversely, suppose the graph of every $p$ in $\\operatornamewithlimits{\\mathsf {basic}}(T)$ is strongly connected.", "Take a IC mechanism $(g,p)$ , with $p$ in the relative interior of $\\mathcal {P}(g)$ .", "As its graph is connected, $\\mathsf {Eig}_0(L^g) = \\lbrace p\\rbrace $ , so $(g,p)$ is RE." ], [ "Acknowledgements", "We wish to thank John Weymark for suggesting to explore possible connections of mechanism design to tropical geometry.", "Our geometric approach is inspired by his work with coauthors in [7].", "Part of this work was written during the Hausdorff Center Conference “Tropical Geometry and Economics”.", "We thank the participants for helpful discussions.", "The Hausdorff Center's role in catalyzing research in mathematical economics is gratefully acknowledged.", "Finally we thank Paul Edelman and John Weymark for pointing out an error in an earlier version, and Alexey Kushnir for generously sharing his manuscript." ] ]
1606.04880
[ [ "Verification of the helioseismic Fourier-Legendre analysis for\n meridional flow measurements" ], [ "Abstract Measuring the Sun's internal meridional flow is one of the key issues of helioseismology.", "Using the Fourier-Legendre analysis is a technique for addressing this problem.", "We validate this technique with the help of artificial helioseismic data.", "The analysed data set was obtained by numerically simulating the effect of the meridional flow on the seismic wave field in the full volume of the Sun.", "In this way, a 51.2-hour long time series was generated.", "The resulting surface velocity field is then analyzed in various settings: Two $360^\\circ \\times 90^\\circ$ halfspheres, two $120^\\circ \\times 60^\\circ$ patches on the front and farside of the Sun (North and South, respectively) and two $120^\\circ \\times 60^\\circ$ patches on the northern and southern frontside only.", "We compare two possible measurement setups: observations from Earth and from an additional spacecraft on the solar farside, and observations from Earth only, in which case the full information of the global solar oscillation wave field was available.", "We find that, with decreasing observing area, the accessible depth range decreases: the $360^\\circ \\times 90^\\circ$ view allows us to probe the meridional flow almost to the bottom of the convection zone, while the $120^\\circ \\times 60^\\circ$ view means only the outer layers can be probed.", "These results confirm the validity of the Fourier-Legendre analysis technique for helioseismology of the meridional flow.", "Furthermore these flows are of special interest for missions like Solar Orbiter that promises to complement standard helioseismic measurements from the solar nearside with farside observations." ], [ "Introduction", "Determining the Sun's internal meridional flow is one of the main challenges of solar physics.", "While the meridional flow amplitude at the solar surface is rather easy to measure from feature tracking  ([26], [14]) and the established local helioseismic methods allow us to probe for the flow in shallow sub-surface layers ([9], [30], [6]), a significantly deeper helioseismic probing of the flow encounters difficulties.", "Since better knowledge about the structure of the meridional flow is desirable for input in to flux-transport dynamo models ([3]), which are essential for understanding the Sun's magnetic activity cycle, it is the deep meridional flow that is of particular importance.", "Only recently, improved techniques of time-distance helioseismology [28] and a new approach of global helioseismology ([25]) have provided first indications of the structure of the deep meridional flow.", "However, today's helioseismic techniques rely on observations of the frontside of the Sun only.", "Within the coming years, this type of data might be complemented by missions like the Solar Orbiter, which provides seismic data from the solar farside.", "In this paper we focus on inferring the Sun's internal meridional flow by using the Fourier-Legendre decomposition (FLD) analysis, which is a further development of the concept proposed by [2].", "We employ numerical simulations to create artificial helioseismic data.", "These data are used to validate the FLD technique and to assess the accuracy of inferences that could be expected from different observational setups.", "The paper is organized as follows: Section 2 describes the numerical simulations, including the flow model and the Fourier-Legendre technique as it is applied to various observational settings; Section 3 presents the results; the conclusions are given in Section 4." ], [ "Methods", "Our study rests on analysing artificial helioseismic data, i.e.", "a time series of a numerically obtained wavefield in the interior of the Sun, which is affected by a large-scale meridional flow." ], [ "Numerical simulation", "The generation of artificial helioseismic data is identical to the method described in [12], which makes use of a numerical code that solves the linearized propagation of helioseismic waves throughout the entire solar interior ([11]).", "This code has been used in previous studies to simulate the effects of various localized sound-speed perturbations, e.g.", "for testing helioseismic farside imaging by simulating the effects of model sunspots on the acoustic field ([13], [15]), for validating time -distance helioseismic measurements of tachocline perturbations ([29]), and for studying the effect of localized subsurface perturbations ([10]).", "As described in [12], the code has been extended to include the effects of mass flows onto the propagation of helioseismic waves.", "In short, the code models solar acoustic oscillations in a spherical domain by using the Euler equations that are linearized around a stationary background state.", "As can be seen in [12], the equations are formulated in a non-rotating frame, and so rotation is accounted for by prescribing an appropriate flow.", "This approach saves computing the usual Coriolis and centrifugal forces that appear in the equations for a rotating frame.", "The excitation of acoustic waves, which is a non-linear process, is not included in the linearized equations.", "Instead, we mimic solar acoustic wave excitation by including in the momentum equation a random function that is only non-zero close to the solar surface.", "Perturbations of the gravitational potential have been neglected and the adiabatic approximation has been used.", "We only consider waves with constant entropy.", "This way, the entropy gradient of the background model does not enter our equations and the model becomes convectively stable.", "Non-reflecting boundary conditions are applied at the upper boundary by means of an absorbing buffer layer with a damping coefficient that is zero in the interior and increases smoothly into the buffer layer.", "A small amount of viscous damping was added for stability reasons.", "The simulation in this study resolves spherical harmonics of angular degree between 0 to 170.", "A staggered Yee scheme ([27]) is used for time integration, with a time step of one second." ], [ "Flow model", "For the background flow, we have imposed a stationary model of the solar meridional circulation according to [12].", "The flow model described there is based on the flow model in [21].", "The meridional flow is a single-cell circulation in each of the meridional quadrants.", "As described in [12], the amplitude was chosen so that the resulting meridional flow has a maximum velocity of 500 m/s.", "But, since these simulations are linear, this kind of an increase does not change the physics but merely increases the signal-to-noise ratio.", "Hence, a simulated time series of 51.2 h length has the same signal-to-noise ratio as a 3.65 year-long time series with realistic speeds." ], [ "Fourier-Legendre decomposition", "The FLD method is the extension to spherical geometry of the Fourier-Hankel decomposition (FHD) which has been used in the past to study wave absorption in sunspots  [1].", "It is a helioseismic technique that is also suited for the measurement of the meridional flow [2].", "Because the FLD method is sensitive to low-degree oscillation modes, it promises probing the average meridional flow in deep layers.", "An early version of this FLD analysis pipeline was tested on GONG (Global Oscillation Network Group) data by [5], who successfully compared near-surface measurements of the meridional flow with results obtained by ring-diagram analyis.", "The time dependent, two-dimensional oscillation signal on the solar surface $V(\\theta ,\\phi ,t)$ can be expressed in terms of a superposition of two wavefields travelling pole- and equatorwards, respectively: $V(\\theta ,\\phi ,t)=\\sum _{nlm} \\left[ A_{lm}H_m^{(1)}(L\\theta )+B_{lm}H_m^{(2)}(L\\theta )\\right]{\\rm e}^{{\\rm i}m\\phi +\\omega _{nl}t}$ with values of $m\\approx 0$ .", "Here $\\theta $ denotes the co-latitude and $\\phi $ the longitude on the solar surface, $\\omega _{nl}$ is the angular frequency of a mode with harmonic degree $l$ , $L=[l(l+1)]^{1/2}$ , and azimuthal order $m$ .", "The quantitities $H_m^{(1,2)}$ are travelling wave associated Legendre functions, described in analogy to [19]: $H_{m}^{(1,2)}= (-1)^m \\frac{(l-m)!}{(l+m)!", "}\\left[P_l^m(\\cos \\theta \\pm \\frac{2\\rm i}{\\pi }Q_l^m(\\cos \\theta )\\right]\\ ,$ where $P_l^m$ and $Q_l^m$ are associated Legendre functions of first and second kind, respectively.", "Due to advection, a meridional flow will result in a frequency shift $\\Delta \\nu _{nl}$ between the poleward and equatorward components [2], [8] $\\Delta \\nu _{nl}=\\frac{l}{\\pi R_\\odot }\\int \\bar{U}_{\\rm mer}(r) K_{nl}(r)\\, \\mathrm {d}r\\ ,$ where $\\bar{U}_{\\rm mer}(r)$ is the averaged meridional flow over the observed region of interest as a function of the position $r$ inside the Sun.", "The sensitivity kernel $K_{nl}(r)$ is the kinetic energy density of a given mode $(n,l)$ which is also a function of the position $r$ inside the Sun.", "This frequency shift $\\Delta \\nu _{nl}$ can be measured from the power spectra of the two series, $A_{lm}$ and $B_{lm}$ , by fitting asymmetric Lorentzian profiles to the individual peaks present.", "The guess frequencies for the fits are obtained from the standard solar Model S ([4]).", "We then employ the SOLA (subtractive optimally localized averages) inversion technique ([20]) to construct localized average inversion kernels at a given depth $r$ ." ], [ "Data analysis", "The simulation results have been tracked with the mean rotation, i.e.", "transformed to a corotating reference frame with the mean rotation period.", "The surface velocity field was projected on a heliographic grid and the region of interest is cut out.", "We study three different observational scenarios: The northern and southern hemispheres, i.e.", "two patches with angular dimensions of $360^\\circ \\times 90^\\circ $ in longitude and latitude, respectively.", "This corresponds to an ideal observational setup, where data from the whole solar surface is available for helioseismic analysis.", "Four patches, each with the dimensions $120^\\circ \\times 60^\\circ $ .", "Two of the patches are on the northern and southern frontside, and two are $180^\\circ $ in longitude apart on the farside, respectively.", "The patches are centred at $\\pm 35^\\circ $ latitude, respectively.", "This corresponds to combined observations in the ecliptic from, e.g.", "Earth and from a spacecraft on the farside of the Sun, as might be possible with missions like the Solar Orbiter [18], [22].", "Two patches on the frontside only, with the dimensions $120^\\circ \\times 60^\\circ $ .", "The patches are located at $\\pm 35^\\circ $ latitude.", "This setup corresponds to the standard case of today's helioseismic observations with one viewpoint only.", "For all three cases, the artificial Dopplergrams are fed into the Fourier-Legendre decomposition procedure as described above.", "This yields time series $A_{lm}(t)$ and $B_{lm}(t)$ , from which power spectra $P_{lm}^{(A)}$ and $P_{lm}^{(B)}$ of north- and southwards travelling waves are generated for each hemisphere, respectively.", "These power spectra are then averaged over the range of azimuthal orders $m=-25,\\dots ,25$ .", "This is meant to include a sufficient number of modes that are affected by the meridional flow, as they have a pole- or equatorward directed component of the wave vector.", "Then, the m-averaged power spectra are smoothed by applying a running boxcar window over three frequency bins.", "The peaks in the averaged power spectra are then fitted using asymmetric Lorentzian profiles to determine the individual central mode frequencies $\\nu _{nl}^{(A)}$ and $\\nu _{nl}^{(B)}$ .", "Finally, frequency differences $\\delta \\nu _{nl}=\\nu _{nl}^{(A)}-\\nu _{nl}^{(B)}$ are calculated.", "The set of frequency differences is then inverted for the meridional flow, as described above." ], [ "Meridional flow measurements", "Figure REF displays the main results of this study.", "In the first case, where both hemispheres of the Sun could be observed, the input meridional flow can be well recovered (Fig.", "REF (top)) down to a depth of 110 Mm.", "This case corresponds to an ideal observational setup, i.e.", "the oscillation signal is available for the full solar globe.", "In the second case, where only part of the solar surface could be observed, i.e.", "two patches per hemisphere, one on the front- the other on the farside of the Sun, the flow could still be recovered well.", "The inversion result agrees well with the input model down to about 30 Mm below the solar surface.", "Below that, the inversion result deviates significantly from the input flow.", "In the third case, where only frontside observations are available, the flow can be reliably recovered down to only 50 Mm.", "At deeper layers the inversion results deviate strongly from the input flow.", "We note that, since the numerical data included only low degree modes, the surface flow was not recovered in all settings.", "The flow measurements were possible at depths greater than 10 Mm." ], [ "Power leaks", "According to theoretical forward calculations by [23] and [7], the frequency shifts introduced by a meridional flow should be negligible.", "Given the results by [7], the meridional flow should introduce a power redistribution among affected modes.", "In fact, this power redistribution comes from the coupling of modes owing to advection, which affects the eigenfunctions ([17], [24]).", "This is an effect that is evaluated in global helioseismology to measure the meridional flow ([25].", "Nevertheless, in this study, we successfully make us of measurable frequency shifts.", "By studying the power spectra of pole- and equatorwards travelling waves in detail, it becomes obvious that the modes are actually not measurably shifted and that this frequency shift must be spurious.", "Two effects need to be considered.", "First, due to the fact that only parts of the solar surface are used for Fourier-Legendre decomposition, the spherical harmonics are no longer orthogonal on these reduced areas of the solar surface.", "As a result, spatial leaks appear.", "Fig.", "REF shows the power spectra for our three observational setups.", "As visible in Fig.", "REF , this number of leaks increases with decreasing observing area, i.e.", "the central peak amplitude decreases and power leaks into a broad range of side lobes.", "Second, the coupling of modes, which are caused by advection, introduces a number of side lobes i.e.", "power leaks, in the mode spectra, too.", "By comparing pole- and equatorward propagating waves, it can be seen that the individual central mode peak and its leaks do not change their position on the frequency axis.", "But, owing to the presence of the meridional flow, power is redistributed among the leaks in a way that the weight of the whole set of coupled modes is shifted in frequency.", "Consequently, smoothing the power spectra over frequency and fitting a common asymmetric Lorentzian profile to the resulting broad peak must yield shifted frequencies.", "This apparent frequency difference between pole- and equatorward travelling waves can then be employed for inversions.", "Making use of the mode kinetic energy, however, needs to be considered as an approximation to proper sensitivity kernels, which are not yet available for this technique.", "Figure: Exemplary power spectra averaged over azimuthal orders m=-25,⋯,25m=-25,\\dots ,25 for the poleward (solid) and equatorward (dashed) travelling waves with harmonic degree l=90l=90 and radial order n=7n=7.", "From top to bottom, the viewing area decreases from 360 ∘ ×90 ∘ 360^\\circ \\times 90^\\circ , over 2×120 ∘ ×60 ∘ 2\\times 120^\\circ \\times 60^\\circ , to 120 ∘ ×60 ∘ 120^\\circ \\times 60^\\circ .", "Power is normalized with the same factor in all three plots." ], [ "Conclusion", "We have validated the Fourier-Legendre decomposition technique as a means of measuring the meridional flow.", "For this we employed artificial helioseismic data.", "The technique was applied to three settings that mimic observation scenarios.", "These range from ideal conditions of full surface observations of the oscillation signal to the grade of scenarios, which are more compatible with observations that are technically possible today, i.e.", "observing parts of the solar hemisphere on the front- and farside only.", "As a main result, we were able to demonstrate that the technique successfully recovers the meridional flow that was included in the simulation.", "This works better when more of the solar surface is available for helioseismic analysis.", "We conclude that in the case that one hemisphere can be fully observed, a meridional flow within of 20 m/s amplitude could be inferred by a 3.65-year long time series with this technique, if the results from the 51.2 hours of artificial data are comparable to 3.65 years of observational data, which is not, a priori, clear, especially as helioseismic techniques work very well on simulations that are linear or convectively stable.", "More advanced simulations should therefore be considered in future for testing this technique, too.", "We note that the technique makes use of an apparent frequency shift, which comes from the fact that power is redistributed between modes that are coupled by the meridional flow.", "This creates asymmetric weights in the power distribution of a mode and its leaks that are close-by in frequency, which can be interpreted as a frequency shift.", "Therefore, we can confirm the theoretical results obtained by [7], but are still able to use the technique to measure the meridional flow down to great depths.", "This works as long as the individual power leaks are not resolved in frequency because then the mode kinetic energy can serve as an approximation of the proper sensitivity kernels for this technique.", "The derivation of these kernels would be part of a future study.", "Thus, making use of the power redistribution would therefore be the next step in developing this technique further because, accounting for the leakage of modes promises a better recovery of the flow at greater depths.", "As a consequence, the results of the FLD analysis will be more comparable to other methods, e.g.", "time-distance helioseismology.", "For example,  [16] analyze this simulation too, using the time-distance technique and obtain more accurate inversion results in deeper layers.", "Still, the method promises advantages in its capability of easy incorporation of data from multiple vantage points, which is a result from the applicability of the method to various patch geometries, in contrast to the ring-diagram technique.", "Concerning multiple vantage points, we conclude that the meridional flow could be detected down to a large percentage of the convection zone using the Fourier-Legendre analysis technique when observational data from front- and farside observations are combined.", "Therefore our study may be interesting when planning upcoming space missions, e.g.", "Solar Orbiter.", "M.R.", "thanks D. Braun for useful discussions.", "The authors thank the unknown referee for valuable comments on the manuscript.", "The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Program (FP/2007-2013) / ERC Grant Agreement no.", "307117." ] ]
1606.05202
[ [ "Chebyshev's bias for products of $k$ primes" ], [ "Abstract For any $k\\geq 1$, we study the distribution of the difference between the number of integers $n\\leq x$ with $\\omega(n)=k$ or $\\Omega(n)=k$ in two different arithmetic progressions, where $\\omega(n)$ is the number of distinct prime factors of $n$ and $\\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity .", "Under some reasonable assumptions, we show that, if $k$ is odd, the integers with $\\Omega(n)=k$ have preference for quadratic non-residue classes; and if $k$ is even, such integers have preference for quadratic residue classes.", "This result confirms a conjecture of Richard Hudson.", "However, the integers with $\\omega(n)=k$ always have preference for quadratic residue classes.", "Moreover, as $k$ increases, the biases become smaller and smaller for both of the two cases." ], [ "Introduction and statement of results", "First, we consider products of $k$ primes in arithmetic progressions.", "Let $\\pi _k(x; q, a)=|\\lbrace n\\le x: \\omega (n)=k, ~n \\equiv a \\bmod q\\rbrace |,$ and $N_k(x; q, a)=|\\lbrace n\\le x: \\Omega (n)=k, ~n \\equiv a \\bmod q\\rbrace |,$ where $\\omega (n)$ is the number of distinct prime divisors of $n$ , and $\\Omega (n)$ is the number of prime divisors of $n$ counted with multiplicity.", "For example, when $k=1$ , $N_1(x; q, a)$ is the number of primes $\\pi (x; q, a)$ in the arithmetic progression $a \\bmod q$ ; and $\\pi _1(x; q, a)$ counts the number of prime powers $p^l\\le x$ for all $l\\ge 1$ in the arithmetic progression $a \\bmod q$ .", "Dirichlet (1837) [3] showed that, for any $a$ and $q$ with $(a, q)=1$ , there are infinitely many primes in the arithmetic progression $a \\bmod q$ .", "Moreover, for any $(a, q)=1$ , $\\pi (x; q, a)\\sim \\frac{x}{\\phi (q) \\log x},$ where $\\phi $ is Euler's totient function [2].", "Analogous asymptotic formulas are available for products of $k$ primes.", "Landau (1909) [12] showed that, for each fixed integer $k\\ge 1$ , $N_k(x):=\\left| \\lbrace n\\le x: \\Omega (n)=k \\rbrace \\right|\\sim \\frac{x}{\\log x}\\frac{(\\log \\log x)^{k-1}}{(k-1)!", "}.$ The same asymptotic is also true for the function $\\pi _k(x):=| \\lbrace n\\le x: \\omega (n)=k \\rbrace |$ .", "For more precise formulas, see [19] (II.", "6, Theorems 4 and 5).", "Using similar methods as in [2] and [19], one can show that, for any fixed residue class $a \\bmod q$ with $(a, q)=1$ , $N_k(x; q, a)\\sim \\pi _k(x; q, a)\\sim \\frac{1}{\\phi (q)} \\frac{x}{\\log x} \\frac{(\\log \\log x)^{k-1}}{(k-1)!", "}.$ For the case of counting primes ($\\Omega (n)=1$ ), Chebyshev (1853) [1] observed that there seem to be more primes in the progression $3\\bmod 4$ than in the progression $1\\bmod 4$ .", "That is, it appears that $\\pi (x; 4,3)\\ge \\pi (x; 4, 1)$ .", "In general, for any $a\\lnot \\equiv b \\bmod q$ and $(a, q)=(b, q)=1$ , one can study the behavior of the functions $\\Delta _{\\omega _k}(x; q, a, b)&:=\\pi _k(x; q, a)-\\pi _k(x; q, b),\\\\\\Delta _{\\Omega _k}(x; q, a, b)&:=N_k(x; q, a)-N_k(x; q, b).$ Denote $\\Delta (x; q, a, b):=\\Delta _{\\Omega _1}(x; q, a, b)$ .", "Littlewood [15] proved that $\\Delta (x; 4, 3, 1)$ changes sign infinitely often.", "Actually, $\\Delta (x; 4, 3, 1)$ is negative for the first time at $x=26, 861$ [14].", "Knapowski and Turán published a series of papers starting with [10] about the sign changes and extreme values of the functions $\\Delta (x; q, a, b)$ .", "And such problems are colloquially known today as \"prime race problems\".", "Irregularities in the distribution, that is, a tendency for $\\Delta (x; q, a, b)$ to be of one sign is known as \"Chebyshev's bias\".", "For a nice survey of such works, see [5] and [7].", "Chebyshev's bias can be well understood in the sense of logarithmic density.", "We say a set $S$ of positive integers has logarithmic density, if the following limit exists: $\\delta (S)=\\lim _{x\\rightarrow \\infty } \\frac{1}{\\log x} \\sum _{\\begin{array}{c}n\\le x\\\\ n\\in S\\end{array}} \\frac{1}{n}.$ Let $\\delta _{f_k}(q; a, b)=\\delta (P_{f_k}(q; a, b))$ , where $P_{f_k}(q; a, b)$ is the set of integers with $\\Delta _{f_k}(n; q, a, b)>0$ , and $f=\\Omega {~\\rm or~} \\omega $ .", "In order to study the Chebyshev's bias and the existence of the logarithmic density, we need the following assumptions: 1) the Extended Riemann Hypothesis (${\\rm ERH_q}$ ) for Dirichlet L-functions modulo $q$ ; 2) the Linear Independence conjecture (${\\rm LI_q}$ ), the imaginary parts of the zeros of all Dirichlet L-functions modulo $q$ are linearly independent over $\\mathbb {Q}$ .", "Under these two assumptions, Rubinstein and Sarnak [18] showed that, for Chebyshev's bias for primes ($\\Omega (n)=1$ ), the logarithmic density $\\delta _{\\Omega _1}(q;a, b)$ exists, and in particular, $\\delta _{\\Omega _1}(4; 3, 1)\\approx 0.996$ which indicates a strong bias for primes in the arithmetic progression $3 \\bmod 4$ .", "Recently, using the same assumptions, Ford and Sneed [6] studied the Chebyshev's bias for products of two primes with $\\Omega (n)=2$ by transforming this problem into manipulations of some double integrals.", "They connected $\\Delta _{\\Omega _2}(x; q, a, b)$ with $\\Delta (x; q, a, b)$ , and showed that $\\delta _{\\Omega _2}(q; a, b)$ exists and the bias is in the opposite direction to the case of primes, in particular, $\\delta _{\\Omega _2}(4; 3, 1)\\approx 0.10572$ which indicates a strong bias for the arithmetic progression $1 \\bmod 4$ .", "By orthogonality of Dirichlet characters, we have $\\Delta _{\\Omega _k}(x; q, a, b)=\\frac{1}{\\phi (q)} \\sum _{\\chi \\ne \\chi _0 \\bmod q} (\\overline{\\chi }(a)-\\overline{\\chi }(b))\\sum _{\\begin{array}{c}n\\le x\\\\ \\Omega (n)=k\\end{array}}\\chi (n),$ and $\\Delta _{\\omega _k}(x; q, a, b)=\\frac{1}{\\phi (q)} \\sum _{\\chi \\ne \\chi _0 \\bmod q} (\\overline{\\chi }(a)-\\overline{\\chi }(b))\\sum _{\\begin{array}{c}n\\le x\\\\ \\omega (n)=k\\end{array}}\\chi (n).$ The inner sums over $n$ are usually analyzed using analytic methods.", "Neither the method of Rubinstein and Sarnak [18] nor the method of Ford and Sneed [6] readily generalizes to handle the cases of more prime factors ($k\\ge 3$ ).", "From the point of view of $L$ -functions, the most natural sum to consider is $\\sum _{\\begin{array}{c}n_1\\cdots n_k\\le x\\\\n_1\\cdots n_k \\equiv a \\bmod q\\end{array}} \\Lambda (n_1)\\cdots \\Lambda (n_k).$ However, estimates for $\\Delta _{\\Omega _k}(x; q, a, b)$ or $\\Delta _{\\omega _k}(x; q, a, b)$ cannot be readily recovered from such an analogue by partial summation.", "Ford and Sneed [6] overcome this obstacle in the case $k=2$ by means of the 2-dimensional integral $\\int _0^\\infty \\int _0^\\infty \\sum _{p_1 p_2\\le x} \\frac{\\chi (p_1 p_2)\\log p_1 \\log p_2}{p_1^{u_1} p_2^{u_2}} du_1 du_2.$ Analysis of an analogous $k$ -dimensional integral leads to an explosion of cases, depending on the relative sizes of the variables $u_j$ , and becomes increasingly messy as $k$ increases.", "We take an entirely different approach, working directly with the unweighted sums.", "We express the associated Dirichlet series in terms of products of the logarithms of Dirichlet $L$ -functions, then apply Perron's formula, and use Hankel contours to avoid the zeros of $L(s, \\chi )$ and the point $s=\\frac{1}{2}$ .", "Using the same assumptions 1) and 2), we show that, for any $k\\ge 1$ , both $\\delta _{\\Omega _k}(q; a, b)$ and $\\delta _{\\omega _k}(q; a, b)$ exist.", "Moreover, we show that, as $k$ increases, if $a$ is a quadratic non-residue and $b$ is a quadratic residue, the bias oscillates with respect to the parity of $k$ for the case $\\Omega (n)=k$ , but $\\delta _{\\omega _k}(q; a, b)$ increases from below $\\frac{1}{2}$ monotonically.", "For some of our results, we need only a much weaker substitute for condition ${\\rm LI_q}$ , which we call the Simplicity Hypothesis (${\\rm SH_q}$ ): $\\forall \\chi \\ne \\chi _0 \\bmod q$ , $L(\\frac{1}{2}, \\chi )\\ne 0$ and the zeros of $L(s, \\chi )$ are simple.", "Let $N(q, a):=\\#\\lbrace u \\bmod q: u^2\\equiv a \\bmod q\\rbrace .$ Then, using the weaker assumptions ${\\rm SH_q}$ and ${\\rm ERH_q}$ , we prove the following theorems.", "Theorem 1 Assume $ERH_q$ and ${SH_q}$ .", "Then, for any fixed $k\\ge 1$ , and fixed large $T_0$ , $\\Delta _{\\Omega _k}(x; q, a, b)&=\\frac{1}{(k-1)!}", "\\frac{\\sqrt{x} (\\log \\log x)^{k-1}}{\\log x}\\Bigg \\lbrace \\frac{(-1)^k}{\\phi (q)}\\sum _{\\chi \\ne \\chi _0}\\left(\\overline{\\chi }(a)-\\overline{\\chi }(b)\\right)\\sum _{\\begin{array}{c}|\\gamma _{\\chi }|\\le T_0\\\\L(\\frac{1}{2}+i\\gamma _{\\chi }, \\chi )=0\\end{array}} \\frac{x^{i\\gamma _{\\chi }}}{\\frac{1}{2}+i\\gamma _{\\chi }}\\nonumber \\\\&\\quad +\\frac{(-1)^k}{2^{k-1}}\\frac{N(q,a)-N(q,b)}{\\phi (q)} +\\Sigma _k(x; q, a, b, T_0)\\Bigg \\rbrace ,$ where $\\nonumber \\limsup _{Y\\rightarrow \\infty }\\frac{1}{Y} \\int _1^Y \\left|\\Sigma _k(e^y; q, a, b, T_0) \\right|^2 dy\\ll \\frac{\\log ^2 T_0}{T_0}.$ Since $\\Delta _{\\Omega _1}(x; q, a, b)=\\Delta (x; q, a, b)$ , we get the following corollary.", "Corollary 1.1 Assume $ERH_q$ and ${SH_q}$ .", "Then, for any fixed $k\\ge 2$ , $\\frac{\\Delta _{\\Omega _k}(x; q, a, b)\\log x}{\\sqrt{x} (\\log \\log x)^{k-1}}&=\\frac{(-1)^{k+1}}{(k-1)!}", "\\left(1-\\frac{1}{2^{k-1}} \\right) \\frac{N(q,a)-N(q, b)}{ \\phi (q)}\\nonumber \\\\&\\quad +\\frac{(-1)^{k+1}}{(k-1)!", "}\\frac{\\Delta (x; q, a, b) \\log x}{ \\sqrt{x}} +\\Sigma ^{\\prime }_k(x; q, a, b), $ where, as $Y\\rightarrow \\infty $ , $\\nonumber \\frac{1}{Y} \\int _1^Y |\\Sigma ^{\\prime }_k(e^y; q, a, b) |^2 dy=o(1).$ In the above theorem, the constant $\\frac{(-1)^k}{2^{k-1}}\\frac{N(q,a)-N(q,b)}{\\phi (q)}$ represents the bias in the distribution of products of $k$ primes counted with multiplicity.", "Richard Hudson conjectured that, as $k$ increases, the bias would change directions according to the parity of $k$ .", "Our result above confirms his conjecture (under $\\rm ERH_q$ and $\\rm SH_q$ ).", "Figures REF and REF show the graphs corresponding to $(q, a, b)=(4, 3, 1)$ for $\\frac{2\\log x}{\\sqrt{x}(\\log \\log x)^2}\\Delta _{\\Omega _3}(x; 4, 3, 1)$ and $\\frac{6\\log x}{\\sqrt{x} (\\log \\log x)^3}\\Delta _{\\Omega _4}(x; 4, 3, 1)$ , plotted on a logarithmic scale from $x=10^3$ to $x=10^8$ .", "In these graphs, the functions do not appear to be oscillating around $\\frac{1}{4}$ and $-\\frac{1}{8}$ respectively as predicted in our theorem.", "This is caused by some terms of order $\\frac{1}{\\log \\log x}$ and even lower order terms, and $\\log \\log 10^8\\approx 2.91347$ and $\\frac{1}{\\log \\log 10^8}\\approx 0.343233$ .", "However, we can still observe the expected direction of the bias through these graphs.", "For the distribution of products of $k$ primes counted without multiplicity, we have the following theorem.", "In this case, the bias will be determined by the constant $\\frac{N(q,a)-N(q,b)}{2^{k-1}\\phi (q)}$ in the theorem below.", "Theorem 2 Assume $ERH_q$ and ${SH_q}$ .", "Then, for any fixed $k\\ge 1$ , and fixed large $T_0$ , $\\Delta _{\\omega _k}(x; q, a, b)&=\\frac{1}{(k-1)!}", "\\frac{\\sqrt{x} (\\log \\log x)^{k-1}}{\\log x}\\Bigg \\lbrace \\frac{(-1)^k}{\\phi (q)}\\sum _{\\chi \\ne \\chi _0}\\left(\\overline{\\chi }(a)-\\overline{\\chi }(b)\\right)\\sum _{\\begin{array}{c}|\\gamma _{\\chi }|\\le T_0\\\\L(\\frac{1}{2}+i\\gamma _{\\chi }, \\chi )=0\\end{array}} \\frac{x^{i\\gamma _{\\chi }}}{\\frac{1}{2}+i\\gamma _{\\chi }}\\nonumber \\\\&\\quad +\\frac{N(q,a)-N(q,b)}{2^{k-1}\\phi (q)} +\\widetilde{\\Sigma }_k(x; q, a, b, T_0)\\Bigg \\rbrace ,$ where $\\nonumber \\limsup _{Y\\rightarrow \\infty } \\frac{1}{Y} \\int _1^Y \\left|\\widetilde{\\Sigma }_k(e^y; q, a, b, T_0) \\right|^2 dy\\ll \\frac{\\log ^2 T_0}{T_0}.$ Corollary 2.1 Assume $ERH_q$ and ${SH_q}$ .", "Then, for any fixed $k\\ge 1$ , $\\frac{\\Delta _{\\omega _k}(x; q, a, b)\\log x}{\\sqrt{x} (\\log \\log x)^{k-1}}&= \\left(\\frac{1}{2^{k-1}}+(-1)^{k+1} \\right) \\frac{N(q,a)-N(q, b)}{(k-1)!", "\\phi (q)}\\nonumber \\\\&\\quad +\\frac{(-1)^{k+1}}{(k-1)!", "}\\frac{\\Delta (x; q, a, b) \\log x}{ \\sqrt{x}} +\\widetilde{\\Sigma }^{\\prime }_k(x; q, a, b), $ where, as $Y\\rightarrow \\infty $ , $\\nonumber \\frac{1}{Y} \\int _1^Y |\\widetilde{\\Sigma }^{\\prime }_k(e^y; q, a, b) |^2 dy=o(1).$ Figure: 2logx x(loglogx) 2 Δ Ω 3 (x;4,3,1)\\frac{2\\log x}{\\sqrt{x}(\\log \\log x)^2}\\Delta _{\\Omega _3}(x; 4, 3, 1)Figure: 6logx x(loglogx) 3 Δ Ω 4 (x;4,3,1)\\frac{6\\log x}{\\sqrt{x}(\\log \\log x)^3}\\Delta _{\\Omega _4}(x; 4, 3, 1)For the distribution of $\\Delta (x; q, a, b)$ , Rubinstein and Sarnak [18] showed the following theorem.", "This is the version from [6].", "Theorem RS Assume $ERH_q$ and $LI_q$ .", "For any $a\\lnot \\equiv b \\bmod q$ and $(a, q)=(b, q)=1$ , the function $\\nonumber \\frac{u\\Delta (e^u; q, a, b)}{e^{u/2}}$ has a probabilistic distribution.", "This distribution i) has mean $\\frac{N(q, b)-N(q, a)}{\\phi (q)}$ , ii) is symmetric with respect to its mean, and iii) has a continuous density function.", "Corollaries REF , REF , and Theorem RS imply the following result.", "Theorem 3 Let $a\\lnot \\equiv b \\bmod q$ and $(a, q)=(b, q)=1$ .", "Assuming $ERH_q$ and $LI_q$ , for any $k\\ge 1$ , $\\delta _{\\Omega _k}(q; a, b)$ and $\\delta _{\\omega _k}(q; a, b)$ exist.", "More precisely, if $a$ and $b$ are both quadratic residues or both quadratic non-residues, then $\\delta _{\\Omega _k}(q; a, b)=\\delta _{\\omega _k}(q; a, b)=\\frac{1}{2}$ .", "Moreover, if $a$ is a quadratic non-residue and $b$ is a quadratic residue, then, for any $k\\ge 1$ , $1-\\delta _{\\Omega _{2k-1}}(q; a, b)<\\delta _{\\Omega _{2k}}(q; a, b)< \\frac{1}{2}<\\delta _{\\Omega _{2k+1}}(q; a, b)<1-\\delta _{\\Omega _{2k}}(q; a, b),$ $\\delta _{\\omega _{k}}(q; a, b)<\\delta _{\\omega _{k+1}}(q; a, b)<\\frac{1}{2},$ $\\delta _{\\Omega _{2k}}(q; a, b)=\\delta _{\\omega _{2k}}(q; a, b), \\qquad \\delta _{\\Omega _{2k-1}}(q; a, b)+\\delta _{\\omega _{2k-1}}(q; a, b)=1.$ Remark 1.", "The above results confirm a conjecture of Richard Hudson proposed years ago in his communications with Ford.", "Borrowing the methods from [18] (Section 4), we are able to calculate $\\delta _{\\Omega _k}(q; a, b)$ and $\\delta _{\\omega _k}(q; a, b)$ precisely for special values of $q$ , $a$ , and $b$ .", "In particular, we record in Tables REF and REF the logarithmic densities up to products of 10 primes for two cases: $q=3$ , $a=2$ , $b=1$ , and $q=4$ , $a=3$ , $b=1$ .", "Table: δ Ω k (4;3,1)\\delta _{\\Omega _{k}}(4; 3, 1) and δ ω k (4;3,1)\\delta _{\\omega _{k}}(4; 3, 1)For fixed $q$ and large $k$ , we give asymptotic formulas for $\\delta _{\\Omega _k}(q; a, b)$ and $\\delta _{\\omega _k}(q; a, b)$ .", "Theorem 4 Assume $ERH_q$ and $LI_q$ .", "Let $A(q)$ be the number of real characters $\\bmod ~q$ .", "Let $a$ be a quadratic non-residue and $b$ be a quadratic residue, and $(a, q)=(b, q)=1$ .", "Then, for any nonnegative integer $K$ , and any $\\epsilon >0$ , $\\delta _{\\Omega _k}(q; a, b)=\\frac{1}{2}+\\frac{(-1)^{k-1}}{2\\pi }\\sum _{j=0}^K \\left(\\frac{1}{2^{k-1}} \\right)^{2j+1}\\frac{(-1)^j A(q)^{2j+1}C_j(q; a,b)}{(2j+1)!}", "+O_{q, K, \\epsilon }\\left(\\frac{1}{(2^{k-1})^{2K+3-\\epsilon }}\\right),$ $\\delta _{\\omega _k}(q; a, b)=\\frac{1}{2}-\\frac{1}{2\\pi }\\sum _{j=0}^K \\left(\\frac{1}{2^{k-1}} \\right)^{2j+1}\\frac{(-1)^j A(q)^{2j+1}C_j(q; a,b)}{(2j+1)!}", "+O_{q, K, \\epsilon }\\left(\\frac{1}{(2^{k-1})^{2K+3-\\epsilon }}\\right),$ where $C_j(q; a, b)$ is some constant depending on $j$ , $q$ , $a$ , and $b$ .", "In particular, for $K=0$ , $\\delta _{\\Omega _k}(q; a, b)=\\frac{1}{2}+(-1)^{k-1}\\frac{A(q)C_0(q; a, b)}{2^k\\pi } +O_{q, \\epsilon }\\left(\\frac{1}{(2^{k})^{3-\\epsilon }}\\right),$ $\\delta _{\\omega _k}(q; a, b)=\\frac{1}{2}-\\frac{A(q)C_0(q; a, b)}{2^k\\pi } +O_{q, \\epsilon }\\left(\\frac{1}{(2^{k})^{3-\\epsilon }}\\right).$ Remark 2.", "We have a formula for $C_j(q; a, b)$ , $C_j(q; a, b)=\\int _{-\\infty }^{\\infty } x^{2j} \\Phi _{q; a, b}(x)dx,$ where $\\Phi _{q; a, b}(z)=\\prod _{\\chi \\ne \\chi _0} \\prod _{\\begin{array}{c}\\gamma _{\\chi }>0\\\\L(\\frac{1}{2}+i\\gamma _{\\chi })=0\\end{array}}J_0\\left(\\frac{2|\\chi (a)-\\chi (b)|z}{\\sqrt{\\frac{1}{4}+\\gamma _{\\chi }^2}} \\right),$ and $J_0(z)$ is the Bessel function, $J_0(z)=\\sum _{m=0}^{\\infty } \\frac{(-1)^m (\\frac{z}{2})^{2m}}{(m!", ")^2}.$ Numerically, $C_0(3; 2, 1)\\approx 3.66043$ and $C_0(4; 3,1)\\approx 3.08214$ .", "When $q$ is large, using the method in [4] (Section 2), we can find asymptotic formulas for $C_j(q; a, b)$ , $\\nonumber C_j(q; a, b)=\\frac{(2j-1)!!", "\\sqrt{2\\pi }}{V(q; a, b)^{j+\\frac{1}{2}}}+O_j\\left(\\frac{1}{V(q; a, b)^{j+\\frac{3}{2}}} \\right),$ where $(2j-1)!", "!=(2j-1)(2j-3)\\cdots 3\\cdot 1$ , $(-1)!", "!=1$ , and $\\nonumber V(q; a, b)=\\sum _{\\chi \\bmod q} |\\chi (b)-\\chi (a)|^2 \\sum _{\\begin{array}{c}\\gamma _{\\chi }\\in \\mathbb {R}\\\\ L(\\frac{1}{2}+i\\gamma _{\\chi }, \\chi )=0\\end{array}}\\frac{1}{\\frac{1}{4}+\\gamma ^2_{\\chi }}.$ By Proposition 3.6 in [4], under ${\\rm ERH_q}$ , $V(q; a, b)\\sim 2\\phi (q)\\log q$ ." ], [ "Formulas for the associated Dirichlet series and orgin of the bias", "Let $\\chi $ be a non-principal Dirichlet character, and denote $F_{f_k}(s,\\chi ):=\\sum _{f(n)=k} \\frac{\\chi (n)}{n^s},$ where $f=\\Omega $ or $\\omega $ .", "The formulas for $F_{f_k}(s, \\chi )$ are needed to analyze the character sums in (REF ) and (REF ).", "The purpose of this section is to express $F_{f_k}(s, \\chi )$ in terms of Dirichlet $L$ -functions, and to explain the source of the biases in the functions $\\Delta _{\\Omega _k}(x; q, a, b)$ and $\\Delta _{\\omega _k}(x; q, a, b)$ .", "Throughout the paper, the notation $\\log z$ will always denote the principal branch of the logarithm of a complex number $z$ ." ], [ "Symmetric functions", "Let $x_1$ , $x_2$ , $\\dots $ be an infinite collection of indeterminates.", "We say a formal power series $P(x_1, x_2, \\dots )$ with bounded degree is a symmetric function if it is invariant under all finite permutations of the variables $x_1$ , $x_2$ , $\\dots $ .", "The $n$ -th elementary symmetric function $e_n=e_n(x_1, x_2, \\dots )$ is defined by the generating function $\\sum _{n=0}^{\\infty } e_n z^n=\\prod _{i=1}^{\\infty } (1+x_i z).$ Thus, $e_n$ is the sum of all square-free monomials of degree $n$ .", "Similary, the $n$ -th homogeneous symmetric function $h_n=h_n(x_1, x_2, \\dots )$ is defined by the generating function $\\sum _{n=0}^{\\infty } h_n z^n=\\prod _{i=1}^{\\infty }\\frac{1}{1-x_i z}.$ We see that, $h_n$ is the sum of all possible monomials of degree $n$ .", "And the $n$ -th power symmetric function $p_n=p_n(x_1, x_2, \\dots )$ is defined to be $p_n=x_1^n+x_2^n+\\cdots .$ The following result is due to Newton or Girard (see [16], Chapter 1, (2.11) and (2.11'), page 23, or [17], Chapter 2, Theorems 2.8 and 2.9).", "Lemma 1 For any integer $k\\ge 1$ , $kh_k=\\sum _{n=1}^k h_{k-n}p_n,$ $ke_k=\\sum _{n=1}^k (-1)^{n-1} e_{k-n}p_n.$" ], [ "Formula for $F_{\\Omega _k}(s, \\chi )$", "For $\\Re (s)>1$ , we define $\\nonumber F(s,\\chi ):=\\sum _{p} \\frac{\\chi (p)}{p^s},$ the sum being over all prime $p$ .", "Since $\\log L(s, \\chi )=\\sum _{m=1}^{\\infty }\\sum _{p~\\text{prime}} \\frac{\\chi (p^m)}{m p^{ms}},$ we then have $F(s,\\chi )=\\log L(s, \\chi )-\\frac{1}{2}\\log L(2s, \\chi ^2)+G(s),$ where $G(s)$ is absolutely convergent for $\\Re (s)\\ge \\sigma _0$ for any fixed $\\sigma _0>\\frac{1}{3}$ .", "Henceforth, $\\sigma _0$ will be a fixed abscissa $>\\frac{1}{3}$ , say $\\sigma _0=0.34$ .", "Because $L(s, \\chi )$ is an entire function for non-principal characters $\\chi $ , formula (REF ) provides an analytic continuation of $F(s, \\chi )$ to any simply-connected domain within the half-plane $\\lbrace s: \\Re (s)\\ge \\sigma _0 \\rbrace $ which avoids the zeros of $L(s, \\chi )$ and the zeros and possible pole of $L(2s, \\chi ^2)$ .", "For any complex number $s$ with $\\Re (s)\\ge \\sigma _0>\\frac{1}{3}$ , let $x_p=\\frac{\\chi (p)}{p^s}$ if $p$ is a prime, 0 otherwise.", "Then, by (REF ) in Lemma REF , we have the following relation $kF_{\\Omega _k}(s, \\chi )=\\sum _{n=1}^k F_{\\Omega _{k-n}}(s, \\chi )F(ns, \\chi ^n).$ For example, for $k=1$ , $F_{\\Omega _1}(s, \\chi )=F(s, \\chi )$ .", "For $k=2$ , $2F_{\\Omega _2}(s, \\chi )=F^2(s, \\chi )+F(2s, \\chi ^2).$ For $k=3$ , $3!", "F_{\\Omega _3}(s, \\chi )&= 2F_{\\Omega _2}(s,\\chi ) F(s,\\chi )+2F(s,\\chi )F(2s, \\chi ^2)+2F(3s, \\chi ^3)\\nonumber \\\\&= F^3(s, \\chi )+3F(s,\\chi )F(2s, \\chi ^2)+2F(3s, \\chi ^3).$ For $k=4$ , $4!", "F_{\\Omega _4}(s, \\chi )&= 3!", "F_{\\Omega _3}(s, \\chi )F(s, \\chi )+3!F_{\\Omega _2}(s,\\chi )F(2s, \\chi ^2)+3!F(s, \\chi )F(3s, \\chi ^2)+3!F(4s, \\chi ^4)\\nonumber \\\\&=F^4(s, \\chi )+6F^2(s, \\chi )F(2s, \\chi ^2)+8F(s,\\chi )F(3s, \\chi ^3)+6F(4s, \\chi ^4)\\nonumber \\\\&\\quad +3F^2(2s, \\chi ^2).$ For any integer $l\\ge 1$ , we define the set $S_{m,l}^{(k)}:=\\lbrace (n_1, \\cdots , n_l) ~|~ n_1+\\cdots +n_l=k-m, 2\\le n_1\\le n_2\\le \\cdots \\le n_l, n_j\\in \\mathbb {N} (1\\le j\\le l) \\rbrace $ Let $S_m^{(k)}=\\bigcup _{l\\ge 1} S_{m, l}^{(k)}.$ Thus any element of $S_m^{(k)}$ is a partition of $k-m$ with each part $\\ge 2$ .", "For any $n=(n_1, n_2, \\cdots , n_l)\\in S_m^{(k)}$ , denote $F(ns, \\chi ):=\\prod _{j=1}^l F(n_j s, \\chi ^{n_j}).$ Hence, by (REF ) and induction on $k$ , we deduce the following result.", "Lemma 2 For $k=1$ , $F_{\\Omega _1}(s, \\chi )=F(s, \\chi )$ .", "For any $k\\ge 2$ , we have $k!", "F_{\\Omega _k}(s, \\chi )=F^k(s, \\chi )+\\sum _{m=0}^{k-2}F^m(s, \\chi ) F_{n_m}(s, \\chi ),$ where $F_{n_m}(s, \\chi )=\\sum \\limits _{n\\in S_m^{(k)}} a^{(k)}_m (n) F(ns, \\chi )$ for some $a^{(k)}_m (n)\\in \\mathbb {N}$ ." ], [ "Formula for $F_{\\omega _k}(s, \\chi )$", "By definition, we have $F_{\\omega _{k}}(s, \\chi )=\\sum _{\\begin{array}{c}p_1<p_2<\\cdots <p_k\\\\ p_i~\\text{prime}\\end{array}} \\prod _{n=1}^k \\left(\\sum _{j=1}^{\\infty } \\frac{\\chi (p_n^j)}{p_n^j}\\right).$ Denote $\\widetilde{F}(s, \\chi ):=\\sum _{p~\\text{prime}} \\left(\\frac{\\chi (p)}{p^s}+\\frac{\\chi (p^2)}{p^{2s}}+\\cdots \\right),$ and for any $u\\in \\mathbb {N}^{+}$ , $\\widetilde{F}(s, \\chi ; u):=\\sum _{p~\\text{prime}} \\left(\\frac{\\chi (p)}{p^s}+\\frac{\\chi (p^2)}{p^{2s}}+\\cdots \\right)^u=\\sum _{p~\\text{prime}} \\sum _{j=u}^{\\infty } \\left(D_u(j)\\frac{\\chi (p^j)}{p^{js}} \\right),$ where $D_u(j)={j-1 \\atopwithdelims ()u-1}$ is the number of ways of writing $j$ as sum of $u$ ordered positive integers.", "By (REF ), we have $\\widetilde{F}(s, \\chi )=\\widetilde{F}(s, \\chi ; 1)=\\sum _{p~\\text{prime}}\\sum _{j=1}^{\\infty } \\frac{\\chi (p^j)}{p^{js}}=\\log L(s, \\chi )+\\frac{1}{2}\\log L(2s,\\chi ^2)+\\widetilde{G}_1(s),$ and $\\widetilde{F}(s, \\chi ; 2)=\\sum _{p~\\text{prime}}\\sum _{j=2}^{\\infty } (j-1) \\frac{\\chi (p^j)}{p^{js}}=\\log L(2s, \\chi ^2)+\\widetilde{G}_2(s),$ where $\\widetilde{G}_1(s)$ and $\\widetilde{G}_2(s)$ are absolutely convergent for $\\Re (s)\\ge \\sigma _0$ .", "Formula (REF ) provides an analytic continuation of $\\widetilde{F}(s, \\chi )$ to any simply-connected domain within the half-plane $\\lbrace s: \\Re (s)\\ge \\sigma _0 \\rbrace $ which avoids the zeros of $L(s, \\chi )$ and the zeros and possible pole of $L(2s, \\chi ^2)$ .", "Moreover, for any fixed $u\\ge 3$ , $\\widetilde{F}(s, \\chi ; u)$ is absolutely convergent for $\\Re (s)\\ge \\sigma _0$ .", "For any complex number $s$ with $\\Re (s)\\ge \\sigma _0$ , take $x_p=\\sum _{j=1}^{\\infty }\\frac{\\chi (p^j)}{p^{js}}$ if $p$ is a prime, 0 otherwise.", "Then by (REF ) in Lemma REF , we get the following formula, $kF_{\\omega _k}(s, \\chi )=F_{\\omega _{k-1}}(s, \\chi )\\widetilde{F}(s, \\chi )-\\sum _{n=2}^k (-1)^n F_{\\omega _{k-n}}(s, \\chi )\\widetilde{F}(s, \\chi ; n).$ For example, for $k=1$ , $F_{\\omega _1}(s, \\chi )=\\widetilde{F}(s, \\chi )$ .", "For $k=2$ , $2F_{\\omega _2}(s, \\chi )=\\widetilde{F}^2(s, \\chi )-\\widetilde{F}(s, \\chi ; 2).$ For $k=3$ , $3!", "F_{\\omega _3}(s, \\chi )&=2F_{\\omega _2}(s,\\chi )\\widetilde{F}(s, \\chi )-2F_{\\omega _1}(s, \\chi )\\widetilde{F}(s, \\chi ; 2)+2\\widetilde{F}(s, \\chi ; 3)\\nonumber \\\\&=\\widetilde{F}^3(s, \\chi )-3\\widetilde{F}(s, \\chi )\\widetilde{F}(s, \\chi ; 2)+2\\widetilde{F}(s, \\chi ; 3).$ For $k=4$ , $4!", "F_{\\omega _4}(s, \\chi )&=3!", "F_{\\omega _3}(s, \\chi )\\widetilde{F}(s, \\chi )-3!F_{\\omega _2}(s, \\chi )\\widetilde{F}(s, \\chi ; 2)+3!\\widetilde{F}(s, \\chi )\\widetilde{F}(s, \\chi ; 3)-3!\\widetilde{F}(s, \\chi ; 4)\\nonumber \\\\&= \\widetilde{F}^4(s, \\chi )-6\\widetilde{F}^2(s, \\chi )\\widetilde{F}(s, \\chi ; 2)+8\\widetilde{F}(s, \\chi )\\widetilde{F}(s, \\chi ; 3)-6\\widetilde{F}(s, \\chi ; 4)+3\\widetilde{F}^2(s, \\chi ; 2).$ Hence, by (REF ) and induction on $k$ , we get the following result.", "Lemma 3 For $k=1$ , $F_{\\omega _1}(s, \\chi )=\\widetilde{F}(s, \\chi )$ .", "For any $k\\ge 2$ , we have $k!F_{\\omega _{k}}(s, \\chi )= \\widetilde{F}^k(s, \\chi )+\\sum _{m=0}^{k-2} \\widetilde{F}^m(s, \\chi ) \\widetilde{F}_{n_m}(s, \\chi ),$ where $\\widetilde{F}_{n_m}(s, \\chi )=\\sum \\limits _{n\\in S_m^{(k)}} b_m^{(k)} (n) \\widetilde{F}(ns,\\chi )$ for some $b_m^{(k)}(n)\\in \\mathbb {Z}$ , and for any $n=(n_1, \\cdots , n_l)\\in S^{(k)}_m$ , $\\widetilde{F}(ns, \\chi ):=\\prod _{j=1}^l \\widetilde{F}(s, \\chi ; n_j).$" ], [ "Origin of the bias", "In this section, we heuristically explain the origin of the bias in our theorems.", "1) Analytical aspect.", "In order to get formulas for $\\Delta _{\\Omega _k}(x; q, a, b)$ and $\\Delta _{\\omega _k}(x; q, a, b)$ , our strategy is to apply Perron's formula to the associated Dirichlet series $F_{\\Omega _k}(s, \\chi )$ and $F_{\\omega _k}(s, \\chi )$ , then we choose special contours to avoid the singularities of these Dirichlet series.", "See Section for the details.", "First, we have a look at the case of counting primes in arithmetic progressions.", "If we only count primes, by (REF ), we have $F_{\\Omega _1}(s, \\chi )=F(s, \\chi )=\\sum _{p}\\frac{\\chi (p)}{p^s}=\\log L(s, \\chi )-\\frac{1}{2}\\log L(2s, \\chi ^2)+G(s).$ The main contributions for $\\Delta _{\\Omega _1}(x; q, a, b)$ are from the first two terms, $\\log L(s, \\chi )-\\frac{1}{2}\\log L(2s, \\chi ^2).$ The first term $\\log L(s, \\chi )$ counts all the primes with weight 1 and prime squares with weight $\\frac{1}{2}$ .", "The higher order powers of primes are negligible since they only contribute $O(x^{\\frac{1}{3}})$ .", "The singularities of $\\log L(s, \\chi )$ , i.e.", "the zeros of $L(s, \\chi )$ , on the critical line contribute the oscillating terms in our result.", "In our proof, we use special Hankel contours to avoid the singularities of $\\log L(s, \\chi )$ and extract these oscillating terms (Lemma REF ).", "See Sections and for the details of how to handle these singularities.", "The second term $-\\frac{1}{2}\\log L(2s, \\chi ^2)$ counts the prime squares with weight $-\\frac{1}{2}$ and contributes the bias term.", "When $\\chi $ is a real character, the point $s=\\frac{1}{2}$ is a pole of $L(2s, \\chi ^2)$ , and hence the integration of $ -\\frac{1}{2}\\log L(2s, \\chi ^2)$ over the Hankel contour around $s=\\frac{1}{2}$ contributes a bias term with order of magnitude $\\frac{\\sqrt{x}}{\\log x}$ .", "Using the orthogonality of Dirichlet characters, and the formula $\\sum _{\\chi ~\\rm real}(\\overline{\\chi }(a)-\\overline{\\chi }(b))=N(q, a)-N(q, b)$ , we get the expected size of the bias.", "Another natural and convenient function to consider is $-\\frac{L^{\\prime }(s, \\chi )}{L(s, \\chi )}=\\sum _{n=1}^{\\infty }\\frac{\\chi (n)\\Lambda (n)}{n^s}$ , which is much easier to analyze than $\\log L(s, \\chi )$ .", "This weighted form is counting each prime $p$ and its powers with weight $\\log p$ .", "Similar to $\\log L(s, \\chi )$ , all the singularities of the function $-\\frac{L^{\\prime }(s, \\chi )}{L(s, \\chi )}$ on the critical line are the non-trivial zeros of $L(s, \\chi )$ and thus there is no bias for this weighted counting function $\\sum _{\\begin{array}{c}n\\le x\\\\ n\\equiv a \\bmod q\\end{array}} \\Lambda (n)-\\sum _{\\begin{array}{c}n\\le x\\\\ n\\equiv b \\bmod q\\end{array}} \\Lambda (n).$ Thus, partial summation is used to extract the sum $\\sum _{\\begin{array}{c}n\\le x\\\\ n\\equiv a \\bmod q\\end{array}}\\frac{\\Lambda (n)}{\\log n}-\\sum _{\\begin{array}{c}n\\le x\\\\ n\\equiv b \\bmod q\\end{array}}\\frac{\\Lambda (n)}{\\log n}$ from the above weighted form, which is possible because $\\log n$ is a smooth function.", "However, there is no way to do this with the analogue (REF ) to recover the unweighted counting function $\\Delta _{\\Omega _k}(x; q, a, b)$ or $\\Delta _{\\omega _k}(x; q, a, b)$ .", "If we count all the prime powers with the same weight 1, by (REF ), we have $ F_{\\omega _1}(s, \\chi )=\\widetilde{F}(s, \\chi )=\\log L(s, \\chi )+\\frac{1}{2}\\log L(2s,\\chi ^2)+\\widetilde{G}_1(s).", "$ In this case, the bias is from the second term $\\frac{1}{2}\\log L(2s,\\chi ^2)$ for real character $\\chi $ which counts the prime squares with positive weight $\\frac{1}{2}$ .", "This is why the bias is opposite to the case of counting only primes.", "For the general case, when we derive the formula for $\\Delta _{\\Omega _k}(x; q, a, b)$ using analytic methods, by (REF ) in Lemma REF , the main contributions for $F_{\\Omega _k}(s, \\chi )$ will be from $\\frac{1}{k!}", "F^k(s, \\chi )$ , which is essentially $\\frac{1}{k!", "}\\left(\\log L(s, \\chi )-\\frac{1}{2}\\log L(2s, \\chi ^2)\\right)^k.$ In the expansion of the above formula, the term $\\frac{1}{k!", "}\\log ^k L(s, \\chi )$ contributes the oscillating terms (see (REF ) and (REF )) $\\frac{(-1)^k}{(k-1)!", "}\\frac{\\sqrt{x}(\\log \\log x)^{k-1}}{\\log x}\\sum _{L(\\frac{1}{2}+i\\gamma _{\\chi }, \\chi )=0} \\frac{x^{i\\gamma _{\\chi }}}{\\frac{1}{2}+i\\gamma _{\\chi }}.$ When $\\chi $ is real, the term $\\frac{1}{k!", "}\\left(-\\frac{1}{2}\\log L(2s, \\chi ^2)\\right)^k=\\frac{(-1)^k}{k!", "2^k}\\left(\\log L(2s, \\chi ^2)\\right)^k$ contributes a bias term (see (REF ) and (REF )) $\\frac{1}{(k-1)!", "}\\frac{(-1)^k}{ 2^{k-1}}\\frac{\\sqrt{x}(\\log \\log x)^{k-1}}{\\log x}.$ Then summing over all the real characters, we get the expected bias term in our formula for $\\Delta _{\\Omega _k}(x; q, a, b)$ .", "The factor $\\frac{(-1)^k}{2^{k-1}}$ explains why the bias has different directions depending on the parity of $k$ and why the bias decreases as $k$ increases.", "Other terms with factors of the form $\\log ^{k-j} L(s, \\chi ) \\log ^j (2s, \\chi ^2)$ for $1\\le j\\le k-1$ only contribute oscillating terms with lower orders of $\\log \\log x$ which can be put into the error term in our formula (see Lemma REF ).", "Similarly, for the case of $\\Delta _{\\omega _k}(x; q, a, b)$ , by (REF ) in Lemma REF , the main contributions for $F_{\\omega _k}(s, \\chi )$ are from $\\frac{1}{k!", "}\\widetilde{F}^k(s, \\chi )=\\frac{1}{k!", "}\\left(\\log L(s, \\chi )+\\frac{1}{2}\\log L(2s,\\chi ^2)+\\widetilde{G}_1(s) \\right)^k.$ The main terms are from the contributions of the terms $\\frac{1}{k!", "}\\log ^k L(s, \\chi )$ and $\\frac{1}{k!}", "\\left(\\frac{1}{2}\\log L(2s, \\chi ^2)\\right)^k$ .", "Thus, the main oscillating terms are the same as that of $\\Delta _{\\Omega _k}(x; q, a, b)$ , and the bias term has the same size without direction change.", "Through the above analysis, we see that the biases are mainly affected by the powers of $\\pm \\frac{1}{2}\\log L(2s, \\chi ^2)$ for real characters which count the products of prime squares.", "2) Combinatorial aspect.", "Instead of giving precise prediction of the size of the bias as above, here we use a simpler combinatorial intuition to roughly explain the behavior of the bias.", "We borrowed this combinatorial explanation from Hudson [8].", "Pick a large number $X$ .", "Let $S_1$ be the set of primes $p\\equiv 1 \\bmod 4$ up to $X$ , and $S_2$ be the set of primes $p\\equiv 3\\bmod 4$ up to $X$ .", "Using these primes, we generate the set $V^{(2)}:=\\lbrace pq: p, q \\in S_1 \\cup S_2, p \\text{~and~} q \\text{~can be the same}\\rbrace $ .", "Let $V_1^{(2)}:=\\lbrace n\\in V^{(2)}: n\\equiv 1\\bmod 4 \\rbrace $ , and $V_2^{(2)}:=\\lbrace n\\in V^{(2)}: n\\equiv 3\\bmod 4 \\rbrace $ .", "Then, the integers in $V_1^{(2)}$ come from either products of two primes from $S_1$ or products of two primes from $S_2$ .", "The integers in $V^{(2)}_2$ are the product of two primes $pq$ with $p\\in S_1$ and $q\\in S_2$ .", "Thus, $|V^{(2)}_1|={|S_1|\\atopwithdelims ()2}+|S_1|+{|S_2|\\atopwithdelims ()2}+|S_2|=\\frac{|S_1|^2+|S_2|^2|}{2}+\\frac{|S_1|+|S_2|}{2},$ and $|V^{(2)}_2|=|S_1|\\cdot |S_2|.$ It is clear that $|V^{(2)}_1|>|V^{(2)}_2|$ .", "Note that $\\frac{|S_1|+|S_2|}{2}$ counts the squares of primes with weight $\\frac{1}{2}$ which makes a crucial difference between $V_1^{(2)}$ and $V_2^{(2)}$ .", "Let $V_1^{(0)}=\\lbrace 1\\rbrace $ and $V_2^{(0)}=\\emptyset $ .", "For any $k\\ge 1$ , denote $V_1^{(k)}:=\\lbrace n=p_1\\cdots p_k: p_j\\in S_1\\cup S_2 \\text{~for all~} 1\\le j\\le k, n\\equiv 1\\bmod 4\\rbrace ,$ $V_2^{(k)}:=\\lbrace n=p_1\\cdots p_k: p_j\\in S_1\\cup S_2 ~\\text{for all~} 1\\le j\\le k, n\\equiv 3\\bmod 4\\rbrace ,$ where the $p_j$ can be the same.", "Note that $V_1^{(1)}=S_1$ and $V_2^{(1)}=S_2$ .", "We give inductive formulas for $|V_1^{(k)}|$ and $|V_2^{(k)}|$ .", "The elements of $V_1^{(k)}$ and $V_2^{(k)}$ are generated by integers of the form $ p^j n_{k-j}$ for $p\\in S_1 \\text{~or~} S_2$ and $n_{k-j}\\in V_1^{(k-j)} \\text{~or~} V_2^{(k-j)} ~(1\\le j\\le k)$ .", "By (REF ) in Lemma REF , we have $k|V_1^{(k)}|=&\\underbrace{\\left(|V_1^{(k-1)}|\\cdot |S_1|+|V_2^{(k-1)}|\\cdot |S_2|\\right)}_{p n_{k-1}}+ \\underbrace{|V_1^{(k-2)}| (|S_1|+|S_2|)}_{p^2 n_{k-2}}\\\\&+\\underbrace{\\left(|V_1^{(k-3)}|\\cdot |S_1|+|V_2^{(k-3)}|\\cdot |S_2|\\right)}_{p^3 n_{k-3}}+\\cdots $ and $k|V_2^{(k)}|=&\\underbrace{\\left(|V_2^{(k-1)}|\\cdot |S_1|+|V_1^{(k-1)}|\\cdot |S_2|\\right)}_{p n_{k-1}}+ \\underbrace{|V_2^{(k-2)}| (|S_1|+|S_2|)}_{p^2 n_{k-2}}\\\\&+\\underbrace{\\left(|V_2^{(k-3)}|\\cdot |S_1|+|V_1^{(k-3)}|\\cdot |S_2|\\right)}_{p^3 n_{k-3}}+\\cdots .$ Thus, $k\\left( |V_1^{(k)}|-|V_2^{(k)}|\\right)=&\\left(|V_1^{(k-1)}|\\cdot |S_1|+|V_2^{(k-1)}|\\cdot |S_2|\\right)-\\left(|V_2^{(k-1)}|\\cdot |S_1|+|V_1^{(k-1)}|\\cdot |S_2|\\right)\\nonumber \\\\&+\\left( |V_1^{(k-2)}|-|V_2^{(k-2)}|\\right) (|S_1|+|S_2|)+\\left(|V_1^{(k-3)}|\\cdot |S_1|+|V_2^{(k-3)}|\\cdot |S_2|\\right)\\nonumber \\\\&-\\left(|V_2^{(k-3)}|\\cdot |S_1|+|V_1^{(k-3)}|\\cdot |S_2|\\right)+\\cdots \\nonumber \\\\=&\\left( |V_1^{(k-1)}|-|V_2^{(k-1)}|\\right) ( |S_1|-|S_2|)+\\left( |V_1^{(k-2)}|-|V_2^{(k-2)}|\\right) (|S_1|+|S_2|)\\nonumber \\\\&+\\left( |V_1^{(k-3)}|-|V_2^{(k-3)}|\\right) ( |S_1|-|S_2|)+\\cdots .$ For $1\\le j\\le k-1$ , suppose $|V_1^{(j)}|<|V_2^{(j)}|$ for odd $j$ and $|V_1^{(j)}|>|V_2^{(j)}|$ for even $j$ .", "Therefore, by (REF ) and induction, we deduce that $|V_1^{(k)}|<|V_2^{(k)}|$ for odd $k$ and $|V_1^{(k)}|>|V_2^{(k)}|$ for even $k$ .", "This provides us a heuristic explanation for the bias oscillation of $\\Delta _{\\Omega _k}(x; q, a, b)$ ." ], [ "Contour integral representation", "In this section, we express the inner sums in (REF ) and (REF ) as integrals over truncated Hankel contours (see Lemma REF below).", "Let $\\psi _{f_k}(x,\\chi ):=\\sum _{\\begin{array}{c}n\\le x\\\\ f(n)=k\\end{array}}\\chi (n),$ where $f=\\Omega $ or $\\omega $ .", "By Perron's formula ([9], Chapter V, Theorem 1), we have the following lemma.", "Lemma 4 For any $T\\ge 2$ , $\\psi _{f_k}(x,\\chi )=\\frac{1}{2\\pi i}\\int _{c-iT}^{c+iT} F_{f_k}(s, \\chi )\\frac{x^s}{s} ds +O\\left(\\frac{x\\log x}{T}+1\\right),$ where $c=1+\\frac{1}{\\log x}$ , and $f=\\Omega $ or $\\omega $ .", "Starting from Lemma REF , we will shift the contour to the left, in a way which avoids the singularities of the integrand.", "We will then require estimates of the integrand along the various parts of the new contour.", "Lemma 5 Assume $ERH_q$ .", "Then, for any $0<\\delta <\\frac{1}{6}$ and for all $\\chi \\ne \\chi _0 \\bmod q$ , there exists a sequence of numbers $\\mathcal {T}=\\lbrace T_n \\rbrace _{n=0}^{\\infty }$ satisfying $n\\le T_n\\le n+1$ such that, for $T\\in \\mathcal {T}$ , $F_{f_k}(\\sigma +iT)=O\\left(\\log ^k T \\right), ~( \\frac{1}{2}-\\delta <\\sigma <2)$ where $f=\\Omega $ or $\\omega $ .", "Proof.", "Using the similar method as in [20] (Theorem 14.16), one can show that, for any $\\epsilon >0$ and for all $\\chi \\ne \\chi _0 \\bmod q$ , there exists a sequence of numbers $\\mathcal {T}=\\lbrace T_n \\rbrace _{n=0}^{\\infty }$ satisfying $n\\le T_n\\le n+1$ such that, $T_n^{-\\epsilon }\\ll |L(\\sigma +iT_n, \\chi )|\\ll T_n^{\\delta +\\epsilon }, ~(\\frac{1}{2}-\\delta <\\sigma <2).$ Hence, by formulas (REF ), (REF ), (REF ), (REF ), and (REF ), we get the conclusion of this lemma.", "$\\Box $ Let $\\rho $ be a zero of $L(s, \\chi )$ , $\\Delta _{\\rho }$ be the distance of $\\rho $ to the nearest other zero, and $D_{\\gamma }:=\\min \\limits _{T\\in \\mathcal {T}}(|\\gamma -T|)$ .", "For each zero $\\rho $ , and $X>0$ , let $\\mathcal {H}(\\rho , X)$ denote the truncated Hankel contour surrounding the point $s=\\rho $ with radius $0<r_{\\rho }\\le \\min (\\frac{1}{x}, \\frac{\\Delta _{\\rho }}{3}, \\frac{D_{\\gamma }}{2}, \\frac{|\\rho -1/2|}{3})$ , which includes the circle $|s-\\rho |=r_{\\rho }$ excluding the point $s=\\rho -r_\\rho $ , and the half-line $(\\rho -X, \\rho -r]$ traced twice with arguments $+\\pi $ and $-\\pi $ respectively.", "Let $\\Delta _0$ be the distance of $\\frac{1}{2}$ to the nearest zero.", "Let $\\mathcal {H}(\\frac{1}{2}, X)$ denote the corresponding truncated Hankel contour surrounding $s=\\frac{1}{2}$ with radius $r_0=\\min (\\frac{1}{x}, \\frac{\\Delta _0}{3})$ .", "Take $\\delta =\\frac{1}{10}$ .", "By Lemma REF , we pull the contour to the left to the line $\\Re (s)=\\frac{1}{2}-\\delta $ using the truncated Hankel contour $\\mathcal {H}(\\rho , \\delta )$ to avoid the zeros of $L(s, \\chi )$ and using $\\mathcal {H}(\\frac{1}{2}, \\delta )$ to avoid the point $s=\\frac{1}{2}$ .", "See Figure REF .", "Figure: Integration contourThen we have the following lemma.", "Lemma 6 Assume $ERH_q$ , and $L(\\frac{1}{2}, \\chi )\\ne 0$ $(\\chi \\ne \\chi _0)$ .", "Then, for any fixed $k\\ge 1$ , and $T\\in \\mathcal {T}$ , $\\psi _{f_k}(x,\\chi )&=\\sum _{|\\gamma |\\le T}\\frac{1}{2\\pi i}\\int _{\\mathcal {H}(\\rho , \\delta )}F_{f_k}(s, \\chi )\\frac{x^s}{s} ds +a(\\chi )\\frac{1}{2\\pi i}\\int _{\\mathcal {H}(\\frac{1}{2}, {\\delta })}F_{f_k}(s, \\chi )\\frac{x^s}{s} ds\\\\&\\quad +O\\left(\\frac{x\\log x}{T}+\\frac{x(\\log T)^k}{T}+x^{\\frac{1}{2}-\\delta } (\\log T)^{k+1}\\right),$ where $a(\\chi )=1$ if $\\chi $ is real, 0 otherwise, and $f=\\omega $ or $\\Omega $ .", "Proof.", "By formulas (REF ) and (REF ), if $\\chi $ is not real, $s=\\frac{1}{2}$ is not a singularity of $F_{f_k}(s, \\chi )$ .", "Hence the second term is zero if $\\chi $ is not real.", "By Lemma REF , the integral on the horizontal line is $\\ll (\\log T)^k\\int _{\\frac{1}{2}-\\delta }^{c}\\frac{x^{\\sigma }}{|\\sigma +iT|} d\\sigma \\ll \\frac{x^c(\\log T)^k}{T} \\ll \\frac{x(\\log T)^k}{T}.$ Under the assumption ${\\rm ERH_q}$ , the integral on the vertical line $\\Re (s)=\\frac{1}{2}-\\delta $ is $\\ll \\int _{-T}^T \\frac{x^{\\frac{1}{2}-\\delta } \\log ^k (|t|+2)}{|\\frac{1}{2}-\\delta +it|}dt\\ll x^{\\frac{1}{2}-\\delta } (\\log T)^{k+1}.$ By (REF ), (REF ), and Lemma REF , we get the desired error term in this lemma.", "$\\Box $" ], [ "Proof of the main theorems ", "In this section, we outline the proof of Theorems REF and REF .", "Let $\\gamma $ be the imaginary part of a zero of $L(s, \\chi )$ in the critical strip.", "We have the following lemma.", "Lemma 7 ([6], Lemma 2.2) Let $\\chi $ be a Dirichlet character modulo $q$ .", "Let $N(T, \\chi )$ denote the number of zeros of $L(s, \\chi )$ with $0<\\Re (s)<1$ and $|\\Im (s)|<T$ .", "Then 1) $N(T, \\chi )=O(T\\log (qT))$ for $T\\ge 1$ .", "2) $N(T, \\chi )-N(T-1, \\chi )=O(\\log (qT))$ for $T\\ge 1$ .", "3) Uniformly for $s=\\sigma +it$ and $\\sigma \\ge -1$ , $\\frac{L^{\\prime }(s, \\chi )}{L(s, \\chi )}=\\sum _{|\\gamma -t|<1} \\frac{1}{s-\\rho } +O(\\log q(|t|+2)).$ For simplicity, we denote $\\frac{1}{\\Gamma _j(u)}:=\\left[\\frac{d^j}{dz^j}\\left(\\frac{1}{\\Gamma (z)}\\right)\\right]_{z=u}.$ The following lemma is the starting lemma to give us the bias terms and oscillating terms in our main theorems.", "This lemma may have independent use, we will give the proof in Section .", "Lemma 8 Let $\\mathcal {H}(a, \\delta )$ be the truncated Hankel contour surrounding a complex number $a \\ (\\Re (a)>2\\delta )$ with radius $0<r\\ll \\frac{1}{x}$ .", "Then, for any integer $k\\ge 1$ , $&\\frac{1}{2\\pi i}\\int _{\\mathcal {H}(a, \\delta )} \\log ^k(s-a)\\frac{x^s}{s}ds\\nonumber \\\\&=\\frac{(-1)^k x^a}{a\\log x}\\left\\lbrace k (\\log \\log x)^ {k-1}+\\sum _{j=2}^k {k \\atopwithdelims ()j} (\\log \\log x)^{k-j} \\frac{1}{\\Gamma _j(0)}\\right\\rbrace +O_k\\left(\\frac{|x^{a-\\delta /3}|}{|a|}\\right)\\nonumber \\\\&\\quad +O_k\\left(\\frac{|x^a|}{|a|^2 \\log ^2 x} (\\log \\log x)^{k-1}\\right)+O_k\\left( \\frac{|x^a|}{|a|^2|\\Re (a)-\\delta |}\\frac{(\\log \\log x)^{k-1}}{(\\log x)^3}\\right).$ Remark.", "By (REF ) in the proof of Lemma REF , one can easily show that $\\left|\\frac{1}{\\Gamma _j(0)}\\right|\\ll \\Gamma (j+1).$ By Lemma REF , we need to examine the integration over the truncated Hankel contours $\\mathcal {H}(\\rho , \\delta )$ and $\\mathcal {H}(\\frac{1}{2}, \\delta )$ .", "By (REF ) and (REF ), and the assumptions of our theorems, on each truncated Hankel contour $\\mathcal {H}(\\rho , \\delta )$ , we integrate the formula (REF ) in Lemma REF to obtain $F(s, \\chi )=\\log (s-\\rho )+H_{\\rho }(s),$ $\\widetilde{F}(s, \\chi )=\\log (s-\\rho )+\\widetilde{H}_{\\rho }(s),$ where $H_{\\rho }(s)&=\\sum _{0<|\\gamma ^{\\prime }-\\gamma |\\le 1} \\log (s-\\rho ^{\\prime })+O(\\log |\\gamma |),\\\\\\widetilde{H}_{\\rho }(s)&=\\sum _{0<|\\gamma ^{\\prime }-\\gamma |\\le 1} \\log (s-\\rho ^{\\prime })+O(\\log |\\gamma |).$ If $\\chi $ is real, $s=\\frac{1}{2}$ is a pole of $L(2s, \\chi ^2)$ .", "So, by (REF ) and (REF ), on the truncated Hankel contour $\\mathcal {H}(\\frac{1}{2}, \\delta )$ , for a real character $\\chi $ , we write $F(s, \\chi )&=\\frac{1}{2}\\log \\left(s-\\frac{1}{2}\\right)+H_B(s),\\\\\\widetilde{F}(s, \\chi )&=-\\frac{1}{2}\\log \\left(s-\\frac{1}{2}\\right)+\\widetilde{H}_B(s),$ where $H_B(s)=O(1)$ and $\\widetilde{H}_B(s)=O(1)$ .", "Denote $I_{\\rho }(x)&:=\\frac{1}{2\\pi i}\\int _{\\mathcal {H}(\\rho ,\\delta )}k!F_{\\Omega _k}(s, \\chi )\\frac{x^s}{s} ds, \\\\I_B(x)&:= \\frac{1}{2\\pi i}\\int _{\\mathcal {H}(\\frac{1}{2},\\delta )}k!F_{\\Omega _k}(s, \\chi )\\frac{x^s}{s} ds,$ and $\\widetilde{I}_{\\rho }(x)&:=\\frac{1}{2\\pi i}\\int _{\\mathcal {H}(\\rho ,\\delta )}k!F_{\\omega _k}(s, \\chi )\\frac{x^s}{s} ds, \\\\\\widetilde{I}_B(x)&:= \\frac{1}{2\\pi i}\\int _{\\mathcal {H}(\\frac{1}{2},\\delta )}k!F_{\\omega _k}(s, \\chi )\\frac{x^s}{s} ds.$ We define a function $T(x)$ as follows: for $T_{n^{\\prime }}\\in \\mathcal {T}$ satisfying $e^{2^{n+1}}\\le T_{n^{\\prime }}\\le e^{2^{n+1}}+1$ , let $T(x)=T_{n^{\\prime }}$ for $e^{2^n}\\le x\\le e^{2^{n+1}}$ .", "In particular, we have $x\\le T(x)\\le 2x^2 \\quad (x\\ge e^2).$ Thus, by Lemma REF , for $T=T(x)$ , $\\psi _{\\Omega _k}(x, \\chi )&=\\frac{1}{k!", "}\\sum _{|\\gamma |\\le T}I_{\\rho }(x)+\\frac{a(\\chi )}{k!", "}I_B(x)+O\\left(x^{\\frac{1}{2}-\\frac{\\delta }{2}} \\right),\\\\\\psi _{\\omega _k}(x, \\chi )&=\\frac{1}{k!", "}\\sum _{|\\gamma |\\le T}\\widetilde{I}_{\\rho }(x)+\\frac{a(\\chi )}{k!", "}\\widetilde{I}_B(x)+O\\left(x^{\\frac{1}{2}-\\frac{\\delta }{2}}\\right).$ We will see later that $ \\sum _{|\\gamma |\\le T}I_{\\rho }(x)$ and $\\sum _{|\\gamma |\\le T}\\widetilde{I}_{\\rho }(x)$ will contribute the oscillating terms, i.e.", "the summation over zeros, in our theorems, and $I_B(x)$ and $\\widetilde{I}_B(x)$ will contribute the bias terms.", "Next, we want to find the main contributions for $I_{\\rho }(x)$ , $I_B(x)$ , $\\widetilde{I}_{\\rho }(x)$ , and $\\widetilde{I}_B(x)$ .", "By (REF ) and (REF ), we have $I_{\\rho }(x)&=\\frac{1}{2\\pi i}\\int _{\\mathcal {H}(\\rho , \\delta )} (\\log (s-\\rho ))^k \\frac{x^s}{s} ds+\\frac{1}{2\\pi i}\\int _{\\mathcal {H}(\\rho , \\delta )} \\sum _{j=1}^k {k \\atopwithdelims ()j} \\left(\\log (s-\\rho )\\right)^{k-j} (H_{\\rho }(s))^j\\frac{x^s}{s}ds\\nonumber \\\\&\\quad +\\frac{1}{2\\pi i}\\int _{\\mathcal {H}(\\rho , \\delta )} \\sum _{m=0}^{k-2} F^m(s ,\\chi ) F_{n_m}(s, \\chi )\\frac{x^s}{s}ds\\nonumber \\\\&=: I_{M_\\rho }(x)+E_{M_\\rho }(x)+E_{R_\\rho }(x),$ and by (REF ) and (REF ), $I_B(x)&=\\frac{1}{2^k} \\frac{1}{2\\pi i} \\int _{\\mathcal {H}(\\frac{1}{2},\\delta )} \\left(\\log \\left(s-\\frac{1}{2}\\right)\\right)^k \\frac{x^s}{s}ds\\nonumber \\\\&\\quad +\\frac{1}{2\\pi i} \\int _{\\mathcal {H}(\\frac{1}{2},\\delta )} \\sum _{j=1}^k {k \\atopwithdelims ()j} \\left(\\frac{1}{2}\\log \\left(s-\\frac{1}{2}\\right) \\right)^{k-j} \\left(H_B(s)\\right)^j \\frac{x^s}{s}ds \\nonumber \\\\&\\quad +\\frac{1}{2\\pi i} \\int _{\\mathcal {H}(\\frac{1}{2},\\delta )} \\sum _{m=0}^{k-2} F^m(s ,\\chi ) F_{n_m}(s, \\chi )\\frac{x^s}{s}ds\\nonumber \\\\&=: B_M(x)+E_B(x)+E_R(x).$ Here, $I_{M_{\\rho }}(x)$ and $B_M(x)$ will make main contributions to $I_{\\rho }(x)$ and $I_B(x)$ , respectively.", "Similarly, by (REF ) and (REF ), we have $\\widetilde{I}_{\\rho }(x)&=\\frac{1}{2\\pi i}\\int _{\\mathcal {H}(\\rho , \\delta )} (\\log (s-\\rho ))^k \\frac{x^s}{s} ds+\\frac{1}{2\\pi i}\\int _{\\mathcal {H}(\\rho , \\delta )} \\sum _{j=1}^k {k \\atopwithdelims ()j} \\left(\\log (s-\\rho )\\right)^{k-j} (\\widetilde{H}_{\\rho }(s))^j\\frac{x^s}{s}ds\\nonumber \\\\&\\quad +\\frac{1}{2\\pi i}\\int _{\\mathcal {H}(\\rho , \\delta )} \\sum _{m=0}^{k-2} \\widetilde{F}^m(s ,\\chi ) \\widetilde{F}_{n_m}(s, \\chi )\\frac{x^s}{s}ds\\nonumber \\\\&=: \\widetilde{I}_{M_\\rho }(x)+\\widetilde{E}_{M_\\rho }(x)+\\widetilde{E}_{R_\\rho }(x),$ and by (REF ) and (), $\\widetilde{I}_B(x)&=\\frac{(-1)^k}{2^k} \\frac{1}{2\\pi i} \\int _{\\mathcal {H}(\\frac{1}{2},\\delta )} \\left(\\log \\left(s-\\frac{1}{2}\\right)\\right)^k \\frac{x^s}{s}ds\\nonumber \\\\&\\quad +\\frac{1}{2\\pi i} \\int _{\\mathcal {H}(\\frac{1}{2},\\delta )} \\sum _{j=1}^k {k \\atopwithdelims ()j} \\left(-\\frac{1}{2}\\log \\left(s-\\frac{1}{2}\\right) \\right)^{k-j} \\left(\\widetilde{H}_B(s)\\right)^j \\frac{x^s}{s}ds \\nonumber \\\\&\\quad +\\frac{1}{2\\pi i} \\int _{\\mathcal {H}(\\frac{1}{2},\\delta )} \\sum _{m=0}^{k-2} \\widetilde{F}^m(s ,\\chi ) \\widetilde{F}_{n_m}(s, \\chi )\\frac{x^s}{s}ds\\nonumber \\\\&=: \\widetilde{B}_M(x)+\\widetilde{E}_B(x)+\\widetilde{E}_R(x).$ Here, $\\widetilde{I}_{M_{\\rho }}(x)$ and $\\widetilde{B}_M(x)$ will make main contributions to $\\widetilde{I}_{\\rho }(x)$ and $\\widetilde{I}_B(x)$ , respectively.", "Applying Lemma REF , we have $I_{M_{\\rho }}(x)=\\widetilde{I}_{M_{\\rho }}(x)&=\\frac{(-1)^k\\sqrt{x}}{\\log x}\\frac{x^{i\\gamma }}{\\frac{1}{2}+i\\gamma }\\left\\lbrace k (\\log \\log x)^ {k-1}+\\sum _{j=2}^k {k \\atopwithdelims ()j} (\\log \\log x)^{k-j} \\frac{1}{\\Gamma _j(0)}\\right\\rbrace \\nonumber \\\\&\\quad +O_k\\left( \\frac{1}{|\\gamma |^2} \\frac{\\sqrt{x}(\\log \\log x)^{k-1}}{(\\log x)^2}\\right)+O_k\\left(\\frac{x^{\\frac{1}{2}-\\frac{\\delta }{3}}}{|\\gamma |}\\right),$ $B_M(x)&=\\frac{(-1)^k\\sqrt{x}}{2^{k-1}\\log x}\\left\\lbrace k (\\log \\log x)^ {k-1}+\\sum _{j=2}^k {k \\atopwithdelims ()j} (\\log \\log x)^{k-j} \\frac{1}{\\Gamma _j(0)}\\right\\rbrace \\nonumber \\\\&\\quad +O_k\\left( \\frac{\\sqrt{x}(\\log \\log x)^{k-1}}{(\\log x)^2}\\right)+O_k\\left(x^{\\frac{1}{2}-\\frac{\\delta }{3}}\\right),$ and $\\widetilde{B}_M(x)=(-1)^k B_M(x).$ For the bias terms, by (REF ), (REF ), (REF ), and (REF ), we have $I_B(x)&=\\frac{(-1)^k\\sqrt{x}}{2^{k-1}\\log x}\\left\\lbrace k (\\log \\log x)^ {k-1}+\\sum _{j=2}^k {k \\atopwithdelims ()j} (\\log \\log x)^{k-j} \\frac{1}{\\Gamma _j(0)}\\right\\rbrace \\nonumber \\\\&\\quad +E_B(x)+E_R(x)+O_k\\left( \\frac{\\sqrt{x}(\\log \\log x)^{k-1}}{(\\log x)^2}\\right)+O_k\\left(x^{\\frac{1}{2}-\\frac{\\delta }{3}}\\right),$ and $\\widetilde{I}_B(x)&=\\frac{\\sqrt{x}}{2^{k-1}\\log x}\\left\\lbrace k (\\log \\log x)^ {k-1}+\\sum _{j=2}^k {k \\atopwithdelims ()j} (\\log \\log x)^{k-j} \\frac{1}{\\Gamma _j(0)}\\right\\rbrace \\nonumber \\\\&\\quad +\\widetilde{E}_B(x)+\\widetilde{E}_R(x)+O_k\\left( \\frac{\\sqrt{x}(\\log \\log x)^{k-1}}{(\\log x)^2}\\right)+O_k\\left(x^{\\frac{1}{2}-\\frac{\\delta }{3}}\\right).$ We will prove the following result in Section .", "Lemma 9 For the bias terms, $I_B(x)=\\frac{(-1)^k k}{2^{k-1}}\\frac{\\sqrt{x}}{\\log x} (\\log \\log x)^{k-1}+O_k\\left( \\frac{\\sqrt{x}(\\log \\log x)^{k-2}}{\\log x}\\right),$ $\\widetilde{I}_B(x)=\\frac{ k}{2^{k-1}}\\frac{\\sqrt{x}}{\\log x} (\\log \\log x)^{k-1}+O_k\\left( \\frac{\\sqrt{x}(\\log \\log x)^{k-2}}{\\log x}\\right).$ Then for the oscillating terms, by (REF ), (REF ), and (REF ), and Lemma REF , for $T=T(x)$ , $\\sum _{|\\gamma |\\le T} I_{\\rho }(x)&=\\frac{(-1)^k k\\sqrt{x}(\\log \\log x)^ {k-1}}{\\log x} \\sum _{|\\gamma |\\le T}\\frac{x^{i\\gamma }}{\\frac{1}{2}+i\\gamma }+\\frac{(-1)^k \\sqrt{x}}{\\log x}\\sum _{j=2}^k {k \\atopwithdelims ()j} \\frac{(\\log \\log x)^{k-j}}{\\Gamma _j(0)}\\sum _{|\\gamma |\\le T}\\frac{x^{i\\gamma }}{\\frac{1}{2}+i\\gamma }\\nonumber \\\\&\\quad + \\sum _{|\\gamma |\\le T} E_{M_{\\rho }}(x) + \\sum _{|\\gamma |\\le T} E_{R_{\\rho }}(x)+O_k\\left(\\frac{\\sqrt{x}(\\log \\log x)^{k-1}}{\\log ^2 x} \\right),$ and $\\sum _{|\\gamma |\\le T} \\widetilde{I}_{\\rho }(x)&=\\frac{(-1)^k k\\sqrt{x}(\\log \\log x)^ {k-1}}{\\log x} \\sum _{|\\gamma |\\le T}\\frac{x^{i\\gamma }}{\\frac{1}{2}+i\\gamma }+\\frac{(-1)^k \\sqrt{x}}{\\log x}\\sum _{j=2}^k {k \\atopwithdelims ()j} \\frac{(\\log \\log x)^{k-j}}{\\Gamma _j(0)}\\sum _{|\\gamma |\\le T}\\frac{x^{i\\gamma }}{\\frac{1}{2}+i\\gamma }\\nonumber \\\\&\\quad + \\sum _{|\\gamma |\\le T} \\widetilde{E}_{M_{\\rho }}(x) + \\sum _{|\\gamma |\\le T} \\widetilde{E}_{R_{\\rho }}(x)+O_k\\left(\\frac{\\sqrt{x}(\\log \\log x)^{k-1}}{\\log ^2 x} \\right).$ The first terms in the above formulas are the main oscillating terms in our theorems.", "We will show in Section that the other terms are small in average.", "For $T=T(x)$ , denote $\\Sigma _1(x; \\chi )&:=\\frac{\\log x}{\\sqrt{x}} \\sum _{|\\gamma |\\le T} E_{M_{\\rho }}(x)=\\log x \\sum _{|\\gamma |\\le T} x^{i\\gamma } E^{\\prime }_{M_{\\rho }}(x), \\\\\\Sigma _2(x; \\chi )&:=\\frac{\\log x}{\\sqrt{x}} \\sum _{|\\gamma |\\le T} E_{R_{\\rho }}(x)= \\log x \\sum _{|\\gamma |\\le T} x^{i\\gamma } E^{\\prime }_{R_{\\rho }}(x), $ where $E^{\\prime }_{M_{\\rho }}(x)=\\frac{E_{M_{\\rho }}(x)}{x^{\\rho }}$ , and $E^{\\prime }_{R_{\\rho }}(x)=\\frac{E_{R_{\\rho }}(x)}{x^{\\rho }}$ .", "Similarly, denote $\\widetilde{\\Sigma }_1(x; \\chi )&:=\\frac{\\log x}{\\sqrt{x}} \\sum _{|\\gamma |\\le T} \\widetilde{E}_{M_{\\rho }}(x)= \\log x \\sum _{|\\gamma |\\le T} x^{i\\gamma } \\widetilde{E}^{\\prime }_{M_{\\rho }}(x), \\\\\\widetilde{\\Sigma }_2(x; \\chi )&:=\\frac{\\log x}{\\sqrt{x}} \\sum _{|\\gamma |\\le T} \\widetilde{E}_{R_{\\rho }}(x)= \\log x \\sum _{|\\gamma |\\le T} x^{i\\gamma } \\widetilde{E}^{\\prime }_{R_{\\rho }}(x), $ where $\\widetilde{E}^{\\prime }_{M_{\\rho }}(x)=\\frac{\\widetilde{E}_{M_{\\rho }}(x)}{x^{\\rho }}$ , and $\\widetilde{E}^{\\prime }_{R_{\\rho }}(x)=\\frac{\\widetilde{E}_{R_{\\rho }}(x)}{x^{\\rho }}$ .", "Then we have the following lemma (see Section REF and Section REF for the proof).", "Lemma 10 For the error terms from the Hankel contours around zeros, we have $\\int _2^Y \\left(\\left| \\Sigma _1(e^y; \\chi )\\right|^2+\\left| \\Sigma _2(e^y; \\chi )\\right|^2 \\right) dy&= o\\left(Y(\\log Y)^{2k-2}\\right),\\\\\\int _2^Y \\left(\\left| \\widetilde{\\Sigma }_1(e^y; \\chi )\\right|^2+\\left| \\widetilde{\\Sigma }_2(e^y; \\chi )\\right|^2 \\right) dy&= o\\left(Y(\\log Y)^{2k-2}\\right).$ Moreover, we also need to bound the lower order sum $S_1(x; \\chi ):=(-1)^k \\sum _{j=2}^k {k \\atopwithdelims ()j} (\\log \\log x)^{k-j} \\frac{1}{\\Gamma _j(0)}\\sum _{|\\gamma |\\le T}\\frac{x^{i\\gamma }}{\\frac{1}{2}+i\\gamma },$ and the error from the truncation by a fixed large $T_0$ , $S_2(x, T_0; \\chi ):=\\sum _{|\\gamma |\\le T}\\frac{x^{i\\gamma }}{\\frac{1}{2}+i\\gamma }-\\sum _{|\\gamma |\\le T_0}\\frac{x^{i\\gamma }}{\\frac{1}{2}+i\\gamma }.$ Then we have the following result (See Section REF for the proof).", "Lemma 11 For the lower order sum and error from the truncation, we have $\\int _2^Y \\left| S_1(e^y; \\chi ) \\right|^2 dy=o\\left( Y(\\log Y)^{2k-2}\\right),$ and for fixed large $T_0$ , $\\int _2^Y |S_2(e^y, T_0; \\chi )|^2 dy\\ll Y \\frac{\\log ^2 T_0}{T_0} +\\log Y \\frac{\\log ^3 T_0}{T_0}+\\log ^5 T_0.$ Combining Lemmas REF , REF , and REF with (REF ), (), (REF ), and (REF ), we get, for fixed large $T_0$ , $\\psi _{\\Omega _k}(x, \\chi )&=\\frac{(-1)^k}{(k-1)!}", "\\frac{\\sqrt{x}}{\\log x} (\\log \\log x)^{k-1}\\left(\\sum _{|\\gamma |\\le T_0}\\frac{x^{i\\gamma }}{\\frac{1}{2}+i\\gamma }+\\Sigma (x, T_0; \\chi )\\right) \\nonumber \\\\&\\quad + a(\\chi )\\frac{(-1)^k}{(k-1)!}", "\\frac{\\sqrt{x}}{\\log x} (\\log \\log x)^{k-1},$ where $\\limsup _{Y\\rightarrow \\infty } \\frac{1}{Y} \\int _1^Y \\left|\\Sigma (e^y, T_0; \\chi ) \\right|^2 dy\\ll \\frac{\\log ^2 T_0}{T_0}.$ Also, $\\psi _{\\omega _k}(x, \\chi )&=\\frac{(-1)^k}{(k-1)!}", "\\frac{\\sqrt{x}}{\\log x} (\\log \\log x)^{k-1}\\left(\\sum _{|\\gamma |\\le T_0}\\frac{x^{i\\gamma }}{\\frac{1}{2}+i\\gamma }+\\widetilde{\\Sigma }(x, T_0; \\chi )\\right) \\nonumber \\\\&\\quad + a(\\chi )\\frac{1}{(k-1)!}", "\\frac{\\sqrt{x}}{\\log x} (\\log \\log x)^{k-1},$ where $\\limsup _{Y\\rightarrow \\infty } \\frac{1}{Y} \\int _1^Y \\left|\\widetilde{\\Sigma }(e^y, T_0; \\chi ) \\right|^2 dy\\ll \\frac{\\log ^2 T_0}{T_0}.$ Note that $\\sum _{\\chi \\ne \\chi _0} (\\overline{\\chi }(a)-\\overline{\\chi }(b))a(\\chi )=N(q, a)-N(q, b).$ Hence, combining (REF ) and (REF ) with (REF ) and (REF ), we get the conclusions of Theorem REF and Theorem REF .", "$\\Box $" ], [ "The bias terms", "In this section, we examine the bias terms and give the proof of Lemma REF ." ], [ "Estimates on the horizontal line", "In order to examine the corresponding integration on the horizontal line in the Hankel contour, we prove the following estimate which we will use many times later to analyze the error terms in our theorems.", "Lemma 12 For any integers $k\\ge 1$ and $m\\ge 0$ , we have $\\int _0^{\\delta } |(\\log \\sigma -i\\pi )^k-(\\log \\sigma +i\\pi )^k|\\sigma ^m x^{-\\sigma }d\\sigma \\ll _{m,k} \\frac{(\\log \\log x)^{k-1}}{(\\log x)^{m+1}}.$ Proof.", "Let $I$ represent the integral in the lemma.", "Then, we have $I\\le 2\\sum _{j=1}^k {k\\atopwithdelims ()j} \\pi ^j \\int _0^{\\delta } |\\log \\sigma |^{k-j}\\sigma ^m x^{-\\sigma }d\\sigma \\ll _k \\sum _{j=1}^k \\int _0^{\\delta } |\\log \\sigma |^{k-j}\\sigma ^m x^{-\\sigma }d\\sigma $ Using a change of variable, $\\sigma \\log x=t$ , we have $&\\int _0^{\\delta } |\\log \\sigma |^{k-j}\\sigma ^m x^{-\\sigma }d\\sigma \\le \\frac{1}{(\\log x)^{m+1}} \\int _0^{\\delta \\log x}|\\log t-\\log \\log x|^{k-j}t^m e^{-t}dt\\nonumber \\\\&\\quad \\le \\frac{1}{(\\log x)^{m+1}} \\sum _{l=0}^{k-j} {k-j \\atopwithdelims ()l} (\\log \\log x)^{k-j-l} \\int _0^{\\delta \\log x} |\\log t|^l t^{m} e^{-t}dt\\nonumber \\\\&\\quad \\ll _k \\frac{1}{(\\log x)^{m+1}} \\sum _{l=0}^{k-j} (\\log \\log x)^{k-j-l} \\int _0^{\\delta \\log x} |\\log t|^l t^{m} e^{-t}dt.$ Next, we estimate $\\int _0^{\\delta \\log x} |\\log t|^l t^m e^{-t}dt\\le \\left(\\int _0^1+\\int _1^{\\infty } \\right) |\\log t|^l t^m e^{-t} dt=: I_{l_1}+I_{l_2}.$ For the first integral in (REF ), $I_{l_1}=\\int _0^1 |\\log t|^l t^m e^{-t} dt\\le \\int _0^1 |\\log t|^l dt\\stackrel{t\\rightarrow \\frac{1}{e^t}}{=}\\int _0^{\\infty } \\frac{t^l}{e^t}dt =\\Gamma (l+1).$ For the second integral in (REF ), $I_{l_2}=\\int _1^{\\infty } \\frac{t^m (\\log t)^l}{e^t} dt \\stackrel{t\\rightarrow e^t}{=} \\int _0^{\\infty } \\frac{t^l}{e^{e^t-(m+1)t}} dt \\ll _m \\Gamma (l+1).$ Then, by (REF )-(REF ), we have $\\int _0^{\\delta } |\\log \\sigma |^{k-j}\\sigma ^m x^{-\\sigma }d\\sigma \\ll _k \\frac{1}{(\\log x)^{m+1}} \\sum _{l=0}^{k-j} (\\log \\log x)^{k-j-l}O_{m,l}(1)\\ll _{m, k}\\frac{(\\log \\log x)^{k-j}}{(\\log x)^{m+1}}.$ Thus, by (REF ), $I\\ll _{m,k} \\frac{(\\log \\log x)^{k-1}}{(\\log x)^{m+1}}.$ Hence, we get the conclusion of this lemma.", "$\\Box $" ], [ "The bias terms", "We have the following estimate for the integral over the truncated Hankel contour $\\mathcal {H}(\\frac{1}{2}, \\delta )$ .", "Lemma 13 Assume the function $f(s)=O(1)$ on $\\mathcal {H}(\\frac{1}{2},\\delta )$ .", "Then, for any integer $m\\ge 0$ , $\\left|\\int _{\\mathcal {H}(\\frac{1}{2},\\delta )} \\left(\\log \\left(s-\\frac{1}{2} \\right) \\right)^{m} f(s) \\frac{x^s}{s}ds\\right| \\ll _m \\frac{\\sqrt{x}(\\log \\log x)^{m-1}}{\\log x}.$ Proof.", "Since the left-hand side is 0 when $m=0$ , we assume $m\\ge 1$ in the following proof.", "By Lemma REF , we have $&\\left|\\int _{\\mathcal {H}(\\frac{1}{2},\\delta )} \\left(\\log \\left(s-\\frac{1}{2} \\right) \\right)^{m} f(s) \\frac{x^s}{s}ds\\right|\\nonumber \\\\& \\le \\left|\\int _{r_0}^{\\delta } \\left((\\log \\sigma -i\\pi )^{m}-(\\log \\sigma +i\\pi )^{m} \\right) f\\left(\\frac{1}{2}-\\sigma \\right) \\frac{x^{\\frac{1}{2}-\\sigma }}{\\frac{1}{2}-\\sigma }d\\sigma \\right|\\nonumber \\\\&\\quad +O \\left(\\int _{-\\pi }^{\\pi } \\frac{\\left(\\log \\frac{1}{r_0}+\\pi \\right)^{m}x^{\\frac{1}{2}+r_0} }{\\frac{1}{2}-r_0} r_0 d\\alpha \\right)\\nonumber \\\\& \\ll \\sqrt{x} \\left(\\int _0^{\\delta } \\left|(\\log \\sigma -i\\pi )^{m}-(\\log \\sigma +i\\pi )^{m} \\right| x^{-\\sigma }d\\sigma +\\frac{(\\log x+\\pi )^{m}}{x}\\right)\\nonumber \\\\&\\ll _m \\frac{\\sqrt{x}(\\log \\log x)^{m-1}}{\\log x}.$ This completes the proof of this lemma.", "$\\Box $ In the following, we prove the asymptotic formulas for the bias terms.", "Proof of Lemma REF .", "Since $H_B(s)=O(1)$ , by (REF ), (REF ), and Lemma REF , $|E_B(x)|&\\ll \\sum _{j=1}^k\\left| \\int _{\\mathcal {H}(\\frac{1}{2},\\delta )} \\left(\\log \\left(s-\\frac{1}{2} \\right) \\right)^{k-j} \\left(H_B(s) \\right)^j \\frac{x^s}{s}ds\\right|\\nonumber \\\\&\\ll \\frac{\\sqrt{x}}{\\log x} \\sum _{j=1}^k (\\log \\log x)^{k-j-1}\\ll _k \\frac{\\sqrt{x}}{\\log x} (\\log \\log x)^{k-2}.$ Similarly, $|\\widetilde{E}_B(x)|\\ll \\sum _{j=1}^k\\left| \\int _{\\mathcal {H}(\\frac{1}{2},\\delta )} \\left(\\log \\left(s-\\frac{1}{2} \\right) \\right)^{k-j} \\left(\\widetilde{H}_B(s) \\right)^j \\frac{x^s}{s}ds\\right|\\ll _k \\frac{\\sqrt{x}}{\\log x} (\\log \\log x)^{k-2}.$ In the following, we estimate $E_R(x)$ in (REF ) and $\\widetilde{E}_R(x)$ in (REF ).", "If $\\chi $ is not real, $E_R(x)=\\widetilde{E}_R(x)=0$ .", "If $\\chi $ is real, by (REF ), on $\\mathcal {H}(\\frac{1}{2}, \\delta )$ , we write $F(2s, \\chi ^2)=-\\log \\left(s-\\frac{1}{2}\\right)+H_2(s).$ On $\\mathcal {H}(\\frac{1}{2}, \\delta )$ , $|H_2(s)|=O(1)$ .", "By (REF ), we have $|E_{R}(x)|\\ll \\sum _{m=0}^{k-2} \\sum _{n\\in S^{(k)}_m}\\left|\\int _{\\mathcal {H}(\\frac{1}{2}, \\delta )} F^m(s, \\chi ) F(ns, \\chi )\\frac{x^s}{s}ds \\right|.$ For each $0\\le m\\le k-2$ , we write $F^m(s, \\chi ) F(ns, \\chi )=F^m(s, \\chi ) F^{m^{\\prime }}(2s,\\chi ^2)G_{n}(s),$ where $m+2m^{\\prime }\\le k$ , and $G_{n}(s)=O(1)$ on $\\mathcal {H}(\\frac{1}{2}, \\delta )$ .", "Thus, by (REF ), (REF ), and Lemma REF , $&&\\left|\\int _{\\mathcal {H}(\\frac{1}{2}, \\delta )} F^m(s, \\chi ) F(ns, \\chi )\\frac{x^s}{s}ds \\right| \\nonumber \\\\&& \\ll \\left| \\int _{\\mathcal {H}(\\frac{1}{2},\\delta )} \\left(\\log \\left(s-\\frac{1}{2} \\right) +H_B(s)\\right)^{m} \\left(\\log \\left(s-\\frac{1}{2} \\right)-H_2(s) \\right)^{m^{\\prime }}G_{n}(s)\\frac{x^s}{s}ds\\right|\\nonumber \\\\&& \\ll \\sum _{j_1=0}^m \\sum _{j_2=0}^{m^{\\prime }} \\left|\\int _{\\mathcal {H}(\\frac{1}{2}, \\delta )}\\left(\\log \\left(s-\\frac{1}{2}\\right) \\right)^{m+m^{\\prime }-j_1-j_2} (H_B(s))^{j_1} (H_2(s))^{j_2} G_{n}(s) \\frac{x^s}{s}ds \\right|\\nonumber \\\\&& \\ll \\sum _{j_1=0}^m \\sum _{j_2=0}^{m^{\\prime }} \\frac{\\sqrt{x}}{\\log x} (\\log \\log x)^{m+m^{\\prime }-j_1-j_2-1} \\ll _k \\frac{\\sqrt{x}}{\\log x} (\\log \\log x)^{k-2}.$ In the last step, we used the conditions $0\\le m\\le k-2$ and $m+2m^{\\prime }\\le k$ .", "Combining (REF ) and (REF ), we deduce that $|E_R(x)|\\ll _k \\frac{ \\sqrt{x}}{\\log x} (\\log \\log x)^{k-2}.$ Similarly, if $\\chi $ is real, by (REF ), we write $\\widetilde{F}(s, \\chi ; 2)=-\\log \\left(s-\\frac{1}{2}\\right)+\\widetilde{H}_2(s),$ where $\\widetilde{H}_2(s)=O(1)$ on $\\mathcal {H}(\\frac{1}{2}, \\delta )$ .", "Using a similar argument as above, by (), (REF ), and Lemma REF , we have $|\\widetilde{E}_R(x)|\\ll \\sum _{m=0}^{k-2} \\sum _{n\\in S^{(k)}_m}\\left|\\int _{\\mathcal {H}(\\frac{1}{2}, \\delta )} \\widetilde{F}^m(s, \\chi ) \\widetilde{F}(ns, \\chi )\\frac{x^s}{s}ds \\right|\\ll _k \\frac{ \\sqrt{x}}{\\log x} (\\log \\log x)^{k-2}.$ By (REF ), (REF ), (REF ), and (REF ), we get $I_B(x)=\\frac{(-1)^k\\sqrt{x}}{2^{k-1}\\log x}\\left\\lbrace k (\\log \\log x)^ {k-1}+\\sum _{j=2}^k {k \\atopwithdelims ()j} (\\log \\log x)^{k-j} \\frac{1}{\\Gamma _j(0)}\\right\\rbrace +O_k\\left(\\frac{\\sqrt{x}(\\log \\log x)^{k-2}}{\\log x}\\right).$ Then, by (REF ), $\\left|\\sum _{j=2}^k {k \\atopwithdelims ()j} (\\log \\log x)^{k-j} \\frac{1}{\\Gamma _j(0)}\\right|\\ll _k (\\log \\log x)^{k-2}.$ Hence, $I_B(x)=\\frac{(-1)^k k}{2^{k-1}}\\frac{\\sqrt{x}}{\\log x} (\\log \\log x)^{k-1}+O_k\\left( \\frac{\\sqrt{x}(\\log \\log x)^{k-2}}{\\log x}\\right).$ Similarly, by (REF ), (REF ), (REF ), and (REF ), we have $\\widetilde{I}_B(x)=\\frac{ k}{2^{k-1}}\\frac{\\sqrt{x}}{\\log x} (\\log \\log x)^{k-1}+O_k\\left( \\frac{\\sqrt{x}(\\log \\log x)^{k-2}}{\\log x}\\right).$ This completes the proof of Lemma REF .", "$\\Box $" ], [ "Average order of the error terms", "In Section REF and Section REF , we examine the error terms from the Hankel contours around zeros and give the proof of Lemma REF .", "In Section REF , we examine the lower order sum and the error from the truncation, and give the proof of Lemma REF ." ], [ "Error terms from the Hankel contours around zeros", "In the section, we give the proof of Lemma REF .", "The following lemma gives an average estimate for the integral over Hankel contours around zeros, which is the key lemma for our proof.", "Lemma 14 Let $\\rho $ be a zero of $L(s, \\chi )$ .", "Assume the function $g(s)\\ll \\left( \\log |\\gamma |\\right)^c$ on $\\mathcal {H}(\\rho , \\delta )$ for some constant $c\\ge 0$ , and $H_{\\rho }(s)=\\sum _{0<|\\gamma ^{\\prime }-\\gamma |\\le 1} \\log (s-\\rho ^{\\prime }) +O\\left(\\log |\\gamma | \\right)\\quad {\\rm on~} \\mathcal {H}(\\rho ,\\delta ).$ For any integers $m, n\\ge 0$ , denote $E(x; \\rho ):=\\int _{\\mathcal {H}(\\rho ,\\delta )} \\left(\\log (s-\\rho )\\right)^{m} \\left(H_{\\rho }(s)\\right)^n g(s) \\frac{x^{s-\\rho }}{s}ds.$ Then, for $T=T(x)$ , we have $\\int _{2}^Y \\left|y \\sum _{|\\gamma |\\le T(e^y)} e^{i\\gamma y} E(e^y; \\rho ) \\right|^2 dy=o\\left( Y(\\log Y)^{2m+2n-2} \\right).$ We will give the proof of Lemma REF in next subsection.", "We use it in this section to prove Lemma REF first.", "Proof of Lemma REF .", "By (REF ), we have $\\left| \\Sigma _1(x; \\chi ) \\right|^2 =\\left|\\log x\\sum _{|\\gamma |\\le T} x^{i\\gamma } E^{\\prime }_{M_{\\rho }}(x)\\right|^2\\ll \\sum _{j=1}^k \\left| \\log x \\sum _{|\\gamma |\\le T} x^{i\\gamma } E_{\\rho , j}(x) \\right|^2,$ where $E_{\\rho , j}(x)=\\int _{\\mathcal {H}(\\rho ,\\delta )} (\\log (s-\\rho ))^{k-j} (H_{\\rho }(s))^j \\frac{x^{s-\\rho }}{s}ds.$ By Lemma REF , take $m=k-j$ , $n=j$ , and $g(s)\\equiv 1$ , (i.e.", "$c=0$ ), $\\int _2^Y \\left|y \\sum _{|\\gamma |\\le T(e^y)} e^{i\\gamma y} E_{\\rho , j}(e^y) \\right|^2 dy=o\\left(Y(\\log Y)^{2k-2}\\right).$ Thus, $\\int _2^Y \\left| \\Sigma _1(e^y; \\chi ) \\right|^2 dy=o\\left(Y(\\log Y)^{2k-2}\\right).$ By definition (REF ) and (), we have $\\left| \\Sigma _2(x; \\chi ) \\right|^2&\\ll \\sum _{m=0}^{k-2} \\sum _{n\\in S^{(k)}_m}\\left| \\log x \\sum _{|\\gamma |\\le T} x^{i\\gamma }\\int _{\\mathcal {H}(\\rho , \\delta )} F^m(s, \\chi )F(ns, \\chi ) \\frac{x^{s-\\rho }}{s}ds \\right|^2\\nonumber \\\\&\\ll \\sum _{m=0}^{k-2} \\sum _{n\\in S^{(k)}_m} \\sum _{j=0}^m \\left|\\log x \\sum _{|\\gamma |\\le T} x^{i\\gamma } E_{m, j}(x, \\chi ; n) \\right|^2,$ where $E_{m,j}(x, \\chi ; n)=\\int _{\\mathcal {H}(\\rho , \\delta )} \\left(\\log (s-\\rho ) \\right)^{m-j} \\left(H_{\\rho }(s) \\right)^j F(ns, \\chi ) \\frac{x^{s-\\rho }}{s}ds.$ Since on $\\mathcal {H}(\\rho , \\delta )$ , we know $F(ns, \\chi )=O\\left((\\log |\\gamma |)^{\\frac{k-m}{2}} \\right)$ , by Lemma REF , we get $\\int _2^Y \\left|y \\sum _{|\\gamma |\\le T(e^y)} e^{i\\gamma y} E_{m, j}(e^y, \\chi ; n) \\right|^2 dy=o\\left( Y(\\log Y)^{2m-2}\\right).$ Hence, by (REF ), we deduce that $\\int _2^Y \\left| \\Sigma _2(e^y; \\chi ) \\right|^2 dy =o\\left(Y(\\log Y)^{2k-2} \\right).$ Combining (REF ) and (REF ), we get the first formula in Lemma REF .", "For $\\widetilde{\\Sigma }_1(x; \\chi )$ and $\\widetilde{\\Sigma }_2(x, \\chi )$ , by (REF ), using a similar argument with Lemma REF , $\\int _2^Y \\left| \\widetilde{\\Sigma }_1(e^y; \\chi ) \\right|^2 dy\\ll \\sum _{j=1}^k \\int _2^Y \\left| y \\sum _{|\\gamma |\\le T(e^y)} e^{i\\gamma y} \\widetilde{E}_{\\rho , j}(e^y) \\right|^2 dy=o\\left(Y(\\log Y)^{2k-2} \\right),$ where $\\widetilde{E}_{\\rho , j}(x)=\\int _{\\mathcal {H}(\\rho ,\\delta )} (\\log (s-\\rho ))^{k-j} (\\widetilde{H}_{\\rho }(s))^j \\frac{x^{s-\\rho }}{s}ds.$ Similarly, by () and Lemma REF , $\\int _2^Y \\left| \\widetilde{\\Sigma }_2(e^y; \\chi ) \\right|^2 dy\\ll \\sum _{m=0}^{k-2} \\sum _{n\\in S^{(k)}_m} \\sum _{j=0}^m \\int _2^Y \\left|y \\sum _{|\\gamma |\\le T(e^y)} e^{i\\gamma y} \\widetilde{E}_{m, j}(e^y, \\chi ; n) \\right|^2 dy= o\\left(Y(\\log Y)^{2k-2} \\right),$ where $\\widetilde{E}_{m,j}(x, \\chi ; n)=\\int _{\\mathcal {H}(\\rho , \\delta )} \\left(\\log (s-\\rho ) \\right)^{m-j} \\left(\\widetilde{H}_{\\rho }(s) \\right)^j \\widetilde{F}(ns, \\chi ) \\frac{x^{s-\\rho }}{s}ds.$ Combining (REF ) and (REF ), we get the second formula in Lemma REF .", "$\\Box $" ], [ "Estimates for integral over Hankel contours around zeros", "We need the following results to finish the proof of Lemma REF .", "Lemma 15 ([6], Lemma 2.4) Assume $L(\\frac{1}{2}, \\chi )\\ne 0$ .", "For $A\\ge 0$ and real $l\\ge 0$ , $\\sum _{\\begin{array}{c}|\\gamma _1|, |\\gamma _2|\\ge A\\\\ |\\gamma _1-\\gamma _2|\\ge 1\\end{array}} \\frac{\\log ^l (|\\gamma _1|+3) \\log ^l (|\\gamma _2| +3)}{|\\gamma _1||\\gamma _2||\\gamma _1-\\gamma _2|}\\ll _l \\frac{(\\log (A+3))^{2l+3}}{A+1}.$ Lemma 16 For any integers $N, j \\ge 1$ , and $0<|\\delta _n|\\le 1$ , we have $\\int _0^{\\delta }\\left| \\sum _{n=1}^N \\log (\\sigma +i\\delta _n)\\right|^j x^{-\\sigma }d\\sigma \\ll _j \\frac{1}{\\log x}\\left\\lbrace \\min \\left(N\\log \\log x, \\log \\frac{1}{\\Delta _N}\\right)+N\\pi \\right\\rbrace ^j,$ where $\\Delta _N=\\prod _{n=1}^N |\\delta _n|$ .", "Proof.", "Let $I$ denote the integral in the lemma.", "We consider two cases: $\\Delta _N\\ge \\left(\\frac{1}{\\log x}\\right)^N$ , and $\\Delta _N< \\left(\\frac{1}{\\log x}\\right)^N$ .", "1) If $\\Delta _N\\ge \\left(\\frac{1}{\\log x}\\right)^N$ , we have $I\\ll \\left(\\log \\frac{1}{\\Delta _N}+N\\pi \\right)^j \\int _0^{\\delta } x^{-\\sigma }d\\sigma \\ll \\frac{1}{\\log x} \\left(\\log \\frac{1}{\\Delta _N}+N\\pi \\right)^j\\ll \\frac{1}{\\log x}(N\\log \\log x+N\\pi )^j.$ 2) If $\\Delta _N< \\left(\\frac{1}{\\log x}\\right)^N$ , we write $I=\\left(\\int _0^{(\\Delta _N)^{\\frac{1}{N}}}+\\int _{(\\Delta _N)^{\\frac{1}{N}}}^{\\frac{1}{\\log x}}+\\int _{\\frac{1}{\\log x}}^{\\delta } \\right)\\left| \\sum _{n=1}^N \\log (\\sigma +i\\delta _n)\\right|^j x^{-\\sigma }d\\sigma =: I_1+I_2+I_3.$ First, we estimate $I_1$ , $I_1\\ll \\left(\\log \\frac{1}{\\Delta _N}+N\\pi \\right)^j \\int _0^{(\\Delta _N)^{\\frac{1}{N}}} x^{-\\sigma }d\\sigma \\ll (\\Delta _N)^{\\frac{1}{N}} \\left(\\log \\frac{1}{\\Delta _N}+N\\pi \\right)^j.$ For $0<t<1$ , consider the function $f(t)=t^{\\frac{1}{N}}\\left(\\log \\frac{1}{t}+N\\pi \\right)^j.$ Since the critical point of $f(t)$ is $t=e^{N(\\pi -1)}>1$ , by (REF ), we have $I_1\\ll f\\left(\\frac{1}{(\\log x)^N}\\right)=\\frac{1}{\\log x}(N\\log \\log x+N\\pi )^j\\ll \\frac{1}{\\log x} \\left(\\log \\frac{1}{\\Delta _N}+N\\pi \\right)^j.$ Next, we estimate $I_3$ .", "Using the change of variable $\\sigma \\log x=t$ , we get $I_3&\\ll & \\int _{\\frac{1}{\\log x}}^{\\delta } \\left(N\\log \\frac{1}{\\sigma }+N\\pi \\right)^j x^{-\\sigma } d\\sigma \\nonumber \\\\&=& \\frac{1}{\\log x}\\int _1^{\\delta \\log x} (N\\log \\log x-N\\log t+N\\pi )^j e^{-t} dt\\nonumber \\\\&=& \\frac{N^j}{\\log x} \\sum _{l=0}^j {j\\atopwithdelims ()l} (\\log \\log x+\\pi )^{j-l} \\int _1^{\\delta \\log x} (-\\log t)^l e^{-t} dt\\nonumber \\\\&\\ll _j& \\frac{N^j}{\\log x} \\sum _{l=0}^j (\\log \\log x+\\pi )^{j-l} \\int _1^{\\infty }\\frac{t^l}{e^t}dt\\nonumber \\\\&\\ll _j& \\frac{ (N\\log \\log x+N\\pi )^j}{\\log x}\\ll \\frac{1}{\\log x} \\left(\\log \\frac{1}{\\Delta _N}+N\\pi \\right)^j.$ For $I_2$ , similar to $I_3$ , using the change of variable $\\sigma \\log x=t$ , we get $I_2&\\ll & \\int _{(\\Delta _N)^{\\frac{1}{N}}}^{\\frac{1}{\\log x}} \\left(N\\log \\frac{1}{\\sigma }+N\\pi \\right)^j x^{-\\sigma } d\\sigma \\nonumber \\\\&=& \\frac{1}{\\log x}\\int _{(\\Delta _N)^{\\frac{1}{N}}\\log x}^1 (N\\log \\log x-N\\log t+N\\pi )^j e^{-t} dt\\nonumber \\\\&=& \\frac{N^j}{\\log x} \\sum _{l=0}^j {j\\atopwithdelims ()l} (\\log \\log x+\\pi )^{j-l} \\int _{(\\Delta _N)^{\\frac{1}{N}}\\log x}^1 (-\\log t)^l e^{-t} dt \\quad (t\\rightarrow \\frac{1}{e^t}) \\nonumber \\\\&\\ll _j& \\frac{N^j}{\\log x} \\sum _{l=0}^j (\\log \\log x+\\pi )^{j-l} \\int _0^{\\infty } \\frac{t^l}{e^t}dt\\nonumber \\\\&\\ll _j& \\frac{ (N\\log \\log x+N\\pi )^j}{\\log x} \\ll \\frac{1}{\\log x} \\left(\\log \\frac{1}{\\Delta _N}+N\\pi \\right)^j.$ Combining (REF ), (REF ), (REF ), with (REF ), we get $I\\ll _j \\frac{ (N\\log \\log x+N\\pi )^j}{\\log x} \\ll _j \\frac{1}{\\log x} \\left(\\log \\frac{1}{\\Delta _N}+N\\pi \\right)^j.$ By (REF ) and (REF ), we get the conclusion of this lemma.", "$\\Box $ In the following, we use the above lemmas to prove Lemma REF .", "Proof of Lemma REF .", "If $m=0$ , $E(x; \\rho )=0$ and hence the integral is 0.", "In the following, we assume $m\\ge 1$ .", "Let $\\Gamma _{\\rho }$ represent the circle in the Hankel contour $\\mathcal {H}(\\rho , \\delta )$ .", "Then, $E(x; \\rho )&=\\int _{\\mathcal {H}(\\rho ,\\delta )} \\left(\\log (s-\\rho )\\right)^{m} \\left(H_{\\rho }(s)\\right)^n g(s) \\frac{x^{s-\\rho }}{s}ds \\nonumber \\\\&= \\int _{r_{\\rho }}^{\\delta } \\left((\\log \\sigma -i\\pi )^{m}-(\\log \\sigma +i\\pi )^{m} \\right) \\left(H_{\\rho }\\left(\\frac{1}{2}-\\sigma +i\\gamma \\right)\\right)^n g\\left(\\frac{1}{2}-\\sigma +i\\gamma \\right)\\nonumber \\\\& \\quad \\quad \\times \\frac{x^{-\\sigma }}{\\frac{1}{2}-\\sigma +i\\gamma }d\\sigma +\\int _{\\Gamma _{\\rho }} \\left(\\log (s-\\rho )\\right)^{m} \\left(H_{\\rho }(s)\\right)^n g(s) \\frac{x^{s-\\rho }}{s}ds.\\nonumber \\\\&=: E_h(x; \\rho )+E_r(x; \\rho ).$ For the second integral in (REF ), since $r_{\\rho }\\le \\frac{1}{x}$ , by Lemma REF , $\\left|E_r(x; \\rho ) \\right|&\\ll \\frac{(\\log |\\gamma |)^c r_{\\rho } x^{r_{\\rho }} }{|\\gamma |} \\left(\\log \\frac{1}{r_{\\rho }} +\\pi \\right)^{m} \\left(\\sum _{0<|\\gamma -\\gamma ^{\\prime }|\\le 1} \\log \\left(\\frac{1}{|\\gamma ^{\\prime }-\\gamma |-r_{\\rho }}\\right) +O(\\log |\\gamma |) \\right)^n\\nonumber \\\\&\\ll \\frac{(\\log |\\gamma |)^c r_{\\rho } x^{r_{\\rho }} }{|\\gamma |} \\left(\\log \\frac{1}{r_{\\rho }} +\\pi \\right)^{m} (\\log |\\gamma |)^n\\left( \\log \\left(\\frac{1}{r_{\\rho }} \\right)+O(1) \\right)^n\\nonumber \\\\&\\ll \\frac{(\\log |\\gamma | )^{n+c}}{|\\gamma |} \\frac{ \\left(\\log (1/r_{\\rho })+\\pi \\right)^{m+n}}{1/r_{\\rho }}\\ll \\frac{(\\log |\\gamma | )^{n+c}}{|\\gamma |} \\frac{1}{x^{1-\\epsilon }}.$ Denote $\\Sigma (x; {\\rm g}):= \\left| \\sum _{|\\gamma |\\le T} x^{i\\gamma } E(x; \\rho ) \\right|^2\\ll \\left| \\sum _{|\\gamma |\\le T} x^{i\\gamma } E_h(x; \\rho ) \\right|^2+\\left| \\sum _{|\\gamma |\\le T} x^{i\\gamma } E_r(x; \\rho ) \\right|^2.$ By (REF ), and $ T(x)\\ll x^2$ , we get $\\left| \\sum _{|\\gamma |\\le T} x^{i\\gamma } E_r(x; \\rho ) \\right|^2\\ll \\frac{1}{x^{2-\\epsilon }}\\left(\\sum _{|\\gamma |\\le T(x)} \\frac{(\\log |\\gamma | )^{n+c}}{|\\gamma |}\\right)^2 \\ll \\frac{1}{x^{2-\\epsilon }}.$ For the first sum in (REF ), $\\left| \\sum _{|\\gamma |\\le T} x^{i\\gamma } E_h(x; \\rho ) \\right|^2&=\\left(\\sum _{\\begin{array}{c}|\\gamma _1-\\gamma _2|\\le 1\\\\|\\gamma _1|, |\\gamma _2|\\le T\\end{array}}+\\sum _{\\begin{array}{c}|\\gamma _1-\\gamma _2|> 1\\\\|\\gamma _1|, |\\gamma _2|\\le T\\end{array}}\\right) x^{i(\\gamma _1-\\gamma _2)} E_h(x; \\rho _1) E_h(x; \\overline{\\rho }_2)\\nonumber \\\\&=:\\Sigma _1(x; {\\rm g})+\\Sigma _2(x; {\\rm g}).$ By (REF ), $\\left| E_h(x; \\rho )\\right| \\ll \\frac{(\\log |\\gamma |)^c}{|\\gamma |} \\sum _{j=1}^{m} \\int _0^{\\delta } |\\log \\sigma |^{m-j} \\left|H_{\\rho }\\left(\\frac{1}{2}-\\sigma +i\\gamma \\right) \\right|^n x^{-\\sigma }d\\sigma .$ Let $S_j(x)&:=\\int _0^{\\delta } |\\log \\sigma |^{m-j} \\left|H_{\\rho }\\left(\\frac{1}{2}-\\sigma +i\\gamma \\right) \\right|^n x^{-\\sigma }d\\sigma \\nonumber \\\\&\\le \\left(\\int _0^{\\delta } |\\log \\sigma |^{2(m-j)} x^{-\\sigma }d\\sigma \\right)^{\\frac{1}{2}} \\left(\\int _0^{\\delta } \\left|H_{\\rho }\\left(\\frac{1}{2}-\\sigma +i\\gamma \\right) \\right|^{2n} x^{-\\sigma }d\\sigma \\right)^{\\frac{1}{2}}.$ By (REF ) in the proof of Lemma REF , $\\int _0^{\\delta } |\\log \\sigma |^{2(m-j)} x^{-\\sigma }d\\sigma \\ll \\frac{(\\log \\log x)^{2(m-j)}}{\\log x}.$ By condition (REF ) and the Cauchy-Schwarz inequality, $\\left|H_{\\rho }\\left(\\frac{1}{2}-\\sigma +i\\gamma \\right) \\right|^{2n}\\ll \\left|\\sum _{0<|\\gamma ^{\\prime }-\\gamma |\\le 1} \\log (\\sigma +i(\\gamma ^{\\prime }-\\gamma ))\\right|^{2n}+(\\log |\\gamma |)^{2n}.$ Then, by Lemma REF , $\\int _0^{\\delta } \\left|H_{\\rho }\\left(\\frac{1}{2}-\\sigma +i\\gamma \\right) \\right|^{2n} x^{-\\sigma }d\\sigma \\ll \\frac{(M_{\\gamma }(x))^{2n}+(\\log |\\gamma |)^{2n}}{\\log x},$ where $M_{\\gamma }(x)=\\min \\left(N(\\gamma )\\log \\log x, \\log \\frac{1}{\\Delta _{N(\\gamma )}} \\right),$ $N(\\gamma )$ is the number of zeros $\\gamma ^{\\prime }$ in the range $ 0<|\\gamma ^{\\prime }-\\gamma |\\le 1 $ , and $\\Delta _{N(\\gamma )}=\\prod \\limits _{0<|\\gamma ^{\\prime }-\\gamma |\\le 1} |\\gamma ^{\\prime }-\\gamma |$ .", "Thus, by (REF ) and (REF ), $S_j(x)\\ll \\frac{(\\log \\log x)^{m-j}}{\\log x}\\left( (M_{\\gamma }(x))^n+(\\log |\\gamma |)^{n}\\right).$ Substituting this into (REF ), we get $\\left| E_h(x; \\rho ) \\right|&\\ll \\frac{(\\log |\\gamma |)^c}{|\\gamma |} \\sum _{j=1}^{m}\\frac{(\\log \\log x)^{m-j}}{\\log x}\\left( (M_{\\gamma }(x))^n+(\\log |\\gamma |)^{n}\\right)\\nonumber \\\\&\\ll \\frac{(\\log |\\gamma |)^c}{|\\gamma |}\\frac{(\\log \\log x)^{m-1}}{\\log x}\\left( (M_{\\gamma }(x))^n+(\\log |\\gamma |)^{n}\\right).$ Then, by Lemma REF , we have $\\left|\\Sigma _1(x; \\rm g) \\right|&\\ll \\sum _{|\\gamma |\\le T} \\log (|\\gamma |) \\left(\\max _{|\\gamma ^{\\prime }-\\gamma |<1}\\left|E_h(x; \\rho ^{\\prime })\\right|\\right)^2\\nonumber \\\\&\\ll \\frac{(\\log \\log x)^{2(m-1)}}{\\log ^2 x} \\sum _{\\gamma }\\frac{(\\log |\\gamma |)^{2c}}{|\\gamma |^2}\\left((M_{\\gamma }(x))^{2n}+\\left(\\log |\\gamma | \\right)^{2n} \\right)\\nonumber \\\\&= \\frac{(\\log \\log x)^{2m+2n-2}}{\\log ^2 x} o(1).$ Thus, for each positive integer $l$ , $\\int _{2^l}^{2^{l+1}} \\Sigma _1(e^y; {\\rm g}) dy=o\\left( \\frac{l^{2m+2n-2}}{2^l} \\right).$ In the following, we examine $\\Sigma _2(x; {\\rm g})$ .", "By (REF ), $\\Sigma _2(x; {\\rm g})=\\sum _{\\begin{array}{c}|\\gamma _1-\\gamma _2|>1\\\\ |\\gamma _1|, |\\gamma _2|\\le T\\end{array}}x^{i(\\gamma _1-\\gamma _2)} E_h(x; \\rho _1) E_h(x; \\overline{\\rho }_2).$ For $e^{2^l}\\le x\\le e^{2^{l+1}}$ , $T=T(x)=T_{l^{\\prime }}$ is a constant, and so we define $J(x; {\\rm g}):= \\sum _{\\begin{array}{c}|\\gamma _1-\\gamma _2|>1\\\\|\\gamma _1|, |\\gamma _2|\\le T_{l^{\\prime }}\\end{array}} x^{i(\\gamma _1-\\gamma _2)} \\int _{r_{\\rho _1}}^{\\delta } \\int _{r_{\\overline{\\rho }_2}}^{\\delta } R_{\\rho _1}(\\sigma _1; x) R_{\\overline{\\rho }_2}(\\sigma _2; x) \\frac{d\\sigma _1 d\\sigma _2}{i(\\gamma _1-\\gamma _2)-(\\sigma _1+\\sigma _2)},$ where $ R_{\\rho }(\\sigma ; x)=\\left((\\log \\sigma -i\\pi )^{m}-(\\log \\sigma +i\\pi )^{m} \\right) H^n_{\\rho }\\left(\\frac{1}{2}-\\sigma +i\\gamma \\right) \\frac{g\\left(\\frac{1}{2}-\\sigma +i\\gamma \\right)x^{-\\sigma }}{\\frac{1}{2}-\\sigma +i\\gamma }.$ Thus, $\\int _{e^{2^l}}^{e^{2^{l+1}}} \\sum _{\\begin{array}{c}|\\gamma _1-\\gamma _2|>1\\\\|\\gamma _1|, |\\gamma _2|\\le T_{l^{\\prime }}\\end{array}} x^{i(\\gamma _1-\\gamma _2)} E_h(x; \\rho _1) E_h(x;\\overline{\\rho }_2)\\frac{dx}{x}=J(e^{2^{l+1}}; {\\rm g})-J(e^{2^l}; {\\rm g}).$ By (REF ), (REF ), and (REF ), and Lemma REF , for $e^{2^l}\\le x\\le e^{2^{l+1}}$ $|J(x; {\\rm g})|&\\ll \\sum _{|\\gamma _1-\\gamma _2|>1} \\frac{(\\log |\\gamma _1|)^c (\\log |\\gamma _2|)^c}{|\\gamma _1||\\gamma _2||\\gamma _1-\\gamma _2|} \\left(\\frac{(\\log \\log x)^{m-1}}{\\log x}\\right)^2 \\nonumber \\\\& \\qquad \\times \\left( (M_{\\gamma _1}(x))^n+(\\log |\\gamma _1|)^{n}\\right)\\left( (M_{\\gamma _2}(x))^n+(\\log |\\gamma _2|)^{n}\\right)\\nonumber \\\\&\\ll \\frac{(\\log \\log x)^{2m+2n-2}}{\\log ^2 x}\\sum _{|\\gamma _1-\\gamma _2|>1} \\frac{(\\log |\\gamma _1|)^{n+c} (\\log |\\gamma _2|)^{n+c}}{|\\gamma _1||\\gamma _2||\\gamma _1-\\gamma _2|}\\ll \\frac{(\\log \\log x)^{2m+2n-2}}{\\log ^2 x}.$ Hence, by (REF ), (REF ), and (REF ), we get, for any positive integer $l$ , $\\int _{2^l}^{2^{l+1}} \\Sigma _2(e^y; {\\rm g}) dy=o\\left( \\frac{l^{2m+2n-2}}{2^l} \\right).$ Therefore, by (REF ), (REF ) and (REF ), $&&\\int _{2}^Y \\left| y \\sum _{|\\gamma |\\le T(e^y)} e^{i\\gamma y} E(e^y; \\rho ) \\right|^2 dy\\ll \\sum _{l\\le \\frac{\\log Y}{\\log 2}+1} 2^{2l}\\int _{2^l}^{2^{l+1}} \\Sigma (e^y; {\\rm g}) dy \\\\&&\\quad \\ll 1+\\sum _{l\\le \\frac{\\log Y}{\\log 2}+1} 2^{2l}\\int _{2^l}^{2^{l+1}} \\left(\\Sigma _1(e^y; {\\rm g}) +\\Sigma _2(e^y; {\\rm g})\\right)dy=o\\left( Y(\\log Y)^{2m+2n-2} \\right).$ This completes the proof of Lemma REF .", "$\\Box $" ], [ "Lower order sum and error from the truncation", "In this section, we examine the lower order sum and the error from the truncation by a fixed large $T_0$ , and give the proof of Lemma REF .", "For the lower order sum, by (REF ), we have $\\int _2^Y \\left| S_1(e^y; \\chi ) \\right|^2 dy\\ll \\sum _{j=2}^k (\\log Y)^{2k-2j} \\int _2^Y \\left| \\sum _{|\\gamma |\\le T(e^y)}\\frac{e^{i\\gamma y}}{\\frac{1}{2}+i\\gamma } \\right|^2 dy.$ For the inner integral, by Lemma REF and Lemma REF , and the definition of $T=T(x)$ , $\\int _2^Y \\left| \\sum _{|\\gamma |\\le T(e^y)}\\frac{e^{i\\gamma y}}{\\frac{1}{2}+i\\gamma } \\right|^2 dy&\\le \\sum _{l\\le \\frac{\\log Y}{\\log 2}+1} \\int _{2^l}^{2^{l+1}}\\left( \\sum _{\\begin{array}{c}|\\gamma _1-\\gamma _2|\\le 1\\\\ |\\gamma _1|, |\\gamma _2|\\le T_{l^{\\prime }}\\end{array}} +\\sum _{\\begin{array}{c}|\\gamma _1-\\gamma _2|> 1\\\\ |\\gamma _1|, |\\gamma _2|\\le T_{l^{\\prime }}\\end{array}} \\right) \\frac{e^{i(\\gamma _1-\\gamma _2)y}}{(\\frac{1}{2}+i\\gamma _1)(\\frac{1}{2}-i\\gamma _2)} dy \\nonumber \\\\&\\ll \\sum _{l\\le \\frac{\\log Y}{\\log 2}+1} \\left(2^l\\sum _{\\gamma } \\frac{\\log |\\gamma |}{|\\gamma |^2}+ \\sum _{\\gamma _1, \\gamma _2} \\frac{1}{|\\gamma _1| |\\gamma _2| |\\gamma _1-\\gamma _2|}\\right)\\ll Y.$ Thus, $\\int _2^Y \\left| S_1(e^y; \\chi ) \\right|^2 dy\\ll \\sum _{j=2}^k Y (\\log Y)^{2k-2j}=o(Y (\\log Y)^{2k-2})).$ Next, we examine $S_2(x, T_0; \\chi )$ .", "For fixed $T_0$ , let $X_0$ be the largest $x$ such that $T=T(x)\\le T_0$ .", "Since $x\\le T(x)\\le 2 x^2$ , $\\log X_0 \\asymp \\log T_0$ .", "By Lemma REF and Lemma REF , $&&\\int _2^Y | S_2(e^y, T_0; \\chi )|^2 dy\\le \\int _2^{\\log X_0} \\left| \\sum _{|\\gamma |\\le T_0} \\frac{1}{|\\gamma |} \\right|^2 dy +\\int _{\\log X_0}^Y \\left| \\sum _{T_0\\le |\\gamma |\\le T(e^y)} \\frac{e^{i\\gamma y}}{\\frac{1}{2}+i\\gamma } \\right|^2 dy \\nonumber \\\\&& \\ll \\log ^5 T_0+\\sum _{\\frac{\\log \\log X_0}{\\log 2}\\le l\\le \\frac{\\log Y}{\\log 2}+1} \\int _{2^l}^{2^{l+1}} \\left( \\sum _{\\begin{array}{c}|\\gamma _1-\\gamma _2|\\le 1\\\\ T_0\\le |\\gamma _1|, |\\gamma _2|\\le T_{l^{\\prime }}\\end{array}} +\\sum _{\\begin{array}{c}|\\gamma _1-\\gamma _2|> 1\\\\ T_0\\le |\\gamma _1|, |\\gamma _2|\\le T_{l^{\\prime }}\\end{array}} \\right) \\frac{e^{i(\\gamma _1-\\gamma _2)y}}{(\\frac{1}{2}+i\\gamma _1)(\\frac{1}{2}-i\\gamma _2)} dy \\nonumber \\\\&&\\ll \\log ^5 T_0+ \\sum _{\\frac{\\log \\log X_0}{\\log 2}\\le l\\le \\frac{\\log Y}{\\log 2}+1} \\left(2^l\\sum _{|\\gamma |\\ge T_0} \\frac{\\log |\\gamma |}{|\\gamma |^2}+ \\sum _{|\\gamma _1|, |\\gamma _2| \\ge T_0} \\frac{1}{|\\gamma _1||\\gamma _2||\\gamma _1-\\gamma _2|}\\right)\\nonumber \\\\&& \\ll Y \\frac{\\log ^2 T_0}{T_0} +\\log Y \\frac{\\log ^3 T_0}{T_0}+\\log ^5 T_0.$ This completes the proof of this lemma.", "$\\Box $" ], [ "Asymptotic formulas for the logarithmic densities", "In this section, we give the proof of Theorem REF .", "For large $q$ , Fiorilli and Martin [4] gave an asymptotic formula for $\\delta _{\\Omega _1}(q; a, b)$ .", "Lamzouri [11] also derived such an asymptotic formula using another method.", "Here, we want to derive asymptotic formulas for $\\delta _{\\Omega _k}(q; a, b)$ and $\\delta _{\\omega _k}(q; a, b)$ for fixed $q$ and large $k$ .", "Let $a$ be a quadratic non-residue $\\bmod q$ and $b$ be a quadratic residue $\\bmod q$ , and $(a, q)=(b, q)=1$ .", "Letting $\\lambda _k=\\frac{1}{2^{k-1}}$ , similar to formula (2.10) of [4], we have, under the assumptions ${\\rm ERH_q}$ and ${ \\rm LI_q}$ , $\\delta _{\\Omega _k}(q; a, b)=\\frac{1}{2}+ \\frac{(-1)^k}{2\\pi }\\int _{-\\infty }^{\\infty } \\frac{\\sin (\\lambda _k (N(q; a)-N(q; b))x}{x} \\Phi _{q; a, b}(x) dx.$ Noting that $N(q, a)-N(q, b)=-A(q)$ , $\\delta _{\\Omega _k}(q; a, b)=\\frac{1}{2}+\\frac{(-1)^{k-1}}{2\\pi } \\int _{-\\infty }^{\\infty }\\frac{\\sin (\\lambda _k A(q) x)}{x} \\Phi _{q; a, b}(x) dx.$ For any $\\epsilon >0$ , $\\int _{-\\infty }^{\\infty }\\frac{\\sin (\\lambda _k A(q) x)}{x} \\Phi _{q; a, b}(x) dx=\\left(\\int _{-\\infty }^{\\frac{1}{\\lambda _k^{\\epsilon }}}+\\int _{-\\frac{1}{\\lambda _k^{\\epsilon }}}^{\\frac{1}{\\lambda _k^{\\epsilon }}}+\\int _{\\frac{1}{\\lambda _k^{\\epsilon }}}^{\\infty } \\right) \\frac{\\sin (\\lambda _k A(q) x)}{x} \\Phi _{q; a, b}(x) dx.$ By Proposition 2.17 in [4], $|\\Phi _{q; a, b}(t)|\\le e^{-0.0454\\phi (q)t}$ for $t\\ge 200$ .", "So for large enough $k$ , $\\int _{\\frac{1}{\\lambda _k^{\\epsilon }}}^{\\infty }\\frac{\\sin (\\lambda _k A(q) x)}{x} \\Phi _{q; a, b}(x) dx\\ll \\lambda _k \\int _{\\frac{1}{\\lambda _k^{\\epsilon }}}^{\\infty } e^{-0.0454\\phi (q)x} dx \\ll _{q, J, \\epsilon }\\lambda _k^{J}, ~\\text{for any}~J>0.$ The integral over $x\\le -\\frac{1}{\\lambda _k^{\\epsilon }}$ is also bounded by $\\lambda _k^{J}$ .", "By Lemma 2.22 in [4], for each nonnegative integer $K$ and real number $C>1$ , we have, uniformly for $|z|\\le C$ , $\\frac{\\sin z}{z}=\\sum _{j=0}^{K} (-1)^j \\frac{z^{2j}}{(2j+1)!", "}+O_{C, K}\\left(|z|^{2K+2} \\right).$ Thus, the second integral in (REF ) is equal to $&&\\lambda _k A(q)\\int _{-\\frac{1}{\\lambda _k^{\\epsilon }}}^{\\frac{1}{\\lambda _k^{\\epsilon }}} \\frac{\\sin (\\lambda _k A(q) x)}{\\lambda _k A(q)x} \\Phi _{q; a, b}(x) dx\\nonumber \\\\&&=\\sum _{j=0}^K \\lambda ^{2j+1}_k\\frac{(-1)^j A(q)^{2j+1}}{(2j+1)!}", "\\int _{-\\frac{1}{\\lambda _k^{\\epsilon }}}^{\\frac{1}{\\lambda _k^{\\epsilon }}} x^{2j} \\Phi _{q; a, b}(x) dx+O_{q, K}\\left(\\lambda _k^{2K+3-\\epsilon } \\right)\\nonumber \\\\&&=\\sum _{j=0}^K \\lambda ^{2j+1}_k\\frac{(-1)^j A(q)^{2j+1}}{(2j+1)!}", "\\int _{-\\infty }^{\\infty } x^{2j} \\Phi _{q; a, b}(x) dx+O_{q, K, \\epsilon }\\left(\\lambda _k^{2K+3-\\epsilon } \\right).$ Combining (REF ), (REF ), and (REF ), we get the asymptotic formula (REF ) for $\\delta _{\\Omega _k}(q; a, b)$ .", "Similarly, or by the results in Theorem REF , we have the asymptotic formula (REF ) for $\\delta _{\\omega _k}(q; a, b)$ .", "$\\Box $" ], [ "The source of main terms and proof of Lemma ", "In this section, we give the proof of the main lemma we used for extracting out the bias terms and oscillating terms from the integrals over Hankel contours.", "Let $\\mathcal {H}(0, X)$ be the truncated Hankel contour surrounding 0 with radius $r$ .", "Lau and Wu [13] proved the following lemma.", "Lemma 17 ([13], Lemma 5) For $X>1$ , $z\\in \\mathbb {C}$ and $j\\in \\mathbb {Z}^{+}$ , we have $\\frac{1}{2\\pi i}\\int _{\\mathcal {H}(0, X)} w^{-z}(\\log w)^j e^w dw=(-1)^j \\frac{d^j}{dz^j}\\left(\\frac{1}{\\Gamma (z)}\\right) +E_{j, z}(X),$ where $|E_{j,z}(X)|\\le \\frac{e^{\\pi |\\Im (z)|}}{2\\pi }\\int _{X}^{\\infty } \\frac{(\\log t+\\pi )^j}{t^{\\Re (z)}e^t}dt.$ Proof of Lemma REF .", "We have the equality $\\frac{1}{s}=\\frac{1}{a}+\\frac{a-s}{a^2}+\\frac{(a-s)^2}{a^2 s}.$ With the above equality, we write the integral in the lemma as $\\frac{1}{2\\pi i}\\int _{\\mathcal {H}(a, \\delta )} \\log ^k(s-a)\\left(\\frac{1}{a}+\\frac{a-s}{a^2}+\\frac{(a-s)^2}{a^2 s}\\right)x^s ds =: I_1+I_2+I_3.$ For $I_3$ , using Lemma REF , we get $&\\int _{\\mathcal {H}(a,\\delta )} \\log ^k(s-a)\\frac{(a-s)^2}{a^2 s} x^s ds\\nonumber \\\\&\\le \\left|\\int _r^{\\delta }\\left((\\log \\sigma -i\\pi )^k-(\\log \\sigma +i\\pi )^k\\right)\\sigma ^2 x^{-\\sigma }\\frac{x^a}{a^2 (a-\\sigma )}d\\sigma \\right|\\nonumber \\\\&\\quad +\\int _{-\\pi }^{\\pi } x^{\\Re (a)+r} \\left(\\log \\frac{1}{r}+\\pi \\right)^k \\frac{r^2}{|a|^2 |\\Re (a)-r|} r d\\alpha \\nonumber \\\\&\\ll \\frac{|x^a|}{|a|^2 |\\Re (a)-\\delta |} \\left(\\int _0^{\\delta } |(\\log \\sigma -i\\pi )^k-(\\log \\sigma +i\\pi )^k|\\sigma ^2 x^{-\\sigma }d\\sigma + \\frac{(\\log \\frac{1}{r}+\\pi )^{k}}{(1/r)^3}\\right)\\nonumber \\\\&\\ll _k \\frac{|x^a|}{|a|^2|\\Re (a)-\\delta |}\\left(\\frac{(\\log \\log x)^{k-1}}{(\\log x)^3}+ \\frac{1}{x^{3-\\epsilon }}\\right)\\ll _k \\frac{|x^a|}{|a|^2|\\Re (a)-\\delta |}\\frac{(\\log \\log x)^{k-1}}{(\\log x)^3}.$ We estimate $I_2$ similarly.", "By Lemma REF , $&&\\int _{\\mathcal {H}(a,\\delta )} \\log ^k(s-a)\\frac{a-s}{a^2} x^s ds\\nonumber \\\\&&\\le \\left|\\int _r^{\\delta }\\left((\\log \\sigma -i\\pi )^k-(\\log \\sigma +i\\pi )^k\\right)\\sigma x^{-\\sigma }\\frac{x^a}{a^2 }d\\sigma \\right|+\\int _{-\\pi }^{\\pi } x^{\\Re (a)+r} \\left(\\log \\frac{1}{r}+\\pi \\right)^k \\frac{r}{|a|^2 } r d\\alpha \\nonumber \\\\&&\\ll \\frac{|x^a|}{|a|^2 } \\left(\\int _0^{\\delta } |(\\log \\sigma -i\\pi )^k-(\\log \\sigma +i\\pi )^k|\\sigma x^{-\\sigma }d\\sigma + \\frac{(\\log \\frac{1}{r}+\\pi )^{k}}{(1/r)^2}\\right)\\nonumber \\\\&&\\ll _k \\frac{|x^a|}{|a|^2}\\left(\\frac{(\\log \\log x)^{k-1}}{(\\log x)^2}+ \\frac{1}{x^{2-\\epsilon }}\\right)\\ll _k \\frac{|x^a|}{|a|^2}\\frac{(\\log \\log x)^{k-1}}{(\\log x)^2}.$ For $I_1$ , using change of variable $(s-a)\\log x=w$ , by Lemma REF , we get $I_1&=\\frac{1}{2\\pi i}\\frac{1}{\\log x}\\int _{\\mathcal {H}(0,\\delta \\log x)}(\\log w-\\log \\log x)^k \\frac{x^a e^w}{a}dw\\nonumber \\\\&=\\frac{x^a}{a\\log x} (-1)^k (\\log \\log x)^k \\frac{1}{2\\pi i}\\int _{\\mathcal {H}(0,\\delta \\log x)} e^w dw \\nonumber \\\\& \\quad +(-1)^{k-1}k \\frac{x^a}{a\\log x}(\\log \\log x)^{k-1} \\frac{1}{2\\pi i}\\int _{\\mathcal {H}(0, \\delta \\log x)}e^w\\log w dw \\nonumber \\\\& \\quad +\\frac{x^a}{a\\log x}\\sum _{j=2}^k {k\\atopwithdelims ()j} \\frac{1}{2\\pi i} \\int _{\\mathcal {H}(0, \\delta \\log x)} (-\\log \\log x)^{k-j} (\\log w)^j e^w dw \\nonumber \\\\&=\\frac{(-1)^k x^a}{a\\log x} \\left\\lbrace k (\\log \\log x)^{k-1}+\\sum _{j=2}^k {k \\atopwithdelims ()j} (\\log \\log x)^{k-j} \\frac{1}{\\Gamma _j(0)} \\right\\rbrace \\nonumber \\\\& \\quad +\\frac{x^a}{a\\log x} \\sum _{j=1}^k {k \\atopwithdelims ()j} E_{j,0}(\\delta \\log x) (-\\log \\log x)^{k-j}.$ By Lemma REF , $|E_{j,0}(\\delta \\log x)|\\le \\frac{1}{2\\pi }\\int _{\\delta \\log x}^{\\infty } \\frac{(\\log t+\\pi )^j}{e^t}dt\\ll _j e^{-\\frac{\\delta \\log x}{2}} \\int _{\\frac{\\delta \\log x}{2}}^{\\infty } \\frac{(\\log t)^j}{e^{t/2}}dt\\ll _j x^{-\\frac{\\delta }{2}}.$ Hence, we get $\\left| \\frac{x^a}{a\\log x} \\sum _{j=1}^k {k \\atopwithdelims ()j} E_{j,0}(\\delta \\log x) (-\\log \\log x)^{k-j} \\right| \\ll _k \\frac{x^{\\Re (a)}}{|a|\\log x} \\sum _{j=1}^k x^{-\\frac{\\delta }{2}} (\\log \\log x)^{k-j}\\ll _k \\frac{|x^{a-\\delta /3|}}{|a|}.$ Combining (REF ), (REF ), (REF ), and (REF ), we get the conclusion of Lemma REF .", "$\\Box $ Acknowledgments.", "This research is partially supported by NSF grants DMS-1201442 and DMS-1501982.", "I would like to thank my advisor, Professor Kevin Ford, for his kindly encouragement, useful suggestions and financial support to finish this project.", "I am grateful for the helpful comments of Dr. Youness Lamzouri.", "I also thank the encouragement of my friend Junjun Cheng.", "The author would like to thank the referee for helpful comments.", "Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA E-mail: [email protected]," ] ]
1606.04877
[ [ "Theory of the Lamb shift and Fine Structure in muonic $\\mathrm{^4He}$\n ions and the muonic $\\mathrm{^3He-^4He}$ Isotope Shift" ], [ "Abstract We provide an up to date summary of the theory contributions to the 2S-2P Lamb shift and the fine structure of the 2P state in the muonic helium ion $(\\mathrm{\\mu^4He})^+$.", "This summary serves as the basis for the extraction of the alpha particle charge radius from the muonic helium Lamb shift measurements at the Paul Scherrer Institute, Switzerland.", "Individual theory contributions needed for a charge radius extraction are compared and compiled into a consistent summary.", "The influence of the alpha particle charge distribution on the elastic two-photon exchange is studied to take into account possible model-dependencies of the energy levels on the electric form factor of the nucleus.", "We also discuss the theory uncertainty which enters the extraction of the $\\mathrm{^3He-^4He}$ isotope shift from the muonic measurements.", "The theory uncertainty of the extraction is much smaller than a present discrepancy between previous isotope shift measurements.", "This work completes our series of $n=2$ theory compilations in light muonic atoms which we have performed already for muonic hydrogen, deuterium, and helium-3 ions." ], [ "Introduction", "The CREMA Collaboration measured both $\\mathrm {2S\\rightarrow {}2P}$ Lamb shift transitions in the muonic helium ion $(\\mu {}^{4}\\mathrm {He})^{+}$ in 2013 and 2014 [1], [2].", "A scheme of the energy levels in muonic helium-4 ions is shown in Fig.", "REF .", "In preparation of a future extraction of nuclear properties from these measurements, such as the nuclear root-mean-square (rms) charge radius $r_\\alpha $ , we provide a careful study of the available calculations of the theory contributions to the involved energy levels, summarizing the results of several theory groups.", "Figure: The 2S\\mathrm {2S} and 2P\\mathrm {2P} energy levels in the muonic helium-4 ion.", "Since the nuclear spin is zero, no hyperfine structure is present.", "The figure is not to scale.Both, the Lamb shift and the fine structure, have been analyzed recently [3], [4], [5], [6] but significant differences between the authors made it necessary to review the individual theory contributions.", "The same was previously done for muonic hydrogen [7], muonic deuterium [8], and muonic helium-3 ions [9].", "Recent measurements of the $\\mathrm {2S\\rightarrow {}2P}$ Lamb shift (LS) in other muonic atoms have already provided the rms charge radii of the proton and the deuteron with unprecedented precision.", "Results from muonic hydrogen measurements provided a proton charge radius of $r_p^{\\mu }~=~0.84087(26)^{\\mathrm {exp}}(29)^{\\mathrm {theo}}~\\mathrm {fm}\\ \\text{\\cite {Pohl:2010:Nature_mup1,Antognini:2013:Science_mup2}}.$ This value is ten times more precise than the CODATA-2014 value of 0.8751(61) fm [12], however also 4 %, or 6 $\\sigma ,$ smaller.", "This discrepancy created the so-called “Proton-Radius-Puzzle” (PRP) [13], [14], [15], [16].", "A recent determination of the deuteron radius from muonic deuterium spectroscopy results in a value of $r_d^{\\mu }~=~2.12562(13)^{\\mathrm {exp}}(77)^{\\mathrm {theo}}~\\mathrm {fm}\\,\\text{\\cite {Pohl:2016:mud}},$ that is also smaller than the CODATA value and hints towards a change in the Rydberg constant [18], [19].", "The determination of the alpha particle charge radius from muonic helium-4 ions, when compared to the radius determinations from electron scattering experiments [20], [21] will provide new input on the existing discrepancies.", "The improved value of $r_\\alpha $ will be used in the near future for tests of fundamental bound state quantum electrodynamics (QED) by measurements of the $\\mathrm {1S\\rightarrow {}2S}$ transition in electronic He$^+$  ions [22], [23].", "Furthermore, the combination of the precise charge radii from muonic helium-3 and helium-4 ions will contribute to a discrepancy between several isotope shift measurements in electronic helium-3 and -4 atoms [24], [25], [26].", "And, finally, in combination with existing isotope shift measurements [27], [28], [29] the nuclear charge radii of the helium-6 and -8 isotopes will be slightly improved.", "A list of previous charge radius determinations is found in Angeli et al.", "[30] from 1999.", "A value from a combined analysis of experimental data is given in their more recent Ref. [31].", "Their value of the alpha particle charge radius $r_\\alpha = 1.6755(28)\\,\\mathrm {fm}$ is dominated by a measurement from Carboni et al.", "[32], [33], which has been excluded by a later measurement from Hauser et al.[34].", "Hence, it should not be used.", "Instead the today established value is $r_{\\alpha }=1.681(4)$  fm, determined by Sick [21] from elastic electron scattering.", "The anticipated accuracy of the CREMA measurement will be about a factor of $\\sim 5$ more precise than the established value.", "The finite size effect that is sensitive on the charge radius of the alpha particle amounts to $\\sim $ 300 meV or 20% of the LS in $(\\mu {}^{4}\\mathrm {He})^{+}$ .", "The frequency uncertainty of the $(\\mu {}^{4}\\mathrm {He})^{+}$ LS measurements is on the order of 15 GHz which corresponds to 0.06 meV1 meV $\\hat{=}$  241.799 GHz.", "In the following, this value serves as accuracy goal for the theory contributions.", "Several contributions have been calculated with uncertainties not much better than our accuracy goal, with the reasoning that the inelastic two-photon exchange (“polarizability”) will anyway dominate the extracted charge radius uncertainty.", "However, once a reliable value for the charge radius exists, e.g.", "from He or He$^+$ spectroscopy, these uncertainties will limit the extracted “muonic polarizability”.", "The paper is structured as follows: Sec.", "discusses the pure QED contributions to the Lamb shift (independent of the charge radius and nuclear structure effects), which are summarized in Tab.", "REF .", "We use the theory calculations of Borie [35] (in this work we refer always to the updated Ref.", "[3], version v7 on the arXiv) and the calculations of the group of Elekina, Faustov, Krutov, and Martynenko et al.", "[4] (for simplicity referred to as “Martynenko” in the rest of the article) that provide a summary of terms contributing to the LS energy.", "Various partial results of QED terms provided by the group of Ivanov, Karshenboim, Korzinin, and Shelyuto [5], [36] (for simplicity referred to as Karshenboim further on) and by Jentschura and Wundt [6] are compared.", "Sec.", "discusses the contributions to the finite size effect, together with higher order corrections that scale with the nuclear charge radius squared $r_\\alpha ^2$ .", "These charge radius dependent terms are summarized in Tab.", "REF .", "We use the works of Borie [3], Martynenko (Krutov et al.", "[4]), and Karshenboim (Karshenboim et al.", "[5]).", "Our summary provides the charge radius coefficient of the LS parameterization needed to extract the charge radius from the experimentally measured transitions.", "In Sec.", "we discuss the two-photon exchange (TPE) in $(\\mu {}^{4}\\mathrm {He})^{+}$ .", "In the first part, Sec.", "REF , we discuss the so-called nuclear and nucleon “Friar moment” contribution, also known as the third Zemach moment contribution.", "In the second part, Sec.", "REF , we discuss the nuclear and nucleon polarizability contributions to the LS.", "The nuclear polarizability, i.e.", "the inelastic part of the TPE stems from the virtual excitation of the nucleus and is related to its excitation spectrum [37].", "It is the least accurately known part of the $(\\mu {}^{4}\\mathrm {He})^{+}$ LS and was investigated by the TRIUMF/Hebrew group in Ji et al.", "[38].", "The fine structure (FS) of the $\\mathrm {2P}$ state in $(\\mu {}^{4}\\mathrm {He})^{+}$ is studied in Sec.", "and summarized in Tab.", "REF .", "Our evaluation is based on the works of Borie [3], Martynenko (Elekina et al.", "[39]), and Karshenboim (Karshenboim et al.", "[5], Korzinin et al.", "[36]).", "In Sec.", ", we discuss the theory contributions with respect to the $^3$ He–$^4$ He isotope shift and extract a value for the uncertainty of the future value from the CREMA measurements in muonic helium ions.", "Here we exploit correlations between model-dependent calculations by the TRIUMF/Hebrew group to significantly reduce the theory uncertainty to the isotope shift.", "Throughout the paper we use the established convention and assign the measured energy differences $\\Delta E({\\rm 2P_{1/2}-2S_{1/2}})$ and $\\Delta E({\\rm 2P_{3/2}-2S_{1/2}})$ a positive sign.", "Labeling of individual terms in LS and FS follows the convention of our previous works [7], [8], [9] in order to maintain comparability.", "Terms that were found to not agree between various sources were averaged for our determination and the resulting value is found in the “Our Choice” column.", "These averaged values are given by the center of the covering band of all values under consideration $\\nu {}_i$ with the uncertainty of their half spread, i.e.", "$\\begin{split}\\mathrm {AVG}~~~=~~~ &\\frac{1}{2}[\\mathrm {MAX}(\\nu {}_i)+\\mathrm {MIN}(\\nu {}_i)]\\\\\\pm {}&\\frac{1}{2}[\\mathrm {MAX}(\\nu {}_i)-\\mathrm {MIN}(\\nu {}_i)]\\end{split}$ The values in the “Our choice” column are followed by the initial of the authors whose results were used to obtain this value (B = Borie, M = Martynenko, K = Karshenboim, J = Jentschura).", "Important abbreviations: $Z$ is the nuclear charge, $\\alpha $ is the fine structure constant.", "“VP”, “SE”, and “RC” refer to vacuum polarization, self energy, and recoil corrections, respectively.", "A proceeding $e$ , $\\mu $ , or $h$ denotes contributions from electrons, muons, or hadrons." ], [ "QED Lamb Shift in $(\\mu {}^{4}\\mathrm {He})^{+}$", "First we consider pure QED terms that do not depend on nuclear properties.", "All terms listed in this section are given in Tab.", "REF .", "As in other muonic atoms, the one-loop electron vacuum polarization (eVP; #1; see Fig.", "REF ) is the largest term contributing to the Lamb shift.", "Martynenko provides a non-relativistic calculation for this Uehling-term (#1) together with a separate term for its relativistic corrections (Breit-Pauli correction, #2) [4].", "The values of the main term plus its correction (#1$+$ #2) agree exactly with the independent calculations of Karshenboim [5] and Jentschura [40], [6] who follow the same procedure.", "Borie's value (#3) already includes relativistic corrections of the order $\\alpha {}(Z\\alpha {})^2$ due to the use of relativistic Dirac wave functions [3].", "The additional $\\alpha {}(Z\\alpha {})^4$ relativistic recoil correction to eVP (#19) already included by the other authors is treated separately in her framework [3].", "The sum of all eVP contributions (#1+#2 or #3+#19) is in agreement between the calculations of all authors.", "The average value of the Uehling contribution yields $\\Delta {}E_{\\mathrm {(1-loop~eVP)}}= 1666.2946\\pm 0.0014\\,\\text{meV}.$ The next largest term in the QED part of the $(\\mu {}^{4}\\mathrm {He})^{+}$ Lamb shift is given by the two-loop electron vacuum polarization in the one-photon interaction of order $\\alpha ^2(Z\\alpha )^4$ (#4).", "This so-called Källén-Sabry (KS) contribution is the sum of three Feynman diagrams as seen in Fig.", "REF , #4.", "It is calculated by Borie and Martynenko, and the agreement between both calculations is still satisfactory, although not as good as for the Uehling term.", "The one-loop eVP contribution with two Coulomb lines (#5, see Fig.", "REF ) is calculated by Martynenko [4] and Jentschura [6] and their results show satisfactory agreement.", "Borie cites a paper of Karshenboim [5], it is however unknown how she obtained the quoted value.", "Karshenboim calculates the sum of both terms (#4+#5) [36].", "The sum is in excellent agreement with the sum of Martynenko's values and also agrees with the calculation of Borie.", "The total contribution from two eVP loops in one and two Coulomb lines is given by the average of 13.2794(26) $\\mathrm {meV}$ .", "Calculations of third order eVP contributions (3-loop eVP, #6+#7 from Martynenko and Karshenboim agree for the required accuracy [3], [4], [36].", "Karshenboim's value is chosen because he showed that the calculation method of Martynenko, first employed by Kinoshita and Nio [41], is not correct [36].", "The value of the third order eVP is given by 0.0740(30) $\\mathrm {meV}$ .", "The size of the third order contribution is comparable in size to our accuracy goal while the uncertainty of the term is even smaller.", "The contribution from two eVP loops in one and two Coulomb lines has additional RC of the order $\\alpha {}^2(Z\\alpha {})^4m$ (#29).", "This correction was calculated by Martynenko and Karshenboim, but the calculations differ by more than a factor of two [4], [36].", "Since similar calculations for $\\mu {}\\mathrm {D}$ are in agreement in previous publications of both authors [42], [36] further investigation is needed.", "For our summary we choose the average value of $0.0039\\pm 0.0018$  meV due to the disagreement between Martynenko and Karshenboim.", "Fortunately, the small discrepancy is not relevant on the level of the accuracy goal.", "The higher order “light-by-light” scattering contribution consists of three individual terms (#9,#10,#9a; see Fig.", "REF ).", "The first, so-called Wichmann-Kroll term, (#9) is in agreement between the works of Karshenboim [43] and Martynenko [4].", "Borie [3] also calculates the term independently and reports a slightly smaller but still agreeing result.", "The remaining Virtual Delbrück (#10) and inverted Wichmann-Kroll (#9a) contributions have been calculated by Karshenboim [43].", "The Virtual Delbrück contribution was calculated by Borie earlier [44] although with larger uncertainty.", "As can be seen from Tab.", "REF , cancellations between the three terms occur.", "We therefore directly adopt Karshenboim's values to include all cancellations.", "A term comparable in size to the Källén-Sabry contribution (but with opposite sign) is given by the effect of muon vacuum polarization ($\\mu $ VP) and muon self energy ($\\mu $ SE) (#20).", "The sum of both terms was calculated by Borie and Martynenko and they show satisfactory agreement [3], [4].", "Figure: Important Feynman diagrams contributing to the QED part of the Lamb shift.", "#1 The Uehling Term; #4 The Källén-Sabry contribution; #5 One loop eVP in two Coulomb lines; #9/9a/10 Light-by-light scattering contributions; #13 Mixed eVP/μ\\mu {}VP; #11 Self energy corr.", "to eVP; #31 Mixed eVP/hadronic VP; #12 eVP loop in SE contribution; #30 Hadr.", "loop in SE contribution; #32 μ\\mu VP loop in SE contribution (included in #21).", "In our summary, terms #5, #9, #10, #9a, #13 (2) ^{(2)} and #31 (2) ^{(2)} also contain their respective cross diagrams.$\\mu $ SE also contributes as correction to the one-loop eVP term (see Fig REF , #11).", "The results of Karshenboim and Jentschura agree very well [43], [6].", "The calculation from Martynenko in Ref.", "[4] provides an incomplete value since he only calculates the second diagram seen in Fig.", "REF , #11$^{(2)}$ .", "He adopts the complete value of Jentschura in his summary.", "Borie only partially calculates this term, as stated in appendix C of her summary [3].", "Therefore our choice is compiled from Karshenboim's and Jentschura's value.", "Insertion of an eVP or hVP loop in the $\\mu $ SE correction leads to corrections of higher order.", "The contribution of the additional eVP loop (#12) was calculated by Borie and Karshenboim and their values agree well.", "The hVP term (#30) was only calculated by Karshenboim whose value we adopt.", "There is also a contribution due to a $\\mu $ VP insertion in the $\\mu $ SE line.", "This contribution is not separately added to our summary, because it is already included in the $\\mu $ SE value.", "The contribution with an eVP and a $\\mu $ VP loop in the one photon interaction is given by the first digram of #13 in Fig.", "REF .", "It was evaluated by Martynenko and Borie and their values agree.", "Karshenboim provides values of this contribution summed with the respective term in the two Coulomb line diagram (second part of #13) [43].", "Both terms are of similar size, therefore the values of Karshenboim and Borie/Martynenko differ by nearly a factor of two.", "Since the total term is small, this uncertainty is not important for the Lamb shift extraction.", "In addition, Karshenboim also calculated the influence of the mixed eVP-hVP diagram in one and two Coulomb lines (Fig.REF , #31).", "Borie only gives a term labeled “higher order correction to $\\mu $ SE and $\\mu $ VP” (#21) that also includes the $\\mu $ VP loop in the SE contribution (previously #32).", "The insertion of a hadronic vacuum polarization (hVP; #14) loop in the one Coulomb-photon interaction leads to another correction calculated by Borie and Martynenko.", "Both values agree within the uncertainty given in Borie's publication [3].", "We use Borie's result [3] as her uncertainty includes Martynenko's value [4].", "Item #17 is the main recoil correction in the Lamb shift, also called the Barker-Glover correction.", "The available calculations of the term by Borie, Martynenko and Karshenboim agree perfectly.", "Item (#18) is the term called “recoil finite size” by Borie [3].", "It is of order $(Z\\alpha )^5\\mathinner {\\langle {r}\\rangle }_{(2)}/M$ and is linear in the first Zemach moment.", "It has first been calculated by Friar  [45] (see Eq.", "F5 in App.", "F) for hydrogen and has later been given by Borie [3] for $\\mu $ d, $(\\mu {}^{3}\\mathrm {He})^{+}$ , and $(\\mu {}^{4}\\mathrm {He})^{+}$ .", "We discard item #18 because it is considered to be included in the elastic TPE [46], [47].", "Further relativistic recoil corrections of the order $(Z\\alpha {})^5$ and $(Z\\alpha {})^6$ are also included in our summary (#22, #23).", "The $(Z\\alpha {})^5$ correction was calculated by Borie, Martynenko and Jentschura and their results agree.", "The $(Z\\alpha {})^6$ term was only determined by Martynenko, but is two orders of magnitude smaller than the term of the previous order.", "Therefore we simply accept his value in our summary.", "Martynenko provides a term called \"radiative correction with recoil of the order $\\alpha (Z\\alpha {})^5$ and $(Z^2\\alpha {})(Z\\alpha {})^4$ \".", "The respective terms are included in Borie's “higher order recoil” together with some additional terms not covered by Martynenko.", "We therefore adopt the more complete value of Borie (#24).", "The total logarithmic recoil in $(\\mu {}^{4}\\mathrm {He})^{+}$ of the order $\\alpha (Z\\alpha {})^5$ (#28) was only calculated by Jentschura [6].", "It includes the dominant seagull-term as well as two more Feynman-diagrams with smaller contributions.", "We directly adopt this result for our summary.", "From the summary given in Tab.", "REF we extract the total nuclear structure independent part of the Lamb shift $\\Delta {}E_{(LS,\\,QED)}= 1668.4892\\pm 0.0135\\,\\mathrm {meV}.$ This value is in agreement with the sum given by Martynenko [4], and agrees also with Borie's value when discarding the recoil finite size term.", "In the case of $(\\mu {}^{4}\\mathrm {He})^{+}$ there is no Darwin-Foldy (DF) term as opposed to $\\mu $ D [8].", "This term normally accounts for the Zitterbewegung of the nucleus but vanishes in $(\\mu {}^{4}\\mathrm {He})^{+}$ due to its zero nuclear spin.", "The uncertainty of the “pure QED” contributions in Eq.", "(REF ) is a factor of 4 smaller than the expected experimental uncertainty and poses no limitation of the charge radius extraction." ], [ "r$^2$ contributions to the Lamb Shift", "The $\\mathrm {2S\\rightarrow {}2P}$ splitting is also affected by the charge radius of the alpha particle.", "This so-called finite size effect dominantly influences S states due to their non-zero wave function at the origin, $\\Psi {}(0)$ .", "The finite size contributions in $(\\mu {}^{4}\\mathrm {He})^{+}$ can be parameterized with the square of the nuclear root-mean-square (rms) charge radius, which is defined as [48], [13] $r_\\alpha ^2 = -6\\frac{dG_{E}}{dQ^2}\\big |_{Q^2=0}\\,$ where $G_{E}$ is the Sachs electric form factor of the nucleus and $Q^2$ is the square of the four-momentum transfer to the nucleus.", "This charge radius definition is consistent with the one used in elastic electron scattering.", "In a simplified, nonrelativistic picture the nuclear charge radius is often referred to as the second moment of the nuclear charge distribution.", "The leading order finite size contribution (#r1) is of order $(Z\\alpha {})^4$ and originates from the one-photon interaction between the muon and the helium nucleus.", "It is calculated by inserting the form factor in the nucleus vertex.", "The coefficient of the leading order finite size effect is provided by Borie [3] and Karshenboim [5], and their results agree.", "Martynenko [4] however only gives absolute energy values for the finite size effect.", "For the leading order contribution he obtains $-295.85\\pm 2.83\\,\\mathrm {meV}$ .", "We have to divide by the square of the rms charge radius of 1.676(8) fm used in his calculations [4], to get the resulting coefficient given in Tab.", "REF .", "Martynenko's value agrees with the other two.", "All authors follow the previous calculations of Friar [45].", "#r1 is given by the average of the three authors as $\\Delta E(\\mathrm {\\#r1})= -105.3210\\pm 0.0020\\,\\,\\mathrm {meV}/\\,\\mathrm {fm}^2\\,r_\\alpha ^2,$ where the uncertainty is far better than our accuracy goal.", "Item #r4 is the one-loop $e$ VP (Uehling) correction of order $\\alpha (Z\\alpha )^4$ , i.e.", "an $e$ VP insertion into the one-photon line.", "It has been calculated by all three groups, Borie [3], Martynenko [4] and Karshenboim [5].", "On p. 31 of [3], Borie notes that she included the correction arising from the Källén-Sabry (KS) potential in her $b_d$ .", "This means that her value already contains item #r6, which is the two-loop $e$ VP correction of order $\\alpha ^2(Z\\alpha )^4$ .", "Item #r6 is given explicitly only by the Martynenko group [4] (No.", "18, Eq. 73).", "The sum of Martynenko et al.", "'s #r4 and #r6 differs by 0.014 $\\mathrm {meV}$ /fm$^2$ from Borie's result.", "Using a charge radius of 1.681 fm this corresponds to roughly 0.04 $\\mathrm {meV}$ and, hence, causes the largest uncertainty in the radius-dependent one-photon exchange part.", "The origin of this difference is not clear [49], [50].", "A clarification of this difference is desired but does not yet limit the extraction of the charge radius.", "As our choice we take the average of the sum (#r4+#r6) of these two groups.", "The resulting average does also reflect the value for #r4 provided by Karshenboim et al. [5].", "Item #r5 is the one-loop $e$ VP (Uehling) correction in second order perturbation theory (SOPT) of order $\\alpha (Z\\alpha )^4$ .", "It has been calculated by all three groups, Borie [3], Martynenko [4] and Karshenboim [5].", "On p. 31 of [3], Borie notes that she included the two-loop corrections to $\\epsilon _{VP2}$ in her $b_e$ .", "This means that her value already contains item #r7, which is the two-loop $e$ VP in SOPT of order $\\alpha ^2(Z\\alpha )^4$ .", "Item #r7 is only given explicitly by the Martynenko group [4] (No. 19).", "The sum of Martynenko et al.", "'s #r5+#r7 differs by 0.01 $\\mathrm {meV}$ from Borie's result.", "As our choice we take the average of the sum (#r5+#r7) of these two groups.", "Again here, our choice reflects the value for #r5 provided by Karshenboim et al. [5].", "The nuclear structure influence on the 2P$_{1/2}$ state is only determined by Borie (#r8) [3].", "We directly adopt her value, but change the sign of #r8 from the original publication to be consistent with our nomenclature of tabulating $\\Delta {}E(\\mathrm {2P-2S})$ .", "Item #r2 is a radiative correction of order $\\alpha (Z\\alpha {})^5$ .", "It has been calculated by Borie [3] and Martynenko [4].", "Their values agree well.", "Martynenko [51] recently calculated an additional term which we denote as #r2'.", "It has a non-trivial dependence of the charge radius and is therefore provided as an absolute value rather than as a coefficient.", "Since it is tiny this procedure does not affect the extracted charge radius.", "Note, that in [51], Martynenko indicates the value for the 1S state, which has to be scaled by $1/2^3$ to account for the 2S state.", "Item #r2b' is a VP correction of order $\\alpha (Z\\alpha )^5$ .", "It is only given by the Martynenko group [4] and not parameterized with the charge radius squared.", "Therefore we adopt their value as a constant.", "This contribution is erroneously not accounted for in our summary for muonic helium-3 [9], we therefore added an appendix to this work, where we update the numbers from [9].", "For the finite size term of the order $(Z\\alpha {})^6$ (#r3) and the same-order correction (#r3'), Borie and Martynenko use different methods of calculation.", "Here, #r3' is given as an absolute value, because of its non-trivial dependence on the charge radius, similar to #r2'.", "A term corresponding to the $\\mathinner {\\langle {\\mathrm {ln}\\,r}\\rangle }$ coefficient is part of term (#r3) for Martynenko and part of (#r3') for Borie, leading to a correlation between both.", "In order to stay consistent with the summary in $\\mu {}\\mathrm {D}$  [8] we decided to average both terms providing $\\#r3=-0.1340\\pm 0.0030$  meV/fm$^2$ and $\\#r3^{\\prime }= 0.067\\pm 0.012$  meV until a clear definition is settled on.", "Note that although the uncertainty of #r3' is still a factor of 5 smaller than the uncertainty goal, it would be helpful if this 20% relative uncertainty in the term could be improved.", "The total $r_\\alpha ^2$ coefficient of the Lamb shift is given by $\\begin{split}\\Delta {}E_{(Fin.\\,size)}= &-106.3536(82)\\,\\mathrm {meV}/\\,\\mathrm {fm}^2\\,r_\\alpha ^2\\\\&+ 0.2054(113)\\,\\mathrm {meV}.\\end{split}$ The uncertainty of the first term corresponds to 0.02 $\\mathrm {meV}$ (for $r_\\alpha =1.681\\,\\mathrm {fm}$ ), already 30% of our uncertainty goal." ], [ "Two-photon exchange", "Important parts of the nuclear structure dependent Lamb shift contributions are created by the two-photon exchange (TPE) between muon and nucleus.", "Two distinct parts can be separated: $\\Delta {}E_{\\mathrm {TPE}}^{\\mathrm {LS}}=\\delta {}E_{\\mathrm {Friar}}^{A+N} + \\delta {}E_{\\mathrm {inelastic}}^{A+N},$ where $\\delta {}E_{\\mathrm {Friar}}^{A+N}$ is the Friar moment contribution The term “Friar moment” has been introduced by Karshenboim et al.", "in [52]., also known as “third Zemach moment contribution”, and $\\delta {}E_{\\mathrm {inelastic}}^{A+N}$ is the inelastic part of the TPE, also called the polarizability contribution.", "Each part is again separated into a nuclear (A) and a nucleon (N) part." ], [ "The Friar moment contribution in $(\\mu {}^{4}\\mathrm {He})^{+}$", "The nuclear Friar moment contribution $\\delta {}E_{\\rm Friar}^{A}$ is an elastic contribution, analog to the finite size effect, but of order $(Z\\alpha )^5$ , i.e.", "in the two-photon interaction (see Fig.", "REF , $(a),(b)$ ).", "In the following we discuss five ways of how the Friar moment can be obtained: Option a: The most modern calculation of the Friar moment contribution is provided by the TRIUMF/Hebrew group in Ji et al.", "[53], updated by Hernandez et al. [54].", "They perform ab initio calculations, using state-of-the-art nuclear potentials.", "Their result of $\\delta E_{\\rm Friar}^A(e) =6.29\\pm 0.28\\,\\,\\mathrm {meV}$ assumes that the sum of their terms $\\delta {}_{Z1}$ and $\\delta {}_{Z3}$ [53] of the nuclear polarizability matches the elastic Friar moment contribution exactly.", "This approach has recently made impressive progress.", "However, compared to the following options below, the uncertainty is still rather large.", "Note, that in the isotope shift (Sec.", "), a large part of this uncertainty cancels.", "Option b: The Friar moment contribution can be parameterized proportional to the Friar moment $\\mathinner {\\langle {r^3}\\rangle }_{(2)}$ of the nucleus' electric charge distribution [45].", "Using $\\mathinner {\\langle {r^3}\\rangle }_{(2)} = 16.73(10)$   $\\mathrm {meV}$ /fm$^{3}$ [55] from measured helium-4 form factors in momentum space, this option yields the most precise value and is furthermore model-independent, as it originates from experimental data.", "From Eq.", "(43a) in [45] we obtain $\\delta E_{\\rm Friar}^A(a) = \\frac{(Z\\alpha )^5 m_r^4}{24} \\mathinner {\\langle {r^3}\\rangle }_{(2)} = 6.695\\pm 0.040\\,\\,\\mathrm {meV}.$ However, expressing the elastic part of the TPE using only the Friar moment does not account for relativistic-recoil corrections [56] (see option d).", "Option c: The Friar moment contribution can also be parameterized proportional to the third power of the nuclear rms charge radius as $C\\times r_\\alpha ^3$ , where $C$ is a factor which depends on the model for the radial charge distribution.", "Borie gives a coefficient of $C = 1.40(4)\\,\\mathrm {meV}/\\,\\mathrm {fm}^3$ (p. 14 of [3]).", "It is valid for a Gaussian charge distribution which is a good assumption for the helium-4 nucleus.", "The given uncertainty is an estimate of possible deviations from the Gaussian charge distribution [49].", "For the nuclear charge distribution Borie uses the charge radius $r_\\alpha = 1.681(4)\\,\\mathrm {fm}$ from Sick [21] and obtains $\\begin{split}\\delta E_{\\rm Friar}^A(b) =& ~1.40\\pm 0.04\\,\\mathrm {meV}/\\,\\mathrm {fm}^3\\, r_\\alpha ^3\\\\=& ~6.650\\pm 0.190\\,\\mathrm {meV}.\\end{split}$ In principle one can benefit from this option by using the charge radius as a free parameter which will be determined by the measurement of the Lamb shift.", "This has initially been done in $\\mu $ p [10].", "The limiting uncertainty in $(\\mu {}^{4}\\mathrm {He})^{+}$ , however, comes from the coefficient which is why this option is not attractive until a better value for the coefficient is available.", "Option d: The Friar moment contribution can be calculated by directly using form factor (FF) parameterizations in momentum space.", "Martynenko did the calculation for Gaussian and Dipole electric FF parameterizations due to their closed analytical form, following the work of Friar [57].", "Using a charge radius of $r_\\alpha = 1.676(8)$  fm, Martynenko with Eqs.", "(62) and (63) in [4] determines an energy contribution of $\\delta E_{\\rm Friar}^A(c) = 6.61\\pm 0.07\\,\\mathrm {meV}$ for a Gaussian charge distribution.", "For the less realistic dipole parameterization, Martynenko obtains a value of $7.1958\\,\\mathrm {meV}$ , which differs by $\\sim 0.6\\,\\mathrm {meV}$ from Eq.", "(REF ), illustrating the sensitivity of the Friar moment contribution to the shape of the charge distribution.", "In his table, however, Martynenko uses the Gaussian charge distribution only.", "Option e: Similar to option d, but instead of assuming the FF to be Gaussian we use FFs based on measured data.", "Sick provided us with an improved parameterization of the charge distribution from current world data on elastic electron scattering on $^{4}\\mathrm {He}$ by means of a sum of Gaussians charge distribution [58].", "By numerical Fourier transformation we obtain the electric FF which is used in Eqs.", "(62) and (63) in [4].", "In Fig.", "REF we compare the FF obtained with the parameterization from Sick, with other parameterizations.", "With the FF parameterization from Sick we obtain a Friar moment contribution of $\\delta {}E_{\\rm Friar}^A(d) = 6.65\\pm 0.09\\,\\mathrm {meV}.$ This value is in agreement with the value reported by Martynenko (option d) for a simple Gaussian FF.", "The value in Eq.", "(REF ) has a slightly larger uncertainty than the one reported in Eq.", "(REF ).", "The advantage of option e, however, is its model-independence due to the experimentally measured FF.", "In contrast to option b, the value in Eq.", "(REF ) accounts for relativistic recoil corrections and is the recommended option to be used [56].", "All five options presented above are in agreement, whereas their uncertainties differ a lot.", "Hence, a simple average of all the options as we do with other contributions doesn't seem to be adequate in this case.", "Since the dominating uncertainty arises from the inelastic contributions and not from the Friar moment contribution, the different options do not influence the total TPE contribution significantly.", "We decided to use option e for the determination of $r_\\alpha $ from the “muonic” Lamb shift measurements for two reasons: first, option e is model-independent because it includes measured form factors and second, it includes the relativistic corrections.", "This choice is also recommended by Carlson [56].", "Note, that in the case of muonic helium-3 ions, less data is available, which is the reason why for $(\\mu {}^{3}\\mathrm {He})^{+}$[9], option e was not considered.", "From option e we obtain as nuclear Friar moment contribution $\\delta E_{\\rm Friar}^A = \\delta {}E_{\\rm Friar}^A(d) = 6.65 \\pm 0.09 \\,\\mathrm {meV}.$ Figure: Parameterizations of the alpha particle electric form factor (FF) which are used in the calculations of the Friar moment contribution.", "Shown are Gaussian (red, solid) and dipole functions (green, dash-dotted) together with two experimentally deduced FF parameterizations (purple and blue, dashed) , on a linear (Top) and logarithmic scale (Bottom).", "The Gaussian and dipole FF reproduce r α =1.681r_\\alpha = 1.681 fm.", "The experimental fitting results agree with the Gaussian shape, and significantly diverge from the dipole parameterization at momentum transfers ≥0.2 GeV \\ge 0.2\\,\\mathrm {GeV}.", "We base our analysis on the SOG fit .In addition to the nuclear Friar moment, also the contribution of the individual nucleons, $\\delta {}E_{\\rm Friar}^{N}$ , has to be considered.", "The neutron Friar moment is found to be negligible [59].", "For the proton, we follow [60], [61], [62] and obtain its value in $(\\mu {}^{4}\\mathrm {He})^{+}$ by using the proton's Friar moment contribution in muonic hydrogen $\\delta {}E_{\\mathrm {Friar}}^{(p)}(\\mu {}\\mathrm {H})=0.0247(13)$  meV provided in [63].", "We scale it with the wavefunction overlap, that depends on the reduced mass ($m_{r}$ ) and proton number ($Z$ ) scaling to the third power.", "We account for the different number of protons in both systems with an additional $Z$ ratio.", "Another reduced mass scaling factor enters from the third term in Eq.", "(11) of [61] according to [64].", "We obtain for the total nucleon Friar moment In Eq.", "(12) of Ref.", "[8], we used a scaling of the nucleon TPE contribution by the reduced mass ratio to the third power, which is only correct for $\\delta E^N_{\\rm inelastic}$ .", "$\\delta E^N_{\\rm Friar}$ should be scaled with the fourth power [61], [60].", "This is due to an additional $m_r$ scaling factor compared to the proton polarizability term.", "This mistake has no consequences for $\\mu $ d yet, as the nuclear uncertainty is much larger, but the correct scaling is relevant for $(\\mu {}^{3}\\mathrm {He})^{+}$ and $(\\mu {}^{4}\\mathrm {He})^{+}$ .", "$\\begin{split}\\delta {}E_{\\mathrm {Friar}}^{N} = &\\biggl (\\frac{m_r(\\mu {}^4\\mathrm {He})}{m_r(\\mu {}\\mathrm {H})}\\frac{Z(\\mu {}^4\\mathrm {He})}{Z(\\mu {}\\mathrm {H})}\\biggr )^{4}\\delta {}E_{\\mathrm {Friar}}^{(p)}(\\mu {}\\mathrm {H})\\\\[7pt]= &~0.541\\pm 0.028\\,\\text{meV}.\\end{split}$ ($m_r$ , $N$ and $Z$ values provided in footnote $m_r(\\mu {}\\mathrm {H})=185.84\\,m_e$ , $m_r(\\mu {}\\mathrm {D})=195.74\\,m_e$ , $m_r(\\mu {}^4\\mathrm {He})=201.07\\,m_e$ , $Z(\\mu {}\\mathrm {H}) = 1$ , $Z(\\mu {}\\mathrm {D}) = 1$ , $Z(\\mu {}^4\\mathrm {He}) = 2$ , $N(\\mu {}\\mathrm {D}) = 2$ , $N(\\mu {}^4\\mathrm {He}) = 4$).", "This value agrees with the result reported in [65]." ], [ "$(\\mu {}^{4}\\mathrm {He})^{+}$ Polarizability", "The second component of the two-photon exchange in Eq.", "(REF ) is given by the inelastic nuclear “polarizability” contribution, $\\delta {}E_{\\rm inelastic}^{A}$ , that stems from the virtual excitation of the nucleus in the two-photon interaction.", "The initial calculation of the polarizability was done by Joachain [66] in 1961.", "Rinker [67] in 1976 and Friar [57] in 1977 improved the calculation and obtained a value of 3.1(6) meV [57].", "Martynenko used this value in his summary [4], as did Borie in previous versions of hers [3].", "Recently, a more accurate calculation of the $(\\mu {}^{4}\\mathrm {He})^{+}$ nuclear polarizability was done by Ji et al.", "[38] using two parameterizations of the nuclear potential.", "Their calculation uses the AV18 nucleon-nucleon (NN) force plus the UIX three-nucleon (NNN) force, as well as NN forces plus NNN forces from chiral effective field theory ($\\chi $ EFT) to calculate the terms up to the order $(Z\\alpha {})^5$ .", "It provides an energy contribution of 2.47(15) meV that is in agreement with Friar's value but four times more precise.", "Borie adopted the value of Ji et al.", "[38] in the newest version of her summary.", "In muonic deuterium, Pachucki [68] found that the elastic part is exactly canceled by a part of the inelastic.", "These terms are called $\\delta {}_{Z1}^{1}$ and $\\delta {}_{Z3}^{1}$ in Ji et al. [38].", "We assumed this cancellation to be exact in our muonic deuterium theory summary [8].", "Here we treat both parts of the TPE separately and do not use the cancellation.", "For the nuclear polarizability contribution we adopt the most recent value of Hernandez et al.", "[54] $\\delta {}E_{\\mathrm {inelastic}}^{A}= 2.36\\pm 0.14\\,\\mathrm {meV}.$ Next, we account for the contribution due to the polarizability of the individual nucleons $\\delta E_{\\rm inelastic}^N$ .", "In [54], the TRIUMF/Hebrew group provides a value of $\\delta E_{\\rm inelastic}^N({\\rm Hernandez}) = 0.38\\pm 0.22\\,\\mathrm {meV}.$ This value is obtained by scaling the contribution for a single proton of 0.0093(11) $\\mathrm {meV}$  The contribution for a single proton is the sum of an inelastic term $0.0135\\,\\mathrm {meV}$ [69] and a subtraction term $\\delta ^{p}_{\\rm subtr} = -0.0042(10)\\,\\mathrm {meV}$ [70].", "by the number of protons and neutrons Assuming isospin symmetry, the value of the neutron polarizability contribution is the same as the one of the proton, but, as in [60], an additional uncertainty of 20% is added, motivated by studies of the nucleon polarizabilities [71]., as well as with the wavefunction overlap, according to Eq.", "(19) of Ref. [60].", "It also includes a 29% correction for estimated medium effects and possible nucleon-nucleon interferences.", "A smaller uncertainty for $\\delta E_{\\rm inelastic}^N$ could be achieved starting with the inelastic contribution for a proton-neutron pair in muonic deuterium, which was determined by Carlson et al.", "to be $\\delta {}E_{\\mu {}\\mathrm {D}}^{\\mathrm {hadr}}=0.028(2)$  meV [69].", "We scale Carlson's value with the nucleon number $N$ and the wavefunction overlap with the nucleus to obtain $\\begin{split}\\delta {}E_{\\mathrm {\\rm \\mu 4He^+}}^{\\rm hadr} = &~\\frac{N(\\mu {}^4\\mathrm {He})}{N(\\mu {}\\mathrm {D})}\\biggl (\\frac{m_r(\\mu {}^4\\mathrm {He})}{m_r(\\mu {}\\mathrm {D})}\\frac{Z(\\mu {}^4\\mathrm {He})}{Z(\\mu {}\\mathrm {D})}\\biggr )^{3} \\delta {}E_{\\mu {}\\mathrm {D}}^{\\mathrm {hadr}} \\\\[7pt]= &~0.486\\pm 0.069\\,\\text{meV}.\\end{split}$ The uncertainty of this value was chosen a factor of two larger than the one that would be obtained by scaling the $\\mu {}\\mathrm {D}$ value.", "This accounts for possible shadow effects that could exist in the $^{4}\\mathrm {He}$ nucleus.", "This uncertainty estimate has been confirmed by Bacca, Carlson and Gorchtein [72], until better results for $(\\mu {}^{4}\\mathrm {He})^{+}$ and $(\\mu {}^{3}\\mathrm {He})^{+}$ are available.", "We have to add to Eq.", "(REF ) the contribution due to the subtraction term for protons and neutrons (see footnotes  REF and REF) and obtain by scaling from $\\mu $ H $\\begin{split}\\delta {}E_{\\mathrm {sub}} = &~\\biggl (\\frac{m_r(\\mu {}^4\\mathrm {He})}{m_r(\\mu {}\\mathrm {H})}\\frac{Z(\\mu {}^4\\mathrm {He})}{Z(\\mu {}\\mathrm {H})}\\biggr )^{3}\\\\[7pt]& \\times {}2(\\delta ^{p}_{\\rm subtr}+\\delta ^{n}_{\\rm subtr})\\\\[7pt]= &~-0.170\\pm 0.032\\,\\text{meV}.\\end{split}$ The sum of Eqs.", "(REF ) and (REF ) yields an alternative value to Eq.", "(REF ) of $\\delta E_{\\rm inelastic}^N({\\rm Carlson}) = 0.316\\pm 0.076\\,\\mathrm {meV}$ which is in good agreement with Eq.", "(REF ) but three times more precise.", "Summarizing, the nuclear polarizability can be scaled either from the proton, or from the deuteron.", "It is not clear which option is the better one.", "We therefore decided to average Eq.", "(REF ) and (REF ), which yields $\\delta E^N_{\\rm inelastic} = 0.35 \\pm 0.22 \\,\\mathrm {meV}.$ For the total TPE contribution, which is the sum of the nuclear and the nucleon Friar moment contributions (Eqs.", "(REF ), (REF )) and the nuclear and the nucleon polarizability contributions (Eqs.", "(REF ), (REF )) we obtain $\\Delta E^{\\rm LS}_{\\rm TPE} = 9.90 \\pm 0.28\\,\\mathrm {meV}.$ Here, the nuclear and nucleon polarizability contribute 0.14 $\\mathrm {meV}$ and 0.22 $\\mathrm {meV}$ to the uncertainty, respectively.", "The value of Eq.", "(REF ) agrees well with the TPE contribution of 9.58(38) $\\mathrm {meV}$ obtained through ab initio calculations [54].", "A dispersive approach as given for helium-3 [73] is required also for helium-4 in order to crosscheck the value provided here.", "The uncertainty of the total TPE contribution is about 4.5 times larger than the value given as uncertainty goal due to the achieved experimental precision.", "An improvement of this value will directly improve the extraction of the charge radius." ], [ "$(\\mu {}^{4}\\mathrm {He})^{+}$ Fine Structure", "The $\\mathrm {2P}$ fine structure splitting (FS) has been calculated by Borie [3] and Martynenko (Elekina et al.", "[39]) (see Tab.", "REF ).", "Both determinations agree within 0.020 $\\mathrm {meV}$ .", "The leading order contribution to Borie's fine structure is given by the Dirac term of the order $(Z\\alpha {})^4$ (#f1).", "Borie provides additional recoil corrections to her Dirac value (#f2) not covered by her relativistic Dirac wavefunction approach.", "These corrections are already included in Martynenko's term (#f3).", "Martynenko separately calculates corrections to his term like the $(Z\\alpha {})^6$ contribution (#f4a), as well as an additional correction of the order $(Z\\alpha {})^6m_1/m_2$ (#f4b).", "The latter term is new in our FS summary and was not accounted for in $\\mu {}$ D [8] due to its negligible size.", "We sum the Dirac contributions of both authors, including all given corrections and take the average for our summary.", "Doing this, we find an unexpected difference of 0.025 $\\mathrm {meV}$ in the Dirac-term calculation between both authors, as opposed to the much better agreement in the leading order Uehling term of the Lamb shift.", "This might be an indication for inconsistencies between the two methods.", "Further corrections to the FS are given by eVP insertions in the FS interaction.", "Borie, Martynenko and Karshenboim have calculated the 1-loop eVP in both, one- (#f5a) and two Coulomb lines (#f5b) and get matching results for the sum of both terms.", "Only Martynenko calculated the two-loop Källén-Sabry-type diagrams (#f6a) (corresponding to Fig.", "REF , #4 in the Lamb shift).", "The consecutive two-loop correction in two Coulomb lines (#f6b) has been calculated by Martynenko, Borie, and Karshenboim.", "We use the value from Karshenboim, as they included some higher order terms as well.", "The corrections of the order $\\alpha {}^2(Z\\alpha {})^4m$ (#f7) are only calculated by Karshenboim and are included in our summary.", "The correction of order $\\alpha (Z\\alpha )^6$ (#f11*) is only provided by Martynenko whose value we adopt.", "Martynenko also calculates the value of the one loop $\\mu $ VP contribution to the FS that we adopt for our summary (#f12*).", "Contributions of the muon anomalous magnetic moment to the fine structure were provided by Borie and Martynenko and agree perfectly (#f8,#f9).", "The first order finite size contribution has a minor influence on the $\\mathrm {2P}$ FS interval because the $2P_{1/2}$ level has a small, yet non-zero wavefunction at the origin.", "Calculations of the first order term (#f10a) by Borie and Martynenko agree very well and we use the average.", "The term also appears as nuclear structure dependent part of the Lamb shift (#r8) and has to be accounted for in both the Lamb shift and the FS.", "The radius dependence is neglected in the FS since it only provides a minor influence to the total energy difference.", "For the second order contribution (#f10b) we adopt the only available calculation by Martynenko.", "Using the named values, the total $\\mathrm {2P}$ FS in $(\\mu {}^{4}\\mathrm {He})^{+}$ is given by $\\Delta {}E_{2P_{3/2}-2P_{1/2}}= 146.1949\\pm 0.0121\\,\\text{meV}.$ This is in reasonable agreement with the independently published values of Borie [3] and Martynenko [39].", "The uncertainty is dominated by the difference of the leading order Dirac term between Borie and Martynenko.", "A clarification of this difference is desirable.", "The uncertainty of the total fine structure is still 5 times better than our uncertainty goal." ], [ "Summary for $(\\mu {}^{4}\\mathrm {He})^{+}$", "We provided a summary of the Lamb shift and $\\mathrm {2P}$ -fine structure in $(\\mu {}^{4}\\mathrm {He})^{+}$ that will be used for the extraction of the alpha particle charge radius from the measurements performed at PSI [2], [1].", "We compared the calculations of Borie [3], the Martynenko group [4], [39], the Karshenboim group [5], [36], the Jentschura group [6], and the TRIUMF/Hebrew group [38], [53], [54].", "After sorting and comparing all individual terms we found some discrepancies between the different sources (see Tabs REF -REF ): Two-loop eVP (#4,#5), $\\alpha ^2(Z\\alpha )^4m$ contribution (#29) and higher orders (#12-#21) in the radius-independent Lamb shift contributions, different finite size contributions (#r1,#r3,#r3',#r4,#r5), the elastic Friar moment contribution (Eq.", "(REF ) and following), as well as the sum of the Dirac contributions in the FS (#f1-f4).", "Several contributions are only single-authored (#r2',#r2b',#r6,#r7,#r8, and others).", "In order to have a reliable theoretical prediction of the Lamb shift we encourage the theory groups to perform independent calculations to crosscheck the terms calculated by others.", "In summary, we obtain the total energy difference of the $\\mathrm {2S}_{1/2}\\rightarrow \\mathrm {2P}_{1/2}$ Lamb shift transition in $(\\mu {}^{4}\\mathrm {He})^{+}$ as a function of the nuclear charge radius $r_\\alpha $ as $\\Delta {}E_{\\rm (2P_{1/2}-2S_{1/2})} &= \\Delta {}E_{(QED+Recoil)}~+~\\Delta {}E_{(Finite~Size)}~+~\\Delta {}E_{\\rm TPE}^{\\rm LS}\\\\[4pt]&= 1668.489(14)~\\,\\mathrm {meV}\\nonumber \\\\&~\\hspace{82.0pt}~-~106.354(8)~\\,\\mathrm {meV}/\\,\\mathrm {fm}^2\\times {}r_\\alpha ^2 + 0.205(11)\\,\\mathrm {meV}\\\\[4pt]&~\\hspace{165.0pt}~+~9.900(280)~\\,\\mathrm {meV}\\nonumber \\\\&= 1678.594(281) - 106.354(8)\\,\\mathrm {meV}/\\,\\mathrm {fm}^2\\times r_\\alpha ^2.\\\\[5pt](\\text{for comparison}&\\text{ use:}~r_\\alpha ^2\\approx {}2.83\\,\\,\\mathrm {fm}^2)\\nonumber $ The $\\Delta {}E_{\\rm (2P_{1/2}-2S_{1/2})}$ energy difference corresponds to the sum of Eqs.", "(REF ), (REF ), and (REF ).", "The $\\Delta {}E_{\\rm (2P_{3/2}-2S_{1/2})}$ energy difference is obtained by including the fine structure from Eq.", "(REF ) which yields $\\begin{split}\\Delta {}E_{\\rm (2P_{3/2}-2S_{1/2})} &= 1824.789(281)\\\\ &\\hphantom{=~}- 106.358(7)\\,\\mathrm {meV}/\\,\\mathrm {fm}^2\\times \\,r_\\alpha ^2.\\end{split}$ The currently limiting factor in the $(\\mu {}^{4}\\mathrm {He})^{+}$ theory originates from the two-photon exchange (TPE) contribution, where mainly the uncertainty of the inelastic nucleon polarizability contribution (Eq.", "(REF )) is dominating.", "Improving the terms which constitute the TPE will directly improve the value of the alpha particle charge radius determination.", "Using the prediction of the Lamb shift from the compiled theory of this work, we derive a theoretical uncertainty of the $^{4}\\mathrm {He}$ nuclear charge radius from the laser spectroscopy measurement in muonic helium-4 ions of $<0.0008\\,\\mathrm {fm}$ .", "Compared to this value the expected experimental uncertainty will be small.", "Hence, with the theory presented in this compilation and the to-be-published measurement, we expect an improvement of the previous best value by a factor of $\\sim 5$ ." ], [ "The ", "The \"isotope shift\" (IS) refers to the squared charge radius difference, e.g.", "$r_h^2 - r_\\alpha ^2$ for $^3$ He and $^4$ He.", "This is the quantity that is directly measured for the He isotopes in electronic He atoms [24], [25], [26].", "For the CREMA measurements in muonic $^3$ He and $^4$ He, one could calculate the IS from the absolute charge radii, determined using Eq.", "(18) in [9] and Eq.", "(REF ) in here.", "The accuracy of both muonic radii is limited by the uncertainty in the nuclear and nucleon two-photon exchange (TPE) contribution.", "However, the uncertainties due to nuclear model-dependence and the ones due to the single nucleons, respectively, are strongly correlated between the two isotopes.", "Here we attempt to reduce the uncertainty in the theory of the TPE contributions to the IS in muonic $^3$ He and $^4$ He, taking into account these correlations.", "For both helium isotopes ($i=h,\\alpha $ ), the transition energy is given in the form $\\Delta E_{\\rm LS}^{(i)} = E_{r-\\rm ind.", "}^{(i)}+ c_i r_i^2 + E_{\\rm TPE}^{(i)},$ where $E_{r-\\rm ind.", "}^{(i)}$ is the radius-independent QED part of the Lamb shift, the second term is the radius-dependent part of the Lamb shift with the charge radius $r_i$ and the coefficient $c_i$ , and $E_{\\rm TPE}^{(i)}$ is the two-photon exchange.", "Using the measured transition energy $h\\nu ^{(i)}_{\\rm LS}$ of isotope $i$ and Eq.", "(REF ), the square of the charge radius $r_i$ is $r_i^2 = \\frac{1}{c_i} \\left(h\\nu ^{(i)}_{\\rm LS} - E_{r-\\rm ind.", "}^{(i)} - E_{\\rm TPE}^{(i)}\\right).$ The value of each charge radius will be limited by the theory uncertainty from the two-photon exchange (TPE) contribution.", "In order to exploit correlations, we use for the IS the TPE results from the TRIUMF/Hebrew group [38], [60], [54] alone.", "The TRIUMF/Hebrew group has consistently calculated the TPE for both, $(\\mu {}^{4}\\mathrm {He})^{+}$ and $(\\mu {}^{3}\\mathrm {He})^{+}$ .", "This means using option a for the Friar moment contribution, described in Sec.", "REF , which is not what we chose for the extraction of the alpha charge radius.", "However, due to correlations in the IS, the large uncertainty of option a partly cancels.", "The uncertainty of the charge radius $r_i$ is obtained by propagating the uncertainties from the experimental value $h\\nu _{\\rm LS}^{(i)}$  To be published.", "The experimental uncertainty will be dominated by statistics.", "and the theory values $E_{r-\\rm ind.", "}^{(i)}$ , $c_i$ , and $E_{\\rm TPE}^{(i)}$  The uncertainty in the TPE contribution dominates by far the total uncertainty in the charge radius..", "The isotope shift is then given by $r_h^2 - r_\\alpha ^2 &= \\frac{h\\nu ^{(h)}_{\\rm LS} - E_{r-\\rm ind.", "}^{(h)} - E_{\\rm TPE}^{(h)}}{c_h} \\nonumber \\\\& \\hphantom{=~} - \\frac{ h\\nu ^{(\\alpha )}_{\\rm LS} - E_{r-\\rm ind.", "}^{(\\alpha )} - E_{\\rm TPE}^{(\\alpha )}}{c_\\alpha }\\\\&= \\frac{h\\nu ^{(h)}_{\\rm LS}}{c_h} - \\frac{h\\nu ^{(\\alpha )}_{\\rm LS}}{c_\\alpha }- \\left(\\frac{E_{r-\\rm ind.", "}^{(h)}}{c_h}- \\frac{E_{r-\\rm ind.", "}^{(\\alpha )}}{c_\\alpha }\\right)\\nonumber \\\\&\\hphantom{=~} -\\left(\\frac{E_{\\rm TPE}^{(h)}}{c_h}- \\frac{E_{\\rm TPE}^{(\\alpha )}}{c_\\alpha }\\right)\\\\&= \\Delta _\\nu - \\Delta _{r-\\rm ind.}", "- \\Delta _{\\rm TPE},$ where $\\Delta _\\nu $ contains the experimental Lamb shift transition energies ($h\\nu _{\\rm LS}^{(h)}$ and $h\\nu _{\\rm LS}^{(\\alpha )}$ ).", "All other contributions including their uncertainties are listed in Tab.", "REF .", "A simple Gaussian propagation of these uncertainties is only correct for uncorrelated terms.", "In the following we discuss the correlations in the uncertainties of the TPE terms and show how to get rid of the uncertainty correlations which arise due to nuclear modeling and due to scaling the nucleon contributions.", "Table: The values and uncertainties of the terms which appear in the isotope shift (Eq.", "()).The dominating uncertainties arise from the TPE terms of both helium isotopes.", "Note that their uncertainties are correlated and should not simply be propagated for the isotope shift.", "For a detailed discussion, see text.", "The values are given in units of   meV \\mathrm {meV}.Since the uncertainties in the TPE terms are by far the dominating uncertainty in the extraction of the isotope shift, we restrict the discussion of correlations to the TPE contributions only and have a closer look into the TPE term $\\Delta _{\\rm TPE}$ from Eq.", "() $\\begin{split}\\Delta _{\\rm TPE} &= \\frac{E_{\\rm TPE}^{(h)}}{c_h} - \\frac{E_{\\rm TPE}^{(\\alpha )}}{c_\\alpha }\\\\&= \\frac{\\delta E^{A, (h)}_{\\rm Friar} + \\delta E^{A, (h)}_{\\rm inel.}", "+ a_h\\delta E^{(p)}_{\\rm Friar} + b_h\\delta E^{(p)}_{\\rm inel.", "}}{c_h}\\\\& \\hphantom{=~}- \\frac{\\delta E^{A, (\\alpha )}_{\\rm Friar} + \\delta E^{A, (\\alpha )}_{\\rm inel.}", "+ a_\\alpha \\delta E^{(p)}_{\\rm Friar} + b_\\alpha \\delta E^{(p)}_{\\rm inel.", "}}{c_\\alpha }\\\\&= \\frac{\\delta E^{A, (h)}_{\\rm TPE}}{c_h} - \\frac{\\delta E^{A, (\\alpha )}_{\\rm TPE}}{c_\\alpha }\\\\& \\hphantom{=~}+ (\\frac{a_h}{c_h}-\\frac{a_\\alpha }{c_\\alpha })\\delta E^{(p)}_{\\rm Friar} + (\\frac{b_h}{c_h}-\\frac{b_\\alpha }{c_\\alpha })\\delta E^{(p)}_{\\rm inel.", "},\\\\\\end{split}$ where, following Eq.", "(REF ), we break down the TPE contribution into its constituents and explicitly write the nucleon terms as a scaling factor $a_{h/\\alpha }$ , $b_{h/\\alpha }$ times the respective contribution from the proton.", "The values of the TPE terms and the scaling factors in Eq.", "(REF ) are listed in Tab.", "REF .", "For the scaling factors $a_{h/\\alpha }$ and $b_{h/\\alpha }$ compare Eqs.", "(17) and (19) in [60].", "In the last line we sum up the nuclear terms for both isotopes, respectively, and we reorder the nucleon terms to make the cancellations visible which appear between the scaling factors.", "Table: The values and uncertainties of the TPE terms which appear in the term Δ TPE \\Delta _{\\rm TPE} (Eq. ()).", "The upper part represents the nuclear and the lower part the nucleon terms.", "The nuclear terms are calculated using the AV18 nuclear potential and χ\\chi EFT.", "The values from the two approaches are used in order to determine the uncertainty due to nuclear model-dependence.", "The values given under “avg.” are the ones which are then used to infer a value of the IS calculation.", "The “avg.” values are not necessarily the true average of the model-dependent numbers given here, since they have been updated .", "In contrast to Ref.", ", here the “avg.” values do not show the uncertainty due to the nuclear model-dependence.", "The given uncertainties are only provided for i=hi=h.", "The uncertainty for i=αi=\\alpha , labeled with * ^* is scaled from helium-3 with the ratio of the contributions.We discuss first the nuclear uncertainties and then the nucleon part.", "The nuclear part.", "The nuclear TPE contribution (nuclear Friar moment + nuclear polarizability) for both muonic helium isotopes has been calculated by the TRIUMF/Hebrew group [38], [60], [54] using the AV18 potential and using $\\chi $ EFT For the Friar moment, the calculation of the TRIUMF/Hebrew group corresponds to option a, discussed in Sec. .", "The difference between the numbers of the two calculations serves as estimate for the uncertainty due to nuclear model-dependence.", "Using the same method for the isotope shift in Eq.", "(REF ), i.e.", "inserting the different values given in Tab.", "REF one after the other, a good estimate for the nuclear model uncertainty is obtained.", "It amounts to 0.0008 $\\mathrm {meV}$ .", "The TRIUMF/Hebrew group also provides an uncertainty due to sources other than the nuclear model which are not further specified.", "This uncertainty is only given for muonic helium-3 ions [60] (Eq.", "(16)) and amounts to 0.27 $\\mathrm {meV}$ .", "For muonic helium-4 ions this uncertainty is not provided.", "In the following we assume it to scale with the value of the nuclear TPE contribution in the respective isotope and obtain 0.16 $\\mathrm {meV}$ for muonic helium-4 ions, see also Tab.", "REF .", "A propagation of these two uncertainties via Eq.", "(REF ) leads to 0.0030 $\\mathrm {meV}$ .", "The nucleon part.", "Similar to the nuclear part, the nucleon TPE contribution for the two isotopes consists of the nucleon Friar moment contribution and the nucleon polarizability contribution.", "The nucleon Friar moment is obtained using the Friar moment from muonic hydrogen of $\\delta E^{(p)}_{\\rm Friar} = 0.0247(13)\\,\\mathrm {meV}$ [63].", "This value is multiplied with a scaling factor as it is done in Eq.", "(REF ).", "The scaling factor for $(\\mu {}^{3}\\mathrm {He})^{+}$ is $a_h=21.1511$ , for $(\\mu {}^{4}\\mathrm {He})^{+}$ $a_\\alpha =21.9247$ .", "Propagating the uncertainty of 0.0013 $\\mathrm {meV}$ via Eq.", "(REF ) leads to $2\\times 10^{-6}\\,\\mathrm {meV}$ , which is negligible.", "The nucleon polarizability contribution is scaled from the single proton polarizability which (including the subtraction term) amounts to $\\delta E^{(p)}_{\\rm inel.}", "= 0.0093(11)\\,\\mathrm {meV}{}$ (see text below Eq.", "(REF )).", "The scaling works according to Eq.", "(19) in [60] which results in scaling factors of $b_h=29.5884$ and $b_\\alpha =40.5284$ for muonic helium-3 and -4, respectively.", "Propagating the uncertainty from the nucleon polarizability of 0.0011 $\\mathrm {meV}$ via Eq.", "(REF ) we obtain an uncertainty for the isotope shift of 0.0001 $\\mathrm {meV}$ .", "The total TPE uncertainty.", "In total, the TPE contributions to the isotope shift lead to an uncertainty of 0.0031 $\\mathrm {meV}$ , which results from the uncertainties added in quadrature.", "It is dominated by the above discussed uncertainty from the nuclear parts.", "The uncertainties due to the radius-independent QED terms $E_{r-\\rm ind.", "}^{(i)}$ and the coefficients $c_i$ are used as presented in Tab.", "REF and propagated via Gaussian propagation of uncertainties through Eq.", "(REF ).", "We obtain isotope shift uncertainties of 0.0002 $\\mathrm {meV}$ and 0.0004 $\\mathrm {meV}$ from the radius-independent terms and the coefficients, respectively.", "Inserting the numbers from Tab.", "REF into Eq.", "(REF ), the $r_h^2-r_\\alpha ^2$ isotope shift from muonic helium spectroscopy is then given by $\\begin{split}r_h^2-r_\\alpha ^2 &= \\left(\\frac{h\\nu _{\\rm LS}^{(h)}/\\,\\mathrm {meV}}{103.518} - \\frac{h\\nu _{\\rm LS}^{(\\alpha )}/\\,\\mathrm {meV}}{106.354}\\right)\\,\\mathrm {fm}^2\\\\&\\hphantom{=~} + 0.1980\\,\\mathrm {fm}^2 + 0.0593\\,\\mathrm {fm}^2\\\\&\\hphantom{=~} (\\pm 0.0002^{\\rm QED} \\pm 0.0004^{\\rm coeff.}", "\\pm 0.0031^{\\rm TPE})\\,\\mathrm {fm}^2\\\\[4pt]&= \\left(\\frac{h\\nu _{\\rm LS}^{(h)}/\\,\\mathrm {meV}}{103.518} - \\frac{h\\nu _{\\rm LS}^{(\\alpha )}/\\,\\mathrm {meV}}{106.354}\\right)\\,\\mathrm {fm}^2\\\\&\\hphantom{=~} + 0.2572\\,\\mathrm {fm}^2 \\pm 0.0031^{\\rm theo} \\,\\mathrm {fm}^2.\\end{split}$ Note that without taking into account the correlations and using instead the TPE contributions from Eq.", "(17) in Ref.", "[9] and from Eq.", "(REF ) in this work, the last line of Eq.", "(REF ) would read $0.2527(57)\\,\\mathrm {fm}^2$ , which is in good agreement, but with a twice larger uncertainty.", "Since the experimental uncertainty is expected to be small compared to the theory uncertainty of $0.0031\\,\\mathrm {fm}^2$ , the extraction of the isotope shift from muonic helium ions will compete with the uncertainty from previous measurements [24], [25], [26] and may shed new light on the discrepancy between those." ], [ "Conclusion", "In the first part of this work we have summarized and compiled all available theory contributions to the $\\mathrm {2S\\rightarrow {}2P}$ Lamb shift in $(\\mu {}^{4}\\mathrm {He})^{+}$ , which is necessary in order to extract the alpha charge radius from laser spectroscopy measurements in muonic helium-4 ions.", "The result of our compilation is shown in Eq.", "(REF ).", "In the second part, we studied the theory uncertainties which enter the value of the 3He–4He isotope shift that can be extracted from the CREMA measurement in $(\\mu {}^{3}\\mathrm {He})^{+}$ and $(\\mu {}^{4}\\mathrm {He})^{+}$ .", "We obtain a total theory uncertainty of $0.0031\\,\\mathrm {fm}^2$ , see Eq.", "(REF ).", "This uncertainty is much smaller than a discrepancy between previous isotope shift measurements in electronic helium atoms.", "The value of the isotope shift from the CREMA collaboration will therefore test the discrepancy with a completely independent method.", "Acknowledgments: We thank E. Borie, A. P. Martynenko, S. Karshenboim, S. Bacca, N. Barnea, M. C. Birse, C. E. Carlson, M. I. Eides, J. Friar, M. Gorchtein, F. Hagelstein, U. Jentschura, C. Ji, J. A.", "McGovern, N. Nevo Dinur, C. Pachucki, V. Pascalutsa, and M. Vanderhaeghen for fruitful discussions in the creation of this summary.", "We specially thank A. P. Martynenko for providing the tools to repeat their calculation of the Friar moment, as well as I. Sick for providing us with a sum of Gaussians FF parameterization of the world data from elastic electron scattering.", "The authors acknowledge support from the European Research Council (ERC) through StG.", "#279765 and CoG.", "#725039, the Excellence Cluster PRISMA of the Unversity of Mainz, and the Swiss National Science Foundation SNF, Project 200021_165854.", "*" ], [ "Update on muonic helium-3 theory", "In our muonic helium-3 theory summary [9] we are lacking a contribution which we add in this appendix.", "In this way we make sure that we treat both isotopes in the same way.", "The contribution is a VP correction of order $\\alpha (Z\\alpha )^5$ .", "It is calculated by the Martynenko group in Krutov et al.", "[4] (see Tab.", "1, line 20).", "In [9] we added the higher order correction just to Martynenko's Friar moment contribution.", "Since we did not use Martynenko's value for the Friar moment, the correction is missing in the final value.", "In this work, for $(\\mu {}^{4}\\mathrm {He})^{+}$ , we add this correction which amounts to 0.1270(13) $\\mathrm {meV}$ to the nuclear structure contributions as item #r2b', in Tab.", "REF .", "In the case of muonic helium-3 ions the correction amounts to 0.2214(22) $\\mathrm {meV}$ [4].", "Including this correction, the updated nuclear structure contributions, originally given in Eq.", "(12) of Ref.", "[9], result in $\\begin{split}\\Delta E^{\\rm LS}_{r-\\rm dep.", "}(\\mathrm {\\mu ^3He^+}) &= -103.5184(98)\\,\\mathrm {meV}/\\,\\mathrm {fm}^2 r_h^2 \\\\&\\hphantom{=~} + 0.3568(40)\\,\\mathrm {meV}.\\end{split}$ With that, we update Eq.", "(18) in [9] and obtain a new energy for the $\\mathrm {2S\\rightarrow {}2P}{}$ Lamb shift of $\\Delta E_{(\\mathrm {2S}_{1/2}\\rightarrow \\mathrm {2P}_{1/2})}(\\mathrm {\\mu ^3He^+})\\\\\\begin{split}=&~ 1644.3466(146)\\,\\mathrm {meV}\\\\ &+~0.3568(40)\\,\\mathrm {meV}-103.5184(98)\\,\\mathrm {meV}/\\,\\mathrm {fm}^2 r_h^2\\\\ &+15.3000(5200)\\,\\mathrm {meV}\\\\=&~1660.00(52)\\,\\mathrm {meV}-103.518(10)\\, r_h^2\\,\\mathrm {meV}/\\,\\mathrm {fm}^2,\\end{split}$ where $r_h$ is the helion charge radius.", "The contributions for fine- and hyperfine structure remain unchanged." ] ]
1606.05231
[ [ "Tuning of Near- and Far-Field Properties of All-dielectric Dimer\n Nanoantennas via Ultrafast Electron-Hole Plasma Photoexcitation" ], [ "Abstract Achievement of all-optical ultrafast signal modulation and routing by a low-loss nanodevice is a crucial step towards an ultracompact optical chip with high performance.", "Here, we propose a specifically designed silicon dimer nanoantenna, which is tunable via photoexcitation of dense electron-hole plasma with ultrafast relaxation rate.", "Basing on this concept, we demonstrate the effect of beam steering up to 20 degrees via simple variation of incident intensity, being suitable for ultrafast light routing in an optical chip.", "The effect is demonstrated both in the visible and near-IR spectral regions for silicon and germanium based nanoantennas.", "We also reveal the effect of electron-hole plasma photoexcitation on local density of states (LDOS) in the dimer gap and find that the orientation averaged LDOS can be altered by 50\\%, whereas modification of the projected LDOS can be even more dramatic: almost 500\\% for transverse dipole orientation.", "Moreover, our analytical model sheds light on transient dynamics of the studied nonlinear nanoantennas, yielding all temporal characteristics of the proposed ultrafast nanodevice.", "The proposed concept paves the ways to creation of low-loss, ultrafast, and compact devices for optical signal modulation and routing." ], [ "Introduction", "The ability to control scattering of light by nanostructures is crucial for the development of functional nanodevices for data processing and information transfer.", "Nanoantenna switching usually implies control of extinction and absorption cross-sections [1], [2], scattering patterns [3], [4], [5], [6], [7] and near field distribution [8], [9].", "Such alteration of optical properties can be achieved via an external impact, e.g.", "electro-optic effect [10], [11], magneto-optic effect [12], thermo-optical effect [13] or carriers injection [14].", "Alternatively, one may utilize the nonlinear response of the structure materials and control scattering with intensity of incident light [15], [16], [17].", "Noble metals offer Kerr nonlinearity which can be employed for all-optical variation of scattering behavior [18].", "However, plasmonic nanoantennas suffer from high Joule losses and heating, which limit the tuning capabilities of such systems.", "Silicon, on the other hand, has become a promising platform for implementation of nonlinear photonic devices thanks to a broad range of optical nonlinearities such as Kerr effect, two-photon absorption, and electron-hole plasma (EHP) excitation [19].", "Silicon nanoantennas demonstrate a damage threshold far exceeding that of their plasmonic counterparts, thus enabling higher degree of tuning.", "Recently, enhancement of optical nonlinearities in silicon has been demonstrated at the scale of single nanoparticles [20], [21], [22], [23], [24].", "In particular, photoexcitation of EHP was employed for tuning of silicon nanoantenna optical properties in the IR and visible regions [21], [22].", "Figure: A schematic view of beam steering via EHP photoexcitation in silicon nanoparticles.In this paper, we explore the capabilities of silicon nanoparticle dimers for nonlinear optical tuning enabled by photoexcitation of EHP.", "In particular, we demonstrate nonlinear beam steering in an asymmetric dimer.", "The main direction of scattered light is controlled via intensity of incident pulse.", "For a realistic 200 fs pulse with a peak intensity of about 40 GW/cm$^2$ we observe steering of the scattering direction as large as 20 degrees compared to a weak reference pulse.", "Apart from the far-field properties of a nanoantenna manifested in its scattering diagram, we investigate how the near-field behavior, namely, local density of optical states (LDOS) can be controlled in the vicinity of the nanodimer via EHP excitation.", "We observe almost two-fold variation of LDOS in the dimer gap when 40 GW/cm$^2$ is applied.", "Our findings provide an additional tool for controlling light scattering at the nanoscale and prove the potential of silicon and germanium for the development of nanoscale all-optical devices." ], [ "Model", "The proposed system for nonlinear beam steering as well as its operation principle are schematically shown in Fig.", "REF .", "The two high-index dielectric nanoparticles of radii $R_1$ and $R_2$ surrounded by vacuum and separated by distance $L$ comprise the asymmetric dimer.", "Incident optical pulse enables photoexcitation of EHP within nanoparticles affecting their optical resonances and therefore scattering behavior.", "Silicon (Si) is chosen as a high-index material due to its high two-photon absorption at optical frequencies resulting in efficient EHP excitation [25].", "On the other hand, germanium (Ge) may become the optimal choice in the near-IR due to its attractive nonlinear properties.", "Figure: (a) Scattering cross-section of isolated nanoparticles with increasing EHP density in the resonant particle.", "Dashed curve shows cross-section of the non-resonant particle.", "(b) Scattering diagram for the asymmetric dimer for different EHP densities in the resonant nanoparticle.In order to simulate nonlinear scattering of an optical pulse we adopt our analytical model developed in Ref. [25].", "Each spherical nanoparticle is modeled as a combination of electric ($\\mathbf {p}$ ) and magnetic ($\\mathbf {m}$ ) dipole moments.", "The temporal dynamics of slowly varying amplitudes of these dipoles is governed by oscillator equations: ${\\begin{array}{c}i\\frac{{\\partial \\alpha _e^{ - 1}}}{{\\partial \\omega }}\\frac{{d{\\mathbf {p}}}}{{dt}} + \\alpha _e^{ - 1}{\\mathbf {p}} = {{\\mathbf {E}}_{{\\text{inc}}}}\\left( t \\right), \\hfill \\\\i\\frac{{\\partial \\alpha _m^{ - 1}}}{{\\partial \\omega }}\\frac{{d{\\mathbf {m}}}}{{dt}} + \\alpha _m^{ - 1}{\\mathbf {m}} = {{\\mathbf {H}}_{{\\text{inc}}}}\\left( t \\right) \\hfill \\\\\\end{array}}$ where dipole polarizabilities are expressed in terms of the Mie coefficients $a_1$ and $b_1$ as ${\\alpha _e} = \\frac{{3i}}{{2{k^3}}}{a_1}$ and ${\\alpha _m} = \\frac{{3i}}{{2{k^3}}}{b_1}$ , respectively,[26] with $k=\\omega /c$ being the free space wave vector.", "${{\\mathbf {E}}_{{\\text{inc}}}}$ and ${{\\mathbf {H}}_{{\\text{inc}}}}$ denote slowly varying amplitudes of incident electric and magnetic fields.", "The equations (REF ) fully describe dipole moments dynamics of a single nanoparticle provided that EHP density, which determines permittivity of photoexcited silicon, is known at all times.", "However, EHP excitation is driven by optical absorption within silicon due to the electric fields of induced dipoles.", "The dynamics of volume-averaged EHP density is described via the following rate equation [25]: $\\frac{{d{\\rho _{{\\text{eh}}}}}}{{dt}} = - \\Gamma {\\rho _{{\\text{eh}}}} + \\frac{{{W_1}}}{{\\hbar \\omega }} + \\frac{{{W_2}}}{{2\\hbar \\omega }}.$ Here, $W_{1,2}$ are the volume-averaged dissipation rates due to one- and two-photon absorption, and $\\Gamma $ is the phenomenological EHP relaxation rate constant which depends on EHP density [27].", "The absorption rates are written in the usual form as ${W_1} = \\frac{\\omega }{{8\\pi }} \\left\\langle {{{\\left| {{\\mathbf {\\tilde{E}_{\\rm in}}}} \\right|}^2}} \\right\\rangle {\\rm Im} (\\varepsilon )$ and ${W_2} = \\frac{\\omega }{{8\\pi }} \\left\\langle {{{\\left| {{\\mathbf {\\tilde{E}_{\\rm in}}}} \\right|}^4}} \\right\\rangle \\operatorname{Im} {\\chi ^{(3)}}$ , where $\\left\\langle {...} \\right\\rangle $ denotes averaging over the nanoparticle volume, and $\\operatorname{Im} {\\chi ^{(3)}} = \\frac{{\\varepsilon {c^2}}}{{8\\pi \\omega }}\\beta $ with $\\beta $ being two-photon absorption coefficient.", "These averaged fields should be related to the instantaneous values of electric and magnetic dipole moments.", "This is done by integrating the total field of the two spherical harmonics corresponding to the given values of $\\mathbf { p}$ and $\\mathbf {m}$ .", "The relaxation rate of EHP in c-Si is dominated by Auger recombination [28]: $\\Gamma = \\Gamma _{\\text{A}} \\rho _{\\rm eh}^2$ with $\\Gamma _{\\rm A}=4 \\cdot 10^{ - 31}$  s$^{-1}$ cm$^6$ (Ref. [29]).", "In germanium, EHP relaxation is again mediated by Auger mechanism with $\\Gamma _{\\rm A}=7 \\cdot 10^{ - 33}$  s$^{-1}$ cm$^6$ (Ref. [30]).", "The system of equations (REF ) and (REF ) should be completed by the expression relating permittivity of excited material $\\varepsilon $ to EHP density $\\rho _{\\rm eh}$ .", "For silicon, this dependence is represented as the following expression:[31], [22] ${\\varepsilon }\\left( {\\omega ,{\\rho _{{\\text{eh}}}}} \\right) = \\varepsilon _{\\text{0}}+ \\Delta \\varepsilon _{\\rm bgr}+ {\\Delta \\varepsilon _{{\\text{bf}}}} + {\\Delta \\varepsilon _{{\\text{D}}}}$ where ${\\varepsilon _{{\\text{0}}}}$ is the permittivity of non-excited material, whereas $\\Delta \\varepsilon _{\\text{bgr}}$ , ${\\Delta \\varepsilon _{{\\text{bf}}}}$ , and $\\Delta \\varepsilon _{{\\text{D}}}$ are the contributions from band gap renormalization, band filling, and Drude term.", "The detailed expressions for all contributions in Eq.", "(REF ) are given in Supporting Information.", "Turning to permittivity of photoexcited germanium, we note that in the IR range it is dominated by Drude contribution, expression for which is adopted from Ref.", "[32] (see Supporting Information for details).", "Figure: (a) Time-dependent real part of permittivity of the two dimer nanoparticles under 40 GW/cm 2 ^2 pulse irradiation.", "Shaded area represents the incident pulse intensity.", "(b) The steering angle δφ\\delta \\phi as a function of time upon irradiation by pulses of different peak intensities.", "(c) The scattering diagrams of the photoexcited dimer corresponding to circles in panel (b).", "(d) - (f) The same as (a) - (c) but for ss-polarized incident wave.When the nanoparticles form a nanodimer, the incident electric and magnetic fields ${{\\mathbf {E}}_{{\\text{inc}}}}\\left( t \\right)$ , ${{\\mathbf {H}}_{{\\text{inc}}}}\\left( t \\right)$ in Eq.", "(REF ) must include the field of the incident plane wave as well as the local fields due to induced dipoles of the adjacent particle: ${\\begin{array}{c}{\\mathbf {E}}_{i.j}^{{\\text{inc}}}\\left( t \\right) = {{\\mathbf {E}}_0}\\left( {{{\\mathbf {r}}_{i,j}}} \\right) + {k^2}{\\mathbf {\\hat{G}}}\\left( {{{\\mathbf {r}}_{i,j}},{{\\mathbf {r}}_{j,i}}} \\right){{\\mathbf {p}}_{j,i}}\\left( t \\right) + \\hfill \\\\ik\\nabla \\times {\\mathbf {\\hat{G}}}\\left( {{{\\mathbf {r}}_{i,j}},{{\\mathbf {r}}_{j,i}}} \\right){{\\mathbf {m}}_{j,i}}\\left( t \\right), \\hfill \\\\{\\mathbf {H}}_{i.j}^{{\\text{inc}}}\\left( t \\right) = {{\\mathbf {H}}_0}\\left( {{{\\mathbf {r}}_{i,j}}} \\right) + {k^2}{\\mathbf {\\hat{G}}}\\left( {{{\\mathbf {r}}_{i,j}},{{\\mathbf {r}}_{j,i}}} \\right){{\\mathbf {m}}_{j,i}}\\left( t \\right) - \\hfill \\\\ik\\nabla \\times {\\mathbf {\\hat{G}}}\\left( {{{\\mathbf {r}}_{i,j}},{{\\mathbf {r}}_{j,i}}} \\right){{\\mathbf {p}}_{j,i}}\\left( t \\right), \\hfill \\\\\\end{array}}$ Here ${\\mathbf {\\hat{G}}}\\left( {{\\mathbf {r}},{\\mathbf {r^{\\prime }}}} \\right) = \\left( {{\\mathbf {1}} + \\frac{1}{{{k^2}}}\\nabla \\otimes \\nabla } \\right)\\frac{{{e^{ik\\left| {{\\mathbf {r}} - {\\mathbf {r^{\\prime }}}} \\right|}}}}{{\\left| {{\\mathbf {r}} - {\\mathbf {r^{\\prime }}}} \\right|}}$ is the electric Green tensor, ${{\\mathbf {E}}_0}\\left( {\\mathbf {r}} \\right) = \\mathbf {E}_0{e^{i{\\mathbf {kr}}}}$ , ${{\\mathbf {H}}_0}({\\mathbf {r}}) = {\\mathbf {k}} \\times {{\\mathbf {E}}_0}({\\mathbf {r}})/k$ , and $\\mathbf {k}$ is the incident wavevector.", "The scattering pattern is determined via calculation of the time-averaged Poynting vector of the scattered field in the far-field zone: ${{\\mathbf {S}}_{{\\text{scat}}}} = \\frac{c}{{8\\pi }}\\operatorname{Re} \\left( {{{\\mathbf {E}}_{{\\text{scat}}}} \\times {\\mathbf {H}}_{{\\text{scat}}}^*} \\right)$ where the scattered fields in the far-field zone are given by ${\\begin{array}{c}{{\\mathbf {H}}_{{\\text{scat}}}} = {k^2}\\sum \\limits _j {{\\mathbf {\\hat{G}}}\\left( {{\\mathbf {r}},{{\\mathbf {r}}_j}} \\right){{\\mathbf {m}}_j}} - ik\\sum \\limits _j {\\nabla \\times {\\mathbf {\\hat{G}}}\\left( {{\\mathbf {r}},{{\\mathbf {r}}_j}} \\right){{\\mathbf {p}}_j}} , \\hfill \\\\{{\\mathbf {E}}_{{\\text{scat}}}} \\approx - \\frac{1}{k}{{\\mathbf {k}}_{{\\text{scat}}}} \\times {{\\mathbf {H}}_{{\\text{scat}}}}, \\hfill \\\\\\end{array}}$ with $\\mathbf {k}_{\\rm scat}$ being wave vector in the scattering direction of interest." ], [ "Beam steering", "We now apply our model to demonstrate the effect of efficient nonlinear beam steering.", "We consider scattering of an $p-$ polarized optical pulse of $\\lambda =600$  nm wavelength with its wave vector $\\mathbf {k}$ normal to the dimer axis.", "The dimer is formed by a magnetic dipole (MD) resonant particle of radius $R_1=74$  nm and non-resonant particle of radius $R_2=68$  nm separated by $L=220$  nm.", "This geometry enables favorable conditions for beam steering (see Supporting Information).", "Wavelength of 600 nm is chosen due to large two-photon absorption allowing to reduce intensities required for considerable switching.", "The mechanism of steering can be understood by considering the optical properties of isolated particles comprising the dimer.", "The scattering cross-section spectra of the isolated particles in the linear regime are shown in Fig.", "REF (a).", "The larger particle is tuned to the MD resonance at 600 nm, while the smaller particle is off resonance and close to the Kerker condition of unidirectional scattering, where $\\alpha _{e}=\\alpha _{m}$  [33].", "This results in a strongly asymmetric radiation pattern of the dimer shown in Fig.", "REF (b).", "The recent observation of directional scattering form silicon dimers [34] suggests that the concept of EHP excitation in silicon nanoparticles can be employed for tailoring the radiation pattern at a constant wavelength but instead with varying incident intensity.", "Indeed, when a strong pulse is incident on the dimer, it will enable dense EHP photoexcitation in the larger resonant particle, while the smaller non-resonant particle will be almost unaffected by the pulse.", "This will cause the MD resonance of the larger particle shift to shorter wavelengths due to refractive index decrease induced by EHP.", "Eventually, at certain level of photoexcited EHP density the effective resonance curves of the two particles may overlap leading to nearly symmetric forward scattering at $\\lambda =600$  nm.", "This evolution of scattering patterns is shown in Fig.", "REF (b) for different values of EHP density in the resonant particle.", "Such modification of optical response for single nanoparticles was recently demonstrated in Refs.", "[22], [23].", "This behavior is confirmed in numerical modeling of the transient nonlinear dynamics of the dimer.", "Time-dependent permirttivities of photoexcited Si in each of the two particles are shown in Fig.", "REF (a) for a 200 fs Gaussian pulse with 40 GW/cm$^2$ peak intensity.", "Near the pulse center at $t \\approx 400$  fs EHP induced permittivity correction in the resonant particle is nearly 5 times larger than that in the non-resonant particle.", "The attainable degree of steering depends on intensity of incident pulse.", "Moreover, since EHP excitation and relaxation are not instantaneous processes, it is useful to investigate dynamics of steering during the pulse action.", "This dynamics for a series of peak intensities is presented in Fig.", "REF (b), where the angle of the main lobe direction $\\delta \\phi $ is shown as a funtion of time (the scattering direction in the cold regime is taken as 0).", "Steering of the scattered radiation is maximal near the pulse center and slowly decreases afterwards owing to ps-scale EHP decay in silicon particles.", "The corresponding scattering diagrams are shown in Fig.", "REF (c), demonstrating nearly $20 ^\\circ $ steering in comparison with the cold dimer.", "Overall, similar dependencies are observed for $s$ -polarized incident wave.", "The corresponding results for permittivity dynamics and scattering patterns are presented in Fig.", "REF (d,e,f) assuming the same dimer geometry and incident pulse parameters as before.", "Again, only permittivity of the MD resonant particle is significantly affected, Fig.", "REF (d).", "A smaller value of steering of about $15 ^\\circ $ degrees is observed for 40 GW/cm$^2$ pulse, Fig.", "REF (e).", "At the same time pulses of lower intensities result in larger steering in contrast to $p-$ polarized incidence: here, 10 GW/cm$^2$ pulse produces $\\approx 10^\\circ $ degrees rotation of the main lobe, while for s polarization the scattering pattern is almost unaffected.", "Unfortunately, there is significant backward scattering in the unexcited regime due to different picture of magnetic and electric dipoles interference.", "Figure: Normalized intensity of radiation scattered by a Si nanodimer along the two directions as a function of time.In order to gain better impression of a nonlinear Si nanodimer as a light router, in Fig.", "REF we present time dependent scattered intensity (normalized by the instantaneous incident intensity) emitted in the two scattering channels: along the cold direction ($\\delta \\phi =0 ^\\circ $ ) and along the forward direction ($\\delta \\phi =20 ^\\circ $ ) for $p-$ polarized 40 GW/cm$^2$ pulse.", "It clearly shows that plasma induced nonlinearity enables redistribution of the emitted radiation between the two channels with the relative difference of about 20%.", "Figure: (a) Time-dependent permittivity of photoexcited Ge in the two dimer particles irradiated by a 200 fs pulse at 2 μ\\mu m wavelength.", "(b) The steering angle form a Ge dimer as a function of time for a 200 fs pulse with different peak intensities.", "Dashed curve shows steering from a Si dimer for the same incident pulse.While silicon exhibits attractive characteristics for its use at optical frequencies, it has weak two-photon absorption in near-IR ($\\beta <2$  cm/GW) and therefore is not preferable for tunable nanoantennas mediated by EHP excitation.", "To extend the idea of EHP-controlled dimer nanoantenna beyond the visible region, we note that germanium is a promising candidate for all-dielectric nanophotonics in the near-IR [35], where it is almost lossless with refractive index $\\approx 4$ .", "Remarkably, it shows huge two-photon absorption ($\\beta \\approx 80$  cm/GW at 2.9 $\\mu $ m, Ref.", "[36]).", "Along with the fact that lower EHP densities are required for nanoantenna tuning at longer wavelengths due to increasing free electrons contribution to $\\varepsilon $ (which scales as $1/\\omega $ ), high TPA coefficient allows for decreasing of incident intensity required for beam steering.", "The results for nonlinear beam steering in a Ge dimer at $\\lambda =2~\\mu $ m are shown in Fig.", "REF for $p-$ polarized incident wave.", "Following the same strategy as before, we compose the dimer of a MD resonant particle (240 nm radius) and a particle obeying the Kerker condition (220 nm radius); the particles are separated by $L=800$  nm.", "In agreement with our expectations, 2 GW/cm$^2$ pulse of 200 fs duration causes excitation of EHP density in the resonant particle resulting in nearly $20 ^\\circ $ degrees steering, Fig.", "REF (b).", "To justify the choice of Ge over Si in near-IR, we show the steering angle in a Si dimer composed of MD resonant and Kerker particles (at 2 $\\mu $ m) in Fig.", "REF (b), which demonstrates negligible tuning as compared to the Ge dimer." ], [ "Near-field tuning", "Above we have demonstrated the effect of optical EHP excitation on the far field properties of the dimer nanoantenna, i.e., its scattering pattern.", "The plasma nonlinearity, however, can be also employed for tuning of the near field behavior, which can be characterized by the local density of states (LDOS) in the gap of the dimer.", "Using the coupled dipoles approximation we calculate the orientation averaged electric LDOS ${\\rho _{\\text{E}}} = \\frac{{2\\omega }}{{3\\pi {c^2}}} \\times \\operatorname{Im} \\left( {{\\text{Tr}}\\left[ {{\\mathbf {\\hat{G}}}({{\\mathbf {r}}_0},{{\\mathbf {r}}_0};\\omega )} \\right]{\\text{ }}} \\right)$ at the center of the symmetric Si dimer.", "The LDOS spectrum for a cold dimer without EHP is shown in Fig.", "REF (a) exhibiting the well-known series of peaks associated with resonances of the dimer [37], [38].", "The time-dependent LDOS enhancement with respect to the free space LDOS ${\\rho _0} = \\frac{{{\\omega ^2}}}{{3{\\pi ^2}{c^3}}}$ is plotted in Fig.", "REF (b) for different wavelengths assuming 200 fs pulse with 40 GW/cm$^2$ peak intensity as that used in the previous section for a Si dimer.", "The results demonstrate that a 200 fs optical pulse can induce nearly 50% change of LDOS at the dimer center for specific wavelength.", "After the pulse action LDOS returns to its initial value during the ps-scale EHP relaxation.", "Our calculations also indicate that for a fixed transverse orientation of a dipole in the dimer gap the projected LDOS $\\rho _{E,\\mathbf {u}}$ modification can be even more dramatic reaching five-fold enhancement or suppression.", "Importantly, photoexcitation of EHP can transform the system from enhanced state with $\\rho _{E,\\mathbf {u}}>\\rho _{0}$ to the state with suppressed projected LDOS $\\rho _{E,\\mathbf {u}}<\\rho _{0}$ .", "Demonstrated tunability of LDOS in a silicon nanodimer allows for selective enhancement or suppression of various optical effects whose strength is dependent on LDOS.", "Those include not only the well-known Purcell effect, but also thermal emission [39] and nonlinear optical effects [40].", "Figure: (a) Spectrum of the orientation averaged electric LDOS ρ E {\\rho _{\\text{E}}} at the dimer center in the unexcited state.", "(b) Electric LDOS as a function of time upon irradiation with a 40 GW/cm 2 ^2 pulse.", "Shaded area represents the incident pulse intensity." ], [ "Discussion and Conclusion", "We finally add a remark regarding the modulation rate which may be attained with the proposed structure.", "Figures REF and REF demonstrate that the direct switching from the unexcited to the photoexcited state of a dimer takes $\\sim 200$  fs, whereas Auger recombination in c-Si enables ultrafast recombination time down to sub-10-ps level for intensities higher than 20 GW/cm$^{2}$ .", "The fastest relaxation can be achieved with nc-Si nanoparticles, for which 2.5 ps EHP relaxation has been demonstrated [25] resulting in approximately 400 Gbit/s bandwidth.", "This large modulation speed may enable additional applications in the area of non-reciprocal emission and absorption at the nanoscale [41], [42].", "To conclude, we have explored the potential of all-dielectric nanoparticle dimers for nonlinear manipulation of the near and far electromagnetic fields via electron-hole plasma photoexcitation.", "In particular, we have demonstrated nonlinear steering of light scattered from an asymmetric dimer of silicon nanoparticles, where the steering angle is controlled via the intensity of incident optical pulse.", "Excitation with a 200 fs pulse of 40 GW/cm$^2$ peak intensity allows to achieve $20 ^\\circ $ steering.", "The concept was also applied in the near-IR for a germanium dimer.", "Plasma excitation also enables control of the near fields manifested in the local density of states in the vicinity of a dimer.", "We have shown that excitation of plasma can induces 50% variation of orientation averaged LDOS and even more dramatic change of projected LDOS for a fixed orientation.", "This variation may be employed for transient selective control of LDOS-sensitive effects such as spontaneous and thermal emission.", "This work was supported by the Ministry of Education and Science of the Russian Federation (project No 14.584.21.0009 with unique identificator RFMEFI58414X0009).", "D.G.B.", "acknowledges support from the Russian Foundation for Basic Research (project No 16-32-00444)." ] ]
1606.05199
[ [ "Gender differences in altruism: Expectations, actual behaviour and\n accuracy of beliefs" ], [ "Abstract Previous research shows that women are more altruist than men in dictator game experiments.", "Yet, little is known whether women are expected to be more altruist than men.", "Here we elicit third-parties' beliefs about dictators' donations conditional on knowing the gender of the dictator.", "Our data provide evidence of three main findings: (i) women are expected to be more altruist than men; (ii) both men and women have correct beliefs about the level of altruism among men; and (iii) both men and women overestimate the level of altruism among women.", "In doing so, our results uncover a perception gap according to which, although women are more altruist than men, they are expected to be even more altruist than they actually are." ], [ "Introduction", "2 The exploration of gender differences in decision making has a long tradition in behavioural economics and other social sciences, and has touched several research areas, including risk-aversion, competitive behaviour, and social preferences.", "For example, a classical study by Eckel and Grossman () has shown that women are more risk averse than men, while an equally classical study by Niederle and Vesterlund () has shown that women are less competitive than men.", "In terms of social preferences, results are more mixed: while previous research has not uncovered any obvious gender difference in cooperative behaviour (see for a review), experimental studies have repeatedly found that women are, on average, more altruist than men (, , , , , , , ).", "More recently, have extended this line of research by showing, through a meta-analysis of 22 studies, that promoting intuition versus reflection increases altruistic behaviour among women, but not among men, suggesting that women but not men have internalised altruism as their spontaneous reaction.", "Other studies have shown that women tend to be more altruistic than men when investing in human capital for children.", "For example, women allocate more resources for women's and children's clothing relative to men's clothing (), invest more in health and nutrition for children (), and spend more on child goods and small scale livestock () than men.", "Despite the vast research in gender differences, the literature has mostly neglected the inverse question of whether people have gender stereotypes in specific decisional settings.", "Understanding whether people have correct beliefs about others' behaviour is an important question per se, because the standard equilibrium analysis assumes that people strategise on their beliefs about their counterparts' behaviour (); and it becomes even more important when it comes to gender differences, since one of the dominant explanations for gender differences in decision-making relies on the assumption that the behaviour of men and women is governed by stereotypes regarding their social roles (, ).", "In sum, understanding whether there is a correspondence between stereotypes of men and women and their actual behaviour is an important question, with potential consequences in economic and psychological modelling.", "Given the aforementioned literature showing that women are more altruistic than men, here we ask whether this gender difference in behaviour corresponds to a gender difference in stereotypes.", "We move a first step into this research area by starting from a simple question: are women expected to be more altruistic than men?", "To the best of our knowledge, only a handful of papers have approached this question, and mostly did so from a psychological perspective.", "For example, Heilman and Chen () showed that work-related altruism is less optional for women than for men, and Heilman and Okimoto () showed that penalties for women's success in male domains result from the perceived violation of gender-stereotypic prescriptions.", "From an economic perspective, we are aware of only one study devoted to eliciting participants' beliefs about the level of altruism in men and women ().", "In this lab experiment, subjects were presented with two boxes, A and B, where A contained donations left by men and B contained donations left by women.", "Subjects were informed that they could choose only one of the two boxes and one donation would be taken at random from the selected box and used to pay them.", "Results showed that subjects were more likely to select donations from the \"women\" box, indicating that women were indeed expected to be more generous than men.", "Although it represents an important first step towards understanding whether women are expected to be more altruistic than men, the work by Aguiar et al.", "() has two important limitations.", "First of all, while it shows that women are expected to be more generous than men, it does not show whether people have correct beliefs about the behaviour of men and women.", "Thus, it remains unclear whether people have correct stereotypes regarding each gender's level of altruism.", "Second, it is only one study: the recent outbreak of the replicability crisis (Open Science Collaboration, ) calls for more studies.", "In the current work, we wish to fill these gaps by: (i) replicating the result that women are expected to be more generous than men; and (ii) giving a quantitative version of this result.", "This allows to answer the question: do men and women fulfil people's expectations about altruistic behaviour?", "The rest of the paper is organised as follows.", "Next section is devoted to methods; section focuses on results and discussion; last section concludes." ], [ "Subject pool", "Subjects were living in the US at the time of the experiment and were recruited using Amazon Mechanical Turk (, , , ) to play a standard Dictator Game (, ).", "In the Dictator Game, one player acts in the role of the dictator and the other one in the role of the receiver.", "Dictators are given a certain amount of money and are asked how much, if any, they want to give to the receiver.", "Receivers have no choice and only get what the dictators decide to give.", "Since dictators have no incentives to give money, a payoff-maximising dictator would donate nothing.", "For this reason, dictators' donations are taken as a measure of individual's altruism, or inequity aversion (, , , , )." ], [ "Protocol ", "In our experiment, subjects were randomly divided between dictators and receivers.", "Dictators: They were given $\\$0.20$ and were asked to decide how much, if any, to give to the receiver.", "Before making their decision, dictators were asked two comprehension questions.", "Specifically, they were asked which choice would maximise their payoff and which choice would maximise the receiver's payoff.", "Subjects failing any comprehension question were automatically excluded from the survey.", "This screening procedure had the effect that we had fewer dictators ($N=456$ ) than receivers ($N=530$ ).", "Thus, the computation of receivers' payoffs is not straightforward, since there is no one-to-one correspondence between dictators and receivers.", "To address this problem, receivers were sequentially paired with a randomly selected dictator; in case a dictator was already used to pay another receiver, we paid the current recipient `out of our pocket', and not using the donation of that dictator, because that donation had already been used.", "This procedure is doable on Amazon Mechanical Turk, because participants are matched only after the end of the experiment.", "Receivers: A part from potentially receiving money from dictators, receivers played also as guessers.", "Specifically, they were asked to predict the donation that another dictator would make to another receiver.", "They would receive, on top of the actual donation, $\\$0.20$ reward for correct guesses.", "Since they do not guess their own donation there is no opportunity to hedge ().", "To elicit recipients' expectations, we designed four treatments: O$_{\\text{n}}$ : recipients were presented with the same screenshots shown to dictators and they were asked to guess the dictator's decision ($N = 134$ ); O$_{\\text{mow}}$ : was identical to O$_{\\text{n}}$ with the only difference that recipients were informed that the dictator was either a man or a woman ($N=140$ ).", "O$_{\\text{m}}$ : was identical to O$_{\\text{n}}$ with the only difference that recipients were informed that the dictator was a man ($N=124$ ); O$_{\\text{w}}$ : was identical to O$_{\\text{m}}$ with the only difference that recipients were informed that the dictator was a woman ($N=132$ ).", "We need both O$_{\\text{n}}$ and O$_{\\text{mow}}$ baselines for two reasons: on the one hand, by comparing O$_{\\text{m}}$ and O$_{\\text{w}}$ with O$_{\\text{mow}}$ , separately, we may investigate the effect of making one particular gender salient versus making both genders salient; on the other hand, by comparing dictators' donations with O$_{\\text{n}}$ , we can explore whether people have correct beliefs about the level of altruism in anonymous strangers." ], [ "Descriptive statistics", "A total of 986 subjects (56% men, mean age = 34.5 years) participated in our experiment.", "The average donation was 27.3% of the total endowment, which is very close to the average donation reported in Engel's meta-analysis of 616 Dictator game experiments conducted in the standard physical laboratory (28.3%, ).", "This confirms the reliability of data collected on Amazon Mechanical Turk using very small stakes, a fact that was already observed in the context of the Dictator Game by d'Adda et al.", "().", "Although the pie size was $\\$0.20$ data are normalised such that the donations correspond to 0-10.", "Next we pass to the analysis of treatment effects." ], [ "Gender framed vs non-framed treatments", "As a preliminary step we start by looking at framing effects on recipients' beliefs.", "Both treatments O$_{\\text{n}}$ and O$_{\\text{mow}}$ report similar averages ($2.79$ and $3.16$ , resp.).", "Table REF shows no significant differences between O$_{\\text{n}}$ and O$_{\\text{mow}}$ (t-test, $p = 0.24$ ; z-test, $p=0.23$ ).", "Similarly, we do not find significant differences between O$_{\\text{m}}$ $\\cup $ O$_{\\text{w}}$ and O$_{\\text{mow}}$ (t-test, $p = 0.89$ ; z-test, $p=0.84$ ).", "Hence, the sum of `men' and `women' frames is equal to the treatment in which `both' genders are mentioned.", "Therefore, we may conclude that mentioning `gender' does not frame recipients expectations." ], [ "Are women expected to be more generous than men?", "To answer this question we compare treatment O$_{\\text{m}}$ with O$_{\\text{mow}}$ and treatment O$_{\\text{w}}$ with O$_{\\text{mow}}$ .", "Figure REF shows the distribution of beliefs by treatment.", "Figures 1a, 1b and 1c show the histograms for O$_{\\text{m}}$ , O$_{\\text{mow}}$ and O$_{\\text{w}}$ , respectively.", "While the modal values for expected behavior of males is 0 (giving nothing) the modal for women (i.e., O$_{\\text{w}}$ ) and, to a lesser extent, for women or men (i.e., O$_{\\text{mow}}$ ) is the equal split.", "Average values reflect the same result: the mean expected altruism in O$_{\\text{m}}$ is 2.33, while the mean for O$_{\\text{mow}}$ is 3.18 (t-test, $p = 0.01$ ; z-test, $p = 0.01$ ).", "Conversely, when the dictator is a `woman', the mean expected generosity in O$_{\\text{w}}$ is 4.05, which is significantly larger than the mean for O$_{\\text{mow}}$ (t-test, $p = 0.01$ ; z-test, $p = 0.00$ ).", "Comparing the expected level of generosity among males and females, O$_{\\text{w}}$ vs O$_{\\text{m}}$ , we observe than the average and median differences are 1.72 and 4 units, respectively.", "The top part of Table REF shows the relevant tests.", "Figure $1a$ to Figure $1c$ provide visual evidence that we can reject the hypothesis that men are expected to be as altruistic as the average person and that women are expected to be as altruistic as the average person.", "Figure $1d$ focuses on the CDFs (cumulative distribution functions).", "While males' CDF is closer to the top right – more selfish – Females CDF is closer to the bottom left –more generous.", "It is easy to check that O$_{\\text{w}}$ stocastically dominates O$_{\\text{m}}$ which is consistent with the test shown in Table REF .", "The entire distribution of O$_{\\text{w}}$ is always toward the right of the rest of distributions.", "In sum, women are expected to me more altruistic than men.", "Result 1 Women are expected to be more altruistic than men." ], [ "Are women actually more generous than men?", "In the previous subsection, we have shown that women are expected to be more altruistic than men in a Dictator Game.", "Is this expectation grounded or not?", "Figure 2a and Figure 2b respectively compare the distribution of donations for both men and women, and provides visual evidence that women are, on average, more altruist than men (means: $3.04$ vs $2.49$ ; t-test, $p = 0.03$ ; z-test, $p = 0.01$ ).", "In fact, giving nothing is the modal value for males (49.6% gave 0) while giving the equal split is the modal value for women (48.3% gave half).", "In sum, not only women are expected to be more generous than men, but they are de facto more generous than men.", "Result 2 Women are more altruistic than men." ], [ "Do subjects have correct beliefs about each gender's average level of generosity?", "In the previous subsections, we have shown that women are expected to be more altruistic than men and that this expectation is grounded, in the sense that women are actually more altruistic than men.", "Now, we ask whether people have correct beliefs about each gender's average level of generosity.", "We begin by observing that subjects have, on average, correct beliefs about the average level of altruism.", "Specifically, the mean level of altruism across the experiment (both males and females) is 2.735, while the mean level of expected generosity in the O$_{\\text{n}}$ condition is 2.798 (t-test, $p = 0.81$ ; z-test, $p = 0.83)$ , see Table 1 bottom).", "Hence subjects have correct beliefs about average level of generosity, which in turn means that we do not observe either wishful thinking or pessimism.", "Next we analyse whether subjects have correct beliefs about men's average level of altruism.", "Figures $3a$ analyses accuracy of beliefs for men and shows that there is no discrepancies since both expectations and actual behavior are almost identical (CDFs are on parallel).", "Controlling for the gender of the recipient, we also find that both men and women have, on average, correct beliefs about men's level of altruism (t-test, guess by men $ p = 0.21 $ and guess by women $ p = 0.60$ ; z-test, guess by men $ p = 0.22 $ and guess by women $ p = 0.44$ , see Table 1 bottom).", "Result 3 Both men and women have correct beliefs about average level of generosity in men.", "However Figure $3b$ shows strong discrepancies between current behaviour and expectations for women: females are not as generous as they are expected to be (O$_{\\text{w}}$ CDF dominates the D$_{\\text{w}}$ CDF).", "This remains true also after controlling for gender.", "Both men and women overestimate women's average level of generosity (t-test, both p-values $<0.03$ ; z-test, both p-values $<0.02$ , see Table 1).", "Result 4 Both men and women overestimate the level of generosity in women.", "Table: Accuracy of beliefs for women" ] ]
1606.04900
[ [ "The Turbulent Boundary Layer on a Horizontally Moving, Partially\n Submerged, Surface-Piercing Vertical Wall" ], [ "Abstract The complex interactions between turbulence and the free surface, including air entrainment processes, in boundary layer shear flows created by vertical surface-piercing plates are considered.", "A laboratory-scale device was built that utilizes a surface-piercing stainless steel belt that travels in a loop around two vertical rollers, with one length of the belt between the rollers acting as a horizontally-moving flat wall.", "The belt is operated both as a suddenly-started plate to reproduce boundary layer flow or at steady state in the presence of a stationary flat plate positioned parallel to the belt to create a Couette flow with a free surface.", "Surface profiles are measured with a cinematic laser-induced fluorescence system in both experiments and air entrainment events and bubble motions are observed with stereo underwater white-light movies in the suddenly started belt experiment.", "It is found that the RMS surface height fluctuations, $\\eta$, peak near the boundaries of the flows and increase approximately linearly with belt speed.", "In the Couette flow experiments, a dominant peak in the spectrum of $\\partial\\eta/\\partial t$ is found at the dimensionless frequency $fH/U\\approx 0.7$.", "In the suddenly started belt experiment, surface fluctuations appear to be strongly influenced by sub-surface turbulence within the boundary layer, while ripples propagate freely when farther from the belt.", "Additionally, some mechanisms for air entrainment are observed and comparisons are made to predictions of incipient entrainment conditions in Brocchini and Pregrine (J. Fluid Mech., 499, 225-254, 2001)." ], [ "1. Introduction", "Turbulent boundary layers near the free surface along ship hulls and surface-piercing flat plates have been explored by a number of authors, see for example , , and .", "However, even though it has long been observed that there is a layer of white water next to the hulls of naval combatant ships moving at high speed, see for example the photograph in figure REF , the entrainment of air at the free surface in ship boundary layers has received relatively little attention.", "It is not known whether this white water is the result of active spray generation and air entrainment due to turbulence in the boundary layer along the ship hull or the result of spray and air bubbles that are generated upstream in the breaking bow wave and then swept downstream with the flow.", "In the free surface boundary layer, the air entrainment process is controlled by the ratios of the turbulent kinetic energy to the gravitational potential energy and the turbulent kinetic energy to the surface tension energy.", "The ratio of the turbulent kinetic energy to the gravitational potential energy is given by the square of the turbulent Froude number ($Fr^2 = q^2 / (g L)$ ) and the ratio of turbulent kinetic energy to surface tension energy is given by the Weber number ($We = \\rho q^2 L/ \\sigma $ ), where $g$ is the acceleration of gravity, $\\rho $ is the density of water, $\\sigma $ is the surface tension of water, $q$ is the characteristic magnitude of the turbulent velocity fluctuations and $L$ is the length scale of this turbulence.", "Several authors have applied theory and numerical methods to explore the interaction of turbulence and a free surface, see for example , , and .", "have used scaling arguments to predict the critical Froude and Weber numbers above which air entrainment and spray generation will occur due to strong free-surface turbulence.", "Figure REF , which is from their paper, shows the boundaries of various types of surface undulations on a plot of $q$ versus $L$ .", "The upper region of the plot is the region of air entrainment and droplet generation.", "We have used classical boundary layer correlations to make estimates of $q$ (taken as the root-mean-square vertical component of the turbulent velocity fluctuations) and $L$ (taken as the boundary layer thickness) at three streamwise positions in a ship boundary layer and plotted these points on the $q$ -$L$ map in figure REF .", "As can be seen from the figure, the points are clearly in the air entrainment region of the plot, especially the points near the bow.", "Thus, air entrainment due to strong turbulent fluid motions in the hull boundary layer at the free surface is a likely cause of the layer of white water.", "The difficulty with laboratory experiments on bubble entrainment and spray stems from the fact that the experiments are performed in the same gravitational field as found in ship flows and that the only practical liquid available is water, as is also found in the ocean.", "Thus, with $g$ , $\\rho $ , and $\\sigma $ the same in the field and in the laboratory, one must attempt to achieve full-scale flow speeds in order to obtain Froude and Reynolds similarity with field conditions.", "Also, even if full scale-values of $q$ and $L$ were obtained by towing a surface piercing flat plat with the length of the ship at high speed in a ship model basin, the free surface flow would include a bow wave which would obfuscate the source of the bubbles and spray.", "Another problem is that in order to obtain realistic entrainment/spray conditions and bubble/droplet size distributions, these experiments should be performed in salt water which is not typically used in ship model basins.", "Figure: Photograph of naval combatant ship showing zone of white water next to the hull.In view of the above difficulties in simulating air entrainment due to the turbulent boundary layer, we have built a novel device that produces an approximation of a full scale ship boundary layer in the laboratory.", "This device, called the Ship Boundary Layer Simulator (SBLS) generates a temporally evolving boundary layer on a surface piercing flat surface.", "The surface consists of a stainless steel belt loop that is 1.0 m wide and about 15 m long.", "The belt is mounted on two vertically oriented drums as shown in figure REF .", "The drums are driven by hydraulic motors and the entire device is placed in a large open-surface water tank as shown in the figure.", "Before each experimental run, the belt and the water in the tank are stationary.", "The water level is set just below the top edge of the belt and the flow outside the belt loop on one of the long lengths between the rollers is studied.", "The belt speed can be set to values between 2.5 and 15 m/s.", "Uncertainty in measured belt speed varies depending on the desired velocity, but typically remains below 5 percent for the conditions presented below.", "Two experiments are discussed in this paper.", "In the first experiment, a Couette flow with a free surface is created by placing a fixed long surface-piercing flat plate parallel to and about 4 cm away from the belt test length.", "In this case, measurements of the free surface height fluctuations with the belt running in a steady state conditions are performed.", "In the second experiment, the belt is accelerated from rest with a 3$g$ acceleration until it reaches a pre-defined speed which is held steady for a short time.", "The flow on the surface of the belt in this case is a simulation of the flow seen by a stationary observer in the ocean as a ship, that makes no waves, passes by at constant speed.", "The distance $x$ along the ship hull corresponding to any time $t$ after the belt begins to move is essentially the distance traveled by the belt, $x = Ut$ , where $U$ is the ship/belt speed.", "The remainder of this paper is divided into three sections.", "The experimental setup is described in the following section 2.", "The results and discussion are given in Section 3 and the conclusions of this study are presented in Section 4.", "Figure: Regions of various types of surface motions for free surface turbulence with velocity fluctuation magnitude qq (vertical axis) and length scale LL (horizontal axis), from .", "Air entrainment and spray production occur in the upper region, above the uppermost curved line.", "The three data points are values obtained for the turbulent boundary layer on a flat plate with qq taken as the rms of the spanwise (which is vertical for the boundary layer along a ship hull) velocity fluctuation (w ' w^{\\prime }) and L taken as the boundary layer thickness (δ\\delta )." ], [ "2. Experimental Details", "The experiments are performed in an open-surface water tank that is 13.34 m long, 2.37 m wide and 1.32 m deep, see figure REF .", "The inner surfaces of the tank walls and bottom are composed of 31.8-mm-thick clear Acrylic panels which are supported by a steel frame.", "The top of the tank is open, offering an unobstructed view of the water surface.", "Two floor-mounted circular steel pads pierce the bottom of the tank and are used to support the Ship Boundary Layer Simulator (SBLS).", "The tank includes a water filtration system consisting of a 2.3-m-long skimmer at one end of the tank, a diatomaceous-earth filter and associated pipes and valves.", "The output line of the skimmer can be directed to the drain or to the filter and from there to return to the tank at the end opposite to the skimmer.", "Figure: Perspective view of the Ship Boundary Layer Simulator (SBLS) and the water tank.Figure: Plan view of the SBLS and water tank for (a) the steady state Couette flow experiment and (b) the suddenly started belt experiment.The main functional component of the SBLS is a one-meter-wide 0.8-mm-thick endless stainless steel belt which is driven by two 0.46-meter-diameter, 1.1-meter-long drums whose rotation axes are vertically oriented and separated by a horizontal distance of approximately 7.5 meters.", "The drums are each driven by two bent-axis hydraulic motors via toothed belt and pulley systems.", "Each drum along with the motors and drive systems form single drive units that are attached to a welded steel frame that maintains the separation between and relative parallel orientation of the drums.", "The drum drive unit on the left in figure REF is attached to the steel frame via two hydraulic pistons, positioned at the top and bottom of the frame.", "The vertical position of the belt on the drum is controlled actively during each experimental run by measuring the belt position with a light sensor and tilting the left drum with differential motion of the hydraulic pistons.", "Tilting the left drum clockwise (counter clockwise) by small amounts causes the belt move up (down).", "During a typical run, the drum position varies by no more than 5 mm.", "The assembled SBLS is placed in a stainless steel sheet metal box (called the dry box) as shown in figure REF .", "The box keeps the assembly essentially dry, while one of the two roller-to-roller sections of the belt exits the box through a set of seals and travels to the second set of seals near the opposite roller where the belt re-enters the box.", "This box was deemed necessary to keep water from corroding the SBLS mechanism and to keep water from being dragged in between the belt and the roller where it might cause the belt to hydroplane off the roller.", "The lone straight section exposed to water is approximately 6 meters long and pierces the free surface with approximately 0.33 meters of freeboard for the water level used in the present experiments.", "At the location where the belt leaves the dry box and enters the water, a sheet metal fairing is installed to reduce the flow separation caused by the backwards-facing step associated with the shape of the dry box at this location.", "This SBLS device is used in two experiments as depicted in the schematic drawings shown in figure REF .", "The first of these experiments is a steady state Couette flow with a free surface.", "This flow is created by placing a flat, stationary plate parallel to the belt surface with a distance of 4 cm in the gap between the two surfaces.", "Like the belt, the top part of the plate pierces the water free surface.", "In this experiment, the belt is turned on for a sufficient time for the flow to reach steady state, after which data is collected.", "The leading edge of the stationary plate includes a fairing to minimize flow entrance effects and the measurement location is positioned approximately 100 gap widths downstream of the leading edge of the plate, in order to obtain a fully developed flow.", "In these experiments, steady-state belt speeds between 2.8 m/s and 4.0 m/s are investigated.", "Figure: Schematic drawing showing the set up for the cinematic LIF measurements of the free surface shape.In the second set of experiments, the belt is started from rest and the stationary plate is placed 1 m from the belt surface.", "Each run lasts for less than 10 s, and at the end of a run the boundary layer thickness is substantially smaller than the distance from the belt to the fixed plate, thus free field conditions are simulated.", "When launched from rest, the belt is capable of acceleration equal to 3 times gravitational acceleration.", "The time to reach constant belt speed varies depending on the final speed, but is typically less than 0.25 seconds for the experimental conditions used here.", "Throughout these transient experiments, the belt travel is analogous to the passage of a flat-sided ship that makes no bow waves; the length along the hull is equivalent to the total distance traveled by the belt.", "Belt speeds ranging from 3 to 5 m/s were used and measurements were continued until a belt length of 30 m had passed by the measurement site.", "Therefore, at 3 m/s, experiments run for 10 seconds, while at 5 m/s, experiments run for 6 seconds.", "To study the water surface deformation in both experiments, a cinematic Laser Induced Fluorescence (LIF) technique was utilized, see figure REF .", "In this technique, a continuous-wave Argon Ion laser beam is converted to a thin sheet using a system of spherical and cylindrical lenses.", "This sheet is projected vertically down onto the water surface in an orientation with the plane of the light sheet normal to the plane of the belt.", "This laser emits light primarily at wavelengths of 488 nm and 512 nm.", "The water in the tank is mixed with fluorescein dye at a concentration of about 5 ppm and dye within the light sheet fluoresces.", "Two cameras view the intersection of the light sheet and the water surface from both upstream and downstream with viewing angles of approximately 20 degrees from horizontal.", "The cameras (Phantom V642 by Vision Research, Inc.) capture 4-Mpixel 12-bit black-and-white images at frame rates up to 1500 Hz.", "A long-wavelength-pass optical filter is placed in front of each camera lens.", "These filters block out the laser light and transmit the light from the fluorescing dye, thus preventing specular reflections of the laser light from the water surface from entering the camera lenses.", "The use of two cameras allows for more accurate surface measurement in the event that the intersection of the light sheet and the water surface is blocked by large deformations of the free surface between the plane of the light sheet and either camera.", "The images seen by these cameras shows a sharp line at the intersection of the light sheet with the free surface.", "Using image processing, instantaneous surface profiles can be extracted from these images.", "In addition to the above-described surface profile measurements, observation of the entrainment and motions of air bubbles were undertaken with a cinematic stereo camera system in the experiments with the belt starting from rest, see schematic in figure REF .", "In this system, the cameras view the flow just under the water free surface adjacent to the belt.", "The cameras are set up along the side wall of the tank that is parallel to the belt with horizontal viewing directions and lines of sight rotated $\\pm 30^\\circ $ about a vertical plane that is normal to the belt surface.", "A photoflood light with a translucent screen is placed next to each camera.", "Since the belt surface is moderately reflective, each light provides the illumination for the opposite camera.", "The cameras (lights) view (illuminate) the scene through water-filled prism boxes that are set to create a straight line of sight that is perpendicular to the side wall of the prism tank.", "Each camera is also equipped with a Scheimpflug lens mount, which inclines the sensor plane relative to the lens plane so that each camera may focus on a plane parallel to the belt despite viewing from a displaced angle.", "Using two cameras at offset angles allows for a more accurate characterization of the non-axisymmetric free surface features found during entrainment events.", "Additionally, utilizing the principles of parallax, three dimensional bubble positions can be determined.", "Though quantitative measurements are planned using this system in the near future, in the present paper, only qualitative observations will be described.", "Figure: A schematic of the underwater stereo bubble measurement system" ], [ "3. Results and Discussion", "In this section, the results for both the Couette flow and the suddenly started belt experiments are presented and discussed.", "The surface profile measurements from the Couette flow experiments are addressed first and then followed by a description of the surface profile measurements and air entrainment observations in the suddenly started belt experiments.", "Finally, a discussion of air-entrainment boundaries for both experiments is given." ], [ "Surface Profiles", "In this set of experiments, the belt speed is varied between 2.8 m/s to 4.0 m/s which corresponds to a Froud number range of 3.01 to 4.31 based on the definition $Fr=U/(2gH)^{1/2}$ where U is the belt speed, g the gravitational acceleration and H the gap width.", "Single LIF images taken from the high-speed movies for belt speeds of 2.8, 3.2, 3.6 and 4.0 m/s are shown in figures REF (a), (b), (c) and (d), respectively.", "These images were taken with the upstream camera; the belt is on the left side of the image and the fixed plate is on the right.", "The belt is moving away from the camera, i.e., into the page in the images.", "The light sheet is projected from above and the glowing dye at the intersection of the light sheet and the water surface is seen as a high-contrast boundary between the upper dark region and the lower light region in each image.", "The shape of this boundary is measured quantitatively using image processing techniques.", "(The automated version of these techniques worked reliably except for regions within about 4 mm of the belt or wall.", "In these regions, the profiles were determined by eye.)", "The bright area below the boundary is created by the glowing fluorescent in the underwater portion of the light sheet.", "The complex light intensity pattern here is created by a combination of the refraction of the laser light sheet as it passes down through the water surface and the refraction of the light from the glowing underwater dye as the light passes up through the water surface between the light sheet and the camera, on its way to the lens.", "This part of the image cannot be quantitatively analyzed.", "As can be seen from the images, the surface height fluctuations increase steadily with belt speed, but wave breaking and air entrainment are not evident, even at the highest speed.", "Instantaneous surface profiles were extracted from 5,000 images (corresponding to 10 s in real time) from one of the high-speed movies at each belt speed.", "A series of about 100 surface profiles for each of three belt speeds is shown in figure REF .", "In the plots, each profile is shifted up by 2 mm from the previous profile to reduce overlap and so that the spatio-temporal evolution of surface features can be shown.", "The horizontal axis, $y/H$ , is the distance from the belt surface normalized by the gap width, $H$ ; the fixed plate is located at $y/H=1.0$ .", "If one were to draw lines connecting surface profile features like the local crests from one profile to the next, the slopes of these lines would correspond to the horizontal speed of these features in a direction normal to the belt.", "It can be seen from the plots that the visually dominant features travel from the belt side to the plate side and the surface height fluctuations become bigger as the belt speed is increased.", "Figure: Four images from separate Couette flow experiments, depicting steady state belt speeds of (a) 2.8 m/s (b) 3.2 m/s (c) 3.6 m/s and (d) 4.0 m/s.", "The horizontal field of view for each image is approximately 5 cm.Figure: A series of profile history plots, with each depicting a surface profile history in time throughout the Couette flow experiment for (a) 3.0 m/s, (b) 3.4 m/s, and (c) 3.8 m/s.", "In each plot, profiles are offset vertically by 2 mm from the previous profile so that the evolution of features can be seen." ], [ "Surface Fluctuations", "Surface profiles extracted from the images were used to analyze the temporal and spatial characteristics of the surface height fluctuations.", "Though each high-speed LIF movie from which the profiles were extracted was taken long after the belt started from rest and had reached a steady state speed, there were some low-frequency overall oscillations in the surface profile records.", "The cause of these oscillations is at present unknown, but is being explored in ongoing experiments.", "Since the surface ripples have a relatively high frequency, it was decided to remove the low-frequency oscillation from the data records by high-pass filtering before further processing.", "Thus, for each pixel column across the Couette flow gap in the images, a fast Fourier transform of a sequence of 5000 data points was obtained.", "It was observed that for all cases the background oscillations had a frequency of about 0.4 Hz, thus the cut off for the high-pass filter was taken as 0.5 Hz.", "Curves of the temporal average of the RMS of the filtered surface height fluctuations versus normalized position, $y/H$ , from the belt surface for each of the belt speeds used in this study are shown in figure REF (a).", "The moving wall is located at position $y/H = 0$ and the stationary plate is at $y/H = 1$ .", "In the regions next to the belt and the fixed wall, the water surface profile data has not yet been extracted for all of these belt speeds and so is not plotted for any cases.", "It is observed that the RMS height fluctuation is higher on the belt side of the gap, but it increases locally at both walls.", "The average over the gap width of the RMS surface data plotted in figure REF (a) is plotted versus belt speed in figure REF (b).", "In general, the RMS value increases with the belt speed as expected.", "It is also found that with the current amount of processed data, the curves are still a bit noisy.", "Additional measurements and data processing are planned for the near future in order to eliminate this problem.", "Figure: (a) Surface height RMS values depicted across the gap for a range of belt speeds.", "Each profile is averaged across 5000 frames.", "(b) Total surface height RMS averaged over the gap width for a range of belt speeds.Figure: (a) Fourier spectrum of time derivative of height fluctuations averaged over the gap width.", "(b) Frequency of the maximum amplitude versus the belt speedFigure: (a) Fourier spectrum of time derivative of height fluctuations averaged over the gap width for U=4.0U=4.0 m/s.", "(b) Non-dimensional frequency of the maximum amplitude (f max H/Uf_{max}H/U) versus non-dimensional position (y/Hy/H) across the gap and averaged over the seven belt speeds.The spectrum of the temporal derivative of the surface height fluctuations ($\\partial \\eta /\\partial t$ ) was computed at each pixel location across the gap width and then averaged over the gap width.", "Plots of these averaged spectra for all seven speeds are given in figure REF (a).", "Though the spectral noise level is a bit high with the amount of data processed so far, it can be seen than the spectral level generally increases with belt speed and that the spectral amplitude has a peak in the range of frequencies from 40 to 50 Hz.", "In order to explore how this frequency varies with belt speed, a 6th order polynomial was fitted to each spectrum and the dimensionless frequency ($f_{max}H/U$ ) of the peak of each polynomial is plotted versus belt speed in figure REF (b).", "The values of $f_{max}H/U$ vary from about 0.7 at the lowest belt speed to about 0.6 at the highest belt speed.", "The 6th order polynomial fit to the spectrum for $U=4.0$  m/s as a function of position across the channel width is shown in figure REF (a).", "In this plot, the vertical axis is non-dimensional frequency ($fH/U$ ), the horizontal axis in non-dimensional position across the gap width, $y/H$ , and the spectral intensity is given by the color according to the scale on the right.", "The spectrum varies across the gap width with non-dimensional frequency of the spectral peak decreasing from about 0.9 near the belt to about 0.5 near the fixed wall.", "For reference, it should be kept in mind that a dimensionless frequency of about 1 is consistent with the model of surface disturbances of stream wise length scale $H$ being convected past the measure station at the belt speed, $U$ .", "The peak spectral intensity is highest at the belt surface and reaches a minimum at about $y/H=0.7$ .", "The reason for this lack of symmetry of the spectra across the channel width, including the possibility that at the measurement location along the channel length (about $x/H = 100$ ) the flow is not yet fully developed, is presently being explored with additional experiments.", "A plot of the non-dimensional frequency ($f_{max}H/U$ ) of the maximum of the spectral intensity versus position across the gap and averaged over all belt speeds is given in figure REF (b).", "In this plot, the red curve is the average and the grey area above and below the curve is the standard deviation over the belt speeds.", "As can be seen from the plot, the non-dimensional frequency varies from about 0.9 near the belt surface to about 0.5 near the fixed wall.", "While the level of noise precludes a discussion of the exact shape of this curve, the asymmetry across the gap is evident." ], [ "Surface Profiles", "Five LIF images of the water free surface next to the belt for an experimental run with $U=5$  m/s are shown in figure REF .", "Though the camera recorded 1,000 frames per second, the five images in the figure are spaced by an interval of 1 s with the first image (a) taken at 0.0 s. Here and in the following, rather than refer to images and data by the time after the belt started moving, we refer to them by the distance, $x=Ut$ , from the leading edge of an equivalent flat plate moving at constant speed $U$ .", "Thus, the images in figure REF (a), (b), (c), (d) and (e) correspond to $x=$ 0.0, 5.0, 10.0, 15.0 and 20.0 m, respectively.", "As discussed in the experimental details section, the plane of the vertical light sheet is oriented normal to the belt surface and the cameras look parallel to the belt surface and down at the water surface at a small angle from horizontal.", "The images in figure REF are from the upstream camera so the belt is near the left side of each image and is moving into the page.", "The sharp boundary between the upper dark and lower bright regions of the images is the intersection of the light sheet and the water surface.", "The upper regions of the later images contain light scattered from roughness features on the water surface behind the light sheet.", "These roughness features include bubbles that appear to be floating on the water surface and moving primarily in the direction of the belt motion.", "The position of the belt is marked on the left side of image (a) and the intensity pattern to the left of this location, is a reflection of the light pattern on the right, in the smooth surface of the belt.", "This line of symmetry gives a good indication of the position of the belt in each image.", "The the high-contrast boundary where the light sheet intersects the free surface is analyzed quantitatively to extract instantaneous surface shapes.", "In the experimental run corresponding to the images in figures REF , the belt launches from rest and reaches a steady state velocity of 5 m/s, after traveling for only about 0.2 s, equivalent to $x= 1.0$  m. It can be seen from these images that surface height fluctuations (ripples) are created close to the belt surface, at the left side of each image, and propagate away from the belt (to the right).", "As time passes, the surface height fluctuations grow dramatically and eventually surface breaking and air entrainment events begin to occur, resulting in bubble and droplet production.", "These events are more easily observed in high-speed LIF movies.", "An example series of water surface profiles from a run with $U=3.0$  m/s over a range of $x$ from 21 m to 21.3 m is shown in figure REF .", "The horizontal axis in the plot is horizontal distance, $y$ , from the belt surface and the vertical axis is water surface height above the mean water level.", "The profiles are equally spaced in $x$ by 0.012 m and each successive profile is plotted 1.5 mm above the previous profile so that overlap is reduced and the evolution of surface features can be seen.", "As in the Couette flow experiment, surface features like ripple crests can be tracked over a number of successive profiles and the slopes of imaginary lines connecting these features are an indication of their horizontal speed away from the belt surface.", "It is clear that close to the belt surface, the ripple features last for only a few, say about five, profiles and their paths have a high slope relative to horizontal, indicating relatively slow motion away from the belt.", "This region appears to be more chaotic than the region to the right.", "In this outer region, surface features remain visible over many frames giving support to the idea that they are freely propagating ripples.", "The red line in the figure, which was drawn by eye to approximate that slope of the imaginary lines connection the ripple crests in the outer region, corresponds to a velocity of about 34 cm/s.", "It should be kept in mind that this is only the $y$ -component of the phase speed.", "Information about the component of the phase speed in the $x$ direction will be obtained in future experiments with the LIF system oriented to record water surface profiles in planes parallel to the belt surface.", "As is clear from the profiles in figure REF and will be seen below in analysis of the RMS surface height fluctuations, the surface motion amplitude is highest close to the belt.", "Surface breaking events are more prevalent in this region of high-amplitude surface motion, and, as is discussed below, this appears to be the primary location for air entrainment.", "Additionally, due to increased surface roughness and air entrainment at higher belt speeds, automation of surface profile processing is more difficult.", "Because of this, only data for $U=3.0$  m/s has been fully processed at 1,000 frames per second for 5 runs.", "For belt speeds of 4.0 m/s and 5.0 m/s, every tenth image was processed, resulting in a time interval of 10 ms between surface profiles.", "While 5 such runs have been processed for $U=4.0$  m/s, only 2 runs have been processed for $U=5.0$  m/s." ], [ "Surface Fluctuations", "In order to characterize these free surface motions, the instantaneous surface profiles are analyzed to determine RMS surface height fluctuations.", "A mean profile is first determined by averaging the time series of surface height data for each wall-normal position throughout a given run.", "This average surface height profile is then subtracted from each individual profile and RMS fluctuations about the mean are calculated from the resulting data.", "(It would be more useful to obtain a temporally evolving mean surface profile by averaging profiles at a given time after the belt begins over many experimental runs.", "Unfortunately, this requires a larger data set than was available at the time of writing this paper.)", "The RMS surface height is first averaged across both wall-normal position and equivalent ship length to quantify the overall mean fluctuation amplitude for each belt velocity, as shown in figure REF .", "Because these profiles are averaged across every surface height measured for each speed, the RMS values are the most statistically significant of the quantities presented here.", "The RMS height fluctuations appear to increase linearly with increasing belt speed, but additional data including more belt speeds will be required to gain confidence in this conclusion.", "More detail about the RMS height fluctuation can be found by examining its distributions in $y$ and $x$ .", "By averaging the RMS over the time series of points for each wall-normal position, the average RMS versus distance from the belt can be determined, as shown in figure REF .", "While the surface fluctuations increase steadily from one speed to the next at all values of $y$ , some differences can be seen in the shape of the distributions.", "Each surface height fluctuation profile reaches a maximum at the belt, but in the highest speed case the values decrease more gradually with increasing distance from the belt than in the lower speed cases.", "In future experiments, turbulent velocity fluctuations under the water surface will be measured and used to shed light on the reasons for the above-described features in the RMS surface height distributions.", "In order to quantify variation of the RMS surface height fluctuations with distance along the “hull\" ($x$ ), the RMS fluctuations were averaged over the wall-normal position for each instantaneous surface profile and plotted versus Reynolds number based on equivalent ship length, $x=Ut$ , in figure REF .", "As can be seen in the plot, there is a large noise level in the curves.", "It was expected that the curves would show a monotonic increase in surface fluctuation amplitude with increasing $Re_x$ and would collpase together at corresponding $Re_x$ values for different speeds.", "Unfortunately, the noise level is too high with the present data set to confirm this hypothesis.", "For a belt speed of 5 m/s, large spikes in RMS can be seen to reach 2 to 3 times the height of the lower speeds.", "Figure: The spatial fourier transform of the free surface fluctuations was computed for each instantaneous surface profile throughout a run.", "This plot shows the spectral power vs wavenumber and equivalent ship length using a 20 ms moving average and averaging over 5 runs at 3 m/s.The spectral content of the free surface profiles was studied using Fourier transform techniques.", "As with the RMS surface fluctuations, a mean surface height for each wall normal position is first subtracted from each profile.", "Next, a straight line that connects the first and last point is subtracted from each profile.", "For each resulting instantaneous surface profile, a discrete Fourier transform is calculated, thus converting the signal from the spatial domain $y$ to the wavenumber domain $k$ .", "After repeating this process for each profile, a moving average over 20 profiles is performed for each run and the resulting data from 5 runs is averaged together.", "In this way, spectral power versus wavenumber and distance along an equivalent ship hull is calculated and shown in figure REF .", "The spectral content is plotted on a log scale because of the large variation in this quantity.", "Before approximately $x=1.5$  m there is very little surface fluctuation and the spectrum is quite narrow.", "For 1.5 m$ <x<$ 5 m, the spectrum grows rapidly in width.", "This corresponds to a Reynolds number based on equivalent ship length of between $4.5 \\times 10^6$ and $15 \\times 10^6$ .", "After this initial sharp increase in surface fluctuations, the spectral shape change becomes in much more gradual." ], [ "Air Entrainment", "From both surface profile movies and underwater white light movies, a variety of air entrainment events can be seen.", "Perhaps the most prominent of these mechanisms begins with the development of deep narrow troughs that are elongated in a quasi-streamwise direction.", "Some such troughs become so deep that the sides collapse together.", "When the sides of the trough meet, air is often entrapped beneath the free surface and bubbles begin to pinch off and convect downstream.", "Another air entrainment mechanism is caused by ejections from the free surface.", "These ejections can fold over the surface and trap a pocket of air, similar to the process of air entrainment in plunging breaking waves.", "From the LIF movies, some indications of both types of air entrainment events can be seen.", "Figure REF contains two sets of images to help illustrate the mechanisms for air entrainment as seen from above the free surface.", "In the left column, a trough collapse is depicted.", "In images (a) and (c), a deep narrow trough can be seen clearly toward the left side of the image.", "In these first images, the left side of the trough is beginning to turn over.", "In image (e), the two sides of the trough meet and a droplet is ejected into the air.", "In the right column, the second type of air entrainment event can be seen.", "In the first image, (b), a jet can be seen forming near the contact point of the free surface with the belt, toward the left side of the image.", "This jet ejects from the surface and becomes nearly horizontal by image (f).", "In image (j), the jet makes contact with the free surface, trapping any air that remained below it during the formation of the jet.", "Since the camera is above the free surface and because of blockage of the camera line of sight by other free surface features between the camera and the light sheet, the LIF images do not give us a clear indication of whether or not air was actually entrained by either of these particular events.", "Another way to visualize these air entrainment events is by recording white light movies beneath the free surface.", "This arrangement reduces line of sight blockage by other surface features and allows for a more detailed study of how the air is entrained rather than just showing the surface features that are likely to produce bubbles.", "Figure REF shows two sets of images from two views of the same event.", "Each image contains both the feature of interest and its shadow projected onto the belt by the light source for each camera and can be seen especially well in image pair (d)-(j).", "In image d, the camera is on the left side, with the light source coming from the right side, so that the actual trough marked as object 2, casts a shadow shown in object 1.", "In image j, the camera is on the right side with the light source to the left, so the same trough is shown as object 4 with its shadow projected to the right, shown as object 5.", "The relative horizontal distance between and object and its shadow is an indication of the distance of the object from the belt, the smaller the distance, the closer the object is to the belt surface.", "In the first three sets of images, the trough can be seen to become deep and narrow.", "In the subsequent images, the left side of the trough can be seen turning over and contacting the opposite side of the trough, as it pinches off and becomes a large bubble in the final image pair.", "These events, where one side turns over completely, seem to primarily form large bubbles when entraining pockets of air.", "A second type of air entrainment event can be seen in figure REF .", "As with the previous figure, this event is shown from two camera views in two columns.", "This event begins, as shown in image (b) with the two sides of the trough closest and farthest from the right camera meeting at a single point in the middle of the trough and forming a round hole in the trough on the top left of the image.", "Following this event, the hole spreads rapidly, propagating outward from the meeting point and forming a large ligament of air that can be seen in image (d).", "In the following images, this ligament can be seen retracting towards the surface as it deposits a series of small bubbles into the flow.", "Figure: Values of RMS fluctuations of ∂η/∂t\\partial \\eta /\\partial t for both the couette flow and suddenly started belt experiments plotted on the previously discussed chart from using ∂η/∂t\\partial \\eta /\\partial t from the experiments in place of the rms vertical velocity fluctuation, qq, used in the theory.", "The large circular data points are from the Couette flow experiments with the highest point corresponding to the highest belt speed and the length scale LL taken as the gap width HH.", "The green line is from the suddenly started belt experiment for a belt speed of 3 m/s.", "For this curve, the RMS ∂η/∂t\\partial \\eta /\\partial t data are values at each yy averaged over all xx and the length scale L=yL =y, the distance from the belt surface.The free surface height records at each $y$ location can be differentiated in time to determine the rate of change of surface height ($\\partial \\eta /\\partial t$ ).", "In the following, we use this quantity in place of the vertical component of the fluid velocity ($w$ ) at the surface to compare our results with the entrainment boundary predictions in the theory of as shown in figure REF .", "By the kinematic boundary condition $w = \\frac{\\partial \\eta }{\\partial t} + u\\frac{\\partial \\eta }{\\partial x}+v\\frac{\\partial \\eta }{\\partial y} \\mbox{ on } z = \\eta (x,y,t),$ thus, the RMS values of $w$ (called $q$ in Brocchini and Peregrine) and $\\partial \\eta /\\partial t$ are not strictly equivalent.", "In figure REF , we have plotted data from the present experiments on the original plot of entrainment regions from Brocchini and Peregrine.", "For the suddenly started belt experiment, only data for $U=3.0$  m/s is presented because this is the only data set for which surface profiles have been processed to a sufficient time resolution to calculate $\\partial \\eta /\\partial t$ .", "The points plotted in varying shades of red/black in figure REF are from the Couette flow experiments.", "For these points, $q$ is taken as the RMS $\\partial \\eta /\\partial t$ data averaged over the gap width and the length scale $L$ is taken as the gap width, $H$ .", "As these points transition from black to red, the velocity increases from 2.8 m/s to 4.0 m/s.", "Little or no air entrainment occurred for the belt speeds tested in the Couette flow experiments.", "As can be seen in figure REF , all of the data points fall on or below the upper curve in the plot.", "According to the theory, for flows that correspond to points above this boundary air entrainment may occur.", "Note that the position of the data points may indicate that a slightly higher belt speed will result in air entrainment.", "The green line in figure REF is data from the suddenly started belt experiments.", "The RMS $\\partial \\eta /\\partial t$ data are values at each $y$ averaged over all $x$ and the length scale $L$ is taken as $y$ , the distance from the belt surface.", "The data points are on or below the air entrainment boundary with the exception of the points in the range between $y=L=0.1$  cm and about $1.5$  cm, which are slightly above the boundary.", "In the experiment, $U=3.0$  m/s is just below the entrainment boundary, in approximate agreement with the Brocchini and Peregrine prediction.", "In future experiments, the stereo imagery described above will be used to determine the range of $y$ where the air entrainment occurs." ], [ "4. Conclusions", "A laboratory-scale device is used to study the interaction of turbulent boundary layer shear flows with a free surface, including air entrainment.", "Two experiments are performed.", "In the first experiment, a stationary flat plate is placed parallel to and at a distance $H$ from a vertical wall that moves horizontally in its own plane.", "Both the moving wall and the fixed plate pierce the water surface.", "The wall is run at constant speed until a steady-state Couette flow is created.", "In the second experiment, the wall motion consists of 3$g$ acceleration followed by a steady speed, $U$ .", "In this case, the fixed plate is absent and the flow represents a temporally evolving free surface boundary layer.", "Via the transformation $x=Ut$ , this experiment mimics the boundary layer on a surface piercing plate moving horizontal at constant speed.", "Water surface profiles are recorded with a cinematic LIF system in both experiments to study the generation of surface height fluctuations by the subsurface turbulence.", "Stereo, underwater white-light movies are recorded in the suddenly started belt experiment to determine the mechanisms for air entrainment.", "In the Couette flow experiment, it is found that surface fluctuations increase sharply near both the moving belt and stationary wall and that there is a peak in the spectra of the rate of change of surface height at a non-dimensional frequency ($fH/U$ ) of about 0.7.", "In the suddenly started belt experiment, it is found that after a relatively calm beginning portion of each experimental run, surface height fluctuations increase sharply before leveling off later in the run.", "From underwater white-light movies, different types of air entrainment events are detected at small distances away from the belt, entraining both large and small bubbles.", "Further experiments are planned to increase the statistical significance of measured quantities, to extend the range of wall speeds, obtain quantitative data from the stereo bubble entrainment movies and to obtain subsurface flow field data using cinematic, stereo particle image techniques." ], [ "Acknowledgements", "The authors gratefully acknowledge the support of the Office of Naval Research under grant N000141110029, Program Manger Dr. Ki-Han Kim." ] ]
1606.05035
[ [ "Networked Control under Random and Malicious Packet Losses" ], [ "Abstract We study cyber security issues in networked control of a linear dynamical system.", "Specifically, the dynamical system and the controller are assumed to be connected through a communication channel that face malicious attacks as well as random packet losses due to unreliability of transmissions.", "We provide a probabilistic characterization for the link failures which allows us to study combined effects of malicious and random packet losses.", "We first investigate almost sure stabilization under an event-triggered control law, where we utilize Lyapunov-like functions to characterize the triggering times at which the plant and the controller attempt to exchange state and control data over the network.", "We then provide a look at the networked control problem from the attacker's perspective and explore malicious attacks that cause instability.", "Finally, we demonstrate the efficacy of our results with numerical examples." ], [ "Introduction", "Cyber security has become a critical problem in industrial processes, since nowadays they incorporate information and communication technologies that are prone to cyber threats.", "Cyber attacks can disrupt the normal operation of services that are critical to the society as they can cause financial losses and environmental damages.", "It is thus essential to ensure cyber security of existing infrastructures and design new cyber-attack-resilient ones.", "Literature on cyber security points out cyber threats against industrial control systems utilized in many fields (see [1] and the references therein).", "Vulnerabilities of the channels used for transmission of measurement and control data pose a critical issue for the security of control systems.", "This is because the channels are recently connected via the Internet or wireless communications [2], [3].", "Communication channels, for instance, may face jamming attacks initiated by malicious agents [4], [5].", "Such attacks block the communication link and effectively prevent transmission of packets between the plant and the controller.", "It is mentioned in [5] that jamming attacks pose a major security threat, as they can be easily performed with devices that target various wireless communication protocols.", "In recent works [6], [7], [8], [9], [10], [11], [12], [13], networked control problems under jamming attacks were investigated using control and/or game-theoretic methods.", "However, jamming may not be the only cause of malicious packet losses.", "Compromised routers in a network may also intentionally drop packets [14], [15].", "The work [16] explored the control problem over a multihop network with malicious nodes that intentionally stop forwarding packets or alter packet contents.", "In addition to actions of malicious agents, state measurement and control input packets may also fail to be transmitted at times due to network congestion or errors in communication.", "Stochastic models provide accurate characterization of such nonmalicious network issues [17], [18].", "In the literature, unreliability of a network is often characterized through random models for packet loss events [19], [20].", "For instance, in [21], [22], [23], Bernoulli processes are used for modeling packet losses in a network.", "Furthermore, in [24], [25], packet loss events are characterized in a more general way by employing Markov chains.", "In those studies, a variety of control methods are also proposed to ensure stability of networked control systems that face random packet losses.", "In this paper, we propose a stochastic representation of packet transmission failures in a network between a plant and a controller.", "Our proposed model is sufficiently general and allows us to explore some of the existing random and malicious packet loss scenarios in a unified manner.", "At the core of this characterization, we have a tail probability condition on the average number of state measurement and control input packet failures in the network.", "We demonstrate that random packet losses, malicious attacks, as well as the combination of those two phenomena satisfy the condition with different parameters.", "We model random losses by using a binary-valued time-inhomogeneous Markov chain.", "Furthermore, to characterize malicious attacks, we use a model similar to the one in [10].", "Specifically, this model allows attacks to happen arbitrarily as long as the total number of packet exchange attempts that face malicious attacks are almost surely bounded by a certain ratio of the number of total packet exchange attempts between the plant and the controller.", "The almost sure bound used in our model in fact allows not only deterministic strategies but also stochasticity in the generation of malicious attacks.", "As a result, the model captures attacks that are generated based on randomly varying information such as state and control input or the random packet losses.", "Besides, an attacker may also intentionally use randomness to imitate packet losses that occur due to congestion or channel noise.", "Through our malicious attack model, we consider scenarios where the attacker targets the network only when the plant and the controller attempt to exchange packets.", "In a jamming attack scenario, our characterization, hence, can be considered as a model for reactive jamming discussed in [4] for wireless networks.", "The classification in [4] divides attackers into two groups: active and reactive ones.", "An active jamming attacker tries to block a communication channel regardless of whether the channel is being used or not, whereas a reactive attacker continuously monitors the channel and attacks only when there is transmission.", "It is mentioned in [4] that it may be harder to detect a reactive jamming attacker as packets may also be lost due to nonmalicious network issues and hence the reason for packet losses may not be known with certainty.", "A similar issue where packet losses occur due to both malicious and nonmalicious reasons exists also in the context of multihop networks.", "For instance [15] investigates combined effects of malicious packet drops and nonmalicious channel errors.", "Motivated by the scenarios mentioned above, we utilize our probabilistic characterization also to investigate networks that are subject to the combination of random transmission errors due to unreliability of the channel and attacks conducted by malicious agents.", "In our analysis, we consider two cases: (i) when the attacks and random packet losses are modeled as independent processes and (ii) when the attack strategy is dependent on the random packet losses.", "The dependent case is essential to model the situation where the attacker has information of the random packet losses in the communication channel and utilizes this information in the attack strategy.", "Furthermore, we may also consider situations when the attacker decides to attack based on the content of packets.", "In the case of jamming attacks, this corresponds to selective jamming discussed in [26], [27], where the intelligent jamming attacker listens to the communication channel and decides whether to interfere or not depending on the packet being transmitted.", "For example, a jamming attacker may decide not to interfere with the communication when the packet being transmitted is already corrupted by channel noise.", "Moreover, in a network of multiple nodes malicious ones may intentionally drop certain packets based on their content [28].", "The main theoretical challenge in dealing with the combination of random packet losses and malicious attacks stems from the fact that these two phenomena are of different nature and hence have different models.", "By utilizing a tail probability inequality for the sum of processes that represent random packet losses and malicious attacks, we show that our proposed probabilistic characterization allows us to deal with both independent and dependent loss cases.", "By utilizing our probabilistic packet transmission model, we investigate the networked control problem of a linear plant through an event-triggered framework.", "Event-triggered control methods have recently been employed in many studies (see [29], [30], [31] and the references therein).", "We follow the approach in [32], [33] and utilize Lyapunov-like functions to determine the triggering times at which the plant and the controller attempt to exchange state and control input information.", "The triggering conditions that we propose ensure that the value of a Lyapunov-like function of the state stays within certain limits.", "Packet exchanges are attempted only before the value of the Lyapunov-like function is predicted to exceed the limit.", "In a successful packet exchange scenario, state measurements are sent from the plant to the controller, which computes a control input and sends it back to the plant.", "However, state measurement or control input packets may fail to be transmitted due to random packet losses and malicious attacks.", "Our packet failure characterization and control system analysis differ from those of the recent studies [34], [35], [36], which also investigate the event-triggered control problem under packet losses.", "Specifically, in [34], the number of consecutive packet losses is assumed to be upper-bounded, and a deterministic Lyapunov function approach is used for the closed-loop stability analysis.", "Moreover, in [35], [36] the packet losses are modeled by a Bernoulli process.", "The stability analysis in [36] is based on investigating the evolution of the expectation of a Lyapunov function.", "Despite the similarity to our malicious attack model, our stability analysis also differs from that of [10], where the analysis relies on a deterministic approach for obtaining an exponentially decreasing upper bound for the norm of the state.", "Our approach for stability analysis is related to obtaining an upper bound on the top Lyapunov exponent (see [37], [38], [39]) of the system and in that sense it is more similar to the stability analysis conducted in [40], [23] for networked systems without event-triggering.", "Specifically, we find a stochastic upper bound for a Lyapunov-like function and show that this stochastic upper bound tends to zero under certain conditions indicating almost sure asymptotic stability.", "In addition to stability analysis, we also address the question of finding instability conditions under which the state of the closed-loop system diverges almost surely.", "We observe that an attack strategy that causes sufficiently frequent packet losses can destabilize the closed-loop dynamics.", "This instability result allows us to investigate effects of potential malicious attacks on a networked control system.", "The rest of the paper is organized as follows.", "In Section , we describe the networked control problem under random and malicious packet losses.", "We present an event-triggered control framework and provide sufficient conditions for almost sure asymptotic stability of the closed-loop system in Section .", "In Section IV, we look at the networked problem from the attacker's perspective and provide conditions for instability of the system.", "We present illustrative numerical examples in Section .", "Finally, in Section VI, we conclude the paper.", "We note that part of the results in Sections  and appeared without proofs in our preliminary report [41].", "Here, we provide a more detailed discussion with complete proofs.", "We use a fairly standard notation in the paper.", "Specifically, we denote positive and nonnegative integers by $\\mathbb {N}$ and $\\mathbb {N}_{0}$ , respectively.", "Moreover, $\\Vert \\cdot \\Vert $ denotes the Euclidean vector norm and $\\left\\lfloor \\cdot \\right\\rfloor $ denotes the largest integer that is less than or equal to its real argument.", "The notation $\\mathrm {\\mathbb {P}}[\\cdot ]$ denotes the probability on a probability space $(\\Omega ,\\mathcal {F},\\mathbb {P})$ with filtration $\\lbrace \\mathcal {F}_{i}\\rbrace _{i\\in \\mathbb {N}_{0}}$ such that $\\mathcal {F}_{i_{1}}\\subset \\mathcal {F}_{i_{2}}\\subset \\mathcal {F}$ for $i_{1},i_{2}\\in \\mathbb {N}_{0}$ with $i_{1}<i_{2}$ ." ], [ "Networked Control Problem and Characterization of Network with Random\nand Malicious Packet Losses ", "In this section we introduce the networked control problem and present a characterization for a network with random packet losses and those caused by malicious agents." ], [ "Networked Control System", "Consider the linear dynamical system $x(t+1) & =Ax(t)+Bu(t),\\quad x(0)=x_{0},\\quad t\\in \\mathbb {N}_{0},$ where $x(t)\\in \\mathbb {R}^{n}$ and $u(t)\\in \\mathbb {R}^{m}$ denote the state and the control input, respectively; furthermore, $A\\in \\mathbb {R}^{n\\times n}$ and $B^{n\\times m}$ are the state and input matrices, respectively.", "In our networked control problem, the plant and the controller exchange information packets over a communication channel to achieve stabilization of the zero solution $x(t)\\equiv 0$ .", "We consider the case where packets are transmitted without delay, but they may get lost.", "In a successful packet exchange scenario, at a certain time instant, measured plant states are transmitted to the controller, which generates a control input signal and sends it to the plant.", "The transmitted control input is applied at the plant side.", "In the case of an unsuccessful packet exchange attempt, either the measured state packet or the control input packet may get dropped, and in such cases control input at the plant side is set to 0, which is a common approach in the literature (e.g., [40], [20], [24], [25]).", "In this setup, the plant is informed about a packet exchange failure by the lack of an incoming control input.", "Specific acknowledgement messages are thus not needed.", "This allows the practical implementation by using a UDP-like communication protocol discussed in [19].", "We use $\\tau _{i}\\in \\mathbb {N}_{0},i\\in \\mathbb {N}_{0}$ , (with $\\tau _{i}<\\tau _{i+1}$ ) to denote the time instants at which packet exchanges between the plant and the controller are attempted.", "In this paper, we consider both the case where packet exchanges are attempted at all time instants and the case where an event-triggering mechanism decides the successive packet exchange attempt times.", "In both cases, the control input $u(t)$ applied to the plant is given by $u(t) & \\triangleq \\left(1-l(i)\\right)Kx(\\tau _{i}),\\,t\\in \\lbrace \\tau _{i},\\ldots ,\\tau _{i+1}-1\\rbrace ,$ where $K\\in \\mathbb {R}^{m\\times n}$ denotes the feedback gain and $\\lbrace l(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ is a binary-valued process that characterizes success or failure of packet exchange attempts.", "When $l(i)=0$ , the packet exchange attempt at time $\\tau _{i}$ is successful and the piecewise-constant control input at the plant side is set to $u(\\tau _{i})=Kx(\\tau _{i})$ .", "On the other hand, $l(i)=1$ indicates that either the packet sent from the plant or the packet sent from the controller is lost at time $\\tau _{i}$ .", "Again, in such situations, control input at the plant side is set to 0.", "We emphasize that the framework described above allows us to deal with dropouts in both state and control input channels of the network illustrated in Fig.", "REF .", "In particular, the process $l(\\cdot )$ is an overall indicator of the packet exchange failures over these channels." ], [ "Network Characterization", "Packet transmission failures in a network may have different reasons.", "In what follows we characterize the effects of certain stochastic and malicious packet loss models in a unified manner by exploring dynamical evolution of the total number of packet exchange failures.", "First, we define a nonnegative integer-valued process $\\lbrace L(k)\\in \\mathbb {N}_{0}\\rbrace _{k\\in \\mathbb {N}}$ by $L(k) & \\triangleq \\sum _{i=0}^{k-1}l(i),\\quad k\\in \\mathbb {N}.$ Note that $L(k)$ denotes the total number of failed packet exchange attempts during the time interval $[0,\\tau _{k-1}]$ , where $k$ attempts have been made.", "In our packet loss model, we place a bound on the ratio of failed attempts in a probabilistic and asymptotic sense.", "Assumption 2.1 There exists a scalar $\\rho \\in [0,1]$ such that $\\sum _{k=1}^{\\infty }\\mathbb {P}[L(k)>\\rho k] & <\\infty .$ The condition (REF ) provides a probabilistic characterization of the evolution of the total number of packet exchange failures through the scalar $\\rho \\in [0,1]$ , representing their average ratio.", "Note also that (REF ) describes a condition on the tail probability $\\mathbb {P}[L(k)>\\rho k]=\\mathbb {P}[\\frac{L(k)}{k}>\\rho ]$ of loss ratio $\\frac{L(k)}{k}$ .", "This condition is sufficiently general and includes some of the existing packet loss models in the literature.", "We illustrate its generality by establishing that condition (REF ) holds for four different cases: random packet losses, malicious packet losses, combination of the two losses in 1) and 2) when they are independent, and finally combination but when they are dependent.", "Note that for any packet loss model, Assumption REF is trivially satisfied with $\\rho =1$ , since $\\mathbb {P}[L(k)>k]=0$ .", "On the other hand, as we see below, for certain random and malicious packet loss models, $\\rho $ can be obtained to be strictly smaller than 1.", "A closely related characterization for packet dropouts is presented in [23]; the scalar $\\rho $ in (REF ) corresponds to the notion of dropout rate discussed there." ], [ "Random Packet Losses", "To characterize nonmalicious network issues such as packet drops due to network congestion or communication errors, we utilize time-inhomogeneous Markov chains.", "Specifically, let $\\lbrace l_{\\mathrm {R}}(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ be a time-inhomogeneous Markov chain adapted to filtration $\\lbrace \\mathcal {F}_{i}\\rbrace _{i\\in \\mathbb {N}_{0}}$ .", "Here, the $\\sigma $ -algebra $\\mathcal {F}_{i}$ contains all random packet transmission success/failure events for the first $i+1$ packet exchange attempt times $\\lbrace \\tau _{0},\\tau _{1},\\ldots ,\\tau _{i}\\rbrace .$ The Markov chain $\\lbrace l_{\\mathrm {R}}(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ is characterized by initial distributions $\\vartheta _{q}\\in [0,1]$ , $q\\in \\lbrace 0,1\\rbrace $ , and time-varying transition probabilities $p_{q,r}\\colon \\mathbb {N}_{0}\\rightarrow [0,1]$ , $q,r\\in \\lbrace 0,1\\rbrace $ , such that $\\begin{array}{c}\\mathbb {P}[l_{\\mathrm {R}}(0)=q]=\\vartheta _{q},\\\\\\mathbb {P}[l_{\\mathrm {R}}(i+1)=r|l_{\\mathrm {R}}(i)=q]=p_{q,r}(i),\\quad i\\in \\mathbb {N}_{0}.\\end{array}$ The state $l_{\\mathrm {R}}(i)=1$ indicates that the network faces random packet losses at time $\\tau _{i}$ , and hence the packet exchange attempt at $\\tau _{i}$ results in failure.", "Here, success/failure of a packet exchange attempt depends on the states of the previous packet exchange attempts.", "Furthermore, transition probabilities between success ($l_{\\mathrm {R}}(i)=0$ ) and failure ($l_{\\mathrm {R}}(i)=1$ ) states are time-dependent.", "It is important to note that the time-inhomogeneous Markov chain characterization with time-varying transition probabilities allows us to take into account the variation in the network between consecutive packet transmission instants.", "Furthermore, this characterization generalizes the Bernoulli and time-homogeneous Markov chain models that are often used in the literature.", "In what follows we show that Assumption REF is satisfied when the network faces random packet losses described by time-inhomogeneous Markov chains.", "In characterization of the scalar $\\rho $ used in Assumption REF we use upper-bounds for transmission failure and success probabilities denoted respectively by $p_{1}\\in [0,1]$ and $p_{0}\\in [0,1]$ such that $p_{q,1}(i) & \\le p_{1},\\\\p_{q,0}(i) & \\le p_{0},\\quad q\\in \\lbrace 0,1\\rbrace ,\\quad i\\in \\mathbb {N}_{0}.$ Note that even though $p_{q,r}(i)$ provide precise information about the transitions between the states of random packet losses, this information cannot be utilized when the network faces the combination of malicious attacks and random packet losses (discussed in Sections REF and REF ).", "In such cases, information about the probability of malicious attacks for each transmission attempt is not available, and as a result, transition probabilities $p_{q,r}(i)$ for random packet losses cannot be utilized to obtain the overall packet exchange failure probabilities.", "On the other hand, we can employ the upper-bounds $p_{1}$ and $p_{0}$ when we show that the overall packet exchange failures satisfy Assumption REF .", "Lemma 2.1 For the time-inhomogeneous process $\\lbrace l_{\\mathrm {R}}(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ with transmission failure probability upper-bound $p_{1}\\in (0,1)$ that satisfy (REF ), we have $\\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{i=0}^{k-1}l_{\\mathrm {R}}(i)>\\rho _{\\mathrm {R}}k] & <\\infty ,$ for all $\\rho _{\\mathrm {R}}\\in (p_{1},1)$ .", "We use Lemma REF in the Appendix to prove this result.", "Specifically, let $\\tilde{p}=p_{1}$ , $\\tilde{w}=1$ , and define the processes $\\lbrace \\xi (i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ and $\\lbrace \\chi (i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ with $\\xi (i)=l_{\\mathrm {R}}(i)$ and $\\chi (i)=1$ , $i\\in \\mathbb {N}_{0}$ .", "Since the conditions in (REF ) and () are satisfied, it follows from Lemma REF that $\\mathbb {P}[\\sum _{i=0}^{k-1}l_{\\mathrm {R}}(i)>\\rho _{\\mathrm {R}}k]\\le \\psi _{k}$ , where $\\psi _{k}\\triangleq \\phi ^{-\\rho _{\\mathrm {R}}k+1}\\frac{\\left((\\phi _{1}-1)p_{1}+1\\right)^{k}-1}{(\\phi -1)p_{1}}$ with $\\phi \\triangleq \\frac{\\rho _{\\mathrm {R}}(1-p_{1})}{p_{1}(1-\\rho _{\\mathrm {R}})}$ , and $\\sum _{k=1}^{\\infty }\\psi _{k}<\\infty $ , which implies (REF ).", "Lemma REF indicates that when packet exchange failures occur due to random packet losses (i.e., $l(i)=l_{\\mathrm {R}}(i)$ ), Assumption REF holds for all $\\rho \\in (p_{1},1)$ ." ], [ "Packet Losses Due to Malicious Activity", "Packet transmissions in a channel may get interrupted due to malicious activities.", "For example, a compromised router in a network may deny to forward incoming packets.", "In addition, packet losses may also be caused by jamming attacks.", "A model for the attack strategy of a malicious agent has been proposed in [10].", "In that study, the sum of the length of attack durations is assumed to be bounded by a certain ratio of total time.", "By following the approach of [10], let $\\lbrace l_{\\mathrm {M}}(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ denote the state of attacks.", "The state $l_{\\mathrm {M}}(i)=1$ indicates that the packet transmission faces an attack at time $\\tau _{i}$ .", "We consider the case where the number of packet exchange attempts that face attacks are upper bounded almost surely by a certain ratio of the total number of packet exchange attempts, that is, $\\lbrace l_{\\mathrm {M}}(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ satisfies $\\mathbb {P}\\big [\\sum _{i=0}^{k-1}l_{\\mathrm {M}}(i)\\le \\kappa +\\frac{k}{\\tau }\\big ]=1,\\quad k\\in \\mathbb {N},$ where $\\kappa \\ge 0$ and $\\tau >1$ .", "In this characterization, among $k$ packet exchange attempts, at most $\\kappa +\\frac{k}{\\tau }$ of them are affected by attacks.", "Note that when $\\kappa =0$ , (REF ) implies no attack in the beginning: $l_{\\mathrm {M}}(i)=0$ , $i\\in \\lbrace 0,\\ldots ,\\lfloor \\tau \\rfloor \\rbrace $ , almost surely.", "Scenarios that involve possible attacks during the first few packet exchange attempts can be modeled by setting $\\kappa >0$ .", "In what follows, we would like to highlight the relations of the malicious packet loss model in (REF ) to those in the literature.", "First, since the attacks only happen at packet exchange attempt instants, the characterization in (REF ) can be considered as a reactive jamming model [4], where the attacker attacks the channel only when there is a packet being transmitted.", "To avoid being detected, an attacker may refrain from causing all packets to be lost.", "The ratio $\\frac{1}{\\tau }$ in (REF ) characterizes the average portion of the packet transmission attempts that face attacks.", "Furthermore, in the case of jamming attacks, in addition to avoid being detected, the attacker may also need to take into account the energy requirements of jamming.", "The ratio $\\frac{1}{\\tau }$ in this case corresponds to the notion jamming rate discussed in [42], and it is related to the energy usage of the jammer.", "Remark 2.2 A packet loss model that may be used to capture behavior of an intelligent attacker is also discussed in [21], where transmissions between the plant and the controller are attempted at all time instants and the proposed model allows packet losses to occur arbitrarily as long as the lengths of intervals between consecutive successful packet transmissions are not more than a given fixed length.", "A similar model has also been used in [34], where an event-triggered control method is used and the number of consecutive packet losses is assumed to be upper-bounded by a constant.", "Note that the packet loss model discussed in [21], [34] can be described within the framework provided by (REF ) through setting $\\tau =\\frac{s+1}{s}$ , where $s\\ge 1$ denotes the upper-bound on the number of consecutive packet losses.", "Under this setting, the condition (REF ) provides more freedom to the attacker as it does not necessarily require lengths of intervals between consecutive successful packet transmission times to be upper-bounded by a fixed constant.", "In fact for any $\\tau >1$ , (REF ) allows the attacker to cause any number of consecutive packet losses after waiting sufficiently long without attacking.", "Notice that the number of consecutive packet losses is not restricted to be bounded also in the case of random packet loss models (see Section REF , as well as [20], [24], [43]).", "As pointed out in [10], the condition (REF ) also shares some similarities with the socalled average dwell time condition [44] utilized in switched systems.", "In switched systems, the average dwell time condition requires the number $N(k_{2},k_{1})$ of switches in between times $k_{1}$ and $k_{2}\\ge k_{1}$ to satisfy $N(k_{2},k_{1}) & \\le \\kappa +\\frac{k_{2}-k_{1}}{\\tau },\\quad k_{2}\\ge k_{1}\\ge 0,$ where $\\tau >0$ denotes the average dwell time.", "The inequality (REF ) guarantees that the switches occur slowly on average.", "In this study, we do not require a condition on the number of switches between packet exchange success and failure states.", "Rather than that we utilize (REF ), which is a condition on the total number of packet exchange failures due to attacks.", "The condition (REF ) guarantees that attacks happen rarely on average.", "Note also that when $N(k_{2},k_{1})$ is defined to denote the number of packet exchange failures due to attacks over all packet exchange attempts at times $\\tau _{k_{1}},\\tau _{k_{1}+1},\\ldots ,\\tau _{k_{2}-1}$ , (REF ) implies (REF ).", "Specifically, (REF ) reduces to (REF ) by setting $N(k_{2},k_{1})\\triangleq \\sum _{i=k_{1}}^{k_{2}-1}l_{\\mathrm {M}}(i)$ , $k_{1}=0$ , and $k_{2}=k$ .", "As we have observed so far, the attack model in (REF ) is sufficiently general to cover known models.", "We further generalize it, because even though the model in (REF ) allows stochasticity in the generation of $l_{\\mathrm {M}}(\\cdot )$ , it is not enough to characterize certain stochastic attacks.", "An example is the case where each packet exchange attempt faces an attack with a fixed probability (e.g., $\\lbrace l_{\\mathrm {M}}(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ is a Bernoulli process).", "To cover such stochastic attacks as well as attacks characterized in (REF ), we consider a model where $\\lbrace l_{\\mathrm {M}}(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ is given through conditions similar to (REF ).", "Specifically, we assume that there exists a scalar $\\rho _{\\mathrm {M}}\\in [0,1]$ such that $\\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{i=0}^{k-1}l_{\\mathrm {M}}(i)>\\rho _{\\mathrm {M}}k] & <\\infty .$ The following lemma shows that the characterization with (REF ) is more general than the one provided by (REF ).", "Lemma 2.3 Suppose the binary-valued process $\\lbrace l_{\\mathrm {M}}(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ satisfies (REF ) with $\\kappa \\ge 0$ and $\\tau >1$ .", "Then (REF ) holds for all $\\rho _{\\mathrm {M}}\\in (\\frac{1}{\\tau },1)$ .", "Using Markov's inequality we obtain $& \\mathbb {P}[\\sum _{i=0}^{k-1}l_{\\mathrm {M}}(i)>\\rho _{\\mathrm {M}}k]\\le \\mathbb {P}[\\sum _{i=0}^{k-1}l_{\\mathrm {M}}(i)\\ge \\rho _{\\mathrm {M}}k]\\nonumber \\\\& \\quad =\\mathbb {P}[e^{\\sum _{i=0}^{k-1}l_{\\mathrm {M}}(i)}\\ge e^{\\rho _{\\mathrm {M}}k}]\\le e^{-\\rho _{\\mathrm {M}}k}\\mathbb {E}[e^{\\sum _{i=0}^{k-1}l_{\\mathrm {M}}(i)}]$ for $k\\in \\mathbb {N}$ .", "By (REF ), we have $\\mathbb {E}[e^{\\sum _{i=0}^{k-1}l_{\\mathrm {M}}(i)}]\\le \\mathbb {E}[e^{\\kappa +\\frac{k}{\\tau }}]=e^{\\kappa +\\frac{k}{\\tau }}$ .", "Therefore, it follows from (REF ) that $\\mathbb {P}[\\sum _{i=0}^{k-1}l_{\\mathrm {M}}(i)>\\rho _{\\mathrm {M}}k]\\le e^{\\kappa -(\\rho _{\\mathrm {M}}-\\frac{1}{\\tau })k},$ $k\\in \\mathbb {N}$ .", "Thus, for all $\\rho _{\\mathrm {M}}\\in (\\frac{1}{\\tau },1)$ , $& \\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{i=0}^{k-1}l_{\\mathrm {M}}(i)>\\rho _{\\mathrm {M}}k]\\le \\sum _{k=1}^{\\infty }e^{\\kappa -(\\rho _{\\mathrm {M}}-\\frac{1}{\\tau })k}\\\\& \\quad =e^{\\kappa }e^{-(\\rho _{\\mathrm {M}}-\\frac{1}{\\tau })}\\left(1-e^{-(\\rho _{\\mathrm {M}}-\\frac{1}{\\tau })}\\right)^{-1}<\\infty ,$ which completes the proof.", "Thus, if the only cause of packet losses is attacks (i.e., $l(i)=l_{\\mathrm {M}}(i)$ ), then Assumption REF holds with $\\rho =\\rho _{\\mathrm {M}}$ ." ], [ "Combination of Random and Malicious Packet Losses (independent case)", "In order to model the case where the network is subject to both random and malicious packet losses, we define $\\lbrace l(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ by $l(i) & ={\\left\\lbrace \\begin{array}{ll}1,\\quad & l_{\\mathrm {R}}(i)=1\\,\\,\\mathrm {or}\\,\\,l_{\\mathrm {M}}(i)=1,\\\\0,\\quad & \\mathrm {otherwise},\\end{array}\\right.", "}\\,\\,\\,i\\in \\mathbb {N}_{0},$ where $\\lbrace l_{\\mathrm {R}}(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ is a time-inhomogeneous Markov chain given in (REF ) characterizing random packet losses (from Section REF ) and $\\lbrace l_{\\mathrm {M}}(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ satisfying (REF ) is a binary-valued process that represents attacks of a malicious agent (from Section REF ).", "Proposition REF below provides a range of values for $\\rho \\in (0,1)$ that satisfy Assumption REF in the case where the network faces both random and malicious packet losses.", "Proposition 2.4 Consider the packet exchange failure indicator process $\\lbrace l(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ given by (REF ) where $\\lbrace l_{\\mathrm {R}}(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ and $\\lbrace l_{\\mathrm {M}}(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ are mutually independent.", "Assume $p_{1}+p_{0}\\rho _{\\mathrm {M}} & <1,$ where $p_{1},p_{0}\\in (0,1)$ are scalars that satisfy (REF ), ().", "Then (REF ) holds for all $\\rho \\in (p_{1}+p_{0}\\rho _{\\mathrm {M}},1)$ .", "From (REF ), the overall loss process can be given by $l(i) & =l_{\\mathrm {R}}(i)+(1-l_{\\mathrm {R}}(i))l_{\\mathrm {M}}(i),\\quad i\\in \\mathbb {N}_{0},$ and hence, by (REF ), $L(k) & =\\sum _{i=0}^{k-1}l_{\\mathrm {R}}(i)+\\sum _{i=0}^{k-1}(1-l_{\\mathrm {R}}(i))l_{\\mathrm {M}}(i),\\quad k\\in \\mathbb {N}.$ Now, let $\\epsilon \\triangleq \\rho -p_{1}-p_{0}\\rho _{\\mathrm {M}}$ , $\\epsilon _{2}\\triangleq \\min \\lbrace \\frac{\\epsilon }{2},\\frac{\\rho _{\\mathrm {M}}-p_{0}\\rho _{\\mathrm {M}}}{2}\\rbrace $ , $\\epsilon _{1}\\triangleq \\epsilon -\\epsilon _{2}$ , and define $\\rho _{1}\\triangleq p_{1}+\\epsilon _{1}$ , $\\rho _{2}\\triangleq p_{0}\\rho _{\\mathrm {M}}+\\epsilon _{2}$ .", "Furthermore, let $L_{1}(k)\\triangleq \\sum _{i=0}^{k-1}l_{\\mathrm {R}}(i)$ and $L_{2}(k)\\triangleq \\sum _{i=0}^{k-1}(1-l_{\\mathrm {R}}(i))l_{\\mathrm {M}}(i)$ .", "We then have $\\mathbb {P}[L(k)>\\rho k] & =\\mathbb {P}[L_{1}(k)+L_{2}(k)>\\rho _{1}k+\\rho _{2}k]\\nonumber \\\\& \\le \\mathbb {P}[\\left\\lbrace L_{1}(k)>\\rho _{1}k\\right\\rbrace \\cup \\left\\lbrace L_{2}(k)>\\rho _{2}k\\right\\rbrace ]\\nonumber \\\\& \\le \\mathbb {P}[L_{1}(k)>\\rho _{1}k]+\\mathbb {P}[L_{2}(k)>\\rho _{2}k].$ In the following we will show that the series $\\sum _{k=1}^{\\infty }\\mathbb {P}[L_{1}(k)>\\rho _{1}k]$ and $\\sum _{k=1}^{\\infty }\\mathbb {P}[L_{2}(k)>\\rho _{2}k]$ are convergent.", "First, note that $\\rho _{1} & =p_{1}+\\epsilon -\\epsilon _{2}=\\max \\lbrace p_{1}+\\frac{\\epsilon }{2},p_{1}+\\epsilon -\\frac{\\rho _{\\mathrm {M}}-p_{0}\\rho _{\\mathrm {M}}}{2}\\rbrace \\nonumber \\\\& =\\max \\lbrace \\frac{p_{1}+\\rho -p_{0}\\rho _{\\mathrm {M}}}{2},\\rho -p_{0}\\rho _{\\mathrm {M}}-\\frac{\\rho _{\\mathrm {M}}-p_{0}\\rho _{\\mathrm {M}}}{2}\\rbrace \\nonumber \\\\& =\\max \\lbrace \\frac{p_{1}+\\rho -p_{0}\\rho _{\\mathrm {M}}}{2},\\frac{2\\rho -\\rho _{\\mathrm {M}}(1+p_{0})}{2}\\rbrace .$ As $\\frac{p_{1}+\\rho -p_{0}\\rho _{\\mathrm {M}}}{2}<1$ and $\\frac{2\\rho -\\rho _{\\mathrm {M}}(1+p_{0})}{2}<1$ , it holds from (REF ) that $\\rho _{1}\\in (p_{1},1)$ .", "Consequently, $\\sum _{k=1}^{\\infty }\\mathbb {P}[L_{1}(k)>\\rho _{1}k]<\\infty $ follows from Lemma REF with $\\rho _{\\mathrm {R}}$ replaced with $\\rho _{1}$ .", "Next, we will use Lemma REF to show that $\\sum _{k=1}^{\\infty }\\mathbb {P}[L_{2}(k)>\\rho _{2}k]<\\infty $ .", "To obtain this result, we first observe that $\\rho _{2}>p_{0}\\rho _{\\mathrm {M}}$ , since $\\epsilon _{2}>0$ .", "Moreover, $\\rho _{2} & =p_{0}\\rho _{\\mathrm {M}}+\\min \\lbrace \\frac{\\epsilon }{2},\\frac{\\rho _{\\mathrm {M}}-p_{0}\\rho _{\\mathrm {M}}}{2}\\rbrace \\le p_{0}\\rho _{\\mathrm {M}}+\\frac{\\rho _{\\mathrm {M}}-p_{0}\\rho _{\\mathrm {M}}}{2}\\\\& <p_{0}\\rho _{\\mathrm {M}}+\\rho _{\\mathrm {M}}-p_{0}\\rho _{\\mathrm {M}}=\\rho _{\\mathrm {M}},$ and hence, we have $\\rho _{2}\\in (p_{0}\\rho _{\\mathrm {M}},\\rho _{\\mathrm {M}})$ .", "As a consequence of (REF ), conditions (REF ), () in the Lemma REF hold with $\\tilde{p}=p_{0}$ and $\\tilde{w}=\\rho _{\\mathrm {M}}$ , together with processes $\\lbrace \\xi (i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ and $\\lbrace \\chi (i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ defined by setting $\\xi (i)=1-l_{\\mathrm {R}}(i)$ , $\\chi (i)=l_{\\mathrm {M}}(i)$ , $i\\in \\mathbb {N}_{0}$ .", "Now, we have $L_{2}(k)=\\sum _{i=0}^{k-1}\\xi (i)\\chi (i)$ and hence, Lemma REF implies $\\sum _{k=1}^{\\infty }\\mathbb {P}[L_{2}(k)>\\rho _{2}k]<\\infty $ .", "Finally, by (REF ), we arrive at $& \\sum _{k=1}^{\\infty }\\mathbb {P}[L(k)>\\rho k]\\\\& \\quad \\le \\sum _{k=1}^{\\infty }\\mathbb {P}[L_{1}(k)>\\rho _{1}k]+\\sum _{k=1}^{\\infty }\\mathbb {P}[L_{2}(k)>\\rho _{2}k]<\\infty ,$ which completes the proof." ], [ "Combination of Random and Malicious Packet Losses (dependent case)", "So far, in Proposition REF , we assumed that packet exchange attempt failures due to attacks are independent of those due to random packet losses.", "Next, we consider the case where the two processes $\\lbrace l_{\\mathrm {R}}(i)\\rbrace _{i\\in \\mathbb {N}_{0}}$ and $\\lbrace l_{\\mathrm {M}}(i)\\rbrace _{i\\in \\mathbb {N}_{0}}$ may be dependent.", "This is clearly the case when the attacker has information of the random packet losses in the channel.", "Furthermore, as we discussed in the Introduction, the attacker may decide to attack based on the content of packets.", "In such cases $l_{\\mathrm {M}}(\\cdot )$ would depend on state and control input, which in turn depend on $l_{\\mathrm {R}}(\\cdot )$ .", "Proposition REF below deals with such cases.", "Proposition 2.5 Consider the packet exchange failure indicator process $\\lbrace l(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ .", "Assume $p_{1}+\\rho _{\\mathrm {M}} & <1,$ where $p_{1}\\in (0,1)$ is a scalar that satisfies (REF ).", "Then (REF ) holds for all $\\rho \\in (p_{1}+\\rho _{\\mathrm {M}},1)$ .", "It follows from (REF ) that $L(k) & \\le \\sum _{i=0}^{k-1}l_{\\mathrm {R}}(i)+\\sum _{i=0}^{k-1}l_{\\mathrm {M}}(i),\\quad k\\in \\mathbb {N}.$ Now, using arguments similar to the ones used for obtaining (REF ) in the proof of Proposition REF , we have $& \\mathbb {P}[L(k)>\\rho k]\\le \\mathbb {P}[\\sum _{i=0}^{k-1}l_{\\mathrm {R}}(i)+\\sum _{i=0}^{k-1}l_{\\mathrm {M}}(i)>\\rho k]\\nonumber \\\\& \\,\\,\\le \\mathbb {P}[\\sum _{i=0}^{k-1}l_{\\mathrm {R}}(i)>\\rho _{1}k]+\\mathbb {P}[\\sum _{i=0}^{k-1}l_{\\mathrm {M}}(i)>\\rho _{2}k],\\,\\,k\\in \\mathbb {N},$ and consequently $& \\sum _{k=1}^{\\infty }\\mathbb {P}[L(k)>\\rho k]\\nonumber \\\\& \\,\\,\\le \\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{i=0}^{k-1}l_{\\mathrm {R}}(i)>\\rho _{1}k]+\\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{i=0}^{k-1}l_{\\mathrm {M}}(i)>\\rho _{2}k],$ where $\\rho _{1}\\triangleq p_{1}+\\frac{\\epsilon }{2}$ , $\\rho _{2}\\triangleq \\rho _{\\mathrm {M}}+\\frac{\\epsilon }{2}$ , and $\\epsilon \\triangleq \\rho -p_{1}-\\rho _{\\mathrm {M}}$ .", "Observe that $\\rho _{1}=p_{1}+\\frac{\\rho -p_{1}-\\rho _{\\mathrm {M}}}{2}=\\frac{\\rho +p_{1}-\\rho _{\\mathrm {M}}}{2}.$ Since $\\frac{\\rho +p_{1}-\\rho _{\\mathrm {M}}}{2}<1$ and $\\epsilon >0$ , we have $\\rho _{1}\\in (p_{1},1)$ .", "By using Lemma REF with $\\rho _{\\mathrm {R}}=\\rho _{1}$ , we obtain $\\sum _{k=1}^{\\infty }\\mathbb {P}[L_{1}(k) & >\\rho _{1}k]<\\infty .$ Furthermore, note that $\\rho _{2}=\\rho _{\\mathrm {M}}+\\frac{\\rho -p_{1}-\\rho _{\\mathrm {M}}}{2}=\\frac{\\rho +\\rho _{\\mathrm {M}}-p_{1}}{2}.$ Also, by $\\frac{\\rho +\\rho _{\\mathrm {M}}-p_{1}}{2}<1$ and $\\epsilon >0$ , we have $\\rho _{2}\\in (\\rho _{\\mathrm {M}},1)$ .", "Since $\\rho _{2}>\\rho _{\\mathrm {M}}$ , by the characterization of $\\lbrace l_{\\mathrm {M}}(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ , $\\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{i=0}^{k-1}l_{\\mathrm {M}}(i)>\\rho _{2}k] & \\le \\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{i=0}^{k-1}l_{\\mathrm {M}}(i)>\\rho _{\\mathrm {M}}k]<\\infty .$ The result then follows from (REF )–(REF ).", "In comparison with Proposition REF , the result above provides a more restricted range of values for $\\rho $ that satisfies Assumption REF .", "This is because in Proposition REF we find $\\rho $ for the worst case scenario where the attacker may be knowledgeable about all random packet losses in the network and may have access to the information of the transmitted state and control input vectors.", "An example scenario is where the attacker avoids placing malicious attacks when there is already a random packet loss, increasing the total number of packet exchange failures, which is clearly to the disadvantage of the controller to maintain closed-loop stability.", "We note that the condition (REF ) guarantees that the range $\\rho \\in (p_{1}+\\rho _{\\mathrm {M}},1)$ identified in Proposition REF is well defined.", "If $p_{1}+\\rho _{M}\\ge 1$ , then Assumption REF holds with $\\rho =1$ .", "We also note that Proposition REF may introduce some conservativeness when it is applied to other scenarios where malicious attacks and random packet losses are dependent, but not as in the worst case scenario mentioned above.", "In such cases additional information about the malicious attacks and random packet losses may be employed to show that Assumption REF holds with $\\rho <1$ even if $p_{1}+\\rho _{\\mathrm {M}}\\ge 1$ .", "Remark 2.6 There may be situations where the attacker has limited knowledge.", "For instance, the attacker may have access only to certain entries of the state and control input vectors.", "This situation arises in a multi-hop network with multiple paths (see, e.g., [16], [45]); different parts of the state and control input vectors may be sent over different paths on the network and the attacker may have access to the data only on some of those paths.", "In this case the attacker would need an estimation mechanism to have information about the state/control input vectors.", "Note that the operator may also utilize encryption methods to prevent the attacker gain any information about the system behavior.", "In the situations where the attacker is not knowledgeable about the random packet losses and has no information of state and control input vectors, Proposition REF can be used." ], [ "Event-Triggered Control Design ", "In this section we investigate event-triggered control of (REF ) over an unreliable and potentially attacked network characterized through Assumption REF .", "As a first step, we introduce the event-triggering scheme for communication between the plant and the controller.", "This scheme will determine the time instants $\\tau _{i}\\in \\mathbb {N}_{0}$ , $i\\in \\mathbb {N}_{0}$ , at which packet exchanges are attempted.", "For this purpose, we utilize the quadratic Lyapunov-like function $V\\colon \\mathbb {R}^{n}\\rightarrow [0,\\infty )$ given by $V(x)\\triangleq x^{\\mathrm {T}}Px$ , where $P>0$ .", "Letting $\\tau _{0}=0$ , we describe $\\tau _{i}$ , $i\\in \\mathbb {N}$ , by $\\tau _{i+1} & \\triangleq \\min \\Big \\lbrace t\\in \\lbrace \\tau _{i}+1,\\tau _{i}+2,\\ldots \\rbrace \\colon t\\ge \\tau _{i}+\\theta \\nonumber \\\\& \\quad \\quad \\quad \\mathrm {or}\\,\\,\\,V(Ax(t)+Bu(\\tau _{i}))>\\beta V(x(\\tau _{i}))\\Big \\rbrace ,$ where $\\beta \\in (0,1)$ , $\\theta \\in \\mathbb {N}$ .", "The triggering condition (REF ) involves two parts.", "The part $V(Ax(t)+Bu(\\tau _{i}))>\\beta V(x(\\tau _{i}))$ ensures that after a successful packet exchange attempt at $\\tau _{i}$ , the value of $V(\\cdot )$ stays below the level $\\beta V(x(\\tau _{i}))$ until the next packet exchange attempt.", "Furthermore, the triggering condition $t\\ge \\tau _{i}+\\theta $ ensures that two consecutive packet exchange attempt instants are at most $\\theta \\in \\mathbb {N}$ steps apart, that is, $\\tau _{i+1}-\\tau _{i}\\le \\theta $ , $i\\in \\mathbb {N}_{0}$ .", "Although the specific value of $\\theta $ does not affect the results developed below, the boundedness of packet exchange attempt intervals guarantees that $\\tau _{i}$ (and hence $V(x(\\tau _{i}))$ ) is well-defined for each $i\\in \\mathbb {N}$ .", "In practice, the value of $\\theta $ can be selected considering how frequent the plant state is desired to be monitored by the controller side.", "Figure: [Top] Networked control system with successful (left) and failed(right) packet transmissions.", "[Bottom] Response of theLyapunov-like function.The operation of the event-triggered networked control system is illustrated in Fig.", "REF .", "The triggering condition (REF ) is checked at the plant side at each step $t\\in \\mathbb {N}_{0}$ .", "At times $t=\\tau _{i}$ , $i\\in \\mathbb {N}$ , the triggering condition is satisfied and packet exchanges are attempted.", "In this example, a packet exchange is attempted at time $t=\\tau _{1}$ , since $V(Ax(t)+Bu(\\tau _{0}))>\\beta V(x(\\tau _{0}))$ .", "At this time instant, the plant and the controller successfully exchange state and control input packets over the network, and as a result, control input on the plant side is updated to $Kx(\\tau _{1})$ .", "Note that packet exchange attempts are not always successful, and may fail due to loss of packets in the network.", "In the figure, the packet exchange attempt at time $\\tau _{2}$ fails.", "In this case, it follows from (REF ) with $l(2)=1$ that the control input at the plant side is set to 0 at time $\\tau _{2}$ , which results in an unstable behavior.", "A packet exchange is attempted again at the very next time step $\\tau _{3}$ , since the triggering condition is also satisfied at that time instant." ], [ "Stability Analysis", "Next, we investigate stability of the closed-loop event-triggered networked control system (REF ), (REF ), (REF ), which is a stochastic dynamical system due to the probabilistic characterization of packet losses.", "Below we define almost sure asymptotic stability for stochastic dynamical systems.", "Definition 3.1 The zero solution $x(t)\\equiv 0$ of the stochastic system (REF ), (REF ), and (REF ) is almost surely stable if, for all $\\epsilon >0$ and $\\bar{p}>0$ , there exists $\\delta =\\delta (\\epsilon ,\\bar{p})>0$ such that if $\\Vert x(0)\\Vert <\\delta $ , then $\\mathbb {P}[\\sup _{t\\in \\mathbb {N}_{0}}\\Vert x(t)\\Vert >\\epsilon ] & <\\bar{p}.$ Moreover, the zero solution $x(t)\\equiv 0$ is asymptotically stable almost surely if it is almost surely stable and $\\mathbb {P}[\\lim _{t\\rightarrow \\infty }\\Vert x(t)\\Vert =0] & =1.$ In our stability analysis for the networked control system (REF ), (REF ), we utilize an upper bound for the long run average of the total number of failed packet exchanges.", "The following result is a direct consequence of the Borel-Cantelli lemma (see [46]) and shows that under Assumption REF , the long run average of the total number of failed packet exchanges is upper bounded by $\\rho $ characterized in (REF ).", "Lemma 3.2 If there exists a scalar $\\rho \\in [0,1]$ such that (REF ) holds, then $\\limsup _{k\\rightarrow \\infty }\\frac{L(k)}{k} & \\le \\rho ,$ almost surely.", "In Propositions REF and REF , we obtained a range of values for $\\rho $ that satisfy (REF ).", "In those results the range was given as an open interval.", "In the following result we show that when Assumption REF holds for a range of values, then (REF ) also holds with $\\rho $ given as the infimum of the range.", "Lemma 3.3 Suppose (REF ) is satisfied for all $\\rho \\in (\\underline{\\rho },1)$ where $\\underline{\\rho }\\in [0,1)$ .", "Then (REF ) holds with $\\rho =\\underline{\\rho }$ , almost surely.", "The proof resembles the sufficiency part of the proof of Proposition 5.6 in [47].", "First, by Lemma REF , $\\mathbb {P}[\\limsup _{k\\rightarrow \\infty }\\frac{L(k)}{k}-\\underline{\\rho }>\\epsilon ] & =0,$ for any $\\epsilon >0$ .", "Now, it follows from (REF ) that $& \\mathbb {P}[\\limsup _{k\\rightarrow \\infty }\\frac{L(k)}{k}-\\underline{\\rho }>0]=\\mathbb {P}[\\cup _{j=1}^{\\infty }\\lbrace \\limsup _{k\\rightarrow \\infty }\\frac{L(k)}{k}-\\underline{\\rho }>\\frac{1}{j}\\rbrace ]\\\\& \\quad \\quad \\le \\sum _{j=1}^{\\infty }\\mathbb {P}[\\limsup _{k\\rightarrow \\infty }\\frac{L(k)}{k}-\\underline{\\rho }>\\frac{1}{j}]=0,$ which implies that $\\mathbb {P}[\\limsup _{k\\rightarrow \\infty }\\frac{L(k)}{k}\\le \\underline{\\rho }]=1$ .", "Remark 3.4 Note that the term $\\limsup _{k\\rightarrow \\infty }\\frac{L(k)}{k}$ in (REF ) corresponds to the “discrete event rate” used in [48], [49] for deterministic systems, when $\\lim _{k\\rightarrow \\infty }\\frac{L(k)}{k}$ exists.", "In this paper, Assumption REF allows the binary-valued process $\\lbrace l(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ to be a non-ergodic stochastic process, for which $\\lim _{k\\rightarrow \\infty }\\frac{L(k)}{k}$ may not be equal for all sample paths.", "For instance, let $l(i)\\triangleq l_{\\mathrm {M}}(i),i\\in \\mathbb {N}_{0}$ , and $l_{\\mathrm {M}}(i) & \\triangleq {\\left\\lbrace \\begin{array}{ll}1,\\quad & i\\in \\lbrace \\alpha ,2\\alpha ,3\\alpha \\ldots \\rbrace ,\\\\0,\\quad & \\mathrm {otherwise},\\end{array}\\right.", "}$ where $\\alpha :\\Omega \\rightarrow \\lbrace 2,4\\rbrace $ is a random variable with $\\mathbb {P}[\\alpha =2]=\\mathbb {P}[\\alpha =4]=\\frac{1}{2}$ .", "In this setting, the attacker decides the period of attacks based on a random variable $\\alpha :\\Omega \\rightarrow \\lbrace 2,4\\rbrace $ .", "Depending on the value of $\\alpha $ , malicious packet losses occur either at every 2 packet exchange attempts or at every 4 packet exchange attempts.", "Thus, the discrete event rate would be a random variable that depends on the value of $\\alpha $ .", "On the other hand, regardless of the value of $\\alpha $ , (REF ) is satisfied with $\\tau =2$ , and hence Lemmas REF and REF imply that $\\limsup _{k\\rightarrow \\infty }\\frac{L(k)}{k}\\le \\frac{1}{2},$ almost surely.", "Note that here $\\frac{1}{2}$ represents the worst-case upper bound for the long run average of the total number of failed packet exchanges.", "We are now ready to state the main result of this paper.", "It provides a sufficient condition for almost sure asymptotic stability of the networked control system (REF ), (REF ) with packet exchange failure indicator $\\lbrace l(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ satisfying (REF ).", "Theorem 3.5 Consider the linear dynamical system (REF ).", "Suppose that the process $\\lbrace l(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ characterizing packet exchange failuresWe set (REF ) as a condition for packet exchange failures as it allows more generality in comparison to Assumption REF .", "Note that by Lemma REF , Assumption REF implies (REF ).", "Furthermore, Lemma REF shows that (REF ) also holds when $\\rho $ is given as the infimum of an open interval where all values satisfy Assumption REF .", "in the network satisfies (REF ) with scalar $\\rho \\in [0,1]$ .", "If there exist a matrix $K\\in \\mathbb {R}^{m\\times n}$ , a positive-definite matrix $P\\in \\mathbb {R}^{n\\times n}$ , and scalars $\\beta \\in (0,1),$ $\\varphi \\in [1,\\infty )$ such that $& \\left(A+BK\\right)^{\\mathrm {T}}P\\left(A+BK\\right)-\\beta P\\le 0,\\\\& A^{\\mathrm {T}}PA-\\varphi P\\le 0,\\\\& (1-\\rho )\\ln \\beta +\\rho \\ln \\varphi <0,$ then the event-triggered control law (REF ), (REF ) guarantees almost sure asymptotic stability of the zero solution $x(t)\\equiv 0$ of the closed-loop system dynamics.", "The proof is composed of three steps.", "In the initial step, we obtain an inequality concerning the evolution of the Lyapunov-like function $V(x)\\triangleq x^{\\mathrm {T}}Px$ , $x\\in \\mathbb {R}^{n}$ .", "Then, we will establish almost sure stability, and then finally we show almost sure asymptotic stability of the closed-loop system.", "First, we use (REF ) and (REF ) together with $V(\\cdot )$ to obtain $V(x(\\tau _{i}+1)) & =x^{\\mathrm {T}}(\\tau _{i})\\left(A+\\left(1-l(i)\\right)BK\\right)^{\\mathrm {T}}P\\nonumber \\\\& \\quad \\,\\cdot \\left(A+\\left(1-l(i)\\right)BK\\right)x(\\tau _{i}),\\,i\\in \\mathbb {N}_{0}.$ Now, for the case $l(i)=0$ , (REF ) and (REF ) imply $V(x(\\tau _{i}+1)) & =x^{\\mathrm {T}}(\\tau _{i})\\left(A+BK\\right)^{\\mathrm {T}}P\\left(A+BK\\right)x(\\tau _{i})\\nonumber \\\\& \\le \\beta x^{\\mathrm {T}}(\\tau _{i})Px(\\tau _{i}).$ Since $\\tau _{i+1}\\ge \\tau _{i}+1$ , it follows from (REF ) and (REF ) that $V(x(t)) & \\le \\beta x^{\\mathrm {T}}(\\tau _{i})Px(\\tau _{i})\\nonumber \\\\& =\\beta V(x(\\tau _{i})),\\quad t\\in \\lbrace \\tau _{i}+1,\\ldots ,\\tau _{i+1}\\rbrace .$ On the other hand, for the case $l(i)=1$ , we have from () and (REF ) that $V(x(\\tau _{i}+1)) & =x^{\\mathrm {T}}(\\tau _{i})A^{\\mathrm {T}}PAx(\\tau _{i})\\le \\varphi x^{\\mathrm {T}}(\\tau _{i})Px(\\tau _{i}).$ Now if $\\tau _{i+1}=\\tau _{i}+1$ , we have $V(x(\\tau _{i+1}))\\le \\varphi V(x(\\tau _{i}))$ due to (REF ).", "Otherwise, that is, if $\\tau _{i+1}>\\tau _{i}+1$ , it means that $V(x(t))\\le \\beta V(x(\\tau _{i}))$ for $t\\in \\lbrace \\tau _{i}+2,\\ldots ,\\tau _{i+1}\\rbrace $ .", "Therefore, since $\\beta \\le \\varphi $ , $V(x(t)) & \\le \\varphi V(x(\\tau _{i})),\\quad t\\in \\lbrace \\tau _{i}+1,\\ldots ,\\tau _{i+1}\\rbrace .$ Using (REF ) and (REF ) we obtain $V(x(\\tau _{i+1})) & \\le (1-l(i))\\beta V(x(\\tau _{i}))+l(i)\\varphi V(x(\\tau _{i})),$ for $i\\in \\mathbb {N}_{0}$ .", "Note that the inequality given in (REF ) provides an upper bound on $V(\\cdot )$ .", "Now, let $\\eta (k)\\triangleq \\prod _{i=0}^{k-1}\\left[(1-l(i))\\beta +l(i)\\varphi \\right]$ .", "Then, by (REF ), $V(x(\\tau _{k})) & \\le \\eta (k)V(x(0)),\\quad k\\in \\mathbb {N}.$ Furthermore, since $\\ln \\left[(1-q)\\beta +q\\varphi \\right]=(1-q)\\ln \\beta +q\\ln \\varphi $ for $q\\in \\lbrace 0,1\\rbrace $ , we have $\\ln \\eta (k) & =\\sum _{i=0}^{k-1}\\ln \\left[(1-l(i))\\beta +l(i)\\varphi \\right]\\\\& =\\sum _{i=0}^{k-1}(1-l(i))\\ln \\beta +\\sum _{i=0}^{k-1}l(i)\\ln \\varphi \\\\& =(k-L(k))\\ln \\beta +L(k)\\ln \\varphi ,$ where $L(k)=\\sum _{i=0}^{k-1}l(i)$ by (REF ).", "Now by $\\beta \\in (0,1)$ , and $\\varphi \\in [1,\\infty )$ , it follows from (REF ) and () that $\\limsup _{k\\rightarrow \\infty }\\frac{\\ln \\eta (k)}{k} & =\\limsup _{k\\rightarrow \\infty }\\frac{1}{k}\\left[(k-L(k))\\ln \\beta +L(k)\\ln \\varphi \\right]\\\\& \\le (1-\\rho )\\ln \\beta +\\rho \\ln \\varphi <0,$ almost surely.", "As a consequence, $\\lim _{k\\rightarrow \\infty }\\ln \\eta (k)=-\\infty $ , and hence, $\\lim _{k\\rightarrow \\infty }\\eta (k)=0$ , almost surely.", "Thus, for any $\\epsilon >0$ , $\\lim _{j\\rightarrow \\infty }\\mathbb {P}[\\sup _{k\\ge j}\\eta (k)>\\epsilon ^{2}]=0$ .", "Therefore, for any $\\epsilon >0$ and $\\bar{p}>0$ , there exists a positive integer $N(\\epsilon ,\\bar{p})$ such that $\\mathbb {P}[\\sup _{k\\ge j}\\eta (k) & >\\epsilon ^{2}]<\\bar{p},\\quad j\\ge N(\\epsilon ,\\bar{p}).$ In what follows, we employ (REF ) and (REF ) to show almost sure stability of the closed-loop system.", "Note that (REF ), (REF ), and $\\varphi \\ge 1>\\beta $ imply that $V(x(t+1))\\le \\varphi V(x(t)),\\,t\\in \\lbrace \\tau _{i},\\ldots ,\\tau _{i+1}-1\\rbrace ,$ $i\\in \\mathbb {N}_{0}$ .", "Since $\\Vert x\\Vert ^{2}\\le \\frac{1}{\\lambda _{\\min }(P)}V(x)$ and $V(x)\\le \\lambda _{\\max }(P)\\Vert x\\Vert ^{2}$ , $x\\in \\mathbb {R}^{n}$ , we have $\\Vert x(t)\\Vert ^{2} & \\le \\varphi \\nu \\Vert x(\\tau _{i})\\Vert ^{2},\\,t\\in \\lbrace \\tau _{i},\\ldots ,\\tau _{i+1}-1\\rbrace $ for $i\\in \\mathbb {N}_{0}$ , where $\\nu \\triangleq \\frac{\\lambda _{\\max }(P)}{\\lambda _{\\min }(P)}$ .", "Now, let $\\mathcal {T}_{k}\\triangleq \\lbrace \\tau _{k},\\ldots ,\\tau _{k+1}-1\\rbrace $ , $k\\in \\mathbb {N}_{0}$ .", "Then by using (REF ) and (REF ), we obtain $\\eta (k)\\ge \\frac{V(x(\\tau _{k}))}{V(x(0))}\\ge \\frac{\\lambda _{\\min }(P)}{\\lambda _{\\max }(P)}\\frac{\\Vert x(\\tau _{k})\\Vert ^{2}}{\\Vert x(0)\\Vert ^{2}}\\ge \\frac{1}{\\nu ^{2}\\varphi }\\frac{\\Vert x(t)\\Vert ^{2}}{\\Vert x(0)\\Vert ^{2}}$ for all $t\\in \\mathcal {T}_{k}$ , $k\\in \\mathbb {N}$ .", "Hence, $\\eta (k)\\ge \\frac{1}{\\nu ^{2}\\varphi }\\frac{\\max _{t\\in \\mathcal {T}_{k}}\\Vert x(t)\\Vert ^{2}}{\\Vert x(0)\\Vert ^{2}}$ , $k\\in \\mathbb {N}$ .", "By (REF ), it follows that for all $\\epsilon >0$ and $\\bar{p}>0$ , $& \\mathbb {P}[\\sup _{k\\ge j}\\max _{t\\in \\mathcal {T}_{k}}\\Vert x(t)\\Vert >\\epsilon \\nu \\sqrt{\\varphi }\\Vert x(0)\\Vert ]\\\\& \\quad =\\mathbb {P}[\\sup _{k\\ge j}\\max _{t\\in \\mathcal {T}_{k}}\\Vert x(t)\\Vert ^{2}>\\epsilon ^{2}\\nu ^{2}\\varphi \\Vert x(0)\\Vert ^{2}]\\\\& \\quad =\\mathbb {P}[\\sup _{k\\ge j}\\frac{1}{\\nu ^{2}\\varphi }\\frac{\\max _{t\\in \\mathcal {T}_{k}}\\Vert x(t)\\Vert ^{2}}{\\Vert x(0)\\Vert ^{2}}>\\epsilon ^{2}]\\\\& \\quad \\le \\mathbb {P}[\\sup _{k\\ge j}\\eta (k)>\\epsilon ^{2}]<\\bar{p},\\quad j\\ge N(\\epsilon ,\\bar{p}).$ We now define $\\delta _{1}\\triangleq \\frac{1}{\\nu \\sqrt{\\varphi }}$ .", "Note that if $\\Vert x(0)\\Vert \\le \\delta _{1}$ , then (since $\\nu \\sqrt{\\varphi }\\Vert x(0)\\Vert \\le 1$ ) for all $j\\ge N(\\epsilon ,\\bar{p})$ , we have $& \\mathbb {P}[\\sup _{k\\ge j}\\max _{t\\in \\mathcal {T}_{k}}\\Vert x(t)\\Vert >\\epsilon ]\\nonumber \\\\& \\quad \\le \\mathbb {P}[\\sup _{k\\ge j}\\max _{t\\in \\mathcal {T}_{k}}\\Vert x(t)\\Vert >\\epsilon \\nu \\sqrt{\\varphi }\\Vert x(0)\\Vert ]<\\bar{p}.$ On the other hand, since $\\varphi \\ge 1>\\beta $ , it follows from (REF ) that $V(x(\\tau _{k}))\\le \\varphi ^{k}V(x(0))\\le \\varphi ^{N(\\epsilon ,\\bar{p})-1}V(x(0))$ for all $k\\in \\lbrace 0,1,\\ldots ,N(\\epsilon ,\\bar{p})-1\\rbrace $ .", "Therefore, $\\Vert x(\\tau _{k})\\Vert ^{2}\\le \\varphi ^{N(\\epsilon ,\\bar{p})-1}\\frac{\\lambda _{\\max }(P)}{\\lambda _{\\min }(P)}\\Vert x(0)\\Vert ^{2}=\\varphi ^{N(\\epsilon ,\\bar{p})-1}\\nu \\Vert x(0)\\Vert ^{2}$ .", "Furthermore, as a result of (REF ), $\\max _{t\\in \\mathcal {T}_{k}}\\Vert x(t)\\Vert ^{2} & \\le \\varphi \\nu \\Vert x(\\tau _{k})\\Vert ^{2}\\le \\nu ^{2}\\varphi ^{N(\\epsilon ,\\bar{p})}\\Vert x(0)\\Vert ^{2},$ and hence, $\\max _{t\\in \\mathcal {T}_{k}}\\Vert x(t)\\Vert \\le \\nu \\sqrt{\\varphi ^{N(\\epsilon ,\\bar{p})}}\\Vert x(0)\\Vert $ for all $k\\in \\lbrace 0,1,\\ldots ,N(\\epsilon ,\\bar{p})-1\\rbrace $ .", "Let $\\delta _{2}\\triangleq \\epsilon \\nu ^{-1}\\sqrt{\\varphi ^{-N(\\epsilon ,\\bar{p})}}$ .", "Now, if $\\Vert x(0)\\Vert \\le \\delta _{2}$ , then $\\max _{t\\in \\mathcal {T}_{k}}\\Vert x(t)\\Vert \\le \\epsilon $ , $k\\in \\lbrace 0,1,\\ldots ,N(\\epsilon ,\\bar{p})-1\\rbrace $ , which implies $\\mathbb {P}[\\max _{k\\in \\lbrace 0,1,\\ldots ,N(\\epsilon ,\\bar{p})\\rbrace }\\max _{t\\in \\mathcal {T}_{k}}\\Vert x(t)\\Vert >\\epsilon ] & = & 0.$ It follows from (REF ) and (REF ) that for all $\\epsilon >0$ , $\\bar{p}>0$ , $& \\mathbb {P}[\\sup _{t\\in \\mathbb {N}_{0}}\\Vert x(t)\\Vert >\\epsilon ]=\\mathbb {P}[\\sup _{k\\in \\mathbb {N}_{0}}\\max _{t\\in \\mathcal {T}_{k}}\\Vert x(t)\\Vert >\\epsilon ]\\\\& \\quad =\\mathbb {P}[\\lbrace \\max _{k\\in \\lbrace 0,1,\\ldots ,N(\\epsilon ,\\bar{p})-1\\rbrace }\\max _{t\\in \\mathcal {T}_{k}}\\Vert x(t)\\Vert >\\epsilon \\rbrace \\\\& \\quad \\quad \\quad \\cup \\,\\lbrace \\sup _{k\\ge N(\\epsilon ,\\bar{p})}\\max _{t\\in \\mathcal {T}_{k}}\\Vert x(t)\\Vert >\\epsilon \\rbrace ]\\\\& \\quad \\le \\mathbb {P}[\\max _{k\\in \\lbrace 0,1,\\ldots ,N(\\epsilon ,\\bar{p})-1\\rbrace }\\max _{t\\in \\mathcal {T}_{k}}\\Vert x(t)\\Vert >\\epsilon ]\\\\& \\quad \\quad +\\mathbb {P}[\\sup _{k\\ge N(\\epsilon ,\\bar{p})}\\max _{t\\in \\mathcal {T}_{k}}\\Vert x(t)\\Vert >\\epsilon ]<\\bar{p},$ whenever $\\Vert x(0)\\Vert <\\delta \\triangleq \\min (\\delta _{1},\\delta _{2})$ , which implies almost sure stability.", "Finally, in order to establish almost sure asymptotic stability of the zero solution, it remains to show (REF ).", "To this end, observe that $\\mathbb {P}[\\lim _{k\\rightarrow \\infty }\\eta (\\tau _{k})=0]=1$ .", "It follows from (REF ) that $\\mathbb {P}[\\lim _{k\\rightarrow \\infty }V(x(\\tau _{k}))=0]=1$ , which implies (REF ).", "Hence the zero solution of the closed-loop system (REF ), (REF ), (REF ) is asymptotically stable almost surely.", "Theorem REF provides a sufficient condition under which the event-triggered control law (REF ), (REF ) guarantees almost sure asymptotic stability of the system (REF ) for the case of packet losses satisfying Assumption REF .", "Note that the scalars $\\beta \\in (0,1)$ and $\\varphi \\in [1,\\infty )$ in conditions (REF ) and () characterize upper bounds on the growth of the Lyapunov-like function, and they are also related to closed-loop and open-loop bounds utilized in [40], [36].", "Specifically, when a packet exchange attempt between the plant and the controller is successful at time $\\tau _{i}$ , the condition (REF ) together with (REF ) guarantees that $V(x(\\tau _{i+1}))\\le \\beta V(x(\\tau _{i}))$ .", "On the other hand, if a packet exchange is unsuccessful at time $\\tau _{i}$ , it follows from (REF ) and () that $V(x(\\tau _{i+1}))\\le \\varphi V(x(\\tau _{i}))$ .", "If successful packet exchanges are sufficiently frequent such that () is satisfied, then the closed-loop stability is guaranteed.", "We remark that the analysis for the closed-loop system stability in the proof above is technically involved partly due to the general characterization in Assumption REF , which captures not only random packet losses but attacks as well.", "If we consider only random packet losses, we may employ methods from discrete-time Markov jump systems theory [50] for obtaining conditions of stability.", "Furthermore, in the case $\\lbrace l(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ is an ergodic process, the results presented in [40] can be directly employed to show stability.", "On the other hand packet losses due to attacks (Section REF ) cannot be described using Markov processes and they may not be ergodic.", "Stability of a system under denial-of-service attacks is explored in [10], where the analysis relies on a deterministic approach for obtaining an exponentially decreasing upper bound for the norm of the state.", "In contrast, in our analysis, we use probabilistic approaches similar to [40], [23] to show almost sure asymptotic stability.", "Specifically, we use tools from probability theory to find a stochastic upper bound for a Lyapunov-like function and show that this bound tends to zero even though it may increase at certain times.", "This approach is related to obtaining an upper bound on the top Lyapunov exponent (see [37], [38], [39]) of a stochastic system.", "Theorem REF provides conditions that guarantee both (REF ) and (REF ) implying almost sure asymptotic stability.", "In this stability definition, (REF ) is concerned with the convergence of solutions to zero, while (REF ) ensures that states sufficiently close to the origin are likely to stay close to the origin.", "However note that (REF ) allows states to leave any given ball in a finite time with positive (even if small) probability.", "For instance, if many consecutive packet transmission attempts fail, the state magnitude may grow due to lack of control action.", "We emphasize that Assumption REF and hence (REF ) ensure packet failures to be statistically rare so that the state eventually converges to the origin.", "Remark 3.6 In addition to almost sure stability, there are other stochastic stability and performance notions that are useful for the analysis of networked control systems.", "In particular, moment stability and moment-based performance notions have been utilized when random packet losses are considered (see [20], [19] and the references therein).", "In comparison with those works, in our problem setting, we must take into account also the effect of malicious attacks.", "We remark that in contrast with random packet losses, precise information of the probabilities of malicious attacks is not available.", "Hence, it is difficult to characterize the evolution of the moments of the state and establish moment stability.", "On the other hand, both random packet losses and malicious attacks, as well as their combination provide us information about the asymptotic ratio of packet exchange failures, which can be employed in the analysis when we consider almost sure asymptotic stability.", "In the following corollary of Theorem REF , we discuss the special case of random packet losses described with time-homogeneous Markov chains.", "Corollary 3.7 Consider the linear dynamical system (REF ).", "Suppose that the process $\\lbrace l(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ is an irreducible time-homogeneous Markov chain with constant transition probabilities $p_{q,r}\\in [0,1]$ , $q,r\\in \\lbrace 0,1\\rbrace $ .", "If there exist a matrix $K\\in \\mathbb {R}^{m\\times n}$ , a positive-definite matrix $P\\in \\mathbb {R}^{n\\times n}$ , and scalars $\\beta \\in (0,1),$ $\\varphi \\in [1,\\infty )$ such that (REF ), () and () hold with $\\rho \\triangleq \\frac{p_{0,1}}{p_{0,1}+p_{1,0}}$ , then the event-triggered control law (REF ), (REF ) guarantees almost sure asymptotic stability of the zero solution $x(t)\\equiv 0$ of the closed-loop system dynamics.", "By the ergodic theorem for irreducible Markov chains [51], we have $\\lim _{k\\rightarrow \\infty }\\frac{L(k)}{k}=\\rho $ .", "Now, since (REF ) holds, the result follows from Theorem REF .", "When we consider transmission attempts at all times by setting $\\theta =1$ in (REF ), Corollary REF recovers a specialization of the result in [40] for linear systems.", "Furthermore, if we consider $\\lbrace l(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ to be a Bernoulli process, then $\\rho $ in Corollary REF is given by $\\rho =p$ , where $p=p_{0,1}=p_{1,1}$ denotes the packet loss probability.", "In this setting, the almost sure stability condition in Corollary REF is tighter than the second-moment stability condition in [36].", "Specifically, for this problem setting, the results in [36] can be used to obtain the second-moment stability condition $(1-\\rho )\\beta +\\rho \\varphi <1$ or equivalently $\\ln [(1-\\rho )\\beta +\\rho \\varphi ]<0$ .", "In comparison to this condition, the stability condition () in Corollary REF is tighter.", "This is because $(1-\\rho )\\ln \\beta +\\rho \\ln \\varphi <\\ln [(1-\\rho )\\beta +\\rho \\varphi ]$ by Jensen's inequality, since $\\beta <\\varphi $ and $\\rho \\notin \\lbrace 0,1\\rbrace $ ." ], [ "Feedback Gain Design for Event-Triggered Control ", "In the following, we outline a numerical method for designing the feedback gain $K\\in \\mathbb {R}^{m\\times n}$ , as well as the positive-definite matrix $P\\in \\mathbb {R}^{n\\times n}$ and the scalar $\\beta \\in (0,1)$ used in the event-triggered control law (REF ), (REF ).", "Corollary 3.8 Consider the linear dynamical system (REF ).", "Suppose that the process $\\lbrace l(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ characterizing packet exchange failures in the network satisfies (REF ) with scalar $\\rho \\in [0,1]$ .", "If there exist a matrix $M\\in \\mathbb {R}^{m\\times n}$ , a positive-definite matrix $Q\\in \\mathbb {R}^{n\\times n}$ , and scalars $\\beta \\in (0,1),$ $\\varphi \\in [1,\\infty )$ such that (), $\\left[\\begin{array}{cc}\\beta Q & \\left(AQ+BM\\right)^{\\mathrm {T}}\\\\AQ+BM & Q\\end{array}\\right] & \\ge 0,\\\\\\left[\\begin{array}{cc}\\varphi Q & (AQ){}^{\\mathrm {T}}\\\\AQ & Q\\end{array}\\right] & \\ge 0,$ hold, then the event-triggered control law (REF ), (REF ) with $P\\triangleq Q^{-1}$ and $K\\triangleq MQ^{-1}$ guarantees almost sure asymptotic stability of the zero solution $x(t)\\equiv 0$ of the closed-loop system dynamics.", "Using Schur complements (see [52]), we transform (REF ) and (), respectively, into $\\beta Q-\\left(AQ+BM\\right)^{\\mathrm {T}}Q^{-1}\\left(AQ+BM\\right) & \\ge 0,\\\\\\varphi Q-(AQ)^{\\mathrm {T}}Q^{-1}AQ & \\ge 0.$ By multiplying both sides of (REF ) and () from left and right by $Q^{-1}$ , we obtain (REF ) and () with $P=Q^{-1}$ and $K=MQ^{-1}$ .", "Thus, the result follows from Theorem REF .", "Figure: Region for β∈(0,1)\\beta \\in (0,1) and ϕ∈[1,∞)\\varphi \\in [1,\\infty ) that satisfy() for ρ=0.4\\rho =0.4We remark that the matrix inequalities (REF ) and () are linear in $M\\in \\mathbb {R}^{m\\times n}$ and $Q\\in \\mathbb {R}^{n\\times n}$ for fixed $\\beta \\in (0,1)$ and $\\varphi \\in [1,\\infty )$ .", "In our method we seek feasible solutions $M$ and $Q$ for linear matrix inequalities (REF ) and () by iterating over a set of values for $\\beta \\in (0,1)$ and $\\varphi \\in [1,\\infty )$ restricted by the condition ().", "It is however noted that we do not need to search $\\beta $ and $\\varphi $ in the entire range characterized by ().", "It turns out to be sufficient to check for larger values of $\\beta $ and $\\varphi $ that are close to the boundary of the range identified by $(1-\\rho )\\ln \\beta +\\rho \\ln \\varphi =0$ .", "Specifically, we set $\\Delta >0$ as a small positive real number, and then we iterate over a set of values for $\\beta $ in the range $(0,e^{-\\frac{\\Delta }{1-\\rho }}]$ to look for feasible solutions $M$ and $Q$ for the linear matrix inequalities (REF ) and () with $\\varphi =e^{-\\frac{(1-\\rho )\\ln \\beta +\\Delta }{\\rho }}$ .", "In this approach, we use only $\\beta \\in (0,1)$ , $\\varphi \\in [1,\\infty )$ that are on the curve $(1-\\rho )\\ln \\beta +\\rho \\ln \\varphi =-\\Delta $ .", "We illustrate this curve with the solid red line in Fig.", "REF , where the shaded region corresponds to $\\beta $ and $\\varphi $ that satisfy ().", "Note that picking smaller values for $\\Delta >0$ moves the curve towards the boundary.", "Also, there is no conservatism in not considering $\\beta $ and $\\varphi $ such that $(1-\\rho )\\ln \\beta +\\rho \\ln \\varphi <-\\Delta $ .", "This is because if there exist $M$ and $Q$ that satisfy (REF ) and () for values $\\beta =\\tilde{\\beta }$ and $\\varphi =\\tilde{\\varphi }$ , then the same $M$ and $Q$ satisfy (REF ) and () also for larger values $\\beta >\\tilde{\\beta }$ and $\\varphi >\\tilde{\\varphi }$ ." ], [ "Attacker's Perspective ", "In order to design cyber-secure control systems, it is essential to understand the risks in networked operation.", "In this regard, it may be useful to consider the control problem from the perspective of an attacker.", "An attacker knowledgeable about the networked control system may generate an attack strategy that causes sufficiently frequent packet losses which can result in instability of the closed-loop dynamics.", "However, the attacker may want to keep the number of attacks as small as possible.", "One reason in the case of jamming attacks is that monitoring the channel and producing jamming signals consume energy [4].", "Moreover, the attacks should be kept minimal to make them less detectable by the system operators.", "In this section, we address the question of finding conditions under which the state diverges almost surely (i.e., $\\mathbb {P}[\\lim _{t\\rightarrow \\infty }\\Vert x(t)\\Vert =\\infty ]=1$ ).", "For the discussions and results presented in this section, we consider the case where the plant and the controller attempt to exchange packets at all time instants, that is, $\\tau _{i}=i$ , $i\\in \\mathbb {N}_{0}$ .", "In the event-triggered scheme, this corresponds to the case with $\\theta =1$ in (REF ).", "First, we obtain a lower-bound for the long run average number of packet exchange failures by utilizing a characterization that is complementary to (REF ) in Assumption REF .", "Lemma 4.1 If there exists a scalar $\\sigma \\in [0,1]$ such that $\\sum _{k=1}^{\\infty }\\mathbb {P}[L(k)<\\sigma k] & <\\infty ,$ where $L(k)\\triangleq \\sum _{i=0}^{k-1}l(t)$ , $k\\in \\mathbb {N}$ , then $\\liminf _{k\\rightarrow \\infty }\\frac{L(k)}{k} & \\ge \\sigma ,$ almost surely.", "Using (REF ), we obtain $\\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{t=0}^{k-1}(1-l(t))>(1-\\sigma )k]<\\infty $ , and hence, by Borel-Cantelli lemma (see [46]), $\\limsup _{k\\rightarrow \\infty }\\frac{1}{k}\\sum _{t=0}^{k-1}(1-l(t)) & \\le 1-\\sigma ,$ almost surely.", "Noting that $\\limsup _{k\\rightarrow \\infty }\\frac{1}{k}\\sum _{t=0}^{k-1}(1-l(t))=1-\\liminf _{k\\rightarrow \\infty }\\frac{1}{k}\\sum _{t=0}^{k-1}l(t)$ , we obtain (REF ) from (REF ).", "The inequality (REF ) can be considered as a complementary characterization to (REF ) in Assumption REF .", "Observe that by Lemma REF , $\\rho \\in [0,1]$ in (REF ) characterizes an upper-bound on the long run average number of packet exchange failures.", "In comparison, as implied by (REF ), the scalar $\\sigma \\in [0,1]$ in (REF ) provides a lower-bound on the long run average number of packet exchange failures.", "Notice that a large $\\sigma \\in [0,1]$ in (REF ) indicates that due to random losses and malicious attacks, packet exchange failures happen statistically frequently.", "In such cases, the overall dynamics may become unstable.", "As mentioned earlier, since malicious attacks often consume energy, the attacker would want to disrupt normal operation and cause unstable behavior with a fewer number of attacks.", "In the case of jamming attacks, recent works considered game-theoretic methods to investigate the optimal strategy of an attacker when the jamming energy is a constraint in the problem [13] and when it is part of the attacker's cost function [53], [12].", "The results obtained there are not directly applicable here, as we investigate sufficient attack rates that cause divergence of the state rather than finding optimal attack strategies.", "Our next result indicates how frequently the attacker should cause packet exchange failures to induce instability.", "Theorem 4.2 Consider the linear networked control system (REF ), (REF ) where packet exchanges between the plant and the controller are attempted at all time instants.", "Suppose that the process $\\lbrace l(t)\\in \\lbrace 0,1\\rbrace \\rbrace _{t\\in \\mathbb {N}_{0}}$ characterizing packet exchange failures in the network satisfies (REF ) with $\\sigma \\in [0,1]$ .", "If there exist a positive-definite matrix $\\hat{P}\\in \\mathbb {R}^{n\\times n}$ and scalars $\\hat{\\beta }\\in (0,1),$ $\\hat{\\varphi }\\in [1,\\infty )$ such that $& \\left(A+BK\\right)^{\\mathrm {T}}\\hat{P}\\left(A+BK\\right)-\\hat{\\beta }\\hat{P}\\ge 0,\\\\& A^{\\mathrm {T}}\\hat{P}A-\\hat{\\varphi }\\hat{P}\\ge 0,\\\\& (1-\\sigma )\\ln \\hat{\\beta }+\\sigma \\ln \\hat{\\varphi }>0,$ then $\\lim _{t\\rightarrow \\infty }\\Vert x(t)\\Vert =\\infty $ , almost surely.", "Consider the Lyapunov-like function $V(\\cdot )$ given by $V(x)\\triangleq x^{\\mathrm {T}}\\hat{P}x$ , $x\\in \\mathbb {R}^{n}$ .", "For the case $\\tau _{i}=i$ , $i\\in \\mathbb {N}_{0}$ , by (REF ), (REF ), we have $V(x(t+1)) & =x^{\\mathrm {T}}(t)\\left(A+\\left(1-l(t)\\right)BK\\right)^{\\mathrm {T}}\\hat{P}\\nonumber \\\\& \\quad \\,\\cdot \\left(A+\\left(1-l(t)\\right)BK\\right)x(t),\\,t\\in \\mathbb {N}_{0}.$ From (REF ), (), and (REF ), this can be bounded by $V(x(t+1)) & \\ge (1-l(t))\\hat{\\beta }V(x(t))+l(t)\\hat{\\varphi }V(x(t))$ for $t\\in \\mathbb {N}_{0}$ .", "Now, let $\\eta (k)\\triangleq \\prod _{t=0}^{k-1}\\left((1-l(t))\\hat{\\beta }+l(t)\\hat{\\varphi }\\right)$ .", "It follows from (REF ) that $V(x(k)) & \\ge \\eta (k)V(x(0))$ for $k\\in \\mathbb {N}$ .", "Furthermore, since $\\ln \\left[(1-q)\\hat{\\beta }+q\\hat{\\varphi }\\right]=(1-q)\\ln \\hat{\\beta }+q\\ln \\hat{\\varphi }$ for $q\\in \\lbrace 0,1\\rbrace $ , we have $\\ln \\eta (k) & =\\sum _{t=0}^{k-1}(1-l(t))\\ln \\hat{\\beta }+\\sum _{t=0}^{k-1}l(t)\\ln \\hat{\\varphi }$ Now since $\\hat{\\beta }\\in (0,1)$ , we have $\\ln \\hat{\\beta }<0$ , and hence by (REF ), $& \\liminf _{k\\rightarrow \\infty }\\frac{1}{k}\\sum _{t=0}^{k-1}(1-l(t))\\ln \\hat{\\beta }=(\\ln \\hat{\\beta })\\limsup _{k\\rightarrow \\infty }\\frac{1}{k}\\sum _{t=0}^{k-1}(1-l(t))\\\\& \\,\\,=(\\ln \\hat{\\beta })(1-\\liminf _{k\\rightarrow \\infty }\\frac{1}{k}\\sum _{t=0}^{k-1}l(t))\\ge (\\ln \\hat{\\beta })(1-\\sigma ).$ Furthermore, since $\\ln \\hat{\\varphi }\\ge 0$ , it follows from (REF ) that $\\liminf _{k\\rightarrow \\infty }\\frac{1}{k}\\sum _{t=0}^{k-1}l(t)\\ln \\hat{\\varphi }\\ge \\sigma \\ln \\hat{\\varphi }$ .", "Consequently, by (), $& \\liminf _{k\\rightarrow \\infty }\\frac{\\ln \\eta (k)}{k}\\ge \\liminf _{k\\rightarrow \\infty }\\frac{1}{k}\\sum _{t=0}^{k-1}(1-l(t))\\ln \\hat{\\beta }\\\\& \\quad +\\liminf _{k\\rightarrow \\infty }\\frac{1}{k}\\sum _{t=0}^{k-1}l(t)\\ln \\hat{\\varphi }\\ge (1-\\sigma )\\ln \\hat{\\beta }+\\sigma \\ln \\hat{\\varphi }>0,$ almost surely.", "As a consequence, $\\lim _{k\\rightarrow \\infty }\\ln \\eta (k)=\\infty $ , and hence, $\\lim _{k\\rightarrow \\infty }\\eta (k)=\\infty $ , almost surely.", "Thus, it follows from (REF ) that $\\mathbb {P}[\\lim _{t\\rightarrow \\infty }V(x(t))=\\infty ]=1$ , which implies that $\\lim _{t\\rightarrow \\infty }\\Vert x(t)\\Vert =\\infty $ , almost surely.", "Theorem REF provides sufficient conditions (REF )–() to assess instability of the closed-loop system (REF ), (REF ).", "These conditions are complementary to the stability conditions (REF )–() in Theorem REF .", "This point is further illustrated by focusing on the scalar systems case.", "Example 4.3 Consider the scalar system (REF ) with $A,B\\in \\mathbb {R}$ .", "Then, conditions (REF ), () as well as (REF ), () can be satisfied by $\\hat{P}=P=1$ , $\\hat{\\beta }=\\beta =(A+BK)^{2}$ , and $\\hat{\\varphi }=\\varphi =A^{2}$ .", "Now, if $\\lim _{k\\rightarrow \\infty }\\frac{L(k)}{k}$ exists and is a fixed constant, we can set $\\sigma =\\rho =\\lim _{k\\rightarrow \\infty }\\frac{L(k)}{k}$ in () and () to obtain the stability condition $(1-\\lim _{k\\rightarrow \\infty }\\frac{L(k)}{k})(A+BK)^{2}+\\lim _{k\\rightarrow \\infty }\\frac{L(k)}{k}A^{2} & <0,$ and the instability condition $(1-\\lim _{k\\rightarrow \\infty }\\frac{L(k)}{k})(A+BK)^{2}+\\lim _{k\\rightarrow \\infty }\\frac{L(k)}{k}A^{2} & >0.$ The limit $\\lim _{k\\rightarrow \\infty }\\frac{L(k)}{k}$ is a fixed constant for example when the packet losses are Bernoulli-type or periodic.", "In those cases, (REF ) and (REF ) indicate that Theorems REF and REF provide tight stability/instability conditions for scalar systems.", "On the other hand, for multi-dimensional systems, scalars $\\beta $ and $\\hat{\\beta }$ as well as $\\varphi $ and $\\hat{\\varphi }$ may not always be selected equal to obtain tight results.", "Furthermore, under random packet losses and malicious attacks, $\\lim _{k\\rightarrow \\infty }\\frac{L(k)}{k}$ may not always exist and hence there may be a discrepancy between $\\rho $ and $\\sigma $ in (REF ) and (REF ).", "Proposition REF below provides a range of values for $\\sigma $ that satisfy (REF ) in the case where the network faces random and malicious packet losses.", "Proposition 4.4 Consider the packet exchange failure indicator process $\\lbrace l(t)\\in \\lbrace 0,1\\rbrace \\rbrace _{t\\in \\mathbb {N}_{0}}$ given by (REF ) where $\\lbrace l_{\\mathrm {R}}(t)\\in \\lbrace 0,1\\rbrace \\rbrace _{t\\in \\mathbb {N}_{0}}$ and $\\lbrace l_{\\mathrm {M}}(t)\\in \\lbrace 0,1\\rbrace \\rbrace _{t\\in \\mathbb {N}_{0}}$ are mutually independent.", "Suppose there exists $\\sigma _{\\mathrm {M}}\\in (0,1)$ such that $\\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{t=0}^{k-1}l_{\\mathrm {M}}(t)<\\sigma _{\\mathrm {M}}k] & <\\infty .$ Furthermore, suppose $\\lbrace l_{\\mathrm {R}}(t)\\in \\lbrace 0,1\\rbrace \\rbrace _{t\\in \\mathbb {N}_{0}}$ satisfies () with $p_{0}\\in (0,1)$ .", "Then (REF ) holds for all $\\sigma \\in (0,1-p_{0}(1-\\sigma _{\\mathrm {M}}))$ .", "First, by using (REF ), we obtain $& \\mathbb {P}[\\sum _{t=0}^{k-1}l(t)<\\sigma k]=\\mathbb {P}[\\sum _{t=0}^{k-1}(1-l(t))>(1-\\sigma )k]\\nonumber \\\\& \\,\\,=\\mathbb {P}[\\sum _{t=0}^{k-1}(1-l_{\\mathrm {R}}(t))(1-l_{\\mathrm {M}}(t))>(1-\\sigma )k],\\,\\,k\\in \\mathbb {N}.$ Furthermore, it follows from (REF ) that $\\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{t=0}^{k-1}(1-l_{\\mathrm {M}}(t))>(1-\\sigma _{\\mathrm {M}})k]=\\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{t=0}^{k-1}l_{\\mathrm {M}}(i)<\\sigma _{\\mathrm {M}}k]<\\infty .$ Hence, $\\lbrace \\chi (i)\\triangleq \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ defined by $\\chi (i)=1-l_{\\mathrm {\\mathrm {M}}}(i),i\\in \\mathbb {N}_{0}$ , satisfies () with $\\tilde{w}=1-\\sigma _{\\mathrm {M}}<1$ .", "Furthermore, $\\lbrace \\xi (i)\\triangleq \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ defined by $\\xi (i)=1-l_{\\mathrm {R}}(i),i\\in \\mathbb {N}_{0}$ , satisfies (REF ) with $\\tilde{p}=p_{0}\\in (0,1)$ .", "We then have from Lemma REF that $& \\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{t=0}^{k-1}(1-l_{\\mathrm {R}}(t))(1-l_{\\mathrm {M}}(t))>\\varrho k]<\\infty ,$ for all $\\varrho \\in (p_{0}(1-\\sigma _{\\mathrm {M}}),1-\\sigma _{\\mathrm {M}})$ .", "In the rest of the proof, we will show that (REF ) holds also for $\\varrho \\in [1-\\sigma _{\\mathrm {M}},1)$ .", "To this end, let $\\varrho ^{\\prime }\\triangleq \\frac{p_{0}(1-\\sigma _{\\mathrm {M}})+1-\\sigma _{\\mathrm {M}}}{2}$ .", "Since $\\varrho ^{\\prime }\\in (p_{0}(1-\\sigma _{\\mathrm {M}}),1-\\sigma _{\\mathrm {M}})$ , by (REF ), we get $\\lambda \\triangleq \\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{t=0}^{k-1}(1-l_{\\mathrm {R}}(t))(1-l_{\\mathrm {M}}(t))>\\varrho ^{\\prime }k]<\\infty $ .", "Furthermore, for all $\\varrho \\in [1-\\sigma _{\\mathrm {M}},1)$ we have $\\varrho \\ge \\varrho ^{\\prime }$ and hence $& \\mathbb {P}[\\sum _{t=0}^{k-1}(1-l_{\\mathrm {R}}(t))(1-l_{\\mathrm {M}}(t))>\\varrho k]\\\\& \\quad \\le \\mathbb {P}[\\sum _{t=0}^{k-1}(1-l_{\\mathrm {R}}(t))(1-l_{\\mathrm {M}}(t))>\\varrho ^{\\prime }k],\\quad k\\in \\mathbb {N}.$ Thus, for $\\varrho \\in [1-\\sigma _{\\mathrm {M}},1)$ , $\\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{t=0}^{k-1}(1-l_{\\mathrm {R}}(t))(1-l_{\\mathrm {M}}(t))>\\varrho k]\\le \\lambda <\\infty $ .", "Therefore, (REF ) holds for all $\\varrho \\in (p_{0}(1-\\sigma _{\\mathrm {M}}),1)=(p_{0}(1-\\sigma _{\\mathrm {M}}),1-\\sigma _{\\mathrm {M}})\\cup [1-\\sigma _{\\mathrm {M}},1)$ .", "Now since $\\sigma =1-\\varrho $ , it follows from (REF ) that (REF ) holds for all $\\sigma \\in (0,1-p_{0}(1-\\sigma _{\\mathrm {M}}))$ .", "Proposition REF shows that when malicious attacks are independent of the random losses and they satisfy (REF ), the inequalities (REF ) and (REF ) (due to Lemma  REF ) hold for a range of values of $\\sigma $ .", "This result indicates the effects of independent random packet losses and malicious attacks on the asymptotic ratio of packet exchange attempt failures over all attempts.", "The next result is concerned with the scenarios where random packet losses and malicious attacks need not be independent.", "Proposition 4.5 Consider the packet exchange failure indicator process $\\lbrace l(t)\\in \\lbrace 0,1\\rbrace \\rbrace _{t\\in \\mathbb {N}_{0}}$ given by (REF ).", "Suppose there exists $\\sigma _{\\mathrm {M}}\\in (0,1)$ such that (REF ) holds.", "Furthermore, suppose $\\lbrace l_{\\mathrm {R}}(t)\\in \\lbrace 0,1\\rbrace \\rbrace _{t\\in \\mathbb {N}_{0}}$ satisfies () with $p_{0}\\in (0,1)$ .", "Then (REF ) holds for all $\\sigma \\in (0,\\max \\lbrace 1-p_{0},\\sigma _{\\mathrm {M}}\\rbrace )$ .", "We will show that (REF ) holds for the cases: 1) $\\max \\lbrace 1-p_{0},\\sigma _{\\mathrm {M}}\\rbrace =1-p_{0}$ and 2) $\\max \\lbrace 1-p_{0},\\sigma _{\\mathrm {M}}\\rbrace =\\sigma _{\\mathrm {M}}$ .", "First, if $\\max \\lbrace 1-p_{0},\\sigma _{\\mathrm {M}}\\rbrace =1-p_{0}$ , then noting that $\\sum _{t=0}^{k-1}l_{\\mathrm {R}}(t)\\le \\sum _{t=0}^{k-1}l(t)$ , we obtain $\\mathbb {P}[\\sum _{t=0}^{k-1}l(t)<\\sigma k] & =\\mathbb {P}[\\sum _{t=0}^{k-1}(1-l(t))>(1-\\sigma )k]\\nonumber \\\\& \\le \\mathbb {P}[\\sum _{t=0}^{k-1}(1-l_{\\mathrm {R}}(t))>(1-\\sigma )k],$ for $k\\in \\mathbb {N}$ .", "Now, $\\lbrace \\chi (i)\\triangleq \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ with $\\chi (i)=1,i\\in \\mathbb {N}_{0}$ , satisfies () with $\\tilde{w}=1$ .", "Furthermore, $\\lbrace \\xi (i)\\triangleq \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ with $\\xi (i)=1-l_{\\mathrm {R}}(i),i\\in \\mathbb {N}_{0}$ , satisfies (REF ) with $\\tilde{p}=p_{0}\\in (0,1)$ .", "Since $1-\\sigma >p_{0}$ , we have from Lemma REF that $\\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{t=0}^{k-1}(1-l_{\\mathrm {R}}(t))>(1-\\sigma )k]<\\infty $ .", "Hence, by (REF ), we have (REF ).", "Next, if $\\max \\lbrace 1-p_{0},\\sigma _{\\mathrm {M}}\\rbrace =\\sigma _{\\mathrm {M}}$ , then since $\\sum _{t=0}^{k-1}l_{\\mathrm {M}}(t)\\le \\sum _{t=0}^{k-1}l(t)$ and $\\sigma <\\sigma _{\\mathrm {M}}$ , we get $\\mathbb {P}[\\sum _{t=0}^{k-1}l(t)<\\sigma k] & \\le \\mathbb {P}[\\sum _{t=0}^{k-1}l_{\\mathrm {M}}(t)<\\sigma _{\\mathrm {M}}k],\\quad k\\in \\mathbb {N}.$ Consequently, (REF ) and (REF ) imply (REF ).", "Proposition REF provides a range for $\\sigma $ in (REF ) when we consider the case where random packet losses and malicious attacks may be dependent.", "This range is smaller in comparison to the one provided in Proposition REF for the independent case.", "This is because Proposition REF deals with scenarios including the worst case from the perspective of the attacker.", "In that scenario, the malicious attacks and random packet losses happen at the same time instants, and hence, the statistical frequency of the overall packet exchange failures cannot exceed the maximum of the frequencies of malicious attacks and random packet losses.", "We remark that there are other scenarios where the attacks depend on the random packet losses.", "For instance, the attacker may intentionally avoid attacking when there is already a random packet loss.", "This scenario is characterized in the mathematical setting by $l_{\\mathrm {R}}(t)l_{\\mathrm {M}}(t)=0$ , $t\\in \\mathbb {N}_{0}$ .", "For this scenario, the following proposition provides a range of $\\sigma $ that satisfy (REF ).", "Proposition 4.6 Consider the packet exchange failure indicator process $\\lbrace l(t)\\in \\lbrace 0,1\\rbrace \\rbrace _{t\\in \\mathbb {N}_{0}}$ given by (REF ).", "Suppose $\\lbrace l_{\\mathrm {R}}(t)\\in \\lbrace 0,1\\rbrace \\rbrace _{t\\in \\mathbb {N}_{0}}$ satisfies () with $p_{0}\\in (0,1)$ .", "Furthermore, suppose $l_{\\mathrm {R}}(t)l_{\\mathrm {M}}(t)=0$ , $t\\in \\mathbb {N}_{0}$ , and there exists $\\sigma _{\\mathrm {M}}\\in (0,1)$ such that (REF ) holds.", "If $1-p_{0}+\\sigma _{\\mathrm {M}}\\le 1$ , then (REF ) holds for all $\\sigma \\in (0,1-p_{0}+\\sigma _{\\mathrm {M}})$ .", "First let $\\epsilon \\triangleq 1-p_{0}+\\sigma _{\\mathrm {M}}-\\sigma $ , and define $\\sigma _{1}\\triangleq \\max \\lbrace 0,1-p_{0}-\\frac{\\epsilon }{2}\\rbrace $ , $\\sigma _{2}\\triangleq \\max \\lbrace 0,\\sigma _{\\mathrm {M}}-\\frac{\\epsilon }{2}\\rbrace $ .", "Note that $\\sigma _{1}+\\sigma _{2}\\ge \\sigma $ .", "Now since $l_{\\mathrm {R}}(t)l_{\\mathrm {M}}(t)=0$ , $t\\in \\mathbb {N}_{0}$ , we have from (REF ) that $\\sum _{t=0}^{k-1}l(t)=\\sum _{t=0}^{k-1}l_{\\mathrm {R}}(t)+\\sum _{t=0}^{k-1}l_{\\mathrm {M}}(t)$ .", "As a result $& \\mathbb {P}[\\sum _{t=0}^{k-1}l(t)<\\sigma k]=\\mathbb {P}[\\sum _{t=0}^{k-1}l_{\\mathrm {R}}(t)+\\sum _{t=0}^{k-1}l_{\\mathrm {M}}(t)<\\sigma k]\\nonumber \\\\& \\,\\,\\le \\mathbb {P}[\\sum _{t=0}^{k-1}l_{\\mathrm {R}}(t)+\\sum _{t=0}^{k-1}l_{\\mathrm {M}}(t)<\\sigma _{1}k+\\sigma _{2}k]\\nonumber \\\\& \\,\\,\\le \\mathbb {P}[\\sum _{t=0}^{k-1}l_{\\mathrm {R}}(t)<\\sigma _{1}k]+\\mathbb {P}[\\sum _{t=0}^{k-1}l_{\\mathrm {M}}(t)<\\sigma _{2}k],\\,\\,k\\in \\mathbb {N}.$ If $\\sigma _{1}=0$ , then $\\mathbb {P}[\\sum _{t=0}^{k-1}l_{\\mathrm {R}}(t)<\\sigma _{1}k]=0$ , and hence $\\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{t=0}^{k-1}l_{\\mathrm {R}}(t)<\\sigma _{1}k]=0<\\infty $ .", "If, on the other hand, $\\sigma _{1}>0$ , then we can utilize Lemma REF .", "Specifically, $\\lbrace \\chi (i)\\triangleq \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ with $\\chi (i)=1,i\\in \\mathbb {N}_{0}$ , satisfies () with $\\tilde{w}=1$ .", "Furthermore, $\\lbrace \\xi (i)\\triangleq \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ with $\\xi (i)=1-l_{\\mathrm {R}}(i),i\\in \\mathbb {N}_{0}$ , satisfies (REF ) with $\\tilde{p}=p_{0}\\in (0,1)$ .", "Since $\\sigma _{1}>0$ , it means that $\\sigma _{1}=1-p_{0}-\\frac{\\epsilon }{2}$ .", "Now, since $\\epsilon >0$ , we have $(1-\\sigma _{1})\\in (p_{0},1)$ .", "Consequently, we obtain from Lemma REF that $\\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{t=0}^{k-1}(1-l_{\\mathrm {R}}(t))>(1-\\sigma _{1})k]<\\infty $ , and hence, $& \\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{t=0}^{k-1}l_{\\mathrm {R}}(t)<\\sigma _{1}k]\\nonumber \\\\& \\quad =\\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{t=0}^{k-1}(1-l_{\\mathrm {R}}(t))>(1-\\sigma _{1})k]<\\infty .$ Similarly, if $\\sigma _{2}=0$ , then $\\mathbb {P}[\\sum _{t=0}^{k-1}l_{\\mathrm {M}}(t)<\\sigma _{2}k]=0$ , and hence $\\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{t=0}^{k-1}l_{\\mathrm {M}}(t)<\\sigma _{2}k]=0<\\infty $ .", "On the other hand, if $\\sigma _{2}=\\sigma _{\\mathrm {M}}-\\frac{\\epsilon }{2}>0$ , since $\\epsilon >0$ , we have $\\sigma _{2}<\\sigma _{\\mathrm {M}}$ .", "Thus, $\\mathbb {P}[\\sum _{t=0}^{k-1}l_{\\mathrm {M}}(t)<\\sigma _{2}k]\\le \\mathbb {P}[\\sum _{t=0}^{k-1}l_{\\mathrm {M}}(t)<\\sigma _{\\mathrm {M}}k]$ .", "It then follows from (REF ) that $& \\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{t=0}^{k-1}l_{\\mathrm {M}}(t)<\\sigma _{2}k]\\le \\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{t=0}^{k-1}l_{\\mathrm {M}}(t)<\\sigma _{\\mathrm {M}}k]<\\infty .$ Finally, (REF ) follows from (REF )–(REF ).", "An attacker that is knowledgeable about the random packet losses in the network may avoid placing malicious attacks when random packet losses occur.", "Proposition REF provides a range of values of $\\sigma $ such that the inequality (REF ) holds when the attacker follows this strategy.", "Compared to the case where attacks and random packet losses are independent, this strategy would increase the overall number of packet exchange failures, even though the number of attacks may be the same.", "The reason is that in the independent case, the attacks and random packet losses may occasionally happen at the same time, reducing the total packet failure count.", "Noe that the range of $\\sigma $ in Proposition REF is larger than that in Proposition REF , where the attacks and random packet losses are independent, even though in both results the malicious attacks satisfy (REF ) with the same $\\sigma _{\\mathrm {M}}\\in (0,1)$ .", "In Section -B, we discuss and compare two attack strategies independent/dependent on random packet losses.", "Both strategies cause instability for certain feedback gain and event-triggering mechanism parameters.", "It is important to note that particular choices of the controller parameters may result in instability when $\\sigma \\in [0,1]$ is large.", "If the packet exchange failures are known to happen statistically frequently, that is, if $\\sigma $ is large, then the feedback gain $K$ and the event-triggering mechanism parameters $\\beta $ and $P$ should be redesigned to ensure stability.", "In such cases, Theorem REF and Corollary REF can be employed with $\\rho \\ge \\sigma $ that satisfies (REF ) or (REF )." ], [ "Numerical Examples ", "In this section we present numerical examples to illustrate our results provided in Sections –." ], [ "A) Example 1", "We consider the system (REF ) with $A\\triangleq \\left[\\begin{array}{cc}1 & 0.1\\\\-0.5 & 1.1\\end{array}\\right], & \\quad B\\triangleq \\left[\\begin{array}{c}0.1\\\\1.2\\end{array}\\right].$ We use the event-triggering control law (REF ), (REF ) for stabilization of (REF ) over a network that faces independent random packet losses and malicious attacks.", "Specifically, random packet losses are assumed to be characterized by the Markov chain $\\lbrace l_{\\mathrm {R}}(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ with initial distribution $\\vartheta _{0}=0$ , $\\vartheta _{1}=1$ , and transition probabilities $p_{0,1}(i)\\triangleq 0.2+0.03\\sin ^{2}(0.1i),$ $p_{1,1}(i)\\triangleq 0.2+0.03\\cos ^{2}(0.1i)$ , and $p_{q,0}(i)=1-p_{q,1}(i)$ , $q\\in \\lbrace 0,1\\rbrace $ , $i\\in \\mathbb {N}_{0}$ .", "Note that $\\lbrace l_{\\mathrm {R}}(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ satisfies (REF ) and () with $p_{1}=0.23$ and $p_{0}=0.8$ .", "Furthermore, the network is subject to jamming attacks that is independent of $\\lbrace l_{\\mathrm {R}}(i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ and satisfies (REF ) with $\\kappa =2$ and $\\tau =5$ .", "By Lemma REF , (REF ) holds with $\\rho _{\\mathrm {M}}=0.21$ since $\\rho _{\\mathrm {M}}=0.21>\\frac{1}{\\tau }=0.2$ .", "Furthermore, note that $p_{1}+p_{0}\\rho _{\\mathrm {M}}<0.4$ .", "Hence, it follows from Proposition REF that for $\\rho =0.4$ , (REF ) of Assumption REF holds, which implies (REF ) through Lemma REF .", "We designed the controller based on the procedure in Section REF and obtained the matrices $Q & =\\left[\\begin{array}{cc}0.618 & -2.119\\\\-2.119 & 28.214\\end{array}\\right],\\,\\,M=\\left[\\begin{array}{cc}0.202 & -20.405\\end{array}\\right],$ and scalars $\\beta =0.55$ , $\\varphi =2.4516$ satisfy (REF ), (), and () with $\\rho =0.4$ .", "Hence, it follows from Corollary REF that the event-triggered control law (REF ), (REF ) with $P=Q^{-1}$ and $K=MQ^{-1}$ guarantees almost sure asymptotic stabilization.", "Figure: Sample paths of the state normFigure: A sample path of Lyapunov-like function V(·)V(\\cdot )We generated 250 sample state trajectories using the same initial condition $x_{0}=\\left[1,\\,1\\right]^{\\mathrm {T}}$ and the event-triggering mechanism parameter $\\theta =1000$ , but with different sample paths for $l_{\\mathrm {R}}(\\cdot )$ and $l_{\\mathrm {M}}(\\cdot )$ .", "We can check in Fig.", "REF that all state trajectories go to the origin.", "The same is true for the Lyapunov-like function $V(x(t))$ .", "We show a single sample trajectory of $V(\\cdot )$ in Fig.", "REF .", "The Lyapunov-like function $V(\\cdot )$ converges to zero, but notice that it is not monotonically decreasing.", "The Lyapunov-like function $V(\\cdot )$ increases in two situations.", "First, when packet exchange attempts fail, $V(\\cdot )$ may grow and take a larger value at the next packet exchange attempt instant due to unstable dynamics of the uncontrolled system.", "Second, $V(\\cdot )$ may also increase some time after a successful packet exchange between the plant and the controller.", "This is because the constant control input updated with the packet exchange becomes ineffective after some time.", "Note that eventually a new packet exchange attempt is triggered before $V(\\cdot )$ leaves the bound identified in the event-triggering condition (REF ).", "Our goal in this example is to illustrate effects of different attack strategies discussed in Section .", "Here, we consider a scalar linear system (REF ) with $A=2$ and $B=1$ .", "Its initial state is set to $x_{0}=1$ .", "Furthermore, the feedback gain and the event-triggering mechanism parameters in (REF ) and (REF ) are given by $K=-1.75$ , $\\beta =0.0625$ , $P=1$ .", "We set the packet exchange events to be triggered at all time instants.", "This is done with $\\theta =1$ in (REF ).", "The random packet losses in the network are characterized by the Markov chain $\\lbrace l_{\\mathrm {R}}(t)\\in \\lbrace 0,1\\rbrace \\rbrace _{t\\in \\mathbb {N}_{0}}$ with initial distribution $\\vartheta _{0}=0$ , $\\vartheta _{1}=1$ , and transition probabilities $p_{0,1}(t)\\triangleq 0.4+0.01\\cos (0.1t),$ $p_{1,1}(t)\\triangleq 0.4+0.01\\sin (0.1t)$ , and $p_{q,0}(t)=1-p_{q,1}(t)$ , $q\\in \\lbrace 0,1\\rbrace $ , $t\\in \\mathbb {N}_{0}$ .", "Note that $l_{\\mathrm {R}}(\\cdot )$ satisfies (REF ) and () with $p_{1}=0.41$ and $p_{0}=0.61$ .", "We consider two attack strategies described by (REF ) and discuss stability properties of the closed-loop system.", "Figure: Sample paths of lnV(x(·))\\ln V(x(\\cdot )) under malicious attack ()with τ=3\\tau =3Figure: Sample paths of the average number of packet exchange attempt failures(L(k)/kL(k)/k) under malicious attack () with τ=3\\tau =3(i) Random-Loss-Independent Attack Strategy: We consider the strategy given by $l_{\\mathrm {M}}(0)\\triangleq 0$ , and $l_{\\mathrm {M}}(t)\\triangleq {\\left\\lbrace \\begin{array}{ll}1, & \\mathrm {if}\\,\\,\\sum _{i=0}^{t-1}l_{\\mathrm {M}}(i)\\le \\frac{t+1}{\\tau }-1,\\\\0, & \\mathrm {otherwise},\\end{array}\\right.", "}\\quad & t\\in \\mathbb {N}.$ Note that (REF ) satisfies (REF ) with $\\kappa =0$ .", "In this strategy, the attacker uses the total count of all attacks prior to time $t$ to check whether placing an attack at time $t$ would meet the requirement in (REF ) or not.", "The attacker causes a packet exchange failure at time $t$ if (REF ) still holds at time $t$ (i.e., $\\sum _{i=0}^{t}l_{\\mathrm {M}}(i)\\le \\frac{t+1}{\\tau }$ ).", "Under this strategy, attacks are independent of random packet losses and the attack times become periodic with period $\\tau $ when $\\tau $ is an integer.", "We will assess stability/instability of the closed-loop system with two different values of $\\tau $ .", "Figure: Sample paths of the average number of packet exchange attempt failures(L(k)/kL(k)/k) under malicious attack () with τ=2\\tau =2First, we consider $\\tau =3$ , that is, the attacker prevents packet exchanges once in every 3 steps.", "In this case the closed-loop system is stable despite the attack.", "We use Theorem REF to show stability as follows.", "By Lemma REF , (REF ) holds with $\\rho _{\\mathrm {M}}=0.3334>\\frac{1}{\\tau }=\\frac{1}{3}$ .", "Now, note that $p_{1}+p_{0}\\rho _{\\mathrm {M}}<0.62$ .", "Since $l_{\\mathrm {R}}(\\cdot )$ and $l_{\\mathrm {M}}(\\cdot )$ are independent, it follows from Proposition REF that (REF ) in Assumption REF holds for $\\rho =0.62$ , which implies (REF ) through Lemma REF .", "Further, (REF )–() hold with $P=1$ , $\\beta =0.0625$ , and $\\varphi =4$ .", "By Theorem REF , the event-triggered control law (REF ), (REF ) with $P=1$ , $\\beta =0.0625$ , and $K=-1.75$ guarantees almost sure asymptotic stabilization.", "Fig.", "REF shows 50 sample trajectories of $\\ln V(x(\\cdot ))$ where $V(x(t))\\triangleq x^{2}(t)$ .", "These trajectories are obtained under malicious attack (REF ) but with different sample paths for $\\lbrace l_{\\mathrm {R}}(t)\\in \\lbrace 0,1\\rbrace \\rbrace _{t\\in \\mathbb {N}_{0}}$ .", "Note that all trajectories of $\\ln V(x(\\cdot ))$ approach $-\\infty $ , indicating convergence of the state to 0.", "Moreover, in Fig.", "REF , we show sample trajectories of the average number of packet exchange attempt failures.", "Observe that the long run average number of packet failures is small enough to guarantee stability ($\\limsup _{k\\rightarrow \\infty }\\frac{L(k)}{k}\\le \\rho =0.62$ ).", "Next, we consider (REF ) with $\\tau =2$ , i.e., the malicious attacker prevents every other packet exchange attempt.", "With $\\tau =2$ , the closed-loop system becomes unstable.", "We can show this through Theorem REF as follows.", "First, note that in this case, (REF ) implies (REF ) with $\\sigma _{\\mathrm {M}}\\triangleq 0.49$ .", "To see this, we observe that $\\sum _{t=0}^{k-1}l_{\\mathrm {M}}(t)\\ge \\frac{k}{\\tau }-1=\\frac{k}{2}-1$ for all $k\\in \\mathbb {N}$ .", "Next, using Markov's inequality we obtain $& \\mathbb {P}[\\sum _{t=0}^{k-1}l_{\\mathrm {M}}(t)<\\sigma _{\\mathrm {M}}k]=\\mathbb {P}[\\sum _{t=0}^{k-1}(1-l_{\\mathrm {M}}(t))>(1-\\sigma _{\\mathrm {M}})k]\\\\& \\quad \\le \\mathbb {P}[e^{\\sum _{t=0}^{k-1}(1-l_{\\mathrm {M}}(t))}\\ge e^{(1-\\sigma _{\\mathrm {M}})k}]\\\\& \\quad \\le e^{-(1-\\sigma _{\\mathrm {M}})k}\\mathbb {E}[e^{\\sum _{t=0}^{k-1}(1-l_{\\mathrm {M}}(t))}]\\le e^{-(1-\\sigma _{\\mathrm {M}})k}e^{1+\\frac{k}{2}},$ for $k\\in \\mathbb {N}$ .", "Consequently, since $\\sigma _{\\mathrm {M}}<\\frac{1}{\\tau }=\\frac{1}{2}$ , we have $\\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{t=0}^{k-1}l_{\\mathrm {M}}(t)<\\sigma _{\\mathrm {M}}k]\\le \\sum _{k=1}^{\\infty }e^{-(1-\\sigma _{\\mathrm {M}})k}e^{1+\\frac{k}{2}}=e^{\\frac{1}{2}+\\sigma _{\\mathrm {M}}}(1-e^{\\sigma _{\\mathrm {M}}-\\frac{1}{2}})^{-1}<\\infty ,$ which implies (REF ).", "Now, note that $1-p_{0}(1-\\sigma _{\\mathrm {M}})>0.68$ .", "Hence, by Proposition REF , we have (REF ) for $\\sigma =0.68$ .", "Consequently, by Lemma REF , (REF ) holds for $\\sigma =0.68$ .", "Furthermore, inequalities (REF )–() hold with $\\hat{P}=1$ , $\\hat{\\beta }=0.0625$ , and $\\hat{\\varphi }=4$ .", "It follows from Theorem REF that the closed-loop system with $K=-1.75$ is unstable.", "This example shows that an attacker can destabilize the system by reducing $\\tau $ from 3 to 2 and hence causing higher number of packet losses on average.", "As illustrated in Fig.", "REF , when $\\tau =2$ , in the long run, the average number of packet failures becomes larger ($\\liminf _{k\\rightarrow \\infty }\\frac{L(k)}{k}\\ge \\sigma =0.68$ ) compared to the case with $\\tau =3$ in Fig.", "REF .", "(ii) Selective Attack Strategy: Next, we consider the case where the attacker is knowledgeable about the random packet losses in the network.", "To describe this strategy we let $\\,l_{\\mathrm {M}}(t)\\triangleq {\\left\\lbrace \\begin{array}{ll}1, & \\mathrm {if}\\,\\,l_{\\mathrm {R}}(t)=0\\,\\,\\mathrm {and}\\,\\,\\sum _{i=0}^{t-1}l_{\\mathrm {M}}(i)\\le \\frac{t+1}{\\tau }-1,\\\\0, & \\mathrm {otherwise},\\end{array}\\right.", "}$ for $t\\in \\mathbb {N}$ and $l_{\\mathrm {M}}(0)=0$ .", "This strategy is similar to the one given by (REF ) in that it satisfies (REF ) with $i=t$ and $\\kappa =0$ .", "However, an attacker following (REF ) utilizes random packet loss information at time $t$ , by not placing an attack when $l_{\\mathrm {R}}(t)=1$ (indicating packet failures due to random errors).", "Figure: Sample paths of lnV(x(·))\\ln V(x(\\cdot )) under malicious attack ()with τ=3\\tau =3Figure: Sample paths of the average number of packet exchange attempt failures(L(k)/kL(k)/k) under malicious attack () with τ=3\\tau =3To compare, we set $\\tau =3$ , under which the first strategy (REF ) cannot destabilize the system.", "From the simulations, we notice that with $\\tau =3$ , the selective attack strategy causes the system state to diverge (see Fig.", "REF for 50 sample paths of $\\ln V(x(t))$ with $V(x(t))\\triangleq x^{2}(t)$ ).", "In fact, we observe from Fig.", "REF that (REF ) holds with $\\sigma =0.7$ .", "This $\\sigma $ satisfies condition () of Theorem REF , indicating instability.", "Next, we consider an extension to the attack model in (REF ).", "In this model, the attacker places attacks whenever $l_{\\mathrm {R}}(t)=0$ , $\\sum _{i=0}^{t-1}l_{\\mathrm {M}}(i)\\le \\frac{t+1}{\\tau }-1$ , and $\\ln V(x(t))\\le \\zeta $ .", "From the simulations with $\\zeta =50$ and $\\tau =3$ , we see that the state does not diverge, but the attacker is able to keep it around the level identified with $\\ln V(x(t))=\\zeta $ (see Fig.", "REF ).", "We also observe that in the long run, the average number of packet failures approaches $\\frac{2}{3}$ .", "We remark that $\\frac{2}{3}$ is a critical value for this example in the sense that $\\rho <\\frac{2}{3}$ in () implies convergence of the state, and $\\sigma >\\frac{2}{3}$ in () implies divergence.", "Figure: Sample paths of lnV(x(·))\\ln V(x(\\cdot )) under state-dependent attacksFinally, we show that by redesigning the feedback gain $K$ , we can reensure closed-loop system stability.", "To this end we set $K=-1.9$ .", "It follows from Theorem REF that the event-triggered control law (REF ), (REF ) with $P=1$ , $\\beta =0.01$ , guarantees almost sure asymptotic stability of the closed-loop system.", "This can be checked as follows.", "By Lemma REF , the attack strategy (REF ) and its extension above satisfy (REF ) with $\\rho _{\\mathrm {M}}=0.3334$ .", "Now, note that $p_{1}+\\rho _{\\mathrm {M}}<0.744$ .", "Since $l_{\\mathrm {R}}(\\cdot )$ and $l_{\\mathrm {M}}(\\cdot )$ are not independent, it follows from Proposition REF that (REF ) of Assumption REF holds for $\\rho =0.744$ , which implies (REF ) through Lemma REF .", "Notice that Proposition REF provides a tight bound ($\\rho =0.744$ ) for the average number of packet exchange attempt failures for the attack strategy (REF ) (Fig.", "REF ).", "Moreover, (REF )–() hold with $P=1$ , $\\beta =0.01$ , and $\\varphi =4$ ." ], [ "Conclusion ", "In this paper, we explored control of linear dynamical systems over networks that face random packet losses and malicious attacks.", "We proposed a probabilistic characterization of the evolution of the total number of packet exchange failures.", "Based on this characterization, we obtained sufficient conditions for almost sure asymptotic stabilization and presented a method for finding a stabilizing feedback gain and parameters for our proposed event-triggered control framework.", "Furthermore, to investigate potential cyber risks in networked control operations, we studied the problem from the perspective of an attacker.", "We obtained conditions under which combined effects of random and malicious packet losses can destabilize the closed-loop system.", "The framework developed in this paper has been utilized to investigate the output feedback control problem in [54].", "The probabilistic characterization developed in this paper is utilized there for modeling random and malicious packet losses in transmission of the output information from the plant sensors to the estimator in the controller side.", "A direction for future research is to explore the networked control problem when wireless communication is used.", "There, several communication nodes and routers can be involved, and some of them may be compromised by adversaries.", "Our proposed network model can be incorporated to describe random failures and malicious attacks observed in such problems [55].", "Furthermore, there are other important issues discussed in the networked control literature such as system and measurement noise [20], transmission delays [19], [56], and the modeling of the communication protocol [19], [45].", "Investigation of these issues within our framework remains as a future work.", "Lemma REF below provides upper bounds on the tail probabilities of sums involving a binary-valued Markov chain.", "Lemma F.1 Let $\\lbrace \\xi (i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ be a time-inhomogeneous Markov chain with transition probabilities $p_{q,r}\\colon \\mathbb {N}_{0}\\rightarrow [0,1]$ , $q,r\\in \\lbrace 0,1\\rbrace $ .", "Furthermore, let $\\lbrace \\chi (i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ be a binary-valued process that is independent of $\\lbrace \\xi (i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ .", "Assume $& p_{q,1}(i)\\le \\tilde{p},\\,\\,q\\in \\lbrace 0,1\\rbrace ,\\,\\,i\\in \\mathbb {N}_{0},\\\\& \\sum _{k=1}^{\\infty }\\mathbb {P}[\\sum _{i=0}^{k-1}\\chi (i)>\\tilde{w}k]<\\infty ,$ where $\\tilde{p}\\in (0,1)$ , $\\tilde{w}\\in (0,1]$ .", "We then have for $\\rho \\in (\\tilde{p}\\tilde{w},\\tilde{w})$ , $\\mathbb {P}[\\sum _{i=0}^{k-1}\\xi (i)\\chi (i)>\\rho k] & \\le \\psi _{k},\\quad k\\in \\mathbb {N},$ where $\\psi _{k}\\triangleq \\tilde{\\sigma }_{k}+\\phi ^{-\\rho k+1}\\frac{\\left((\\phi -1)\\tilde{p}+1\\right)^{\\tilde{w}k}-1}{(\\phi -1)\\tilde{p}}$ with $\\phi \\triangleq \\frac{\\frac{\\rho }{\\tilde{w}}(1-\\tilde{p})}{\\tilde{p}(1-\\frac{\\rho }{\\tilde{w}})}$ , $\\tilde{\\sigma }_{k}\\triangleq \\mathbb {P}[\\sum _{i=0}^{k-1}\\chi (i)>\\tilde{w}k],k\\in \\mathbb {N}$ .", "Moreover, $\\sum _{k=1}^{\\infty }\\psi _{k}<\\infty .$ In the proof of Lemma REF , by following the approach used for obtaining Chernoff-type tail distribution inequalities for sums of independent random variables (see Appendix B of [57] and Section 1.9 of [58]) we use Markov's inequality.", "Specifically, let $y$ denote the sum of a number of random variables, and consider the tail probability $\\mathbb {P}[y>\\varsigma ]$ , where $\\varsigma \\in \\mathbb {R}$ .", "In obtaining a bound for this tail probability, Markov's inequality is utilized to obtain $\\mathbb {P}[y & >\\varsigma ]\\le \\mathbb {P}[y\\ge \\varsigma ]=\\mathbb {P}[\\phi ^{y}\\ge \\phi ^{\\varsigma }]\\le \\phi ^{-\\varsigma }\\mathbb {E}[\\phi ^{y}],$ for $\\phi >1$ .", "Chernoff bound is then given by $\\min _{\\phi >1}\\phi ^{-\\varsigma }\\mathbb {E}[\\phi ^{y}]$ .", "In the proof of Lemma REF we do not provide the details of the minimization process to obtain $\\phi $ that gives the optimum bound.", "Instead, we show that the tail probability inequality (REF ) holds with $\\psi _{k},k\\in \\mathbb {N},$ and that $\\sum _{k=1}^{\\infty }\\psi _{k}<\\infty $ .", "To obtain this result, in addition to Markov's inequality, some additional key steps (including Lemma REF below) are also required due to the fact that in Lemma REF we consider sums of (not necessarily independent) random variables composed of the product of states of a time-inhomogeneous Markov chain and a binary-valued process that satisfy ().", "Lemma F.2 Let $\\lbrace \\xi (i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ be an $\\mathcal {F}_{i}$ -adapted binary-valued Markov chain with transition probability functions $p_{q,r}\\colon \\mathbb {N}_{0}\\rightarrow [0,1]$ , $q,r\\in \\lbrace 0,1\\rbrace $ .", "Then for all $\\phi >1$ , $s\\in \\mathbb {N}$ , and $\\tilde{p}\\in [0,1]$ such that $p_{q,1} & (i)\\le \\tilde{p},\\,\\,q\\in \\lbrace 0,1\\rbrace ,\\,\\,i\\in \\mathbb {N}_{0},$ we have $\\mathbb {E}[\\phi ^{\\sum _{j=1}^{s}\\xi (i_{j})}] & \\le \\phi \\left((\\phi -1)\\tilde{p}+1\\right)^{s-1},$ where $i_{1},i_{2},\\ldots ,i_{s}\\in \\mathbb {N}_{0}$ denote indices such that $0\\le i_{1}<i_{2}<\\ldots <i_{s}$ .", "We show by induction.", "First, for the case $s=1$ , $\\mathbb {E}[\\phi ^{\\sum _{j=1}^{s}\\xi (i_{j})}] & =\\mathbb {E}[\\phi ^{\\xi (i_{1})}]\\le \\phi .$ For the case $s=2$ , the random variable $\\xi (i_{1})$ is $\\mathcal {F}_{i_{2}-1}$ -measurable (because $i_{1}\\le i_{2}-1$ ), and thus we have $\\mathbb {E}[\\phi ^{\\sum _{j=1}^{s}\\xi (i_{j})}] & =\\mathbb {E}[\\phi ^{\\xi (i_{1})}\\phi ^{\\xi (i_{2})}]\\nonumber \\\\& =\\mathbb {E}[\\mathbb {E}[\\phi ^{\\xi (i_{1})}\\phi ^{\\xi (i_{2})}\\mid \\mathcal {F}_{i_{2}-1}]]\\nonumber \\\\& =\\mathbb {E}[\\phi ^{\\xi (i_{1})}\\mathbb {E}[\\phi ^{\\xi (i_{2})}\\mid \\mathcal {F}_{i_{2}-1}]].$ Noting that $\\lbrace \\xi (i)\\in \\lbrace 0,1\\rbrace \\rbrace _{i\\in \\mathbb {N}_{0}}$ is a Markov chain, we obtain $\\mathbb {E}[\\phi ^{\\xi (i_{2})}\\mid \\mathcal {F}_{i_{2}-1}]=\\mathbb {E}[\\phi ^{\\xi (i_{2})}\\mid \\xi (i_{2}-1)]$ .", "Consequently, $& \\mathbb {E}[\\phi ^{\\sum _{j=1}^{s}\\xi (i_{j})}]=\\mathbb {E}[\\phi ^{\\xi (i_{1})}\\mathbb {E}[\\phi ^{\\xi (i_{2})}\\mid \\xi (i_{2}-1)]]\\nonumber \\\\& \\,\\,=\\mathbb {E}\\Big [\\phi ^{\\xi (i_{1})}\\Big (\\phi \\mathbb {P}[\\xi (i_{2})=1\\mid \\xi (i_{2}-1)]\\nonumber \\\\& \\,\\,\\quad \\quad +\\mathbb {P}[\\xi (i_{2})=0\\mid \\xi (i_{2}-1)]\\Big )\\Big ]\\nonumber \\\\& \\,\\,=\\mathbb {E}\\Big [\\phi ^{\\xi (i_{1})}\\Big (\\phi \\mathbb {P}[\\xi (i_{2})=1\\mid \\xi (i_{2}-1)]\\nonumber \\\\& \\,\\,\\quad \\quad +1-\\mathbb {P}[\\xi (i_{2})=1\\mid \\xi (i_{2}-1)]\\Big )\\Big ]\\nonumber \\\\& \\,\\,=\\mathbb {E}\\Big [\\phi ^{\\xi (i_{1})}\\Big ((\\phi -1)\\mathbb {P}[\\xi (i_{2})=1\\mid \\xi (i_{2}-1)]+1\\Big )\\Big ].$ Then by using (REF ) and (REF ), we arrive at $& \\mathbb {E}[\\phi ^{\\sum _{j=1}^{s}\\xi (i_{j})}]\\le \\mathbb {E}\\Big [\\phi ^{\\xi (i_{1})}\\Big ((\\phi -1)\\tilde{p}+1\\Big )\\Big ]\\nonumber \\\\& \\quad =\\mathbb {E}[\\phi ^{\\xi (i_{1})}]((\\phi -1)\\tilde{p}+1)\\le \\phi ((\\phi -1)\\tilde{p}+1).$ Hence, we have that (REF ) is satisfied for $s\\in \\lbrace 1,2\\rbrace $ .", "Now, suppose that (REF ) holds for $s=\\tilde{s}>2$ , that is, $\\mathbb {E}[\\phi ^{\\sum _{j=1}^{\\tilde{s}}\\xi (i_{j})}] & \\le \\phi \\left((\\phi -1)\\tilde{p}+1\\right)^{\\tilde{s}-1}.$ We must show that (REF ) holds for $s=\\tilde{s}+1$ .", "Using arguments similar to those used for obtaining (REF )–(REF ), we obtain $\\mathbb {E}[\\phi ^{\\sum _{j=1}^{\\tilde{s}+1}\\xi (i_{j})}] & =\\mathbb {E}[\\phi ^{\\sum _{j=1}^{\\tilde{s}}\\xi (i_{j})}\\phi ^{\\xi (i_{\\tilde{s}+1})}]\\nonumber \\\\& =\\mathbb {E}[\\mathbb {E}[\\phi ^{\\sum _{j=1}^{\\tilde{s}}\\xi (i_{j})}\\phi ^{\\xi (i_{\\tilde{s}+1})}\\mid \\mathcal {F}_{i_{\\tilde{s}+1}-1}]]\\nonumber \\\\& =\\mathbb {E}[\\phi ^{\\sum _{j=1}^{\\tilde{s}}\\xi (i_{j})}\\mathbb {E}[\\phi ^{\\xi (i_{\\tilde{s}+1})}\\mid \\mathcal {F}_{i_{\\tilde{s}+1}-1}]]\\nonumber \\\\& =\\mathbb {E}[\\phi ^{\\sum _{j=1}^{\\tilde{s}}\\xi (i_{j})}\\mathbb {E}[\\phi ^{\\xi (i_{\\tilde{s}+1})}\\mid \\xi (i_{\\tilde{s}+1}-1)]]\\nonumber \\\\& \\le \\mathbb {E}[\\phi ^{\\sum _{j=1}^{\\tilde{s}}\\xi (i_{j})}]((\\phi -1)\\tilde{p}+1).$ Using (REF ) and (REF ), we arrive at (REF ) with $s=\\tilde{s}+1$ .", "Proof of Lemma REF : First, let $\\overline{\\xi }(k) & \\triangleq [\\xi (0),\\xi (1),\\ldots ,\\xi (k-1)]^{\\mathrm {T}},\\\\\\overline{\\chi }(k) & \\triangleq [\\chi (0),\\chi (1),\\ldots ,\\chi (k-1)]^{\\mathrm {T}},\\quad k\\in \\mathbb {N}.$ Now let $F_{s,k} & \\triangleq \\lbrace \\overline{\\chi }\\in \\lbrace 0,1\\rbrace ^{k}\\colon \\overline{\\chi }^{\\mathrm {T}}\\overline{\\chi }=s\\rbrace ,\\,\\,s\\in \\lbrace 0,1,\\ldots ,k\\rbrace ,\\,k\\in \\mathbb {N}.$ It is clear that $F_{s_{1},k}\\cap F_{s_{2},k}=\\emptyset $ , $s_{1}\\ne s_{2}$ ; moreover, $\\mathbb {P}[\\overline{\\chi }(k)\\in \\cup _{s=0}^{k}F_{s,k}] & =1,\\quad k\\in \\mathbb {N}.$ It then follows that for all $\\rho \\in (\\tilde{p}\\tilde{w},1)$ and $k\\in \\mathbb {N}$ , $& \\mathbb {P}[\\sum _{i=0}^{k-1}\\xi (i)\\chi (i)>\\rho k]=\\mathbb {P}[\\overline{\\xi }^{\\mathrm {T}}(k)\\overline{\\chi }(k)>\\rho k]\\nonumber \\\\& \\quad =\\sum _{s=0}^{k}\\sum _{\\overline{\\chi }\\in F_{s,k}}\\mathbb {P}[\\overline{\\xi }^{\\mathrm {T}}(k)\\overline{\\chi }(k)>\\rho k\\mid \\overline{\\chi }(k)=\\overline{\\chi }]\\nonumber \\\\& \\quad \\quad \\cdot \\mathbb {P}[\\overline{\\chi }(k)=\\overline{\\chi }].$ Due to the mutual independence of $\\xi (\\cdot )$ and $\\chi (\\cdot )$ , $\\mathbb {P}[\\overline{\\xi }^{\\mathrm {T}}(k)\\overline{\\chi }(k)>\\rho k\\mid \\overline{\\chi }(k)=\\overline{\\chi }] & =\\mathbb {P}[\\overline{\\xi }^{\\mathrm {T}}(k)\\overline{\\chi }>\\rho k].$ As a result, it follows from (REF ) and (REF ) that for $k\\in \\mathbb {N}$ , $& \\mathbb {P}[\\sum _{i=0}^{k-1}\\xi (i)\\chi (i)>\\rho k]\\nonumber \\\\& \\,\\,=\\sum _{s=0}^{k}\\sum _{\\overline{\\chi }\\in F_{s,k}}\\mathbb {P}[\\overline{\\xi }^{\\mathrm {T}}(k)\\overline{\\chi }>\\rho k]\\mathbb {P}[\\overline{\\chi }(k)=\\overline{\\chi }]\\nonumber \\\\& \\,\\,=\\sum _{s=0}^{\\lfloor \\tilde{w}k\\rfloor }\\sum _{\\overline{\\chi }\\in F_{s,k}}\\mathbb {P}[\\overline{\\xi }^{\\mathrm {T}}(k)\\overline{\\chi }>\\rho k]\\mathbb {P}[\\overline{\\chi }(k)=\\overline{\\chi }]\\nonumber \\\\& \\,\\,\\quad +\\sum _{s=\\lfloor \\tilde{w}k\\rfloor +1}^{k}\\sum _{\\overline{\\chi }\\in F_{s,k}}\\mathbb {P}[\\overline{\\xi }^{\\mathrm {T}}(k)\\overline{\\chi }>\\rho k]\\mathbb {P}[\\overline{\\chi }(k)=\\overline{\\chi }].$ In the following, we will find upper-bounds for the two summation terms in (REF ).", "First, for the second term, since $\\mathbb {P}[\\overline{\\xi }^{\\mathrm {T}}(k)\\overline{\\chi }>\\rho k]\\le 1$ , $k\\in \\mathbb {N}$ , we have $& \\sum _{s=\\lfloor \\tilde{w}k\\rfloor +1}^{k}\\sum _{\\overline{\\chi }\\in F_{s,k}}\\mathbb {P}[\\overline{\\xi }^{\\mathrm {T}}(k)\\overline{\\chi }>\\rho k]\\mathbb {P}[\\overline{\\chi }(k)=\\overline{\\chi }]\\nonumber \\\\& \\quad \\le \\sum _{s=\\lfloor \\tilde{w}k\\rfloor +1}^{k}\\sum _{\\overline{\\chi }\\in F_{s,k}}\\mathbb {P}[\\overline{\\chi }(k)=\\overline{\\chi }]\\nonumber \\\\& \\quad =\\mathbb {P}[\\sum _{i=0}^{k-1}\\chi (i)>\\tilde{w}k]=\\tilde{\\sigma }_{k},\\quad k\\in \\mathbb {N}.$ Next, we look at the first term in (REF ).", "Note that $\\mathbb {P}[\\overline{\\xi }^{\\mathrm {T}}(k)\\overline{\\chi }>\\rho k]=0$ for $\\overline{\\chi }\\in F_{0,k}$ .", "Hence, for all $k\\in \\mathbb {N}$ such that $\\lfloor \\tilde{w}k\\rfloor =0$ , we have $\\sum _{s=0}^{\\lfloor \\tilde{w}k\\rfloor }\\sum _{\\overline{\\chi }\\in F_{s,k}}\\mathbb {P}[\\overline{\\xi }^{\\mathrm {T}}(k)\\overline{\\chi }>\\rho k]\\mathbb {P}[\\overline{\\chi }(k)=\\overline{\\chi }] & =0.$ Furthermore, for all $k\\in \\mathbb {N}$ such that $\\lfloor \\tilde{w}k\\rfloor \\ge 1$ , we have $& \\sum _{s=0}^{\\lfloor \\tilde{w}k\\rfloor }\\sum _{\\overline{\\chi }\\in F_{s,k}}\\mathbb {P}[\\overline{\\xi }^{\\mathrm {T}}(k)\\overline{\\chi }>\\rho k]\\mathbb {P}[\\overline{\\chi }(k)=\\overline{\\chi }]\\nonumber \\\\& \\,=\\sum _{s=1}^{\\lfloor \\tilde{w}k\\rfloor }\\sum _{\\overline{\\chi }\\in F_{s,k}}\\mathbb {P}[\\overline{\\xi }^{\\mathrm {T}}(k)\\overline{\\chi }>\\rho k]\\mathbb {P}[\\overline{\\chi }(k)=\\overline{\\chi }].$ Now, for $s\\in \\lbrace 1,2,\\ldots ,\\lfloor \\tilde{w}k\\rfloor \\rbrace $ , let $i_{1}(\\overline{\\chi }),i_{2}(\\overline{\\chi }),\\ldots ,i_{s}(\\overline{\\chi })$ denote the indices of the nonzero entries of $\\overline{\\chi }\\in F_{s,k}$ such that $i_{1}(\\overline{\\chi })<i_{2}(\\overline{\\chi })<\\cdots <i_{s}(\\overline{\\chi })$ .", "Consequently, $\\mathbb {P}[\\overline{\\xi }^{\\mathrm {T}}(k)\\overline{\\chi }>\\rho k] & =\\mathbb {P}[\\sum _{j=1}^{s}\\overline{\\xi }_{i_{j}(\\bar{\\chi })}(k)>\\rho k]\\nonumber \\\\& =\\mathbb {P}[\\sum _{j=1}^{s}\\xi (i_{j}(\\overline{\\chi })-1)>\\rho k],$ for $\\overline{\\chi }\\in F_{s,k}$ , $s\\in \\lbrace 1,2,\\ldots ,\\lfloor \\tilde{w}k\\rfloor \\rbrace $ , and $k\\in \\mathbb {N}$ such that $\\lfloor \\tilde{w}k\\rfloor \\ge 1$ .", "Now note that $\\phi >1$ , since $\\rho \\in (\\tilde{p}\\tilde{w},\\tilde{w})$ .", "We use Markov's inequality to obtain $\\mathbb {P}[\\overline{\\xi }^{\\mathrm {T}}(k)\\overline{\\chi }>\\rho k] & \\le \\mathbb {P}[\\sum _{j=1}^{s}\\xi (i_{j}(\\overline{\\chi })-1)\\ge \\rho k]\\nonumber \\\\& =\\mathbb {P}[\\phi ^{\\sum _{j=1}^{s}\\xi (i_{j}(\\overline{\\chi })-1)}\\ge \\phi ^{\\rho k}]\\nonumber \\\\& \\le \\phi ^{-\\rho k}\\mathbb {E}[\\phi ^{\\sum _{j=1}^{s}\\xi (i_{j}(\\overline{\\chi })-1)}].$ It follows from Lemma REF that $\\mathbb {E}[\\phi ^{\\sum _{j=1}^{s}\\xi (i_{j}(\\overline{\\chi })-1)}]\\le \\phi \\left((\\phi -1)\\tilde{p}+1\\right)^{s-1}$ .", "Using this inequality together with (REF ) and (REF ), for all $k\\in \\mathbb {N}$ such that $\\lfloor \\tilde{w}k\\rfloor \\ge 1$ , we obtain $& \\sum _{s=0}^{\\lfloor \\tilde{w}k\\rfloor }\\sum _{\\overline{\\chi }\\in F_{s,k}}\\mathbb {P}[\\overline{\\xi }^{\\mathrm {T}}(k)\\overline{\\chi }>\\rho k]\\mathbb {P}[\\overline{\\chi }(k)=\\overline{\\chi }]\\nonumber \\\\& \\,\\le \\sum _{s=1}^{\\lfloor \\tilde{w}k\\rfloor }\\sum _{\\overline{\\chi }\\in F_{s,k}}\\phi ^{-\\rho k}\\phi \\left((\\phi -1)\\tilde{p}+1\\right)^{s-1}\\mathbb {P}[\\overline{\\chi }_{k}=\\overline{\\chi }]\\nonumber \\\\& \\,=\\phi ^{-\\rho k+1}\\sum _{s=1}^{\\lfloor \\tilde{w}k\\rfloor }\\left((\\phi -1)\\tilde{p}+1\\right)^{s-1}\\sum _{\\overline{\\chi }\\in F_{s,k}}\\mathbb {P}[\\overline{\\chi }_{k}=\\overline{\\chi }]\\nonumber \\\\& \\,=\\phi ^{-\\rho k+1}\\sum _{s=1}^{\\lfloor \\tilde{w}k\\rfloor }\\left((\\phi -1)\\tilde{p}+1\\right)^{s-1}\\mathbb {P}[\\overline{\\chi }_{k}\\in F_{s,k}]\\nonumber \\\\& \\,\\le \\phi ^{-\\rho k+1}\\sum _{s=1}^{\\lfloor \\tilde{w}k\\rfloor }\\left((\\phi -1)\\tilde{p}+1\\right)^{s-1},$ where we also used the fact that $\\mathbb {P}[\\overline{\\chi }_{k}\\in F_{s,k}]\\le 1$ to obtain the last inequality.", "Here, we have $\\sum _{s=1}^{\\lfloor \\tilde{w}k\\rfloor }\\left((\\phi -1)\\tilde{p}+1\\right)^{s-1} & =\\frac{\\left((\\phi -1)\\tilde{p}+1\\right)^{\\lfloor \\tilde{w}k\\rfloor }-1}{\\left((\\phi -1)\\tilde{p}+1\\right)-1}\\nonumber \\\\& \\le \\frac{\\left((\\phi -1)\\tilde{p}+1\\right)^{\\tilde{w}k}-1}{(\\phi -1)\\tilde{p}}.$ Hence, (REF ) and (REF ) imply $& \\sum _{s=0}^{\\lfloor \\tilde{w}k\\rfloor }\\sum _{\\overline{\\chi }\\in F_{s,k}}\\mathbb {P}[\\overline{\\xi }^{\\mathrm {T}}(k)\\overline{\\chi }>\\rho k]\\mathbb {P}[\\overline{\\chi }(k)=\\overline{\\chi }]\\nonumber \\\\& \\quad \\le \\phi ^{-\\rho k+1}\\frac{\\left((\\phi -1)\\tilde{p}+1\\right)^{\\tilde{w}k}-1}{(\\phi -1)\\tilde{p}},$ for all $k\\in \\mathbb {N}$ such that $\\lfloor \\tilde{w}k\\rfloor \\ge 1$ .", "Because the right-hand side of this inequality is zero if $\\lfloor \\tilde{w}k\\rfloor =0$ , (REF ) holds for all $k\\in \\mathbb {N}$ .", "Now, this fact together with (REF ), (REF ) leads us to (REF ).", "Our next goal is to show $\\sum _{k=1}^{\\infty }\\psi _{k}<\\infty $ .", "To this end, first note that $& \\sum _{k=1}^{\\infty }\\phi ^{-\\rho k+1}\\frac{\\left((\\phi -1)\\tilde{p}+1\\right)^{\\tilde{w}k}-1}{(\\phi -1)\\tilde{p}}\\nonumber \\\\& \\quad =\\frac{\\phi }{(\\phi -1)\\tilde{p}}\\sum _{k=1}^{\\infty }\\phi ^{-\\rho k}\\left((\\phi -1)\\tilde{p}+1\\right)^{\\tilde{w}k}\\nonumber \\\\& \\quad \\quad -\\frac{\\phi }{(\\phi -1)\\tilde{p}}\\sum _{k=1}^{\\infty }\\phi ^{-\\rho k}.$ We will show that the series on the far right-hand side of (REF ) are both convergent.", "First, since $\\phi >1$ , we have $\\phi ^{-\\rho }<1$ , and thus, the geometric series $\\sum _{k=1}^{\\infty }\\phi ^{-\\rho k}$ converges, that is, $\\sum _{k=1}^{\\infty }\\phi ^{-\\rho k} & <\\infty .$ Next, we show $\\phi ^{-\\rho }\\left((\\phi -1)\\tilde{p}+1\\right)^{\\tilde{w}}<1$ .", "We obtain $\\phi ^{-\\rho }\\left((\\phi -1)\\tilde{p}+1\\right)^{\\tilde{w}} & =\\left(\\phi ^{-\\frac{\\rho }{\\tilde{w}}}\\left((\\phi -1)\\tilde{p}+1\\right)\\right)^{\\tilde{w}}.$ Furthermore, $& \\phi ^{-\\frac{\\rho }{\\tilde{w}}}\\left((\\phi -1)\\tilde{p}+1\\right)\\\\& \\quad =\\left(\\frac{\\frac{\\rho }{\\tilde{w}}(1-\\tilde{p})}{\\tilde{p}(1-\\frac{\\rho }{\\tilde{w}})}\\right)^{-\\frac{\\rho }{\\tilde{w}}}\\left(\\left(\\frac{\\frac{\\rho }{\\tilde{w}}(1-\\tilde{p})}{\\tilde{p}(1-\\frac{\\rho }{\\tilde{w}})}-1\\right)\\tilde{p}+1\\right)\\\\& \\quad =\\left(\\frac{\\tilde{p}\\tilde{w}}{\\rho }\\right)^{\\frac{\\rho }{\\tilde{w}}}\\left(\\frac{1-\\tilde{p}}{1-\\frac{\\rho }{\\tilde{w}}}\\right)^{-\\frac{\\rho }{\\tilde{w}}}\\left(\\frac{1-\\tilde{p}}{1-\\frac{\\rho }{\\tilde{w}}}\\right)\\\\& \\quad =\\left(\\frac{\\tilde{p}\\tilde{w}}{\\rho }\\right)^{\\frac{\\rho }{\\tilde{w}}}\\left(\\frac{1-\\tilde{p}}{1-\\frac{\\rho }{\\tilde{w}}}\\right)^{1-\\frac{\\rho }{\\tilde{w}}}.$ Note that $\\frac{\\tilde{p}\\tilde{w}}{\\rho },\\frac{1-\\tilde{p}}{1-\\frac{\\rho }{\\tilde{w}}}\\in (0,1)\\cup (1,\\infty )$ .", "Since $\\ln v<v-1$ for any $v\\in (0,1)\\cup (1,\\infty )$ , we have $& \\ln \\left(\\phi ^{-\\frac{\\rho }{\\tilde{w}}}\\left((\\phi -1)\\tilde{p}+1\\right)\\right)\\\\& \\quad =\\frac{\\rho }{\\tilde{w}}\\ln \\left(\\frac{\\tilde{p}\\tilde{w}}{\\rho }\\right)+(1-\\frac{\\rho }{\\tilde{w}})\\ln \\left(\\frac{1-\\tilde{p}}{1-\\frac{\\rho }{\\tilde{w}}}\\right)\\\\& \\quad <\\frac{\\rho }{\\tilde{w}}\\left(\\frac{\\tilde{p}\\tilde{w}}{\\rho }-1\\right)+(1-\\frac{\\rho }{\\tilde{w}})\\left(\\frac{1-\\tilde{p}}{1-\\frac{\\rho }{\\tilde{w}}}-1\\right)\\\\& \\quad =\\tilde{p}-\\frac{\\rho }{\\tilde{w}}+\\frac{p}{\\tilde{w}}-\\tilde{p}\\,=\\,0,$ which implies that $\\phi ^{-\\frac{\\rho }{\\tilde{w}}}\\left((\\phi -1)\\tilde{p}+1\\right)<1$ , and hence by (REF ), $\\phi ^{-\\rho }\\left((\\phi -1)\\tilde{p}+1\\right)^{\\tilde{w}}<1$ .", "Therefore, $\\sum _{k=1}^{\\infty }\\phi ^{-\\rho k}\\left((\\phi -1)\\tilde{p}+1\\right)^{\\tilde{w}k}<\\infty .$ Finally, (REF ), (REF ), and (REF ) imply $\\sum _{k=1}^{\\infty }\\psi _{k}<\\infty $ .", "$\\hfill \\square $" ] ]
1606.05245
[ [ "On The Fourier And Wavelet Analysis Of Coronal Time Series" ], [ "Abstract Using Fourier and wavelet analysis, we critically re-assess the significance of our detection of periodic pulsations in coronal loops.", "We show that the proper identification of the frequency dependence and statistical properties of the different components of the power spectra provies a strong argument against the common practice of data detrending, which tends to produce spurious detections around the cut-off frequency of the filter.", "In addition, the white and red noise models built into the widely used wavelet code of Torrence & Compo cannot, in most cases, adequately represent the power spectra of coronal time series, thus also possibly causing false positives.", "Both effects suggest that several reports of periodic phenomena should be re-examined.", "The Torrence & Compo code nonetheless effectively computes rigorous confidence levels if provided with pertinent models of mean power spectra, and we describe the appropriate manner in which to call its core routines.", "We recall the meaning of the default confidence levels output from the code, and we propose new Monte-Carlo-derived levels that take into account the total number of degrees of freedom in the wavelet spectra.", "These improvements allow us to confirm that the power peaks that we detected have a very low probability of being caused by noise." ], [ "MOTIVATION", "Following the early detection of an individual case by [18]Even though these authors attribute this event to a filament, our re-analysis indicates that the locations of significant Fourier power correspond to coronal arcades., [3] presented evidence that coronal active region loops frequently undergo episodes of periodic pulsations with periods of several hours and lasting several days.", "This statistical study was based on 13 years of quasi-continuous observations at a cadence of 12 min in the 19.5 nm channel of the Solar and Heliospheric Observatory [16] Extreme-ultraviolet Imaging Telescope [14].", "Recently, [20] published a detailed analysis of three of these pulsation events observed simultaneously in the six coronal passbands of the Solar Dynamics Observatory [49] Atmospheric Imaging Assembly [33].", "Using a combination of the time-lag analysis method of [60] and of the differential emission measure (DEM) diagnostics developed by [24], [25], [23], Froment et al.", "argued that the observed pulsations are the signatures of incomplete evaporation–condensation cycles similar to those arising in numerical simulations of highly stratified heating of coronal loops [35], [40], [62], [34].", "Our detections add to an already vast bibliography indicating that periodic phenomena in general seem ubiquitous in the solar corona.", "In fact, they have been reported to occur in most coronal structures: bright points [59], [54], loops [1], [41], [52], prominences [50], [6], [63], polar plumes [13], [46], flares [28], [44], [15], open and closed field large-scale structures [53], etc.", "The physical processes involved are diverse and the interest of their study is manifold.", "For example, whether found to be sufficient [37] or insufficient [56] to compensate for the coronal losses, the energy carried by Alfvén waves is at the heart of the long-standing coronal heating debate, and their detection is thus critical.", "Also, coronal seismology techniques use the characteristics of wave and oscillatory phenomena to derive the properties of the ambient plasma and magnetic field [45], [10].", "However, recent papers have questioned the validity of several accounts of quasi-periodic pulsations (QPPs) in solar flares [22], [27], [29].", "These authors point out that, in many cases, the fundamental power-law dependence of the power spectra of coronal time series has not been recognized, resulting in erroneous confidence levels.", "After the re-analysis of 19 time series, [27] concluded that coherent oscillatory power is necessary to explain the observed Fourier spectra in only one case.", "Beyond the specific case of QPPs, their results cast doubt on a number of previously published reports and suggest that the prevalence of oscillatory phenomena in the corona may be artificial.", "In this context, even though we properly accounted for the power-law nature of the coronal power spectra in our previous works, we decided to critically re-evaluate the statistical significance of the events presented by [20].", "Indeed, several sometimes subtle effects can cause false positives in Fourier or wavelet analyses.", "For example, Appendix A of [3] describes how the shadow cast on the detector by the mesh grid holding the focal filters in SOHO/EIT [12], [5] produces spurious frequencies in the range of those detected in coronal loops.", "This effect could not be found in identical processing of AIA images, which is explained by the much smaller amplitude of the mesh grid pattern in this latter instrument.", "In the present paper, before re-interpreting our previous detections, we present possible sources of false positives in the methods of spectral (Fourier and wavelet) analysis: section  describes the effect of time series detrending on the power spectra, section  discusses the choice of noise model, and section  treats confidence levels in wavelet spectra.", "As an example, one of the periodic pulsation events studied by [20] is then re-analyzed in section  in the light of these considerations.", "Our conclusions and recommendations for Fourier and wavelet analysis of coronal time series are summarized in section ." ], [ "DETRENDING", "Detrending is often applied to time series before analysis of their frequency content in order to filter out the low frequencies and thus to enhance the periodic signals potentially present in the original data [18], [59], [28], [54], [19], [43], [15].", "However, several authors [8], [31], [22] have shown that detrending, combined with an improper accounting for the frequency dependence of the power spectrum, can lead to overestimating the claimed significance levels or even to false detections.", "Our analysis confirms and generalizes these results.", "We demonstrate that detrending not only can but generally does lead to false-detections in Fourier or wavelet analyses, although it produces visually convincing filtered versions of the time series.", "Detrending is achieved by removing from the original time series $s_o(t)$ a smooth version of it, which is commonly obtained with high-pass filtering.Other methods can be used – like approximation by polynomials – but in all cases the power spectrum will be affected.", "Some authors compute the detrended time series $s_d(t)$ by subtracting the smooth estimate from the original data [59], [28], [43], [15] $s_d(t) = s_o(t) - s_o(t) * f(t),$ where $s_o(t) * f(t)$ represents the smoothed data obtained by convolution of $s_o(t)$ with the filter kernel $f(t)$ .", "Other authors divide the original data by the smooth estimate and subtract 1 [17], [18], [19] to obtain $s_d(t) = \\frac{s_o(t)}{s_o(t) * f(t)} - 1.$ If the variations $\\delta s_o(t)$ of the signal are small with respect to its global average $\\overline{s_o}$ , equation REF can be expressed as $s_d(t) \\approx \\frac{\\delta s_o(t) - \\delta s_o(t) * f(t)}{\\overline{s_o(t)}},$ which is identical to equation REF for the relative variations of the signal.", "These two methods are thus equivalent and have the same effect on the Fourier or wavelet spectra.", "If the variations of the signal are not small compared to its average, the second method modifies the power spectra more profoundly than described below, which renders their proper interpretation impossible.", "Using the convolution theorem on Equation REF , we obtain the power spectrum of the detrended time series $\\Psi (\\nu )=\\left|S_d(\\nu )\\right|^2=\\left|S_o(\\nu )\\right|^2\\left|1-F(\\nu )\\right|^2,$ where capital letters are used to denote the Fourier transforms.", "The most commonly used high-pass filter is a running boxcar of width $\\Delta t$ , in which case the power spectrum becomesSimilar expressions could be derived for other high-pass filters, resulting in similar distortions of the power spectrum.", "$\\Psi (\\nu )=\\left|S_o(\\nu )\\right|^2\\left(1 - \\frac{\\sin \\left(\\pi \\nu \\Delta t\\right)}{\\pi \\nu \\Delta t}\\right)^2,$ which is the power spectrum of the original time series multiplied by a filtering function that attenuates the low frequencies while preserving the high frequencies (the black curve in Figure REF ).", "As recently emphasized by [27] and [29], the mean Fourier power spectra of extreme-ultraviolet emission from active region cores, loop footpoints, and the quiet Sun follow a power-law-like behaviour.", "This property was already noted by [3] for all types of on-disk structures and for most of the time series that they analyzed, and [26] also reports power-law spectra off-disk in polar plumes.", "A power-law is actually what is expected from line of sight superimposition of many exponentially decaying emission pulses [29], [2], which is consistent with the idea that the corona is heated by many small scale impulsive events.", "This indicates that a power-law is the most likely spectral shape for coronal time series.", "From Equation REF , the expected power spectra of detrended time series can thus be described by $\\Psi (\\nu ) = a\\nu ^s\\left(1 - \\frac{\\sin \\left(\\pi \\nu \\Delta t\\right)}{\\pi \\nu \\Delta t}\\right)^2.$ $\\Psi (\\nu )$ is represented in Figure REF for integer values of the exponent $s$ from -1 to -5, which covers the range of observed slopes reported by [27].", "In addition, the shape of the filter itself is given by the $s=0$ case.", "For $s\\leqslant -4$ , the maximum of $\\Psi (\\nu )$ is at $\\nu =0$ .", "For $-1>s>-4$ , which is the case for the majority of the reported power-laws for coronal time series, we determined numerically that the position of the maximum can be approximated (to within better than 1%) by $\\frac{1}{\\Delta t}\\sqrt{\\frac{4+s}{2}}.$ The maximum is located near the cut-off frequency $1/\\Delta t$ of the high-pass filter, and exactly at the cut-off for $s=-2$ .", "The right panel of Figure REF , which uses a linear axis for the power, clearly illustrates the formation of a peak of dominant frequencies around the cut-off by the filtering out of the power-laws below $1/\\Delta t$ .", "The corresponding detrended time series are thus strongly chromatic, which can be incorrectly interpreted as periodicity in the original data.", "In order to demonstrate the effect in practice, using an independently derived version of the algorithm described by [55], we simulated a random time series of $N=512$ data points whose power spectral distribution (PSD) is a power-law of exponent -2.", "The original time series is the gray curve in the top left panel of Figure REF and its $\\Delta t=30$ time steps wide running boxcar detrended version is in magenta.", "Both curves are normalized to their respective standard deviations $\\sigma _0$ .", "The results of the spectral analysis (wavelet and Fourier) of the detrended and original series are represented respectively in the middle and bottom rows, respectively.", "A periodicity is clearly visible in the detrended time series even though the original data are completely random.", "This periodicity is real, as shown by the narrow peak of power in the log–linear fast Fourier transform power spectrum of the central panel.", "A narrow band of power is also visible at the same frequencies in the Morlet wavelet spectrum (left panel) computed with the code of [57].", "Fourier (gray line) and wavelet (blue contours and lines) white noise 95% confidence levels could be used to argue in favor of the significance of the peak.", "However, as revealed by the log–log representation of the right panel, this periodicity is only due to the high-pass filtering of a power-law-like spectrum.", "The Fourier (gray histogram) and time-averaged wavelet (black curve) power spectra follow the theoretical curve given by Equation REF (in cyan, i.e.", "the $s=-2$ curve of Figure REF ).", "As expected, the peak of power is located around the cut-off frequency $1/\\Delta t$ .", "At higher frequencies, the power spectrum is quasi-unaffected and resembles that of the original series (lower right panel).", "As we will see in the next section, not recognizing the power-law nature of the original spectrum will lead to incorrect conclusions regarding the significance of the observed peak of power." ], [ "BACKGROUND NOISE MODELS", "The determination of confidence levels requires an appropriate model of the background power, i.e.", "an estimate of the expected value $\\sigma (\\nu )$ of the power, and of its probability distribution, at each frequency in the absence of oscillatory phenomena.As mentioned by [21], the commonly adopted notation $\\sigma $ for the expected power is not strictly correct, and it is not to be confused with the variance $\\sigma _0^2$ of the time series, even though the two are equal for white noise.", "This is a fundamental step in wavelet or Fourier analysisIt is worth noting that randomization methods [47] do not replace the assumption of a noise model since they are in fact equivalent to comparing to white noise..", "If we assume white noise, the expected power $\\sigma $ is constant (dark blue lines in the two right-most panels of the middle row of Figure REF ) and equal to 1 since, following TC98, we normalized the Fourier and wavelet spectra by $N/\\sigma _0^2$ , which gives a measure of power relative to white noise.", "For a normally distributed random variable, the Fourier power at each frequency being distributed as $\\sigma \\chi _2^2/2$ [21]The notation $\\sigma \\chi _d^2/d$ , as used by TC98, means that the probability distribution function (PDF) of the power $p$ at the frequency $\\nu $ is $f\\left(p\\ d/\\sigma ;d\\right)d/\\sigma $ , where $f(x;d)$ is the PDF of the chi-square distribution of degree $d$ .", "This distribution of power has mean $\\sigma $ ., the probability that the power is greater than $m$ times the mean $\\sigma $ for at least one of any of the $N/2$ frequencies is given by $1-\\left(1-\\text{e}^{-m}\\right)^{N/2}$ [51], [21], from which we obtain that the peaks of power above $m\\sigma = -\\log \\left(1 - 0.95^{1/256}\\right)\\sigma = 8.51\\sigma $ have only a 5% chance of occurring from white noise (95% global confidence levels, the gray lines in the two right-most panels).", "Equivalently, we derived empirically (see § ) the 95% global confidence levels above which white noise power only has a 5% probability to lie for at least one of any of the points of the wavelet and time-averaged wavelet spectra (light blue contours and curves).", "The less stringent local confidence levels built-in the TC98 code (medium blue contours and curves) give the 95% probability that greater power at each frequency and/or time is not due to white noise.", "Although perfectly rigorous for white noise, these confidence levels lead to erroneous conclusions because this background model is obviously inadequate for either the original or the detrended time series.", "The red noise model of [57] (red curves) is not a good approximation of the expected spectrum of the detrended series either.", "It is based on the lag-1 auto-regressive (AR(1), or Markov) process: $x_n=\\alpha x_{n-1}+z_n,$ where $\\alpha $ is the lag-1 autocorrelation, $x_0=0$ and $z_n$ is taken from Gaussian white noise.", "The normalized expected Fourier power spectrum of the resulting time series is given by: $\\sigma (\\nu )=\\frac{1-\\alpha ^2}{1+\\alpha ^2-2\\alpha \\cos \\left(2\\pi \\nu \\delta t\\right)},$ where $\\delta t$ is the sampling interval.", "Since $x_n$ is a normally distributed random variable, its power is distributed as white noise of mean $\\sigma (\\nu )$ in each sufficiently narrow frequency band centered on $\\nu $ .", "Thus, the Fourier and wavelet spectra are distributed as $\\sigma (\\nu )\\chi _2^2/2$ at every frequency and timeThis expression is valid for complex wavelets such as the Morlet and Paul wavelets used in the present paper.", "(see TC98 and references therein) and the corresponding confidence levels are simply those derived for white noise multiplied by $\\sigma (\\nu )$ .", "As was the case for white noise, using the red noise AR(1) model to study the power spectrum of the detrended time series of Figure REF naturally yields misleading results because the model cannot represent the expected spectrum.", "The detrended time series has a lag-1 autocorrelation coefficient of 0.83.", "The corresponding mean power (red curves in the right panels of the middle row) underestimates the expected value (cyan curves) around the cut-off $1/\\Delta t$ and largely overestimates it at lower frequencies.", "Consequently, while the Fourier power is everywhere below the 95% confidence level (gray curves), both the wavelet and the time-averaged wavelet spectra exceed the TC98 95% local confidence levels (orange contours and curves).", "Conversely, the AR(1) model can adequately represent the mean power of the non-detrended time series (red curves in the bottom right panels) since this latter has a lag-1 autocorrelation coefficient of 0.995.", "Indeed, for $\\alpha \\approx 1$ and small $\\nu $ , using a second order Taylor expansion of the cosine function, Equation REF can be approximated by $\\sigma (\\nu )\\approx \\frac{1-\\alpha }{2\\left(\\pi \\nu \\delta t\\right)^2},$ which is a power-law of exponent -2, i.e.", "by construction the expected PSD of the original data.", "As a result of the appropriateness of the model, none of the bins of the Fourier and time-averaged wavelet spectra are above the associated 95% global confidence levels (gray and yellow curves respectively).", "However, at least one point of the time-averaged wavelet spectrum is above the TC98 95% local confidence level (orange curves), but this should not be taken as evidence for significant oscillatory power, for it is in fact likely to occur by chance: while the probability for the noise power at each frequency to be above that level is only 5%, the probability for any of the 256 frequencies to be above is much higher (see § ).", "Likewise, several orange contours (TC98 95% local confidence levels) appear by chance on the wavelet spectrum (bottom left panel).", "This effect was mentioned in TC98 (see their Fig.", "4) and the authors caution against over-interpretation on their website.http://paos.colorado.edu/research/wavelets/faq.html On the other hand, no points of the wavelet or time-averaged spectra are above the 95% global confidence levels derived in §  (yellow contours and curves).", "The AR(1) red noise model is nonetheless strongly limited in that, as shown by Equation REF , it can only satisfactorily approximate power-law-like spectra of exponent -2, while the values reported in the literature for coronal time series span at least the -1.72 to -4.95 range [27], [29].", "It is thus likely that the AR(1) model is in most cases not pertinent for solar data.", "Figure REF is similar to Figure REF for a random time series having an expected power-law spectrum of exponent -3.", "The detrended time series (in magenta) presents strong oscillations, for the same reasons as described above.", "Its spectral analysis (middle row) thus presents characteristics similar to those illustrated in Figure REF for the $s=-2$ series.", "The false detections appear to be even more significant because in this case the AR(1) noise model (with $\\alpha =0.92$ ) is not a valid approximation of the PSD anywhere.", "The non-detrended data have a lag-1 autocorrelation coefficient of 0.998.", "The mean power spectrum given by the AR(1) model is thus close to a power-law of exponent -2.", "It intersects the true PSD at mid-frequencies, and the corresponding 95% confidence levels erroneously lead to the conclusion that there is significant excess power at low frequencies.", "Supposing that we ignore how the time series was built, the log–log representation of its PSD (bottom right panel) would naturally suggest choosing a power-law of the variable exponent as a noise model.", "The thus fitted expected power is given by the dark green line.", "The Fourier spectrum is not above the corresponding 95% global confidence level (gray line) anywhere.", "Wavelet confidence levels can in fact be computed with the TC98 code for any given mean power spectrum (see the Appendix for practical details).", "Some points of the wavelet and time-averaged wavelet spectra are above the resulting local 95% confidence levels (medium green contours and curves) for the same reason as described above: these levels account only for the local number of degrees of freedom of the spectra, and thus power will randomly surpass them for 5% of the frequencies and/or times even if the noise model is appropriate.", "In the next Section, we derive global confidence levels that account for this effect." ], [ "Fourier Confidence Levels", "At each frequency $\\nu $ , the Fourier or (wavelet) power spectrum of a random time series is distributed as [21]This expression is valid for complex wavelets.", "The 1/2 factor would be removed for real wavelets (see TC98) $\\frac{1}{2}\\sigma (\\nu )\\chi _2^2$ around the mean power $\\sigma (\\nu )$ .", "The probability for one point of a spectrum to have a power greater than $m$ times the mean $\\sigma $ is thus $P(m)=e^{-m}$ , and the associated confidence level as defined by TC98 is $1-P$ .", "For example, points above the TC98 95% confidence level have a probability $P=0.05$ (5%) of having a power greater than $m=-\\log (0.05)\\sigma (\\nu )=2.99\\sigma (\\nu )$ .", "However, in most practical situations, the probability of having power greater than these confidence levels in at least one of the bins is close to one.", "The reason is that the confidence levels above which peaks of power have a given probability to occur by chance depend on the number of points in the spectrum.", "For a random time series of $N$ points, $1-P(m)$ is the probability for each of the $N/2$ frequency bins of the Fourier spectrum to have a power lower than $m\\sigma (\\nu )$ .", "Since the bins are independent, $(1-P(m))^{N/2}$ is the probability for the power to be lower than $m\\sigma (\\nu )$ in all bins, and the probability that at least one bin has a power greater than $m\\sigma (\\nu )$ is thus $P_g(m)=1 - (1 - P(m))^{N/2} = 1 - (1 - e^{-m})^{N/2}.$ We refer to $P_g$ as the global probability (and corresponding confidence levels) over the whole spectrum as opposed to the local probability associated with individual bins.The term global refers to the taking into account of the total number of degrees of freedom in the spectrum and is not to be confused with the global wavelet spectrum term used in TC98.", "To avoid confusion, in this paper we always use time-averaged spectrum to denote what TC98 also call the global wavelet spectrum.", "For example, for $N=512$ , the global probability to have at least one bin of the Fourier spectrum above the 95% local confidence level is $1 - (1 - 0.05)^{256}=0.999998$ .", "Conversely, using Equation REF , one can properly derive the value of $m$ corresponding to the 95% global confidence level, i.e.", "$m=-\\log (1-0.95^{1/256})= 8.51$ .", "The derivation of global confidence levels for a wavelet spectrum is more complex because its bins are not statistically independent.", "Without providing a quantitative test, Torrence & Compo suggest that the significance of a peak of power be judged by comparing its duration with the decorrelation time, which is given by the width of the cone of influence (COI) at the corresponding frequency.", "This approach is followed by, e.g., [32].", "But there is a non-zero probability for two or more random peaks to be close enough together to form structures longer than the decorrelation time.", "Figure REF shows an example in the wavelet spectrum of a 1024 data-point-long Gaussian white noise time series.", "The structure visible around the frequency $12/N\\delta t$ remains above the 95% local confidence level (dark blue contours) for about 4 times the decorrelation time.", "As shown below, the probability of occurrence of such a structure at a given frequency (or scale) can be estimated using the binomial distribution.", "TC98 showed that the time-averaged wavelet power is distributed as $\\sigma (\\nu )\\chi _d^2/d$ , with $d$ being the number of degrees of freedom at each scale $s$ $d=2\\sqrt{1+\\left(\\frac{n_a\\delta t}{\\gamma s}\\right)^2},$ where $n_a$ is the total number of points $N$ minus half the number of those in the COI, and $\\gamma $ is an empirically derived decorrelation factor equal to 2.32 for the Morlet wavelet.", "At each scale, the wavelet spectrum thus behaves as $l=d/2$ statistically independent $\\chi _2^2$ distributed bins.", "Defining a success as a peak of power surpassing the 95% local confidence level, which has probability $p=0.05$ , the probability of obtaining exactly $k$ successes in $l$ trials is given by $P_b(k;l,p)=\\frac{l!}{k!(l-k)!", "}p^k(1-p)^{l-k}.$ In practice, $l$ and $k$ do not have to be integers; in this case the Gamma function has to be used instead of factorials.", "We have $l=4.5$ at the scale of the long structure of Figure REF and $k=1.8$ given that it lasts for 40% of the total duration.", "The probability to obtain the same number of bins above the 95% local confidence level by chance is thus $2\\times 10^{-2}$ .", "Since the binomial distribution accounts for all possible arrangements of $k$ successes in $n$ trials, this value is in fact an upper-limit for the probability of occurrence of that particular structure, which may thus be considered unlikely.", "This structure corresponds to a peak of power of the time-averaged spectrum (the black curve in the right panel) that lies above the 95% local confidence level (dark blue curve).", "Knowing that the time-averaged spectrum is distributed as $\\sigma (\\nu )\\chi _d^2/d$ (TC98), the probability associated with this maximum of power is $10^{-3}$ , which is also unlikely.", "In fact, it is intuitive that there must be a correlation between the probability associated with a power level in the time-averaged spectrum and the probability to have the observed number of bins above the local 95% confidence level at the corresponding scale.", "Figure REF shows, for 100,000 wavelet spectra of Gaussian white noise time series, the joint probability density of these two quantities.", "The spread around the diagonal comes from the binomial approximation and from the fact that at a given scale, several temporal power profiles can produce the same time-averaged power while having different number of bins above the 95% local confidence level.", "The observed correlation demonstrates that estimating the probability of occurrence of a given coherent structure simply amounts to estimating the probability associated with the corresponding time-averaged power.", "In order to do this, one must nonetheless take into account the total number of degrees of freedom in the spectrum.", "In the example of Figure REF , while the time-averaged wavelet power exceeds the 95% local confidence level, the 95% global Fourier confidence level given by Equation REF (gray line) correctly rejects the corresponding peaks in the Fourier spectrum (gray histogram).", "Likewise, it is possible to define robust global confidence levels for both wavelet and time-averaged wavelet spectra by generalizing the principles described  REF for Fourier spectra." ], [ "Global wavelet confidence levels", "Equation REF , which was derived for Fourier spectra, cannot be used for wavelet spectra by simply replacing $N/2$ with the total number of bins, for these are not statistically independent.", "Nonetheless, as shown below, the relation between $P_g(m)$ and $P(m)$ can in practice be approximated by a modified version of Equation  REF $P_g(m)=1 - (1 - P(m)^a)^n$ where $a$ and $n$ are empirically derived coefficients.", "By analogy with Equation REF , $n$ is expected to be proportional to the number of bins in the spectrum, but limited to those outside the COI since the local confidence levels corresponding to $P(m)$ are not relevant for the bins inside the COI.", "Indeed, for a zero-padded time series, the mean power inside the COI decreases as $1-e^{-2t/\\tau _s}/2$ , where $\\tau _s$ is the e-folding time (equal to $\\sqrt{2}s$ for the Morlet wavelet) and $t$ is the time from either the beginning or end of the spectrum.", "Thus, bins inside the COI have a reduced probability of being above the chosen confidence level compared to those outside.", "Finally, the total number of bins outside the COI should be normalized to the resolution in scale because the probability of having at least one bin above a given confidence level should be independent of the resolution.", "In order to determine $a$ and $n$ , we constructed 100,000 Gaussian white noise time seriesNote that since, as shown by Equation REF , the power is always distributed as $\\chi _2^2$ around the mean power $\\sigma (\\nu )$ , the Monte-Carlo results would be identical for, e.g., AR(1) or power-law noise.", "along with their associated Morlet wavelet power spectra, from which we estimated the probability $P_g(m)$ to have at least one bin (outside the COI) above the power threshold $m\\sigma (\\nu )$ and corresponding to the local probability $P(m)$ .", "We repeated this procedure for different numbers of data points $N$ (powers of 2 from $2^6$ to $2^{14}$ ) and scale resolutions $\\delta j$ (from $1/2$ to $1/2^8$ ).", "The wavelet spectra were all computed using zero padding over $J+1$ scales $s_j=s_02^{j\\delta j}$ with $J=\\log _2(N\\delta t/s_0)/\\delta j$ and the minimum resolvable scale $s_0=2\\delta t$ (see TC98 for details on the meaning of these parameters).", "The total number of bins in each spectrum is $N(J+1)$ .", "Figure: Monte-Carlo-derived probabilities P g (m)P_g(m) that at least one bin of a Morlet wavelet spectrum has power greater than mm times the mean power σ(ν)\\sigma (\\nu ), as a function of the probability P(m)P(m) for each bin to have a power greater than mσ(ν)m\\sigma (\\nu ), for several lengths NN of the input random time series.", "The dashed curves correspond to the parameterization given by Equations ,  and .Figure: The same as figure  for time-averaged wavelet spectra.", "The dashed curves correspond to the parameterization given by Equations ,  and .The results are shown in Figure REF .", "The histogram-style curves represent the Monte-Carlo-derived relations between $P_g(m)$ and $P(m)$ for $\\delta j=1/8$ and varying values of $N$ .", "The curves saturate at one, and this for smaller $P(m)$ as $N$ is large, reflecting the fact that the probability to have power greater than the threshold in at least one bin increases quickly with the number of bins.", "Almost identical curves are obtained for values of $\\delta j$ smaller than 1/8, but they differ somewhat for larger values.", "TC98 mention that 1/2 is the largest $\\delta j$ that still gives adequate sampling in scale for the Morlet wavelet.", "Our tests indicate that, at least from the point of view of the global confidence levels, $\\delta j=1/8$ should be the maximum used.", "We fitted the $\\delta j\\le 1/8$ curves with Equation REF and found that they can be described by the following parameterization of the fitted $a$ and $n$ (dashed curves in Figure REF ) $a&=&0.810\\ (N_{out}\\delta j)^{0.011}\\\\n&=&0.491\\ (N_{out}\\delta j)^{0.926},$ where $N_{out}$ is the number of bins outside the COI.", "The exponent 0.926 reflects the slightly slower increase of $n$ with $N_{out}\\delta j$ compared to the expected linear relationship.", "The coefficient $a$ varies slowly with $N_{out}\\delta j$ , which corresponds to the fact that the curves are parallel to each other.", "Furthermore it is close to 1, which implies that it only introduces small perturbations to $P(m)$ .", "Inverting Equation REF and using the parameterization given by Equations REF and , one can now compute the local probability (or confidence level) that should be used to achieve a chosen global probability (or confidence level) $P(m)=\\left(1 - \\left(1 - P_g(m)\\right)^{1/n}\\right)^{1/a}.$ For example, assuming the $N=1024$ data points time series of Figure REF , we have $a=0.892$ and $n=1595$ with the wavelet parameters used in this paper.", "The 95% global confidence level ($P_g(m)=0.05$ ) thus corresponds to $P(m)=(1-(1-0.05)^{1/1595})^{1/0.892}=9\\times 10^{-6}$ , i.e.", "to a 99.999% local confidence level.", "This value can in turn be used as input to the TC98 code (see the Appendix for practical details), and the structure visible in the bottom left panel of Figure REF is now correctly rejected (no light blue contours)." ], [ "Global Time-averaged Confidence Levels", "Using the same Monte-Carlo simulations, we derived global confidence levels for time-averaged Morlet wavelet spectra.", "As shown by TC98, time-averaging over the $\\chi _2^2$ distributed points of the wavelet spectrum results in the averaged power being distributed as $\\sigma (\\nu )\\chi _d^2/d$ , with $d$ being the number of degrees of freedom at each scale given by Equation REF .", "For a given local confidence level, the corresponding threshold $m$ is thus a function of scale.", "But since all of the bins of the time-averaged spectrum have the same probability $P(m(s))$ to be above $m(s)\\sigma (\\nu )$ by chance, the global probability for at least one bin to be above $m(s)\\sigma (\\nu )$ should again follow a relation described by Equation REF , and we now expect $n$ to be proportional to $S_{out}\\delta j$ , the number of scales for which at least one bin is outside the COI, normalized by the resolution.", "Figure REF is the equivalent of Figure REF for time-averaged spectra.", "As previously, the curves are all similar for $\\delta j\\le 1/8$ .", "The Monte-Carlo results can be described by the following parameterization of $a$ and $n$ $a&=&0.805+0.45\\times 2^{-S_{out}\\delta j}\\\\n&=&1.136\\ (S_{out}\\delta j)^{1.2}.$ Instead of the expected linear relationship, we had to raise $S_{out}\\delta j$ to the power 1.2 in order to reproduce the fitted values of $n$ .", "The dependence of $a$ with $S_{out}\\delta j$ is not intuitive but it is always close to 1.", "As before, we can now use the above expressions to compute for a time-averaged spectrum the local confidence level that should be used to achieve a chosen global confidence level.", "For Figure REF , we have $a=0.807$ and $n=12.7$ .", "The 95% global confidence level ($P_g(m)=0.05$ ) thus corresponds to $P(m)=(1 - (1-0.05)^{1/12.7})^{1/0.807}= 1\\times 10^{-3}$ , i.e.", "to a 99.89% local confidence level.", "This value can in turn be used as input to the TC98 code.", "We also ran Monte-Carlo simulations for the Paul wavelet and found results similar to those presented above for the Morlet wavelet: the coefficients $n$ are close to being proportional to the number of bins outside the COI and the coefficients $a$ are close to 1.", "We can use parameterizations of the same form as before, i.e.", "for the wavelet spectrum $a&=&0.817\\ (N_{out}\\delta j)^{0.011}\\\\n&=&0.320\\ (N_{out}\\delta j)^{0.926},$ and for the time-averaged spectrum $a&=&1.02 + 0.70\\times 1.42^{-S_{out}\\delta j}\\\\n&=&1.0 + 0.56\\ (S_{out}\\delta j)^{1.2}.$ As a final note, it is important to realize that a peak of power may be above the global confidence level in the time-averaged wavelet spectrum while the power may be below the global confidence level at all times in the wavelet spectrum at the corresponding frequency.", "By analogy with the Fourier case, we derived global confidence levels to test individual peaks, not the probabilities of occurrence of extended structures.", "As shown in § REF , this latter is simply given by the probability associated with the power in the time-averaged spectrum.", "Thus, while surpassing the global confidence level confirms the significance of a peak of power in the wavelet spectrum, the converse is not true.", "Before discarding as insignificant a peak in the wavelet spectrum that is below the global confidence level , one should check whether or not a corresponding peak is present and above the global confidence level in the time-averaged wavelet spectrum." ], [ "Re-analysis of AIA time series", "We present a re-analysis of the 33.5 nm SDO/AIA time series corresponding to one of the events studied by [20].", "Figure REF shows the middle image of the one minute cadence, 9202 frame-long sequence starting 2012 June 3 at 18:00 UT and ending 2012 June 10 at 04:29 UT, remapped to the heliographic coordinates system used to compensate the solar differential rotation (see [4] for details on the projection method).", "The white contour delimits the region automatically detected in the outer part of NOAA AR 11499 by the algorithm described in [3].", "It clearly delineates a bundle of loops whose length can estimated from magnetic field extrapolations to be about 280 Mm.", "The black box defines the area manually selected by [20] for detailed analysis, which encloses the region where the maximum of power is observed.", "The dashed box delineates a nearby reference region of identical surface area chosen outside the white contour or excess power.", "The time series of intensities averaged over these boxes, normalized to their standard deviation $\\sigma _0$ , are shown in the top left panels of Figure REF and REF , respectively.", "Data gaps, defined as the intervals during which no data exist within 30 s of an integer number of minutes since the beginning, represent 0.7% of the sequence and are represented by the vertical gray bars, the height of which also represents the range of variation of the intensity.", "The gaps have been filled with linear interpolations between the nearest data points.", "Since we used a one minute-cadence sample of the original 12 s cadence AIA data, the remainder of the time series was considered evenly spaced and thus kept as-is.", "The TC98 code zero-pads the times-series up to the next-higher power of two ($2^{14}$ in this case), while for Fourier analysis we apodized them using the Hann window.", "Figure: Fourier and wavelet analysis of the intensity time series averaged over the black box of Figure .", "The bottom left panel shows the whitened wavelet spectrum (see §).", "The peak of Fourier power labeled h1 at 30 μ\\mu Hz (9 hr) has a 1.7×10 -8 1.7\\times 10^{-8} probability of occurrence.", "The corresponding Fourier component is over-plotted on the time series in magenta.", "The equivalent peak in the time-averaged spectrum is also well above the global confidence level (yellow curve).", "It corresponds to the elongated structure visible at the same frequency in the wavelet spectrum.", "Such a long-lived structure has a 7×10 -11 7\\times 10^{-11} probability of occurrence at this frequency.", "The vertical streak of wavelet power above 2×10 2 2\\times 10^2 μ\\mu Hz around 118 hr is caused by the two small short-lived impulsive events visible in the time series.Figure: Same as Figure  for the reference region of Figure  (dashed box).", "Taking into account all the degrees of freedom, no significant power surpasses the global confidence levels in the Fourier, time-averaged wavelet or wavelet spectra.", "To facilitate comparison, the power spectra use the same scaling.In Figures REF and REF , the right-hand panel gives the Fourier (histogram-style) and time-averaged wavelet (black solid line) spectra on a log-log scale.", "As discussed in § , the power spectra of solar coronal time series generally exhibit a power-law dependence with frequency caused by a background of stochastic fluctuations.", "However, while the power spectra of Figures REF and REF exhibit an overall power-law behavior, they depart significantly from a true power-law, as do many of the spectra studied by [3].", "Several effects indeed commonly produce complex spectral shapes.", "If different regimes of turbulence are present in the observed plasma, one can expect the spectrum to have one or several breaks due to the different power-law exponents in different frequency ranges, like in the solar wind [7].", "In addition, the superimposition on the background emission of one or a few sporadic transients also creates humps in the spectrum.", "In practice, the envelope of the power spectrum of many types of pulses can be modeled by a kappa function $\\text{K}_\\rho (\\nu )=\\left(1+\\frac{\\nu ^2}{\\kappa \\rho ^2}\\right)^{-\\frac{\\kappa +1}{2}},$ $\\rho $ being the width of the PSD and $\\kappa $ defining the extent of its high-frequency wing.", "We thus chose to fit the Fourier spectrum with a background model of the form $\\sigma (\\nu )=A\\nu ^s+B\\text{K}_\\rho (\\nu )+C.$ The first term represents the power-law dependence caused by the background of stochastic fluctuations present in most solar coronal time series.", "The second term is a kappa function that accounts for the possible presence of pulses in the time series.", "Finally, the constant $C$ corresponds to the high-frequency white noise component expected from photon statistics.", "In the right-hand panels of Figures REF and REF , the solid red curve shows the resulting models of mean power $\\sigma (\\nu )$ , and the three dashed red curves correspond to the individual components.", "The two power spectra could be satisfactorily fitted with the same model, the reduced chi-squares being 1.7 and 1.8 respectively.", "In both cases, the white noise components dominate above 2 $\\mu $ Hz and the power-law slopes $s$ are similar.", "The kappa function components create humps around 3 $\\mu $ Hz in both cases because they have similar widths, which is the signature of transients of similar duration in the two time series.", "At this stage, it is convenient (and justified from Equation REF ) to normalize the spectra to $\\sigma (\\nu )$ in order to better visualize possible deviations from the random $\\chi _2^2$ distributed variations around the mean power.", "The middle panels display the same information as the right panels, but for the whitened power spectra.", "Given the number of data points, the 95% global Fourier confidence level is at 11.4 $\\sigma $ (Equation REF , gray lines).", "In Figure REF , the peak at 30 $\\mu $ Hz labeled h1 reaches 26.3 $\\sigma $ , which corresponds to a global probability of occurrence of $1.7\\times 10^{-8}$ (Equation REF ), while the Fourier power spectrum of the reference time series is everywhere below the 95% global confidence level.", "At 5.8 $\\sigma $ , the time-averaged wavelet spectrum also peaks well above the corresponding 95% global confidence level (yellow curve, 2.9 $\\sigma $ at 30 $\\mu $ Hz from Equations REF and  REF -).", "The associated global probability is $6\\times 10^{-7}$ , i.e.", "35 times larger than that derived from the Fourier spectrum, yet still extremely small.", "The reference time-averaged wavelet power exceeds the 95% local confidence level at 20 $\\mu $ Hz (orange curve), but this nearby peak is excluded at the 95% global confidence level.", "The single Fourier component corresponding to the h1 peak is plotted in magenta in the top left panel of Figure REF .", "Comparison with the time series indicates that, except for the last pulse, the repetition period of 9 hr is very regular.", "The bottom left panels of Figures REF and REF show the whitened Morlet wavelet power spectra, i.e.", "normalized at each time step by the estimated mean Fourier power (red curves in the right-hand panels).Note that by using the same background at each time step, one assumes the stationarity of the random process against which the significance of the observed power is tested (TC98).", "They are the counterparts of the whitened spectra of the middle panels.", "From Equation REF , the 95% local confidence level is at 3 $\\sigma $ and, as expected given the large number of points in the spectrum, many points outside the COI lie above it for both time series (orange contours).", "The 95% global confidence level is at 14 $\\sigma $ (Equations REF and REF -REF ).", "In the reference wavelet spectrum, the power outside the COI does not exceed 8 $\\sigma $ .", "In contrast, in the bottom left panel Figure REF , the main peak of power, around 30 $\\mu $ Hz and 35 hours, reaches 16.3 $\\sigma $ .", "This peak has a $4.7\\times 10^{-3}$ global probability to occur by chance (Equations REF and REF ).", "Around this frequency, the power remains above the 95% local confidence level during most of the time series, except between 100 and 115 hr.", "Using Equation REF , the probability of occurrence of a structure of this length at a given frequency is $7.3\\times 10^{-11}$ , which is comparable to the local probability associated with the corresponding peak in the time-averaged spectrum, as expected from § REF and Figure REF ." ], [ "SUMMARY", "The wavelet code described in TC98 is widely used in a variety of scientific fields, as indicated by the 1741 citations of the paper referenced in the Astrophysics Data System as of 2016 March 21.", "It is used both for the analysis of observations [30], [9], [61], [11], [38], [36] and of numerical simulations [42], [39], [48].", "The popularity of this code is due to its ease of use, its ability to produce clear and convincing graphics, as well as its output of rigorous quantitative confidence levels.", "However, it should not be used as a black box, for the confidence levels are linked to background models and, as we have demonstrated in § , the white and red noise models built into the code generally cannot represent the power spectra of solar coronal time series.", "The problem is potentially worse if detrending is applied to the time series, for this pre-processing distorts its power spectrum (section ).", "In addition, even assuming an adequate background model, the confidence levels are local, i.e.", "they do not take into account the total number of degrees of freedom in the wavelet and time-averaged wavelet spectra.", "In most cases, it is thus likely that at least one bin of the spectrum lies above the TC98 confidence levels.", "Both effects – improper background model and local confidence levels – are prone to produce false positives.", "These limitations of the TC98 code can nevertheless be easily overcome.", "First, a noise model suited to the considered data set must be found.", "We propose in §  a function (Equation REF ) that satisfactorily fits the power spectra of many time series, but the adequacy of the chosen model should always be verified.", "Generalizing the principle introduced by [51], global confidence levels taking into account the total number of degrees of freedom can then be computed for the Fourier, wavelet and time-averaged wavelet spectra, as described in § .", "The Appendix describes the practical details.", "Note that the empirically derived coefficients are valid only for the Morlet and Paul wavelets and for time series of up to $2^{14}$ samples.", "Other Monte-Carlo simulations should be carried out to determine their equivalents for other wavelets and/or longer series.", "Following this methodology, we re-analyzed in §  one of the SDO/AIA intensity time series in which [20] detected 9 hr period pulsations.", "The Fourier and time-averaged wavelet spectra both exhibit a strong peak at 30 $\\mu $ Hz with associated global probabilities of occurrence below $10^{-6}$ .", "The corresponding structure in the wavelet spectrum lasts for most of the sequence and also peaks above the 95% global confidence level.", "In contrast, no significant power could be detected in a nearby reference region, which implies a sharp spatial boundary of the detected periodic phenomenon, as was already visible in the power maps (Figure 4) of [20].", "The present analysis of the Fourier and wavelet confidence levels, combined with our previous investigation of potential instrumental and geometrical artefacts [3], lead us to conclude beyond reasonable doubt that the detected pulsations are of solar origin.", "The authors would like to thank John Leibacher for relentlessly trying to demonstrate the instrumental origin of the pulsation phenomena reported here.", "The authors acknowledge the use of the wavelet code by [57].", "The authors acknowledge the use of SDO/AIA data.", "This work used data provided by the MEDOC data and operations centre (CNES/CNRS/Univ.", "Paris-Sud), http://medoc.ias.u-psud.fr/.", "Following the design principles advocated by [58], we aimed to maximize the data-ink ratio of our graphics." ], [ "HOW TO USE THE TC98 CODE WITH CUSTOM NOISE MODELS AND CONFIDENCE LEVELS", "Typos in the original version of the paper (as published in ApJ) prevented the code snippets provided in the Annex from running properly.", "These typos have been corrected in the present version of the paper.", "In addition, a full demonstration code is available at https://idoc.ias.u-psud.fr/MEDOC/wavelets_tc98 Here we provide practical details on how to use the TC98 code with any noise model (e.g.", "the one described in § , Equation REF ) instead of the built-in white or red noises, and with the global confidence levels introduced in § .", "Let data be an n element long 1D floating point array containing a time series of step dt.", "The Morlet wavelet power spectrum is obtained with the following Interactive Data Language commands mother = 'Morlet' s0 = 2*dt dj = 1/8.0 j1 = FIX(alog(n/2.0)/alog(2)/dj) wave = WAVELET(data,dt,PERIOD=period,S0=s0,PAD=1,DJ=dj,$                                 J=j1,MOTHER=mother,COI=coi,SCALE=scale) power = ABS(wave)^2 Let us now assume that we have fitted the Fourier power spectrum of the time series with, e.g., Equation REF and that the result as a function of the period values returned by the above call to the WAVELET function is stored in the 1D floating point array background_fit_period.", "The trick to compute the corresponding confidence levels is to exploit the GWS keyword that is normally provided in the code to use the time-averaged wavelet spectrum as a background noise model.", "For a local confidence level of 95% we use local_siglvl = 0.95 local_signif = WAVE_SIGNIF(data,dt,scale,0, $                            SIGLVL=local_siglvl,GWS=background_fit_period,MOTHER=mother) local_signif = REBIN(TRANSPOSE(local_signif),n,j1+1) The last line replicates the local_signif 1D array at every time step to form an array of the same size as power.", "The points of the wavelet spectrum above the 95% local confidence level then correspond to the elements of the power array greater than local_signif.", "Now for a global confidence level of 95%, we first use Equations REF -REF , to compute the corresponding local confidence level global_siglvl = 0.95 jcoi = ALOG(coi/1.033/s0)/ALOG(2)/dj Nout = TOTAL(jcoi>0) a_coeff = 0.810*(Nout*dj)^0.011 n_coeff = 0.491*(Nout*dj)^0.926 local_siglvl = 1 - (1 - global_siglvl^(1/n_coeff))^(1/a_coeff) Note that we estimate Nout – the number of bins of the spectrum outside the COI – by using the coi array returned by the WAVELET function, with 1.033 being the Fourier factor for the Morlet wavelet (see TC98).", "Then we use the same syntax as above to call the WAVE_SIGNIF function, the only difference being the different value of the local_siglvl variable global_signif = WAVE_SIGNIF(data,dt,scale,0, $                             SIGLVL=local_siglvl,GWS=background_fit_period,MOTHER=mother) global_signif = REBIN(TRANSPOSE(global_signif),n,j1+1) The points of the wavelet spectrum above the 95% global confidence level then correspond to the elements of the power array greater than global_signif.", "Similarly, we can compute the 95% global confidence level for the time-averaged wavelet spectrum after using Equations REF , REF and  to compute the corresponding local significance level Sout = MAX(jcoi) a_scl_coeff = 0.805+0.45*2^(-Sout*dj) n_scl_coeff = 1.136*(Sout*dj)^1.2 time_avg_local_siglvl = 1 - (1 - global_siglvl^(1/n_scl_coeff))^(1/a_scl_coeff) dof = n - scale/dt time_avg_global_signif = WAVE_SIGNIF(data,dt,scale,1, $                                      SIGLVL=time_avg_local_siglvl, $                                      GWS=background_fit_period,DOF=dof,MOTHER=mother) The WAVE_SIGNIF function is now called with setting the sigtest argument to 1 and setting the DOF keyword to the number of data points minus half the number of those in the COI.", "The resulting array has the same number of elements as the time averaged spectrum and is used to determine which bins are above the 95% global confidence level." ] ]
1606.05251
[ [ "HierarchicalForecast: A Reference Framework for Hierarchical Forecasting\n in Python" ], [ "Abstract Large collections of time series data are commonly organized into cross-sectional structures with different levels of aggregation; examples include product and geographical groupings.", "A necessary condition for coherent decision-making and planning, with such datasets, is for the dis-aggregated series' forecasts to add up exactly to the aggregated series forecasts, which motivates the creation of novel hierarchical forecasting algorithms.", "The growing interest of the Machine Learning community in cross-sectional hierarchical forecasting systems states that we are in a propitious moment to ensure that scientific endeavors are grounded on sound baselines.", "For this reason, we put forward the HierarchicalForecast library, which contains preprocessed publicly available datasets, evaluation metrics, and a compiled set of statistical baseline models.", "Our Python-based framework aims to bridge the gap between statistical, econometric modeling, and Machine Learning forecasting research.", "Code and documentation are available in https://github.com/Nixtla/hierarchicalforecast." ], [ "Introduction", "Large collections of time series organized into structures at different aggregation levels often require their forecasts to follow their aggregation constraints, which poses the challenge of creating novel algorithms capable of coherent forecasts.", "A fundamental component in developing practical methods is extensive empirical evaluations and comparing newly proposed forecasting methods with state-of-the-art and well-established baselines; regarding this, Machine Learning research on hierarchical forecasting faces two obstacles: Statistical Baseline's Absence: Eventhough Python continues to grow in popularity among the Machine Learning community [35], it lacks a lot of statistical and econometric model packages, for which its forecasting research struggles with access to these baselines.", "This problem exacerbates by the most recent advancements in the hierarchical forecasting literature.", "Statistical Baseline's Computational Efficiency: The Python global interpreter lock limits its programs to use a single thread, which translates into a lost opportunity to speed up algorithms by leveraging the available CPU resources.", "When implemented naively, statistical baselines in Python take an excessively long execution, even surpassing those of more complex methods, which discourages their use.", "We tackle these challenges, by putting forward BlueVioletHierarchicalForecast, an open-source benchmark for hierarchical forecasting tasksLicense: CC-by 4.0, see BlueViolethttps://creativecommons.org/licenses/by/4.0/.", "Code and documentation available in BlueViolethttps://github.com/Nixtla/hierarchicalforecast.", ".", "Our work builds upon the fastest (to the best of our knowledge) ETS/ARIMA open-source Python implementations to improve the availability of hierarchical forecasting baselines and provide utilities for evaluating and forecasting hierarchical time series systems." ], [ "Hierarchical Forecasting Notation", "We denote a cross-sectional hierarchical time series by the vector $\\mathbf {y}_{[a,b],\\tau } = [\\,\\mathbf {y}^{\\intercal }_{[a],\\tau }\\,|\\,\\mathbf {y}^{\\intercal }_{[b],\\tau }\\,]^{\\intercal } \\in \\mathbb {R}^{N_{a}+N_{b}}$ , for the time step $\\tau $ , where $[a],[b]$ denote respectively the aggregate and bottom level indices.", "The total number of series in the hierarchy is $|[a,b]| = (N_{a}+N_{b})$ .", "We distinguish between the time indices $[t]$ and the $h$ steps ahead indices $[t+1:t+h]$ , and bottom and aggregate indexes $\\beta \\in [b]$ , $\\alpha \\in [a]$ .", "At any time $\\tau \\in [t]$ , a hierarchical time series is defined by the following aggregation constraints in matrix representation: $\\mathbf {y}_{[a,b],\\tau } = \\mathbf {S} \\mathbf {y}_{[b], \\tau }\\quad \\Leftrightarrow \\quad \\begin{bmatrix}\\mathbf {y}_{[a],\\tau }\\\\\\mathbf {y}_{[b],\\tau }\\end{bmatrix}= \\begin{bmatrix}\\mathbf {A}_{[a][b]}\\\\\\mathbf {I}_{[b][b]}\\end{bmatrix}\\mathbf {y}_{[b],\\tau }$ Under this notation the summing matrix $\\mathbf {S} \\in \\mathbb {R}^{(N_{a}+N_{b}) \\times N_{b}}$ controls the aggregation of the bottom series to the upper levels, and it is composed of an aggregate matrix $\\mathbf {A}_{[a][b]} \\in \\mathbb {R}^{N_{a} \\times N_{b}}$ and the identity matrix $\\mathbf {I}_{[b][b]} \\in \\mathbb {R}^{N_{b} \\times N_{b}}$ .", "In Figure REF example $N_{a}=3$ , $N_{b}=4$ , and $\\begin{aligned}y_{\\mathrm {Total},\\tau } = y_{\\beta _{1},\\tau }+y_{\\beta _{2},\\tau }+y_{\\beta _{3},\\tau }+y_{\\beta _{4},\\tau }\\qquad \\qquad \\qquad \\qquad \\qquad \\\\\\mathbf {y}_{[a],\\tau }=\\left[y_{\\mathrm {Total},\\tau },\\; y_{\\beta _{1},\\tau }+y_{\\beta _{2},\\tau },\\;y_{\\beta _{3},\\tau }+y_{\\beta _{4},\\tau }\\right]^{\\intercal }\\qquad \\mathbf {y}_{[b],\\tau }=\\left[ y_{\\beta _{1},\\tau },\\; y_{\\beta _{2},\\tau },\\; y_{\\beta _{3},\\tau },\\; y_{\\beta _{4},\\tau } \\right]^{\\intercal }\\end{aligned}$ Figure: Three-level time series hierarchical structure example, with four bottom level variables, marked with a gray background.", "In this description, each node represents non-overlapping series for a single point in time.The constraints matrix associated to Figure REF and Equations (REF ) is: $\\mathbf {S}=\\begin{bmatrix}\\\\\\mathbf {A}_{\\mathrm {[a][b]}} \\\\\\\\ \\\\\\mathbf {I}_{\\mathrm {[b][b]}} \\\\\\\\\\end{bmatrix}=\\begin{bmatrix}\\;1 & 1 & 1 & 1 \\\\\\;1 & 1 & 0 & 0 \\\\\\;0 & 0 & 1 & 1 \\\\ \\hline \\;1 & 0 & 0 & 0 \\\\\\;0 & 1 & 0 & 0 \\\\\\;0 & 0 & 1 & 0 \\\\\\;0 & 0 & 0 & 1 \\\\\\end{bmatrix}$" ], [ "Hierarchical Forecasting Baselines", "The classic approach to hierarchical forecasting has been a two-stage process where base level forecasts are produced $\\hat{\\mathbf {y}}_{[a,b],\\tau }$ and later reconciled into coherent forecasts $\\tilde{\\mathbf {y}}_{[a,b],\\tau }$ [17].", "Reconciliation can be represented in the following notation: $\\tilde{\\mathbf {y}}_{[a,b],\\tau } = \\mathbf {S} \\mathbf {P} \\hat{\\mathbf {y}}_{[a,b],\\tau }.$ Where $\\mathbf {S} \\in \\mathbb {R}^{(N_{a}+N_{b}) \\times N_{b}}$ is the hierarchical constraints matrix, $\\mathbf {P} \\in \\mathbb {R}^{N_{b}\\times (N_{a}+N_{b})}$ is a matrix that maps base forecasts into bottom level, specified by the reconciliation strategies.", "Bottom-Up: This method constrains the base-level predictions to the bottom-level series, which are usually treated as independent [31].", "The reconciliation of the method is a simple addition to the upper levels.", "Its $\\mathbf {P}$ matrix is defined $\\mathbf {P}_{\\text{BU}}=[\\mathbf {0}_{\\mathrm {[b][a]}}\\,|\\,\\mathbf {I}_{\\mathrm {[b][b]}}]$ , where the first columns collapses the aggregate level predictions and the identity picks only the bottom-level forecasts.", "Top-Down: The second method constrains the base-level predictions to the top-most aggregate-level serie and then distributes it to the disaggregate series through the use of proportions $\\mathbf {p}_{[b]}$ .", "Its $\\mathbf {P}$ matrix is defined $\\mathbf {P}_{\\text{TD}}=[\\mathbf {p}_{\\mathrm {[b]}}\\,|\\,\\mathbf {0}_{\\mathrm {[b][a+b-1]}}]$ , the rest of the columns zero-out the base forecast below the highest aggregation.", "Several variants of the methods emerge depending on the strategy for the computation of $\\mathbf {p}_{[b]}$ [11], [9].", "Alternative: Recent hierarchical reconciliation strategies, transcend the base forecasts' single level origin, and define the $\\mathbf {P}$ optimally under reasonable assumptions.", "Middle Out: This method is only available for strictly hierarchical structures.", "It anchors the base predictions in a middle level.", "The levels above the base predictions use the bottom-up approach, while the levels below use a top-down.", "Minimum Trace: [46] computed the reconciliation matrix $\\mathbf {P}$ to minimize the total forecast variance of the space of coherent forecasts, with the Minimum Trace reconciliation.", "Under unbiasedness assumptions for the base forecasts, in this method, $\\mathbf {P}_{\\text{MinT}}=(\\mathbf {S}^{\\intercal }\\mathbf {W}^{-1}_{\\tau }\\mathbf {S})^{-1}\\mathbf {S}^{\\intercal }\\mathbf {W}^{-1}_{\\tau }$ and $\\mathbf {W}_{\\tau }=\\mathrm {Var}[\\,(\\mathbf {y}_{[a,b],\\tau }-\\hat{\\mathbf {y}}_{[a,b],\\tau })\\,]$ is the base predictions' variance-covariance matrix.", "Empirical Risk Minimization: The ERM approach relaxes the unbiasedness assumption and optimizes the reconciliation matrix minimizing an L1 regularized objective [4].", "The method also has a closed-form solution considering only the reconciliation quadratic errors.", "$\\mathbf {P}_{\\text{ERM}} = \\text{argmin}_{\\mathbf {P}} ||\\mathbf {y}-\\mathbf {S} \\mathbf {P} \\hat{\\mathbf {y}} ||^{2}_{2} + \\lambda ||\\mathbf {P}-\\mathbf {P}_{\\text{BU}}||_{1}$ The BlueVioletHierarchicalForecast package contains a curated collection of reference hierarchical forecasting algorithms through the BottomUp, TopDown, MiddleOut, MinTrace, and ERM model classes.", "Currently, the collection is restricted to point and mean forecasting methods, but we are working on implementing coherent probabilistic algorithms." ], [ "Dependencies", "The BlueVioletHierarchicalForecast library is built with minimal dependencies using NumPy for linear algebra and array operations [13], Pandas for data manipulation [26] and sklearn [34] for data processing.", "Additionally, the auto-ARIMA and auto-ETS [18] base forecast models depend on NumBa and open-source just-in-time compiler that translates Python and NumPy code into C++ currently these are Python's fastest implementations." ], [ "Evaluation Metrics", "For the evaluation of the forecasting algorithms we provide access to the following initial prediction accuracy metrics that have been used in past hierarchical forecasting literature.", "Mean/Point Forecasting Accuracy: To measure the point forecasts accuracy, we summarize the forecast errors into the mean absolute scaled error (MASE), and the mean squared scaled error (MSSE), using $\\hat{y}^{^{\\prime }}_{i,\\tau }$ the Naive1's in the denominator errors, their definition is inspired by [19]: $\\mathrm {MASE}(\\mathbf {y}_{i}, \\mathbf {\\hat{y}}_{i}, \\mathbf {\\hat{y}^{^{\\prime }}}_{i})&= \\frac{\\sum ^{t+H}_{\\tau =t+1} |y_{i,\\tau } - \\hat{y}_{i,\\tau }|}{\\sum ^{t+H}_{\\tau =t+1} |y_{i,\\tau } - \\hat{y}^{^{\\prime }}_{i,\\tau }|} \\qquad \\mathrm {MSSE}(\\mathbf {y}_{i}, \\mathbf {\\hat{y}}_{i}, \\mathbf {\\hat{y}^{^{\\prime }}}_{i})= \\frac{\\sum ^{t+H}_{\\tau =t+1} (y_{i,\\tau } - \\hat{y}_{i,\\tau })^{2}}{\\sum ^{t+H}_{\\tau =t+1} (y_{i,\\tau } - \\hat{y}^{^{\\prime }}_{i,\\tau })^{2}}$ Quantile Forecasting Accuracy: Transitioning from point predictions, towards probabilistic predictions we measure the accuracy of individual quantiles $\\hat{y}^{(q)}_{i,\\tau } = \\hat{F}^{-1}_{i,\\tau }(\\,q\\,)$ of a predictive distribution $\\hat{F}_{i,\\tau }$ using the quantile loss (QL) defined as: $\\mathrm {QL}(\\hat{y}^{(q)}_{i,\\tau }, y_{i,\\tau }) =\\mathrm {QL}(\\hat{F}^{-1}_{i,\\tau }(q), y_{i,\\tau }) =2\\left(\\mathbb {1}\\lbrace y_{i,\\tau }\\le \\hat{y}^{(q)}_{i,\\tau }\\rbrace -q\\right)\\left(\\hat{y}^{(q)}_{i,\\tau }-y_{i,\\tau }\\right)$ Probabilistic Forecasting Accuracy: For a fully probabilistic prediction, we use the continuous ranked probability score (CRPS) [24]The CRPS uses a left Riemann numeric approximation of the integral and averages a discrete set of uniformly distanced quantile losses., that summarizes the accuracy of the entire predictive distribution $\\hat{F}_{i,\\tau }$ , the CRPS definition is: $\\mathrm {CRPS}(\\hat{F}_{i,\\tau }, y_{i,\\tau }) =\\int ^{1}_{0}\\mathrm {QL}(\\hat{F}^{-1}_{i,\\tau }(q), y_{i,\\tau }) dq$" ], [ "Datasets", "We are making the following preprocessed five publicly available hierarchical datasets easily accessible through the Pandas and NumPy libraries.", "Each dataset is accompanied by metadata capturing its seasonality/frequency, the forecast horizon used in previous hierarchical forecast publications, its corresponding hierarchical aggregation constraints matrix, and the names of its levels.", "Hierarchical forecasting studies have used these datasets in the past.", "We briefly describe the datasets in Table REF .", "In more detail, the available datasets are: Traffic measures the occupancy of 963 traffic lanes in the Bay Area, the data is grouped into a year of daily observations and organized into a 207 hierarchical structure [8].", "Tourism-S consists of 89 Australian location quarterly visits series; it covers from 1998 to 2006.", "Several hierarchical forecasting studies have used this dataset in the past [44].", "Tourism-L summarizes an Australian visitor survey managed by the Tourism Research Australia agency, the dataset contains 555 monthly series from 1998 to 2016, and it is organized into geographic and purpose of travel [45].", "Labour reports monthly Australian employment from February 1978 to December 2020.", "It contains a hierarchical structure built by the labour categories [3].", "Wiki2 contains the daily views of 145,000 Wikipedia articles from July 2015 to December 2016.", "The dataset is filtered and processed into 150 bottom series and 49 aggregate series [2], [4]." ], [ "Dataset Class", "Each dataset class contains load and preprocessing methods that output readily available hierarchical time series as well as its aggregation's constraints matrix, and hierarchical indexes for each level Additionally, each dataset contains metadata that includes the frequency and seasonality of the data, hierarchical indexes for the convenient evaluation across the levels of the hierarchical or grouped structures, and the dates." ], [ "Base Predictions and Reconciliation", "In this subsection, we demonstrate the use of the BlueVioletHierarchicalForecast library to predict twelve months of the 555 series of the Tourism-L dataset using auto-ARIMA base model and later reconcile using the BottomUp , MinTrace , and ERM classes.", "from datasetsforecast.hierarchical import HierarchicalData from hierarchicalforecast.core import HierarchicalReconciliation from hierarchicalforecast.methods import BottomUp, MinTrace, ERM from statsforecast.core import StatsForecast from statsforecast.models import autoarima Load TourismL dataset Ydf, S, tags = HierarchicalData.load('./data', 'TourismLarge') Ydf = Ydf.setindex('uniqueid') Compute base auto-ARIMA predictions fcst = StatsForecast(df=Ydf, models=[(autoarima,12)], freq='M', njobs=-1) Yhatdf = fcst.forecast(h=12) Reconcile the base predictions reconcilers = [ BottomUp(), MinTrace(method='ols'), MinTrace(method='wlsstruct'), MinTrace(method='wlsvar'), MinTrace(method='mintshrink'), ERM(method='lasso'), ] hrec = HierarchicalReconciliation(reconcilers=reconcilers) Yrecdf = hrec.reconcile(Yhatdf, Ydf, S, tags)" ], [ "Hierarchical Forecast Evaluation", "Our library facilitates a complete evaluation across the levels of the hierarchical structure.", "We evaluate the reconciliation strategies' predictions using the MASE metric in this example.", "In a complete hierarchical forecasting pipeline example is available in this BlueVioletjupyter notebook, we perform a thorough evaluation with a rolling window cross-validation hyperparameter selection on the available hierarchical forecasting methods for the Traffic, Labour, Wiki2, Tourism-S, and Tourism-L datasets.", "from hierarchicalforecast.evaluation import HierarchicalEvaluation, mase evaluator = HierarchicalEvaluation(evaluators=[mase]) evaluator.evaluate(Yh=Yhatdf, Ytest=Ytest, tags=tags, benchmark='naive')" ], [ "Conclusion", "We presented BlueVioletHierarchicalForecast, an open-source python library dedicated to hierarchical time series forecasting.", "The library integrates publicly available processed datasets, evaluation metrics, and a curated set of highly efficient statistical baselines.", "We provide usage examples and references to extensive experiments where we showcase the baseline's use and evaluate the accuracy of their predictions.", "With this work, we hope to contribute to Machine Learning forecasting by bridging the gap to statistical and econometric modeling, as well as providing tools for the development of novel hierarchical forecasting algorithms rooted in a thorough comparison of these well-established models.", "We intend to continue maintaining and increasing the repository, promoting collaboration across the forecasting community." ], [ "Acknowledments", "The authors want to thank Alejandro Alvarez, Shibo Zhou, and José Morales for their contributions.", "Additionally we thank Chirag Nagpal and for the insightful conversations.", "Appendix Open-source Python Forecasting Libraries The Python forecasting ecosystem continues to grow at a steady pace.", "Regarding classic statistical and econometric models and implementations, like ARIMA, ETS, GARCH among others, statsmodels, tf_sts, kats, and pyflux have low-level implementations based on NumPy.", "While other libraries like gluonts, tf_sts, flowforecast, and neuralforecast.", "The gluonts has Python API connections to R forecasting libraries that enable comparisons with these well-established methods, with the cost of R dependencies frictions.", "Finally, higher-level libraries like darts, sktime, tslearn, pmdarima, seglearn, have outstandingly curated, improved usability, and made available a large variety of models.", "We refer to [20] and [38] for complete surveys.", "Table: Summary of some popular open-source forecasting libraries.", "Open-source Hierarchical Forecasting Libraries Hierarchical Forecasting continues to grow in popularity as shown by the international competitions (GEFCOM2012; [15]) and (M5; [23]), and the growing interest of the Machine Learning community on the topic [36], [33], [12].", "As mentioned in Section , for a long time, there has been an absence of reliable statistical and econometric methods that exacerbates with forecasting-subfields with recent innovations.", "One possible explanation is the poor computational efficiency of previous Python implementations of ARIMA and ETS methods that could not leverage the full multi-core capabilities of the machines.", "The forecasting community has made an effort and started the adoption of recently published NumBa implementations of the methods available in the statsforecast package, like darts, and sktime two of the most popular open-source forecasting frameworks in Python.", "The work of gluonts, darts, scikit-hts, sktime, and pyhts spearheaded the Python's hierarchical forecasting availability.", "A summary is available in Table REF .", "With our work, we seize the opportunities to increase the available hierarchical methods beyond those valuable contributions and to perform a robust validation of the performance of said implementations to ensure that the Python community has access to efficient and reliable baselines.", "Table: Hierarchical forecasting libraries.", "Hierarchical Forecasting Evaluation To complement the examples from Section .", "Here we perform a thorough hierarchical forecasting experiment on the Labour, Tourism-L, Tourism-S, Traffic, and Wiki2 datasets, comparing the predictions accuracy of BottomUp, MiddleOut, MinTrace, and ERM methods using the mean absolute scaled error (MASE) using $\\hat{y}^{^{\\prime }}_{i,\\tau }$ , the Naive1 forecast in the denominator.", "$\\mathrm {MASE}(\\mathbf {y}_{i}, \\mathbf {\\hat{y}}_{i}, \\mathbf {\\hat{y}^{^{\\prime }}}_{i}) = \\frac{\\sum ^{t+H}_{\\tau =t+1} |y_{i,\\tau } - \\hat{y}_{i,\\tau }|}{\\sum ^{t+H}_{\\tau =t+1} |y_{i,\\tau } - \\hat{y}^{^{\\prime }}_{i,\\tau }|} $ We use two settings for the experiments; one uses a rolling window approach where we fit and predict a horizon of length defined in Table REF , using the last 25% of the dataset as the test set.", "For the second experiment, we restrict the evaluation to the last window of the forecast's horizon length.", "Table REF reports the rolling window evaluation, and Table REF reports the single window evaluation.", "Table: Empirical evaluation of hierarchically coherent forecasts.", "Mean absolute scaled error (MASE).", "These experiments use a rolling window evaluation where we use the last 25% of the available observations to test.Table: Empirical evaluation of hierarchically coherent forecasts.", "Mean absolute scaled error (MASE).", "These experiments use a single window evaluation where we use the last available observations to test." ], [ "Open-source Python Forecasting Libraries", "The Python forecasting ecosystem continues to grow at a steady pace.", "Regarding classic statistical and econometric models and implementations, like ARIMA, ETS, GARCH among others, statsmodels, tf_sts, kats, and pyflux have low-level implementations based on NumPy.", "While other libraries like gluonts, tf_sts, flowforecast, and neuralforecast.", "The gluonts has Python API connections to R forecasting libraries that enable comparisons with these well-established methods, with the cost of R dependencies frictions.", "Finally, higher-level libraries like darts, sktime, tslearn, pmdarima, seglearn, have outstandingly curated, improved usability, and made available a large variety of models.", "We refer to [20] and [38] for complete surveys.", "Table: Summary of some popular open-source forecasting libraries." ], [ "Open-source Hierarchical Forecasting Libraries", "Hierarchical Forecasting continues to grow in popularity as shown by the international competitions (GEFCOM2012; [15]) and (M5; [23]), and the growing interest of the Machine Learning community on the topic [36], [33], [12].", "As mentioned in Section , for a long time, there has been an absence of reliable statistical and econometric methods that exacerbates with forecasting-subfields with recent innovations.", "One possible explanation is the poor computational efficiency of previous Python implementations of ARIMA and ETS methods that could not leverage the full multi-core capabilities of the machines.", "The forecasting community has made an effort and started the adoption of recently published NumBa implementations of the methods available in the statsforecast package, like darts, and sktime two of the most popular open-source forecasting frameworks in Python.", "The work of gluonts, darts, scikit-hts, sktime, and pyhts spearheaded the Python's hierarchical forecasting availability.", "A summary is available in Table REF .", "With our work, we seize the opportunities to increase the available hierarchical methods beyond those valuable contributions and to perform a robust validation of the performance of said implementations to ensure that the Python community has access to efficient and reliable baselines.", "Table: Hierarchical forecasting libraries." ], [ "Hierarchical Forecasting Evaluation", "To complement the examples from Section .", "Here we perform a thorough hierarchical forecasting experiment on the Labour, Tourism-L, Tourism-S, Traffic, and Wiki2 datasets, comparing the predictions accuracy of BottomUp, MiddleOut, MinTrace, and ERM methods using the mean absolute scaled error (MASE) using $\\hat{y}^{^{\\prime }}_{i,\\tau }$ , the Naive1 forecast in the denominator.", "$\\mathrm {MASE}(\\mathbf {y}_{i}, \\mathbf {\\hat{y}}_{i}, \\mathbf {\\hat{y}^{^{\\prime }}}_{i}) = \\frac{\\sum ^{t+H}_{\\tau =t+1} |y_{i,\\tau } - \\hat{y}_{i,\\tau }|}{\\sum ^{t+H}_{\\tau =t+1} |y_{i,\\tau } - \\hat{y}^{^{\\prime }}_{i,\\tau }|} $ We use two settings for the experiments; one uses a rolling window approach where we fit and predict a horizon of length defined in Table REF , using the last 25% of the dataset as the test set.", "For the second experiment, we restrict the evaluation to the last window of the forecast's horizon length.", "Table REF reports the rolling window evaluation, and Table REF reports the single window evaluation.", "Table: Empirical evaluation of hierarchically coherent forecasts.", "Mean absolute scaled error (MASE).", "These experiments use a rolling window evaluation where we use the last 25% of the available observations to test.Table: Empirical evaluation of hierarchically coherent forecasts.", "Mean absolute scaled error (MASE).", "These experiments use a single window evaluation where we use the last available observations to test." ] ]
2207.03517
[ [ "Code Translation with Compiler Representations" ], [ "Abstract In this paper, we leverage low-level compiler intermediate representations (IR) to improve code translation.", "Traditional transpilers rely on syntactic information and handcrafted rules, which limits their applicability and produces unnatural-looking code.", "Applying neural machine translation (NMT) approaches to code has successfully broadened the set of programs on which one can get a natural-looking translation.", "However, they treat the code as sequences of text tokens, and still do not differentiate well enough between similar pieces of code which have different semantics in different languages.", "The consequence is low quality translation, reducing the practicality of NMT, and stressing the need for an approach significantly increasing its accuracy.", "Here we propose to augment code translation with IRs, specifically LLVM IR, with results on the C++, Java, Rust, and Go languages.", "Our method improves upon the state of the art for unsupervised code translation, increasing the number of correct translations by 11% on average, and up to 79% for the Java -> Rust pair with greedy decoding.", "With beam search, it increases the number of correct translations by 5.5% in average.", "We extend previous test sets for code translation, by adding hundreds of Go and Rust functions.", "Additionally, we train models with high performance on the problem of IR decompilation, generating programming source code from IR, and study using IRs as intermediary pivot for translation." ], [ "Introduction", "Automatic code translation allows to port old codebases to new frameworks, or high-level (but slow) languages to low-level (and fast) ones.", "Current industry solutions, known as transpilers or transcompilershttps://en.wikipedia.org/wiki/Source-to-source_compiler, rely on handcrafted rules that are applied systematically.", "They produce unidiomatic translations that prove hard to read for human programmers.", "This is a serious limitation: the translated code should be easy to read and understand, as it will eventually be maintained by human developers.", "In recent years, Neural Machine Translation (NMT) was proposed as an alternative to rule-based code translation [34], [42], [43].", "These models, trained from existing human-readable code, produce idiomatic, easy to understand, translations.", "Unfortunately, neural transpilers are unreliable, and often produce code that would not even compile or execute.", "This is, again, a serious limitation: if the human work saved by the transpiler has to be reinvested debugging its output, there is little interest in using it.", "We propose to improve the reliability of NMT by leveraging information from compiler toolchains.", "When processing source code, compilers create Intermediary Representations (IR): language-agnostic pseudocode that describes the semantics of the program.", "Augmenting training data with the corresponding IR can benefit a Neural Transpiler in two ways: it helps align embeddings for different languages, and serves as a pivot for translation between two languages.", "As shown in Figure REF , this can greatly improve the semantic quality of neural translations.", "Decompilation, translating from a low-level representation of a program (e.g.", "assembly) to a high-level one (source code), is another application of IR-augmented Neural Machine Translation.", "A compiler is used to produce the low-level IR , and the model is trained to invert the mapping from source code to low-level representation.", "In this work, we leverage LLVM [25] to augment source code with corresponding Intermediary Representation and train models for code translation and decompilation.", "We experiment with four languages: C++ Java, Rust and Go, and show that adding IR to the training data allows for an average increase of 11% in the number of correct translations.", "Our main contributions are: We implement and evaluate two methods leveraging LLVM IRs for code translation, improved representations with IR and using IR as pivot: we show that improved representations allow us to increase the number of correct translations generated by TransCoder for C++, Java, Go and Rust by 11%, we study the pivot method to recast the problem of translation as a problem of decompilation of the source's IR in a new language, we extend the parallel evaluation dataset of 852 functions in C++, Java and Python from [34] with 343 more functions in Go and 280 more in Rust with corresponding test cases.", "We achieve 78% accuracy when decompiling LLVM IRs to C++, a large improvement over existing tools, and achieve a similar performance across the four languages we consider.", "Figure: Improvements over TransCoder.", "The first example shows a translation from C++ to rust, where TransCoder generates code using unsigned instead of signed integers.", "In the second example, a translation from Java to Go, it generates a function with the wrong return type.", "In the third example, which is also a translation from Java to Go, the model outputs a function that looks similar to the correct solution but it confuses > with >> and closes an expression with a parenthesis too early.In these cases and many others, TransCoder makes mistakes that are small in terms of edit distance, but have a large impact on the semantics of the code.Using the IR to ground the representations to the semantics often helps solving these issues." ], [ "Related Works", "Source-to-Source Translation.", "Many rule-based methods are available for transpilation, an inventory of which can be found at REF .", "In particular, C2Rust https://github.com/immunant/c2rust and CxGo https://github.com/gotranspile/cxgo, along with manual corrections, were central for us in translating evaluation tests to Go and Rust (See Section REF ).", "Similarly, 2to3https://docs.python.org/2/library/2to3.html, a Python library porting Python 2 code to Python 3, was used in [1] to create a parallel dataset and train a machine learning model.", "Neural Machine Translation for code is hampered by the lack of parallel data between programming languages.", "Indeed, apart from a few language pairs, such as Java-C# [31], [7], and specific domains (e.g.", "competitive programming code), it is difficult to collect large datasets of semantically equivalent code in different languages.", "TransCoder  [34] aims to bridge this gap by introducing Unsupervised Machine Translation, taking advantage of large monolingual code bases to learn to translate between C++, Python and Java with high performance.", "Later, DOBF  [22] improved the model pre-training method used in TransCoder, and [35] used automatically generated unit tests to improve translation performance between Java, C++ and Python.", "Recently, large language models trained on code, such as Davinci Codex[6] and PALM [8], have been used for unsupervised code translation.", "Starting from the outputs from Transcoder, [42] and [43] survey the links between humans and NMT methods for code translation, and view neural translation methods as aids to programmers and not as a final goal.", "In this context, they demonstrate that even imperfect models can improve the quality of an engineer's work for code translation, and plead for an improvement of human-machine interfaces.", "Decompilation.", "Like transpilation, decompilation is usually performed using rule-based methods that rely on pattern matching to parse the control flow structure of the program.", "RetDec, an open source decompiler created by Avast [20], can decompile an executable to C and a Python-like language via LLVM IR, and took seven years to be completed by a team of 24 developers https://blog.fpmurphy.com/2017/12/avast-retargetable-decompiler-ida-plugin.html.", "Other tools exist, such as the Hex-Rays Decompiler https://hex-rays.com/decompiler/ and [5], and a thorough review of rule-based methods can be found in papers such as [26] and [18].", "With these methods, decompilation can fail if the code is too convoluted, or if it contains language features that were not explicitly translated.", "Most methods also produce unstructured programs, relying on a large number of goto statements to simulate the control flow of the lower level programming languages.", "This is semantically correct, but very rarely found in human-written code.", "There has been only few works on neural decompilation, using sequence-to-sequence neural networks.", "[18] uses LSTM networks to decompile LLVM IRs and assembly code to C. Their approach generates code templates, based on the IRs, that will determine the structure of the output, then fills them with correct variable assignments and numerical values.", "In the same vein, [13] tries to address limitations of neural decompilation, with two sequential phases: code sketch generation and iterative error correction.", "Finally, [27] use a method close to ours, and train Transformer models to translate between binary code and C. Intermediate representations are almost as old as compiler design.", "The first IR, UNCOL [38] was introduced in the mid-1950s, together with the idea of reusing the same compiler for several languages and machines.", "In 1960, NELIAC (a variant of ALGOL) [15] was the first retargetable compiler, portable to different architectures.", "Feldman [11] describes how a compiler for Fortran 77 can be added to the C compilers of Johnson [16] and Ritchie [33].", "GCC [37] introduces Register Transfer Language (RTL) a low-level IR inspired by [9], and then GENERIC and GIMPLE [30], precursors of the IR used in LLVM [25].", "Figure: A bird's eye view of a compiler toolchain, exemplified with LLVM.The unoptimized version (-O0) is shown here for illustration.", "In practice we used the size-optimized version (-Oz) of the IR as boxed, which does the compile time optimization of computing the addition of 26 and 16." ], [ "Intermediate representations in compilers", "Compilers translate programs written in a computer language into executable code for a specific machine.", "Most compilers consist of a front-end taking source code as input, and a back-end, which produces machine binary code.", "The front-end lexes (tokenizes) and parses the program.", "Then, it produces an abstract syntax tree (AST), and translates it into some Intermediate Representation (IR).", "The back-end converts the IR into machine-specific executable code.", "In modern compilers, like LLVM [25], the IR is generic across different input languages (and thus different front-ends).", "This allows transformations and target agnostic optimizations to be applied to the IR, in a middle-end module independent from the source language and target machine.", "This results in an efficient compiler structure: new languages can be implemented by rewriting the front-end, and new target machines by rewriting the back-end.", "Several IRs usually co-exist in a compiler: each stage in the toolchain (Figure REF ) introduces a new representation.", "Early stage IRs are language-dependent (e.g.", "ASTs mirror the syntax of the source language).", "Late stage IRs replace named variables by registers and reflect the specifics of the target architecture.", "In this work, we are interested in middle-end IRs, which are independent from the target machine, and similar for all source languages (like dialects in natural languages)." ], [ "Training Data", "Our monolingual training data was extracted with Google BigQuery, which indexes over 2.8 million open source repositories from GitHub https://console.cloud.google.com/marketplace/details/github/github-repos.", "We selected projects whose license explicitly permits re-distribution of parts, and extracted all individual C++, Java, Rust and Go functions.", "To learn to decompile IRs, we also used the CodeNet dataset [32], a repository of 14 million competitive programming solutions in 55 languages.", "Our models work at function level: this reduces compilation failures over missing dependencies, while keeping sequence length short.", "Table: Dataset coverage across languages, in number of standalone functions." ], [ "Generating Intermediate Representations", "While the LLVM ecosystem is large, not every language has an LLVM front-end, and not every front-end can produce LLVM IR out-of-the-box.", "We use clang++ https://clang.llvm.org/ [25] from the established LLVM C++ compilation toolchain, JLang https://polyglot-compiler.github.io/JLang/ for Java, Gollvm https://go.googlesource.com/gollvm/ for Go and rustc [29] for Rust.", "For the same program, written in different languages, different front-ends may produce different IR.", "To minimize these variations, we process the source code as follows.", "First, we generate the most size-optimized IR (-Oz flag), which makes the IR more uniform across languages.", "Second, we strip all unnecessary information (e.g.", "header and footer with attributes, debug information, comments).", "Finally, block names are canonicalized and symbol names demangled to facilitate their recovery.", "The functions that fail to compile at this point (e.g.", "because of missing dependencies) are not included in the parallel dataset, as seen in the last row of Table REF ." ], [ "Evaluation", "Traditional NMT evaluation relies on metrics such as BLEU, that measure the semantic similarity between pieces of text.", "However, when dealing with programming languages, syntax and in particular compilation and computation outputs can be very different despite minor changes in the code.", "Conversely, semantically equivalent code, that differ only in variable names or order of operations can have a low BLEU score.", "To take this into account, we use and enhance the computational accuracy test suite from [34], that contains 852 parallel competitive programming solutions in C++, Java and Python.", "Using C2Rust, CxGo and some manual code cleaning, we translated 280 functions and test suites in Rust and 343 in Go to measure the performance of our models in these languages.", "We measure our performance using the computational accuracy (CA@1) metric [21], [34], which considers that a translation is correct if it passes a all tests." ], [ "Training objectives", "Unsupervised machine translation consists of two tasks: training language embeddings for each language and aligning them [24].", "We now present the training objectives associated with those tasks.", "In Section REF , we briefly describe the objective functions of our baseline NMT system, TransCoder.", "These objectives are also used for training all of our models.", "In Section REF , we present our methods for training an IR decompiler and a code translation model using the IR as a pivot.", "In Section REF , we present three new objectives leveraging LLVM IRs to improve multilingual representations of source code, and the translation capabilities of machine translation models.", "Compared to the pivot, this method does not require to generate IRs at test or inference time.", "More formally, we denote $x = x_{1}\\dots x_{N_{so}}$ the source sentence, $z^{(x)} = z^{(x)}_{1}\\dots z^{(x)}_{N_{ir}}$ the corresponding IR, and $y = y_{1}\\dots y_{N_{ta}}$ the target sentence.", "Let $\\mathcal {L}_{CE}(\\hat{x}, x) = \\sum _i \\ell _{CE}(\\hat{x_i}, x_i)$ , with $\\ell _{CE}(\\hat{x_i}, x_i)$ the pairwise cross-entropy loss of $\\hat{x_i}$ and $x_i$ , and $\\mathcal {L}_{MT}(x, y)$ be the machine translation loss (also named seq2seq loss) from $x$ to $y$ (the lengths of $x$ and $y$ can differ).", "$\\mathcal {L}_{MT}$ is the negative log-likelihood of tokens in $y$ given $x$ and previous tokens in $y$ : $\\mathcal {L}_{MT}(x, y)=-\\sum _i \\log \\left(P(y_i | x, y_{1}\\dots y_{i-1})\\right)$" ], [ "Common Objective Functions", "TransCoder  [34] learns to translate between programming languages using three unsupervised objectives developed for natural language processing [10]: Masked Language Modeling (MLM).", "The Masked Language Modeling (MLM) trains an encoder to recover randomly masked inputs.", "It is commonly used to pre-train embeddings for natural [10], [28] and programming languages [17], [12].", "Using this objective on code allows the model to quickly learn the syntax and achieve lower losses than with natural languages.", "Other objectives have also been developed for programming languages [14], [22], [2], [41], but MLM remains quite effective and easy to use on a wide range of programming languages.", "It uses the following loss: $\\mathcal {L}_{MLM} = \\mathcal {L}_{CE}\\left(enc(mask(x)), x\\right).$ Denoising Auto Encoding (AE).", "The denoising Auto-Encoding (AE) objective corrupts a sequence and trains a seq2seq model to retrieve the original sequence.", "Sequence corruption is done by randomly masking spans of tokens, removing and shuffling tokens.", "The loss is: $\\mathcal {L}_{AE} = \\mathcal {L}_{MT}\\left(noise(x), x\\right).$ Back-Translation (BT).", "Back-Translation [36] uses the existing model to generate a noisy translation of the input sentence.", "Then, it trains the model to regenerate an input from the noisy translation.", "It is a simple yet powerful objective for unsupervised machine translation [24], [3].", "More formally, for a every code snippet $x$ , we generate a noisy translation $\\hat{y}$ in other languages and train the model to reverse it with the loss: $\\mathcal {L}_{BT} = \\mathcal {L}_{MT}\\left(\\hat{y}, x\\right)$" ], [ "IR decompilation.", "In this supervised task, we use LLVM to generate IR from source code, and train a language model to reverse the process, i.e.", "learn to predict the source code from the IR.", "Models are pre-trained using the MLM and AE objectives, then decompilation is learned using the loss: $\\mathcal {L}_{Decomp} = \\mathcal {L}_{MT}\\left(z^{(x)}, x\\right)$ Figure: IR Decompilation objective.", "Here, we generate the IR corresponding to each function and train a model to decompile it.The IR pivot model uses this objective, as well as back-translation objectives, allowing it generalize to IRs generated from any language.s" ], [ "IR Pivot.", "In this task, we use the IR as a pivot for code translation.", "For instance, to translate from Rust to C++, we first translate a Rust program into IR and then from IR to C++.", "The IR corresponding to isolated functions is generated by the LLVM compiler.", "Then, we translate the IR into source code in the target language, by a process similar to neural IR decompilation.", "Naive implementations of this task prove challenging.", "Simply training a neural decompiler model on all (IR, language) pairs leads to poor performance, because slight variations exist between the IR generated for different languages (i.e.", "C++-IR and Rust-IR behave like dialects of the LLVM IR).", "To mitigate this, instead of using a single IR language embedding (like we do in all other tasks), we create separate embeddings for each IR dialect (i.e.", "IRs generated from different languages).", "We then use back-translation to align the embeddings for different dialects, i.e.", "train models to translate from any IR into any programming language.", "More formally, for every code snippet $x$ in language $l_{so}$ , we generate a noisy translation $\\hat{z}^{(y)}$ in the IR dialect corresponding to language $l_{ta}$ ($l_{ta}$ IR) and train with $\\mathcal {L}_{MT}\\left(\\hat{z}^{(y)}, x\\right)$ .", "For every $l_{so}$ IR code fragment $z^{(x)}$ , we generate a noisy translation $\\hat{y}$ into language $l_{ta}$ and train with $\\mathcal {L}_{MT}\\left(\\hat{y}, z^{(x)}\\right)$ .", "We use the encoder of the IR decompilation model to pre-train the encoder for this model.", "For the decoder, pre-training from the IR decompilation decoder often leads to training failure, because the model “learns” back-translation by translating a language into itself.", "Instead, we initialize the decoder randomly and use denoising auto-encoding (AE) for every source language and IR dialect at train time." ], [ "IR for code representations", "Leveraging IR to help train translation models is a promising idea, but providing the IR and source code as the input to our model creates practical difficulties.", "At inference, before it can be translated, every code snippet would need to be compiled with LLVM, to generate the IR.", "To leverage LLVM IR at train time, without needing the IR at inference, we add these three objectives to those presented in Section REF .", "Figure: IR for code representation objectives.", "We show examples of masking (used in TLM and TAE) and IR generation used to improve code representations with IRs.The masking objective in TLM or TAE makes the model understand the relationship between code and IR.The IR generation objective helps the model to build semantic representations of the code.For instance, another C++ function computing 39+3\\texttt {39 + 3} would result in the same IR.A Go function that returns 42 would also have a similar LLVM IR.Therefore, the IR Generation objective encourages the model to build similar representations for these three semantically equivalent functions." ], [ "Translation Language Modeling (TLM).", "First introduced in [23], the translation language modeling objective aims to generate common representations for parallel sentences in different languages.", "It is similar to the masked language modeling (MLM) objective, as it trains an encoder to recover random masks.", "The main difference is that TLM is trained on parallel sentences which are concatenated in the same sequence.", "In our case, we concatenate (symbolized by $|$ ) standalone functions and their IRs and train to recover random masks.", "It allows the model to find relevant information in the IR to fill the blanks and vice versa.", "The tokens at the beginning of the sequence are given token embeddings corresponding to the source code language, and the next tokens (corresponding to the IR) are given the language embeddings of the IR.", "$\\mathcal {L}_{TLM} = \\mathcal {L}_{CE}\\left(mask(x|z^{(x)}), x|z^{(x)}\\right)$" ], [ "Translation Auto-Encoding (TAE).", "The translation auto-encoding (TAE) objective is a generalization of the TLM objective to auto-encoding.", "Here, we add noise and masks on the source code and it's corresponding IRs and concatenate the two modified sequences.", "Like for TLM, we make sure that the token embeddings of each token correspond to whether it is a token from the IR or from the source code.", "$\\mathcal {L}_{TAE} = \\mathcal {L}_{MT}\\left(noise(x)|noise(z^{(x)}), x|z^{(x)}\\right)$" ], [ "IR Generation (MT).", "We use machine translation (MT) steps to learn to generate LLVM IRs from the corresponding source code.", "Basically we train a model to compile into IR.", "The goal is to make the encoder learn representations of the source code based on the semantics contained in the IR.", "$\\mathcal {L}_{IRGen} = \\mathcal {L}_{MT}\\left(x, z^{(x)}\\right)$ These objectives require parallel data of code and corresponding IRs.", "Since we could only compile a fraction of the functions and files, we also train our models on the full monolingual data using the MLM and AE objectives described above.", "In this setup, the back-translation (BT) objective is the same as in [34]." ], [ "Experimental details", "For TransCoder, we consider a sequence-to-sequence (seq2seq) transformer model [40] with attention [4], [39] Our model has 12 layers (6 in the encoder and 6 in the decoder), 8 attention heads, and a hidden dimension of 1024.", "For the objectives that add noise and masks to the input sentence, such as MLM, TLM, AE, and TAE, we choose the masked tokens and noise randomly on the fly at each epoch.", "We mask 15% of the tokens in MLM and TLM.", "In AE and TAE, we mask 20% and drop 10% of the tokens.", "MLM is trained on streams of data, while the other objectives are trained at function level.", "We use the Adam optimizer [19] and an inverse squared-root learning rate scheduler, with an initial learning rate of $10^{-5}$ in most of our experiments.", "Our models are implemented in PyTorch using half-precision floats.", "The pre-trained models were trained until convergence.", "The translation models presented in Tables REF and REF were trained for a week on 32 NVIDIA V100 GPUs." ], [ "Decompilation", "We tried two separate configurations for decompilation: a shared decoder with 6 layers for all language / IR pairs, or four separate decoders of with two layers each (one per language).", "Using a shared decoder improves the performance for all languages, and particularly when the data is scarce (e.g.", "Rust).", "See results in Table REF .", "As a baseline, we use RetDec [20] that is a standard tool to decompile LLVM IR language to C. It obtains a computational accuracy (see Section REF ) of 38.1 on our C++ dataset and a BLEU score of 8.54, which is far below our model that obtains a computational accuracy of 77.9 and a BLEU score of 63.6 in the same setting.", "In general, the computation accuracy of the rule-based methods is lower because they fail to decompile many of the files.", "Table: Performance of LLVM IRs Decompilation.", "This table shows the computational accuracy (CA@1) of our neural decompiler and the RetDec C++ rule-based decompiler.Our neural decompiler outperforms RedDec on C++ and is more broadly applicable.Table: Translation performance (CA@1).", "See Table  in the appendix for more detailed results.", "All combinations of the TLM, MLM and TAE objectives improve the performance of the model.", "Our best results are obtained when using all three at the same time." ], [ "Translation", "We compare our results to the performance of a TransCoder  [34] MLM model trained on the same dataset.", "Our new TLM, TAE, and IR generation (MT) objectives are able to use the IR to improve the semantic representations of the model.", "As shown on Table REF , they lead to improved performances compared to the TransCoder baseline (pre-trained with MLM) when translating from and to any of our four languages.", "In average, the improvement reaches 4.4% points, or an 11% relative improvement.", "They are especially useful in the low data regime: with relative improvements reaching 19.3% when translating to Rust and 25.6% when translating from it.", "Qualitatively, it helps the model translate types correctly when the source and target types are represented by different tokens (e.g.", "int and i32 in Figure REF ) and to better translate the actual semantics of the input function.", "Improving code representations using the IR with the TLM, TAE and MT objectives leads to models that are just as simple to use as TransCoder: computing the IR is not required at test time and the model generates the translation directly from the source function.", "Pivot translation achieves non-trivial performances for translating to and from any language.", "The IR is computed using a compiler, and the model only needs to generate code in the target language from the IR.", "Hence, the IR pivot performs relatively well when translating from low-resource to high-resource languages (e.g.", "from Rust), and badly when translating to low-resource languages (e.g.", "to Rust).", "However, the methods using IRs to improve code representations outperform the IR pivot method, even for translating to Rust." ], [ "Pivot vs Embedding", "TransCoder is an unsupervised model that learns to align code representations and translate code from one language into another.", "It does not use the IRs for translation.", "The pivot method uses automatically generated parallel sentences to learn to decompile IRs, and back-translation to adapt to different IR dialects.", "It performs relatively well when little data is available for the source language, because the IR can be computed using a rule-based compiler.", "However, it requires to compute IRs at test time, which can be cumbersome.", "This method learns to translate using only IR-level similarities, and does not use the source code itself to learn to translate.", "Adding the TLM, TAE, and MT objectives to the objectives generally used for unsupervised code translation allows the model to get the best of both worlds.", "It can learn multilingual representations of source code from similarities in the IR and in the source code itself, and outperforms both TransCoder and the pivot.", "At the same time, this model does not require to compute IRs at test time, and is as easy to use as TransCoder.", "Using both types of input allows our model to outperform both TransCoder and the pivot method, as shown in Table REF (last row)." ], [ "Different IR and interpreted languages", "The four languages considered in this work have front-ends that can output LLVM Intermediary Representation.", "LLVM presently covers more than 30 computer languages.", "Using IR as pivot requires that the source and destination language have front-ends that use the same IR.", "This rules out some widely-used languages (e.g.", "Python).", "Using the IR to improve embeddings is less restrictive: the source and destination language can be trained on different IR, and aligned with back-translation.", "In this paper, we focus on compiled languages, but it is important to note that Intermediary Representations are usually available for interpreted languages as well: modern interpreters translate the source code into byte-code, that can serve as an IR." ], [ "Conclusion", "In this paper, we leverage LLVM IRs to improve neural machine translation for source code.", "The IR provides a common semantically-rich language, into which C++, Go, Java and Rust code can all be compiled.", "We develop three objectives, designed to leverage IRs for better multilingual representations of source code, which lead to a 11% relative average improvement for code translation.", "We also show that sequence-to-sequence transformers perform well for neural decompilation, and use this for pivot translation.", "We only worked with the LLVM IR, but our approach is broadly applicable to any pair of languages that share a common Intermediate Representation.", "More generally any IR can help improve the code representations by tying them to the semantics.", "Another limitation is the scale of our current source and target sequences.", "As future work, LLVM IRs could be generated at a larger scale by compiling entire projects, which would greatly improve the percentage of successful IR compilations in Table REF .", "More languages and IRs could be used, and those extensions could be powered by larger models." ] ]
2207.03578
[ [ "Deep Learning to Jointly Schema Match, Impute, and Transform Databases" ], [ "Abstract An applied problem facing all areas of data science is harmonizing data sources.", "Joining data from multiple origins with unmapped and only partially overlapping features is a prerequisite to developing and testing robust, generalizable algorithms, especially in health care.", "We approach this issue in the common but difficult case of numeric features such as nearly Gaussian and binary features, where unit changes and variable shift make simple matching of univariate summaries unsuccessful.", "We develop two novel procedures to address this problem.", "First, we demonstrate multiple methods of \"fingerprinting\" a feature based on its associations to other features.", "In the setting of even modest prior information, this allows most shared features to be accurately identified.", "Second, we demonstrate a deep learning algorithm for translation between databases.", "Unlike prior approaches, our algorithm takes advantage of discovered mappings while identifying surrogates for unshared features and learning transformations.", "In synthetic and real-world experiments using two electronic health record databases, our algorithms outperform existing baselines for matching variable sets, while jointly learning to impute unshared or transformed variables." ], [ "Introduction", "A perennial problem in modern data science is integrating multiple data sources.", "We focus on the problem in which similar or identical concepts are measured in distinct data sources, commonly referred to as the schema matching problem [10].", "Manually reconciling large scale databases where the “columns” (database attributes) have not been previously mapped to a common ontology is an expensive and error prone process.", "Our motivating example comes from healthcare, where electronic health records (EHR) store thousands of distinct types of observations, many of them using redundant or ambiguous names.", "Combining these databases is a necessity for the development of large-scale registries, generalizable clinical decision support tools, and measuring performance in small subpopulations.", "Where possible, matching the meta-data from two sources (such as column names) is a useful approach, reviewed briefly below.", "However, in many cases it is not sufficient and using summary statistics of a column as a fingerprint (instance-based matching) is both necessary and powerful [35].", "The EHR context is particularly challenging for instance-based matching using summary statistics for several reasons.", "First, a large fraction of EHR data columns are binary, meaning that other than the frequency and missingness ratio no univariate summary statistics are possible.", "All binary columns with similar frequencies are indistinguishable potential matches in a univariate approach.", "Second, a shift in variable distributions is the norm when concatenating databases from distinct contexts.", "This can occur both naturally, for example, when populations in two hospitals have different weight and age distributions, and artificially when systems use incompatible units to record observations.", "For example, in EHR data originating in the United States, weight, height, pressure, and concentration could be recorded in SI or US conventional units.", "More generally, up to a unit change, any approximately normally distributed set of variables will be possible matches.", "Finally, some columns may be simple but nonlinear re-parameterizations of others; for example, body surface area of height and weight.", "This final problem confounds many prior approaches to matching the distribution of features from multiple databases.", "We propose a novel deep-learning based solution to the schema matching problem inspired by distributional semantics.", "That is, the relationships between entries for a given column and other database columns define the semantics of that column.", "For example, indicators for a patient having diabetes, overweight, high blood pressure, and hyperlipidemia are a cluster known clinically as “metabolic syndrome,\" and this cluster of correlation could be identified in each dataset.", "Relationships between variables are plausibly unchanged across databases even with substantial variable shift, especially if they are causal relationships.", "In some special cases, meaningful matching can be performed with no prior information, but we focus on the more realistic case where a few columns in two databases are known to be semantically identical.", "That is, we assume that at least some features are mapped by auxiliary knowledge or manual verification (known-mapped columns).", "We then use autoencoders to compress the data to a relatively low-dimension latent space, using the known-mapped columns to anchor the latent spaces of the autoencoders together.", "Those autoencoders contain the pattern of dependency between columns (the semantics) in each database.", "To extract human-readable interpretations from the semantics of the autoencoders, we take a novel knowledge distillation approach.", "We create “chimeric” encoders that transform from the format of the first database to the second.", "To identify additional matching variables, we then compute cross-format associations between the original and transformed data.", "We use a simple matching algorithm to pair columns from the first database to the second, maximizing the sum of correlations between proposed matches.", "A step-down procedure is used to reject low correlation proposed matches while controlling the overall false discovery rate.", "The discovery of low-dimensional transformations is facilitated by graphical analysis and mutual information between the original and transformed data.", "These chimeric encoders also function to impute features between databases where an exact match is not found.", "This is a very common occurrence in medical databases where a feature is not measured in some sources, but strong correlates are.", "For example, in our real dataset, the diagnosis of atrial fibrillation (a common abnormal heart rhythm) may not be explicitly noted, but the presence of medications related to atrial fibrillation allow it to be inferred reasonably well.", "Remarkably, the algorithm does not require the “imputed” variable to be present in both databases.", "A complete pipeline of our approach is presented in Figure REF .", "To summarize, we make the following contributions: We propose a novel summary statistic (fingerprint) method that solves the schema matching problem with moderate accuracy and high efficiency.", "Our algorithm identifies surrogates for features that are not shared by both databases but have a closely-related concept available.", "Our chimeric encoders learn to jointly impute and transform samples from one database to another when correlates or re-parameterizations of unshared features are present.", "Figure: An illustration of the KMF initialized chimeric schema matching algorithm with module-wise description at the start of Section .", "It also depicts how the input datasets could be differently sized with few mapped features (first three blue coloured columns with common feature names) and majority unmapped (pink and orange coloured columns with ambiguous feature names)." ], [ "Terminology.", "The literature on related problems has used a confusing variety of terms.", "Each data source will be represented by a database which is a combination of a dataset (a set of tuples each of which represents the information on a single object e.g.", "a patient) and schema information, meta-data, relationships, and restrictions.", "Datasets are also organized by features (also commonly called attributes), which in the simplest cases correspond to a column index in a data matrix.", "We will occasionally refer to a column as the slice of all data for a feature in a dataset.", "Each feature is a measurement of a real-world variable, and features in multiple databases can represent the same variable.", "We refer to these features as mapping to one another.", "Meta-data is information about a feature, such as its name, data type, and ontology tags.", "A fingerprint of a feature is a set of summary statistics about that column; this is also commonly called an embedding.", "The process of discovering features across databases that reflect the same variable is schema matching; features that share a variable are maps of one another.", "When two databases have exactly the same set of variables, they have a bijective map or are permutations of one another.", "When the variables in one database are a proper subset of another, they have an onto map.", "When the two share some but not all variables, they have a partial map.", "Organization.", "In Section REF , we start with a brief review of related literature in schema matching and autoencoders.", "In Section REF , we provide details about our motivating example involving EHRs.", "We present the proposed schema matching algorithm in Section REF .", "In Section , we compare the performance of proposed algorithms to baselines and simple alternatives on multiple synthetic and real-data examples.", "Finally, in Section we discuss the results, alternatives, and agenda for future work." ], [ "Related work", "Prior approaches to schema matching fall into a few categories; a recent review is provided in [1].", "First, meta-data about each feature (such as names) can be compared across databases.", "Modern deep-learning approaches to language embedding can be used to combine text descriptions and other meta-data of a feature before sequence-pair classification, as in [22].", "Properties of the schema (such as restrictions) can narrow the set of potential mappings and fit within this category.", "Second, if specific examples are present in multiple databases (e.g.", "shared patients across hospitals), these can be used to match schema using entity matching algorithms [18].", "Third, fingerprints of each feature can be compared across databases to find the best matches (instance-based matching).", "Authors in [19] provide a comprehensive review of the above categories with pros and cons of the algorithms available in each category and present an interface that can evaluate different types of schema matching methods on a common metric.", "Practical implementations combine these approaches to complement each other [23].", "The use of meta data for schema matching is the first approach to be used if the schema has clear and comprehensible attribute names.", "Authors in [5] review many of the element-level matchers that use the name and description of a column.", "They further demonstrate that a hybrid of these matchers (combining different similarity measures between the entity pairs) using machine learning outperforms the individual matchers.", "Recent work by [30] uses RoBERTa, a transformer-based deep learning model to obtain the embeddings for the attribute names from a given schema and then computes similarity on the generated embeddings.", "Another attention-based deep learning solution is proposed by [39] that only uses attribute names and descriptions.", "Instance-based matching requires a set of summaries of a data column (a fingerprint); one of many matching algorithms is then used to select the best mapping of features of multiple databases based on the similarity of fingerprints.", "Much of this literature uses string data types, where formatting patterns, word frequencies, and neural embedding can identify the semantics of a feature.", "For quantitative data, simple characteristics such as the mean, variance, mode, and fraction missing are commonly used [35], [29].", "Others have proposed more complex fingerprints for continuous data.", "Dhamankar and colleagues [10] computed the Kullback-Leibler divergence between each pair of quantitative columns (that is, using the distribution function as a high dimensional summary).", "Jaiswal and colleagues [13] similarly fingerprint a feature by fitting a Gaussian mixture model to its distribution and compared the fit pairwise to candidate variables.", "Mueller and Smola [25] used a corpus of labeled correspondences between datasets to learn a neural-network based embedding of distributions.", "Because of the large (infinite) dimension of possible embedding of continuous columns, variable reduction has been proposed using penalized regression [3] and random projections [4].", "Notably, the above proposed fingerprints are functions of a single column.", "However, there is good reason to believe that associations between variables (and hence features) will tend to be more reproducible across databases.", "For example, although the fraction of men at a given hospital can vary, the relationship between sex and height is likely very similar.", "Several authors have proposed using the dependencies between columns to define the fingerprint.", "Kang and colleagues [15] proposed a 3 step process: first, in both datasets compute the mutual information of every pair of columns; second, filter potential pairs from across databases as potential mappings based on univariate entropy (or other summary statistics); third, search over assignments from the second database to the first maximizing the similarity of the mutual information matrices.", "They later compared optimization criteria and exhaustive versus heuristic methods for searching over mapping assignments in [16].", "Cruz and colleagues [9] (details unpublished) combined the entropy and mutual information characteristics with a privacy preserving set intersection method to replace graph alignment.", "Authors in [40] presented a similar proposal that replaced mutual information with the earth movers distance and took a stepwise approach to correlation clustering and matching the similarity of clusters.", "Rabinovich and Last [27] modified the Kang approach by filtering potential assignments on absolute correlation (or an equivalent measure for binary variables) and added noise (or knock-off) columns to force the two databases to have the same number of columns.", "In Kang and related approaches, any features which are known to map are fixed during the initialization of the graph search.", "Translating between two databases (i.e.", "estimating for a given row in database A, a row which approximately matches the semantics of database B) is a similar problem, which has been approached by one prior method.", "RadialGAN [36] attempts to generate samples from auxiliary databases in the format of a primary database to support training a classifier.", "Briefly, for each database an autoencoder (AE) is fitted.", "To encourage the latent spaces to have the same semantics, two modification to a standard AE are taken.", "First a cycle consistency loss is added while training the autoencoders by comparing encoded examples to re-encoded translated examples (Mapping A to B, then back to A).", "Second, a discriminator is trained to distinguish true samples from generated samples, and the loss of that discriminator maximized, similar to a GAN except using the latent representation of alternative databases to supply the generator (the decoder) rather than noise.", "After training (when the discriminator can no longer differentiate generated from authentic samples), the generated examples are concatenated to the primary dataset for training a classifier.", "The RadialGAN method does not propose to match the schemas directly, but it produces mechanically similar translations to our chimeric encoders and can be used as input to matching methods.", "The process of transforming estimated similarities between columns into proposed matches and evaluating the result can itself be a complicated process.", "Shraga, Gal, and Roitman review current approaches to this step of the algorithm, including the use of neural networks to learn transformation which optimally adjust estimated similarities [31].", "Another approach that improves the matches is proposed by [38] where algorithms generate questions for crowd-sourced workers to maximize reduction in uncertainty of the matches.", "Authors in [37] also consider this issue while matching multiple schema by finding pairwise schema overlap using some existing similarity measure based schema matches.", "These approaches have substantial deficiencies in our motivating example.", "For example, metadata may be absent or misleading.", "The most common and useful metadata is often the feature name.", "In the Epic EHR, many data elements (“smartdata”) have short, meaningless names assigned differently at different sites; some examples are available in Table REF in Appendix REF .", "Where present, names that appear similar may be distinct concepts, such as “Ur Creatinine” and “POC Creatinine” or may contain ambiguous abbreviations such as “PHTN” representing “pulmonary hypertension,” “portal hypertension,” or “pre-hypertension” which are themselves distinct from “HTN” or “hypertension” as a concept.", "True database schema properties may not be available or preserved, as data is often exported to a flat format.", "Finally, in the healthcare setting, few identical records from the same source will exist between databases in most cases.", "That is, from two hospitals in different states or time periods we expect few contemporaneous records of the same patient.", "However, we fully expect these methods to produce some successful high-confidence matches, and regard them as one source of known-mapped columns in our approach.", "As mentioned above, instance-based fingerprints are also very limited by binary variables and distributional shifts between data sources.", "For example, indicator variables for two diseases with roughly the same frequency would have very similar fingerprints and be vulnerable to mis-mapping.", "Generative adversarial networks (GANs) can fulfil some of the same application needs as schema matching.", "For example, realistic synthetic data in the format of a target database can augment training a classifier.", "Our method uses autoencoders (AE), and many researchers have adapted GANs to improve the training of AE or enforce desired properties; for example, [24] use a discriminator network to make the encoder output similar in distribution to spherical Gaussian data.", "After jointly training the AE and discriminator network, Gaussian random variables can be fed into the AE's decoder to generate new data points.", "Variational AEs such as $\\alpha $ -GAN [28] can be used to generate synthetic data with additional distributional requirements.", "However, a discriminator network may not be necessary for these models.", "The Adversarial Generator-Encoder Networks (AGE) [33] model uses the divergence of two induced latent distributions as a kind of pre-made discriminator.", "Expanding this approach with multiple discriminators and generators/encoders such as done in [20] can stabilize the mapping of data points to a latent space and back, perhaps reducing the mode collapse and training difficulty of many GANs.", "Even though the above methods have the potential to generate augmenting data, they do not directly address all the use cases of a schema-matched dataset and do not exploit the relationships between features across databases." ], [ "ACTFAST", "Clinicians and informaticists in the Anesthesiology department at Washington University School of Medicine developed a database of surgical patients including preoperative clinical characteristics, laboratory data, high-frequency intraoperative monitor data, and postoperative outcomes.", "This database was developed by manually linking multiple existing sources for applications in clinical epidemiology [17] and clinical decision support tools [11].", "It was also linked to an existing ontology as part of a large registry project [8].", "However, in 2018 the hospital switched from the prior combination of EHRs (MetaVision, Allscripts) to Epic.", "Although many variables were able to be mapped using manual exploration, ontology links, and hierarchical representations within Epic, others were not.", "A data query revealed thousands of potential mapping variables, making exhaustive manual review impossible." ], [ "MIMIC", "The Medical Information Mart for Intensive Care (MIMIC) dataset includes routinely collected EHR and administrative data on a large number of critically ill patients, and is the most widely used dataset for experimentation with machine learning in critical care [14].", "One interesting feature of MIMIC is that during the study, the Carevue EHR was replaced with the Metavision EHR.", "Data from many systems which indirectly feed into the EHR, such as laboratory values, has a consistent representation over the entire study, but discrete data which is manually charted (the chartevents table) have inconsistent identification numbers (called d_items) after the change.", "Prior projects 1) https://github.com/USC-Melady/Benchmarking_DL_MIMICIII/blob/master/Codes/mimic3_mvcv/10_get_99plus-features-raw.ipynb 2) https://github.com/YerevaNN/mimic3-benchmarks/blob/master/mimic3benchmark/resources/itemid_to_variable_map.csv have mapped many chartevent d_items from the two eras using column names, units of measure, data distributions, and contextual clues.", "However, many d_items have no known mapping between the two eras.", "In practice, because the most important d_items are believed to have been mapped, machine learning experiments usually ignore the unmapped features or learn distinct patterns in the two eras.", "Because of changes in documentation practice related to the new EHR and temporal changes in patient makeup or medical practices, some shift in feature distributions between these eras is expected." ], [ "Notation", "Assume that we have multiple databases $\\lbrace D^1, D^2, \\ldots , D^I \\rbrace $ to be schema-matched.", "Indexing databases by $i$ , each has $p^i$ features.", "To simplify the notation, we will focus on the case where $I=2, i \\in \\lbrace A , B \\rbrace $ , avoiding the need to index many quantities that depend on the pair of databases.", "When more than two databases need to be joined, one could directly apply pairwise methods in a round-robin fashion.", "The $I=2$ case supports the most common application for medicine: a new database ($D^B$ ) joins an existing consortium ($D^A$ ).", "We distinguish the database $D^i$ (which also contains meta-data) from the contained data $x^i$ .", "We will focus on the case that databases contain only numeric data (i.e., they are matrices in $\\mathbb {R}^2$ ) although extensions to higher-order (tensor) databases and embedding complex objects are possible.", "Denote an encoder for database $i$ as $f^i$ and a decoder $g^i$ ; that is for any example $j$ , $x^i_{j} \\approx \\left(g^i \\circ f^i\\right) (x^i_{j}) $ .", "Denote by $L(,)$ a loss function appropriate to the data type at hand.", "Because $f$ and $g$ operate on row-vectors, we will write them as functions of both tuples ($x^i_{j}$ ) and matrices ($x^i$ ) where unambiguous.", "We require the output spaces of $f^A$ and $f^B$ to be identical in dimension, and we refer to $g^A \\circ f^B$ as the chimeric encoder from $B$ to $A$ .", "We also compute chimeric dependence denoted by $r$ and defined as the dependence between every pair of features from $x^i$ and the output of chimeric encoder (when $x^i$ is given as an input)." ], [ "Proposed algorithms", "This section presents the modules involved in the algorithm in order.", "The complete algorithm is presented in Figure REF .", "The overall flow is as follows: Pre-process both databases to create a tabular representation without explicit missingness.", "Identify a small number of features known to correspond across the two databases using any of the other approaches to schema matching.", "KMF Module.", "Compute an association vector between each unmapped feature and the set of mapped features.", "Use the cosine similarity of these vectors to create a similarity matrix between unmapped features in the two databases.", "Gale-Shapley Matching Module.", "Transform the above similarity matrix to a discrete set of proposed mappings using the rankings of similarities as an affinity between two features.", "Chimeric autoencoder module.", "Using the features mapped in the pre-processing and KMF modules, fit a pair of AE with a common latent space.", "Using the encoder from one dataset and decoder from the other, estimate paired samples of original data and data transformed to the format of the other database.", "Discover mappings in transformed data.", "Create a similarity matrix between known-mapped features and transformed features using a bivariate association measure.", "Use the Gale-Shapley module to identify additional matches, surrogates, and transformed variables." ], [ "Pre-processing", "In our real-data example, we use several pre-processing steps.", "First, we assume that a subset of variables are pre-mapped in the two databases.", "This can be based on the multiple sources of metadata discussed in the “related work” section above.", "For continuous variables, although unit conversions can cause unexpected failures, nearly matching ranges, means, or variance are likely to be identical and this information can be included in the pre-matching stage.", "Similarly, the numerous methods for matching text variables can be applied and that text can be transformed into continuous variables via neural embedding.", "In practice, this would likely be followed by a manual verification stage.", "Even if exploring all potential matches is impossible, checking some initial guesses with a content expert can likely be accomplished quickly.", "We assume that $K$ variables are matched with certainty weights $w_k$ with $k \\in [1, K]$ .", "Also, for clarity in indexing, the known-mapped variables are moved to the front of each database in the same order.", "We one-hot encode categorical variables in each dataset.", "Our justification for this step is that categorical variables are semi-arbitrarily split and combined in real data sets, so forcing a $1:1$ match between databases may fail unexpectedly.", "We unit-norm quantitative variables because, as mentioned above, we suspect that unit conversions occur commonly and because the mean and scale information are expected to have been used as part of the initialization of known-mapped columns.", "This step also simplifies the development of our neural network-based procedures, which can be adversely affected by inputs on very different scales.", "Although not required, we greatly simplify the presentation by assuming that preliminary exploration (for example, using a single autoencoder) has been performed to detect and merge features that are identical concepts within a single dataset.", "This assumption is necessary to define a “correct” mapping rather than an equivalence-class of features.", "We also assume that missing data in each database has already been imputed; however, modifications to the proposal to allow missing data are straightforward.", "In our example, we use imputation with predictive mean matching [6] to fill in missing data.", "For higher-dimensional objects like time series, an approach like Gaussian process adapters [21] can embed irregularly sampled data." ], [ "Known-Map Association Fingerprints (KMF)", "As a fast alternative or initialization for the Chimeric encoder, we use a simple correlation-based approach.", "We create a fingerprint of each unmapped column using the $K$ -vector of correlation to the mapped columns.", "We use the Pearson correlation coefficient because it is invariant to scale and centering, which are affected by unit of measure changes and assumed to have already been used in the verification of known mapped columns.", "The matrix of cosine similarity between fingerprints can then be estimated across databases and maximized by the matching algorithm below." ], [ "Mapping via Gale-Shapley", "After creating a similarity matrix between features, many methods have been used to infer a set of mappings, and there is no uniformly optimal way to do so [31].", "The Gale-Shapley algorithm is a widely-used algorithm for matching preferences of “applicants” and “reviewers” which has the advantage of depending only on the ranks of $r$ rather than absolute values [12].", "Before running Gale-Shapley, a-priori impossible matches (such as those with different types or extremely different marginal distributions) can be removed, as are variables with known matches.", "Because of the extensive prior work on creating univariate summaries to use as a filter reviewed in Section REF , we do not recapitulate that development here.", "Gale-Shapley produces locally optimal matches (no pairwise trades are mutually beneficial) and is robust to misrepresentation of preferences by applicants, meaning that matches are never degraded by improving the optimization of $r$ or addition of irrelevant alternatives.", "In the examples where the number of features is identical in $D^A$ and $D^B$ , the “stable marriage” version of Gale-Shapley is used; otherwise, the “hospital-resident” version is used with a single acceptance slot per “hospital”.", "After creating proposed mappings, we determine a threshold at which matches are so weak that they are likely false positives.", "Because sample correlation has well-known statistical properties, we compute approximate $p$ -values for each proposed mapping in a hold-out sample.", "We use the Benjamini–Yekutieli procedure [2] on the $p$ -values to set an acceptance threshold which controls the false discovery rate under arbitrary correlation, as implemented by the Pingouin Python module [34].", "Since, KMF is computationally inexpensive, we identify the proposals (new matches) above a threshold of similarity or a top fraction of proposals and add them to the “known map” set before running the chimeric encoder in Section REF ." ], [ "Chimeric translation", "Even though KMF is simple and works efficiently for schema matching, it does not have the capability to identify surrogates for features with no match or to transform features.", "Therefore, we propose chimeric encoders that in addition to matching, can jointly transform and impute features between two databases.", "The overall approach is illustrated in Figure REF .", "Define an autoencoder (AE) for each database using the notations in Section REF : $ \\hat{x}^i = g^i( f^i( x^i) ) ,$ and define a chimeric encoder: $ z^{i,i^{\\prime }} = g^{i^{\\prime }}( f^i( x^i) ) .$ In the 2-database case, we omit the redundant index and label $z^{i,i^{\\prime }}$ as $z^{i^{\\prime }}$ .", "With databases with very different numbers of features, a shared output dimension of the encoders can have unacceptably poor reconstruction accuracy, in which case an adapter function would need to be applied between $f^i$ and $g^{i^{\\prime }}$ in equation (REF ).", "During training, the total loss is the reconstruction loss of each AE ($\\sum _i L(\\hat{x}^i , x^i)$ ) plus any regularization losses, plus a cross-reconstruction loss (encode, decode to other space) on the chimeric encoder which considers only the $K$ known-mapped features: $ \\sum _{k=1}^K w_k L(z^{B}_{k} , x^B_{k}),$ plus the symmetric expression in $A$ .", "We also add a cycle-consistency loss (encode, decode to the other space, re-encode, decode to original space): $ L( g^B( f^A( g^{A}( f^B( x^B) ) ) ), x^B ),$ plus the symmetric expression reconstructing $X^A$ .", "The paired samples $(x^i,z^{i^{\\prime }})$ are the key inputs to the next step of the algorithm.", "The rationale for the loss in equation (REF ) is that encoded space elements which are important to reconstruct the $K$ mapped features will also tend to be important in reconstructing the unmapped variables, similar to the latent components in principal components analysis.", "Requiring that the two encoded spaces have similar relationships to the known-mapped columns will hopefully encourage that they have similar relationships to the unmapped columns.", "The cycle consistency loss (REF ) also encourages the encoded spaces to have similar or identical semantics because they are used to cross-decode across databases.", "Unlike the cross-reconstruction loss (REF ), the cycle consistency loss includes structure within the unmapped columns.", "We consider regularization losses including weight decay and orthogonalization of the latent space [7], but we do not present here due to space constraints.", "The encoders are given substantial noise injection (Bernoulli or Gaussian dropout) and a narrow latent dimension to encourage the latent space to collapse the data to a shared representation (i.e.", "requiring information from each component to be used for reconstructing multiple columns)." ], [ "Chimeric mapping", "Having created paired samples $(x^i, z^{i^{\\prime }})$ , we then have the task of identifying feature correspondence.", "We propose to create a similarity matrix using chimeric dependence, $r_{kl}$ , for each pair $(k,l)$ of columns in $(x^i, z^{i^{\\prime }})$ .", "As explained earlier, we use Pearson correlation as $r$ for our experiments to detect features which are mapped without any transformation.", "Alternatives such as mutual information are considered in the discussion section and are more appropriate to detect features which are transformed.", "We then use the same Gale-Shapley algorithm on the chimeric dependence as we described for KMF.", "A step by step procedure including both Chimeric encoding and Gale Shapley matching is given in REF .", "We considered a threshold in the absolute similarity in addition to the Benjamini–Yekutieli based stopping; however, we found that the high degree of compression in our autoencoder networks tended to degrade these correlations from the ideal and that a single threshold could not easily be selected.", "Gale-Shapley is an asymmetric matching process; a given application may provide a rationale to favor one database over the other (such as $D^A$ being much larger); alternatively, in a symmetric problem this procedure can be repeated with $(x^i, z^{i^{\\prime }})$ from each $i$ , and either the similarity matrices are combined prior to mapping or the mappings are compared for consistency.", "The optimal way to combine similarity matrices is itself a topic of active research [31].", "In our examples, we arbitrarily use the $(x^A, z^{B})$ mapping for simplicity.", "[!htbp] algorithm Chimeric schema matching with $k$ pre-mapped features [1] Mini-batch inner loop within epochs not shown for readability.", "Also omitted any regularization losses such as weight decay and orthogonalization.", "Input: Training data: $x^A$ (feature set $\\mathcal {F}_A$ ), $x^B$ (feature set $\\mathcal {F}_B$ ), List of mapped features $\\mathcal {F}_m$ of size $k$ Require: Learning rate $\\alpha $ , max epochs $n_e$ , Weight for cross loss $w_c$ , Weight for cycle consistency loss $w_{cy}$ , initial parameters $\\theta _A$ , $\\theta _B$ , Matching algorithm GSM.", "Output: Mappings between $\\mathcal {F}_A$ and $\\mathcal {F}_B$ , i.e., $M_{x^A}$ , $M_{x^B}$ .", "Let $(f^A, g^A)$ and $(f^B,g^B)$ be the encoder and decoder of $x^A$ and $x^B$ $t = 0,\\cdots , n_{e}$ $AE_{A}(\\theta _A) = \\mathbf {L}(x^A,g^A(f^A(x^A)))$ ,    $AE_{B}(\\theta _B) = \\mathbf {L}(x^B,g^B(f^B(x^B)))$ $CE_{A}(\\theta _A,\\theta _B) = \\mathbf {L}(x^A[,1:k],g^B(f^A(x^A)))[,1:k]$ ,    $CE_{B}(\\theta _A,\\theta _B) = \\mathbf {L}(x^B[,1:k],g^A(f^B(x^B))[,1:k])$ $CY_{A}(\\theta _A,\\theta _B) = \\mathbf {L}(x^A,g^A(f^B(g^B(f^A(x^A)))))$ ,    $CY_{B}(\\theta _A,\\theta _B) = \\mathbf {L}(x^B,g^B(f^A(g^A(f^B(x^B)))))$ loss $= (AE_A + AE_B) + w_c(CE_A + CE_B) +w_{cy}(CY_A + CY_B) $ $\\theta _A = \\theta _A - \\alpha *\\nabla (\\textrm {loss}, \\theta _A)$ $\\theta _B = \\theta _B - \\alpha *\\nabla (\\textrm {loss}, \\theta _B)$ $CC_{x^A}$ = Cor$(x^A,g^B(f^A(x^A)))$ , $CC_{x^B}$ = Cor$(x^B,g^A(f^B(x^B)))$ $M_{x^A} = \\mathbf {GSM}(CC_{x^A})$ , $M_{x^B} = \\mathbf {GSM}(CC_{x^B})$" ], [ "Hyperparameter tuning", "We recommend a leave-one-out approach to select the architecture of encoders and decoders, regularization strength, learning rate, and other hyperparameters in REF .", "That is, after verifying the $K$ known mapped features, mark only $K-1$ as known, run REF , and evaluate the mapping accuracy on the held-out feature in $K$ runs.", "As in any hyperparameter tuning, prior knowledge from related tasks and iterative evaluation of the results in easily verified cases are likely to narrow the search space and reduce computational costs." ], [ "Federated learning", "We note that the above chimeric schema matching algorithm does not require any party to have direct access to more than one database.", "Each database can be hosted on a distinct server and pass current parameters of $f$ and $g$ for parameter updates, gradient calculations, or partial loss function calculation using the other databases." ], [ "The final schema matching procedure is a combination of the above two methods where we use KMF as an initialization step to increase the number of pre-mapped features and then run the chimeric encoder to obtain the final mapping and transformations.", "A schematic diagram of the above combination is depicted in Figure REF .", "We envision that the above process would be iterative.", "Having created candidate matches, these could be verified using alternative data sources.", "With these new “known” variables the algorithm can be restarted, bootstrapping to identify more matches.", "Having assigned maps to most features from $D^A$ to $D^B$ , the translated value would be an element-wise combination of the mapped variable (where defined) and the chimerically encoded value otherwise.", "We also evaluated a simpler supervised learning approach to generating paired $(x^i, z^{i^{\\prime }})$ samples.", "First, fit a classifier or regression function for the unmapped features in each dataset as a function of the mapped features only, $x^{i^{\\prime }} \\sim h^{i^{\\prime }}(x^{i^{\\prime }}_{1:k})$ , second use the fitted function from one dataset on samples from the other dataset $h^{i^{\\prime }}(x^i_{1:k})$ to generate $z^{i^{\\prime }}$ by parametric simulation or predictive mean matching to a sample in $x^{i^{\\prime }}$ .", "This approach does not take advantage of any structure within the unmapped features, but is much easier to fit.", "However, we did not find any performance advantages over the full chimeric encoder, and its results are not reported." ], [ "Synthetic Data generation", "We consider three synthetic data-generating mechanisms: multivariate Gaussian, 2-cluster multivariate Gaussian, and a binarized version of the 2-cluster multivariate Gaussian.", "The mixture case is intended to mimic data derived from a case-control study.", "Details of the simulation can be found in the Appendix REF and a code repository in REF .", "Samples for each database are drawn independently from these distributions.", "We consider simulations where the two column spaces are identical, one is a subset of the other, and where they only partially overlap by randomly dropping columns from one or both datasets respectively.", "Known mapped columns are assigned randomly from the set of shared columns.", "The remainder (unmapped columns) have their order randomly permuted.", "In some experiments, we transform a randomly selected column by squaring it, making it zero correlation with the raw data.", "Finally, to check the null behavior of our models, we generate data according to independent Gaussians, from which no information for matching should be available.", "The results for this experiment are provided in Appendix REF , but in no settings did the algorithm produce less than the maximal number of mismatches." ], [ "Real data description", "We used the MIMIC-III v1.3 dataset [14] to demonstrate our algorithm for mapping features between two eras, Carevue (CV) and Metavision (MV).", "We included patients who were 18 years and older at the time of hospital admission and survived beyond the first 24 hours of their first ICU stay.", "Where more than 1 ICU stay occurred, we included only the first.", "We treated the laboratory table (labevents) as the known-mapped columns, and used the nurse-charted data table (chartevents) as the unmapped columns.", "Features were identified as belonging to the CV or MV era based on the value of their d_items index.", "A small number of features appeared to substantially overlap the two eras and were removed from the unmapped set.", "Some labevents data is directly copied into chartevents, and these features were removed from chartevents.", "Because both tables are time series, we selected the last observation in the first 24 hours for each patient, excluding any observations tagged as errors.", "We excluded features with no data in more than $90\\%$ of patients in labevents and $80\\%$ in chartevents.", "After these restrictions, the CV dataset had 26130 patients, and the MV dataset had 21125 patients.", "Both datasets had 71 chartevents features and 58 labevents features.", "Of the included chartevents features, 49 have been mapped by prior work, and these mappings are treated as a “gold standard” to evaluate the algorithm.", "The remaining 22 features are included in the procedure, but the results are not included in the evaluation.", "We treat mapping a feature with a gold-standard pairing to another feature as an error, but mapping between two features with no gold standard are ignored.", "In experiments varying the number of mapped features, features are selected at random from the labevents table.", "Labevents features not included as mapped are included in the evaluation set of unmapped features.", "We used the MIMIC dataset to validate our proposal for hyperparameter selection for the chimeric encoder method.", "Because the number of mapped features ($k$ ) in MIMIC dataset was large, we chose to randomly split the sample of lab features into half used for training (marked as known in the procedure) and hold-out (used to evaluate F1 score).", "This cross-validation split was chosen instead of leave-one-out because with increase in $k$ , using leave-one out strategy for hyperparameter tuning would become computationally expensive and the estimates would have higher variance.", "10 such cross-validation folds were evaluated for each hyperparameter set (grid search) and the true accuracy on the chartevents table compared between the cross-validation selected hyperparameters and the global optimizer.", "For the ACTFAST dataset, we included all binary or categorical features from the preoperative evaluation.", "The dataset had 97068 patients with 31 binary features and 9 categorical features.", "Feature names are available in Table REF in Appendix REF .", "Meta data and more details about the features can be found in [11].", "We always select the mapped features from the set of categorical features, since it seems more realistic that these text values would be pre-mapped by meta data.", "All categorical features are one hot encoded before analysis and therefore the total number of features is 81 and the number of binary mapped features can vary with the selection of categorical features.", "Most of these binary variables have a low frequency of positive values, and no missing data was present in these features.", "Two experimental databases were created by randomly partitioning the patients and randomly dropping and permuting unmapped columns, simulating two “eras” in an identical manner to the synthetic data experiments." ], [ "Implementation and Baseline Methods", "Details of the neural network model architecture for algorithm REF and hyperparameter search for each experiment are contained in Appendix REF .", "We use PyTorch to implement all the neural network steps, and code is made available in Appendix REF .", "We compare Algorithm REF and the simplified version stopping after KMF initialization to two baselines in instance-based schema matching.", "First, the mutual-information-similarity method of Kang and colleagues [16] is labeled “Kang” in plots and tables.", "Unfortunately, there was no publicly available implementation of the Kang methods and we received no reply from the corresponding author.", "We therefore include a re-implementation of their method in our code.", "Second, RadialGAN [36] transforms samples between database formats.", "Although it does not explicitly match features, we can feed its output into the same similarity-matrix and mapping steps in REF .", "The authors of RadialGAN also declined to share code or data, and we have done our best to re-implement their approach." ], [ "Metrics", "Our primary evaluation metric is F1 score computed on the accepted mappings between previously unmapped features.", "The average F1 scores from the proposed methods and the baseline are compared with the Wilcoxon rank sum test.", "In experiments where columns are intentionally dropped or transformed, we quantify the results where no match exists (imputation) or a transformation occurs with the correlation between the reconstructed value and the true masked value.", "Because we generally think of the method as being applied to matching a large local pool of candidate variables to a smaller number in an existing consortium, we report matching with the larger number of columns as “reviewers” and the smaller number as “applicants”, optimizing the “preferences” of the consortium more.", "Alternatively, the two sets of proposals could be combined or compared.", "We considered augmenting classifier training data as an evaluation metric similar to Yoon and colleagues [36]; however, we found this to require repeated tuning on the sample sizes, classifier complexity, and problem difficulty to prevent the classifier from effectively saturating its learning and therefore not improving much from additional data.", "If the primary dataset is small, the results on combined data quickly converge to simply using the external data.", "For example, logistic regression weights (coefficients) converge to population values with error $\\propto n^{-\\frac{1}{2}}$ , and therefore at realistic sample sizes only modest changes occur with additional data." ], [ "Synthetic data results", "Table REF presents the performance evaluations with statistical tests over all experiments.", "Figure REF shows the results in the simplest synthetic dataset Multivariate Gaussian simulated (20-D).", "Figure REF shows the performance as the number of mapped features varies.", "First, the RadialGAN method performs poorly under all conditions, and we will not repeat this finding with each scenario.", "Second, the KMF and chimeric encoder quickly saturate the task, and Kang also performs well once more than 25% of the features are mapped.", "The high standard deviation in Figure REF is due to the selection of different mapped features across different trials.", "As can be seen in Figure REF , there are only a few features with strong correlation to the rest of the features, and if the mapped feature set includes them then the F1 score for that trial is high.", "In Figure REF , we vary the sample size of the total dataset with the number of mapped features fixed at 4.", "For KMF and chimeric encoder methods, performance stabilizes around 5000 samples, suggesting the variation that we observe in other experiments (using 10000 samples) is the large sample behavior.", "The chimeric encoder performs worse than KMF at small sample sizes, which is expected given the potential for its autoencoders to overfit.", "The sample-size dependence of Kang is less easy to explain.", "In Figure REF , we evaluate onto mapping (one feature set is a subset of the other) varying the number of unshared features.", "The chimeric encoder slightly outperformed others, but differences were small.", "Interestingly, increasing the number of candidate features had minimal effect on the F1 score.", "Figure: Comparison between KMF (red), KMF initialized chimeric encoder (blue), Kang (brown) and RadialGAN (green) on Multivariate Gaussian simulated (20-D) dataset (a) Correlation structure among features obtained after spectral clustering with 5 clusters (b) Performance variation as the number of mapped features increase and as expected the average F1 score increases with more prior information (c) Performance variation with increase in the total sample size with 4 pre-mapped features (d) Performance variation as the feature size increase in x B x^B (onto map case) with 2 pre-mapped features.Figure REF shows the performance on the somewhat more complex mixture distribution synthetic data.", "Figure REF shows that KMF, chimeric, and Kang methods are all successful with mixture data.", "The F1 score is little smaller when these features are transformed to binary data (Figure REF ), but again all three methods work well.", "Figure REF shows that transforming variables overall degrades performance.", "As expected, KMF and chimeric (which use Pearson correlation in this setup) are more affected than Kang, which is using mutual information as its dependence measure and should therefore be minimally affected by transformations.", "Figure REF presents an example of chimeric encoder learning an inverse square transformation for a feature from $x^A$ .", "We attempted to generate a similar plot for RadialGAN to set as a baseline.", "However, RadialGAN was unable to learn the transformation for similar settings (as shown in Figure REF in Appendix REF ).", "Figure REF is the more challenging setup of incomplete overlap (columns randomly dropped from both datasets).", "We vary the total number of features in $x^B$ with the number of pre-mapped features fixed at 4.", "While for all methods there is a decrease in F1 score values, the KMF and chimeric encoder methods are substantially better than the baseline.", "Figure REF displays the unobserved true values on an unshared feature and the reconstruction from chimeric encoder, showing the method's ability to learn to impute unobserved features from those that act as surrogate.", "We observed that the Gale-Shapley matching step's performance was much worse when reversing the direction of application (large database applying to small database).", "This was evident as the difference between the size of feature set of $x^A$ and $x^B$ increases, the gap between the performance of two directions also increases.", "An illustration of this phenomenon is presented in Figure REF in Appendix REF .", "Our initial experiments also suggested that the choice of the encoder output dimension in the chimeric method has a significant effect on its accuracy.", "An example for 2-cluster Gaussian simulated (20-D) with dim$=5$ and dim$=10$ is provided in Figure REF of Appendix REF and clearly, as the dimension decreases, the chimeric AE gets better at mapping between databases.", "This affirms that our algorithm relies on the compression power of the autoencoders.", "Figure: Average F1 score for the KMF initialized chimeric encoder based schema matching in comparison to Kang method and KMF method on synthetic datasets.", "2-cluster Gaussian simulated (20-D): (a) x A x^A and x B x^B have equal number of columns but they are permuted, (b) both x A x^A and x B x^B have been binarized by thresholding all values at 0 after the original dataset has been normalized.Multivariate Gaussian simulated (50-D): (c) The plot shows the performance comparison when x A x^A has 15 features and there is a partial map between the features of x A x^A and x B x^B (with 4 pre-mapped features) (d) an illustration of chimeric encoder being able to learn a feature's representation (not present while training) using the surrogate correlations.", "2-cluster Gaussian simulated (20-D): (e) x A x^A and x B x^B have equal number of columns but 5 random features in unmapped set in x B x^B were square transformed, (f) an illustration of chimeric encoder learning an inverse square transformation when 6 features were pre-mapped.Table: Table showing the average F1 score for different methods in different scenarios.", "Bold values denote the method with a statistically significant higher average F1 score in comparison to others based on the Wilcoxon test." ], [ "Figure REF and REF show the correlograms for the Carevue and Metavision eras respectively.", "Although the number of mapped features is fairly high, the correlation between individual mapped and unmapped features is generally low.", "Figure REF shows the matching performance while varying the number of included mapped laboratory features.", "The performance is overall lower (correctly mapping at most 22 of 49 features), as we might expect given the weak dependence between features and the large fraction of features not shared between the two datasets.", "Even after choosing the optimal parameters for the baseline method (Kang), its performance is extremely poor compared to our KMF and chimeric encoder methods.", "The cross-validation hyperparameter tuning strategy when applied to the MIMIC dataset yielded an average F1 score of $0.40$ , which was modestly lower than the global optimum of $0.45$ .", "On an average, the proposed algorithms are able to correctly match 22 out of 49 (KMF) and 20 out of 49 (KMF initialized Chimeric) unknown chartevents features when 58 features are known-mapped.", "Figure: Performance of chimeric encoder in terms of average F1 score on MIMIC-III dataset.", "Correlation structure within features (labevents and chartevents combined) of MIMIC-III data in (a) Carevue (CV) era and (b) Metavision (MV) era.", "(c) Average F1 score for schema matching between CV and MV era data by chimeric encoder and baselines." ], [ "Figure REF presents a correlogram for the ACTFAST dataset.", "Some features like ASA, Anesthesia type, and Functional capacity are substantially correlated with many variables.", "In Figure REF , we present F1 score while varying the number of mapped features, and in Figure REF we vary the sample size.", "KMF is consistently the best performing algorithm, and RadialGAN is consistently the worst.", "Chimeric encoding is superior to the Kang baseline when a small number of features are mapped, and the two are largely equivalent otherwise.", "As we saw before, the chimeric encoder is more sample-size dependent than any other method because of its complexity.", "Although the sample size of ACTFAST dataset is larger than the other setting we consider, the frequency of the binary features is much less, and as a result the performance has not saturated with the entire dataset used.", "The large standard errors are a function of the random selection of variables to treat as pre-mapped, since a few variables are very informative.", "Figure REF displays mapping accuracy as a function of the number of unshared features in the case where one column space is a subset of the other, and Figure REF displays the same when two column spaces have unique features too.", "When the number of features in the consortium is fixed at 50 and the size of the new database interested in joining is increased, we observe that the F1 score does not show a substantial decrease.", "The Kang baseline has a statistically significant edge in performance averaged over settings in this experiment (Table REF ), but the absolute difference performance is small with all 3 methods performing about the same.", "The findings are similar in the partial mapping experiment in Figure REF .", "We investigated the lower F1 scores in the chimeric and KMF methods, and found that the chimeric and KMF methods were declaring incorrect matches to exist when a statistically significant surrogate was identified.", "Given that some of the clinical entities in the assessment are closely correlated (seen in Figure REF ), we feel that this represents more of a problem with the evaluation metric.", "Figure REF in Appendix REF depicts a histogram of density of the true correlations on the GS matches for `no-match' features that were declared significant by step down procedure.", "Most of these mistakes are due to the `no-match' features being matched to correlated surrogates as seen on the right half of the plot.", "However, the surrogate feature matches have the advantage of allowing one to create composite measures or reconstruct features that are not present in one dataset using the chimeric encoder.", "Figure REF shows such an example, displaying the cumulative distribution function of the reconstruction of a binary feature “Outpatient insulin” that was not present in $x^A$ , stratified based on the true unobserved value.", "The distribution of this feature is very skewed with only $0.052$ positive ratio.", "The distribution functions for the two populations are well separated, which is an evidence for good reconstruction of a binary feature at a variety of potential thresholds.", "Figure: Performance of chimeric encoder in terms of average F1 score on ACTFAST dataset (a) Correlation structure within features (after one-hot encoding) of ACTFAST data obtained after spectral clustering with 10 clusters (b) Comparison between proposed methods and the baselines (c) Performance variation with increase in the total sample size when number of mapped features were fixed at 5 categorical variables (d) Effect of feature overlap extent on F1 score when the consortium size is only 50 features (onto map case, 3 pre-mapped features) (e) Effect of feature overlap extent on F1 score when the consortium size is only 50 features (partial map case, 3 pre-mapped features) (f) Empirical cumulative distribution function of reconstructed binary feature using chimeric encoder that had `no-match'; difference in two curves show that the reconstructed feature values ({0,1}\\lbrace 0,1\\rbrace ) are well discriminated." ], [ "Discussion", "The problem of combining two data sources with shared features is complex and multi-step in applied work.", "Recently, major software firms have put forward proposals to match healthcare databases 1) https://cloud.google.com/healthcare 2) https://aws.amazon.com/healthlake/.", "Most approaches to this task either rely on extensive meta-data or columns with very distinct distributions, such as text fields.", "We envision our contribution as addressing a specific step in these applications.", "After identifying easily mapped features using the above sources of information, we focus on the challenging case of binary or continuous features without substantial metadata.", "We further assume that variable shift or rescaling in continuous variables make recognizing univariate distributions unreliable.", "In that setting, we proposed a procedure that jointly learns to match features between databases, identify transformed features, and impute missing features when a surrogate is present.", "Our method has several novel contributions.", "First, our use of Gale-Shapley matching provides a locally optimal transformation of a similarity matrix to a set of mappings.", "The procedure is fast compared to the heuristic stochastic search proposal of Kang and colleagues [16] and provides candidates for the next-best matches, should the reviewer find a proposal to be incorrect.", "In practice, we would expect the final review stage of matching the two databases to consider a small number of mappings, so the incorrect modifications by the chimeric encoder are of little applied consequence.", "Because Gale-Shapley is very fast, if any corrections are made by users it can be interactively re-run with those corrections enforced.", "Gale-Shapley does have some disadvantages.", "First, it does not identify global optima, only a local one.", "Second, the results depend on the selection of proposal direction.", "Third, Gale-Shapley on its own does not produce any measure of uncertainty and requires an ad-hoc secondary step to identify when to stop accepting matches.", "The procedure of matching between inputs and chimeric outputs can be seen as a form of knowledge distillation; we are searching over the space of permutations (and discarding some features) for a function that approximates the chimeric encoder.", "Search strategies built into the neural network optimization are certainly possible, but we defer this to future work.", "Second, we are the first, to our knowledge, to use false discovery rate control to define a cutoff for partial mapping.", "In our experiments with low dimensional latent classes or spaces, we find that this cutoff is generally liberal (allows too many matches) because a surrogate variable related to the same latent class is identified with low but statistically non-zero correlation.", "By ranking the proposed matches by the estimated correlation, users can detect when the chimeric correlation is no longer meaningful.", "As discussed below, a leave-k-out or cross-validation approach to validation is also possible, which can help define a threshold.", "We are also the first to combine the schema matching problem with learning transformations, including multivariable transformations used to impute missing features.", "The chimeric encoder reveals the transformation both graphically and quantitatively with the highly-ranked matches.", "We regard identifying semantically identical features as a distinct task from transformed features, and recommend this as a multi-stage procedure.", "We used Pearson correlation to generate a similarity matrix between features using the chimeric data, which preferentially matches native versus transformed features.", "As discussed above, this aspect of the proposal (linking a feature to a surrogate) probably explains its lower than expected F1 score in the partial mapping case.", "Non-parametric associations like distance correlation [32] or mutual information expand the notion of “matching” to any one-to-one univariate transformation, which would modify the problem definition for KMF.", "Multiple association measures per pair of features (for example, if the conditional relationship is not homogeneous and there is substantial distributional shift) are also possible.", "Our work is naturally compared to that of Kang [15] and others following the same theme.", "Kang relies upon the association between features to form a fingerprint and searches the space of permutations between databases to optimize the overall similarity of association matrices.", "Interestingly, our simple initialization procedure performed better in almost all cases.", "In the Gaussian case, the pairwise covariance matrix completely defines the dependency between features, and the method's poor performance was surprising.", "We observed that in datasets with binary features the Kang method retains good performance.", "We identified several relative weaknesses of the Kang method.", "First, it was sensitive to the choice of “normal metric” versus “Euclidean metric” (data not shown).", "Second, it performed poorly in the more realistic and difficult partial mapping case even after tuning over its free parameter.", "Third, we observed it to have occasional dramatic failures resulting in wide standard deviations, presumably related to difficulty with its stochastic search over permutations.", "One advantage of the Kang method is that it can work without any prior knowledge matching features (or a very small number); our method is engineered assuming this prior data is available.", "In such cases, one can replace the correlation-fingerprint initialization we propose with Kang's method and rely on the chimeric encoder for transformation.", "Our method also resembles the work of Yoon and colleagues in RadialGAN [36].", "They have a similar approach to transformation between datasets, but they replace the known-mapped chimeric loss with a discriminator.", "RadialGAN has no natural way to take advantage of prior information mapping some variables.", "The RadialGAN authors explicitly do not attempt to match schemas, but there is no reason not to use their network to estimate similarity matrices across datasets.", "We found RadialGAN to be conceptually problematic, because if there is feature shift between databases (such as different mixes of patient classes), then the algorithm will be forced to awkwardly transform patients from one class to another (otherwise the discriminator will easily notice the differences in distribution).", "We note that our cycle consistency loss is slightly different from the same term in RadialGAN, $L(g^A(f^B(z^B)), x^A )$ versus $L(f^B(z^B), f^A(x^A) )$ where $z^B \\equiv g^B(f^A(x^A))$ .", "The RadialGAN cycle consistency loss shifts the comparison to the encoded latent space, which has artificial neural network output in both terms.", "During development, we found that this version of cycle consistency requirement leads to difficulty in training similar to mode collapse in other GANs.", "That is, various undesirable solutions (like a constant value in the encoded space) maximize the discriminator loss.", "Because our cycle consistency is anchored to real data, these mode collapse problems did not occur.", "We found the RadialGAN method to suffer slow and inconsistent training leading to poor evaluation metrics despite multiple attempts to tune it for this purpose.", "However, the inclusion of both cross-reconstruction and cycle consistency losses improved matching performance in the early development of the chimeric encoder approach, and we suspect cycle consistence is a valuable component encouraging the unmapped features to be effectively represented.", "Unfortunately, the RadialGAN authors were not able to share code or data for us to further understand the differences between their experiments and ours." ], [ "Limitations", "Our method assumes that the data has a true latent representation (effective compression) with a substantial association between the known-mapped features and unmapped features to allow the latent spaces to be aligned.", "We found that the results were sensitive to the degree of compression in the encoder architecture.", "For example, with too much compression the associations between features and their chimeric decoding are unreliable, making the matching stage struggle.", "On the other hand, with too little compression there is no shared representation to align the latent spaces; in the limiting case, the encoder could simply learn an identity transformation.", "The encoder hyperparameters like dropout are also important, as dropout encourages the encoder to spread out influence for the latent representation.", "The addition of nonlinearity was also important; permutation is a linear operator, and so a network without nonlinearity can learn such a transformation effectively.", "However, without nonlinearity, the encoder struggles with nonlinear representation in the latent space and nonlinearly transformed variables.", "There was tension when choosing hyperparameters between the quality of the mapping and imputation (requires more compression) and quality of transformation reconstructions (favors less compression).", "This suggests that a multistage procedure with distinct hyperparameters for the mapping and transformation stages can further improve this work.", "We expect that users seeing output like in Figure REF would fit a polynomial or similar relationship to the $(x, z)$ pairs and update the data to include the transformation.", "Although the chimeric encoder may lose a substantial amount of accuracy in reconstruction with heavy compression, we stress that this is unimportant if a mapping or transformation is ultimately discovered.", "Selecting these hyperparameters is difficult.", "As demonstrated for the MIMIC-III dataset, we can withhold some prior-mapped inputs from the algorithm and estimate its performance in matching them.", "This is easy when the number of mapped features is large and they have saturated their contribution to learn the encoded latent space; however, if the number of mapped features available is small, this would be a very noisy estimate.", "In practice, we examined the performance of the autoencoders and cycle consistency outputs with typical validation samples to narrow the search space, although these do not guarantee good mapping performance.", "To be fair to the baselines, we tuned our method and Kang method on the same dataset in synthetic data case and the presented results in Figure REF -REF are optimal for all methods.", "In a real example, verification of the proposed mapping with other sources would also give an indication of whether the selected hyperparameters had failed.", "We find that there is a substantial tradeoff between the chimeric encoder's ability to schema match and its ability to impute.", "Heuristically, this occurs because schema matching using autoencoders requires a substantial amount of compression.", "Only by compression to a low-dimensional latent space using diffuse inputs for each latent variable are we able to closely align the semantics of the two databases.", "Otherwise, one can easily imagine a latent space that “separates” into a term for the known-mapped features and unmapped features.", "However, with mapping mostly accomplished, learning to impute and transform benefits from a lower level of compression and regularization.", "Fortunately, the simple initialization with KMF seems to get most of the “easy” cases and so we can let the chimeric encoder optimize for imputation and transformation.", "We illustrated our proposal with numeric and binary features, or categorical features easily transformed to those types.", "Many real databases will have more complex data types, which will require modification of this approach.", "For example, lists of diagnostic codes using the ICD10, ICD9, or SNOMED CT ontologies would need an appropriate embedding applied.", "Because these structured data are recognizable by their formats and have fully-developed equivalence maps, we assume that in practice they would be “known-mapped” features if present in both databases.", "Applying the same (pre-trained) embedding to both databases would create comparable numeric columns.", "If structured data was present in only one database, the ability to map features to the existing ontology would depend on the embedding chosen and creating a suitable loss function for the hierarchy of entities in that ontology.", "For example, the AHRQ CCS system creates a moderate number of categorical variables representing less fine-grained detail in the ICD9 and ICD10 ontologies, and these would be suitable for mapping.", "Mapping features containing text documents, for example, determining that a pair of text features were both “discharge summaries,” is likely better handled by techniques specific to that task.", "However, if text columns are mapped, they can be embedded to a numeric space and treated as known-mapped features in our proposal.", "Missing data creates both logistic and conceptual difficulties for transformation between databases.", "The correct transformation of informatively missing but easily imputed data could be either “missing” or the unobserved value.", "For example, many patients do not have invasive arterial blood gas measurements, because they are measured only under special clinical concerns, but they do have non-invasive measurements which are closely related.", "Whether or not the two ought to be mapped will depend on the goals of the project.", "To simplify our presentation, we assumed an imputation model was already available to fill in sporadic missing data, but it is straightforward to combine masking strategies used in imputing autoencoders [26] with our pipeline for a more integrated algorithm.", "Finally, we observed the optimization problem behind the chimeric encoder to be a difficult one.", "The achieved loss functions were often far from the optimum in cases where it performed poorly, and the optimization was very dependent on batch size and other learning hyperparameters.", "This behaviour suggests that further experimentation with initialization and optimization strategies could meaningfully improve its performance.", "As these details are likely to be problem- and architecture-specific, we have not exhaustively explored them." ], [ "Conclusion", "We study the problem of schema matching across databases using Electronic Health Records (EHRs) from different hospitals or from different time points as a motivating example.", "We address this problem in two stages: first, we find many easy or exact matches using a pairwise correlation-based fingerprint method.", "Second, we improve on those matches and transform or impute features without a direct match using deep learning.", "Our proposed method relies on the dependencies between a few known mapped features and the rest of the unmapped features." ], [ "Code", "Code for the algorithm and synthetic data experiments are available at github.com/sandhyat/KMFChimericE_SchMatch.", "Neither MIMIC nor ACTFAST is available without restrictions, and so the raw data is not included.", "Processing code for MIMIC and ACTFAST are included." ], [ "Synthetic Data", "We generate covariance matrices by $\\mathbf {W}\\mathbf {W}^T + D$ where $\\mathbf {W}$ is a spherical Gaussian matrix of size $(n,k)$ and $D$ is a diagonal matrix with $n$ integer values randomly drawn from the interval $[1,20]$ .", "Here, $k$ denotes the underlying factor dimension of the generated data.", "For the 2-cluster Gaussian case, we use common covariance matrix and mean vectors sampled from a uniform distribution.", "For each data generating mechanism, we sample 5 datasets of $10,000$ examples.", "For the case where the two databases have equal number of columns and onto mapping where smaller set of column is always a subset of the larger column set, we demonstrate the performance of proposed algorithm on samples from following two data generating distributions.", "2-cluster Gaussian simulated (20-D) : Mixture of two 20-dim Gaussians with mean vectors randomly sampled from $[10,20]$ separately.", "The true factor dimension is taken to be $k=10$ .", "Multivariate Gaussian simulated (20-D): 20-dim Gaussian with covariance as above.", "For the partial map example, where the number of columns in two databases are not equal and there is an incomplete overlap between the feature sets, we use the following data generating mechanism: Multivariate Gaussian simulated (50-D): 50-dim Gaussian with covariance matrix generated as above.", "To account for randomness while partitioning the dataset and selecting the set of mapped variables, for a fixed number of mapped features, we repeat the experiment $n_t$ times with different samples of pre-mapped features.", "Also, to account for the randomness in permutation or the features that are added/removed, for a fixed set of mapped features, we repeat the experiment $n_p$ times.", "So, for a given number of pre-mapped features, we repeat the experiment $n_t*n_p$ times." ], [ "ACTFAST data details", "Feature names and their types for ACTFAST data that were used in this study are presented in Table REF .", "The pre-mapped feature set was chosen from the categorical features that were further one-hot encoded.", "Table REF shows how the metadata is not always useful for mapping two databases.", "Table: Feature names for ACTFAST data used in Section Table: Table showing example features where metadata (feature names) can not be used by language model algorithms for schema matching." ], [ "Null case result verification on synthetic data", "In this section, we present the null behaviour of our model.", "To achieve this, we generated a 20-dimensional Gaussian dataset with covariance as identity matrix.", "As in this case, there is no correlation among the features, choosing any number of mapped features cannot provide the information about the others and hence can not be used by the decoder of other AE.", "The results from this experiment are illustrated in Figure REF .", "Clearly, even when the number of features are as high as 10, the fraction of mistakes is still close to 1." ], [ "Effect of latent representation size", "In Figure REF , we consider 2-cluster Gaussian simulated (20-D) dataset to demonstrate the observation that smaller latent dimension is beneficial as far as matching features is concerned.", "Figure: (a) Performance of chimeric AE on a synthetic dataset Multivariate Gaussian IID simulated (20-D) in which there is no inter-feature dependence within the feature set.", "As expected, the method performs badly in the absence of any correlations between the mapped and unmapped features.", "(b) chimeric encoder's performance variation on 2-cluster Gaussian simulated (20-D) w.r.t.", "the size (L dim) of the latent representation; higher compression (small L dim) leads to good matching performance." ], [ "Additional results for learning some non-linear transformations and partial map experiments with step down procedure", "We first present a scatter plot that demonstrates the ability of RadialGAN to learn an inverse square transformation.", "As can be seen in Figure REF , RadialGAN is not even able to learn the sign correctly, whereas chimeric encoder could learn an inverted parabola as seen in Figure REF .", "Moving to the incomplete feature overlap case, we provide an example in Figure REF that shows the difference between the fraction of mistakes obtained from the two different direction of proposals as the size of one of the databases is increased.", "As explained earlier this is due to the inability of chimeric encoder (that generates $z_A$ ) to reconstruct the latent representation (of $x_B$ ) containing information of larger number of features than $x_A$ and hence the performance worsens (red curve ) as the number of features in $x_B$ increase.", "This phenomenon is not true vice versa as the latent representation of $x_A$ has more flexibility when the chimeric encoder transforms it to $z_B$ and hence the performance is invariant (blue curve) to increase in feature size.", "Next we present the example where the match for an originally `no-match' feature is a surrogate and we demonstrate it by providing the density of the true correlations on the GS matches for `no-match' features that were declared significant by step down (mistakes) in Figure REF .", "Some examples of such GS matched feature pairs used to obtain the above mentioned density along with their frequency and the corresponding correlation are reported in Table REF .", "Figure: 2-cluster Gaussian simulated (20-D) (a) RadialGAN transformed feature data; actual transformation to be learnt is an inverse square transformation which RadialGAN fails to learn for the same settings as in Figure .ACTFAST data (b) Example to show the difference in the performance in case of onto or partial map case when two different proposing directions are used in GS algorithm.", "(c) Histogram of true correlation between `no-match' features and their proposed match using GS algorithm.", "These matches were wrongly accepted as correct by step down procedure as they act as surrogate matches due to high correlation.Table: List of `no-match' features and their corresponding match from chimeric encoder across different permutations and different trials.", "The high value of true correlation implies there are features that have surrogate matches that FDR step down procedure is not able to identify as mistakes." ], [ "Model hyperparameters", "In all the experiments, we chose both the encoders and decoders to be multi-layer perceptrons with two hidden layers each.", "For synthetic and ACTFAST datasets, hidden units for first and second layer is 80 and 40 respectively for encoder and vice versa for the decoder.", "For MIMIC experiments, the hidden layer neurons were chosen to be 120 and 70.", "For the optimizer, we used Adam optimizer with weight decay of $10^{-5}$ along with a learning rate scheduler.", "The initial learning rate of the scheduler for synthetic and ACTFAST experiments is fixed at $10^{-2}$ , but it was tuned in the MIMIC experiments.", "For the synthetic data experiments, we used tanh activation and dropout after second hidden layer of encoder and after first hidden layer of the decoder.", "We used batch size of 64 for both $x^A$ and $x^B$ .", "Number of epochs for all synthetic data experiments was kept at 40 for chimeric AE.", "We encouraged orthogonalization of the latent space with a regularization loss $\\Vert f^i(x^i)^Tf^i(x^i) - \\mathbb {I}_l\\Vert _2$ evaluated within batches with weighing parameter value $w_o = 0.01$ .", "For the binarized synthetic data experiment in Figure REF , we use the same settings as above except for no tanh activation within the network and add sigmoid activation at the end of the encoders.", "Hyperparameter optimization was done on one of the synthetic datasets (Multivariate Gaussian simulated (20-D)), and the set with best matching performance was chosen and used in all other Gaussian synthetic dataset examples.", "We compared dropout rate $p \\in \\lbrace 0.5,0.6,0.7 \\rbrace $ , encoder dimension $l \\in \\lbrace 5,8,10,12\\rbrace $ , and the weights for different loss terms from $ \\lbrace 0.5,0.8,0.9, 1.0, 1.1, 1.2 \\rbrace $ in cross-validation.", "We set the FDR value in the mapping step at $0.05$ .", "For MIMIC data experiments, we used tanh activation and dropout after the second hidden layer of the encoder and after the first hidden layer of the decoder.", "For other hyperparameters, we performed a grid search as follows: dropout rate $p \\in \\lbrace 0.4,0.5,0.6,0.7, 0.8\\rbrace $ , latent representation size $l \\in \\lbrace 20,30,40\\rbrace $ , batch size $\\in \\lbrace 32, 64\\rbrace $ , learning rate $\\in \\lbrace 10^{-2}, 10^{-3}\\rbrace $ and the weights for different loss terms $\\in \\lbrace 0.5,0.7,0.8, 1.0, 1.1, 1.2, 1.4 \\rbrace $ .", "Number of epochs was set at 50 for chimeric encoders.", "For the ACTFAST data experiments, we add sigmoid activation at the end of the encoders.", "We used ReLu activation (only in onto and partial map case) and dropout with dropout rate $p \\in \\lbrace 0.5,0.6,0.7 \\rbrace $ after second hidden layer of encoder and after first hidden layer of the decoder.", "We used batch size of 32 for both $x^A$ and $x^B$ .", "The output layer of encoder (latent space) was tuned over $l \\in \\lbrace 10,20,30,40\\rbrace $ .", "Number of epochs for all data experiments were set at 50 for chimeric encoders.", "Other settings remain same as for the synthetic experiments described above.", "Hyperparameter tuning was done separately for permutation case and onto map case with matching performance as the criterion.", "The best set from onto map experiment was also used in partial map training except for latent space dimension which was 20 for onto and 10 for partial map.", "In all 3 experiments for ACTFAST dataset, we set number of trials $n_t=4$ and number of permutations within a trial as $n_p=3$ .", "For the Kang method we optimize Euclidean distance for the permutation map case and Normal distance for the onto and partial map case with $\\alpha $ tuned in the set $\\lbrace 1,1.2,1.4,1.6,1.8,2,,4,5,10\\rbrace $ .", "3000 iterations were used in the Kang method search for Euclidean distance and 5000 for Normal distance.", "To choose the best value of $\\alpha $ , we ran the experiments over 2 trials and 2 permutations and then for the final experiment, use the value that has the highest F1 score.", "For the number of iterations, we gradually increased the number of iterations, until the average performance saturated and used the corresponding number in our experiments.", "For RadialGAN, we use the same architecture settings and hyperparameter tuning range as suggested by the authors and run the experiment for 100 epochs (no changes in performance were noted out to 1000 epochs in initial experiments)." ] ]
2207.03536
[ [ "Dynamics of free time-dependent effective mass" ], [ "Abstract The consensus is that an object with a large mass will not manifest quantum behavior.", "Therefore, we expect that the quantumness of a time-dependent effective mass (TDEM) will erase after a long time when the mass profile continuously grows with time.", "However, the present article depicts that the Wigner quasi-probability distribution (WQD) will manifest an entanglement behavour forever for two spatially separated free TDEM.", "The time-dependent Schr\\\"{o}dinger equation for a free particle with TDEM is solved with the help of Lewis-Riesenfeld phase space invariant method.", "WQD for the system of two identical TDEM with quadratically increasing mass profile shows that the particles are never separated.", "In particular, their reminiscent is present at the origin of the phase-space forever." ], [ "Introduction", "The idea of the existence of ideal free particles (FP) is rooted in the philosophy of Descartes, who had prescribed eliminating “the influences from far away”to remove the plague of medieval sciences [1].", "Modern science adopted the concepts of the existence of free particles in a similar fashion [2].", "The notion of FP is widely accepted to be fruitful for the approximate behavior of a physical system.", "However, the quantum entanglement phenomenon seems to defy the strict meaning of FP [3].", "Therefore, it seems reasonable to claim that the problem of FP deserves attention in its own right.", "In this article, we have revisited the aspects of a time-dependent effective mass, which is moving as an FP.", "The time-dependent effective mass (TDEM) appears to be relevant for a diverse domain of Physics, namely in the energy density functional approach to many-body problems [4], [5], in the study of electronic properties of condensed matter systems [6], [7], [8], [9], [10], [11], [12], [13], the Schrödinger equation in curved space under the shed of deformed algebras [14], nonlinear optical properties in quantum well [15], [16], and even the cosmological models to quantum information theory [17], [18], [19], [20], [21], [22] and many more [23], [24], [25], [26], [27], [28], [29], [30].", "For instance, the “asymmetric shape of crackling noise pulses emitted by a diverse range of noisy systems”is modeled with the help of TDEM in [32], [33].", "For a TDEM, the system becomes non-conservative [34], [35].", "In particular, if a quantum system interacts with time-varying environments such as temperature, pressure, stress, and energy, the effective masses will be modified by some TDEM [34], [35], [36], [37], [38], [39].", "A first principle calculation for the viable Hamiltonian of a position and time-dependent effective mass was proposed in [31].", "The non-perturbative solution of the time-dependent Schrödinger equation (TDSE) for non-conservative systems (in particular for TDEM) is a tricky one (if not impossible).", "However, if a class of phase-space invariant operators (PSIO) corresponding to the time-dependent Hamiltonian ($\\hat{H}(t)$ ) exists, then the system is exactly solvable.", "For example, the quadratic (both in position and momentum) Hamiltonians are solvable by some factorization method [40], [41], [42].", "Lewis-Riesenfeld phase-space invariant method (LRIM) is of the classic formalisms for solving a TDSE with time-dependent parameters (e.g., mass $m(t)$ ) [43], [44], [45], [46], [47], [48], [49], [50].", "The idea behind the LRIM is to construct a PSIO corresponding to the Hamiltonian $\\hat{H}(t)$ .", "Up to a time-dependent phase factor, the eigenstates of the PSIO ( $\\hat{\\mathcal {I}}$ ) will satisfy the TDSE corresponding to $\\hat{H}(t)$ [51], [52], [53], [54], [55], [56], [57], [58].", "LRIM for various classes of TDSE had been studied extensively in the literature.", "Such as the harmonic oscillators, a particle moving in a time-dependent electromagnetic field, and a particle moving in non-commutative (NC) space with time-dependent NC parameters have been solved with the help of LRIM [59], [60], [61], [62], [63].", "M Maamache et al.", "constructed a class of Lewis-Riesenfeld invariant operators (LRIO) for a free particle with TDEM [56].", "However, there is a gap in the literature on the general study of LRIO for an FP with TDEM.", "The present paper aims to fulfill this gap.", "The construction of the most general quadratic LRIO for an FP with TDEM is one of the motivations behind the present article.", "Moreover, the restrictions on the parameters of the LRIO are determined so that the diagonalization of $\\hat{\\mathcal {I}}$ with the help of a similarity transformation (symplectic group $sp(2,\\mathbb {R})$ ) is possible.", "It turns out that, in diagonal representation, $\\hat{\\mathcal {I}}$ can be factorized in terms of annihilation ($\\hat{a}$ ) and creation ($\\hat{a}^\\dagger $ ) operators.", "The eigenstates along with the corresponding eigenvalues of $\\hat{\\mathcal {I}}$ are then determined from the eigenstates of $\\hat{a}$ .", "To obtain the complete solution of TDSE, we have computed the time-dependent phase factor (both the geometrical and dynamical phases).", "With the help of the uncertainty relation in $\\hat{x}$ and $\\hat{p}$ , we have shown that the ground state is a squeezed coherent state (CS).", "Being the minimum uncertainty state, a CS mostly resembles the classical states [64], [65], [66], [67].", "On the other hand, the Wigner quasiprobability distributions (WQD) provide a phase-space representation of QM.", "it is customary to study the Wigner quasiprobability distributions (WQD) for CS.", "WQD is generally felt to offer several advantages for use in modeling the behavior of physical processes.", "For instance, being a phase-space formulation, WQD involves both real space and the momentum space variables, distinctly different from SE [68], [69].", "It is believed that one can conceptually identify where quantum corrections enter a problem by comparing it with the classical version.", "However, philosophical debates in this regard are inevitable, which we shall avoid in the present article.", "Rather, we shall indulge ourselves in the computation of WQD for our problem, keeping in mind that, being entirely real, WQD simplifies both the calculation and the interpretation of results.", "For this reason, it was a natural choice for the simulation of quantum transport in devices such as the resonant tunneling diode [68], [69].", "In the present paper, we have considered a system of two noninteracting free particles with TDEM, moving in the opposite direction to each another.", "For the constant mass case, it is well known that this type of bipartite state will produce an entanglement term at the origin of the phase-space [68], [69], [70].", "Since amplitude depends on the mass, it is expected that the entanglement term will be diminished as time flows.", "However, we have shown that the amplitude of WQD remains intact at the origin of the phase-space.", "The organization of the article is as follows.", "At first, a class of quadratic LRIO corresponding to a free particle with TDEM is constructed.", "Then the eigenstates and corresponding eigenvalues of the invariant operator are constructed with the help of the factorization method.", "The geometric and dynamic phase factors are determined in a closed form, which enables us to write down the exact solutions of TDSE corresponding to our system.", "It is shown that the states are indeed squeezed CS.", "Finally, we have constructed the WQD, which corresponds to the bipartite CS." ], [ "Lewis-Risenfeld Invariant operator", "Lewis-Riesenfeld (LR) theorem [58] states that for a system described by a time-dependent (TD) Hamiltonian $\\hat{H}(t)$ , a particular solution of the associated TD Schrödinger equation (SE), $\\hat{H}\\psi =i\\hbar \\frac{\\partial \\psi }{\\partial t},$ is given by the eigenstate $\\vert n,t\\rangle $ of a TD invariant $\\mathcal {\\hat{I}}$ defined by the equation $\\frac{d \\hat{\\mathcal {I}}}{dt}= \\frac{\\partial \\hat{\\mathcal {I}}}{\\partial t} + \\frac{1}{i \\hbar }[ \\hat{\\mathcal {I}}(t),\\hat{H}(t)] =0,$ apart from a TD phase factor $e^{i\\theta (t)}$ .", "Assuming the eigenvalue equation $\\hat{\\mathcal {I}}(t)\\vert n,t \\rangle = \\lambda _n \\vert n,t \\rangle $ for a discrete spectrum, $n=0,1,2,...$ , one can verify that $\\dot{\\lambda }=0$ , and $\\hbar \\theta _n =\\int _0^t\\langle n,\\tau \\vert (i\\hbar \\frac{\\partial }{\\partial t}-\\hat{H}(t))\\vert n,\\tau \\rangle d\\tau .$ Here dot denotes the derivative with respect to time.", "The general solution of the TDSE is given by the superposition state $\\vert \\psi (t)\\rangle = \\sum _{n} c_n e^{i\\theta _n} \\vert n,t\\rangle , \\; \\mbox{with}\\; c_n= \\langle n \\vert \\psi (0)\\rangle .$ We shall apply the LR theorem for the following Hamiltonian of a free particle.", "$\\hat{H}(t)=\\frac{\\hat{p}^2}{2m(t)}.$ At first, we observe that the set of operators $\\mathcal {A} =\\left\\lbrace \\hat{p}^2, \\hat{x}^2, \\left\\lbrace \\hat{x},\\hat{p}\\right\\rbrace \\right\\rbrace $ forms a closed quasi-algebra with respect to $\\hat{H}$ of  (REF ).", "In particular, $[\\hat{H},\\hat{p}^2] = 0,\\; [\\hat{H}, \\hat{x}^2] = -\\frac{i\\hbar }{m} \\lbrace \\hat{x},\\hat{p}\\rbrace , \\;[\\hat{H}, \\lbrace \\hat{x},\\hat{p}\\rbrace ] = -\\frac{2i\\hbar }{m}\\hat{p}^2.$ The algebraic relations  (REF ) suggest the following ansatz for the LR-invariant operator.", "$\\hat{\\mathcal {I}}(t) = \\alpha (t) \\hat{p}^2 + \\beta (t)\\hat{x}^2 + \\gamma (t)\\left\\lbrace \\hat{x},\\hat{p}\\right\\rbrace +\\delta (t)\\hat{\\mathbb {I}},$ where $\\hat{\\mathbb {I}}$ is the identity operator.", "Using  (REF ) and  (REF ) in (REF ), we get the following set of first order coupled differential equations.", "$\\dot{\\beta }&=&\\dot{\\delta }=0.", "\\\\\\dot{\\gamma }&=&-\\frac{1}{m}\\beta .", "\\\\\\dot{\\alpha }&=&-\\frac{2}{m}\\gamma .", "$ According to equation  (REF ), $\\beta $ and $\\delta $ are constants.", "Since, $\\delta $ is the constant coefficient of the identity operator, without loss of generality, we can set it to zero.", "Moreover, using equations  () and  (), we can see that $\\frac{d}{dt}(\\gamma ^2 - \\beta \\alpha )=0 \\Rightarrow \\gamma ^2 - \\beta \\alpha = -\\kappa _0^2,$ where $\\kappa _0^2$ is a real constant.", "Solving equations  () and  (), we get $\\gamma (t)&=& \\gamma _0 -\\beta \\mu (t), \\\\\\alpha (t)&=& \\alpha _0 -2\\gamma _0 \\mu (t)+\\beta \\mu ^2(t) , \\\\&& \\mbox{with} \\; \\dot{\\mu }=\\frac{1}{m(t)}.$ $\\alpha _0$ and $\\gamma _0$ are integration constants, subject to the constraint (using equation (REF )) $\\gamma _0^2 - \\beta \\alpha _0 = -\\kappa _0^2.$ For future convenience, let us define an auxiliary real-valued function $ \\alpha (t)=\\sigma ^2(t)$ .", "Then, the system  ()- () is equivalent to the auxiliary equation $\\dot{\\sigma }^2 = \\dot{\\mu }^2 (\\beta + \\kappa _0^2\\sigma ^{-2}).$ In terms of a real solution of $\\sigma (t)$  (REF ), the invariant operator (REF ) takes the form $\\hat{\\mathcal {I}}= \\sigma ^2 \\hat{p}^2 + \\beta \\hat{x}^2 + \\sqrt{\\beta \\sigma ^2 -\\kappa _0^2}\\lbrace \\hat{x},\\hat{p}\\rbrace .$" ], [ "Constraint on parameters for quadratic form", "If the integration constants in () and  (REF ) are considered to be zero ($\\alpha _0=\\gamma _0=0$ ), then  (REF ) can be written in the quadratic form $\\hat{\\mathcal {I}}=\\frac{1}{2}X^T \\hat{\\mathcal {H}}_0X,$ where $\\hat{\\mathcal {H}}_0=2\\beta \\left(\\begin{array}{cc}1 & -\\mu \\\\-\\mu & \\mu ^2\\end{array}\\right), \\; X=(\\hat{x},\\hat{p})^T.$ One can identify the intrinsic symplectic structure (symplectic group Sp$(2,\\mathbb {R})$ ) $\\left[\\hat{\\mathcal {I}}, X\\right] = -i \\hbar \\hat{\\Omega }X = \\hbar \\Sigma _y \\hat{\\mathcal {H}}_0 X,$ with $\\hat{\\Omega }=2\\beta \\left( \\begin{array}{cc}-\\mu & \\mu ^2 \\\\-1 & \\mu \\end{array}\\right),\\;\\Sigma _y =\\left( \\begin{array}{cc}0 & -i \\\\i & 0\\end{array}\\right).$ To have an equivalent coordinate system, in which $\\hat{\\mathcal {I}} $ is diagonalized, we have to construct a similarity transformation (symplectic group Sp($2,\\mathbb {R}$ )).", "The characteristic polynomial ($P_\\lambda = Det(\\hat{\\Omega }-\\lambda \\hat{\\mathbb {I}})$ ) of $\\hat{\\Omega }$ , has trivial roots $\\lambda =0,0$ .", "That means $\\hat{\\mathcal {I}}$ can not be diagonalized by a canonical transformation keeping the Sp($2,\\mathbb {R}$ ) structure intact for the parameter values $ \\alpha _0= \\gamma _0= 0$ .", "However, we can diagonalize the system for the parameter values $\\alpha _0\\ne 0,\\; \\gamma _0\\ne 0,\\; \\kappa _0\\ne 0$ , which is considered throughout this paper." ], [ "Diagonalization of $\\hat{\\mathcal {I}}$ and evaluation of eigenstates", "By direct observation, it is not difficult to rewrite  (REF ) as $\\hat{\\mathcal {I}} = \\left(\\sigma \\hat{p}-m\\dot{\\sigma }\\hat{x}\\right)^2+\\kappa ^2x^2 ,\\;\\;\\mbox{with}\\;\\; \\kappa ^2(t)= \\frac{\\kappa _0^2}{\\alpha (t)}.$  (REF ) suggests the following form for the annihilation operator $\\hat{a}$ and the corresponding creation operator $\\hat{a}^\\dagger $ .", "$\\hat{a} = \\frac{1}{\\sqrt{\\hbar \\omega }}\\left( \\sigma \\hat{p} -m\\dot{\\sigma }\\hat{x}-i\\kappa \\hat{x}\\right),\\\\\\hat{a}^\\dagger = \\frac{1}{\\sqrt{\\hbar \\omega }}\\left( \\sigma \\hat{p} -m\\dot{\\sigma }\\hat{x}+i\\kappa \\hat{x}\\right).$ The constraint $[\\hat{a},\\hat{a}^\\dagger ]=1 \\; \\Rightarrow \\omega = 2\\kappa _0.$ We can factorize  (REF ) straightforwardly as $\\hat{\\mathcal {I}}= \\left(\\hat{a}^\\dagger \\hat{a} +\\frac{1}{2}\\right)\\hbar \\omega .$ $\\hat{a}$ and $\\hat{a}^\\dagger $ act on the eigenstates ($\\vert n\\rangle $ ) of $\\hat{\\mathcal {I}}$ as follows.", "$\\hat{a}\\vert n\\rangle &=& \\sqrt{n}\\vert n-1\\rangle , \\;\\hat{a}^\\dagger \\vert n\\rangle =\\sqrt{n+1}\\vert n+1\\rangle , \\\\\\vert n\\rangle &=& \\frac{1}{\\sqrt{n!}}", "\\left(\\hat{a}^\\dagger \\right)^n \\vert 0\\rangle ,\\; \\; n=0,1,2,.... $ The ground state (vacuum) is given by the solution of $\\hat{a}\\vert 0\\rangle =0.$ In position representation ($\\lbrace \\vert x\\rangle \\rbrace $ ), the explicit solution of  (REF ) reads $\\langle x\\vert 0\\rangle = \\phi _0(x)= \\@root 4 \\of {\\kappa /(\\pi \\hbar \\sigma )} \\exp \\left(-K(t)x^2\\right) ,\\\\\\mbox{with}\\; K(t)= \\frac{1}{2\\hbar \\sigma }(\\kappa - im\\dot{\\sigma }).$ If the variance $\\Delta \\hat{\\mathcal {O}}\\vert _{\\phi }$ of an observable $\\hat{\\mathcal {O}}$ on the normalized state $\\phi $ is given by $\\Delta \\hat{\\mathcal {O}}\\vert _{\\phi }=\\sqrt{\\langle \\hat{\\mathcal {O}}^2\\rangle _{\\phi } - \\langle \\hat{\\mathcal {O}}\\rangle _{\\phi }^2},$ through the expectation value $\\langle \\hat{\\mathcal {O}}\\rangle _{\\phi }= \\langle \\phi \\vert \\hat{\\mathcal {O}}\\vert \\phi \\rangle $ , then we have the uncertainty relation $\\Delta \\hat{x}\\Delta \\hat{p}\\vert _{\\phi _0} = \\frac{\\hbar }{2}\\left(1+ \\frac{m^2\\dot{\\sigma }^2}{\\kappa ^2}\\right)^{\\frac{1}{2}}.$ Moreover, $\\Delta \\hat{x}$ and $\\Delta \\hat{p}$ satisfy the equation of an ellipse $\\frac{(\\Delta \\hat{x})^2}{s_1^2} + \\frac{(\\Delta \\hat{p})^2}{s_2^2} =1,$ where $s_1^2=\\frac{\\hbar \\sigma }{\\kappa }=\\frac{\\hbar }{\\kappa _0}\\alpha (t),\\;\\; s_2^2= \\frac{\\hbar \\beta }{\\sigma \\kappa }=\\frac{\\hbar }{\\kappa _0}\\beta .$ The axis of $\\Delta \\hat{x}$ evolves with time, whereas the axis of $\\Delta \\hat{p}$ is time-independent.", "The semi-major axis and the semi-minor axis of the ellipse are interchanged with the time evolution.", "The ellipse becomes a circle for $\\sigma ^2=\\beta $ (the trivial case of a time-independent system), for which the ground state is indeed a coherent state (CS).", "However, in general, the state is a squeezed CS." ], [ "Solution of the time-dependent Schrödinger equation", "If $\\phi _n(x,t)$ is an eigenstate of $\\hat{\\mathcal {I}}$ , then $\\psi _n(x,t)=\\phi _n(x,t)\\exp (i\\theta _n(t))$ will satisfy the TDSE  (REF ) for some $\\theta _n(t)$ , which is given by  (REF ).", "Let us express the phase factor ($\\theta _n$ ) as a sum of the geometric ($\\theta _n^g$ ) and dynamical ($\\theta _n^d$ ) phases, i.e., $\\theta _n=\\theta ^d_n + \\theta _n^g$ .", "Using  (REF ), we get $\\hbar \\dot{\\theta }_n^d &=& -\\langle n \\vert \\hat{H}\\vert n\\rangle , \\\\\\hbar \\dot{\\theta }_n^g &=& i\\hbar \\langle n\\vert \\frac{\\partial }{\\partial t}\\vert n\\rangle .", "$ To obtain the matrix element $h_{nn}=\\langle n \\vert \\hat{H}\\vert n\\rangle $ , it is convenient to rewrite  (REF ) in terms of $\\hat{a}$ and $\\hat{a}^\\dagger $ .", "In particular, $\\hat{H}= \\frac{\\hbar }{2m\\omega }\\left[ k_\\alpha ^2 \\hat{a}^2+ (k^*_\\alpha )^2 (\\hat{a}^\\dagger )^2 + \\vert k_\\alpha \\vert ^2(\\hat{a}^\\dagger \\hat{a}+\\hat{a}\\hat{a}^\\dagger )\\right],\\;\\;\\mbox{with} \\; k_\\alpha =\\kappa + im\\dot{\\sigma }.$ Using  (REF ) in  (REF ), and utilizing  (REF ), we get $h_{nn}= \\frac{\\hbar \\beta }{2\\kappa _0}(n+1/2)\\dot{\\mu }.$ Using  (REF ) in  (REF ) we can determine the dynamical phases $\\theta _n^d$ , which reads $\\theta _n^d =\\int _0^t \\dot{\\theta }_n^d(\\tau )d\\tau = - \\frac{\\beta }{2\\kappa _0}(n+1/2)\\mu (t).$ To calculate $\\theta _n^g$ from the equation  (), we first note that $\\frac{\\partial \\hat{a}^\\dagger }{\\partial t} = \\Lambda _1 \\hat{a} + \\Lambda _2 \\hat{a}^\\dagger ,$ where $\\Lambda _1 = -\\frac{1}{2\\kappa ^2}\\frac{d}{dt}(\\kappa \\kappa _\\alpha ),\\;\\;\\Lambda _2 = \\frac{\\dot{\\kappa }}{\\kappa }- \\frac{1}{2\\kappa ^2}\\frac{d}{dt}(\\kappa \\kappa _\\alpha ^*).$ On the other hand, using  (REF ) and  (REF ), we can write $\\frac{\\partial \\hat{\\mathcal {I}}}{\\partial t} \\vert 0\\rangle = \\left(\\frac{1}{2}\\hbar \\omega -\\hat{\\mathcal {I}}\\right) \\frac{\\partial \\vert 0\\rangle }{\\partial t}.$ Moreover, using the explicit form  (REF ), we get $\\langle 0\\vert \\frac{\\partial }{\\partial t}\\vert 0\\rangle = \\frac{\\dot{\\zeta }}{\\zeta } -\\frac{\\dot{\\Lambda }}{4\\Lambda _r},$ with $\\zeta = \\@root 4 \\of {\\kappa /(\\pi \\hbar \\sigma )},\\;\\; \\Lambda = \\frac{1}{2\\hbar \\sigma }(\\kappa -im\\dot{\\sigma }),\\;\\; \\Lambda _r=\\operatorname{Re}(\\Lambda ).$ Integrating  (REF ), we can easily obtain $\\theta _0^g$ .", "For the general $\\theta _n^g$ , one can use  (REF ), along with  (REF ) and  () in  () to get $\\langle n\\vert \\frac{\\partial }{\\partial t}\\vert n\\rangle = \\frac{\\dot{\\zeta }}{\\zeta } - \\frac{\\dot{\\Lambda _r}}{4\\Lambda _r} + \\frac{n\\dot{\\sigma }}{2\\sigma } -\\frac{i}{2\\kappa _0}(n+1)m\\dot{\\sigma }^2 + \\frac{i}{4\\kappa _0}\\frac{d}{dt}(m\\sigma \\dot{\\sigma }).$ Using  (REF ) in  () and integrating, we get the explicit form of the geometric phase, which reads $\\theta _n^g=-\\frac{1}{2\\kappa _0}(n+1/2)\\gamma + \\frac{i}{4}\\ln (2\\sigma ^{2n}/\\pi ) +\\frac{1}{2}(n+1)\\tan ^{-1}(\\gamma /\\kappa _0).$ Using  (REF ) and  (REF ), we can write the total phase as $\\theta _n (t) = -\\frac{\\gamma _0}{2\\kappa _0}(n+1/2) + \\frac{i}{4} \\ln (2\\sigma ^{2n}/\\pi ) + \\frac{1}{2}(n+1) \\tan ^{-1} (\\gamma /\\kappa _0).$ Thus we can write the general time-dependent states of the system by $\\psi _n(x,t)=\\phi _n(x,t)e^{i\\theta _n(t)}$ .", "For example, the ground state reads $\\psi _0(x,t)= \\@root 4 \\of {\\kappa _0/(2\\hbar \\sigma ^2)} \\exp \\left( -Kx^2 -\\frac{i\\gamma _0}{4\\kappa _0}+\\frac{i}{2}\\tan ^{-1}(\\gamma /\\kappa _0)\\right).$" ], [ "Wigner distribution", "The characteristic function $\\hat{M}(\\tilde{\\tau },\\tilde{\\theta })$ of a random variable $X=(\\hat{x},\\hat{p})$ is defined by $\\hat{M}(\\tilde{\\tau },\\tilde{\\theta })= e^{i\\mbox{Trace}(\\tilde{t}^T X)},$ with the parameter $\\tilde{t}=(\\tilde{\\tau },\\tilde{\\theta })$ .", "The Fourier transformation of the expectation value of the characteristic function ($\\hat{M}$ ) is known as the Wigner quasiprobability distribution (WQD) [71], [72], [73].", "In particular, WQD is related to the quantum wave function, which is obtained from the Schrödinger equation, through the following integral transform [68], [69], [70], [71], [72], [73].", "$W(x,p,t)= \\int _{-\\infty }^{\\infty } \\psi ^*(x+\\frac{x^{\\prime }}{2})\\psi (x-\\frac{x^{\\prime }}{2})e^{\\frac{ipx^{\\prime }}{\\hbar }}dx^{\\prime }.$ We would now like to turn to a wave function which is composed of two wave packets of the form  (REF ), which we write as $\\psi _T(x,t)= \\frac{1}{\\sqrt{2}}[\\psi _0(x-x_0,t)+\\psi _0(x+x_0,t)].$ This is thus one portion of the wave function centered at $x_0$ , and a second portion centered at $-x_0$ .", "The probability density ($\\rho _T=\\psi _T^*\\psi _T$ ) for the composite system  (REF ) reads $\\rho _T(x,t)=\\frac{1}{2}\\sqrt{\\kappa _0/(2\\hbar \\sigma ^2)}[e^{-2k_r(x-x_0)^2} + e^{-2k_r(x+x_0)^2} + 2e^{-2k_r(x^2+x_0^2)}\\cos (4k_ix_0x)],$ where $k_r(t) = \\operatorname{Re}(K)$ and $k_i(t)=\\operatorname{Im}(K)$ .", "WQD  (REF ) for the composite system  (REF ) reads $W(x,p,t)=\\sqrt{\\frac{\\pi }{2}}[ e^{-\\frac{2\\vert K\\vert ^2 }{k_r}(x-x_0)^2 -\\frac{2k_i}{\\hbar k_r}p(x-x_0)}+e^{-\\frac{2\\vert K\\vert ^2 }{k_r}(x+x_0)^2 -\\frac{2k_i}{\\hbar k_r}p(x+x_0)}\\nonumber \\\\+2 e^{-\\frac{2\\vert K\\vert ^2 }{k_r}x^2 -\\frac{2k_i}{\\hbar k_r}px} \\cos (2x_0 p/\\hbar )]e^{-\\frac{p^2}{2\\hbar ^2 k_r}}.$ It is worth noting that $k_i=0$ implies $\\beta =\\gamma =0$ and $\\alpha =\\alpha _0$ constant.", "In other words, $K\\in \\mathbb {R}$ corresponds to the constant mass system ($\\hat{\\mathcal {I}}=\\alpha _0\\hat{p}^2$ ).", "Using $k_i=0$ in  (REF ), one can verify that it is consistent with the well known WQD corresponding to the constant mass cat-state [68].", "A toy model is illustrated graphically in the next section." ], [ "Toy Model", "As a toy model, we shall consider the TDEM $m(t)=m_0(1+bt)^2,$ which appears to be effective in modeling the transport phenomena in the electronic band structure [74].", "In particular,  (REF ) is incorporated to account for phenomena such as electron-phonon scattering that orchestrate relaxation in charge carrier energy observed in nanostructures [74], [75].", "(REF ) implies $\\mu (t)= \\frac{t}{m_0(1+bt)}.$ One can retrieve the constant mass ($m_0$ ) case by setting the parameter $b\\rightarrow 0$ .", "For the visual illustration, let us choose the following convenient values of the parameters.", "$m_0=1, \\; \\beta =1,\\; \\alpha _0=2,\\; \\gamma _0=1.$ Using the value  (REF ) of the parameters, we get $\\gamma =1-\\mu ,\\; \\alpha = 1+\\gamma ^2,\\\\k_i= \\gamma k_r,\\;k_r = \\frac{1}{2\\hbar (1+\\gamma ^2)} .$ The uncertainty measure (through the variance of the observables) reads as $\\Delta \\hat{x}\\Delta \\hat{p}=\\frac{\\hbar }{2}\\sqrt{1+ \\gamma ^2}.$ From nowon we shall consider $b=0.5$ and $\\hbar =1$ throughout our discussion.", "For the choice (REF ), the FIG.", "REF represents the probability density corresponding to $\\psi _0(x,t)$ .", "Figure: probability density ρ(x,t)=ψ 0 * ψ 0 \\rho (x,t)=\\psi _0^*\\psi _0 .The uncertainty measure ($\\Delta \\hat{x}\\Delta \\hat{p}$ ) is on the FIG.", "REF , which indicates that the state is a squeezed CS.", "Figure: Variance with respect to time.", "Clearly Δx ^Δp ^≥1 2\\Delta \\hat{x}\\Delta \\hat{p}\\ge \\frac{1}{2}.The composite system  (REF ) is considered with $x_0=4$ .", "The probability density ($\\rho _T= \\psi _T^*\\psi _T$ ) corresponding to  (REF ) is shown in FIG.", "REF .", "Figure: Probability density (ρ T =|ψ T | 2 \\rho _T=\\vert \\psi _T\\vert ^2)for the composite system (Considering x 0 =4x_0=4) From FIG.", "REF , we can envisage the time variation of WQD corresponding to the composite system $\\psi _T$ .", "Figure: NO_CAPTIONFigure: blue Wigner distributions show that the correlation at the origin remains forever.", "Since $\\mu (t) \\rightarrow 1/(m_0b)$ , the amplitude of WQD remains almost unaltered for large $t$ .", "$W(x,p,2)$ and $W(x,p,20)$ on FIG.", "REF are showing this feature.", "For small $t$ , the amplitude changes notably, which can be seen from the figure of $W(x,p,1)$ .", "FIG.", "REF depicts that the entanglement at the origin remains forever.", "In other words, the particles are never separated.", "They leave their trace at the origin of the phase-space.", "Since our mass function is a monotone increasing function in time, we expect that the quantum effect will goes off after some time.", "However, contrary to this, we have seen that the quantumness of the system remains intact for all time." ], [ "Conclusions", "Our study uncovers two aspects.", "Firstly, we have developed the scope of Lewis-Riesenfeld (LR)- invariant operators for a quantum system of free-particle (FP) with time-dependent(TD) effective mass (EM).", "Secondly, we have studied the entanglement behavior of such a bipartite system.", "On the way of diagonalizing the quadratic LR-invariant operator (LRIO), we have obtained the constraint on the free parameters so that LRIOs can be diagonalized with a unitary transformation (symplectic group Sp($4,\\mathbb {R}$ )).", "It turns out that the LRIOs can be factorized with the annihilation and creation operators, which enable us to obtain the eigenfunctions and eigenvalues of the LRIOs in closed form.", "Moreover, we have shown that the ground state eigenfunctions of the LRIOs are indeed squeezed coherent states.", "Both the geometric and dynamical phase (time-dependent) corresponding to the general $n^{th}$ state have been determined.", "Thus we have completely solved the TD-Schrödinger equation for an FP with general TDEM.", "In the next part, we have considered a bi-partite system, which is composed of two wave packets, one portion of which is centered at $x_0$ , and a second portion is centered at $-x_0$ .", "The Wigner quasiprobability distribution for such a spatially separated state is constructed.", "It turns out that, the particles are never separated, even for a TDEM.", "It is often argued that the quantumness of the system will be diminished for an object with a large mass.", "Since our toy model TDEM is a monotone increasing function of time, one can expect that the quantumness of the system will go off after a long time.", "However, this is not observed in our phase-space distribution.", "Therefore, the long-standing expectation of achieving the classical theory as a limiting case of quantum theory might not be materialized for such a straightforward phase-space distribution approach.", "Rather, it demands a fresh study on the interpretation of the quantum world and classical world." ], [ "Acknowledgement", "The authors AC and MJ are grateful to Brahmananda Keshab Chandra College for the hospitality.", "We are grateful to the anonymous referee for the fruitful suggestions." ], [ "Data Availability Statement", "The present manuscript has no associated data." ] ]
2207.03549
[ [ "Individual Preference Stability for Clustering" ], [ "Abstract In this paper, we propose a natural notion of individual preference (IP) stability for clustering, which asks that every data point, on average, is closer to the points in its own cluster than to the points in any other cluster.", "Our notion can be motivated from several perspectives, including game theory and algorithmic fairness.", "We study several questions related to our proposed notion.", "We first show that deciding whether a given data set allows for an IP-stable clustering in general is NP-hard.", "As a result, we explore the design of efficient algorithms for finding IP-stable clusterings in some restricted metric spaces.", "We present a polytime algorithm to find a clustering satisfying exact IP-stability on the real line, and an efficient algorithm to find an IP-stable 2-clustering for a tree metric.", "We also consider relaxing the stability constraint, i.e., every data point should not be too far from its own cluster compared to any other cluster.", "For this case, we provide polytime algorithms with different guarantees.", "We evaluate some of our algorithms and several standard clustering approaches on real data sets." ], [ "Introduction", "Clustering, a foundational topic in unsupervised learning, aims to find a partition of a dataset into sets where the intra-set similarity is higher than the inter-set similarity.", "The problem of clustering can be formalized in numerous ways with different ways of measuring similarity within and between sets, such as centroid-based formulations like $k$ -means, $k$ -median and $k$ -center [9], hierarchical partitionings [30], or spectral clustering [66].", "All these formulations have also been considered under additional constraints [67]; in particular, a recent line of work studies clustering under various fairness constraints [27].", "Most formulations of clustering are NP-hard, which has led to both the understanding of approximation algorithms and conditions under which (nearly) optimal clusterings can be found in computationally efficient ways.", "Such conditions that have been studied in recent years are variants of perturbation stability, i.e., small perturbations to the distances do not affect the optimal clustering [2], [11], [20], or are variants of approximation stability, i.e., approximately optimal clusterings according to some objective are all close to each other [15], [60].", "In this work, we introduce a distinct notion of stability in clustering, individual preference stability of a clustering, which measures whether data points can reduce their individual objective by reassigning themselves to another cluster.", "The study of individual preference (IP) stability, and clusterings which are IP-stable, has a number of motivations behind it, both in applications and connections to other concepts in computing.", "For example, clustering can be used to design curricula for a collection of students, with the goal that each student is assigned to a cluster with the most similar learning needs.", "One might also design personalized marketing campaigns where customers are first clustered–the campaign's personalization will be more effective if customers have most affinity to the clusters to which they are assigned.", "More generally, if one first partitions a dataset and then designs a collection of interventions or treatments for each subset, IP stability ensures individuals are best represented by the cluster they belong to.", "This notion has connections to several other concepts studied in computing, both for clustering and for other algorithmic problems.", "Suppose each data point “belongs” to an individual, and that individual chooses which cluster she wants to join to minimize her average distance to points in that cluster.", "If her features are fixed (and she cannot change them), an IP-stable clustering will correspond to a Nash equilibrium of this game.", "In this sense, IP-stability is related to the concept of stability from matching theory [62], [41].", "IP-stability is also a natural notion of individual fairness for clustering, asking that each individual prefers the cluster they belong to over any other cluster (and directly captures envy-freeness of a clustering such as has been recently studied within the context of classification [16]).", "Finally, the study of IP-stability (whether such a clustering exists, whether an objective-maximizing clustering is also IP-stable) may open up a new set of conditions under which approximately optimal clusterings can be found more efficiently than in the worst case." ], [ "Our Contributions", "We propose a natural notion of IP-stability clustering and present a comprehensive analysis of its properties.", "Our notion requires that each data point, on average, is closer to the points in its own cluster than to the points in any other cluster (Section ).", "We show that, in general, IP-stable clusterings might not exist, even for Euclidean data sets in $\\mathbb {R}^2$ (Section ).", "Moreover, we prove that deciding whether a given data set has an IP-stable $k$ -clustering is NP-hard, even for $k=2$ and when the distance function is assumed to be a metric (Section ).", "Here, and in the following, $k$ is the desired number of clusters.", "This naturally motivates the study of approximation algorithms for the problem and the study of the problem in special metric spaces.", "A clustering is called $t$ -approximately IP-stable if the average distance of each data point to its own cluster is not more than $t$ times its average distance to the points in any other cluster.", "By exploiting the techniques from metric embedding, in Section REF , first we provide a polynomial time algorithm that finds an $\\mathcal {O}(\\log ^2{n}/\\varepsilon )$ -approximation IP-stable clustering for $(1-\\varepsilon )$ -fraction of points in any metric space.", "Second, by designing a modified single-linkage approach as a pre-processing step, in Section REF we provide an “efficient” approximation algorithm when the input has a “well-separated” IP-stable clustering.", "More precisely, if the input has an IP-stable clustering with clusters of size at least $\\alpha \\cdot n$ , then our algorithm will find an $\\tilde{\\mathcal {O}}(\\frac{1}{\\alpha })$ -approximation IP-stable clustering of the input in polynomial time.", "When the data set lies on the real line, surprisingly, we show that an IP-stable $k$ -clustering always exists, and we design an efficient algorithm with running time $\\mathcal {O}(kn)$ to find one, where $n$ is the number of data points (Section REF ).", "In addition to finding a IP-stable clustering, one might want to optimize an additional global objective; we study such a problem in Appendix , where we minimize the deviation of clusters' sizes from desirable target sizes.", "We propose a DP solution which runs in $\\mathcal {O}(n^3 k)$ time for this problem.", "Moreover, we show how to efficiently find an IP-stable 2-clustering when the distance function is a tree metric (Section REF ).", "We in fact conjecture that the result on the line can be extended to any tree, and while we show a positive result only for $k=2$ , we believe that a similar result holds for any $k$ .", "Finally, we perform extensive experiments on real data sets and compare the performance of several standard clustering algorithms such as $k$ -means++ and $k$ -center w.r.t.", "IP-stability (Section ).", "Although in the worst-case the violation of IP-stability by solutions produced by these algorithms can be arbitrarily large (as we show in Section  / Appendix ), some of these algorithms perform surprisingly well in practice.", "We also study simple heuristic modifications to make the standard algorithms more aligned with the notion of IP-stability, in particular for linkage clustering (Appendix )." ], [ "Individual Preference Stability", "Our notion of individual preference stability applies to a data set $\\mathcal {D}$ together with a given dissimilarity function $d$ that measures how close two data points are.", "We use the terms dissimilarity and distance synonymously.", "We assume $d:\\mathcal {D}\\times \\mathcal {D}\\rightarrow \\mathbb {R}_{\\ge 0}$ to be symmetric with $d(x,x)=0$ , but not necessarily to be a metric (i.e., to additionally satisfy the triangle inequality and $d(x,y)=0 \\Leftrightarrow x=y$ ).", "Our stability notion defines what it means that a data point is IP-stable in a clustering of $\\mathcal {D}$ ; namely: a data point is IP-stable if the average distance to the points in its own cluster (the point itself excluded) is not greater than the average distance to the points in any other cluster.", "Then a clustering of $\\mathcal {D}$ is said to be IP-stable if every data point in $\\mathcal {D}$ is stable.", "For the rest of the paper we assume $\\mathcal {D}$ to be finite.", "Our definition of IP-stability for clustering can then be formally stated as follows (for $l\\in \\mathbb {N}$ , we let $[l]=\\lbrace 1,\\ldots ,l\\rbrace $ ): Definition 1 (Individual preference (IP) stability) Let $\\mathcal {C}=(C_1,\\ldots ,C_k)$ be a $k$ -clustering of $\\mathcal {D}$ , that is $\\mathcal {D}=C_1\\dot{\\cup }\\ldots \\dot{\\cup } C_k$ and $C_i\\ne \\emptyset $ for $i\\in [k]$ .", "For $x\\in \\mathcal {D}$ , we write $C(x)$ for the cluster $C_i$ that $x$ belongs to.", "We say that $x\\in \\mathcal {D}$ is IP-stable if either $C(x)=\\lbrace x\\rbrace $ or $\\frac{1}{|C(x)|-1}\\sum _{y\\in C(x)}d(x,y)\\le \\frac{1}{|C_i|}\\sum _{y\\in C_i} d(x,y)$ for all $i\\in [k]$ with $C_i\\ne C(x)$ .", "The clustering $\\mathcal {C}$ is an IP-stable $k$ -clustering if every $x\\in \\mathcal {D}$ is IP-stable.", "For brevity, instead of IP-stable we may only say stable.", "Figure: A data seton the real line with more than one IP-stable clustering.", "Top: The data set and the distances between the points.Bottom: The same data set with twoIP-stable2-clusterings (one encoded by color: red vs blue / one encoded by frames: solid vs dotted boundary).We discuss some important observations about IP-stability as defined in Definition REF : if in a clustering all clusters are well-separated and sufficiently far apart, then this clustering is IP-stable.", "An example of such a scenario is provided in the left part of Figure REF .", "Hence, at least for such simple clustering problems with an “obvious” solution, IP-stability does not conflict with the clustering goal of partitioning the data set such that “data points in the same cluster are similar to each other, and data points in different clusters are dissimilar” [24].", "However, there are also data sets for which no IP-stable $k$ -clustering exists (for a fixed $k$ and a given distance function $d$ ).Of course, the trivial 1-clustering $\\mathcal {C}=(\\mathcal {D})$ or the trivial $|\\mathcal {D}|$ -clustering that puts every data point in a singleton are stable, and for a trivial distance function $d\\equiv 0$ , every clustering is stable.", "This can even happen for Euclidean data sets and $k=2$ , as the right part of Figure REF shows.", "If a data set allows for an IP-stable $k$ -clustering, there might be more than one IP-stable $k$ -clustering.", "An example of this is shown in Figure REF .", "This example also illustrates that IP-stability does not necessarily work towards the aforementioned clustering goal.", "Indeed, in Figure REF the two clusters of the clustering encoded by the frames, which is stable, are not even contiguous.", "These observations raise a number of questions such as: when does an IP-stable $k$ -clustering exist?", "Can we efficiently decide whether an IP-stable $k$ -clustering exists?", "If an IP-stable $k$ -clustering exists, can we efficiently compute it?", "Can we minimize some (clustering) objective over the set of all IP-stable clusterings?", "How do standard clustering algorithms such as Lloyd's algorithm (aka $k$ -means; [55]) or linkage clustering [64] perform in terms of IP-stability?", "Are there simple modifications to these algorithms in order to improve their stability?", "In this paper, we explore some of these questions as outlined in Section ." ], [ "Clustering Stability", "There is a large body of work on the design of efficient clustering algorithms, both in the worst case and under various stability notions [10].", "Some works have studied stability notions called “average stability” that are similar to our notion of IP-stability.", "The work of [14] studies properties of a similarity function that are sufficient in order to approximately recover (in either a list model or a tree model) an unknown ground-truth clustering.", "One of the weaker properties they consider is the average attraction property, which requires inequality (REF ) with $t=1$ to hold for the ground-truth clustering, but with an additive gap of $\\gamma >0$ between the left and the right side of (REF ).", "[14] show that the average attraction property is sufficient to successfully cluster in the list model, but with the length of the list being exponential in $1/\\gamma $ , and is not sufficient to successfully cluster in the tree model.", "The works discussed above utilize strong forms of average stability to bypass computational barriers and recover the ground-truth clustering.", "We, on the other hand, focus on both this problem and the complementary question of when does such stability property even hold, either exactly or approximately.", "Related notions of clustering stability such as perturbation stability and approximation stability have also been studied.", "However, the goal in these works is to approximate a clustering objective such as $k$ -means or $k$ -median under stability assumptions on the optimizer of the objective [2], [11], [20], [15], [13], [57].", "Closely related to our work is the paper by [29].", "They show that if there is an unknown ground-truth clustering with cluster size at least $\\alpha n$ satisfying a slightly stronger requirement than IP-stability, i.e., each point, on average, is at least a factor 3 closer to the points in its own cluster than to the points in any other cluster, then they can find an $\\mathcal {O}(1)$ -approximately IP-stable clustering.", "However, their algorithm runs in time $n^{\\Omega (\\log 1/\\alpha )} = \\Omega (n^{\\log k})$ ." ], [ "Individual Fairness and Fairness in Clustering", "In the fairness literature, fairness notions are commonly categorized into notions of individual fairness or group fairness.", "The latter ensures fairness for groups of individuals (such as men vs women), whereas the former aims to ensure fairness for single individuals.", "Our proposed notion of IP-stability can be interpreted as a notion of individual fairness for clustering.", "Individual fairness was originally proposed by [34] for classification, requiring that similar data points (as measured by a given task-specific metric) should receive a similar prediction.", "It has also been studied in the setting of online learning [50], [51], [44].", "Recently, two notions of individual fairness for clustering have been proposed.", "The first notion, proposed by [52] and also studied by [56], [25] and [65], asks that every data point is somewhat close to a center, where “somewhat” depends on how close the data point is to its $\\lceil n/k \\rceil $ nearest neighbors.", "[52] motivate their notion from the perspective of facility location, but it can also be seen in the context of data points that strive to be well represented by being close to a center similarly to the notion of [26] discussed above.", "Note that our proposed notion of IP-stability defines “being well represented” in terms of the average distance of a data point to the other points in its cluster, rather than the distance to a center, and hence is not restricted to centroid-based clustering.", "The second notion, proposed by [8], is analogous to the notion of individual fairness by [34] for classification.", "It considers probabilistic clustering, where each data point is mapped to a distribution over a set of centers, and requires that similar data points are mapped to similar distributions.", "Interestingly, [8] allow the similarity measure used to define fairness and the similarity measure used to evaluate clustering quality to be the same or different.", "However, individual fairness as introduced by [34] has often been deemed impractical due to requiring access to a task-specific metric [49], and it is unclear where a fairness similarity measure that is different from the clustering similarity measure should come from.", "Note that both notions are different from our notion: (1) there exists clusterings that are IP-stable, but will achieve $\\alpha = \\infty $ according $\\alpha $ -fairness notion of [52] and (2) the distributional property of [8] is satisfied with uniform distribution whereas our stability clusterings do not always exist." ], [ "Hedonic Games", "A related line of work is the class of hedonic games as a model of coalition formation [32], [21], [35].", "In a hedonic game the utility of a player only depends on the identity of players belonging to her coalition.", "[40] study clustering from the perspective of hedonic games.", "Our work is different from theirs in the sense that in our model the data points are not selfish players." ], [ "NP-Hardness", "We show that in general metrics, the problem of deciding whether an IP-stable $k$ -clustering exists is NP-hard.", "For such a result it is crucial to specify how an input instance is encoded: we assume that a data set $\\mathcal {D}$ together with a distance function $d$ is represented by the distance matrix $(d(x,y))_{x,y\\in \\mathcal {D}}$ .", "Under this assumption we can prove the following theorem: Theorem 1 (NP-hardness of IP-stable clustering) Deciding whether a data set $\\mathcal {D}$ together with a distance function $d$ has an IP-stable $k$ -clustering is NP-hard.", "This holds even if $k=2$ and $d$ is a metric distance.", "Due to space limitation, the proof of this theorem and all other missing proofs are deferred to the appendix.", "The proof is via a reduction from a variant of 3-SAT where the number of clauses is equal to the number of variables and each variable occurs in at most three clauses.", "Unless $\\text{P}=\\text{NP}$ , Theorem REF implies that for general data sets, even when being guaranteed that an IP-stable $k$ -clustering exists, there cannot be any efficient algorithm for computing such an IP-stable clustering.", "However, as with all NP-hard problems, there are two possible remedies.", "The first remedy is to look at approximately IP-stable clusterings: Definition 2 (Approximate IP-stability) Let $\\mathcal {C}=(C_1,\\ldots , C_k)$ be a $k$ -clustering of $\\mathcal {D}$ and for $x\\in \\mathcal {D}$ let $C(x)$ be the cluster $C_i$ that $x$ belongs to.", "We say that for some $t\\ge 1$ , $x\\in \\mathcal {D}$ is $t$ -approximately IP-stable if either $C(x)=\\lbrace x\\rbrace $ or $\\frac{1}{|C(x)|-1}\\sum _{y\\in C(x)}d(x,y)\\le \\frac{t}{|C_i|}\\sum _{y\\in C_i} d(x,y)$ for every $C_i\\ne C(x)$ .", "The clustering $\\mathcal {C}$ is $t$ -approximately IP-stable if every $x\\in \\mathcal {D}$ is $t$ -approximately IP-stable.", "An alternative way of defining approximate IP-stability, which is explored in Section REF , would be to allow a violation of inequality (REF ) in Definition REF for a certain number of points.", "In our experiments in Section  we will actually consider both these notions of approximation.", "The second remedy is to restrict our considerations to data sets with some special structure.", "This is what we do in Section  where we consider the 1-dimensional Euclidean metric and tree metrics.", "To complement Theorem REF , we show theoretical lower bounds for the standard clustering algorithms $k$ -means++, $k$ -center, and single linkage.", "Their violation of IP-stability can be arbitrarily large.", "theorembadexamples For any $\\alpha >1$ , there exist separate examples where the clusterings produced by $k$ -means++, $k$ -center, and single linkage algorithms are $t$ -approximately IP-stable only for $t\\ge \\alpha $ ." ], [ "Approximation Algorithms for IP-Stability", "In this section, we provide two algorithms for finding approximately IP-stable clustering.", "Our first algorithm finds a partially IP-stable clustering and the second one finds an approximately IP-stable clustering if the input point set admits a clustering that satisfies a set of requirements that are slightly stronger than the requirements of IP-stability." ], [ "Partially Stable Clustering", "Here, we show that if we only want to cluster $(1-\\varepsilon )$ -fraction of the points, it is possible to find a $\\mathcal {O}(\\frac{\\log ^2 n}{\\varepsilon })$ -approximation for IP-stable $k$ -clustering in any metric space.", "We first define hierarchically well-separated trees (HSTs).", "Definition 3 ([17]) A $t$ -hierarchically well-separated tree ($t$ -HST) is defined as a rooted weighted tree with the following properties: The edge weight from any node to each of its children is the same.", "The edge weight along any path from the root to a leaf are decreasing by a factor of at least $t$ .", "[37] shows that it is possible to have a tree embedding where distortion is $\\mathcal {O}(\\log n)$ in expectation.", "Recently, [46] proposes an algorithm that give a worst-case distortion, which they call partial tree embeddings if it is allowed to remove a constant fraction of points from the embedding.", "While their work concerns hop-constrained network design, we provide a simplified version of their result which is useful in our context.", "Theorem 2 ([46]) Given weighted graph $G=(V,E,w)$ , $0 < \\varepsilon < \\frac{1}{3}$ and root $r \\in V$ , there is a polynomial-time algorithm which samples from a distribution over partial tree embeddings whose trees are 2-HST and rooted at $r$ with exclusion probabilityLet $\\mathcal {D}$ be distribution of partial tree metrics of $G$ .", "$\\mathcal {D}$ has exclusion probability $\\epsilon $ if for all $v \\in V$ , we have $\\text{Pr}_{d \\sim \\mathcal {D}}[v~\\in ~V_d]~\\ge ~1-\\epsilon $ $\\varepsilon $ and worst-case distance stretch $\\mathcal {O}(\\frac{\\log ^2 n}{\\varepsilon })$ .", "Claim 3 Without loss of generality, we can assume the following two properties from the construction in thm:maintreeembedding: (1) $V$ is the set of leaves of $T$ , and (2) $\\textrm {depth}(u) = \\textrm {depth}(v)$ for any $u,v \\in V$ .", "Our remaining task is to find an IP-stable clustering of leaves (according to the tree metric distance).", "If we can do that, then it follows that we have an $\\mathcal {O}(\\log ^2{(n)}/\\varepsilon )$ -approximately IP-stable $k$ -clustering.", "Claim 4 There exists a $k$ -clustering $\\mathcal {C} = (C_1,\\ldots , C_k)$ of leaves on $t$ -HST for any $t \\ge 2$ such that $\\mathcal {C}$ is IP-stable for any leaf node.", "Moreover, for $u,v \\in C_i$ and $w \\in C_j$ , $d_T(u,v) \\le d_T(u,w)$ .", "By combining the tree embedding result and our clustering on HSTs we have the following theorem.", "Theorem 5 Let $(V,d)$ be a metric.", "There is a randomized, polynomial time algorithm that produces a clustering $\\mathcal {C}=(C_1,\\ldots , C_k)$ for $V^{\\prime } \\subseteq V$ , where $V^{\\prime }$ is taken from $V$ with exclusion probability $\\varepsilon $ such that, for any node $u~\\in ~C_i, j\\ne ~i$ , $\\overline{d}(u,C_i)~\\le ~\\mathcal {O}(\\frac{\\log ^2 n}{\\varepsilon }) \\overline{d}(u, C_{j })$ We let $\\overline{d}(u,C) = \\sum _{v \\in C} \\frac{d(u,v)}{|C \\setminus \\lbrace u\\rbrace |}$ be the average distance from $u$ to cluster $C$ .", "If $|C| = \\lbrace u\\rbrace $ , then $\\overline{d}(u,C) = 0$ .. We use Theorem REF to embed $(V,d)$ into a 2-HST $(V^{\\prime }, T)$ .", "We then apply Claim REF to produce $(C_1,\\ldots , C_k)$ that is IP-stable in $T$ .", "Since $T$ is a partial tree embedding with worst-case distortion $\\mathcal {O}(\\frac{\\log ^2n}{\\varepsilon })$ , it follows that $\\overline{d}(u,C_i)~\\le ~\\overline{d}_T(u,C_i)~\\le ~\\overline{d}_T(u,C_j)~\\le ~\\mathcal {O}(\\frac{\\log ^2 n}{\\varepsilon })\\overline{d}(u,C_j)$ ." ], [ "Instances with Stable Clustering", "Next, we show an algorithm that finds an approximately IP-stable clustering, if there is a well-separated underlying clustering.", "Let us first define a well-separated clustering.", "A similar notion is also defined by [29].", "Definition 4 ($(\\alpha ,\\gamma )$ -clustering) Let $\\mathcal {C}=(C_1, \\ldots , C_k)$ be a $k$ -clustering of $\\mathcal {D}$ , that is $\\mathcal {D}=C_1\\dot{\\cup }\\ldots \\dot{\\cup } C_k$ and $C_i\\ne \\emptyset $ for $i\\in [k]$ .", "For $x\\in \\mathcal {D}$ , we write $C(x)$ for the cluster $C_i$ that $x$ belongs to.", "We say that $\\mathcal {C}$ is $(\\alpha ,\\gamma )$ -clustering if For all $C_i$ , $|C_i| \\ge \\alpha \\cdot n$ , where $n = |\\mathcal {D}|$ , and For all $i\\ne j$ and $x \\in C_i$ , $\\overline{d}(x,C_j) \\ge \\gamma \\cdot \\overline{d}(x,C_i)$ .", "Lemma 1 ([29]) Let $\\mathcal {C} = (C_1, \\ldots , C_k)$ be an $(\\alpha ,\\gamma )$ -clustering, and let $i \\ne j$ .", "Then For every[29] use the term almost every to avoid sets of measurement zero.", "$x \\in C_i, y \\in C_j$ , $\\frac{\\gamma -1}{\\gamma } \\overline{d}(y,C_i) \\le d(x,y) \\le \\frac{\\gamma ^2+1}{\\gamma (\\gamma -1)} \\overline{d}(y,C_i)$ , and For every $x,y \\in C_i$ , $d(x,y) \\le \\frac{2}{\\gamma -1} \\overline{d}(x,C_j)$ .", "Here, we show that, for large enough $\\gamma $ , the edges inside a cluster will always be smaller than edges between clusters.", "Claim 6 Let $\\mathcal {C} = (C_1, \\ldots , C_k)$ be an $(\\alpha ,\\gamma )$ -clustering.", "Let $i \\ne j$ , and let $x,y \\in C_i$ and $z \\in C_j$ .", "If $\\gamma \\ge 2+\\sqrt{3}$ , then $d(x,y) \\le d(y,z)$ .", "Theorem 7 Let $\\gamma \\ge 2+\\sqrt{3}$ .", "If there exists an $(\\alpha ,\\gamma )$ -clustering, then there is an algorithm that finds such a clustering in time $\\mathcal {O}( n^2 \\log n + n \\cdot \\left(\\frac{1}{\\alpha }\\right)^k)$ .", "Let us consider a modification to single-linkage algorithm: We consider edges in non-decreasing order and only merge two clusters if at least one of them has size at most $\\alpha n$ .", "clm:maingoodedgeproperty suggests that the edges within an underlying cluster will be considered before edges between two clusters.", "Since we know that any cluster size is at least $\\alpha n$ , when considering an edge, it is safe to use this condition to merge them.", "If it is not the case, we can ignore the edge.", "By this process, we will end up with a clustering where each cluster has size at least $\\alpha n$ .", "Hence, there are at most $\\mathcal {O}(1/\\alpha )$ such clusters.", "We then can enumerate over all possible clusterings, as there are at most $\\mathcal {O}( \\frac{1}{\\alpha ^k})$ such clusterings, the running time follows.", "Note that the algorithm described in the above theorem has a high running time (especially if $\\alpha $ is small).", "Notice that, before the enumeration, we do not make any mistakes when we run the modified single-linkage, and we get a clustering where each cluster has size at least $\\alpha n$ .", "We further show it is possible to define a metric over this clustering and apply thm:mainpartialtree to the metric.", "The clustering we get will be approximately IP-stable." ], [ "Conditioning the clusters via single-linkage", "When we run the single-linkage algorithm, in addition to the size of any pair of clusters, we can also consider the ratio between each pair of points in two different clusters.", "We want it so that distances for every pairs of points in two clusters are roughly the same.", "Moreover, the distance from a point $x$ to its own cluster should not be large compared to the distance from $x$ to any other clusters.", "We formally define this condition below.", "Claim 8 Let $\\mathcal {C} = (C_1, \\ldots , C_k)$ be an $(\\alpha ,\\gamma )$ -clustering.", "Let $D \\subseteq C_i$ and $D^{\\prime } \\subseteq C_j$ be two set of points from different clusters.", "Then for $x \\in D$ and $y,y^{\\prime } \\in D^{\\prime }$ , $\\frac{d(x,y)}{d(x,y^{\\prime })} \\le \\frac{\\gamma ^2+1}{(\\gamma -1)^2}$ .", "Corollary 1 For two subsets $D,D^{\\prime }$ from different underlying clusters, let $x,x^{\\prime } \\in D$ and $y,y^{\\prime } \\in D^{\\prime }$ .", "Then $\\frac{d(x,y)}{d(x^{\\prime },y^{\\prime })} \\le {\\left( \\frac{\\gamma ^2+1}{(\\gamma -1)^2} \\right)}^2.$ Claim 9 Let $D,D^{\\prime }$ be two clusters we consider in the single-linkage algorithm.", "Let $e = (x,y)$ be an edge that we merge in an arbitrary step of single-linkage algorithm where $x \\in D, y \\in D^{\\prime }$ , then $D$ and $D^{\\prime }$ must belong to the same underlying clustering if one of the followings is true.", "$|D| < \\alpha \\cdot n$ or $|D^{\\prime }| < \\alpha \\cdot n$ , $\\frac{\\max _{x\\in D, y\\in D^{\\prime }} d(x,y)}{\\min _{x\\in D, y\\in D^{\\prime }} d(x,y) } > {\\left( \\frac{\\gamma ^2+1}{(\\gamma -1)^2} \\right)}^2$ , There exists $x^{\\prime } \\in D$ such that $d(x,x^{\\prime }) > \\frac{2\\gamma }{(\\gamma -1)^2} d(x,y)$ .", "Theorem 10 Let $\\alpha > 0 ,\\gamma \\ge 2+\\sqrt{3}$ .", "If there exists an $(\\alpha , \\gamma )$ -clustering, then there is a randomized algorithm that finds a $\\mathcal {O}(\\frac{\\log ^2{(1/\\alpha )}}{\\alpha })$ -approximately IP-stable k-clustering in polynomial time and constant success probability.", "In the first phase, we run the modified single-linkage algorithm: As in the standard single-linkage algorithm, we consider edges in a non-decreasing order of length and merge two clusters using the criteria of clm:mainmergecriteria.", "When this phase finishes, we get a clustering $(D_1, \\ldots , D_{\\ell })$ with $k \\le \\ell \\le \\mathcal {O}(\\frac{1}{\\alpha })$ , where (1) $|D_i| \\ge \\alpha n$ , (2) For any $x \\in D_i, y \\in D_j$ , $d(x,y)$ approximates $\\overline{d}(D_i,D_j)$$\\overline{d}(C,C^{\\prime }) = \\sum _{x \\in C, y\\in C^{\\prime }} \\frac{d(x,y)}{|C||C^{\\prime }|}$ .", ", and (3) $\\overline{d}(x,D_i) = \\mathcal {O}( \\overline{d}(x,D_j))$ .", "This implies that we can pick $\\mathcal {R} = \\lbrace r_i, \\ldots , r_\\ell \\rbrace $ where each $r_i \\in D_i$ is the representative of $D_i$ .", "Then, we show an IP-stable clustering of $\\mathcal {R}$ yields an IP-stable clustering of the initial point set as well.", "We apply thm:maintreeembedding to $\\mathcal {R}$ with exclusion probability $\\varepsilon < \\alpha $ to produce a partial tree embedding $T$ .", "By the choice of $\\varepsilon $ , with constant probability, not a single point in $\\mathcal {R}$ is excluded We then apply claim:mainclusteringhst to produce a clustering $\\mathcal {C} = ( C_1, \\ldots , C_k)$ of $\\mathcal {R}$ .", "Let $\\mathcal {F} = (F_1, \\ldots , F_k)$ where $F_i = \\bigcup _{r_j \\in C_i} D_j$ be the final clustering.", "Next, we show that $\\mathcal {F}$ is an approximately IP-stable clustering.", "For every $x \\in F_i$ , let $y$ be the point furthest to $x$ in $F_i$ and $z$ be the point closest to $x$ in $F_j$ .", "If we show that $d(x,y) = \\mathcal {O}( \\frac{\\log ^2{(1/\\alpha )}}{\\alpha } ) d(x,z)$ , then this proves the claim as $\\overline{d}(x,F_i) = \\mathcal {O}( d(x,y))$ and $\\overline{d}(x,F_j) = \\Omega (d(x,z)).$ Let $D_x,D_y,D_z$ be the clusters $x,y,z$ belong to after the merging phases terminates, respectively.", "Also, let $r_x, r_y, r_z$ be the representatives of $D_x,D_y,D_z$ .", "By claim:mainclusteringhst, since $r_x,r_y$ belong to the same cluster $C$ , and since $r_z$ belongs to another cluster $C^{\\prime }$ , $d(r_x,r_y) \\le d_T(r_x,r_y) \\le d_T(r_x,r_z) \\le \\mathcal {O}( \\frac{\\log ^2{(1/\\alpha )}}{\\alpha } ) d(r_x,r_z)$ where the last inequality holds since $T$ is a 2-HST embedding for the representative points with worst-case distortion guarantee of $\\mathcal {O}( \\frac{\\log ^2{(1/\\alpha )}}{\\alpha } )$ .", "Because of clm:mainmergecriteria, $d(x,y) = \\Theta (d(r_x,r_y))$ and $d(x,z) = \\Theta (d(r_x,r_z))$ , it follows that $d(x,y) = \\mathcal {O}( \\frac{\\log ^2{(1/\\alpha )}}{\\alpha } ) d(x,z)$ .", "Hence, the proof is complete." ], [ "Remark", "Our analysis relies on the fact that the partial embedding provides a worst-case guarantee.", "However, to the best of our knowledge, a probabilistic tree embedding does not seem to work because the distortion guarantee is on expectation.", "In other words, there could be an edge $e$ such that the distortion is bad, and having only one such edge is enough to break the stability of any $k$ -clustering on the tree embedding." ], [ "Special Metrics", "Another approach to circumvent the NP-hardness of IP-stable clustering is to restrict our considerations to data sets with some special structure.", "Here, we consider 1-dimensional Euclidean metrics and tree metrics." ], [ "1-dimensional Euclidean Case", "Here we study the special case of $\\mathcal {D}\\subseteq \\mathbb {R}$ where $|\\mathcal {D}|=n$ , and $d$ is the Euclidean metric.", "We provide an $\\mathcal {O}(kn)$ algorithm that finds an IP-stable $k$ -clustering.", "We show the following Theorem REF holds." ], [ "High-level Idea", "We start with a specific clustering where all but one cluster are singletons.", "By the definition, only nodes in the non-singleton cluster may want to leave the cluster (i.e., the IP-stability condition may be violated for a subset of points in that cluster).", "Lemma REF shows that if a node $v$ of a cluster $C$ desires to leave $C$ , then a boundary node $v^{\\prime }$ of $C$ will want to leave $C$ as well.", "This allows us to focus only on boundary vertices.", "There can be at most two vertices per cluster, as long as our clustering is contiguous.", "We further observe that a cluster will become unstable only when an additional vertex joins the cluster.", "When this happens, since the vertex that just joined, which has become a boundary vertex, does not desire to leave the cluster, it must be the boundary vertex on the other end that wants to leave the cluster.", "Since we maintain the order and the monotonicity of clusters and boundary vertices that we inspect, we end up with $\\mathcal {O}(kn)$ time algorithm.", "We show that the following theorem holds.", "Theorem 11 Let $\\mathcal {D}\\subseteq \\mathbb {R}$ be a point set of size $n$ .", "There exists an algorithm that for any $1\\le k\\le n$ , gives an IP-stable $k$ -clustering of $\\mathcal {D}$ , and has a time complexity of $\\mathcal {O}(kn)$ .", "It is well-known that for any set of $n$ points in a metric space, there exists an embedding to one-dimensional Euclidean space with distortion $\\mathcal {O}(n)$ .", "Hence: Corollary 2 Let $\\mathcal {D}$ be a point set of size $n$ in an arbitrary metric space.", "There exists a polynomial time algorithm that returns an $\\mathcal {O}(n)$ -approximately IP-stable $k$ -clustering of $\\mathcal {D}$ .", "In Appendix , we consider an extension of the 1-dimensional Euclidean case where a target size for each cluster is given, and the goal is to find an IP-stable $k$ -clustering that minimizes the total violation from the target cluster sizes.", "Formally, the goal is to solve $\\min _{\\begin{array}{c}\\mathcal {C}=(C_1,\\ldots ,C_k):~\\mathcal {C}~\\text{is an}\\\\ \\text{IP-stable clustering of $\\mathcal {D}$}\\\\ \\text{with contiguous clusters}\\end{array}}~ \\Vert (|C_1|-t_1,\\ldots ,|C_k|-t_k)\\Vert _p,$ where $t_1,\\ldots ,t_k\\in [n]$ with $\\sum _{i=1}^k t_i=n$ are given target cluster sizes, $p\\in \\mathbb {R}_{\\ge 1}\\cup \\lbrace \\infty \\rbrace $ and $\\Vert \\cdot \\Vert _p$ denotes the $\\ell _p$ -norm.", "In Appendix , we provide an $\\mathcal {O}(n^3k)$ dynamic programming approach for this problem." ], [ "IP-Stable Clusterings on Trees", "In this section, we show how to find an IP-stable 2-clustering when $d$ is a tree metric.", "Let $T = (V,E)$ be a given weighted tree rooted at $r$ ($r$ can be arbitrarily chosen).", "Our construction will first pick a boundary edge among edges adjacent to $r$ .", "We then rotate the boundary edge in a systematic manner until the clustering we have, which are two connected components we get by removing the boundary edge $e$ from $T$ , is IP-stable.", "While it could be the case that non-contiguous IP-stable clustering exists, we only consider contiguous clustering in our algorithm.", "Our algorithm implies that such a clustering exists.", "For any graph $G=(V,E)$ and $v \\in V$ , let $C_G(v)$ be the set of vertices reachable from $v$ in $G$ .", "For any tree $T =(V,E)$ and an edge $e \\in E$ , let $T\\setminus e = (V,E \\setminus \\lbrace e\\rbrace )$ be the subgraph of $T$ obtained by removing $e$ .", "Let $N(u)$ denote the set of neighbors of $u$ in $T$ .", "We say that $u^f \\in N(u)$ is the furthest neighbor of $u$ if $\\overline{d}(u, C_{T{\\setminus (u,u^f)}}(u^f)) \\ge \\overline{d}(u,C_{T{\\setminus (u,w)}}(w))$ for any $w \\in N(u)$ .", "Next, we define a rotate operation which is crucial for our algorithm and show that the following property holds: Claim 12 Suppose the current boundary edge is $(u,v)$ , we define the operation $\\textrm {rotate}(u)$ to be an operation that moves the boundary edge from $(u,v)$ to $(u,u^f)$ .", "After $\\textrm {rotate}(u)$ is called, the clustering defined by the boundary edge $(u,u^f)$ is stable for $u$ ." ], [ "Algorithm", "Now we are ready to describe our algorithm.", "First, pick an edge $(b_0=r,v)$ arbitrarily among edges adjacent to the root $r$ , and call $\\textrm {rotate}(b_0)$ .", "After rotation, let $e_1 = (b_0, b_1)$ denote the boundary edge.", "By Claim REF , the clustering defined by $e_1$ is stable for $b_0$ .", "If it is also stable for $b_1$ , then we have a stable clustering.", "If not, suppose the boundary edge is now $e_i = (b_{i-1}, b_i)$ .", "By induction, we assume that the clustering is stable for $b_{i-1}$ .", "If this clustering is not stable, then it is not stable for $b_i$ .", "We call $\\textrm {rotate}(b_i)$ to move the boundary edge to $e_{i+1} = (b_i, b_{i+1})$ .", "$b_{i+1}$ cannot be $b_{i-1}$ , otherwise, the previous clustering would already be stable for $b_i$ .", "We can repeat this process until we find a stable clustering.", "The process will terminate because the boundary edge is moved further away from the root at every step.", "Eventually, we will reach the point where $b_{i}^f = b_{i-1}$ .", "This might happen at a leaf.", "Once it happens, then we have a stable clustering for the boundary nodes.", "In appendix:proof-boundary-nodes-stability-suffices, we show that when considering a contiguous clustering, if the boundary nodes are stable, then every node is stable.", "In other words: Claim 13 Let $e = (u,v)$ be the boundary edge of a 2-clustering of $T$ .", "If $u$ and $v$ are stable, then the clustering is stable for every node $x \\in V$ .", "Finally, since the boundary edge is moving away from a fixed vertex, we call $\\textrm {rotate}$ at most $T$ times." ], [ "Experiments", "We ran a number of experiments.Code available on https://github.com/amazon-research/ip-stability-for-clustering Our experiments are intended to serve as a proof of concept.", "They do not focus on the running times of the algorithms or their applicability to large data sets.", "Hence, we only use rather small data sets of sizes 500 to 1885.", "Let us define some quantities.", "For any point $x$ , let $\\text{Vi}(x) = \\max _{C_i\\ne C(x)}\\frac{\\frac{1}{|C(x)|-1}\\sum _{y\\in C(x)} d(x,y)}{\\frac{1}{|C_i|}\\sum _{y\\in C_i} d(x,y)},$ where we use the convention that $\\frac{0}{0}=0$ .", "A point $x$ is IP-stable if and only if $\\text{Vi}(x)\\le 1$ .", "Let $U=\\lbrace x\\in \\mathcal {D}: \\text{$x$ is not stable}\\rbrace $ .", "We measure the extent to which a $k$ -clustering $\\mathcal {C}=(C_1,\\ldots ,C_k)$ of a dataset $\\mathcal {D}$ is (un-)stable by $\\text{\\#\\,Uns}= |U|$ (“number of unstable”), $\\text{MaxVi}=\\max _{x\\in \\mathcal {D}} \\text{Vi}(x)$ (“maximum violation”), and $\\text{MeanVi}$ (“mean violation”) defined as $\\text{MeanVi}&={\\left\\lbrace \\begin{array}{ll}\\frac{1}{|U|}\\sum _{x \\in U} \\text{Vi}(x) &\\quad U\\ne \\emptyset \\\\0 &\\quad U=\\emptyset \\end{array}\\right.", "}.$ The clustering $\\mathcal {C}$ is IP-stable if and only if $\\text{\\#\\,Uns}=0$ and $\\text{MaxVi}\\le 1$ .", "$\\text{MaxVi}$ is the smallest value of $t$ such that $\\mathcal {C}$ is a $t$ -approximately IP-stable clustering.", "We measure the quality of $\\mathcal {C}$ w.r.t.", "the goal of putting similar points into the same cluster by $\\text{Co}$ (“cost”), defined as the average within-cluster distance.", "Formally, $\\text{Co}=\\sum _{i=1}^k \\frac{1}{{|C_i| \\atopwithdelims ()2}}\\sum _{\\lbrace x,y\\rbrace \\in C_i\\times C_i}d(x,y).$ We performed all experiments in Python.", "We used the standard clustering algorithms from Scikit-learnhttps://scikit-learn.org/ or SciPyhttps://scipy.org/ with all parameters set to their default values." ], [ "General Data Sets", "We performed the same set of experiments on the first 1000 records of the Adult data set, the Drug Consumption data set (1885 records), and the Indian Liver Patient data set (579 records), which are all publicly available in the UCI repository [33].", "As distance function $d$ we used the Euclidean, Manhattan or Chebyshev metric.", "Here we only present the results for the Adult data set on the Euclidean metric, the other results are provided in Appendix .", "Our observations are largely consistent between the different data sets and metrics." ], [ "(Un-)Stability of Standard Algorithms", "Working with the Adult data set, we only used its six numerical features (e.g., age, hours worked per week), normalized to zero mean and unit variance, for representing records.", "We applied several standard clustering algorithms as well as the group-fair $k$ -center algorithm of [53] (referred to as $k$ -center GF) to the data set ($k$ -means++; $k$ -medoids) or its distance matrix ($k$ -center using the greedy strategy of [45]; $k$ -center GF; single / average / complete linkage clustering).", "In order to study the extent to which these methods produce (un-)stable clusterings, for $k=2,5,10, 15, 20,30,\\ldots ,100$ , we computed $\\text{\\#\\,Uns}$ , $\\text{MaxVi}$ and $\\text{MeanVi}$ as defined above for the resulting $k$ -clusterings.", "For measuring the quality of the clusterings we computed $\\text{Co}$ as defined in (REF ).", "Figure: bad hbox\\text{\\#\\,Uns} (top-left),Co\\text{Co} (top-right),MaxVi\\text{MaxVi} (bottom-left)and MeanVi\\text{MeanVi} (bottom-right)for the clusterings produced by thevariousalgorithms as a function ofkk.", "kk-center GF denotes the group-fair kk-center algorithm of.Figure REF shows the results, where we removed the single linkage algorithm since it performs significantly worse than the other algorithms.", "For $k$ -means++, $k$ -medoids, $k$ -center and $k$ -center GF we show average results obtained from running them for 25 times since their outcomes depend on random initializations.", "We can see that, in particular for large values of $k$ , $k$ -center, $k$ -center GF, and the linkage algorithms can be quite unstable with rather large values of $\\text{\\#\\,Uns}$ and $\\text{MaxVi}$ .", "In contrast, $k$ -means++ produces quite stable clusterings with $\\text{\\#\\,Uns}\\le 90$ and $\\text{MaxVi}\\le 1.4$ even when $k$ is large.", "For a baseline comparison, for a random clustering in which every data point was assigned to one of $k=100$ clusters uniformly at random we observed $\\text{\\#\\,Uns}= 990$ , $\\text{MaxVi}=11.0$ , and $\\text{MeanVi}=2.5$ on average.", "The $k$ -medoids algorithm performs worse than $k$ -means++, but better than the other algorithms.", "The clusterings produced by $k$ -center GF, which we ran with the constraint of choosing $\\lfloor k/2\\rfloor $ female and $\\lceil k/2\\rceil $ male centers, are slightly more stable than the ones produced by $k$ -center.", "However, note that it really depends on the data set whether group-fairness correlates with IP-stability or not (see Appendix REF ).", "All methods perform similarly w.r.t.", "the clustering cost $\\text{Co}$ ." ], [ "Heuristics to Improve Standard Algorithms", "One might wonder whether we can modify the standard clustering algorithms in order to make them more stable.", "A natural idea to make any clustering more stable is to change it locally and to iteratively pick a data point that is not stable and assign it to the cluster that it is closest to.", "However, this idea turned out not to work as we observed that usually we can only pick a very small number of data points whose reassignment does not cause other data points that are initially stable to become unstable after the reassignment (see Appendix REF ).", "Another idea that we explored is specifically tied to linkage clustering.", "We provide the details in Appendix REF ." ], [ "1-dimensional Euclidean Data Sets", "In Appendix  we present experiments with our DP approach on real world 1-dimensional Euclidean data sets." ], [ "Discussion", "We proposed a notion of IP-stability that aims at data points being well represented by their clusters.", "Formally, it requires every point, on average, to be closer to the points in its own cluster than to the points in any other cluster.", "IP-stability is not restricted to centroid-based clustering and raises numerous questions, some of which we addressed: in a general metric space, we showed it is NP-hard to decide whether an IP-stable $k$ -clustering exists and we provided approximation algorithms for the problem when the input contains a “well-separated” clustering or when a small fraction of inputs can be excluded in the output.", "For one-dimensional Euclidean data sets we can compute an IP-stable clustering in polynomial time.", "For the case of tree metrics, we showed how to find an IP-stable 2-clustering in polynomial time.", "We proved that in the worst-case scenario some standard clustering algorithms including $k$ -means++ provide arbitrarily unstable clusterings.", "However, our experiments show that $k$ -means++ works quite well in practice.", "While our works focus mostly on the upper bound side, understanding the lower bound, e.g., hardness of approximation for IP-stability is one important open question.", "It is a natural direction to explore whether our ideas for IP-stable 2-clustering on trees can be extended to $k$ -clustering for $k>2$ ?", "Finally, it would be very interesting to provide theoretical evidence supporting our empirical findings that show the surprising effectiveness of $k$ -means++ for our proposed notion of IP-stability, in spite of the existence of bad worst-case scenarios." ], [ "Acknowledgements", "SA is supported by the Simons Collaborative grant on Theory of Algorithmic Fairness, and the National Science Foundation grant CCF-1733556.", "Part of the research was done when the author was visiting Northwestern University.", "JM is supported by fundings from the NSF AI Institute for the Foundations of Machine Learning (IFML), an NSF Career award, and the Simons Collaborative grant on Theory of Algorithmic Fairness.", "AV is supported by NSF award CCF-1934843." ], [ "Other notions of fairness for clustering", "Fair clustering was first studied in the seminal work of [27].", "It is based on the fairness notion of disparate impact [39], which says that the output of a machine learning algorithm should be independent of a sensitive attribute, and asks that each cluster has proportional representation from different demographic groups.", "[27] provide approximation algorithms that incorporate their notion into $k$ -median and $k$ -center clustering, assuming that there are only two demographic groups.", "Several follow-up works extend this line of work to other clustering objectives such as $k$ -means or spectral clustering, multiple or non-disjoint groups, some variations of the fairness notion, to address scalability issues or to improve approximation guarantees [61], [63], [4], [7], [12], [18], [19], [48], [54], [5], [6], [3], [36], [47].", "The recent work of [31] shows that for two groups, when given any clustering, one can efficiently compute the fair clustering (fair according to the notion of [27]) that is most similar to the given clustering using linear programming.", "[31] also show that it is NP-hard to decide whether a data set allows for a fair clustering that additionally satisfies some given must-link constraints.", "They mention that such must-link constraints could be used for encoding individual level fairness constraints of the form “similar data points must go to the same cluster”.", "However, for such a notion of individual fairness it remains unclear which pairs of data points exactly should be subject to a must-link constraint.", "Alternative fairness notions for clustering are tied to centroid-based clustering such as $k$ -means, $k$ -median and $k$ -center, where one chooses $k$ centers and then forms clusters by assigning every data point to its closest center: (i) Motivated by the application of data summarization, [53] propose that the various demographic groups should be proportionally represented among the chosen centers.", "(ii) [26] propose a notion of proportionality that requires that no sufficiently large subset of data points could jointly reduce their distances from their closest centers by choosing a new center.", "This notion is similar to our notion of IP-stability in that it assumes that an individual data point strives to be well represented (in the notion of [26] by being close to a center) and it also does not rely on demographic group information.", "However, while our notion aims at ensuring stability for every single data point, the notion of [26] only looks at sufficiently large subsets.", "[59] further studied the proportionally fair clustering in Euclidean space and on graph metrics.", "(iii) [22], [23] introduce the notions of pairwise fairness and community preserving fairness and incorporate them into $k$ -center clustering.", "(iv) [43] and [1] study a fair version of $k$ -means clustering where the goal is to minimize the maximum average cost that a demographic group incurs (the maximum is over the demographic groups and the average is over the various data points within a group).", "[58] and [28] designed optimal algorithms for minimizing the global clustering cost of a clustering (e.g., $k$ -median and $k$ -means cost) with respect to this notion of fairness and its generalization known as clustering with cascaded norms." ], [ "Proof of Theorem ", "We show NP-hardness of the IP-stable clustering decision problem (with $k=2$ and $d$ required to be a metric) via a reduction from a variant of 3-SAT.", "It is well known that deciding whether a Boolean formula in conjunctive normal form, where each clause comprises at most three literals, is satisfiable is NP-hard.", "NP-hardness also holds for a restricted version of 3-SAT, where each variable occurs in at most three clauses [42].", "Furthermore, we can require the formula to have the same number of clauses as number of variables as the following transformation shows: let $\\Phi $ be a formula with $m$ clauses and $n$ variables.", "If $n>m$ , we introduce $l=\\lfloor \\frac{n-m+1}{2}\\rfloor $ new variables $x_1,\\ldots ,x_l$ and for each of them add three clauses $(x_i)$ to $\\Phi $ (if $n-m$ is odd, we add only two clauses $(x_l)$ ).", "The resulting formula has the same number of clauses as number of variables and is satisfiable if and only if $\\Phi $ is satisfiable.", "Similarly, if $n<m$ , we introduce $l=\\lfloor 3\\cdot \\frac{m-n}{2}+\\frac{1}{2}\\rfloor $ new variables $x_1,\\ldots ,x_l$ and add to $\\Phi $ the clauses $(x_1\\vee x_2\\vee x_3),(x_4\\vee x_5\\vee x_6),\\ldots ,(x_{l-2}\\vee x_{l-1}\\vee x_l)$ (if $m-n$ is odd, the last clause is $(x_{l-1}\\vee x_l)$ instead of $(x_{l-2}\\vee x_{l-1}\\vee x_l)$ ).", "As before, the resulting formula has the same number of clauses as number of variables and is satisfiable if and only if $\\Phi $ is satisfiable.", "So let $\\Phi =C_1 \\wedge C_2\\wedge \\ldots \\wedge C_n$ be a formula in conjunctive normal form over variables $x_1,\\ldots ,x_n$ such that each clause $C_i$ comprises at most three literals $x_j$ or $\\lnot x_j$ and each variable occurs in at most three clauses (as either $x_j$ or $\\lnot x_j$ ).", "We construct a metric space $(\\mathcal {D},d)$ in time polynomial in $n$ such that $\\mathcal {D}$ has an IP-stable 2-clustering with respect to $d$ if and only if $\\Phi $ is satisfiable (for $n$ sufficiently large).", "We set $\\mathcal {D}=\\lbrace True, False,\\star ,\\infty ,C_1,\\ldots ,C_n,x_1,\\lnot x_1,\\ldots ,x_n,\\lnot x_n\\rbrace $ and $d(x,y)=\\left[d^{\\prime }(x,y)+{1}\\lbrace x\\ne y\\rbrace \\right]+{1}\\lbrace x\\ne y\\rbrace \\cdot \\max _{x,y\\in \\mathcal {D}}\\left[d^{\\prime }(x,y)+1\\right],\\quad x,y\\in \\mathcal {D},$ for some symmetric function $d^{\\prime }:\\mathcal {D}\\times \\mathcal {D}\\rightarrow \\mathbb {R}_{\\ge 0}$ with $d^{\\prime }(x,x)=0$ , $x\\in \\mathcal {D}$ , that we specify in the next paragraph.", "It is straightforward to see that $d$ is a metric.", "Importantly, note that for any $x\\in \\mathcal {D}$ , inequality (REF ) holds with respect to $d$ if and only if it holds with respect to $d^{\\prime }$ .", "We set $d^{\\prime }(x,y)=0$ for all $x,y\\in \\mathcal {D}$ except for the following: $d^{\\prime }(True,False)&=A,\\\\d^{\\prime }(True,\\star )&=B,\\\\d^{\\prime }(\\star ,False)&=C,\\\\d^{\\prime }(C_i,False)&=D,\\quad i=1,\\ldots ,n,\\\\d^{\\prime }(C_i,\\star )&=E,\\quad i=1,\\ldots ,n,\\\\d^{\\prime }(\\infty ,True)&=F,\\\\d^{\\prime }(\\infty ,False)&=G,\\\\d^{\\prime }(\\infty ,\\star )&=H,\\\\d^{\\prime }(C_i,\\infty )&=J,\\quad i=1,\\ldots ,n,\\\\d^{\\prime }(x_i,\\lnot x_i)&=S,\\quad i=1,\\ldots ,n,\\\\d^{\\prime }(C_i,\\lnot x_j)&=U, \\quad (i,j)\\in \\lbrace (i,j)\\in \\lbrace 1,\\ldots ,n\\rbrace ^2:x_j\\text{~appears in~}C_i\\rbrace ,\\\\d^{\\prime }(C_i, x_j)&=U, \\quad (i,j)\\in \\lbrace (i,j)\\in \\lbrace 1,\\ldots ,n\\rbrace ^2:\\lnot x_j\\text{~appears in~}C_i\\rbrace ,$ where we set $\\begin{split}A &= n, \\qquad F = n^2, \\qquad B = 2F=2n^2, \\qquad E = \\frac{5}{2}F = \\frac{5}{2} n^2, \\\\J &= E+\\log n=\\frac{5}{2} n^2+\\log n, \\qquad D = J + \\log ^2 n=\\frac{5}{2} n^2+\\log n+\\log ^2 n,\\\\U &=3J=\\frac{15}{2} n^2+3\\log n, \\qquad H= n D +E=\\frac{5}{2} n^3+\\frac{5}{2} n^2+n\\log n+n\\log ^2 n,\\\\G &= H+2n^2-n-2n\\log ^2 n=\\frac{5}{2} n^3+\\frac{9}{2} n^2-n+n\\log n-n\\log ^2 n,\\\\S &= (3n+3)U=\\frac{45}{2} n^3+\\frac{45}{2} n^2+9n\\log n+9\\log n,\\\\C &= \\frac{A + G + n D}{2}=\\frac{5}{2} n^3+\\frac{9}{4} n^2+n\\log n.\\end{split}$ We show that for $n\\ge 160$ there is a satisfying assignment for $\\Phi $ if and only if there is an IP-stable 2-clustering of $\\mathcal {D}$ .", "[leftmargin=*] “Satisfying assignment $\\Rightarrow $ IP-stable 2-clustering” Let us assume we are given a satisfying assignment of $\\Phi $ .", "We may assume that if $x_i$ only appears as $x_i$ in $\\Phi $ and not as $\\lnot x_i$ , then $x_i$ is true; similarly, if $x_i$ only appears as $\\lnot x_i$ , then $x_i$ is false.", "We construct a clustering of $\\mathcal {D}$ into two clusters $V_1$ and $V_2$ as follows: $V_1&=\\lbrace True,\\infty ,C_1,\\ldots ,C_n\\rbrace \\cup \\lbrace x_i: x_i\\text{~is true in sat.", "ass.", "}\\rbrace \\cup \\lbrace \\lnot x_i: \\lnot x_i\\text{~is true in sat.", "ass.", "}\\rbrace ,\\\\V_2&=\\lbrace False,\\star \\rbrace \\cup \\lbrace x_i: x_i\\text{~is false in satisfying assignment}\\rbrace \\cup \\lbrace \\lnot x_i: \\lnot x_i\\text{~is false in sat.", "ass.", "}\\rbrace .$ It is $|V_1|=2+2n$ and $|V_2|=2+n$ .", "We need show that every data point in $\\mathcal {D}$ is stable.This is equivalent to verifying that the following inequalities are true: Points in $V_1$ : $True:~~~~&\\frac{1}{1+2n}\\sum _{v\\in V_1}d^{\\prime }(True,v)=\\frac{F}{1+2n}\\le \\frac{A+B}{2+n}=\\frac{1}{2+n}\\sum _{v\\in V_2}d^{\\prime }(True,v)\\\\\\infty :~~~~&\\frac{1}{1+2n}\\sum _{v\\in V_1}d^{\\prime }(\\infty ,v)=\\frac{F+nJ}{1+2n}\\le \\frac{G+H}{2+n}= \\frac{1}{2+n}\\sum _{v\\in V_2}d^{\\prime }(\\infty ,v)\\\\C_i:~~~~& \\frac{1}{1+2n}\\sum _{v\\in V_1}d^{\\prime }(C_i,v)\\le \\frac{J+2U}{1+2n}\\le \\frac{U+D+E}{2+n}\\le \\frac{1}{2+n}\\sum _{v\\in V_2}d^{\\prime }(C_i,v)\\\\x_i:~~~~&\\frac{1}{1+2n}\\sum _{v\\in V_1}d^{\\prime }(x_i,v)\\le \\frac{2U}{1+2n}\\le \\frac{S}{2+n}\\le \\frac{1}{2+n}\\sum _{v\\in V_2}d^{\\prime }(x_i,v)\\\\\\lnot x_i:~~~~&\\frac{1}{1+2n}\\sum _{v\\in V_1}d^{\\prime }(\\lnot x_i,v)\\le \\frac{2U}{1+2n}\\le \\frac{S}{2+n}\\le \\frac{1}{2+n}\\sum _{v\\in V_2}d^{\\prime }(\\lnot x_i,v)$ Points in $V_2$ : $False:~~~~&\\frac{1}{1+n}\\sum _{v\\in V_2}d^{\\prime }(False,v)=\\frac{C}{1+n}\\le \\frac{A+G+nD}{2+2n}=\\frac{1}{2+2n}\\sum _{v\\in V_1}d^{\\prime }(False,v)\\\\\\star :~~~~&\\frac{1}{1+n}\\sum _{v\\in V_2}d^{\\prime }(\\star ,v)=\\frac{C}{1+n}\\le \\frac{B+H+nE}{2+2n}=\\frac{1}{2+2n}\\sum _{v\\in V_1}d^{\\prime }(\\star ,v)\\\\x_i:~~~~&\\frac{1}{1+n}\\sum _{v\\in V_2}d^{\\prime }(x_i,v)=0\\le \\frac{S}{2+2n}\\le \\frac{1}{2+2n}\\sum _{v\\in V_1}d^{\\prime }(x_i,v)\\\\\\lnot x_i:~~~~&\\frac{1}{1+n}\\sum _{v\\in V_2}d^{\\prime }(\\lnot x_i,v)=0\\le \\frac{S}{2+2n}\\le \\frac{1}{2+2n}\\sum _{v\\in V_1}d^{\\prime }(\\lnot x_i,v)$ It is straightforward to check that for our choice of $A,B,C,D,E,F,G,H,J,S,U$ as specified in (REF ) all inequalities (REF ) to () are true.", "“IP-stable 2-clustering $\\Rightarrow $ satisfying assignment” Let us assume that there is an IP-stable clustering of $\\mathcal {D}$ with two clusters $V_1$ and $V_2$ .", "For any partitioning of $\\lbrace C_1,\\ldots ,C_n\\rbrace $ into two sets of size $l$ and $n-l$ ($0\\le l\\le n$ ) we denote the two sets by $\\mathcal {C}_l$ and $\\widetilde{\\mathcal {C}}_{n-l}$ .", "We first show that $x_i$ and $\\lnot x_i$ cannot be contained in the same cluster (say in $V_1$ ).", "This is because if we assume that $x_i,\\lnot x_i \\in V_1$ , for our choice of $S$ and $U$ in (REF ) we have $\\frac{1}{|V_2|}\\sum _{v\\in V_2}d^{\\prime }(x_i,v)\\le U< \\frac{S}{3n+2}\\le \\frac{1}{|V_1|-1}\\sum _{v\\in V_1}d^{\\prime }(x_i,v)$ in contradiction to $x_i$ being stable.", "As a consequence we have $n\\le |V_1|,|V_2|\\le 2n+4$ .", "Next, we show that due to our choice of $A,B,C,D,E,F,G,H,J$ in (REF ) none of the following cases can be true: $\\lbrace True,\\infty \\rbrace \\cup \\mathcal {C}_l\\subset V_1$ and $\\widetilde{\\mathcal {C}}_{n-l}\\cup \\lbrace False,\\star \\rbrace \\subset V_2$ for any $0\\le l< n$ In this case, $False$ would not be stable since for all $0\\le l< n$ , $\\frac{1}{|V_1|}\\sum _{v\\in V_1}d^{\\prime }(False,v)= \\frac{A+G+lD}{l+2+n}<\\frac{C+(n-l)D}{n-l+1+n}=\\frac{1}{|V_2|-1}\\sum _{v\\in V_2}d^{\\prime }(False,v).$ $\\lbrace True\\rbrace \\cup \\mathcal {C}_l\\subset V_1$ and $\\widetilde{\\mathcal {C}}_{n-l}\\cup \\lbrace False,\\star ,\\infty \\rbrace \\subset V_2$ for any $0\\le l\\le n$ In this case, $False$ would not be stable since for all $0\\le l\\le n$ , $\\frac{1}{|V_1|}\\sum _{v\\in V_1}d^{\\prime }(False,v)=\\frac{A+lD}{l+1+n}<\\frac{C+G+(n-l)D}{n-l+2+n}=\\frac{1}{|V_2|-1}\\sum _{v\\in V_2}d^{\\prime }(False,v).$ $\\lbrace False,\\infty \\rbrace \\cup \\mathcal {C}_l\\subset V_1$ and $\\widetilde{\\mathcal {C}}_{n-l}\\cup \\lbrace True,\\star \\rbrace \\subset V_2$ for any $0\\le l\\le n$ In this case, $True$ would not be stable since for all $0\\le l\\le n$ , $\\frac{1}{|V_1|}\\sum _{v\\in V_1}d^{\\prime }(True,v)= \\frac{A+F}{l+2+n}<\\frac{B}{n-l+1+n}=\\frac{1}{|V_2|-1}\\sum _{v\\in V_2}d^{\\prime }(True,v).$ $\\lbrace False\\rbrace \\cup \\mathcal {C}_l\\subset V_1$ and $\\widetilde{\\mathcal {C}}_{n-l}\\cup \\lbrace True,\\star ,\\infty \\rbrace \\subset V_2$ for any $0\\le l\\le n$ In this case, $True$ would not be stable since for all $0\\le l\\le n$ , $\\frac{1}{|V_1|}\\sum _{v\\in V_1}d^{\\prime }(True,v)=\\frac{A}{l+1+n}<\\frac{B+F}{n-l+2+n}=\\frac{1}{|V_2|-1}\\sum _{v\\in V_2}d^{\\prime }(True,v).$ $\\lbrace \\star ,\\infty \\rbrace \\cup \\mathcal {C}_l\\subset V_1$ and $\\widetilde{\\mathcal {C}}_{n-l}\\cup \\lbrace False,True\\rbrace \\subset V_2$ for any $0\\le l\\le n$ In this case, $\\star $ would not be stable since for all $0\\le l\\le n$ , $\\frac{1}{|V_2|}\\sum _{v\\in V_2}d^{\\prime }(\\star ,v)= \\frac{B+C+(n-l)E}{n-l+2+n}<\\frac{H+lE}{l+1+n}=\\frac{1}{|V_1|-1}\\sum _{v\\in V_1}d^{\\prime }(\\star ,v).$ $\\lbrace \\star \\rbrace \\cup \\mathcal {C}_l\\subset V_1$ and $\\widetilde{\\mathcal {C}}_{n-l}\\cup \\lbrace False,True,\\infty \\rbrace \\subset V_2$ for any $0\\le l\\le n$ In this case, $\\infty $ would not be stable since for all $0\\le l\\le n$ , $\\frac{1}{|V_1|}\\sum _{v\\in V_1}d^{\\prime }(\\infty ,v)=\\frac{H+lJ}{l+1+n}<\\frac{F+G+(n-l)J}{n-l+2+n}=\\frac{1}{|V_2|-1}\\sum _{v\\in V_2}d^{\\prime }(\\infty ,v).$ $\\mathcal {C}_l\\subseteq V_1$ and $\\widetilde{\\mathcal {C}}_{n-l}\\cup \\lbrace True,False,\\star ,\\infty \\rbrace \\subseteq V_2$ for any $0\\le l\\le n$ In this case, $True$ would not be stable since for all $0\\le l\\le n$ , $\\frac{1}{|V_1|}\\sum _{v\\in V_1}d^{\\prime }(True,v)=0<\\frac{A+B+F}{3+(n-l)+n}=\\frac{1}{|V_2|-1}\\sum _{v\\in V_2}d^{\\prime }(True,v).$ $\\lbrace \\infty \\rbrace \\cup \\mathcal {C}_l\\subseteq V_1$ and $\\widetilde{\\mathcal {C}}_{n-l}\\cup \\lbrace True,False,\\star \\rbrace \\subseteq V_2$ for any $0\\le l\\le n$ In this case, $True$ would not be stable since for all $0\\le l\\le n$ , $\\frac{1}{|V_1|}\\sum _{v\\in V_1}d^{\\prime }(True,v)=\\frac{F}{1+l+n}<\\frac{A+B}{2+(n-l)+n}=\\frac{1}{|V_2|-1}\\sum _{v\\in V_2}d^{\\prime }(True,v).$ Of course, in all these cases we can exchange the role of $V_1$ and $V_2$ .", "Hence, $True,\\infty , C_1,\\ldots ,C_n$ must be contained in one cluster and $\\star ,False$ must be contained in the other cluster.", "W.l.o.g., let us assume $True,\\infty , C_1,\\ldots ,C_n\\in V_1$ and $\\star ,False\\in V_2$ and hence $|V_1|=2n+2$ and $|V_2|=n+2$ .", "Finally, we show that for the clause $C_i=(l_j)$ or $C_i=(l_j\\vee l_{j^{\\prime }})$ or $C_i=(l_j\\vee l_{j^{\\prime }} \\vee l_{j^{\\prime \\prime }})$ , with the literal $l_j$ equaling $x_j$ or $\\lnot x_j$ , it cannot be the case that $C_i,\\lnot l_j$ or $C_i,\\lnot l_j$ , $\\lnot l_{j^{\\prime }}$ or $C_i,\\lnot l_j$ , $\\lnot l_{j^{\\prime }}$ , $\\lnot l_{j^{\\prime \\prime }}$ are all contained in $V_1$ .", "This is because otherwise $\\frac{1}{|V_2|}\\sum _{v\\in V_2}d^{\\prime }(C_i,v)=\\frac{D+E}{n+2}<\\frac{U+J}{2n+1}\\le \\frac{1}{|V_1|-1}\\sum _{v\\in V_1}d^{\\prime }(C_i,v)$ for our choice of $D,E,J,U$ in (REF ) and $C_i$ would not be stable.", "Consequently, since $x_j$ and $\\lnot x_j$ are not in the same cluster, for each clause $C_i$ at least one of its literals must be in $V_1$ .", "Hence, if we set every literal $x_i$ or $\\lnot x_i$ that is contained in $V_1$ to a true logical value and every literal $x_i$ or $\\lnot x_i$ that is contained in $V_2$ to a false logical value, we obtain a valid assignment that makes $\\Phi $ true.", "$\\square $" ], [ "Proof of clm:hst-extend", "The idea is to extend nodes corresponding to actual points in $V$ so that they are the leaves and that they are at the same depth.", "The distortion of the modified tree will be worse by a constant factor, but that is fine for our purpose as the construction has $\\mathcal {O}(polylog(n))$ distortion.", "Let $(V^{\\prime },d_T)$ be a given partial tree metric from the construction in thm:mainpartialtree and let $\\ell = \\textrm {depth}(d_T)$ be the depth of the tree.", "If a node $v \\in V$ is not a leaf of $T$ , we augment $T$ by replacing $v$ with $v^{\\prime }$ and connecting $v^{\\prime }$ to $v$ with a path $v^{\\prime }, v_1, v_2, \\ldots , v_{\\ell -\\textrm {depth}_T(v) -1}, v$ .", "Let $T^{\\prime }$ be the tree after our modification.", "By this augmentation, $v$ is now a leaf in $T^{\\prime }$ and $\\textrm {depth}_{T^{\\prime }}(v) = \\ell - 1$ .", "Let $(V^{\\prime \\prime },d_{T^{\\prime }})$ be the new tree metric.", "For any $u\\in V$ , we show that $d_{T^{\\prime }}(u,v) \\le 3 d_{T}(u,v)$ .", "In words, this is because the weight of the added path between $v^{\\prime }$ and $v$ is a telescopic sum, and this sums up to at most the weight of the edge connecting $v$ to its parent in $T$ , i.e.", "$p_T(v)$ .", "That is, $d_{T^{\\prime }}(u,v) = d_{T^{\\prime }}(u,v^{\\prime }) + d_{T^{\\prime }}(v^{\\prime },v) = d_{T}(u,v) + d_{T^{\\prime }}(v^{\\prime },v) \\le d_{T}(u,v) + d_{T^{\\prime }}(p_{T^{\\prime }}(v^{\\prime }),v^{\\prime }) = d_{T}(u,v) + d_{T}(p_{T}(v),v)\\le 3d_{T}(u,v)$ .", "Since we can apply this argument to any node, the claim holds true." ], [ "Proof of claim:mainclusteringhst", "Let $T$ be our HST tree rooted at $r$ .", "Let $S = \\lbrace v_1, v_2 \\ldots , v_{k}\\rbrace $ be a set of nodes that we are going to pick.", "Suppose they are sorted in such a way that for every $1\\le i<k$ , $\\textrm {depth}(v_i) \\ge \\textrm {depth}(v_{i+1})$ .", "We let $\\textrm {depth}(r)=0$ , and for every node $u$ , $\\textrm {depth}(u) = \\textrm {depth}(p(u)) + 1$ where $p(u)$ is the parent of $u$ in $T$ .", "To find $S$ , consider the lowest depth, i.e., furthest from the root, $\\ell $ such that the number of nodes at depth $\\ell $ is at most $k$ .", "Let $V_\\ell $ be the set of nodes at depth $\\ell $ .", "Initially, $S = \\emptyset $ .", "For each $v \\in V_\\ell $ , if $|S| + |\\textrm {children}(v)| + |V_\\ell | - 1 < k$ , then add $\\textrm {children}(v)$ to $S$ and remove $v$ from $V_\\ell $ and continue.", "Otherwise, if $|S| + |\\textrm {children}(v)| + |V_\\ell | - 1 = k$ , then add $\\textrm {children}(v)$ and $V_\\ell - \\lbrace v\\rbrace $ to $S$ .", "Since $|S| = k$ after this operation, we are done.", "In the last case where $|S| + |\\textrm {children}(v)| + |V_\\ell | - 1 > k$ , select a subset of $\\textrm {children}(v)$ of size $ k - |S| + |V_\\ell |$ .", "Let this subset be $V_v$ .", "Let $S = S \\cup V_\\ell \\cup V_v$ .", "An example is given in Figure REF for $k=4$ .", "Figure: IP-stable 4-clustering on a 2-HST.", "SS is the set of the red nodes.For $i \\le k$ , let $T_i$ to be the subtree of $T \\setminus \\left( \\bigcup _{j<i} T_j \\right)$ rooted at $v_i$ .", "Let $C_i$ denote the $i^{th}$ cluster consisting of the leaves of $T_i$ .", "We argue that $\\mathcal {C} = (C_1, C_2, \\ldots , C_k)$ is an IP-stable clustering of $T$ .", "Let $u,v$ be two leaves in $T$ .", "Let $x$ be the lowest common ancestor (lca) of $u$ and $v$ .", "Since $T$ is an HST tree, and since all leaves are at the same depth, $d_T(u,v) = d_T(u,x) + d_T(x,v) = 2 d(u,x)$ .", "For any pair of $C_i\\in \\mathcal {C}, u\\in C_i$ where $u$ is a leaf, as we show above, the distance from $u$ to any other leaf node $v\\in C_i$ is at most $2 d_T(u,v_i)$ .", "This is because the lca of $u$ and $v$ can either be $v_i$ or its descendant.", "Hence, we can say that $\\frac{1}{|C_i| -1} \\sum _{w \\in C_i}d_T(u, w) \\le 2 d_T(u, v_i)$ .", "Moreover, for any leaf node $v^{\\prime } \\in C_{j \\ne i}$ , we have that $d_T(u, v^{\\prime }) \\ge 2 d_T(u, v_i)$ .", "This is because the lca of $u$ and $v^{\\prime }$ can either be $v_i$ or its ancestor.", "Hence, $\\frac{1}{|C_j|}\\sum _{w \\in C_j} d_T(u, w) \\ge 2 d_T(u, v_i)$ .", "Therefore, $\\frac{1}{|C_i| -1} \\sum _{w \\in C_i}d(u, w)\\le \\frac{1}{|C_j|}\\sum _{w \\in C_j} d_T(u, w)$ , and $\\mathcal {C}$ is a stable clustering for $T$ .", "We end the proof by noting that for any $u,v \\in C_i$ and $w \\in C_j$ , $d_T(u,v) \\le 2 d_T(u,v_i) \\le d_T(u, v_j) \\le d_T(u,w) $ ." ], [ "Proof of thm:mainexactbf", "Let us consider a modification to single-linkage algorithm: We consider edges in non-decreasing order and only merge two clusters if one (or both) size is consisting of at most $\\alpha n$ points.", "clm:maingoodedgeproperty suggests that the edges within an underlying cluster will be considered before edges between two clusters.", "Since we know that any cluster size is at least $\\alpha n$ , when considering an edge, it is safe to use this condition to merge them.", "If it is not the case, we can ignore the edge.", "By this process, we will end up with a clustering where each cluster has size at least $\\alpha n$ .", "There are at most $\\mathcal {O}(1/\\alpha )$ such clusters.", "We then can enumerate over all possible clusterings, as there are at most $\\mathcal {O}( \\frac{1}{\\alpha ^k})$ such clusterings, the running time follows." ], [ "Proof of clm:maingoodedgeproperty2", "From lem:maindanielystructure, $\\frac{\\gamma -1}{\\gamma } \\overline{d}(x,C_j) \\le d(x,y),d(x,y^{\\prime }) \\le \\frac{\\gamma ^2+1}{\\gamma (\\gamma -1)} \\overline{d}(x,C_j)$ .", "Hence, $\\frac{d(x,y)}{d(x,y^{\\prime })} \\le \\frac{\\gamma ^2+1}{\\gamma (\\gamma -1)} \\cdot \\frac{\\gamma }{\\gamma -1} = \\frac{\\gamma ^2+1}{(\\gamma -1)^2}.$" ], [ "Proof of clm:mainmergecriteria", "$(1)$ follows from clm:maingoodedgeproperty and the fact that we consider edges in non-decreasing order, hence, when we consider $e$ , if $D,D^{\\prime }$ belong to different underlying clusters, and if $|D| < \\alpha n$ , then an edge $e^{\\prime }$ within the cluster $C_D$ must already be considered.", "At that point, $D$ should be merged into another cluster, hence a contradiction.", "$(2)$ follows from col:mainboundedlength.", "If $\\frac{\\max _{x\\in D, y\\in D^{\\prime }} d(x,y)}{\\min _{x\\in D, y\\in D^{\\prime }} d(x,y) } > {\\left( \\frac{\\gamma ^2+1}{(\\gamma -1)^2} \\right)}^2$ , then there exists $x,x^{\\prime } \\in D$ and $y,y^{\\prime } \\in D^{\\prime }$ that violate the inequality in col:mainboundedlength, so $D$ and $D^{\\prime }$ must belong to the same underlying cluster.", "It remains to show $(3)$ .", "Suppose $D$ belongs to the underlying cluster $C_D$ and $D^{\\prime }$ belongs to another underlying cluster $C_{D^{\\prime }}$ , then $d(x,y) \\ge \\frac{\\gamma -1}{\\gamma } \\overline{d}(x,C_{D^{\\prime }})$ ($(1)$ in lem:maindanielystructure).", "However, $d(x,x^{\\prime }) &> \\frac{2 \\gamma }{(\\gamma -1)^2} d(x,y)> \\frac{2 \\gamma }{(\\gamma -1)^2} \\cdot \\frac{\\gamma -1}{\\gamma } \\overline{d}(x,C_{D^{\\prime }})\\\\&= \\frac{2}{\\gamma -1} \\overline{d}(x,C_{D^{\\prime }}).$ Since we know that $x,x^{\\prime }$ belong to the same underlying cluster, this is a contradiction." ], [ "Proof of Theorem ", "Initially, start with the leftmost cluster containing $n-k+1$ points, and all remaining $k-1$ clusters having one single point.", "Imagine clusters are separated with some separators.", "An example is given in fig:initializationintervalclustering for $k=4$ .", "We show that by only moving the separators to the left, an IP-stable $k$ -clustering can be found.", "Figure: Initialization for k=4k=4 and n=8n=8.In this proof, we mainly argue about the stability of boundary nodes of clusters; however, Lemma REF in Appendix  shows that stability of boundary nodes implies stability of all the nodes.", "Since all the clusters but the first one are singletons, and singleton clusters are IP-stable, the only separator that might want to move is the first separator.", "The first separator moves to the left until the first cluster becomes IP-stable.", "In the next step, the only cluster that is not IP-stable is the second cluster since the first cluster has just been made IP-stable, and the rest are singletons.", "The second cluster is unstable due to some nodes on its right hand side that want to join the third cluster by moving the second separator to the left (note that the leftmost node of the second cluster is stable since the first separator has just moved).", "If the second cluster shrinks, some rightmost nodes of the first cluster might get unstable, and want to join the second cluster by moving the first separator to the left.", "The same scenario repeats over and over again until the first two clusters get stable by moving the first two separators only to the left.", "Using the same argument, at each step $i$ , the only cluster that is not IP-stable is the $i^{th}$ cluster, and the first $i$ clusters can be made stable by moving the first $i$ separators only to the left.", "Consider the last, i.e.", "$k^{th}$ step.", "If the $k^{th}$ cluster is a singleton, then it is IP-stable, and our algorithm has converged to an IP-stable $k$ -clustering.", "Assume the $k^{th}$ cluster is not a singleton.", "The reason that it is not a singleton is that the points on its left hand side had preferred to move from the $(k-1)^{th}$ cluster to the $k^{th}$ cluster.", "If the $k^{th}$ cluster is a better cluster for its leftmost point, so is a better cluster for all other points inside it.", "Therefore, the $k^{th}$ cluster is an IP-stable cluster, and our algorithm converges to an IP-stable $k$ -clustering.", "By using the fact that if a separator moves it only moves to the left, and therefore, each separator moves at most $n$ times, in Lemma REF in Appendix REF , we show the running time of the algorithm is $\\mathcal {O}(kn)$ ." ], [ "Running Time of the Algorithm for 1-dimensional Euclidean Metric", "Lemma 2 The running time of the algorithm described for the 1-d Euclidean metric case is $\\mathcal {O}(kn)$ .", "First, we argue that if the following information are maintained for each cluster, the stability of each boundary node can be checked in $\\mathcal {O}(1)$ .", "sum of distances from the left-most and right-most nodes to all other nodes in the cluster the number of nodes in the cluster Suppose that $u$ is the right-most node of a cluster $C_i$ .", "In order to check if $u$ is stable, we compare the average distance from $u$ to its own cluster ($C_i$ ) with the average distance from $u$ to the adjacent cluster, say ($C_{i+1}$ ).", "Let $v$ be the node adjacent to $u$ in $C_{i+1}$ , then $\\overline{d}(u, C_{i+1}) = d(u,v) + \\overline{d}(v, C_{i+1})$ .", "Since $d(u,C_i), d(v,C_{i+1}), |C_i|, |C_{i+1}|$ are all maintained, we can check if $\\overline{d}(u,C_i) \\le \\overline{d}(u,C_{i+1})$ in $\\mathcal {O}(1)$ .", "The argument for the case that $u$ is the leftmost node of a cluster $C_i$ holds in a similar way.", "At the beginning of the algorithm, since our initialization has only one big cluster, coming up with the needed information for two boundary nodes of the big cluster takes $\\mathcal {O}(n)$ .", "Since the only operation that our algorithm does is to move a separator to the left, we show whenever such an operation happens, all the information stored that need to get updated, can get updated in $\\mathcal {O}(1)$ .", "Suppose when a separator moves to the left, the right-most node of $C_i$ , let's say $u$ , moves to $C_{i+1}$ .", "Let $s,t,v,w$ be the left-most node of $C_i$ , the node next to $u$ in $C_i$ , the node next to $u$ in $C_{i+1}$ , and the right-most node of $C_{i+1}$ before the move, respectively.", "Also, let $\\tilde{C}_i = C_i \\setminus \\lbrace u\\rbrace $ and $\\tilde{C}_{i+1} = C_{i+1} \\cup \\lbrace u\\rbrace $ denote the updated clusters $C_i$ and $C_{i+1}$ after $u$ joins the cluster on its right.", "As $|\\tilde{C}_i| = |C_i| -1$ and $|\\tilde{C}_{i+1}| = |C_{i+1}| + 1$ , updating the numbers of nodes in two clusters is trivial.", "Now we show how to update other information, namely, $d(s,\\tilde{C}_i), d(t, \\tilde{C}_i), d(u, \\tilde{C}_{i+1}), d(w, \\tilde{C}_{i+1})$ .", "$d(s,\\tilde{C}_i) & = d(s, C_i) - d(s,u) \\\\d(t, \\tilde{C}_i) & = d(u, C_i) - |\\tilde{C}_i| d(t,u) \\\\d(u, \\tilde{C}_{i+1}) & = d(u, C_{i+1}) = d(v,C_{i+1}) + |\\tilde{C}_{i+1}|d(u,v) \\\\d(w, \\tilde{C}_{i+1}) & = d(w, C_{i+1}) + d(u,w)$ As the right-hand sides of the relationships above are maintained each update can be performed in $\\mathcal {O}(1)$ .", "Since in the algorithm each separator only moves to the left, the total number of times that separators are moved is at most $kn$ .", "Each time that a separator wants to move, the stability of the node on its right is checked in $\\mathcal {O}(1)$ .", "Also, if a separator moves, updating the information stored takes $\\mathcal {O}(1)$ .", "Hence, the total time complexity of the algorithm is $\\mathcal {O}(kn)$ ." ], [ "Proof of thm:boundary-nodes-stability-suffices", "Let $C_1,C_2$ be the clusters containing $u$ and $v$ , respectively.", "Wlog, suppose $|C_1| > 1$ and $x \\in C_1$ .", "Our goal is to show that $\\overline{d}(x,C_1) \\le \\overline{d}(x,C_2)$ .", "Since the path from $x$ to any $y \\in C_2$ has to go through $u$ , we have that $\\overline{d}(x,C_2) = d(x,u) + \\overline{d}(u,C_2)$ .", "As $u$ is stable, $\\overline{d}(u,C_1) \\le \\overline{d}(u,C_2)$ .", "If we can show that $\\overline{d}(x,C_1) \\le d(x,u) + \\overline{d}(u,C_1)$ , then we are done.", "This is true as $\\overline{d}(x,C_1)&=\\frac{1}{|C_1|-1} \\sum _{y \\in C_1, y\\ne x } d(x,y) \\\\&=\\frac{1}{|C_1|-1} \\sum _{y \\in C_1, y \\ne x } d(x,y) \\\\&\\le \\frac{1}{|C_1|-1} \\sum _{y \\in C_1, y \\ne x }\\left( d(x,u) + d(u,y) \\right)\\\\&=d(x,u) + \\frac{1}{|C_1|-1} \\sum _{y \\in C_1, y \\ne x } d(u,y) \\\\&\\le d(x,u) + \\frac{1}{|C_1|-1} \\sum _{y \\in C_1 } d(u,y) \\\\&= d(x,u) + \\overline{d}(u,C_1).$" ], [ "Hard Instances for $k$ -means++, {{formula:ddd4167e-6134-409c-a6e4-f653c59ff474}} -center, and single linkage", "In this section, we prove Theorem  by showing hard instances for $k$ -means++, $k$ -center, and single linkage clustering algorithms." ], [ "Hard Instances for $k$ -means++", "Here, we show that for any given approximation factor $\\alpha >1$ , there exists an instance $\\mathcal {I}_\\alpha $ and a target number of clusters $k_\\alpha $ such that with constant probability $k$ -means++ outputs a $k_\\alpha $ -clustering of $\\mathcal {I}_\\alpha $ that violates the requirement of IP-stable clustering by a factor of $\\alpha $ ." ], [ "High-level description of $k$ -means++.", "$k$ -means++ is an algorithm for choosing the initial seeds for the Lloyd's heuristic that provides a provable approximation guarantee [9].", "The high-level intuition is to spread out the initial set of $k$ centers.", "The first center is picked uniformly at random, after which each subsequent center is picked from the remaining set points according to the probability distribution proportional to the squared distance of the points to the so-far-selected set of centers.", "Once the seeding phase picks $k$ centers, $k$ -means++ performs Lloyd's heuristic.", "Before stating the argument formally, we show how to construct the hard instance $\\mathcal {I}_\\alpha $ for any given approximation parameter $\\alpha >1$ for the IP-stability requirement." ], [ "Structure of the hard instance.", "Given the approximation parameter $\\alpha $ , $\\mathcal {I}_\\alpha $ is constructed as follows.", "The instance consists of $n_\\alpha $ copies of block $I$ such that each adjacent blocks are at distance $D_\\alpha $ from each other; see Figure REF .", "As we explain in more detail, the solution $\\mathrm {SOL}$ returned by $k$ -means++ violates the IP-stability by a factor of $\\alpha $ if there exists a block $I_j\\in \\mathcal {I}_\\alpha $ such that $\\mathrm {SOL}\\cap I_j = \\lbrace v_j, u_j\\rbrace $ .", "In the rest of this section, our goal is to show that with constant probability this event happens when $k_\\alpha =\\frac{13}{12}n_{\\alpha }$ .", "Figure: The construction of hard instance for a given parameter α\\alpha .", "(a) shows the structure of a building block II and (b) shows a hard instance which constitutes of n α n_\\alpha blocks I 1 ,...,I n α I_1, \\ldots , I_{n_\\alpha }.Claim 14 Suppose that $v$ and $u$ are picked as the initial set of centers for 2-clustering of $I$ .", "Then, the solution returned by $k$ -means++ violates the IP-stability requirement by a factor of $\\alpha $ .", "It is straightforward to verify that once $v$ and $u$ are picked as the initial set of centers, after a round of Lloyd's heuristic $v$ and $u$ remain the cluster centers and $k$ -means++ stops.", "This is the case since when $v$ and $u$ are centers initially the clusters are $\\lbrace z, z^{\\prime }, v\\rbrace $ and $\\lbrace u\\rbrace $ .", "The centroids of these two clusters are respectively $v$ and $u$ .", "Hence, $k$ -means++ outputs $\\lbrace z, z^{\\prime }, v\\rbrace $ and $\\lbrace u\\rbrace $ as the 2-clustering of $I$ .", "The maximum violation of the IP-stable requirement for this 2-clustering corresponds to $v$ which is equal to $\\frac{(d(v, z) + d(v, z^{\\prime }))/2}{d(v,u)} = \\alpha $ .", "In this section, the squared distance function is denoted by $\\Delta $ .", "Here is the main theorem of this section.", "Theorem 15 For any given parameter $\\alpha $ , there exists an instance $\\mathcal {I}_\\alpha $ such that with constant probability the solution returned by $k$ -means++ on $(\\mathcal {I}_\\alpha , k_\\alpha = \\frac{13}{12}n_\\alpha )$ is violating the requirement of IP-stable clustering by a factor of $\\alpha $ —the maximum violation over the points in $\\mathcal {I}_\\alpha $ is at least $\\alpha $ .", "To prove the above theorem, we show that by setting the values of $D_\\alpha $ and $n_{\\alpha }$ properly, $k$ -means++ with constant probability picks the points corresponding to $u,v$ from at least one of the copies $I_j$ (and picks no other points from $I_j$ ).", "Lemma 3 Let $\\mathcal {S}_1$ denote the set of centers picked by $k$ -means++ on $(\\mathcal {I}_\\alpha , \\frac{13}{12}n_\\alpha )$ after the first $n_\\alpha $ iterations in the seeding phase where $n_\\alpha \\ge 18700$ .", "If $D_\\alpha > \\alpha r \\sqrt{\\frac{3n_{\\alpha }}{\\varepsilon }}$ where $\\varepsilon <\\frac{1}{100 n_\\alpha }$ , then with probability at least $0.98$ , $\\mathcal {S}_1$ contains exactly one point from each block; $\\forall I\\in \\mathcal {I}_\\alpha , |\\mathcal {S}_1 \\cap I| = 1$ , and In at least $\\frac{n_\\alpha }{10}$ blocks, copies of $v$ are picked; $|j\\in [n_\\alpha ]\\;|\\; I_j\\cap \\mathcal {S}_1 = \\lbrace v_j\\rbrace | \\ge \\frac{n_\\alpha }{10}$ .", "We start with the first property.", "Consider iteration $i\\le n_\\alpha $ of $k$ -means++ and let $S_{i-1}$ denote the set of points that are selected as centers in the first $i-1$ iterations.", "Let $\\mathcal {I}^{+}_{i-1} := \\lbrace I_j \\;|\\; I_j \\cap S_{i-1} \\ne \\emptyset \\rbrace $ and $\\mathcal {I}^{-}_{i-1} :=\\lbrace I_j \\;|\\; I_j \\cap S_{i-1} = \\emptyset \\rbrace $ .", "Since for any pair of points $p\\in I, p^{\\prime }\\in I^{\\prime }$ where $I\\ne I^{\\prime }$ , $\\Delta (p,p^{\\prime }) \\ge D^2_\\alpha $ , for any point $p\\in \\mathcal {I}^{-}_{i-1}$ , $\\Delta (p, S_{ i-1}) \\ge D_\\alpha ^2$ .", "Moreover, by the construction of the building block (see Figure REF -(a)), for any point $p^{\\prime }\\in \\mathcal {I}^{+}_{i-1}$ , $\\Delta (p^{\\prime }, S_{i-1}) \\le 4\\alpha ^2 r^2\\le \\frac{4\\varepsilon }{3n_{\\alpha }} D_\\alpha ^2$ .", "Hence, the probability that in iteration $i$ the algorithm picks a point from $\\mathcal {I}^{-}_{i-1}$ is $\\frac{\\sum _{p\\in \\mathcal {I}^{-}_{i-1}}\\Delta (p, S_{i-1})}{\\sum _{p\\in \\mathcal {I}^{-}_{i-1}}\\Delta (p, S_{i-1}) + \\sum _{p\\in \\mathcal {I}^{+}_{i-1}}\\Delta (p, S_{i-1})} \\ge \\frac{4D^2_\\alpha }{4D^2_\\alpha + 3n_\\alpha \\cdot (4\\alpha ^2 r^2)} \\ge \\frac{1}{1+\\varepsilon } \\ge 1-\\varepsilon $ Thus, the probability that $i$ -th point is picked from $\\mathcal {I}^+_{i-1}$ is at most $\\varepsilon $ .", "By union bound over the first $n_\\alpha $ iterations of $k$ -means++, the probability that $\\mathcal {S}_1$ picks exactly one point from each block in $\\mathcal {I}_{\\alpha }$ is at least $1 - \\sum _{i=1}^{n_\\alpha } \\Pr [\\text{$i$-th point is picked from $\\mathcal {I}^+_{i-1}$}] \\ge 1 - \\varepsilon \\cdot n_\\alpha \\ge 0.99$ Next, we show that the second property also holds with high probability.", "Consider iteration $i$ of the algorithm.", "By the approximate triangle inequality for $\\Delta $ function$\\forall u,v,w\\in P, \\Delta (v,w) \\le 2(\\Delta (v,u) + \\Delta (u,w))$ .", "and since the distance of any pair of points within any block is $2\\alpha r$ , for a point $p\\in I_j \\in \\mathcal {I}^-_{i-1}$ , $\\eta ^2 \\le \\Delta (p, S_{i-1}) \\le 2(\\eta ^2 + 4\\alpha ^2r^2)$ where $\\eta \\ge D_\\alpha $ .", "In particular, for each block $I_j\\in \\mathcal {I}^{-}_{i-1}$ , $\\frac{\\Delta (v_j, S_{i-1})}{\\sum _{p\\in I_j}\\Delta (p, S_{i-1})} \\ge \\frac{\\eta ^2}{7\\eta ^2 + 24\\alpha ^2 r^2}\\ge 1/(8+\\frac{\\varepsilon }{n_\\alpha })$ Let $X_i$ be a random variable which indicates that the $i$ -th point belongs to $\\lbrace v_j | j \\in [n_\\alpha ]\\rbrace $ ; $X_i = 1$ if the $i$ -th point belongs to $\\lbrace v_j | j \\in [n_\\alpha ]\\rbrace $ and is equal to zero otherwise.", "$\\Pr [X_i = 1]&\\ge \\frac{\\sum _{I_j\\in \\mathcal {I}^{-}_{i-1}} \\Delta (v_j, S_{i-1})}{\\sum _{I_j\\in \\mathcal {I}^{-}_{i-1}} \\sum _{p\\in I_j}\\Delta (p, S_{i-1}) + \\sum _{p\\in \\mathcal {I}^{+}_{i-1}}\\Delta (p, S_{i-1})} \\\\&\\ge \\frac{D^2_\\alpha }{(8+\\varepsilon /n_\\alpha ) \\cdot D^2_\\alpha + \\sum _{p\\in \\mathcal {I}^{+}_{i-1}}\\Delta (p, S_{i-1})} \\quad \\rhd \\text{by Eq.~(\\ref {eq:v-pick})}\\\\&\\ge \\frac{D^2_\\alpha }{(8+\\varepsilon /n_\\alpha ) \\cdot D^2_\\alpha + 3n_\\alpha \\cdot (4\\alpha ^2 r^2)} \\quad \\rhd \\forall p\\in \\mathcal {I}^+_{i-1}, \\Delta (p, S_{i-1})\\le 4\\alpha ^2 r^2\\\\&\\ge \\frac{D^2_\\alpha }{(8+\\varepsilon /n_\\alpha ) \\cdot D^2_\\alpha + \\varepsilon \\cdot D^2_\\alpha } > 1/9$ Note that we showed that $\\Pr [X_i=1] \\ge 1/9$ independent of the algorithm's choices in the first $i-1$ iterations of the algorithm.", "In particular, random variables $X_1, \\ldots , X_{n_\\alpha }$ are stochastically dominated by independent random variable $Y_1, \\ldots , Y_{n_{\\alpha }}$ where each $Y_i =1$ with probability $1/9$ and is zero otherwise.", "Hence, the random variable $X:=X_1 + \\ldots + X_{n_{\\alpha }}$ is stochasitcally dominated by $Y$ which is $\\mathrm {B}(n_\\alpha , 1/9)$ , the binomial distribution with $n_\\alpha $ trials and success probability $1/9$ .", "Hence, by an application of Hoeffding's inequality on $Y$ , $\\Pr [X \\le \\frac{n_\\alpha }{10}] \\le \\Pr [Y\\le \\frac{n_\\alpha }{10}]&\\le \\exp (-2n_{\\alpha }(\\frac{1}{9}-\\frac{1}{10})^2) \\le 0.01 &&\\rhd \\text{for $n_\\alpha >18700$}$ Thus, by Eq.", "(REF ) and (REF ), with probability at least $0.98$ both properties hold.", "Lemma 4 Let $\\mathcal {S}$ be the set of centers picked at the end of the seeding phase of $k$ -means++ on $(\\mathcal {I}_\\alpha , k_\\alpha = \\frac{13}{12}n_\\alpha )$ where $n_\\alpha >\\max (9000, 240\\alpha ^2)$ .", "Then, with probability at least $0.33$ , there exists a block $I_j\\in \\mathcal {I}_\\alpha $ such that $\\mathcal {S}\\cap I_j = \\lbrace v_j, u_j\\rbrace $ .", "Let $\\mathcal {T}_1$ denote the event that $\\mathcal {S}_1$ , the set of centers picked in the first $n_\\alpha $ iterations, picks exactly one point from each block and in at least $n_{\\alpha }/10$ blocks the $v$ -type points are picked in $\\mathcal {S}_1$ .", "Note that by an application of Lemma REF , $\\Pr [\\mathcal {T}_1] \\ge 0.98$ —with probability at least $0.98$ , $\\mathcal {T}_1$ holds.", "First we show the following.", "Let $m_\\alpha = n_{\\alpha }/20$ .", "With constant probability, there exists an iteration $i^*\\in [n_\\alpha +1, n_\\alpha + m_\\alpha ]$ in which $k$ -means++ picks a $u$ -type point form $I_j$ such that $I_j \\cap S_{i^*-1} = \\lbrace v_j\\rbrace $ .", "Let $\\mathcal {E}_i$ denote the event that in iteration $i+n_\\alpha $ where $i \\in [m_\\alpha ]$ , $k$ -means++ picks the point $u_j$ from $I_j$ such that $S_{i-1} \\cap I_j = \\lbrace v_j\\rbrace $ .", "Note that for each block in $\\lbrace I_j \\; | \\; \\mathcal {S}_1 \\cap I_j = \\lbrace v_j\\rbrace \\rbrace $ , $\\sum _{p\\in I_j} \\Delta (p, \\mathcal {S}_1) = (2\\alpha ^2+1) r^2$ ; otherwise, $\\sum _{p\\in I_j} \\Delta (p, \\mathcal {S}_1) \\le (5\\alpha ^2+1) r^2$ .", "Hence, for each $i\\in [m_{\\alpha }]$ , conditioned on $\\mathcal {T}_1$ and independent of the prior choices of the algorithm in iterations $n_\\alpha +1, \\ldots , i^*:=n_{\\alpha }+i-1$ , $\\Pr [\\mathcal {E}_i | \\mathcal {T}_1]&= \\frac{\\sum _{\\lbrace I_j \\; | \\; I_j\\cap S_{i^*} = \\lbrace v_j\\rbrace \\rbrace } \\Delta (u_j, S_{i^*})}{\\sum _{\\lbrace I_j \\; | \\; I_j\\cap S_{i^*} = \\lbrace v_j\\rbrace \\rbrace } \\sum _{p\\in I_j}\\Delta (p, S_{i^*}) + \\sum _{\\lbrace I_j \\; | \\; I_j\\cap S_{i^*} \\ne \\lbrace v_j\\rbrace \\rbrace } \\sum _{p\\in I_j}\\Delta (p, S_{i^*})} \\\\&\\ge \\frac{(\\frac{n_\\alpha }{10} - m_\\alpha )\\cdot r^2}{(\\frac{n_\\alpha }{10} - m_\\alpha )\\cdot (2\\alpha ^2 + 1) r^2 + (n_\\alpha - (\\frac{n_\\alpha }{10} -m_\\alpha ))\\cdot (5\\alpha ^2+1)r^2} \\\\&\\ge \\frac{m_\\alpha }{ m_\\alpha (2\\alpha ^2+1) + (n_\\alpha -m_\\alpha )(5\\alpha ^2+1)} \\\\&\\ge \\frac{1}{2\\alpha ^2 + 1 + 20(5\\alpha ^2 +1)}\\\\&\\ge \\frac{1}{123\\alpha ^2}$ For each $i\\in [m_\\alpha ]$ , let random variable $X_i := 1_{\\mathcal {E}_i|\\mathcal {T}_1}$ .", "Since $X_1, \\ldots , X_{m_\\alpha }$ are stochastically dominated by independent random variables $Y_i$ such that $Y_i = 1$ with probability $1/(123\\alpha ^2)$ and zero otherwise, we can bound the probability that none of $\\mathcal {E}_1, \\ldots , \\mathcal {E}_{m_{\\alpha }}$ happens as follows.", "$\\Pr [\\lnot \\mathcal {E}_1 \\wedge \\ldots \\wedge \\lnot \\mathcal {E}_{m_\\alpha } | \\mathcal {T}_1]&= \\Pr [\\sum _{i\\le m_\\alpha } X_i < 1 | \\mathcal {T}_1] \\nonumber \\\\&\\le \\Pr [\\sum _{i\\le m_\\alpha } Y_i < 1] \\nonumber \\\\&= (1- \\frac{1}{123\\alpha ^2})^{m_\\alpha } \\le \\exp (-\\frac{m_\\alpha }{123\\alpha ^2}) $ Hence, with probability at least $1- \\exp (-\\frac{m_\\alpha }{123\\alpha ^2})$ , there exists an iteration $i^*\\in [n_\\alpha +1, n_\\alpha + m_\\alpha ]$ and block $I^*_j$ such that $I^*_j\\cap S_{i^*} = \\lbrace v_j, u_j\\rbrace $ .", "Next, we show that with constant probability no more points is picked from $I^*_j$ in the remaining iterations of the seeding phase $k$ -means++.", "Let $\\mathcal {T}_2$ denote the event that there exists $(I^*_j, i^*)$ such that $I^*_j \\cap S_{i^*} = \\lbrace v_j, u_j\\rbrace $ ; $\\mathcal {T}_2 := \\mathcal {E}_1 \\vee \\ldots \\vee \\mathcal {E}_{m_{\\alpha }}$ .", "For any $i\\in [1, n_\\alpha +m_\\alpha - i^*]$ , let $\\mathcal {F}_i$ denote the event that in iteration $i^* + i$ , $k$ -means++ picks another point from $I^*_j$ .", "$\\Pr [\\mathcal {F}_i | \\mathcal {T}_1 \\wedge \\mathcal {T}_2]&< \\frac{2\\alpha ^2 r^2}{\\sum _{\\lbrace I_j \\; | \\; I_j\\cap S_{i-1} = \\lbrace v_j\\rbrace \\rbrace } \\sum _{p\\in I_j}\\Delta (p, S_{i-1})} \\nonumber \\\\&\\le \\frac{2\\alpha ^2 r^2}{(\\frac{n_\\alpha }{10}-m_\\alpha )\\cdot (2\\alpha ^2 + 1) r^2} = \\frac{2}{3m_{\\alpha }}$ Let $t_\\alpha := n_\\alpha + m_\\alpha - i^*$ and define $\\mathcal {T}_3$ to denote the event that none of $\\mathcal {F}_1, \\ldots , \\mathcal {F}_{t_\\alpha }$ happens.", "For each $i\\in [t_\\alpha ]$ , let random variable $\\tilde{X}_i := 1_{\\mathcal {F}_i|\\mathcal {T}_1 \\wedge \\mathcal {T}_2}$ .", "Since $\\tilde{X}_1, \\ldots , \\tilde{X}_{m_\\alpha }$ are stochastically dominated by independent random variables $\\tilde{Y}_i$ such that $\\tilde{Y}_i = 1$ with probability $1/(123\\alpha ^2)$ and zero otherwise, we can lower bound the probability that event $\\mathcal {T}_3$ holds as follows.", "$\\Pr [\\mathcal {T}_3 | \\mathcal {T}_1 \\wedge \\mathcal {T}_2] = \\Pr [\\lnot \\mathcal {F}_1 \\wedge \\ldots \\wedge \\lnot \\mathcal {F}_{t_{\\alpha }} | \\mathcal {T}_1 \\wedge \\mathcal {T}_2]&\\ge \\Pr [\\sum _{i\\le t_\\alpha } \\tilde{Y}_i < 1 | \\mathcal {T}_1 \\wedge \\mathcal {T}_2] \\nonumber \\\\&\\ge (1-\\frac{2}{3m_\\alpha })^{t_\\alpha } &&\\rhd \\text{by Eq.~(\\ref {eq:bad-event})} \\nonumber \\\\&\\ge (1-\\frac{2}{3m_\\alpha }) \\cdot \\exp (-\\frac{2}{3}) &&\\rhd t_\\alpha \\le m_\\alpha $ Hence, Eq.", "(REF ) and Eq.", "(REF ) together imply that the probability that there exists no block $I^*_j$ such that $|I^*_j \\cap \\mathcal {S}| = \\lbrace v_j, u_j\\rbrace $ is at most $\\Pr [\\lnot \\mathcal {T}_1] + \\Pr [\\lnot \\mathcal {T}_2 | \\mathcal {T}_1] + \\Pr [\\lnot \\mathcal {T}_3 | \\mathcal {T}_1 \\wedge \\mathcal {T}_2]&\\le 0.02 + e^{-\\frac{m_\\alpha }{123\\alpha ^2}} + 1 - (1-\\frac{2}{3m_\\alpha })e^{-\\frac{2}{3}}\\\\&\\le 0.02 + e^{-\\frac{m_\\alpha }{123\\alpha ^2}} + \\frac{1}{2}+\\frac{1}{3m_\\alpha } \\;\\rhd m_\\alpha \\ge 240\\alpha ^2 \\\\&\\le 0.02 + 0.145 + 0.5 + 0.005 < 0.67$ Thus, with probability at least $0.33$ , at the end of the seeding phase, there exists a block $I^*_j$ such that $I^*_j \\cap \\mathcal {S}= \\lbrace v_j, u_j\\rbrace $ .", "Now, we are ready to prove Theorem REF .", "[Proof of Theorem REF ] Note that Lemma REF together with Lemma REF implies that with constant probability, at the end of the seeding phase, at least one point is picked from each block in $\\mathcal {I}_{\\alpha }$ and there exist a block $I_j$ such that $I_j\\cap S = \\lbrace v_j, u_j\\rbrace $ .", "Given the structure of $\\mathcal {I}_{\\alpha }$ , after each step of the Lloyd's heuristic, no points from two different blocks will be in the same cluster.", "Furthermore, by Claim REF , the clusters of $I_j$ which are initially $\\lbrace z_j, z^{\\prime }_j, v_j\\rbrace , \\lbrace u_j\\rbrace $ remain unchanged in the final solution of $k$ -means++.", "However, such clustering is violating the requirement of IP-stable clustering for $v_j$ by a factor of $\\alpha $ .", "Thus, with constant probability, the solution of $k$ -means++ on $(\\mathcal {I}_\\alpha , k_\\alpha )$ violates the requirement of IP-stable clustering by a factor of at least $\\alpha $ ." ], [ "Hard Instances for $k$ -center", "Figure REF shows a hard instance for $k$ -center.", "Each of the balls $B_1$ and $B_2$ have radius $\\epsilon $ .", "The center of $B_1$ has a distance of $1+\\epsilon $ from $c_1$ , and a distance of $1-\\epsilon $ from $c_2$ .", "The center of $B_2$ has a distance of $1-\\epsilon $ from $c_1$ , and a distance of $1+\\epsilon $ from $c_2$ .", "The nodes of the graph are $c_1$ , $c_2$ , $n$ points on the surface of $B_1$ , and $n$ points on the surface of $B_2$ .", "Consider $k$ -center using the greedy strategy by [45].", "Suppose $k=2$ , and the first center picked is $c_1$ .", "The furthest node from $c_1$ is $c_2$ , and hence it is the second center picked.", "Since all the points get assigned to the closest center, all the points on $B_1$ get assigned to $c_2$ , except the point $p$ that has the same distance from $c_1$ and $c_2$ and gets assigned to $c_1$ .", "All the points on $B_2$ get assigned to $c_1$ .", "We show such a clustering is unstable for $p$ within a factor of $n/8$ .", "The average distance of $p$ to the nodes in its own cluster is at least $\\frac{n(1/2-2\\epsilon )+1}{n+1}$ .", "The average distance of $p$ to the nodes assigned to $c_2$ is at most $\\frac{2{\\epsilon }(n-1)+1}{n}$ .", "For $\\epsilon \\le 1/2n$ , the instability for $p$ is at least within a multiplicative factor of $n/8$ .", "By setting $n>8\\alpha $ , $p$ is not $t$ -approximately IP-stable for any value $t<\\alpha $ ." ], [ "Hard Instances for Single Linkage", "Figure REF shows a bad example for single linkage clustering when $k=2$ .", "A possible implementation of single linkage first merges $v_2$ and $v_3$ .", "Next, it merges $v_4$ and $\\lbrace v_2, v_3\\rbrace $ , and repeatedly at each iteration $i$ adds $v_{i+2}$ to the set $\\lbrace v_2, v_3, \\ldots , v_{i+1}\\rbrace $ .", "When there are only two clusters $\\lbrace v_2, v_3, \\ldots , v_n\\rbrace $ and $v_1$ left, the algorithm terminates.", "By setting $\\epsilon = 1/2$ , $v_2$ gets unstable by a factor of $(n-1)/4$ .", "By setting $(n-1)/4 > \\alpha $ , $v_2$ is not $t$ -approximately IP-stable for any value of $t<\\alpha $ .", "Figure: A hard instance for single linkage clustering." ], [ "Dynamic Programming Approach for Solving (", "In the following, we propose an efficient DP approach to find a solution to (REF ).", "Let $\\mathcal {D}=\\lbrace x_1,\\ldots ,x_n\\rbrace $ with $x_1\\le \\ldots \\le x_n$ .", "Our approach builds a table $T\\in (\\mathbb {N}\\cup \\lbrace \\infty \\rbrace )^{n\\times n\\times k}$ with $T(i,j,l)=\\min _{(C_1,\\ldots ,C_l)\\in \\mathcal {H}_{i,j,l}} \\Vert (|C_1|-t_1,\\ldots ,|C_l|-t_l)\\Vert _p^p$ for $i\\in [n]$ , $j\\in [n]$ , $l\\in [k]$ , where $&\\mathcal {H}_{i,j,l}=\\big \\lbrace \\mathcal {C}=(C_1,\\ldots ,C_l):~\\text{$\\mathcal {C}$ is a stable $l$-clustering of $\\lbrace x_1,\\ldots ,x_i\\rbrace $ with $l$ non-empty}\\\\&~~~~~~~~~~~~~~~~~~~~~\\text{contiguous clusters such that the right-most cluster $C_l$ contains exactly $j$ points}\\big \\rbrace $ and $T(i,j,l)=\\infty $ if $\\mathcal {H}_{i,j,l}=\\emptyset $ .", "Here, we consider the case $p\\ne \\infty $ .", "The modifications of our approach to the case $p=\\infty $ are minimal and are described later.", "The optimal value of (REF ) is given by $\\min _{j\\in [n]} T(n,j,k)^{1/p}$ .", "Below, we will describe how to use the table $T$ to compute an IP-stable $k$ -clustering solving (REF ).", "First, we explain how to build $T$ .", "We have, for $i,j\\in [n]$ , $\\begin{split}&T(i,j,1)={\\left\\lbrace \\begin{array}{ll}|i-t_1|^p,& j=i, \\\\\\infty , &j\\ne i\\\\\\end{array}\\right.", "},\\qquad T(i,j,i)={\\left\\lbrace \\begin{array}{ll}\\sum _{s=1}^i|1-t_s|^p,& j=1, \\\\\\infty , &j\\ne 1\\\\\\end{array}\\right.", "},\\\\&T(i,j,l)=\\infty ,\\quad j+l-1>i,\\end{split}$ and the recurrence relation, for $l>1$ and $j+l-1\\le i$ , $\\begin{split}T(i,j,l)=|j-t_l|^p+\\min \\left\\lbrace T(i-j,s,l-1): s\\in [i-j-(l-2)],\\frac{\\sum _{f=1}^{s-1}|x_{i-j}-x_{i-j-f}|}{s-1}\\le \\right.\\\\\\left.\\frac{\\sum _{f=1}^{j}|x_{i-j}-x_{i-j+f}|}{j},\\frac{\\sum _{f=2}^{j}|x_{i-j+1}-x_{i-j+f}|}{j-1}\\le \\frac{\\sum _{f=0}^{s-1}|x_{i-j+1}-x_{i-j-f}|}{s}\\right\\rbrace ,\\end{split}$ where we use the convention that $\\frac{0}{0}=0$ for the fractions on the left sides of the inequalities.", "First, we explain relation (REF ).", "Before that we need to show the following lemma holds: Lemma 5 (Stable boundary points imply stable clustering) Let $\\mathcal {C}=(C_1,\\ldots ,C_k)$ be a $k$ -clustering of $\\mathcal {D}=\\lbrace x_1,\\ldots ,x_n\\rbrace $ , where $x_1\\le x_2\\le \\ldots \\le x_n$ , with contiguous clusters $C_1=\\lbrace x_1,\\ldots ,x_{i_1}\\rbrace ,C_2=\\lbrace x_{i_1+1},\\ldots ,x_{i_2}\\rbrace ,\\ldots ,C_k=\\lbrace x_{i_{k-1}+1},\\ldots ,x_n\\rbrace $ , for some $1\\le i_1<\\ldots <i_{k-1}<n$ .", "Then $\\mathcal {C}$ is IP-stable if and only if all points $x_{i_l}$ and $x_{i_l+1}$ , $l\\in [k-1]$ , are stable.", "Furthermore, $x_{i_l}$ ($x_{i_l+1}$ , resp.)", "is stable if and only if its average distance to the points in $C_l\\setminus \\lbrace x_{i_l}\\rbrace $ ($C_{l+1}\\setminus \\lbrace x_{i_l+1}\\rbrace $ , resp.)", "is not greater than the average distance to the points in $C_{l+1}$ ($C_{l}$ , resp.).", "We assume that $\\mathcal {D}=\\lbrace x_1,\\ldots ,x_n\\rbrace \\subseteq \\mathbb {R}$ with $x_1\\le x_2\\le \\ldots \\le x_n$ and write the Euclidean metric $d(x_i,x_j)$ between two points $x_i$ and $x_j$ in its usual way $|x_i-x_j|$ .", "If $\\mathcal {C}$ is IP-stable, then all points $x_{i_l}$ and $x_{i_l+1}$ , $l\\in [k-1]$ , are stable.", "Conversely, let us assume that $x_{i_l}$ and $x_{i_l+1}$ , $l\\in [k-1]$ , are stable.", "We need to show that all points in $\\mathcal {D}$ are stable.", "Let $\\tilde{x}\\in C_l=\\lbrace x_{i_{l-1}+1},\\ldots ,x_{i_l}\\rbrace $ for some $l\\in \\lbrace 2,\\ldots ,k-1\\rbrace $ and $l^{\\prime }\\in \\lbrace l+1,\\ldots ,k\\rbrace $ .", "Since $x_{i_l}$ is stable, we have $\\frac{1}{|C_l|-1}\\sum _{y\\in {C_{l}}} (x_{i_l}-y)= \\frac{1}{|C_l|-1}\\sum _{y\\in {C_{l}}} |x_{i_l}-y|\\le \\frac{1}{|C_{l^{\\prime }}|}\\sum _{y\\in {C_{l^{\\prime }}}} |x_{i_l}-y|=\\frac{1}{|C_{l^{\\prime }}|}\\sum _{y\\in {C_{l^{\\prime }}}} (y-x_{i_l})$ and hence $\\frac{1}{|C_l|-1}\\sum _{y\\in {C_{l}}} |\\tilde{x}-y|&\\le \\frac{1}{|C_l|-1}\\sum _{y\\in C_{l}\\setminus \\lbrace \\tilde{x}\\rbrace } (|\\tilde{x}-x_{i_l}|+|x_{i_l}-y|)\\\\&= (x_{i_l}-\\tilde{x})+\\frac{1}{|C_l|-1}\\sum _{y\\in C_{l}\\setminus \\lbrace \\tilde{x}\\rbrace } (x_{i_l}-y)\\\\&\\le (x_{i_l}-\\tilde{x})+\\frac{1}{|C_{l^{\\prime }}|}\\sum _{y\\in {C_{l^{\\prime }}}} (y-x_{i_l})\\\\&=\\frac{1}{|C_{l^{\\prime }}|}\\sum _{y\\in {C_{l^{\\prime }}}} (y-\\tilde{x})\\\\&=\\frac{1}{|C_{l^{\\prime }}|}\\sum _{y\\in {C_{l^{\\prime }}}} |\\tilde{x}-y|.$ Similarly, we can show for $l^{\\prime }\\in \\lbrace 1,\\ldots ,l-1\\rbrace $ that $\\frac{1}{|C_l|-1}\\sum _{y\\in {C_{l}}} |\\tilde{x}-y|\\le \\frac{1}{|C_{l^{\\prime }}|}\\sum _{y\\in {C_{l^{\\prime }}}} |\\tilde{x}-y|,$ and hence $\\tilde{x}$ is stable.", "Similarly, we can show that all points $x_1,\\ldots ,x_{{i_1}-1}$ and $x_{i_{k-1}+2},\\ldots ,x_n$ are stable.", "For the second claim observe that for $1\\le s\\le l-1$ , the average distance of $x_{i_l}$ to the points in $C_s$ cannot be smaller than the average distance to the points in $C_l\\setminus \\lbrace x_{i_l}\\rbrace $ and for $l+2\\le s\\le k$ , the average distance of $x_{i_l}$ to the points in $C_s$ cannot be smaller than the average distance to the points in $C_{l+1}$ .", "A similar argument proves the claim for $x_{i_l+1}$ .", "It follows from Lemma REF that a clustering $(C_1,\\ldots ,C_l)$ of $\\lbrace x_1,\\ldots ,x_i\\rbrace $ with contiguous clusters and $C_l=\\lbrace x_{i-j+1},\\ldots ,x_i\\rbrace $ is IP-stable if and only if $(C_1,\\ldots ,C_{l-1})$ is a stable clustering of $\\lbrace x_1,\\ldots ,x_{i-j}\\rbrace $ and the average distance of $x_{i-j}$ to the points in $C_{l-1}\\setminus \\lbrace x_{i-j}\\rbrace $ is not greater than the average distance to the points in $C_l$ and the average distance of $x_{i-j+1}$ to the points in $C_{l}\\setminus \\lbrace x_{i-j+1}\\rbrace $ is not greater than the average distance to the points in $C_{l-1}$ .", "The latter two conditions correspond to the two inequalities in (REF ) (when $|C_{l-1}|=s$ , where $s$ is a variable).", "By explicitly enforcing these two constraints, we can utilize the first condition and rather than minimizing over $\\mathcal {H}_{i,j,l}$ in (REF ), we can minimize over both $s\\in [i-j-(l-2)]$ and $\\mathcal {H}_{i-j,s,l-1}$ (corresponding to minimizing over all IP-stable $(l-1)$ -clusterings of $\\lbrace x_1,\\ldots ,x_{i-j}\\rbrace $ with non-empty contiguous clusters).", "It is $\\min _{\\begin{array}{c}s\\in [i-j-(l-2)]\\\\ (C_1,\\ldots ,C_{l-1})\\in \\mathcal {H}_{i-j,s,l-1}\\end{array}}\\Vert (|C_1|-t_1,\\ldots ,|C_{l-1}|-t_{l-1})\\Vert _p^p =\\min _{s\\in [i-j-(l-2)]} T(i-j,s,l-1),$ and hence we end up with the recurrence relation (REF ).", "It is not hard to see that using (REF ), we can build the table $T$ in time $\\mathcal {O}(n^3k)$ .", "Once we have $T$ , we can compute a solution $(C_1^*,\\ldots ,C_k^*)$ to (REF ) by specifying $|C_1^*|,\\ldots ,|C_k^*|$ in time $\\mathcal {O}(nk)$ as follows: let $v^*=\\min _{j\\in [n]} T(n,j,k)$ .", "We set $|C_k^*|=j_0$ for an arbitrary $j_0$ with $v^*=T(n,j_0,k)$ .", "For $l=k-1,\\ldots ,2$ , we then set $|C_l^*|=h_0$ for an arbitrary $h_0$  with (i) $T(n-\\sum _{r=l+1}^k |C_r^*|,h_0,l)+\\sum _{r=l+1}^k ||C_r^*|-t_r|^p=v^*$ , (ii) the average distance of $x_{n-\\sum _{r=l+1}^k |C_r^*|}$ to the closest $h_0-1$ many points on its left side is not greater than the average distance to the points in $C_{l+1}^*$ , and (iii) the average distance of $x_{n-\\sum _{r=l+1}^k |C_r^*|+1}$ to the other points in $C_{l+1}^*$ is not greater than the average distance to the closest $h_0$ many points on its left side.", "Finally, it is $|C_1^*|=n-\\sum _{r=2}^k |C_r^*|$ .", "It follows from the definition of the table $T$ in (REF ) and Lemma REF that for $l=k-1,\\ldots ,2$ we can always find some $h_0$ satisfying (i) to (iii) and that our approach yields an IP-stable $k$ -clustering $(C_1^*,\\ldots ,C_k^*)$ of $\\mathcal {D}$ .", "Hence we have shown the following theorem: Theorem 16 (Efficient DP approach solves (REF )) By means of the dynamic programming approach (REF ) to (REF ) we can compute an IP-stable clustering solving (REF ) in running time $\\mathcal {O}(n^3k)$ .", "Let us first explain the recurrence relation (REF ): because of $\\Vert (x_1,\\ldots ,x_l)\\Vert _p^p=\\Vert (x_1,\\ldots ,x_{l-1})\\Vert _p^p+|x_l|^p$ and for every clustering $(C_1,\\ldots ,C_l)\\in \\mathcal {H}_{i,j,l}$ it is $|C_l|=j$ , we have $T(i,j,l)=|j-t_l|^p+\\min _{(C_1,\\ldots ,C_l)\\in \\mathcal {H}_{i,j,l}} \\Vert (|C_1|-t_1,\\ldots ,|C_{l-1}|-t_{l-1})\\Vert _p^p.$" ], [ "$p=\\infty $", "Now we describe how to modify the dynamic programming approach of Section REF to the case $p=\\infty $ : in this case, we replace the definition of the table $T$ in (REF ) by $T(i,j,l)=\\min _{(C_1,\\ldots ,C_l)\\in \\mathcal {H}_{i,j,l}} \\Vert (|C_1|-t_1,\\ldots ,|C_l|-t_l)\\Vert _{\\infty },\\quad i\\in [n], j\\in [n], l\\in [k],$ and $T(i,j,l)=\\infty $ if $\\mathcal {H}_{i,j,l}=\\emptyset $ as before.", "The optimal value of (REF ) is now given by $\\min _{j\\in [n]} T(n,j,k)$ .", "Instead of (REF ), we have, for $i,j\\in [n]$ , $T(i,j,1)={\\left\\lbrace \\begin{array}{ll}|i-t_1|,& j=i, \\\\\\infty , &j\\ne i\\\\\\end{array}\\right.", "},\\hspace{19.91692pt}T(i,j,i)={\\left\\lbrace \\begin{array}{ll}\\max _{s=1,\\ldots ,i}|1-t_s|,& j=1, \\\\\\infty , &j\\ne 1\\\\\\end{array}\\right.", "}$ and $T(i,j,l)=\\infty ,\\quad j+l-1>i,$ and the recurrence relation (REF ) now becomes, for $l>1$ and $j+l-1\\le i$ , $&T(i,j,l)=\\max \\Bigg \\lbrace |j-t_l|,\\min \\bigg \\lbrace T(i-j,s,l-1): s\\in [i-j-(l-2)],\\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\frac{1}{s-1}\\,{\\sum _{f=1}^{s-1}|x_{i-j}-x_{i-j-f}|}\\le \\frac{1}{j}\\,{\\sum _{f=1}^{j}|x_{i-j}-x_{i-j+f}|},\\\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\frac{1}{j-1}\\,{\\sum _{f=2}^{j}|x_{i-j+1}-x_{i-j+f}|}\\le \\frac{1}{s}\\,{\\sum _{f=0}^{s-1}|x_{i-j+1}-x_{i-j-f}|}\\bigg \\rbrace \\Bigg \\rbrace .$ Just like before, we can build the table $T$ in time $\\mathcal {O}(n^3k)$ .", "Computing a solution $(C_1^*,\\ldots ,C_k^*)$ to (REF ) also works similarly as before.", "The only thing that we have to change is the condition (i) on $h_0$ (when setting $|C_l^*|=h_0$ for $l=k-1,\\ldots ,2$ ): now $h_0$ must satisfy $\\max \\left\\lbrace T\\left(n-\\sum _{r=l+1}^k |C_r^*|,h_0,l\\right),\\max _{r=l+1,\\ldots ,k} ||C_r^*|-t_r|\\right\\rbrace =v^*$ or equivalently $T\\left(n-\\sum _{r=l+1}^k |C_r^*|,h_0,l\\right)\\le v^*.$" ], [ "\nAddendum to Section ", "In this section, we provide supplements to Section REF : In Section REF , we present a simple example that shows that it really depends on the data set whether a group-fair clustering is IP-stable or not.", "In Section REF , we provide an example illustrating why the local search idea of Section REF does not work.", "In Section REF , we present a heuristic approach to make linkage clustering more aligned with IP-stability.", "In Section REF , we present the experiments of Section REF on the Adult data set [33]: we study the performance of various standard clustering algorithms as a function of the number of clusters $k$ when the underlying metric is either the Euclidean (as in the plot of Figure REF ), Manhattan or Chebyshev metric.", "We also study our heuristic approach of Appendix REF for making linkage clustering more stable: just as for the standard algorithms, we show $\\text{\\#\\,Uns}$ , $\\text{MaxVi}$ , $\\text{MeanVi}$ and $\\text{Co}$ as a function of $k$ for ordinary average / complete / single linkage clustering and their modified versions using our heuristic approach in its both variants ($\\#$ U denotes the variant based on $\\text{\\#\\,Uns}$ and MV the variant based on $\\text{MaxVi}$ ).", "In Section REF , we show the same set of experiments on the Drug Consumption data set [38].", "We used all 1885 records in the data set, and we used all 12 features describing a record (e.g., age, gender, or education), but did not use the information about the drug consumption of a record (this information is usually used as label when setting up a classification problem on the data set).", "We normalized the features to zero mean and unit variance.", "When running the standard clustering algorithms on the data set, we refrained from running spectral clustering since the Scikit-learn implementation occasionally was not able to do the eigenvector computations and aborted with a LinAlgError.", "Other than that, all results are largely consistent with the results for the Adult data set.", "In Section REF , we show the same set of experiments on the Indian Liver Patient data set [33].", "Removing four records with missing values, we ended up with 579 records, for which we used all 11 available features (e.g., age, gender, or total proteins).", "We normalized the features to zero mean and unit variance.", "Again, all results are largely consistent with the results for the Adult data set.", "Figure: An example illustrating why the local search idea outlined in Section  does not work.Top left: 12 points in ℝ 2 \\mathbb {R}^2.Top right: A kk-means clustering of the 12 points (encoded by color) with two points that are not IP-stable (surrounded by a circle).Bottom row: After assigning one of the two points that are not stable in the kk-means clustering to its closest cluster, that point is stable.", "However,now some points that were initially stable are not stable anymore." ], [ "Compatibility of Group Fairness and IP-Stability", "By means of a simple example we want to illustrate that it really depends on the data set whether group fairness and IP-stability are compatible or at odds with each other.", "Here we consider the prominent group fairness notion for clustering of [27], which asks that in each cluster, every demographic group is approximately equally represented.", "Let us assume that the data set consists of the four 1-dimensional points 0, 1, 7 and 8 and the distance function $d$ is the ordinary Euclidean metric.", "It is easy to see that the only IP-stable 2-clustering is $\\mathcal {C}=(\\lbrace 0,1\\rbrace , \\lbrace 7,8\\rbrace )$ .", "Now if there are two demographic groups $G_1$ and $G_2$ with $G_1=\\lbrace 0,7\\rbrace $ and $G_2=\\lbrace 1,8\\rbrace $ , the clustering $\\mathcal {C}$ is perfectly fair according to the notion of [27].", "But if $G_1=\\lbrace 0,1\\rbrace $ and $G_2=\\lbrace 7,8\\rbrace $ , the clustering $\\mathcal {C}$ is totally unfair according to the their notion." ], [ "Why Local Search Does not Work", "Figure REF presents an example illustrating why the local search idea outlined in Section REF does not work: assigning a data point that is not IP-stable to its closest cluster (so that that data point becomes stable) may cause other data points that are initially stable to not be stable anymore after the reassignment." ], [ "Heuristic Approach to Make Linkage Clustering More Stable", "Linkage clustering builds a binary tree that represents a hierarchical clustering with the root of the tree corresponding to the whole data set and every node corresponding to a subset such that a parent is the union of its two children.", "The leaves of the tree correspond to singletons comprising one data point [64].", "If one wants to obtain a $k$ -clustering of the data set, the output of a linkage clustering algorithm is a certain pruning of this tree.", "With IP-stability being our primary goal, we propose to construct a $k$ -clustering / a pruning of the tree as follows: starting with the two children of the root, we maintain a set of nodes that corresponds to a clustering and proceed in $k-2$ rounds.", "In round $i$ , we greedily split one of the $i+1$ many nodes that we currently have into its two children such that the resulting $(i+2)$ -clustering minimizes, over the $i+1$ many possible splits, $\\text{\\#\\,Uns}$ (the number of non-stable datapoints; cf.", "the beginning of Section ).", "Alternatively, we can split the node that gives rise to a minimum value of $\\text{MaxVi}$ (cf.", "the beginning of Section ).", "Algorithm REF provides the pseudocode of our proposed strategy.", "[H] Algorithm to greedily prune a hierarchical clustering [1] Input: binary tree $T$ representing a hierarchical clustering obtained from running a linkage clustering algorithm; number of clusters $k\\in \\lbrace 2,\\ldots ,|\\mathcal {D}|\\rbrace $ ; measure $meas\\in \\lbrace \\text{\\#\\,Uns},\\text{MaxVi}\\rbrace $ that one aims to optimize for Output: a $k$ -clustering $\\mathcal {C}$ # Conventions: [leftmargin=*] for a node $v\\in T$ , we denote the left child of $v$ by $Left(v)$ and the right child by $Right(v)$ for a $j$ -clustering $\\mathcal {C}^{\\prime }=(C_1,C_2,\\ldots ,C_j)$ , a cluster $C_l$ and $A,B\\subseteq C_l$ with $A\\dot{\\cup } B =C_l$ we write $\\mathcal {C}^{\\prime }|_{C_l\\hookrightarrow A,B}$ for the $(j+1)-$ clustering that we obtain by replacing the cluster $C_l$ with two clusters $A$ and $B$ in $\\mathcal {C}^{\\prime }$ Let $r$ be the root of $T$ and initialize the clustering $\\mathcal {C}$ as $\\mathcal {C}=(Left(r),Right(r))$ $i=1$ to $k-2$ by 1 Set $v^\\star =\\operatornamewithlimits{argmin}_{v: v\\text{ is a cluster in $\\mathcal {C}$ with }|v|>1} meas(\\mathcal {C}|_{v\\hookrightarrow Left(v),Right(v)})$ and $\\mathcal {C}=\\mathcal {C}|_{v^\\star \\hookrightarrow Left(v^\\star ),Right(v^\\star )}$   $\\mathcal {C}$" ], [ "\nExperiments on the Adult Data Set", " Figure: Adult data set — plots of Figure :bad hbox\\text{\\#\\,Uns} (top-left),Co\\text{Co} (top-right),MaxVi\\text{MaxVi} (bottom-left)and MeanVi\\text{MeanVi} (bottom-right)for the clusterings produced by the various standard algorithms as a function ofkk.", "Figure: Adult data set — similar plots as in Figure ,but for the Manhattan metric.Figure: Adult data set — similar plots as in Figure , but for the Chebyshev metric.Figure: Adult data set with Euclidean metric:bad hbox\\text{\\#\\,Uns} (top-left),Co\\text{Co} (top-right),MaxVi\\text{MaxVi} (bottom-left),and MeanVi\\text{MeanVi} (bottom-right)for the clusterings produced byaverage linkage clusteringand the two variants of our heuristic of Appendix to improve it: the first (#\\#U in the legend) greedily chooses splits as to minimize bad hbox\\text{\\#\\,Uns}, the second(MV)as to minimize MaxVi\\text{MaxVi}.Figure: Adult data set with Euclidean metric:the various measuresfor the clusterings produced bycomplete linkage clusteringand the two variants of our heuristicapproach.Figure: Adult data set with Euclidean metric:the various measuresfor the clusterings produced bysingle linkage clusteringand the two variants of our heuristicapproach.Figure: Adult data set with Manhattan metric:the various measuresfor the clusterings produced byaverage linkage clusteringand the two variants of our heuristicapproach.Figure: Adult data set with Manhattan metric:the various measuresfor the clusterings produced bycomplete linkage clusteringand the two variants of our heuristicapproach.Figure: Adult data set with Manhattan metric:the various measuresfor the clusterings produced bysingle linkage clusteringand the two variants of our heuristicapproach.Figure: Adult data set with Chebyshev metric:the various measuresfor the clusterings produced byaverage linkage clusteringand the two variants of our heuristicapproach.Figure: Adult data set with Chebyshev metric:the various measuresfor the clusterings produced bycomplete linkage clusteringand the two variants of our heuristicapproach.Figure: Adult data set with Chebyshev metric:the various measuresfor the clusterings produced bysingle linkage clusteringand the two variants of our heuristicapproach." ], [ "Experiments on the Drug Consumption Data Set", " Figure: Drug Consumption data set with Euclidean metric:bad hbox\\text{\\#\\,Uns} (top-left),Co\\text{Co} (top-right),MaxVi\\text{MaxVi} (bottom-left)and MeanVi\\text{MeanVi} (bottom-right)for the clusterings produced by the various standard algorithms as a function ofthe number of clusters kk.kk.", "Figure: Drug Consumption data set — similar plots as in Figure ,but for the Manhattan metric.Figure: Drug Consumption data set — similar plots as in Figure , but for the Chebyshev metric.Figure: Drug Consumption data set with Euclidean metric:bad hbox\\text{\\#\\,Uns} (top-left),Co\\text{Co} (top-right),MaxVi\\text{MaxVi} (bottom-left),and MeanVi\\text{MeanVi} (bottom-right)for the clusterings produced byaverage linkage clusteringand the two variants of our heuristic of Appendix to improve it: the first (#\\#U in the legend) greedily chooses splits as to minimize bad hbox\\text{\\#\\,Uns}, the second(MV)as to minimize MaxVi\\text{MaxVi}.Figure: Drug Consumption data set with Euclidean metric:the various measuresfor the clusterings produced bycomplete linkage clusteringand the two variants of our heuristicapproach.Figure: Drug Consumption data set with Euclidean metric:the various measuresfor the clusterings produced bysingle linkage clusteringand the two variants of our heuristicapproach.Figure: Drug Consumption data set with Manhattan metric:the various measuresfor the clusterings produced byaverage linkage clusteringand the two variants of our heuristicapproach.Figure: Drug Consumption data set with Manhattan metric:the various measuresfor the clusterings produced bycomplete linkage clusteringand the two variants of our heuristicapproach.Figure: Drug Consumption data set with Manhattan metric:the various measuresfor the clusterings produced bysingle linkage clusteringand the two variants of our heuristicapproach.Figure: Drug Consumption data set with Chebyshev metric:the various measuresfor the clusterings produced byaverage linkage clusteringand the two variants of our heuristicapproach.Figure: Drug Consumption data set with Chebyshev metric:the various measuresfor the clusterings produced bycomplete linkage clusteringand the two variants of our heuristicapproach.Figure: Drug Consumption data set with Chebyshev metric:the various measuresfor the clusterings produced bysingle linkage clusteringand the two variants of our heuristicapproach." ], [ "Experiments on the Indian Liver Patient Data Set", " Figure: Indian Liver Patient data set with Euclidean metric:bad hbox\\text{\\#\\,Uns} (top-left),Co\\text{Co} (top-right),MaxVi\\text{MaxVi} (bottom-left)and MeanVi\\text{MeanVi} (bottom-right)for the clusterings produced by the various standard algorithms as a function ofthe number of clusters kk.kk.", "Figure: Indian Liver Patient data set — similar plots as in Figure ,but for the Manhattan metric.Figure: Indian Liver Patient data set — similar plots as in Figure , but for the Chebyshev metric.Figure: Indian Liver Patient data set with Euclidean metric:bad hbox\\text{\\#\\,Uns} (top-left),Co\\text{Co} (top-right),MaxVi\\text{MaxVi} (bottom-left),and MeanVi\\text{MeanVi} (bottom-right)for the clusterings produced byaverage linkage clusteringand the two variants of our heuristic of Appendix to improve it: the first (#\\#U in the legend) greedily chooses splits as to minimize bad hbox\\text{\\#\\,Uns}, the second(MV)as to minimize MaxVi\\text{MaxVi}.Figure: Indian Liver Patient data set with Euclidean metric:the various measuresfor the clusterings produced bycomplete linkage clusteringand the two variants of our heuristicapproach.Figure: Indian Liver Patient data set with Euclidean metric:the various measuresfor the clusterings produced bysingle linkage clusteringand the two variants of our heuristicapproach.Figure: Indian Liver Patient data set with Manhattan metric:the various measuresfor the clusterings produced byaverage linkage clusteringand the two variants of our heuristicapproach.Figure: Indian Liver Patient data set with Manhattan metric:the various measuresfor the clusterings produced bycomplete linkage clusteringand the two variants of our heuristicapproach.Figure: Indian Liver Patient data set with Manhattan metric:the various measuresfor the clusterings produced bysingle linkage clusteringand the two variants of our heuristicapproach.Figure: Indian Liver Patient data set with Chebyshev metric:the various measuresfor the clusterings produced byaverage linkage clusteringand the two variants of our heuristicapproach.Figure: Indian Liver Patient data set with Chebyshev metric:the various measuresfor the clusterings produced bycomplete linkage clusteringand the two variants of our heuristicapproach.Figure: Indian Liver Patient data set with Chebyshev metric:the various measuresfor the clusterings produced bysingle linkage clusteringand the two variants of our heuristicapproach." ], [ "Experiments on 1-dimensional Euclidean data sets\n", "In this section we present experiments with our dynamic programming (DP) approach of Appendix  for 1-dim Euclidean data sets.", "We used the German Credit data set [33].", "It comprises 1000 records (corresponding to human beings) and for each record one binary label (good vs. bad credit risk) and 20 features.", "In our first experiment, we clustered the 1000 people according to their credit amount, which is one of the 20 features.", "A histogram of the data can be seen in the left plot of Figure REF .", "We were aiming for $k$ -clusterings with clusters of equal size (i.e., target cluster sizes $t_i=\\frac{1000}{k}$ , $i\\in [k]$ ) and compared our DP approach with $p=\\infty $ to $k$ -means clustering as well as a naive clustering that simply puts the $t_1$ smallest points in the first cluster, the next $t_2$ many points in the second cluster, and so on.", "We considered two initialization strategies for $k$ -means: we either used the medians of the clusters of the naive clustering for initialization (thus, hopefully, biasing $k$ -means towards the target cluster sizes) or we ran $k$ -means++ [9].", "For the latter we report average results obtained from running the experiment for 100 times.", "In addition to the four quantities $\\text{\\#\\,Uns}$ , $\\text{MaxVi}$ , $\\text{MeanVi}$ and $\\text{Co}$ considered in the experiments of Section REF / Appendix REF  - REF (defined at the beginning of Section ), we report $\\text{Obj}$ (“objective”), which is the value of the objective function of (REF ) for $p=\\infty $ .", "Note that $k$ -means / $k$ -means++ yields contiguous clusters and $\\text{Obj}$ is meaningful for all four clustering methods that we consider.", "The results are provided in Table REF , where we consider $k=5$ , $k=10$ , $k=20$ or $k=50$ .", "As expected, for the naive clustering we always have $\\text{Obj}=0$ , and for our DP approach (DP) we have $\\text{\\#\\,Uns}=0$ , $\\text{MaxVi}\\le 1$ and $\\text{MeanVi}=0$ .", "Most interesting to see is that both versions of $k$ -means yield almost perfectly stable clusterings when $k$ is small and moderately stable clusterings when $k=50$ (with $k$ -means++ outperforming $k$ -means).", "In our second experiment, we used the first 500 records of the data set to train a multi-layer perceptron (MLP) for predicting the label (good vs. bad credit risk).", "We then applied the MLP to estimate the probabilities of having a good credit risk for the other 500 people.", "A histogram of the probability estimates is shown in the right plot of Figure REF .", "We used the same clustering methods as in the first experiment to cluster the 500 people according to their probability estimate.", "We believe that such a clustering problem may arise frequently in practice (e.g., when a bank determines its lending policy) and that IP-stability is highly desirable in this context.", "Table REF and Table REF provide the results.", "In Table REF , we consider uniform target cluster sizes $t_i=\\frac{500}{k}$ , $i\\in [k]$ , while in Table REF we consider various non-uniform target cluster sizes.", "The interpretation of the results is similar as for the first experiment of Section REF .", "Most notably, $k$ -means can be quite unstable with up to 33 data points being unstable when $k$ is large, whereas $k$ -means++ produces very stable clusterings with not more than three data points being unstable.", "However, $k$ -means++ performs very poorly in terms of $\\text{Obj}$ , which can be almost ten times as large as for $k$ -means and our dynamic programming approach (cf.", "Table REF , $k=50$ ).", "The MLP that we used for predicting the label (good vs. bad credit risk) in the second experiment of Section REF has three hidden layers of size 100, 50 and 20, respectively, and a test accuracy of 0.724.", "Table: Experiment on German credit data set.", "Clustering the second 500 people according to their estimated probability of having a good credit risk.Target cluster sizes t i =500 kt_i=\\frac{500}{k}, i∈[k]i\\in [k].Naive==naive clustering that matches the target cluster sizes, DP==dynamic programming approach of Appendix ,kk-means=k=k-means initialized with medians of theclusters of the naive clustering, kk-me++=k=k-means++.Results forkk-me++averaged over 100 runs.", "Best values in bold.Table: Experiment on German credit data set.", "Clustering the second 500 people according to their estimated probability of having a good credit risk.Variousnon-uniformtarget cluster sizes.Naive==naive clustering that matches the target cluster sizes, DP==dynamic programming approach of Appendix , kk-means=k=k-means initialized with medians of theclusters of the naive clustering, kk-me++=k=k-means++.Results forkk-me++averaged over 100 runs.", "Best values in bold." ] ]
2207.03600
[ [ "Non-isomorphic smooth compactifications of the moduli space of cubic\n surfaces" ], [ "Abstract The moduli space of complex cubic surfaces has three different, but isomorphic, compact realizations: as a GIT quotient, as a Baily--Borel compactification of a ball quotient, and as a compactified $K$-moduli space.", "From all three perspectives, there is a unique boundary point corresponding to non-stable surfaces.", "From the GIT point of view, to deal with this point, it is natural to consider the Kirwan blowup, while from the ball quotient point of view it is natural to consider the toroidal compactification.", "Both these spaces have the same cohomology and and it is therefore natural to ask whether they are isomorphic.", "Here we show that this is in fact not the case.", "Indeed, we show the more refined statement that both spaces are equivalent in the Grothendieck ring, but not $K$-equivalent.", "Along the way, we establish a number of results and techniques for dealing with singularities and canonical classes of Kirwan blowups and toroidal compactifications of ball quotients." ], [ "Introduction", "The four-dimensional moduli space $\\mathcal {M}$ of smooth complex cubic surfaces is one of the gems of classical algebraic geometry.", "As a moduli space of hypersurfaces, it comes with a GIT model ${\\mathcal {M}}^{\\operatorname{GIT}}$ , well-studied via the classical invariant theory (see [19], [21]).", "Through an auxiliary construction with cubic threefolds, there is a Hodge-theoretic ball quotient model ${\\mathcal {B}_4/\\Gamma }$ of the moduli space $\\mathcal {M}$ , and its Baily–Borel compactification ${(\\mathcal {B}_4/\\Gamma )^*}$ [1] (see also [22] for an alternative construction of the ball quotient model, involving $K3$ surfaces instead).", "Finally, as cubic surfaces are Fano, there is a $K$ -stable compactification [37].", "These three models of $\\mathcal {M}$ , each of a totally different nature, turn out to be isomorphic; in particular ${\\mathcal {M}}^{\\operatorname{GIT}}\\cong {(\\mathcal {B}_4/\\Gamma )^*}$ by [1].", "The moduli space $\\mathcal {M}$ is open in ${\\mathcal {M}}^{\\operatorname{GIT}}$ , and the complement ${\\mathcal {M}}^{\\operatorname{GIT}}-\\mathcal {M}$ , which we refer to as the boundary or, equivalently, the discriminant, is an irreducible divisor.", "All of the points in the boundary, with the exception of one, parameterize cubic surfaces with $A_1$ singularities, which are GIT stable.", "The remaining point is the unique GIT boundary point $\\Delta _{3A_2}\\in {\\mathcal {M}}^{\\operatorname{GIT}}$ which, from the GIT (and also $K$ -moduli) point of view, corresponds to the unique strictly polystable orbit of cubic surfaces with $3A_2$ singularities.", "From the ball quotient perspective, under the identification ${\\mathcal {M}}^{\\operatorname{GIT}}\\cong {(\\mathcal {B}_4/\\Gamma )^*}$ , $\\Delta _{3A_2}$ is the unique cusp $c_{3A_2}$ of ${(\\mathcal {B}_4/\\Gamma )^*}$ .", "For the general GIT setup, in the case where one is taking a GIT quotient of a smooth projective variety, Kirwan has introduced a procedure, usually called the Kirwan blowup of the GIT moduli space, which we denote ${\\mathcal {M}}^{\\operatorname{K}}\\rightarrow {\\mathcal {M}}^{\\operatorname{GIT}}$ in our case.", "The crucial point is that in general a Kirwan blowup is always a desingularization in the sense that it only has finite quotient singularities; alternatively, in the language of stacks, the natural quotient stack structure for the Kirwan blowup gives a smooth Deligne–Mumford stack.", "Similarly, to improve the singularities of the Baily–Borel compactification, one considers toroidal compactifications.", "Additionally, in some cases, one can attach modular meaning to some toroidal compactifications (e.g., [5]).", "In general, a toroidal compactification depends on some choice of a fan, but for the case of ball quotients, the choice is unique, and we denote the toroidal compactification in our particular case by ${\\overline{\\mathcal {B}_4/\\Gamma }}\\rightarrow {(\\mathcal {B}_4/\\Gamma )^*}$ .", "In our particular situation, it is a coincidence that ${\\mathcal {M}}^{\\operatorname{GIT}}\\cong {(\\mathcal {B}_4/\\Gamma )^*}$ has only finite quotient singularities.", "However, it still makes sense to consider the Kirwan blow-up ${\\mathcal {M}}^{\\operatorname{K}}$ and the toroidal compactification ${\\overline{\\mathcal {B}_4/\\Gamma }}$ respectively, in the sense that they have a better chance of having modular interpretations (see e.g., [40] for some related work) that would give rise to natural Deligne–Mumford moduli stacks.", "In our case, both ${\\mathcal {M}}^{\\operatorname{K}}$ and ${\\overline{\\mathcal {B}_4/\\Gamma }}$ are blowups of the same point $\\Delta _{3A_2}(=c_{3A_2})$ on the same space ${\\mathcal {M}}^{\\operatorname{GIT}}\\cong {(\\mathcal {B}_4/\\Gamma )^*}$ .", "In [11], we studied the cohomology of various compactifications for moduli of cubic threefolds and cubic surfaces.", "In particular, we showed that ${\\mathcal {M}}^{\\operatorname{K}}$ and ${\\overline{\\mathcal {B}_4/\\Gamma }}$ have the same cohomology, and we asked if ${\\mathcal {M}}^{\\operatorname{K}}$ and ${\\overline{\\mathcal {B}_4/\\Gamma }}$ were in fact isomorphic.", "This seems very plausible especially in the context of the recent work of Gallardo–Kerr–Schaffler [24] on the moduli of marked cubic surfaces.", "Namely, recall that the moduli space of cubic surfaces has a natural $W(E_6)$ -cover $\\mathcal {M}_m$ obtained by labeling the 27 lines on the cubic.", "Naruki [35] proved that this marked moduli space $\\mathcal {M}_m$ admits a smooth normal crossing compactification $\\overline{\\mathcal {N}}$ (see also [25] for further modular interpretations attached to $\\overline{\\mathcal {N}}$ ).", "On the other hand, Allcock–Carlson–Toledo [1] noticed that the ball quotient model ${(\\mathcal {B}_4/\\Gamma )^*}$ comes with a natural marked cover ${(\\mathcal {B}_4/\\Gamma _m)^*}$ (with $\\Gamma _m\\unlhd \\Gamma $ and $\\Gamma /\\Gamma _m\\cong W(E_6) \\times \\lbrace \\pm 1 \\rbrace $ ) and that the associated toroidal compactification ${\\overline{\\mathcal {B}_4/\\Gamma _m}}$ behaves similarly to $\\overline{\\mathcal {N}}$ .", "This was further clarified by [24], who showed that in fact the Naruki and the marked toroidal compactifications agree: $\\overline{\\mathcal {N}}\\cong {\\overline{\\mathcal {B}_4/\\Gamma _m}}$ .", "Returning to the unmarked case, the main result of our paper is that, surprisingly, ${\\mathcal {M}}^{\\operatorname{K}}$ and ${\\overline{\\mathcal {B}_4/\\Gamma }}$ are not isomorphic.", "Furthermore, we investigate and explain the geometry underlying this phenomenon.", "Theorem 1.1 Neither the birational map $f:{\\mathcal {M}}^{\\operatorname{K}}\\dashrightarrow {\\overline{\\mathcal {B}_4/\\Gamma }}$ from the Kirwan blowup of the GIT moduli space of cubic surfaces to the toroidal compactification of the ball quotient model, which is the identity on the moduli space $\\mathcal {M}$ of smooth cubic surfaces, nor its inverse $f^{-1}$ , extend to a morphism of the compactifications.", "We provide a quick proof of this theorem by showing that the strict transform in ${\\overline{\\mathcal {B}_4/\\Gamma }}$ of the discriminant divisor (the closure of the locus of cubics with an $A_1$ singularity) in ${(\\mathcal {B}_4/\\Gamma )^*}={\\mathcal {M}}^{\\operatorname{GIT}}$ meets the exceptional divisor in ${\\overline{\\mathcal {B}_4/\\Gamma }}$ generically transversally, whereas this is not the case in ${\\mathcal {M}}^{\\operatorname{K}}$ .", "This shows a priori that the period map $f$ does not extend to an isomorphism.", "Using straightforward arguments from higher-dimensional birational geometry, using that both ${\\mathcal {M}}^{\\operatorname{K}}$ and ${\\overline{\\mathcal {B}_4/\\Gamma }}$ are $\\mathbb {Q}$ -factorial, one can deduce from this that neither the period map $f$ , nor its inverse $f^{-1}$ , extends to a morphism.", "While T:mainNonIso shows that the Kirwan and toroidal compactifications are not naturally isomorphic, one could still wonder if the agreement of the Betti numbers, observed in [11], could be explained by an abstract isomorphism as complex projective varieties.", "To this end we strengthen T:mainNonIso by showing that ${\\mathcal {M}}^{\\operatorname{K}}$ and ${\\overline{\\mathcal {B}_4/\\Gamma }}$ are not abstractly isomorphic, and in fact not even $K$ -equivalent.", "Recall that two birational normal projective $\\mathbb {Q}$ -Gorenstein varieties $X$ and $Y$ are said to be $K$ -equivalent if there exists a smooth variety $Z$ dominating birationally both of them ${&Z[ld]_g [rd]^h&\\\\X@{<-->}[rr]&&Y}$ such that $g^*K_X \\sim _{\\mathbb {Q}} h^*K_Y$ .", "Two facts about $K$ -equivalent varieties that motivate our interest in this question are the following.", "First, it is known that smooth $K$ -equivalent varieties have the same Betti numbers [7] (see for instance [38] for the case where $X$ and $Y$ are not assumed to be Calabi–Yau).", "Second, birational normal projective $\\mathbb {Q}$ -Gorenstein varieties with canonical singularities, such that their canonical bundles are nef, are known to be $K$ -equivalent [26].", "We are, however, in neither of these situations: first, the Kirwan and toroidal compactifications are singular (with finite quotient singularities), and second, the canonical bundles of the compactifications are far from being nef, as the spaces are birational to the $\\mathbb {Q}$ -Fano variety ${\\mathcal {M}}^{\\operatorname{GIT}}$ .", "As it turns out, these two compactifications are not $K$ -equivalent.", "Theorem 1.2 The Kirwan compactification ${\\mathcal {M}}^{\\operatorname{K}}$ and the toroidal compactification ${\\overline{\\mathcal {B}_4/\\Gamma }}$ of the moduli space of smooth cubic surfaces $\\mathcal {M}$ are not $K$ -equivalent.", "We prove T:mainNonK by showing that the top self-intersection numbers of the canonical bundles of the two compactifications are different.", "Clearly, T:mainNonK implies T:mainNonIso (since it implies that $f$ does not extend to an isomorphism).", "However, we prefer first to give an independent proof of T:mainNonIso in sec:ballquoitient, as this also helps elucidate the geometry of the compactifications.", "This approach can also be used in other contexts such as cubic threefolds or other Deligne–Mostow spaces.", "A further major ingredient of our proof is an explicit geometric recipe for computing the canonical class of a Kirwan blowup of a GIT quotient.", "This works in full generality, so long as the strictly semi-stable locus has codimension at least 2, which we explain in sec:KKirwan, and which may be of independent interest; see also R:S-BGLM regarding the singularities of these spaces.", "In the absence of odd cohomology (as is the case here), another possible explanation for the equality of Betti numbers of ${\\mathcal {M}}^{\\operatorname{K}}$ and ${(\\mathcal {B}_4/\\Gamma )^*}$ could be their $\\mathbb {L}$ -equivalence in the Grothendieck ring of varieties.", "We prove that in fact the classes of these compactifications are equal in the Grothendieck ring of varieties.", "Theorem 1.3 The Kirwan compactification ${\\mathcal {M}}^{\\operatorname{K}}$ and the toroidal compactification ${\\overline{\\mathcal {B}_4/\\Gamma }}$ of the moduli space of smooth cubic surfaces $\\mathcal {M}$ are equivalent in the Grothendieck ring of varieties.", "The structure of the paper is as follows.", "We start in sec:example by discussing a simple explicit example, for motivation.", "This example is of two birational surfaces, which are equivalent in the Grothendieck ring, but not $K$ -equivalent.", "These are obtained as different blowups of $\\mathbb {P}^2$ supported at one point.", "While this is not a GIT problem, the example demonstrates some of the features similar to our case of the moduli of cubic surfaces, and serves as motivation.", "In sec:calM we recall the constructions and the geometry of the compactifications of the moduli space of cubic surfaces, and give the formulas for the discriminant and boundary divisors in local coordinates, as well as the crucial computation of the finite stabilizers of points on them.", "Some of the proofs are by detailed explicit computations, which are given in sec:app.", "In particular, in sec:calM we obtain the resulting non-transversality results for the Kirwan blowup needed for proving T:mainNonIso.", "In sec:ballquoitient we discuss the geometry of the ball quotient model in detail, obtain the resulting transversality results for the toroidal compactification, and thereby give a direct proof of T:mainNonIso.", "In sec:canonicalbundle we provide details on the canonical bundles of the ball quotient models.", "Going further, in sec:KKirwan we develop the general machinery for relating the canonical bundle of a GIT compactification and its Kirwan desingularization.", "We then specialize to our case, and compute the canonical bundle of ${\\mathcal {M}}^{\\operatorname{K}}$ .", "Finally, we prove our main results in sec:proofnonKequiv, where we establish the non-$K$ -equivalence of the Kirwan blowup ${\\mathcal {M}}^{\\operatorname{K}}$ and the toroidal compactification ${\\overline{\\mathcal {B}_4/\\Gamma }}$ , and in sec:proofLequiv, where we show their equivalence in the Grothendieck ring.", "The first requires a discussion of the top self-intersection numbers of canonical bundles on these varieties, the second relies on a concrete description of the boundary of the Kirwan blowup as a quotient of a toric variety by a finite group.", "In sec:app we collect a number of results whose proofs are based on explicit computations in the Luna slice.", "Remark 1.4 One of our motivations for this paper is our systematic investigation of the moduli space of cubic threefolds and its compactifications (e.g., [11] and references within).", "In that case as well, there is a toroidal compactification of a ball quotient model and a Kirwan blowup of the natural GIT quotient.", "Furthermore, as here, the two models have the same cohomology.", "Methods similar to those used in the proof of T:mainNonIso show that for moduli of cubic threefolds the natural period map does not extend to an isomorphism, either.", "However, the more refined results (T:mainNonK and T:mainL) seem at the moment out of reach for cubic threefolds.", "We also note that there are other related moduli spaces where these techniques apply, such as the moduli space of cubic surfaces with a line, see rem:cubics with a line.", "Again, we will not pursue this here." ], [ "Acknowledgements", "We thank Dan Abramovich for asking about the $K$ -equivalence of the moduli spaces.", "We are grateful to Damiano Fulghesu for a conversation on equivariant Chow rings.", "We thank John Christian Ottem for asking about the $\\mathbb {L}$ -equivalence of the compactifications ${\\mathcal {M}}^{\\operatorname{K}}$ and ${\\overline{\\mathcal {B}_4/\\Gamma }}$ .", "We are grateful to David Rydh for discussions and explanations regarding the Kirwan desingularization as a blowup of an Artin stack, which helped to clarify our thinking and to Daniel Allcock for answering questions on the ball quotient picture.", "Our thanks also go to Bert van Geemen who helped us clear up a question concerning the ring of invariants.", "We thank Mathieu Dutour Sikiric for a computation with polytopes.", "This work was partially supported by the Swedish Research Council under grant no.", "2016-06596 while S. Grushevsky, K. Hulek, and R. Laza visited the Institut Mittag-Leffler in Djursholm, Sweden during the fall of 2021.", "S. Grushevsky thanks the Weizmann Institute of Science for its hospitality in spring 2022 when this paper was finished." ], [ "A motivating example", "Before discussing the case of the moduli of cubic surfaces, we present an elementary example that captures some of the essential aspects of our arguments.", "While we are not aware of a global moduli interpretation of the example discussed here, the motivation and the construction of the example comes from the local study of semi-stable reduction for curves with an $A_2$ singularity (see esp.", "[14]), and it is at least morally related to the moduli of cubic surfaces discussed here (e.g., see [12]).", "Namely, consider the pair $(M,D)$ consisting of $M\\cong \\mathbb {P}^2$ together with a cuspidal cubic $D$ .", "Let $o\\in D$ be the cusp point.", "Define $M^{\\prime }$ to be the (standard) blowup of $M\\cong \\mathbb {P}^2$ at $o$ , with the exceptional divisor $E^{\\prime }\\subseteq M^{\\prime }$ , and let $\\widehat{M}$ be the standard log resolution of the cusp, obtained by 2 further blowups of $M^{\\prime }$ , with exceptional divisors $E_1,E_2,E_3$ over $M$ .", "We label the exceptional divisors on $\\widehat{M}$ in such a way that $(E_i)^2=i-4$ ; i.e., $E_1$ is the strict transform of $E^{\\prime }$ .", "We contract $E_1$ and $E_2$ on $\\widehat{M}$ and obtain $\\overline{M}$ , which will have two quotient singularities of types $\\frac{1}{2}(1,1)$ (or equivalently $A_1$ ) and $\\frac{1}{3}(1,1)$ (e.g., simply note that $E_1^2=-3$ , $E_2^2=-2$ , and that $E_1$ and $E_2$ do not intersect) and we denote by $\\overline{E}\\subseteq \\overline{M}$ the exceptional divisor of the blowdown $\\overline{M}\\rightarrow M$ , so that $\\overline{E}$ is the image of $E_3\\subseteq \\widehat{M}$ .", "We obtain the following diagram: ${&\\widehat{M} _{\\pi _1}[dl]@{->}^{\\bar{\\pi }}[dr]&\\\\M^{\\prime } [dr]_{\\epsilon ^{\\prime }}@{-->}[rr]^{f}&&\\overline{M}[dl]^{\\bar{\\epsilon }}\\\\ &M}$ Clearly, $\\overline{M}$ is a blowup of $M$ at $o$ , and in fact it is a single weighted blowup of $M$ at $o$ .", "In conclusion, both $M^{\\prime }$ and $\\overline{M}$ are blowups of $M$ at $o$ .", "Both are $\\mathbb {Q}$ -factorial with klt singularities, but they are non-isomorphic (e.g., $M^{\\prime }$ is smooth, while $\\overline{M}$ is singular).", "More in line with our arguments, we note that the birational map $f:M^{\\prime }\\dashrightarrow \\overline{M}$ does not extend to an isomorphism, since the exceptional divisors $E^{\\prime }\\subseteq M^{\\prime }$ and $\\overline{E}\\subseteq \\overline{M}$ , and the strict transforms of $D$ (denote them $D^{\\prime }\\subseteq M^{\\prime }$ and $\\overline{D}\\subseteq \\overline{M}$ ) are non-transversal in $M^{\\prime }$ , but are transversal in $\\overline{M}$ .", "In fact, note that $(\\overline{M}, \\overline{D}+\\overline{E})$ is dlt, while $(M^{\\prime }, D^{\\prime }+E^{\\prime })$ is not.", "This shows that even though $\\overline{M}$ is singular, when accounting for the “discriminant” $D$ , the “correct” resolution of $M$ is in fact $\\overline{M}$ and not $M^{\\prime }$ (which is smooth!).", "Clearly, $M^{\\prime }$ and $\\overline{M}$ have the same Betti numbers, and are equal in the Grothendieck ring of varieties.", "On the other hand, an elementary computation shows that $M^{\\prime }$ and $\\overline{M}$ are not $K$ -equivalent.", "Namely, $K_{M^{\\prime }}^2=8$ (as $M^{\\prime }$ is a single regular blowup of $\\mathbb {P}^2$ ), while $K_{\\overline{M}}^2=6+\\frac{1}{3}$ , since $K_{\\widehat{M}}=\\bar{\\pi }^*K_{\\overline{M}}-\\frac{1}{3} E_1$ (as the discrepancy of a quotient singularity of type $\\frac{1}{n}(1,1)$ is $\\frac{2-n}{n}$ ).", "We note that the weighted blowup $\\overline{M}\\rightarrow M$ is motivated by [14], which discusses the simultaneous semi-stable reduction for curves with certain singularities.", "Indeed, locally near the cusp $o$ , the pair $(M,D)$ can be identified with the versal deformation for $A_2$ singularities, with $D$ being the discriminant.", "From this perspective, it is natural to consider (locally near $o$ ) the $W(A_2)$ cover $\\widetilde{M}$ of $M$ branched over $D$ .", "On this cover, the discriminant $D$ pulls back to the $A_2$ hyperplane arrangement.", "The standard blowup $\\widehat{\\widetilde{M}}$ at $\\widetilde{o}\\in \\widetilde{M}$ (the preimage of $o$ ) leads to a normal crossing discriminant (see [14] for the general construction and discussion).", "Clearly $W(A_2)$ acts on $\\widehat{\\widetilde{M}}$ , and $\\overline{M}$ can be recovered as the quotient $\\widehat{\\widetilde{M}}/W(A_2)$ (again, the discussion is meant locally near $o$ ).", "The main point of this construction is that it takes into account the monodromy around the discriminant divisor, and thus it leads to a “modular” resolution $\\overline{M}$ of the pair $(M,D)$ (see [14] for precise statements).", "Similarly, returning to our setup in the current paper, the toroidal compactification of the moduli space of cubic surfaces is a “modular” blowup of the Baily–Borel compactification, which takes into account the monodromy around the discriminant divisor.", "In contrast, the Kirwan resolution of the GIT quotient does not see the monodromy, and consequently the Kirwan blowup behaves more like the standard blowup $M^{\\prime }\\rightarrow M$ .", "What we see is that while the GIT compactification and the Baily–Borel compactification for the moduli of cubic surfaces are isomorphic as projective varieties, the natural stack structure (even at certain stable points) is different.", "Namely, at the generic point of the discriminant divisor, corresponding to a cubic with an $A_1$ singularity, on the GIT side there are no extra automorphisms (i.e., the stabilizer of the generic point of the discriminant is the diagonal $\\mu _4$ in $\\operatorname{SL}(4,\\mathbb {C})$ , which induces the trivial automorphism of the cubic surface), while on the Hodge theoretic side there is an extra automorphism given by the Picard–Lefschetz transformation corresponding to the nodal degenerations (compare [36])." ], [ "The GIT models for the moduli of cubic surfaces", "As is the case in general for hypersurfaces, the moduli of cubic surfaces has a natural compact model: the GIT compactification ${\\mathcal {M}}^{\\operatorname{GIT}}$.", "For cubic surfaces, ${\\mathcal {M}}^{\\operatorname{GIT}}$ is well understood via classical invariant theory; in particular, ${\\mathcal {M}}^{\\operatorname{GIT}}\\cong \\mathbb {P}(1,2,3,4,5)$ .", "Here, we are interested in the Kirwan blowup ${\\mathcal {M}}^{\\operatorname{K}}\\rightarrow {\\mathcal {M}}^{\\operatorname{GIT}}$, which is obtained by blowing up the unique GIT strictly polystable boundary point $\\Delta \\in {\\mathcal {M}}^{\\operatorname{GIT}}$ according to a general procedure due to Kirwan [27].", "After reviewing ${\\mathcal {M}}^{\\operatorname{GIT}}$ and ${\\mathcal {M}}^{\\operatorname{K}}$ , we discuss the local structure of the Kirwan blowup ${\\mathcal {M}}^{\\operatorname{K}}$ along the exceptional divisor $D_{3A_2}$ .", "We conclude in P:discMK that the exceptional divisor $D_{3A_2}$ and the strict transform $\\widetilde{D}_{A_1}$ of the discriminant divisor do not meet, even generically, transversally in ${\\mathcal {M}}^{\\operatorname{GIT}}$ ." ], [ "Preliminaries on moduli of cubic surfaces", "We denote by $\\mathcal {M}\\mathbb {P}H^0(\\mathbb {P}^3,\\mathcal {O}_{\\mathbb {P}^3}(3))^\\circ /\\operatorname{SL}(4,\\mathbb {C})$ the four-dimensional moduli space of smooth (complex) cubic surfaces, where $\\mathbb {P}H^0(\\mathbb {P}^3,\\mathcal {O}_{\\mathbb {P}^3}(3))^\\circ $ denotes the locus of smooth cubic surfaces embedded in $\\mathbb {P}^3$ .", "We denote by ${\\mathcal {M}}^{\\operatorname{GIT}}\\mathbb {P}H^0(\\mathbb {P}^3,\\mathcal {O}_{\\mathbb {P}^3}(3))/\\!\\!/_{\\mathcal {O}(1)} \\operatorname{SL}(4,\\mathbb {C})$ the GIT compactification, and by $\\pi :{\\mathcal {M}}^{\\operatorname{K}}\\longrightarrow {\\mathcal {M}}^{\\operatorname{GIT}}$ the Kirwan resolution of ${\\mathcal {M}}^{\\operatorname{GIT}}$ .", "The GIT stability for cubic surfaces has been completely described (see e.g., [33]).", "For a cubic surface $S\\subseteq \\mathbb {P}^3$ : $S$ is stable if and only if it has at worst $A_1$ singularities, $S$ is semi-stable if and only if it is stable, or has at worst $A_2$ singularities, and does not contain the axes of the $A_2$ singularities, $S$ is strictly polystable if and only if it is projectively equivalent to the so-called $3A_2$ cubic surface $S_{3A_2}\\lbrace x_0x_1x_2+x_3^3=0\\rbrace \\,,$ which has exactly 3 singular points, each of which is an $A_2$ singularity.", "It is a classical result (see [22]) that the GIT compactification ${\\mathcal {M}}^{\\operatorname{GIT}}\\cong \\mathbb {P}(1,2,3,4,5)$ is a weighted projective space.", "We will denote by $D_{A_1}\\subseteq {\\mathcal {M}}^{\\operatorname{GIT}}$ the so-called discriminant divisor, that is the closure of the locus of (stable) cubics with an $A_1$ singularity.", "This divisor is irreducible, as the locus of corresponding cubics in $\\mathbb {P}H^0(\\mathbb {P}^3,\\mathcal {O}_{\\mathbb {P}^3}(3))$ is irreducible; the general point is given by the orbit of the locus of cubics of the form $x_0q(x_1,x_2,x_3)+f(x_1,x_2,x_3)$ with $q(x_1,x_2,x_3)$ a smooth quadric, and $f(x_1,x_2,x_3)$ a cubic.", "We denote by $R\\subseteq {\\mathcal {M}}^{\\operatorname{GIT}}$ the so-called Eckardt divisor, which generically parameterizes smooth cubic surfaces having an Eckardt point — a point that lies on three lines contained in the cubic surface.", "This divisor is also irreducible, as the corresponding locus in $\\mathbb {P}H^0(\\mathbb {P}^3,\\mathcal {O}_{\\mathbb {P}^3}(3))$ is irreducible; the locus of smooth Eckardt cubics is the orbit of the locus of cubics of the form $x_0^2\\ell (x_1,x_2,x_3)+f(x_1,x_2,x_3)$ with $\\ell (x_1,x_2,x_3)$ a linear form and $f(x_1,x_2,x_3)$ a cubic, with the line and cubic meeting transversally.", "In the coordinates of the previous equation, the point $(1:0:0:0)$ is then an Eckardt point, and the three lines through the Eckardt point are the ones determined by the Eckardt point and the intersection $\\lbrace \\ell =f=0\\rbrace $ in the hyperplane at infinity.", "Such an Eckardt cubic in these coordinates has an involution given by $x_0\\mapsto -x_0$ .", "Note that a general smooth cubic surface with an Eckardt point has a unique Eckardt point, and has automorphism group $\\mathbb {Z}_2$ , and moreover, the Eckardt divisor $R$ contains the locus of all smooth cubic surfaces that have any non-trivial automorphism (see e.g., [16]).", "From the description of stability given above, it follows that ${\\mathcal {M}}^{\\operatorname{GIT}}$ contains a unique strictly polystable point $\\Delta _{3A_2}\\in {\\mathcal {M}}^{\\operatorname{GIT}}$ corresponding to the orbit of the $3A_2$ cubic $S_{3A_2}$ , and consequently the Kirwan resolution ${\\mathcal {M}}^{\\operatorname{K}}$ of ${\\mathcal {M}}^{\\operatorname{GIT}}$ is a blowup with center supported at $\\Delta _{3A_2}$ (see e.g., [28] and [40]); we denote by $D_{3A_2}\\subseteq {\\mathcal {M}}^{\\operatorname{K}}$ the exceptional divisor, which is irreducible (as is the case for exceptional divisors in Kirwan blowups, being the quotient of an open subset of the blowup of a smooth irreducible subvariety of a smooth irreducible variety).", "Note that ${\\mathcal {M}}^{\\operatorname{K}}$ and $D_{3A_2}$ are, by construction, smooth up to finite quotient singularities.", "We will denote by $\\widetilde{D}_{A_1}\\subseteq {\\mathcal {M}}^{\\operatorname{K}}$ the strict transform of the discriminant divisor $D_{A_1}\\subseteq {\\mathcal {M}}^{\\operatorname{GIT}}$ , and by $\\widetilde{R}\\subseteq {\\mathcal {M}}^{\\operatorname{K}}$ the strict transform of the Eckardt divisor $R\\subseteq {\\mathcal {M}}^{\\operatorname{GIT}}$ ." ], [ "Geometry of ${\\mathcal {M}}^{\\operatorname{GIT}}$ as a weighted projective space", "In full generality, consider a weighted projective space $\\mathbb {P}(q_0,\\dots , q_n)=\\operatorname{Proj}\\mathbb {C}[x_0,\\dots ,x_n]=(\\mathbb {C}^{n+1}-\\lbrace 0\\rbrace )/\\mathbb {C}^*$ , where the weight of $x_i$ (resp.", "the weight of the action of $\\mathbb {C}^*$ on $x_i$ ) is $q_i$ for $i=0,\\dots ,n$ , where, without loss of generality, we assume that $\\gcd (q_0,\\dots , q_n)=1$ [17].", "Denoting $G_{\\vec{q}}\\mu _{q_0} \\times \\dots \\times \\mu _{q_n}$ , where $\\mu _{\\ell }$ is the multiplicative group of roots of unity of order $\\ell $ , and letting the group $G_{\\vec{q}}$ act diagonally on $\\mathbb {P}^n$ , we can express the weighted projective space as the quotient $\\mathbb {P}(q_0,\\dots , q_n) = \\mathbb {P}^n / G_{\\vec{q}}\\,,$ with quotient map $h:\\mathbb {P}^n \\longrightarrow \\mathbb {P}(q_0,\\dots , q_n)\\,.$ This is the map of spaces associated to the identification of the graded ring $\\mathbb {C}[x_0,\\dots ,x_n]$ as the subring $\\mathbb {C}[y_0^{q_1},\\dots ,y_n^{q_n}]\\subseteq \\mathbb {C}[y_0,\\dots ,y_n]$ of the standard graded polynomial ring, which can be viewed as the invariant ring for the group $G_{\\vec{q}}$ acting diagonally.", "While $G_{\\vec{q}}$ need not be cyclic, the weighted projective space is locally a cyclic quotient.", "It is covered by the open sets $U_i=D_+(x_i)\\lbrace \\mathfrak {p}\\in \\mathbb {P}(q_0,\\dots ,q_n): x_i\\notin \\mathfrak {p}\\rbrace $ (i.e., $U_i$ is the image of $\\lbrace x_i \\ne 0\\rbrace \\subseteq \\mathbb {C}^{n+1}-\\lbrace 0\\rbrace $ under the quotient map $h$ ), and one has $U_i \\cong \\mathbb {C}^n/\\mu _{q_i}=\\mathbb {C}^n/\\langle (\\zeta _{q_i}^{q_0}, \\dots ,\\zeta _{q_i}^{q_{i-1}}, \\zeta _{q_i}^{q_{i+1}}, \\dots ,\\zeta _{q_i}^{q_{n}}) \\rangle \\,,$ where $\\zeta _{q_i}$ is a primitive root of unity of order $q_i$ .", "This identification comes from the identification $U_i=\\operatorname{Spec}(\\mathbb {C}[x_0,\\dots ,x_n]_{(x_i)})$ , where $\\mathbb {C}[x_0,\\dots ,x_n]_{(x_i)}$ is the subring of elements of degree 0 in the localized ring $\\mathbb {C}[x_0,\\dots ,x_n]_{x_i}$ , and the identification of the ring $\\mathbb {C}[x_0,\\dots ,x_n]_{(x_i)}$ with the subring of $\\mathbb {C}[z_0,\\dots ,z_{i-1},z_{i+1},\\dots ,z_n]^{\\mu _{q_i}} \\subseteq \\mathbb {C}[z_0,\\dots ,z_{i-1},z_{i+1},\\dots ,z_n]$ of invariants for the diagonal action of the cyclic group $\\mu _{q_i}=\\langle (\\zeta _{q_i}^{q_0}, \\dots ,\\zeta _{q_i}^{q_{i-1}}, \\zeta _{q_i}^{q_{i+1}}, \\dots ,\\zeta _{q_i}^{q_{n}}) \\rangle $ .", "It is now straightforward to apply the Reid–Shepherd-Barron–Tai criterion to ${\\mathcal {M}}^{\\operatorname{GIT}}$ , by local computations in these charts.", "Lemma 3.1 The space ${\\mathcal {M}}^{\\operatorname{GIT}}\\cong \\mathbb {P}(1,2,3,4,5)$ has canonical singularities.", "We first note that $U_0 \\cong \\mathbb {C}^4$ .", "To explain the other charts we first treat the chart $U_4= \\mathbb {C}^4/\\langle g_4 \\rangle \\,,$ where $g_4(\\zeta _{5}, \\zeta _{5}^2, \\zeta _{5}^3, \\zeta _{5}^4)\\,.$ There is only one fixed point of this finite group action, namely the origin.", "Here the RSBT criterion tells us that we have to check for all non-trivial powers $g_4^k$ that the inequality $\\lfloor \\tfrac{k}{5} \\rfloor + \\lfloor \\tfrac{2k}{5} \\rfloor + \\lfloor \\tfrac{3k}{5} \\rfloor + \\lfloor \\tfrac{4k}{5} \\rfloor \\ge 1$ holds.", "Altogether we find one singularity in this chart, namely the point $P_4=(0:0:0:0:1)$ ; i.e., the image of the point $(0,0,0,0,1)\\in \\mathbb {C}^{n+1}-\\lbrace 0\\rbrace $ under the quotient map $h$ .", "The singularity at this point is canonical.", "The other open sets $U_i$ can be treated similarly.", "The situation for $U_2$ is completely analogous, and we find one further canonical singularity, namely $P_2=(0:0:1:0:0)$ .", "For $U_3$ we have to consider $U_3= \\mathbb {C}^4 / \\langle g_4 \\rangle = \\mathbb {C}^4 / \\langle i,-1,-i,i \\rangle \\,.$ Once again, we find one canonical singular point, namely $P_3=(0:0:0:1:0)$ .", "Finally in the chart $U_1= \\mathbb {C}^4 / \\langle g_2 \\rangle = \\mathbb {C}^4 / \\langle -1,1,-1,-1 \\rangle $ we have a 1-dimensional fixed locus, namely the line $L\\lbrace x_0=x_2=x_4=0\\rbrace \\,.$ We also note that $g_4^2=g_2$ and that $P_3 \\in L$ .", "Outside $P_3$ we have a transversal singularity along $L$ of type $\\mathbb {C}^3/\\langle (-1,-1,-1) \\rangle $ .", "In the proof of the Lemma we have also verified that $\\operatorname{Sing}{\\mathcal {M}}^{\\operatorname{GIT}}= \\lbrace P_2\\rbrace \\cup \\lbrace P_4\\rbrace \\cup L\\,,$ as already stated in [22].", "We note that the space ${\\mathcal {M}}^{\\operatorname{GIT}}=\\mathbb {P}(1,2,3,4,5)$ is $\\mathbb {Q}$ -factorial (since it only has finite quotient singularities), and is thus $\\mathbb {Q}$ -Gorenstein, but it is not Gorenstein.", "Indeed, the line bundle $\\mathcal {O}_{\\mathbb {P}^n}(1)$ descends under the cover $h$ (REF ) to a $\\mathbb {Q}$ -Cartier divisor on $\\mathbb {P}(q_0,\\dots ,q_n)$ , which by abuse of notation we denote by $\\mathcal {O}_{\\mathbb {P}(q_0,\\dots ,q_n)}(1)$ .", "The lowest multiple of $\\mathcal {O}_{\\mathbb {P}(q_0,\\dots ,q_n)}(1)$ which is Cartier is then $\\mathcal {O}_{\\mathbb {P}(q_0,\\dots ,q_n)}\\left(\\operatorname{lcm}(q_0, \\dots ,q_n)\\right)$ (e.g., [23] or [10]).", "For the canonical bundle, using the fact that the weighted projective space is a toric variety, and that the covering map $h$ is ramified along the toric divisors, one obtains the standard formula (e.g., [17]) $K_{\\mathbb {P}(q_0,\\dots ,q_n)}=\\left(-\\sum q_i \\right)\\mathcal {O}_{\\mathbb {P}(q_0,\\dots ,q_n)}(1)\\,.$ Thus in our case $K_{\\mathbb {P}(1,2,3,4,5)}=-15\\mathcal {O}_{\\mathbb {P}(1,2,3,4,5)}(1)\\,,$ and its smallest multiple that is Cartier is $4K_{\\mathbb {P}(1,2,3,4,5)}$ .", "Classical invariant theory for cubic surfaces explicitly identifies the geometric divisors $D_{A_1}$ (the nodal or discriminant) and $R$ (the Eckardt) divisors inside ${\\mathcal {M}}^{\\operatorname{GIT}}\\cong \\mathbb {P}(1,2,3,4,5)$ .", "For further use, we review this.", "By [21] the discriminant $D_{A_1}$ is given by the equation $(I_8^2-2^6 I_{16})^2= 2^{14}(I_{32} + 2^{-3}I_8I_{24})\\,,$ where $I_8,I_{16}, I_{24}, I_{32}, I_{40},I_{100}$ are the standard generators of the ring of invariants of the action of $\\operatorname{SL}(4,\\mathbb {Z})$ on the space of cubics, and the subscripts denote their degrees.", "As $I_{100}^2$ is a polynomial in the other invariants listed, these degrees show that we are working with $\\mathbb {P}(8,16,24,32,40)\\cong \\mathbb {P}(1,2,3,4,5)$ .", "Moreover, the Eckardt divisor is given by $I_{100}^2=0$ (e.g., [21]).", "Pulling back the defining equation (REF ) of $D_{A_1}$ to $\\mathbb {P}^4$ under the quotient map $h$ given by (REF ), we obtain $h^*D_{A_1}=\\left\\lbrace (y_0^2-2^6 {y_1}^2)^2= 2^{14}(y_3^4 + 2^{-3}y_0y_2^3)\\right\\rbrace \\,,$ where $y_0, \\dots , y_4$ are homogeneous coordinates on $\\mathbb {P}^4$ .", "Furthermore, [22] gives the coordinates of the point $\\Delta _{3A_2}\\in \\mathbb {P}(1,2,3,4,5)$ as $\\Delta _{3A_2}=(8:1:0:0:0) \\in D_{A_1}\\,,$ which in particular is a smooth point of ${\\mathcal {M}}^{\\operatorname{GIT}}$ .", "It is also a smooth point of $D_{A_1}\\subseteq \\mathbb {P}(1,2,3,4,5)$ , as in the local coordinates on the open chart $U_0\\cong \\mathbb {C}^4$ the defining equation (REF ) of the discriminant divisor becomes $(1-2^6z_1)^2 = 2^{14}(z_3 +2^{-3} z_2)$ , and $\\Delta _{3A_2}$ corresponds to the point $(1/8^2,0,0,0)$ , so that taking partial derivatives of this equation at the point $\\Delta _{3A_2}$ gives smoothness of $D_{A_1}$ at $\\Delta _{3A_2}$ .", "We can perform a similar analysis for the Eckardt divisor.", "As mentioned above, the Eckardt divisor $R$ is defined by $I_{100}^2$ , which is an irreducible polynomial in $I_8,\\dots ,I_{40}$ .", "The exact expression due to Salmon for $I_{100}^2$ in terms of $I_8,\\dots ,I_{40}$ no longer seems to be easily accessible in the literature.", "Dardanelli and van Geemen recently rederived it for their paper [21], and provided us with the expression, which we have omitted to save space; for reference, it is now available on the first author's website.", "From that description, one can easily see that $h^*R=\\mathcal {O}_{\\mathbb {P}^4}(25).$ As already noted, ${\\mathcal {M}}^{\\operatorname{GIT}}$ has Picard number 1.", "For further reference, we collect here the classes of various Weil divisors on ${\\mathcal {M}}^{\\operatorname{GIT}}=\\mathbb {P}(1,2,3,4,5)$ : $\\begin{aligned}K_{{\\mathcal {M}}^{\\operatorname{GIT}}}&=\\mathcal {O}_{\\mathbb {P}(1,2,3,4,5)}(-15) & \\\\D_{A_1}&=\\mathcal {O}_{\\mathbb {P}(1,2,3,4,5)}(4)& \\text{(Discriminant divisor)}\\\\R& =\\mathcal {O}_{\\mathbb {P}(1,2,3,4,5)}(25)& \\text{(Eckardt divisor)}\\end{aligned}$ These come from (REF ), (REF ), and (REF ), respectively.", "In particular, we note that the following relation holds in $\\operatorname{Pic}({\\mathcal {M}}^{\\operatorname{GIT}})_\\mathbb {Q}$ : $ K_{\\mathbb {P}(1,2,3,4,5)}= - \\frac{15}{4} D_{A_1}\\,.$ Remark 3.2 There is an important subtle point that we emphasize here.", "The divisor $\\mathcal {O}_{\\mathbb {P}^{19}}(1)$ on $(\\mathbb {P}^{19})^{ss}$ descends as a $\\mathbb {Q}$ -divisor to the divisor $\\frac{1}{8}\\mathcal {O}_{\\mathbb {P}(1,2,3,4,5)}(1)$ .", "The discriminant in $(\\mathbb {P}^{19})^{ss}$ has degree 32 (the discriminant has degree $(n+2)(d-1)^{n+1}$ for degree $d$ hypersurfaces in  $\\mathbb {P}^{n+1}$ ) and descends to the Weil divisor $D_{A_1}=\\mathcal {O}_{\\mathbb {P}(1,2,3,4,5)}(4)$ on $\\mathbb {P}(1,2,3,4,5)$ .", "Similarly, the Eckardt divisor on $(\\mathbb {P}^{19})^{ss}$ has degree 100 (see, e.g., [21], or [15] for an approach that works for more general Eckardt loci) and descends as a $\\mathbb {Q}$ -divisor to the divisor $\\frac{1}{2}\\mathcal {O}_{\\mathbb {P}(1,2,3,4,5)}(25)$ .", "However, this is not the Eckardt divisor $R$ on $\\mathbb {P}(1,2,3,4,5)$ , which, as explained above, has class $\\mathcal {O}_{\\mathbb {P}(1,2,3,4,5)}(25)$ .", "In other words, it is twice the Eckardt divisor on $(\\mathbb {P}^{19})^{ss}$ that descends to the Eckardt divisor $R$ on $\\mathbb {P}(1,2,3,4,5)$ .", "One can view this as a reflection of the fact that generic Eckardt cubic surfaces have an extra automorphism, as opposed to the case of generic cubic surfaces with an $A_1$ singularity, which do not have any extra automorphisms." ], [ "Local structure of the Kirwan blowup along the exceptional divisor", "We recall some relevant computations from [11].", "First, to employ the Luna Slice Theorem, we will want to understand the stabilizer of the $3A_2$ cubic surface $S_{3A_2}$ , as well as its action on a Luna slice.", "To begin, given a cubic surface $S\\subseteq \\mathbb {P}^3$ , we denote by $\\operatorname{Aut}(S)$ the automorphisms of $S$ (which are automorphisms of $S$ as a subvariety of $\\mathbb {P}^3$ , and therefore we naturally have $\\operatorname{Aut}(S)\\subseteq \\operatorname{PGL}(4,\\mathbb {C})$ ).", "From our GIT setup, we are also interested in $\\operatorname{Stab}(S)\\subseteq \\operatorname{SL}(4,\\mathbb {C})$ , the stabilizer subgroup, and, since it is sometimes easier to work with, we will also consider the stabilizer $\\operatorname{GL}(S)\\subseteq \\operatorname{GL}(4,\\mathbb {C})$ .", "We recall from [11] that the former two of these stabilizer groups for $S_{3A_2}$ are 2-dimensional, but we also want to work out the finite parts explicitly.", "To this end, we denote $D\\lbrace \\operatorname{diag}(\\lambda _0,\\lambda _1,\\lambda _2,\\lambda _3): \\lambda _0\\lambda _1\\lambda _2=\\lambda _3^3\\rbrace \\subseteq \\operatorname{GL}(4,\\mathbb {C})$ an auxiliary group, and observe that there is an isomorphism $\\mathbb {T}^3\\cong D$ given by $(\\lambda _1,\\lambda _2,\\lambda _3)\\mapsto \\operatorname{diag}(\\lambda _1^{-1}\\lambda _2^{-1}\\lambda _3^3,\\lambda _1,\\lambda _2,\\lambda _3)$ .", "We also want $D^{\\prime }D\\cap \\operatorname{SL}(4,\\mathbb {C}) = \\lbrace \\operatorname{diag}(\\lambda _0,\\lambda _1,\\lambda _2,\\lambda _3): \\lambda _0\\lambda _1\\lambda _2=\\lambda _3^3, \\ \\lambda _0\\lambda _1\\lambda _2\\lambda _3=1 \\rbrace \\subseteq \\operatorname{SL}(4,\\mathbb {C})\\,,$ and note the isomorphism $\\mathbb {T}^2\\times \\mu _4\\cong D^{\\prime }$ given by $(\\lambda _1,\\lambda _2,i^j)\\mapsto \\operatorname{diag}(\\lambda _1^{-1}\\lambda _2^{-1}i^{3j},\\lambda _1,\\lambda _2,i^j)$ .", "To see that this map is an isomorphism, note that this certainly gives an inclusion $\\mathbb {T}^2\\times \\mu _4\\hookrightarrow D^{\\prime }$ , and given $\\operatorname{diag}(\\lambda _0,\\lambda _1,\\lambda _2,\\lambda _3)\\in D^{\\prime }$ , the two equations together give $\\lambda _3^4=1$ , so that $\\lambda _3=i^j$ for some $j$ .", "Finally, we denote $D^{\\prime \\prime }\\lbrace \\operatorname{diag}(\\lambda _0,\\lambda _1,\\lambda _2,1): \\ \\lambda _0\\lambda _1\\lambda _2=1 \\rbrace \\subseteq \\operatorname{SL}(4,\\mathbb {C})\\,,$ and note the isomorphism $\\mathbb {T}^2\\cong D^{\\prime \\prime }$ given by $(\\lambda _1,\\lambda _2)\\mapsto \\operatorname{diag}(\\lambda _1^{-1}\\lambda _2^{-1},\\lambda _1,\\lambda _2,1)$ .", "We use the notation $\\mathbb {S}_3$ for the group of matrices obtained from the group of invertible diagonal $3\\times 3$ complex matrices by applying all possible permutations of the columns.", "The determination of the relevant stabilizer groups is parallel to the case of the $3D_4$ cubic threefold, treated in [11], and proceeds by an explicit computation, which we put in S:R3A2Norm2.", "Lemma 3.3 ($3A_2$ -stabilizer) We have: The group $\\operatorname{Stab}(S_{3A_2})$ is equal to $\\operatorname{Stab}(S_{3A_2}))=\\left\\lbrace \\left(\\begin{array}{c|c}\\mathbb {S}_3&\\\\ \\hline &\\mathbb {C}^*\\\\\\end{array}\\right)\\in \\operatorname{SL}(4,\\mathbb {C}): \\lambda _0\\lambda _1\\lambda _2=\\lambda _3^3\\right\\rbrace ,$ where the $\\lambda _i$ is the unique non-zero element in the $i$ -th row, and the group $\\operatorname{GL}(S_{3A_2})$ is equal to $\\operatorname{GL}(S_{3A_2}))=\\left\\lbrace \\left(\\begin{array}{c|c}\\mathbb {S}_3&\\\\ \\hline &\\mathbb {C}^*\\\\\\end{array}\\right)\\in \\operatorname{GL}(4,\\mathbb {C}): \\lambda _0\\lambda _1\\lambda _2=\\lambda _3^3\\right\\rbrace .$ There are central extensions $1\\rightarrow \\mu _4\\rightarrow \\operatorname{Stab}(S_{3A_2}) \\rightarrow \\operatorname{Aut}(S_{3A_2})\\rightarrow 1\\,,$ $1\\rightarrow \\mathbb {C}^*\\rightarrow \\operatorname{GL}(S_{3A_2})\\rightarrow \\operatorname{Aut}(S_{3A_2})\\rightarrow 1\\,.$ There are short exact sequences ${1 [r]& D^{\\prime } [r] @{^(->}[d]& \\operatorname{Stab}(S_{3A_2}) @{^(->}[d] [r] & S_3[r] @{=}[d]& 1\\\\1 [r]& D [r]& \\operatorname{GL}(S_{3A_2}) [r] & S_3[r]& 1\\\\}$ with the second being split, so that there is an isomorphism $\\operatorname{GL}(S_{3A_2})\\cong D\\rtimes S_3\\,,$ where the action of $S_3$ on $D$ is to permute the first three entries $\\lambda _0,\\lambda _1,\\lambda _2$ .", "The connected components of the groups above are $\\operatorname{Stab}(S_{3A_2})^\\circ = D^{\\prime \\prime }\\cong \\mathbb {T}^2$ , $\\operatorname{GL}(S_{3A_2})^\\circ = D\\cong \\mathbb {T}^3$ , and $\\operatorname{Aut}(S_{3A_2})^\\circ \\cong \\mathbb {T}^2$ .", "There is a short exact sequence $1\\rightarrow \\mu _4\\rightarrow \\operatorname{Stab}(S_{3A_2})/\\operatorname{Stab}(S_{3A_2})^\\circ \\rightarrow S_3 \\rightarrow 1\\,,$ and we have $\\operatorname{GL}(S_{3A_2})/\\operatorname{GL}(S_{3A_2})^\\circ \\cong \\operatorname{Aut}(S_{3A_2})/\\operatorname{Aut}(S_{3A_2})^\\circ \\cong S_3$ .", "$\\Box $ We now describe the action of the stabilizer of the $S_{3A_2}$ cubic on the Luna slice.", "The description of the Luna slice appears in [40], but for clarity, particularly for the action of the stabilizer, we include a discussion here as well.", "Lemma 3.4 ($3A_2$ -Luna slice) A Luna slice for $S_{3A_2}$ , normal to the orbit $\\operatorname{SL}(4,\\mathbb {C})\\cdot [S_{3A_2}] \\subseteq \\mathbb {P}^{19}$ , is isomorphic to $\\mathbb {C}^6$ , spanned by the 6 monomials $x_0^3,\\ x_1^3,\\ x_2^3,\\ x_0^2x_3,\\ x_1^2x_3, \\ x_2^2x_3$ in the tangent space $H^0(\\mathbb {P}^3,\\mathcal {O}_{\\mathbb {P}^3}(3))$ .", "The Luna slice can be projectively completed to give a $\\mathbb {P}^6$ : $\\mathbb {P}^6 = \\lbrace \\alpha _0x_0^3 +\\alpha _1x_1^3 +\\alpha _2x_2^3+\\alpha _{\\widehat{0}}x_0^2x_3+\\alpha _{\\widehat{1}}x_1^2x_3 +\\alpha _{\\widehat{2}}x_2^2x_3 +\\alpha _{3A_2}(x_0x_1x_2+x_3^3) \\rbrace $ $\\subseteq \\mathbb {P}H^0(\\mathbb {P}^3,\\mathcal {O}_{\\mathbb {P}^3}(3))=\\mathbb {P}^{19}\\,.$ The action of $\\operatorname{Stab}(S_{3A_2})$ and $\\operatorname{GL}(S_{3A2})$ on the projectively completed Luna slice is given by their inclusion into the groups $\\operatorname{SL}(4,\\mathbb {C})$ and $GL(4,\\mathbb {C})$ , respectively, with the given actions of those groups on $H^0(\\mathbb {P}^3,\\mathcal {O}_{\\mathbb {P}^3}(3))$ from the GIT setup.", "The action of $\\operatorname{Stab}(S_{3A_2})$ and $\\operatorname{GL}(S_{3A2})$ on the Luna slice is the natural induced action.", "In terms of L:R3A2Norm2(3) and the description in (REF ), the action of an element $\\operatorname{diag}(\\lambda _0,\\lambda _1,\\lambda _2,\\lambda _3) $ in $D$ or $D^{\\prime }$ is given by $\\begin{aligned}(\\lambda _0,\\lambda _1,\\lambda _2,\\lambda _3)&\\cdot (\\alpha _0,\\alpha _1,\\alpha _2,\\alpha _{\\widehat{0}},\\alpha _{\\widehat{1}},\\alpha _{\\widehat{2}})=\\\\ &=\\left(\\left(\\frac{\\lambda _0}{\\lambda _3}\\right)^3\\alpha _0,\\left(\\frac{\\lambda _1}{\\lambda _3}\\right)^3\\alpha _1,\\left(\\frac{\\lambda _2}{\\lambda _3}\\right)^3\\alpha _2,\\left(\\frac{\\lambda _0}{\\lambda _3}\\right)^2\\alpha _{\\widehat{0}}, \\left(\\frac{\\lambda _1}{\\lambda _3}\\right)^2\\alpha _{\\widehat{1}}, \\left(\\frac{\\lambda _2}{\\lambda _3}\\right)^2\\alpha _{\\widehat{2}}\\right)\\,,\\end{aligned}$ and the action of $\\sigma \\in S_3\\subseteq \\operatorname{GL}(S_{3A_2})$ is given by $\\sigma \\cdot (\\alpha _0,\\alpha _1,\\alpha _2,\\alpha _{\\widehat{0}},\\alpha _{\\widehat{1}},\\alpha _{\\widehat{2}})=(\\alpha _{\\sigma (0)},\\alpha _{\\sigma (1)},\\alpha _{\\sigma (2)},\\alpha _{\\widehat{\\sigma (0)}},\\alpha _{\\widehat{\\sigma (1)}},\\alpha _{\\widehat{\\sigma (2)}})\\,.$ For $S_{3A_2}=\\lbrace F_{3A_2}=0\\rbrace $ , the matrix $DF_{3A_2}$ , whose entries span the tangent space to the orbit of the $3A_2$ cubic, is given by (see [11] for similar computations for the $3D_4$ cubic threefold) $DF_{3A_2}=\\begin{pmatrix}x_0x_1x_2&x_1^2x_2&x_1x_2^2&x_1x_2x_3\\\\x_0^2x_2&x_0x_1x_2&x_0x_2^2&x_0x_2x_3\\\\x_0^2x_1&x_0x_1^2&x_0x_1x_2&x_0x_1x_3\\\\3x_0x_3^2&3x_1x_3^2&3x_2x_3^2&3x_3^3\\\\\\end{pmatrix}\\,.$ Since all entries of this matrix are monomial, the only possible linear relations are pairwise equalities, up to a constant factor.", "One sees that the only monomial that repeats more than once is $x_0x_1x_2$ , and thus all linear relations satisfied by the entries of $DF_{3A_2}$ are $(DF_{3A_2})_{00}=(DF_{3A_2})_{11}=(DF_{3A_2})_{22}\\,.$ This means that the normal space to the orbit ($\\dim \\mathbb {P}^{19}-\\dim \\text{ orbit } = 19-(16-3)=6$ ) is spanned by the 6 monomials $x_0^3,\\ x_1^3,\\ x_2^3,\\ x_0^2x_3,\\ x_1^2x_3, \\ x_2^2x_3\\,.$ The given action of the stabilizer on the Luna slice can be seen for instance from taking the Luna slice to be the affine space $\\alpha _0x_0^3 +\\alpha _1x_1^3 +\\alpha _2x_2^3+\\alpha _{\\widehat{0}}x_0^2x_3+\\alpha _{\\widehat{1}}x_1^2x_3 +\\alpha _{\\widehat{2}}x_2^2x_3 +(x_0x_1x_2+x_3^3)\\subseteq H^0(\\mathbb {P}^3,\\mathcal {O}_{\\mathbb {P}^3}(3))\\,,$ then using the fact that $\\lambda _0\\lambda _1\\lambda _2=\\lambda _3^3$ , and dividing through the natural action by $\\lambda _3^3$ , to fix the cubic form $x_0x_1x_2+x_3^3$ in the affine space.", "We now turn to the Eckardt divisor whose generic point parameterizes smooth cubics with a non-trivial automorphism, and determine the multiplicity with which it contains $S_{3A_2}$ .", "The proof is by an elaborate lengthy explicit computation using the explicit form of the action on the Luna slice, and is given in S:Eck-no-3A2.", "Lemma 3.5 The Eckardt divisor $R\\subseteq \\mathbb {P}H^0(\\mathbb {P}^3,\\mathcal {O}_{\\mathbb {P}^3}(3))=\\mathbb {P}^{19}$ contains the $3A_2$ orbit $\\operatorname{SL}(4,\\mathbb {C}) \\cdot [S_{3A_2}]$ with multiplicity $\\mu =15$ ; i.e., if $\\tilde{\\pi }: \\operatorname{Bl}_{\\operatorname{SL}(4,\\mathbb {C}) \\cdot [S_{3A_2}]}(\\mathbb {P}^{19})^{ss}\\rightarrow (\\mathbb {P}^{19})^{ss}$ is the blowup of the $3A_2$ orbit, with exceptional divisor $D_{3A_2}$ , then $\\tilde{\\pi }^*R= \\widetilde{R}+15D_{3A_2}$ , where $\\widetilde{R}\\subset {\\overline{\\mathcal {B}_4/\\Gamma }}$ is the strict transform of $R$ .", "$\\Box $ One can see from the above that for the action of $\\operatorname{Stab}(S_{3A_2})$ on the normal space, the stabilizer of a general line will be trivial.", "In fact, when we blow up $(\\mathbb {P}^{19})^{ss}$ along the orbit $\\operatorname{SL}(4,\\mathbb {C}) \\cdot [S_{3A_2}]$ in the Kirwan blowup process, then in the Luna slice we are blowing up the origin in $\\mathbb {C}^6$ , with exceptional divisor $\\mathbb {P}^5$ .", "The Lemma above gives the action of the stabilizer on this $\\mathbb {P}^5$ .", "For future use, we first describe the semi-stable locus for this action of the stabilizer on $\\mathbb {P}^5$ .", "Lemma 3.6 Denoting by $(T_0:T_1:T_2:T_{\\widehat{0}}:T_{\\widehat{1}}:T_{\\widehat{2}})$ the homogeneous coordinates on the exceptional divisor $\\mathbb {P}^5$ of the Kirwan blowup $\\operatorname{Bl}_0\\mathbb {C}^6\\rightarrow \\mathbb {C}^6$ of the Luna slice described above, the unstable locus of the action of $\\operatorname{GL}(S_{3A_2})$ is the union of the three codimension two loci $\\lbrace T_0=T_{\\widehat{0}}=0\\rbrace $ , $\\lbrace T_1=T_{\\widehat{1}}=0\\rbrace $ , $\\lbrace T_2=T_{\\widehat{2}}=0\\rbrace $ in $\\mathbb {P}^5$ .", "The action on the Luna slice given by (REF ) gives the following action of $\\mathbb {T}^2\\simeq D^{\\prime \\prime }=\\operatorname{Aut}(S_{3A_2})^\\circ $ on the $\\mathbb {C}^6$ with coordinates $T$ , of which the exceptional divisor is the projectivization: $\\begin{aligned}(T_0,T_1,T_2,T_{\\widehat{0}},T_{\\widehat{1}},T_{\\widehat{2}})&\\mapsto (\\lambda _0^3T_0,\\lambda _1^3 T_1,\\lambda _2^3 T_2, \\lambda _0^2T_{\\widehat{0}},\\lambda _1^2T_{\\widehat{1}},\\lambda _2^2T_{\\widehat{2}})\\\\&=(\\lambda _1^{-3}\\lambda _2^{-3} T_0,\\lambda _1^3 T_1,\\lambda _2^3 T_2, \\lambda _1^{-2}\\lambda _2^{-2}T_{\\widehat{0}},\\lambda _1^2T_{\\widehat{1}},\\lambda _2^2T_{\\widehat{2}})\\end{aligned}$ (here we are acting by $\\operatorname{diag}(\\lambda _0,\\lambda _1,\\lambda _2,1)$ with $\\lambda _0\\lambda _1\\lambda _2=1$ , and thus expressing $\\lambda _0=\\lambda _1^{-1}\\lambda _2^{-1}$ ).", "The action is by multiplying each coordinate by a monomial in $\\lambda _1,\\lambda _2$ , and thus $\\mathbb {C}^6$ is decomposed into a direct sum of 6 one-dimensional torus representations.", "Plotting the weights of each monomial in $\\mathbb {R}^2$ , a point in $\\mathbb {C}^6$ is stable if and only if the convex hull of the set of weights corresponding to non-zero coordinates contains the origin in $\\mathbb {R}^2$ .", "The weight diagram consists of 2 points on each of 3 rays from the origin.", "Thus a convex hull of some subset of these 6 weights contains the origin if and only if this subset contains at least one weight from each ray.", "This is to say, a point is stable if and only if at least one of its two coordinates $T_0$ and $T_{\\widehat{0}}$ is non-zero, etc.", "Thus the unstable points in $\\mathbb {C}^4\\subseteq \\mathbb {C}^6$ are precisely those given by a pair of equations $T_i=T_{\\widehat{i}}=0$ for some $i$ .", "We note that, as is the case for any Kirwan desingularization, there are no strictly semi-stable points on this exceptional divisor.", "We now describe the finite stabilizers along the exceptional divisor.", "This is another detailed explicit computation using the stabilizer computed in L:R3A2Norm2, and we give it in S:stab–ex.", "We note that the proof actually allows us to determine all possible stabilizers, but we will not need this information.", "Proposition 3.7 (Stabilizers along the exceptional divisor) Let $x\\in D_{3A_2}\\subseteq {\\mathcal {M}}^{\\operatorname{K}}$ be a point in the exceptional divisor, and let $S_x\\subseteq \\operatorname{Stab}(S_{3A_2})\\subseteq \\operatorname{SL}(4,\\mathbb {C})$ be its stabilizer; i.e., the stabilizer of a point in the exceptional divisor of $\\operatorname{Bl}_{\\operatorname{SL}(4,\\mathbb {C}) \\cdot [S_{3A_2}]}(\\mathbb {P}^{19})^{ss}$ with orbit corresponding to $x$ .", "For $x\\in D_{3A_2}$ general, $S_x=\\mu _4$ (the diagonal subgroup of $\\operatorname{SL}(4,\\mathbb {C})$ ).", "For any $x\\in D_{3A_2}$ , the order of $S_x$ is not divisible by 5.", "$\\Box $ Remark 3.8 The proof of P:discMK, given in S:discMK, will provide another proof of the claim (1) of P:stab–ex above.", "There, we will even show that this assertion holds for a general point of the intersection of the strict transform $\\widetilde{D}_{A_1}$ of the discriminant with the exceptional divisor $D_{3A_2}$ .", "We note also that part (2) above is what will enable us to argue that the top self-intersection numbers of the canonical class on ${\\mathcal {M}}^{\\operatorname{K}}$ and ${\\overline{\\mathcal {B}_4/\\Gamma }}$ are different.", "We conclude the section with the following non-transversality result: Proposition 3.9 At a generic point of the intersection $\\widetilde{D}_{A_1}\\cap D_{3A_2}\\subseteq {\\mathcal {M}}^{\\operatorname{K}}$ , these two divisors do not meet transversally.", "The proof of this is by a detailed computation in local coordinates in charts of the blowup.", "To help the reader and the flow of the paper, we only summarize the key steps of the arguments here, postponing further details until S:discMK, where the proof will also benefit from building upon the explicit setup developed in the Appendix prior to that proof.", "We first observe that the three $A_2$ singularities can be deformed independently.", "In the Luna slice the deformation space of each of the $A_2$ singularities is $\\mathbb {C}^2$ , within which the discriminant divisor $D_{A_1}$ is a cuspidal curve.", "Thus altogether in the $\\mathbb {C}^6$ Luna slice near the $S_{3A_2}$ cubic surface, the discriminant divisor is the product of the three equations of cubics, in three disjoint pairs of coordinates, one of which has the form $27\\alpha _0^2+4\\alpha _{\\widehat{0}}^3=0$ .", "To determine the local structure of the Kirwan blowup, one considers the blowup $\\operatorname{Bl}_0\\mathbb {C}^6$ of the origin in the Luna slice, and then studies the action of $\\mathbb {T}^2$ (the connected component of the stabilizer of $S_{3A_2}$ on this blowup).", "Identifying the explicit 4-dimensional Luna slice for this action, one writes down the equation of the discriminant divisor in this Luna slice explicitly, in charts on the projective space.", "In a suitable chart this discriminant divisor (that is, of $\\widetilde{D}_{A_1}\\subseteq {\\mathcal {M}}^{\\operatorname{K}}$ ) is locally a union of a number of hypersurfaces, one of which has the form $27t_0^2+4\\alpha _{\\widehat{0}}$ , where $\\alpha _{\\widehat{0}}=0$ is the local equation of the exceptional divisor of the blowup, that is of $D_{3A_2}$ .", "This intersection is manifestly non-transverse, except that extra care is needed to take care of the finite part of the stabilizer.", "Indeed, in principle a quotient of a non-transverse intersection under a finite group may become transverse, and thus we need to ensure that the finite part of the stabilizer of $S_{3A_2}$ does not influence this (this is a local computation weaker than what is needed for the proof of P:stab–ex)." ], [ "The ball quotient model and the first proof of T:mainNonIso", "Allcock–Carlson–Toledo [1] have constructed a ball quotient model ${\\mathcal {B}_4/\\Gamma }$ for the moduli of cubic surfaces.", "They proved that the Baily–Borel compactification ${(\\mathcal {B}_4/\\Gamma )^*}$ is isomorphic to the GIT model ${\\mathcal {M}}^{\\operatorname{GIT}}$ and that under this identification the unique cusp of ${(\\mathcal {B}_4/\\Gamma )^*}$ corresponds to the GIT boundary point $\\Delta _{3A_2}\\in {\\mathcal {M}}^{\\operatorname{GIT}}$ .", "By the general theory, one has a toroidal compactification ${\\overline{\\mathcal {B}_4/\\Gamma }}$ of ${\\mathcal {B}_4/\\Gamma }$ , unique in this situation, which can be described as a blowup of ${(\\mathcal {B}_4/\\Gamma )^*}$ at the unique cusp.", "Similarly to the previous section, we study the intersection of the exceptional divisor $T_{3A_2}$ of the blowup ${\\overline{\\mathcal {B}_4/\\Gamma }}\\rightarrow {(\\mathcal {B}_4/\\Gamma )^*}$ with the strict transform $\\widetilde{D}_n\\subseteq {\\overline{\\mathcal {B}_4/\\Gamma }}$ of the discriminant (Heegner) divisor $D_n\\subseteq {(\\mathcal {B}_4/\\Gamma )^*}$ .", "Here, in contrast with the Kirwan blowup where P:discMK gives non-transversality, we show in P:discTor that $T_{3A_2}$ and $\\widetilde{D}_n$ meet generically transversally.", "We thus obtain a first proof of T:mainNonIso that the isomorphism $ {\\mathcal {M}}^{\\operatorname{GIT}}-\\lbrace \\Delta _{3A_2}\\rbrace \\cong {\\mathcal {B}_4/\\Gamma }$ does not extend to an isomorphism of the compactifications ${\\mathcal {M}}^{\\operatorname{K}}$ and ${\\overline{\\mathcal {B}_4/\\Gamma }}$ , despite both spaces being the blowup of the same point in ${\\mathcal {M}}^{\\operatorname{GIT}}\\cong {(\\mathcal {B}_4/\\Gamma )^*}$ .", "One key differentiating aspect of the ball quotient model (vs. the GIT model) is the functorial behavior with respect to marking all the lines on the cubic surfaces (i.e., with respect to the natural $W(E_6)$ cover $\\mathcal {M}_m\\rightarrow \\mathcal {M}$ ).", "This allows us to use Naruki's compactification $\\overline{\\mathcal {N}}$ [35], which is a smooth normal crossing model for the marked moduli space, in order to understand the structure of ${\\overline{\\mathcal {B}_4/\\Gamma }}$ , by applying the isomorphism $\\overline{\\mathcal {N}}\\cong \\overline{{\\mathcal {B}_4/\\Gamma }_m}$ previously established by [24]." ], [ "Preliminaries on the ball quotient model", "We will now describe the compactifications of the ball quotient model of the moduli space of cubic surfaces.", "Before delving into the specifics for cubic surfaces, we first recall the compactification of ball quotients in general, referring to [6] for the general details of the constructions.", "Let $\\mathcal {B}_n\\lbrace z\\in \\mathbb {C}^{n}: \\sum |z_i|^2 < 1\\rbrace $ be an $n$ -dimensional ball.", "Alternatively, we can realize $\\mathcal {B}_n$ as follows.", "Let $\\mathcal {O}$ be the ring of integers of an imaginary quadratic field $\\mathbb {Q}(\\sqrt{d})$ , and let $\\Lambda $ be a free $\\mathcal {O}$ -module equipped with a hermitian form $h$ of signature $(1,n)$ .", "Then $\\mathcal {B}_n=\\left\\lbrace [z] \\in \\mathbb {P}(\\Lambda \\otimes \\mathbb {C}): h(z) >0 \\right\\rbrace \\,.$ For an arithmetic subgroup $\\Gamma \\subseteq \\operatorname{SU}(1,n)$ , there is a quotient quasi-projective analytic space $\\mathcal {B}_n/\\Gamma $ , which by construction has at worst finite quotient singularities.", "This quotient admits a projective Baily–Borel compactification $(\\mathcal {B}_n/\\Gamma )^*$ defined as the $\\operatorname{Proj}$ of the ring of automorphic forms with respect to $\\Gamma $ .", "Geometrically, the boundary $(\\mathcal {B}_n/\\Gamma )^*-\\mathcal {B}_n/\\Gamma =c_{F_1}\\sqcup \\dots \\sqcup c_{F_r}$ consists of a finite number of points, called cusps.", "These are in 1-to-1 correspondence with the $\\Gamma $ -orbits of isotropic lines in $\\Lambda _{\\mathbb {Q}(\\sqrt{d})}$ .", "There are no higher dimensional cusps since the signature is $(1,n)$ , implying that no other isotropic subspaces exist.", "In the case of ball quotients there exists a unique toroidal compactification $\\overline{\\mathcal {B}_n/\\Gamma }$ .", "Uniqueness follows since all tori involved have rank 1.", "More precisely, after dividing by the unipotent radical of the parabolic subgroup that stabilizes a given cusp, the quotient locally looks like an open set in $D^* \\times \\mathbb {C}^{n-1} \\subseteq \\mathbb {C}^* \\times \\mathbb {C}^{n-1}$ that contains $\\lbrace 0\\rbrace \\times \\mathbb {C}^{n-1}$ in its closure, where here $D^*$ is the punctured unit disk.", "The toroidal compactification is then simply obtained by adding the divisor $\\lbrace 0\\rbrace \\times \\mathbb {C}^{n-1}$ .", "As a result, the boundary $\\overline{\\mathcal {B}_n/\\Gamma }-\\mathcal {B}_n/\\Gamma =T_{F_1}\\sqcup \\dots \\sqcup T_{F_r}$ consists of a finite disjoint union of smooth (up to finite quotient singularities) irreducible divisors, each of which is in fact a finite quotient of an abelian variety.", "The natural map $p:\\overline{\\mathcal {B}_n/\\Gamma }\\rightarrow (\\mathcal {B}_n/\\Gamma )^*$ simply contracts each divisor $T_{F_i}$ to the cusp $c_{F_i}$ (see P:MarkedTor below for a detailed discussion of the case relevant in this paper).", "With this setup, we return to the case of cubic surfaces, and describe the period map to the ball quotient.", "Specifically, considering the triple cover of $\\mathbb {P}^3$ branched along a cubic surface, one obtains a cubic threefold, and via the period map for cubic threefolds, taking the $\\mathbb {Z}_3$ action into account, one obtains a period map to a 4-dimensional ball quotient, $\\mathcal {M}\\rightarrow {\\mathcal {B}_4/\\Gamma }$ (see [1]).", "This is an open embedding, and the complement of the image is the Heegner divisor $D_n=\\mathcal {D}_n/\\Gamma \\subseteq {\\mathcal {B}_4/\\Gamma }$ .", "It turns out that in this case the rational period map ${\\mathcal {M}}^{\\operatorname{GIT}}\\dashrightarrow {(\\mathcal {B}_4/\\Gamma )^*}$ to the Baily–Borel compactification extends to an isomorphism, taking the discriminant divisor $D_{A_1}\\subseteq {\\mathcal {M}}^{\\operatorname{GIT}}$ to the (closure of the) Heegner divisor $D_n\\subseteq {(\\mathcal {B}_4/\\Gamma )^*}$ (which is denoted this way for “nodal\").", "Under this isomorphism, the unique strictly polystable point $\\Delta _{3A_2}\\in {\\mathcal {M}}^{\\operatorname{GIT}}$ corresponding to the $3A_2$ cubic is identified with the sole cusp $c_{3A_2}=\\partial {(\\mathcal {B}_4/\\Gamma )^*}$ .", "The natural map $p:{\\overline{\\mathcal {B}_4/\\Gamma }}\\rightarrow {(\\mathcal {B}_4/\\Gamma )^*}$ contracts the irreducible boundary divisor $T_{3A_2}$ to $c_{3A_2}$ .", "From now on, we will write $D_{A_1}=D_n$ for the discriminant divisor, where we use $D_{A_1}$ when we are thinking of it from the GIT point of view, and $D_n$ when thinking of the ball quotient — to emphasize the context we are in.", "In summary, for the case of cubic surfaces we have a diagram ${{\\mathcal {M}}^{\\operatorname{K}}_{\\pi }[d]@{-->}^f[r]&{\\overline{\\mathcal {B}_4/\\Gamma }}^p[d]\\\\ {\\mathcal {M}}^{\\operatorname{GIT}}[r]^{\\sim }&{(\\mathcal {B}_4/\\Gamma )^*}}$ where $f$ is a birational map that restricts to an isomorphism $f:{\\mathcal {M}}^{\\operatorname{K}}-D_{3A_2} \\cong {\\overline{\\mathcal {B}_4/\\Gamma }}- T_{3A_2}\\,.$ In [28] and [40] the (intersection) Betti numbers of the spaces ${\\mathcal {M}}^{\\operatorname{GIT}}\\cong {(\\mathcal {B}_4/\\Gamma )^*}$ and ${\\mathcal {M}}^{\\operatorname{K}}$ were computed.", "In [11], the Betti numbers of ${\\overline{\\mathcal {B}_4/\\Gamma }}$ were computed, and they turned out to be the same as for ${\\mathcal {M}}^{\\operatorname{K}}$ , which served as motivation for our query as to whether these two compactifications are isomorphic, which is the main subject of the current paper." ], [ "The toroidal compactification via marked cubic surfaces", "An indispensable tool in the study of cubic surfaces is the group $W(E_6)$ , which is the automorphism group for the configuration of the 27 lines on a smooth cubic.", "The moduli space of cubic surfaces $\\mathcal {M}$ admits a natural $W(E_6)$ cover $\\mathcal {M}_m$ parameterizing marked smooth cubic surfaces, i.e., cubics together with a labeling of the 27 lines.", "Since the automorphism group of a smooth cubic surface acts faithfully on the primitive cohomology, it follows that $\\mathcal {M}_m$ is in fact smooth.", "Furthermore, Naruki [35] constructed a smooth normal crossing compactification $\\overline{\\mathcal {N}}$ of $\\mathcal {M}_m$ , which admits various geometric interpretations (see e.g., [25]).", "In this section, we use the geometry of the Naruki model $\\overline{\\mathcal {N}}$ to get a good hold on the space of interest in our paper, ${\\overline{\\mathcal {B}_4/\\Gamma }}$ .", "By construction, the marked moduli space $\\mathcal {M}_m$ is a Galois cover of $\\mathcal {M}$ with Galois group $W(E_6)$ .", "This cover is compatible with the ball quotient construction of Allcock–Carlson–Toledo [1].", "Specifically, the monodromy group $\\Gamma $ for cubic surfaces contains a normal subgroup $\\Gamma _m\\unlhd \\Gamma $ with $\\Gamma /\\Gamma _m\\cong W(E_6) \\times \\lbrace \\pm 1\\rbrace .$ (see [1]).", "We observe that $-1 $ acts trivially on the ball $\\mathcal {B}^4$ .", "This leads to a $W(E_6)$ cover ${\\mathcal {B}_4/\\Gamma }_m\\rightarrow {\\mathcal {B}_4/\\Gamma }$ .", "Furthermore, this cover extends to the Baily–Borel compactifications (compare [1]), and then also to the toroidal compactifications — essentially because in the ball quotient case, the toroidal compactification is canonical, and thus there is an automatic extension.", "In summary, the following holds: Proposition 4.1 With notation as above, we have the following diagram: ${\\mathcal {M}_m @{->>}[d]@{^{(}->}[r]&{\\mathcal {B}_4/\\Gamma }_m @{->>}[d]@{^{(}->}[r]@/^1.3pc/@{^{(}->}[rr]&({\\mathcal {B}_4/\\Gamma }_m)^*@{->>}[d]&\\overline{{\\mathcal {B}_4/\\Gamma }_m}[l]@{->>}[d]\\\\ \\mathcal {M}@{^{(}->}[r]&{\\mathcal {B}_4/\\Gamma }@{^{(}->}[r]@{^{(}->}[r]@/^1.3pc/@{^{(}->}[rr]&({\\mathcal {B}_4/\\Gamma })^*&{\\overline{\\mathcal {B}_4/\\Gamma }}[l]}$ where all the spaces in the top row admit a $W(E_6)$ action, and the morphisms are $W(E_6)$ -equivariant, while the spaces in the bottom row are the quotients with respect to this $W(E_6)$ action.", "$\\Box $ Remark 4.2 The GIT construction does not admit a natural $W(E_6)$ cover (e.g., it involves taking a quotient by $\\mathrm {PGL}(4,\\mathbb {C})$ , which has no natural connection to $W(E_6)$ ).", "Remark 4.3 We note that there is another very natural moduli space $\\mathcal {M}_\\ell $ parameterizing smooth cubics together with a chosen line.", "It is a degree 27 non-Galois cover $\\mathcal {M}_\\ell \\rightarrow \\mathcal {M}$ , which is in turn covered via $\\mathcal {M}_m\\rightarrow \\mathcal {M}_\\ell $ .", "This moduli of cubics with a line is of particular interest as it has a model as a Deligne–Mostow moduli space $DM(2^5,1^2)$ of points on a line.", "As such, $\\mathcal {M}_\\ell $ has both a GIT compactification, and the corresponding Kirwan blowup, as of a configuration of points, and a toroidal compactification covering ${\\overline{\\mathcal {B}_4/\\Gamma }}$ .", "It seems likely that our methods from this and previous works would make it possible to determine whether the corresponding Kirwan blowup and toroidal compactification are naturally isomorphic, but we will not pursue it here.", "It was shown recently that the marked toroidal and Naruki compactifications coincide.", "Theorem 4.4 ([24]) The Naruki compactification $\\overline{\\mathcal {N}}$ is isomorphic to the toroidal compactification ${\\overline{\\mathcal {B}_4/\\Gamma _m}}$ .", "More precisely, there is a $W(E_6)$ -equivariant commutative diagram ${\\mathcal {M}_m @{^{(}->}[r]@{^{(}->}[d]&{\\mathcal {B}_4/\\Gamma }_m@{^{(}->}[d]\\\\{\\overline{\\mathcal {N}}}[r]^{\\sim }&{\\overline{\\mathcal {B}_4/\\Gamma _m}}}$ Notation 4.5 In what follows, we use freely the identification given by ThmGallardo.", "Motivated by the compatibility given by diagram (REF ), we will use for ${\\mathcal {B}_4/\\Gamma }_m$ and its compactifications the same notation as for ${\\mathcal {B}_4/\\Gamma }$ , simply adding the subscript $m$ .", "In particular, we will consider the divisors $\\widetilde{D}_{n,m}$ and $T_{3A_2,m}$ on ${\\overline{\\mathcal {B}_4/\\Gamma _m}}$ .", "We will informally refer to $\\widetilde{D}_n$ (and $\\widetilde{D}_{n,m}$ ) as the nodal divisor, to $\\widetilde{R}$ (and $\\widetilde{R}_m$ on ${\\overline{\\mathcal {B}_4/\\Gamma _m}}$ ) as the Eckardt divisor, and to $T_{3A_2}$ (and $T_{3A_2,m}$ ) as the (toroidal) boundary divisor.", "Note that while $\\widetilde{D}_n$ and $\\widetilde{R}$ are irreducible divisors in ${\\overline{\\mathcal {B}_4/\\Gamma }}$ , the corresponding divisors in the marked case have several irreducible components transitively permuted by the natural $W(E_6)$ action.", "The Naruki compactification has a well-understood structure, mostly due to Naruki [35], with some further later clarifications by other authors.", "Theorem 4.6 (Naruki) The spaces and maps above have the following descriptions: ${\\overline{\\mathcal {B}_4/\\Gamma _m}}(\\cong \\overline{\\mathcal {N}})$ is smooth, and the complement of the locus $\\mathcal {M}_m$ of smooth marked cubic surfaces is the simple normal crossing divisor $T_{3A_2,m}\\cup \\widetilde{D}_{n,m}$ .", "a) The Baily–Borel compactification ${(\\mathcal {B}_4/\\Gamma _m)^*}$ has 40 cusps, permuted transitively by the $W(E_6)$ action, each of which lies over the unique cusp $c_{3A_2}\\in {(\\mathcal {B}_4/\\Gamma )^*}$ .", "Near each of the 40 cusps, ${(\\mathcal {B}_4/\\Gamma _m)^*}$ is locally isomorphic to the cone over $(\\mathbb {P}^1)^{\\times 3}$ embedded in $\\mathbb {P}^7$ via $\\mathcal {O}(1,1,1)$ .", "b) The boundary divisor $T_{3A_2,m}\\subseteq {\\overline{\\mathcal {B}_4/\\Gamma _m}}$ has 40 disjoint irreducible components, each isomorphic to $(\\mathbb {P}^1)^{\\times 3}$ .", "The deck transformation group $W(E_6)$ of the cover ${\\overline{\\mathcal {B}_4/\\Gamma _m}}\\rightarrow {\\overline{\\mathcal {B}_4/\\Gamma }}$ acts transitively on the set of these irreducible components.", "a) The $W(E_6)$ cover ${(\\mathcal {B}_4/\\Gamma _m)^*}\\rightarrow {(\\mathcal {B}_4/\\Gamma )^*}$ is branched along the discriminant divisor $D_{A_1}$ and the Eckardt divisor $R$ , with ramification index 2 along each.", "b) The $W(E_6)$ cover ${\\overline{\\mathcal {B}_4/\\Gamma _m}}\\rightarrow {\\overline{\\mathcal {B}_4/\\Gamma }}$ is generically étale along the toroidal boundary divisors $T_{3A_2,m}$ .", "The stabilizer $S_{3A_2,m}\\subseteq W(E_6)$ of an irreducible component of the divisor $T_{3A_2,m}$ fits in an extension $1\\rightarrow (S_3)^{\\times 3}\\rightarrow S_{3A_2,m}\\rightarrow S_3\\rightarrow 1\\,.$ Under the identification of the irreducible component with $(\\mathbb {P}^1)^{\\times 3}$ , the stabilizer $S_{3A_2,m}$ acts as follows: the normal subgroup $(S_3)^{\\times 3}$ acts diagonally on $(\\mathbb {P}^1)^{\\times 3}$ , while the residual $S_3$ acts on the quotient $(\\mathbb {P}^1)^{\\times 3}/(S_3)^{\\times 3}\\cong (\\mathbb {P}(2,3))^{\\times 3}\\cong (\\mathbb {P}^1)^{\\times 3}$ by permuting the factors.", "Item (0) is the main result of Naruki [35].", "Naruki also proved that the toroidal boundary $T_{3A_2,m}$ consists of 40 irreducible components (permuted transitively by $W(E_6)$ ), each isomorphic to $(\\mathbb {P}^1)^{\\times 3}$ .", "The stabilizer of a boundary components and its action are discussed in [35].", "In particular, items (1b), (2b), and (3) follow.", "As an aside, we note that each of the 40 components corresponds to a choice of embedding of the $A_2\\times A_2\\times A_2$ lattice into the $E_6$ lattice, and that intrinsically $S_{3A_2,m}$ is the normalizer $N_{W(E_6)}(3A_2)$ of such an embedding (recall that $N_{W(E_6)}(3A_2)$ is the unique, up to conjugacy, index 40 subgroup of $W(E_6)$ ; compare [9]).", "The ramification statement (2a) is clear for geometric reasons: the branch divisor consists of the nodal locus (the Picard–Lefschetz transformations act as reflections in $W(E_6)$ ), and the locus of smooth cubics with extra automorphisms, which coincides with the Eckardt divisor.", "While the occurrence of nodal degenerations is quite general, the presence of the Eckardt component is special to cubic surfaces: it is rare for the locus of objects with extra automorphisms to form a divisor in the moduli space, and secondly it reflects the fact that the automorphism group of a cubic surface embeds into $W(E_6)$ .", "The ramification statement is also worked out in detail (from the ball quotient perspective) in [1] for the nodal locus, and [1] for the Eckardt locus.", "For the interested reader, we point out that these two types of ramification are associated, respectively, to short and long roots in the Eisenstein lattice used to construct the ball quotient model of [1].", "Finally, it remains to discuss the structure at the boundary of the Baily–Borel compactification and the relation to the toroidal compactification.", "First, the Baily–Borel compactification is a contraction of the toroidal boundary $T_{3A_2,m}$ in ${\\overline{\\mathcal {B}_4/\\Gamma _m}}(\\cong \\overline{\\mathcal {N}})$ .", "Thus, (1b) implies that ${(\\mathcal {B}_4/\\Gamma _m)^*}$ has 40 cusps (see also [22], where this fact is proved without reference to the toroidal compactification).", "Since $({\\mathcal {B}_4/\\Gamma })^*$ has a unique cusp, and the two Baily–Borel compactifications are compatible as in (REF ), the first part of (1a) follows.", "It remains to determine the local structure near the cusps of ${(\\mathcal {B}_4/\\Gamma _m)^*}$ .", "Naruki [35] constructed a contraction $ \\overline{\\mathcal {N}}\\rightarrow \\mathcal {N}^*$ of the 40 irreducible components of $T_{3A_2,m}$ to 40 singularities of the type described in (1a).", "It was then noted in [22] that $\\mathcal {N}^*$ coincides with ${(\\mathcal {B}_4/\\Gamma _m)^*}$ ; this completes the proof.", "With these preliminaries, we can extract a series of immediate consequences on the boundary of $\\mathcal {M}\\subseteq {\\overline{\\mathcal {B}_4/\\Gamma }}$ .", "First, Prop-compatible and P:MarkedTor(0) together show that this is a normal crossing compactification in a stack sense: Corollary 4.7 The boundary $\\widetilde{D}_n\\cup T_{3A_2}$ in ${\\overline{\\mathcal {B}_4/\\Gamma }}$ is a normal crossings divisor, up to finite quotients.", "$\\Box $ For further reference, we also record the structure of the toroidal boundary divisor: Corollary 4.8 The toroidal boundary divisor $T_{3A_2}\\subseteq {\\overline{\\mathcal {B}_4/\\Gamma }}$ is isomorphic to $\\mathbb {P}^3$ .", "The boundary divisor $T_{3A_2}$ is the quotient of a fixed component of $T_{3A_2,m}$ by the relevant stabilizer group.", "Using P:MarkedTor, we get $T_{3A_2}\\cong (\\mathbb {P}^1)^{\\times 3}/S_{3A_2,m}\\cong \\operatorname{Sym}^3\\mathbb {P}^1\\cong \\mathbb {P}^3$ ." ], [ "Proof of T:mainNonIso", "At this point, we are able to establish one of our main results, namely that the period map does not extend to an isomorphism between the Kirwan compactification and the toroidal compactification of the ball quotient model.", "We have seen by P:discMK that the nodal and boundary divisors do not meet transversally, even generically, in the Kirwan model, while an immediate consequence of the above discussion is that they do so in the toroidal model: Proposition 4.9 The discriminant $\\widetilde{D}_n$ and boundary $T_{3A_2}$ divisors in ${\\overline{\\mathcal {B}_4/\\Gamma }}$ meet generically transversally along an irreducible surface.", "This follows easily from the geometry explicitly described in P:MarkedTor for the marked case.", "We first observe that the intersection of $\\widetilde{D}_n$ and $T_{3A_2}$ is irreducible and hence it will be enough to consider a generic point $P$ of some component of the intersection $ \\widetilde{D}_{n,m}\\cap T_{3A_2,m}$ .", "To see this we recall that, by P:MarkedTor (1)(b), the Weyl group $W(E_6)$ acts transitively on the cups and hence the toroidal boundary components.", "We fix a component $T$ of $T_{3A_2,m}$ .", "We claim that the stabilizer $S_T(\\cong S_{3A_2,m}$ see P:MarkedTor (3)) of $T$ in $W(E_6)$ permutes all components of $T \\cap \\widetilde{D}_{n,m}$ .", "For this we recall that $T\\cong (\\mathbb {P}^1)^{\\times 3}$ and use the description of the action of the stabilizer $S_T$ given in P:MarkedTor (3).", "This stabilizer contains a normal subgroup isomorphic to $(S_3)^{\\times 3}$ acting independently on each $\\mathbb {P}^1$ factor.", "There is also a residual $S_3$ factor, which acts by permuting the 3 copies of $\\mathbb {P}^1$ in $T$ .", "The three copies of $(S_3)^{\\times 3}$ each act on one of the copies of $(\\mathbb {P}^1)^{\\times 3}$ by the projectivization of the standard action of $S_3$ on $\\mathbb {C}^2$ , or more intrinsically here, the projectivization of the action of $W(A_2)$ on $A_2\\otimes _{\\mathbb {Z}}\\mathbb {C}$ , and trivially on the other two factors.", "Choosing suitable coordinates we can assume that $S_3$ permutes the points $0,1,\\infty $ .", "By [35] the intersection $T \\cap \\widetilde{D}_{n,m}$ consists of 9 components, namely the surfaces $\\lbrace 0\\rbrace \\times (\\mathbb {P}^1)^{\\times 2}$ , $\\lbrace 1\\rbrace \\times (\\mathbb {P}^1)^{\\times 2}$ and $\\lbrace \\infty \\rbrace \\times (\\mathbb {P}^1)^{\\times 2}$ and their translates under the group $S_3$ interchanging the three factors of $(\\mathbb {P}^1)^{\\times 3}$ .", "Clearly these components are permuted under $S_{T}$ .", "We can now work with the component $\\lbrace 0\\rbrace \\times (\\mathbb {P}^1)^{\\times 2}$ and recall that in ${\\overline{\\mathcal {B}_4/\\Gamma _m}}$ , the divisor $\\widetilde{D}_{n,m}\\cup T_{3A_2,m}$ has simple normal crossings.", "The stabilizer of a point $P=(0,*_1,*_2)$ with $*_i \\ne 0,1,\\infty $ and $*_1 \\ne *_2$ has order 2; its non-trivial element is the involution in the first factor of $(S_3)^{\\times 3}$ which fixes 0 and interchanges 1 and $\\infty $ .", "This defines a reflection in $W(E_6)$ whose fixed locus is the component of $ \\widetilde{D}_{n,m}$ passing through $P$ .", "This involution also fixes the component $T$ of $T_{3A_2,m}$ (as a set, not pointwise).", "Thus we can choose local analytic coordinates $(x_1,x_2,x_3,x_4)$ on ${\\overline{\\mathcal {B}_4/\\Gamma _m}}$ near $P$ such that $\\widetilde{D}_{n,m}$ and $T_{3A_2,m}$ are the zero loci of coordinates $x_1$ and $x_2$ , respectively, and such that the stabilizer acts by $x_1 \\mapsto -x_1$ , leaving all other coordinates fixed.", "On the quotient we can therefore take local analytic coordinates $y_1=x_1^2, y_i=x_i, i= 2,3,4$ .", "Then locally $\\widetilde{D}_n$ and $T_{3A_2}$ are given by $y_1=0$ and $y_2=0$ respectively, and the claim follows.", "Remark 4.10 In rem:intersectionwithboundarylocal we shall provide a different proof for the fact that $W(E_6)$ acts transitively on the components of $\\widetilde{D}_{n,m}\\cup T_{3A_2,m}$ .", "This follows immediately from P:discMK and P:discTor.", "Indeed, these Propositions show a priori that the period map does not extend to an isomorphism.", "However, this is enough to show that the period map does not extend to a morphism in either direction.", "Indeed, since $f$ gives an isomorphism ${\\mathcal {M}}^{\\operatorname{K}}-D_{3A_2} \\cong {\\overline{\\mathcal {B}_4/\\Gamma }}- T_{3A_2}$ , if $f$ extended to a morphism, it would have to send the irreducible divisor $D_{3A_2}$ to the irreducible divisor $T_{3A_2}$ .", "Thus, if $f$ were to extend to a morphism, which was not an isomorphism, it would have to be a small contraction.", "For a small contraction of complex varieties $f:Y\\rightarrow X$ , if $Y$ is quasi-projective and $X$ is normal, then $X$ is not $\\mathbb {Q}$ -factorial.", "This would contradict the fact that ${\\overline{\\mathcal {B}_4/\\Gamma }}$ is $\\mathbb {Q}$ -factorial, having only finite quotient singularities.", "A similar argument holds for the rational map $f^{-1}$ , since ${\\mathcal {M}}^{\\operatorname{K}}$ is also $\\mathbb {Q}$ -factorial.", "As a corollary of T:mainNonIso, we have the following result, which contrasts with C:oBGNCb for the toroidal compactification, and strengthens P:discMK (which was itself used in the proof of T:mainNonIso) for the Kirwan compactification.", "Corollary 4.11 The boundary $\\widetilde{D}_{A_1}\\cup D_{3A_2}$ in ${\\mathcal {M}}^{\\operatorname{K}}$ is not a normal crossings divisor, even up to finite quotients.", "If the boundary $\\widetilde{D}_{A_1}\\cup D_{3A_2}$ in ${\\mathcal {M}}^{\\operatorname{K}}$ were a normal crossings divisor, up to finite quotients, then the standard extension theorems for period maps to toroidal compactifications [6] would imply that the period map $f:{\\mathcal {M}}^{\\operatorname{K}}\\dashrightarrow {\\overline{\\mathcal {B}_4/\\Gamma }}$ extended to a morphism, contradicting T:mainNonIso.", "Remark 4.12 We emphasize that cor:quickproofnontransversal depends on T:mainNonIso, which in turn depends on P:discMK, so that the proof of cor:quickproofnontransversal above does not provide an alternate proof of P:discMK.", "On the other hand, T:mainNonIso follows from T:mainNonK, whose proof is independent of P:discMK.", "In other words, one can give an alternate proof of P:discMK by first proving T:mainNonK, which implies T:mainNonIso, and then proving cor:quickproofnontransversal." ], [ "The canonical bundles of the ball quotient models", "While T:mainNonIso says that the period map does not extend to an isomorphism between ${\\mathcal {M}}^{\\operatorname{K}}$ and ${\\overline{\\mathcal {B}_4/\\Gamma }}$ , a priori it is possible that ${\\mathcal {M}}^{\\operatorname{K}}$ and ${\\overline{\\mathcal {B}_4/\\Gamma }}$ might be abstractly isomorphic.", "To distinguish them, we study their canonical classes.", "We start with a fairly complete discussion of divisor classes (in particular the canonical class) on the ball quotient side.", "Similar computations occur in [13] for the ball quotient model for cubic threefolds.", "However, due to the simpler structure of the moduli of cubic surfaces, we are able to obtain sharper results here." ], [ "Divisors and relations in ${(\\mathcal {B}_4/\\Gamma )^*}$", "The ball quotients come equipped with a natural $\\mathbb {Q}$ -line bundle $\\lambda $ , the so-called Hodge line bundle, that gives the polarization for the Baily–Borel compactification (and pulls back to a big and nef bundle on the toroidal compactification).", "Since ${(\\mathcal {B}_4/\\Gamma )^*}\\cong {\\mathcal {M}}^{\\operatorname{GIT}}\\cong \\mathbb {P}(1,2,3,4,5)$ has Picard rank 1, the divisors $\\lambda $ , $ D_n,$ and $R$ must all be proportional.", "It turns out that one can express the classes of these divisors in terms of $\\lambda $ as a consequence of the work of Borcherds.", "Specifically, the following holds: Theorem 5.1 ([2]) There is an automorphic form $\\chi _4$ of weight 4 whose divisor in $\\mathcal {B}_4$ is the sum of all short mirrors, each with multiplicity 1.", "Similarly, there is an automorphic form $\\chi _{75}$ of weight 75 whose divisor in $\\mathcal {B}_4$ is the sum of all long mirrors, each with multiplicity 1.", "Taking into account the ramification orders 6 for the discriminant divisor, which corresponds to the sum of all short mirrors, and 2 for Eckardt divisor, which corresponds to the sum of all long mirrors (see esp.", "[1]), we obtain the following equalities in $\\operatorname{Pic}_\\mathbb {Q}({(\\mathcal {B}_4/\\Gamma )^*})$ : $4\\lambda =\\frac{1}{6} D_n;\\qquad 75\\lambda =\\frac{1}{2} R\\,.$ Similarly, in the marked case, where the map $\\mathcal {B}_4\\rightarrow {(\\mathcal {B}_4/\\Gamma _m)^*}$ is only ramified along the nodal locus with index 3 (e.g., [1]), the following holds in $\\operatorname{Pic}_\\mathbb {Q}(({\\mathcal {B}_4/\\Gamma }_m)^*)$ : $4\\lambda _m=\\frac{1}{3} D_{n,m};\\qquad 75\\lambda _m= R_m\\,.$ Under the identification ${(\\mathcal {B}_4/\\Gamma )^*}\\cong \\mathbb {P}(1,2,3,4,5)$ , we have seen (REF ) that $D_n=\\mathcal {O}_{\\mathbb {P}(1,2,3,4,5)}(4)$ and $R=\\mathcal {O}_{\\mathbb {P}(1,2,3,4,5)}(25)$ (which gives $R=\\frac{25}{4} D_n$ , agreeing with (REF )), either of which combined with (REF ) give the following relation between the natural polarizations on ${(\\mathcal {B}_4/\\Gamma )^*}$ and $\\mathbb {P}(1,2,3,4,5)$ : $\\mathcal {O}_{\\mathbb {P}(1,2,3,4,5)}(1)=6\\lambda \\,.$ We now turn to the canonical divisors.", "Taking into account the ramification, by the general theory of ball quotients (e.g., [34], which uses modular forms for the canonical automorphy factor given by the Jacobian, which have weight “$\\dim +1$ ”, and then taking into account ramification as in [4]), we obtain $K_{{(\\mathcal {B}_4/\\Gamma )^*}}=5\\lambda - \\frac{5}{6} D_n - \\frac{1}{2} R =-90\\lambda \\,.$ By (REF ), we see that $K_{{(\\mathcal {B}_4/\\Gamma )^*}}=\\mathcal {O}_{\\mathbb {P}(1,2,3,4,5)}(-15)$ , which agrees with the canonical bundle of $\\mathbb {P}(1,2,3,4,5)$ (e.g., (REF )).", "In the marked case, as the map $\\mathcal {B}_4\\rightarrow {(\\mathcal {B}_4/\\Gamma _m)^*}$ is only ramified along the nodal locus with index 3, we thus obtain $K_{{(\\mathcal {B}_4/\\Gamma _m)^*}}=5\\lambda - \\frac{2}{3} D_{n,m}\\,.$ The same general considerations give the formulas for the canonical bundles of the toroidal compactifications.", "Proposition 5.2 The following hold: $K_{{\\overline{\\mathcal {B}_4/\\Gamma }}}=5\\lambda - \\frac{5}{6} \\widetilde{D}_n - \\frac{1}{2} \\widetilde{R} - T_{3A_2}$  , $K_{{\\overline{\\mathcal {B}_4/\\Gamma _m}}}=5\\lambda _m - \\frac{2}{3}\\widetilde{D}_{n,m} -T_{3A_2,m}$  .", "This is a standard computation, a consequence of Mumford's Hirzebruch proportionality theorem (see e.g., [4])." ], [ "Computations of discrepancies", "In order to enable us to compute the top self-intersection of the canonical class, in this section we compute how divisors on the Baily–Borel compactifications compare to divisors on the toroidal compactifications.", "Corollary 5.3 In the notation above, the canonical class is given by $K_{{\\overline{\\mathcal {B}_4/\\Gamma _m}}}=p_m^*K_{{(\\mathcal {B}_4/\\Gamma _m)^*}}+T_{3A_2,m}\\,.$ By P:MarkedTor, locally near a cusp of ${(\\mathcal {B}_4/\\Gamma _m)^*}$ , the map ${\\overline{\\mathcal {B}_4/\\Gamma _m}}\\rightarrow {(\\mathcal {B}_4/\\Gamma _m)^*}$ is the standard blowup of the cone over $(\\mathbb {P}^1)^{\\times 3}\\hookrightarrow \\mathbb {P}^7$ .", "The claim then follows by a standard computation for the blowup.", "To descend to the unmarked case, we need to understand the intersection of the boundary divisor $T_{3A_2}$ with the two ramification divisors $\\widetilde{D}_{n,m}$ and $\\widetilde{R}_{m}$ .", "Proposition 5.4 The normal bundle to the marked toroidal boundary divisor $T_{3A_2,m}$ , restricted to each irreducible boundary component, is equal to $\\mathcal {O}_{(\\mathbb {P}^1)^{\\times 3}}(-1,-1,-1)$ .", "The restriction $\\widetilde{D}_{n,m}|_{T_{3A_2,m}}$ of the marked discriminant divisor to each irreducible component of the marked toroidal boundary divisor is isomorphic to $\\mathcal {O}_{(\\mathbb {P}^1)^{\\times 3}}(3,3,3)$ .", "The restriction $\\widetilde{R}_{m}|_{T_{3A_2,m}}$ of the marked Eckardt divisor to each irreducible component of the marked toroidal boundary divisor is isomorphic to $\\mathcal {O}_{(\\mathbb {P}^1)^{\\times 3}}(12,12,12)$ .", "The same argument as in cor:discr gives the first item.", "As discussed in P:MarkedTor, the divisors $\\widetilde{D}_{n,m}$ and ${T_{3A_2,m}}$ intersect transversely, and [35] describes this intersection precisely.", "For a fixed irreducible component $T_0$ of $\\widetilde{T}_{3A_2}$ , which we have seen is isomorphic to $(\\mathbb {P}^1)^{\\times 3}$ , the intersection of $\\widetilde{D}_{n,m}$ with it consists of 9 irreducible components of type $\\lbrace \\textrm {pt}\\rbrace \\times \\mathbb {P}^1\\times \\mathbb {P}^1\\subseteq \\mathbb {P}^1\\times \\mathbb {P}^1\\times \\mathbb {P}^1$ , and thus of class $\\mathcal {O}_{(\\mathbb {P}^1)^{\\times 3}}(1,0,0)$ (up to permuting the coordinates).", "The claim (2) thus follows.", "The stabilizer $S_{3A_2,m}\\subseteq W(E_6)$ of $T_0$ was identified in P:MarkedTor.", "Taking $U_m$ to be a suitable invariant neighborhood of $T_0$ , locally near $T_0$ , the map ${\\overline{\\mathcal {B}_4/\\Gamma _m}}\\rightarrow {\\overline{\\mathcal {B}_4/\\Gamma }}$ is simply the quotient $U_m\\rightarrow U_m/S_{3A_2,m}\\cong U$ , with $U$ a neighborhood of the toroidal boundary $T_{3A_2}\\subseteq {\\overline{\\mathcal {B}_4/\\Gamma }}$ .", "Since $S_{3A_2,m}$ contains a normal subgroup $(S_3)^{\\times 3}$ , we can take the intermediate quotient $U^{\\prime }=U_m/(S_3)^{\\times 3}$ and obtain a diagram ${U_m @{->}[r]^{/(S_3)^{\\times 3}}_{\\alpha }&U^{\\prime }@{->}[r]^{/S_3}_{\\beta }&U\\\\(\\mathbb {P}^1)^{\\times 3}[r]@{^{(}->}[u]&(\\mathbb {P}^1)^{\\times 3}@{^{(}->}[u][r]&\\ \\ T_{3A_2}\\cong \\mathbb {P}^3@{^{(}->}[u]}$ compatible with P:MarkedTor(3) and C:T3A2.", "Let $T^{\\prime }\\subseteq U^{\\prime }$ be the quotient $T_0/((S_3)^{\\times 3})\\cong (\\mathbb {P}(2,3))^{\\times 3}\\cong (\\mathbb {P}^1)^{\\times 3}$ .", "To describe $\\widetilde{R}_{m}\\cap T_0$ , first recall that $U_m\\rightarrow U$ is ramified along the Eckardt and nodal loci, with order 2 along each.", "The factorization $U_m\\xrightarrow{} U^{\\prime }\\xrightarrow{} U$ has the property that $\\alpha $ is ramified along the nodal locus, while $\\beta $ is ramified along the Eckardt locus.", "Indeed, in the language of [1], the subgroup $(S_3)^{\\times 3}\\subseteq S_ {3A_2,m}$ is generated by short roots (corresponding to the 9 nodal components meeting $T_0$ ), and the residual $S_3=S_ {3A_2,m}/(S_3)^{\\times 3}$ is generated by classes of long roots (corresponding to the Eckardt locus, and geometrically to cubics with an extra involution).", "In this description, it is clear that the restriction of the Eckardt locus to $T^{\\prime }$ is simply the sum of the 3 small diagonals (each of type $\\mathcal {O}_{(\\mathbb {P}^1)^{\\times 3}}(1,1,0)$ , up to permutation), and thus of class $\\mathcal {O}_{(\\mathbb {P}^1)^{\\times 3}}(2,2,2)$ .", "The pullback via $\\alpha _{\\mid T_0}$ of this class will be of type $\\mathcal {O}_{(\\mathbb {P}^1)^{\\times 3}}(12,12,12)$ .", "Since $\\alpha $ is not ramified along the Eckardt locus, this will be also the class of the reduced divisor $\\widetilde{R}_{m}|_{T_0}$ .", "Remark 5.5 The last two claims in P:MarkDivRestrict can be obtained also by a purely arithmetic argument.", "As alluded to in the proofs above, they follow by counting the short and long roots incident to a fixed cusp in ${(\\mathcal {B}_4/\\Gamma _m)^*}$ .", "To explain this, we consider the vector space $\\mathbb {F}_3^5$ , equipped with the standard orthogonal form of signature $(4,1)$ .", "It is well known that the orthogonal group can then be identified as $\\operatorname{O}(\\mathbb {F}_3^5) \\cong W(E_6) \\times \\lbrace \\pm 1 \\rbrace .$ An element $[w] \\in \\mathbb {P}(\\mathbb {F}_3^5)$ is called isotropic, short or long, depending on whether the norm of $w$ equals $0,1$ or 2 (this does not depend on the chosen representative in $\\mathbb {F}_3^5$ ).", "By [2] (also [1]) these elements enumerate the cusps, the components of the discriminant divisor $\\widetilde{D}_{n,m}$ , and of the Eckardt divisor $\\widetilde{R}_{m}$ , respectively.", "We further know from [2] that there are 40 isotropic, 36 short, and 45 long elements in $ \\mathbb {P}(\\mathbb {F}_3^5)$ , and that the Weyl group $W(E_6)$ acts transitively on each of these three sets.", "Counting the number of components of the discriminant divisor $\\widetilde{D}_{n,m}$ and of the Eckardt divisor $\\widetilde{R}_{m}$ intersecting an irreducible component $T_0$ of $T_{3A_2,m}$ as above is then an easy enumeration.", "Indeed, the choice of the cusp, and of $T_0$ , means fixing an isotropic element $h\\in \\mathbb {P}(\\mathbb {F}_3^5)$ , and a straightforward count shows that there are 9 short and 18 long vectors orthogonal to $h$ , which counts the number of irreducible components of $\\widetilde{D}_{n,m}$ and $\\widetilde{R}_m$ that intersect $T_0$ .", "To describe the intersection of these components of $\\widetilde{D}_{n,m}$ and $\\widetilde{R}_{m}$ with $T_0$ , note that since the stabilizer subgroup of $h$ in $W(E_6)$ acts transitively on the set of all sort (and also on the set of all long) vectors orthogonal to $h$ , by symmetry it is enough to understand the intersection with $T_0$ of only one component of $\\widetilde{D}_{n,m}$ and one component of $\\widetilde{R}_{m}$ with $T_0$ , which can be seen, e.g., from the standard local coordinates near a boundary component.", "All we need is then to understand the involutions which account for the degree 2 ramification along $\\widetilde{D}_n$ and $\\widetilde{R}$ .", "In the case of the nodal divisor, this is, up to symmetry, an involution on a factor $\\mathbb {P}^1$ which fixes some point $p\\in \\mathbb {P}^1$ , and then the restriction of the component of $\\widetilde{D}_{n,m}$ to $T_0$ , fixed under this involution, is $p \\times (\\mathbb {P}^1)^{\\times 2}$ , which gives the divisor class $ \\mathcal {O}_{(\\mathbb {P}^1)^{\\times 3}}(1,0,0)$ .", "Adding up all 9 components of $\\widetilde{D}_{n,m}$ that meet $T_0$ we obtain $ \\mathcal {O}_{(\\mathbb {P}^1)^{\\times 3}}(3,3,3)$ .", "In the case of the Eckardt divisor, the involution is given by interchanging two of the factors of $(\\mathbb {P}^1)^{\\times 3}$ , while fixing the third factor.", "In the case of interchanging the first two factors, the fixed locus is then equal to $\\Delta _{3A_2,12} \\times \\mathbb {P}^1$ , and has class $ \\mathcal {O}_{(\\mathbb {P}^1)^{\\times 3}}(1,1,0)$ .", "Summing over all 18 components of $\\widetilde{R}_m$ which meet $T_0$ we obtain $\\mathcal {O}_{(\\mathbb {P}^1)^{\\times 3}}(12,12,12)$ .", "We can now compare the discrepancies in the pullbacks of the discriminant and Eckardt divisors for the moduli of marked cubic surfaces: Corollary 5.6 Let $p_m^*:{\\overline{\\mathcal {B}_4/\\Gamma _m}}\\rightarrow {(\\mathcal {B}_4/\\Gamma _m)^*}$ be the natural map.", "The following hold: $p_m^* D_{n,m} = \\widetilde{D}_{n,m}+3T_{3A_2,m}$  ; $p_m^* R_m = \\widetilde{R}_m + 12 T_{3A_2,m}$  .", "This follows immediately from P:MarkDivRestrict, by restricting to $T_{3A_2,m}$ .", "Indeed, a priori we have $p_m^* D_{n,m} = \\widetilde{D}_{n,m}+aT_{3A_2,m}$ for some $a$ , and restricting to the component $T_0$ of $T_{3A_2,m}$ gives $0= \\widetilde{D}_{n,m}|_{T_0}+ aT_{3A_2,m}|_{T_0}$ , which gives $a=3$ by parts (1) and (2) of P:MarkDivRestrict.", "The computation for the Eckardt divisor is identical using parts (1) and (3).", "Remark 5.7 It is interesting to note that the formulas above are compatible with those of pro:canonicalbundleontoroidals that were obtained by general considerations.", "Specifically, using (REF ), cor:discr and C:mu, we get $K_{{\\overline{\\mathcal {B}_4/\\Gamma _m}}}&=&p_m^*K_{{(\\mathcal {B}_4/\\Gamma _m)^*}}+T_{3A_2,m}\\\\&=& 5\\lambda -\\frac{2}{3} (\\widetilde{D}_{n,m}+3T_{3A_2,m})+T_{3A_2,m}\\\\&=&5\\lambda -\\frac{2}{3} \\widetilde{D}_{n,m}-T_{3A_2,m}\\,,$ agreeing indeed with pro:canonicalbundleontoroidals(2).", "The main result of this Section is the computation of the discrepancy of $p^*:{\\overline{\\mathcal {B}_4/\\Gamma }}\\rightarrow {(\\mathcal {B}_4/\\Gamma )^*}$ in the unmarked case, as this coefficient will be crucial for computing the top self-intersection number of $K_{{\\overline{\\mathcal {B}_4/\\Gamma }}}$ .", "Proposition 5.8 The canonical bundle of the toroidal compactification of the ball quotient model of the moduli space of cubic surfaces is given by the formula: $K_{{\\overline{\\mathcal {B}_4/\\Gamma }}}= p^*K_{{(\\mathcal {B}_4/\\Gamma )^*}}+16T_{3A_2}\\,.$ Since in (REF ) we computed the discrepancy for the finite cover ${\\overline{\\mathcal {B}_4/\\Gamma _m}}\\rightarrow {(\\mathcal {B}_4/\\Gamma _m)^*}$ , a standard computation (see e.g., [30]) gives $K_{{\\overline{\\mathcal {B}_4/\\Gamma }}}= p^*K_{{(\\mathcal {B}_4/\\Gamma )^*}}+aT_{3A_2}$ for $a= \\frac{1}{r(T_{3A_2,m})}\\left((1+\\mu _{\\widetilde{D}_{n,m}}+\\mu _{\\widetilde{R}_m})+1)\\right)-1\\,,$ where $r(T_{3A_2,m})=1$ is the ramification index for the cover ${\\overline{\\mathcal {B}_4/\\Gamma _m}}\\rightarrow {\\overline{\\mathcal {B}_4/\\Gamma }}$ along the toroidal boundary divisor (P:MarkedTor(4)), and $\\mu _{\\widetilde{D}_{n,m}}=3$ and $\\mu _{\\widetilde{R}_m}=12 $ are defined so that $p_m^* D_{n,m} = \\widetilde{D}_{n,m}+\\mu _{\\widetilde{D}_{n,m}}T_{3A_2,m}$ and $p_m^* R_m = \\widetilde{R}_m + \\mu _{\\widetilde{R}_m} T_{3A_2,m}$ (C:mu).", "We conclude that $a= ((1+3+12) +1)-1=16$ as claimed.", "Remark 5.9 For completeness, we note that the analogue of C:mu in the unmarked case is $p^*D_n&=&\\widetilde{D}_n+5T_{3A_2}\\,;\\\\p^*R&=&\\widetilde{R}+24T_{3A_2}\\,.$ Similarly to Rem-doublecheck, these formulas are compatible with P:Torcan, pro:canonicalbundleontoroidals(1), and (REF ), giving a double check of our computations." ], [ "Self-intersection numbers for the toroidal compactification", "Using the fact that the Baily–Borel compactification is a weighted projective space ${(\\mathcal {B}_4/\\Gamma )^*}\\cong {\\mathcal {M}}^{\\operatorname{GIT}}\\cong \\mathbb {P}(1,2,3,4,5)$ , and (REF ) we conclude: Corollary 5.10 On ${(\\mathcal {B}_4/\\Gamma )^*}$ , the following holds: $\\left(K_{{(\\mathcal {B}_4/\\Gamma )^*}}\\right)^4=\\frac{(-15)^4}{5!", "}=\\frac{15^3}{2^3}=\\frac{3375}{8}$ and $\\lambda ^4=\\frac{1}{5!\\cdot 6^4}=\\frac{1}{2^73^55}=\\frac{1}{25,920}$ .$\\Box $ Using the description of toroidal boundary given by P:MarkedTor, we obtain: Lemma 5.11 The self-intersection numbers of the toroidal boundary divisors are $(T_{3A_2,m})^4&=&-240\\quad (\\hbox{on }{\\overline{\\mathcal {B}_4/\\Gamma _m}})\\\\(T_{3A_2})^4&=& -\\frac{1}{6^3}\\quad (\\hbox{on }{\\overline{\\mathcal {B}_4/\\Gamma }})$ By P:MarkedTor, each component of $T_{3A_2,m}$ is the exceptional divisor of the standard blowup of the cone over the Segre embedding $(\\mathbb {P}^1)^{\\times 3}\\hookrightarrow \\mathbb {P}^7$ .", "It follows that the self-intersection of each such component in ${\\overline{\\mathcal {B}_4/\\Gamma _m}}$ is $-6$ .", "Taking into account that there are 40 disjoint such components, the first item follows.", "The degree of the map ${\\overline{\\mathcal {B}_4/\\Gamma _m}}\\rightarrow {\\overline{\\mathcal {B}_4/\\Gamma }}$ is $51,840=|W(E_6)|$ , and since this covering map is unramified along $T_{3A_2}$ , the second claim follows.", "Finally, we can compute the top degree self-intersection of the canonical class on the toroidal compactification.", "Theorem 5.12 The top self-intersection number of the canonical class on ${\\overline{\\mathcal {B}_4/\\Gamma }}$ is $(K_{{\\overline{\\mathcal {B}_4/\\Gamma }}})^4=\\frac{25,589}{2^33^3}=\\frac{25,589}{216}\\,.$ From P:Torcan, we get $(K_{{\\overline{\\mathcal {B}_4/\\Gamma }}})^4=(K_{{(\\mathcal {B}_4/\\Gamma )^*}})^4+16^4(T_{3A_2})^4\\,.$ Substituting the numbers from cor-BBG and self-t3a2, the conclusion follows." ], [ "The canonical bundle of Kirwan desingularizations", "For addressing the issue of $K$ -equivalence of the compactifications, we now need to perform the computations on the GIT side parallel to the ball quotient computations in the previous Section.", "These turn out to be more involved, and we devote this Section to the general setup and results on computing the canonical bundle of the Kirwan resolution of a GIT quotient.", "This does not seem to be available in the literature, and may be of independent interest.", "After discussing the general case, we specialize to the case of cubic surfaces in SS:KMK." ], [ "Setup", "We start with the general setup of a GIT triple $(X,L,G)$ , where $X$ is a scheme of finite type (we will continue to work over $\\mathbb {C}$ to simplify notation, but the argument works over any algebraically closed field of characteristic zero, and, with minor adjustments, in positive characteristic, as well), $L$ is an ample line bundle on $X$ , and $G$ is a connected reductive linear algebraic group acting on $X$ , with $L$ being $G$ -linearized.", "We will also make the following assumptions that put us into the setup for Kirwan's work: $X$ is a smooth quasi-projective variety; $X^s \\ne \\emptyset $ , i.e., the stable locus is non-empty.", "Note that from the second condition, and say the Luna Slice Theorem, it follows that there is a Zariski dense open subvariety $U\\subseteq X^s$ and a finite group $G_X$ such that for all $x\\in U$ the stabilizer $G_x\\subseteq G$ is isomorphic (although not necessarily equal) to $G_X$ ; i.e., $G_X$ is the stabilizer of some general point of $X$ .", "We denote by $q: X^{ss}\\longrightarrow YX/\\!\\!/_LG$ the GIT quotient." ], [ "Canonical classes for GIT quotients", "Since $X^{ss}$ is normal, so is $Y$ (e.g, [18]), and so canonical classes $K_{X^{ss}}$ and $K_Y$ are defined.", "Note that while $K_{X^{ss}}$ is Cartier, recall that there are elementary examples of quotients of smooth varieties by reductive groups that are not $\\mathbb {Q}$ -Gorenstein (e.g., [8]); in other words, $K_Y$ need not be $\\mathbb {Q}$ -Cartier." ], [ "The stable locus", "Now let $Y^sX^{s}/G\\subseteq Y= X/\\!\\!/_LG$ be the stable locus, and to fix notation, we have the map $q^s: X^{s}\\longrightarrow Y^s= X^s/G\\,.$ Note that since $X$ is smooth, the Luna Slice Theorem implies that $Y^s$ is étale locally the quotient of a smooth variety $U$ by some finite group $G_i$ .", "In particular, $Y^s$ is $\\mathbb {Q}$ -factorial, and there is a well-defined pullback $q^{s*}$ for $\\mathbb {Q}$ -Weil divisor classes.", "In this situation, we have the following Riemann–Hurwitz lemma: Lemma 6.1 (Riemann–Hurwitz for the stable locus) Let $R^s$ be the divisorial locus in $X^s$ that has stabilizer strictly containing $G_X$ (i.e., the union of codimension 1 irreducible components of the locus of points in $X^s$ where the stabilizer is not isomorphic to $G_X$ ), let $R^s=\\bigcup R_i^s$ be its decomposition into irreducible components, and let $G_{R_i}$ be the stabilizer of a general point of $R_i^s$ .", "Then $K_{X^s}&= q^{s*}K_{Y^s}+\\sum (|G_{R_i}|/|G_X| -1)R^s_i\\,.$ It suffices to check étale locally.", "The Luna Slice Theorem implies that, up to a smooth factor, the quotient $q^s:X^s\\rightarrow Y^s$ is étale locally equivalent to the quotient $U\\rightarrow U/G_i$ for a smooth scheme $U$ and some finite group $G_i$ .", "Computing the canonical bundles is then a standard computation for the ramified cover $U\\rightarrow U/G_i$ (see, e.g., [30])." ], [ "Strictly semi-stable locus of codimension at least 2", "The computations for the stable locus carry over immediately to the general case, as long as the strictly semi-stable locus is of codimension at least 2.", "From now on, we will thus assume that $X^{ss}-X^{s} \\subseteq X^{ss}$ and $Y-Y^s\\subseteq Y$ are codimension at least 2.", "Under this assumption, one can define a pullback $q^*$ on $\\mathbb {Q}$ -Weil divisor classes by restricting to the stable locus $Y^s$ (which, as noted above, is $\\mathbb {Q}$ -factorial), pulling back to $X^s$ , and then extending over the boundary, which is assumed to be of codimension at least 2.", "This immediately yields: Corollary 6.2 (Riemann–Hurwitz) Assume that $X^{ss}-X^{s} \\subseteq X^{ss}$ and $Y-Y^s\\subseteq Y$ have codimension at least 2.", "Then the same conclusion as in L:RH-DM holds: $K_{X^{ss}}&= q^{*}K_{Y}+\\sum (|G_{R_i}|/|G_X| -1)R_i\\,,$ where $R_i$ is the closure of $R_i^s$ in $X^{ss}$ .", "$\\Box $ Remark 6.3 The codimension at least 2 hypothesis above does rule out some standard GIT constructions.", "For instance, C:RH-GIT is not applicable in the case of the GIT moduli space of cubic curves ${\\mathcal {M}}^{\\operatorname{GIT}}_{\\operatorname{curve}}$ , as the locus of strictly semi-stable cubic curves is the locus of $A_1$ cubic curves, which is a codimension 1 locus in the semi-stable locus in the Hilbert scheme $\\mathbb {P}^{9}=\\mathbb {P}H^0(\\mathbb {P}^2,\\mathcal {O}_{\\mathbb {P}^2}(3))$ .", "Of course ${\\mathcal {M}}^{\\operatorname{GIT}}_{\\operatorname{curve}}$ is simply equal to $\\mathbb {P}^1$ , so this particular situation is trivial." ], [ "Canonical bundle of the Kirwan blowup", "Here we consider a step in the Kirwan blowup process: ${F@{^(->}[rr] [d]&&\\widetilde{X}^{ss}[rr]^{\\tilde{\\pi }} [d]^{\\tilde{q}} &&X^{ss} [d]^q &\\\\E@{^(->}[rr]&&\\widetilde{Y}= \\widetilde{X}/\\!\\!/_{\\tilde{L}}G [rr]^{\\pi }&&Y= X/\\!\\!/_LG&\\\\}$ We refer the reader to Kirwan's various papers on the topic for the details on Kirwan blowups, or to our paper [11] for a summary.", "For the convenience of the reader, recall that this includes the data of $\\operatorname{Stab}^\\circ \\le G$ , a maximal dimensional connected component of a stabilizer, and the associated locus $Z_{\\operatorname{Stab}^\\circ }^{ss}\\lbrace x\\in X^{ss}: \\operatorname{Stab}^\\circ \\text{ fixes } x\\rbrace \\,,$ which is a smooth closed subvariety of $X^{ss}$ .", "Note that in Kirwan's papers and in [11], $\\operatorname{Stab}^\\circ $ is denoted by “$R$ ”; this would conflict with our notation here, that $R$ is the Eckardt (ramification) divisor, and so we use the (possibly) more transparent $\\operatorname{Stab}^\\circ $ in the current paper.", "Lemma 6.4 Assume that $X^{ss}-X^{s} \\subseteq X^{ss}$ and $Y-Y^s\\subseteq Y$ are codimension at least 2.", "Let $x\\in Z_{\\operatorname{Stab}^\\circ }^{ss}$ be a general point, let $\\mathcal {N}_x$ be the fiber of the normal bundle to $G\\cdot Z_{\\operatorname{Stab}^\\circ }^{ss}$ in $X^{ss}$ at $x$ , let $\\ell _x\\subseteq \\mathcal {N}_x$ be a general line through the origin, and let $c=\\operatorname{codim}_X (G\\cdot Z_{\\operatorname{Stab}^\\circ }^{ss})$ .", "Denote by $\\widetilde{R}_i$ the strict transform of the ramification divisor $R_i$ in $X^{ss}$ , denote by $G_{F}\\subseteq G_x$ the stabilizer of the general line $\\ell _x$ , and denote by $\\mu _i$ the coefficient defined by $\\tilde{\\pi }^* R_i=\\widetilde{R}_i+\\mu _i F$ .", "In terms of these invariants, the canonical bundles admit the following expressions: $K_{\\widetilde{X}^{ss}}&= \\tilde{q}^{*}K_{\\widetilde{Y}}+\\sum (|G_{ R_i}|/|G_X| -1)\\widetilde{R}_i + (|G_{F}|/|G_X| -1)F = \\tilde{\\pi }^* K_{X^{ss}} + (c-1) F\\,, \\\\K_{\\widetilde{Y}}& = \\pi ^*K_Y+\\left(\\frac{c+\\sum (|G_{ R_i}|/|G_X| -1)\\mu _i }{|G_F/G_X|}-1\\right)E, \\ \\ \\ \\text{ if $Y$ is $\\mathbb {Q}$-Gorenstein.", "}$ The first equality in (REF ) just follows from C:RH-GIT, using that $F$ is the projectivized normal bundle to the orbit $G\\cdot Z^{ss}_{\\operatorname{Stab}^\\circ }\\subseteq X^{ss}$ , and the stabilizer of a generic point of $F$ is therefore the stabilizer of the projectivized normal space at a generic point.", "The second equality in (REF ) is just the formula for the canonical bundle of the blowup of a smooth variety along a smooth subvariety.", "For (), we use (REF ).", "Indeed, substituting the expressions $K_{\\widetilde{Y}}= \\pi ^*K_Y +a(E,Y,0)E$ and $K_{X^{ss}}= q^* K_Y+ \\sum (|G_{ R_I}|/|G_X| -1)R_i$ , we obtain $K_{\\widetilde{X}^{ss}}&= \\tilde{q}^{*}\\pi ^*K_{Y}+ a(E,Y,0)|G_F/G_X|F+\\sum (|G_{ R_i}|/|G_X| -1)\\widetilde{R}_i + (|G_{F}|/|G_X| -1)F\\\\&= \\tilde{q}^{*} \\pi ^*K_{Y}+ \\sum (|G_{ R_i}|/|G_X| -1)\\widetilde{R}_i + \\left((a(E,Y,0)+1)|G_{F}|/|G_X| -1\\right)F\\\\K_{\\widetilde{X}^{ss}}& = \\tilde{\\pi }^* q^*K_{Y} + \\sum (|G_{ R_i}|/|G_X| -1)\\widetilde{R}_i +\\sum (|G_{ R_i}|/|G_X| -1)\\mu _i F+ (c-1) F \\\\&= \\tilde{\\pi }^* q^*K_{Y} + \\sum (|G_{ R_i}|/|G_X| -1)\\widetilde{R}_i + \\left(c-1 +\\sum (|G_{ R_i}|/|G_X| -1)\\mu _i \\right)F\\,.$ Solving for $a(E,Y,0)$ gives the result.", "Remark 6.5 In order to effectively compute invariants on a Kirwan blowup (e.g., cohomology, canonical bundles, etc.", "), it is useful to be able to make computations directly on $X$ , where one in principle has good control of the geometry and group action (as opposed to on the blowups of $X$ ).", "The invariants $\\mu _i$ and $|G_F|$ in L:KKirw are chosen for this reason; note that these can be computed at general points of the strata in $X$ in Kirwan's setup for the Kirwan blowup.", "Remark 6.6 From (), one can see that $\\mathbb {Q}$ -Gorenstein GIT quotients, in our restricted setup, have klt singularities.", "This is a special case of a much more general result due to Schoutens [39].", "We also note that one can conclude from this that certain moduli spaces of $K$ -stable Fano manifolds are klt; this is again a special case of much deeper results of Braun et al.", "[8] and [31].", "Remark 6.7 (Kirwan blowups with boundary) Still under the assumption that $X^{ss}-X^{s} \\subseteq X^{ss}$ and $Y-Y^s\\subseteq Y$ are codimension at least 2, if we consider the case of a boundary $(Y,\\Delta _Y)$ and assume that $K_Y+\\Delta _Y$ is $\\mathbb {Q}$ -Cartier, then setting $\\Delta _{X^{ss}}q^*\\Delta _Y$ , and letting $\\Delta _{\\widetilde{Y}}$ be the strict transform of $\\Delta _Y$ in $\\widetilde{Y}$ , we have $K_{\\widetilde{Y}}+\\Delta _{\\widetilde{Y}}& = \\pi ^*(K_Y+\\Delta _Y)+\\left(\\frac{a(F,X^{ss},\\Delta _{X^{ss}})+1+\\sum (|G_{ R_i}|/|G_X| -1)\\mu _i }{|G_F/G_X|}-1\\right)E\\,.$ Indeed, setting $\\Delta _{\\widetilde{X}^{ss}}$ to be the strict transform of $\\Delta _{X^{ss}}$ in $\\widetilde{X}^{ss}$ , we have $\\tilde{q}^*\\Delta _{\\widetilde{Y}}= \\Delta _{\\widetilde{X}^{ss}}$ ; in our situation, both the strict transform and pullback are defined by restricting to the locus where the morphisms are either isomorphisms or étale, and then taking closures, and so the strict transform and pullback commute.", "Then the same analysis as above using $K_{\\widetilde{Y}}+\\Delta _{\\widetilde{Y}}= \\pi ^*(K_Y+\\Delta _Y) +a(E,Y,\\Delta _Y)E$ $K_{\\widetilde{X}^{ss}}+\\Delta _{\\widetilde{X}^{ss}}=\\tilde{\\pi }^*(K_{X^{ss}}+\\Delta _{X^{ss}}) +a(F,X^{ss},\\Delta _{X^{ss}})F$ $K_{X^{ss}}+\\Delta _{X^{ss}}= q^*(K_Y+\\Delta _Y)+ \\sum (|G_{ R_i}|/|G_X| -1)R_i$ gives (REF ).", "Note that from (REF ), it follows that if $(X^{ss}, \\Delta _{X^{ss}} = q^*\\Delta _Y)$ is klt, then so is $(Y,\\Delta _Y)$ ; we emphasize that we started by assuming $K_Y+\\Delta _Y$ was $\\mathbb {Q}$ -Cartier." ], [ "Computation of $K_{{\\mathcal {M}}^{\\operatorname{K}}}$", "We now specialize the general discussion of the previous section to the particular case of the moduli of cubic surfaces.", "We want to apply C:RH-GIT to compute $K_{{\\mathcal {M}}^{\\operatorname{GIT}}}$ , and then $K_{{\\mathcal {M}}^{\\operatorname{K}}}$ .", "To do this for ${\\mathcal {M}}^{\\operatorname{GIT}}$ , recall that the locus of unstable points in $\\mathbb {P}^{19}$ has codimension $\\ge 2$ , and the locus of strictly semi-stable points has codimension $\\ge 2$ in the semi-stable locus, both in $(\\mathbb {P}^{19})^{ss}$ as well as in ${\\mathcal {M}}^{\\operatorname{GIT}}$ .", "Similarly, for ${\\mathcal {M}}^{\\operatorname{K}}$ , the locus of unstable points in the blowup of $(\\mathbb {P}^{19})^{ss}$ has codimension $\\ge 2$ , and the locus of strictly semi-stable points has codimension $\\ge 2$ within the semi-stable locus, both in the blowup of $(\\mathbb {P}^{19})^{ss}$ as well as in ${\\mathcal {M}}^{\\operatorname{K}}$ (see [11]).", "We are thus in the setup of the previous Section.", "Recall from sec:calM that the locus of cubics in ${\\mathcal {M}}^{\\operatorname{GIT}}$ with non-trivial stabilizers is the irreducible Eckardt divisor $R$ , and the cubic surface parameterized by a general point of $R$ has automorphism group $\\mathbb {Z}_2$ .", "Thus, the Riemann–Hurwitz formula (C:RH-GIT) in this case gives $K_{(\\mathbb {P}^{19})^{ss}} = q^* K_{{\\mathcal {M}}^{\\operatorname{GIT}}} + R\\,.$ The class of the Eckardt divisor is known (see rem:Eckardt): $R=\\mathcal {O}_{\\mathbb {P}^{19}}(100)$ , where $\\mathcal {O}_{\\mathbb {P}^{19}}(1)$ is the restriction to $(\\mathbb {P}^{19})^{ss}$ of the hyperplane section in $\\mathbb {P}^{19}$ .", "Thus we have $-\\mathcal {O}_{\\mathbb {P}^{19}}(20) = q^*K_{{\\mathcal {M}}^{\\operatorname{GIT}}} + \\mathcal {O}_{\\mathbb {P}^{19}}(100)\\,,$ giving $q^*K_{{\\mathcal {M}}^{\\operatorname{GIT}}} = -\\mathcal {O}_{\\mathbb {P}^{19}}(120)\\,.$ Recall also (e.g., rem:Eckardt) that the discriminant has class $\\mathcal {O}_{\\mathbb {P}^{19}}(32)$ in our situation.", "In other words, as the discriminant in $(\\mathbb {P}^{19})^{ss}$ descends to give the Weil divisor $D_{A_1}$ on ${\\mathcal {M}}^{\\operatorname{GIT}}$ , the divisor $\\mathcal {O}_{\\mathbb {P}^{19}}(1)$ descends to ${\\mathcal {M}}^{\\operatorname{GIT}}$ (as a $\\mathbb {Q}$ -divisor) to give the $\\mathbb {Q}$ -Cartier divisor $\\frac{1}{32}D_{A_1}$ .", "This finally gives $K_{{\\mathcal {M}}^{\\operatorname{GIT}}} = -\\frac{120}{32}D_{A_1} = -\\frac{15}{4}D_{A_1}\\,,$ agreeing with the computation in (REF ).", "We now compute $K_{{\\mathcal {M}}^{\\operatorname{K}}}$ using the same general machinery.", "Recall that the Kirwan desingularization for the case of cubic surfaces is obtained by a single blowup, supported at the point $\\Delta _{3A_2}\\in {\\mathcal {M}}^{\\operatorname{GIT}}$ corresponding to the orbit of the $3A_2$ cubic surface $S_{3A_2}$ .", "Corollary 6.8 $K_{{\\mathcal {M}}^{\\operatorname{K}}} = \\pi ^* K_{{\\mathcal {M}}^{\\operatorname{GIT}}} + 20 D_{3A_2}\\,.$ With the notation as above, L:KKirw and P:stab–ex(1) together give that $K_{{\\mathcal {M}}^{\\operatorname{K}}} = \\pi ^* K_{{\\mathcal {M}}^{\\operatorname{GIT}}} +(c+\\mu -1)D_{3A_2}$ , where $c$ is the codimension of the $3A_2$ orbit, and $\\mu $ is the multiplicity of the Eckardt locus along the $3A_2$ orbit in the sense of L:KKirw.", "Since we have $c=6$ (e.g., L:3A2slice) and $\\mu =15$ by L:Eck-no-3A2, the result follows.", "Remark 6.9 The coefficient 20 for $D_{3A_2}$ (or at least the fact that the coefficient is divisible by 5) is crucial for our computation and proof of non-$K$ -equivalence.", "Note that it follows immediately that while $\\pi :{\\mathcal {M}}^{\\operatorname{K}}\\rightarrow {\\mathcal {M}}^{\\operatorname{GIT}}$ is a blowup supported at the smooth point $\\Delta _{3A_2}\\in {\\mathcal {M}}^{\\operatorname{GIT}}$ , it is not the standard blowup of the point, since otherwise $K_{{\\mathcal {M}}^{\\operatorname{K}}}$ would equal $\\pi ^*K_{{\\mathcal {M}}^{\\operatorname{GIT}}}+3D_{3A_2}$ ." ], [ "Intersection numbers for divisors on DM stacks", "On DM stacks, the top self-intersection numbers of canonical classes are rational numbers.", "We need the following statement regarding the denominators that may appear, which is essentially [3]: Proposition 6.10 Let $\\mathcal {X}$ be a smooth proper DM stack over a field $K$ of characteristic 0 with coarse moduli space $\\Phi :\\mathcal {X}\\rightarrow X$ of dimension $d$ .", "For each geometric point $p:\\operatorname{Spec}\\overline{K} \\rightarrow \\mathcal {X}$ , denote by $e_p$ the exponent of the automorphism group of $p$ , and denote by $e$ the least common multiple of the numbers $e_p$ for all geometric points of $\\mathcal {X}$ .", "For any divisor classes $D_1,\\dots ,D_d\\in \\operatorname{CH}_{d-1}(X)$ , we have $D_1\\cdots D_d \\in \\frac{1}{e^d}\\mathbb {Z}\\,.$ We first recall that for line bundles $M_1,\\dots ,M_d$ on $X$ , with divisor classes $[M_1], \\dots , [M_d]\\in \\operatorname{CH}_{d-1}(X)$ , we have by definition $[M_1]\\cdots [M_d] \\int _X c_1(M_1) \\cdots c_1(M_d)\\cap [X] \\in \\mathbb {Z}\\,.$ Using the fact that $X$ is $\\mathbb {Q}$ -factorial, this allows us to define the intersection number $D_1\\cdots D_d$ as a rational number for any divisor classes $D_1,\\dots ,D_d\\in \\operatorname{CH}_{d-1}(X)$ .", "To prove the bound on the denominators, we argue as follows.", "For any divisor $D\\in \\operatorname{CH}_{d-1}(X)$ , the pullback $\\Phi ^*D$ is the divisor class associated to some line bundle $\\mathcal {L}$ on $\\mathcal {X}$ (given an étale presentation $P:U\\rightarrow \\mathcal {X}$ , the pullback $P^* \\Phi ^*D$ is the class of a line bundle on $U$ , which descends to $\\mathcal {X}$ ).", "The fact that $\\mathcal {L}^{\\otimes e}$ descends to a line bundle $M$ on $X$ is [3].", "Thus we have $(eD_1) \\cdots (e D_d) \\in \\mathbb {Z}$ , completing the proof." ], [ "Proof of the non-$K$ -equivalence", "We can now conclude that ${\\mathcal {M}}^{\\operatorname{K}}$ and ${\\overline{\\mathcal {B}_4/\\Gamma }}$ are not $K$ -equivalent, which is one of our main results.", "All the work for this proof has been already done, and we just gather the pieces here.", "We recall that the top self-intersection numbers of the canonical class on $K$ -equivalent varieties are equal — this follows from say [29], which implies that if $f:Y\\rightarrow X$ is a morphism of schemes of dimension $d$ over a field, such that $f_*(\\mathcal {O}_Y) = \\mathcal {O}_X$ , and $D_i$ are $\\mathbb {Q}$ -Cartier divisors on $X$ , then $f^*D_1\\cdots f^*D_d = D_1\\cdots D_d$ .", "Consequently, given two $K$ -equivalent birational normal projective $\\mathbb {Q}$ -Gorenstein varieties $X$ and $Y$ of dimension $d$ , one has from the definition of $K$ -equivalence that, in the notation of diagram (REF ), $K_X^d=g^*K_X^d= K_Z^d=h^*K_Y^d=K_Y^d$ .", "We will thus prove T:mainNonK by showing that the top self-intersection numbers of the canonical classes on the Kirwan and toroidal compactifications are not equal to each other.", "Showing that the top self-intersection numbers of $K_{{\\mathcal {M}}^{\\operatorname{K}}}$ and of $K_{{\\overline{\\mathcal {B}_4/\\Gamma }}}$ are not equal to each other is greatly simplified by the fact that both of these spaces admit blowdown maps ($\\pi $ and $p$ respectively) to the same space ${(\\mathcal {B}_4/\\Gamma )^*}={\\mathcal {M}}^{\\operatorname{GIT}}$ , with the exceptional divisors of $\\pi $ and $p$ both contracted to a single point $\\Delta _{3A_2}$ .", "Thus, as computed above, the canonical class in each case is the pullback of the canonical bundle $K_{{\\mathcal {M}}^{\\operatorname{GIT}}}$ plus the exceptional divisor of the blowup with a suitable multiplicity.", "Since the exceptional divisors of $\\pi $ and $p$ are both contracted to a point, the top self-intersection numbers of $K_{{\\mathcal {M}}^{\\operatorname{K}}}$ and $K_{{\\overline{\\mathcal {B}_4/\\Gamma }}}$ will each equal to the top self-intersection number of $K_{{\\mathcal {M}}^{\\operatorname{GIT}}}$ plus a suitable multiple of the top self-intersection number of the corresponding exceptional divisor.", "By the above discussion, it suffices to show that $K_{{\\mathcal {M}}^{\\operatorname{K}}}^4\\ne K_{{\\overline{\\mathcal {B}_4/\\Gamma }}}^4$ .", "To this end, recall from E:MKcan and P:Torcan that $K_{{\\mathcal {M}}^{\\operatorname{K}}}& = \\pi ^* K_{ {\\mathcal {M}}^{\\operatorname{GIT}}}+20D_{3A2}\\,,\\\\K_{{\\overline{\\mathcal {B}_4/\\Gamma }}} & = p^* K_{{\\mathcal {M}}^{\\operatorname{GIT}}}+16T_{3A_2}\\,.$ The top self-intersection number for the latter is then $K_{{\\overline{\\mathcal {B}_4/\\Gamma }}}^4 =K_{{\\mathcal {M}}^{\\operatorname{GIT}}}^4-\\tfrac{16^4}{6^3}$ by self-t3a2.", "We then compute $K_{{\\mathcal {M}}^{\\operatorname{K}}}^4= K_{{\\mathcal {M}}^{\\operatorname{GIT}}}^4+20^4D_{3A_2}^4=K_{{\\mathcal {M}}^{\\operatorname{GIT}}}^4+5^42^8D_{3A_2}^4\\,.$ By P:AGV we have $D_{3A_2}^4\\in \\frac{1}{e}\\mathbb {Z}$ for some $e$ that is not divisible by 5; indeed, $D_{3A_2}^4= (D_{3A_2}|_{D_{3A_2}})^3$ , where we are using that ${\\mathcal {M}}^{\\operatorname{K}}$ is $\\mathbb {Q}$ -factorial to define the restriction.", "By P:stab–ex, no stabilizers along $D_{3A_2}$ have order divisible by 5.", "Consequently, the two intersection numbers $K_{{\\mathcal {M}}^{\\operatorname{K}}}^4$ and $K_{{\\overline{\\mathcal {B}_4/\\Gamma }}}^4$ cannot be the same since if we write $5^42^8D_{3A_2}^4$ as a product of nonzero powers of distinct primes, 5 will appear with positive exponent, while it does not appear with positive exponent in $-\\tfrac{16^4}{6^3}=-\\tfrac{2^{13}}{3^3}$ ." ], [ "Proof of equality in the Grothendieck ring of varieties", "In this section we show that ${\\mathcal {M}}^{\\operatorname{K}}$ and ${\\overline{\\mathcal {B}_4/\\Gamma }}$ have the same class in the Grothendieck ring of varieties, which is our last main result to be proven.", "Due to the fact that moduli spaces of cubic surfaces have no odd degree cohomology, this will turn out to give another, conceptual rather than computational, proof that these two compactifications have the same cohomology (C:BettiEq).", "Since ${\\mathcal {M}}^{\\operatorname{K}}- D_{3A_2}\\cong {\\overline{\\mathcal {B}_4/\\Gamma }}- T_{3A_2}$ , by additivity of the class of the disjoint union of varieties in the Grothendieck ring, it suffices to show that the exceptional divisors $D_{3A_2}$ and $T_{3A_2}$ , as varieties, are equivalent in the Grothendieck ring.", "Since we have seen that $T_{3A_2}$ is isomorphic to $\\mathbb {P}^3$ , it thus suffices to show that $D_{3A_2}$ is equivalent to $\\mathbb {P}^3$ in the Grothendieck ring.", "This is accomplished below in L:D3A2class (for an alternative approach and proof, see L:D3A2toric and R:D3A2class).", "We now describe explicitly the geometry of the exceptional divisor $D_{3A_2}\\subseteq {\\mathcal {M}}^{\\operatorname{K}}$ , by thinking of it as the GIT quotient of the exceptional divisor $\\mathbb {P}^5$ of the blowup of the Luna slice by the group $\\operatorname{Aut}(S_{3A_2})$ .", "Recall that $\\operatorname{Aut}(S_{3A_2})$ is the semidirect product of $\\mathbb {T}^2$ and $S_3$ , as described in L:R3A2Norm2.", "Furthermore, the $\\mathbb {C}^6$ Luna slice is described in L:3A2slice, and the action of $\\operatorname{Aut}(S_{3A_2})$ on it is given by (REF ).", "We are interested in the action of $\\operatorname{Aut}(S_{3A_2})$ on the exceptional divisor of the blowup $\\operatorname{Bl}_0\\mathbb {C}^6\\rightarrow \\mathbb {C}^6$ of the Luna slice at the origin.", "Recall that the action of $S_3$ on the exceptional $\\mathbb {P}^5$ is simply by permuting the homogeneous coordinates $(T_0:T_1:T_2:T_{\\widehat{0}}:T_{\\widehat{1}}:T_{\\widehat{2}})$ pairwise.", "Our goal is to describe all semi-stable orbits, up to the action of $S_3$ .", "Recall that the set of unstable orbits has already been determined in L:LunaStability.", "We will describe sets of various orbits of the same type, where by type we will simply mean which of the homogeneous coordinates $T$ vanish (and we will write 0 for them), and which of the coordinates $T$ do not vanish (and we will write $*$ for an arbitrary non-zero complex number).", "For such a given type of an orbit, we will parameterize the orbits of such a type by a number of copies of $\\mathbb {C}^*$ , for the non-zero coordinates, quotiented by a finite group.", "Indeed, for all orbits of a given type we can set one of the non-zero (labeled $*$ ) homogeneous coordinates $T$ to be equal to 1 by projectivizing, and then set two more of the non-zero homogeneous coordinates to be equal to 1 by acting by $\\mathbb {T}^2\\subseteq \\operatorname{Aut}(S_{3A_2})$ .", "However, a given $\\mathbb {T}^2$ orbit may contain more than one point where these three chosen coordinates are equal to 1, due to finite stabilizers that we will describe in detail in L:D3A2class below.", "To enumerate possible types of orbits, up to the $S_3$ action, we will always permute the coordinates $T_0,T_1,T_2$ in such a way that all zeroes precede all non-zero coordinates.", "If possible to do so while keeping this condition for $T_0,T_1,T_2$ , we will further permute $T_{\\widehat{0}},T_{\\widehat{1}},T_{\\widehat{2}}$ to also put all zeroes before all the non-zero coordinates.", "Recalling from L:LunaStability that semi-stable points on $\\mathbb {P}^5$ are those where $T_i\\ne 0$ or $T_{\\widehat{i}}\\ne 0$ , for each $i=0,1,2$ , we thus enumerate the types of semi-stable (in fact all of them stable) $\\mathbb {T}^2$ orbits on $\\mathbb {P}^5$ , and their contributions to the class $[D_{3A_2}]$ in the Grothendieck ring of varieties in F:1 (where we number the types of orbits, for easy reference).", "Table: Stable 𝕋 2 \\mathbb {T}^2 orbits on ℙ 5 \\mathbb {P}^5Here we note that the quotient of $\\mathbb {C}^*$ by any finite subgroup is still a $\\mathbb {C}^*$ , so that contributions from (4) and (6) are equal to $[\\mathbb {C}^*]$ , which is also to the class of any quotient of $\\mathbb {C}^*$ by a finite subgroup.", "Thus to compute the class of $D_{3A_2}$ in the Grothendieck ring of varieties it remains to compute the classes of the higher-dimensional quotients, for which we need to understand the actions of $S_2$ and $S_3$ appearing in orbits of types (1),(2),(3) above.", "Lemma 8.1 The divisor $D_{3A_2}\\subseteq {\\mathcal {M}}^{\\operatorname{K}}$ is equivalent to $\\mathbb {P}^3$ in the Grothendieck ring of varieties.", "From the table of contributions (F:1), and referring to the types of orbits by those numbers, we compute the class in the Grothendieck ring to be $[D_{3A_2}]=[(1)]+[(2)]+[(3)]+3[\\mathbb {C}^*]+4[\\mathbb {C}^0]\\,,$ where the first three summands denote the classes in the Grothendieck ring of varieties of the orbits of the corresponding types from F:1.", "We start with locus (1), noticing first that any $\\mathbb {T}^2$ orbit of any such point contains a point whose homogeneous coordinates are $(*:*:*:1:1:1)$ (i.e., such that $T_{\\widehat{0}}=T_{\\widehat{1}}=T_{\\widehat{2}}=1$ ).", "Moreover, looking at the explicit action of $\\operatorname{diag}(\\lambda _0,\\lambda _1,\\lambda _2)\\in D^{\\prime \\prime }\\simeq \\mathbb {T}^2$ (with $\\lambda _0\\lambda _1\\lambda _2=1$ ) given by (REF ), if $(T_0:T_1:T_2:1:1:1)$ and $(T^{\\prime }_0:T^{\\prime }_1:T^{\\prime }_2:1:1:1)$ lie on the same $\\mathbb {T}^2$ orbit, it means that there must exist $(\\lambda _0,\\lambda _1,\\lambda _2)\\in \\mathbb {T}^2$ such that $\\lambda _0^2=\\lambda _1^2=\\lambda _2^2$ and such that $T_i=\\lambda _i^3 T^{\\prime }_i$ for $i=0,1,2$ .", "This means that each of the three $\\lambda _i$ must be equal to some $\\sigma _i=\\pm 1$ , subject to the condition that the product of the three signs is equal to $+1$ , and the values of $T_i$ and $T_i^{\\prime }$ must then differ by the corresponding signs.", "Thus the set of orbits of this form is equal to $(\\mathbb {C}^*)^3/G\\,,$ where $G$ is the subgroup of $\\mu _2\\times \\mu _2\\times \\mu _2$ given by the condition $\\sigma _0\\sigma _1\\sigma _2=1$ , and the action is by diagonal multiplication.", "The action of $S_3$ on $(\\mathbb {C}^*)^3/G$ is induced by permuting the coordinates on $(\\mathbb {C}^*)^3$ , and we note that it does not commute with the action of $G$ .", "For example, acting by signs $(-1,1,-1)$ maps $T_0:T_1:T_2$ to $-T_0:T_1:-T_2$ , and then permuting $0\\leftrightarrow 1$ gives $T_1:-T_0:-T_2$ , while first permuting and then acting by signs gives $-T_1:T_0:-T_2$ .", "By taking the squares of the coordinates, we observe that the quotient $(\\mathbb {C}^*)^3/(\\mu _2\\times \\mu _2\\times \\mu _2)$ , where the action is by multiplication by three independent signs, is isomorphic to $(\\mathbb {C}^*)^3$ .", "Furthermore, recall that the quotient $\\mathbb {C}^3/S_3$ , under the action that permutes coordinates, is identified with $\\mathbb {C}^3$ by taking elementary symmetric polynomials, i.e., the bijection $\\mathbb {C}^3/S_3\\simeq \\mathbb {C}^3$ is given in coordinates by $(x_1,x_2,x_3)\\mapsto (x_1x_2x_3\\,,\\ x_1x_2 + x_2x_3 + x_1x_3\\,,\\ x_1 + x_2 + x_3)\\,.$ By inspection, the image of $(\\mathbb {C}^*)^3\\subseteq \\mathbb {C}^3$ under this bijection is $\\mathbb {C}^*\\times \\mathbb {C}^2\\simeq (\\mathbb {C}^*)^3/S_3$ .", "Altogether, this means that the map $(y_1,y_2,y_3)\\mapsto (y_1^2y_2^2y_3^2\\,,\\ y_1^2y_2^2 + y_2^2y_3^2 + y_1^2y_3^2\\,,\\ y_1^2 + y_2^2 + y_3^2)$ identifies the quotient $(\\mathbb {C}^*)^3/(\\mu _2^{\\times 3})\\rtimes S_3$ with $\\mathbb {C}^*\\times \\mathbb {C}^2$ .", "The contribution $[(1)]$ to the class $[D_{3A_2}]$ is $[(\\mathbb {C}^*)^3/G\\rtimes S_3]$ .", "We now claim that the double cover $(\\mathbb {C}^*)^3/G\\rtimes S_3\\rightarrow (\\mathbb {C}^*)^3/\\mu _2^{\\times 3}\\rtimes S_3$ is étale.", "Indeed, to prove this we need to check that no element of $((\\mu _2)^{\\times 3}\\rtimes S_3)- (G\\rtimes S_3)$ stabilizes any point in the domain of this map.", "Indeed, up to renumbering the coordinates, we need to worry about the permutation being the identity, an involution $0\\leftrightarrow 1$ or the cycle $0\\mapsto 1\\mapsto 2\\mapsto 0$ , and the signs can either all be minus, or just one sign can be minus.", "We thus check case by case that there are no fixed points: $\\begin{aligned}(T_0,T_1,T_2)&=(-T_0,T_1,T_2)&\\Rightarrow T_0=0\\,\\,\\\\(T_0,T_1,T_2)&=(-T_0,-T_1,-T_2)&\\Rightarrow T_0=T_1=T_2=0\\,\\,\\\\(T_0,T_1,T_2)&=(-T_1,T_0,T_2)&\\Rightarrow T_0=-T_1=-T_0\\Rightarrow T_0=T_1=0\\,\\,\\\\(T_0,T_1,T_2)&=(T_1,T_0,-T_2)&\\Rightarrow T_2=0\\,\\,\\\\(T_0,T_1,T_2)&=(-T_1,-T_0,-T_2)&\\Rightarrow T_2=0\\,\\,\\\\(T_0,T_1,T_2)&=(-T_1,T_2,T_0)&\\Rightarrow T_0=-T_1=-T_2=-T_0\\Rightarrow T_0=0\\,\\,\\\\(T_0,T_1,T_2)&=(-T_1,-T_2,-T_0)&\\Rightarrow T_0=-T_1=T_2=-T_0\\Rightarrow T_0=0\\,,\\\\\\end{aligned}$ so that in each case we deduce that some coordinate must be zero, and thus the fixed point set in $(\\mathbb {C}^*)^3$ is empty.", "Since the only connected étale double cover of $\\mathbb {C}^*\\times \\mathbb {C}^2$ is topologically itself (covering along the $\\mathbb {C}^*$ factor), it follows that $[(1)]=[\\mathbb {C}^*\\times \\mathbb {C}^2]$ in the Grothendieck ring.", "The contributions to the class in the Grothendieck ring of the orbits of types (2) and (3) are simpler.", "For (2), similarly to the previous case, we can always find a representative with homogeneous coordinates of the form $(T_0:T_1:0:1:1:1)$ , and such a point lies on the same $\\mathbb {T}^2$ orbit as $(T^{\\prime }_0:T^{\\prime }_1:0:1:1:1)$ if and only if $T_0=\\pm T^{\\prime }_0$ and $T_1=\\pm T^{\\prime }_1$ , with the signs chosen independently (as the signs can be compensated by choosing the suitable sign for $\\lambda _2$ , multiplying by which fixes the zero coordinate $T_2$ anyway).", "Thus the set of such orbits is $(\\mathbb {C}^*/\\mu _2)^2\\simeq (\\mathbb {C}^*)^2$ , where the explicit isomorphism is given by squaring each coordinate.", "Then the action of the coordinate interchange involution $0\\leftrightarrow 1$ in these coordinates, as an action on $(\\mathbb {C}^*)^2$ , is simply the interchange of the two coordinates, i.e.", "the restriction to $(\\mathbb {C}^*)^2\\subseteq \\mathbb {C}^2$ of the usual action of $S_2$ interchanging the two coordinates.", "Under the bijection $\\mathbb {C}^2/S_2\\leftrightarrow \\mathbb {C}^2$ given explicitly by $(x_1,x_2)\\mapsto (x_1x_2,x_1+x_2)\\,,$ the image of $(\\mathbb {C}^*)^2$ is equal to $\\mathbb {C}^*\\times \\mathbb {C}$ , and thus $[(2)]=[\\mathbb {C}^*\\times \\mathbb {C}]$ .", "Finally, for orbits of type (3), each orbit has a representative of the form $(T_0:1:T_2:1:0:1)$ (we choose this form as it is preserved by the involution $0\\leftrightarrow 2$ ), and such a point lies on the same $\\mathbb {T}^2$ orbit as $(T^{\\prime }_0:1:T^{\\prime }_2:1:0:1)$ if and only if they are mapped to each other by the action of $\\operatorname{diag}(\\lambda _0,\\lambda _1,\\lambda _2)$ , which means we must have $\\lambda _0\\lambda _1\\lambda _2=1$ and $\\lambda _0^2=\\lambda _2^2=\\lambda _1^3=1$ .", "This is to say $\\lambda _0=\\lambda _2=\\sigma =\\pm 1$ and $\\lambda _1=1$ , and thus the set of such orbits is $(\\mathbb {C}^*)^2/\\mu _2$ , where the action is by multiplying both coordinates by $-1$ simultaneously.", "Similarly to orbits of type (2), this action of $\\mu _2$ commutes with the action of the coordinate interchange involution, and thus the contribution to the Grothendieck ring of varieties is $[(\\mathbb {C}^*)^2/\\mu _2\\times S_2]$ .", "To determine this class, we first identify, as above, $(\\mathbb {C}^*)^2/S_2\\simeq \\mathbb {C}^*\\times \\mathbb {C}$ by using the elementary symmetric functions.", "Then $\\mu _2$ action $(x_1,x_2)\\mapsto (-x_1,-x_2)$ acts on elementary symmetric polynomials via $(x_1x_2,x_1+x_2)\\mapsto (x_1x_2,-x_1-x_2)$ , and thus finally $[(3)]=[(\\mathbb {C}^*)^2/\\mu _2\\times S_2]=[\\mathbb {C}^*\\times (\\mathbb {C}/\\mu _2)]=[\\mathbb {C}^*\\times \\mathbb {C}]\\,$ where $\\mu _2$ acts on $\\mathbb {C}$ by sign, and the quotient is readily identified with $\\mathbb {C}$ by taking the square of the coordinate.", "Altogether, we thus compute $\\begin{aligned}[D_{3A_2}]&=[(1)]+[(2)]+[(3)]+3[\\mathbb {C}^*]+4[\\mathbb {C}^0] =[\\mathbb {C}^*\\times \\mathbb {C}^2]+2[\\mathbb {C}^*\\times \\mathbb {C}]+3[\\mathbb {C}^*]+4[\\mathbb {C}^0]\\\\&=[\\mathbb {C}^*]\\cdot \\left([\\mathbb {C}^*]^2+2[\\mathbb {C}^*]+[\\mathbb {C}^0]\\right)+2[\\mathbb {C}^*]\\cdot \\left([\\mathbb {C}^*]+[\\mathbb {C}^0]\\right)+3[\\mathbb {C}^*]+4[\\mathbb {C}^0]\\\\ &=[\\mathbb {C}^*]^3+4[\\mathbb {C}^*]^2+6[\\mathbb {C}^*]+4[\\mathbb {C}^0]\\,,\\end{aligned}$ which is equal to the class $[\\mathbb {P}^3]$ , as can be seen by decomposing $\\mathbb {P}^3$ in the usual toric way, as it is the toric variety associated to a tetrahedron, which has 1 highest-dimensional cell, 4 faces, 6 edges, and 4 vertices.", "One can interpret the computations above as giving a description of the exceptional divisor $D_{3A_2}$ as the quotient of a toric threefold by the action of $S_3$ .", "Rather than going into the geometry of this action in detail, we sketch an alternative direct toric approach.", "Using this alternative approach, we can in fact identify the polytope of this toric threefold and the action of $S_3$ explicitly.", "This gives us the added information that the toric threefold is simplicial, and provides an alternate proof of T:mainL (see R:D3A2class).", "We first note that by general theory the GIT quotient $\\mathbb {P}^5/\\!\\!/\\mathbb {T}^2$ is itself a toric variety, see [10].", "We now describe the polytope giving us this toric variety, and the $S_3$ action on it.", "Lemma 8.2 (Toric polytope) Let $P_a$ be the polytope in $\\mathbb {Z}^3\\otimes _\\mathbb {Z}\\mathbb {R}$ defined by the columns of the following matrix: $P_a \\longleftrightarrow \\left(\\begin{array}{rrrrrrrr}-\\frac{3}{8}&0&\\frac{3}{8}&0&\\frac{3}{7}&0&0&-\\frac{3}{7}\\\\1&\\frac{1}{7}&-\\frac{7}{8}&0&-\\frac{8}{7}&-\\frac{1}{8}&0&1\\\\2&\\frac{20}{7}&\\frac{11}{4}&2&\\frac{20}{7}&2&3&2\\\\\\end{array}\\right)\\ $ and consider the action of $S_3$ on $\\mathbb {Z}^3$ given by the transposition $\\tau $ and the 3-cycle $\\sigma $ : $\\tau =\\left(\\begin{array}{rrr}-1&0&0\\\\5&1&0\\\\-2&0&1\\\\\\end{array}\\right), \\ \\ \\sigma =\\left(\\begin{array}{rrr}-8&-3&0\\\\19&7&0\\\\-16&-6&1\\\\\\end{array}\\right).$ The polytope $P_a$ is a combinatorial cube, the associated toric variety $X_{P_a}$ is simplicial, and the quotient of $X_{P_a}$ by the induced action of $S_3$ is isomorphic to $D_{3A_2}$ ; i.e., $D_{3A_2}\\cong X_{P_a}/S_3\\,.$ Remark 8.3 Strictly speaking, to define $X_{P_a}$ we must first clear denominators to obtain a convex lattice polytope, but scaling the polytope does not affect the abstract variety, it only affects the polarization.", "From the Kirwan construction, and the Luna Slice Theorem, identifying the exceptional divisor in the blowup of the Luna slice at the origin with $\\mathbb {P}^5$ gives $D_{3A_2}\\cong \\mathbb {P}^5/\\!\\!/_{\\mathcal {O}_{\\mathbb {P}^5}(1)}\\operatorname{Stab}(S_{3A_2})$ (see L:3A2slice and L:R3A2Norm2).", "Note that in principle, from the Kirwan construction, one only has that the action of the stabilizer lifts to give a linearization on some positive tensor power of $\\mathcal {O}_{\\mathbb {P}^5}(1)$ , but the lift of the action to $\\mathcal {O}_{\\mathbb {P}^5}(1)$ is evident from the given explicit formula for the action, and of course which positive tensor power one uses will not change the GIT quotient.", "From the description of the stabilizer in L:R3A2Norm2, we can conclude that $D_{3A_2}\\cong (\\mathbb {P}^5/\\!\\!/_{\\mathcal {O}_{\\mathbb {P}^5}(1)} (\\mathbb {C}^*)^2)/S_3$ .", "In order to keep track of the $S_3$ action, we prefer to use the linearization on $\\mathcal {O}_{\\mathbb {P}^5}(3)$ .", "Our first goal therefore is to describe the toric variety $\\mathbb {P}^5/\\!\\!/_{\\mathcal {O}_{\\mathbb {P}^5}(3)} (\\mathbb {C}^*)^2$ .", "To start, we claim that the action of $(\\mathbb {C}^*)^2$ on $\\mathbb {P}^5$ is induced by the natural action on the Luna slice $\\mathbb {C}^6$ , so that fixing the inclusion of tori $\\mathbb {C}^*\\times (\\mathbb {C}^*)^2\\rightarrow (\\mathbb {C}^*)^6$ given by the matrix $\\gamma =\\left(\\begin{array}{rrrrrr}1&1&1&1&1&1\\\\-3&3&0&-2&2&0\\\\-3&0&3&-2&0&2\\\\\\end{array}\\right)\\,,$ and the character $\\chi :\\mathbb {C}^*\\times (\\mathbb {C}^*)^2\\rightarrow \\mathbb {C}^*$ defined by $\\chi (t,\\lambda _1,\\lambda _2)=t^3$ , one has an identification of GIT quotients $\\mathbb {C}^6/\\!\\!/_\\chi (\\mathbb {C}^*\\times (\\mathbb {C}^*)^2) = \\mathbb {P}^5/\\!\\!/_{\\mathcal {O}_{\\mathbb {P}^5}(3)} (\\mathbb {C}^*)^2$ (invariant sections of tensor powers of the trivial line bundle $\\mathbb {C}^6\\times \\mathbb {C}$ over $\\mathbb {C}^6$ with respect to the character $\\chi $ are canonically identified with the invariant sections of tensor powers of $\\mathcal {O}_{\\mathbb {P}^5}(3)$ ; see e.g., [10]).", "Using the technique in [10], one sees that $\\mathbb {C}^6/\\!\\!/_\\chi (\\mathbb {C}^*\\times (\\mathbb {C}^*)^2)$ is the toric variety associated to a 3-dimensional polytope obtained in the following way.", "One defines a lattice $M$ by the exact sequence ${0[r]& M[r]& \\mathbb {Z}^6 [r]^\\gamma & \\mathbb {Z}^3\\,.", "}$ Then considering the composition $\\mathbb {C}^*\\times (\\mathbb {C}^*)^2\\rightarrow (\\mathbb {C}^*)^6 \\rightarrow \\mathbb {C}^*$ , where the first map is the inclusion of tori determined by $\\gamma $ , and the second map of tori is determined by the matrix $a=\\left(\\begin{array}{rrrrrr}-2&-2&-2&3&3&3\\\\\\end{array}\\right),$ one sees that the composition is the character $\\chi $ given above.", "From [10], one sees that the quotient $\\mathbb {C}^6/\\!\\!/_\\chi (\\mathbb {C}^*\\times (\\mathbb {C}^*)^2)$ is the toric variety associated to the lattice $M$ and the polytope $P_a\\lbrace m\\in M\\otimes _{\\mathbb {Z}}\\mathbb {R}: e_i(m)\\ge -a_i, \\ i=1,\\dots ,6 \\rbrace \\,,$ where $e_i$ is the standard dual coordinate.", "The $S_3$ action on $\\mathbb {C}^6$ (see L:3A2slice) induces the $S_3$ action on $(\\mathbb {C}^*)^6$ , and therefore on $\\mathbb {Z}^6$ , and in turn determines an $S_3$ action on $M$ .", "The columns of the following matrix give an integral basis of $M$ : $\\left(\\begin{array}{rrr}0&0&1\\\\-2&0&1\\\\-16& -6&1\\\\-3&-1&-1\\\\0&-1&-1\\\\21&8&-1\\end{array}\\right)$ and therefore, identifying $M$ with $\\mathbb {Z}^3$ using this basis, we may identify $P_a$ as the set of $(a,b,c)\\in \\mathbb {R}^3$ such that $\\begin{array}{rcrcrcr}&&&&c&\\ge &2\\\\-2a&&&+&c&\\ge &2\\\\-16a&-&6b&+&c&\\ge &2\\\\-3a&-&b&-&c&\\ge &-3\\\\&&-b&-&c&\\ge &-3\\\\21a&+&8b&-&c&\\ge &-3\\\\\\end{array}$ From this one can identify $P_a$ with the convex hull of eight vertices, given by the columns of the matrix given in (REF ).We thank Mathieu Dutour Sikirić for computing this for us.", "The polytope $P_a$ is combinatorially a cube; for instance the first four columns and last four columns give “top” and “bottom” faces of the combinatorial cube, respectively, and the vectors with last coordinate equal to 2 give a “side” face of the cube.", "Considering the dictionary between polytopes and fans (e.g., [10]), it is elementary to check that the toric variety associated to $P_a$ , being a combinatorial cube, is simplicial.", "Finally, in these coordinates, the $S_3$ action on $M$ , identified with $\\mathbb {Z}^3$ , is given by the transposition $\\tau $ and 3-cycle $\\sigma $ given in (REF ).", "Remark 8.4 (Class in the Grothendieck ring) In the terminology of L:D3A2toric, we note that the vectors $(0,1,-1)$ , $(3,-8,-1)$ , $(-3,7,-7)$ define a rank 3, index 27 sublattice of $\\mathbb {Z}^3$ on which $S_3$ acts via $\\tau $ and $\\sigma $ by the standard permutation of vectors.", "Moreover, one can check directly that the action of $S_3$ on the vertices of the combinatorial cube defining $P_a$ corresponds to the standard action of $S_3$ on a cube, fixing two antipodal vertices, and in particular acts by toric automorphisms.", "In this situation, it is the antipodal vertices $(0,0,3)$ and $(0,0,2)$ of $P_a$ that are fixed.", "At the same time, taking the basis $(0,0,1)$ , $(-1,3,0)$ , $(1,-2,2)$ for $\\mathbb {Z}^3$ , we have that the action of $S_3$ in these coordinates is given by the matrices $\\tau =\\left(\\begin{array}{ccc}0&1&0\\\\1&0&0\\\\0&0&1\\\\\\end{array}\\right), \\ \\ \\sigma =\\left(\\begin{array}{rrr}0&1&0\\\\-1&-1&0\\\\0&0&1\\\\\\end{array}\\right).$ From this, using the dictionary between faces of polytopes and torus orbits, one can easily work out the action of $S_3$ on all the torus orbits of $X_{P_a}$ , as well as their quotients, and consequently, re-derive the class of $X_{P_a}/S_3\\cong D_{3A_2}$ in the Grothendieck ring.", "For instance, it is immediate from the action of the matrices above that the quotient of the maximal torus $(\\mathbb {C}^*)^3/S_3$ is $\\mathbb {C}^2\\times \\mathbb {C}^*$ , and similarly, that the two-dimensional tori contribute $(\\mathbb {C}^*)^2/S_2$ with quotient $\\mathbb {C}\\times \\mathbb {C}^*$ , and that the one-dimensional tori contribute quotients of $\\mathbb {C}^*$ by subgroups, each giving $\\mathbb {C}^*$ .", "Adding up all of the contributions in the Grothendieck ring gives the same class as $\\mathbb {P}^3$ .", "This gives an alternate proof of L:D3A2class, and therefore of T:mainL.", "It turns out that one can use T:mainL and L:D3A2toric to recover the fact that the Betti numbers of ${\\mathcal {M}}^{\\operatorname{K}}$ and ${\\overline{\\mathcal {B}_4/\\Gamma }}$ are equal.", "Of course, the Betti numbers of ${\\mathcal {M}}^{\\operatorname{K}}$ were already computed in [28] and [40] (see also [11] to reconcile the numbers in those two papers), and the Betti numbers of ${\\overline{\\mathcal {B}_4/\\Gamma }}$ were computed in our paper [11], but the proof of the following corollary helps to give a more intuitive reason for the agreement of the Betti numbers.", "Corollary 8.5 The Kirwan compactification ${\\mathcal {M}}^{\\operatorname{K}}$ and the toroidal compactification ${\\overline{\\mathcal {B}_4/\\Gamma }}$ of the moduli space of smooth cubic surfaces $\\mathcal {M}$ have the same Betti numbers.", "First, we claim that ${\\mathcal {M}}^{\\operatorname{K}}$ and ${\\overline{\\mathcal {B}_4/\\Gamma }}$ have no odd cohomology.", "This follows from the fact that ${\\mathcal {M}}^{\\operatorname{GIT}}$ has no odd cohomology, together with the decomposition theorem and the fact that the exceptional divisors for $\\pi :{\\mathcal {M}}^{\\operatorname{K}}\\rightarrow {\\mathcal {M}}^{\\operatorname{GIT}}$ and $p:{\\overline{\\mathcal {B}_4/\\Gamma }}\\rightarrow {\\mathcal {M}}^{\\operatorname{GIT}}$ are quotients of simplicial toric varieties by finite groups [10] (for ${\\mathcal {M}}^{\\operatorname{K}}$ L:D3A2toric shows that the exceptional divisor $D_{3A_2}$ is a finite quotient of a simplicial toric variety, while for ${\\overline{\\mathcal {B}_4/\\Gamma }}$ the exceptional divisor $T_{3A_2}$ is simply equal to $\\mathbb {P}^3$ , by C:T3A2).", "The decomposition theorem also gives $\\dim H^2({\\mathcal {M}}^{\\operatorname{K}})=\\dim H^2({\\mathcal {M}}^{\\operatorname{GIT}})+1= \\dim H^2({\\overline{\\mathcal {B}_4/\\Gamma }})$ .", "Finally, it remains to determine the cohomology in the middle degree 4, and the agreement $\\dim H^4({\\mathcal {M}}^{\\operatorname{K}})=\\dim H^4({\\overline{\\mathcal {B}_4/\\Gamma }})$ then follows from the fact that the topological Euler characteristic for cohomology with compact supports is well-defined on the Grothendieck ring.", "Remark 8.6 Note that this also gives a short method of computing the rational cohomology of ${\\overline{\\mathcal {B}_4/\\Gamma }}$ and ${\\mathcal {M}}^{\\operatorname{K}}$ .", "Indeed, ${\\mathcal {M}}^{\\operatorname{GIT}}$ is a weighted projective space, and so has the rational cohomology of $\\mathbb {P}^4$ .", "From the decomposition theorem, and the fact that ${\\overline{\\mathcal {B}_4/\\Gamma }}\\rightarrow {\\mathcal {M}}^{\\operatorname{GIT}}$ has exceptional divisor equal to $\\mathbb {P}^3$ , it follows that $\\dim H^0({\\overline{\\mathcal {B}_4/\\Gamma }})=\\dim H^8({\\overline{\\mathcal {B}_4/\\Gamma }})= 1$ , $\\dim H^2({\\overline{\\mathcal {B}_4/\\Gamma }})=\\dim H^6({\\overline{\\mathcal {B}_4/\\Gamma }}) = 2$ , and $\\dim H^4({\\overline{\\mathcal {B}_4/\\Gamma }})=2$ .", "This determines the cohomology of ${\\mathcal {M}}^{\\operatorname{K}}$ via C:BettiEq." ], [ "Luna slice computations for the GIT model for cubic surfaces", "In this Appendix, we give the detailed proofs of some of the statements from sec:calM.", "While the results of these computations are crucial for our argument, the method is by explicit computations in local charts on the exceptional divisors, and we have put the calculations here in order not to interrupt the line of thought of our arguments in sec:calM." ], [ "Proof of L:R3A2Norm2", "While this proof is parallel to the case of the $3D_4$ cubic threefold, which was treated in [11], we still give the complete details, as the careful identification of the finite groups involved is essential in our argument.", "We first determine the stabilizer group $\\operatorname{GL}(S_{3A_2})\\subseteq \\operatorname{GL}(4,\\mathbb {C})$ .", "To begin, it is clear that the group $\\left\\lbrace \\left(\\begin{array}{c|c}\\mathbb {S}_3&\\\\ \\hline &\\mathbb {C}^*\\\\\\end{array}\\right): \\lambda _0\\lambda _1\\lambda _2=\\lambda _3^3\\right\\rbrace \\subseteq \\operatorname{GL}(4,\\mathbb {C})$ stabilizes $S_{3A_2}$ .", "We wish to show that the stabilizer is equal to this group.", "For this, we observe that any symmetry must permute the 3 singularities of the cubic $S_{3A_2}$ , i.e., the points $(1:0:0:0)$ , $(0:1:0:0)$ and $(0:0:1:0)$ .", "This forces a matrix stabilizing $S_{3A_2}$ to be of the form $\\left(\\begin{array}{c|c}\\mathbb {S}_3&*\\\\ \\hline 0&\\lambda _3\\\\\\end{array}\\right)\\,.$ Such a transformation sends the monomial $x_0x_1x_2$ to $(\\lambda _0 x_0+*x_3)\\cdot (\\lambda _1 x_1+*x_3)\\cdot (\\lambda _2 x_2+*x_3)$ , where all the $\\lambda $ 's are non-zero, and $*$ are the entries of the unknown $1\\times 3$ block of the matrix.", "Furthermore, $x_3$ is sent to $\\lambda _3 x_3$ .", "Thus all entries $*$ must be equal to zero, or otherwise applying this transformation to $S_{3A_2}$ would give a cubic with non-zero coefficient of some monomial $x_ax_b x_3$ with $0\\le a<b\\le 2$ .", "Thus we have deduced that the matrix stabilizing $S_{3A_2}$ must actually be of the form $\\left(\\begin{array}{c|c}\\mathbb {S}_3&0\\\\ \\hline 0&\\lambda _3\\\\\\end{array}\\right)\\,.$ Finally it is obvious that any element of the stabilizer satisfies the condition $\\lambda _0\\lambda _1\\lambda _2=\\lambda _3^3$ .", "This completes the proof that the stabilizer group $\\operatorname{GL}(S_{3A_2}) \\subseteq \\operatorname{GL}(4,\\mathbb {C})$ is as claimed.", "The description of $\\operatorname{Stab}(S_{3A_2}) \\subseteq \\operatorname{SL}(4,\\mathbb {C})$ follows immediately.", "(2) This is immediate since $\\operatorname{Aut}(S_{3A_2})$ is naturally a subgroup of $\\operatorname{PGL}(4,\\mathbb {C})$ .", "(3) We now want to describe the structure of the stabilizer group $\\operatorname{GL}(S_{3A_2})\\subseteq \\operatorname{GL}(4,\\mathbb {C})$ more precisely.", "There is clearly a short exact sequence $1\\rightarrow D\\rightarrow \\operatorname{GL}(S_{3A_2})\\rightarrow S_3\\rightarrow 1\\,,$ where $D$ is the subgroup of diagonal matrices in $\\operatorname{GL}(S_{3A_2})$ , and the map to $S_3$ is the one taking a generalized permutation matrix to the associated permutation.", "There is an obvious section $S_3\\rightarrow \\operatorname{GL}(S_{3A_2})$ , viewing $S_3$ as block diagonal permutation matrices.", "This gives the identification $\\operatorname{GL}(S_{3A_2})\\cong D\\rtimes S_3\\,,$ where the action of $S_3$ on $D$ is to permute the first three entries.", "The surjection of $\\operatorname{Stab}(S_{3A_2})$ onto $S_3$ can be seen by the matrices $\\left(\\begin{array}{cccc}0&1&0&0\\\\1& 0&0&0\\\\0&0&\\zeta _8^{3}&0\\\\0&0&0&\\zeta _8\\end{array}\\right) \\qquad \\left(\\begin{array}{cccc}0& 0&1&0\\\\1& 0&0&0\\\\0&1&0&0\\\\0&0&0&1\\end{array}\\right)\\,,$ where $\\zeta _8$ is a primitive 8-th root of unity.", "(4) The identification $\\operatorname{GL}(S_{3A_2})^\\circ = D$ comes from (REF ) and the identification $D\\cong \\mathbb {T}^3$ .", "We then obtain the isomorphism $\\operatorname{GL}(S_{3A_2})/\\operatorname{GL}(S_{3A_2})^\\circ \\cong S_3$ from (REF ), as well.", "The isomorphism $\\operatorname{Aut}(S_{3A_2})/\\operatorname{Aut}(S_{3A_2})^\\circ \\cong S_3$ then comes from (REF ).", "Indeed, taking connected components of the identity we have ${1 [r]& \\mathbb {C}^* [r] @{=}[d]& \\operatorname{GL}(S_{3A_2})^\\circ @{^(->}[d] [r] & \\operatorname{Aut}(S_{3A_2})^\\circ [r] @{^(->}[d]& 1\\\\1 [r]& \\mathbb {C}^* [r]& \\operatorname{GL}(S_{3A_2}) [r] & \\operatorname{Aut}(S_{3A_2})[r]& 1\\\\}$ The surjection on the right in the top row is standard (say coming from looking at the dimensions of the connected components), and the identification of the kernels of the two rows is elementary in this case since $\\mathbb {C}^*$ is connected.", "Then one applies the Snake Lemma.", "The short exact sequence (REF ) follows from (REF ) and the description $D^{\\prime }\\cong \\mathbb {T}^2\\times \\mu _4$ .", "$\\Box $" ], [ "Proof of L:Eck-no-3A2", "The Luna Slice Theorem implies that we can compute in the Luna slice.", "More precisely, we mean the following.", "First, in order to have shorter, more standard notation (e.g., to match the discussion of the Luna Slice Theorem in [32]), let us set $X\\mathbb {P}^{19}$ , $G\\operatorname{SL}(4,\\mathbb {C})$ , $x\\in X$ the point corresponding to the $3A_2$ cubic $S_{3A_2}$ , $G_x\\operatorname{Stab}(S_{3A_2})$ , and $W\\subseteq X$ the Luna slice.", "Then there is an open affine neighborhood $U\\subseteq X$ of $x$ such that $U$ is étale equivalent to $(G\\times W)/G_x$ , and since $G\\times W\\rightarrow (G\\times W)/G_x$ is a principal $G_x$ -bundle (e.g., [32]), it follows that, up to a smooth factor, $U$ is étale equivalent to $G\\times W$ .", "In other words, we have a diagram ${G\\times W [d]_{\\text{$G_x$-bundle}} [rd]^{\\text{smooth}}&\\\\(G\\times W)/G_x [r]^<>(0.5){\\text{ét}}& U\\subseteq X}$ Since the Eckardt divisor $R\\subseteq X$ is an irreducible effective divisor, preserved by the action of $G$ , it is elementary to check that $R|_U$ is the image of $G\\times (R\\cap W)$ under the map $G\\times W\\rightarrow U$ , where we take the reduced induced scheme structure on $R\\cap W$ .", "In other words, up to a smooth factor, $R|_U$ is étale equivalent to $G\\times (R\\cap W)$ .", "Then, since $G\\times (R\\cap W)$ is, up to a smooth factor, étale equivalent to $R\\cap W$ , we conclude that $\\mu $ is the multiplicity of $R\\cap W$ at the origin.", "Returning now to the Luna slice in our example, we know that the Eckardt divisor is the divisorial locus containing all smooth cubic surfaces with extra automorphisms, so it suffices to describe the locus of cubic surfaces in the Luna slice that have extra automorphisms.", "Translating into the action of $\\operatorname{SL}(4,\\mathbb {C})$ , it suffices to describe the locus of cubic surfaces in the Luna slice with stabilizer group strictly containing the diagonal $\\mu _4$ .", "Since we are only interested in this locus in a neighborhood of the $3A_2$ cubic surface, we can use the fact from the Luna Slice Theorem that in a neighborhood of the origin in the Luna slice, the stabilizer groups must be subgroups of $\\operatorname{Stab}(S_{3A_2})$ .", "Since the general point of the Eckardt divisor parameterizes cubic surfaces with a $\\mathbb {Z}_2$ automorphism group, we will first describe all subgroups of $\\operatorname{Stab}(S_{3A_2})$ whose image in $\\operatorname{PGL}(4,\\mathbb {C})$ is of order 2.", "Given such a subgroup, there is a matrix $A$ in the group such that the image of $A$ in $\\operatorname{PGL}(4,\\mathbb {C})$ has order 2.", "Let us classify these matrices.", "First, considering the sequence $0\\rightarrow \\mu _4\\rightarrow \\operatorname{Stab}(S_{3A_2})\\rightarrow \\operatorname{PGL}(4,\\mathbb {C})\\,,$ we have $\\langle A\\rangle \\cap \\mu _4$ is one of $\\lbrace Id\\rbrace $ , $\\lbrace Id,-Id\\rbrace $ , or $\\mu _4$ , so that $|A|= 2,4,8\\,.$ Then, considering the diagram ${&\\langle A\\rangle @{^(->}[r]&\\operatorname{Stab}(S_{3A_2}) @{^(->}[d] [rd]&&\\\\1[r]&D [r]&\\operatorname{GL}(S_{3A_2})[r]&S_3[r]&1}$ we see that $A$ must map to an element of $S_3$ of order a power of 2; i.e., either the identity or a transposition.", "Thus there are two cases.", "We have $\\text{Case I: }\\ \\ \\ \\ A=\\left(\\begin{array}{cccc}\\lambda _0& 0&0&0\\\\0&\\lambda _1&0&0\\\\0&0&\\lambda _2&0\\\\0&0&0&\\lambda _3\\end{array}\\right)\\ \\ \\ \\ \\lambda _0\\lambda _1\\lambda _2=\\lambda _3^3, \\ \\ \\lambda _0\\lambda _1\\lambda _2\\lambda _3=1$ and, up to changing indices for the transposition, $\\text{Case II: }\\ \\ \\ \\ A=\\left(\\begin{array}{cccc}0&\\lambda _1&0&0\\\\\\lambda _0& 0&0&0\\\\0&0&\\lambda _2&0\\\\0&0&0&\\lambda _3\\end{array}\\right)\\ \\ \\ \\ \\lambda _0\\lambda _1\\lambda _2=\\lambda _3^3, \\ \\ \\lambda _0\\lambda _1\\lambda _2\\lambda _3=-1\\,.$ We will first consider Case I.", "In this case, combining the two equations for the $\\lambda _i$ , we see that $\\lambda _3^4=1$ .", "Now let us further subdivide Case I by the order of $A$ ; we will denote by $\\zeta _n$ a primitive $n$ -th root of unity.", "If $|A|=2$ , we have (recalling that we are always assuming that $A$ generates a subgroup of $\\operatorname{PGL}(4,\\mathbb {C})$ of order 2) $\\begin{aligned}&|A|=2 \\Rightarrow A=\\left(\\begin{array}{cccc}\\pm 1& 0&0&0\\\\0&\\pm 1&0&0\\\\0&0&\\pm 1&0\\\\0&0&0&\\pm 1\\end{array}\\right)&\\text{two entries $1$ and two entries $-1$}\\\\&|A|=4 \\Rightarrow A=\\left(\\begin{array}{cccc}i^{i_0}& 0&0&0\\\\0&i^{i_1}&0&0\\\\0&0&i^{3i_3-i_0-i_1}&0\\\\0&0&0&i^{i_3}\\end{array}\\right) &\\text{not all entries $i$ or all $-i$, not all indices even}\\\\&|A|=8 \\Rightarrow A=\\left(\\begin{array}{cccc}\\zeta _8^{i_0}& 0&0&0\\\\0&\\zeta _8^{i_1}&0&0\\\\0&0&\\zeta _8^{6i_3-i_0-i_1}&0\\\\0&0&0&\\zeta _8^{2i_3}\\end{array}\\right)&\\text{$i_0$ and $i_1$ not both even}\\end{aligned}$ In Case II, combining the two equations for the $\\lambda _i$ , we see that $\\lambda _3^4=-1$ , so that $\\lambda _3$ must be a primitive 8-th root of unity.", "Consequently, we can rule out the cases $|A|=2$ and $|A|=4$ , since then $\\lambda _3^4=1$ .", "Thus we must have $|A|=8$ , and we start by observing $\\left(\\begin{array}{cccc}0&\\lambda _1&0&0\\\\\\lambda _0& 0&0&0\\\\0&0&\\lambda _2&0\\\\0&0&0&\\lambda _3\\end{array}\\right)\\cdot \\left(\\begin{array}{cccc}0&\\lambda _1&0&0\\\\\\lambda _0& 0&0&0\\\\0&0&\\lambda _2&0\\\\0&0&0&\\lambda _3\\end{array}\\right)=\\left(\\begin{array}{cccc}\\lambda _0\\lambda _1& 0&0&0\\\\0&\\lambda _0\\lambda _1&0&0\\\\0&0&\\lambda _2^2&0\\\\0&0&0&\\lambda _3^2\\end{array}\\right)\\,.$ We know that $\\lambda _3=\\zeta _8$ is a primitive 8-th root of unity.", "The above shows that $\\lambda _0\\lambda _1$ is a 4-th root of unity, so we must have $\\lambda _0\\lambda _1=\\zeta _8^{2n}$ for some $n$ .", "Combining this with $\\lambda _0\\lambda _1\\lambda _2=\\lambda _3^3$ , we see that $A$ must be of the form $A=\\left(\\begin{array}{cccc}0&\\lambda _0&0&0\\\\\\zeta _8^{2n}/\\lambda _0& 0&0&0\\\\0&0&\\zeta _8^{3-2n}&0\\\\0&0&0&\\zeta _8\\end{array}\\right)\\,.$ Concretely, we have four options (recalling that we are always assuming that $A$ generates a subgroup of $\\operatorname{PGL}(4,\\mathbb {C})$ of order 2): $A=\\left(\\begin{array}{cccc}0&\\lambda _0&0&0\\\\1/\\lambda _0& 0&0&0\\\\0&0&\\zeta _8^{3}&0\\\\0&0&0&\\zeta _8\\end{array}\\right)$ $A=\\left(\\begin{array}{cccc}0&\\lambda _0&0&0\\\\\\zeta _8^2/\\lambda _0& 0&0&0\\\\0&0&\\zeta _8&0\\\\0&0&0&\\zeta _8\\end{array}\\right)$ $A=\\left(\\begin{array}{cccc}0&\\lambda _0&0&0\\\\\\zeta _8^4/\\lambda _0& 0&0&0\\\\0&0&\\zeta _8^{-1}&0\\\\0&0&0&\\zeta _8\\end{array}\\right)$ $A=\\left(\\begin{array}{cccc}0&\\lambda _0&0&0\\\\\\zeta _8^6/\\lambda _0& 0&0&0\\\\0&0&\\zeta _8^{-3}&0\\\\0&0&0&\\zeta _8\\end{array}\\right)$ We now consider when there can be a divisor with generic point fixed by any family of the matrices above.", "In Case I, the matrices $A$ form discrete families and so we must just show that each such $A$ has fixed locus of codimension 2 or more.", "This is a case by case analysis.", "We recall that the action is given by (REF ).", "For the case $|A|=2$ , the last three entries are preserved, and because exactly two of the $\\lambda _i/\\lambda _3$ are equal to $-1$ , we see that the fixed locus is the intersection of the two coordinate hyperplanes given by those $\\alpha _i=0$ , thus of codimension at least two.", "For the case $|A|=4$ , we have a matrix $A=\\left(\\begin{array}{cccc}i^{i_0}& 0&0&0\\\\0&i^{i_1}&0&0\\\\0&0&i^{3i_3-i_0-i_1}&0\\\\0&0&0&i^{i_3}\\end{array}\\right)\\,,$ where exactly two of the entries are real, and two are imaginary.", "Again, looking at the action, we see that the fixed locus is contained in intersections of more than two coordinate hyperplanes and we are done.", "The case $|A|=8$ is similar.", "Indeed, suppose without loss of generality that $i_0$ is odd.", "Then $2i_0\\lnot \\equiv 0\\mod {4}$ , and $3i_0\\lnot \\equiv 0 \\mod {2}$ , and thus since $\\lambda _3$ is a power of the fourth (not eighths) root of unity, both $(\\lambda _0/\\lambda _3)^2$ and $(\\lambda _0/\\lambda _3)^3$ are not equal to identity, so this means the fixed locus must have $\\alpha _0=\\alpha _{\\widehat{0}}=0$ .", "In other words, there are no divisors in the Luna slice that have a general point fixed by a matrix $A$ in the form of Case I.", "We now consider Case II.", "Recall that in this case the action (similar to (REF )) is given by $A\\cdot (\\alpha _0,\\alpha _1,\\alpha _2,\\alpha _{\\widehat{0}},\\alpha _{\\widehat{1}},\\alpha _{\\widehat{2}})=\\left(\\left(\\frac{\\lambda _1}{\\lambda _3}\\right)^3\\alpha _1,\\left(\\frac{\\lambda _0}{\\lambda _3}\\right)^3\\alpha _0, \\left(\\frac{\\lambda _2}{\\lambda _3}\\right)^3\\alpha _2,\\left(\\frac{\\lambda _1}{\\lambda _3}\\right)^2\\alpha _{\\widehat{1}}, \\left(\\frac{\\lambda _0}{\\lambda _3}\\right)^2\\alpha _{\\widehat{0}}, \\left(\\frac{\\lambda _2}{\\lambda _3}\\right)^2\\alpha _{\\widehat{2}}\\right)\\,.$ Now assume that we have a general point $(\\alpha _0,\\dots ,\\alpha _{\\widehat{2}})$ of some divisor, fixed by a matrix $A$ in one of the four subcases of Case II.", "We see that $\\alpha _0=0$ if and only if $\\alpha _1=0$ .", "Since this is codimension 2 (and could not sweep out a divisor while moving $\\lambda _0$ ), we can assume that $\\alpha _0$ and $\\alpha _1$ are nonzero.", "Then looking in the first coordinate we have $\\alpha _0/\\alpha _1=(\\lambda _1/\\lambda _3)^3$ , and in the second coordinate, $\\alpha _1/\\alpha _0=(\\lambda _0/\\lambda _3)^3$ .", "Multiplying gives us $\\left(\\frac{\\lambda _1}{\\lambda _3}\\cdot \\frac{\\lambda _0}{\\lambda _3}\\right)^3=1$ , so that $(\\lambda _0\\lambda _1)^3=\\lambda _3^6$ .", "Looking at the list of cases, Case II(i)–(iv), the only option is Case II(ii).", "Now focusing on Case II(ii), we see that $\\lambda _2/\\lambda _3=1$ .", "Note that one also sees that $\\alpha _{\\widehat{0}}/\\alpha _{\\widehat{1}}=(\\lambda _1/\\lambda _3)^2$ , and $\\alpha _{\\widehat{1}}/\\alpha _{\\widehat{0}}=(\\lambda _0/\\lambda _3)^2$ , which would imply that $(\\lambda _0\\lambda _1)^2=\\lambda _3^4$ , which also holds for Case II(ii).", "In other words, if we have a general point $(\\alpha _0,\\dots ,\\alpha _{\\widehat{2}})$ of some divisor that is fixed by a matrix $A$ in the form of Case II, then we must be in Case II(ii), and we must have $\\alpha _0/\\alpha _1=(\\lambda _1/\\lambda _3)^3, \\ \\ \\alpha _1/\\alpha _0=(\\lambda _0/\\lambda _3)^3, \\text{ and }\\alpha _{\\widehat{0}}/\\alpha _{\\widehat{1}}=(\\lambda _1/\\lambda _3)^2, \\ \\ \\alpha _{\\widehat{1}}/\\alpha _{\\widehat{0}}=(\\lambda _0/\\lambda _3)^2\\,.$ Since in Case II(ii) we have $(\\lambda _1/\\lambda _3)=(\\lambda _0/\\lambda _3)^{-1}$ , we see that the above only constitutes two conditions, and cuts out a codimension 2 locus.", "As we vary $\\lambda _0$ , this indeed sweeps out a divisor.", "More precisely, consider the following divisor: $\\lbrace \\alpha _1^2\\alpha _{\\widehat{0}}^3 - \\alpha _0^2\\alpha _{\\widehat{1}}^3=0\\rbrace \\,,$ i.e., $\\left(\\frac{\\alpha _1}{\\alpha _0}\\right)^2 = \\left(\\frac{\\alpha _{\\widehat{1}}}{\\alpha _{\\widehat{0}}}\\right)^3$ .", "Then the general point $(\\alpha _0,\\alpha _1,\\alpha _2,\\alpha _{\\widehat{0}}, \\alpha _{\\widehat{1}}, \\alpha _{\\widehat{2}})$ of this divisor is stabilized by a matrix of the form $A_{\\lambda _0}=\\left(\\begin{array}{cccc}0 & \\lambda _0& 0& 0\\\\\\frac{\\zeta _8^2}{\\lambda _0}& 0 & 0& 0\\\\0&0&\\zeta _8 & 0\\\\0&0&0&\\zeta _8\\end{array}\\right)\\in \\operatorname{Stab}(S_{3A_2})\\,,$ where $\\zeta _8$ is a primitive 8-th root of unity, and $\\lambda _0$ is chosen so that $(\\frac{\\lambda _0}{\\zeta _8})^2= \\frac{\\alpha _{\\widehat{1}}}{\\alpha _{\\widehat{0}}}$ .", "There are two choices of $\\lambda _0$ , and one is also free to choose another primitive 8-th root of unity, but these matrices generate the same subgroup of $\\operatorname{Stab}(S_{3A_2})$ ; note that since the cyclic group $\\langle A_{\\lambda _0}\\rangle $ contains the diagonal $\\mu _4$ , the image of $\\langle A_{\\lambda _0}\\rangle $ in $\\operatorname{PGL}(4,\\mathbb {C})$ is isomorphic to $\\mathbb {Z}_2$ .", "In other words, the generic point of the divisor above corresponds to a cubic surface with automorphism group isomorphic to $\\mathbb {Z}_2$ .", "Permuting the indices, this gives 3 divisors of the same type.", "Each has degree 5, and is a cone through the origin, and so has multiplicity 5 at the origin.", "There are no other divisors in a neighborhood of 0 in the Luna slice parameterizing cubic surfaces with extra automorphisms.", "This implies that the multiplicity of the restriction of the Eckardt divisor to the Luna slice, as a reduced variety, is 15.", "This completes the proof.", "$\\Box $ Some of the computations below will use related notation to the above.", "Specifically, we will perform computations in charts on the blowup $\\operatorname{Bl}_0\\mathbb {C}^6\\rightarrow \\mathbb {C}^6$ of a Luna slice as above.", "Recall that the blowup is embedded in $\\mathbb {C}^6\\times \\mathbb {P}^5$ , with coordinates $\\alpha $ on the $\\mathbb {C}^6$ and homogeneous coordinates $T$ on the $\\mathbb {P}^5$ .", "By using the $S_3$ action on $\\mathbb {C}^6$ and its extension to the blowup, it will be enough to work only with two charts on $\\operatorname{Bl}_0\\mathbb {C}^6$ .", "The first chart is the chart $U_0$ where $T_0\\ne 0$ , and the local coordinates in this chart are then $(\\alpha _0,t_1,t_2,t_{\\widehat{0}},t_{\\widehat{1}},t_{\\widehat{2}})\\quad \\hbox{where}\\quad t_i=T_i/T_0,\\, t_{\\widehat{i}}=T_{\\widehat{i}}/T_0,\\, \\alpha _i=\\alpha _0t_i,\\,\\alpha _{\\widehat{i}}=\\alpha _0t_{\\widehat{i}}\\,,$ and similarly for the chart $U_{\\widehat{0}}$ where $T_{\\widehat{0}}\\ne 0$ ." ], [ "Proof of P:stab–ex", "Let $x\\in D_{3A_2}\\subseteq {\\mathcal {M}}^{\\operatorname{K}}$ be a point in the exceptional divisor.", "We want to compute the stabilizer $S_x\\subseteq \\operatorname{Stab}(S_{3A_2})\\subseteq \\operatorname{SL}(4,\\mathbb {C})$ ; i.e., the stabilizer of a point in the exceptional divisor of $\\operatorname{Bl}_{\\operatorname{SL}(4,\\mathbb {C}) \\cdot [S_{3A_2}]}(\\mathbb {P}^{19})^{ss}$ with orbit corresponding to $x$ .", "By construction, it suffices to compute the stabilizers of points in the exceptional divisor of the blowup of the Luna slice, with respect to the action of $\\operatorname{Stab}(S_{3A_2})$ .", "Indeed, the $\\operatorname{SL}(4,\\mathbb {C})$ orbit of each point in the exceptional divisor of $\\operatorname{Bl}_{\\operatorname{SL}(4,\\mathbb {C}) \\cdot [S_{3A_2}]}(\\mathbb {P}^{19})^{ss}$ intersects the blowup of the Luna slice, and since the blowups are equivariant, one can check that the stabilizer for the $\\operatorname{SL}(4,\\mathbb {C})$ action on $\\operatorname{Bl}_{\\operatorname{SL}(4,\\mathbb {C}) \\cdot [S_{3A_2}]}(\\mathbb {P}^{19})^{ss}$ is just the stabilizer for a corresponding point on the Luna slice, with respect to the action of $\\operatorname{Stab}(S_{3A_2})$ .", "Therefore, we work with the 6-dimensional Luna slice from L:3A2slice.", "In order to obtain the Kirwan blowup we have to blow up the Luna slice at the origin, and we denote the exceptional $\\mathbb {P}^5$ by $E$ .", "The group $\\operatorname{Stab}(S_{3A_2})$ acts on the blowup and we have to analyze the stabilizers of the action of $\\operatorname{Stab}(S_{3A_2})$ on the semi-stable locus $E^{ss}$ .", "For this we recall that the connected component of $\\operatorname{Stab}(S_{3A_2})$ is the torus given by $\\operatorname{diag}(\\lambda _1, \\lambda _2, \\lambda _3,1)$ with $\\lambda _1 \\lambda _2 \\lambda _3=1$ .", "From L:R3A2Norm2, we have an exact sequence $1 \\rightarrow \\mathbb {T}^2 \\rightarrow \\operatorname{Stab}(S_{3A_2}) \\rightarrow G(3A_2) \\rightarrow 1\\,,$ where $G(3A_2)$ is a finite group which is an extension of the form $1 \\rightarrow \\mu _4 \\rightarrow G(3A_2) \\rightarrow S_3 \\rightarrow 1\\,,$ where $S_3$ acts by permuting the coordinates $x_0,x_1,x_2$ .", "We shall now analyze the stabilizers of the action of $\\operatorname{Stab}(S_{3A_2})$ on $E^{ss}$ .", "(1) This claim follows from L:Eck-no-3A2.", "Indeed, $E$ is the projectivization of the Luna slice, and in L:Eck-no-3A2 it is shown that the general point of the Luna slice has stabilizer $\\mu _4$ (more precisely, it is trivial that every point of the Luna slice has stabilizer containing the diagonal $\\mu _4$ , and in L:Eck-no-3A2 it is shown that there is a neighborhood of the origin so that the points with stabilizer group strictly containing $\\mu _4$ form a divisor in this neighborhood).", "(2) Since the order of the group $G(3A_2)$ is $2^3\\cdot 3$ , which is not divisible by 5, it is enough to analyze the stabilizers of the connected component $D^{\\prime \\prime }\\cong \\mathbb {T}^2$ acting on $E$ , with the action as given in (REF ).", "There the action is on the Luna slice, but of course this gives the action on the projectivized Luna slice.", "Since it will be convenient in the next proof in this Appendix, we prefer to describe the action of the connected component $\\operatorname{Stab}(S_{3A_2})^\\circ =D^{\\prime \\prime } \\cong \\mathbb {T}^2$ on affine charts for the blowup of $\\mathbb {C}^6$ at the origin.", "Using the $S_3$ symmetry we can assume that either $T_0 \\ne 0$ or $T_{\\widehat{0}}\\ne 0$ , and thus only deal with the two charts $U_0$ and $U_{\\widehat{0}}$ on the blowup.", "We shall start with the chart $U_0$ , with coordinates (REF ), and focus on the exceptional divisor $\\alpha _0=0$ within it.", "In the chart $U_0$ the action (REF ) is given by $\\begin{aligned}(\\lambda _0,\\lambda _1,\\lambda _2,\\lambda _3)&\\cdot (\\alpha _0,t_1,t_2,t_{\\widehat{0}}, t_{\\widehat{1}}, t_{\\widehat{2}})=\\\\ &=\\left(\\left(\\frac{\\lambda _0}{\\lambda _3}\\right)^3\\alpha _0,\\left(\\frac{\\lambda _1}{\\lambda _0}\\right)^3t_1,\\left(\\frac{\\lambda _2}{\\lambda _0}\\right)^3t_2,\\left(\\frac{\\lambda _3}{\\lambda _0}\\right)t_{\\widehat{0}}, \\left(\\frac{\\lambda _1^2\\lambda _3}{\\lambda _0^3}\\right) t_{\\widehat{1}}, \\left(\\frac{\\lambda _2^2\\lambda _3}{\\lambda _0^3}\\right)t_{\\widehat{2}}\\right)&\\quad (\\hbox{for $D^{\\prime }$})\\,\\,\\\\&=\\left(\\lambda _0^3\\alpha _0,\\lambda _0^{-3}\\lambda _1^{3}t_1, \\lambda _0^{-6}\\lambda _1^{-3}t_2,\\lambda _0^{-1}t_{\\widehat{0}},\\lambda _0^{-3}\\lambda _1^{2}t_{\\widehat{1}}, \\lambda _0^{-5}\\lambda _1^{-2}t_{\\widehat{2}}\\right)& (\\hbox{for $D^{\\prime \\prime }$})\\,,\\end{aligned}$ where in the last equality we are using that in $\\operatorname{Stab}(S_{3A_2})^\\circ =D^{\\prime \\prime } \\cong \\mathbb {T}^2$ we have $\\lambda _3=1$ , and $\\lambda _2=\\lambda _0^{-1}\\lambda _1^{-1}$ .", "We note in passing that this chart for the blowup is not equivariant with respect to the full stabilizer group $\\operatorname{Stab}(S_{3A_2})$ , as any element of $S_3$ that does not fix 0 would not preserve it.", "The proof of (1) now becomes a case by case check, determining the group of $(\\lambda _0,\\lambda _1)\\subseteq \\mathbb {T}^2$ that stabilizes a given point on the exceptional divisor, i.e., with coordinate $\\alpha _0=0$ .", "Recall that L:LunaStability described the unstable locus on this exceptional divisor, and we are only interested in semi-stable (which are in fact all stable, as this is the Kirwan blowup) orbits.", "First, we consider the locus where $t_{\\widehat{0}}\\ne 0$ .", "Such a point can only be stabilized if $\\lambda _0^{-1}=1$ .", "If all four coordinates $t_1,t_2,t_{\\widehat{1}},t_{\\widehat{2}}$ are non-zero, then the stabilizer is trivial.", "If $t_1=t_2=0$ , then stabilizer consists of $\\lambda _1$ such that $\\lambda _1^2=\\lambda _1^{-2}=1$ , i.e., is the group $\\mathbb {Z}_2$ .", "If $t_{\\widehat{1}}=t_{\\widehat{2}}=0$ , then the stabilizer is $\\mathbb {Z}_3$ .", "We now deal with the points where the coordinate $t_{\\widehat{0}}=0$ .", "For such a point to be stable, L:LunaStability implies that one of the following pairs of coordinates must both be non-zero: $(t_1,t_2)$ , $(t_1,t_{\\widehat{2}})$ , $(t_{\\widehat{1}},t_2)$ or $(t_{\\widehat{1}},t_{\\widehat{2}})$ .", "For any pair of non-zero coordinates, the action of $\\mathbb {T}^2$ on this pair of coordinates is given by multiplying them by $(\\lambda _0^a\\lambda _1^b,\\lambda _0^c\\lambda _1^d)$ .", "If $t_1t_2\\ne 0=t_{\\widehat{1}}=t_{\\widehat{2}}$ , then the stabilizer must satisfy $\\lambda _0^{-3}\\lambda _1^3=1$ and $\\lambda _0^{-6}\\lambda _1^{-3}=1$ .", "Thus $\\lambda _0=\\lambda _1\\rho $ from the first equation, for $\\rho $ a third root of unity, and then the second equation gives $\\lambda _1^{-6}\\lambda _1^{-3}=1$ , so that $\\lambda _1$ can then be an arbitrary ninth root of unity.", "Altogether the stabilizer group is then $\\mathbb {Z}_3\\times \\mathbb {Z}_9$ .", "For the other three cases, the powers $(a,b)$ and $(c,d)$ are linearly independent, and at least one of them is a primitive integral vector.", "Thus in each of these cases the stabilizer of the pair where these are the only two coordinates is a cyclic group of order $|ad-bc|$ .", "For $t_1t_{\\widehat{2}}\\ne 0=t_2=t_{\\widehat{1}}$ , the powers are $(-3,3)$ and $(-5,-2)$ , and thus we obtain $\\mathbb {Z}_{21}$ .", "For $t_{\\widehat{1}}t_2\\ne 0=t_1=t_{\\widehat{2}}$ , the powers are $(-6,-3)$ and $(-3,2)$ , and thus we again obtain $\\mathbb {Z}_{21}$ (as we should, since the coordinate interchange $(t_1,t_{\\widehat{1}})\\leftrightarrow (t_{\\widehat{1}},t_{\\widehat{2}})$ interchanges this with the previous case).", "Finally, for $t_{\\widehat{1}}t_{\\widehat{2}}\\ne 0=t_1=t_2$ , the powers are $(-3,2)$ and $(-5,-2)$ , and thus we obtain $\\mathbb {Z}_{16}$ .", "The stabilizers for any case where more than two of the coordinates are non-zero is a subgroup of one of these listed groups, and thus also has order not divisible by 5.", "It remains to consider the chart $U_{\\widehat{0}}$ where $T_{\\widehat{0}}\\ne 0$ , so that the coordinates on this chart are $(t_0,t_1,t_2,\\alpha _{\\widehat{0}},t_{\\widehat{1}},t_{\\widehat{2}})$ with $\\alpha _i=t_i\\alpha _{\\widehat{0}}$ .", "Writing down the action (REF ) in these coordinates gives $\\begin{aligned}(\\lambda _0,\\lambda _1,\\lambda _2,\\lambda _3)&\\cdot (t_0,t_1,t_2,\\alpha _{\\widehat{0}},t_{\\widehat{1}},t_{\\widehat{2}})=\\\\&=\\left(\\tfrac{\\lambda _0}{\\lambda _3}t_0,\\tfrac{\\lambda _1^3}{\\lambda _0^2\\lambda _3}t_1,\\tfrac{\\lambda _2^3}{\\lambda _0^2\\lambda _3}t_2,\\tfrac{\\lambda _0^2}{\\lambda _3^2}\\alpha _{\\widehat{0}},\\tfrac{\\lambda _1^2}{\\lambda _0^2}t_{\\widehat{1}}, \\tfrac{\\lambda _2^2}{\\lambda _0^2}t_{\\widehat{2}}\\right)&\\quad (\\hbox{for $D^{\\prime }$})\\,\\,\\\\&=\\left(\\lambda _0 t_0,\\lambda _0^{-2}\\lambda _1^{3}t_1,\\lambda _0^{-5}\\lambda _1^{-3}t_2,\\lambda _0^2\\alpha _{\\widehat{0}},\\lambda _0^{-2}\\lambda _1^{2}t_{\\widehat{1}}, \\lambda _0^{-4}\\lambda _1^{-2}t_{\\widehat{2}}\\right)&\\quad (\\hbox{for $D^{\\prime \\prime }$})\\,.\\end{aligned}$ The only points in the exceptional divisor in the chart $U_{\\widehat{0}}$ whose $S_3$ orbit is disjoint from the chart $U_0$ are those where $t_0=t_1=t_2=0$ .", "For such a point to be stable, we must then have $t_{\\widehat{1}}t_{\\widehat{2}}\\ne 0$ .", "In this case the action is by $\\lambda _0^{-2}\\lambda _1^2$ and $\\lambda _0^{-4}\\lambda _1^{-2}$ .", "From $\\lambda _0^{-2}\\lambda _1^2=1$ it then follows that $\\lambda _0=\\lambda _1\\sigma $ for some $\\sigma \\in \\pm 1$ , and then from the second equation $\\lambda _1$ is a 6th root of unity, so we obtain the stabilizer $\\mathbb {Z}_2\\times \\mathbb {Z}_6$ .", "$\\Box $" ], [ "Proof of P:discMK", "We will work in the Luna slice identified in L:3A2slice, with the action of the toric part of the stabilizer on the affine space $\\mathbb {C}^6$ with coordinates $\\alpha _0,\\alpha _1,\\alpha _2, \\alpha _{\\widehat{0}},\\alpha _{\\widehat{1}},\\alpha _{\\widehat{2}}$ given by (REF ), and $S_3$ permuting pairs of coordinates with the same index.", "We first claim that in this Luna slice the discriminant divisor $D_{A_1}$ locally near the origin is given by the equation $(27\\alpha _0^2+4\\alpha _{\\widehat{0}}^3)\\cdot (27\\alpha _1^2+4\\alpha _{\\widehat{1}}^3)\\cdot (27\\alpha _2^2+4\\alpha _{\\widehat{2}}^3)=0\\,.$ More precisely, it is known [20] that the global deformations of the $3A_2$ cubic surfaces versally (and independently) unfold the three $A_2$ singularities (i.e., the global-to-local restriction of deformations is surjective).", "Since the Luna slice is smooth of dimension 6, it follows that locally analytically (or étale locally) the Luna slice is just a product of three copies of the standard deformation space for the $A_2$ singularity (see e.g., [12]).", "In particular, it follows that the divisor $D_{A_1}$ in the Luna slice is locally the union of three (divisorial) components, and we denote it $HH_0\\cup H_1\\cup H_2$ , with $S_3$ permuting the components.", "This shows that locally analytically there are some coordinates for the Luna slice so that (REF ) is the equation of the discriminant.", "We claim that (REF ) is in fact the local equation of the discriminant in our given coordinates from L:3A2slice.", "By the discussion above, it suffices to show that the hypersurface in the Luna slice given by the equation $27\\alpha _0^2+4\\alpha _{\\widehat{0}}^3=0$ in our coordinates is contained in the discriminant.", "Taking partial derivatives of our family of cubics parameterized by the Luna slice in L:3A2slice $\\alpha _0x_0^3 +\\alpha _1x_1^3 +\\alpha _2x_2^3+\\alpha _{\\widehat{0}}x_0^2x_3+\\alpha _{\\widehat{1}}x_1^2x_3 +\\alpha _{\\widehat{2}}x_2^2x_3 +(x_0x_1x_2+x_3^3)$ one can check that if $27\\alpha _0^2+4\\alpha _{\\widehat{0}}^3=0$ , then the associated cubic has a singularity at the point $(1:0:0:-\\tfrac{3\\alpha _0}{2\\alpha _{\\widehat{0}}})$ , unless $\\alpha _{\\widehat{0}}=0$ , in which case $\\alpha _0=0$ , and the associated cubic has a singularity at $(1:0:0:0)$ , establishing the claim.", "The Kirwan desingularization proceeds by blowing up the origin of this Luna slice $\\mathbb {C}^6$ , and then taking the GIT quotient of the blowup by the stabilizer $\\operatorname{Stab}(S_{3A_2})$ given by (REF ).", "We will determine the strict transform of $H$ in the coordinate charts on the blowup $\\operatorname{Bl}_0\\mathbb {C}^6\\subseteq \\mathbb {C}^6\\times \\mathbb {P}^5$ .", "As in the previous proof, by symmetry it is enough to work in the chart $U_0$ given by (REF ), and in the chart $U_{\\widehat{0}}$ where $t_{\\widehat{0}}\\ne 0$ .", "We start with the chart $U_0$ , and use (REF ) to express the proper transform of the divisor $H$ on it as $\\alpha _0^6\\cdot (27+4\\alpha _0 t_{\\widehat{0}}^3)\\cdot (27t_1^2+4\\alpha _0 t_{\\widehat{1}}^3) \\cdot (27 t_2^2+4\\alpha _0 t_{\\widehat{2}}^3)=0\\,.$ In this chart the exceptional divisor of the blowup is given by $\\alpha _0=0$ , and thus the strict transform of the divisor $H$ just omits the $\\alpha _0^6$ factor above.", "To compute the local structure of the Kirwan blowup, we now need the Luna slice for the action of the torus $\\mathbb {T}^2\\subseteq \\operatorname{Stab}(S_{3A_2})$ given by diagonal matrices with $\\lambda _0\\lambda _1\\lambda _2=1=\\lambda _3$ in the chart $U_0$ , which is given in (REF ).", "One can check directly that the $\\mathbb {C}^4$ given by the two equations $t_1=t_{\\widehat{1}}=1$ is the Luna slice for the action of $\\mathbb {T}^2$ in this chart.", "Thus the intersection of the strict transform of the divisor $H$ with this Luna slice is given by $(27+4\\alpha _0 t_{\\widehat{0}}^3)\\cdot (27+4\\alpha _0) \\cdot (27 t_2^2+4\\alpha _0 t_{\\widehat{2}}^3)=0\\,,$ which intersects the exceptional divisor $\\alpha _0=0$ non-transversally.", "Indeed, the last factor in this strict transform intersects the exceptional divisor as the intersection of the loci $\\alpha _0=0$ and $27 t_2^2+4\\alpha _0 t_{\\widehat{2}}^3=0$ , which is non-transversal: they intersect along the codimension 2 space $\\alpha _0=t_2=0$ , but with multiplicity 2.", "To ascertain the non-transversality in the moduli space, one needs to further check that taking the quotient by $\\mathbb {T}^2$ and by $S_3$ does not cause the intersection to become transverse.", "For completeness, and as a cross-check, we will perform this computation in full detail in the chart $U_{\\widehat{0}}$ .", "We now work in the chart $U_{\\widehat{0}}$ with coordinates $t_0,t_1,t_2,\\alpha _{\\widehat{0}},t_{\\widehat{1}},t_{\\widehat{2}}$ , and express $\\alpha _i=\\alpha _{\\widehat{0}}t_i$ .", "Thus the proper transform of the discriminant divisor $H$ becomes $\\alpha _{\\widehat{0}}^6\\cdot (27t_0^2+4\\alpha _{\\widehat{0}})\\cdot (27t_1^2+4\\alpha _{\\widehat{0}}t_{\\widehat{1}}^3) \\cdot (27 t_2^2+4\\alpha _{\\widehat{0}}t_{\\widehat{2}}^3)\\,.$ In this chart the exceptional divisor of the blowup is given by $\\alpha _{\\widehat{0}}=0$ , and thus the strict transform of the divisor $H$ again just omits the $\\alpha _{\\widehat{0}}^6$ factor.", "To compute the local structure of the Kirwan blowup, we need the Luna slice for the action of the torus $\\mathbb {T}^2\\subseteq \\operatorname{Stab}(S_{3A_2})$ given by diagonal matrices with $\\lambda _0\\lambda _1\\lambda _2=1=\\lambda _3$ in the chart $U_{\\widehat{0}}$ , where the action is given by (REF ).", "We claim that the $\\mathbb {C}^4$ given by the two equations $t_{\\widehat{1}}=t_{\\widehat{2}}=1$ is a Luna slice for this action of $\\mathbb {T}^2$ on $U_{\\widehat{0}}=\\mathbb {C}^6$ .", "Indeed, for any point $t_0,t_1,t_2,\\alpha _{\\widehat{0}},t_{\\widehat{1}},t_{\\widehat{2}}\\in U_{\\widehat{0}}$ with $t_{\\widehat{1}}t_{\\widehat{2}}\\ne 0$ , there exist $\\lambda _0,\\lambda _1$ such that $\\lambda _0^{-2}\\lambda _1^2=t_{\\widehat{1}}^{-1}$ and $\\lambda _0^{-4}\\lambda _1^{-2}=t_{\\widehat{2}}^{-1}$ , and thus acting by these $(\\lambda _0,\\lambda _1)$ shows that the orbit contains a point with $t_{\\widehat{1}}=t_{\\widehat{2}}=1$ .", "The same computation also shows that the stabilizer within $\\mathbb {T}^2$ of any point on this $\\mathbb {C}^4$ slice is at most finite.", "Restricting the strict transform of the divisor $H$ to this Luna slice for the $\\mathbb {T}^2$ gives $(27t_0^2+4\\alpha _{\\widehat{0}})\\cdot (27t_1^2+4\\alpha _{\\widehat{0}}) \\cdot (27 t_2^2+4\\alpha _{\\widehat{0}})=0\\,.$ The intersection of the first factor with the exceptional locus $\\alpha _{\\widehat{0}}=0$ is along the codimension two locus $t_0=\\alpha _{\\widehat{0}}=0$ , along which the first factor intersects the exceptional divisor non-transversally (we observe that by L:LunaStability a generic point in the chart $U_{\\widehat{0}}$ with coordinates $t_0=\\alpha _{\\widehat{0}}=0$ is stable, as indeed $T_{\\widehat{0}}\\ne 0$ ).", "What remains to check non-transversality is to handle the finite part of the stabilizer $\\operatorname{Stab}(S_{3A_2})$ , as in principle the quotient of a non-transversal intersection by a finite group may be transversal.", "We will thus verify that the subgroup of $\\operatorname{Stab}(S_{3A_2})$ that fixes a generic point of the intersection of the strict transform of $H$ in chart $U_{\\widehat{0}}$ with the exceptional divisor $\\alpha _{\\widehat{0}}=0$ is trivial.", "Indeed, such a generic point of intersection has coordinates $(t_0=0,t_1,t_2,\\alpha _{\\widehat{0}}=0,t_{\\widehat{1}},t_{\\widehat{2}})$ , as discussed above.", "If an element of the group $\\operatorname{Stab}(S_{3A_2})$ as described in (REF ) fixes this point, then we claim that the image of this element in $S_3$ must be either the identity or the involution $1\\leftrightarrow 2$ , which permutes coordinates $(t_1,t_{\\widehat{1}})\\leftrightarrow (t_2,t_{\\widehat{2}})$ (this statement also appears in the proof of L:Eck-no-3A2).", "To see this, note that at a generic point the only zero coordinate in $\\mathbb {P}^5$ is $t_0$ , and thus the image in $S_3$ of an element of a stabilizer fixing a generic point must fix $0\\in \\lbrace 0,1,2\\rbrace $ .", "For an element of $D^{\\prime }\\subseteq \\operatorname{Stab}(S_{3A_2})$ (so that its image in $S_3$ is the identity), the action (REF ) on the locus $\\alpha _{\\widehat{0}}=t_0=0$ restricts to $(t_1,t_2,t_{\\widehat{1}},t_{\\widehat{2}})\\mapsto (\\lambda _1^3\\lambda _0^{-2}\\lambda _3^{-1}t_1,\\lambda _2^3\\lambda _0^{-2}\\lambda _3^{-1}t_2,\\lambda _1^2\\lambda _0^{-2}t_{\\widehat{1}}, \\lambda _2^2\\lambda _0^{-2}t_{\\widehat{2}})\\,.$ If a general point $t_1,t_2,t_{\\widehat{1}},t_{\\widehat{2}}$ is mapped to itself, then from $t_{\\widehat{1}}$ coordinate being preserved it follows that $\\lambda _1=\\sigma _1\\lambda _0$ for some $\\sigma _1\\in \\lbrace \\pm 1\\rbrace $ , and then from the preservation of $t_1$ coordinate it follows that $\\lambda _1=\\lambda _3$ .", "Similarly from the preservation of the $t_{\\widehat{2}}$ coordinate it follows that $\\lambda _2=\\sigma _2\\lambda _0$ with $\\sigma _2\\in \\lbrace \\pm 1\\rbrace $ , and then from the preservation of the $t_2$ coordinate it follows that $\\lambda _2=\\lambda _3$ .", "Thus finally $\\lambda _1=\\lambda _2=\\lambda _3=\\sigma _1 \\lambda _0$ .", "Furthermore, equation $\\lambda _0\\lambda _1\\lambda _2=\\lambda _3^3$ in the description of $\\operatorname{Stab}(S_{3A_2})$ in (REF ) gives $\\sigma _1=1$ , so that finally $\\lambda _0=\\lambda _1=\\lambda _2=\\lambda _3$ , and since the matrix is in $\\operatorname{SL}(4,\\mathbb {C})$ , they must all be equal to the same fourth root of unity, so that as an element of $\\operatorname{PGL}(4,\\mathbb {C})$ the matrix is equal to the identity.", "Finally, for an element of $\\operatorname{Stab}(S_{3A_2})$ whose image in $S_3$ is the involution $1\\leftrightarrow 2$ , the action is $(t_1,t_2,t_{\\widehat{1}},t_{\\widehat{2}})\\mapsto (\\lambda _2^3\\lambda _0^{-2}\\lambda _3^{-1}t_2,\\lambda _1^3\\lambda _0^{-2}\\lambda _3^{-1}t_1,\\lambda _2^2\\lambda _0^{-2}t_{\\widehat{2}},\\lambda _1^2\\lambda _0^{-2}t_{\\widehat{1}})\\,.$ If this action preserves a point, then from the coordinate $t_{\\widehat{1}}$ being preserved we see that $\\lambda _2^2t_{\\widehat{2}}=\\lambda _0^2 t_{\\widehat{1}}$ .", "Fix a square root $x=(t_{\\widehat{1}}/t_{\\widehat{2}})^{1/2}$ , so that then $\\lambda _2=\\sigma _2\\lambda _0 x$ for some $\\sigma _2\\in \\lbrace \\pm 1\\rbrace $ .", "From the coordinate $t_{\\widehat{2}}$ being preserved we obtain $\\lambda _1^2\\lambda _0^{-2}t_{\\widehat{1}}=t_{\\widehat{2}}$ and thus we see that $\\lambda _1=\\sigma _1\\lambda _0 x^{-1}$ for some $\\sigma _1\\in \\lbrace \\pm 1\\rbrace $ .", "From the product of coordinates $t_1t_2$ being preserved we see that $\\lambda _2^3\\lambda _1^3\\lambda _0^{-4}\\lambda _3^{-2}=1$ .", "Substituting here our expressions for $\\lambda _1$ and $\\lambda _2$ yields $\\sigma _1\\sigma _2\\lambda _0^2=\\lambda _3^2$ , so that $\\lambda _3=\\gamma _3\\lambda _0$ with $\\gamma _3^2=\\sigma _1\\sigma _2$ .", "Furthermore substituting the expressions for all $\\lambda $ 's in terms of $\\lambda _0$ in the condition $\\lambda _0\\lambda _1\\lambda _2=\\lambda _3^3$ yields then $\\sigma _1\\sigma _2=\\gamma _3^3=\\sigma _1\\sigma _2\\gamma _3$ , so that $\\gamma _3=1$ and thus $\\sigma _1\\sigma _2=1$ .", "Finally computing the determinant, the condition for the matrix to lie in $\\operatorname{SL}(4,\\mathbb {C})$ gives $\\lambda _0\\lambda _1\\lambda _2\\lambda _3=\\lambda _0^4(\\sigma _1\\sigma _2)=-1$ (Note that this is indeed a minus, as the matrix is not diagonal, but includes a transposition!", "Recall furthermore that we are thinking about the action of $S_3$ , and the involution $1\\leftrightarrow 2$ interchanges two pairs of coordinates, $t_1 \\leftrightarrow t_2$ and $t_{\\widehat{1}}\\leftrightarrow t_{\\widehat{2}}$ ).", "Thus finally $\\lambda _0=\\lambda _3$ must be some eighth roots of unity, but in this case $t_1$ is mapped to some eighth root of unity times $x^3$ times $t_2$ .", "Since $t_1,t_2,t_{\\widehat{1}},t_{\\widehat{2}}$ , and thus also $x$ , were general, $t_1$ cannot be equal to such a product, and thus there is no stabilizer of this form.", "$\\Box $" ] ]
2207.03533
[ [ "Rotated ansatz for approximate counterdiabatic driving" ], [ "Abstract Approximate counterdiabatic (CD) protocols are a powerful tool to enhance quantum adiabatic processes that allow to reliably manipulate quantum systems on short time scales.", "However, implementing CD protocols entails the introduction of additional control fields in the Hamiltonian, often associated with highly non-local multi-body interactions.", "Here, we introduce a novel variational rotated ansatz (RA) to systematically generate experimentally accessible approximate CD protocols.", "We numerically benchmark our approach on state preparation and adiabatic quantum computing algorithms, and find that using RA protocols significantly enhances their performances." ], [ "Limitations of the variational counterdiabatic approach", "This section highlights some limitations of the traditional variational CD approach [1] (without frame rotations).", "We consider the case where the two Hamiltonians $\\widehat{H}_a$ and $\\widehat{H}_b$ generate the space of allowed physical operators [2], [3].", "Without any further assumption on the form of $\\widehat{H}_a$ and $\\widehat{H}_b$ , we consider the parametric Hamiltonian $\\widehat{H}_0(t) = A_0(\\lambda )\\widehat{H}_a + B_0(\\lambda )\\widehat{H}_b\\;,$ where $A_0(\\lambda )$ and $B_0(\\lambda )$ are arbitrary real schedule functions, describing the original UA driving.", "Within the space of allowed operators, the most general prametrization for the variational AGP reads $\\hat{\\mathcal {A}}=\\alpha _a (\\lambda ) \\widehat{H}_a + \\alpha _b (\\lambda ) \\widehat{H}_b\\;,$ where $\\alpha _a (\\lambda )$ and $\\alpha _b (\\lambda )$ are variational parameters.", "The CD Hamiltonian is $\\widehat{H}_{\\mathrm { \\scriptscriptstyle CD}}(t)=\\widehat{H}_0(\\lambda ) + \\dot{\\lambda }\\hat{\\mathcal {A}}=A_{\\mathrm { \\scriptscriptstyle CD}}(t)\\widehat{H}_a + B_{\\mathrm { \\scriptscriptstyle CD}}(t)\\widehat{H}_b\\;,$ where $A_{\\mathrm { \\scriptscriptstyle CD}}(t) = A_0(t)+\\dot{\\lambda }\\alpha _a(t)$ and $B_{\\mathrm { \\scriptscriptstyle CD}}(t) = B_0(t)+\\dot{\\lambda }\\alpha _b(t)$ are the control fields of the CD protocol.", "Inserting Eq.", "(REF ) and Eq.", "(REF ) into Eq.", "() of the main text, we get the following expression for the action $\\mathcal {S}(\\hat{\\mathcal {A}})&=\\operatorname{Tr}\\Big (\\widehat{G}_{\\lambda }^2\\Big ) =\\operatorname{Tr}\\left((\\partial _\\lambda A_0 \\widehat{H}_a + \\partial _\\lambda B_0 \\widehat{H}_b - i[A_0\\widehat{H}_a + B_0\\widehat{H}_b, \\alpha _a \\widehat{H}_a + \\alpha _b\\widehat{H}_b])^2\\right) \\nonumber \\\\&= \\operatorname{Tr}\\Big ((\\partial _\\lambda \\widehat{H}_0)^2\\Big ) + \\operatorname{Tr}\\Big ([\\widehat{H}_a,\\widehat{H}_b]^2 \\Big )(A_0\\alpha _b - B_0 \\alpha _a)^2\\;.$ Eq.", "(REF ) implies that, the trivial choice $\\alpha _a(\\lambda )=\\alpha _b(\\lambda )=0$ is a global minimum of the action.", "Hence, the traditional CD variational approach, without frame rotations, returns the trivial result $A_{\\mathrm { \\scriptscriptstyle CD}}(t) = A_0(t) $ and $B_{\\mathrm { \\scriptscriptstyle CD}}(t) = B_0(t)$ , which gives no performance improvement.", "However, as we showed in the main text, the RA method can overcome this performance limitation." ], [ "Boundary conditions for the rotated protocols", "In this section, we show that by choosing a ramp $\\lambda (t)$ such that $\\dot{\\lambda }(0)=\\dot{\\lambda }(\\tau )=0$ we can enforce the time boundary condition, required by RA protocols.", "We recall that for the RA driving to result in an alternative STA, we need the time boundary condition, given in Eq.", "() of the main text to hold for the initial and final times $t=0,\\tau $ .", "Namely, we must require $|\\langle {\\epsilon (\\lambda )|\\mathrm {e}^{ -\\frac{i}{\\hbar } \\widehat{\\mathcal {Q}}}|\\psi (t)} \\rangle |&=|\\langle {\\epsilon (\\lambda )|\\psi (t)} \\rangle |\\qquad \\mathrm {for}\\; t = 0,\\tau $ We now consider the specific case of $\\dot{\\lambda }\\rightarrow 0$ for $t=0,\\tau $ .", "We scale the action by the non negative (time-dependent) factor $\\dot{\\lambda }^2$ .", "The scaled action $\\bar{\\mathcal {S}}(\\hat{\\mathcal {A}})$ is $\\bar{\\mathcal {S}}(\\hat{\\mathcal {A}}) = \\dot{\\lambda }^2 \\mathcal {S}(\\hat{\\mathcal {A}})&= \\Big ((\\dot{\\lambda }\\partial _\\lambda \\widehat{H}_{\\mathrm { 0}}- \\frac{i}{\\hbar } [\\widehat{H}_{\\mathrm { 0}}, \\dot{\\lambda }\\hat{\\mathcal {A}}])^2\\Big ) \\nonumber \\\\&= \\Big ((\\dot{\\lambda }\\partial _\\lambda \\widehat{H}_{\\mathrm { 0}}- \\frac{i}{\\hbar } [\\widehat{H}_{\\mathrm { 0}}, \\mathrm {e}^{ \\frac{i}{\\hbar } \\widehat{\\mathcal {Q}}}(\\widehat{H}_0 + \\widehat{\\mathcal {K}})\\mathrm {e}^{-\\frac{i}{\\hbar } \\widehat{\\mathcal {Q}}}])^2\\Big )$ For $\\dot{\\lambda }^2> 0$ , the optimal auxiliary potentials also minimize $\\bar{\\mathcal {S}}(\\hat{\\mathcal {A}})$ .", "Assuming that $\\partial _\\lambda \\widehat{H}_{\\mathrm { 0}}$ is bounded, we get $\\lim _{\\dot{\\lambda } \\rightarrow 0} \\bar{\\mathcal {S}}(\\hat{\\mathcal {A}}) =\\frac{1}{\\hbar ^2}\\Big ([\\widehat{H}_{\\mathrm { 0}}, \\mathrm {e}^{ \\frac{i}{\\hbar } \\widehat{\\mathcal {Q}}}(\\widehat{H}_0 + \\widehat{\\mathcal {K}})\\mathrm {e}^{-\\frac{i}{\\hbar } \\widehat{\\mathcal {Q}}}])^2\\Big )\\;.$ In particular, if $\\dot{\\lambda }=0$ for $t=0,\\tau $ , by taking $\\widehat{\\mathcal {K}}$ and $\\widehat{\\mathcal {Q}}$ such that $\\mathrm {e}^{ -\\frac{i}{\\hbar } \\widehat{\\mathcal {Q}}} = \\hat{1}, \\qquad \\mathrm { and }\\; \\widehat{\\mathcal {K}}= \\hat{0}$ we simultaneously minimize $\\bar{\\mathcal {S}}(\\hat{\\mathcal {A}})$ and satisfy Eq.", "(REF ).", "Hence, we can enforce the time boundary condition by choosing $\\lambda (t)$ such that $\\dot{\\lambda }(0)=\\dot{\\lambda }(\\tau )=0$ ." ], [ "Sequential local optimization of the action", "In the following, we describe the numerical algorithm we use to optimize the RA action and compute the RA driving protocols.", "To simplify the description of the algorithm, we explicitly consider the case where the two Hamiltonians $\\widehat{H}_a$ and $\\widehat{H}_b$ generate the space of allowed physical operators [2], [3].", "Without any further assumption on the form of $\\widehat{H}_a$ and $\\widehat{H}_b$ , we consider the parametric Hamiltonian $\\widehat{H}_0(t) = A_0(\\lambda )\\widehat{H}_a + B_0(\\lambda )\\widehat{H}_b\\;,$ where $A_0(\\lambda )$ and $B_0(\\lambda )$ are arbitrary real schedule functions, describing the original UA driving.", "Within the space of allowed operators, we parameterize the auxiliary RA potentials as $\\widehat{\\mathcal {Q}}&=\\gamma \\widehat{H}_a\\\\\\widehat{\\mathcal {K}}&=\\beta \\widehat{H}_b$ where $\\beta (t)$ and $\\gamma (t)$ are variational parameters.", "The corresponding AGP and RA driving Hamiltonian read $\\dot{\\lambda }\\hat{\\mathcal {A}}&= \\mathrm {e}^{ \\frac{i}{\\hbar } \\widehat{\\mathcal {Q}}}\\left(\\widehat{H}_0 + \\widehat{\\mathcal {K}}\\right) \\mathrm {e}^{ -\\frac{i}{\\hbar } \\widehat{\\mathcal {Q}}}- \\widehat{H}_0 = (B_0 + \\beta ) \\mathrm {e}^{ i \\gamma \\widehat{H}_a }\\widehat{H}_b \\mathrm {e}^{ -i \\gamma \\widehat{H}_a }- B_0\\widehat{H}_b\\\\\\widehat{H}_{\\mathrm { \\scriptscriptstyle RA}}&= \\widehat{H}_{\\mathrm { 0}}+\\widehat{\\mathcal {K}}+ \\dot{\\widehat{\\mathcal {Q}}} = A_{\\mathrm { \\scriptscriptstyle RA}}(t)\\widehat{H}_a + B_{\\mathrm { \\scriptscriptstyle RA}}(t)\\widehat{H}_b $ where $A_{\\mathrm { \\scriptscriptstyle RA}}(t) = A_0(t) + \\dot{\\gamma }(t)$ and $B_{\\mathrm { \\scriptscriptstyle RA}}(t) = B_0(t)+\\beta (t)$ are the control fields of the RA protocol.", "The expression of the AGP leads to the following action $\\mathcal {S}(\\hat{\\mathcal {A}})&=\\operatorname{Tr}\\Big (\\widehat{G}_{\\lambda }^2\\Big ) =\\operatorname{Tr}\\left((\\partial _\\lambda A_0 \\widehat{H}_a + \\partial _\\lambda B_0 \\widehat{H}_b - i[A_0\\widehat{H}_a + B_0\\widehat{H}_b, (B_0 + \\beta ) \\mathrm {e}^{ i \\gamma \\widehat{H}_a }\\widehat{H}_b \\mathrm {e}^{ -i \\gamma \\widehat{H}_a }- B_0\\widehat{H}_b])^2\\right) \\;.", "$ We observe that the trivial choice $\\beta (t)=\\gamma (t)=0$ is generally not a global minimum of the action.", "Hence, to find the functions $\\gamma (t)$ and $\\beta (t)$ , we must explicitly minimize Eq.", "(REF ).", "Since $A_{\\mathrm { \\scriptscriptstyle RA}}(t) = A_0(t) + \\dot{\\gamma }(t)$ , we are specifically looking for differentiable functions and we are not necessarily interested in finding the global optimum of the action for each $t\\in [0,\\tau ]$ .", "We, therefore, use a sequential local optimization algorithm to compute the functions in the interval $t\\in [0,\\tau ]$ .", "We first consider the discrete time problem of finding $\\beta ^{(m)}=\\beta (t^{(m)} )$ and $\\gamma ^{(m)}=\\gamma (t^{(m)} )$ , for $t^{(m)} =\\frac{m-1}{M}\\tau $ and $m=1, \\dots M+1$ .", "Here the integer $M$ indicates the number of discrete points considered.", "Since we are ultimately interested in generating smooth differentiable functions, we can assueme that $\\beta ^{(m)}$ and $\\gamma ^{(m)}$ are close to $\\beta ^{(m+1)}$ and $\\gamma ^{(m+1)}$ respectively.", "This observation suggests to use sequential local optimizations to solve the discrete problem.", "Starting from given initial values, eg.", "$\\gamma ^{(0)}=\\mathbf {0}$ , $\\beta ^{(0)}=\\mathbf {0}$ , we use the BFGS algorithm [4] to minimize the action at time 0, which returns the optimal values $\\beta ^{(1)}$ and $\\gamma ^{(1)}$ .", "Then, we proceed iteratively, using $\\beta ^{(m)}$ and $\\gamma ^{(m)}$ as initial guess to find $\\beta ^{(m+1)}, \\gamma ^{(m+1)}$ .", "Finally, we interpolate the resulting $M$ points to get the functions $\\beta (t)$ and $\\gamma (t)$ , which we, then, use to obtain the new schedules $A_{\\mathrm { \\scriptscriptstyle RA}}(t)$ and $B_{\\mathrm { \\scriptscriptstyle RA}}(t)$ .", "The algorithm's generalization to multiple variational parameters case is straightforward.", "The computation of RA protocols $A_{\\mathrm { \\scriptscriptstyle RA}}(t)$ and $B_{\\mathrm { \\scriptscriptstyle RA}}(t)$ does not require spectral information and can run on a classical CPUs.", "The algorithm's bottleneck is the evaluation of the action, which can still be exponentially hard, as it involves the multiplication of large matrices.", "In particular, in a system of $N$ spins, the matrix multiplication's complexity scales exponentially with $N$ .", "However, as reported in the main text, for various physically relevant Hamiltonians, the complexity of computing RA action is only polynomial in the system size $N$ ." ], [ "Rotated ansatz protocol for a two-level system", "This section provides additional details on the two-spin system studied in the main text.", "Refs.", "[5], [1] used the two-level nature of the system to compute an analytical expression for the adiabatic gauge potential.", "Here, we use the same problem to illustrate the RA approach.", "We use natural units and set $\\hbar =1$ , we indicate with $\\dot{x}(t)$ the time derivative of a given function $x(t)$ .", "The parametric Hamiltonian of the two-spin system considered in Eq.", "() of the main text is $\\widehat{H}_{\\mathrm { 0}}(\\lambda ) = h_0(\\lambda ) \\widehat{H}_a + J_0(\\lambda )\\widehat{H}_b\\;,$ where $\\widehat{H}_a &= -\\hat{\\sigma }^z_1 - \\hat{\\sigma }^z_2, \\qquad \\widehat{H}_b = \\hat{\\sigma }^x_1\\hat{\\sigma }^x_2 +\\hat{\\sigma }^z_1\\hat{\\sigma }^z_2$ generate the space of allowed physical operators.", "We parametrize the auxiliary potentials as $\\widehat{\\mathcal {Q}}&=\\gamma \\widehat{H}_a\\\\\\widehat{\\mathcal {K}}&=\\beta \\widehat{H}_b$ where $\\beta (t)$ and $\\gamma (t)$ are variational parameters.", "The corresponding RA driving Hamiltonian reads $\\widehat{H}_{\\mathrm { \\scriptscriptstyle RA}}&= \\widehat{H}_{\\mathrm { 0}}+\\widehat{\\mathcal {K}}+ \\dot{\\widehat{\\mathcal {Q}}} = h_{\\mathrm { \\scriptscriptstyle RA}}(t)\\widehat{H}_a + J_{\\mathrm { \\scriptscriptstyle RA}}(t)\\widehat{H}_b\\;.", "$ where $h_{\\mathrm { \\scriptscriptstyle RA}}(t) = h_0(t) + \\dot{\\gamma }(t)$ and $J_{\\mathrm { \\scriptscriptstyle RA}}(t) = J_0(t)+\\beta (t)$ are the control fields of the RA protocol.", "To obtain the optimal values of $\\beta $ and $\\dot{\\gamma }$ , we must minimize the RA action." ], [ "RA adiabatic potential and RA action ", "To simplify the computation of the RA Adiabatic gauge potential, we exploit the two-level nature of the system.", "Indeed, Eq.", "(REF ) describes a two-level system where $\\widehat{H}(t)$ only couples the states $|{\\uparrow \\uparrow }\\rangle $ and $|{\\downarrow \\downarrow }\\rangle $ .", "We, therefore, introduce the operators $\\hat{s}^z=\\frac{1}{4}(\\hat{\\sigma }^z_1+\\hat{\\sigma }^z_2), \\qquad \\hat{s}^x=\\frac{1}{4}( \\hat{\\sigma }^x_1\\hat{\\sigma }^x_2 - \\hat{\\sigma }^y_1\\hat{\\sigma }^y_2), \\qquad \\hat{s}^y =\\frac{1}{4}(\\hat{\\sigma }^x_1\\hat{\\sigma }^y_2 + \\hat{\\sigma }^y_1\\hat{\\sigma }^x_2), \\qquad \\hat{o} &= \\frac{1}{2}(\\hat{\\sigma }^x_1\\hat{\\sigma }^x_2 +\\hat{\\sigma }^y_1\\hat{\\sigma }^y_2)+\\hat{\\sigma }^z_1\\hat{\\sigma }^z_2\\;$ The operators $\\hat{s}^x$ , $\\hat{s}^y$ and $\\hat{s}^z$ obey spin algebra, and $\\hat{o}$ is such that $\\hat{o}\\hat{s}^{\\nu }=\\hat{s}^{\\nu } \\hat{o}=\\hat{s}^{\\nu }$ for $\\nu =x,y,z$ .", "In particular, in the subspace generated by $\\lbrace |{\\uparrow \\uparrow }\\rangle ,|{\\downarrow \\downarrow }\\rangle \\rbrace $ , the operators $\\hat{s}^x$ , $\\hat{s}^y$ and $\\hat{s}^z$ are representet by spin-$1/2$ matrices, while $\\hat{o}$ is the identity matrix.", "Using these operators, the two Hamiltonian terms read $\\widehat{H}_a = -4\\hat{s}^z$ and $\\widehat{H}_b =2\\hat{s}^x + \\hat{o}$ .", "Inserting these expressions in the definition of the RA adiabatic gauge potential, we get $\\dot{\\lambda }\\hat{\\mathcal {A}}&= \\mathrm {e}^{ \\frac{i}{\\hbar } \\widehat{\\mathcal {Q}}}\\left(\\widehat{H}_0 + \\widehat{\\mathcal {K}}\\right) \\mathrm {e}^{ -\\frac{i}{\\hbar } \\widehat{\\mathcal {Q}}}- \\widehat{H}_0 = 2(\\beta +J_0 )\\mathrm {e}^{ -i 4\\gamma \\hat{s}^z } \\hat{s}^x\\mathrm {e}^{ i 4\\gamma \\hat{s}^z} +\\beta \\hat{o} - 2J_0 \\hat{s}^x \\nonumber \\\\&= \\beta \\hat{o} + 2(\\beta +J_0 )(\\cos (4\\gamma )\\hat{s}^x - \\sin (4\\gamma )\\hat{s}^y) - 2J_0\\hat{s}^x \\nonumber \\\\&= \\beta \\hat{o} + 4\\alpha _{xx}\\hat{s}^x + 4\\alpha _{xy}\\hat{s}^y \\nonumber \\\\&= \\beta \\hat{o} + \\alpha _{xx}\\hat{\\sigma }^x_1\\hat{\\sigma }^x_2 + \\alpha _{yy}\\hat{\\sigma }^y_1\\hat{\\sigma }^y_2 + \\alpha _{xy}(\\hat{\\sigma }^x_1\\hat{\\sigma }^y_2 + \\hat{\\sigma }^y_1\\hat{\\sigma }^x_2)$ where we introduced the AGP coefficients $\\alpha _{xx} = -\\alpha _{yy} = \\frac{(\\beta +J_0 )\\cos (4\\gamma ) - J_0}{2}\\,,\\qquad \\alpha _{xy} = -\\frac{(\\beta +J_0 )}{2}\\sin (4\\gamma )\\;.$ Eq.", "(REF ) shows that the frame rotation introduces in the AGP a new term proportional to $\\hat{\\sigma }^x_1\\hat{\\sigma }^y_2 + \\hat{\\sigma }^y_1\\hat{\\sigma }^x_2$ .", "We also compute the operator $\\widehat{G}$ , obtaining $\\dot{\\lambda } \\widehat{G}&= \\dot{\\widehat{H}}_0 +\\frac{i}{\\hbar }\\left[\\dot{\\lambda }\\hat{\\mathcal {A}},\\widehat{H}_0\\right]\\nonumber \\\\&= \\dot{J}_0(2\\hat{s}^x + \\hat{o})- 4\\dot{h}_0 \\hat{s}^z +i\\left[ 4\\alpha _{xx}\\hat{s}^x + 4\\alpha _{xy}\\hat{s}^y,2 J_0 \\hat{s}^x - 4h_0\\hat{s}^z\\right]\\nonumber \\\\&= \\dot{J}_0\\hat{o} + 2(\\dot{J}_0+8h_0 \\alpha _{xy})\\hat{s}^x - 16 h_0\\alpha _{xx}\\hat{s}^y + 4( 2J_0 \\alpha _{xy} -\\dot{h}_0 )\\hat{s}^z\\;.$ Finally, we insert Eq.", "(REF ) into the definition of the action, leading to $\\mathcal {S}&= \\operatorname{Tr}(\\widehat{G}^2) = 6\\dot{J}^2_0 + 2(\\dot{J}_0+8h_0 \\alpha _{xy})^2+ 2^7 h^2_0\\alpha _{xx}^2 + 2^3( 2 J_0 \\alpha _{xy} -\\dot{h}_0 )^2$" ], [ "RA action minimization and RA protocols", "To minimize the action we use the equations $\\frac{\\partial \\mathcal {S}}{\\partial \\alpha _{xx}} = \\frac{\\partial \\mathcal {S}}{\\partial \\alpha _{xy}}=0$ .", "The quadratic function is minimized at the point ${\\left\\lbrace \\begin{array}{ll}\\alpha _{xx} &= 0\\\\\\alpha _{xy} &= -\\frac{1}{2}\\varphi _0\\end{array}\\right.", "}\\Rightarrow {\\left\\lbrace \\begin{array}{ll}\\cos (4\\gamma ) &= \\frac{J_0}{J_0 + \\beta }\\\\\\sin (4\\gamma ) &= \\frac{\\dot{J}_0 h_0-J_0 \\dot{h}_0}{(J_0 + \\beta )(4h_0^2 + J_0^2)} = \\frac{\\varphi _0}{J_0 + \\beta }\\end{array}\\right.", "}$ where $\\varphi _0 = \\frac{\\dot{J}_0 h_0-J_0 \\dot{h}_0}{4h_0^2 + J_0^2}\\;.", "$ If we assume $J_0+\\beta >0$ and $4\\gamma \\in ( -\\frac{\\pi }{2}, \\frac{\\pi }{2})$ , we can write $\\beta = \\sqrt{J_0^2 + \\varphi _0^2} - J_0,\\qquad \\gamma = \\frac{1}{4}\\arctan \\left(\\frac{\\varphi _0}{J_0}\\right)$ By substitution Eq.", "(REF ), we get that the optimal RA $\\frac{\\partial \\mathcal {S}}{\\partial \\alpha _{xx}} = \\frac{\\partial \\mathcal {S}}{\\partial \\alpha _{xy}}=0\\Rightarrow \\hat{\\mathcal {A}}=-\\frac{1}{2}\\varphi _0 (\\hat{\\sigma }^x_1\\hat{\\sigma }^y_2 + \\hat{\\sigma }^y_1\\hat{\\sigma }^x_2)+\\left(\\sqrt{J_0^2 + \\varphi _0^2} - J_0\\right)\\hat{o}\\;.$ This result is compatible, up to an irrelevant term proportional to $\\hat{o}$ , with the well-known exact AGP $\\hat{\\mathcal {A}}^{\\mathrm {exact}} = -2\\varphi _0 \\hat{s}_y$ for two level systems [1].", "Hence, the RA Hamiltonian for the two spin problem implements an exact counterdiabatic protocol in the rotated frame, which tracks the instantaneous rotated ground state $|{\\tilde{\\epsilon }_0(\\lambda )}\\rangle $ with perfect unit $\\tilde{F}(t)=\\big |\\langle {\\tilde{\\epsilon }_0(\\lambda )| \\psi (t)} \\rangle \\big |^2 = 1$ .", "The analytical expressions in Eq.", "(REF ) are a specific feature of two level systems.", "We Eq.", "(REF ) to benchmark the numerical sequential local optimization algorithm on the two spin problem.", "As in the main text we start from the UA protocol $J_0(t)=-1$ , $h_0(t)=5(1-\\lambda )$ , of total duration $\\tau =1$ .", "In Fig.", "REF a and Fig.", "REF b we compare the $M=100$ sequential local optimization data and the analytical expression for $\\beta (t)$ and $\\gamma (t)$ .", "The results show that the sequential local optimization points agree with the analytical expression.", "In Fig.", "REF c and Fig.", "REF d we show the original UA control fields and the resulting RA control fields respectively.", "In Fig.", "REF e we compare the instantaneous ground state's fidelity of an UA and a RA protocol of duration $\\tau =1$ .", "The diabatic transitions in the UA dynamics monotonically decreases the fidelity $F_{\\mathrm {\\scriptscriptstyle UA}}(t)$ (black dashed line) up to the final value $F_{\\mathrm {\\scriptscriptstyle UA}}(\\tau )=0.66$ .", "The RA protocol's fidelity $F_{\\mathrm { \\scriptscriptstyle RA}}(t)$ (solid blue line), has a non monotonic time dependence.", "In particular, although for intermediate times $F_{\\mathrm { \\scriptscriptstyle RA}}(t)$ is lower $F_{\\mathrm {\\scriptscriptstyle UA}}(t)$ , the RA protocol enhances the final fidelity up to $F_{\\mathrm { \\scriptscriptstyle RA}}(\\tau )=1$ .", "Indeed, the implemented RA driving, tracks the instantaneous rotated ground state $|{\\tilde{\\epsilon }_0(\\lambda )}\\rangle $ with perfect fidelity $\\tilde{F}_{\\mathrm { \\scriptscriptstyle RA}}(t)=1$ (light blue dot-dashed line)." ], [ "Rotated ansatz protocol for the quantum Ising chain ", "This section provides additional details on the transnational invariant quantum spin chain system studied in the main text (see Eq. ()).", "Refs.", "[1] used the least action principle to construct approximate local CD drivings for the spin chain.", "Here, we use the least action principle to design an approximate RA protocol.", "We use natural units and set $\\hbar =1$ , we indicate with $\\dot{x}(t)$ the time derivative of a given function $x(t)$ .", "The parametric Hamiltonian of the spin chain considered in Eq.", "() of the main text is $\\widehat{H}_{\\mathrm {spin-chain}} = J_0(\\lambda )\\widehat{H}_a + h_0(\\lambda )\\widehat{H}_b + b_0(\\lambda )\\widehat{H}_c \\;,$ with $\\widehat{H}_a = -\\sum _{j=1}^{N}\\hat{\\sigma }_j^z\\hat{\\sigma }_{j+1}^z, \\qquad \\widehat{H}_b = -\\sum _{j=1}^{N}\\hat{\\sigma }^x_j, \\qquad \\widehat{H}_c = - \\sum _{j=1}^{N}\\hat{\\sigma }^z_j\\;.$ We parametrize the auxiliary potentials as $\\widehat{\\mathcal {Q}}&=\\gamma \\widehat{H}_a + \\phi \\widehat{H}_c \\\\\\widehat{\\mathcal {K}}&=\\beta \\widehat{H}_b$ where $\\beta (t)$ , $\\gamma (t)$ and $\\phi (t)$ are variational parameters.", "The corresponding RA driving Hamiltonian reads $\\widehat{H}_{\\mathrm { \\scriptscriptstyle RA}}&= \\widehat{H}_{\\mathrm { 0}}+\\widehat{\\mathcal {K}}+ \\dot{\\widehat{\\mathcal {Q}}} = J_{\\mathrm { \\scriptscriptstyle RA}}(t)\\widehat{H}_a + h_{\\mathrm { \\scriptscriptstyle RA}}(t)\\widehat{H}_b + b_{\\mathrm { \\scriptscriptstyle RA}}(t) \\widehat{H}_c$ where $J_{\\mathrm { \\scriptscriptstyle RA}}(t) = J_0(t) + \\dot{\\gamma }(t)$ , $b_{\\mathrm { \\scriptscriptstyle RA}}(t) = b_0(t) + \\dot{\\phi }(t)$ and $h_{\\mathrm { \\scriptscriptstyle RA}}(t) = h_0(t)+\\beta (t)$ are the control fields of the RA protocol.", "To obtain the optimal values of $\\beta $ , $\\dot{\\phi }$ and $\\dot{\\gamma }$ , we must minimize the RA action." ], [ "RA adiabatic potential and RA action ", "We consider systems of $N\\ge 4$ spins.", "Using Pauli's matrices algebra to compute the RA adiabatic gauge potential, we get $\\dot{\\lambda }\\hat{\\mathcal {A}}&= \\mathrm {e}^{ \\frac{i}{\\hbar } \\widehat{\\mathcal {Q}}} \\left(\\widehat{H}_0 + \\widehat{\\mathcal {K}}\\right) \\mathrm {e}^{ -\\frac{i}{\\hbar } \\widehat{\\mathcal {Q}}} - \\widehat{H}_0=- (J_0+\\beta )\\sum _{j=1}^{N}\\, \\mathrm {e}^{ i\\gamma (\\hat{\\sigma }^z_{j-1}\\hat{\\sigma }^z_{j}+\\hat{\\sigma }^z_{j}\\hat{\\sigma }^z_{j+1}) +i\\phi \\hat{\\sigma }^z_{j}}\\hat{\\sigma }_j^x \\mathrm {e}^{ -i\\gamma (\\hat{\\sigma }^z_{j-1}\\hat{\\sigma }^z_{j}+\\hat{\\sigma }^z_{j}\\hat{\\sigma }^z_{j+1}) -i\\phi \\hat{\\sigma }^z_{j}} + h_0\\sum _{j=1}^{N} \\hat{\\sigma }_j^x\\nonumber \\\\&=\\alpha _x\\sum _{j=1}^{N}\\hat{\\sigma }_j^x+\\alpha _y\\sum _{j=1}^{N}\\hat{\\sigma }_j^y+\\alpha _{xz}\\sum _{j=1}^{N}(\\hat{\\sigma }_{j}^z\\hat{\\sigma }_{j+1}^x + \\hat{\\sigma }_j^x\\hat{\\sigma }_{j+1}^z)+\\alpha _{yz}\\sum _{j=1}^{N}(\\hat{\\sigma }_{j}^z\\hat{\\sigma }_{j+1}^y + \\hat{\\sigma }_j^y\\hat{\\sigma }_{j+1}^z)\\nonumber \\\\&\\hspace{28.45274pt}+\\alpha _{zxz}\\sum _{j=1}^{N}\\hat{\\sigma }_{j-1}^z\\hat{\\sigma }_j^x\\hat{\\sigma }_{j+1}^z+\\alpha _{zyz}\\sum _{j=1}^{N}\\hat{\\sigma }_{j-1}^z\\hat{\\sigma }_j^y\\hat{\\sigma }_{j+1}^z$ where we introduced the coefficients $\\alpha _x &= h_0-\\frac{h_0+\\beta }{2}\\cos 2\\phi \\,(\\cos 4\\gamma +1), \\qquad \\alpha _y = \\frac{h_0+\\beta }{2}\\sin 2\\phi \\,(\\cos 4\\gamma +1), \\qquad \\alpha _{xz} = \\frac{h_0+\\beta }{2}\\sin 2\\phi \\sin 4 \\gamma ,\\\\\\alpha _{yz} &= \\frac{h_0+\\beta }{2}\\cos 2\\phi \\sin 4\\gamma , \\qquad \\alpha _{zxz} = -\\frac{h_0+\\beta }{2} \\cos 2\\phi \\, (\\cos 4\\gamma -1), \\qquad \\alpha _{zyz} = \\frac{h_0+\\beta }{2} \\sin 2\\phi \\, (\\cos 4\\gamma -1) \\,.$ This RA adiabatic gauge potential explicitly contains terms proportional to $\\sum _{j}\\hat{\\sigma }_j^y$ , $\\sum _{j}(\\hat{\\sigma }_{j}^z\\hat{\\sigma }_{j+1}^y + \\hat{\\sigma }_j^y\\hat{\\sigma }_{j+1}^z)$ and $\\sum _{j}\\hat{\\sigma }_{j-1}^z\\hat{\\sigma }_j^y\\hat{\\sigma }_{j+1}^z$ , which are not available in the original parametric Hamiltonian.", "Finally, for $N\\ge 4$ , the action reads $\\frac{\\mathcal {S}(\\hat{\\mathcal {A}})}{N2^N}&=\\frac{\\operatorname{Tr}(\\widehat{G}_{\\lambda }^2)}{N2^N} =\\left(4 b^2+8 J^2\\right) \\alpha _x^2+\\left(4 b_0^2+8J_0^2\\right) \\alpha _y^2+\\alpha _{\\text{xz}}^2 \\left(8 b_0^2+8 h_0^2+32 J_0^2\\right)+\\alpha _{\\text{yz}}^2 \\left(8 b_0^2+32 h_0^2+32 J_0^2\\right)\\nonumber \\\\&\\hspace{28.45274pt}+\\left(4 b_0^2+8 h_0^2+8 J_0^2\\right)\\alpha _{\\text{zxz}}^2+ \\left(4 b_0^2+12h_0^2+8 J_0^2\\right)\\alpha _{\\text{zyz}}^2+\\alpha _y\\left(\\left(8 b_0^2+16 J_0^2\\right) \\alpha _x+32 b_0 J_0 \\alpha _{\\text{xz}}+16 J_0^2 \\alpha _{\\text{zxz}}\\right)\\nonumber \\\\&\\hspace{28.45274pt}+\\alpha _x \\left(32 b_0 J_0 \\alpha _{\\text{xz}}+16J_0^2 \\alpha _{\\text{zxz}}\\right)+32 b_0 J_0 \\alpha _{\\text{xz}} \\alpha _{\\text{zxz}}+32 b_0 J_0 \\alpha _{\\text{yz}} \\alpha _{\\text{zyz}}+\\left(8 h_0 \\dot{J}_0-8 \\dot{h}_0 J_0\\right) \\alpha _{\\text{yz}}\\nonumber \\\\&\\hspace{28.45274pt}+\\dot{b}_0^2+\\dot{h}_0^2+\\dot{J}_0^2\\,.$ Eq.", "(REF ) reveals that, due to the Hamiltonian's translation symmetry and locality, the ratio $\\frac{\\mathcal {S}^{\\mathrm { \\scriptscriptstyle RA}}(\\widehat{\\mathcal {Q}},\\widehat{\\mathcal {K}})}{N2^N}$ is not $N$ dependent.", "This implies that the optimal RA coupling functions $J^{\\mathrm { \\scriptscriptstyle RA}}(t)$ , $b^{\\mathrm { \\scriptscriptstyle RA}}(t)$ and $h^{\\mathrm { \\scriptscriptstyle RA}}(t)$ are also $N$ independent, and that the RA protocol can be computed for arbitrarily large systems (with $N\\rightarrow \\infty $ )." ], [ "RA action minimization and RA protocols", "The numerical minimization of the action Eq.", "(REF ) with respect to $\\beta $ , $\\gamma $ and $\\phi $ leads to the results presented in the main text." ], [ "Rotated ansatz protocol for quantum annealing", "In this section, we compute the RA action for transverse field quantum annealing [6] for a system of $N$ $\\frac{1}{2}$ -spins.", "We can write the parametric Hamiltonian for transverse field quantum annealing as $\\widehat{H}_{\\mathrm { 0}}(\\lambda ) = \\widehat{H}_z(\\lambda ) -\\sum _{j=1}^{N}h_{0,j}(\\lambda )\\hat{\\sigma }^x_j\\,$ where $h_{0,j}(\\lambda )$ are local control fields of the parametric Hamiltonian and $\\widehat{H}_z(\\lambda )$ is a $\\lambda $ -depend operator, which is diagonal in the computational $z$ -basis.", "Here, and in the following, we say that an operator $\\widehat{D}_z$ is diagonal if $[\\widehat{D}_z, \\hat{\\sigma }^z_j]=0$ for $j=1,2,\\dots , N$ .", "We parameterize the auxiliary potentials as $\\widehat{\\mathcal {Q}}&= \\widehat{\\mathcal {Q}}_z\\\\\\widehat{\\mathcal {K}}(\\beta ) &= -\\sum _{j=1}^{N}\\beta _j\\hat{\\sigma }^x_j\\;.$ where $\\beta _j(t)$ , for $j=1,2,\\dots , N$ , are $N$ real time-dependent variational parameters and $\\widehat{\\mathcal {Q}}_z(t)$ is an operator diagonal in the computational $z$ -basis.", "Me make no further assumption on $\\widehat{\\mathcal {Q}}_z(t)$ .", "The corresponding RA driving Hamiltonian reads $\\widehat{H}_{\\mathrm { \\scriptscriptstyle RA}}= \\widehat{H}_{\\mathrm { 0}}+\\widehat{\\mathcal {K}}+ \\dot{\\widehat{\\mathcal {Q}}}= \\widehat{H}_{\\mathrm { \\scriptscriptstyle RA},z}(t) - \\sum _{j=1}^{N} h_{\\mathrm { \\scriptscriptstyle RA},j}(t)\\hat{\\sigma }^x_j$ where $h_{\\mathrm { \\scriptscriptstyle RA}, j}(t) = h_{0,j}(t) + \\beta _{j}(t)$ are local control fields of the RA protocol and $\\widehat{H}_{\\mathrm { \\scriptscriptstyle RA},z}(t) = \\widehat{H}_z(t) + \\dot{\\widehat{\\mathcal {Q}}}_z(t)$ is the RA time dependent diagonal Hamiltonian.", "To obtain the optimal values of $\\beta _j$ and $\\dot{\\gamma }$ , we must minimize the RA with respect to the optimal values of $\\beta ^{j}(t)$ and $\\widehat{H}_{\\mathrm { \\scriptscriptstyle RA},z}(t)$ we must minimize the RA action with respect to the variational parameters." ], [ "RA adiabatic potential and RA action ", "To compute the action, we rely on a convenient decomposition of diagonal operators.", "Let $\\widehat{D}_z$ be an arbitrary diagonal operator.", "We define $\\widehat{D}^{[j]}_z &\\equiv \\frac{1}{2} \\hat{\\sigma }_j^z\\widehat{D}_z +i\\frac{1}{4}\\left(\\hat{\\sigma }_j^x\\widehat{D}_z\\hat{\\sigma }_j^y - \\hat{\\sigma }_j^y\\widehat{D}_z\\hat{\\sigma }_j^x\\right)\\qquad \\mathrm {for}\\; j=1, 2\\dots , N \\\\\\widehat{D}^{[-j]}_z&\\equiv \\frac{1}{2} \\widehat{D}_z +\\frac{1}{2}\\hat{\\sigma }_j^x\\widehat{D}_z\\hat{\\sigma }_j^x\\qquad \\mathrm {for}\\; j=1, 2\\dots , N\\;.$ The operators $\\widehat{D}^{[j]}_z$ and $\\widehat{D}^{[j]}_z$ then satisfy properties (i) $\\widehat{D}_z = \\widehat{D}_z^{[j]}\\hat{\\sigma }_j^z + \\widehat{D}_z^{[-j]}$ for $j=1, 2, \\dots , N$         (desired decomposition) (ii) $[ \\widehat{D}_{z}^{[\\pm j]},\\hat{\\sigma }_k^z]=0$ for $k=1, 2, \\dots , N$        ($\\widehat{D}^{[\\pm j]}_z$ is are diagonal operators) (iii) $[ \\widehat{D}_{z}^{[\\pm j]},\\hat{\\sigma }_j^x]=[ \\widehat{D}_{z}^{[\\pm j]},\\hat{\\sigma }_j^y]=[ \\widehat{D}_{z}^{[\\pm j]},\\hat{\\sigma }_j^z]=0$        ($\\widehat{D}^{[\\pm j]}_z$ do not involve $\\hat{\\sigma }_j^z$ ) The properties (i),(ii) and (iii) additionally imply the following identities (iv) $[\\widehat{D}_z,\\hat{\\sigma }_j^x]= i2\\widehat{D}_z^{[j]}\\hat{\\sigma }_j^y$ and $ [\\widehat{D}_z,\\hat{\\sigma }_j^y] = -i2\\widehat{D}_z^{[j]}\\hat{\\sigma }_j^x\\;.", "$ (v) $\\big (\\cos \\widehat{D}_z\\big )^{[k]}=-\\sin \\big (\\widehat{D}_z^{[k]} \\big )\\sin \\big (\\widehat{D}_z^{[-k]} \\big )$ and $\\big (\\sin \\widehat{D}_z\\big )^{[k]} =\\sin \\big (\\widehat{D}_z^{[k]} \\big )\\cos \\big (\\widehat{D}_z^{[-k]} \\big )$ (vi) $\\widehat{D}_z^{[j][k]} = \\widehat{D}_z^{[k][j]}$ and $\\widehat{D}_z^{[j][j]} = \\hat{0}\\;.", "$ The rationale behind the decomposition is to explicitly factor out the $\\hat{\\sigma }_j^z$ from the diagonal operator.", "Using (i)-(iv), the RA for the gauge potential (defined in Eq.", "() of the main text) reads ${\\dot{\\lambda }}\\hat{\\mathcal {A}}&=\\mathrm {e}^{ \\frac{i}{\\hbar } \\widehat{\\mathcal {Q}}} \\left(\\widehat{H}_{\\mathrm { 0}}+\\widehat{\\mathcal {K}}\\right)\\mathrm {e}^{ -\\frac{i}{\\hbar } \\widehat{\\mathcal {Q}}}- \\widehat{H}_{\\mathrm { 0}}=-\\sum _{j=1}^{N}(h_{0,j}+\\beta _j)\\mathrm {e}^{ i\\widehat{\\mathcal {Q}}_z }\\hat{\\sigma }_j^x\\mathrm {e}^{ -i \\widehat{\\mathcal {Q}}_z}+\\sum _{j=1}^{N}h_{0,j}\\hat{\\sigma }_j^x\\\\&=-\\sum _{j=1}^{N}\\tilde{\\beta }_j \\cos \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\,\\hat{\\sigma }_j^x + \\sum _{j=1}^{N}\\tilde{\\beta }_j\\sin \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\,\\hat{\\sigma }_j^y +\\sum _{j=1}^{N}h_{0,j}\\hat{\\sigma }_j^x\\;,$ where we introduced the variables $\\tilde{\\beta }_j(t)=\\beta _j(t)+h_{0,j}(t)$ .", "Similarly, using (iv) and (v), the operator $\\widehat{G}$ reads $\\dot{\\lambda } \\widehat{G}&= \\dot{\\widehat{H}}_0 +\\frac{i}{\\hbar }\\left[\\dot{\\lambda }\\hat{\\mathcal {A}},\\widehat{H}_0\\right]\\nonumber \\\\&=\\dot{\\widehat{H}}_z - \\sum _{j=1}^{N}\\dot{h}_{0,j}\\hat{\\sigma }^x_j +i\\left[\\sum _{j=1}^{N}h_{0,j}\\hat{\\sigma }_j^x+\\sum _{j=1}^{N}\\tilde{\\beta }_j \\left(\\cos \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\hat{\\sigma }_j^x - \\sin \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\hat{\\sigma }_j^y\\right),\\widehat{H}_z-\\sum _{j=1}^{N}h_{0,j}\\hat{\\sigma }^x_j\\right]\\nonumber \\\\&= \\dot{\\widehat{H}}_z - \\sum _{j=1}^{N}\\dot{h}_{0,j}\\hat{\\sigma }^x_j +2\\sum _{j=1}^{N}\\left(h_{0,j}\\hat{\\sigma }_j^y-\\tilde{\\beta }_j\\cos \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\hat{\\sigma }_j^y -\\tilde{\\beta }_j \\sin \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\hat{\\sigma }_j^x\\right)\\widehat{H}_z^{[j]}\\nonumber \\\\&\\hspace{28.45274pt}+2\\sum _{j=1}^{N}\\sum _{k=1}^{N}h_{0,j}\\tilde{\\beta }_k \\left(\\left(\\cos \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\right)^{[j]}\\hat{\\sigma }_j^y\\hat{\\sigma }_k^x-\\left(\\sin \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\right)^{[j]}\\hat{\\sigma }_j^y\\hat{\\sigma }_k^y+\\sin \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\,\\hat{\\sigma }_k^z\\right)\\nonumber \\\\&=\\dot{\\widehat{H}}_z -2\\sum _{j=1}^{N} \\tilde{\\beta }_j h_{0,j} \\sin \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\nonumber \\hat{\\sigma }_j^z- \\sum _{j=1}^{N}\\left( 2\\tilde{\\beta }_j \\widehat{H}_z^{[j]}\\sin \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)+\\dot{h}_{0,j}\\right)\\hat{\\sigma }^x_j\\nonumber \\\\&\\hspace{28.45274pt}+2\\sum _{j=1}^{N}\\left(h_{0,j}\\widehat{H}_z^{[j]}-\\tilde{\\beta }_j\\cos \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\widehat{H}_z^{[j]}\\right)\\hat{\\sigma }_j^y+2\\sum _{j=1}^{N}\\sum _{k=1}^{N}\\tilde{\\beta }_j h_{0,k}\\sin \\left(2\\widehat{\\mathcal {Q}}_z^{[j][k]}\\right)\\sin \\left(2\\widehat{\\mathcal {Q}}_z^{[j][-k]}\\right)\\hat{\\sigma }_k^y\\hat{\\sigma }_j^x\\nonumber \\\\&\\hspace{28.45274pt}+2\\sum _{j=1}^{N}\\sum _{k=1}^{N}\\tilde{\\beta }_j h_{0,k}\\sin \\left(2\\widehat{\\mathcal {Q}}_z^{[j][k]}\\right)\\cos \\left(2\\widehat{\\mathcal {Q}}_z^{[j][-k]}\\right)\\hat{\\sigma }_k^y\\hat{\\sigma }_j^y$ Finally, we use properties (i), (v) and (vi) to compute the RA action $\\mathcal {S}&= \\operatorname{Tr}(\\widehat{G}^2) \\nonumber \\\\&=\\operatorname{Tr}\\Bigg (\\;\\left(\\dot{\\widehat{H}}_z -2\\sum _{j=1}^{N} \\tilde{\\beta }_j h_{0,j} \\sin \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\hat{\\sigma }_j^z\\right)^2+ \\sum _{j=1}^{N}\\left(\\dot{h}_{0,j} +2\\tilde{\\beta }_j \\sin \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\widehat{H}_z^{{[j]}}\\right)^2\\nonumber \\\\&\\hspace{28.45274pt}+4\\sum _{j=1}^{N}\\left(h_{0,j}\\widehat{H}_z^{[j]}-\\tilde{\\beta }_j\\cos \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\widehat{H}_z^{[j]}\\right)^2+4\\sum _{j=1}^{N}\\sum _{k=1}^{N}\\tilde{\\beta }_j^2 h_{0,k}^2\\sin ^2 \\left(2\\widehat{\\mathcal {Q}}_z^{[j][k]}\\right)\\sin ^2 \\left(2\\widehat{\\mathcal {Q}}_z^{[j][-k]}\\right)\\nonumber \\\\&\\hspace{28.45274pt}+4\\sum _{j=1}^{N}\\sum _{k=1}^{N}\\tilde{\\beta }_j^2 h_{0,k}^2\\sin ^2 \\left(2\\widehat{\\mathcal {Q}}_z^{[j][k]}\\right)\\cos ^2 \\left(2\\widehat{\\mathcal {Q}}_z^{[j][-k]}\\right)\\nonumber \\\\&\\hspace{28.45274pt}+4\\sum _{j=1}^{N}\\sum _{k=1}^{N}h_{0,j}\\tilde{\\beta }_j h_{0,k}\\tilde{\\beta }_k \\sin ^2 \\left(2\\widehat{\\mathcal {Q}}_z^{[j][k]}\\right)\\cos \\left(2\\widehat{\\mathcal {Q}}_z^{[j][-k]}\\right)\\cos \\left(2\\widehat{\\mathcal {Q}}_z^{[k][-j]}\\right)\\Bigg )\\nonumber \\\\&=\\operatorname{Tr}\\Bigg (\\left(\\dot{\\widehat{H}}_z \\right)^2 -4\\sum _{j=1}^{N}\\tilde{\\beta }_j h_{0,j} \\dot{\\widehat{H}}_z^{[j]}\\sin \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)+8\\sum _{j=1}^{N}\\sum _{k=1}^{j-1} \\tilde{\\beta }_j h_{0,j}\\tilde{\\beta }_k h_{0,k} \\sin ^2 \\left(2\\widehat{\\mathcal {Q}}_z^{[j][k]}\\right)\\cos \\left(2\\widehat{\\mathcal {Q}}_z^{[j][-k]}\\right)\\cos \\left(2\\widehat{\\mathcal {Q}}_z^{[k][-j]}\\right)\\nonumber \\\\&\\hspace{28.45274pt}+4\\sum _{j=1}^{N} \\tilde{\\beta }_j^2 h_{0,j}^2 \\sin ^2 \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right) + 4\\sum _{j=1}^{N} \\dot{h}_{0,j}\\tilde{\\beta }_j \\widehat{H}_z^{{[j]}}\\sin \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right) + 4\\sum _{j=1}^{N} \\tilde{\\beta }^2_j \\left(\\widehat{H}_z^{{[j]}}\\right)^2\\sin ^2 \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\nonumber \\\\&\\hspace{28.45274pt}+4\\sum _{j=1}^{N}\\left(\\widehat{H}_z^{[j]}\\right)^2\\bigg (\\tilde{\\beta }_j^2\\cos ^2\\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)-2h_{0,j}\\tilde{\\beta }_j\\cos \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)+h_{0,j}^2\\bigg )+4\\sum _{j=1}^{N}\\sum _{k=1}^{N}\\tilde{\\beta }_j^2 h_{0,k}^2\\sin ^2 \\left(2\\widehat{\\mathcal {Q}}_z^{[j][k]}\\right)\\nonumber \\\\&\\hspace{28.45274pt}+4\\sum _{j=1}^{N}\\sum _{k=1}^{N}h_{0,j}\\tilde{\\beta }_j h_{0,k}\\tilde{\\beta }_k \\sin ^2 \\left(2\\widehat{\\mathcal {Q}}_z^{[j][k]}\\right)\\cos \\left(2\\widehat{\\mathcal {Q}}_z^{[j][-k]}\\right)\\cos \\left(2\\widehat{\\mathcal {Q}}_z^{[k][-j]}\\right)\\;\\Bigg )$ After using property (vi) and the identity $1 - \\cos X=2\\sin ^2X$ to rearrange the terms in the sum, the final expression for the action reads $\\mathcal {S}&= \\operatorname{Tr}\\Bigg (\\dot{\\widehat{H}}_z^2 +\\sum _{j=1}^{N}\\dot{h}_{0,j}^2 +4\\sum _{j=1}^{N} \\beta _j^2 \\left(\\widehat{H}_z^{{[j]}}\\right)^2+16\\sum _{j=1}^{N}h_{0,j}\\tilde{\\beta }_j\\left(\\widehat{H}_z^{[j]}\\right)^2\\sin ^2\\left(\\widehat{\\mathcal {Q}}_z^{[j]}\\right)+4\\sum _{j=1}^{N}h_{0,j}^2 (\\beta _j+h_{0,j})^2 \\sin ^2 \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\nonumber \\\\&\\hspace{28.45274pt}-4\\sum _{j=1}^{N}(\\beta _j+h_{0,j}) \\left(h_{0,j}\\dot{\\widehat{H}}_z^{[j]} - \\dot{h}_{0,j}\\widehat{H}_z^{{[j]}}\\right) \\sin \\left(2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)+4\\sum _{j=1}^{N}\\sum _{k=1}^{N}(\\beta _j+h_{0,j})^2 h_{0,k}^2\\sin ^2 \\left(2\\widehat{\\mathcal {Q}}_z^{[j][k]}\\right)\\nonumber \\\\&\\hspace{28.45274pt}-\\sum _{j=1}^{N}\\sum _{k=1}^{N}h_{0,j}h_{0,k}(\\beta _j+h_{0,j}) (\\beta _k+h_{0,k}) \\sin ^2 \\left(2\\widehat{\\mathcal {Q}}_z^{[j][k]}\\right)\\cos \\left(2\\widehat{\\mathcal {Q}}_z^{[j][-k]}\\right)\\cos \\left(2\\widehat{\\mathcal {Q}}_z^{[k][-j]}\\right)\\Bigg )\\;.$" ], [ "RA action minimization and RA protocols", "Eq.", "(REF ) is the starting point to compute the RA action in various models.", "However, a numerical evaluation of the action involves computing traces of various $2^N\\times 2^N$ diagonal matrices.", "To appreciate the complexity of evaluating the action we use the identity $\\mathrm {e}^{iX}=\\cos X + i\\sin X$ into Eq.", "(REF ), which leads to $\\mathcal {S}&=\\Re \\operatorname{Tr}\\Bigg (\\dot{\\widehat{H}}_z^2 +\\sum _{j=1}^{N}\\dot{h}_{0,j}^2 +4\\sum _{j=1}^{N} \\beta _j^2 \\left(\\widehat{H}_z^{{[j]}}\\right)^2+8\\sum _{j=1}^{N}h_{0,j}\\tilde{\\beta }_j\\left(\\widehat{H}_z^{[j]}\\right)^2\\left(1-\\exp \\left(i2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\right)\\nonumber \\\\&\\hspace{28.45274pt}+2\\sum _{j=1}^{N}h_{0,j}^2 (\\beta _j+h_{0,j})^2 \\left( 1 - \\exp \\left(i4\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\right)+i4\\sum _{j=1}^{N}(\\beta _j+h_{0,j}) \\left(h_{0,j}\\dot{\\widehat{H}}_z^{[j]} - \\dot{h}_{0,j}\\widehat{H}_z^{{[j]}}\\right) \\exp \\left(i2\\widehat{\\mathcal {Q}}_z^{[j]}\\right)\\nonumber \\\\&\\hspace{28.45274pt}+2\\sum _{j=1}^{N}\\sum _{k=1}^{N}(\\beta _j+h_{0,j})^2 h_{0,k}^2\\left( 1 - \\exp \\left(i4\\widehat{\\mathcal {Q}}_z^{[j][k]}\\right)\\right)\\nonumber \\\\&\\hspace{28.45274pt}-\\sum _{j=1}^{N}\\sum _{k=1}^{N}h_{0,j}h_{0,k}(\\beta _j+h_{0,j}) (\\beta _k+h_{0,k}) \\sum _{s,u,v=\\pm 1}s \\exp \\left(i2\\left((1+s) \\widehat{\\mathcal {Q}}_z^{[j][k]}+u \\widehat{\\mathcal {Q}}_z^{[j][-k]}+v\\widehat{\\mathcal {Q}}_z^{[k][-j]}\\right)\\right)\\Bigg )\\;,\\;,$ where $\\Re (x)$ indicates the real part of the complex number $x$ .", "There are at least two relevant cases for which the RA action (REF ) can be computed efficiently: (a) quadratic Hamiltonians: $\\widehat{H}_z$ and $\\widehat{\\mathcal {Q}}_z$ contain only single-spin and two-spin terms.", "(b) finite range Hamiltonians: both in $\\widehat{H}_z$ and in $\\widehat{\\mathcal {Q}}_z$ each spin interacts with at most $\\ell $ other spins, with $\\ell $ being an $N$ independent natural number.", "The quadratic Hamiltonian realized on DWAVE annealers [7] belongs to (a), while the Hamiltonian associated with parity (or LHZ) architectures [8] belongs to (b).", "In the following, we will discuss the two cases separately." ], [ "Quadratic architectures", "The relevant Hamiltonian terms for the QUBO annealing schedules are $\\widehat{H}_p = -\\frac{1}{2}\\sum _{j=1}^{N}\\sum _{k=1}^{N}J_{jk}\\hat{\\sigma }^z_j\\hat{\\sigma }_k^z-\\sum _{j=1}^{N}J_{j0}\\hat{\\sigma }^z_j , \\qquad \\widehat{H}_x = -\\sum _{j=1}^{N}\\hat{\\sigma }^x_j,$ where the couplings $J_{jk}$ , for $j,k=0,1,\\dots ,N$ encode the optimization problem.", "Without loss of generality we assume that $J_{jk}=J_{kj}$ and $J_{jk}=0$ .", "The parametric QA Hamiltonian for the QUBO problem is $\\widehat{H}_{\\mathrm {QUBO}} = A_0(\\lambda )\\widehat{H}_p + B_0(\\lambda )\\widehat{H}_x\\;,$ where $A_0$ and $B_0$ are the QA control fields.", "We parametrize the auxiliary fields as $\\widehat{\\mathcal {Q}}=\\gamma \\widehat{H}_p$ and $\\widehat{\\mathcal {K}}=\\beta \\widehat{H}_x$ .", "Then, the corresponding RA driving Hamiltonian reads $\\widehat{H}_{\\mathrm { \\scriptscriptstyle RA}}= \\widehat{H}_{\\mathrm { 0}}+\\widehat{\\mathcal {K}}+ \\dot{\\widehat{\\mathcal {Q}}}= A_{\\mathrm { \\scriptscriptstyle RA}}(t)\\widehat{H}_p + B_{\\mathrm { \\scriptscriptstyle RA}}(t)\\widehat{H}_x$ where $A_{\\mathrm { \\scriptscriptstyle RA}}(t) = A_0(t) + \\dot{\\gamma }(t)$ and $B_{\\mathrm { \\scriptscriptstyle RA}}(t) = B_0(t)+\\beta (t)$ are the control fields of the RA protocol.", "To compute the RA action, we first substitute $\\widehat{\\mathcal {Q}}_z=\\gamma (t)\\widehat{H}_p $ , $ h_{0,j}(t) = B_0(t)$ , $\\widehat{H}_z(t)=A_0(t)\\widehat{H}_p$ and $\\beta _j = \\beta (t)$ in Eq.", "(REF ), The final expression of the RA action for the QUBO problem reads $\\mathcal {S}(\\beta ,\\gamma )&=2^{N}\\Bigg (\\frac{1}{2}\\dot{A}^2_0\\sum _{j=0}^{N}\\sum _{k=0}^{N}J_{jk}^2+N\\dot{B}^2_0 + 4\\beta ^2A_0^2\\sum _{j=1}^{N}\\sum _{k=0}^{N} J_{jk}^2 +8A_0^2B_0(\\beta +B_0)\\sum _{j=1}^{N}\\sum _{k=0}^{N} J_{jk}^2\\left(1 - f_j(\\gamma )\\right)\\nonumber \\\\&\\hspace{28.45274pt} +16A_0^2B_0(\\beta +B_0)\\sum _{j=1}^{N}\\sum _{k=0}^{N}\\sum _{l=0}^{k-1}J_{jk}J_{jl}\\tan (2\\gamma J_{jk})\\tan (2\\gamma J_{jl})f_j(\\gamma )+2 B_0^2(\\beta +B_0)^2\\sum _{j=1}^{N} \\left(1-f_j(2\\gamma )\\right)\\nonumber \\\\&\\hspace{28.45274pt}-4(\\beta +B_0) (B_0\\dot{A}_0 -\\dot{B}_0A_0)\\sum _{j=0}^{N}\\sum _{k=0}^{N}J_{jk} \\tan (2 J_{jk}\\gamma )f_j(\\gamma )+4(\\beta +B_0)^2B_0^2\\sum _{j=1}^{N}\\sum _{k=1}^{N}\\sin ^2 \\left(2 J_{jk}\\gamma \\right)\\nonumber \\\\&\\hspace{28.45274pt}+4(\\beta +B_0)^2B_0^2\\sum _{j=1}^{N}\\sum _{k=1}^{N}\\tan ^2 \\left(2J_{jk}\\gamma \\right)\\left(g_{+,jk}(\\gamma )+g_{-,jk}(\\gamma )\\right)\\Bigg )\\;,$ where we defined the functions $f_{j}(\\gamma ) = \\prod _{m=0}^{N}\\cos (2 J_{jm}\\gamma )$ and $ g_{\\pm ,jk}(\\gamma ) = \\prod _{m=0}^{N}\\cos (2J_{jm}\\gamma \\pm 2J_{km}\\gamma )$ , and we used the following identity $\\frac{1}{2^N}\\operatorname{Tr}\\left(\\mathrm {e}^{ i\\sum _{m=1}^{N}\\theta _m\\hat{\\sigma }_m^z}\\right) = \\prod _{m=1}^{N}\\cos (\\theta _m)$ .", "Using Eq.", "(REF ), the computational complexity of evaluating the RA action is as $\\mathcal {O}(N^3)$ .", "Thus, computing RA protocol numerically (with a sequential local optimization) will only result in a polynomial overhead in the number of qubits $N$ .", "We benchmark the RA approach on a spin-glass problem with uniform random couplings $J_{jk}\\in [-1,1]$ (for $j\\ne k$ ).", "We start from the UA protocol, shown in Fig.", "REF (a).", "We first analyze a single problem instance of $N=8$ qubits.", "Figure REF (b) shows the optimal RA control fields obtained by minimizing the action in Eq.", "(REF ).", "In Fig.", "REF (c), we compare the instantaneous ground state's fidelity of the UA, the local CD, and the RA protocols on the same problem instance.", "We find that the RA tracks the rotated ground states with higher fidelity than the UA and local CD drivings.", "Fig.", "REF (d) shows the relative improvement of the RA and local CD, averaged over an ensemble of 100 random problems with different qubits numbers $N=3,4,5,\\dots ,15$ .", "The data suggest that the features observed for $N=6$ hold for larger systems.", "As observed in Ref.", "[9], the local CD driving shows no scaling advantage over the UA driving.", "However, the RA protocol provides a scaling advantage over the UA protocol.", "Figure: Quantum annealing on the quadratic architecture.", "(a) and (b) control fields for the UA and RA protocols.", "(c) ground state fidelity F(t)F(t) for the UA, local CD and RA protocols, and rotated ground state fidelity F ˜(t)\\tilde{F}(t) for the RA protocol.", "The data in (a), (b) and (c) refers to a system of N=8N=8 spins with random couplings J jk ∈[-1,1]J_{jk}\\in [-1,1], and to protocols of duration τ=1\\tau =1.", "The inset of (c) shows the fully connected quadratic architecture.", "(d) shows the the relative improvement the local CD F local - CD (τ)/F UA (τ)F_{\\mathrm { \\scriptscriptstyle local-CD}}(\\tau )/F_{UA}(\\tau ) and for the RA protocol F RA (τ)/F UA (τ)F_{\\mathrm { \\scriptscriptstyle RA}}(\\tau )/F_{UA}(\\tau ) as a function of the number of qubits.", "The points in (d) were averaged over 100 different instances random couplings J jk ∈[-1,1]J_{jk}\\in [-1,1].", "The shaded region represents the points between the 25th and 75th percentiles.", "The protocols duration is τ=1\\tau =1." ], [ "Parity architectures", "The relevant Hamiltonian terms for the LHZ annealing schedules are $\\widehat{H}_p = -\\sum _{k=1}^NJ_k\\hat{\\sigma }_k^z, \\qquad \\widehat{H}_c=-\\sum _{l}^{L}\\hat{H}_{\\vspace{-1.75003pt}[line cap=round, line join=round]{[black] (0ex,0ex) -- (0ex,0.8ex) -- (0.8ex,0.8ex) -- (0.8ex,0ex) -- cycle;[color=black, fill=black] (0ex,0ex) circle (0.175ex);[color=black, fill=black] (0ex,0.8ex) circle (0.175ex);[color=black, fill=black] (0.8ex,0ex) circle (0.175ex);[color=black, fill=black] (0.8ex,0.8ex) circle (0.175ex);},l} \\qquad \\widehat{H}_x = -\\sum _{j=1}^{N}\\hat{\\sigma }^x_j,$ where the couplings $J_{j}$ , for $j=0,1,\\dots ,N$ encode the optimization problem.", "The parametric QA Hamiltonian for the LHZ driving is $\\widehat{H}_{\\mathrm {LHZ}} = A_0(\\lambda )\\widehat{H}_p + B_0(\\lambda )\\widehat{H}_x+C_0(t)\\widehat{H}_c\\;,$ Where $A_0$ ,$B_0$ and $C_0$ are the QA control fields.", "We parametrize the auxiliary fields as $\\widehat{\\mathcal {Q}}=\\gamma \\widehat{H}_p+\\phi \\widehat{H}_c$ and $\\widehat{\\mathcal {K}}=\\beta \\widehat{H}_x$ .", "Then, the corresponding RA driving Hamiltonian reads $\\widehat{H}_{\\mathrm { \\scriptscriptstyle RA}}= \\widehat{H}_{\\mathrm { 0}}+\\widehat{\\mathcal {K}}+ \\dot{\\widehat{\\mathcal {Q}}}= A_{\\mathrm { \\scriptscriptstyle RA}}(t)\\widehat{H}_p + B_{\\mathrm { \\scriptscriptstyle RA}}(t)\\widehat{H}_x+C_{\\mathrm { \\scriptscriptstyle RA}}(t)\\widehat{H}_c$ where $A_{\\mathrm { \\scriptscriptstyle RA}}(t) = A_0(t) + \\dot{\\gamma }(t)$ , $C_{\\mathrm { \\scriptscriptstyle RA}}(t) = C_0(t) + \\dot{\\phi }(t)$ and $B_{\\mathrm { \\scriptscriptstyle RA}}(t) = B_0(t)+\\beta (t)$ are the control fields of the RA protocol.", "To compute the RA action, we first substitute $\\widehat{\\mathcal {Q}}_z=\\gamma (t)\\widehat{H}_p +\\phi \\widehat{H}_c$ , $ h_{0,j}(t) = B_0(t)$ , $\\widehat{H}_z(t)=A_0(t)\\widehat{H}_p +C_0(t)\\widehat{H}_c$ and $ \\beta _j(t) = \\beta (t)$ in Eq.", "(REF ).", "The final expression of the RA action for the QUBO problem reads $\\mathcal {S}(\\beta ,\\gamma ,\\phi )&=2^N\\Bigg (\\dot{A}^2_0\\sum _{\\mu =1}^{N}J_{\\mu }^2+N\\dot{B}_0^2+L\\dot{C}_0^2 +4\\left(B_{0}^2+(\\beta +B_0)^2\\right)\\sum _{\\mu =1}^{N} \\left(A^2_0J_{\\mu }^2+L^{[\\mu ]} C^2_0\\right)\\nonumber \\\\&\\hspace{28.45274pt}-8B_0(\\beta +B_0)\\sum _{\\mu =1}^{N}\\cos (2J_\\mu \\gamma )\\left(A_0^2J_\\mu ^2+C_0^2\\left(L^{[\\mu ]}-L^{[\\mu ]}(L^{[\\mu ]}-1)\\tan ^2(2\\phi ) \\right)\\right)\\cos ^{L^{[\\mu ]}} (2\\phi )\\nonumber \\\\&\\hspace{28.45274pt}+16A_0B_0C_0(\\beta +B_0)\\sum _{\\mu =1}^{N}J_\\mu L^{[\\mu ]}\\sin (2J_\\mu \\gamma )\\tan (2\\phi )\\cos ^{L^{[\\mu ]}} (2\\phi )\\nonumber \\\\&\\hspace{28.45274pt}+2(\\beta +B_0)^2 B_{0}^2\\sum _{\\mu =1}^{N} \\left( 1 - \\cos (4J_\\mu \\gamma )\\cos ^{L^{[\\mu ]}} (4\\phi )\\right)\\nonumber \\\\&\\hspace{28.45274pt}-4\\left(B_{0}\\dot{C}_0 - \\dot{B}_{0}C_0\\right)(\\beta +B_0)\\sum _{\\mu =1}^{N} L^{[\\mu ]}\\cos (2J_\\mu \\gamma )\\tan (2\\phi )\\cos ^{L^{[\\mu ]}} (2\\phi )\\nonumber \\\\&\\hspace{28.45274pt}-4\\left(B_{0}\\dot{A}_0 - \\dot{B}_{0}A_0\\right)(\\beta +B_0)\\sum _{\\mu =1}^{N} J_\\mu \\sin (2J_\\mu \\gamma )\\cos ^{L^{[\\mu ]}} (2\\phi ) +(\\beta +B_0)^2 B_{0}^2\\sum _{\\langle \\mu ,\\nu \\rangle }\\left( 1 - \\cos ^{L^{[\\mu ][\\nu ]}} (4\\phi )\\right)\\nonumber \\\\&\\hspace{28.45274pt}+2B_{0}^2(\\beta +B_0)^2\\sum _{\\langle \\mu ,\\nu \\rangle } \\cos (2J_\\mu \\gamma +2J_\\nu \\gamma ) \\left(1-\\cos ^{L^{[\\mu ][\\nu ]}} (4\\phi )\\right)( \\cos 2\\phi )^{L^{[\\mu ][-\\nu ]}+L^{[\\nu ][-\\mu ]}}\\Bigg )\\;,$ Here $L$ is the total number of constrain terms, $L^{[\\mu ]}$ is the number of constrain terms involving site $\\mu $ , $L^{[\\mu ][\\nu ]}$ is the number of constrain terms involving both sites $\\mu $ and $\\nu $ , and $L^{[\\mu ][-\\nu ]}$ is the number of constrain terms involving site $\\mu $ but not site $\\nu $ .", "Consequently, natural numbers $L$ , $L^{[\\mu ]}$ and $L^{[\\mu ][\\nu ]}$ depend only on the architecture, not on the problem instance.", "The notation $\\sum _{\\langle \\mu ,\\nu \\rangle }$ indicates a sum over all pairs of interacting spins in the LHZ architecture.", "To obtain Eq.", "(REF ), we used the identity $\\frac{1}{2^N}\\operatorname{Tr}\\left( \\mathrm {e}^{i\\sum _{l=1}^L\\theta _l\\hat{H}_{\\vspace{-1.75003pt}[line cap=round, line join=round]{[black] (0ex,0ex) -- (0ex,0.8ex) -- (0.8ex,0.8ex) -- (0.8ex,0ex) -- cycle;[color=black, fill=black] (0ex,0ex) circle (0.175ex);[color=black, fill=black] (0ex,0.8ex) circle (0.175ex);[color=black, fill=black] (0.8ex,0ex) circle (0.175ex);[color=black, fill=black] (0.8ex,0.8ex) circle (0.175ex);},l}^{[\\mu ]} }\\right) = \\prod _{l \\in \\mathcal {C}_\\mu }{\\cos \\theta _l}\\;,$ where $\\mathcal {C}_\\mu $ is the set of constrains involving the $\\hat{\\sigma }^z_\\mu $ .", "Using Eq.", "(REF ), the computational complexity of computing the RA action for the LHZ architecture is $\\mathcal {O}(N)$ .", "Thus, computing RA protocol numerically (with a sequential local optimization) will only result in a polynomial overhead in the number of qubits $N$ .", "The numerical minimization of the action given in Eq.", "(REF ), for a spin-glass problem leads to the results presented in the main text." ] ]
2207.03553
[ [ "On Non-Linear operators for Geometric Deep Learning" ], [ "Abstract This work studies operators mapping vector and scalar fields defined over a manifold $\\mathcal{M}$, and which commute with its group of diffeomorphisms $\\text{Diff}(\\mathcal{M})$.", "We prove that in the case of scalar fields $L^p_\\omega(\\mathcal{M,\\mathbb{R}})$, those operators correspond to point-wise non-linearities, recovering and extending known results on $\\mathbb{R}^d$.", "In the context of Neural Networks defined over $\\mathcal{M}$, it indicates that point-wise non-linear operators are the only universal family that commutes with any group of symmetries, and justifies their systematic use in combination with dedicated linear operators commuting with specific symmetries.", "In the case of vector fields $L^p_\\omega(\\mathcal{M},T\\mathcal{M})$, we show that those operators are solely the scalar multiplication.", "It indicates that $\\text{Diff}(\\mathcal{M})$ is too rich and that there is no universal class of non-linear operators to motivate the design of Neural Networks over the symmetries of $\\mathcal{M}$." ], [ "Introduction", "Given a physical domain $\\mathcal {M}$ and measurements $f:\\mathcal {M}\\rightarrow \\mathcal {Y}$ observed over it, one is often interested in processing intrinsic information from $f$ , i.e.", "consistent with the symmetries of the domain.", "In words, if two measurements $f$ , $\\tilde{f}=g.f$ are related by a symmetry $g$ of the domain, like a rigid motion on an observed molecular compound, we would like our processed data $M(f)$ and $M(\\tilde{f})$ to be related by the same symmetry — thus that $M(g.f)=g.M(f)$ or equivalently that $M$ commutes with the symmetry transformation of the domain.", "The study of operators that satisfy such symmetry constraints has played a long and central role in the history of physics and mathematics, motivated by the inherent symmetries of physical laws.", "More recently, such importance has also extended to the design of machine learning systems, where symmetries improve the sample complexity  [22], [3].", "For instance, Convolutional Neural Networks build translation symmetry, whereas Graph Neural Networks build permutation symmetry, amongst other examples coined under the `Geometric Deep Learning' umbrella [5], [4].", "Lie groups of transformations are of particular interest, because there exists a precise and systematic framework to build such intrinsic operators.", "Indeed, for a locally compact group $G$ , it is possible to define a Haar measure which is invariant to the action of $G$  [2]; then a simple filtering along the orbit of $G$ allows to define a class of linear operators that commute with the group action.", "Examples of locally compact groups are given by specific Lie groups acting on $\\mathbb {R}^d$ , such as the translations or the rotations $O_d(\\mathbb {R})$ .", "Often these Lie groups $G$ only act on a manifold $\\mathcal {M}$ , and one tries to average along the orbit induced by $G$ .", "Note that it is possible, beyond invariance, to linearize more complex groups of variability like diffeomorphisms $\\text{Diff}(\\mathcal {M})$  [7].", "While the description of such linear intrinsic structures is of central mathematical importance and forms the basis of Representation theory [27], in itself is not sufficient to bear fruit in the context of Representation learning using Neural Networks [11].", "Indeed, linear operators do not have the capacity to extract rich information needed to solve challenging high-dimensional learning problems.", "It is therefore necessary to extend the systematic construction and classification of intrinsic operators to the non-linear case.", "With that purpose in mind, our work aims at studying the class of (non-linear) operators $M$ which commute with the action of the group $\\text{Diff}(\\mathcal {M})$ , the diffeomorphisms over $\\mathcal {M}$ .", "This approach will lead to a natural class of non-linear intrinsic operators.", "Indeed, any group $G$ of symmetries is, by definition, a subgroup of $\\text{Diff}(\\mathcal {M})$ , and thus commutes with such $M$  [21].", "Consequently, obtaining a non-linear invariant to a symmetry group $G$ could be done by using a cascade of interlacing non-linear operators which commute with $\\text{Diff}(\\mathcal {M})$ and linear operators which commute with $G$ .", "A notable example of linear operators that are covariant to the Lie group of translations is a given by the convolutions along the orbit of the group.", "These can be constructed thanks to the canonical Haar measure [28].", "However, such an approach fails for infinite dimensional groups, like our object of interest: contrary to Lie groups, $\\text{Diff}(\\mathcal {M})$ is not locally compact and it is thus not possible to define a Haar measure on this group.", "Our first contribution is to demonstrate that the non-linear operators which act on vector fields (elements of $L^p_\\omega (\\mathcal {M},T\\mathcal {M})$ ) and which commute with the group of diffeomorphisms, are actually just scalar multiplications.", "This implies that $\\text{Diff}(\\mathcal {M})$ is too rich to obtain non-trivial operators.", "Our second contribution is to demonstrate that non-linear operators acting on signals in $L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ are pointwise non-linearities.", "This fills a gap in the results of [7], and a fortiori justifies the use of point-wise non-linearities in geometric Deep Learning [4].", "Our paper is structured as follows: Sec.", "introduces the necessary formalism, that we use through this paper: in particular, we formally define the action of diffeomorphism.", "Then, we state and discuss our theorems in Sec.", "REF and sketch their proofs in Sec.", "REF .", "Rigorous proofs of each statement can be found in the Appendix." ], [ "Related work and motivation", "In this section, we discuss the notion of intrinsic operators, invariant and covariant non-linear operators and linear representation over standard symetry groups.", "Then, we formally state our objective." ], [ "Intrinsic Operators", "As discussed above, in this work we are interested in intrinsic operators $M:L^p(\\mathcal {M}, E) \\rightarrow L^p(\\mathcal {M}, E)$ , where $\\mathcal {M}$ is a Riemannian manifold, and $E=\\mathbb {R}$ or $E=T\\mathcal {M}$ , capturing respectively the setting of scalar signals and vector fields over $\\mathcal {M}$ .", "Here the notion of `intrinsic' means that $M$ is consistent with an equivalence class induced by a symmetry group $G$ in $L^p(\\mathcal {M}, E)$ : if $f,\\tilde{f} \\in L^p(\\mathcal {M}, E)$ are related by a transformation $g \\in G$ (in which case we write $f =g.", "\\tilde{f}$ ), then $M(f) =g.", "M(\\tilde{f})$ .", "Naturally, a stronger equivalence class imposes a stronger requirement towards $M$ , and consequently restrains the complexity of $M$ .", "We now describe the plausible techniques used to design such operators $M$ ." ], [ "GM-Convolutions", "The notion of $GM$ -convolutions [30] is an example of linear covariant operators which commute with the reparametrization of a manifold.", "In practice, this implies that the weights of a $GM$ -convolution are shared and the action of $GM$ -convolutions is local – two properties that facilitate implementation and point out the similarity with Lie groups.", "Another example of symmetry group corresponds to the isometry group of a Riemaniann manifold, whose pushforward preserves the tensor metric.", "In this case, it is well known that isometries [29] are the only diffeomomorphism which commute with a manifold Laplacian.", "Thus, any linear operators which commute with isometries is stabilized by Laplacian's eigenspaces.", "However, little is known on the non-linear counterpart of the symmetry-covariant operators.", "In this work, we characterize non-linear operators which commute with $\\text{Diff}(\\mathcal {M})$ .", "We will see that such operators are intrinsically defined by $\\text{Diff}(\\mathcal {M})$ and could be combined with any linear operators covariant with a symmetry group $G$ ." ], [ "Non-linear operators", "It has been shown that Convolutional Neural Networks are dense in the set of non-linear covariant operators [31].", "The recipe of the corresponding proof is an extension of the proof of the universal approximation theorem [13].", "The Scattering Transform [6], [20] is also an example of a well-understood non-linear operator which corresponds to a cascade of complex wavelet transforms followed by a point-wise modulus non-linearity.", "This representation provably linearizes small deformations." ], [ "Compact Lie Groups", "In the context of geometric Machine Learning [5], there are several relevant notions of equivalence.", "For instance, we can consider a compact Lie Group $G$ acting on $\\mathcal {M}$ , and an associated representation in $\\mathcal {F}=\\lbrace f: \\mathcal {M}\\rightarrow \\mathbb {R}\\rbrace $ : Given $g \\in G$ and $f \\in \\mathcal {F}$ , then $g.f(x)\\triangleq f(g^{-1}.x)$ for $x \\in \\mathcal {M}$ .", "We then consider $f \\sim \\tilde{f}$ , related by this group action: $\\tilde{f}=g.f$ for some $g \\in G$ .", "The operators $M$ which are compatible with such group action are referred as being $G$ -equivariant (or covariant to the action of $G$ ) in the ML literature [12], [4].", "Such groups are typically of finite and small dimension, e.g.", "the Euclidean transformations of $\\mathcal {M}=\\mathbb {R}^d$ , with $d=2$ for computer vision applications, or $d=3$ for computational biology/chemistry applications.", "In this case, it is possible to characterize all linear intrinsic operators $M$ as group convolutions [18], leading to a rich family of non-linear intrinsic operators by composing such group convolutions with element-wise non-linear operators, as implemented in modern Neural Networks.", "We highlight that stability to symetries via non-linear operators finds useful application, in particular for flat manifolds [7]." ], [ "Isometries", "Riemanian manifolds $\\mathcal {M}$ come with a default equivalence class, which is given by isometries.", "If $m_u: T_u\\mathcal {M} \\times T_u\\mathcal {M} \\rightarrow \\mathbb {R}$ denotes the Riemannian metric tensor at point $u\\in \\mathcal {M}$ , a diffeomorphism $\\psi : \\mathcal {M} \\rightarrow \\mathcal {M}$ is an isometry if $g_u( v, w) = g_{\\psi (u)}( d\\psi _u(v), d\\psi _u(w) )$ for any $u \\in \\mathcal {M}$ and $v, w \\in T_u \\mathcal {M}$ .", "In words, isometries are changes of variables that preserve the local distances in the domain.", "The ensemble of all isometries forms a Lie Group which is locally compact [24].", "In this case, one can also build a rich class of intrinsic operators by following the previously explained `blueprint', namely composing linear intrinsic operators with element-wise non-linearities.", "As a representative example, the Laplace-Beltrami operator of $\\mathcal {M}$ only depends on intrinsic metric properties [29]: as said above, isometries preserve the invariant subspaces of a Laplacian." ], [ "Beyond Isometries", "While isometries are the `natural' transformations of the geometric domain, they cannot express high-dimensional sources of variability; indeed, if $\\mathcal {M}$ is a $d$ -dimensional complete connected Riemannian manifold, its isometry group has dimension at most $d(d+1)/2$ [9].", "This raises the question whether one can characterize intrinsic operators relative to a broader class of transformations.", "Another class of important symmetries corresponds to the ones which are gauge invariant, i.e.", "which leads to transformations which preserve the change of parametrization and which are used in [10], [30] through the notion of $G$ -structure.", "In this work, we consider the class of transformations given by $\\text{Diff}(\\mathcal {M})$ , the diffeomorphisms over $\\mathcal {M}$ .", "As shown in the Appendix, compactly supported deformations $\\psi :\\mathcal {M}\\rightarrow \\mathcal {M}$ define bounded linear operators $L_\\psi $ acting on $L^p(\\mathcal {M},E) \\rightarrow L^p(\\mathcal {M},E)$ , and constitute a far broader class of transformations than isometries.", "Our proof is mainly based on the use of compactly supported diffeomorphisms.", "Our objective is to characterize the (non-linear) operators $M$ such that $\\forall \\phi \\in \\text{Diff}(\\mathcal {M}),L_\\phi M=ML_\\phi \\,.$ In other words, we aim to understand continuous operators $M$ that commute with deformations.", "We will show that such operators are act locally and that they can be descriped explicitly, with simple formula.", "The commutation condition is visualized in the following diagram: $ {f [r]^{L_{\\phi }} [d]^M @{}[rd]|{\\circlearrowleft }& g [d]^M \\\\M f [r]^{L_{\\phi }} & M g}$" ], [ "Notations", "We will now formally introduce the mathematical objects of interest in this document.", "Let $(\\mathcal {M},g)$ be an orientable, connected, Riemannian manifold, of finite dimension $d\\in \\mathbb {N}^*$ , with $g\\in \\Gamma ( T^*M \\otimes T^* M)$ a section of symmetric definite positive bilinear forms on the tangent bundle of $M$ .", "Fix $p\\in [1,+\\infty [$ .", "For any volume form $\\omega \\in \\Gamma ( \\bigwedge ^d T^*M)$ let us define $L^p_\\omega (\\mathcal {M},T\\mathcal {M})$ , the space of $L^p$ vector fields, defined as the subspace of measurable functions $f:\\mathcal {M}\\rightarrow T\\mathcal {M}$ such that $f(x)\\in T_xM$ almost everywhere and $\\Vert f \\Vert _p^p \\triangleq \\int _{x\\in \\mathcal {M}} g_x(f(x),f(x))^{\\frac{p}{2}}\\,d\\omega (x) <+\\infty \\,.$ We will also consider $L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ the space of measurable scalar functions $f:\\mathcal {M}\\rightarrow \\mathbb {R}$ that fulfill $\\Vert f\\Vert _p^p\\triangleq \\int _{x\\in \\mathcal {M}} |f(x)|^p\\,d\\omega (x)<+\\infty \\,.$ We may write $\\Vert \\cdot \\Vert $ instead of $\\Vert \\cdot \\Vert _p$ when there is no ambiguity.", "For a $C^\\infty $ diffeomorphism $\\phi \\in \\text{Diff}(\\mathcal {M})$ , we will consider the action of $L_\\phi :L^p_\\omega (\\mathcal {M},T\\mathcal {M})\\rightarrow L^p_\\omega (\\mathcal {M},T\\mathcal {M})$ which we define for for any $f\\in L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ as $L_\\phi f(u)\\triangleq d\\phi ( u)^{-1}.f(\\phi (u))\\,.$ Note that this action is contravariant: $L_{\\psi \\circ \\phi }f(u)=d(\\psi \\circ \\phi )^{-1}.f(\\psi \\circ \\phi (u))=L_\\phi L_\\psi f(u)$ For scalar function $f\\in L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ , we define the action of $\\phi $ via $L_\\phi f(u)\\triangleq f(\\phi (u))\\,.$ This latter operator is also contravariant.", "If there is no ambiguity, we will use the same notation $L_\\phi $ , whether we apply it to $L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ or $L^p_\\omega (\\mathcal {M},T\\mathcal {M})$ .", "Throughout the article we restrict ourselves to $\\phi $ such that $L_\\phi $ is a bounded operator.", "Write $\\text{supp}(\\phi )=\\lbrace x, \\phi (x)\\ne x\\rbrace $ for the support of $\\phi $ and say that $\\phi $ has a compact support if $\\text{supp}(\\phi )$ is compact.", "We denote by $\\text{Diff}_c(\\mathcal {M})\\subset \\text{Diff}(\\mathcal {M})$ the set of compactly supported diffeomorphisms.", "Recall that since a $\\mathcal {M}$ is second-countable, $\\mathcal {C}^\\infty _c(\\mathcal {M})$ is dense in $L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ and $\\mathcal {C}^\\infty _c(\\mathcal {M},T\\mathcal {M})$ is dense in $L^p_\\omega (\\mathcal {M},T\\mathcal {M})$ .", "Finally, denote by $O_d(\\mathbb {R})$ the set of unitary operators on $\\mathbb {R}^d$ .", "Throughout the article, we might not write explicitly that equalities hold almost everywhere, since this is the default in $L^p$ spaces.", "As mentioned earlier, compactly supported diffeomorphisms lead to continuous operators, which is made rigorous by the following lemma whose proof is in the appendix.", "Lemma 1 If $\\text{supp}(\\phi )$ is compact, then $L_\\phi $ is bounded." ], [ "Main theorems", "In this section we present our main results.", "We first show that any (non-linear) deformation-equivariant operator acting on scalar fields must be point-wise (Theorem REF ), and then establish that any deformation-equivariant operator acting on vector fields corresponds to a multiplication by a scalar (Theorem REF )." ], [ "Theorem statements", "Now, we are ready to state our two main theorems: Theorem 1 (Scalar case) Let $\\mathcal {M}$ be a connected and orientable manifold of dimension $d\\ge 1$ .", "We consider a Lipschitz continuous operator $M:L^p_\\omega (\\mathcal {M,\\mathbb {R}})\\rightarrow L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ , where $1\\le p< \\infty $ .", "Then, $\\forall \\, \\phi \\in \\text{Diff}(\\mathcal {M}):\\; ML_\\phi =L_\\phi M$ is equivalent to the existence of a Lipschitz continuous function $\\rho :\\mathbb {R}\\rightarrow \\mathbb {R}$ that fulfills $M[f](m)=\\rho (f(m)) \\quad \\text{ a.e.", "}$ In that case, we have $\\rho (0)=0$ if $\\omega (\\mathcal {M})=\\infty $ .", "Theorem 2 (Vector case) Let $\\mathcal {M}$ be a connected and orientable manifold of dimension $d\\ge 1$ .", "We consider a continuous operator $M:L^p_\\omega (\\mathcal {M},T\\mathcal {M})\\rightarrow L^p_\\omega (\\mathcal {M},T\\mathcal {M})$ , where $1\\le p< \\infty $ .", "Then, $\\forall \\, \\phi \\in \\text{Diff}(\\mathcal {M}):\\; ML_\\phi =L_\\phi M$ is equivalent to the existence of a scalar $\\lambda \\in \\mathbb {R}$ such that $\\forall f\\in L^p_\\omega (\\mathcal {M},T\\mathcal {M}):\\, M[f](m)=\\lambda f(m) \\quad \\text{a.e.", "}$ We highlight that our theorems are quite generic in the sense that they apply to the manifolds usually used in applications or theory, $\\mathbb {R}^d$ in particular.", "Remark 1 The scalar case allows to recover standard operators which are exploited for Deep Neural Networks architectures.", "However, Theorem REF indicates that the group of diffeomorphism is too rich to obtain non-trivial non-linear operators.", "Remark 2 The case $p=\\infty $ leads to different results.", "For instance, in the scalar case we may consider the operator $Mf(x)=\\sup _y |f(y)|$ which fulfills $L_\\phi Mf=ML_\\phi f$ but is not pointwise.", "Remark 3 The condition “$\\omega (\\mathcal {M})=\\infty \\, \\Rightarrow \\, \\rho (0)=0$ ” in Theorem REF is necessary, since in the case $\\mathcal {M}=\\mathbb {R}$ , the operator $Mf(x)\\triangleq e^{if(x)}$ is not in $L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ .", "Remark 4 The Lipschitz condition in Theorem REF is crucial, otherwise, $Mf(x)=\\rho (f(x))$ might not be an operator of $L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ .", "For instance, if $p=2$ , $\\mathcal {M}=[0,1]$ and $Mf(x)=\\sqrt{f(x)}$ , we see that in this case, let $f(x)=x$ , then $f\\in L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ and $Mf\\notin L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ Remark 5 If $M$ is not Lipschitz, we can find an example which is not even continuous.", "The following example holds in both cases, the scalar case and the vector case.", "In both cases $f\\in L^p(M,\\mathbb {R})$ , the only thing that changes is the action of $L_\\phi $ on $f$ .", "$\\mathcal {M}=\\mathbb {R}$ , let for all $f\\in L^p(M,\\mathbb {R})$ : $Mf(x)=1_{\\lbrace z,\\lim _{y\\rightarrow z}f(y)=f(z)\\rbrace }(x) f(x).$ It is a measurable function.", "Let us show that this $M$ is a counterexample to the vector case: for any $\\phi \\in \\text{Diff}(\\mathcal {M})$ and $x\\in \\mathbb {R}$ , one has $ML_\\phi f(x)&=1_{\\lbrace z,\\lim _{y\\rightarrow z}f(\\phi (y))=f(\\phi (z))\\rbrace }(x)\\quad d\\phi (x)^{-1}f(\\phi (x))\\\\&=1_{\\lbrace z,\\lim _{y\\rightarrow \\phi (z)}f(y)=f(\\phi (z))\\rbrace }(x)\\quad d\\phi (x)^{-1}f(\\phi (x))\\\\&=1_{\\lbrace z,\\lim _{y\\rightarrow z}f(y)=f(z)\\rbrace }(\\phi (x))\\quad d\\phi (x)^{-1}f(\\phi (x))\\\\&=L_\\phi Mf(x)\\,.$ However, $M$ is not continuous as changing any function to 0 on $\\mathbb {Q}$ does not change its norm but changes the set where the limits exists.", "More precisely let $c>0$ be a strictly positive scalar, $M[c]=c$ ; let $f=c1[x\\notin \\mathbb {Q}]$ , $M[f]=0$ as $\\lbrace z,\\exists \\lim _{y\\rightarrow z}f(\\phi (y))\\rbrace =\\emptyset $ .", "However $c=f$ almost everywhere but $M[c]\\ne M[f]$ therefore $M$ is not continuous." ], [ "Proof Sketch", "We now describe the main ideas for proving the Theorems REF and REF .", "The appendix contains complete formal arguments and technical lemmata which we omit here due to lack of space.", "The two proofs share quite some similarities despite substantially different final results.", "Three ideas guide our proofs: First, we prove that it is possible to localize $M$ on a certain class of open sets which behaves nicely with the manifold structure, the strongly convex sets which we denote as $\\mathcal {O}_1$ .", "This is closely related to the notion of pre-sheaf [14].", "Secondly, we characterize $M$ on small open-sets.", "In the scalar case, we will study the representation of locally constant functions.", "In the vector case, we will show that locally, the image $M(1_Uc)$ of a vector field $c$ is co-linear to $c$ provided that $U$ is small enough.", "We will also show that those local properties are independent of the position on the manifold $\\mathcal {M}$ via a connectedness argument.", "Thirdly and finally, we combine a compacity and a density argument to extend this characterization to $\\mathcal {M}$ , which is developed in Sec.", "REF .", "Throughout the presentation, we will use the following definitions and theorems obtained from other works: Definition 1 (Strong convexity, from [17]) Let $\\mathcal {O}_1$ be the collection of open sets which are bounded and strongly convex, i.e.", "such that any points $p,q$ in such a set can be joined by a geodesic contained in the set.", "Furthermore let $\\dot{\\mathcal {O}_1}=\\lbrace A\\in \\mathcal {O}_1: \\, \\exists B\\in \\mathcal {O}_1, \\bar{A}\\subset B\\text{ and }\\omega (\\bar{A}\\backslash A)=0\\rbrace $ .", "The intuition behind the definition of $\\dot{\\mathcal {O}_1}$ is that all of its elements are contained in a `security' open set,which avoids degenerated effects on the manifold.", "In particular, this allows to control the boundary of a given open set.", "Theorem 3 (theorem adapted from [16], [17]) (1) $\\dot{\\mathcal {O}_1}$ is a system of neighborhoods.", "(2) Any element of $\\mathcal {O}_1$ is diffeomorph to $\\mathbb {R}^d$ .", "(3) Both $\\mathcal {O}_1$ and $\\dot{\\mathcal {O}_1}$ are stable by intersection.", "Theorem 4 (Flowbox theorem, as stated in [8]) Let $f,g\\in \\mathcal {C}^\\infty _c(\\mathcal {M},T\\mathcal {M})$ .", "For any $m\\in \\mathcal {M}$ with $f(m)\\ne 0$ and $g(m)\\ne 0$ , there exists an open set $U \\subset \\mathcal {M}$ and $\\phi \\in \\text{Diff}(\\mathcal {M})$ such that $\\phi (m)=m$ and $L_\\phi (1_Uf)=1_{\\phi (U)}g$ .", "We will now present some lemmata that are necessary for the proofs of theorems REF and REF .", "As a first step, we argue that one may assume $M(0)=0$ where 0 denotes the constant 0-function.", "This is because in the appendix we show that $M(0)$ is a constant function $C$ , with $C = 0$ if $\\omega (\\mathcal {M})=\\infty $ .", "Therefore, we may substract $C$ from $\\rho $ and $\\lambda $ , leaving us with having to show the theorems only for $M(0) = 0$ .", "Next, a key idea of the proof is to exploit the flexibility of the deformation equivariance to localise the input, i.e.", "to show that the image of compactly supported functions is also compactly supported.", "To do so, the following lemma provides a way of collapsing an open ball into a singleton while maintaining a good control on the support of the diffeomorphism.", "Lemma 2 (Key lemma) Let $\\epsilon >0$ .", "There exists a sequence of diffeomorphisms $\\phi _n:\\mathbb {R}^d\\rightarrow \\mathbb {R}^d$ , compactly supported in $\\mathcal {B}(0,1+\\epsilon )$ such that: $\\phi _n(\\mathcal {B}(0,1))=\\mathcal {B}(0,\\frac{1}{n})\\,,$ and $\\sup _{u\\in \\mathcal {B}(0,1)}\\Vert d\\phi _n(u)\\Vert \\le \\frac{1}{n}~.$ Set $\\phi _n(u)=f_n(\\Vert u\\Vert )u$ , where $f_n(r)={\\left\\lbrace \\begin{array}{ll}\\frac{1}{n}&\\text{, if }|r|\\le 1\\\\1&\\text{, if }|r|\\ge 1+\\epsilon \\,,\\end{array}\\right.", "}$ and $f_n$ is smoothly interpolated for $|r|\\in [1,1+\\epsilon ]$ in a way that it remains nondecreasing.", "It is then clear that $\\phi _n$ fulfills the desired properties.", "We will often use that if the support of $\\phi \\in \\text{Diff}(\\mathcal {M})$ is such that $\\text{supp}(\\phi )\\cap U=\\emptyset $ , then for any $f\\in L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ one has $1_Uf=L_\\phi (1_Uf)$ .", "This implies the following important lemma, for which a rigorous proof can be found in the appendix: Lemma 3 Let $U\\in \\dot{\\mathcal {O}_1}$ and $M$ as in Theorem REF or Theorem REF .", "Then, for any $f\\in E$ , where $E=L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ or $E=L^p_\\omega (\\mathcal {M},T\\mathcal {M})$ respectively, we have: $M[f1_U]=1_UM[f]\\,.$ Furthermore, if $U$ is any closed set, the same conclusion applies.", "Equipped with this result, our proof will characterize the image of functions of the type $c1_U$ where either $c\\in \\mathbb {R}$ , or $c$ is a vector field which can be straightened (isomorphic to a constant vector), via the following Lemma.", "In the Vector case: Lemma 4 (Image of localized vector field) For $M$ as in Theorem REF there is $U\\in \\dot{\\mathcal {O}_1}$ , and $\\lambda (U)$ such that for any $f\\in L^p_\\omega (M,TM)$ : $M[f1_U]= 1_U\\lambda (U) f\\,.$ Here is the scalar case: Lemma 5 (Image of constant functions, scalar case) Let $M$ as in Theorem REF .", "For any $U\\in \\dot{\\mathcal {O}_1}$ and $c\\in \\mathbb {R}$ , then: $M(c1_U)=h(c,U)1_U$ .", "Furthermore, $c\\rightarrow h(c,U)$ is Lipschitz for any $U\\in \\dot{\\mathcal {O}_1}$ .", "At this stage, we note that both representations are point-wise, and the next steps of the proofs will be identical both for the scalar and vector cases.", "The extension to $L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ or $L^p_\\omega (\\mathcal {M},T\\mathcal {M})$ will be done thanks to: Lemma 6 (Image of a disjoint union of opensets) Let $U_1,...,U_n\\in \\mathcal {O}_1$ and $M$ as in Theorem REF or Theorem REF , s.t.", "$\\forall i\\ne j, \\overline{U_i}\\cap \\overline{U_j}=\\emptyset $ .", "Then for any $f\\in L^p_\\omega (\\mathcal {M},T\\mathcal {M})$ : $M[\\sum _{i=1}^n1_{U_i}f]=\\sum _{i=1}^nM[1_{U_i}f]\\,.$ This lemma states that we can completely characterize $M$ on disjoint union of simple sets.", "We will then need an argument similar to Vitali covering Lemma in order to \"glue\" those open sets together, which shows that simple functions with disjoint support can approximate any elements of $L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ or $L^p_\\omega (\\mathcal {M},T\\mathcal {M})$ (we only state the lemma for $L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ as our proof on $L^p_\\omega (\\mathcal {M},T\\mathcal {M})$ does not necessarily need this result): Lemma 7 (Local Vitali) For $f\\in \\mathcal {C}^\\infty _c(\\mathcal {M})$ and $m\\in \\mathcal {M}$ , there exists $U\\in \\dot{\\mathcal {O}_1}$ with $m\\in U$ , such that for any $\\epsilon >0$ , there exist subsets $U_1,...,U_n\\in \\dot{\\mathcal {O}_1}$ with $U_i\\subset U$ and numbers $c_1,...,c_n\\in \\mathbb {R}$ such that: $\\Vert \\sum _n 1_{U_n}c_n-1_Uf\\Vert <\\epsilon \\,.$ Note that this type of covering is not possible on any open set without further assumptions on the manifold, such as bounds on its Ricci curvature [19].", "Fortunately, we will only need a local version which is true because charts are locally bi-Lipschitz.", "Both Lemma REF and Lemma REF imply that: Proposition 1 Consider $M$ from either Theorem REF or REF .", "Assume that there exists $U\\in \\dot{\\mathcal {O}_1}$ such that $M(c1_V)=h(c,V)1_V$ for any $V\\subset U$ , with $V\\in \\dot{\\mathcal {O}_1}$ , where $c$ is either a vector field in the case $E=L^p_\\omega (\\mathcal {M},T\\mathcal {M})$ or a constant scalar in the case $E=L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ .", "If we further assume that $c\\rightarrow h(c,U)$ is $L$ -Lipschitz, then $\\forall f\\in E, \\forall m\\in \\mathcal {M}, M[1_Uf](m)=1_Uh(f(m),U)\\,.$ Furthermore, it does not depend on $U$ , meaning that for any other such $\\tilde{U}$ , we have: $\\forall f\\in E, \\forall m\\in U\\cap \\tilde{U}, M[1_{\\tilde{U}}f](m)=1_{ U} h(f(m),U)\\,.$ We briefly discuss the intuition behind Theorem REF .", "It is linked to the idea that the operators $M$ at hand have to commute with local rotations, and this even for locally constant vector fields.", "We reduce the characterisation of deformation-equivariant vector operators using an invariance to symmetry argument: functions which are invariant to rotations are multiples of a scalar.", "The intuition is contained in the following lemma, which is commonly used in physics: Lemma 8 (Invariance to rotation) Let $f:\\mathbb {R}^d\\rightarrow \\mathbb {R}^d$ such that for any $W\\in O_d(\\mathbb {R})$ and $x\\in \\mathbb {R}^d$ , one has $f(Wx)= W f(x)$ .", "Then, there is $\\lambda :\\mathbb {R}^d\\rightarrow \\mathbb {R}, f(x)=\\lambda (\\Vert x\\Vert ) x$ .", "We write $f(x)=\\lambda (x) x +x^{\\perp }$ , with $x^{\\perp }(m)\\ne 0$ and $x^{\\perp }\\perp x$ .", "Then, we introduce $W\\in O_d(\\mathbb {R})$ such that $Wx^{\\perp }(m)=-x^{\\perp }(m)$ and $Wx(m)=x(m)$ .", "From $f(x)=f(Wx)=Wf(x)$ we deduce that $x^{\\perp }=0$ .", "Next, $\\lambda (Wx)=\\lambda (x)$ thus $\\lambda (x)=\\lambda (x^{\\prime })$ for any $\\Vert x\\Vert =\\Vert x^{\\prime }\\Vert $ ." ], [ "Distinction between scalar and vector case", "The scalar case is simpler to handle than the vector case: there are several more steps for the proof of Theorem REF , one needs to show that the point-wise non-linearity is actually a scalar multiplication.", "We also highlight that the non-linearity is fully defined by its image on locally constant functions.", "Finally, we conclude the proof of the theorem by appealing to a common density argument of the functions smooth with compact support, combing all the lemmata we have just presented in Sec.", "REF ." ], [ "Proofs conclusions (common to the scalar and vector case)", "In this section, we prove that the local properties of $M$ can be extended globally on $\\mathcal {M}$ .", "The main idea is to exploit the well-known Poincaré's formula, which states that: $1_{\\cup _i U_i}=\\sum _{k=1}^n(-1)^k\\sum _{i_1<...<i_k}1_{U_{i_1}\\cap U_{i_2}\\cap ...\\cap U_{i_k}}\\,,$ and to localize the action of $M$ on each $U_{i_1}\\cap U_{i_2}\\cap ...\\cap U_{i_k}\\in \\dot{\\mathcal {O}_1}$ thanks to Lemma REF .", "[Proof of Theorem REF and Theorem REF ] Let $f$ be a smooth and compactly supported function.", "Further consider $\\cup _{i\\le n}U_i$ a finite covering of its support with $U_i\\in \\dot{\\mathcal {O}_1}$ .", "Using an inclusion-exclusion formula together with Lemma REF , we obtain $1_{\\cup _i U_i}M[f]&=\\sum _{k=1}^n(-1)^k\\sum _{i_1<...<i_k}1_{U_{i_1}\\cap U_{i_2}\\cap ...\\cap U_{i_k}}M[f]\\\\&=\\sum _{k=1}^n(-1)^k\\sum _{i_1<...<i_k}M[f 1_{U_{i_1}\\cap U_{i_2}\\cap ...\\cap U_{i_k}}]\\,,$ where we used that $U_{i_1}\\cap U_{i_2}\\cap ...\\cap U_{i_k}\\in \\dot{\\mathcal {O}_1}$ .", "Now, the support of $f$ is closed and included in $\\cup _iU_i$ .", "Thus using Lemma REF : $M[f]=\\sum _{k=1}^n(-1)^k\\sum _{i_1<...<i_k}M[f 1_{U_{i_1}\\cap U_{i_2}\\cap ...\\cap U_{i_k}}],$ Note that if $\\rho $ is a pointwise operator with $\\rho (0)=0$ , then $\\rho (1_Uf)=1_U\\rho (f)$ and $\\sum _{k=1}^n(-1)^k\\sum _{i_1<...<i_k}\\rho (f 1_{U_{i_1}\\cap U_{i_2}\\cap ...\\cap U_{i_k}})&=\\sum _{k=1}^n(-1)^k\\sum _{i_1<...<i_k} 1_{U_{i_1}\\cap U_{i_2}\\cap ...\\cap U_{i_k}}\\rho (f)\\\\&=1_{\\cup _i U_i}\\rho (f)=\\rho (f)\\,.$ Thus, $Mf=\\rho (f)$ where $\\rho $ is obtained from Lemma REF or REF combined with Prop REF .", "We conclude by density in $L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ or $L^p_\\omega (\\mathcal {M},T\\mathcal {M})$ respectively.", "This ends the proof." ], [ "Remarks and conclusion", "In this work, we have fully characterized non-linear operators which commute under the action of smooth deformations.", "In some sense, it settles the intuitive fact that commutation with the whole diffeomorphism group is too strong a property, leading to a small, nearly trivial family of non-linear intrinsic operators.", "While on their own they have limited interest for geometric deep representation learning, they can `upgrade' any family of linear operators associated with any group $G \\subset \\text{Diff}(\\mathcal {M})$ into a powerful non-linear class — the so-called GDL Blueprint in [4].", "Also, this result is a first step towards characterizing the non-linear operators which commute with Gauge transformations and could give useful insights for specifying novel Gauge invariant architectures.", "We now state a couple of unsolved questions and future work." ], [ "On the commutativity assumption:", "For $\\mathcal {M}=\\mathbb {R}^d$ , it is unclear which type of non-linear operators commute with smaller groups of symmetry such as the Euclidean group.", "In fact, a generic question holds for manifolds: for a given symmetry group $G$ , what is elementary non-linear building block of a Neural Network?", "This could be, for instance, useful to design Neural Networks which are Gauge invariant.", "It is an open question for future work which would be relevant many applications in physics [15]." ], [ "Example of vector operators for $L^\\infty $", "It is slightly unclear how the vector case $p=\\infty $ can be handled in our framework, yet [1] seems to have interesting insights toward this direction." ], [ "Linearization of $\\text{Diff}(\\mathcal {M})$", "In this work, we considered an exact commutation between operators and a symmetries: however, it is unclear which operators approximatively commute with a given symmetry group.", "Such operators would be better to linearize a high-dimensional symmetry group like $\\text{Diff}(\\mathcal {M})$ .", "An important instance of non-linear operators that are non-local and that `nearly' commute with diffeomorphisms is the Wavelet Scattering representation [20], [7], [25].", "EO was supported by the Project ANR-21-CE23-0030 ADONIS and EMERG-ADONIS from Alliance SU.", "GSP was also supported by France Relance and Median Technologies." ], [ "Checklist", " For all authors... Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?", "Did you describe the limitations of your work?", "Did you discuss any potential negative societal impacts of your work?", "Have you read the ethics review guidelines and ensured that your paper conforms to them?", "If you are including theoretical results... Did you state the full set of assumptions of all theoretical results?", "Did you include complete proofs of all theoretical results?", "If you ran experiments... Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?", "Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?", "Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?", "Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?", "If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...", "If your work uses existing assets, did you cite the creators?", "Did you mention the license of the assets?", "Did you include any new assets either in the supplemental material or as a URL?", "Did you discuss whether and how consent was obtained from people whose data you're using/curating?", "Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content?", "If you used crowdsourcing or conducted research with human subjects... Did you include the full text of instructions given to participants and screenshots, if applicable?", "Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?", "Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?" ], [ "Technical Lemmata", "[Proof of Lemma REF ] We simply exihibit the proof for $E=L^2_\\omega (\\mathcal {M},T\\mathcal {M})$ .", "Indeed, let $f\\in L^2_{\\omega }(\\mathcal {M},T\\mathcal {M})$ , then: $\\Vert L_\\phi f\\Vert ^2&=\\int g(L_\\phi f,L_\\phi f)d\\omega \\\\&=\\int _{\\text{supp}(\\phi )}g(L_\\phi f,L_\\phi f)d\\omega +\\int _{\\mathcal {M}\\backslash \\text{supp}(\\phi )}g(L_\\phi f,L_\\phi f)d\\omega \\\\&=\\int _{\\phi (\\text{supp}(\\phi ))}g(d\\phi ^{-1}.f,d\\phi ^{-1}.f)\\det (J\\phi ^{-1})d\\omega ^{\\prime }+\\int _{\\mathcal {M}\\backslash \\text{supp}(\\phi )}g(f,f)d\\omega \\\\&\\le \\int _{\\text{supp}(\\phi )}g(f,f)\\Vert d\\phi ^{-1}\\Vert ^2\\det (J\\phi ^{-1})d\\omega ^{\\prime }+\\Vert f\\Vert ^2\\\\&\\le (\\sup _{\\omega \\in \\text{supp}(\\phi )} \\Vert d\\phi ^{-1}(\\omega )\\Vert ^{2(d+1)}+1)\\Vert f\\Vert ^2<\\infty \\\\$ Thus, $L_\\phi $ is bounded." ], [ "A remark on the Flowbox theorem", "Usually, the Flowbox Theorem (here Theorem REF ) is stated for a (often local) diffeomorphism.", "If $c(m)\\ne 0, \\tilde{c}(m)\\ne 0$ , then there exists $U,V$ and $\\phi :U\\rightarrow V$ a diffeomorphism such that $m\\in U\\cap V$ and $L_{\\phi }(1_Uc)=1_V\\tilde{c}$ .", "However, we note that thanks to Theorem 4 of [26], it is possible to find $\\tilde{U}$ smaller such that there exists $\\tilde{\\phi }:\\mathcal {M}\\rightarrow \\mathcal {M}$ which is a global diffeomorphism and $\\forall m\\in \\tilde{U}, \\tilde{\\phi }(m)=\\phi (m)$ .", "In this case, $\\tilde{\\phi }, \\tilde{U}$ and $\\tilde{V}=\\tilde{\\phi }(\\tilde{U})$ are the candidates of our statement in Theorem REF .", "As this is quite technical and rather intuitive, we skipped this remark in the main paper." ], [ "Spatial localization (common to the scalar and vector case)", "We now explain how to localize our operator $M$ .", "Equipped with Lemma REF , we can extend our contraction result on $\\mathbb {R}^d$ to $\\mathcal {M}$ as follow: Corollary 1 (Contraction of an openset) For any $U\\in \\mathcal {O}_1$ and $W$ openset such that $\\bar{U}\\subset W\\subset \\mathcal {M}$ , there exists $\\phi _n$ supported on $W$ such that for any $f\\in L^p_\\omega (\\mathcal {M},T\\mathcal {M})$ : $L_{\\phi _n}(1_Uf)\\rightarrow 0\\,.$ We prove first the result for $U=\\mathcal {B}(0,1)$ and $\\bar{U}\\subset W$ .", "In this case, it is possible to find $\\epsilon >0$ such that $\\mathcal {B}(0,1+\\epsilon )\\subset W$ .", "Now, taking $\\phi _n^{-1}$ as in Lemma REF , we get: $\\int _{\\mathbb {R}^d} \\Vert L_{\\phi _n}(1_\\mathcal {B}(0,1)f)(u)\\Vert ^p\\,du&=\\int _{\\mathbb {R}^d} \\Vert 1_\\mathcal {B}(0,1)(\\phi _n^{-1}(u))d\\phi _n(u).f(\\phi _n^{-1}(u))\\Vert ^p\\,du\\\\&=\\int _{\\mathbb {R}^d}\\Vert 1_{\\mathcal {B}(0,\\frac{1}{n})}(u)d\\phi _n(u).f(nu)\\Vert ^p\\,du\\\\&=\\frac{1}{n^d}\\int _{\\mathbb {R}^d}1_{\\mathcal {B}(0,1)}(u)\\Vert d\\phi _n(\\frac{u}{n}).f(u)\\Vert ^p \\,du\\\\&\\le \\frac{1}{n^{d+1}}\\Vert 1_{\\mathcal {B}(0,1)} f\\Vert ^p\\rightarrow 0$ Next, getting back to the manifold, we know that if $U\\in \\dot{\\mathcal {O}_1}$ , there is $V\\in \\mathcal {O}_1$ such that $\\bar{U}\\subset V$ .", "We can thus find an openset $\\mathcal {B}\\subset V$ , such that in the chart of $V$ , $\\mathcal {B}$ is an open ball, and $U\\subset \\mathcal {B}\\subset W$ .", "We can thus apply the technique derived above to get $\\phi _n:V\\rightarrow V$ , compactly supported, which contracts $\\mathcal {B}$ (and thus $U$ ) to 0 and supported in $W$ .", "Since it is smooth, compactly supported on $W$ , we can extend it on $\\mathcal {M}$ and we get the result.", "Next, this technique can be used to build a sequence of contraction, which allows to explicitly localize the image of a compactly supported function, as follow: Lemma 9 ( Lemma REF restated for closed sets) Let $F\\subset \\mathcal {M}$ a closed set.", "Then, for any $f\\in L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ , we have: $M[f1_F]=1_FM[f]$ Because $\\mathcal {M}$ is a manifold, it is second countable and thus there is a countable collection of opens such that $\\mathcal {M}\\backslash F=\\cup _{i\\ge 0}U_i$ with $U_i\\in \\mathcal {O}_1$ .", "We use Lemma REF and, we apply the dominated convergence theorem to $f_n=1_{\\cup _{i\\le n}U_i}f$ to conclude.", "[Proof of Lemma REF ] We note that if $U\\in \\dot{\\mathcal {O}_1}$ , then $\\omega (\\bar{U}\\backslash U)=0$ and we can thus use the Corollary REF to conclude." ], [ "Action on locally constant functions, for the scalar and vector cases", "We now prove the part specific to the vector field setting, i.e., that the action of $M$ is locally a multiplication by a scalar.", "[Proof of Lemma REF ] Step 1:$M(1_Uc)(m)=1_V\\lambda (m,U,c)c$ such that $c(m)\\ne 0, \\forall m\\in U$ .", "Let $c\\in C_c^\\infty (\\mathcal {M}, T\\mathcal {M})$ .", "For $U\\in \\dot{\\mathcal {O}_1}$ , $m_0\\in U$ , fix a chart $\\psi :U\\rightarrow \\mathbb {R}^d$ , $\\psi (m_0)=0$ and $c$ is constant in $\\psi $ denoted $c^\\psi \\in \\mathbb {R}^d$ , which is possible thanks to the Theorem REF .", "This can also be written as for $m$ in a neighborhood of $m_0$ : $d\\psi (m).c(m)=c^\\psi \\,.$ Following the strategy in Lemma REF ,there is $W\\in \\mathcal {O}_d$ such that $Wc^\\psi = c^\\psi $ and $Wv= -v$ for any vector $v$ orthogonal to $c^\\psi $ .", "By compacity, we can find $A$ an open set small enough, with boundary of measure 0, such that $0\\in A$ , and $\\mathcal {W}A\\subset \\psi (U)$ for any $\\mathcal {W}\\in \\mathcal {O}_d$ .", "Now, setting $\\tilde{\\phi }=\\psi ^{-1}\\circ W\\circ \\psi $ , which is well defined on the open $\\cup _{\\mathcal {W}\\in \\mathcal {O}_d}\\mathcal {W} A$ , using Theorem 4 of [26](see remark Sec.", "REF of the appendix), we can can extend $\\phi $ globally such that on a local neighborhood, $\\forall m\\in \\tilde{U}, \\phi (m)=\\tilde{\\phi }(m)$ .", "Now, up to taking $A$ even smaller, we can use: $V=\\overline{\\psi ^{-1}(\\cup _{n\\in \\mathbb {Z}}W^nA)}\\subset U$ , which is closed with a measure 0 boundary(we have a countable union).", "We get: $L_{\\phi }(1_{V}c)(m_0)&=[d\\psi ^{-1}(m_0)\\circ W\\circ d\\psi (m_0)]c( m_0)1_{V}\\\\&=1_{V}c(m_0)\\,.$ Let us denote $p_{c^\\psi }^\\perp $ the orthogonal projection (with respect to the Euclidean scalar product) on the orthogonal plane to $c^\\psi $ .", "As $V\\subset U$ , $V$ is closed and $U\\in \\dot{\\mathcal {O}_1}$ from Lemma REF , we know that: $M(c)(m_0)=M(1_U c)(m_0)=M(1_V c)(m_0)=\\lambda (m_0,c,U)d\\psi ^{-1}(0)c^\\psi +d\\psi ^{-1}(0)p_{c^\\psi }^\\perp M(1_Vc)(m_0)$ Yet, on the other hand: $L_{\\phi }M(1_Vc)(m_0)&=\\lambda (m_0,c,U)d\\psi ^{-1}(0)c_\\psi -d\\psi ^{-1}(0)p_{c^\\psi }^\\perp M(1_Vc)(m_0)$ As this is true for any $m_0$ , we thus proved that: $M(1_U c)(m)=1_U\\lambda (m,U,c)c$ Step 2: In fact, $\\lambda (m,c,U)=\\lambda (m,U)$ if $c$ does not cancel on $U$ and $m\\in U$ .", "Let $c,\\tilde{c}$ be two vector fields as above and defined on $U$ both not equal to 0, and $m\\in U$ .", "Using the Theorem REF combined with the remark of Sec.", "REF of the appendix, there exists $\\phi :\\mathcal {M}\\rightarrow \\mathcal {M}$ a diffeomorphism and $\\tilde{V}, V\\subset U$ and $m\\in \\tilde{V}\\cap V$ , such that $L_\\phi (1_{ V}c(m))=1_{\\tilde{V}}\\tilde{c}(m)$ and $\\phi (m)=m$ Now, we could take a smaller closed set $V\\subset U$ with measure 0 boundary, so that $M[1_{V}c](m)=M[1_{U}c](m)=M[c](m)$ , which would lead to, following a similar argument to above: $\\lambda (m,\\tilde{c},U)\\tilde{c}(m)=M[1_{\\tilde{V}}\\tilde{c}(m)]=L_{\\phi }M[1_{ V} c](m)=L_{\\phi }(\\lambda (.,c,U)c)(m)=\\lambda (m,c,U)\\tilde{c}(m)$ and then locally $\\lambda $ is independent of the choice of a vector field, which implies the desired property.", "Step 3: In fact, $\\lambda (m,U)=\\lambda (U)$ .", "Indeed, let $m,m_0\\in V$ and $\\phi \\in \\text{Diff}(\\mathcal {M})$ such that $\\phi (m)=m_0$ (as $V$ is connex, by using Lemma REF ).", "Now, along the same line as above: $\\lambda (m,U)=\\lambda (m_0,U)$ The previous results hold when the vector field can be locally straightened, however the vector fields that take value 0 on some points of $U$ can not be straightened.", "We will now show that vector fields that can be straightened on $U\\in \\dot{\\mathcal {O}_1}$ are dense dense in $C^\\infty (U, T U)$ for the $L^p_\\omega $ norm.", "Let $f\\in C^\\infty (U, T U)$ , let $A=\\lbrace x\\in U\\vert f(x)=0\\rbrace $ , and $A^\\epsilon =\\lbrace x\\in U\\vert \\Vert f(x)\\Vert \\le \\epsilon \\rbrace $ for $\\epsilon >0$ .", "By Urysohn's lemma there is $\\chi ^\\epsilon : U\\rightarrow \\mathbb {R}$ be such that $\\chi |_{A^\\epsilon }=1$ and $\\chi |_{U\\setminus A^{2\\epsilon }}=0$ .", "Let, $f^\\epsilon = f + 2\\epsilon \\chi ^\\epsilon $ For any $x\\in U$ , $\\Vert f^\\epsilon (x) \\Vert \\ge \\vert \\Vert f(x)\\Vert - 2\\epsilon \\chi ^\\epsilon (x) \\vert $ and by construction $\\vert \\Vert f(x)\\Vert - 2\\epsilon \\chi ^\\epsilon (x) \\vert >0$ .", "Therefore, $M[f^\\epsilon 1_U]= \\lambda (U) f^\\epsilon $ Furthermore for all $0<\\epsilon \\le 1$ , $\\Vert f^\\epsilon \\Vert $ is bounded by $\\Vert f \\Vert +2$ that is integrable, so by dominated convergence theorem, $f^\\epsilon \\overset{L^p_\\omega }{\\underset{\\epsilon \\rightarrow 0}{\\longrightarrow }} f$ .", "So, $M[f 1_U]= \\lambda (U) f$ .", "To end the proof, one remarks that $C_c^\\infty (\\mathcal {M}, T\\mathcal {M})$ is dense in $L^p_\\omega (\\mathcal {M},T\\mathcal {M})$ .", "The next Lemma shows that, in the scalar case, we can consider $\\tilde{M}f\\triangleq Mf-M(0)$ for $f\\in L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ without losing in generality.", "Lemma 10 Under the assumptions of Theorem REF , $M(0)$ is constant, and if $\\omega (\\mathcal {M})=\\infty $ , then $M(0)=0$ .", "Following the Theorem 1 of [23], for any $m,m_0\\in \\mathcal {M}$ , we can find $\\phi $ global diffeomorphism such that $\\phi (m)=m_0$ .", "We note that $L_\\phi (0)=0$ and thus for any $m\\in \\mathcal {M}$ : $M(0)(m)=M[L_\\phi (0)]=L_\\phi M(0)(m)=M(0)(m_0)$ Thus, $M(0)$ is constant, and if $\\omega (\\mathcal {M})=\\infty $ , it is necessary that $M(0)=0$ .", "The corresponding Lemma in the scalar case is substantially simpler, as strongly convex sets are connex: [Proof of Lemma REF ] Fix $m_0\\in V$ , and let $m\\in V$ , using Lemma REF (because $V\\in \\dot{\\mathcal {O}_1}$ is connex, we can apply a connexity argument or the transitivity argument of Theorem 1 of [23] for compactly supported diffeomorphisms), we can find $\\phi $ supported in $V$ such that $\\phi (m_0)=m$ .", "Thus, $L_\\phi f=f$ and $Mf(m_0)=ML_\\phi f(m_0)=L_\\phi Mf(m_0)=Mf(m)$ .", "Thus, $M(c1_V)=h(c,V)1_V$ .", "The Lipschitz aspect is inherited from the fact that $M$ is Lipschitz." ], [ "Extrapolation to any good open sets (common to the scalar and vector case)", "In this section, we use the fact that we want to prove that both scalar and vector operators correspond to point-wise non-linearity, which are locally Lipschitz due to the regularity assumptions that we used.", "[Proof of Proposition REF ] Step 1:Fix $c$ , for any $m\\in U$ such that $V\\subset U$ , then $h(c,U)(m)=h(c,V)(m)$ Indeed, we note that for $m\\in U$ , where we used Lemma REF : $M(1_Vf)(m)=1_V(m)M(f)(m)=1_U(m)M(f)(m)=M(1_U f)(m)$ Thus, $h(c,V)|_V=h(c,U)|_V$ for any $V\\subset U$ .", "Step 2: extension by density, for any $f$ , $M(f1_U)=1_Uh(f,U)$ for any $f\\in L^p_\\omega (\\mathcal {M,\\mathbb {R}})$.", "Using Lemma REF , consider $f\\in \\mathcal {C}^\\infty _c(E)$ , $f_n=\\sum _n 1_{U_n}c_n$ , where $c_n$ is either a constant scalar, either a vector field, with disjoint support such that $\\Vert 1_Uf-1_Uf_n\\Vert <\\epsilon $ .", "We know that, from Lemma REF that: $M(1_Uf_n)=M[\\sum _n 1_{U_n}c_n]=\\sum _n 1_{U_n}M[1_{U_n}c_n]=\\sum _n 1_{U_n} h(c_n,U)$ Next, we note that: $\\Vert M1_Uf-1_Uh(f,U)\\Vert &\\le \\Vert 1_U(Mf_n-Mf)\\Vert +\\Vert 1_UMf_n-1_U h(f_n,U)\\Vert \\\\&+\\Vert 1_U(h(f_n,U)-h(f,U))\\Vert \\\\&\\le 2L\\Vert 1_U(f_n-f)\\Vert $ and from this, given that $h(.,U)$ is $L$ -Lipschitz, we conclude by density of $\\mathcal {C}^\\infty _c(\\mathcal {M})$ in $L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ .", "Step 3: Independence from $U$ Step 1 allows for the following definition of a global $h$ from local $h_U$ : let $m\\in \\mathcal {M}$ , pose, $\\forall U\\in \\dot{\\mathcal {O}_1}\\quad h(f(m)):= h(f(m),U)$ In the scalar case and in the vector case, one can build a scalar function and vector function such that, $f(m)= \\mu \\in \\mathbb {R}$ or $f(m)= c \\in T_x M$ (as shown in Step 3 of proof of REF ).", "Therefore in the scalar case $h$ is a function from $\\mathbb {R}$ to $\\mathbb {R}$ and in the vector case for any $x\\in \\mathcal {M}$ and $v\\in T_x\\mathcal {M}$ , $h(x)=\\lambda x$ .", "We only prove the Vitali version for $L^p_\\omega (\\mathcal {M,\\mathbb {R}})$ , as the proof for $L^p_\\omega (\\mathcal {M},T\\mathcal {M})$ would be identical, replacing solely the scalar by constant vector fields in their local parametrization.", "[Proof of Lemma REF ] We consider $U$ small enough such that $U\\in \\dot{\\mathcal {O}_1}$ , $m\\in U$ and $\\exp _m:\\mathcal {B}\\rightarrow U$ is locally a diffeomorphism from $\\mathcal {B}\\subset T\\mathcal {M}_m$ , and let $U_i=\\exp _m(\\mathcal {B}_i)$ with $\\mathcal {B}(x_i,r_i)\\subset \\mathcal {B}$ , which is strongly convex and thus $U_i\\in \\dot{\\mathcal {O}}_1$ .", "We remind that $\\exp _m$ is bi-Lipschitz on the bounded set $U$ .", "In this case, there is $C_1,C_2>0$ such that for any $x_i,r_i$ with $\\mathcal {B}(x_i,r_i)\\subset \\mathcal {B}$ , we have $r_i^d\\le \\lambda (\\mathcal {B}(x_i,r_i))\\le C_1\\omega (U_i)\\le C_2\\lambda (\\mathcal {B}(x_i,r_i))\\le C_dr^d$ .", "By Vitali's lemma, we have for any $\\epsilon >0$ and $r>0$ , the existence of some $x_i,r_i<r$ : $\\Vert 1_\\mathcal {B}-\\sum _{i=1}^n1_{\\mathcal {B}(x_i,r_i)}\\Vert ^p\\le \\epsilon ^p$ For $f$ smooth, let: $\\Vert f(x)1_U-\\sum _{i=1}^nf(x_i)1_{U_i}\\Vert ^p&\\le \\Vert \\sum _{i=1}^n (f(x)-f(x_i))1_{U_i}\\Vert ^p+\\Vert 1_{U\\backslash (\\cup _i U_i)}f(x)\\Vert ^p$ Now, as $\\exp _m$ is bi-Lipschitz, we get a $r$ small enough such that $|f(x)-f(x_i)|<\\epsilon $ .", "Next, because the sets are disjoint: $\\Vert \\sum _{i=1}^n (f(x)-f(x_i))1_{U_i}\\Vert ^p&=\\sum _{i=1}^n \\int _{U_i}|f(x)-f(x_i)|^p\\\\&\\le \\sum _{i=1}^n \\omega (U_i)\\epsilon ^p\\\\&\\le \\epsilon ^p\\omega (U))\\,.$ Now, using $|f(x)|\\le \\Vert f\\Vert _\\infty $ , we get: $\\Vert 1_{U\\backslash (\\cup _i U_i)}f(x)\\Vert ^p\\le \\Vert f\\Vert _\\infty \\epsilon ^p$ And: $\\Vert f-\\sum _{i=1}^n f(x_i)1_{U_i}\\Vert <(1+\\omega (U))^{1/p}\\epsilon \\,.$ The following Lemma allows to build diffeomorphism with compact support - we give this proof for the sake of completeness, at it is proved in [23].", "Lemma 11 Fix $\\rho >0$ , and $x_0,x_1\\in \\mathcal {B}(0,\\rho )$ , there exists $\\phi $ diffeomorphism, such that $\\phi (x_0)=x_1$ and $\\text{supp}(\\phi )\\subset \\mathcal {B}(0,\\rho )$ .", "Consider $f$ , smooth, supported in $[-1,1]$ and such that $f(0)=1$ .", "We will use a connexity argument: let us fix $x_0\\in \\mathcal {B}(0,\\rho )$ .", "Let's consider $\\Gamma =\\lbrace x \\in \\mathcal {B}(0,\\rho ): \\exists \\phi \\text{ diffeomorphism }\\phi (x)=x_0, \\text{ supp}(\\phi )\\subset \\mathcal {B}(0,\\rho )\\rbrace $ .", "Let $x_1\\in \\Gamma $ , then there is $\\eta <\\frac{1}{2}$ , $\\mathcal {B}(x_1,\\eta )\\subset \\mathcal {B}(0,\\rho )$ .", "For $x_2$ such that $\\Vert x_1-x_2\\Vert \\le \\frac{\\eta }{4\\sup |f^{\\prime }|}$ , we introduce: $\\tau (x)=(x_2-x_1)f(\\frac{\\Vert x-x_1\\Vert ^2}{\\eta ^2})\\,.$ We have that $\\text{supp}(\\mathbf {I}-\\tau )\\subset \\mathcal {B}(x_1,\\eta )$ , and: $\\frac{\\partial \\tau }{\\partial x}(x)=2\\frac{(x_2-x_1)\\langle x-x_1,x_1\\rangle }{\\eta ^2} f^{\\prime }(\\frac{\\Vert x-x_1\\Vert ^2}{\\eta ^2})$ leading to: $\\Vert \\frac{\\partial \\tau }{\\partial x}(x)\\Vert < \\frac{1}{2}$ This implies that the spectrum of $\\partial \\tau $ is in $[0,1[$ and thus, $\\mathbf {I}-\\partial \\tau $ is invertible.", "Now, by assumption, we know there is $\\phi $ such that $\\phi (x_1)=x_0$ , compactly supported in $\\Omega $ .", "Introducing $\\phi ^{\\prime }=\\phi \\circ (\\mathbf {I}-\\tau )$ , then $\\phi ^{\\prime }$ is a diffeomorphism, compactly supported in $\\Omega $ and $\\phi ^{\\prime }(x_2)=\\phi (x_1)=x$ , thus $x_2\\in \\Gamma $ .", "This shows $\\Gamma $ is open.", "But also $\\Gamma $ is closed (otherwiwe, we can make a path ...).", "Thus, by connexity $\\Gamma =\\Omega $ .", "The next Lemma is crucial in our proof, and allows to characterize union of well behaving opensets: Lemma 12 Let $n\\ge 0$ , $\\lbrace U_i\\rbrace _{i\\le n}\\subset \\dot{\\mathcal {O}_1}$ and $F$ a closed set such that $\\bar{U}_i\\cap F=\\emptyset , \\forall i$ .", "Then for any $f\\in L^p_\\omega (\\mathcal {M},T\\mathcal {M})$ : $1_FM[(1_F+1_{\\cup _{i\\le n}U_i})f]=1_FM[1_Ff]$ We work by induction on $n$ .", "For $n=0$ , the result is true.", "Then, let's write $U_{n+1}^\\epsilon =\\lbrace x,d(U_{n+1},x)<\\epsilon \\rbrace $ .", "It's an openset which contains $\\bar{U}_{n+1}$ , and by assumption we can pick $\\epsilon $ small enough such that $U_{n+1}^\\epsilon \\cap F=\\emptyset $ .", "Next, let's apply Lemma REF to $U_{n+1}$ and $W=U_{n+1}^\\epsilon $ .", "Then: $1_FM[(1_F+1_{(\\cup _{i\\le n}U_i\\backslash U_{n+1}^\\epsilon )\\cup U_{n+1}})f]&=L_{\\phi _n^{-1}}1_FM[(1_F+1_{(\\cup _{i\\le n}U_i\\backslash U_{n+1}^\\epsilon )\\cup U_{n+1}})f]\\\\&=1_FM[L_{\\phi _n}(1_Ff +1_{(\\cup _{i\\le n}U_i\\backslash U_{n+1}^\\epsilon )\\cup U_{n+1}}f)]\\\\&\\rightarrow 1_FM[1_Ff+1_{(\\cup _{i\\le n}U_i\\backslash U_{n+1}^\\epsilon )}f]$ Now, we remark that: $1_FM[1_Ff+1_{(\\cup _{i\\le n}U_i\\backslash U_{n+1}^\\epsilon )}f]=1_FM[1_Ff+1_{\\cup _{i\\le n}U_i}(1_{\\mathcal {M}\\backslash U_{n+1}^\\epsilon }f)]$ And we apply the induction hypothesis to $(1_{\\mathcal {M}\\backslash U_{n+1}^\\epsilon }f)$ .", "The next Lemma is crucial in our proof, and allows to characterize disjoint union of well behaving opensets: [Proof of lemma REF ] We note that $\\cup _{i=1}^n\\overline{U_i}=\\overline{\\cup _{i=1}^nU_i}$ .", "Thus, using Lemma REF , given this union is closed and disjoint and as for any closed set $F$ , $M[f 1_F]1_{F^c}= M[0]1_{F^c} =0$ the following linearity property holds, $M[\\sum _{i=1}^n1_{\\bar{U}_i}f]=\\sum _{i=1}^n1_{\\bar{U}_i}M[f]=\\sum _{i=1}^nM[1_{\\bar{U}_i}f]$ Now, we conclude as the boundaries have measure 0." ] ]
2207.03485
[ [ "Gravitational Waves from Current-Carrying Cosmic Strings" ], [ "Abstract Cosmic strings are predicted by many Standard Model extensions involving the cosmological breaking of a symmetry with nontrivial first homotopy group and represent a potential source of primordial gravitational waves (GWs).", "Present efforts to model the GW signal from cosmic strings are often based on minimal models, such as, e.g., the Nambu-Goto action that describes cosmic strings as exactly one-dimensional objects without any internal structure.", "In order to arrive at more realistic predictions, it is therefore necessary to consider nonminimal models that make an attempt at accounting for the microscopic properties of cosmic strings.", "With this goal in mind, we derive in this paper the GW spectrum emitted by current-carrying cosmic strings (CCCSs), which may form in a variety of cosmological scenarios.", "Our analysis is based on a generalized version of the velocity-dependent one-scale (VOS) model, which, in addition to the mean velocity and correlation length of the string network, also describes the evolution of a chiral (light-like) current.", "As we are able to show, the solutions of the VOS equations imply a temporarily growing fractional cosmic-string energy density, $\\Omega_{\\rm cs}$.", "This results in an enhanced GW signal across a broad frequency interval, whose boundaries are determined by the times of generation and decay of cosmic-string currents.", "Our findings have important implications for GW experiments in the Hz to MHz band and motivate the construction of realistic particle physics models that give rise to large currents on cosmic strings." ], [ "Introduction", "Cosmic defects are a common prediction in many models of physics beyond the Standard Model (BSM), especially, in the context of grand unified theories (GUTs) [1], [2].", "Cosmological phase transitions associated with the spontaneous breaking of new global or local symmetries can in particular give rise to various types of topological and nontopological defects [3], [4], including domain walls [5], cosmic strings [6], and monopoles [7].", "Evidence for any type of cosmic defect would clearly point to new physics, since the two phase transitions predicted by the Standard Model, the QCD crossover [8], [9], [10] and the electroweak crossover [11], [12], [13], are both expected to yield no observational signal from cosmic defects.", "In this paper, we are going to focus on cosmic strings, which, unlike monopoles and domain walls, do not threaten to overclose the Universe, as long as the string network reaches its characteristic scaling regime.", "Cosmic strings contribute to the primordial scalar power spectrum and can hence be probed in observations of the cosmic microwave background [14].", "Another appealing avenue for the detection of cosmic strings is the search for primordial gravitational waves (GWs) [15].", "The GW signal from a string network is dominated by the emission from closed string loops [16], which are formed when long strings in the network intersect with each other or with themselves.", "String loops oscillate under their own tension and hence emit GWs.", "In addition, localized features on closed loops, such as cusps and kinks, emit bursts of GW radiation, which all together results in a stochastic GW background (SGWB) over a large frequency range.", "An interesting property of this SGWB signal is that it acts as a logbook of the expansion history of the early Universe.", "Indeed, the GW signal emitted by cosmic strings encodes information on the state of the Universe when cosmic string loops were formed and emitted GWs [17].", "This dependence on the expansion rate notably results in a flat GW spectrum across many orders of magnitude in frequency that is associated with the era of radiation domination in the early Universe.", "Similarly, loops produced during the matter era result in a steeply falling contribution to the GW spectrum when going from higher to lower frequencies [15].", "In this sense, cosmic strings can be used to probe nonstandard expansion histories of the early Universe, such as early matter domination [18], [19] and kination [20], [21], see also Ref. [22].", "Figure: Benchmark GW spectra emitted by current-carrying cosmic strings.", "For spectra (A), (C), (E), we assume a cosmic-string tension of Gμ=0.5×10 -10 G\\mu =0.5\\times 10^{-10}, while for spectra (B) and (E), we assume Gμ=0.5×10 -17 G\\mu =0.5\\times 10^{-17}.", "In scenarios (A) and (B), the current flowing on cosmic strings decays around the time of the QCD crossover, while in scenarios (C) and (D), it decays around the time of the electroweak crossover.", "In scenario (E), the current decays at even earlier times (see text for details).", "For each scenario, the lower and upper edges of the predicted spectrum respectively correspond to k=1k=1 and k=10 6 k=10^6 harmonic string modes accounted for in the computation of the GW spectrum.", "We also show standard (i.e., no current) spectra for string networks with Gμ=0.5×10 -10 G\\mu =0.5\\times 10^{-10} and Gμ=0.5×10 -17 G\\mu =0.5\\times 10^{-17} (black lines), together with existing constraints and future sensitivities of present and planned GW experiments .", "All spectra for Gμ=0.5×10 -10 G\\mu =0.5\\times 10^{-10} can explain the PTA signal at nHz frequencies.Recently, GWs from cosmic strings attracted a lot of attention in consequence of the first hints for a SGWB signal at nHz frequencies reported by the NANOGrav, PPTA, EPTA, and IPTA pulsar timing array (PTA) collaborations [24], [25], [26], [27].", "As demonstrated in Refs.", "[28], [29], [30], [31], [32], GWs from cosmic strings represent a plausible interpretation of the PTA data and hence compete with the astrophysical interpretation in terms of inspiraling supermassive black-hole binaries [33].", "At present, the true nature of the new PTA signal is still unclear, as the characteristic interpulsar quadrupole correlations that would clearly indicate a GW origin have not yet been confirmed.", "However, in anticipation of near-future PTA data, cosmic strings remain one of the leading candidates for the source of a SGWB at nHz frequencies, which calls for refined theoretical predictions of the expected GW signal.", "The precise description of a string network requires large-scale numerical simulations [34], which are typically either based on the Nambu–Goto action [35], [36], [37], [38], [39] or a field-theoretic description in terms of the Abelian Higgs model [40], [41], [42], [43], [44], [45]; for a comparison of Abelian Higgs and Nambu–Goto strings, we refer to Sec.", "3.4 of Ref. [15].", "In the Nambu–Goto approximation, cosmic strings are treated as exactly one-dimensional featureless objects whose only relevant property is their tension or energy per unit length, $\\mu $ .", "In order to obtain more realistic predictions, it is therefore necessary to go beyond this minimal description and consider, e.g., extensions of the simplest Nambu–Goto model featuring additional worldsheet degrees of freedom.", "Such degrees of freedom can be currents induced by charge carriers that propagate along cosmic strings [46], [47], or, e.g., short-wavelength propagation modes known as wiggles [48].", "In this paper, we will focus on the former possibility and consider neutral currents, i.e., currents of particles that are only charged under the Abelian gauge group whose spontaneous breaking results in the formation of the string network but that do not interact with any other gauge force.", "This notably ensures that strings cannot lose energy via their coupling to massless vector bosons, as would be the case if they carried electromagnetic currents and which would in fact lead to a highly suppressed GW emission [49].", "Current-carrying cosmic strings (CCCSs) were first introduced by Witten [50], who studied both bosonic and fermionic charge carriers.", "More recently, CCCSs were investigated in Refs.", "[51], [52], [53], [54], [55], [56], [57].", "The evolution of standard Nambu–Goto strings can be conveniently described by the so-called velocity-dependent one-scale (VOS) model [58], [59], which allows one to track the evolution of two important properties of the string network: its correlation length and the root-mean-square velocity of long strings.", "By comparison, a network of CCCSs features on top a third property that is not described by the VOS model: the strength of the current propagating on cosmic strings.", "Recently, the authors of Ref.", "[47] generalized the standard VOS model in order to account for the time evolution of this current.", "In our analysis, we will make use of these results and solve the generalized VOS equations derived in Ref.", "[47] for a CCCS network featuring a chiral (light-like) current.", "This corresponds to charge carriers that are massive far away from the string core, but allow for massless zero modes along the string, where the symmetry is restored.", "In combination with the fact that we assume neutral currents, a well-motivated CCCS scenario appears to be right-handed neutrinos (RHNs) propagating along cosmic $B\\!-\\!L$ cosmic strings, which can be easily embedded in grand unified theories (GUTs) based on $SO\\left(10\\right)$ .", "A first study of RHN scattering and capture by cosmic strings has been carried out in Ref. [60].", "On the other hand, RHN currents can shut off at the time of the electroweak phase transition, when the Higgs vacuum expectation value gives the RHN zero modes propagating along the string cores a nonvanishing mass [61].", "As we will see below, the decay of RHN currents at the time of electroweak symmetry breaking can then be instrumental in avoiding a potential overclosure problem that may arise if the currents carried by the string network are too long-lived.", "Overall, we, however, emphasize that the process of current generation and quenching requires further work, especially, in the context of specific GUT models.", "In this paper, we will not address this question and defer any further top-down model building to future work.", "Instead, we will adopt a bottom-up approach and carry out a phenomenological analysis based on the recently proposed generalized VOS model focusing on a general scenario where currents are present only for a certain amount of time between the instant of network formation and today.", "As can be seen in fig:money, this approach leads to very promising results, in particular, a boosted SGWB signal over large ranges in frequency that is related to a temporary growth in the fractional CCCS energy density, $\\Omega _{\\rm cs}$ .", "We therefore hope that our analysis is going to motivate further model-building efforts aiming at the construction of realistic particle physics models that give rise to large currents on cosmic strings.", "The remainder of this paper is organized as follows.", "First, we will discuss the generalized VOS model in sec:vos.", "We will start by introducing the generalized VOS equations in sec:eqs, which include the new terms accounting for the presence of a current.", "We will then derive the attractor solutions of these equations, first in the case of zero current in sec:vanish and subsequently in the case of nonvanishing current in sec:steady, which is the scenario of our interest.", "In sec:loopnumber, we will employ these results in order to derive the loop number density as well as the energy density of a CCCS network.", "The SGWB signal is computed in sec:gw, where we compare our predictions to the existing and projected sensitivities of current and planned GW experiments.", "sec:conclusions, finally, contains our conclusions." ], [ "Generalized VOS model with chiral currents", "The energy density of a network of long strings is $\\rho _\\infty = \\frac{\\mu }{L^2}\\,,$ where $L$ is the characteristic correlation length.", "A network of Nambu–Goto strings in the expanding Universe with no worldsheet degrees of freedom and intercommutation probability $P=1$ is known to reach a scaling solution, $L\\propto t$ , soon after formation.", "The existence of such a solution can be derived analytically within the VOS framework [58], [59], [15], which in its standard form contains two coupled differential equations describing the evolution of $L$ and the root-mean-square (RMS) velocity $v$ ." ], [ "Nonlinear autonomous system", "Using $t$ to denote cosmic time, the generalized VOS equations for superconducting strings carrying a chiral current read [47]Note that we express these equations in terms of cosmic time and physical length whereas in Ref.", "[47] conformal time and wavenumber are used.", "$ \\dot{L} & = \\frac{\\dot{a}}{a} \\frac{L}{1+Y}(1+v^{2}+2Y) + \\frac{g \\tilde{c}}{2\\sqrt{1+Y}}v\\,, \\\\\\dot{v} & = \\frac{1-v^{2}}{1+Y} \\left[\\frac{(1-Y) k(v)}{L \\sqrt{1+Y}} - 2 v \\frac{\\dot{a}}{a}\\right]\\,, \\\\\\dot{Y} & = 2Y \\left[\\frac{v k(v)}{L\\sqrt{1+Y}} - \\frac{\\dot{a}}{a} \\right] - \\frac{v}{L} \\tilde{c} (g-1) \\sqrt{1+Y}\\,, \\\\k(v) & = \\frac{2\\sqrt{2}}{\\pi } \\frac{1-8v^{6}}{1+8v^{6}}(1-v^{2})(1+2\\sqrt{2}v^{3})\\,.", "$ Here, $Y$ denotes the current strength defined as $Y=(Q^2+J^2)/2$ , where $Q$ and $J$ are the total charge and current energy density of the system, respectively.", "The three quantities $Y$ , $Q^2$ , and $J^2$ are dimensionless and understood to describe the properties of the current carried by the string network in units of the string tension $\\mu $ .", "Chiral (or light-like) currents are characterized by a vanishing state parameter $K = Q^2 - J^2$ , meaning that in our scenario of interest $K= 0$ and hence $Y = Q^2 = J^2$ .", "Further, $a$ in the above set of equations denotes the cosmic scale factor.", "$\\tilde{c}$ is the so-called chopping parameter, which describes the efficiency of chopping off loops from the long-string network.", "In absence of dedicated numerical simulations of CCCS networks, we fix $\\tilde{c}\\approx 0.23$ , the typical value found in numerical simulations of standard uncharged Nambu–Goto strings [59].", "We also set $g=\\sqrt{1+Y}$ , in which case the energy loss of the long-string network due to loop formation is qualitatively described in the same way as in the current-less case; see Eqs.", "(33), (38), and (39) in Ref. [47].", "Finally, the function $k(v)$ is known as the momentum parameter; again we adopt the standard expression for uncharged Nambu-Goto strings [15], [47].", "This system of ordinary differential equations can be rewritten in terms of the scaling length $\\alpha (t) = L(t) / t$ and derivatives with respect to logarithmic time ${\\alpha }{\\ln t} & = \\frac{\\alpha \\nu }{1+Y}(1+v^{2}+2Y) - \\alpha + \\frac{g \\tilde{c}}{2\\sqrt{1+Y}}v\\,, \\\\{v}{\\ln t} & = \\frac{1-v^{2}}{1+Y} \\left[\\frac{(1-Y) k(v)}{\\alpha \\sqrt{1+Y}} - 2 v \\nu \\right]\\,, \\\\{Y}{\\ln t} & = 2Y \\left[\\frac{v k(v)}{\\alpha \\sqrt{1+Y}} - \\nu \\right] - \\frac{v}{\\alpha } \\tilde{c} (g-1) \\sqrt{1+Y}\\,, $ where $\\nu =1/2$ ($\\nu =2/3$ ) for radiation (matter) dominated Universe since we defined $a(t)\\sim t^\\nu $ .", "Notice that the right-hand side does not depend explicitly on time, meaning that this set of equations is an autonomous system and the vector $X(\\ln t) = (\\alpha , v, Y)(\\ln t)$ describing the solution of this system is an integral curve.", "We showcase in fig:streamtube the integral curves for $X(\\ln t)$ in three-dimensional space.", "Figure: Streamlines derived from the autonomous system of ordinary differential equations (), () and () for c ˜=0.23\\tilde{c}=0.23 and ν=1/2\\nu =1/2.", "All the trajectories start from the plane α=0.27\\alpha = 0.27 and eventually fall into one of the two different attractors, one at Y=0Y=0 and another one at Y∼1Y\\sim 1.Looking at fig:streamtube, we identify two attractor solutions.", "First, there is an attractor solution where the current vanishes and the parameters $\\alpha $ and $v$ reach a constant value, identical to the predictions of the standard VOS framework.", "We discuss it in more details in sec:vanish.", "A second attractor solution exists when $Y$ is large enough, in which case the current remains close to $Y=1$ and the parameters $\\alpha $ and $v$ feature a power-law decay.", "We cover this situation in sec:steady." ], [ "Vanishing current attractor", "In this section, we study the autonomous system of Eqs.", "(REF ), () and () around the attractor appearing at $Y=0$ .", "In what follows we will first determine the exact location of the attractor, i.e.", "the fixed point for this system of equations around $Y=0$ .", "Then we will determine its characteristics, such as its stability and the efficiency with which the system converges to $Y= 0$ ." ], [ "Fixed point at vanishing current", "First, we note that setting $Y=0$ trivially satisfies Eq.", "().", "Then, imposing that $*{\\alpha }{\\ln t} = 0$ , we obtain that $\\alpha = \\frac{\\tilde{c} v_\\infty }{2(1-\\nu -\\nu v_\\infty ^2)}\\,,$ where $v_\\infty $ is the RMS velocity at the fixed point.", "Further, if we impose that $*{v}{\\ln t} = 0$ , we find that the velocity has to satisfy the polynomial equation $k(v_\\infty ) = \\frac{\\tilde{c} \\nu v_\\infty ^2}{1 - \\nu - \\nu v_\\infty ^2}\\,,$ where $k(v)$ is defined in Eq.", "(); the solution of eq:k cannot be determined analytically.", "We provide numerical values for $v_\\infty $ in table:vanish." ], [ "Characteristics of the $Y=0$ attractor solution", "One can assess the behaviour of $Y$ around the $Y=0$ fixed point as follows: Let us divide Eq.", "() by $Y$ , then expand for $Y \\rightarrow 0$ and evaluate at the fixed point.", "We thus find a power-law behavior $Y \\propto t^\\beta $ , with the coefficient $\\beta $ determined by $\\beta \\equiv \\frac{1}{Y} {Y}{\\ln t} = \\frac{2 k(v_\\infty ) v_\\infty }{\\alpha } - \\frac{c v_\\infty }{2 \\alpha } - 2\\nu = -1 + (5 v_\\infty ^2 - 1) \\nu \\,.$ Numerical values for $\\beta $ are also given in table:vanish.", "In order to prove that this fixed point is indeed an attractor we utilize that the system of Eqs.", "(REF ), () and () is an autonomous system of the form $\\dot{X} = f(X)\\,,$ with a fixed point $X_p = (\\alpha , v_\\infty , 0)$ , i.e.", "$f(X_p) = 0$ .", "The Jacobian of $f(X)$ evaluated at $X_p$ reads $\\begin{pmatrix}-1 + \\nu (1 + v_\\infty ^2) & \\tilde{c} [\\frac{1}{2} + \\frac{\\nu v_\\infty ^2}{1 - \\nu (1+v_\\infty ^2)}] & \\frac{\\tilde{c} \\nu v_\\infty }{2 [1 + \\nu (1+v_\\infty ^2)]}(1-v_\\infty ^2) \\\\- \\frac{4 \\nu }{\\tilde{c}} (1 - v_\\infty ^2)[1 - \\nu (1+v_\\infty ^2)] & - 2 \\nu (1 - v_\\infty ^2) & -3 \\nu v_\\infty ( 1 - v_\\infty ^2) \\\\0 & 0 & -1 +\\nu (5v_\\infty ^2 - 1).\\end{pmatrix}\\,.$ Looking at the last row of the Jacobian, we identify one of its eigenvalues, dubbed $\\lambda _3 = -1 +\\nu (5v_\\infty ^2 - 1)$ which matches $\\beta $ from Eq.", "(REF ).", "For the fixed point to be an attractor, the three eigenvalues $\\lambda _1, \\lambda _2$ and $\\lambda _3$ are required to have a negative real part.", "From the trace and determinant of the Jacobian, we find $\\lambda _1 + \\lambda _2 & = -1 +\\nu (3 v_\\infty ^2 - 1) \\in \\mathbb {R}\\,, \\\\\\lambda _1 \\cdot \\lambda _2 & = 4 \\nu (1 - v_\\infty ^2) (1 - \\nu ) > 0 \\,.$ Hence, $\\lambda _1$ and $\\lambda _2$ are complex conjugate and $\\Re {\\lambda _1} = \\Re {\\lambda _2} = \\frac{1}{2} [-1 +\\nu (3 v_\\infty ^2 - 1)]\\,.$ Numerical values for $\\Re {\\lambda _1}$ are shown in table:vanish for several cases; for all values of $\\tilde{c}$ we have considered, the fixed point $Y=0$ is a stable attractor.", "Table: Values of parameters α\\alpha , v ∞ v_\\infty and β\\beta around the vanishing current attractor of sec:vanish for ν=1/2\\nu = 1/2." ], [ "Steady current attractor", "Now we focus our attention on the attractor solution around $Y=1$ identified in fig:streamtube.", "In fig:streamplots, we show slices of this streamplot across the planes $\\alpha = 0.1$ and $Y=1$ .", "Clearly, around this attractor solution, the parameters $\\alpha $ , $v$ and $\\epsilon \\equiv 1 - Y$ converge to 0.", "In what follows, we describe the attractor solution and discuss its stability." ], [ "Description of the attractor solution", "We start again from Eqs.", "(REF ), () and () and expand around $Y=1$ , such that ${\\epsilon (t)} \\ll 1$ .", "We obtain the following nonlinear autonomous system of differential equations ${\\alpha }{\\ln t} & = \\nu \\frac{\\alpha }{2}(3+v^{2} - \\frac{2}{\\nu }) + \\frac{\\tilde{c}}{2}v\\,, \\\\{v}{\\ln t} & = \\frac{1-v^{2}}{2} [\\frac{\\epsilon k(v)}{\\alpha (t) \\sqrt{2}} - 2 v \\nu ]\\,, \\\\- {\\epsilon }{\\ln t} & = 2 [\\frac{v k(v)}{\\alpha \\sqrt{2}} - \\nu ] - \\frac{v}{\\alpha } \\tilde{c} (2 - \\sqrt{2})\\,.$ We make the additional assumption that $\\alpha (t) \\ll 1$ and $v(t) \\ll 1$ after a certain amount of time, thus reducing the system of differential equations to ${\\alpha }{\\ln t} & = \\frac{3\\nu \\alpha }{2} - \\alpha + \\frac{\\tilde{c}}{2} v\\,, \\\\{v}{\\ln t} & = \\frac{\\epsilon }{\\alpha \\pi } - \\nu v\\,, \\\\{\\epsilon }{\\ln t} & = 2\\nu + \\frac{v}{\\alpha } \\tilde{c} (2 - \\sqrt{2} - \\frac{4}{\\tilde{c}\\pi })\\,.", "$ One should note that even though we numerically observe that $(\\alpha , v, \\epsilon ) = (0, 0, 0)$ is an attractor in our dynamical system, it is not a fixed point.", "Indeed, Eqs.", "() and () are singular when $\\alpha = 0$ .", "In the following, we determine the attractor trajectory when $\\alpha (t) \\rightarrow 0$ , such that Eqs.", "(REF ), () and () converge to 0.", "Imposing ${*{\\epsilon }{\\ln t}} \\ll 1$ , we find that $\\alpha $ and $v$ are proportional to each other $\\kappa \\equiv \\frac{v}{\\alpha } = -\\frac{2\\nu }{\\tilde{c}(2-\\sqrt{2})-4 / \\pi }\\,.$ At this point, if we impose that ${*{\\alpha }{\\ln t}} \\ll 1$ , we recover ${\\alpha } \\ll 1$ .", "However, we can go further and show that $\\alpha $ follows a power-law decay, $\\alpha \\propto t^\\delta $ , around the attractor with coefficient $\\delta \\equiv \\frac{1}{\\alpha }{\\ln \\alpha }{\\ln t} = \\frac{3\\nu }{2}-1 - \\frac{\\nu }{(2-\\sqrt{2})-4 / (\\tilde{c}\\pi )}\\,.$ Similarly, if we divide eq:steady-velocity by $v$ , considering that $v$ is proportional to $\\alpha $ , we obtain that $\\delta = \\frac{1}{v} {v}{\\ln t} = \\frac{\\epsilon }{\\alpha v \\pi } - \\nu \\,.$ Thus, we deduce that $\\epsilon $ is proportional to $\\alpha ^2$ $\\frac{\\epsilon }{\\alpha ^2} =(\\delta +\\nu ) \\pi \\frac{v}{\\alpha }\\,.$ The numerical values for all these coefficients are in table:steady.", "In the left panel of fig:transition we show the evolution of $\\alpha $ , $v$ and $Y$ in the scenario where the current suddenly increases from $Y=0$ to $Y=1$ at a particular time.", "Given $\\alpha \\propto t^\\delta $ and derived eq:voveralpha,eq:epsilonoveralpha, in the right panel of fig:transition we show the evolution of expressions that should be approximately time-independent for steady current attractor.", "Figure: Left panel: Evolution of the parameters α\\alpha , v ∞ v_\\infty and YY when the current is artificially sourced to 1 at t=10 -18 t=10^{-18} s, for c ˜=0.23\\tilde{c} = 0.23 and ν=1/2\\nu =1/2.", "Right panel: Evolution of αt -δ \\alpha t^{-\\delta }, v/αv/\\alpha and ϵ/α 2 \\epsilon /\\alpha ^2 that should be approximately time independent for a steady current attractor.Table: Values of parameters v ∞ /αv_\\infty / \\alpha and δ\\delta determining the steady current attractor of sec:steady for ν=1/2\\nu =1/2." ], [ "Stability of the attractor solution", "Unlike in the case of the $Y=0$ attractor, the system of differential equations is not well defined at the steady current attractor $(\\alpha , v, Y) = (0, 0, 1)$ , and we cannot straightforwardly perform the same stability analysis.", "First, the existence of the steady current attractor requires that $\\delta < 0$ which imposes $\\nu < \\frac{8 - 2 ( 2 - \\sqrt{2}) \\tilde{c} \\pi }{12 + (3 \\sqrt{2} - 4) \\tilde{c} \\pi }\\,.$ If $\\tilde{c}=0.23$ , then $\\nu < 0.588$ , meaning that this attractor does not exist in the matter era where $\\nu =2/3$ .", "Second, we estimate numerically the basin of attraction of the steady current attractor.", "Assuming that the system starts from the vanishing current attractor and the current is instantaneously switched on, we compute the minimal value of the current, $Y_\\mathrm {min}$ , needed in order for the system to fall into the steady current attractor.", "Numerical values for $Y_\\mathrm {min}$ are collected in table:steady.", "Our detailed analysis in this section is supported by the previous findings in the literature.", "While the behaviour of the vanishing current attractor solution follows from the standard VOS model, the behavior of the steady current solution has previously been discussed in Ref.", "[46] (see also [55]); our findings are consistent with those in Ref.", "[46]." ], [ "Loop number density", "Let us now derive the number density of cosmic string loops for the steady current attractor scenario, discussed in sec:steady.", "This will allow us to calculate the gravitational-wave spectra (see sec:gw)." ], [ "General expressions", "The length, $L(t)$ , defined in Eq.", "(REF ), is directly related to the energy density of the long-string network.", "The energy lost by the long string reads ${\\rho _{\\infty }}{t}= \\frac{-2 \\mu }{L^{3}} {L}{t}= \\frac{-2 \\mu }{L^{3}} [ \\frac{\\dot{a}}{a} \\frac{L}{1+Y}(1+v^{2}+2Y) + \\frac{\\tilde{c} v}{2} ] \\,.$ The chopping of loops, driven by the last term of eq:energy-lost, induces the following transfer of energy density to the cosmic string loops $\\dot{\\rho }_{\\infty , \\,\\text{loops}} = - \\frac{\\tilde{c}\\mu v(t)}{L^3(t)}\\,.$ Assuming that a fraction, $\\mathcal {F}$ , of this energy goes into large loops with boost factor $\\gamma $ yields $\\mathcal {F} \\dot{\\rho }_{\\infty , \\,\\mathrm {loops}}= - \\gamma \\mu \\int {\\ell } \\ell \\mathcal {P}(\\ell , t)\\,.$ Here, the loop production function $\\mathcal {P}(\\ell , t)$ is the number of loops of sizes in the range $(\\ell , \\ell +{\\ell })$ produced in the time interval $(t, t+{t})$ .", "Furthermore, we assume that all the loops are produced at the same scale with a fraction $\\alpha _L$ of $L(t)$ .", "The loop production function then reads $\\mathcal {P}(\\ell , t) = \\frac{\\tilde{c} \\mathcal {F} v(t)}{\\gamma \\alpha _{L}\\alpha ^{4}(t)} t^{-5} \\delta _D[\\alpha _L \\alpha (t) - \\frac{\\ell }{t}],$ where $\\delta _D$ is the Dirac delta distribution.", "The loop number density is obtained by integrating the loop production function over the possible loop formation times $n(\\ell , t) = \\frac{1}{a^{3}(t)} \\int _{t_{\\mathrm {ini}}}^{t} {t^{\\prime }} a^{3}(t^{\\prime }) \\, \\mathcal {P}(\\ell ^{\\prime }(t^{\\prime }), t^{\\prime })\\,,$ where $\\ell ^{\\prime }(t^{\\prime }) = \\ell + \\Gamma G\\mu (t-t^{\\prime })$ is the hypothetical length of the loop at the time of formation.", "The delta distribution in $\\mathcal {P}$ imposes a unique choice for the loop formation time $t_\\star $ satisfying $\\alpha _L \\alpha (t_{\\star })t_{\\star } + \\Gamma G\\mu t_{\\star } = \\ell + \\Gamma G\\mu t\\,.$ The loop number density is then given by $t^{4} n(\\ell , t) = \\frac{\\tilde{c} \\mathcal {F} v(t_\\star )}{\\gamma \\alpha _{L}\\alpha ^{4}(t_\\star )} \\frac{\\Theta (t_{\\star } - t_{\\mathrm {ini}}) \\Theta (t - t_{\\star })}{\\Gamma G\\mu + \\alpha _L \\alpha (t_{\\star }) + \\alpha _L \\alpha ^{\\prime }(t_{\\star }) t_{\\star }} [\\frac{a(t_{\\star })}{a(t)}]^{3} (\\frac{t}{t_{\\star }})^{4}\\,,$ where $t_{\\mathrm {ini}}$ is the time when the current is switched on.", "Let us stress that there does not exist a closed form for $t_\\star $ and we evaluate eq:tstar numerically for all practical purposes.", "We, however, point out the existence of an analytical approximation (see sec:approx) for the loop number density that works rather well and which helps in gaining an analytical understanding." ], [ "Analytical approximations", "The equation for the loop number density in Eq.", "(REF ) can be simplified if we consider the limit in which $\\alpha _L \\alpha (t)\\gg \\Gamma G\\mu $ for all the duration of the current-carrying phase.", "This approximation allows us to solve for the loop formation time, $t_\\star $ , in a closed form $\\ell + \\Gamma G\\mu t =[\\alpha _L \\alpha (t_\\star )+\\Gamma G \\mu ]\\, t_\\star \\quad \\Longrightarrow \\quad t_\\star \\simeq t_{\\mathrm {ini}} \\left(\\frac{\\ell +\\Gamma G\\mu t}{\\alpha _L \\alpha _s t_{\\mathrm {ini}}}\\right)^{\\frac{1}{1+\\delta }},$ where we employed $\\alpha (t)=\\alpha _s \\, (t/t_{\\mathrm {ini}})^\\delta $ parametrization according to the asymptotic power–law behaviour described in sec:steady; here $\\alpha _s=\\alpha (t_{\\mathrm {ini}})$ .", "We then obtain $t^4 n(\\ell ,t) \\simeq \\frac{\\tilde{c} \\mathcal {F} \\kappa \\alpha _L^2}{(1+\\delta )\\gamma }\\frac{a(t_\\star )^3}{a(t)^3}\\left(x + \\Gamma G\\mu \\right)^{-4}\\Theta (t_{\\star }-t_{\\mathrm {ini}})\\Theta (t - t_{\\star })\\,,$ where we have defined $x=\\ell /t$ , and $\\kappa \\equiv v(t_\\star )/\\alpha (t_\\star )$ is an $\\mathcal {O}(1)$ number within the current-carrying attractor solution, as shown in table:steady.", "The expression above can be further simplified if the time at which the loop number density is evaluated and the loop formation time belong to the same cosmological era described by $a(t)\\propto t^\\nu $ .", "Then we obtain $t^4 n(\\ell ,t) \\simeq \\frac{ \\tilde{c} \\mathcal {F} \\kappa \\alpha _L^2}{(1+\\delta )\\gamma (\\alpha _L \\alpha _s)^{\\frac{3\\nu }{1+\\delta }}}(\\frac{t}{t_{\\mathrm {ini}}})^{-\\frac{3\\nu \\delta }{1+\\delta }}\\left(x + \\Gamma G\\mu \\right)^{-4+\\frac{3\\nu }{1+\\delta }}\\,\\Theta (t_{\\star }-t_{\\mathrm {ini}})\\Theta (t - t_{\\star })\\,.$ A couple of comments are in order.", "First, if we set $\\delta =0$ we see that $n(\\ell ,t) t^4$ agrees with the usual scaling in which $\\ell $ and $t$ only enter through the dimensionless ratio $x$ .", "We then recover $n(\\ell ,t)t^4 \\propto (x + \\Gamma G \\mu )^\\beta $ with $\\beta = -5/2 \\, (-2)$ in radiation (matter) domination, see e.g.", "Ref. [15].", "On the other hand, the loop number density does not scale for $\\delta \\ne 0$ , namely we still have an explicit dependence on $t$ due to the presence of $t_{\\mathrm {ini}}$ .", "In particular, since $\\delta \\in (-1, 0)$ the loop number density increases with the duration of the current-carrying phase, $t/t_{\\mathrm {ini}}$ .", "Figure: t 4 n(ℓ,t)t^4 n(\\ell ,t) for different times tt as a functionof x=ℓ/tx=\\ell /t.", "Left panel: The black dashed line correspondsto the standard scaling.", "It can be seen that during the current-carrying phase more loops are produced at shorter lengths.", "In the calculation, we have taken Gμ=10 -10 G\\mu =10^{-10} and assumed that the transition starts at redshift z=10 15 z=10^{15} and ends at z=10 9 z=10^{9}.Right panel: t 4 n(ℓ,t)t^4 n(\\ell ,t) after the end of the current-carrying phase is shown.", "A gap develops in the loop distribution, see text for details.We present the loop number density in the current-carrying phase in fig:n (left panel).", "Comparing to the standard case, we can infer that during the current-carrying phase more loops are produced at shorter lengths.", "This already suggests interesting implications for the calculation of the gravitational-wave spectrum (sec:gw), which depends on the loop number density.", "The right panel of fig:n features a gap which arises due to the fact that we assume an instantaneous switch from the steady current attractor to the standard attractor for the long string network.", "This means that the loop size at formation is suddenly changed from $\\alpha _L \\alpha (t_\\text{last})$ to $\\alpha _L \\alpha $ , with $\\alpha > \\alpha (t_\\text{last})$ .", "In the realistic case of a current gradually turning on and off, the gap will be partially filled, thus increasing the gravitational-wave signal coming from the current–carrying phase.", "In this sense, our assumption of instantaneous turn off yields a conservative estimate of the spectrum." ], [ "Energy density of the network", "Let us now move on to discuss how the different components of the energy density related to the cosmic string network behave compared to the critical density.", "In the standard scaling regime ($\\delta =0$ ), both the long strings and the loops constitute a constant and subdominant component of the energy density at all times.", "This is, however, no longer true in the presence of currents.", "For the long strings, we have $\\frac{\\rho _\\infty }{\\rho _c} \\sim G t^2 \\frac{\\mu }{L^2}= \\frac{G \\mu }{\\alpha (t)^2}\\simeq \\frac{G \\mu }{\\alpha _s^2}\\left(\\frac{t}{t_{\\mathrm {ini}}}\\right)^{-2\\delta }\\,,$ where we have taken the critical density as $\\rho _c \\sim 1/Gt^2$ .", "This shows that, given negative $\\delta $ for steady current attractor (see again sec:steady), the energy density in the long strings has a power-law growth with the duration of the steady-current phase $t/t_{\\mathrm {ini}}$ .", "The time evolution of the energy density in loops is given by the following integral $\\frac{\\rho _L}{\\rho _c} \\sim G t^2 \\int {\\ell } \\,\\mu \\, \\ell \\, n(\\ell ,t)\\,,$ which can be evaluated assuming $\\alpha _L\\alpha (t)\\gg \\Gamma G \\mu $ by using (REF ) for the radiation era.", "The result becomes particularly simple when we look at sufficiently late times, $t\\gg t_{\\mathrm {ini}}/\\Gamma G \\mu $ , for which we find$\\endcsname $Notice that this is compatible with $\\alpha _L\\alpha (t)\\gg \\Gamma G \\mu $ , which requires $t \\ll t_{\\mathrm {ini}}/(\\Gamma G \\mu )^{1/|\\delta |}$ .", "$\\frac{\\rho _L}{\\rho _c} \\sim \\frac{\\tilde{c} \\mathcal {F} \\kappa \\alpha _L^2}{(1+\\delta )(\\alpha _L \\alpha _s)^{\\frac{3 \\nu }{1+\\delta }}}\\frac{(G \\mu )^{3+p} \\Gamma ^{2+p}}{(1+p)(2+p)} \\left(\\frac{t}{t_{\\mathrm {ini}}}\\right)^{-\\frac{3 \\nu \\delta }{1+\\delta }}\\,.$ Here, $p=-4+3\\nu /(1+\\delta )\\approx -2.2$ .", "When comparing the energy in loops with the energy in the long strings we obtain $\\frac{\\rho _\\infty }{\\rho _c} \\left(\\frac{\\rho _L}{\\rho _c}\\right)^{-1}\\sim (G \\mu )^{-2-p} \\left(\\frac{t}{t_{\\mathrm {ini}}}\\right)^{-2\\delta + 3 \\nu \\frac{\\delta }{1+\\delta }}\\sim (G \\mu )^{0.214\\dots }\\left(\\frac{t}{t_{\\mathrm {ini}}}\\right)^{0.034\\dots }\\,.$ We thus conclude that given the very mild time growth in (REF ) and the small prefactor, the energy density in loops will generically be dominant with respect to the long string network.", "For completeness, note that in the case with no currents and $\\delta =0$ , we recover the standard results $\\rho _\\infty /\\rho _c \\sim G \\mu $ and $\\rho _L/\\rho _c \\sim (G \\mu /\\Gamma )^{1/2}\\,$ .", "Finally, let us evaluate the energy density in gravitational waves during the current-carrying phase.", "Each loop is expected to emit with a constant power $P=\\Gamma G\\mu ^2$ during its lifetime, so that the total amount of energy in gravitational waves at the time $t$ is given by $\\frac{\\rho _\\mathrm {GW}}{\\rho _c} = \\Gamma (G\\mu )^2 t^2\\int _{t_{\\mathrm {ini}}}^t \\left[\\frac{a(t^\\prime )}{a(t)}\\right]^4 \\mathcal {N}(t^\\prime ) \\,{t^\\prime },$ where $\\mathcal {N}(t)$ is the total number density obtained after integrating $n(\\ell ,t)$ over all possible loop lengths, $\\mathcal {N}(t) = \\int n(\\ell ,t) \\,{\\ell }\\,$ .", "The lower integration bound, $t_{\\mathrm {ini}}$ , ensures that we select contributions only from loops formed after the occurrence of the current.", "Loops formed prior to the current carrying phase, as well as loops formed afterwards, can be accounted for in the standard way.", "Direct evaluation of (REF ) at times $t\\gg t_{\\mathrm {ini}}/\\Gamma G \\mu $ shows that the energy density in loops and in gravitational waves are proportional to each other and of the same order, $\\frac{\\rho _\\mathrm {GW}}{\\rho _L}\\simeq \\frac{(1+\\delta )(-2-p)}{-2-2\\delta +4\\nu +\\delta \\nu } \\sim \\mathcal {O}(1)\\,.$ A summary plot with the behavior of the different components of the energy density associated to the cosmic strings is shown in fig:rhos.", "We conclude that the most relevant contribution to the total energy density comes from the population of small loops and it grows in time.", "This can still be a subdominant component provided that the right hand side of (REF ) is much smaller than unity at the time at which the current-carrying phase has ended, providing agreement with standard cosmology.", "The disappearance of the current can occur because of a change in the microscopic properties of the system, e.g.", "the occurrence of a phase transition (see the comment on RHN currents in the Introduction), or due to the onset of matter domination in which the steady current attractor no longer exists.", "Figure: Energy density in small loops, ρ L \\rho _L,gravitational waves, ρ GW \\rho _\\mathrm {GW}, and long string network,ρ ∞ \\rho _\\infty , normalized to the critical density ρ c ∼1/Gt 2 \\rho _c \\sim 1/Gt^2, as a function of time after the onset of the current–carrying phase, t ini t_\\text{ini}." ], [ "Gravitational wave signal", "Equipped with the modified loop number density during the current-carrying phase we are now ready to present the gravitational wave spectra emitted by CCCSs.", "We work with the assumption that the current-carrying phase does not match the time of string network formation.", "In particular, it is expected that the current does not form at the time of the phase transition, but only at a later time when the temperature of the thermal bath has become sufficiently low.", "This is because the temperature needs to be smaller than the difference in current-carrier mass inside and outside of the string.", "We describe the gravitational wave signal as the superposition of three different contributions, $\\Omega _\\mathrm {GW}^{tot}= \\Omega _\\mathrm {GW}^0(t_F,t_{\\mathrm {ini}}) +\\Omega _\\mathrm {GW}(t_{\\mathrm {ini}},t_\\text{end})+\\Omega _\\mathrm {GW}^0(t_\\text{end},t_0)\\,,$ stemming from loops whose formation time is $t_F<t<t_{\\mathrm {ini}}$ , where $t_F$ is the formation time of the cosmic string network (which can be approximately taken as the time of the corresponding phase transition) and $t_{\\mathrm {ini}}$ is the time at which the current first appears.", "For these loops we adopt the standard loop production function ; loops formed during the current-carrying phase, for which the loop number density is given in (REF ).", "This will be the case for $t_{\\mathrm {ini}}<t<t_\\text{end}$ , where $t_\\text{end}$ corresponds to the time at which the current on the network disappears either because of leakage effects or because the Universe entered matter domination era (as discussed in sec:steady-stability, there is no steady current attractor solution during such times) ; loops formed after the end of the current carrying phase, $t>t_\\text{end}$ , for which the production function is again taken to be the standard one as in $(i)$  .", "Figure: GW spectrum as a function of the time at which the current appears, t ini t_{\\mathrm {ini}}, and disappears, t end t_\\text{end}, from the networkfor a string tension of Gμ=10 -11 G \\mu = 10^{-11}.", "The parameter region labeled LVK is disfavored by searches for the stochastic gravitational wave background at ground-based interferometers .We excluded GW signals with h 2 Ω GW >5×10 -9 h 2 h^2 \\Omega _\\mathrm {GW} > 5\\times 10^{-9} h^2 at 20 Hz corresponding to the limits set on a flat power spectrum with a log-uniform prior.For contributions from $(i)$ and $(iii)$ we use the standard formula for energy density in gravitational waves, $\\Omega _\\mathrm {GW}^0$ , see for instance Eq.", "(27) in Ref. [22].", "Hence, we will only show here the expression for gravitational wave energy density during current-carrying phase.", "Note that the average power spectrum for GW emission reads $P_k=\\Gamma /(k^q \\zeta (q))$ , and we focus on cusps for which $q=4/3$ .", "Utilizing results from sec:loopnumber, we obtain the following formula for $k$ -th mode contribution to the gravitational wave spectrum $\\Omega _\\mathrm {GW}^{k}(t_{\\mathrm {ini}},t_\\text{end}) = \\frac{F}{G \\rho _c}\\frac{2 k}{f} \\Gamma ^k (G\\mu )^2 \\int _{t_{\\mathrm {ini}}}^{t_0} {t}\\frac{\\theta (t_*^k-t_{\\mathrm {ini}})\\theta (t_\\text{end}-t_*^k)}{\\alpha _L \\alpha _s [\\alpha _L \\alpha _s(1+\\delta )(t_*^k/t_{\\mathrm {ini}})^\\delta + \\Gamma G\\mu ]}\\left[\\frac{a(t)}{a(t_0)}\\right]^5\\left[\\frac{a(t_*^k)}{a(t)}\\right]^3\\frac{C_\\mathrm {eff}}{(t_*^k)^4}\\,,$ where $t_0$ denotes the present time, $t_*^k$ is the formation time of the loop contributing at the time $t$ to the production of gravitational waves at frequency $f$ , and it is the solution of $\\frac{2 k a(t)}{f a(t_0)} = \\alpha _L \\alpha (t_{\\mathrm {ini}})(t_*^k/t_{\\mathrm {ini}})^\\delta t_*^k -\\Gamma G \\mu (t-t_*^k)\\,,$ for which a good approximation is given by $t_*^k \\approx \\left\\lbrace \\frac{t_{\\mathrm {ini}}^\\delta }{\\alpha _L \\alpha (t_{\\mathrm {ini}})}\\left[\\frac{a(t)}{a(t_0)} \\frac{2 k}{f} + \\Gamma G \\mu t\\right]\\right\\rbrace ^{\\frac{1}{1+\\delta }}.$ In passing, let us stress that, as expected, the general expression in eq:GWspectrum matches the standard one in the limit $\\delta \\rightarrow 0$ .", "In order to compute the GW spectrum, we sum over $10^6$ modes, namely, $\\Omega _\\mathrm {GW}(t_{\\mathrm {ini}},t_\\text{end})=\\sum _{k=1}^{10^6}\\Omega _\\mathrm {GW}^{k}(t_{\\mathrm {ini}},t_\\text{end})\\,.$ We stress that the current is expected to cause a backreaction on string loops.", "This can lead to the emission of current carriers from strings cusps, which would result in the smoothing of the cusps.", "We, therefore, in addition to the full contribution from all cusps also compute the case where string loops oscillate at their fundamental frequency (only taking mode $k=1$ ).", "Hence, for each of the five benchmark cases in fig:money we show both $k=1$ and $k\\lesssim 10^6$ contributions and this is what gives a “width” to the predicted gravitational wave spectra.", "What is immediately obvious by looking at each of the spectral predictions is an enhancement in $\\Omega _\\mathrm {GW}^{tot}$ arising from the period when the current is on.", "This enhancement can be explained by looking at the denominator of eq:GWspectrum.", "For $-1<\\delta <0$ , ${\\alpha _L \\alpha _s [\\alpha _L \\alpha _s(1+\\delta ) (t_*^k/t_{\\mathrm {ini}})^\\delta + \\Gamma G\\mu ]}$ would attain smaller values than in the standard $\\delta =0$ case, leading to the rise in the spectrum.", "The origin of the enhancement is the increased loop number density in presence of a current when compared to the standard case.", "In fig:money we consider five different scenarios (denoted with (A)–(E)), corresponding to different values of $t_{\\mathrm {ini}}$ and $t_\\text{end}$ and this is what causes the rise of the spectrum in different frequency regions.", "Benchmark points (A), (C) and (E) are computed for $G\\mu =0.5\\times 10^{-10}$ .", "For benchmark point (A), $t_\\text{end}$ is the time of QCD phase transition; further, the current for point (C) ends at the time of electroweak phase transition while for (E) the current stops flowing at the time that is $10^{-4}$ times smaller than for (C).", "As the current gets turned off earlier, the spectral enhancement shifts to higher frequency.", "Benchmark points (A) and (C) are receiving current-induced spectral modification in the frequency region where ground-based interferometers such as LIGO [63], [64] are most sensitive.", "In addition, (A) can also be tested at forthcoming space-based interferometers such as BBO [65] and DECIGO [66].", "Benchmark point (E), on the other hand, is interesting in the context of experiments sensitive to GWs in MHz region [67].", "The two remaining benchmark points are calculated for $G\\mu =0.5\\times 10^{-17}$ .", "For (B) and (D), the current stops at QCD and electroweak phase transition, respectively.", "As $G\\mu $ gets smaller, we observe that for a fixed $t_\\text{end}$ the spectral enhancement is shifted to higher frequencies; we can see that by comparing (A) and (B) as well as (C) and (D) for which current stops flowing at the same time.", "We point out that the enhancement in the spectrum in the region where PTA experiments such as NANOGrav [24] are sensitive is not supported by standard cosmology; the same holds for LISA [68], [69].", "Namely, to make those experiments sensitive to the considered CCCS scenario, one would need to require that currents last for a very long time, which can easily result in an overclosure problem or unwanted modifications of the CMB power spectrum.", "In addition, it is not clear whether any microscopic theory of current generation and decay would realistically allow one to delay the time of current quenching to very late times.", "Naively, we would expect phase transitions in the string network to occur much earlier.", "We nevertheless observe that a portion of the parameter space of cosmic string tension that is in the standard case considered to be unreachable by ground-based detectors such as LIGO, becomes testable in the scenario where cosmic strings feature currents.", "For all benchmark points in fig:money we use $t_{\\mathrm {ini}}=10^{-35}$ s. In fig:time-time we show the dependence of the total gravitational wave spectrum as a function of $t_{\\mathrm {ini}}$ and $t_\\text{end}$ fixing the string tension as $G \\mu = 10^{-11}$ .", "The region excluded by present ground-based interferometers is indicated in the upper left corner as “LVK”." ], [ "Conclusions", "The emission of gravitational waves by cosmic-string networks in the early Universe is typically studied in terms of simple models, e.g., the Nambu–Goto action or the Abelian Higgs model.", "In this work, we made an attempt to go beyond the standard Nambu–Goto approximation by investigating the implications for the gravitational-wave spectrum in the presence of additional worldsheet degrees of freedom, namely, charge carriers that give rise to a current flowing on cosmic strings.", "We specifically focused on neutral currents, i.e., currents composed of particles that do not interact with any long-range force field, in order to avoid energy loss of the string network via other decay channels on top of the emission of gravitational waves.", "At the same time, we restricted ourselves to the case of chiral currents, which are characterized by a particularly simple state parameter $K = Q^2 - J^2 = 0$ and which appear to be well motivated in view of the possibility of chiral RHN currents on cosmic strings in $SO(10)$ GUT models.", "In order to study the evolution of a current-carrying cosmic string (CCCS) network, we employed the generalized velocity-depdendent one-scale model formulated in Ref.", "[47], which allows one to track the time dependence of three characteristic properties of the network: its correlation length, the RMS velocity of long strings, and the strength of the current carried by the network.", "After rewriting them as a nonlinear autonomous system, we analyzed and solved the generalized VOS equations, which allowed us to identfity two attractor solutions: one solution in the vicinity of the standard VOS attractor solution featuring a decaying current, $Y \\rightarrow 0$ ; as well as a new solution in the large-current regime, $Y \\rightarrow 1$ , featuring a decaying dimensionless correlation length $\\alpha = L/t$ as well as a decaying RMS velocity $v$ .", "The second attractor solution can be reached whenever the process of current formation on cosmic strings results in a large initial current that is of $\\mathcal {O}\\left(1\\right)$ right from the start; see e.g.", "the $Y_{\\rm min}$ values in Tab.", "REF .", "In future work, it will therefore be interesting to identify concrete microscopic CCCS models that can give rise to such large currents on cosmic strings, i.e., physical current energy densities as large as $\\mathcal {O}\\left(\\mu \\right)$ .", "For the purposes of this paper, we decided to restrict ourselves to a bottom-up phenomenological analysis and focus on the implications of the second attractor solution of the generalized VOS model for the gravitational-wave spectrum.", "To this end, we computed the loop number density as well as energy density of a CCCS network based on the $Y\\rightarrow 1$ attractor solution, which led us to several interesting observations.", "First of all, the shrinking dimensioneless correlation length $\\alpha = L/t$ during the current-carrying phase results in the production of smaller loops, which boosts the loop number density in the small-loop regime and causes a gap in the loop number density after the end of of the current-carrying phase; see Fig.", "REF .", "At the same, the steadily decreasing value of $\\alpha $ results in a growing fractional energy density of long strings, $\\Omega _{\\infty } = \\rho _\\infty / \\rho _c \\propto \\alpha ^{-2}$ , which eventually translates into growing fractional energy densities of loops and gravitational waves; see Fig.", "REF .", "The extra energy in gravitational waves notably manifests itself in a peak-like enhancement of the standard gravitational-wave spectrum from cosmic strings whose boundaries are controlled by the times of current generation and decay; see Fig.", "REF .", "The boosted gravitational-wave signal across large ranges in frequency implies interesting prospects for current and upcoming gravitational-wave experiments.", "Values of the cosmic-string tension that are out of reach of existing experiments may e.g.", "be probed in the near future if currents should be responsible for an enhanced signal in the right frequency band; see e.g.", "benchmark points (A) and (C).", "At the same, we observe that gravitational waves from CCCSs are an interesting target for planned gravitational-wave searches at high frequencies, reaching up to the MHz regime; see e.g.", "benchmark point (E).", "Finally, we caution that our results come with a sizable uncertainty, which is, at least to some extent, reflected in the width of the different bands in Fig.", "REF .", "Ultimately, our results call for an independent confirmation based on numerical simulations.", "The validity of the standard VOS model has been well confirmed by simulations; similarly, simulations of CCCS networks might allow one to confirm the validity of the generalized VOS model, and hence our promising predictions for the associated gravitational-wave signal." ], [ "Acknowledgements", "We thank Patrick Peter, Christophe Ringeval, Paul Shellard and Lara Sousa for useful discussions.", "Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under Contract No.", "DE-AC02-07CH11359.", "The work of PA is partially supported by the Wallonia-Brussels Federation Grant ARC № 19/24-103." ] ]
2207.03510
[ [ "Generalization Guarantee of Training Graph Convolutional Networks with\n Graph Topology Sampling" ], [ "Abstract Graph convolutional networks (GCNs) have recently achieved great empirical success in learning graph-structured data.", "To address its scalability issue due to the recursive embedding of neighboring features, graph topology sampling has been proposed to reduce the memory and computational cost of training GCNs, and it has achieved comparable test performance to those without topology sampling in many empirical studies.", "To the best of our knowledge, this paper provides the first theoretical justification of graph topology sampling in training (up to) three-layer GCNs for semi-supervised node classification.", "We formally characterize some sufficient conditions on graph topology sampling such that GCN training leads to a diminishing generalization error.", "Moreover, our method tackles the nonconvex interaction of weights across layers, which is under-explored in the existing theoretical analyses of GCNs.", "This paper characterizes the impact of graph structures and topology sampling on the generalization performance and sample complexity explicitly, and the theoretical findings are also justified through numerical experiments." ], [ "Introduction", "Graph convolutional neural networks (GCNs) aggregate the embedding of each node with the embedding of its neighboring nodes in each layer.", "GCNs can model graph-structured data more accurately and compactly than conventional neural networks and have demonstrated great empirical advantage in text analysis [14], [17], [29], [23], computer vision [26], [31], [15], recommendation systems [34], [28], physical reasoning [2], [25], and biological science [10].", "Such empirical success is often achieved at a cost of higher computational and memory costs, especially for large graphs, because the embedding of one node depends recursively on the neighbors.", "To alleviate the exponential increase of computational cost in training deep GCNs, various graph topology sampling methods have been proposed to only aggregate the embeddings of a selected subset of neighbors in training GCNs.", "Node-wise neighbor-sampling methods such as GraphSAGE [14], VRGCN [4], and Cluster-GCN [6] sample a subset of neighbors for each node.", "Layer-wise importance sampling methods such as FastGCN [3] and LADIES [39] sample a fixed number of nodes for each layer based on the estimate of node importance.", "Another line of works such as [36], [19], [5] employ graph sparsification or pruning to reduce the computational and memory cost.", "Surprisingly, these sampling methods often have comparable or even better testing performance compared to training with the original graph in many empirical studies [3], [5].", "In contrast to the empirical success, the theoretical foundation of training GCNs with graph sampling is much less investigated.", "Only [7] analyzes the convergence rate of graph sampling, but no generalization analysis is provided.", "One fundamental question about training GCNs is still vastly open, which is: Under what conditions does a GCN learned with graph topology sampling achieve satisfactory generalization?", "Our contributions: To the best of our knowledge, this paper provides the first generalization analysis of training GCNs with graph topology sampling.", "We focus on semi-supervised node classification problems where, with all node features and partial node labels, the objective is to predict unknown node labels.", "We summarize our contributions from the following dimensions.", "First, this paper proposes a training framework that implements both stochastic gradient descent (SGD) and graph topology sampling, and the learned GCN model with Rectified Linear Unit (ReLU) activation is guaranteed to approach the best generalization performance of a large class of target functions.", "Moreover, as the number of labeled nodes and the number of neurons increase, the class of target function enlarges, indicating improved generalization.", "Second, this paper explicitly characterizes the impact of graph topology sampling on the generalization performance through the proposed effective adjacency matrix ${A}^*$ of a directed graph that models the node correlations.", "${A}^*$ depends on both the given normalized graph adjacency matrix in GCNs and the graph sampling strategy.", "We provide the general insights that (1) if a node is sampled with a low frequency, its impact on other nodes is reduced in ${A}^*$ compared with ${A}$ ; (2) graph sampling on a highly-unbalanced ${A}$ , where some nodes have a dominating impact in the graph, results in a more balanced ${A}^*$ .", "Moreover, these insights apply to other graph sampling methods such as FastGCN [3].", "We show that learning with topology sampling has the same generalization performance as training GCNs using ${A}^*$ .", "Therefore, a satisfactory generalization can still be achieved even when the number of sampled nodes is small, provided that the resulting ${A}^*$ still characterizes the data correlations properly.", "This is the first theoretical explanation of the empirical success of graph topology sampling.", "Third, this paper shows that the required number of labeled nodes, referred to as the sample complexity, is a polynomial of $\\Vert {A}^*\\Vert _\\infty $ and the maximum node degree, where $\\Vert \\cdot \\Vert _\\infty $ measures the maximum absolute row sum.", "Moreover, our sample complexity is only logarithmic in the number of neurons $m$ and consistent with the practical over-parameterization of GCNs, in contrast to the loose bound of poly($m$ ) in [35] in the restrictive setting of two-layer (one-hidden-layer) GCNs without graph topology sampling." ], [ "Related Works", "Generalization analyses of GCNs without graph sampling.", "Some recent works analyze GCNs trained on the original graph.", "[32], [7] characterize the expressive power of GCNs.", "[33] analyzes the convergence of gradient descent in training linear GCNs.", "[21], [20], [12], [22] characterize the generalization gap, which is the difference between the training error and testing error, through Rademacher complexity.", "[30], [7], [38] analyze the generalization gap of training GCNs using SGD via the notation of algorithmic stability.", "To analyze the training error and generalization performance simultaneously, [9] uses the neural tangent kernel (NTK) approach, where the neural network width is infinite and the step size is infinitesimal, shows that the training error is zero, and characterizes the generalization bound.", "[35] proves that gradient descent can learn a model with zero population risk, provided that all data are generated by an unknown target model.", "The result in [35] is limited to two-layer GCNs and requires a proper initialization in the local convex region of the optimal solution.", "Generalization analyses of feed-forward neural networks.", "The NTK approach was first developed to analyze fully connected neural networks (FCNNs), see, e.g., [16].", "The works of [37], [11], [18] analyze one-hidden-layer neural networks with Gaussian input data.", "[8] analyzes multi-layer FCNNs but focuses on training the last layer only, while the changes in the hidden layers are negligible.", "[1] provides the optimization and generalization of three-layer FCNNs.", "Our proof framework is built upon [1] but makes two important technical contributions.", "First, this paper provides the first generalization analysis of graph topology sampling in training GCNs, while [1] considers FCNNs with neither graph topology nor graph sampling.", "Second, [1] considers i.i.d.", "training samples, while this paper considers semi-supervised GCNs where the training data are correlated through graph convolution." ], [ "Notations", "Vectors are in bold lowercase, matrices and tensors in are bold uppercase.", "Scalars are in normal fonts.", "For instance, ${Z}$ is a matrix, and ${z}$ is a vector.", "$z_i$ denotes the $i$ -th entry of ${z}$ , and $Z_{i,j}$ denotes the $(i,j)$ -th entry of ${Z}$ .", "$[K]$ ($K>0$ ) denotes the set including integers from 1 to $K$ .", "${I}_d\\in \\mathbb {R}^{d\\times d}$ and ${e}_i$ represent the identity matrix in $\\mathbb {R}^{d \\times d}$ and the $i$ -th standard basis vector, respectively.", "We denote the column $\\ell _p$ norm for ${W}\\in \\mathbb {R}^{d\\times N}$ (for $p\\ge 1$ ) as $\\Vert {W}\\Vert _{2,p}=(\\sum _{i\\in [m]}\\Vert {w}_i\\Vert _2^p)^\\frac{1}{p}$ Hence, $\\Vert {W}\\Vert _{2,2}=\\Vert {W}\\Vert _F$ is the Frobenius norm of ${W}$ .", "We use ${w}_i$ ($\\tilde{{w}}_i$ ) to denote the $i$ -th column (row) vector of ${W}$ .", "We follow the convention that $f(x)=O(g(x))$ (or $\\Omega (g(x))$ , $\\Theta (g(x)))$ means that $f(x)$ increases at most (or at least, or in the same, respectively,) order of $g(x)$ .", "With high probability (w.h.p.)", "means with probability $1-e^{-c\\log ^2(m_1, m_2)}$ for a sufficient large constant $c$ where $m_1$ and $m_2$ are the number of neurons in the two hidden layers.", "Function complexity.", "For any smooth function $\\phi (z)$ with its power series representation as $\\phi (z)=\\sum _{i=0}^\\infty c_i z^i$ , define two useful parameters as follows, $\\mathcal {C}_\\epsilon (\\phi , R)=\\sum _{i=0}^\\infty \\Big ((C^*R)^i+(\\frac{\\sqrt{\\log (1/\\epsilon )}}{\\sqrt{i}}C^*R)^i\\Big )|c_i|$ $\\mathcal {C}_s(\\phi , R)=C^*\\sum _{i=0}^\\infty (i+1)^{1.75} R^i|c_i|$ where $R\\ge 0$ and $C^*$ is a sufficiently large constant.", "These two quantities are used in the model complexity and sample complexity, which represent the required number of model parameters and training samples to learn $\\phi $ up to $\\epsilon $ error, respectively.", "Many population functions have bounded complexity.", "For instance, if $\\phi (z)$ is $\\exp (z)$ , $\\sin (z)$ , $\\cos (z)$ or polynomials of $z$ , then $\\mathcal {C}_\\epsilon (\\phi ,O(1))\\le O(\\text{poly}(1/\\epsilon ))$ and $\\mathcal {C}_s(\\phi , O(1))\\le O(1)$ .", "The main notations are summarized in Table REF in Appendix." ], [ "Training GCNs with Topology Sampling: Formulation and Main Components", "GCN setup.", "Let $\\mathcal {G}=\\lbrace \\mathcal {V},\\mathcal {E}\\rbrace $ denote an un-directed graph, where $\\mathcal {V}$ is the set of nodes with size $|\\mathcal {V}|=N$ and $\\mathcal {E}$ is the set of edges.", "Let $\\tilde{{A}}\\in \\lbrace 0,1\\rbrace ^{N\\times N}$ be the adjacency matrix of $\\mathcal {G}$ with added self-connections.", "Let ${D}$ be the degree matrix with diagonal elements $D_{i,i}=\\sum _{j} \\tilde{A}_{i, j}$ and zero entries otherwise.", "${A}$ denotes the normalized adjacency matrix with ${A}={D}^{-\\frac{1}{2}}\\tilde{{A}}{D}^{-\\frac{1}{2}}$ .", "Let ${X}\\in \\mathbb {R}^{N\\times d}$ denote the matrix of the features of $N$ nodes, where the $n$ -th row of ${X}$ , denoted by $\\tilde{{x}}_n \\in \\mathbb {R}^{1\\times d}$ , represents the feature of node $n$ .", "Assume $\\Vert \\tilde{{x}}_n\\Vert =1$ for all $n$ without loss of generality.", "$y_n\\in \\mathcal {Y}$ represents the label of node $n$ , where $\\mathcal {Y}$ is a set of all labels.", "$y_n$ depends on not only ${x}_n$ but the neighbors.", "Let $\\Omega \\subset \\mathcal {V}$ denote the set of labeled nodes.", "Given ${X}$ and labels in $\\Omega $ , the objective of semi-supervised node-classification is to predict the unknown labels in $\\mathcal {V}/\\Omega $ .", "Learner network We consider the setting of training a three-layer GCN $F: \\mathbb {R}^N\\times \\mathbb {R}^{N\\times d}\\rightarrow \\mathbb {R}^{1\\times K}$ with $\\begin{aligned}F_{{A}}({e}_g,{X};{W},{V})&=\\bf {e}_g^\\top {A}\\sigma ( {r}+{B}_2){C}~\\text{~and~} \\\\{r}&= {A}\\sigma ({A}{X}{W}+{B}_1){V}\\end{aligned}$ where $\\sigma (x)= \\max (x,0)$ is the ReLU activation function, ${W}\\in \\mathbb {R}^{d\\times m_1}$ and ${V}\\in \\mathbb {R}^{m_1\\times m_2}$ represent the weights of $m_1$ and $m_2$ hidden nodes in the first and second layer, respectively.", "${B}_1\\in \\mathbb {R}^{N\\times m_1}$ and ${B}_2\\in \\mathbb {R}^{m_1\\times m_2}$ represent the bias matrices.", "${C}\\in \\mathbb {R}^{m\\times K}$ is the output weight vector.", "${e}_g\\in \\mathbb {R}^N$ belongs to $\\lbrace {e}_i\\rbrace _{i=1}^N$ and selects the index of the node label.", "We write $F$ as $F_{{A}}({e}_g,{X}; {W}, {V})$ , because we only update ${W}$ and ${V}$ in training, and ${A}$ represents the graph topology.", "Note that in conventional GCNs such as [17], ${C}$ is a learnable parameter, and ${B}_1$ and ${B}_2$ can be zero.", "Here for the analytical purpose, we consider a slightly different model where ${C}$ , ${B}_1$ and ${B}_2$ are fixed as randomly selected values.", "Consider a loss function $L: \\mathbb {R}^{1\\times k}\\times \\mathcal {Y} \\rightarrow \\mathbb {R}$ such that for every $y\\in \\mathcal {Y}$ , the function $L(\\cdot , y)$ is nonnegative, convex, 1-Lipschitz continuous and 1-Lipschitz smooth and $L(0, y)\\in [0, 1]$ .", "This includes both the cross-entropy loss and the $\\ell _2$ -regression loss (for bounded $\\mathcal {Y}$ ).", "The learning problem solves the following empirical risk minimization problem: $\\min _{{W}, {V}}L_{\\Omega }({W}, {V})= \\frac{1}{|\\Omega |}\\sum _{i \\in \\Omega } L(F_{A}({e}_i,{X}; {W}, {V}), y^i)$ where $L_{\\Omega }$ is the empirical risk of the labeled nodes in $\\Omega $ .", "The trained weights are used to estimate the unknown labels on $\\mathcal {V}/\\Omega $ .", "Note that the results in this paper are distribution-free, and no assumption is made on the distributions of $\\tilde{x}_n$ and $y_n$ .", "Training with SGD.", "In practice, (REF ) is often solved by gradient type of methods, where in iteration $t$ , the currently estimations are updated by subtracting the product of a positive step size and the gradient of $L_{\\Omega }$ evaluated at the current estimate.", "To reduce the computational complexity in estimating the gradient, an SGD method is often employed to compute the gradient of the risk of a randomly selected subset of $\\Omega $ rather than using the whole set $\\Omega $ .", "However, due to the recursive embedding of neighboring features in GCNs, see the concatenations of ${A}$ in (REF ), the computation and memory cost of computing the gradient can be high.", "Thus, graph topology sampling methods have been proposed to further reduce the computational cost.", "Graph topology sampling.", "A node sampling method randomly removes a subset of nodes and the incident edges from $\\mathcal {G}$ in each iteration independently, and the embedding aggregation is based on the reduced graph.", "Mathematically, in iteration $s$ , replace ${A}$ in (REF ) withHere we use the same sampled matrix ${A}^{s}$ in all three layers in (REF ) to simplify the representation.", "Our analysis applies to the more general setting that each layer uses a different sampled adjacency matrix, i.e., the three ${A}$ matrices in (REF ) are replaced with ${A}^{s(1)}={A}{P}^{s(1)},\\ {A}^{s(2)}={A}{P}^{s(2)},\\ {A}^{s(3)}={A}{P}^{s(3)}$ , respectively, as in [39], [24], where ${P}^{s(1)}$ , ${P}^{s(2)}$ , and ${P}^{s(3)}$ are independently sampled following the same sampling strategy.", "${A}^s={A}{P}^s$ , where ${P}^s$ is a diagonal matrix, and the $i$ th diagonal entry is 0, if node $i$ is removed in iteration $s$ .", "The non-zero diagonal entries of ${P}^s$ are selected differently based on different sampling methods.", "Because ${A}^s$ is much more sparse than ${A}$ , the computation and memory cost of embedding neighboring features is significantly reduced.", "This paper will analyze the generalization performance, i.e., the prediction accuracy of unknown labels, of our algorithm framework that implements both SGD and graph topology sampling to solve (REF ).", "The details of our algorithm are discussed in Section REF -REF , and the generalization performance is presented in Section REF ." ], [ "Informal Key Theoretical Findings", "We first summarize the main insights of our results before presenting them formally.", "1.", "A provable generalization guarantee of GCNs beyond two layers and with graph topology sampling.", "The learned GCN by our Algorithm REF can approach the best performance of label prediction using a large class of target functions.", "Moreover, the prediction performance improves when the number of labeled nodes and the number of neurons $m_1$ and $m_2$ increase.", "This is the first generalization performance guarantee of training GCNs with graph topology sampling.", "2.", "The explicit characterization of the impact of graph sampling through the effective adjacency matrix ${A}^*$.", "We show that training with graph sampling returns a model that has the same label prediction performance as that of a model trained by replacing ${A}$ with ${A}^*$ in (REF ), where ${A}^*$ depends on both ${A}$ and the graph sampling strategy.", "As long as ${A}^*$ can characterize the correlation among nodes properly, the learned GCN maintains a desirable prediction performance.", "This explains the empirical success of graph topology sampling in many datasets.", "3.", "The explicit sample complexity bound on graph properties.", "We provide explicit bounds on the sample complexity and the required number of neurons, both of which grow as the node correlation increase.", "Moreover, the sample complexity depends on the number of neurons only logarithmically, which is consistent with the practical over-parameterization.", "To the best of our knowledge, [35] is the only existing work that provides a sample complexity bound based on the graph topology, but in the non-practical and restrictive setting of two-layer GCNs.", "Moreover, the sample complexity bound by [35] is polynomial in the number of neurons.", "4.", "Tackling the non-convex interaction of weights between different layers.", "The convexity plays a critical role in many exiting analyses of GCNs.", "For instance, the analyses in [35] require a special initialization in the local convex region of the global minimum, and the results only apply to two-layer GCNs.", "The NTK approach in [9] considers the limiting case that the interactions across layers are negligible.", "Here, we directly address the non-convex interaction of weights ${W}$ and ${V}$ in both algorithmic design and theoretical analyses." ], [ "Graph Topology Sampling Strategy", "Here we describe our graph topology sampling strategy using ${A}^s$ , which we randomly generate to replace ${A}$ in the $s$ th SGD iteration.", "Although our method is motivated for analysis and different from the existing graph sampling strategies, our insights generalize to other sampling methods like FastGCN [3].", "The outline of our algorithmic framework of training GCNs with graph sampling is deferred to Section REF .", "Suppose the node degrees in $\\mathcal {G}$ can be divided into $L$ groups with $L \\ge 1$ , where the degrees of nodes in group $l$ are in the order of $d_l$ , i.e., between $cd_l$ and $Cd_l$ for some constants $c \\le C$ , and $d_l$ is order-wise smaller than $d_{l+1}$ , i.e., $d_l=o(d_{l+1})$ .", "Let $N_l$ denote the number of nodes in group $l$ .", "Graph sampling strategyHere we discuss asymmetric sampling as a general case.", "The special case of symmetric sampling is introduced in Section REF.. We consider a group-wise uniform sampling strategy, where $S_l$ out of $N_l$ nodes are sampled uniformly from each group $l$ .", "For all unsampled nodes, we set the corresponding diagonal entries of a diagonal matrix ${P}^s$ to be zero.", "If node $i$ is sampled in this iteration and belongs to group $l$ for any $i$ and $l$ , the $i$ th diagonal entry of ${P}^s$ is set as $p^*_lN_l/S_l$ for some non-negative constant $p^*_l$ .", "Then ${A}^s={A}{P}^s$ .", "$N_l/S_l$ can be viewed as the scaling to compensate for the unsampled nodes in group $l$ .", "$p^*_l$ can be viewed as the scaling to reflect the impact of sampling on nodes with different importance that will be discussed in detail soon.", "Effective adjacency matrix ${A}^*$ by graph sampling.", "To analyze the impact of graph topology sampling on the learning performance, we define the effective adjacency matrix as follows: ${A}^{*}={A}{P}^*$ where ${P}^*$ is a diagonal matrix defined as ${P}^*_{ii}=p^*_l \\quad \\textrm { if node } i \\textrm { belongs to degree group } l$ Therefore, compared with ${A}$ , all the columns with indices corresponding to group $l$ are scaled by a factor of $p^*_l$ .", "We will formally analyze the impact of graph topology sampling on the generalization performance in Section REF , but an intuitive understanding is that our graph sampling strategy effectively changes the normalized adjacency matrix ${A}$ in the GCN network model (REF ) to ${A}^*$ .", "${A}^*$ can be viewed as an adjacency matrix of a weighted directed graph $\\mathcal {G^{\\prime }}$ that reflects the node correlations, where each un-directed edge in $\\mathcal {G}$ corresponds to two directed edges in $\\mathcal {G^{\\prime }}$ with possibly different weights.", "${A}^*_{ji}$ measures the impact of the feature of node $i$ on the label of node $j$ .", "If $p^*_l$ is in the range of $(0,1)$ , the corresponding entries of columns with indices in group $l$ in ${A}^*$ are smaller than those in ${A}$ .", "That means the impact of a node in group $l$ on all other nodes is reduced from those in ${A}$ .", "Conversely, if $p^*_l>1$ , then the impact of nodes in group $l$ in ${A}^*$ is enhanced from that in ${A}$ .", "Parameter selection and insights (1) The scaling factor $p^*_l$ should satisfy $0 \\le p^*_l \\le \\frac{c_1}{L\\psi _l}, \\quad \\forall l$ for a positive constant $c_1$ that can be sufficiently large.", "$\\psi _l$ is defined as follows, $\\psi _l := \\frac{\\sqrt{d_Ld_l}{N}_l}{\\sum _{i=1}^L d_i {N}_i} \\quad \\quad \\forall l\\in [L]$ Note that (REF ) is a minor requirement for most graphs.", "To see this, suppose $L$ is a constant, and every $N_l$ is in the order of $N$ .", "Then $\\psi _l$ is less than $O(1)$ for all $l$ .", "Thus, all constant values of $p^*_{\\hat{l}}$ satisfy (REF ) with $\\psi _l$ from (REF ).", "A special example is that $p^*_l$ are all equal, i.e., ${A}^*=c_2{A}$ for some constant $c_2$ .", "Because one can scale ${W}$ and ${V}$ by $1/c_2$ in (REF ) without changing the results, ${A}^*$ is equivalent to ${A}$ in this case.", "The upper bound in (REF ) only becomes active in highly unbalanced graphs where there exists a dominating group $\\hat{l}$ such that $\\sqrt{d_{\\hat{l}}} N_{\\hat{l}}\\gg \\sqrt{d_l}N_l$ for all other $l$ .", "Then the upper bound of $p^*_{\\hat{l}}$ is much smaller than those for other $p^*_l$ .", "Therefore, the columns of ${A}^*$ that correspond to group $\\hat{l}$ are scaled down more significantly than other columns, indicating that the impact of group $\\hat{l}$ is reduced more significantly than other groups in ${A}^*$ .", "Therefore, the takeaway is that graph topology sampling reduces the impact of dominating nodes more than other nodes, resulting in a more balanced ${A}^*$ compared with ${A}$.", "(2) The number of sampled nodes shall satisfy $\\frac{S_l}{N_l} \\ge \\ (1+ \\frac{c_1 \\text{poly}(\\epsilon )}{Lp^*_l \\psi _l})^{-1} \\quad \\quad \\forall l\\in [L]$ where $\\epsilon $ is a small positive value.", "The sampling requirement in (REF ) has two takeaways.", "First, the higher-degree groups shall be sampled more frequently than lower-degree groups.", "To see this, consider a special case that $p^*_l=1$ , and $N_l=N/L$ for all $l$ .", "Then (REF ) indicates that $S_l$ is larger in a group $l$ with a larger $d_l$ .", "This intuition is the same as FastGCN [3], which also samples high-degree nodes with a higher probability in many cases.", "Therefore, the insights from our graph sampling method also apply to other sampling methods such as FastGCN.", "We will show the connection to FastGCN empirically in Section REF .", "Second, reducing the number of samples in group $l$ corresponds to reducing the impact of group $l$ in ${A}^*$ .", "To see this, note that decreasing $p^*_l$ reduces the right-hand side of (REF )." ], [ "The Algorithmic Framework of Training GCNs", "Because (REF ) is non-convex, solving it directly using SGD can get stuck at a bad local minimum in theory.", "The main idea in the theoretical analysis to address this non-convexity is to add weight decay and regularization in the objective of (REF ) such that with a proper regularization, any second-order critical point is almost a global minimum.", "[t] Training with SGD and graph topology sampling [1] Input: Normalized adjacency matrix ${A}$ , node features ${X}$ , known node labels in $\\Omega $ , the step size $\\eta $ , the number of inner iterations $T_w$ , the number of outer iterations $T$ , $\\sigma _w$ , $\\sigma _v$ , $\\lambda _w$ , $\\lambda _v$ .Initialize ${W}^{(0)}$ , ${V}^{(0)}$ , ${B}_1$ , ${B}_2$ , ${C}$ .", "${W}_0=0$ , ${V}_0=0$ .", "$t=0,1,\\cdots ,T-1$ Apply noisy SGD with step size $\\eta $ on the stochastic objective $\\hat{L}_\\Omega (\\lambda _{t};{W},{V})$ in (REF ) for $T_w$ steps.", "To generate the stochastic objective in each step $s$ , randomly sample a batch of labeled nodes $\\Omega ^s$ from $\\Omega $ ; generate ${A}^{s}$ using graph sampling; randomly generate ${W}^\\rho $ , ${V}^\\rho $ and $\\Sigma $ .", "Let the starting point be ${W}={W}_t$ , ${V}={V}_t$ and suppose it reaches ${W}_{t+1}$ and ${V}_{t+1}$ .", "$\\lambda _{t+1}=\\lambda _t\\cdot (1-\\eta )$ .", "Output: ${W}^{(out)}=\\sqrt{\\lambda _{T-1}}({W}^{(0)}+{W}^{\\rho }+{W}_{T} \\Sigma )$ ${V}^{(out)}=\\sqrt{\\lambda _{T-1}}({V}^{(0)}+{V}^{\\rho }+ \\Sigma {V}_{T})$ .", "Specifically, for initialization, entries of ${W}^{(0)}$ are i.i.d.", "from $\\mathcal {N}(0,\\frac{1}{m_1})$ , and entries of ${V}^{(0)}$ are i.i.d.", "from $\\mathcal {N}(0,\\frac{1}{m_2})$ .", "${B}_1$ (or ${B}_2$ ) is initialized to be an all-one vector multiplying a row vector with i.i.d.", "samples from $\\mathcal {N}(0,\\frac{1}{m_1})$ (or $\\mathcal {N}(0, \\frac{1}{m_2})$ ).", "Entries of ${C}$ are drawn i.i.d.", "from $\\mathcal {N}(0, 1)$ .", "In each outer loop $t=0, ..., T-1$ , we use noisy SGDNoisy SGD is vanilla SGD plus Gaussian perturbation.", "It is a common trick in the theoretical analyses of non-convex optimization [13] and is not needed in practice.", "with step size $\\eta $ for $T_w$ iterations to minimize the stochastic objective function $\\hat{L}_\\Omega $ in (REF ) with some fixed $\\lambda _{t-1}$ , where $\\lambda _0=1$ , and the weight decays with $\\lambda _{t+1}=(1-\\eta )\\lambda _{t}$ .", "$\\begin{aligned}&\\hat{L}_\\Omega (\\lambda _{t};{W},{V})\\\\=&L_\\Omega ( \\sqrt{\\lambda _{t}}({W}^{(0)}+{W}^\\rho +{W}\\Sigma ), \\sqrt{\\lambda _{t}}({V}^{(0)}+{V}^\\rho +\\Sigma {V}))\\\\&+\\lambda _w\\Vert \\sqrt{\\lambda _t}{W}\\Vert _{2,4}^4+\\lambda _v\\Vert \\sqrt{\\lambda _t}{V}\\Vert _F^2\\end{aligned}$ $\\hat{L}_\\Omega (\\lambda _{t};{W},{V})$ is stochastic because in each inner iteration $s$ , (1) we randomly sample a subset $\\Omega ^s$ of labeled nodes; (2) we randomly sample ${A}^s$ from the graph topology sampling method in Section REF ; (3) ${W}^{\\rho }$ and ${V}^{\\rho }$ are small perturbation matrices with entries i.i.d.", "drawn from $\\mathcal {N}(0, \\sigma _w^2)$ and $\\mathcal {N}(0, \\sigma _v^2)$ , respectively; and (4) $\\Sigma \\in \\mathbb {R}^{m_1\\times m_1}$ is a random diagonal matrix with diagonal entries uniformly drawn from $\\lbrace 1, -1\\rbrace $ .", "${W}^{\\rho }$ and ${V}^{\\rho }$ are standard Gaussian smoothing in the literature of theoretical analyses of non-convex optimization, see, e.g.", "[13], and are not needed in practice.", "$\\Sigma $ is similar to the practical Dropout [27] technique that randomly masks out neurons and is also introduced for the theoretical analysis only.", "The last two terms in (REF ) are additional regularization terms for some positive $\\lambda _w$ and $\\lambda _v$ .", "As shown in [1], $\\Vert \\cdot \\Vert _{2,4}$ is used for the analysis to drive the weights to be evenly distributed among neurons.", "The practical regularization $\\Vert \\cdot \\Vert _F$ has the same effect in empirical results, while the theoretical justification is open.", "Algorithm REF summarizes the algorithm with the parameter selections in Table REF .", "Let ${W}^{out}$ and ${V}^{out}$ denote the returned weights.", "We use $F_{{A}^*}({e}_i,{X};{W}^{out},{V}^{out})$ to predict the label of node $i$ .", "This might sound different from the conventional practice which uses ${A}$ in predicting unknown labels.", "However, note that ${A}^*$ only differs from ${A}$ by a column-wise scaling as from (REF ).", "Moreover, ${A}^*$ can be set as ${A}$ in many practical datasets based on our discussion after (REF ).", "Here we use the general form of ${A}^*$ for the purpose of analysis.", "We remark that our framework of algorithm and analysis can be easily applied to the simplified setup of two-layer GCNs.", "The resulting algorithm is much simplified to a vanilla SGD plus graph topology sampling.", "All the additional components above are introduced to address the non-convex interaction of ${W}$ and ${V}$ theoretically and may not be needed for practical implementation.", "We skip the discussion of two-layer GCNs in this paper.", "Table: Parameter choices for Algorithm" ], [ "Generalization Guarantee", "Our formal generalization analysis shows that our learning method returns a GCN model that approaches the minimum prediction error that can be achieved by the best function in a large concept class of target functions, which have two important properties: (1) the prediction error decreases as size of the function class increases; and (2) the concept class uses ${A}^*$ in (REF ) as the adjacency matrix of the graph topology.", "Therefore, the result implies that if ${A}^*$ accurately captures the correlations among node features and labels, the learned GCN model can achieve a small prediction error of unknown labels.", "Moreover, no other functions in a large concept class can perform better than the learned GCN model.", "To formalize the results, we first define the target functions as follows.", "Concept class and target function $F^*$.", "Consider a concept class consisting of target functions $F^*: \\mathbb {R}^N\\times \\mathbb {R}^{N\\times d}\\rightarrow \\mathbb {R}^{1\\times K}$ : $\\begin{aligned}F^*_{{A}^*} ({e}_g, {X})={e}_g^\\top {A}^*\\big (\\Phi ({r}_1) \\odot {r}_2\\big ) {C}^* \\\\{r}_1= {A}^* \\phi _1 ({A}^*{X}{W}_1^*){V}_1^*\\\\{r}_2={A}^* \\phi _2({A}^* {X}{W}_2^*){V}_2^*\\end{aligned}$ where $\\phi _{1}$ , $\\phi _{2}$ , $\\Phi $ : $\\mathbb {R}\\rightarrow \\mathbb {R}$ all infinite-order smoothWhen $\\Phi $ is operated on a matrix ${r}_1$ , $\\Phi ({r}_1)$ means applying $\\Phi $ on each entry of ${r}_1$ .", "In fact, our results still hold for a more general case that a different function $\\Phi _j$ is applied to every entry of the $j$ th column of ${r}_1$ , $j\\in [p_2]$ .", "We keep the simpler model to have a more compact representation.", "The similar arguments hold for $\\phi _{1}$ , $\\phi _{2}$ ..", "The parameters ${W}_{1}^*, {W}_{2}^*\\in \\mathbb {R}^{d \\times p_2}$ , ${V}_{1}^*, {V}_{2}^*\\in \\mathbb {R}^{p_2\\times p_1}$ , ${C}^*\\in \\mathbb {R}^{p_1 \\times k}$ satisfy that every column of ${W}_{1}^*$ , ${W}_{2}^*$ , ${V}_{1}^*$ , ${V}_{2}^*$ is unit norm, and the maximum absolute value of ${C}^*$ is at most 1.", "The effective adjacency matrix ${A}^*$ is defined in (REF ).", "Define $\\mathcal {C}_\\epsilon (\\phi , R) = \\max \\big (\\mathcal {C}_\\epsilon (\\phi _1, R),\\mathcal {C}_\\epsilon (\\phi _2, R)\\big ),\\\\\\mathcal {C}_s(\\phi , R) =\\max \\big (\\mathcal {C}_s(\\phi _1, R),\\mathcal {C}_2(\\phi _1, R)\\big ).$ We focus on target functions where the function complexity $\\mathcal {C}_\\epsilon (\\Phi , R)$ , $\\mathcal {C}_s(\\Phi , R)$ , $\\mathcal {C}_\\epsilon (\\phi , R)$ , $\\mathcal {C}_s(\\phi , R)$ , defined in (REF )-(REF ), (REF )-(), as well as $p_1$ and $p_2$ , are all bounded.", "(REF ) is more general than GCNs.", "If ${r}_2$ is a constant matrix, (REF ) models a GCN, where ${W}^*_1$ and ${V}^*_1$ are weight matrices in the first and second layer, respectively, and $\\phi _1$ and $\\Phi $ are the activation functions in each layer.", "Modeling the prediction error of unknown labels.", "We will show that the learned GCN by our method performs almost the same as the best function in the concept class in (REF ) in predicting unknown labels.", "Because the practical datasets usually contain noise in features and labels, we employ a probabilistic model to model the data.", "Note that our result is distribution-free , and the following distributions are introduced for the presentation of the results.", "Specifically, let $\\mathcal {D}_{\\tilde{x}_n}$ denote the distribution from which the feature $\\tilde{x}_n$ of node $n$ is drawn.", "For example, when the noise level is low, $\\mathcal {D}_{\\tilde{x}_n}$ can be a distribution centered at the observed feature of node $n$ with a small variance.", "Similarly, let $\\mathcal {D}_{y_n}$ denote the distribution from which the label $y_n$ at node $n$ is drawn.", "Let ${e}_g$ be uniformly selected from $\\lbrace {e}_i\\rbrace _{i=1}^N\\in \\mathbb {R}^N$ .", "Let $\\mathcal {D}$ denote the concatenation of these distributions of a data point $z=({e}_g, {X}, y)\\in \\mathbb {R}^N \\times \\mathbb {R}^{N\\times d}\\times \\mathcal {Y}.$ Then the given feature matrix ${X}$ and partial labels in $\\Omega $ can be viewed as $|\\Omega |$ identically distributed but correlated samples from $\\mathcal {D}$ .", "The correlation results from the fact that the label of node $i$ depends on not only the feature of node $i$ but also neighboring features.", "This model of correlated samples is different from the conventional assumption of i.i.d.", "samples in supervised learning and makes our analyses more involved.", "Let $\\mathrm {OPT}_{{A}^*}= \\mathop {\\rm {\\min }}_{{W}^*_1,\\ {W}^*_2, \\atop {V}^*_1,\\ {V}^*_2,\\ {C}^*}\\mathbb {E}_{({e}_g, {X}, y)\\sim \\mathcal {D}} L (F^*_{{A}^*}({e}_g, {X}), y)$ be the smallest population risk achieved by the best target function (over the choices of ${W}^*_1$ , ${W}^*_2$ , ${V}^*_1$ , ${V}^*_2$ , ${C}^*$ ) in the concept class $F^*_{{A}^*}$ in (REF ).", "$\\textrm {OPT}_{{A}^*}$ measures the average loss of predicting the unknown labels if the estimates are computed using the best target function in (REF ).", "Clearly, $\\textrm {OPT}_{{A}^*}$ decreases as the size of the concept increases, i.e., when $p_1$ and $p_2$ increase.", "Moreover, if ${A}^*$ indeed models the node correlations accurately, $\\textrm {OPT}_{{A}^*}$ can be very small, indicating a desired generalization performance.", "We next show that the population risk of the learned GCN model by our method can be arbitrarily close to $\\textrm {OPT}_{{A}^*}$ .", "Theorem 3.1 For every $\\gamma \\in (0,\\frac{1}{4}]$ , every $\\epsilon _0\\in (0,\\frac{1}{100}]$ , every $\\epsilon \\in (0,(Kp_1p_2^2\\mathcal {C}_s(\\Phi , p_2\\mathcal {C}_s(\\phi ,O(1)))\\mathcal {C}_s(\\phi ,O(1))\\\\\\cdot \\Vert {A}^*\\Vert _\\infty ^2)^{-1}\\epsilon _0)$ , as long as $\\begin{aligned}&m_1=m_2=m\\\\\\ge &\\text{poly}\\Big (\\mathcal {C}_\\epsilon \\big (\\Phi , \\mathcal {C}_\\epsilon (\\phi ,O(1))\\big ),p_2, \\Vert {A}^*\\Vert _\\infty , \\frac{1}{\\epsilon }\\Big )\\end{aligned}$ $\\begin{aligned}|\\Omega | \\ge &\\Theta (\\epsilon _0^{-2}\\Vert {A}^*\\Vert _\\infty ^8K^6(1+p_1^4p_2^5\\mathcal {C}_\\epsilon (\\Phi , \\sqrt{p_2}\\mathcal {C}_\\epsilon (\\phi , O(1)))\\\\&\\cdot \\mathcal {C}_\\epsilon (\\phi ,O(1))(\\Vert {A}^*\\Vert _\\infty +1)^4)(1+\\delta )^4\\log N \\log m),\\end{aligned}$ (REF ) and (REF ) hold, there is a choice $\\eta =1/\\text{poly}(\\Vert {A}^*\\Vert _\\infty , K, m)$ and $T=\\text{poly}(\\Vert {A}^*\\Vert _\\infty , K, m)$ such that with probability at least $ 0.99$ , $\\begin{aligned}&\\mathbb {E}_{({e}_g,{X},y)\\in \\mathcal {D}}L(F_{{A}^*}({e}_g,{X};{W}^{(out)}, {V}^{(out)}),y)\\\\\\le &(1+\\gamma )\\mathrm {OPT}_{{A}^*}+\\epsilon _0,\\end{aligned}$ where ${A}^*$ is the effective adjacency matrix in (REF ).", "Theorem REF shows that the required sample complexity is polynomial in $\\Vert {A}^*\\Vert $ and $\\delta $ , where $\\delta $ is the maximum node degree without self-connections in ${A}$ .", "Note that condition (REF ) implies that $\\Vert {A}^*\\Vert _\\infty $ is $O(1)$ .", "Then as long as $\\delta $ is $O(N^\\alpha )$ for some small $\\alpha $ in $(0,1)$ , say $\\alpha =1/5$ , then one can accurately infer the unknown labels from a small percentage of labeled nodes.", "Moreover, our sample complexity is sufficient but not necessary.", "It is possible to achieve a desirable generalization performance if the number of labeled nodes is less than the bound in (REF ).", "Graph topology sampling affects the generalization performance through ${A}^*$ .", "From the discussion in Section REF , graph sampling reduces the node correlation in ${A}^*$ , especially for dominating nodes.", "The generalization performance does not degrade when $\\textrm {OPT}_{{A}^*}$ is small, i.e., the resulting ${A}^*$ is sufficient to characterize the node correlation in a given dataset.", "That explains the empirical success of graph sampling in many datasets." ], [ " Numerical Results", "To unveil how our theoretical results are aligned with GCN's generalization performance in experiments, we will focus on numerical evaluations on synthetic data where we can control target functions and compare with ${A}^*$ explicitly.", "We also evaluate both our graph sampling method and FastGCN [3] to validate that insights for our graph sampling method also apply to FastGCN.", "We generate a graph $\\mathcal {G}$ with $N=2000$ nodes.", "$\\mathcal {G}$ has two degree groups.", "Group 1 has $N_1$ nodes, and every node degree approximately equals $d_1$ .", "Group 2 has $N_2$ nodes, and every node degree approximately equals $d_2$ .", "The edges between nodes are randomly selected.", "${A}$ is the normalized adjacency matrix of $\\mathcal {G}$ .", "The node labels are generated by the target function $y=(\\sin (\\hat{{A}}{X}{W}^*)\\odot \\tanh (\\hat{{A}}{X}{W}^*)){C}^*,$ where $\\hat{{A}}\\in \\mathbb {R}^{N\\times N}$ , ${X}\\in \\mathbb {R}^{N\\times d}$ , ${W}^*\\in \\mathbb {R}^{d\\times p}$ and ${C}^*\\in \\mathbb {R}^{p\\times K}$ .", "The feature dimension $d=10$ .", "$p=10$ , and $K=2$ .", "${X}$ , ${W}^*$ and ${C}^*$ are all randomly generated with each entry i.i.d.", "from $\\mathcal {N}(0,1)$ .", "We consider a regression task with the $\\ell _2$ -regression loss function.", "A three-layer GCN as defined in (REF ) with $m$ neurons in each hidden layer is trained on a randomly selected set $\\Omega $ of labeled nodes.", "The rest $N-|\\Omega |$ labels are used for testing.", "The learning rate $\\eta = 10^{-3}$ .", "The mini-batch size is 5, and the dropout rate as $0.4$ .", "The total number of iterations is $TT_w=4|\\Omega |$ .", "Our graph topology sampling method samples $S_1=0.9 N_1$ and $S_2=0.9 N_2$ nodes for both groups in each iteration." ], [ "Sample Complexity and Neural Network Width with respect to $\\Vert {A}^*\\Vert _\\infty $", "We fix $N_1=100$ , $N_2=1900$ and vary ${A}$ by changing node degrees $d_1$ and $d_2$ .", "In the graph topology sampling method, $p^*_1=0.7$ and $p^*_2=0.3$ .", "For every fixed ${A}$ , the effective adjacency matrix ${A}^*$ is computed based on (REF ) using $p^*_1$ and $p^*_2$ .", "Synthetic labels are generated based on (REF ) using ${A}^*$ as $\\hat{{A}}$ .", "Figure REF shows the testing error decreases as the number of labeled nodes $|\\Omega |$ increases, when the number of neurons per layer $m$ is fixed as 500.", "Moreover, as $\\Vert {A}^*\\Vert _\\infty $ increases, the required number of labeled nodes increases to achieve the same level of testing error.", "This verifies our sample complexity bound in (REF ).", "Figure REF shows the testing error decreases as $m$ increases when $|\\Omega |$ is fixed as 1500.", "Moreover, as $\\Vert {A}^*\\Vert _\\infty $ increases, a larger $m$ is needed to achieve the same level of testing error.", "This verifies our bound on the number of neurons in (REF ).", "Figure: The testing error when |Ω||\\Omega | and ∥A * ∥ ∞ \\Vert {A}^*\\Vert _\\infty change.", "m=500m=500Figure: The testing error when mm and ∥A * ∥ ∞ \\Vert {A}^*\\Vert _\\infty change.", "|Ω|=1500|\\Omega |=1500." ], [ "Graph Sampling Affects ${A}^*$", "Here we fix ${A}$ and the graph sampling strategy, and evaluate the prediction performance on datasets generated by (REF ) using different $\\hat{{A}}$ .", "We generate $\\hat{{A}}$ from $\\hat{{A}}={A}\\hat{{P}}$ , where $\\hat{{P}}$ is a diagonal matrix with $\\hat{ {P}}_{ii}=\\hat{p}_1$ for nodes $i$ in group 1 and $\\hat{ {P}}_{ii}=\\hat{p}_2$ for nodes $i$ in group 2.", "We vary $\\hat{p}_1$ and $\\hat{p}_2$ to generate three different datasets from (REF ).", "We consider both our graph sampling method in Section REF and FastGCN [3].", "In Figure REF , $N_1=100$ and $N_2=1900$ .", "$d_1=10$ and $d_2=1$ .", "Figure REF (a) shows the testing performance of a learned GCN by Algorithm 1, where $p_1^*=0.9$ and $p_2^*=0.1$ .", "the method indeed performs the best on Dataset 1 when $\\hat{{A}}$ is generated using $\\hat{p}_1=0.9$ and $\\hat{p}_2=0.1$ , in which case ${A}^*=\\hat{{A}}$ .", "This verifies our theoretical result that graph sampling affects ${A}^*$ in the target functions, i.e., it achieves the best performance if ${A}^*$ is the same as $\\hat{{A}}$ in the target function.", "Figure: Generalization performance of learned GCNs on datasets generated from different A ^\\hat{{A}} by (a) our graph sampling strategy and (b) FastGCN.", "A{A} is very unbalanced.Fig.", "REF (b) shows the performance on the same three datasets where in each iteration of Algorithm 1, the graph sampling strategy is replaced with FastGCN [3].", "The method also performs the best in Dataset 1 when ${A}^*$ is generated using $\\hat{p}_1=0.9$ and $\\hat{p}_2=0.1$ .", "The reason is that the graph topology is highly unbalanced in the sense that $\\sqrt{d_2}N_2 \\gg \\sqrt{d_1}N_1$ , which means group 2 has a much higher impact on other nodes in group 1 in ${A}$ .", "The graph sampling reduces the impact of group 2 nodes more significantly than group 1 nodes, as discussed in Section REF .", "To further illustrate this, in Figure REF we change the graph topology by setting $N_1=1000$ and $N_2=1000$ , and all the other settings remain the same.", "In this case, the graph is balanced because $\\sqrt{d_2}N_2$ and $\\sqrt{d_1}N_1$ are in the same order.", "We generate different datasets using the new ${A}$ following the same method and evaluate the performance of both our graph sampling method and FastGCN.", "Both methods perform the best in Dataset 3 when $\\hat{{A}}$ is generated using $\\hat{p}_1=0.5$ and $\\hat{p}_2=0.5$ .", "That is because on a balanced graph, graph sampling reduces the impact of both groups equally.", "Figure: Generalization performance of learned GCNs on datasets generated from different A * {A}^* by (a) our graph sampling strategy and (b) FastGCN.", "A{A} is balanced." ], [ "Conclusion", "This paper provides a new theoretical framework for explaining the empirical success of graph sampling in training GCNs.", "It quantifies the impact of graph sampling explicitly through the effective adjacency matrix and provides generalization and sample complexity analyses.", "One future direction is to develop active graph sampling strategies based on the presented insights and analyze its generalization performance.", "Other potential extension includes the construction of statistical-model-based characterization of ${A}^*$ and fitness to real-world data, and the generalization analysis of deep GCNs, graph auto-encoders, and jumping knowledge networks." ], [ "Acknowledgements", "This work was supported by AFOSR FA9550-20-1-0122, ARO W911NF-21-1-0255, NSF 1932196 and the Rensselaer-IBM AI Research Collaboration (http://airc.rpi.edu), part of the IBM AI Horizons Network (http://ibm.biz/AIHorizons).", "We thank Ruisi Jian, Haolin Xiong at Rensselaer Polytechnic Institute for the help in formulating numerical experiments.", "We thank all anonymous reviewers for their constructive comments." ], [ "Preliminaries", "Lemma A.1 $\\Vert \\tilde{{a}}_n{X}\\Vert \\le \\Vert {A}\\Vert _\\infty $ .", "Proof: $\\begin{aligned}\\Vert \\tilde{{a}}_n{X}\\Vert &=\\Vert \\sum _{k=1}^N a_{n,k}\\tilde{{x}}_k\\Vert \\\\&= \\Vert \\sum _{k=1}^N \\frac{a_{n,k}}{\\sum _{k=1}^N a_{n,k}}\\tilde{{x}}_k\\Vert \\cdot \\sum _{k=1}^N a_{n,k}\\\\&\\le \\sum _{k=1}^N \\frac{a_{n,k}}{\\sum _{k=1}^N a_{n,k}}\\Vert \\tilde{{x}}_k\\Vert \\cdot \\Vert {A}\\Vert _\\infty \\\\&= \\Vert {A}\\Vert _\\infty \\end{aligned}$ where the second to last step is by the convexity of $\\Vert \\cdot \\Vert $ .", "Lemma A.2 Given a graph $\\mathcal {G}$ with $L(\\ge 1)$ groups of nodes, where the group $i$ with node degree $d_i$ is denoted as $\\mathcal {N}_i$ .", "Suppose that in iteration $t$ , ${A}^t$ (or any of ${A}^{t(1)}$ , ${A}^{t(2)}$ , ${A}^{t(3)}$ in the general setting) is generated from the sampling strategy in Section REF , if the number of sampled nodes satisfies $l_i\\ge |\\mathcal {N}_i|/(1+\\frac{c_1\\text{poly}(\\epsilon )}{L p^*_i\\Psi _i})$ , we have $\\Vert {A}^t-{A}^*\\Vert _\\infty \\le \\text{poly}(\\epsilon )$ Proof: From Section REF , we can rewrite that $\\tilde{{a}}_n^{t}={\\left\\lbrace \\begin{array}{ll}\\frac{|\\mathcal {N}_k|}{l_k}p^*_k A_{n,j}, &\\text{if the nodes }n, j\\text{ are connected and }j\\text{ is selected and }j\\in \\mathcal {N}_k\\\\0, &\\text{else}\\end{array}\\right.", "}$ $\\tilde{{a}^*}_n={\\left\\lbrace \\begin{array}{ll}p^*_k A_{n,j}, &\\text{if the nodes }n, j\\text{ are connected and }j\\in \\mathcal {N}_k\\\\0, &\\text{else}\\end{array}\\right.", "}$ Let ${A}^*=(\\tilde{{a}^*}_1^\\top ,\\tilde{{a}^*}_2^\\top ,\\cdots ,\\tilde{{a}^*}_{n})^\\top $ .", "Since that we need that $\\sum _{j=1}^N A_{n,j}^*\\le O(1)$ , we require $p^*_i\\sum _{j\\in \\mathcal {N}_i}A_{n,j}\\le O(1/L), \\text{ holds for any }i\\in [L], n\\in [N]\\\\$ We first roughly compute the ratio of edges that one node is connected to the nodes in another group.", "For the node with degree $\\text{deg}(i)$ , it has $\\text{deg}(i)-1$ open edges except the self-connection.", "Hence, the group with degree $\\text{deg}(j)$ has $(\\text{deg}(j)-1)|\\mathcal {N}_j|$ open edges except self-connections in total.", "Therefore, the ratio of the edges connected to the group $j$ to all groups is $\\frac{(\\text{deg}(j)-1)|\\mathcal {N}_j|}{\\sum _{l=1}^L (\\text{deg}(l)-1)|\\mathcal {N}_l|}\\approx \\frac{d_j|\\mathcal {N}_j|}{\\sum _{l=1}^L d_l|\\mathcal {N}_l|}$ Define $\\Psi (n,i)=\\sqrt{\\frac{d_n}{d_{i}}}\\cdot \\frac{d_i|\\mathcal {N}_i|}{\\sum _{l=1}^L d_l|\\mathcal {N}_l|}$ Then, as long as $p^*_i\\sum _{j\\in |\\mathcal {N}_i|}A_{n,j}\\approx p^*_i\\frac{1}{\\sqrt{d_{i} d_{n}}}\\cdot \\frac{d_i|\\mathcal {N}_i|}{\\sum _{l=1}^L d_l|\\mathcal {N}_l|} d_{n}\\lesssim p^*_i\\Psi (n,i)\\le O(1/L)$ i.e., $p^*_i\\le \\frac{c_1}{L\\cdot \\max _{n\\in [L]}\\lbrace \\Psi (n,i)\\rbrace }=\\frac{c_1}{L\\cdot \\Psi (L,i)}=\\frac{c_1}{L}\\sqrt{\\frac{d_i}{d_L}}\\frac{\\sum _{l=1}^L d_l |\\mathcal {N}_l|}{d_i|\\mathcal {N}_i|}$ for some constant $c_1>0$ , we can obtain that $\\Vert {A}^*\\Vert _\\infty \\le O(1)$ .", "Since that $\\sum _{j\\in \\mathcal {S}_k}A_{n,j}\\approx \\frac{1}{\\sqrt{d_{i} d_{n}}}\\cdot \\frac{d_i|\\mathcal {N}_i|}{\\sum _{l=1}^L d_l|\\mathcal {N}_l|} d_{n}\\frac{l_k}{|\\mathcal {N}_k|}\\approx \\sum _{j\\in \\mathcal {N}_k}A_{n,j}\\frac{l_k}{|\\mathcal {N}_k|}$ $\\sum _{j\\notin \\mathcal {S}_k}A_{n,j}\\approx \\frac{1}{\\sqrt{d_{i} d_{n}}}\\cdot \\frac{d_i|\\mathcal {N}_i|}{\\sum _{l=1}^L d_l|\\mathcal {N}_l|} d_{n}(1-\\frac{l_k}{|\\mathcal {N}_k|})\\approx \\sum _{j\\in \\mathcal {N}_k}A_{n,j}(1-\\frac{l_k}{|\\mathcal {N}_k|}),$ the difference between $\\tilde{{a}}_n^{t}$ and $\\tilde{{a}^*}_n$ can then be derived as $\\begin{aligned}&\\Vert \\tilde{{a}}_n^{t}-\\tilde{{a}^*}_{n}\\Vert _1\\\\=&\\Big | \\sum _{k=1}^L\\sum _{j\\in \\mathcal {S}_k}A_{n,j} p^*_k(\\frac{|\\mathcal {N}_k|}{l_k}- 1)+\\sum _{k=1}^L \\sum _{j\\notin \\mathcal {S}_k}A_{n,j} p^*_k\\Big |\\\\\\lesssim &\\sum _{k=1}^L(p^*_k(\\frac{|\\mathcal {N}_k|}{l_k}-1)\\frac{l_k}{|\\mathcal {N}_k|}\\sum _{j\\in \\mathcal {N}_k} A_{n,j}+ (1-\\frac{l_k}{|\\mathcal {N}_k|})p^*_k\\sum _{j\\in \\mathcal {N}_k} A_{n,j})\\\\\\lesssim & \\text{poly}(\\epsilon )\\sum _{k=1}^L \\frac{1}{L\\Psi (L,k)} \\sum _{j\\in \\mathcal {N}_k}A_{n,j}\\\\:=&\\text{poly}(\\epsilon )\\Gamma ({A}^*)\\end{aligned}$ where the first inequality is by (REF , REF ) and the second inequality holds as long as $l_i\\ge |\\mathcal {N}_i|/(1+\\frac{c_1\\text{poly}(\\epsilon )}{L p^*_i\\Psi (L,i)})$ .", "Combining (REF ), we have $\\sum _{i=1}^L p^*_i\\sum _{j\\in \\mathcal {N}_i}A_{n,j}\\lesssim \\sum _{i=1}^L \\frac{1}{L \\Psi (L,i)}\\sum _{j\\in \\mathcal {N}_i}A_{n,j}=\\Gamma ({A}^*)\\le O(1)$ Hence, (REF ) can be bounded by $\\text{poly}(\\epsilon )$ ." ], [ "Symmetric graph sampling method", "We provide and discuss a symmetric graph sampling method in this section.", "The insights behind this version of sampling strategy is the same as in Section REF .", "Similar to the asymmetric construction in Section REF , we consider a group-wise uniform sampling strategy, where $S_l$ nodes are sampled uniformly from $N_l$ nodes.", "For all unsampled nodes, we set the corresponding diagonal entries of a diagonal matrix ${P}^s$ to be zero.", "If node $i$ is sampled in this iteration and belongs to group $l$ for any $i$ and $l$ , the $i$ th diagonal entry of ${P}^s$ is set as $\\sqrt{p^*_lN_l/S_l}$ for some non-negative constant $p^*_l$ .", "Then ${A}^s={P}^s{A}{P}^s$ .", "Based on this symmetric graph sampling method, we define the effective adjacency matrix as ${A}^*={P}^*{A}{P}^*,$ where ${P}^*$ is a diagonal matrix defined as ${P}_{ii}^*=\\sqrt{p_l^*}\\ \\ \\ \\ \\text{if node }i\\text{ belongs to degree group }l$ The scaling factor $p^*_l$ should satisfy $0 \\le p^*_l \\le \\frac{c_2}{L^2\\psi _l^2}, \\quad \\forall l$ for a positive constant $c_2$ that can be sufficiently large.", "$\\psi _l$ is defined in (REF ).", "The number of sampled nodes shall satisfy $\\frac{S_l}{N_l} \\ge \\ (1+ \\frac{c_2 \\text{poly}(\\epsilon )}{L\\sqrt{p^*_l} \\psi _l})^{-2} \\quad \\quad \\forall l\\in [L]$ where $\\epsilon $ is a small positive value.", "Lemma A.3 Given a graph $\\mathcal {G}$ with $L(\\ge 1)$ groups of nodes, where the group $i$ with node degree $d_i$ is denoted as $\\mathcal {N}_i$ .", "Suppose ${A}^{t}$ (or any of ${A}^{t(1)}$ , ${A}^{t(2)}$ , ${A}^{t(3)}$ in the general setting) is generated from the sampling strategy in Section REF , if the number of sampled nodes satisfies $l_i\\ge |\\mathcal {N}_i|/(1+\\frac{c_2\\text{poly}(\\epsilon )}{L p^*_i\\Psi _i})$ , then we have $\\Vert {A}^{t}-{A}^*\\Vert _\\infty \\le \\text{poly}(\\epsilon )$ Proof: From Section REF , we can rewrite that $\\tilde{{a}}_n^{t}={\\left\\lbrace \\begin{array}{ll}\\sqrt{\\frac{|\\mathcal {N}_k||\\mathcal {N}_u|}{l_k l_u}p^*_k p^*_u} A_{n,j}, &\\text{if the nodes }n, j\\text{ are connected and }j\\text{ is selected and }n\\in \\mathcal {N}_u,\\ j\\in \\mathcal {N}_k\\\\0, &\\text{else}\\end{array}\\right.", "}$ $\\tilde{{a}^*}_n={\\left\\lbrace \\begin{array}{ll}\\sqrt{p^*_k p^*_u} A_{n,j}, &\\text{if the nodes }n, j\\text{ are connected and }n\\in \\mathcal {N}_u, j\\in \\mathcal {N}_k\\\\0,\\ &\\text{else}\\end{array}\\right.", "}$ Then ${A}^*=(\\tilde{{a}^*}_1^\\top ,\\tilde{{a}^*}_2^\\top ,\\cdots ,\\tilde{{a}^*}_{n})^\\top $ .", "Then, for $n\\in \\mathcal {N}_u$ , as long as $\\sum _{j\\in |\\mathcal {N}_i|}\\sqrt{p_i^* p_u^*}A_{n,j}\\approx \\sqrt{p_u^*p^*_i}\\frac{1}{\\sqrt{d_{i} d_{n}}}\\cdot \\frac{d_i|\\mathcal {N}_i|}{\\sum _{l=1}^L d_l|\\mathcal {N}_l|} d_{n}\\lesssim \\sqrt{p^*_u p^*_i}\\Psi (n,i)\\le \\sqrt{ p^*_i}\\Psi (n,i)\\le O(1/L)$ i.e., $\\sqrt{p^*_i}\\le \\frac{c_2}{L\\cdot \\max _{n\\in [L]}\\lbrace \\Psi (n,i)\\rbrace }=\\frac{c_1}{L\\cdot \\Psi (L,i)}=\\frac{c_2}{L}\\sqrt{\\frac{d_i}{d_L}}\\frac{\\sum _{l=1}^L d_l |\\mathcal {N}_l|}{d_i|\\mathcal {N}_i|}$ for some constant $c_2>0$ , we can obtain that $\\Vert {A}^*\\Vert _\\infty \\le O(1)$ .", "The difference between $\\tilde{{a}}_n^{t}$ and $\\tilde{{a}^*}_n$ can then be derived as $\\begin{aligned}&\\Vert \\tilde{{a}}_n^{t}-\\tilde{{a}^*}_{n}\\Vert _1\\\\=&\\Big | \\sum _{k=1}^L\\sum _{j\\in \\mathcal {S}_k}A_{n,j} \\sqrt{p^*_u p^*_k}(\\sqrt{\\frac{|\\mathcal {N}_k||\\mathcal {N}_u|}{l_k l_u}}- 1)+\\sum _{k=1}^L \\sum _{j\\notin \\mathcal {S}_k}A_{n,j} \\sqrt{ p^*_u p^*_k}\\Big |\\\\\\approx &\\Big | \\sum _{k=1}^L\\sum _{j\\in \\mathcal {N}_k}A_{n,j} \\sqrt{p^*_u p^*_k}(\\sqrt{\\frac{|\\mathcal {N}_k||\\mathcal {N}_u|}{l_k l_u}}- 1)\\frac{l_k}{|\\mathcal {N}_k|}+\\sum _{k=1}^L \\sum _{j\\in \\mathcal {N}_k}A_{n,j} \\sqrt{ p^*_u p^*_k}(1-\\frac{l_k}{|\\mathcal {N}_k|})\\Big |\\\\\\lesssim &\\text{poly}(\\epsilon )\\end{aligned}$ as long as $l_i\\ge |\\mathcal {N}_i|/(1+\\frac{c_2\\text{poly}(\\epsilon )}{L \\sqrt{p^*_i}\\Psi (L,i)})^2$ ." ], [ "Node classification for three layers", "In the whole proof, we consider a more general target function compared to (REF ).", "We write $F^*: \\mathbb {R}^N\\times \\mathbb {R}^{N\\times d}\\rightarrow \\mathbb {R}^K$ : $\\begin{aligned}&F^*_{{A}^*}=(f_1^*,f_2^*,\\cdots , f_K^*), \\\\&f_r^*({e}_g, {X})={e}_g^\\top \\sum _{k\\in [p_1]}c_{k,r}^*\\Phi \\Big ({A}^*\\sum _{j\\in [p_2]}v_{1,k,j}^*\\phi _{1,j}({A}^*{X}{w}_{1,j}^*)\\Big )\\odot \\Big ({A}^*\\sum _{l\\in [p_2]}v_{2,k,l}^*\\phi _{2,l}({A}^*{X}{w}_{2,l}^*)\\Big ),\\ \\forall r\\in [K],\\end{aligned}$ where each $\\phi _{1,j}$ , $\\phi _{2,j}$ , $\\Phi _i$ : $\\mathbb {R}\\rightarrow \\mathbb {R}$ is infinite-order smooth.", "Table REF shows some important notations used in our theorem and algorithm.", "Table REF gives the full parameter choices for the three-layer GCN.", "$\\text{ploy}(\\log (m_1 m_2))$ in the following analysis.", "Table: Summary of notationsTable: Full parameter choices for three-layer GCN" ], [ "Function approximation", "To show that the target function can be learnt by the learner network with the Relu function, a good approach is to firstly find a function $h(\\cdot )$ such that the $\\phi $ functions in the target function can be approximated by $h(\\cdot )$ with an indicator function.", "In this section, Lemma REF provides the existence of such $h(\\cdot )$ function.", "Lemma REF and REF are two supporting lemmas to prove Lemma REF .", "Lemma B.1 For every smooth function $\\phi $ , every $\\epsilon \\in (0,\\frac{1}{\\mathcal {C}(\\phi ,a)\\sqrt{a^2+1}})$ , there exists a function $h: \\mathbb {R}^2\\rightarrow [-\\mathcal {C}_\\epsilon (\\phi ,a)\\sqrt{a^2+1}, \\mathcal {C}_\\epsilon (\\phi ,a)\\sqrt{a^2+1}]$ that is also $\\mathcal {C}_\\epsilon (\\phi ,a)\\sqrt{a^2+1}$ -Lipschitz continuous on its first coordinate with the following two (equivalent) properties: (a) For every $x_1\\in [-a,a]$ where $a>0$ : $\\Big |\\mathbb {E}\\Big [\\mathbb {1}_{\\alpha _1x_1+\\beta _1\\sqrt{a^2-x_1^2}+b_0\\ge 0}h(\\alpha _1,b_0)\\Big ]-\\phi (x_1)\\Big |\\le \\epsilon $ where $\\alpha _1,\\beta _1,b_0\\sim \\mathcal {N}(0,1)$ are independent random variables.", "(b) For every ${w}^*, {x}\\in \\mathbb {R}^d$ with $\\Vert {w}^*\\Vert _2=1$ and $\\Vert {x}\\Vert \\le a$ : $\\Big |\\mathbb {E}\\Big [\\mathbb {1}_{{w}{X}+b_0\\ge 0}h({w}^\\top {w}^*,b_0)\\Big ]-\\phi ({{w}^*}^\\top {x})\\Big |\\le \\epsilon $ where ${w}\\sim \\mathcal {N}(0,{I})$ is an d-dimensional Gaussian, $b_0\\sim \\mathcal {N}(0,1)$ .", "Furthermore, we have $\\mathbb {E}_{\\alpha _1,b_0\\sim \\mathcal {N}(0,1)}[h(\\alpha _1,b_0)^2]\\le (\\mathcal {C}_s(\\phi ,a))^2(a^2+1)$ .", "(c) For every ${w}^*, {x}\\in \\mathbb {R}^d$ with $\\Vert {w}^*\\Vert _2=1$ , let $\\tilde{{w}}=({w},b_0)\\in \\mathbb {R}^{d+1}$ , $\\tilde{{x}}=({x},1)\\in \\mathbb {R}^{d+1}$ with $\\Vert \\tilde{{x}}\\Vert \\le \\sqrt{a^2+1}$ , then we have $\\Big |\\mathbb {E}\\Big [\\mathbb {1}_{\\tilde{{w}}^\\top \\tilde{{x}}\\ge 0}h({\\tilde{{w}}[1:d]}^\\top {w}^*,\\tilde{{w}}[d+1])\\Big ]-\\phi ({{w}^*}^\\top \\tilde{{x}}[1:d])\\Big |\\le \\epsilon $ where $\\tilde{{w}}\\sim \\mathcal {N}(0,{I}_{d+1})$ is an d-dimensional Gaussian.", "We also have $\\mathbb {E}_{\\tilde{{w}}\\in \\mathcal {N}(0,{I}_{d+1})}[h({\\tilde{{w}}[1:d]}^\\top {w}^*,\\tilde{{w}}[d+1])^2]\\le (\\mathcal {C}_s(\\phi ,a))^2(a^2+1)$ .", "Proof: Firstly, since we can assume ${w}^*=(1,0,\\cdots ,0)$ without loss of generality by rotating ${x}$ and ${w}$ , it can be derived that ${x}$ , ${w}$ , ${w}^*$ are equivalent to that they are two-dimensional.", "Therefore, proving Lemma REF b suffices in showing Lemma REF a.", "Let ${w}_0=(\\alpha ,\\beta )$ , ${x}=(x_1, \\sqrt{t^2-x_1^2})$ where $\\alpha $ and $\\beta $ are independent.", "Following the idea of Lemma 6.3 in [1], we use another randomness as an alternative, i.e., we write ${x}^\\perp =(\\sqrt{t^2-x_1^2},-x_1)$ , ${w}_0=\\alpha \\frac{{x}}{t}+\\beta \\frac{{x}^\\perp }{t}\\sim \\mathcal {N}(0,{I})$ .", "Then ${w}_0{X}=t\\alpha $ .", "Let $\\alpha _1=w_{01}=\\alpha \\frac{x_1}{t}+\\beta \\sqrt{1-\\frac{x_1^2}{t^2}}$ , where $\\alpha ,\\beta \\sim \\mathcal {N}(0,1)$ .", "Hence, $\\alpha _1\\sim \\mathcal {N}(0,1)$ .", "We first use Lemma REF to fit $\\phi (x_1)$ .", "By Taylor expansion, we have $\\begin{aligned}\\phi (x_1)&=c_0+\\sum _{i=1, \\text{\\ odd\\ } i}^\\infty c_i x_1^i+\\sum _{i=2, \\text{\\ even\\ }i}^\\infty c_i x_1^i\\\\&=c_0+\\sum _{i=1}^\\infty c_i^{\\prime }\\mathbb {E}_{\\alpha ,\\beta \\sim \\mathcal {N}(0,1)}[h_i(\\alpha _1)\\mathbb {1}[q_i(b_0)]\\mathbb {1}[{w}_0{X}+b_0\\ge 0]]\\end{aligned}$ where $h_i(\\cdot )$ is the Hermite polynomial defined in Definition A.5 in [1], and $c_i^{\\prime }=\\frac{c_i}{p_i^{\\prime }},\\ |c_i^{\\prime }|\\le \\frac{200i^2|c_i|}{(i-1)!!", "}\\frac{\\sqrt{t^2+1}}{t^{1-i}} \\text{ and } q_i(b_0)={\\left\\lbrace \\begin{array}{ll}|b_0|\\le t/(2i), & i\\mbox{ is odd} \\\\ 0<-b_0\\le t/(2i), & i\\mbox{ is even}\\end{array}\\right.", "}$ Let $B_i=100i^\\frac{1}{2}+10\\sqrt{\\log (\\frac{1}{\\epsilon }\\frac{\\sqrt{t^2+1}}{t^{1-i}})}$ .", "Define $\\hat{h}_i(\\alpha _1)=h_i(\\alpha _1)\\cdot \\mathbb {1}[|\\alpha _1|\\le B_i]+h_i(\\text{sign}(\\alpha _1)B_i)\\cdot \\mathbb {1}[|\\alpha _1|> B_i]$ as the truncated version of the Hermite polynomial.", "Then we have $\\phi (x_1)=c_0+R(x_1)+\\sum _{i=1}^\\infty c_i^{\\prime }\\mathbb {E}_{\\alpha ,\\beta \\sim \\mathcal {N}(0,1)}[\\hat{h}_i(\\alpha _1)\\mathbb {1}[q_i(b_0)]\\mathbb {1}[{w}_0{X}+b_0\\ge 0]],$ where $R(x_1)=\\sum _{i=1}^\\infty c_i^{\\prime }\\mathbb {E}_{\\alpha ,\\beta \\sim \\mathcal {N}(0,1)}\\Big [\\big (h_i(\\alpha _1)\\cdot \\mathbb {1}[|\\alpha _1|>B_i]-h_i(\\text{sign}(\\alpha _1)B_i\\cdot \\mathbb {1}[|\\alpha |> B_i])\\big )\\mathbb {1}[q_i(b_0)]\\mathbb {1}[{w}_0{X}+b_0\\ge 0]\\Big ]$ Define $h(\\alpha _1,b_0)=c_0+\\sum _{i=1}^\\infty c_i^{\\prime }\\cdot \\hat{h}_i(\\alpha _1)\\cdot \\mathbb {1}[q_i(b_0)]$ Then by Lemma REF , we have $|\\mathbb {E}_{\\alpha ,\\beta ,b_0\\sim \\mathcal {N}(0,1)}[\\mathbb {1}[{w}_0{X}+b_0\\ge 0]\\cdot h(\\alpha _1,b_0)-\\phi (x_1)|\\le |R(x_1)|\\le \\frac{\\epsilon }{4}$ We also have $\\begin{aligned}\\mathbb {E}_{\\alpha _1,b_0\\sim \\mathcal {N}(0,1)}[h(\\alpha _1,b_0)^2]&\\le (\\epsilon ^2+c_0^2)+O(1)\\cdot \\sum _{i=1}^\\infty \\frac{i!\\cdot |c_i|^2 i^3}{((i-1)!!", ")^2}\\cdot (\\frac{\\sqrt{t^2+1}}{t^{1-i}})^2\\\\&\\le (\\epsilon ^2+c_0^2)+\\sum _{i=1}^\\infty i^{3.5}\\cdot |c_i|^2\\cdot (\\frac{\\sqrt{t^2+1}}{t^{1-i}})^2\\\\&\\le (\\epsilon ^2+c_0^2)+\\Big (\\sum _{i=0}^\\infty (i+1)^{1.75}\\cdot |c_i|\\cdot t^i\\sqrt{t^2+1}\\Big )^2\\\\&\\le \\mathcal {C}_s(\\phi ,t)^2(t^2+1)\\end{aligned}$ Lemma B.2 Denote $h_i(x)$ as the degree-i Hermite polynomial as in Definition A.5 in [1].", "For every integer $i\\ge 1$ , there exists constant $p_i^{\\prime }$ with $|p_i^{\\prime }|\\ge \\frac{t^{1-i}}{\\sqrt{t^2+1}}\\frac{(i-1)!!", "}{100i^2}$ such that $\\text{for even }i:\\ \\ x_1^i=\\frac{1}{p_i^{\\prime }}\\mathbb {E}_{{w}_0\\sim \\mathcal {N}(0,{I}), b_0\\sim \\mathcal {N}(0,1)}[h_i(\\alpha _1)\\mathbb {1}[\\alpha \\ge -\\frac{b_0}{t}]\\mathbb {1}[0<-b_0\\le \\frac{t}{2i}]]$ $\\text{for odd }i:\\ \\ x_1^i=\\frac{1}{p_i^{\\prime }}\\mathbb {E}_{{w}_0\\sim \\mathcal {N}(0,{I}), b_0\\sim \\mathcal {N}(0,1)}[h(\\alpha _1)\\mathbb {1}[\\alpha \\ge -\\frac{b_0}{t}]\\mathbb {1}[|b_0|\\le \\frac{t}{2i}]]$ for $\\Vert {x}\\Vert \\le t$ .", "Proof: For even $i$ , by Lemma A.6 in [1], we have $\\mathbb {E}_{{w}_0\\sim \\mathcal {N}(0,{I}), b_0\\sim \\mathcal {N}(0,1)}[h_i(\\alpha _1)\\mathbb {1}[\\alpha \\ge -\\frac{b_0}{t}]\\mathbb {1}[0<-b_0\\le \\frac{t}{2i}]]=\\mathbb {E}_{b_0\\sim \\mathcal {N}(0,1)}[p_i\\cdot \\mathbb {1}[0<-b_0\\le \\frac{t}{2i}]]\\cdot \\frac{x_1^i}{t^i}$ , where $p_i=(i-1)!", "!\\frac{\\exp (-b_0^2/(2t^2))}{\\sqrt{2\\pi }}\\sum _{r=1, r\\text{ odd}}^{i-1}\\frac{(-1)^\\frac{i-1-r}{2}}{r!!", "}\\binom{i/2-1}{(r-1)/2}(-b_0/t)^r$ Define $c_r=\\frac{(-1)^\\frac{i-1-r}{2}}{r!!", "}\\binom{i/2-1}{(r-1)/2}$ .", "Then $\\text{sign}(c_r)=-\\text{sign}(c_{r+2})$ .", "We can derive $\\Big |\\frac{c_{r}(-b_0/t)^{r}}{c_{r-2}(-b_0/t)^{r-2}}\\Big |=\\Big |(\\frac{b_0}{t})^2\\frac{i+1-r}{r(r-1)}\\Big |\\le \\frac{1}{4i}\\le \\frac{1}{4}$ Therefore, $\\Big |\\sum _{r=1, r\\text{ odd}}^{i-1}c_r(-b_0/t)^r\\Big |\\ge \\frac{3}{4}|b_0/t|$ $\\begin{aligned}&|\\mathbb {E}_{b_0\\sim \\mathcal {N}(0,1)}[p_i\\cdot \\mathbb {1}[0\\le -b_0/t\\le 1/(2i)]]|\\cdot t^{-i}\\\\\\ge &|\\mathbb {E}_{b_0\\sim \\mathcal {N}(0,1)}[(i-1)!", "!\\frac{\\exp (-b_0^2/2t^2)}{\\sqrt{2\\pi }}\\cdot \\frac{3}{4}|b_0/t|\\cdot \\mathbb {1}[0\\le -b_0/t\\le 1/(2i)]]|\\cdot t^{-i}\\\\=& t^{-i}\\cdot \\int _{-\\frac{t}{2i}}^0(i-1)!", "!\\frac{\\exp (-\\frac{b_0^2}{2}(1+\\frac{1}{t^2}))}{2\\pi }\\cdot \\frac{3}{4}(-\\frac{b_0}{t})db_0\\\\=& t^{-i}\\cdot \\frac{t}{t^2+1}\\exp (-\\frac{b_0^2}{2}(1+\\frac{1}{t^2}))(i-1)!", "!\\frac{3}{8\\pi }\\Big |^0_{-\\frac{t}{2i}}\\\\=& t^{-i}\\frac{t}{t^2+1}(i-1)!", "!\\frac{3}{8\\pi }\\Big (1-\\exp (-\\frac{t^2+1}{8i^2})\\Big )\\\\\\ge & t^{1 -i}\\frac{(i-1)!!", "}{100i^2}\\\\\\end{aligned}$ For odd $i$ , similarly by Lemma A.6 in [1], we can obtain $\\mathbb {E}_{{w}_0\\sim \\mathcal {N}(0,{I}), b_0\\sim \\mathcal {N}(0,1)}[h(\\alpha _1)\\mathbb {1}[\\alpha \\ge -\\frac{b_0}{t}]\\mathbb {1}[|b_0|\\le \\frac{t}{2i}]]=\\mathbb {E}_{b_0\\sim \\mathcal {N}(0,1)}[p_i\\cdot \\mathbb {1}[|b_0|\\le \\frac{t}{2i}]]\\cdot \\frac{x_1^i}{t^{i}}$ , where $p_i=(i-1)!", "!\\frac{\\exp (-b_0^2/(2t^2))}{\\sqrt{2\\pi }}\\sum _{r=1, r\\text{ even}}^{i-1}\\frac{(-1)^\\frac{i-1-r}{2}}{r!!", "}\\binom{i/2-1}{(r-1)/2}(-b_0/t)^r$ Then we also have $\\Big |\\frac{c_{r}(-b_0/t)^{r}}{c_{r-2}(-b_0/t)^{r-2}}\\Big |=\\Big |(\\frac{b_0}{t})^2\\frac{i+1-r}{r(r-1)}\\Big |\\le \\frac{1}{4i}\\le \\frac{1}{4}$ Therefore, $\\Big |\\sum _{r=1, r\\text{ odd}}^{i-1}c_r(-b_0/t)^r\\Big |\\ge \\frac{3}{4}|c_0|=\\frac{3}{4}\\frac{(\\frac{i}{2}-1)!", "}{\\pi (\\frac{1}{2}\\cdot \\frac{3}{2}\\cdots \\frac{i-1}{2})}\\ge \\frac{3}{4}\\frac{1}{\\pi \\frac{i-1}{2}}\\ge \\frac{3}{2\\pi i}$ $\\begin{aligned}&|\\mathbb {E}_{b_0\\sim \\mathcal {N}(0,1)}[p_i\\cdot \\mathbb {1}[|b_0|/t\\le 1/(2i)]]|\\cdot t^{-i}\\\\\\ge &t^{-i}\\cdot |\\mathbb {E}_{b_0\\sim \\mathcal {N}(0,1)}[(i-1)!", "!\\frac{\\exp (-b_0^2/2t^2)}{\\sqrt{2\\pi }}\\cdot \\frac{3}{2\\pi i}\\cdot \\mathbb {1}[|b_0|/t\\le 1/(2i)]]|\\\\=& t^{-i}\\cdot \\int _{-\\frac{t}{2i}}^{\\frac{t}{2i}}(i-1)!", "!\\frac{\\exp (-\\frac{b_0^2}{2}(1+\\frac{1}{t^2}))}{2\\pi }\\cdot \\frac{3}{2\\pi i}db_0\\\\= & t^{-i}\\cdot (i-1)!", "!\\frac{3}{4\\pi ^2 i}\\cdot \\frac{t}{\\sqrt{t^2+1}}\\cdot \\sqrt{2\\pi }\\cdot \\Big (2\\Phi (\\frac{\\sqrt{t^2+1}}{2i})-1\\Big )\\\\= & t^{-i}\\cdot (i-1)!", "!\\frac{3}{4\\pi ^2 i}\\cdot \\frac{t}{\\sqrt{t^2+1}}\\cdot \\sqrt{2\\pi }\\cdot \\frac{2\\Phi (\\frac{\\sqrt{t^2+1}}{2})-1}{i}\\\\\\ge & \\frac{t^{1-i}}{\\sqrt{t^2+1}}\\frac{(i-1)!!", "}{100i^2}\\end{aligned}$ Lemma B.3 For $B_i=100i^{1/2}+10\\sqrt{\\log (t^{i-1}\\sqrt{t^2+1}/\\epsilon _i^2)}$ where $\\epsilon _i^2= t^{i-1}\\sqrt{t^2+1}\\epsilon ^2$ , we have $\\sum _{i=1}^\\infty |c_i^{\\prime }|\\cdot |\\mathbb {E}_{x\\sim \\mathcal {N}(0,1)}[|h_i(x)|\\cdot \\mathbb {1}[|x|\\ge b]]|\\le \\frac{\\epsilon }{8}\\sqrt{t^2+1}$ $\\sum _{i=1}^\\infty |c_i^{\\prime }|\\cdot |\\mathbb {E}_{x\\sim \\mathcal {N}(0,1)}[|h_i(b)|\\cdot \\mathbb {1}[|x|\\ge b]]|\\le \\frac{\\epsilon }{8}\\sqrt{t^2+1}$ $\\sum _{i=1}^\\infty |c_i^{\\prime }|\\mathbb {E}_{z\\in \\mathcal {N}(0,1)}[|h_i(z)|\\mathbb {1}[|z|\\le B_i]]\\le \\mathcal {C}_\\epsilon (\\phi ,t)\\sqrt{t^2+1}$ $\\sum _{i=1}^\\infty |c_i^{\\prime }|\\mathbb {E}_{z\\in \\mathcal {N}(0,1)}[|\\frac{d}{dz} h_i(z)|\\mathbb {1}[|z|\\le B_i]]\\le \\mathcal {C}_\\epsilon (\\phi , t)\\sqrt{t^2+1}$ Proof: By the definition of Hermite polynomial in Definition A.5 in [1], we have $h_i(x)\\le \\sum _{j=1}^{\\left\\lfloor i/2\\right\\rfloor }\\frac{|x|^{i-2j}i^{2j}}{j!", "}$ Combining (REF ), we can obtain $|c_i^{\\prime }h_i(x)|\\le O(1)|c_i|\\frac{\\sqrt{t^2+1}}{t^{1-i}}\\frac{i^4}{i!!", "}\\sum _{j=1}^{\\left\\lfloor i/2\\right\\rfloor }\\frac{|x|^{i-2j}i^{2j}}{j!", "}$ (1) Let $b=100i^\\frac{1}{2}\\theta _i$ and $\\theta _i=1+\\frac{\\sqrt{\\log (\\frac{1}{\\epsilon _i^2}\\frac{\\sqrt{t^2+1}}{t^{1-i}})}}{10\\sqrt{i}}$ for $\\epsilon _i^2=\\frac{\\sqrt{t^2+1}}{t^{1-i}}\\epsilon ^2$ where $i\\ge 1$ , then we have $\\begin{aligned}(\\theta _i\\cdot e^{-10^2\\theta _i^2})^i= & \\Big (\\big (1+\\frac{\\sqrt{\\log (\\frac{1}{\\epsilon _i^2}\\frac{\\sqrt{t^2+1}}{t^{1-i}})}}{10\\sqrt{i}}\\big )\\cdot e^{-10^2}e^{-2\\cdot 10^2\\frac{\\sqrt{\\log (\\frac{1}{\\epsilon _i^2}\\frac{\\sqrt{t^2+1}}{t^{2-i}})}}{10\\sqrt{i}}}\\cdot e^{-10^2 \\frac{\\log (\\frac{1}{\\epsilon _i^2}\\frac{\\sqrt{t^2+1}}{t^{2-i}})}{100i}}\\Big )^i\\\\=& \\epsilon _i^2\\frac{t^{1-i}}{\\sqrt{t^2+1}}\\cdot e^{-10^2i}\\cdot (1+\\frac{\\sqrt{\\log (\\frac{1}{\\epsilon _i^2}\\frac{\\sqrt{t^2+1}}{t^{1-i}})}}{10\\sqrt{i}})e^{-2\\cdot 10^2 \\frac{\\sqrt{\\log (\\frac{1}{\\epsilon _i^2}\\frac{\\sqrt{t^2+1}}{t^{1-i}}})}{10\\sqrt{i}}}\\\\\\le & \\frac{\\epsilon _i^2}{100000^i}\\frac{t^{1-i}}{\\sqrt{t^2+1}}\\end{aligned}$ where the second step comes from that $(1+s)\\cdot e^{-2\\cdot 10^4\\cdot s}\\le 1$ for any $s>0$ .", "Combining the equation C.6, C.7 in [1] and (REF ), we can derive $\\begin{aligned}&\\sum _{i=1}^\\infty |c_i^{\\prime }|\\cdot \\mathbb {E}_{x\\sim \\mathcal {N}(0,1)}[|h_i(z)|\\cdot \\mathbb {1}[|x|\\ge b]]\\\\\\le & \\sum _{i=1}^\\infty O(1)|c_i|\\frac{\\sqrt{t^2+1}}{t^{1-i}}\\frac{i^4}{i!!", "}\\cdot i^\\frac{i}{2}\\cdot 1200^i\\cdot (\\theta _i \\cdot e^{-10^2\\theta _i^{2}})^i\\\\\\le & \\frac{\\epsilon }{8}\\sqrt{t^2+1}\\end{aligned}$ for any $\\epsilon >0$ and $t\\le O(1)$ .", "(b) Similarly, following (REF ) and (REF ), we have $\\sum _{i=1}^\\infty |c_i^{\\prime }|\\cdot |\\mathbb {E}_{x\\sim \\mathcal {N}(0,1)}[|h_i(b)|\\cdot \\mathbb {1}[|x|\\ge b]]|\\le \\sum _{i=1}^\\infty O(1)\\frac{\\sqrt{t^2+1}}{t^{1-i}}|c_i|\\frac{i^4}{i!!", "}\\cdot e^{-\\frac{b^2}{2}}(3b)^i\\le \\frac{\\epsilon }{8}\\sqrt{t^2+1}$ (c) Similar to (REF ), $\\begin{aligned}\\sum _{i=1}^\\infty |c_i^{\\prime }|\\mathbb {E}_{z\\in \\mathcal {N}(0,1)}[|h_i(z)|\\mathbb {1}[|z|\\le B_i]]& \\le O(1) \\sum _{i=1}^\\infty |c_i|\\frac{i^4}{i!!", "}\\sum _{j=0}^{\\left\\lfloor i/2\\right\\rfloor }\\frac{B_i^{i-2j}i^{2j}}{j!", "}t^{i-1}\\sqrt{t^2+1}\\\\& \\le \\sum _{i=1}^\\infty |c_i|(O(1)\\theta _i)^i t^{i-1}\\sqrt{t^2+1}\\\\&\\le \\mathcal {C}_\\epsilon (\\phi , t)\\sqrt{t^2+1},\\end{aligned}$ where the step follows from Claim C.2 (c) in [1].", "(d) Since we have $|\\frac{d}{dx}h_i(x)|\\le \\sum _{j=0}^{\\left\\lfloor i/2\\right\\rfloor }|x|^{i-2j}i^{2j}$ by Definition A.5 in [1], we can derive $\\sum _{i=1}^\\infty |c_i^{\\prime }|\\mathbb {E}_{z\\in \\mathcal {N}(0,1)}[|\\frac{d}{dz} h_i(z)|\\mathbb {1}[|z|\\le B_i]]\\le \\mathcal {C}_\\epsilon (\\phi , t)\\sqrt{t^2+1}$" ], [ "Existence of a good pseudo network", "We hope to find some good pseudo network that can approximate the target network.", "In such a pseudo network, the activation $\\mathbb {1}_{{x}\\ge 0}$ is replaced by $\\mathbb {1}_{{x}^{(0)}\\ge 0}$ where ${x}^{(0)}$ is the value at the random initialization.", "We can define a pseudo network without bias as $g_r^{(0)}({q},{A},{W},{V}, {B})=\\sum _{n=1}^N {q}^\\top {a}_n\\sum _{i\\in [m_2]}c_{i,r}\\mathbb {1}_{{r}_{n,i}+B_{2(n,i)}\\ge 0}\\sum _{j=1}^N a_{n,j}\\sum _{l\\in [m_1]}v_{i,l}\\mathbb {1}_{{a}_j{X}{w}_l+B_{1(j,l)}}{a}_j{X}{w}_l$ Lemma REF shows the target function can be approximated by the pseudo network with some parameters.", "Lemma REF to REF provides how the existence of such a pseudo network is developed step by step.", "Lemma B.4 For every $\\epsilon \\in (0,\\frac{1}{K\\Vert {q}\\Vert _1 p_1p_2^2\\mathcal {C}_s(\\Phi , p_2\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty ))\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty )\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}})$ , there exists $M=\\text{poly}(\\mathcal {C}_\\epsilon (\\Phi , \\sqrt{p_2}\\mathcal {C}_\\epsilon (\\phi ,\\Vert {A}\\Vert _\\infty )\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}), 1/\\epsilon )$ $C=\\mathcal {C}_\\epsilon (\\phi ,\\Vert {A}\\Vert _\\infty )\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}$ $C^{\\prime }=10 C\\sqrt{p_2}$ $C^{\\prime \\prime }=\\mathcal {C}_\\epsilon (\\Phi , C^{\\prime })\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}$ $C_0=\\tilde{O}(p_1^2p_2K^2 CC^{\\prime \\prime })$ such that with high probability, there exists $\\widehat{{W}}$ , $\\widehat{{V}}$ with $m_1, m_2\\ge M$ , $\\Vert \\widehat{{W}}\\Vert _{2,\\infty }\\le \\frac{C_0}{m_1},\\ \\ \\ \\ \\Vert \\widehat{{V}}\\Vert _{2,\\infty }\\le \\frac{\\sqrt{m_1}}{m_2}$ such that $\\mathbb {E}_{({X},y)\\in \\mathcal {D}}\\Big [\\sum _{r=1}^K|f_r^*({q},{A},{X})-g_r^{(0)}({q},{A},{X},\\widehat{{W}}.\\widehat{{V}})|\\Big ]\\le \\epsilon $ $\\mathbb {E}_{({X},y)\\in \\mathcal {D}}[|L(G^{(0)}({q},{A},{X},\\widehat{{W}},\\widehat{{V}}))|]\\le OPT+\\epsilon $ Proof: For each $\\phi _{2,j}$ , we can construct $h_{\\phi ,j}:\\ \\mathbb {R}^2\\rightarrow [-C,C]$ where $C=\\mathcal {C}_\\epsilon (\\phi ,\\Vert {A}\\Vert _\\infty )\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}$ using Lemma REF satisfying $\\mathbb {E}[h_{\\phi ,j}({{w}_{2,j}^*}^\\top {w}_i^{(0)}, B_{1(n,i)}^{(0)})\\mathbb {1}_{{\\tilde{{a}}_n}{X}{w}_i^{(0)}+B_{1(n,i)\\ge 0}}]=\\phi _{2,j}({\\tilde{{a}}_n}{X}{w}_{2,j})\\pm \\epsilon $ for $i\\in [m_1]$ .", "Consider any arbitrary ${b}\\in \\mathbb {R}^{m_1}$ with $v_i\\in \\lbrace -1,1\\rbrace $ .", "Define $\\widehat{{W}}=\\frac{(C_0C^{\\prime \\prime }/C)^{\\frac{1}{2}}}{\\epsilon _c^2 m_1}(v_i\\sum _{j\\in [p_2]}v_{2,j}^*h_{\\phi ,j}({{w}_{2,j}^*}^\\top {w}_i^{(0)}, B_{1(i)}^{(0)}){e}_d)_{i\\in [m_1]}$ $\\widehat{{V}}=(C_0C^{\\prime \\prime }/C)^{-\\frac{1}{2}}\\sum _{k\\in [p_1]}\\frac{c_k^*}{m_2}({v}h(\\sqrt{m_2}\\sum _{j\\in [p_2]}v_{1,j}^*\\alpha _{i,j}, B_{2(i)}^{(0)})\\sum _{r=1}^K c_{i,r})_{i\\in [m_2]}$ Then, $\\begin{aligned}&g_r^{(0)}({q},{A},\\widehat{{W}},\\widehat{{V}}, {B})\\\\=&\\sum _{n=1}^N {q}^\\top {a}_n\\sum _{i\\in [m_1]}c_{i,r}\\mathbb {1}_{{r}_{n,i}+B_{2(n,i)}\\ge 0}\\sum _{i^{\\prime }\\in [m_2]}\\sum _{j=1}^N a_{n,j}\\mathbb {1}_{{a}_j{X}{w}^{(0)}_i+B_{1(j,i)}\\ge 0}{a}_j{X}\\widehat{{W}}_{i} \\widehat{V}_{i,i^{\\prime }}\\\\=&\\sum _{k\\in [p_1]}\\frac{c_k^*}{m_2\\epsilon _c^2}\\sum _{n=1}^N {q}^\\top {a}_n\\sum _{i\\in [m_1]}c_{i,r}^2\\mathbb {1}_{{r}_{n,i}+B_{2(n,i)}\\ge 0}h(\\sqrt{m_2}\\sum _{j\\in [p_2]}v_{1,j}^*\\alpha _{i,j}, B_{2(i)}^{(0)})\\sum _{j=1}^N a_{n,j}\\sum _{l\\in [p_2]}v_{2,l}^*\\phi _{2,l}({a}_j{X}{w}_{2,l}^*)\\\\=& \\sum _{k\\in [p_1]}\\sum _{n=1}^N {q}^\\top {a}_n c_k^*\\Phi (\\sum _{j\\in [p_2]}v_{1,j}^*\\sum _{m=1}^N a_{m,n}\\phi _{1,j}({a}_m{X}{w}_{1,j}^*))\\sum _{j=1}^N a_{n,j}\\sum _{l\\in [p_2]}v_{2,l}^*\\phi _{2,l}({a}_j{X}{w}_{2,l}^*)\\\\&\\pm O(p_1p_2^2\\mathcal {C}_s(\\Phi , p_2\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty ))\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty )\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}\\epsilon )\\\\=& \\sum _{n=1}^N {q}^\\top {a}_n \\sum _{k\\in [p_1]}c_k^*\\Phi ({\\tilde{{a}}_n}\\sum _{j\\in [p_2]}v_{1,j}^*\\phi _{1,j}({A}{X}{w}_{1,j}^*)){\\tilde{{a}}_n}\\sum _{l\\in [p_2]}v_{2,l}^*\\phi _{2,l}({A}{X}{w}_{2,l}^*)\\\\&\\pm O(\\Vert {q}\\Vert _1 p_1p_2^2\\mathcal {C}_s(\\Phi , p_2\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty ))\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty )\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}\\epsilon )\\\\\\end{aligned}$ where the first step comes from definition of $g^{(0)}$ , the second step is derived from (REF ) and (REF ) and the second to last step is by Lemma REF .", "Lemma B.5 For every smooth function $\\phi $ , every ${w}^*\\in \\mathbb {R}^d$ with $\\Vert {w}^*\\Vert =1$ , for every $\\epsilon \\in (0,\\frac{1}{\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty )\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}})$ , there exists real-valued functions $\\rho ({v}_1^{(0)}, {W}^{(0)}, {B}_{1(n)}^{(0)})$ , $J(\\tilde{{a}}_n{X}, {v}_1^{(0)}, {W}^{(0)}, {B}_{1(n)}^{(0)})$ , $R(\\tilde{{a}}_n{X}, {v}_1^{(0)}, {W}^{(0)}, {B}_{1(n)}^{(0)})$ and $\\phi _\\epsilon (\\tilde{{a}}_n{X})$ such that for every ${X}$ $r_{n,1}({X})=\\rho ({v}_1^{(0)}, {W}^{(0)}, {B}_{1(n)}^{(0)})\\sum _{j=1}^N a_{j,n}\\phi _\\epsilon ({a}_j{X}{w}^*)+J({X}, {v}_1^{(0)}, {W}^{(0)}, {B}_{1(n)}^{(0)})+R({X}, {v}_1^{(0)}, {W}^{(0)}, {B}_{1(n)}^{(0)})$ Moreover, letting $C=\\mathcal {C}_\\epsilon (\\phi ,\\Vert {A}\\Vert _\\infty )\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}$ be the complexity of $\\phi $ , and if $v_{1,i}\\sim \\mathcal {N}(0,\\frac{1}{m_2})$ and $w_{i,j}^{(0)}, {B}_{1(n)}^{(0)}\\sim \\mathcal {N}(0,\\frac{1}{m_1})$ are at random initialization, then we have 1. for every fixed $\\tilde{{a}}_n{X}$ , $\\rho ({v}_1^{(0)}, {W}^{(0)}, {B}_{1(n)}^{(0)})$ is independent of $J(\\tilde{{a}}_n{X}, {v}_1^{(0)}, {W}^{(0)}, {B}_{1(n)}^{(0)})$ .", "2.", "$\\rho ({v}_1^{(0)}, {W}^{(0)}, {B}_{1(n)}^{(0)})\\sim \\mathcal {N}(0,\\frac{1}{100C^2m_2})$ .", "3.", "$|\\phi _\\epsilon (\\tilde{{a}}_n{X}{w}_i^*)-\\phi (\\tilde{{a}}_n{X}{w}_{i}^*)|\\le \\epsilon $ 4. with high probability, $|R({X}, {v}_1^{(0)}, {W}^{(0)}, {B}_{1(n)}^{(0)})|\\le \\tilde{O}(\\frac{\\Vert {A}\\Vert _\\infty }{\\sqrt{m_1m_2}})$ , $|J({X}, {v}_1^{(0)}, {W}^{(0)}, {B}_{1(n)}^{(0)})|\\le \\tilde{O}(\\frac{\\Vert {A}\\Vert _\\infty (1+\\Vert {A}\\Vert _\\infty )}{\\sqrt{m_2}})$ and $\\mathbb {E}[J({X}, {v}_1^{(0)}, {W}^{(0)}, {B}_{1(n)}^{(0)})]=0$ .", "With high probability, we also have $\\tilde{\\rho }(v_1^{(0)})\\sim \\mathcal {N}(0,\\frac{\\tau }{C^2m_2})$ $\\mathcal {W}_2(\\rho |_{{W}^{(0)},{B}_{1(n)}^{(0)}},\\tilde{\\rho })\\le \\tilde{O}(\\frac{1}{C\\sqrt{m_2}})$ Proof: By Lemma REF , we have $\\mathbb {E}_{{w}_i^{(0)}\\sim \\mathcal {N}(0,\\frac{{I}}{m_1}), b_{1(n,i)}\\sim \\mathcal {N}(0,\\frac{1}{m_1})}[h(\\sqrt{m_1}{{w}_i^{(0)}}^\\top {w}^* , b_{1(n,i)})\\mathbb {1}[\\tilde{{a}}_n{X}{w}_i^{(0)}+b_{1(n,i)}\\ge 0]]=\\frac{\\phi _\\epsilon (\\tilde{{a}}_n{X}{w}_i^*)}{C}$ with $|\\phi _\\epsilon (\\tilde{{a}}_n{X}{w}^*)-\\phi (\\tilde{{a}}_n{X}{w}^*)|\\le \\epsilon $ and $|h(\\sqrt{m_1}{{w}_i^{(0)}}^\\top {w}^* , b_{1(n,i)})|\\in [0,1]$ .", "Note that here the $h$ function is rescaled by $1/C$ .", "Then, applying Lemma A.4 of [1], we define $I_i=I(h(\\sqrt{m_1}{{w}_i^{(0)}}^\\top {w}^* , B_{1(n,i)}))\\subset [-2,2]$ $S=\\lbrace i\\in [m_1]: \\sqrt{m_2}v_{1,i}^{(0)}\\in I_i\\rbrace $ $s_i=s(h(\\sqrt{m_1}{{w}_i^{(0)}}^\\top {w}^* , B_{1(n,i)})), \\sqrt{m_2}v_{1,i}^{(0)})$ $u_i={\\left\\lbrace \\begin{array}{ll}\\frac{s_i}{\\sqrt{|S|}}, &\\text{if }i\\in S\\\\0, &\\text{if }s\\notin S\\end{array}\\right.", "}$ where $u_i,\\ i\\in [m_1]$ is independent of ${W}^{(0)}$ .", "We can write ${W}^{(0)}=\\alpha {e}_d{u}^\\top +\\beta ,$ where $\\alpha ={u}^\\top {e}_d^\\top {W}^{(0)}\\sim \\mathcal {N}(0,1/m_1)$ and $\\beta \\in \\mathbb {R}^{d\\times m_1}$ are two independent random variables given ${u}$ .", "We know $\\alpha $ is independent of ${u}$ .", "Since each $i\\in S$ with probability $\\tau $ , we know with high probability, $|S|=\\tilde{\\Theta }(\\tau m_1)$ Since $\\alpha =\\sum _{i\\in S}u_i[{e}_d^\\top {W}^{(0)}]_i$ and $|u_i[{e}_d^\\top {W}^{(0)}]_i|\\le \\tilde{O}(1/\\sqrt{m_1|S|})$ , by (REF ) and the Wasserstein distance bound of central limit theorem we know there exists $g\\sim \\mathcal {N}(0,\\frac{1}{m_1})$ such that $\\mathcal {W}_2(\\alpha |_{{W}^{(0)}, {B}_{1(n)}^{(0)}},g)\\le \\tilde{O}(\\frac{1}{\\sqrt{\\tau }m_1})$ Then, $\\begin{aligned}r_{n,1}({X})&=\\sum _{j=1}^N a_{j,n}\\sum _{i=1}^{m_1}v_{i,1}^{(0)}\\sigma ({a}_j{X}{w}_1^{(0)}+B_{1(n,i)}^{(0)})\\\\&= \\sum _{j=1}^N a_{j,n}\\sum _{i\\notin S}v_{i,1}^{(0)}\\sigma ({a}_j{X}{w}_1^{(0)}+B_{1(n,i)}^{(0)})+\\sum _{j=1}^N a_{j,n}\\sum _{i\\in S}v_{i,1}^{(0)}\\sigma ({a}_j{X}{w}_1^{(0)}+B_{1(n,i)}^{(0)})\\\\&=J_1+\\sum _{j=1}^N a_{j,n}\\sum _{i\\in S}v_{i,1}^{(0)}\\sigma ({a}_j{X}{w}_1^{(0)}+B_{1(n,i)}^{(0)})\\end{aligned}$ $\\begin{aligned}&r_{n,1}({X})-J_1\\\\=& \\sum _{j=1}^N a_{j,n}\\sum _{i\\in S}v_{i,1}^{(0)}\\mathbb {1}[{a}_j{X}{w}_1^{(0)}+B_{1(n,i)}^{(0)}]\\frac{s_i}{2\\sqrt{|S|}}\\alpha +\\sum _{j=1}^N a_{j,n}\\sum _{i\\in S}v_{i,1}^{(0)}\\mathbb {1}[{a}_j{X}{w}_1^{(0)}+B_{1(n,i)}^{(0)}]({a}_j{X}\\beta _i+B_{1(n,i)}^{(0)})\\\\=&P_1+P_2\\end{aligned}$ Here, we know that since $\\mathbb {E}[v_{i,1}^{(0)}\\sigma ({a}_j{X}{w}_1^{(0)}+B_{1(n,i)}^{(0)})]=\\mathbb {E}[v_{i,1}^{(0)}]\\cdot \\mathbb {E}[\\sigma ({a}_j{X}{w}_1^{(0)}+B_{1(n,i)}^{(0)})]=0$ Hence, $\\mathbb {E}[J_1]=\\mathbb {E}[\\sum _{j=1}^N a_{j,n}\\sum _{i\\notin S}v_{i,1}^{(0)}\\sigma ({a}_j{X}{w}_1^{(0)}+B_{1(n,i)}^{(0)})]=0$ Then we can derive $P_1=\\sum _{j=1}^N a_{j,n}\\sum _{i\\in S}\\mathbb {1}[{a}_j{X}{w}_1^{(0)}+B_{1(n,j)}^{(0)}]\\frac{\\alpha }{2\\sqrt{|S|m_2}}h(\\sqrt{m_1}{{w}_i^{(0)}}^\\top {w}^* , B_{1(n,i)}) +R_1$ where $|R_1|\\le \\tilde{O}(\\sqrt{\\frac{|S|}{m_1m_2}})$ .", "We write $P_3=\\frac{P_1-R_1}{\\alpha }$ .", "Then, $|P_3-\\frac{\\sqrt{|S|}}{\\sqrt{m_2}C}\\sum _{j=1}^N a_{j,n}\\phi _\\epsilon ({a}_j{X}{w}^*)|\\le \\tilde{O}(\\Vert {A}\\Vert _\\infty \\frac{1}{\\sqrt{m_2}})$ $|\\frac{C\\sqrt{m_2}}{\\sqrt{\\tau m_1}}P_1-\\sum _{j=1}^N a_{j,n}\\phi _\\epsilon ({a}_j{X}{w}^*)|\\le \\tilde{O}(\\Vert {A}\\Vert _\\infty \\frac{C}{\\sqrt{\\tau m_1}})$ Define $\\rho ({v}_1^{(0)},{W}^{(0)}, {B}_{1(n)}^{(0)})=\\frac{\\sqrt{\\tau m_1}}{C\\sqrt{m_2}}\\alpha \\sim \\mathcal {N}(0, \\frac{\\tau }{C^2 m_2})$ Then, $P_1=\\rho ({v}_1^{(0)},{W}^{(0)}, {B}_{1(n)}^{(0)})\\cdot \\sum _{j=1}^N a_{j,n}\\phi _\\epsilon ({a}_j{X}{w}^*)+R_1+R_2({X}, {v}_1^{(0)},{W}^{(0)}, {B}_{1(n)}^{(0)})$ where $|R_2|\\le \\tilde{O}(\\frac{1}{\\sqrt{m_1m_2}})$ .", "We can also define $\\tilde{\\rho }(v_1^{(0)})=\\frac{\\sqrt{\\tau m_1}}{C\\sqrt{m_2}}g\\sim \\mathcal {N}(0,\\frac{\\tau }{C^2m_2})$ Therefore, $\\mathcal {W}_2(\\rho |_{{W}^{(0)},{B}_{1(n)}^{(0)}},\\tilde{\\rho })\\le \\tilde{O}(\\frac{1}{C\\sqrt{m_2}})$ Meanwhile, ${a}_j{X}{w}_i^{(0)}=\\alpha \\frac{s_i}{\\sqrt{|S|}}{a}_j{X}{e}_d+{a}_j{X}\\beta _i+B_{1(n,i)}^{(0)}={a}_j{X}\\beta _i+B_{1(n,i)}^{(0)}\\pm \\tilde{O}(\\frac{1}{\\sqrt{|S|m_1}})$ we have $P_2=\\sum _{j=1}^N a_{j,n}\\sum _{i\\in S}v_{i,1}^{(0)}\\mathbb {1}[{a}_j{X}\\beta _i+b_{1(n,j)}^{(0)}]({a}_j{X}\\beta _i+b_{1(n,i)}^{(0)})+R_3=J_2+R_3$ $\\mathbb {E}[J_2]=0$ with $|R_3|\\le \\tilde{O}(\\frac{\\Vert {A}\\Vert _\\infty }{\\sqrt{m_1m_2}})$ .", "Let $J=J_1+J_2$ , $R=R_1+R_2+R_3$ .", "Then, w.h.p., $\\mathbb {E}[J]=0$ , $|J|\\le \\tilde{O}(\\frac{\\Vert {A}\\Vert _\\infty (1+\\Vert {A}\\Vert _\\infty )}{\\sqrt{m_2}})$ , $|R|\\le \\tilde{O}(\\frac{\\Vert {A}\\Vert _\\infty }{\\sqrt{m_1m_2}})$ .", "Lemma B.6 For every $\\epsilon \\in (0,\\frac{1}{\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty )\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}})$ , there exists real-valued functions $\\phi _{1,j.\\epsilon }(\\cdot )$ such that $|\\phi _{1,j,\\epsilon }(\\tilde{{a}}_n{X}{w}_{1,j}^*)-\\phi _{1,j}(\\tilde{{a}}_n{X}{w}_{1,j}^*)|\\le \\epsilon $ for $j\\in [p_2]$ .", "Denote by $C=\\mathcal {C}_\\epsilon (\\phi , \\Vert {A}\\Vert _\\infty )\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1},\\ C^{\\prime }=10C\\sqrt{p_2},\\ \\phi _{1,j,\\epsilon }({a}_j{X}{w}_{1,i}^*)=\\frac{1}{C^{\\prime }}\\phi _{1,j,\\epsilon }({a}_j{X}{w}_{1,i}^*)$ For every $i\\in [m_2]$ , there exist independent Gaussians $\\alpha _{i,j}\\sim \\mathcal {N}(0,\\frac{1}{m_2}),\\ \\beta _i({X})\\sim \\mathcal {N}(0, \\frac{1}{m_2}),$ satisfying $\\mathcal {W}_2(r_{n,i}({X}), \\sum _{j\\in [p_2]}\\alpha _{i,j}\\sum _{m=1}^N a_{m,n}\\phi _{1,j,\\epsilon }({a}_m{X}{w}_{1,i}^*)+C_i\\beta _i({X}))\\le \\tilde{O}( \\frac{p_2^\\frac{2}{3}}{m_1^\\frac{1}{6}\\sqrt{m_2}})$ Proof: Define $p_2 S$ many chunks of the first layer with each chunk corresponding to a set $S_{j,l}$ , where $|S_{j,l}|=m_1/(p_2 S)$ for $j\\in [p_2]$ and $l\\in [S]$ , such that $\\mathcal {S}_{j,l}=\\lbrace (j-1)\\frac{m_1}{p_2}+(l-1)\\frac{m_1}{p_2} S+k|k\\in [\\frac{m_1}{p_2 S]}\\rbrace \\subset [m_1]$ By Lemma REF , we have $\\begin{aligned}r_{n,i}({X})=&\\sum _{j\\in [p_2], l\\in [S]}\\rho ({v}_i^{(0)}[j,l], {W}^{(0)}[j,l], {B}_{1(n)}^{(0)}[j,l])\\sum _{m=1}^N a_{m,n}\\phi _\\epsilon ({a}_m{X}{w}_{1,j}^*)\\\\&+\\sum _{j\\in [p_2], l\\in [S]}J_j({X}, {v}_i^{(0)}[j,l], {W}^{(0)}[j,l], {B}_{1(n)}^{(0)}[j,l])+R_j({X}, {v}_i^{(0)}[j,l], {W}^{(0)}[j,l], {B}_{1(n)}^{(0)}[j,l]),\\end{aligned}$ where $\\rho ({v}_i^{(0)}[j,l], {W}^{(0)}[j,l], {B}_{1(n)}^{(0)}[j,l])\\sim \\mathcal {N}(0,\\frac{1}{100C^2m_2p_2S})$ .", "Then $\\rho _j=\\sum _{l\\in [S]}\\rho _{j,l}\\sim \\mathcal {N}(0,\\frac{1}{C^{\\prime 2}m_2})$ for $C^{\\prime }=10C\\sqrt{p_2}$ .", "Define $J_j^S({X})=\\sum _{l\\in [S]}J_j({X}, {v}_i^{(0)}[j,l], {W}^{(0)}[j,l], {B}_{1(n)}^{(0)}[j,l])$ $R_j^S({X})=\\sum _{l\\in [S]}R_j({X}, {v}_i^{(0)}[j,l], {W}^{(0)}[j,l], {B}_{1(n)}^{(0)}[j,l])$ Then there exists Gaussian random variables $\\beta _j({X})$ and $\\beta ^{\\prime }({X})=\\sum _{i\\in [p_2]}\\beta _j({X})$ such that $\\mathcal {W}_2(J_j^S({X}),\\beta _j({X}))\\le \\frac{\\Vert {A}\\Vert _\\infty (1+\\Vert {A}\\Vert _\\infty )}{\\sqrt{m_2 p S}}$ $\\mathcal {W}_2(r_{n,i}({X}), \\sum _{j\\in [p_2]}\\rho _j\\sum _{m=1}^N a_{m,n}\\phi _{1,j,\\epsilon }({a}_m{X}{w}_{1,j}^*)+\\beta ^{\\prime }({X}))\\le \\tilde{O}(\\frac{Sp_2}{\\sqrt{m_1m_2}}+\\frac{\\sqrt{p_2}\\Vert {A}\\Vert _\\infty (1+\\Vert {A}\\Vert _\\infty )}{m_2 S})$ We know there exists a positive constant $C_i$ such that $\\beta ^{\\prime }/ C_i\\sim \\mathcal {N}(0,\\frac{1}{m_2})$ .", "Let $\\alpha _{i,j}=C^{\\prime }\\rho _j$ , $\\beta _i^{\\prime }=\\beta ^{\\prime }/C_i$ .", "Notice that $\\mathbb {E}\\big [\\sum _{l\\in [S], i\\in [p_2]}[J_j^2({X}, {v}_i^{(0)}[j,l], {W}^{(0)}[j,l], b_1^{(0)}[j,l])]\\big ]=\\tilde{O}(\\Vert {A}\\Vert _\\infty ^2(1+\\Vert {A}\\Vert _\\infty )^2/m_2)$ .", "Hence, we have $C_i\\le \\tilde{O}(\\Vert {A}\\Vert _\\infty (1+\\Vert {A}\\Vert _\\infty )$ Let $S=(m_1/p_2)^\\frac{1}{3}$ , we can obtain $\\mathcal {W}_2(r_{n,i}({X}), \\sum _{j\\in [p_2]}\\alpha _{i,j}\\sum _{m=1}^N a_{m,n}\\phi _{1,j,\\epsilon }({a}_m{X}{w}_{1,i}^*)+C_i\\beta _i({X}))\\le \\tilde{O}(\\frac{p_2^\\frac{2}{3}}{m_1^\\frac{1}{6}\\sqrt{m_2}})$ Lemma B.7 There exists function $h: \\mathbb {R}^2\\rightarrow [-C^{\\prime \\prime },C^{\\prime \\prime }]$ for $C^{\\prime \\prime }=\\mathcal {C}_\\epsilon (\\Phi ,C^{\\prime })\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}$ such that $\\begin{aligned}&\\mathbb {E}[\\mathbb {1}_{r_{n,i}({X})+b_{2(n,i)}^{(0)}\\ge 0}h(\\sqrt{m_2}\\sum _{j\\in [p_2]}v_{1,j}^*\\alpha _{i,j}, b_{2(n,i)}^{(0)})(\\sum _{j\\in [p_2]}v_{2,j}^*\\phi _{2,j}(\\tilde{{a}}_n{X}{w}_{2,j}^*))]\\\\=&\\Phi (\\sum _{j\\in [p_2]}v_{1,j}^*\\sum _{m=1}^N a_{m,n}\\phi _{1,j})\\sum _{j\\in [p_2]}v_{2,j}^*\\phi _{2,j}(\\tilde{{a}}_n{X}{w}_{2,j}^*))\\pm \\tilde{O}(p_2^2\\mathcal {C}_s(\\Phi , p_2\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty ))\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty )\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}\\epsilon )\\end{aligned}$ Proof: Choose ${w}=(\\alpha _{i,1},\\cdots ,\\alpha _{i,p_2}, \\beta _i)$ , ${x}=(\\sum _{m=1}^N a_{m,n}\\phi _{1,1,\\epsilon }, \\cdots , \\sum _{m=1}^N a_{m,n}\\phi _{1,p_2,\\epsilon }, C_i)$ and ${w}^*=(v_{1,1}^*,\\cdots , v_{1,p_2}^*,0)$ .", "Then, $\\Vert {x}\\Vert \\le O(\\Vert {A}\\Vert _\\infty ^2+\\Vert {A}\\Vert _\\infty )$ .", "By Lemma REF , there exists $h: \\mathbb {R}^{2}\\rightarrow [-C^{\\prime \\prime },C^{\\prime \\prime }]$ for $C^{\\prime \\prime }=\\mathcal {C}_s(\\Phi ,C^{\\prime })\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}$ such that $\\begin{aligned}&\\mathbb {E}[\\mathbb {1}_{{w}{X}+b_{2(n,i)}^{(0)}\\ge 0}h(\\sqrt{m_2}{w}^\\top {w}^*, b_{2(n,i)}^{(0)})(\\sum _{j\\in [p_2]}v_{2,j}^*\\phi _{2,j}(\\tilde{{a}}_n{X}{w}_{2,j}^*))]\\\\=&\\mathbb {E}_{\\alpha _i,\\beta _i}[\\mathbb {1}_{\\sum _{j\\in [p_2]}\\alpha _{i,j}\\sum _{m=1}^N a_{m,n}\\phi _{1,j,\\epsilon }(\\tilde{{a}}_n{X}{w}_{1,i}^*)+C_i\\beta _i^{\\prime }+b_{2(n,i)}^{(0)}\\ge 0}h(\\sqrt{m_2}{w}^\\top {w}^*, b_{2(n,i)}^{(0)})(\\sum _{j\\in [p_2]}v_{2,j}^*\\phi _{2,j}(\\tilde{{a}}_n{X}{w}_{2,j}^*))]\\\\=&\\Phi (C^{\\prime }\\sum _{j\\in [p_2]}v_{1,j}^*\\sum _{m=1}^N a_{m,n}\\phi _{1,j,\\epsilon })\\sum _{j\\in [p_2]}v_{2,j}^*\\phi _{2,j}(\\tilde{{a}}_n{X}{w}_{2,j}^*))\\pm \\epsilon C^{\\prime \\prime \\prime }\\end{aligned}$ where $C^{\\prime \\prime \\prime }=\\sup |\\sum _{j\\in [p_2]}v_{2,j}^*\\phi _{2,j}(\\tilde{{a}}_n{X}{w}_{2,j}^*)|\\le p_2\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty )\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}$ By Lemma REF , we know $\\mathcal {W}_2(r_{n,i}({X}), \\sum _{j\\in [p_2]}\\alpha _{i,j}\\sum _{m=1}^N a_{m,n}\\phi _{1,j,\\epsilon }({a}_m{X}{w}_{1,i}^*)+C_i\\beta _i({X}))\\le \\tilde{O}( \\frac{p_2^\\frac{2}{3}}{m_1^\\frac{1}{6}\\sqrt{m_2}})$ Denote $\\mathcal {H}=\\lbrace i\\in [m_1]: |\\sum _{j\\in [p_2]}\\alpha _{i,j}\\sum _{m=1}^N a_{m,n}\\phi _{1,j,\\epsilon }(\\tilde{{a}}_n{X}{w}_{1,i}^*)+C_i\\beta _i^{\\prime }|\\ge \\tilde{O}( \\frac{2 p_2^\\frac{2}{3}}{m_1^\\frac{1}{6}\\sqrt{m_2}})\\rbrace $ .", "Then, for every $i\\in [\\mathcal {H}]$ , we have that $\\mathbb {1}_{r_{n,i}({X})+b_{2(n,i)}^{(0)}\\ge 0}=\\mathbb {1}_{\\sum _{j\\in [p_2]}\\alpha _{i,j}\\sum _{m=1}^N a_{m,n}\\phi _{1,j,\\epsilon }(\\tilde{{a}}_n{X}{w}_{1,i}^*)+C_i\\beta _i^{\\prime }+b_{2(n,i)}^{(0)}\\ge 0}$ $\\Pr \\Big (\\big |\\sum _{j\\in [p_2]}\\alpha _{i,j}\\sum _{m=1}^N a_{m,n}\\phi _{1,j,\\epsilon }(\\tilde{{a}}_n{X}{w}_{1,i}^*)+C_i\\beta _i^{\\prime }\\big |\\le \\tilde{O}( \\frac{2 p_2^\\frac{2}{3}}{m_1^\\frac{1}{6}\\sqrt{m_2}})\\Big )\\le \\tilde{O}( \\frac{2 p_2^\\frac{2}{3}}{m_1^\\frac{1}{6}\\sqrt{m_2}})\\cdot \\sqrt{m_2}=\\tilde{O}(\\frac{2 p_2^\\frac{2}{3}}{m_1^\\frac{1}{6}}),$ which implies with probability at least $1-2p_2^{2/3}/m_1^{1/6}$ , (REF ) holds.", "Therefore, $\\begin{aligned}&\\mathbb {E}[\\mathbb {1}_{r_{n,i}({X})+b_{2(n,i)}^{(0)}\\ge 0}h(\\sqrt{m_2}\\sum _{j\\in [p_2]}v_{1,j}^*\\alpha _{i,j}, b_{2(n,i)}^{(0)})(\\sum _{j\\in [p_2]}v_{2,j}^*\\phi _{2,j}(\\tilde{{a}}_n{X}{w}_{2,j}^*))]\\\\=&\\mathbb {E}[\\mathbb {1}_{\\sum _{j\\in [p_2]}\\alpha _{i,j}\\sum _{m=1}^N a_{m,n}\\phi _{1,j,\\epsilon }(\\tilde{{a}}_n{X}{w}_{1,i}^*)+C_i\\beta _i^{\\prime }+b_{2(n,i)}^{(0)}\\ge 0}h(\\sqrt{m_2}\\sum _{j\\in [p_2]}v_{1,j}^*\\alpha _{i,j}, b_{2(n,i)}^{(0)})(\\sum _{j\\in [p_2]}v_{2,j}^*\\phi _{2,j}(\\tilde{{a}}_n{X}{w}_{2,j}^*))]\\\\\\ &\\pm \\mathbb {E}[\\mathbb {1}_{r_{n,i}({X})+b_{2(n,i)}^{(0)}\\ge 0}\\ne \\mathbb {1}_{\\sum _{j\\in [p_2]}\\alpha _{i,j}\\sum _{m=1}^N a_{m,n}\\phi _{1,j,\\epsilon }(\\tilde{{a}}_n{X}{w}_{1,i}^*)+C_i\\beta _i^{\\prime }+b_{2(n,i)}^{(0)}\\ge 0}]O(C^{\\prime \\prime \\prime }C^{\\prime \\prime })\\\\=&\\mathbb {E}[\\mathbb {1}_{\\sum _{j\\in [p_2]}\\alpha _{i,j}\\sum _{m=1}^N a_{m,n}\\phi _{1,j,\\epsilon }(\\tilde{{a}}_n{X}{w}_{1,i}^*)+C_i\\beta _i^{\\prime }+b_{2(n,i)}^{(0)}\\ge 0}h(\\sqrt{m_2}\\sum _{j\\in [p_2]}v_{1,j}^*\\alpha _{i,j}, b_{2(n,i)}^{(0)})(\\sum _{j\\in [p_2]}v_{2,j}^*\\phi _{2,j}(\\tilde{{a}}_n{X}{w}_{2,j}^*))]\\\\\\ &\\pm \\tilde{O}(\\frac{2p_2^{2/3}}{m_1^{1/6}}C^{\\prime \\prime \\prime }C^{\\prime \\prime })\\\\=&\\Phi (\\sum _{j\\in [p_2]}v_{1,j}^*\\sum _{m=1}^N a_{m,n}\\phi _{1,j})\\sum _{j\\in [p_2]}v_{2,j}^*\\phi _{2,j}({x}))\\pm \\tilde{O}(p_2^2\\mathcal {C}_s(\\Phi , p_2\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty ))\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty )\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}\\cdot \\epsilon ),\\end{aligned}$ where the first step is by Lemma REF , the second step is by (REF ) and (REF ) and the last step comes from (REF ) and $m_1\\ge M$ .", "Lemma B.8 $\\begin{aligned}&\\frac{1}{m_2}\\mathbb {E}[\\sum _{i=1}^{m_2}\\frac{c_{i,l}^2}{\\epsilon _c^2}\\mathbb {1}_{r_{n,i}({X})+b_{2(n,i)}^{(0)}\\ge 0}h(\\sqrt{m_2}\\sum _{j\\in [p_2]}v_{1,j}^*\\alpha _{i,j}, b_{2(n,i)}^{(0)})(\\sum _{j\\in [p_2]}v_{2,j}^*\\phi _{2,j}(\\tilde{{a}}_n{X}{w}_{2,j}^*))]\\\\=&\\Phi (\\sum _{j\\in [p_2]}v_{1,j}^*\\sum _{m=1}^N a_{m,n}{a}_m{X}\\delta \\phi _{1,j})\\sum _{j\\in [p_2]}v_{2,j}^*\\phi _{2,j}(\\tilde{{a}}_n{X}{w}_{2,j}^*))\\\\&\\ \\ \\pm \\tilde{O}(p_2^2\\mathcal {C}_s(\\Phi , p_2\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty ))\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty )\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}\\cdot \\epsilon )\\end{aligned}$ Proof: Recall $\\tilde{\\rho }({v}_1^{(0)})\\sim \\mathcal {N}(0,\\frac{\\tau }{C^2m_2})$ .", "Define $\\tilde{\\rho }_{j,l}=\\tilde{\\rho }({v}_1^{(0)}[j,l])$ .", "Therefore, $\\mathcal {W}_2(\\rho _{j,l}|_{{W}^{(0)},{B}_{1(n)}^{(0)}},\\tilde{\\rho }_{j,l})\\le \\tilde{O}(\\frac{1}{C^{\\prime }\\sqrt{m_2}S})$ $\\mathcal {W}_2(\\rho _j|_{{W}^{(0)},{B}_{1(n)}^{(0)}},\\tilde{\\rho }_j)\\le \\tilde{O}(\\frac{1}{C^{\\prime }\\sqrt{m_2}})$ where $\\tilde{\\rho }_j=\\sum _{l\\in [S]}\\rho _{j,l}$ .", "We then define $\\tilde{\\alpha }_{i,j}=C^{\\prime }\\tilde{\\rho }_{j}$ Next modify $r_{n,i}({X})$ .", "Define $\\tilde{r}_{n,i}({X})=\\frac{\\sum _{m=1}^N a_{m,n}\\sum _{j\\in [m_1]}v_{j,i}^{(0)}\\sigma ({a}_m{X}{w}_i^{(0)}+b_{1(n,i)}^{(0)})}{\\Vert {u}\\Vert }\\mathbb {E}[\\Vert {u}\\Vert ]$ where $u=(\\sigma (\\tilde{{a}}_n{X}{w}_j^{(0)}+b_{1(n,j)}^{(0)}))_{j\\in [m_1]}$ .", "By definition, we know $\\tilde{r}_{n,i}\\sim \\mathcal {N}(0, \\frac{\\Vert \\tilde{{a}}_n\\Vert _\\infty ^2}{m_2}\\mathbb {E}[\\Vert {u}\\Vert ]^2)$ Then we have $\\mathcal {W}_2(r_{n,i}({X}), \\tilde{r}_{n,i}({X}))\\le \\tilde{O}(\\frac{\\Vert {A}\\Vert _\\infty \\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}}{\\sqrt{m_2}})$ Combining (REF ), (REF ), (REF ) and Lemma REF , we have $\\begin{aligned}&\\frac{1}{m_2}\\mathbb {E}[\\sum _{i=1}^{m_2}\\frac{c_{i,l}^2}{\\epsilon _c^2}\\mathbb {1}_{r_{n,i}({X})+b_{2(n,i)}^{(0)}\\ge 0}h(\\sqrt{m_2}\\sum _{j\\in [p_2]}v_{1,j}^*\\alpha _{i,j}, b_{2(n,i)}^{(0)})(\\sum _{j\\in [p_2]}v_{2,j}^*\\phi _{2,j}(\\tilde{{a}}_n{X}{w}_{2,j}^*))]\\\\=&\\Phi (\\sum _{j\\in [p_2]}v_{1,j}^*\\sum _{m=1}^N a_{m,n}\\phi _{1,j})\\sum _{j\\in [p_2]}v_{2,j}^*\\phi _{2,j}(\\tilde{{a}}_n{X}{w}_{2,j}^*))\\pm \\tilde{O}(p_2^2\\mathcal {C}_s(\\Phi , p_2\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty ))\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty )(\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}\\cdot \\epsilon )\\end{aligned}$" ], [ "Coupling", "This section illustrates the coupling between the real and pseudo networks.", "We first define diagonal matrices ${D}_{n,{w}}$ , ${D}_{n,{w}}+{D}_{n,{w}}^{\\prime \\prime }$ , ${D}_{n,{w}}+{D}_{n,{w}}^{\\prime }$ for node $n$ as the sign of Relu's in the first layer at weights ${W}^{(0)}$ , ${W}^{(0)}+{W}^\\rho $ and ${W}^{(0)}+{W}^\\rho +{W}^{\\prime }$ , respectively.", "We also define diagonal matrices ${D}_{n,{v}}$ , ${D}_{n,{v}}+{D}_{n,{v}}^{\\prime \\prime }$ , ${D}_{n,{v}}+{D}_{n,{v}}^{\\prime }$ for node $n$ as the sign of Relu's in the second layer at weights $\\lbrace {W}^{(0)}, {V}^{(0)}\\rbrace $ , $\\lbrace {W}^{(0)}+{W}^\\rho , {V}^{(0)}+{V}^\\rho \\rbrace $ and $\\lbrace {W}^{(0)}+{W}^\\rho +{W}^{\\prime }, {V}^{(0)}+{V}^\\rho +{V}^{\\prime }\\rbrace $ , respectively.", "For every $l\\in [K]$ , we then introduce the pseudo network and its semi-bias, bias-free version as $g_l({q},{A},{X},{W},{V})={q}^\\top {A}(({A}({A}{X}{W}+{B}_1)\\odot ({D}_{w}+{D}_{w}^{\\prime }){V}+{B}_2)\\odot ({D}_{{v}}+{D}_{{v}}^{\\prime })){c}_l$ $g_l^{(b)}({q},{A},{X},{W},{V})={q}^\\top {A}(({A}({A}{X}{W}+{B}_1)\\odot ({D}_{w}+{D}_{w}^{\\prime }){V})\\odot ({D}_{{v}}+{D}_{{v}}^{\\prime })){c}_l$ $g_l^{(b,b)}({q},{A},{X},{W},{V})={q}^\\top {A}(({A}({A}{X}{W})\\odot ({D}_{w}+{D}_{w}^{\\prime }){V})\\odot ({D}_{{v}}+{D}_{{v}}^{\\prime })){c}_l$ Lemma REF gives the final result of coupling with added Drop-out noise.", "Lemma REF states the sparse sign change in Relu and the function value changes of pseudo network by some update.", "To be more specific, Lemma REF shows that the sign pattern can be viewed as fixed for the smoothed objective when a small update is introduced to the current weights.", "Lemma REF proves the bias-free pseudo network can also approximate the target function.", "Lemma B.9 Let $F_{A}=(f_1,f_2,\\cdots ,f_K)$ .", "With high probability, we have for any $\\Vert {W}^{\\prime }\\Vert _{2,4}\\le \\tau _w$ , $\\Vert {V}^{\\prime }\\Vert _F\\le \\tau _v$ , such that $\\begin{aligned}&f_l({q},{A},{X},{W}^{(0)}+{W}^{\\prime }\\Sigma ,{V}^{(0)}+\\Sigma {V}^{\\prime })\\\\=& {q}^\\top {A}({A}(({A}{X}{W}^{(0)}+{B}_1^{(0)})\\odot {D}_{{w},{x}}^{(0)}{V}^{(0)}+{B}_2^{(0)})\\odot {D}_{{v},{x}}^{(0)}){c}+{q}^\\top {A}({A}(({A}{X}{W}^{\\prime })\\odot {D}_{{w},{x}}^{(0)}{V}^{\\prime })\\odot {D}_{{v},{x}}^{(0)}){c}_l\\\\&\\pm \\tilde{O}(\\tau _v\\frac{\\sqrt{m_2}}{\\sqrt{m_1}}+m_1^\\frac{9}{5}\\tau _w^\\frac{16}{5}\\sqrt{m_2}+\\tau _w^\\frac{8}{5}m_1^\\frac{9}{10})\\cdot \\Vert {q}^\\top {A}\\Vert _1\\Vert {A}\\Vert _\\infty ^2,\\end{aligned}$ where we use ${D}_{{w},{x}}^{(0)}$ and ${D}_{{v},{x}}^{(0)}$ to denote the sign matrices at random initialization ${W}^{(0)}$ , ${V}^{(0)}$ and we let ${D}_{{w},{x}}^{(0)}+{D}_{{w},{x}}^{\\prime }$ , ${D}_{{v},{x}}^{(0)}+{D}_{{v},{x}}^{\\prime }$ be the sign matrices at ${W}+{W}^{\\prime }\\Sigma $ , ${V}+\\Sigma {V}^{\\prime }$ .", "Proof: Since $\\tilde{{a}}_n{X}{w}_i^{(0)}+B_{1(n,i)}^{(0)}=\\tilde{{a}}_n\\tilde{{X}}\\tilde{{w}}_i^{(0)}$ where $\\tilde{{w}}_i^{(0)}=({w}_i^{(0)},B_{1(n,i)}^{(0)})\\in \\mathbb {R}^{d+1}$ and $\\tilde{{X}}=({X},1)\\in \\mathbb {R}^{N\\times (d+1)}$ , we can ignore the bias term for simplicity.", "Define ${Z}= {A}({A}{X}{W}^{(0)})\\odot {D}_{{w},{x}}^{(0)}$ ${Z}_1= {A}({A}{X}{W}^{\\prime }\\Sigma )\\odot {D}_{{w},{x}}^{(0)}$ ${Z}_2= {A}({A}{X}({W}^{(0)}+{W}^{\\prime })\\Sigma )\\odot {D}_{{w},{x}}^{\\prime }$ Then by Fact C.9 in [1] we have $\\begin{aligned}\\Vert {Z}_n\\Sigma {V}^{\\prime }\\Vert _2^2&\\le \\sum _{i=1}^{m_2}({Z}_n\\Sigma {V}^{\\prime }_i)^2\\le \\sum _{i=1}^{m_2} \\tilde{O}(\\Vert {Z}_n\\Vert _\\infty ^2\\cdot \\Vert {V}^{\\prime }_i\\Vert _2^2)\\\\&\\le \\tilde{O}(\\Vert {A}\\Vert _\\infty ^2 m_1^{-1}\\tau _v^2)\\end{aligned}$ Therefore, we have $\\Vert {Z}_n\\Sigma {V}^{\\prime }\\Vert _2\\le \\tilde{O}(\\Vert {A}\\Vert _\\infty m_1^{-\\frac{1}{2}}\\tau _v)$ .", "Let $s$ be the total number of sign changes in the first layer caused by adding ${W}^{\\prime }$ .", "Note that the total number of coordinated $i$ such that $|\\tilde{{a}}_n{X}{w}^{(0)}_i|\\le s^{\\prime \\prime }=\\frac{2\\tau _w}{s^\\frac{1}{4}}$ is at most $s^{\\prime \\prime } m_1^\\frac{3}{2}$ with high probability.", "Since $\\Vert {W}^{\\prime }\\Vert _{2,4}\\le \\tau _w$ , we must have $s\\le \\tilde{O}(s^{\\prime \\prime }m^\\frac{3}{2})=\\tilde{O}(\\frac{\\tau _w}{s^\\frac{1}{4}}m_1^\\frac{3}{2})$ .", "Therefore, $\\Vert {Z}_{2,n}\\Vert _0\\le s=\\tilde{O}(\\tau _w^\\frac{4}{5} m_1^\\frac{6}{5})$ .", "Then, $\\begin{aligned}\\Vert {Z}_{2,n}\\Vert _2=&\\Vert ({A}({A}{X}({W}^{(0)}+{W}^{\\prime }\\Sigma ))\\odot {D}_{{w},{x}}^{\\prime })_n\\Vert \\\\\\le & \\Big (s\\cdot \\sum _{({D}_{{w},{x}}^{\\prime })_n\\ne 0}({A}{A}{X}{W}^{(0)}+{A}{A}{X}{W}^{\\prime }\\Sigma )_{n,i}^4\\Big )^\\frac{1}{4}\\\\\\le & \\Big (s\\cdot \\sum _{({D}_{{w},{x}}^{\\prime })_n\\ne 0}({A}{A}{X}{W}^{\\prime }\\Sigma )_{n,i}^4\\Big )^\\frac{1}{4}\\\\\\le & s^\\frac{1}{4} \\Vert {A}\\Vert _\\infty \\tau _w\\\\\\le & \\tilde{O}(\\tau _w^\\frac{6}{5}m_1^\\frac{3}{10}\\Vert {A}\\Vert _\\infty )\\end{aligned}$ Then we have $ \\Vert {Z}_{2,n}\\Sigma {V}^{\\prime }\\Vert _2\\le \\tilde{O}(\\tau _v\\tau _w^\\frac{6}{5}m_1^\\frac{3}{10}\\Vert {A}\\Vert _\\infty )$ With high probability, we have $\\sum _{n=1}^N {q}^\\top {a}_n\\sum _{i=1}^{m_2}c_{i,l}(\\sigma (r_{n,i}+r^{\\prime }_{n,i})-\\sigma (r_{n,i}))\\le \\tilde{O}(\\Vert {q}\\Vert \\sqrt{m_2})\\Vert r^{\\prime }_{n,i}\\Vert $ $\\begin{aligned}&f_l({q},{A},{X},{W}^{(0)}+{W}^{\\prime }\\Sigma ,{V}^{(0)}+\\Sigma {V}^{\\prime })\\\\=& \\sum _{n=1}^N {q}^\\top {a}_n\\sum _{i=1}^{m_2}c_{i,l}\\sigma \\Big (({Z}+{Z}_1+{Z}_2)_n^\\top ({V}_i+(\\Sigma )_i{V}^{\\prime })\\Big )\\\\=& \\sum _{n=1}^N {q}^\\top {a}_n\\sum _{i=1}^{m_2}c_{i,l}\\sigma \\Big (({Z}_n+{Z}_{1,n}+{Z}_{2,n})^\\top {V}_i+{Z}_{1,n}^\\top (\\Sigma {V}^{\\prime })_i\\Big )\\pm \\tilde{O}(\\Vert {q}\\Vert \\Vert {A}\\Vert _\\infty \\frac{\\sqrt{m_2}}{\\sqrt{m_1}}\\tau _v+\\sqrt{m_2}\\Vert {q}\\Vert \\Vert {A}\\Vert _\\infty \\tau _w^\\frac{6}{5}m_1^\\frac{3}{10})\\end{aligned}$ We consider the difference between $A_1={q}^\\top {A}((({Z}+{Z}_1+{Z}_2){V}^{(0)}+{Z}_1\\Sigma {V}^{\\prime })\\odot ({D}_{{v},{x}}^{(0)}+{D}_{{v},{x}}^{\\prime \\prime })){c}_l$ $A_2={q}^\\top {A}((({Z}+{Z}_1+{Z}_2){V}^{(0)}+{Z}_1\\Sigma {V}^{\\prime })\\odot {D}_{{v},{x}}^{(0)}){c}_l$ where ${D}_{{v},{x}}^{\\prime \\prime }$ is the diagonal sign change matrix from ${Z}{V}^{(0)}$ to $({Z}+{Z}_1+{Z}_2){V}^{(0)}+{Z}_1\\Sigma {V}^{\\prime }$ .", "The difference includes three terms.", "$\\Vert {Z}_{1,n}{V}^{(0)}\\Vert _\\infty \\le \\tilde{O}(\\Vert {A}\\Vert _\\infty m_1^\\frac{1}{4}\\tau _w m_2^{-\\frac{1}{2}})$ $\\Vert {Z}_{2,n}{V}^{(0)}\\Vert _\\infty \\le \\tilde{O}(\\Vert {Z}_{2,n}\\Vert m_2^{-\\frac{1}{2}}\\sqrt{s})\\le \\tilde{O}(\\Vert {A}\\Vert _\\infty m_1^\\frac{9}{10}\\tau _w^\\frac{8}{5}m_2^{-\\frac{1}{2}})$ $\\Vert {Z}_{1,n}\\Sigma {V}^{\\prime }\\Vert \\le \\tilde{O}(\\tau _v\\Vert {A}\\Vert _\\infty \\tau _w m_1^\\frac{1}{4})$ where (REF ) is by Fact C.9 in [1].", "Then we have $|A_1-A_2|\\le \\Vert {q}^\\top {A}\\Vert _1\\cdot \\tilde{O}(m_2^\\frac{3}{2}\\Vert {A}\\Vert _\\infty ^2(m_1^\\frac{1}{4}\\tau _w m_2^{-\\frac{1}{2}}+m_1^\\frac{9}{10}\\tau _w^\\frac{8}{5}m_2^{-\\frac{1}{2}})^2+m_2^\\frac{1}{2}\\tau _v^\\frac{4}{3}\\Vert {A}\\Vert _\\infty ^\\frac{4}{3}\\tau _w^\\frac{4}{3}m_1^\\frac{1}{3})$ From $A_2$ to our goal $A_3= {q}^\\top {A}({Z}{V}^{(0)}\\odot {D}_{{v},{x}}^{(0)}){c}_l+{q}^\\top {A}({A}({A}{X}{W}^{\\prime }\\odot {D}_{{w},{x}}^{(0)}{V}^{\\prime })\\odot {D}_{{v},{x}}^{(0)}){c}_l$ There are two more terms.", "$|{q}^\\top {A}({Z}_2{V}^{(0)}\\odot {D}_{{v},{x}}^{(0)}){c}_l|\\le \\tilde{O}(\\Vert {q}^\\top {A}\\Vert _1\\Vert {Z}_{2,n}\\Vert \\sqrt{s})\\le \\tilde{O}(\\Vert {q}\\Vert _1\\Vert {A}\\Vert _\\infty \\tau _w^\\frac{8}{5}m_1^\\frac{9}{10})$ $|{q}^\\top {A}({Z}_1{V}^{(0)}\\odot {D}_{{v},{x}}^{(0)}){c}_l|\\le \\tilde{O}(\\Vert {q}^\\top {A}\\Vert _1\\Vert {A}\\Vert _\\infty \\tau _w m_1^\\frac{1}{4})$ Therefore, we have $|A_2-A_3|\\le \\tilde{O}(\\Vert {q}^\\top {A}\\Vert _1\\Vert {A}\\Vert _\\infty \\tau _w^\\frac{8}{5}m_1^\\frac{9}{10}+\\Vert {q}\\Vert _1\\Vert {A}\\Vert _\\infty \\tau _w m_1^\\frac{1}{4}+\\Vert {q}\\Vert _1\\tau _v\\Vert {A}\\Vert _\\infty \\tau _w m_1^\\frac{1}{4})$ Finally, we have $\\begin{aligned}&f_l({q},{A},{X},{W}^{(0)}+{W}^{\\prime }\\Sigma ,{V}^{(0)}+\\Sigma {V}^{\\prime })\\\\=& {q}^\\top {A}({Z}{V}^{(0)}\\odot {D}_{{v},{x}}^{(0)}){c}_l+{q}^\\top {A}({A}({A}{X}{W}^{\\prime }\\odot {D}_{{w},{x}}^{(0)}{V}^{\\prime })\\odot {D}_{{v},{x}}^{(0)}){c}_l\\\\&\\pm \\tilde{O}(\\tau _v\\frac{\\sqrt{m_2}}{\\sqrt{m_1}}+m_1^\\frac{9}{5}\\tau _w^\\frac{16}{5}\\sqrt{m_2}+\\tau _w^\\frac{8}{5}m_1^\\frac{9}{10})\\cdot \\Vert {q}\\Vert _1\\Vert {A}\\Vert _\\infty \\end{aligned}$ Lemma B.10 Suppose $\\tau _v\\in (0,1]$ , $\\tau _w\\in [\\frac{1}{m_1^\\frac{3}{2}},\\frac{1}{m_1^\\frac{1}{2}}]$ , $\\sigma _w\\in [\\frac{1}{m_1^\\frac{3}{2}}, \\frac{\\tau _w}{m_1^\\frac{1}{4}}]$ , $\\sigma _v\\in (0,\\frac{1}{m_2^\\frac{1}{2}})]$ .", "The perturbation matrices satisfies $\\Vert {W}^{\\prime }\\Vert _{2,4}\\le \\tau _w$ , $\\Vert {V}^{\\prime }\\Vert _F\\le \\tau _v$ , $\\Vert {W}^{\\prime \\prime }\\Vert _{2,4}\\le \\tau _w$ , $\\Vert {V}^{\\prime \\prime }\\Vert _F\\le \\tau _v$ and random diagonal matrix $\\Sigma $ has each diagonal entry i.i.d.", "drawn from $\\lbrace \\pm 1\\rbrace $ .", "Then with high probability, we have (1) Sparse sign change $\\Vert {D}_{n,{w}}^{\\prime }\\Vert _0\\le \\tilde{O}(\\tau _w^\\frac{4}{5}m_1^\\frac{6}{5})$ $\\Vert {D}_{n,{v}}^{\\prime }\\Vert _0\\le \\tilde{O}(m_2^\\frac{3}{2}\\sigma _v(\\Vert {A}\\Vert _\\infty +\\Vert {A}\\Vert _\\infty \\tau _w m_1^\\frac{1}{4})+m_2\\Vert {A}\\Vert _\\infty ^\\frac{2}{3}(\\Vert {A}\\Vert _\\infty \\tau _v+\\Vert {A}\\Vert _\\infty \\tau _w m_1^\\frac{1}{4}(1+\\tau _v))^\\frac{2}{3})$ (2) Cross term vanish $\\begin{aligned}&g_r({q},{A},{X},{W}^{(0)}+{W}^\\rho +{W}^{\\prime }+\\eta {W}^{\\prime \\prime }\\Sigma , {V}^{(0)}+{V}^\\rho +{V}^{\\prime }+\\eta \\Sigma {V}^{\\prime \\prime })\\\\= &g_r({q},{A},{X},{W}^{(0)}+{W}^\\rho +{W}^{\\prime }, {V}^{(0)}+{V}^\\rho +{V}^{\\prime })+g_r^{(b,b)}({q},{A},{X},\\eta {W}^{\\prime \\prime }\\Sigma , \\eta \\Sigma {V}^{\\prime \\prime })+g_r^{\\prime }\\end{aligned}$ for every $r\\in [K]$ , where $\\mathbb {E}_{\\Sigma }[g_r^{\\prime }]=0$ and $|g_r^{\\prime }|\\le \\eta \\Vert {q}^\\top {A}\\Vert _1\\Vert {A}\\Vert _\\infty ^2\\tau _v$ .", "Proof: (1) We first consider the sign changes by ${W}^\\rho $ .", "Since $\\tilde{{a}}_n{X}{w}_i^{(0)}+B_{1(n,i)}^{(0)}=\\tilde{{a}}_n\\tilde{{X}}\\tilde{{w}}_i^{(0)}$ where $\\tilde{{w}}_i^{(0)}=({w}_i^{(0)},B_{1(n,i)}^{(0)})\\in \\mathbb {R}^{d+1}$ and $\\tilde{{X}}=({X},1)\\in \\mathbb {R}^{N\\times (d+1)}$ , we can ignore the bias term for simplicity.", "We have $\\tilde{{a}}_n\\tilde{{X}}\\tilde{{w}}_i^{(0)}\\sim \\mathcal {N}(0,\\frac{\\Vert \\tilde{{a}}_n\\tilde{{X}}\\Vert ^2}{m_1})$ $\\tilde{{a}}_n\\tilde{{X}}\\tilde{{w}}_i^\\rho \\sim \\mathcal {N}(0,\\Vert \\tilde{{a}}_n\\tilde{{X}}\\Vert ^2\\sigma _w^2)$ Therefore, $\\frac{\\tilde{{a}}_n\\tilde{{X}}\\tilde{{w}}_i^{(0)}}{\\tilde{{a}}_n\\tilde{{X}}\\tilde{{w}}_i^\\rho }\\sim p(z)=\\frac{1}{\\pi (\\sigma _w\\sqrt{m_1}z^2+\\frac{1}{\\sigma _w\\sqrt{m_1}})}$ $\\begin{aligned}\\Pr [|\\tilde{{a}}_n\\tilde{{X}}\\tilde{{w}}_i^{(0)}|\\le |\\tilde{{a}}_n\\tilde{{X}}\\tilde{{w}}_i^\\rho |]&=\\Pr [|z|\\le 1]\\\\&=\\int _{-1}^1 \\frac{1}{\\pi (\\sigma _w\\sqrt{m_1}z^2+\\frac{1}{\\sigma _w\\sqrt{m_1}})}dz\\\\&= \\int _{-(\\sigma _w^2m_1)^\\frac{1}{2}}^{(\\sigma _w^2m_1)^\\frac{1}{2}} \\frac{1}{\\pi (t^2+1)}dt\\\\&= \\frac{2}{\\pi }\\arctan {\\sigma _w\\sqrt{m_1}}\\\\&\\le \\tilde{O}(\\sigma _w\\sqrt{m_1})\\end{aligned}$ Then, we have $\\Vert {D}_{n,{w}}^{\\prime \\prime }\\Vert _0\\le \\tilde{O}(\\sigma _w m_1^\\frac{3}{2})$ $\\Vert \\tilde{{a}}_n\\tilde{{X}}\\tilde{{W}}^{(0)}{D}_{n,{w}}^{\\prime \\prime }\\Vert _2\\le \\tilde{O}(\\Vert \\tilde{{a}}_n\\tilde{{X}}\\Vert \\sigma _w^\\frac{3}{2} m_1^\\frac{3}{4})$ We then consider the sign changes by ${W}^{\\prime }$ .", "Let $s=\\Vert {D}_{n,{w}}^{\\prime }-{D}_{n,{w}}^{\\prime \\prime }\\Vert _0$ be the total number of sign changes in the first layer caused by adding ${W}^{\\prime }$ .", "Note that the total number of coordinated $i$ such that $|\\tilde{{a}}_n\\tilde{{X}}(\\tilde{{W}}^{(0)}+\\tilde{{W}}^\\rho )_i|\\le s^{\\prime \\prime }=\\frac{2\\tau _w}{s^\\frac{1}{4}}$ is at most $s^{\\prime \\prime } m_1^\\frac{3}{2}$ with high probability.", "Since $\\Vert {W}^{\\prime }\\Vert _{2,4}\\le \\tau _w$ , we must have $s\\le \\tilde{O}(s^{\\prime \\prime }m^\\frac{3}{2})=\\tilde{O}(\\frac{\\tau _w}{s^\\frac{1}{4}}m_1^\\frac{3}{2})$ $\\Vert {D}_{n,{w}}^{\\prime }-{D}_{n,{w}}^{\\prime \\prime }\\Vert _0=s\\le \\tilde{O}(\\tau _w^\\frac{4}{5}m_1^\\frac{6}{5})$ $\\Vert \\tilde{{a}}_n\\tilde{{X}}(\\tilde{{W}}^{(0)}+\\tilde{{W}}^\\rho )({D}_{n,{w}}^{\\prime }-{D}_{n,{w}}^{\\prime \\prime })\\Vert _2\\le \\tilde{O}(s^\\frac{1}{4}\\tau _w)\\le \\tilde{O}(\\tau _w^\\frac{6}{5}m_1^\\frac{3}{10})$ To sum up, we have $\\Vert {D}_{n,{w}}^{\\prime }\\Vert _0\\le \\tilde{O}(\\sigma _w m_1^\\frac{3}{2}+\\tau _w^\\frac{4}{5}m_1^\\frac{6}{5})\\le \\tilde{O}(\\tau _w^\\frac{4}{5}m_1^\\frac{6}{5})$ Denote ${z}_{n,0}=\\tilde{{a}}_n\\tilde{{X}}\\tilde{{W}}^{(0)}{D}_{n,{w}}$ and ${z}_{n,2}= \\tilde{{a}}_n\\tilde{{X}}(\\tilde{{W}}^{(0)}+\\tilde{{W}}^\\rho +{W}^{\\prime })({D}_{n,{w}}+{D}_{n,{w}}^{\\prime })-\\tilde{{a}}_n\\tilde{{X}}\\tilde{{W}}^{(0)}{D}_{n,{w}}$ .", "With high probability, we know $\\begin{aligned}\\Vert {z}_{n,2}\\Vert \\le &\\Vert \\tilde{{a}}_n\\tilde{{X}}{W}^{\\prime }\\Vert +\\Vert \\tilde{{a}}_n\\tilde{{X}}\\tilde{{W}}^\\rho \\Vert \\\\\\le & \\tilde{O}(m_1^\\frac{1}{4}\\tau _w\\Vert {A}\\Vert _\\infty +\\Vert {A}\\Vert _\\infty \\sigma _w m_1^\\frac{1}{2})\\\\\\le & \\tilde{O}(\\Vert {A}\\Vert _\\infty \\tau _w m_1^\\frac{1}{4})\\end{aligned}$ Denote ${Z}_0=({z}_{1,0}^\\top ,\\cdots , {z}_{N,0}^\\top )^\\top \\in \\mathbb {R}^{N\\times m_1}$ , ${Z}_2=({z}_{1,2}^\\top ,\\cdots , {z}_{N,2}^\\top )^\\top \\in \\mathbb {R}^{N\\times m_1}$ .", "The sign change in the second layer is from $\\tilde{{a}}_n{Z}_0{V}^{(0)}$ to $\\tilde{{a}}_n({Z}_0+{Z}_2)({V}^{(0)}+{V}^\\rho +{V}^{\\prime })$ .", "We have $\\Vert \\tilde{{a}}_n({Z}_0+{Z}_2){V}^\\rho \\Vert _\\infty \\le \\tilde{O}(\\sigma _v\\Vert {A}\\Vert _\\infty (\\Vert {z}_{1,0}\\Vert +\\Vert {z}_{1,2}\\Vert ))$ $\\Vert \\tilde{{a}}_n({Z}_0+{Z}_2){V}^{\\prime }+\\tilde{{a}}_n{Z}_2{V}^{(0)}\\Vert _2\\le \\tilde{O}(\\Vert {A}\\Vert _\\infty ((\\Vert {z}_{1,0}\\Vert +\\Vert {z}_{1,2}\\Vert )\\tau _v+\\Vert {z}_{1,2}\\Vert ))$ Combining $\\Vert {z}_{1,0}\\Vert \\le \\tilde{O}(\\Vert {A}\\Vert _\\infty )$ , by Claim C.8 in [1] we have $\\Vert {D}_{n,{v}}^{\\prime }\\Vert _0\\le \\tilde{O}(m_2^\\frac{3}{2}\\sigma _v(\\Vert {A}\\Vert _\\infty ^2+\\Vert {A}\\Vert _\\infty ^2\\tau _w m_1^\\frac{1}{4})+m_2\\Vert {A}\\Vert _\\infty ^\\frac{2}{3}(\\Vert {A}\\Vert _\\infty \\tau _v+\\Vert {A}\\Vert _\\infty \\tau _w m_1^\\frac{1}{4}(1+\\tau _v))^\\frac{2}{3})$ (2) Diagonal Cross terms.", "Denote ${D}_{{w}}=(\\text{diag}({D}_{1,{w}})^\\top ,\\cdots ,\\text{diag}({D}_{N,m_1})^\\top )^\\top \\in \\mathbb {R}^{N\\times m_1}$ and define ${D}_{w}^{\\prime }$ , ${D}_{{w}}^{\\prime \\prime }$ , ${D}_{{v}}$ , ${D}_{{v}}^{\\prime }$ , ${D}_{v}^{\\prime \\prime }$ accordingly.", "Recall $g_r({q},{A},{X},{W},{V})={q}^\\top {A}(({A}({A}{X}{W}+{B}_1)\\odot ({D}_{w}+{D}_{w}^{\\prime }){V}+{B}_2)\\odot ({D}_{{v}}+{D}_{{v}}^{\\prime })){c}_r$ $g_r^{(b)}({q},{A},{X},{W},{V})={q}^\\top {A}(({A}({A}{X}{W}+{B}_1)\\odot ({D}_{w}+{D}_{w}^{\\prime }){V})\\odot ({D}_{{v}}+{D}_{{v}}^{\\prime })){c}_r$ $g_r^{(b,b)}({q},{A},{X},{W},{V})={q}^\\top {A}(({A}({A}{X}{W})\\odot ({D}_{w}+{D}_{w}^{\\prime }){V})\\odot ({D}_{{v}}+{D}_{{v}}^{\\prime })){c}_r$ Then $\\begin{aligned}&g_r({q},{A},{X},{W}^{(0)}+{W}^\\rho +{W}^{\\prime }+\\eta {W}^{\\prime \\prime }\\Sigma , {V}^{(0)}+{V}^\\rho +{V}^{\\prime }+\\eta \\Sigma {V}^{\\prime \\prime })\\\\=& g_r({q},{A},{X},{W}^{(0)}+{W}^\\rho +{W}^{\\prime }, {V}^{(0)}+{V}^\\rho +{V}^{\\prime })+g_r^{(b,b)}({q},{A},{X},\\eta {W}^{\\prime \\prime }\\Sigma , \\eta \\Sigma {V}^{\\prime \\prime })\\\\&+ g_r^{(b)}({q},{A},{X},{W}^{(0)}+{W}^\\rho +{W}^{\\prime }, \\eta \\Sigma {V}^{\\prime \\prime })+g_r^{(b,b)}({q},{A},{X},\\eta {W}^{\\prime \\prime }\\Sigma , {V}^{(0)}+{V}^\\rho +{V}^{\\prime })\\end{aligned}$ where the last two terms are the error terms.", "We know that $\\Vert {W}^{(0)}\\Vert \\le \\max _{\\Vert {a}\\Vert =1}\\Vert {a}^\\top {W}^{(0)}\\Vert \\le \\max _{\\Vert {a}\\Vert =1}\\sqrt{\\sum _{i=1}^{m_1}({a}^\\top {w}_i^{(0)})^2}\\le \\max _{\\Vert {a}\\Vert =1}\\sqrt{\\sum _{i=1}^{m_1}(\\frac{1}{\\sqrt{m_1}})^2}=1$ Therefore, $\\begin{aligned}&|g_r^{(b)}({q},{A},{X},{W}^{(0)}+{W}^\\rho +{W}^{\\prime }, \\eta \\Sigma {V}^{\\prime \\prime })|\\\\=&\\eta \\sum _{n=1}^N \\sum _{i=1}^{m_2}{q}^\\top {a}_n c_{i,r} {D_{n, {v}}}_{i}\\sum _{k=1}^N a_{n,k}\\sum _{l=1}^{m_1} {(\\Sigma {V})}_{l,i}^{\\prime \\prime } {D_{k,{w}}}_{l}({a}_k{X}({W}^{(0)}+{W}^\\rho +{W}^{\\prime })_l+{{B}_1}_{(k,l)})\\\\=&\\eta \\sum _{n=1}^N {q}^\\top {a}_n\\sum _{k=1}^N a_{n,k} (({a}_k{X}({W}^{(0)}+{W}^\\rho +{W}^{\\prime })+{{B}_1}_k)\\odot {{D}_{k,{w}}})\\Sigma {V}^{\\prime \\prime }\\odot {{D}_{n,{v}}}{c}_r\\\\\\le & \\eta \\Vert {q}^\\top {A}\\Vert _1\\Vert {A}\\Vert _\\infty (\\Vert {A}\\Vert _\\infty (m^{-\\frac{1}{2}}+\\sigma _w+\\tau _w)+m^{-\\frac{1}{2}})\\tau _v m_2^\\frac{1}{2}\\\\\\le & \\tilde{O}(\\eta \\Vert {q}^\\top {A}\\Vert _1\\Vert {A}\\Vert _\\infty ^2\\tau _v m_2^\\frac{1}{2}m_1^{-\\frac{1}{2}}),\\\\\\end{aligned}$ where the last step is by the value selection of $\\sigma _w$ , $\\tau _w$ and $\\tau _v$ .", "$\\begin{aligned}&|g_r^{(b,b)}({q},{A},\\eta {W}^{\\prime \\prime }\\Sigma , {X},{V}^{(0)}+{V}^\\rho +{V}^{\\prime })|\\\\=& |\\eta \\sum _{n=1}^N {q}^\\top {a}_n\\sum _{k=1}^N a_{n,k} ({a}_k{X}{W}^{\\prime \\prime }\\Sigma \\odot ({{D}_{k,{w}}}+{{D}_{k,{w}}}^{\\prime }))({V}^{(0)}+{V}^\\rho +{V}^{\\prime \\prime })\\odot ({{D}_{n,{v}}}+{{D}_{n, {v}}}^{\\prime }){c}_r|\\\\\\le & |\\eta \\sum _{n=1}^N {q}^\\top {a}_n\\sum _{k=1}^N a_{n,k} ({a}_k{X}{W}^{\\prime \\prime }\\Sigma \\odot ({{D}_{k,{w}}}+{{D}_{k,{w}}}^{\\prime })){V}^{\\prime }\\odot ({{D}_{n,{v}}}+{{D}_{n,{v}}}^{\\prime }){c}_r|\\\\&+|\\eta \\sum _{n=1}^N {q}^\\top {a}_n\\sum _{k=1}^N a_{n,k} ({a}_k{X}{W}^{\\prime \\prime }\\Sigma \\odot ({{D}_{k,{w}}}+{{D}_{k,{w}}}^{\\prime }))({V}^{(0)}+{V}^\\rho )\\odot {{D}_{n,{v}}}{c}_r|\\\\&+|\\eta \\sum _{n=1}^N {q}^\\top {a}_n\\sum _{k=1}^N a_{n,k} ({a}_k{X}{W}^{\\prime \\prime }\\Sigma \\odot ({{D}_{k,{w}}}+{{D}_{k,{w}}}^{\\prime }))({V}^{(0)}+{V}^\\rho )\\odot {{D}_{n,{v}}}^{\\prime }{c}_r|\\\\\\le & |\\eta \\Vert {q}^\\top {A}\\Vert _1\\Vert {A}\\Vert _\\infty ^2\\tau _w \\tau _v m_2^\\frac{1}{2}|+2|\\eta \\Vert {q}^\\top {A}\\Vert _1\\Vert {A}\\Vert _\\infty ^2\\tau _w m_1^\\frac{1}{2}|\\\\\\le & \\tilde{O}(|\\eta \\Vert {q}^\\top {A}\\Vert _1\\Vert {A}\\Vert _\\infty ^2\\tau _w m_1^\\frac{1}{2}|)\\\\\\end{aligned}$ Lemma B.11 Denote $\\begin{aligned}P_{\\rho ,\\eta }=&F_{A}({q},{X},{W}+{W}^\\rho +\\eta {W}^{\\prime \\prime }\\Sigma , {V}+{V}^\\rho +\\eta \\Sigma {V}^{\\prime \\prime })\\\\=& {q}^\\top {A}({A}({A}{X}({W}+{W}^\\rho +\\eta {W}^{\\prime \\prime }\\Sigma )+{B}_1)\\odot {D}_{{w}, \\rho ,\\eta }({V}+{V}^\\rho +\\eta \\Sigma {V}^{\\prime \\prime })\\odot {D}_{{v},\\rho ,\\eta }){c}_r\\end{aligned}$ $\\begin{aligned}P_{\\rho ,\\eta }^{\\prime }=&G({q},{A},{X},{W}+{W}^\\rho +\\eta {W}^{\\prime \\prime }\\Sigma , {V}+{V}^\\rho +\\eta \\Sigma {V}^{\\prime \\prime })\\\\=& {q}^\\top {A}({A}({A}{X}({W}+{W}^\\rho +\\eta {W}^{\\prime \\prime }\\Sigma )+{B}_1)\\odot {D}_{{w}, \\rho }({V}+{V}^\\rho +\\eta \\Sigma {V}^{\\prime \\prime })\\odot {D}_{{v},\\rho }){c}_r\\end{aligned}$ There exists $\\eta _0=\\frac{1}{\\text{poly}(m_1,m_2)}$ such that for every $\\eta \\le \\eta _0$ , for every ${W}^{\\prime \\prime }$ , ${V}^{\\prime \\prime }$ that satisfies $\\Vert {W}^{\\prime \\prime }\\Vert _{2,\\infty }\\le \\tau _{{w},\\infty }$ , $\\Vert {V}^{\\prime \\prime }\\Vert _{2,\\infty }\\le \\tau _{{v},\\infty }$ , we have $\\mathbb {E}_{{W}^\\rho ,{V}^\\rho }[\\frac{|P_{\\rho ,\\eta }-P_{\\rho ,\\eta }^{\\prime }|}{\\eta ^2}]=\\tilde{O}({q}^\\top {A}1\\Vert {A}\\Vert ^4(\\frac{\\tau ^2_{{w},\\infty }}{\\sigma _w}m_1+\\frac{(\\tau ^2_{{w},\\infty }+\\tau ^2_{{v},\\infty }m_1^{-1})}{\\sigma _v}m_2))+O_p(\\eta ),$ where $O_p$ hides polynomial factor of $m_1$ and $m_2$ .", "Proof: $\\begin{aligned}P_{\\rho ,\\eta }-P_{\\rho ,\\eta }^{\\prime }&= {q}^\\top {A}({A}({A}{X}({W}+{W}^\\rho +\\eta {W}^{\\prime \\prime }\\Sigma )+{B}_1)\\odot ({D}_{{w}, \\rho ,\\eta }-{D}_{{w},\\rho })({V}+{V}^\\rho +\\eta \\Sigma {V}^{\\prime \\prime })\\odot {D}_{{v},\\rho }){c}_r\\\\&\\ +{q}^\\top {A}({A}({A}{X}({W}+{W}^\\rho +\\eta {W}^{\\prime \\prime }\\Sigma )+{B}_1)\\odot {D}_{{w}, \\rho ,\\eta }({V}+{V}^\\rho +\\eta \\Sigma {V}^{\\prime \\prime })\\odot ({D}_{{v},\\rho ,\\eta }-{D}_{{v},\\rho })){c}_r\\end{aligned}$ We write ${Z}={A}({A}{X}({W}+{W}^\\rho +\\eta {W}^{\\prime \\prime }\\Sigma )+{B}_1)\\odot {D}_{{w}, \\rho }$ ${Z}+{Z}^{\\prime }={A}({A}{X}({W}+{W}^\\rho +\\eta {W}^{\\prime \\prime }\\Sigma )+{B}_1)\\odot {D}_{{w}, \\rho ,\\eta }$ Since for all $n\\in [N]$ , $\\Vert \\eta ({A}{A}{X}{W}^{\\prime \\prime }\\Sigma )_n\\Vert _\\infty \\le \\eta \\Vert {A}\\Vert _\\infty \\tau _{{w},\\infty }$ , we have $\\Vert {Z}_n^{\\prime }\\Vert _\\infty \\le \\eta \\Vert {A}\\Vert _\\infty \\tau _{{w},\\infty }$ $\\Pr _{{W}^\\rho }[Z_{n,i}^{\\prime }\\ne 0]\\le \\tilde{O}(\\frac{\\eta \\Vert {A}\\Vert _\\infty \\tau _{{w},\\infty }}{\\sigma _w}),\\ i\\in [m_1]$ Then we have $\\Pr [\\Vert {Z}_n^{\\prime }\\Vert _0\\ge 2]\\le O_p(\\eta ^2)$ Then we only need to consider the case $\\Vert {Z}_n^{\\prime }\\Vert _0=1$ .", "Let $Z_{n,n_i}^{\\prime }\\ne 0$ .", "Then the first term in (REF ), ${q}^\\top {A}({Z}^{\\prime }({V}+{V}^\\rho +\\eta \\Sigma {V}^{\\prime \\prime })\\odot {D}_{{v},\\rho }){c}_r$ should be dealt with separately.", "The term ${q}^\\top {A}({Z}^{\\prime }\\eta \\Sigma {V}^{\\prime \\prime }\\odot {D}_{{v},\\rho }){c}_r)$ contributes to $O_p(\\eta ^3)$ to the whole term.", "Then we have $\\Vert {q}^\\top {A}({Z}^{\\prime }\\eta ({V}+{V}^\\rho )\\odot {D}_{{v},\\rho }){c}_r\\Vert \\le \\tilde{O}(\\eta \\Vert \\Vert {q}^\\top {A}\\Vert _1\\Vert \\Vert {A}\\Vert _\\infty \\tau _{{w},\\infty })$ We also have that $\\tilde{O}((\\frac{\\eta \\Vert {A}\\Vert _\\infty \\tau _{{w},\\infty }}{\\sigma _{w}}m_1)^N)\\le \\tilde{O}(\\frac{\\eta \\Vert {A}\\Vert _\\infty \\tau _{{w},\\infty }}{\\sigma _{w}}m_1)\\le 1$ Therefore, the contribution to the first term is $\\tilde{O}(\\eta ^2\\Vert {q}^\\top {A}\\Vert _1\\Vert {A}\\Vert _\\infty ^2\\frac{\\tau ^2_{{w},\\infty }}{\\sigma _w}m_1)+O_p(\\eta ^3)$ .", "Denote $\\begin{aligned}\\delta =&{A}({A}{X}({W}+{W}^\\rho +\\eta {W}^{\\prime \\prime }\\Sigma )+{B}_1)\\odot {D}_{{w}, \\rho ,\\eta }({V}+{V}^\\rho +\\eta \\Sigma {V}^{\\prime \\prime })\\\\&-{A}({A}{X}({W}+{W}^\\rho )+{B}_1)\\odot {D}_{{w}, \\rho }({V}+{V}^\\rho )\\end{aligned}$ $\\delta \\in \\mathbb {R}^{m_2}$ has the following terms: 1.", "${Z}^{\\prime }({V}+{V}^\\rho +\\eta \\Sigma {V}^{\\prime \\prime })$ .", "We have its n-th row norm bounded by $O_p(\\eta )$ .", "2.", "${Z}\\eta \\Sigma {V}^{\\prime \\prime }$ .", "We have its n-th row infinity norm bounded by $\\tilde{O}(\\Vert {A}\\Vert _\\infty \\eta \\tau _{v,\\infty }m_1^{-\\frac{1}{2}})$ .", "3.", "${A}({A}{X}\\eta {W}^{\\prime \\prime }\\Sigma \\odot {D}_{{w},\\rho })({V}+{V}^\\rho )$ , of which the n-th row infinity is bounded by $\\tilde{O}(\\Vert {A}\\Vert _\\infty \\eta \\tau _{{w},\\infty })$ .", "4.", "$ {A}({A}{X}\\eta ^2{W}^{\\prime \\prime }\\Sigma \\odot {D}_{{w},\\rho ,\\eta }\\Sigma {V}^{\\prime \\prime })$ .", "Bounded by $O_p(\\eta ^2)$ .", "Therefore, $\\Vert \\delta _n\\Vert _\\infty \\le \\tilde{O}(\\Vert {A}\\Vert _\\infty \\eta (\\tau _{{v},\\infty }m_1^{-\\frac{1}{2}}+\\tau _{{w},\\infty }))+O_p(\\eta ^2)$ Similarly, we can derive that the contribution to the second term is $\\tilde{O}(\\eta ^2\\Vert {q}^\\top {A}\\Vert _1\\Vert {A}\\Vert _\\infty ^2\\frac{(\\tau ^2_{{w},\\infty }+\\tau ^2_{{v},\\infty }m_1^{-1})}{\\sigma _v}m_2)+O_p(\\eta ^3)$ .", "Lemma B.12 Let ${F}^*_{{A}}=(f_1^*,\\cdots ,f_K^*)$ .", "Perturbation matrices ${W}^{\\prime }$ , ${V}^{\\prime }$ satisfy $\\Vert {W}^{\\prime }\\Vert _{2,4}\\le \\tau _w,\\ \\ \\Vert {V}^{\\prime }\\Vert _F\\le \\tau _v$ There exists $\\widehat{{W}}$ and $\\widehat{{V}}$ such that $\\Vert \\widehat{{W}}\\Vert _{2, \\infty }\\le \\frac{C_0}{m_1},\\ \\ \\Vert \\widehat{{V}}\\Vert _{2,\\infty }\\le \\frac{K\\sqrt{m_1}}{m_2}$ $\\mathbb {E}[\\sum _{r=1}^{K}|f_r^*({q},{A},{X},\\widehat{{W}}, \\widehat{{V}})-g_r^{(b,b)}({q},{A},{X},\\widehat{{W}}, \\widehat{{V}})|]\\le \\epsilon $ $\\mathbb {E}[G^{(b,b)}({q},{A},{X},\\widehat{{W}}, \\widehat{{V}})]\\le OPT+\\epsilon $ Proof: By Lemma REF , we have $\\Vert {D}_{n,{w}}^{\\prime }\\Vert _0\\le \\tilde{O}(\\tau _w^\\frac{4}{5}m_1^\\frac{6}{5})\\ll \\tilde{O}(m_1)$ $\\Vert {D}_{n,{v}}^{\\prime }\\Vert _0\\le \\tilde{O}(m_2^\\frac{3}{2}\\sigma _v(\\Vert {A}\\Vert _\\infty ^2+\\Vert {A}\\Vert _\\infty ^2\\tau _w m_1^\\frac{1}{4})+m_2\\Vert {A}\\Vert _\\infty ^\\frac{2}{3}(\\Vert {A}\\Vert _\\infty \\tau _v+\\Vert {A}\\Vert _\\infty \\tau _w m_1^\\frac{1}{4}(1+\\tau _v))^\\frac{2}{3})\\le \\tilde{O}(m_2\\Vert {A}\\Vert _\\infty ^2(\\epsilon /C_0)^{\\Theta (1)})$ Applying Lemma REF , we know $\\begin{aligned}&{q}^\\top {A}(({A}({A}{X}\\widehat{{W}})\\odot ({D}_{w}^{\\prime })\\widehat{{V}})\\odot ({D}_{{v}})){c}_r\\\\=& \\sum _{n=1}^N {q}^\\top {a}_n\\sum _{k=1}^N a_{n,k} (({a}_k{X}\\widehat{{W}})\\odot {{D}_{k, {w}}}^{\\prime })\\widehat{{V}}\\odot {{D}_{n,{v}}}{c}_r\\\\\\le & \\Vert {q}^\\top {A}\\Vert _1\\Vert {A}\\Vert _\\infty ^2 m_1^\\frac{3}{10} \\frac{C_0}{m_1}\\frac{K\\sqrt{m_1}}{m_2}\\cdot m_2\\\\\\le & \\epsilon \\end{aligned}$ $\\begin{aligned}&{q}^\\top {A}(({A}({A}{X}\\widehat{{W}})\\odot ({D}_{w})\\widehat{{V}})\\odot ({D}_{{v}}^{\\prime })){c}_r\\\\=& \\sum _{n=1}^N {q}^\\top {a}_n\\sum _{k=1}^N a_{n,k} (({a}_k{X}\\widehat{{W}})\\odot {{D}_{k, {w}}})\\widehat{{V}}\\odot {{D}_{n,{v}}^{\\prime }}{c}_r\\\\\\le & \\Vert {q}^\\top {A}\\Vert _1\\Vert {A}\\Vert _\\infty ^2 m_1^\\frac{1}{2} \\frac{C_0}{m_1}\\frac{K\\sqrt{m_1}}{m_2}\\cdot m_2\\cdot \\Vert {A}\\Vert _\\infty ^2(\\frac{\\epsilon }{C_0})^{\\Theta (1)}\\\\\\le & \\epsilon \\end{aligned}$ Then, the conclusion can be derived." ], [ "Optimization", "This section states the optimization process and convergence performance of the algorithm.", "Lemma REF shows that during the optimization, either there exists an updating direction that decreases the objective, or weight decay decreases the objective.", "Lemma REF provides the convergence result of the algorithm.", "Define $\\begin{aligned}&L^{\\prime }({A}^*, {A}^*, {A}^*, \\lambda _t,{W}_t,{V}_t)\\\\=&\\frac{1}{\\Omega ^t}\\sum _{i=1}^{|\\Omega ^t|}\\mathbb {E}_{{W}^\\rho ,{V}^\\rho , \\Sigma ^{\\prime }}[L(\\lambda _t F_{{A}^*}({e}_g,{X}; {W}^{(0)}+{W}^\\rho +{W}_t\\Sigma ^{\\prime }, {V}^{(0)}+{V}^\\rho +\\Sigma ^{\\prime }{V}_t),y_i)]+R(\\sqrt{\\lambda _t}{W}_t,\\sqrt{\\lambda _t}{V}_t)\\end{aligned}$ where $R(\\sqrt{\\lambda }{W}_t, \\sqrt{\\lambda }{V}_t)=\\lambda _v\\Vert \\sqrt{\\lambda }{V}_t\\Vert _F^2+\\lambda _w\\Vert \\sqrt{\\lambda }{W}_t\\Vert _{2,4}^2$ Lemma B.13 For every $\\epsilon _0\\in (0,1)$ and $\\epsilon \\in (0,\\frac{\\epsilon _0}{K\\Vert {A}\\Vert _\\infty p_1p_2^2\\mathcal {C}_s(\\Phi , p_2\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty ))\\mathcal {C}_s(\\phi ,\\Vert {A}\\Vert _\\infty )\\sqrt{\\Vert {A}\\Vert _\\infty ^2+1}})$ and $\\gamma \\in (0,\\frac{1}{4}]$ , consider any ${W}_t,{V}_t$ with $L^{\\prime }({A}^*, {A}^*,{A}^*, \\lambda _t,{W}_t,{V}_t)\\in [(1+\\gamma )OPT+\\Omega ({q}^\\top {A}^*1\\Vert {A}^*\\Vert _\\infty ^4\\epsilon _0/\\gamma ),\\tilde{O}(1)]$ With high probability on random initialization, there exists $\\widehat{{W}}$ , $\\widehat{{V}}$ with $\\Vert \\widehat{{W}}\\Vert _F\\le 1$ , $\\Vert \\widehat{{V}}\\Vert _F\\le 1$ such that for every $\\eta \\in (0,\\frac{1}{\\text{poly}(m_1,m_2)}]$ , $\\begin{aligned}&\\min \\lbrace \\mathbb {E}_{\\Sigma }[L^{\\prime }({A}^*,{A}^*,{A}^*, \\lambda _t, {W}_t+\\sqrt{\\eta }\\widehat{{W}}\\Sigma , {V}_t+\\sqrt{\\eta }\\Sigma \\widehat{{V}})], L^{\\prime }({A}^*,{A}^*, {A}^*, (1-\\eta )\\lambda _t,{W}_t,{V}_t)\\rbrace \\\\\\le & (1-\\eta \\gamma /4)L^{\\prime }({A}^*, {A}^*, {A}^*, \\lambda _t,{W}_t,{V}_t)\\end{aligned}$ Proof: Recall the pseudo network and the real network for every $r\\in [K]$ as $g_r({q},{A}^*, {X},{W}^{\\prime },{V}^{\\prime })={q}^\\top {A}^*({A}^*({A}^*{X}({W}^{(0)}+{W}^\\rho +{W}^{\\prime })+{B}_1)\\odot {D}_{{w}, \\rho ,t}({V}^{(0)}+{V}^\\rho +{V}^{\\prime })\\odot {D}_{{v},\\rho ,t}){c}_r$ $f_r({q},{A}^*, {X},{W}^{\\prime },{V}^{\\prime })={q}^\\top {A}^*({A}^*({A}^*{X}({W}^{(0)}+{W}^\\rho +{W}^{\\prime })+{B}_1)\\odot {D}_{{w}, \\rho ,{W}^{\\prime }}({V}^{(0)}+{V}^\\rho +{V}^{\\prime })\\odot {D}_{{v},\\rho ,{V}^{\\prime }}){c}_r$ where ${D}_{{w}, \\rho ,t}$ and ${D}_{{v}, \\rho ,t}$ are the diagonal matrices at weights ${W}^{(0)}+{W}^\\rho +{W}_t$ and ${V}^{(0)}+{V}^\\rho +{V}_t$ .", "${D}_{{w}, \\rho ,{W}^{\\prime }}$ and ${D}_{{v}, \\rho ,{V}^{\\prime }}$ are the diagonal matrices at weights ${W}^{(0)}+{W}^\\rho +{W}^{\\prime }$ and ${V}^{(0)}+{V}^\\rho +{V}^{\\prime }$ .", "Denote $G({q},{A}^*,{X}, {W}^{\\prime },{V}^{\\prime })=(g_1,\\cdots , g_K)$ , $F_{{A}^*}({q},{X}, {W}^{\\prime },{V}^{\\prime })=(f_1,\\cdots ,f_K)$ .", "As long as $L^{\\prime }({A}^*, {A}^*, {A}^*, \\lambda _t,{W}_t,{V}_t)\\le \\tilde{O}(1)$ , according to C.32 to C.34 in [1], we have $\\lambda _w\\Vert \\sqrt{\\lambda _t}\\widehat{{W}}\\Vert _{2,4}^4\\le \\epsilon _0$ $\\lambda _v\\Vert \\sqrt{\\lambda _t}\\widehat{{V}}\\Vert _{F}^2\\le \\epsilon _0$ $\\Vert \\widehat{{W}}\\Vert _F\\ll 1$ $\\Vert \\widehat{{V}}\\Vert _F\\ll 1$ The we need to study an update direction $\\widetilde{{W}}={W}_t+\\sqrt{\\eta }\\widehat{{W}}\\Sigma $ $\\widetilde{{V}}={V}_t+\\sqrt{\\eta }\\Sigma \\widehat{{V}}$ Changes in Regularizer.", "Note that here ${W}_t\\in \\mathbb {R}^{d\\times m_1}$ , ${V}_t\\in \\mathbb {R}^{m_1\\times m_2}$ , $\\Sigma \\in \\mathbb {R}^{m_1\\times m_1}$ .", "We know that $\\mathbb {E}_{\\Sigma }[\\Vert {V}_t+\\sqrt{\\eta }\\Sigma \\widehat{{V}}\\Vert _F^2]=\\Vert {V}_t\\Vert _F^2+\\eta \\Vert \\widehat{{V}}\\Vert _F^2$ $\\mathbb {E}_{\\Sigma }[\\Vert {W}_t+\\sqrt{\\eta }\\widehat{{W}}\\Sigma \\Vert _{2,4}^4]=\\sum _{i\\in [m_1]}\\mathbb {E}[\\Vert {w}_{t,i}+\\sqrt{\\eta }\\widehat{{W}}\\Sigma _i\\Vert _2^4]$ For each term $i\\in [m_1]$ , we can bound $\\Vert {w}_{t,i}+\\sqrt{\\eta }\\widehat{{W}}\\Sigma _i\\Vert _2^2=\\Vert {w}_{t,i}\\Vert _2^2+\\eta \\Vert \\widehat{{W}}\\Sigma _i\\Vert _2^2+2\\sqrt{\\eta }{{w}_{t,i}}^\\top \\widehat{{W}}\\Sigma _i$ $\\begin{aligned}\\Vert {w}_{t,i}+\\sqrt{\\eta }\\widehat{{W}}\\Sigma _i\\Vert _2^4&=\\Vert {w}_{t,i}\\Vert _2^4+\\eta ^2\\Vert \\widehat{{W}}\\Sigma _i\\Vert _2^4+4\\eta \\Vert {{w}_{t,i}}^\\top \\widehat{{W}}\\Sigma _i\\Vert ^2+2\\eta \\Vert {w}_{t,i}\\Vert _2^2\\Vert \\widehat{{W}}\\Sigma _i\\Vert _2^2\\\\&\\le \\Vert {w}_{t,i}\\Vert _2^4+6\\eta \\Vert {w}_{t,i}\\Vert _2^2\\Vert \\widehat{{W}}\\Sigma _i\\Vert _2^2+O_p(\\eta ^2)\\end{aligned}$ Therefore, by Cauchy-Schwarz inequality, we have $\\mathbb {E}_{\\Sigma }[\\Vert {W}_t+\\sqrt{\\eta }\\widehat{{W}}\\Sigma \\Vert _{2,4}^4]\\le \\Vert {W}_t\\Vert _{2,4}^4+6\\eta \\Vert {W}_t\\Vert _{2,4}^2\\Vert \\widehat{{W}}\\Vert _{2,4}^2+O_p(\\eta ^2)$ Therefore, by $\\lambda _w\\Vert \\sqrt{\\lambda _t}{W}_t\\Vert _{2,4}^4\\le R(\\sqrt{\\lambda _t}{W}_t,\\sqrt{\\lambda _t}{V}_t)$ , we have $\\begin{aligned}\\mathbb {E}[R(\\sqrt{\\lambda _t}\\widetilde{{W}},\\sqrt{\\lambda _t}\\widetilde{{V}})]\\le & R(\\sqrt{\\lambda _t}{W}_t,\\sqrt{\\lambda _t}{V}_t)+6\\eta \\sqrt{\\epsilon _0}\\sqrt{R(\\sqrt{\\lambda _t}{W}_t,\\sqrt{\\lambda _t}{V}_t)}+\\eta \\epsilon _0\\\\\\le & R(\\sqrt{\\lambda _t}{W}_t,\\sqrt{\\lambda _t}{V}_t)+\\frac{1}{4}\\eta R(\\sqrt{\\lambda _t}{W}_t,\\sqrt{\\lambda _t}{V}_t)+143\\eta \\epsilon _0\\end{aligned}$ Changes in Objective.", "Recall that here $\\widehat{{W}}$ and $\\widehat{{V}}$ satisfy $\\tau _{{w},\\infty }\\le \\frac{1}{m_1^{\\frac{999}{1000}}}$ and $\\tau _{{v},\\infty }\\le \\frac{1}{m_2^\\frac{999}{2000}}$ .", "By Lemma REF , we have for every $r\\in [K]$ $\\begin{aligned}&\\mathbb {E}_{{W}^\\rho ,{V}^\\rho }[|f_r({q},{A}^*,{X},{W}+{W}^\\rho +\\widetilde{{W}}\\Sigma , {V}+{V}^\\rho +\\Sigma ^{\\prime }\\widetilde{{V}})-g_r({q},{A}^*,{X},{W}+{W}^\\rho +\\widetilde{{W}}\\Sigma , {V}+{V}^\\rho +\\Sigma \\widetilde{{V}})|]\\\\&\\le \\tilde{O}(\\Vert {q}^\\top {A}^*\\Vert _1\\Vert {A}^*\\Vert _\\infty ^2\\epsilon _0\\eta )+O_p(\\eta ^{1.5})\\end{aligned}$ By Lemma REF , we have $\\begin{aligned}G({q},{A}^*,{X},\\widetilde{{W}}, \\widetilde{{V}})&= G({q},{A}^*,{X},{W}_t, {V}_t)+\\eta G^{(b,b)}({q},{A}^*,{X},\\widehat{{W}}\\Sigma ,\\Sigma \\widehat{{V}})+\\sqrt{\\eta } G^{\\prime }\\\\&=F_{{A}^*}({q},{X},{W}_t, {V}_t)+\\eta G^{(b,b)}({q},{A}^*,{X},\\widehat{{W}}\\Sigma ,\\Sigma \\widehat{{V}})+\\sqrt{\\eta } G^{\\prime }\\\\&=F_{{A}^*}({q},{X},{W}_t, {V}_t)+\\eta G^{(b,b)}({q},{A}^*,{X},\\widehat{{W}},\\widehat{{V}})+\\sqrt{\\eta } G^{\\prime }\\end{aligned}$ where $\\mathbb {E}_{\\Sigma }[G^{\\prime }]=0$ and $|G^{\\prime }|\\le \\epsilon $ with high probability.", "By C.38 in [1], we have $\\begin{aligned}&\\mathbb {E}_{{W}^\\rho ,{V}^\\rho ,\\Sigma }[L(\\lambda _t F_{{A}^*}({q},{X},\\widetilde{{W}}, \\widetilde{{V}}),y)]\\\\\\le & \\mathbb {E}_{{W}^\\rho ,{V}^\\rho }[L(\\lambda _t F_{{A}^*}({q},{X},\\widetilde{{W}}, \\widetilde{{V}})+\\eta F^*_{{A}^*}({q},{A}^*,{X},\\widetilde{{W}}, \\widetilde{{V}}),y)]+O(\\Vert {q}^\\top {A}^*\\Vert _1\\Vert {A}^*\\Vert _\\infty ^2\\epsilon _0\\eta )+O_p(\\eta ^{1.5})\\end{aligned}$ Following C.40 in [1], we have $\\begin{aligned}&\\mathbb {E}_{{W}^\\rho ,{V}^\\rho }[L(\\lambda _t F_{{A}^*}({q},{X},{W}_t, {V}_t)+\\eta F^*_{{A}^*}({q},{A}^*,{X},{W}_t, {V}_t),y)]\\\\\\le & (1-\\eta )(2L(\\lambda _t F_{{A}^*}({q},{X},{W}_t, {V}_t),y)-L((1-\\eta )\\lambda _t F_{{A}^*}({q},{X},{W}_t, {V}_t),y))+\\eta L( F^*_{{A}^*}, y)+O_p(\\eta ^2)\\end{aligned}$ Putting all of them together.", "Denote $c_1=\\frac{1}{|\\Omega ^t|}\\sum _{i=1}^{|\\Omega |}\\mathbb {E}_{{W}^\\rho ,{V}^\\rho ,\\Sigma ,\\Sigma ^{\\prime }}[L(\\lambda _t F_{{A}^*}({e}_g,{X},{W}^{(0)}+{W}^\\rho +\\widetilde{{W}}\\Sigma ^{\\prime },{V}^{(0)}+{V}^\\rho +\\Sigma ^{\\prime }\\widetilde{{V}}),y_i)]$ $c_1^{\\prime }=\\mathbb {E}_{\\Sigma }[L^{\\prime }({A}^*,{A}^*,{A}^*, \\lambda _t,\\widetilde{{W}},+\\widetilde{{V}})]=c_1+\\mathbb {E}_{\\Sigma }[R(\\sqrt{\\lambda _t} \\widetilde{{W}},\\sqrt{\\lambda }\\widetilde{{V}})]$ $c_2=\\frac{1}{|\\Omega ^t|}\\sum _{i=1}^{|\\Omega |}\\mathbb {E}_{{W}^\\rho ,{V}^\\rho }[L((1-\\eta )\\lambda _t F_{{A}^*}({e}_g,{X},{W}^{(0)}+{W}^\\rho +{W}_t\\Sigma ^{\\prime },{V}^{(0)}+{V}^\\rho +\\Sigma ^{\\prime }{V}_t),y_i)]$ $c_2^{\\prime }=L^{\\prime }({A}^*,{A}^*,{A}^*, (1-\\eta )\\lambda _t,{W}_t,{V}_t)=c_2+R(\\sqrt{(1-\\eta )\\lambda _t} {W}_t,\\sqrt{(1-\\eta )\\lambda _t}{V}_t)$ $c_3=\\frac{1}{|\\Omega ^t|}\\sum _{i=1}^{|\\Omega |}\\mathbb {E}_{{W}^\\rho ,{V}^\\rho }[L(\\lambda _t F_{{A}^*}({e}_g,{X},{W}^{(0)}+{W}^\\rho +{W}_t\\Sigma ^{\\prime },{V}^{(0)}+{V}^\\rho +\\Sigma ^{\\prime }{V}_t),y_i)]$ $c_3^{\\prime }=L^{\\prime }({A}^*, {A}^*,{A}^*, \\lambda _t,{W}_t,{V}_t)=c_3+R(\\sqrt{\\lambda _t} {W}_t,\\sqrt{\\lambda }{V}_t)$ Then following from C.38 to C.42 in [1], we have $c_1^{\\prime }\\le (1-\\eta )(2c_3^{\\prime }-c_2^{\\prime })+\\frac{\\eta \\gamma }{4}c_3^{\\prime }+\\eta (OPT+O(\\Vert {q}^\\top {A}^*\\Vert _1\\Vert {A}^*\\Vert _\\infty ^4\\epsilon _0/\\gamma ))+O_p(\\eta ^{1.5}),$ which implies $\\min \\lbrace c_1^{\\prime },c_2^{\\prime }\\rbrace \\le (1-\\eta \\frac{1}{2}+\\frac{\\eta \\gamma }{8})c_3^{\\prime }+\\eta \\frac{1}{2}OPT+O(\\Vert {q}^\\top {A}^*\\Vert _1\\Vert {A}^*\\Vert _\\infty ^2\\epsilon _0\\eta /\\gamma )+O_p(\\eta ^{1.5})$ As long as $c_3^{\\prime }\\ge (1+\\gamma )OPT+\\Omega (\\Vert {q}^\\top {A}^*\\Vert _1\\Vert {A}^*\\Vert _\\infty ^2\\epsilon _0/\\gamma )$ and $\\gamma \\in [0,1]$ , we have $\\min \\lbrace c_1^{\\prime },c_2^{\\prime }\\rbrace \\le (1-\\eta \\frac{\\gamma }{4})c_3^{\\prime }$ Lemma B.14 Note that the three sampled aggregation matrices in a three-layer learner network can be be different.", "We denote them as ${{A}^{t(1)}}$ , ${A}^{t(2)}$ and ${{A}^{t(3)}}$ .", "Let ${W}_t$ , ${V}_t$ be the updated weights trained using ${A}^*$ and let ${W}_t^{\\prime }$ , ${V}_t^{\\prime }$ be the updated weights trained using ${A}^{t(i)},\\ i\\in [3]$ .", "With probability at least $99/100$ , the algorithm converges in $TT_w=\\text{poly}(m_1,m_2)$ iterations to a point with $\\eta \\in (0,\\frac{1}{\\text{poly}(m_1, m_2, \\Vert {A}^*\\Vert _\\infty ,K)})$ $L^{\\prime }({A}^*, {A}^*, {A}^*, \\lambda _t,{W}_t,{V}_t)\\le (1+\\gamma )OPT+\\epsilon _0$ If $\\begin{aligned}&L^{\\prime }({A}^{t(1)},{A}^{t(2)}, {A}^{t(3)}, \\lambda _t,{W}_t^{\\prime },{V}_t^{\\prime })\\\\=&\\frac{1}{|\\Omega ^t|}\\sum _{i=1}^{|\\Omega ^t|}\\mathbb {E}_{{W}^\\rho ,{V}^\\rho , \\Sigma }[L(\\lambda _t F_{{A}^{(1)},{A}^{(2)},{A}^{(3)}}({q}, {X}_i, {W}^{(0)}+{W}^\\rho +{W}_t^{\\prime }\\Sigma ^{\\prime }, {V}^{(0)}+{V}^\\rho +\\Sigma ^{\\prime }{V}_t^{\\prime }),y_i)]\\\\&+R(\\sqrt{\\lambda _t}{W}_t^{\\prime },\\sqrt{\\lambda _t}{V}_t^{\\prime }),\\end{aligned}$ where $F_{{A}^{t(1)}, {A}^{t(2)},{A}^{t(3)}}({q},{X},{W},{V})={q}^\\top {A}^{t(3)}\\sigma ({A}^{t(2)}\\sigma ({A}^{t(1)}{X}{W}+{B}_1){V}+{B}_2){C},$ we also have $L^{\\prime }({A}^{t(1)},{A}^{t(2)},{A}^{t(3)}, \\lambda _{T-1}, {W}_T^{\\prime }, {V}_T^{\\prime })\\le L^{\\prime }({A}^*, {A}^*, {A}^*, \\lambda _T, {W}_T,{V}_T)+\\lambda _{T-1}\\cdot O(\\text{poly}(\\epsilon ))\\le (1+\\gamma )OPT+\\epsilon _0$ Proof: By Lemma REF , we know that as long as $L^{\\prime }({A}^*, {A}^*,{A}^*, \\lambda _t,{W}_t,{V}_t)\\in [(1+\\gamma )OPT+\\Omega ({q}^\\top {A}^*1\\Vert {A}^*\\Vert _\\infty ^4\\epsilon _0/\\gamma ),\\tilde{O}(1)]$ , then there exists $\\Vert \\widehat{{W}}\\Vert _F\\le 1$ , $\\Vert \\widehat{{V}}\\Vert _F\\le 1$ such that either $\\mathbb {E}_{\\Sigma ,\\Sigma ^{\\prime }}[L^{\\prime }({A}^*, {A}^*, {A}^*, \\lambda _t, {W}_t\\Sigma ^{\\prime }+\\sqrt{\\eta }\\widehat{{W}}\\Sigma \\Sigma ^{\\prime }, \\Sigma ^{\\prime }{V}_t+\\sqrt{\\eta }\\Sigma ^{\\prime }\\Sigma \\widehat{{V}})]\\le (1-\\eta \\gamma /4)L^{\\prime }({A}^*, {A}^*, {A}^*, \\lambda _t,{W}_t,{V}_t)$ or $L^{\\prime }({A}^*, {A}^*, {A}^*, (1-\\eta )\\lambda _t,{W}_t,{V}_t)\\le (1-\\eta \\gamma /4)L^{\\prime }({A}^*, {A}^*, {A}^*, \\lambda _t,{W}_t,{V}_t)$ Denote ${W}={W}^{(0)}+{W}^\\rho +{W}_t\\Sigma ^{\\prime }+\\sqrt{\\eta }\\widehat{{W}}\\Sigma \\Sigma ^{\\prime }$ , ${V}={V}^{(0)}+{V}^\\rho +\\Sigma ^{\\prime }{V}_t+\\sqrt{\\eta }\\Sigma ^{\\prime }\\Sigma \\widehat{{V}}$ .", "Note that $\\frac{\\partial L}{\\partial {w}_j}=\\sum _{i=1}^K \\frac{\\partial L}{\\partial f_i}\\frac{\\partial f_i}{\\partial {w}_j}$ $\\begin{aligned}&\\frac{\\partial }{\\partial {w}_j}f_r({q},{A}^*,{X},{W}^{(0)}+{W}^\\rho +{W}_t\\Sigma ^{\\prime }+\\sqrt{\\eta }\\widehat{{W}}\\Sigma \\Sigma ^{\\prime },{V}^{(0)}+{V}^\\rho +\\Sigma ^{\\prime }{V}_t+\\sqrt{\\eta }\\Sigma ^{\\prime }\\Sigma \\widehat{{V}} )\\\\=& \\sum _{n=1}^N {q}^\\top {a}_n\\sum _{i=1}^{m_2}c_{i,r}\\mathbb {1}_{r_{n,i}+B_{2(n,i)}\\ge 0}\\sum _{k=1}^N a_{n,k}v_{j,i}\\mathbb {1}_{\\tilde{{a}^*}_{n}{X}{w}_k+B_{1(n,k)}\\ge 0}(\\tilde{{a}^*}_{n}{X})^\\top ,\\end{aligned}$ which implies $\\frac{\\partial F}{\\partial {w}_t}$ , $\\frac{\\partial ^2 F}{\\partial {w}_t^2}$ , $\\frac{\\partial ^3 F}{\\partial {w}_t^3}$ are summations of $\\mathbb {1}$ , $\\delta $ , $\\delta ^{\\prime }$ functions and their multiplications.", "It can be found that no $\\delta (x)\\delta ^{\\prime }(x)$ , $\\delta (x)^2$ or $\\delta ^{\\prime 2}(x)$ exist in these terms.", "Therefore, by $\\int _{-\\infty }^{\\infty }\\delta (t)f(t)dt=f(0)$ and $\\int _{-\\infty }^{\\infty }\\delta ^{\\prime }(t)f(t)dt=-f^{\\prime }(0)$ , we can obtain that the value of the third-order derivative w.r.t.", "${W}^\\rho $ of $\\mathbb {E}_{{W}^\\rho ,{V}^\\rho , \\Sigma }[L(\\lambda _t F_{{A}^*}({e}_g, {X}, {W}^{(0)}+{W}^\\rho +{W}_t\\Sigma , {V}^{(0)}+{V}^\\rho +\\Sigma {V}_t),y)]$ is proportional to $\\text{poly}(\\Vert {A}^*\\Vert _\\infty ,K)$ , some certain value of the probability density function of ${W}^\\rho $ and its derivative, i.e., $\\text{poly}(\\sigma _w^{-1})$ .", "Similarly, the value of the third-order derivative w.r.t.", "${W}^\\rho $ of $\\mathbb {E}_{{W}^\\rho ,{V}^\\rho , \\Sigma }[L(\\lambda _t F_{{A}^*}({e}_g, {X}, {W}^{(0)}+{W}^\\rho +{W}_t\\Sigma , {V}^{(0)}+{V}^\\rho +\\Sigma {V}_t),y)]$ is polynomially depend on $\\sigma _v^{-1}$ and $\\Vert {A}^*\\Vert _\\infty $ .", "By the value selection of $\\sigma _w$ and $\\sigma _v$ , we can conclude that $L^{\\prime }$ is $B=\\text{poly}(m_1,m_2, \\Vert {A}^*\\Vert _\\infty ,K)$ second-order smooth.", "By Fact A.8 in [1], it satisfies with $\\eta \\in (0,\\frac{1}{\\text{poly}(m_1,m_2,\\Vert {A}^*\\Vert _\\infty ,K)})$ $\\lambda _{\\min }(\\nabla ^2 L^{\\prime }({A}^*,{A}^*, {A}^*, \\lambda _{t-1}, {W}_t,{V}_t))<-\\frac{1}{(m_1m_2)^8}$ Meanwhile, for $t\\ge 1$ , by the escape saddle point theorem of Lemma A.9 in [1], we know with probability at least $1-p$ , $ \\lambda _{\\min }(\\nabla ^2 L^{\\prime }({A}^*,{A}^*, {A}^*, \\lambda _{t-1}, {W}_t,{V}_t))>-\\frac{1}{(m_1m_2)^8}$ holds.", "Choosing $p=\\frac{1}{100T}$ , then this holds for $t=1,2,\\cdots ,T$ with probability at least 0.999.Therefore, for $t=1,2,\\cdots ,T$ , the first case cannot happen, i.e., as long as $L^{\\prime }({A}^*, {A}^*, {A}^*, \\lambda _t,{W}_t,{V}_t)\\ge (1+\\gamma )OPT+\\Omega ({q}^\\top {A}^*1\\Vert {A}^*\\Vert _\\infty ^4\\epsilon _0/\\gamma )$ , $L^{\\prime }({A}^*,{A}^*, {A}^*,(1-\\eta )\\lambda _t,{W}_t,{V}_t)\\le (1-\\eta \\gamma /4)L^{\\prime }({A}^*, {A}^*, {A}^*, \\lambda _t,{W}_t,{V}_t)$ On the other hand, for $t=1,2,\\cdots ,T-1$ , as long as $L^{\\prime }\\le \\tilde{O}(1)$ , by Lemma A.9 in [1], we have $L^{\\prime }({A}^*, {A}^*, {A}^*, \\lambda _t,{W}_{t+1},{V}_{t+1})\\le L^{\\prime }({A}^*, {A}^*, {A}^*, \\lambda _t,{W}_t,{V}_t)+(m_1m_2)^{-1}$ By $L^{\\prime }({A}^*,{A}^*,{A}^*, \\lambda _1,{W}_0,{V}_0)\\le \\tilde{O}(1)$ with high probability, we have $L^{\\prime }({A}^*,{A}^*,{A}^*, \\lambda _t,{W}_t,,{V}_t)\\le \\tilde{O}(1)$ with high probability for $t=1,2,\\cdots ,T$ .", "Therefore, after $T=\\tilde{\\Theta }(\\eta ^{-1}\\log \\frac{\\log m}{\\epsilon _0})$ rounds of weight decay, we have $L^{\\prime }({A}^*, {A}^*, {A}^*, \\lambda _t,{W}_t,{V}_t)\\le (1+\\gamma )OPT+\\Omega ({q}^\\top {A}^*1\\Vert {A}^*\\Vert _\\infty ^4\\epsilon _0/\\gamma )$ .", "Rescale down $\\epsilon _0$ and we can obtain our final result.", "Consider $L^{\\prime }({A}^{t(1)},{A}^{t(2)},{A}^{t(3)}, \\lambda _t,{W}_t^{\\prime },{V}_t^{\\prime })$ .", "Let ${w}_i$ , ${v}_i$ be the output weights updated with all the aggregation matrices equal to ${A}^*$ , and let ${w}_i^{\\prime }$ , ${v}_i^{\\prime }$ be the output weights updated with our sampling strategy in Section REF .", "We know that $\\begin{aligned}\\Vert {w}_i-{{w}_i}^{\\prime }\\Vert &\\lesssim \\sum _{t=0}^{T-1} \\Vert \\eta \\sum _{l=0}^{T_w-1}\\sum _{n=1}^N {q}^\\top {{a}_n}^* \\sum _{i=1}^{m_2}c_{i,r}\\mathbb {1}[{{a}^*}_n\\sigma ({A}^*{X}{W}){v}_i\\ge 0]\\sum _{k=1}^N {a_{n,k}}^*v_{j,i}\\mathbb {1}[{{a}^*}_k{X}{w}_j\\ge 0]({{a}^*}_k-{{a}_k}^{t(1)}){X}\\Vert \\\\& \\le \\frac{1}{\\text{poly}(m_1, m_2)}\\cdot \\frac{1}{\\text{poly}(\\epsilon )}\\text{poly}(m_1,m_2)\\epsilon _c\\Vert {A}^*\\Vert _\\infty \\cdot \\text{poly}(\\epsilon )=O(\\epsilon )\\end{aligned}$ $\\begin{aligned}\\Vert {v}_i-{v}_i^{\\prime }\\Vert &\\lesssim \\sum _{t=0}^{T-1}\\Vert \\eta \\sum _{l=0}^{T_w-1}\\sum _{n=1}^N {q}^\\top {a}_n^* \\sum _{i=1}^{m_2}c_{i,r}\\mathbb {1}[{{a}^*}_n\\sigma ({A}^*{X}{W}){v}_i\\ge 0]({{a}^*}_n\\sigma ({A}^*{X}{W})-{a}_n^{t(2)}\\sigma ({A}^{t(1)}{X}{W}^{\\prime }))\\Vert \\\\& \\le \\frac{1}{\\text{poly}(m_1, m_2)}\\cdot \\frac{1}{\\text{poly}(\\epsilon )}\\text{poly}(m_1,m_2)\\epsilon _c\\Vert {A}^*\\Vert _\\infty \\cdot \\text{poly}(\\epsilon )=O(\\epsilon )\\end{aligned}$ With a slight abuse of notation, for $r\\in [K]$ , we denote $f_r({q},{A}^{t(1)}, {A}^{t(2)}, {A}^{t(3)}, {X},{W}_t^{\\prime },{V}_t^{\\prime })={q}^\\top {A}^{t(3)}\\sigma ({A}^{t(2)}\\sigma ({A}^{t(1)}{X}{W}+{B}_1){V}+{B}_2){c}_r$ The difference between $f_r({q},{A}^*,{X},{W}_t,{V}_t)$ and $f_r({q},{A}^{t(1)}, {A}^{t(2)}, {A}^{t(3)}, {X},{W}_t^{\\prime },{V}_t^{\\prime })$ is caused by $\\Vert {A}^*-{A}^{t(1)}\\Vert _\\infty $ , $\\Vert {A}^*-{A}^{t(2)}\\Vert _\\infty $ , $\\Vert {A}^*-{A}^{t(3)}\\Vert _\\infty $ , ${w}_i^{(t)}-{{w}_i^{(t)}}^{\\prime }$ and ${v}_i^{(t)}-{{v}_i^{(t)}}^{\\prime }$ .", "Following the proof in Lemma REF , we can easily obtain that if $|p_l-p_l^*|\\le p_l^*\\cdot O(\\text{poly}(\\epsilon ))$ and $l_i\\ge |\\mathcal {N}_i|/(1+\\frac{c_1\\cdot \\text{poly}(\\epsilon )}{p_l^* L \\Phi (L,i)})$ , it can be derived that $\\Vert {A}^*-{A}^{(1)}\\Vert _\\infty \\le O(\\text{poly}(\\epsilon ))$ , $\\Vert {A}^*-{A}^{(2)}\\Vert _\\infty \\le O(\\text{poly}(\\epsilon ))$ and $\\Vert {A}^*-{A}^{(3)}\\Vert _\\infty \\le O(\\text{poly}(\\epsilon ))$ .", "Then, by (REF ) and (REF ), we have $\\begin{aligned}&|{q}^\\top {A}^*\\sigma ({A}^*\\sigma ({A}^*{X}{W}){V}){c}_r-{q}^\\top {A}^*\\sigma ({A}^{(2)}\\sigma ({A}^{(1)}{X}{W}^{\\prime }){V}^{\\prime }){c}_r|\\\\\\le & \\Big |\\sum _{n=1}^N {q}^\\top {{a}^*}_n \\sum _{i=1}^{m_2}c_{i,r}|\\sigma ({{a}^*}_n\\sigma ({A}^*{X}{W}){v}_i)-\\sigma ({{a}_n^{(2)}}\\sigma ({A}^{(1)}{X}{W}^{\\prime }){v}_i^{\\prime })|\\Big |\\\\\\le & \\Big |\\sum _{n=1}^N {q}^\\top {{a}^*}_n \\sum _{i=1}^{m_2}c_{i,r}|{{a}^*}_n\\sigma ({A}^*{X}{W}){v}_i-{{a}_n^{(2)}}\\sigma ({A}^{(1)}{X}{W}^{\\prime }){v}_i^{\\prime }|\\Big |\\\\\\le & \\Big |\\sum _{n=1}^N {q}^\\top {{a}^*}_n \\sum _{i=1}^{m_2}c_{i,r}|({{{a}^*}_n}-{{a}_n^{(2)}})\\sigma ({A}^*{X}{W}){v}_i|\\Big |+\\Big |\\sum _{n=1}^N {q}^\\top {{a}^*}_n \\sum _{i=1}^{m_2}c_{i,r}|{{a}_n^{(2)}}(\\sigma ({A}^*{X}{W}){v}_i-\\sigma ({A}^{(1)}{X}{W}^{\\prime }){v}_i^{\\prime }|)\\Big |\\\\\\le & \\Big |\\sum _{n=1}^N {q}^\\top {{a}^*}_n \\sum _{i=1}^{m_2}c_{i,r}|({{{a}^*}_n}-{{a}_n^{(2)}})\\sigma ({A}^*{X}{W}){v}_i|\\Big |\\\\&\\ \\ +\\Big |\\sum _{n=1}^N {q}^\\top {{a}^*}_n \\sum _{i=1}^{m_2}c_{i,r}|{{a}_n^{(2)}}((\\sigma ({A}^*{X}{W})-\\sigma ({A}^{(1)}{X}{W}^{\\prime })){v}_i+\\sigma ({A}^{(1)}{X}{W}^{\\prime })({v}_i-{v}_i^{\\prime }))|\\Big |\\\\\\le &\\Big |\\sum _{n=1}^N {q}^\\top {{a}^*}_n \\sum _{i=1}^{m_2}c_{i,r}|({{{a}^*}_n}-{{a}_n^{(2)}})\\sigma ({A}^*{X}{W}){v}_i|\\Big |+\\Big |\\sum _{n=1}^N {q}^\\top {{a}^*}_n \\sum _{i=1}^{m_2}c_{i,r}|{{a}_n^{(2)}}\\sigma ({A}^{(1)}{X}{W}^{\\prime })({v}_i-{v}_i^{\\prime })|\\Big |\\\\&+\\Big |\\sum _{n=1}^N {q}^\\top {{a}^*}_n \\sum _{i=1}^{m_2}c_{i,r}\\big |\\sum _{k=1}^N {a_{n,k}^{\\prime \\prime }}\\sum _{l=1}^{m_1}v_{i,l}|({{a}^*}_k-{a}_k^{\\prime }){X}{w}_l+{a}_k^{\\prime }{X}({w}_l-{w}_l^{\\prime })|\\big |\\Big |\\\\\\le & O(\\text{poly}(\\epsilon )).\\end{aligned}$ Hence, $\\begin{aligned}&|{e}_g^\\top {A}^*\\sigma ({A}^*\\sigma ({A}^*{X}{W}){V}){c}_r-{e}_g^\\top {A}^{(3)}\\sigma ({A}^{(2)}\\sigma ({A}^{(1)}{X}{W}^{\\prime }){V}^{\\prime }){c}_r|\\\\\\le &|{e}_g^\\top {A}^*\\sigma ({A}^*\\sigma ({A}^*{X}{W}){V}){c}_r-{e}_g^\\top {A}^*\\sigma ({A}^{(2)}\\sigma ({A}^{(1)}{X}{W}^{\\prime }){V}^{\\prime }){c}_r|+|{e}_g^\\top ({A}^*-{A}^{(3)})\\sigma ({A}^{(2)}\\sigma ({A}^{(1)}{X}{W}){V}){c}_r|\\\\\\le & O(\\text{poly}(\\epsilon )).\\end{aligned}$ which implies $L^{\\prime }({A}^{t(1)},{A}^{t(2)}, {A}^{t(3)}, \\lambda _{T-1}, {W}_T^{\\prime }, {V}_T^{\\prime })\\le L^{\\prime }({A}^*, {A}^*, {A}^*, \\lambda _T, {W}_T,{V}_T)+\\lambda _{T-1}\\cdot O(\\text{poly}(\\epsilon ))\\le (1+\\gamma )OPT+\\epsilon _0$ Proof of Theorem REF : By Lemma REF , we have that the algorithm converges in $TT_w$ iterations to a point $L^{\\prime }({A}^{t(1)}, {A}^{t(2)}, {A}^{t(3)}, \\lambda _t,{W}_t,{V}_t)\\le (1+\\gamma )OPT+\\epsilon _0$ We know w.h.p., among $\\tilde{O}(1/\\epsilon _0^2)$ choices of $j$ , $\\min _j\\lbrace \\mathbb {E}_{{W}^\\rho ,{V}^\\rho ,\\Sigma ,\\ {z}\\in \\Omega }L(\\lambda _{T-1}F_{{A}^*}({e}_g,{X}, {W}^{(0)}+{W}^{\\rho ,j}+{W}_T\\Sigma ,{V}^{(0)}+{V}^{\\rho ,j}+\\Sigma {V}_T)\\rbrace \\le (1+\\gamma )OPT+\\epsilon _0$ Then we have $\\Vert {W}_T\\Vert _{2,4}\\le \\epsilon _0^\\frac{1}{4}\\tau _w^{\\prime }$ $\\Vert {V}_T\\Vert _F\\le \\epsilon _0^\\frac{1}{2}\\tau _v^{\\prime }$ By Lemma REF , we know that $\\begin{aligned}&f_r({e}_g, {A}^*,{X}_i, {W}^{(0)}+{W}^\\rho +{W}_T\\Sigma ,{V}^{(0)}+{V}^\\rho +\\Sigma {V}_T, {B})\\\\=& f_r({e}_g, {A}^*,{X}_i, {W}^{(0)}+{W}^\\rho ,{V}^{(0)}+{V}^\\rho , {B})+g_r^{(b,b)}({e}_g, {A}^*,{X}_i, {W}_T,{V}_T, {B})\\pm \\frac{\\epsilon }{K}\\end{aligned}$ Denote ${r}^{\\prime }={A}^*\\sigma ({A}^*{X}({W}^{(0)}+{W}^\\rho )+{B}_1)({V}^{(0)}+{V}^\\rho )$ .", "Then, $\\begin{aligned}\\Vert {r}^{\\prime }\\Vert \\le \\Vert {A}^*\\Vert (\\Vert {A}^*\\Vert _\\infty \\cdot \\tilde{O}(1))\\cdot \\tilde{O}(1)\\le \\Vert {A}^*\\Vert _\\infty \\end{aligned}$ Therefore, $\\begin{aligned}&|f_r({e}_g, {A}^*,{X}_i, {W}^{(0)}+{W}^\\rho ,{V}^{(0)}+{V}^\\rho , {B})|\\\\=& |{e}_g^\\top {A}^*\\sigma ({r}^{\\prime }+{B}_2){c}_r|\\\\\\le & \\tilde{O}(\\Vert {A}^*\\Vert _\\infty (\\Vert {A}^*\\Vert _\\infty +1)\\epsilon _c)\\end{aligned}$ We also have $\\begin{aligned}&|g_r^{(b,b)}({e}_g, {A}^*,{X}_i, {W}_T,{V}_T, {B})|\\\\\\le &|{e}_g^\\top {A}^*{A}^*({A}^*{X}{W}_T\\odot {D}_{{w},{x}}^{(0)}{V}_T)\\odot {D}_{{v},{x}}^{(0)}{c}_r|\\\\\\le & \\Vert {A}^*\\Vert _\\infty ^2\\tau _v^{\\prime }\\tau _w^{\\prime } m_1^\\frac{1}{4}\\sqrt{m_2}\\epsilon _c\\\\\\le & C_0\\Vert {A}^*\\Vert _\\infty ^2\\end{aligned}$ Hence, $f_r({e}_g, {A}^*,{X}_i, {W}^{(0)}+{W}^\\rho +{W}_T\\Sigma ,{V}^{(0)}+{V}^\\rho +\\Sigma {V}_T, {B})\\le \\tilde{O}(\\Vert {A}^*\\Vert _\\infty ^2(\\epsilon _c+C_0))$ Combining (REF , REF ), we can obtain $f_r({e}_g, {{A}^t}^{(1)},{{A}^t}^{(2)}, {{A}^t}^{(3)}, {X}_i, {W}^{(0)}+{W}^\\rho +{W}_T\\Sigma ,{V}^{(0)}+{V}^\\rho +\\Sigma {V}_T, {B})\\le \\tilde{O}(\\Vert {A}^*\\Vert _\\infty ^2(\\epsilon _c+C_0))$ as long as $\\Vert {A}^*-{{A}^t}^{(1)}\\Vert _\\infty \\le \\text{poly}(\\epsilon )$ , $\\Vert {A}^*-{{A}^t}^{(2)}\\Vert _\\infty \\le \\text{poly}(\\epsilon )$ and $\\Vert {A}^*-{{A}^t}^{(3)}\\Vert _\\infty \\le \\text{poly}(\\epsilon )$ .", "For any given $\\lbrace {X}_i, y_i\\rbrace _{i=1}^{|\\Omega |}$ , the dependency between $y_i$ , $y_j$ , where $i,j\\in |\\Omega |$ can be considered in two steps.", "Figure: (a) Dependency between a s X{a}_s{X} and a p X{a}_p{X} (b) Dependency between y i y_i and y j y_jFigure REF (a) shows ${a}_i{X}$ is dependent with at most $(1+\\delta )^2$ ${a}_j{X}^{\\prime }s$ .", "This is because each ${a}_i{X}$ is determined by at most $(1+\\delta )$ row vector $\\tilde{{x}}_l^{\\prime }s$ , while each $\\tilde{{x}}_l$ is contained by at most $(1+\\delta )$ ${a}_p{X}^{\\prime }s$ .", "Similarly, $y_i$ is determined by at most $(1+\\delta )$ ${a}_p{X}^{\\prime }s$ and by Figure REF (b) we can find $y_i$ is dependent with at most $(1+\\delta )^4$ $y_j$ (including $y_i$ ).", "Since the matrix ${A}^*$ shares the same non-zero entries with ${A}$ , the output with ${A}^*$ indicates the same dependence.", "Denote $u_i=1/|\\Omega ^t|\\sum _{i=1}^{|\\Omega ^t|}|L(\\lambda _{T-1} F_{{A}^*}({e}_g, {X}, {W}^{(0)}+{W}^\\rho +{W}_T\\Sigma ,{V}^{(0)}+{V}^\\rho +\\Sigma {V}_T),y_i)-\\mathbb {E}_{({e}_g,{X},y)\\in \\mathcal {D}}[L(\\lambda _{T-1} F_{{A}^*}({e}_g,{X}, {W}^{(0)}+{W}^\\rho +{W}_T\\Sigma ,{V}^{(0)}+{V}^\\rho +\\Sigma {V}_T),y_i)]$ .", "Then, $\\mathbb {E}[u_i]=0$ .", "Since that $L$ is 1-lipschitz smooth and $L(0^K,y)\\in [0,1]$ , we have $\\begin{aligned}&|L(\\lambda _{T-1} F_{{A}^*}({e}_g, {X}, {W}^{(0)}+{W}^\\rho +{W}_T\\Sigma ,{V}^{(0)}+{V}^\\rho +\\Sigma {V}_T),y_i)-L(0^K,y_i)|\\\\\\le &\\Vert (\\lambda _{T-1} F_{{A}^*}({e}_g, {X}, {W}^{(0)}+{W}^\\rho +{W}_T\\Sigma ,{V}^{(0)}+{V}^\\rho +\\Sigma {V}_T),y_i)-(0^K,y_i)\\Vert \\\\\\le &\\tilde{O}(\\sqrt{K}\\Vert {A}^*\\Vert _\\infty ^2(\\epsilon _c+C_0))\\end{aligned}$ Then, $|u_i|\\le 2\\sqrt{K}\\Vert {A}^*\\Vert _\\infty ^2(\\epsilon _c+C_0)$ $\\mathbb {P}(|u_i|\\ge t)\\le 1\\le \\exp (1-\\frac{t^2}{4K\\Vert {A}^*\\Vert _\\infty ^4(\\epsilon _c+C_0)^2})$ Then, $u_i$ is a sub-Gaussian random variable.", "We have $\\mathbb {E} e^{s u_i}\\le e^{\\Vert {A}^*\\Vert _\\infty ^4(\\epsilon _c+C_0)^2 s^2}$ .", "By Lemma 7 in [35], we have $\\mathbb {E}e^{s\\sum _{i=1}^{|\\Omega |} u_i}\\le e^{(1+\\delta )^4K\\Vert {A}^*\\Vert _\\infty ^4(\\epsilon _c+C_0)^2|\\Omega | s^2}$ Therefore, $\\mathbb {P}\\Big (\\Big |\\sum _{i=1}^{|\\Omega |} \\frac{1}{|\\Omega |}u_i\\Big |\\ge k\\Big )\\le \\exp (\\Vert {A}^*\\Vert _\\infty ^4(\\epsilon _c+C_0)^2K(1+\\delta )^4|\\Omega | s^2-|\\Omega | ks)$ for any $s>0$ .", "Let $s=\\frac{k}{2\\Vert {A}^*\\Vert _\\infty ^4(\\epsilon _c+C_0)^2K(1+\\delta )^4}$ , $k=\\Vert {A}^*\\Vert _\\infty ^4(\\epsilon _c+C_0)^2K\\sqrt{\\frac{(1+\\delta )^4\\log N}{|\\Omega |}}$ , we can obtain $\\mathbb {P}\\Big (\\Big |\\sum _{i=1}^{|\\Omega |} \\frac{1}{|\\Omega |}u_i\\Big |\\ge k\\Big )\\le \\exp (-\\Vert {A}^*\\Vert _\\infty ^4(\\epsilon _c+C_0)^2K\\log N)\\le N^{-K}$ Therefore, with probability at least $1-N^{-K}$ , we have $\\begin{aligned}&\\Big |\\mathbb {E}_{({e}_g,{X},y)\\sim \\mathcal {D}}[L(\\lambda _{T-1} F_{{A}^*}({e}_g,{X}, {W}^{(0)}+{W}^\\rho +{W}_T\\Sigma ,{V}^{(0)}+{V}^\\rho +\\Sigma {V}_T),y_i)]\\\\&-\\frac{1}{|\\Omega ^t|}\\sum _{i=1}^{|\\Omega ^t|} L(\\lambda _{T-1} F_{{A}^*}({e}_g,{X}, {W}^{(0)}+{W}^\\rho +{W}_T\\Sigma ,{V}^{(0)}+{V}^\\rho +\\Sigma {V}_T),y_i)\\Big |\\\\\\le & \\epsilon _0\\end{aligned}$ as long as $|\\Omega |\\ge \\tilde{\\Theta }(\\epsilon _0^{-2}\\Vert {A}^*\\Vert _\\infty ^8(1+p_1^4p_2^5\\mathcal {C}_\\epsilon (\\phi ,\\Vert {A}^*\\Vert _\\infty )\\mathcal {C}_\\epsilon (\\Phi , \\sqrt{p_2}\\mathcal {C}_\\epsilon (\\phi , \\Vert {A}^*\\Vert _\\infty ))(\\Vert {A}^*\\Vert _\\infty +1)^4K^6(1+\\delta )^4\\log N)$ , i.e., $\\mathbb {E}_{({e}_g, {X},y)\\sim \\mathcal {D}}[L(\\lambda _{T-1} F_{{A}^*}({e}_g,{X}, {W}^{(0)}+{W}^\\rho +{W}_T\\Sigma ,{V}^{(0)}+{V}^\\rho +\\Sigma {V}_T),y_i)]\\le OPT+ \\epsilon $" ] ]
2207.03584
[ [ "Negative Mass Black Holes in de-Sitter Space" ], [ "Abstract We show that asymptotically de Sitter black holes of negative mass can exist in Lovelock gravity.", "Such black holes have horizon geometries with non-constant curvature and are known as Exotic Black Holes.", "We explicitly examine the case of Gauss-Bonnet gravity.", "We briefly discuss the positive mass case where we show how the transverse space geometry affects whether a black hole will exist or not.For negative mass solutions we shown how three different black hole spacetimes are possible depending on the transverse space geometry.", "We also provide closed form expressions for the geometric parameters to ensure that a black hole spacetime is observed.", "We close with a discussion of the massless case, where there are many different spacetimes that are permitted." ], [ "Introduction", "In General Relativity, the simplest black hole solution is that provided by the Schwarzschild metric [1], where the only parameter of the solution is the mass $m$ .", "Less then half a century later a more general solution was obtained [2], which describes a black hole with three parameters: mass, angular momentum, and charge.", "The introduction of a cosmological constant admits further solutions, as do other matter sources.", "In 4 dimensions the assumption that the metric has radial symmetry restricts black hole solutions to have horizons of constant curvature in Einstein gravity.", "These can be spherical, planar, or hyperbolic if a negative cosmological constant $\\Lambda < 0$ is introduce [3], [4], [5].", "If the geometry is hyperbolic then these black holes can have negative mass [6] and can form from gravitational collapse [7].", "However if $\\Lambda \\ge 0$ then the situation is notably different.", "While it has recently been shown that de Sitter Schwarzschild spacetimes with negative mass in [8], [9] can exist in Einstein gravity, these solutions consist of spacetime bubbles.", "No negative mass black hole vacuum solutions have been obtained.", "We show in this paper that asymptotically de Sitter black hole vacuum solutions of negative mass exist in Lovelock gravity.", "This somewhat unexpected finding arises from a feature recently pointed out for black holes in higher-curvature gravity: their event horizons need not be of constant curvature [10], [11], [12], [13], [14], [15].", "The transverse space of the (potential) black hole solution need only be an Einstein space, admitting horizon geometries of increasing complexity as the spacetime dimension increases.", "For this reason these have been referred to as Exotic Black Holes (EBHs).", "We demonstrate that static asymptotically de Sitter Lovelock EBHs can have negative mass.", "We explicitly illustrate this for Gauss-Bonnet gravity, but it is clear that this will occur in higher-order Lovelock gravity and we expect in more general higher-curvature theories as well.", "Lovelock gravity is a particular subclass of higher curvature gravity theories, which have garnered much attention since quantum gravity induces such higher-order corrections to the standard Einstein-Hilbert gravitational action [16].", "Furthermore, renormalization properties are improved for such theories [17].", "The Lovelock class [18] is of particular interest: it is regarded as the most natural higher-curvature generalization of Einstein gravity whose field equations are of second order in the metric functions.", "Research into EBHs has been of recent interest particularly in the thermodynamics of asymptotically Anti de Sitter black holes.", "The geometry of event horizon was shown to greatly affect the thermodynamic phenomena [19], [20], [21], [22].", "While investigation of black holes in AdS space has been of great interest since the discovery the AdS/Conformal Field Theory correspondence [23], [24], interest in de Sitter black holes is motivated primarily by cosmological considerations [25], though some motivation comes from considerations of holographic duality [26].", "We begin with a brief review of Lovelock gravity and EBHs before moving onto Gauss-Bonnet solutions.", "We provide general polynomial equations for determining horizon locations as well as the Kretschmann scalar, which will function as a diagnostic for finding spacetime singularties.", "We then discuss the positive mass case and how horizon geometry can dictate whether or not a black hole may exist.", "We then provide an analysis of the negative mass case, where we obtain a bound on the minimum allowed mass for obtaining a black hole.", "We also present a closed form expression that dictates the bounds on the horizon geometry for which a negative mass black hole is admitted as a vacuum solution.", "We have shown that it is possible to obtain massless de Sitter black holes with the proper set of parameters, and discuss this in the penultimate section before summarizing our work." ], [ "Lovelock Gravity & ", "For a Lovelock theory of gravity the Lagrangian density is given by $ \\mathcal {L}=\\frac{1}{16 \\pi G_{N}} \\sum _{k=0}^{K} \\hat{\\alpha }_{k} \\mathcal {L}^{(k)}$ where $\\hat{\\alpha }_{k}$ are the Lovelock coupling constants and $\\mathcal {L}^{(k)}$ are the Euler densities with dimension $2k$ , which are $\\mathcal {L}^{(k)}=\\frac{1}{2^{k}} \\delta _{c_{1} d_{1} \\ldots c_{k} d_{k}}^{a_{1} b_{1} \\ldots a_{k} b_{k}} R_{a_{1} b_{1}}^{c_{1} d_{1}} \\ldots R_{a_{k} b_{k}}^{c_{k} d_{k}}$ with $\\delta $ representing the generalized fully anti-symmetric Kronecker delta.", "The first few terms are $\\mathcal {L}^{(0)}=1 \\qquad \\mathcal {L}^{(1)}=R \\qquad \\mathcal {L}^{(2)}=R^{2}-4 R_{a b } R^{a b }+R_{a b c d} R^{a b c d}$ with $R$ the Ricci Scalar and $\\mathcal {L}^{(2)}$ the Gauss-Bonnet term.", "From the Lagrangian density (REF ) we may construct an action $S=\\int d^{d}x \\sqrt{-g}\\left( \\frac{1}{16 \\pi G_{N}} \\sum _{k=0}^{K} \\alpha _{k} \\mathcal {L}^{(k)}+\\mathcal {L}_{m}\\right)$ including a matter term $\\mathcal {L}_{m}$ .", "Variation with respect to the metric $g^{ab}$ yields the field equations $ \\sum _{k=0}^{K} \\hat{\\alpha }_{(k)} \\mathcal {G}_{a b}^{(k)}=8 \\pi G_{N} T_{ a b}.$ where $T_{ab}$ is the stress energy tensor of the matter field and $\\mathcal {G}_{ab}^{(k)}$ are the Lovelock tensors $\\mathcal {G}_{a b}^{(k)}=-\\frac{1}{2^{(k+1)}} g_{a b} \\delta _{ e_{1} f_{1} \\ldots e_{k} f_{k}}^{ c_{1} d_{1} \\ldots c_{k} d_{k}} R_{c_{1} d_{1}}^{e_{1} f_{1}} \\ldots R_{c_{k} d_{k}}^{e_{k} f_{k}}$ We will consider only vacuum solutions with $\\mathcal {L}_{m}=0$ henceforth, so the field equations will be (REF ) with the stress energy being set to zero.", "The parameter $K \\le \\frac{d-1}{2}$ , where $d$ is the dimension of spacetime, and sets the maximal degree of non-linearity in the curvature.", "Values of $K$ larger than this make no contribution to the field equations and are topological invariants in the action.", "The metric ansatz we will make use of is $ \\begin{aligned}d s^{2} &=\\textsf {g}_{i j} d y^{i} d y^{j}+\\gamma _{\\alpha \\beta } d x^{\\alpha } d x^{\\beta } \\\\&=-f(r) d t^{2}+\\frac{d r^{2}}{f(r)}+r^{2} d \\Sigma _{d-2}^{2}\\end{aligned}$ where $y^{i}=(t,r)$ and $\\textsf {g}_{ij}=\\mathrm {diag}(-f(r),\\frac{1}{f(r)})$ .", "The line element $ d\\Sigma ^{2}_{d-2}$ of the base manifold (or transverse space) has metric $\\gamma _{\\alpha \\beta }$ and coordinates $x^\\alpha $ .", "We shall assume it to be compact, with volume $\\Sigma _{d-2}$ , the analogue of the volume of a unit sphere.", "If this space is not compact then quantities such as $M/\\Sigma _{d-2}$ will be regarded as finite.", "The only other condition on the transverse space is that it is an Einstein space, $R_{\\alpha \\beta }\\propto \\gamma _{\\alpha \\beta }$ .", "The most commonly studied special case is when the base space has constant curvature, with curvature parameter $\\kappa =(-1,0,1)$ , which corresponds to negative, flat and positive curvature respectively.", "Solutions in which this latter condition is relaxed, so that the transverse space is only an Einstein space [11], shall be referred to as Exotic Black Holes [10], [27].", "Introducing this metric into the field equations yields a resulting polynomial equation for $f(r)$ [10] $ \\sum _{n=0}^{K} \\frac{b_{n}}{r^{2 n}}\\left(\\sum _{k=n}^{K} \\alpha _{k}\\binom{k}{n}\\left(\\frac{-f(r)}{r^{2}}\\right)^{k-n}\\right)=\\frac{16 \\pi G_{N} M}{(d-2) \\Sigma _{d-2} r^{d-1}}$ where $M$ represents the mass of the black hole and $\\Sigma _{d-2}$ the volume of the base space.", "The quantities $\\alpha _{k}$ are rescaled Lovelock couplings, defined as $\\alpha _{0}=\\frac{\\hat{\\alpha }_{0}}{(d-1)(d-2)}, \\quad \\alpha _{1}=\\hat{\\alpha }_{1}, \\quad \\alpha _{k}=\\hat{\\alpha }_{k} \\prod _{n=1}^{2 k}(d-n) \\quad \\text{ for } k \\ge 2.$ The cosmological constant is defined through the zeroth Lovelock coupling constant $\\Lambda =-\\frac{\\hat{\\alpha }_{0}}{2}$ .", "Imposing only the condition that the transverse space is an Einstein space admits the following possibilities $\\begin{aligned}\\hat{\\mathcal {G}}_{\\beta }^{(n) \\alpha } &=-\\frac{(d-3) !", "b_{n}}{2(d-2 n-3) !}", "\\delta _{\\beta }^{\\alpha }\\\\\\hat{\\mathcal {L}}^{(n)} &=\\frac{(d-2) !", "b_{n}}{(d-2 n-2) !", "}\\end{aligned}$ for its Lovelock tensor and associated intrinsic Euler density respectively.", "The constants $b_{n}$ we refer to as topological terms, and can take any value in $\\mathbb {R}$ .", "They define the topology of the base manifold and the geometry of the event horizon from the condition $f(r_{+})=0$ that defines the horizon location $r_+$ .", "Without loss of generality we set $b_{0}=1$ ; to recover Einstein gravity in the proper limit of vanishing $\\alpha _{k \\ge 2}$ , we will set $\\alpha _{1}=1$ .", "The mass $M$ is defined through the Hamiltonian formalism, and is the conserved charge of the timelike Killing vector of the background spacetime.", "When the dimension is $d=2K+1$ the transverse space is restricted to be of constant curvature [28], [29].", "The Kretschmann scalar for a metric of the form (REF ) is $R^{a b c d} R_{a b c d}=\\left(\\frac{d^{2} f(r)}{d r^{2}}\\right)^{2}+2 \\frac{(d-2)}{r^{2}}\\left(\\frac{d f(r)}{d r}\\right)^{2}+2 \\frac{(d-2)(d-3) f(r)^{2}}{r^{4}}-4 \\frac{R[\\gamma ] f(r)}{r^{4}}+\\frac{\\mathcal {K}[\\gamma ]}{r^{4}}$ where $R[\\gamma ]$ and $\\mathcal {K}[\\gamma ]$ are the Ricci and Kretschmann scalars of the tranverse space respectively." ], [ "Gauss-Bonnet Solutions", "Setting $K=2$ in (REF ) yields $\\frac{\\alpha _{2} f^{2}}{r^{4}}+\\left(-\\frac{1}{r^{2}}-\\frac{2 b_{1} \\alpha _{2}}{r^{4}}\\right) f+\\alpha _{0}+\\frac{b_{1}}{r^{2}}+\\frac{b_{2} \\alpha _{2}}{r^{4}}=\\frac{16 \\pi M}{(d-2) \\Sigma _{d-2} r^{d-1}}$ with the solutions $f=f_{\\pm }(\\mathrm {m}) \\equiv \\frac{r^{2}+2 b_{1} \\alpha _{2} \\pm \\sqrt{\\left(b_{1}^{2}-b_{2}\\right) 4 \\alpha _{2}^{2}+r^{4}\\left(1-4 \\alpha _{2} \\alpha _{0}\\right)+\\frac{8\\mathrm {m} \\alpha _{2}}{r^{d-5}}}}{2 \\alpha _{2}}$ where we have written $\\mathrm {m} \\equiv \\frac{8 \\pi M}{(d-2) \\Sigma _{d-2}}.$ The two solutions are distinguished by their behaviour in the $\\alpha _{2}\\rightarrow 0$ limit.", "The $f_{-}$ branch is referred to as the Einstein branch as it has a smooth $\\alpha _{2}\\rightarrow 0$ limit, giving $\\lim _{\\alpha _{2} \\rightarrow 0} f_{-}(\\mathrm {m})=\\alpha _{0} r^{2}+b_{1}-\\frac{2 \\mathrm {m}}{r^{d-3}}$ whereas the $f_{+}$ solution does not have a smooth limit as $\\alpha _{2}\\rightarrow 0$ ; it is referred to as the Gauss-Bonnet branch.", "For the remainder of this paper we will only consider the Einstein branch for analysis; we will also choose $b_{1}=1$ , so that as $\\alpha _{2}\\rightarrow 0$ we recover the standard Schwarzschild (Anti) de Sitter solution.", "We will only examine solutions with $d \\ge 6$ , since the $d=5$ case only admits constant curvature solutions in the transverse space.", "The cosmological constant is $\\begin{aligned}\\Lambda &=-\\frac{\\hat{\\alpha }_{0}}{2}=-\\frac{\\alpha _{0}(d-1)(d-2)}{2}\\end{aligned}$ and we shall set $\\alpha _{0}<0$ since we are interested in asymptotically de Sitter solutions.", "This allows us to write $f_{-}&=\\frac{r^{2}+2 \\alpha _{2} -\\sqrt{\\left(1-b_{2}\\right) 4 \\alpha _{2}^{2}+r^{4}\\left(1+4 \\alpha _{2} \\frac{2\\Lambda }{(d-1)(d-2)}\\right)+\\frac{8\\mathrm {m} \\alpha _{2}}{r^{d-5}}}}{2 \\alpha _{2}}\\\\$ where we note $\\Lambda >0$ .", "It is convenient at this point to introduce a set of dimensionless variables $r=x \\sqrt{\\alpha _{2}}, \\quad \\Lambda =\\frac{(d-1)(d-2)z}{2\\alpha _{2}}, \\quad \\mathrm {m}=m \\alpha _{2}^{\\frac{d-3}{2}},\\quad z>0$ so that $ f=1+\\frac{x^2}{2}-\\frac{\\sqrt{(1+4 z) x^{4}+4(1-b_{2})+\\frac{8 m}{x^{d-5}}}}{2}.$ The horizon(s) are located at the roots of $f$ , which can be found from solving the polynomial: $x^{d-1} z- x^{d-3}- b_{2} x^{d-5}+2 m=0 \\quad \\mathrm {with}\\quad d\\ge 6, \\quad z>0$ For a black hole solution we must have $f(x)>0 \\; \\mathrm {for} \\; x>x_{+}$ for some range of $x$ , where $x_{+}$ locates the outermost event horizon of the black hole.", "Superfically it appears that there will be 2 or 0 positive roots to this polynomial from Descartes' rule of signs.", "However if we allow both $b_{2}$ and $m$ to be either positive or negative the number of possible positive roots are then the following: $& \\mathrm {If} \\ m>0 &\\mathrm {then} \\ 2\\; \\textrm {roots} \\rightarrow (x_{c},x_{+}) \\ \\mathrm {or}\\ 0\\ \\textrm {roots for any}\\; b_2 & \\nonumber \\\\& \\mathrm {If} \\ m<0\\ \\textrm {and} \\ b_{2}>0, &\\mathrm {then}\\ 1\\ \\mathrm {root}\\rightarrow x_{c} \\hfill & \\nonumber \\\\ &\\mathrm {If} \\ m<0\\ \\textrm {and} \\ b_{2}<0, &\\mathrm {then}\\ 3\\ \\mathrm {roots} \\rightarrow (x_{-},x_{+},x_{c})\\ \\mathrm {or} \\ 1 \\ \\mathrm {root}\\rightarrow x_{c}.", "&$ where $x_{c}$ is the cosmological horizon, $x_{-}$ is the inner horizon, and $x_{+}$ is the outer horizon of the black hole." ], [ "Positive Mass", "Consider first positive mass.", "We plot the behaviour of $f$ in Figure REF for sample values $b_{2}=-2.5$ , $z=0.05$ , and $d=6$ , varying $m$ .", "For small nonzero $m$ we have an asymptotically de Sitter black hole (left diagram in Figure REF ) where as for sufficiently large $m$ there is no static black hole solution and we have an expanding spacetime with $r$ and $t$ interchanging roles (middle diagram).", "As $m$ increases from small to large values, there is one value of $m$ where $f$ has a double root and the horizons merge, illustrated in the right diagram in Figure REF .", "This is a Gauss-Bonnet generalization of the Nariai solution.", "This structure is the same in all dimensions $d \\ge 6$ .", "Only quantitative differences are seen as we increase the dimension: the spread between $x_{+}$ and $x_{c}$ increases and the maximum value of $f$ slightly increases.", "Consequently we will illustrate our results in 6 dimensions unless otherwise stated.", "Figure: Positive Mass Black Hole b 2 =-2.5,z=0.5,d=6b_{2}=-2.5,z=0.5,d=6.", "Left: Black hole with mass m=1m=1 showing two distinct real roots.", "Center: Mass m=5m=5, showing no roots, and a naked singularity.", "Right: Mass m=4.0638m=4.0638, showing a double root.We may also examine what happens when we hold all the parameters fixed except for $b_{2}$ , which is seen in Figure REF .", "For a fixed mass $m$ and cosmological parameter $z$ increasing the value of $b_{2}$ will shift $f$ upwards.", "In this manner the geometry of the horizon plays an important role regarding the existence of a black hole solution, namely that a minimum allowed value of $b_{2}$ is required.", "This minimum value occurs when $f$ has a double root, corresponding to a Nariai type of black hole solution.", "In de Sitter space this is referred to as the maximum allowed mass a black hole can have, other parameters being fixed.", "To determine the value of $x_+=x_c$ for the Nariai-type solution, the equations $f=0,\\quad \\frac{\\partial f}{\\partial x}=0$ must be simultaneously solved.", "In 6 dimensions there is no closed form solution since the latter equation is quintic polynomal is $x$ .", "For the parameters in Figure 2 we obtain numerically $b_{{2}_{min}} \\approx -1.62$ .", "For $b_{2}$ less than this the spacetime will have a naked singularity at the origin.", "We can also make another observation, shown in Figure REF .", "That is, for a certain set of parameters $f$ can become discontinuous for some range of $x$ .", "This discontinuity is due to the argument inside the square root becoming negative.", "Whenever this occures the Kretschmann scalar (REF ) diverges, yielding a spacetime singularity.", "Solutions to the following polynomial $ \\left(4-4 b_{2}\\right) x^{d-5}+(4 z+1) x^{d-1}+8 m=0$ yield the singularity location(s).", "From Decartses' rule of signs this will only have a root when $b_{2}>1$ .", "This is a necessary but not sufficient condition; as illustrated in the right image of Figure REF , increasing the value of $z$ allows the two discontinuous branches to join, yielding a smooth continuous metric function $f$ .", "If we hold all parameters fixed, increasing $z$ shifts $f$ down vertically and reduces the spread between the black hole and cosmological horizons, allowing the possibility of a smooth metric function between them.", "Figure: Positive Mass m=1,z=0.1,d=6m=1,z=0.1,d=6.", "Left: Naked singularity with b 2 =-2b_{2}=-2 showing no real roots.", "Right: de Sitter black hole with b 2 =2b_{2}=2 show a cosmological horizon and event horizon x c >x + x_{c}>x_{+}Figure: Positve mass m=1m=1,d=6d=6,b 2 =3.5b_{2}=3.5.", "Left:z=0.1z=0.1 Two discontinuous branches with two spacetime singularities.", "Right: z=0.3z=0.3 de Sitter black hole." ], [ "Negative Mass", "Negative mass solutions have more interesting behaviour, since if $b_{2}<0$ there can now be three roots to (REF ).", "We illustrate this in Figure REF .", "The left image depicts three horizons, while the center and right image show the two possible cases where horizons can merge.", "In the center image $x_{c}=x_{+}$ (a Nariai-type solution) and in the right $x_{-}=x_{+}$ (an extremal black hole with a cosmological horizon).", "These latter two extremes constrain the range of allowed values of $b_2 < 0$ that yield black hole solutions of negative mass.", "For the case illustrated in Figure REF , we approximately have $-1.3\\ge b_{2} >-3$ .", "We also find that as $x\\rightarrow 0$ , $f$ does not have a smooth limit, and instead becomes complex at some finite positive value of $x$ .", "For $m<0$ and $b_{2}<0$ there is only one sign change in the coefficients of the terms in (REF ) and hence this equation has only one positive root, at which point the Kretschmann scalar (REF ) diverges.", "For the admissible range of $b_2$ , this singularity is always behind an horizon as can be seen in Figure REF .", "For the more generic 3-horizon case shown in the left diagram in Figure REF , the solution to (REF ) is $x=0.1999637662$ .", "Figure: Negative Mass: m=-0.3m=-0.3,z=0.09z=0.09, d=6d=6 Left:b 2 =-2b_{2}=-2, showing three distinct horizons.", "Center:b 2 =-3b_{2}=-3, showing one double root depicting a Nariai like solutionRight:b 2 =-1.3b_{2}=-1.3, showing an extremal black hole with a distinct cosmological horizon." ], [ "Allowed values of $b_{2}$ for valid negative mass black hole solutions", "To determine the the range of $b_{2}$ that allows for a valid black hole solution of negative mass in any dimension, we must solve the following set of equations $& f=0,\\quad \\frac{\\partial f}{\\partial x}=0\\ \\mathrm {for}\\ b_{2},x \\quad \\mathrm {with} \\quad \\lbrace x,b_{2}|\\in \\mathbb {R} >0 \\rbrace .$ The first equation is given by (REF ) and yields $b_{2}=x_+^{4} z-x_+^{2}+2 x_+^{5-d} m$ where $x_+$ solves $ 2 x_+^{d-1} z-x_+^{d-3}-m(d-5)=0.$ from the latter equation.", "There is no general solution to this set of equations except in certain dimensions.", "For $d=6$ equation (REF ) is a quintic, and there is no analytic solution for arbitrary $m,z$ .", "For the parameters used in Figure REF , we obtain numerically $\\left\\lbrace x_{{1+}}=0.6897087623,\\; b_{{2}_{max}}=-1.325264590\\right\\rbrace ,\\left\\lbrace x_{{2+}}=2.328863088,\\; b_{{2}_{min}}=-3.033847194\\right\\rbrace .$ refining the values given above.", "In 7 dimensions however, we may find a closed form expression.", "The equations become $&b_{2}=\\frac{x_+^{6} z-x_+^{4}+2 m}{x_+^{2}}\\\\&2 x_+^{6} z-x_+^{4}-2 m=0.", "$ which admits analytic solutions for arbitrary $m,z$ .", "The latter equation is a cubic in $x_+^{2}$ ; writing $m=-|\\textbf {m}|$ we get $2 y^{3} z-y^{2}+2 |\\textbf {m}|=0\\quad \\text{with}\\quad y=x_+^2,\\ z>0$ Now through Decartes' rule of signs, the possible number of distinct positive roots is either 2 or zero, while the number of negative roots will always be one; this latter case is inadmissible since we must have $y>0$ .", "To have two distinct positive real roots for $y$ (and thus the same number of distinct positive values for $x_+$ ), we require the discriminant $ \\Delta _{3}=8 |\\textbf {m}|\\left(1-54 z^{2} |\\textbf {m}|\\right) > 0$ for (REF ), implying $ -\\frac{1}{54 z^{2}}<m<0.$ This condition must be satisfied to have three distinct roots.", "Writing (REF ) as a depressed cubic, the solutions (for $x$ ) are easily written as $x_{k}=\\sqrt{\\frac{1}{3 z} \\cos \\left(\\frac{1}{3} \\arccos \\left(1-108 \\mathbf {|m|} z^{2}\\right)-\\frac{2 \\pi k}{3}\\right)-\\frac{1}{6 z}} \\quad k=0,1,2$ For all values of $m$ satisfying (REF ), the only $k = 0,1$ yield real solutions, where $x_{0}>x_{1}$ .", "The value $x_{0}$ corresponds to the location of the location of the Nariai-like black hole solution whereas $x_{1}$ locates the extremal black hole of negative mass.", "From (REF ), the respective minimal and maximal values of $b_2$ are ${\\begin{array}{c}b_{{2}_{min}}=\\frac{8 \\cos (A)^{3}-432 \\mathbf {|m|} z^{2}-36 \\cos (A)^{2}+30 \\cos (A)-7}{72 z^{2} \\cos (A)-36 z} \\\\A=\\frac{1}{3} \\arccos \\left(1-108 \\mathbf {|m|} z^{2}\\right) \\\\\\\\b_{{2}_{max}}=\\frac{8 \\cos (B)^{3}-432 \\mathbf {|m|} z^{2}-36 \\cos (B)^{2}+30 \\cos (B)-7}{72 z^{2} \\cos (B)-36 z}\\\\B=\\frac{1}{3} \\arccos \\left(1-108 \\mathbf {|m|} z^{2}\\right) -\\frac{2 \\pi }{3}\\end{array}}$ and provided $ b_{{2}_{max}} > b_2 > b_{{2}_{min}}$ , there will be a valid negative mass black hole with two horizons and a de Sitter cosmological horizon.", "We can make a further interesting observation about (REF ).", "If we saturate the inequality (REF ) from above, setting set $ m = 0$ , a range of interesting phenomena can occur.", "It is possible to have a black hole, a Nariai like black hole, pure de Sitter space or a naked singularity.", "These scenarios are analyzed in the next section." ], [ "Massless Black Holes", "The saturation of (REF ) from above allows for the possibility of zero mass exotic black holes.", "A intriguing feature of these possible black holes is that they are independent of the dimension of the spacetime.", "Taking (REF ) and setting the mass to zero gives us $f=1+\\frac{x^{2}}{2}-\\frac{\\sqrt{(1+4 z) x^{4}+4-4 b_{2}}}{2}.$ The root equation (REF ) is now $x^{4} z- x^{2}- b_{2}=0$ which is a biquadratic equation in $x^2$ so it can be written as $y^2 z - y -b_{2}=0.$ We may again analyze the roots of this polynomial by Decartses' rule of signs.", "Assuming $z>0$ , then $\\begin{aligned}&\\text{for}\\ b_{2}<0\\qquad 2\\ \\text{or}\\ 0\\ \\text{roots}\\\\&\\text{for}\\ b_{2}\\ge 0\\qquad 1\\ \\text{root}\\end{aligned}$ As a quadratic the discriminant is immediately written down as: $\\Delta _{2}\\equiv 4 z b_{2}+1$ .", "For a valid black hole solution, we must have $\\Delta _{2}\\ge 0$ .", "The saturation of this inequality suggests the Nariai solution.", "The unsaturated inequality gives us, similar to the negative mass case, a condition on $b_{2}$ as a function of $z$ that must be satisfied for a black hole to exist: $-\\frac{1}{4 z}<b_{2}<0$ .", "If $b_{2}$ is not within these bounds there will be a naked singularity located at the origin.", "For $b_{2}>0$ the space is pure de Sitter, provided (REF ) has no solutions so as to avoid singularities.", "These equations are easily solved for any $d$ , yielding $ x=\\frac{\\sqrt{2}\\left(b_{2}-1\\right)^{1 / 4}}{(1+4 z)^{1 / 4}}.$ and so we conclude that we must have $0\\le b_{2}<1$ to obtain a pure de Sitter space without any naked singularities.", "The naked singularity that would occur for $b_{2}\\ge 1$ is at a finite value of $x$ given by (REF ).", "The different scenarios are outlined in Figure REF .", "The 3 leftmost images display negative values of $b_{2}$ respectively showing a naked singularity, Nariai black hole and de Sitter black hole.", "The two right most images are for zero and positive values of $b_{2}$ respectively, with the former being pure de Sitter space and the latter a naked singularity outside of the origin.", "For increasing values of $b_{2}>0$ the termination point of $f$ at $x=0$ moves up the vertical axes until it reaches a maximum point at $f(x,b_{2},z)\\rightarrow f(0,1,0.1)=1$ .", "Figure: Massless Soltuions z=0.1z=0.1 From left to right the b 2 b_{2} values are:-3, -2.5, -1, 0, 2.", "It depicts a naked singularity at the origin, a Nariai like black hole, a de Sitter black hole, pure de Sitter space, and then a non-trivial naked singularity outside of the origin, respectively." ], [ "Conclusion", "We have shown that metrics of the form (REF ), with $\\gamma _{\\alpha \\beta }$ being the metric of an arbitrary Einstein space, can possess both negative mass and massless asymptotically de Sitter black hole solutions in Gauss-Bonnet gravity.", "These $M\\le 0$ black holes are all exotic black holes, and their existence depends on the relative values of the parameters $b_{2}$ , $z$ and $m$ .", "For the massless case, for there is a direct relationship between $b_{2}$ and $z$ with $b_{2}$ being bounded from below by $z$ .", "For $M<0$ black holes there are more restrictions on the parameters.", "The range of $m$ is bounded by $z$ , as given by (REF ).", "There is also a minimum and maximum allowed value of $b_2$ admitting such solutions.", "We close by noting that the existence of such horizon geometries is still an open question.", "This is due to the fact that field equations for the transverse space are an over constrained PDE system, where certain topological parameters $b_{n}$ may not provide a solution.", "However we are not aware of any theorem forbidding the existence of non-trivial solutions to (REF ).", "With the discovery of negative mass black holes in de Sitter space interesting questions arise.", "These include the behaviour of geodesics, formation from gravitational collapse, and possible pair production in the early universe of such objects.", "Their thermodynamic behaviour is also likely to yield interesting surprises in comparison to their Anti de Sitter counterparts [19], [20]." ], [ "Acknowledgements", "This work was supported in part by the Natural Sciences and Engineering Research Council of Canada.", "tocsectionReferences" ] ]
2207.03507
[ [ "Thermal spin dynamics of Kitaev magnets $-$ scattering continua and\n magnetic field induced phases within a stochastic semiclassical approach" ], [ "Abstract The honeycomb magnet $\\alpha-$RuCl$_3$ is a prime candidate material for realizing the Kitaev quantum spin liquid (QSL), but it shows long-range magnetic order at low temperature.", "Nevertheless, its broad inelastic neutron scattering (INS) response at finite frequency has been interpreted as that of a 'proximate QSL'.", "A moderate in-plane magnetic field indeed melts the residual zigzag order, giving rise to peculiar intermediate field phases before the high-field polarized state.", "In INS measurements the low-frequency spin waves disappear, leading to a broad scattering continuum in the field-induced intermediate regime, whose nature is currently under debate.", "Here, we study the magnetic field dependent spin dynamics of the $K-\\Gamma-\\Gamma'-$model within a stochastic semiclassical treatment, which incorporates the effect of finite-temperature fluctuations.", "At temperatures relevant for INS experiments, we show how the excitations of the zigzag phase broaden and that the different intermediate phases all show a similar continuum response.", "We discuss the implications of our results for experiments and highlight the importance of distinguishing finite temperature fluctuations from genuine quantum fractionalization signatures in frustrated magnets." ], [ "Introduction", "The dynamical spin structure factor, as measured in inelastic neutron scattering (INS), is an ideal tool for gaining a comprehensive understanding of quantum magnets.", "However, in sought-after quantum spin liquids (QSL) [1], [2], [3], it only yields a broad continuum response [4], as local spin flip excitations decay into multiple fractionalized excitations.", "This has complicated the unambiguous identification of a genuine QSL in a material.", "In recent years, a number of Mott insulating systems with strong spin orbit coupling [5] have been put forward as candidates for realising the Kitaev honeycomb QSL [6] (see Refs.", "[7], [8], [9], [10], [11] for reviews on the subject).", "The layered honeycomb magnet $\\alpha -$ RuCl$_3$  [12] appears to be one of the most promising, as its INS [13], [14], [15] and Raman spectroscopic response [16], [17], [18] display broad scattering continua at elevated frequency similar to the predictions for the ideal Kitaev model [19], [20], [21], [22], [23], [24].", "Despite the presence of long-range zigzag magnetic order at low temperature [25], [26], these have been interpreted as signatures of fractionalized excitations of a 'proximate QSL' nearby in the phase diagram [7].", "Indeed, via the application of a moderate in-plane magnetic field of about 8 [27], [25], [28], [29], the zigzag order is suppressed, giving rise to unusual intermediate field phases with a whole range of atypical properties [30].", "For example, a thermal Hall response reminiscent of the one predicted for the non-Abelian Kitaev QSL [6] has been reported in Refs.", "[31], [32], which is currently under debate [33], [34], [35], [36].", "INS measurements in a field have shown how the low frequency spin wave excitations of $\\alpha -$ RuCl$_3$ melt as the zigzag order disappears for increasing magnetic fields giving rise to a broad scattering continuum centred around the $\\Gamma $ -point of the Brillouin zone [37].", "Whether the broad scattering response is best understood as a signature of (weakly confined) fractional spin excitations or arises due to nonlinearities beyond harmonic magnon dynamics, for example from magnon-magnon interactions, is again currently under debate [38], [39], [40], [41], [42], [43], [44].", "A further complication arises because the microscopic Hamiltonian describing the low-energy magnetic degrees of freedom of $\\alpha -$ RuCl$_3$ is, as of yet, not known.", "A consensus has been reached that a strong bond-dependent Kitaev exchange is present, but the value of the perturbing Heisenberg and spin-off-diagonal $\\Gamma $ and $\\Gamma ^{\\prime }$ interactions (and further neighbor interactions) remains under discussion [45], [46], [47], [48].", "In that context, the observation of a sub-leading yet sizeable out-of-plane modulation of excitations in INS [49] points to the importance of interlayer couplings.", "Nevertheless, an extended $K-\\Gamma -\\Gamma ^{\\prime }-$ model can capture a number of qualitative features of $\\alpha -$ RuCl$_3$ including magnetic-field induced intermediate phases [50], [51].", "For example, Ref.", "[52] showed that, within a classical Monte Carlo sampling scheme, a whole zoo of intermediate phases appears between the low-field zigzag phase and the high-field polarized state [53], [54], [55].", "Classically, these are characterized as non-collinear/coplanar states with large magnetic unit cells and their weak long-range order is expected to be unstable upon the inclusion of quantum and/or thermal fluctuations [56].", "Here, we study the temperature dependent spin excitations of the $K-\\Gamma -\\Gamma ^{\\prime }-$ model as a function of magnetic field.", "We employ a stochastic semiclassical method [57] which we show can reproduce the intermediate field phases found via classical Monte Carlo sampling [52].", "In addition, our method incorporates the effect of thermal fluctuations on the dynamical response [58], relevant for interpreting INS experiments.", "So far, these have been carried out for temperatures not significantly lower than the bare exchange scales (Ref.", "[37] reports e.g.", "results down to approx.", "${2}{}$ with an estimate of the Kitaev exchange of approx.", "${100}{}$ ).", "Moreover, most theoretical modeling has been restricted to zero temperature quantum calculations except for a few recent exceptions [22], [17], [59].", "Our choice of method is motivated by the remarkable finding of Ref.", "[60] that the semiclassical Landau-Lifshitz dynamics (starting from initial states which are sampled via low-temperature classical Monte Carlo) can capture the salient features of the pure Kitaev model.", "Concretely, the broad frequency continua and weak momentum modulation of the classical Kitaev spin liquid is remarkably similar to the one of the exact QSL at zero temperature.", "Only the low frequency response differs as it is governed by quantum selection rules associated with fractionalized flux excitations [19], [61].", "Moreover, semiclassical dynamics of thermally disordered frustrated magnets have recently been shown to capture the INS response of other QSL candidates like NaCaNi$_2$ F$_7$  [62], MgCr$_2$ O$_4$  [63], Ce$_2$ Zr$_2$ O$_7$  [64] or the $\\Gamma $ -model [65].", "Thus, our work similarly addresses more general questions beyond the concrete example of $\\alpha -$ RuCl$_3$ , namely understanding the broad INS scattering continua of frustrated magnets, and diagnosing genuine quantum fractionalization signatures." ], [ "Model, method and phase diagram", "We describe the honeycomb magnet $\\alpha -$ RuCl$_3$ within a $K-\\Gamma -\\Gamma ^{\\prime }-$ model and focus on the Hamiltonian studied in Ref.", "[66], [52] $\\begin{split}\\mathcal {H} &= \\sum _{\\lambda = x,y,z} \\sum _{\\mathinner {\\langle {ij}\\rangle } \\in \\lambda } \\big [ K S_i^\\lambda S_j^\\lambda + \\Gamma \\left( S_i^\\mu S_j^\\nu + S_i^\\nu S_j^\\mu \\right) \\\\&+ \\Gamma ^\\prime \\left( S_i^\\mu S_j^\\lambda + S_i^\\lambda S_j^\\mu + S_i^\\nu S_j^\\lambda + S_i^\\lambda S_j^\\nu \\right) \\big ] - \\mathbf {h} \\cdot \\sum _i \\mathbf {S}_i,\\end{split}$ which includes the bond-dependent ferromagnetic Kitaev interaction, $K<0$ , and the off-diagonal $\\Gamma $ and $\\Gamma ^\\prime $ interactions.", "In eqn:Hamiltonian, $\\mathinner {\\langle {ij}\\rangle } \\in \\lambda $ denotes the nearest-neighbor pair formed by the spins $\\mathbf {S}_i$ and $\\mathbf {S}_j$ with bond orientation $\\lambda \\in \\left\\lbrace \\mathrm {x}, \\mathrm {y}, \\mathrm {z} \\right\\rbrace $ , as shown in the inset of fig:phasediagram1.", "The off-diagonal interactions are given by cyclic permutations of the spin components $(\\lambda , \\mu , \\nu )$ .", "In the following, we will set $K=-1$ and use dimensionless parameters in units of $|K|$ and fundamental physical constants, e.g.", "frequency $\\omega $ is implicitly expressed in units of $|K|/ \\hbar $ and the temperature $T$ in units of $|K|/k_B$ .", "Figure: Phase diagram of the field induced phases of the K-Γ-Γ ' -K-\\Gamma -\\Gamma ^{\\prime }-model.", "The magnetic orders dependent on the Γ\\Gamma interaction and the [111][111]-oriented magnetic field hh.", "We identify the orders along the Γ=0.2\\Gamma = 0.2 line by their static structure factors (see fig:staticAll) and present their dynamic structure factors at different temperatures in fig:dynamicAll.", "The inset shows the honeycomb lattice with bond-dependent exchange interactions, i.e., each type of bond λ∈x,y,z\\lambda \\in \\left\\lbrace \\mathrm {x}, \\mathrm {y}, \\mathrm {z} \\right\\rbrace is represented by a different color.", "Figure adapted from Ref.", ".The classical Kitaev spin liquid quickly reaches a polarized state in a $[111]$ -oriented magnetic field $\\mathbf {h} \\ne 0$ , but a finite $\\Gamma $ interaction leads to a multitude of intermediate magnetic field induced ordered phases, which posses large unit cells and persist to greater values of $h={\\mathbf {h}}$ for increasing $\\Gamma $ .", "Even more fragile magnetic orders are realised when adding a small $\\Gamma ^\\prime = -0.02$ interaction, which also stabilises the zigzag phase around $h \\approx 0$ .", "The phase diagram as a function of $\\Gamma $ and $[111]$ -oriented magnetic field is depicted in fig:phasediagram1, with the different orders named according to the number of spins in their respective unit cells (adapted from Ref.", "[52]).", "Figure: Static structure factors along the Γ=0.2\\Gamma = 0.2 line of the phase diagram in fig:phasediagram1 at low temperature T=0.0001T=0.0001.", "The sharp peaks are broadened by a Gaussian filter (σ=0.12\\sigma = 0.12) for visibility and the first Brillouin zone is indicated by a dashed line.", "For concreteness, we show results for the points |𝐡|∈{0.1,0.225,0.35,0.386,0.6}|\\mathbf {h}| \\in \\lbrace 0.1,0.225,0.35,0.386,0.6\\rbrace denoted by a to e in fig:phasediagram1 with Γ ' =-0.02\\Gamma ^\\prime = -0.02.", "Note that we normalized the structure factors individually to a maximum of one and scaled the intensity of c to e as S(𝐤)\\sqrt{S(\\mathbf {k})} to enhance the visibility of the peaks.We study the spin dynamics based on the atomistic Landau-Lifshitz-Gilbert (LLG) equation $\\frac{\\partial \\mathbf {S}_i}{\\partial t} = \\frac{- 1}{(1 + \\alpha ^2)} \\left[ \\mathbf {S}_i \\times \\mathbf {H}_i(t) + \\alpha \\mathbf {S}_i \\times (\\mathbf {S}_i \\times \\mathbf {H}_i(t)) \\right],$ which describes the damped precession of the classical spins around a local effective (exchange) field $\\mathbf {H}_i(t) = -\\frac{\\partial \\mathcal {H}_i (\\mathbf {S}_i(t))}{\\partial \\mathbf {S}_i} + \\mathbf {b}_i(t).$ Here, the spins are represented by their normalized magnetic moments $\\mathbf {S}_i$ at site $i$ (with $|\\mathbf {S}_i|=1$ ).", "An effective damping of the dynamics from coupling to lattice and other degrees of freedom is included via the dimensionless parameter $\\alpha $  [58].", "For concreteness, we fix it to a small nonzero value $\\alpha = 0.0075$ to incorporate both fluctuations and dissipation whilst allowing for the propagation of long-lived spin waves [57].", "We include the effects of finite temperature via a stochastic magnetic field $\\mathbf {b}_i(t)$ .", "This thermal noise, which describes the interaction of the system with a thermostat (e.g.", "of a phonon subsystem) obeys $\\mathinner {\\langle {\\mathbf {b}_i(t)}\\rangle }&= 0 \\\\\\mathinner {\\langle {\\mathrm {b}_i^\\nu (t) \\mathrm {b}_j^\\kappa (t^{\\prime })}\\rangle } &= 2 \\alpha T \\delta _{ij} \\delta _{\\nu \\kappa } \\delta (t-t^{\\prime })$ for the three components of the spins $\\nu , \\kappa = \\mathrm {x}, \\mathrm {y}, \\mathrm {z}$ .", "The definition of the stochastic field ensures thermodynamic consistency as it reproduces a stationary Boltzmann probability distribution of the magnetic moments in statistical equilibrium [58].", "The approximation of uncorrelated “white noise” is justified when the autocorrelation time of the stochastic field is much shorter than the response of the system.", "Although this assumption breaks down at very low temperatures [67], it is an efficient way of including the general qualitative effects of thermal fluctuations on the dynamical magnetic response [58].", "We solve the system of nonlinear coupled stochastic differential equations, eq:LLG, with an adaptive Runge-Kutta (RK) method of 4th order [68] on Graphics Processing Units (GPUs) using the parallel computing platform CUDA.", "The parallel architecture allows us to significantly improve performance at two major steps, namely the calculation of the effective magnetic field $\\mathbf {H}_i(t)$ using sparse matrix-vector multiplications and the local spin updates according to eq:LLG, whereby each spin component is mapped to a GPU thread.", "This enables us to explore systems of up to $14 \\, 112$ spins ($84 \\times 84$ unit cells).", "More details about the adaptive RK method employed in this work are provided in app:sec:RKmethod.", "We determine the classical ground state of the system using a simulated annealing prescription (see app:sec:simannealing).", "In short, we start with a ferromagnetic initial state at high temperature (in which the spins are pointed along the $[111]$ direction) and then repeatedly cool and reheat the system until the final temperature is reached.", "The annealing process takes about $t = 10^7-10^8$ time units.", "Our stochastic LLG method is in agreement with the phase diagram of Ref.", "[52], which was obtained previously via a standard classical Monte Carlo sampling.", "We identify the competing intermediate orders along the $\\Gamma = 0.2$ line via the real space spin configurations and by their static structure factors shown in fig:staticAll.", "Figure: The dynamical structure factor obtained from LLG simulations is shown for different points of the phase diagram from fig:phasediagram1, along the Γ=0.2\\Gamma = 0.2 line.", "The three different columns depict results for three different temperatures (measured in units of the Kitaev exchange |K||K|).", "For the highest temperature (right column) the zizgag phase (panel a) is still ordered but the intermediate field phases' long-range order has disappeared.", "While the intermediate field phases (panels b, c and d) have distinct spin wave excitations at low temperature in their long-range order, at intermediate temperatures they show a very similar broad scattering continuum up to the magnetic bandwidth ω≈2.2\\omega \\approx 2.2.", "The intensities of the different subplots were each individually normalized.The main objective of our work is to study the dynamical spin structure factors $\\mathcal {S} \\left( \\omega , \\bf {k} \\right)$ as probed by INS experiments.", "The latter is defined as the Fourier transform of the dynamic spin-spin correlation function $\\mathcal {S} \\left( \\omega , \\mathbf {k} \\right) = \\sum _{i,j,\\nu } \\int \\text{d} t \\ e^{i \\mathbf {q} \\cdot \\left( \\mathbf {r}_i -\\mathbf {r}_j \\right)+ i \\omega t} \\left\\langle S^{\\nu }_i (t) S^{\\nu }_j (0) \\right\\rangle .$ In eqn:dynssfactor, $\\left\\langle \\dots \\right\\rangle $ denotes averaging over different thermodynamic ensembles.", "In practice however, we compute this average in a single simulation run by using the ergodic theorem $\\left\\langle S^{\\nu }_i (t) S^{\\nu }_j (0) \\right\\rangle = \\frac{1}{T_0} \\int _{0}^{T_0} \\text{d} t^{\\prime } S^{\\nu }_i (t+t^{\\prime }) S^{\\nu }_j (t^{\\prime })$ for a sufficiently large time window $T_0$ .", "The static spin structure factor $ \\mathcal {S} \\left( \\mathbf {k} \\right)$ is related to the dynamic one via $\\mathcal {S} \\left( \\mathbf {k} \\right) = \\frac{1}{2 \\pi } \\int _{-\\infty }^{\\infty } \\text{d} \\omega \\mathcal {S} \\left( \\omega , \\mathbf {k} \\right) .$ We solve the stochastic LLG equation starting from an equilibrated spin configuration and employ a time window of $T_0 = 25 \\,000$ divided into $50 \\, 000$ time steps (for $T=0.001$ we extend the time window to $T_0=70 \\, 000$ for better frequency resolution).", "We have checked that a longer time window and more time steps do not change the results.", "Consequently, we find that we can calculate the dynamic structure factors without averaging over initial spin configurations as the stochastic field renders the system self-averaging.", "A well-known problem of classical spin dynamics calculations is that both Monte Carlo sampling, as well as our stochastic method with a white noise field lead to a classical Boltzmann distribution of excitations.", "As a result, the weight of the dynamical structure factor over different frequency components differs compared to the correct quantum calculation (in which harmonic spin excitations obey the Bose-Einstein distribution).", "There are two ways to overcome – at least partially – this problem.", "Within the stochastic LLG approach, one can implement a quantum thermostat via a coloured noise field which fulfills the quantum fluctuation-dissipation theorem leading to the correct Bose-Einstein thermal distribution of the harmonic excitations [67].", "A numerically much cheaper, albeit more phenomenological, alternative for the dynamic structure factor is to simply rescale the intensity [62], [63].", "The key idea is to match the definition of the classical and quantum fluctuation-dissipation relations of the spin structure factor (see Appendix H of Ref.", "[64] for a recent discussion).", "In this work we use white noise and rescale the numerically calculated dynamic structure factor $\\mathcal {S} \\left(\\omega , \\mathbf {k}\\right)$ by a factor of $\\beta \\omega / (1-e^{-\\beta \\omega })$ reducing the spectral weight at small frequencies for low temperatures.", "This way of correcting shortcomings of a purely classical calculation has recently been shown to give quantitatively similar results as the $1/S$ Holstein-Primakoff expansion, with qualitative agreement to INS experiments on frustrated three-dimensional magnets  [62], [63], [64]." ], [ "Results", "In fig:dynamicAll we show the dynamical spin structure factor for five representative points along the $\\Gamma = 0.2$ line of the phase diagram fig:phasediagram1 for three different temperatures (see app:sec:G05results for results along the $\\Gamma = 0.5$ line).", "The low and the high field regimes from fig:dynamicAll:a,fig:dynamicAll:e show sharp spin wave excitations in the ordered phases at lowest temperature (left column).", "In the zigzag phase shown in fig:dynamicAll:a, the main intensity is centered around the $\\Gamma $ point being contributed by the two lowest-frequency modes.", "The latter are broadened and only weakly dispersing (as opposed to the two higher frequency branches).", "As soon as the system enters the field polarized state from fig:dynamicAll:e, sharp spin waves appear, which are robust to thermal fluctuations.", "In contrast, for increasing temperatures the excitations of the zigzag phase broaden significantly.", "For our choice of highest temperature $T=0.035$ the order parameter of the zigzag phase is significantly reduced by thermal fluctuations, but still nonzero.", "Nevertheless, spin excitations are very diffusive and the two low frequency modes quickly merge into one broad mode for increasing temperature which is reminiscent of the weakly-dispersive broad mode measured in $\\alpha -$ RuCl$_3$  [13], [14], [15].", "We find that above the ordering temperature (not shown) the response turns into a scattering continuum over the whole magnetic bandwidth.", "To investigate the thermal broadening of the excitations in the zigzag phase in more detail, we show the response at the $\\Gamma $ -point for four different temperatures in fig:dynamicGammaPoint.", "Indeed, the two-peak structure of the two spin wave modes quickly merges into one mode even in the ordered regime before disappearing into a broad continuum in the disordered phase.", "Next, we turn to the three different intermediate-field-induced phases from fig:dynamicAll:b,fig:dynamicAll:c,fig:dynamicAll:d, which show distinct spin-wave excitations in their low-temperature ordered phases.", "Again, the low-frequency dispersive modes carry most of the intensity.", "Due to the large unit cells, the intermediate-field phases have a large number of spin-wave branches at high frequency.", "For increasing temperature, the fragile orders melt, giving rise to a broad scattering continuum over the entire magnetic bandwidth.", "Remarkably, the three different fields corresponding to the different intermediate phases display a very similar higher frequency continuum response for temperatures when the order has disappeared.", "In fig:dynamicAll:c,fig:dynamicAll:d, the main intensity is centered around the $\\Gamma $ -point which, in conjunction with the broad scattering continuum, is again reminiscent of the INS results for $\\alpha -$ RuCl$_3$  [37].", "Finally, we show the dynamical structure for a fixed frequency $\\omega =0.4$ at elevated temperature $T=0.035$ in fig:finiteomega.", "In the zigzag phase from fig:finiteomega:a, the broad scattering takes the form of a star-like pattern akin to the one found in INS experiments on $\\alpha -$ RuCl$_3$  [14].", "For increasing magnetic field, the region of maximum intensity changes from the $\\Gamma $ point to a ring-like shape, which again is very broad in momentum space because of the short real space correlations resulting from the magnetic frustration and thermal disordering.", "Only in the field polarised state from fig:finiteomega:e does the normal sharp ring expected from spin wave excitations reappear.", "Figure: The dynamical response at the Γ\\Gamma -point of the zigzag phase at four different temperatures.", "With increasing temperature, the two peaks of the low frequency modes merge into a single mode, which eventually disappears into a broad continuum.Figure: The dynamical response for fixed ω=0.4\\omega =0.4 is shown at temperature T=0.035T=0.035 for the different field induced phases.", "The broad star-like scattering feature of the zigzag phase (panel a), which is reminiscent of the INS results for α-\\alpha -RuCl 3 _3 , turns into a broad ring-like feature for increasing magnetic field." ], [ "Discussion and Conclusion", "We have shown that thermal fluctuations drastically affect the dynamical spin response in frustrated spin models relevant for Kitaev materials like $\\alpha -$ RuCl$_3$ .", "Already in the absence of any applied magnetic field, the spin-wave excitations of the zigzag state quickly broaden for increasing temperature, even in the ordered low-temperature phase.", "In contrast, in the high-field spin-polarized phase the spin-wave excitations remain sharp up to high temperature.", "Within our semiclassical description, finite temperature fluctuations appear as a stochastic field in the LLG dynamics.", "Thus, the increased sensitivity of the zigzag phase can be directly traced back to the frustrated interactions of the extended Kitaev model, which allow the stochastic thermal fluctuations to transition between a large number of approximately degenerate spin configurations.", "Most interestingly, we find that the different field-induced intermediate phases from fig:phasediagram1, are even more fragile with respect to thermal fluctuations: their ordering quickly disappears for experimentally relevant temperatures.", "The corresponding INS response shows only a broad scattering continuum which only weakly depends on the magnetic field.", "In connection to $\\alpha -$ RuCl$_3$ , a broad continuum response has been observed in Ref.", "[37] at temperatures down to 2, which is still relatively high in comparison to the ordering temperatures of the fragile large-unit-cell intermediate-field phases.", "Hence, in order to understand the origin of the INS response, more measurements at lower temperature are highly desirable.", "Our work highlights the importance of thermal fluctuations for accurately describing the INS response of QSL-candidate materials.", "To distinguish different scenarios of broad scattering – i.e.", "fractionalized excitations of a genuine QSL [19], [69], nonlinearities of magnon-magnon interactions [39], [70], or thermal fluctuations between approximately degenerate spin configurations – requires a careful comparison between different theoretical predictions and experiments at the lowest possible temperatures.", "Of course, the scenarios are not mutually exclusive, but might be at play simultaneously, which could further complicate the picture.", "The general lesson of the present as well as previous [60], [62], [63], [64] works is that the finite temperature response of a frustrated classical magnet can look surprisingly similar to the one expected from fractionalized excitations in a QSL at zero temperature.", "This makes the unambiguous observation of quantum fractionalization a challenging task for scattering experiments (at least if they are not performed at temperatures several orders of magnitude below the magnetic exchange scales).", "On the positive side, the stochastic LLG equation employed here should be a powerful method for comparing different models to INS data at different temperatures in order to extract the microscopic Hamiltonian parameters [71].", "In the future, it would be worthwhile to study spin and thermal transport of frustrated (Kitaev) magnets within the stochastic LLG approach, to investigate other dynamical probes like inelastic light scattering, and to explore the effect of quenched disorder.", "In general, we expect that a clear diagnostic of genuine quantum spin fractionalization will require complementary experimental measurements at lowest temperatures and comprehensive comparison to different quantum as well as semiclassical methods." ], [ "Acknowledgements", "We thank P. Roy, R. Otxoa, P. McClarty, R. Moessner, S. Nagler, B. Placke for helpful discussions and especially A. Banerjee also for detailed comments on the manuscript.", "JK acknowledges support via the Imperial-TUM flagship partnership.", "The research is part of the Munich Quantum Valley, which is supported by the Bavarian state government with funds from the Hightech Agenda Bayern Plus.", "OF acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 328545488 – TRR 227, project B03.", "AN acknowledges funding by the Royal Society." ], [ "Adaptive Runge-Kutta method", "In this appendix, we provide details on the RK method used to numerically simulate the finite-temperature spin dynamics.", "The stochastic LLG equation from eq:LLG can be rewritten as a generic first-order stochastic differential equation $\\mathrm {d} \\mathbb {S}=f\\left(\\mathbb {S},t\\right)\\mathrm {d} t+g\\left(\\mathbb {S},t\\right) \\mathrm {d}\\mathbb {W},$ where $\\mathbb {S}$ denotes the vector containing all the components of the spins in the lattice, and $\\mathbb {W}$ represents a corresponding vector of Wiener processes.", "The explicit forms of vector functions $f\\left(\\mathbb {S},t\\right)$ and $g\\left(\\mathbb {S},t\\right)$ can be determined directly from eq:LLG.", "For the purpose of explaining the RK numerical integration method, we leave their forms unspecified.", "The spin values at a give time-step $t_{n+1}$ can be obtained from the ones from the previous time-step $t_{n}$ by employing a RK approximation of order $p=4$ $\\mathbb {S}_{n+1} = \\mathbb {S}_{n} + \\sum ^{p}_{i=1} b_i \\mathbb {K}_i + \\dfrac{1}{2} \\left( \\eta _1 + \\eta _2 \\right),$ where we have defined $\\eta _1&= g\\left(\\mathbb {S},t_n\\right) \\sqrt{\\Delta t_n},\\nonumber \\\\\\mathbb {K}_1 &= f\\left(\\mathbb {S},t_n\\right) \\Delta t_n,\\nonumber \\\\\\eta _2&= g\\left(\\mathbb {S}+\\mathbb {K}_1 + \\eta _1,t_n+\\Delta t_n\\right) \\sqrt{\\Delta t_n},\\nonumber \\\\\\mathbb {K}_i &= f\\left(\\mathbb {S} + c_i \\eta _1 + \\sum ^{i-1}_{j=1}a_{ij} \\mathbb {K}_j, t_n + c_i \\Delta t_n \\right) \\Delta t_n, i>1, $ and $\\Delta t_n=t_{n+1}-t_{n}$ .", "The coefficients $a_{ij}$ , $b_i$ and $c_i$ are tabulated and depend on the specific RK method used.", "Eq.", "(REF ) represents a generalization of the RK4-Heun [72] method, where the deterministic part of the differential equation is computed by a $p$ -th order RK method.", "Because the stochastic term of the equation is evaluated at both ends of the interval, the result will converge to the Stratonovich solution [72].", "In order to obtain an estimate for the error after each time-step, the $p$ -th order RK method is complemented by another one of order $p+1$ .", "The coefficients $a_{ij}$ , $b_i$ and $c_i$ corresponding to each method are chosen in such a way as to ensure a minimal number of evaluations of the function $f$ .", "This is achieved by using the same set of intermediate values $\\lbrace \\mathbb {K}_i \\rbrace $ .", "Once an estimate of the error ($\\mathbb {E}_{n+1}$ ) is found, the next time interval, $\\Delta t_{n+1}$ , can be adjusted according to [73] $\\Delta t_{n+1}=0.9\\left(\\dfrac{\\delta }{\\mathbb {E}_{n+1}}\\right)^{\\frac{1}{p}+1}.$ This ensures that the algorithm uses the largest time interval that keeps the truncation error below a given tolerance, $\\delta $ .", "For the specific case of the sLLG equation, we employ the method RK5(4)7S [73], that was adapted according to Eq.", "(REF ) to account for the stochastic magnetic field." ], [ "Simulated annealing procedure", "In this appendix, we outline the procedure used to determine the classical ground states of the system used as initial conditions for computing the static and dynamic structure factors.", "It should be noted that the simulated annealing procedure is necessary precisely because using a random spin configuration as an initial condition at low temperatures will trap the system into a meta-stable state.", "To prevent the formation of such meta-stable states, we start from a temperature much higher than the ordering temperatures and then progressively cool the system until the target temperature (i.e.", "the temperature at which we simulate the spin structure factors) is reached.", "There are multiple options one can employ for the cooling protocols.", "In this work, we repeatedly linearly cool and slightly reheat the system until we reach the target temperature.", "At each cooling step we encourage the system to settle in the lowest-energy state.", "The reheating steps destroy any meta-stable states that could form during the cooling stage.", "Our annealing procedure starts from an initial temperature $T_{\\mathrm {init}}$ and consists of $N_{\\mathrm {a}}$ cycles each lasting a time $\\Delta t_{\\mathrm {a}}$ .", "During one cycle, the temperature is first linearly decreased by a factor $f_{\\mathrm {c}}$ during a time $\\Delta t_{\\mathrm {h}}$ (we use the convention in which $f_{\\mathrm {c}} < 1$ denotes a net cooling).", "The cooling stage is immediately followed by a reheating step in which the temperature is raised by a factor $f_{\\mathrm {h}}>1$ (with $f_{\\mathrm {h}} f_{\\mathrm {c}}<1$ ) in a time $\\Delta t_{\\mathrm {h}}$ (such that $\\Delta t_{\\mathrm {a}} = \\Delta t_{\\mathrm {c}} + \\Delta t_{\\mathrm {h}}$ ).", "As such, the temperature evolution is given by $\\frac{T \\left( n \\Delta t_{\\mathrm {a}} + t \\right)}{T_{\\mathrm {init}}} = \\left( f_{\\mathrm {h}}f_{\\mathrm {c}} \\right)^n \\left[ \\frac{t \\left( f_{\\mathrm {c}} - 1 \\right) }{\\Delta t_{\\mathrm {c}}} + 1 \\right],$ for $0\\le t \\le \\Delta t_{\\mathrm {c}}$ , and $0 \\le n < N_a $ ($n \\in \\mathbb {Z}$ ), during the cooling stage.", "Similarly, during the heating stage $\\frac{T \\left( n \\Delta t_{\\mathrm {a}} + \\Delta t_{\\mathrm {c}} + t \\right)}{T_{\\mathrm {init}}} = \\left( f_{\\mathrm {h}}f_{\\mathrm {c}} \\right)^n f_{\\mathrm {c}} \\left[ \\frac{t \\left( f_{\\mathrm {h}} - 1 \\right) }{\\Delta t_{\\mathrm {h}}} + 1 \\right],$ for $0\\le t \\le \\Delta t_{\\mathrm {h}}$ .", "For the simulations, we use $N_{\\mathrm {a}} = 45 $ cycles with $f_{\\mathrm {c}} \\approx 0.1$ , $f_{\\mathrm {h}} \\approx 8$ , $\\Delta t_{\\mathrm {c}} \\approx 200 \\, 000$ , and $\\Delta t_{\\mathrm {h}} \\approx 20 \\, 000$ .", "The temperature effectively decreases by a factor $f_{\\mathrm {h}} f_{\\mathrm {c}}<1$ during each cycle." ], [ "Results for $\\Gamma = 0.5$", "Analogous to the results of the main text, in this appendix, we show the static and dynamic structure factors at different temperatures along the $\\Gamma = 0.5$ line of the phase diagram from fig:phasediagram1, shown in more detail in fig:phasediagram2.", "We obtain all structure factors except the first zigzag phase ($h = 0.01$ ), as described in app:sec:simannealing.", "For the latter, we employ the high-field zigzag phase ($h = 1.108$ ) as the initial configuration and do not use an annealing procedure.", "For the low-field zigzag phase, we find that the spin configuration obtained through simulated annealing is modulated and features additional low-intensity spin waves.", "As for the $\\Gamma = 0.2$ line, we identify the different orders via their real space spin configurations and by their static structure factors, which are shown in fig:staticAllG05.", "Figure: Detailed phase diagram of the Γ=0.5\\Gamma =0.5 line in fig:phasediagram1, as obtained by Ref. .", "The various phases are labeled as in fig:phasediagram1.We observe a thermal broadening of the spin wave excitations, which are, however, more stable against thermal fluctuations than before, due to the stronger exchange interaction.", "Whereas the intermediate phases in fig:dynamicAll at temperature $T=0.035$ show a similarly broad (almost) continuum response, the different phases for $\\Gamma =0.5$ are clearly distinguishable at least up to $T=0.05$ (not shown here).", "For even higher temperatures, however, the dynamic response of all phases again looks very similar as is shown in the right panel of fig:dynamicAllG05.", "We further note that the magnetic bandwidth increases with stronger magnetic field from $h \\approx 2.3$ to $h \\approx 2.5$ for the 32-site order.", "Figure: The dynamical structure factor obtained from LLG simulations is shown for different points of the phase diagram, see fig:phasediagram2 and fig:phasediagram1 along the Γ=0.5\\Gamma = 0.5 line.", "The three different columns depict results for three different temperatures (measured in units of the Kitaev exchange |K||K|).", "The intermediate orders host a multitude of distinct spin wave excitations and are distinguishable up to higher temperatures than the orders along the Γ=0.2\\Gamma = 0.2 line.Thermal fluctuations have a particularly interesting effect on both the 50-site and 98-site order.", "With increasing temperature, the modes around the K-points become symmetric, which might be attributed to different magnetic domain realizations, and the temperature causes a split of the low frequency mode at the M-points on a path between K-points.", "The similarity between these two orders is not surprising because the 98-site order appears like an augmented 50-site order, as already observed in Ref.", "[52].", "The dynamic response at fixed $\\omega = 0.9$ seen in fig:finiteomegaG05 again resembles the star-shaped scattering feature expected from INS experiments of $\\alpha -$ RuCl$_3$ [14] and we observe the ring-like feature around the $\\Gamma $ -point for increasing fields.", "Figure: The dynamical response for fixed ω=0.9\\omega =0.9 is shown at temperature T=0.035T=0.035 for the different field induced phases.", "As in fig:finiteomega, we reproduce a star-like scattering feature for the ZZ phases." ] ]
2207.03515
[ [ "Killing Tensors in Koutras-McIntosh Spacetimes" ], [ "Abstract The Koutras-McIntosh family of metrics include conformally flat pp-waves and the Wils metric.", "It appeared in a paper of 1996 by Koutras-McIntosh as an example of a pure radiation spacetime without scalar curvature invariants or infinitesimal symmetries.", "Here we demonstrate that these metrics have no \"hidden symmetries\", by which we mean Killing tensors of low degrees.", "For the particular case of Wils metrics we show the nonexistence of Killing tensors up to degree 6.", "The technique we use is the geometric theory of overdetermined PDEs and the Cartan prolongation-projection method.", "Application of those allows to prove the nonexistence of polynomial in momenta integrals for the equation of geodesics in a mathematical rigorous way.", "Using the same technique we can completely classify all lower degree Killing tensors and, in particular, prove that for generic pp-waves all Killing tensors of degree 3 and 4 are reducible." ], [ "Formulation and motivation", "Polynomial integrals of Hamiltonian ODEs were actively studied in the XIX$^\\text{th}$ century classical mechanics; in particular the existence of quadratic integral for the metric of the ellipsoid allowed Jacobi in 1836 to find an explicit formula for geodesics in terms of elliptic functions.", "This problem also appeared in general relativity: the famous metrics of Schwarzschild, Gödel and Kerr admit polynomial integrals allowing to describe geodesics of the corresponding spacetimes in detail.", "Often integrals are conserved quantities related to Killing vectors via Noether's theorem, but sometimes there are higher degree integrals, known as Killing tensors.", "One of those is the Carter constant [3], [26] reducing the geodesic motion to quadratures.", "There exist obstructions to the existence of polynomial integrals: according to [14] a generic metric $g$ admits no such integrals even locally.", "It is thus important to realize the existence/nonexistence of Killing tensors for concrete metrics from applications, see [6], [8], [9], [13], [25].", "The following is the Koutras–McIntosh family of spacetimes for $(a,b)\\ne (0,0)$ : $g = 2(ax+b)\\,du\\,dw -2aw\\,dx\\,du +\\bigl (f(u)(ax+b)(x^2+y^2)-a^2w^2\\bigr )\\,du^2 -dx^2 -dy^2.$ These metrics were shown in [11] to possess neither invariants nor symmetries.", "The first property means that all polynomial curvature invariants, i.e., complete contractions of tensor products of the Riemann tensor and its covariant derivatives $\\nabla _{i_1}\\!\\cdots \\nabla _{i_s}R_{abcd}$ , vanish and so cannot be used to distinguished $g$ from the Minkowski metric.", "These are so-called VSI (vanishing scalar invariants) spaces that received considerable attention in recent time [21].", "They belong to a more general class of spacetimes not separated by their scalar curvature invariants [4], [5], which in dimension 4 were proven to be of degenerate Kundt type.", "Note that Kundt spaces can be distinguished by their Cartan [19] or differential [16] invariants, see [15] for a comparisson.", "The second property above means there are no Killing vectors, or linear integrals, for (REF ).", "In this paper we show that it also does not possess “hidden symmetries”, by which we mean Killing tensors of low degrees.", "Note that the nonexistence of Killing tensors is important in several applications.", "For instance, it is necessary for linearization stability of Einstein’s equations [1] and also for the inverse problem in tensor tomography [20].", "Thus, even though Killing tensors do not have direct geometric interpretation (as noticed by Penrose and Walker [26], see however [2]) their existence or nonexistence carries certain dynamical implications." ], [ "Main results", "Metric (REF ) is conformally flat (but nonflat for $f\\ne 0$ ) and describes pure radiation, satisfying Einstein’s field equations of the type $R_{ab}=\\phi \\, l_al_b$ for a null vector field $l$ and a scalar field $\\phi $ .", "Metric (REF ) for $a=0,b=1$ is a pp-wave, possessing 6 Killing vectors and 1 homothety except for special cases $f(u)=c$ and $f(u)=c/u^2$ , where the number of Killing vectors increases to 7 [22], [8] and the homothety persists.", "We will examine the existence of higher order Killing tensors (up to degree 4) and for specific cases $f(u)=cu^m$ , $m=0,1,2,-2$ we prove that there is only one such irreducible quadratic tensor.", "Metric (REF ) for $a=1,b=0$ defines the Wils spacetime [27].", "This metric is known to have no Killing vectors or homotheties for general functional parameter $f(u)$ , so we examine it for the existence of higher order Killing tensors.", "It turns out that up to order 6 no irreducible Killing tensors exist (that is with the exception of powers of the Hamiltonian and combinations with Killing vectors when they exist).", "These results are presented in Section .", "For the general Koutras–McIntosh family we deduce the following statement: Theorem 1 For generic numerical parameters $a,b$ and functional parameter $f(u)$ the spacetime (REF ) possesses no Killing tensors up to degree 6 except for energy and its powers $H$ , $H^2$ and $H^3$ .", "Here and below genericity of $f(u)$ is understood in $C^{k+1}$ topology, where $k$ is the prolongation level determined by Algorithm 1 of §REF , where the matrix $M_k$ depends on the jet $j^{k+1}f$ .", "Table 2 shows values of $k$ for degrees $d\\le 6$ .", "We can be more specific on the exceptional values of the involved parameters.", "To find those that allow Killing vectors one may follow the general approach with metric invariants via the Cartan-Karlhede algorithm [7], however our method with counting compatibility conditions via the coefficient matrix of the prolonged PDE system gives an alternative and implies the following results.", "Theorem 2 Metrics (REF ) possess Killing vectors if and only if either $a=0$ (then rescale $b\\rightarrow 1$ ), so that the spacetime is a plane wave, or $b=0$ (then rescale $a\\rightarrow 1$ ), so that $g$ is Wils metric with $f(u)=(c_0+c_1u+c_2u^2)^{-2}$ .", "The same approach but with much heavier computations yields the following results.", "Theorem 3 Metrics (REF ) possess Killing 2-tensors different from the Hamiltonian $H$ in the same range of parameters as for the Killing vectors, i.e., either $a=0$ or $b=0$ , $f(u)=(c_0+c_1u+c_2u^2)^{-2}$ .", "The proofs and further specifications will be given in Section .", "The Maple & LinBox worksheets, which demonstrate our computations, can be found in a supplement to the arXiv version of this paper." ], [ "Geometric Theory of PDEs", "We start with the general setup.", "Let $(N^n,g)$ be a pseudo-Riemannian manifold.", "In this section we formalize searching for Killing tensors (or polynomial integrals of the geodesic flow on the tangent bundle but we work on the cotangent bundle using raising/lowering indices with the metric $g$ ) via a compatibility analysis of an overdetermined PDE system and discuss the prolongation-projection technique." ], [ "Hamiltonian formalism", "The energy function $H=\\frac{1}{2}\\Vert p\\Vert ^2_g$ writes in local coordinates $H(x,p) = \\frac{1}{2} g^{ij}(x) p_i p_j \\hspace{28.45274pt} [g^{ij}]=[g_{ij}]^{-1}.$ It is well-known that geodesics of $g$ are projections to the base $N$ of trajectories of the corresponding Hamiltonian vector field $X_H=\\omega ^{-1}dH$ on $T^*N\\stackrel{g}{\\simeq }TN$ , where $\\omega $ is the canonical symplectic form on the cotangent bundle.", "Integrating the equations of geodesics requires conserved quantities for this Hamiltonian system.", "A function $I:T^*N\\rightarrow {\\mathbb {R}}$ is an integral (of motion) $X_H(I)=0$ if it Poisson commutes with the Hamiltonian: $\\lbrace H,I\\rbrace =\\sum _{i=1}^n \\left(\\frac{\\partial H}{\\partial p_i}\\frac{\\partial I}{\\partial x^i}-\\frac{\\partial H}{\\partial x^i}\\frac{\\partial I}{\\partial p_i}\\right)=0.$ The natural action of the isometry group on the cotangent bundle $T^{*}N$ is Hamiltonian and it preserves the energy $H$ .", "Thus, the isometries represent infinitesimal symmetries of the geodesic flow given, by virtue of Noether's theorem, by linear in momenta integrals of motion.", "Explicitly, if $X = X^i(x)\\partial _{x^i}\\in \\mathfrak {iso}(N,g)$ is a Killing vector field then the corresponding integral is $I(x,p) =\\langle X,p\\rangle = X^i(x) p_i$ .", "More generally, a Killing tensor of degree $d$ corresponds to a homogeneous in momenta polynomial $I_d:= a^{i_1\\cdots i_d}(x)\\ p_{i_1}\\cdots p_{i_d},$ which Poisson commutes with $H$ , and is thus a polynomial integral.", "Since the Hamiltonian is quadratic in momenta, for any (REF ) the Poisson bracket $\\lbrace H,I_{d}\\rbrace $ is of degree $d+1$ in momenta.", "Consequently, Killing $d$ -tensors correspond to solutions of a system of differential equations formed by vanishing of $p$ -coefficients of the Poisson bracket, which we call the Killing equation, $\\mathcal {E}_d:= \\lbrace F = 0 : F \\in \\text{coeffs}_p(\\lbrace H, I_d\\rbrace ) \\rbrace .$ This is an overdetermined system of linear first order PDEs on the coefficients $a^{i_1\\cdots i_d}(x)$ of the Killing tensor.", "Actually, there are ${n+d\\atopwithdelims ()d+1}$ equations on ${n+d-1\\atopwithdelims ()d}$ unknown functions.", "Denote solutions to this system – the linear space of all Killing $d$ -tensors – by $K_d$ ." ], [ "Jet spaces and equations", "The notion of jet-space formalizes the computational device of truncated Taylor polynomials; we refer for details to [12].", "If $x^i$ are local coordinates on $N$ then the jet-space $J^kN$ of $k$ -jet of functions $u:N\\rightarrow {\\mathbb {R}}$ has local coordinates $(x^i,u_\\sigma )$ for multi-indices $\\sigma =(i_1,\\dots ,i_n)$ , $i_s\\ge 0$ , $|\\sigma |=\\sum i_s\\le k$ .", "Similarly are defined jets of vector-valued functions, sections, etc.", "For a bundle $\\pi :E\\rightarrow N$ the jet-space of its sections is denoted by $J^k(N,E)$ .", "The space of $k$ -jets of maps $u:{\\mathbb {R}}^n\\rightarrow {\\mathbb {R}}^m$ will be simply denoted by $J^k(n,m)$ .", "It is a bundle of rank $m\\cdot {n+k-1\\atopwithdelims ()k}$ over $n$ -dimensional base.", "Any map $u=(u^j):{\\mathbb {R}}^n\\rightarrow {\\mathbb {R}}^m$ lifts to the jet-section $j^ku:{\\mathbb {R}}^n\\rightarrow J^k(n,m)$ given by $x^i\\mapsto u^j_\\sigma =\\partial u^j(x)/\\partial x^\\sigma $ .", "Definition 4 (Geometric PDE) A partial differential equation of order $k$ is a submanifold $\\mathcal {E} \\subseteq J^k(n,m)$ .", "A solution of the PDE is defined to be a function $u:{\\mathbb {R}}^n\\rightarrow {\\mathbb {R}}^m$ such that its $k$ -jet $j^ku$ takes values in $\\mathcal {E}$ .", "A local solution is the same but defined on a domain $U\\subset {\\mathbb {R}}^n$ .", "We denote by $\\mathop {\\rm Sol}\\nolimits (\\mathcal {E})$ the space of all (local) solutions of $\\mathcal {E}$ .", "Elements of a $k$ 'th order geometric PDE $\\mathcal {E} \\subseteq J^k(n,m)$ are solutions up to order $k$ (at a point).", "To find the solutions of the PDE $\\mathcal {E}$ up to order $k+1$ and higher, we have to differentiate the defining equations.", "To encode the chain rule, we define the $q$ 'th total derivative of $F:J^k\\rightarrow {\\mathbb {R}}^s$ to be a vector-function on $J^{k+1}$ given by $D_qF:= \\frac{\\partial F}{\\partial x^q}+ \\sum _{j=1}^m \\sum _{|\\sigma |\\le k}\\frac{\\partial F}{\\partial u^j_\\sigma }\\cdot u^j_{\\sigma +1_q}.$ (Here we use the notation $\\sigma + 1_q$ for the multi-index obtained by adding 1 to the $q$ 'th entry of $\\sigma $ .)", "Now, a point $(x^i,u^j_\\sigma )\\in J^{k+1}$ is said to be a solution of $\\mathcal {E}$ up to order $k+1$ if it satisfies the following system of equations: $\\mathcal {E}^{(1)} := \\Bigl \\lbrace F(x^i,u^j_\\sigma )= 0,\\ (D_qF)(x^i,u^j_\\alpha )= 0\\ \\forall \\ q=1,\\dots , n\\Bigr \\rbrace .$ The resulting system of equations is called the first prolongation of $\\mathcal {E}$ .", "By construction, a solution of the prolongation $\\mathcal {E}^{(1)}$ is still a solution of $\\mathcal {E}$ .", "We inductively define the $l$ 'th prolongation by $\\mathcal {E}^{(l)} = (\\mathcal {E}^{(l-1)})^{(1)}\\subset J^{k+l}$ .", "It corresponds to solutions up to order $k+l$ (at a point).", "Definition 5 (Finite Type) A PDE $\\mathcal {E}\\subseteq J^k(n,m)$ is called of finite type $l$ if after $l$ prolongations all the highest order derivatives of the dependent variables can be expressed algebraically in terms of the lower order derivatives.", "A PDE is called of Frobenius type if it is of finite type 0.", "Given a PDE $\\mathcal {E}$ of finite type, it is readily seen that the space of formal solutions $\\mathcal {E}^{(\\infty )}$ is necessarily finite-dimensional.", "This implies that (under some regularity conditions) the solution space $\\text{Sol}(\\mathcal {E})$ is finite-dimensional.", "The Killing PDE is represented by a first order system $\\mathcal {E}_d\\subset J^1(N,S^dTN)$ .", "The following fundamental result is well-known, cf.", "[24] and [28].", "Theorem 6 (Killing PDE is of Finite Type) The PDE $\\mathcal {E}_d$ defining a Killing $d$ -tensor is a first order linear PDE of finite type $d$ with $\\mathop {\\rm Sol}\\nolimits (\\mathcal {E}_d)=K_d$ .", "This equation and its prolongations possess no compatibility conditions before achieving Frobenius type." ], [ "Prolongation-projection", "Let $\\mathcal {E}=\\lbrace F(x^i,u^j_\\alpha )=0\\rbrace \\subseteq J^k(n,m)$ be a PDE of order $k$ .", "Its solution up to order $k$ can be extended to order $(k+l)$ if and only if it belongs to the projection of the prolongation $\\pi _{k+l,k}(\\mathcal {E}^{(l)})\\subseteq \\mathcal {E}$ .", "In the case of equality here, every $k$ -jet solution can be extended to a $(k+l)$ -jet solution.", "In the opposite case, there is a linear combination of iterated total derivatives up to order $l$ , $\\Box (F)=\\sum _{|\\tau |\\le l} a^\\tau D_\\tau F$ , which has order $k$ .", "Definition 7 (Compatibility) A compatibility condition of $\\mathcal {E}$ is an equation defining $\\pi _{k+l,k}(\\mathcal {E})$ that is algebraically independent of $F$ and that is satisfied by all formal solutions.", "Associated to a PDE is the Cartan distribution.", "Solutions arise as integral manifolds of this distribution, cf.", "[12].", "Therefore, in the PDE setting, Frobenius theorem implies: Theorem 8 (Frobenius Theorem) Solutions of a PDE $\\mathcal {E} \\subseteq J^k(n,m)$ of finite type $l$ are determined uniquely by their $(k+l-1)$ -jets.", "If in addition $\\mathcal {E}$ has no compatibility conditions, then for every $\\xi \\in \\mathcal {E}^{(l)}$ there exists a local solution $u\\in \\mathop {\\rm Sol}\\nolimits (\\mathcal {E})$ satisfying $j_x^{k+l}u=\\xi $ .", "Note that if a PDE is of finite type, then all of its prolongations are finite type as well.", "There is the following more general claim, which holds also true in infinite type case, under the assumption of analyticity of the equation.", "Theorem 9 (Cartan's Involution) There exists $q\\in \\mathbb {N}$ such that $\\mathcal {E}^{(q)}$ is compatible.", "Thus, in regular domains, there are only finitely many compatibility conditions.", "However to find them explicitly is generally difficult, and bringing to involution in practice is a formidable computation.", "We therefore substitute searching for involution by the following criterion.", "For the finite type $l$ case: If $\\pi :\\mathcal {E}^{(r)}\\rightarrow \\mathcal {E}^{(r-1)}$ is surjective for some $r>l$ , then $\\mathcal {E}^{(r-1)}$ is compatible.", "This is especially simple for linear overdetermined PDEs: over regular domains $U\\subset N$ such $\\mathcal {E}$ are vector bundles and on each step of the prolongation-projection a compatibility condition reduces its rank; once this rank is stabilized for one step, then by Theorem REF the system is compatible, so the involution level $q$ of Theorem REF is achieved." ], [ "Algorithmic implementation", "The above criterion allows for an effective implementation of evaluation of $\\dim K_d$ for a given metric $g$ using computer algebra systems.", "The Killing PDE $\\mathcal {E}_d$ as well as its prolongations $\\mathcal {E}_d^{(k)}$ are linear in $(k+1)$ -jets of the dependent variables.", "We convert this linear system of equations to a matrix-valued function $M_k(x)$ on the spacetime.", "For our class of metrics $g$ the entries are polynomials with rational coefficients.", "Hence to make use of computer algebra software, we insert a rational point $x_0\\in N$ to obtain a matrix with rational coefficients (in this case computer calculations are exact!).", "The first thing to do is to find the points that work nicely with Cartan's prolongation-projection method.", "We call a point $x_0\\in N$ regular if the function $x\\mapsto \\mathop {\\rm rank}\\nolimits (M_k(x))$ attains its maximum at $x_0$ for all $k \\ge 0$ , that is, at each step we find the maximal number of compatibility conditions.", "Note that a regular point is generic, i.e., the set of regular points is an open dense subset of $N$ .", "A point is singular if it is not regular.", "Algorithm 1.", "(Cartan's Prolongation Method for Geodesic Flow).", "(Input: A nonnegative integer $d$ , a regular point $x_0$ .)", "Step 1.)", "Compute the Poisson bracket $\\lbrace H, I_d\\rbrace $ of a polynomial in momenta $p$ function $I_d$ with the Hamiltonian $H$ .", "Step 2.)", "Collect the coefficients of $\\lbrace H, I_d\\rbrace $ with respect to the momentum variables.", "Define the first order linear PDE $\\mathcal {E}_d:=\\lbrace F = 0: F \\in \\text{coeffs}_{p}(\\lbrace H, I_d\\rbrace ) \\rbrace $ .", "Step 3.)", "Set $k:= 0$ .", "Convert the linear system of equations $\\mathcal {E}_d^{(k)}$ w.r.t.", "the variables $\\mathcal {V}_{k+1, d} :=\\lbrace a_{\\alpha }^{i_1 \\cdots i_d} : | \\alpha | \\le k +1 \\rbrace $ into a matrix $M_k(x)$ that depends on the $x$ -coordinates.", "Substitute $x_0$ to obtain a matrix $M_k:=M_k(x_0)$ , the $k$ 'th prolongation matrix.", "Set $\\delta _k := \\text{columns}(M_k) - \\text{rank}(M_k)$ .", "If $(k\\le d)$ or ($k>d$ and $\\delta _k \\ne \\delta _{k-1}$ ), increase $k$ by 1 and repeat Step 3.", "Step 4.)", "Return $(\\delta _k, k)$ .", "(Output: The dimension of the space of Killing $d$ -tensors is $\\dim K_d = \\delta _k$ .", "The integer $k$ indicates the number of prolongations necessary to find all compatibility conditions of $\\mathcal {E}$ .)", "Proposition 10 Algorithm 1 is correct and it terminates.", "Termination is clear.", "We now justify correctness, i.e., that the algorithm computes the number of Killing $d$ -tensors.", "Turn the prolongation matrix $M_k$ into row reduced echelon form.", "The equation $\\mathcal {E}_d^{(k)}$ is linear and its rank as a bundle over $N$ , equal to $\\text{columns}(M_k)$ , counts the number of $(k+1)$ -jets of dependent variables.", "Rows of the matrix represent equations defining $\\mathcal {E}_d^{(k)}$ , so they consist of the original Killing PDE, their differential corollaries and compatibility conditions.", "Consequently, $\\delta _k$ is number of free jets (coordinates on fibers of the equation $\\mathcal {E}_d\\rightarrow N$ ).", "In view of the Frobenius theorem, each free variable corresponds to a $(k+1)$ jet-solution of the Killing PDE.", "Now consider the conditions in step 3 determining termination of the loop.", "The first part $(k \\le d)$ addresses whether the prolongation has achieved Frobenius type, see Theorem REF .", "The second part ($k>d$ and $\\delta _k\\ne \\delta _{k-1}$ ) checks whether all compatibility conditions have been computed, as guaranteed by the criterion after Theorem REF .", "Thus every $(k+1)$ jet yields a local solution." ], [ "Syzygies and Irreducible Killing Tensors", "The pointwise multiplication of functions gives rise to a linear map $K_{d_1}\\otimes K_{d_2}\\mapsto K_{d_1+d_2},\\ I_{d_1}\\otimes I_{d_2}\\mapsto I_{d_1}\\cdot I_{d_2}.$ A relation (syzygy) among Killing tensors of rank $d_1$ and $d_2$ with $d_1\\ne d_2$ is an element of the kernel of the map $K_{d_1} \\otimes K_{d_2} \\rightarrow K_{d_1 + d_2}.$ If $d_1=d_2=:d$ a relation is given by an element in the kernel of the map $S^2K_d \\rightarrow K_{2d}$ .", "A Killing $d$ -tensor ($d \\ge 2$ ) is irreducible if it is not a linear combination of the symmetric product of lower rank Killing tensors.", "The number of irreducible Killing $d$ -tensors can be found using the number of syzygies.", "We demonstrate this for Killing 2-tensors.", "The space of irreducible Killing 2-tensors can be identified with the cokernel of map $\\iota _2: S^2K_1\\rightarrow K_2$ , fitting into a short exact sequence $0 \\longrightarrow \\text{Ker}\\ \\iota _2 \\longrightarrow S^2K_1 \\rightarrow K_2\\longrightarrow \\text{Coker}\\ \\iota _2 \\longrightarrow 0.$ The space of irreducible Killing 3-tensors can be identified with the cokernel of the map $\\iota _3: K_1 \\otimes K_2\\rightarrow K_3$ , etc.", "The number of syzygies among Killing tensors is found as follows.", "(We use the notation $\\text{Taylor}(a(x), x_0, k)$ for the Taylor polynomial of the function $a$ around $x_0$ up to order $k$ .)", "Algorithm 2.", "(Finding Relations among Killing Tensors).", "(Input: Nonnegative integers $d_1, d_2$ , a regular point $x_0$ .)", "Step 1.)", "For $s=1,2$ : run algorithm 1 obtain $\\dim K_{d_s}$ and the number of prolongation $k_s$ needed to achieve compatibility.", "Consider the polynomial $I_{k_s+1, d_s} := \\text{Taylor}(a^{i_1 \\cdots i_{d_s}}, x_0,k_s+1)\\cdot p_{i_1} \\cdots p_{i_{d_s}}$ .", "Step 2.)", "Consider the linear algebraic system of equations $\\lbrace \\text{Taylor}(c, x_0, k_s) = 0: c \\in \\text{coeffs}_p(\\lbrace H, I_{k_s+1, d_s}\\rbrace ) \\rbrace $ on the variables $\\mathcal {V}_{k_s+1, d_s}(x_0):= \\lbrace a^{i_1 \\cdots i_{d_s}}_{\\alpha }(x_0) : |\\alpha | \\le k_s + 1 \\rbrace $ for $s=1,2$ .", "Solve these linear equations and substitute the corresponding solutions into $I_{k_s +1,d_s}$ to obtain the truncated integrals $I_{k_s +1 , d_s}^j$ for $1 \\le j \\le \\dim K_{d_s}$ .", "Step 3.)", "Set $T:= \\sum _{l_1=1}^{\\dim K_{d_1}}\\sum _{l_2=1}^{\\dim K_{d_2}} c_{l_1,l_2}\\ I_{k_1+1,d_1}^{l_1}\\cdot I_{k_2+1,d_2}^{l_2}.$ Define $S :=\\lbrace \\text{Taylor}(c, x_0, d_1+d_2) : c \\in \\text{coeffs}_p(T)\\rbrace )$ .", "Step 4.)", "Solve the linear algebraic system of equations $ \\lbrace F = 0 : F\\in \\text{coeffs}_{x}(S) \\rbrace $ in terms of the coefficients $c_{l_1,l_2}$ , and denote the resulting solution space $R$ .", "Step 5.)", "Return $R$ and $\\dim R$ .", "(Output: Relations among Killing tensors of rank $d_1$ and $d_2$ ; $\\#$ (indep) syzygies $=\\dim R$ .)", "Proposition 11 Algorithm 2 is correct and it terminates.", "For $d \\ge 1$ , consider the Killing PDE $\\mathcal {E}_d\\subseteq J^1$ .", "A $(k+1)$ -jet $j_{x_0}^{k+1}u$ of a vector-function $u= (a^{i_1\\cdots i_d}(x))$ can be identified with the Taylor polynomial $I_{k+1,d} =\\text{Taylor}(a^{i_1 \\cdots i_d}, x_0, k+1) p_{i_1} \\cdots p_{i_d}$ .", "Under this correspondence, we have that $j_{x_0}^{k+1}u\\in \\mathcal {E}^{(k)}$ if and only if $\\lbrace H,I_{k+1,d}\\rbrace $ vanishes up to order $k$ at $x_0$ .", "These observations explain steps 1 and 2.", "By thm:killingfinitetype $K_d$ is determined by $d$ -jets in the sense that we can compute all jets of a Killing $d$ -tensor at a point if we know its $d$ -jet.", "Thus, in step 3 we must include the jets up to order $d_1 + d_2$ in order to determine uniquely the corresponding $(d_1 + d_2)$ -tensor.", "Application.", "In practice we apply algorithm 2 as follows.", "First, using algorithm 1 we compute the dimensions of $S^2K_1$ and $K_2$ .", "Then we use algorithm 2 to determine the dimension of the kernel $\\text{Ker}\\ \\iota _2$ .", "Finally, the number of (lin.", "independent) irreducible Killing 2-tensors is given by $\\dim \\text{Coker}\\ \\iota _2 = \\dim K_2 - \\dim S^2K_1 + \\text{Ker}\\ \\iota _2.$ This method can be readily generalized to higher order Killing tensors.", "Regular and singular points.", "Even though the regular points are dense, it is difficult to verify (in practice) that a given point is regular.", "Thus, we must be careful in order to get rigorous results.", "For a singular point, algorithm 1 gives an upper bound on the number of Killing tensors.", "The number of syzygies imply lower bounds on the number of Killing tensors (indeed, the syzygies imply the number of reducible Killing tensors).", "Thus, whenever the algorithms suggest the existence of an irreducible Killing tensor it is important to find it explicitly.", "(For our metrics $g$ it turns out to be possible to find the irreducible Killing tensors explicitly using Maple's pdsolve.)" ], [ "Note on the computability of the algorithm.", "We briefly discuss the computational difficulties associated with the proposed method and how we deal with them.", "Dimension of the prolongation matrix $M_k$ from algorithm 1 equals $\\text{rows}(M_k) = {n+d \\atopwithdelims ()d+1}\\cdot {n+k\\atopwithdelims ()n},\\quad \\text{columns}(M_k) = {n+d-1\\atopwithdelims ()d}\\cdot {n+k+1\\atopwithdelims ()n}.$ In particular, we see that the number of rows grows faster with $k$ than the number of columns.", "We highlight several elements that have made the computer implementation more efficient: [leftmargin=*] (LinBox).", "The LinBox package [18] in Sage allows for incredibly fast rank computations of large sparse integer matrices.", "For example, computing the rank of the quartic prolongation matrix $M_{19}$ for metric 2 with size $(495880) \\times (371910) $ took less than an hour.", "In comparison, rank computations of smaller matrices (say 50000 by 40000) would take several days in Maple or not give a result at all.", "Thanks to LinBox, the time to compute the ranks is negligible.", "Generating a prolongation matrix takes by far the longest time of the steps in algorithm 1.", "(Exploiting Sparsity.)", "The prolongation matrices $M_k$ that we encounter here are sparse (with density $<$ 0.001).", "It is important that the generation of the matrix reflects this.", "We generate the initial matrix with all entries zeroes and then substitute the nonzero values.", "(Combinatorial Description of Prolongations.)", "For the quartic case, we used a combinatorial description of the prolongation equations.", "We demonstrate this for metric 2.", "Since $I_4$ is of degree 4, we have that $\\lbrace H, I_4\\rbrace $ is of degree 5 in momenta.", "Thus, we can write $\\lbrace H, I_4\\rbrace = \\text{coeff}_{\\tau } p^{\\tau }$ where $p^{\\tau } =p_1^{\\tau _1}p_2^{\\tau _2}p_3^{\\tau _3}p_4^{\\tau _4}$ .", "Given a multi-index $\\tau $ of length 5, we obtain the $p^{\\tau }$ -coefficient in terms of the coefficients of $I$ : $\\begin{split}\\hphantom{aaa}\\text{coeff}_{\\tau }(\\lbrace H, I_4\\rbrace ) & = 2 \\partial _1(a^{\\tau - 1_1}) + 2\\partial _2(a^{\\tau - 1_2}) - 2 \\partial _4(a^{\\tau - 1_3}) -2\\partial _3(a^{\\tau - 1_4}) \\\\& + 4x^3((x^1)^2 + (x^2)^2) \\partial _4(a^{\\tau - 1_4}) - 2((x^1)^2 +(x^2)^2) \\frac{(\\tau + 1_3 - 2 \\cdot 1_4)!", "}{(\\tau - 2\\cdot 1_4)} \\\\& -4 x^1 x^3 \\frac{(\\tau +1_1 - 2 \\cdot 1_4)!", "}{(\\tau - 2\\cdot 1_4)!", "}a^{\\tau + 1_1 - 2\\cdot 1_4} -4 x^2 x^3 \\frac{(\\tau +1_2 - 2 \\cdot 1_4)!", "}{(\\tau - 2\\cdot 1_4)!}", "a^{\\tau + 1_2 - 2\\cdot 1_4}.\\end{split}$ Using the Leibniz rule for multi-index notation, we can subsequently determine the general expression for the derivative $\\partial ^{\\alpha }(\\text{coeff}_{\\tau }(\\lbrace H, I_4\\rbrace ))$ , where $\\alpha $ is a multi-index.", "In this way we obtain the equations of the prolongation as a function of the multi-indices $\\tau $ and $\\alpha $ .", "This combinatorial description significantly reduces the time needed to generate the equations in Maple, especially as the order increases.", "This approach is most beneficial for Hamiltonians which are polynomials of low order in the independent $x$ -variables.", "(For the Kerr metric, for example, these combinatorics would be unfeasible.)" ], [ "Computations and results", "Here we discuss the results of concrete computations with the above algorithms.", "We begin with investigations of the special cases of pp-waves and Wils metric and then discuss the general case." ], [ "Conformally Flat pp-Waves", "These are given by the following formula: $g= 2dx^3dx^4 + \\bigl (f(x^3)((x^1)^2+(x^2)^2)\\bigr )(dx^3)^2 -(dx^1)^2 -(dx^2)^2$ Sippel and Goenner classified pp-waves in terms of their isometry groups [22].", "For conformally flat pp-waves there are three classes: $f(x^3)=c$ , $f(x^3)=c(x^3)^{-2}$ and the generic case with $\\dim K_1=6$ .", "We apply our prolongation-projection algorithm to the following four metrics (rescaling of $f$ does not play a role for the first three metrics): ${\\rm (i)}\\ f(x^3)=1,\\quad {\\rm (ii)}\\ f(x^3)=x^3,\\quad {\\rm (iii)}\\ f(x^3)=(x^3)^{2},\\quad {\\rm (iv)}\\ f(x^3)=2 (x^3)^{-2}.$ If two subsequent values $\\delta _k$ , $\\delta _{k+1}$ are equal (with $k \\ge d$ ), the sequence of $\\delta $ -values stabilizes and we can read off the number of Killing $d$ -tensors.", "In the table this is shown by circling this $\\delta $ -value.", "Table: NO_CAPTIONTable: Left up: Metric (i) f(x 3 )=1f(x^3) = 1;            Right up: Metric (ii) f(x 3 )=x 3 f(x^3)=x^3.Left down: Metric (iii) f(x 3 )=(x 3 ) 2 f(x^3)=(x^3)^{2};               Right down: Metric (iv)f(x 3 )=2(x 3 ) -2 f(x^3)=2(x^3)^{-2}.We see that metrics 1 and 4 have 7-dimensional isometry, which is consistent with the classification by Sippel and Goenner.", "Note that for the quartic case of metrics 2 and 3 we have to go all the way to the 19'th prolongation of the Killing PDE.", "The number of equations and variables at this stage are so large that it is unlikely that we can compute the number of Killing 5-tensors with present computational powers." ], [ "Syzygies and irreducible Killing tensors for pp-waves", "In order to find the number of irreducible Killing tensors, we have to take into account the number of syzygies among the Killing tensors.", "We demonstrate this for metric 2 (the other cases are similar).", "Consider the following short exact sequence $0 \\longrightarrow \\underbrace{\\text{Ker}\\ \\iota _2}_{\\text{1syzygy}} \\rightarrow \\underbrace{S^2K_1}_{21-\\text{dim.", "}}\\xrightarrow{} \\underbrace{K_2}_{22-\\text{dim}.}", "\\longrightarrow \\underbrace{\\text{Coker}\\ \\iota _2}_{\\text{2 irreducible Killing$2$-tensors}} \\rightarrow 0$ Algorithm 2 gives $\\dim \\text{Ker}\\ \\iota _2 = 1$ , and so there are 2 irreducible Killing 2-tensors.", "Next, we consider $0 \\longrightarrow \\underbrace{\\text{Ker}\\ \\iota _3}_{\\text{70 syzygies}} \\rightarrow \\underbrace{K_1 \\otimes K_2}_{132-\\text{dim.", "}}\\xrightarrow{} \\underbrace{K_3}_{62-\\text{dim}.}", "\\longrightarrow \\underbrace{\\text{Coker}\\ \\iota _3}_{\\text{0 irreducible Killing$3$-tensors}} \\rightarrow 0$ Algorithm 2 gives $\\dim \\text{Ker}\\ \\iota _3 = 70$ and so there are no irreducible Killing 3-tensors.", "Since there are no irreducible Killing 3-tensors, the source space of $\\iota _4$ is the second symmetric power $S^2K_2$ .", "(If there were irreducible Killing 3-tensors, then the source space would be $K_1\\otimes K_3\\oplus S^2K_2$ .)", "Thus we obtain the short exact sequence $0 \\longrightarrow \\underbrace{\\text{Ker}\\ \\iota _2}_{\\text{105 syzygies}} \\rightarrow \\underbrace{S^2K_2}_{253-\\text{dim.", "}}\\xrightarrow{} \\underbrace{K_4}_{148-\\text{dim}.}", "\\longrightarrow \\underbrace{\\text{Coker}\\ \\iota _4}_{\\text{0 irreducible Killing$4$-tensors}} \\rightarrow 0$ There are 105 syzygies, it follows that there are no irreducible Killing 4-tensors.", "For metrics ${\\rm (i)}$ , ${\\rm (ii)}$ and ${\\rm (iii)}$ we obtain that there exists one irreducible Killing 2-tensor, in addition to the Hamiltonian.", "Actually, we can explicitly write this Killing 2-tensor as follows: $J:= -x^3 H + x^1 p_1 p_4 + x^2 p_2 p_4 + 2 x^4 p_4^2.$ For metric ${\\rm (iv)}$ the only irreducible Killing 2-tensor is the Hamiltonian $H$ , i.e., the Killing 2-tensor $J$ is reducible in this case (due to the existence of an extra Killing vector).", "In the general case (REF ) for $f(u)\\ne c,cu^{-2}$ the Killing vectors are the following: $I_1= p_1x^2-p_2x^1,\\ I_2=p_4,\\ I_{3,4}=a_{1,2}(x^3)p_1+a_{1,2}^{\\prime }(x^3)x^1p_4,\\ I_{5,6}=a_{1,2}(x^3)p_2+a_{1,2}^{\\prime }(x^3)x^2p_4,$ where $a_i$ ($i=1,2$ ) are fundamental solutions of the linear second order ODE $a^{\\prime \\prime }+fa=0$ , i.e., solutions satisfying the initial conditions $a_1(0)=1,a^{\\prime }_1(0)=0$ , $a_2(0)=0,a^{\\prime }_2(0)=1$ .", "The Hamiltonian is equal to $H=2p_3p_4-p_1^2-p_2^2-((x^1)^2+(x^2)^2)f(x^3)p_4^2$ and the other quadratic integral $J$ is given by (REF ) (also for general $f$ ).", "These results are consistent with the following theorem by Keane and Tupper [8] that was proven using the Koutras algorithm [10] (our approach is different).", "Theorem 12 ([8]) A conformally flat pp-wave with $\\dim K_1=6$ or with $f(x^3)=c$ admits an irreducible Killing 2-tensor, independent of the (irreducible) Hamiltonian $H$ .", "By using our computational algorithm we can also establish the nonexistence results of higher order Killing tensors for these conformally flat pp-waves.", "Theorem 13 A conformally flat pp-wave (REF ) with $f(u)=cu^m$ , $m=0,1,2$ , or $f(u) = 2u^{-2}$ , admits no irreducible Killing 3- and 4-tensors.", "The result follows straightforwardly from the above computations and a rescaling argument.", "Corollary 14 For a generic conformally flat pp-wave (REF ) all 3- and 4- Killing tensors are combinations of Killing vectors (REF ), the Hamiltonian $H$ (REF ) and the Killing 2-tensor $J$ (REF ).", "Here $f$ is generic in $C^{13}$ topology for Killing 3-tensors and in $C^{19}$ topology for Killing 4-tensors, see Table 1 for $k=k_d$ , however we believe that also holds in lower regularity by the approach of [14].", "It follows from our computations and algebraic dependence of the matrix $M_k$ on $j^{k+1}f$ that $\\dim K_i$ ($i=2,3,4$ ) is upper semi-continuous in this jet.", "Hence, for a generic $f(x^3)$ the dimension of $K_2,K_3,K_4$ are as indicated in the third term of the above short exact sequences.", "Due to full control of $K_1,K_2$ the second terms have dimensions as indicated.", "Dimension of the first term is also upper semi-continuous, so for a generic $f(x^3)$ we have at most the indicated number of syzygies.", "In fact, this number is realizable as follows.", "In the case of Killing 2-tensor (first short exact sequence) the only syzygy is (verifying this exploits constancy of the Wronskian of $a_1,a_2$ ) $\\mathfrak {S}_2:\\quad I_1I_2+I_3I_6-I_4I_5=0.$ For Killing 3-tensor (second short exact sequence) the only 6 syzygies are $I_j\\cdot \\mathfrak {S}_2$ ($1\\le j\\le 6$ ).", "To explain dimension 70 of the first term, note that kernel of the symmetrization operator $K_1\\otimes S^2K_1/K_1\\otimes \\mathfrak {S}_2\\rightarrow S^3K_1$ is 64-dimensional.", "Similarly one justifies the case of Killing 4-tensor (third short exact sequence).", "Actually, we can also obtain the claim from the fact that the functional rank of 8 functions $I_j,H,J$ is 7, while that of $I_j$ is 5.", "Thus no syzygies can involve $H,J$ and the only syzygy among 6 Killing vectors $I_i$ is given by $\\mathfrak {S}_2$ ." ], [ "Absence of Killing Tensors for the Wils Metric", "The Wils metric is given by $g = 2x^1dx^3dx^4 -2x^4dx^1dx^3 +\\bigl (f(x^3)x^1((x^1)^2+(x^2)^2)-(x^4)^2\\bigr )(dx^3)^2 -(dx^1)^2-(dx^2)^2.$ We apply our prolongation-projection algorithm to the following three cases: $f(u)=u^m$ , $m=0,1,2$ .", "The results are displayed in the following table.", "Table: Metric (i) f(x 3 )=1f(x^3)=1;        Metric (ii) f(x 3 )=x 3 f(x^3)=x^3;                Metric (iii) f(x 3 )=(x 3 ) 2 f(x^3)=(x^3)^2.Theorem 15 The Wils metric (REF ) for $f(u)=u^m$ , $m=0,1,2$ , admits no Killing tensors up to degree 6 except for powers of the Hamiltonian.", "This statement follows directly from Table 2.", "It also implies that for generic values of the functional parameter $f$ there are no lower degree Killing tensors.", "Now we want to be more specific on those exceptional parameters.", "Theorem 16 The Wils metric admits Killing vectors if and only if $f$ is of the form $f(x^3) = (c_0+c_1x^3+c_2(x^3)^2)^{-2}.$ In this case the Killing vector is unique up to scale and is given by the formula $X:= (c_0+c_1x^3+c_2(x^3)^2)\\,\\partial _{x^3}-(2c_2x^1+c_1x^4+2c_2x^3 x^4)\\ \\partial _{x^4}.$ In order to simplify the calculations we evaluate at $x^1=1, x^2=2, x^4=4$ but leave $x^3$ general.", "Step 1 and 2.)", "Using, the equations defining the PDE $\\mathcal {E}$ , we express the 1-jets $a^1_1$ , $a^1_2$ , $a^1_3$ , $a^1_4$ , $a^2_3$ , $a^2_4$ , $a^3_2$ , $a^3_4$ , $a^4_1$ , $a^4_2$ in terms of the free variables $a^1$ , $a^2$ , $a^3$ , $a^4$ , $a^2_1$ , $a^2_2$ , $a^3_1$ , $a^3_3$ , $a^4_3$ , $a^4_4$ and the function $f(x^3)$ .", "Step 3.)", "For the first prolongation $\\mathcal {E}^{(1)}$ , we can express all 2-jets in terms of lower order jets without making any assumptions on $f$ .", "Step 4.)", "Consider $\\mathcal {E}^{(2)}$ .", "If we assume that $f \\ne 0$ , we obtain the following compatibility conditions: $a^3_1=0,\\ a^3_3=-\\frac{a^1f+f^{\\prime }a^3}{2f},\\ a^4_4=0.$ We are left with 7 free jet variables.", "For $\\mathcal {E}^{(3)}$ , we obtain the additional compatibility conditions: $a^1=0,\\ a^2_2=a^4_3,\\ a^2_1=\\frac{2a^2f^2-4a^3ff^{\\prime }+2a^3ff^{\\prime \\prime }-3a^3(f^{\\prime })^2}{6f^2}.$ We are left with 4 free variables.", "The prolongation $\\mathcal {E}^{(4)}$ gives three additional compatibility conditions: $a^4_{3} =0$ and two expressions for $a^2$ and $a^4$ .", "Only 1 free variable $a^3$ remains, and the next prolongation $\\mathcal {E}^{(5)}$ does not give an additional compatibility condition if and only if $f$ is a solution of the ODE $f^{\\prime \\prime \\prime } = \\frac{18ff^{\\prime }f^{\\prime \\prime }-15(f^{\\prime })^3}{4f^2}.$ Resolving this ODE gives the required formula (REF ).", "Expression (REF ) follows.", "The following theorem is proven in the same manner, but the number of steps is larger, so the proof is omitted.", "Theorem 17 The Wils metric has Killing 2-tensors if and only if it has nontrivial Killing vectors.", "This happens only for the functional parameter (REF ); in this case, denoting $I_1=\\langle X,p\\rangle $ the linear integral corresponding to (REF ), the general quadratic integral is a linear combination $k_1I_1^2+k_2H$ ." ], [ "General Koutras-McIntosh metrics", "Investigation of the general metric (REF ) follows the same scheme.", "First of all, the computation in the previous section implies that the matrix $M_k$ of the prolonged Killing PDE for degree $d\\le 6$ tensors has minimal possible value for $\\delta _k$ , i.e., 0 for odd $d$ and 1 for even $d$ .", "This implies Theorem REF .", "To obtains Theorems REF and REF we can perform general computation with symbolic matrix for the prolongation $\\mathcal {E}^{(6)}$ when $d=1$ and $\\mathcal {E}^{(7)}$ when $d=2$ .", "The matrix $M_k$ has size $1260\\times 840$ for $d=1$ and $4200\\times 3300$ for $d=2$ .", "To compute its rank we use the idea exploited in [17], namely successively identifying rows or columns with few non-zero terms (this means $\\le 2$ for $d=1$ and $\\le 8$ for $d=2$ ) and doing Gauss elimination, while storing the involved factors to check their vanishing separately.", "This gives the splitting $a=0$ or $b=0$ and the rest follows from Theorem REF .", "In fact, this computation also yields equation (REF ).", "The exceptional functional parameters $f(u)$ in (REF ) up to transformations $u\\rightarrow ku+b$ (change of coordinates: $x^1\\mapsto \\lambda x^1$ , $x^2\\mapsto \\lambda x^2$ , $x^3\\mapsto kx^3+b$ , $x^4\\mapsto \\lambda x^4/k$ , $g\\mapsto \\lambda ^2 g$ , $f\\mapsto f/(\\lambda k^2)$ ) give the following different cases $f(u)=1,\\quad f(u)=u^{-1},\\quad f(u)=c u^{-2},\\quad f(u)=u^{-4},\\quad f(u)=|u^2\\pm 1|^{-2}.$ In each of these cases one can directly verify there are no irreducible Killing 3- or 4-tensors (for the middle case this was only verified for a generic parameter $c$ ), i.e.", "all of them are algebraic combinations of $I_1$ and $H$ ." ], [ "Outlook", "In this paper we obtain the nonexistence of Killing tensors of degrees $d$ up to 6 for the Koutras-McIntosh spacetimes for generic parameters.", "This complements the previous result on the nonexistence of Killing vectors [11].", "The problem of existence of higher order $d>6$ Killing tensors remains open.", "The size of the involved matrices ($163800 \\times 152880$ for $d=6$ ) does not allow further computational progress, and we have to stress that our success for metrics (REF ) is related to sparsity of the corresponding matrices $M_k$ and rationality of their entries in coordinates and parameters.", "The complexity of computations carried here is much higher than that in preceeding works [13], [17], [25]; actually those possessed Killing vectors allowing to reduce the PDE setup to that on a 2-dimensional manifold, while our setup here is fully 4-dimensional (that is why the size of the matrix $M_k$ of $\\mathcal {E}_d^{(k)}$ grows much faster).", "Other works [6], [8], [9], addressing Killing 2-tensors, have in similar vein reductions to ODEs (that is, differential equations on a 1-dimensional manifold), so our work on higher degree $d$ Killing tensors is apparently novel.", "One may envision that the following approach is feasible for large $d$ .", "Consider the Killing PDE $\\mathcal {E}_d$ with $\\mathop {\\rm Sol}\\nolimits (\\mathcal {E}_d)=K_d$ .", "This is an overdetermined system and a compatibility analysis gives the dimension of $K_d$ depending on certain rank invariants.", "Those depend on vanishing of some relative invariants.", "Since the construction involves only invariant algebraic operations and all absolute polynomial invariants vanish, there are only few possibilities and the answer for higher $d$ might be the same as that for $d=1$ .", "This is indeed confirmed by what we have investigated.", "The nonexistence of polynomial integrals of low degree raises the question whether the geodesic flow of metrics (REF ) is integrable.", "Depending on the class of admissible integrals the methods to approach this problem are: differential Galois theory, Painlevé test, numerical simulations.", "None of these have been done yet.", "Acknowledgment.", "The authors thank Simon King from the University of Jena for the crucial suggestion to use the LinBox package.", "BK thanks Vladimir Matveev for useful discussion and collaboration within RCN-DAAD project 2020-2021 “Differential-Geometric Structures: Invariants and Integrals”.", "WS thanks his fellow student Alessandro Schena for help regarding the implementation of the `Exploiting Sparsity' point in Maple.", "The research leading to our results has received funding from the Norwegian Financial Mechanism 2014-2021 (project registration number 2019/34/H/ST1/00636), the Polish National Science Centre (NCN grant number 2018/29/B/ST1/02583), and the Tromsø Research Foundation (project “Pure Mathematics in Norway”).", "This work is an extension and elaboration of [23]." ] ]
2207.03474
[ [ "Automatic Synthesis of Neurons for Recurrent Neural Nets" ], [ "Abstract We present a new class of neurons, ARNs, which give a cross entropy on test data that is up to three times lower than the one achieved by carefully optimized LSTM neurons.", "The explanations for the huge improvements that often are achieved are elaborate skip connections through time, up to four internal memory states per neuron and a number of novel activation functions including small quadratic forms.", "The new neurons were generated using automatic programming and are formulated as pure functional programs that easily can be transformed.", "We present experimental results for eight datasets and found excellent improvements for seven of them, but LSTM remained the best for one dataset.", "The results are so promising that automatic programming to generate new neurons should become part of the standard operating procedure for any machine learning practitioner who works on time series data such as sensor signals." ], [ "Introduction", "Time series or sequence data is abundant in machine learning and range from signal processing to stock prices and more complex data such as natural language.", "Indeed, the entire life of a human being is also a time series dataset generated by the five senses.", "This paper focuses on improving LSTM neurons [17] since they both are theoretically interesting and also one of the most popular and effective machine learning tools.", "By using the ADATE automatic programming system [25], we generated new neurons for each specific dataset and found that they were up to three times better than LSTM as measured by cross entropy on test data.", "We used eight datasets with between 22 and 500 timesteps and found the biggest improvement for the 500 timestep dataset.", "The main novelties of the ADATE Recursive Neurons (ARNs) are as follows.", "ARNs often contain skip connections like in Highway nets and ResNet, but through the time dimension.", "Up to four memory states per neuron instead of just one as in the LSTM.", "Novel activation functions built from, for example, small quadratic forms and sequences of a set of three predefined and well known activation functions.", "A given neuron may use linear combinations of the states of other neurons.", "Differentiating between recurrent output as well as state values from “self” and “all others”.", "We will now give an overview of the paper.", "First, we present related work on automatic programming and evolution of recurrent neurons.", "The next section describes how to represent ARNs in a small and purely functional subset of the Standard Meta Language (SML) [23] and also gives a compact and purely functional definition of the classic LSTM as an illustration.", "We then provide experimental results for eight medium size datasets that typically are found in practical sensor data analysis applications.", "We finish by some conclusions and a glimpse of the almost infinite possibilities for future work that are enabled by the technology in this paper.", "The appendix contains seven examples of ARNs to exhibit some of their novelties." ], [ "Related work", "Long short-term memory (LSTM) was originally presented in Sepp Hochreiter's masters thesis [17] with Jürgen Schmidhuber as the advisor and made more well known by Hochreiter and Schmidhuber [17].", "It remains a quite popular and competitive technology also 25 years later, which is remarkable in a rapidly developing field such as machine learning.", "We assume that the reader already is familiar with LSTMs and will not describe them here.", "Some newer and prominent recurrent neurons that sometimes outperform LSTM include Gated Recurrent Units [8] and Independently Recurrent Neural Nets [22]." ], [ "Using evolutionary computation to synthesize recurrent neurons", "Galvan and Mooney [12] provide an extensive review of neuroevolution and show that the field primarily deals with using evolutionary methods for hyperparameter or architecture and topology search for entire neural nets using a predefined set of neuron types.", "They do not mention any work that evolves the neurons themselves.", "An early attempt to evolve neurons using ADATE was made by Berg et.", "al.", "[6], but these neurons were a special type of spiking neurons for segmentation of gray scale images containing noisy rectangles.", "Although reasonable improvements were found, the results are not practically competitive and have been superseded by modern CNN based nets for image processing.", "A key distinction between different approaches to neuron evolution is whether the goal is to find a general neuron that is better for a large class of datasets or whether it is to produce more specialized neurons that work best for an application specific class of datasets.", "Our goal is to produce superior and application specific neurons whereas Bayer et.", "al  [5] and Jozefowicz et.", "al.", "[18] try to produce more general replacements for the LSTM, which may be much more difficult to achieve.", "Of course, the disadvantage with application specific neurons, for example neurons specialized for high frequency trading, is that it is more difficult to replace the neuron itself in the well known neural toolboxes than it is to just optimize the hyperparameters and the architecture.", "A possible advantage though is that much bigger application specific improvements may be found.", "Thus, we argue that all three of overall architecture, hyperparameters and neuron design need to be optimized for the best results.", "Bayer et.", "al  [5] mention another distinction, namely if a very indirect representation such as DNA is employed to represent neurons or whether transformations such as so-called “mutations” are performed directly on the neuron circuits.", "They use the latter approach and define a number of well designed transformations on a directed acyclic graph representation of neurons.", "Thus, they build an evolutionary system from scratch expressly for evolving LSTM replacements.", "Our approach is a bit different since we use the universal ADATE automatic programming system and do not code any evolutionary search nor any transformations specifically for neuron evolution.", "Jozefowicz et.", "al.", "[18] also build their evolutionary computation system from scratch and excel in neural architecture design and hyperparameter search.", "It is obvious from the paper that they have deep expertise in these areas.", "In contrast to us, they perform hyperparameter search on-the-fly during the evolution.", "As a result of this and other experimental choices, they evaluate a number of new neuron candidates that is about five orders of magnitude smaller than we did with ADATE." ], [ "Automatic design of algorithms through evolution", "Automatic design of algorithms through evolution (ADATE) [25] was originally developed for automatic synthesis of symbolic and recursive functional programs, but a few years later also given simple mechanisms for optimizing floating point constants.", "It writes programs in a very small subset of SML [23], which was conceived by Robin Milner as a meta language for symbolic logic [14].", "As we will see in this paper, it is also quite suitable for meta machine learning.", "Some other and fundamentally different approaches to inductive programming, that are more suitable for explainable machine learning than ADATE, are discussed in the JMLR special issue edited by [20].", "ADATE has the ability to invent help functions as they are needed and to perform so-called lifting and distribution transformations on case- and let-expressions as well as many other semantics preserving and useful program transformations [25].", "However, the details of these are beyond the scope of the current paper.", "ADATE needs a specification of the problem to be solved.", "A specification contains the following primary ingredients.", "Algebraic datatypes, for examples various lists and trees.", "A signature for the function f that is to be synthesized and optionally also an initial definition of that function.", "An evaluation function that tests a synthesized function on a number of training inputs.", "Note that f may occur as a small part of a big program and that calculating f typically means to run this program on the training inputs.", "Thus, ADATE performs symbolic reinforcement (meta-)learning.", "Given the specification, ADATE runs on a cluster where many programs are transformed and evaluated in parallel.", "The result is a Pareto front of gradually more syntactically complex and better programs.", "Syntactic complexity is the sum of the base-2 logarithm of the occurrence probabilities of the symbols in all nodes in the syntax tree of a program.", "It falls on the human operator of ADATE to select a suitable program from the Pareto front.", "Below, we will make it simple and always choose the one that has the lowest mean squared error or cross entropy on validation data." ], [ "How to represent ARNs as functional programs", "We will first explain how to represent neurons and then explain how a neuron is a transition function from one timestep to the next.", "Then, we will give a functional program corresponding to an LSTM neuron as an example." ], [ "Datatypes", "A key ingredient in neural nets is linear combination of a set of floating point values and we will start by explaining how to represent this using algebraic datatypes.", "For example, consider a linear combination of two values $x_1$ and $x_2$ using weights $w_1$ and $w_2$ and bias $b$ .", "$w_1 x_1 + w_2 x_2 + b$ In our algebraic datatype, we abstract away the weights and represent a linear combination using the SML type linComb, defined below, with two constructors bias and cons.", "The * is Cartesian product and the | introduces a sum type.", "datatype linComb = bias | cons of real * linComb This is essentially the type of lists that always contain at least one value, namely a bias.", "The type real represents floating point numbers in spite of its SML name.", "The linear combination above would correspond to the following list.", "cons( x1, cons( x2, bias ) ) At this point, the reader perhaps wonder how on earth the weights will be specified.", "The answer is that we only allow a limited number of weight mappings, where each mapping is indicated by a function called lc$i$ for $i = 0, 1, \\ldots , n$ for $n+1$ mappings.", "The current implementation has $n=4$ , which means that there will be five different sets of forward, recurrent and peep weight matrices.", "The expression below will be translated by our compiler to the linear combination above assuming that mapping 0 has weights $w_1$ and $w_2$ and bias $b$ .", "lc0( cons( x1, cons( x2, bias ) ) ) However, if we were to use say lc1 instead of lc0, another bias and set of weights would be used after compilation." ], [ "Transition function", "It is now time to start explaining how to represent recursive neural net neurons and their internal and external connections.", "To introduce our notation, let us first consider the transition function $f$ of type $\\mathbb {R}^2 \\rightarrow \\mathbb {R} $ for a simple RNN neuron with input $ x^{(t)} $ and output $ y^{(t)} $ at time $t$ .", "Thus, the inputs and outputs of $f$ are as follows.", "$f( x^{(t)}, y^{(t)} ) = y^{(t+1)}$ An LSTM neuron has an internal state $s$ which means that its type is $\\mathbb {R}^3 \\rightarrow \\mathbb {R}^2 $ .", "$f( x^{(t)}, s^{(t)}, y^{(t)} ) = ( s^{(t+1)}, y^{(t+1)} )$ An ADATE recursive neuron (ARN) has four internal states instead of just one which implies that it could be a function with the following signature.", "$f( x^{(t)}, s_0^{(t)}, s_1^{(t)}, s_2^{(t)}, s_3^{(t)}, y^{(t)} ) =( s_0^{(t+1)}, s_1^{(t+1)}, s_2^{(t+1)}, s_3^{(t+1)}, y^{(t+1)} )$ However, we will differentiate between the output from the neuron itself at timestep $t$ and the linear combination of the outputs of the other neurons in the same layer.", "This “recognition of self” turns out to be an important feature of ARNs as shown by the experimental results in Section REF .", "Experimentally, we also found that skip connections through time were invented by ADATE and it is easy to see that this is enabled by having many internal states instead of just one.", "For example, if we wish the output at timestep $t$ to be available as an input at timestep $t+2$ , we can choose $s_1^{(t+1)}$ to $y^{(t)}$ and $s_2^{(t+1)}$ to $s_1^{(t)}$ .", "Many other and much more intricate ways of curating information through time are possible and have been invented by ADATE." ], [ "Functional programming definition of an ARN", "In order to explain the signature of an ARN, we will switch to functional programming notation and start by explaining the input parameters.", "InputsLC : linComb.", "The list of inputs.", "SelfPeep0 : real.", "State $s_0^{(t)}$ .", "SelfPeep1 : real.", "State $s_1^{(t)}$ .", "SelfPeep2 : real.", "State $s_2^{(t)}$ .", "SelfPeep3 : real.", "State $s_3^{(t)}$ .", "SelfOutput : real.", "The current output value $y_{t}$ .", "OtherPeepsLC : linComb.", "The list of the $s_0^{(t)}$ values of all other nodes in the layer.", "OtherOutputsLC : linComb.", "The list of the $y^{(t)}$ values of all other nodes in the layer.", "Finally, we will explain how variables are bound to values of expressions in the small subset of SML used in ADATE.", "The syntax is a bit unusual and relies on case-expressions for this.", "For example, the case-expression below means that the variable V will represent the value of the expression E1 and that its scope is E2.", "case E1 of V => E2 Of course, these case expressions are used to construct functional programs that correspond to directed acyclic graphs (DAGs) in neural circuit diagrams.", "Consider the usual LSTM neuron with peepholes in Figure REF that is a direct translation of the LSTM circuit diagram in [16].", "We now have everything that is needed to understand the functional programming definition of this neuron.", "Figure REF contains five case-expressions before giving the output tuple in which only the first and the last fields are used and the three in the middle just contain dummy values, in this case $0.0$ , since they are not needed for LSTM.", "Recall that the output quintuple first contains the four next states and then the next output as its last field.", "Since all our datasets are centered and scaled, it turns out that biases are almost totally redundant in the LSTM, which means that they have been omitted in Figure REF .", "Figure: The LSTM neuron with peepholes as a functional program.As another example, we will explain a rather simple neuron syntesized by ADATE and that comes from the Pareto front for Double pendulum discussed later in Section REF .", "The program corresponding to the neuron is given in Figure REF .", "The auxiliary function g has been invented by ADATE.", "Its purpose is to bind S0 to the tanh expression.", "The return value of g is a quintuple where the last field is a quadratic polynomial that gives the output from the neuron.", "The first field is just S0, which will become SelfPeep0 for the next timestep.", "A linear combination of these values for the other neurons than “self” will become OtherPeepsLC.", "Figure: A small neuron for modeling a double pendulum.Figure: A circuit diagram for the neuron in Figure .Figure REF shows a more traditional representation of the neuron with one copy of the neuron for timestep $t$ and another for $t+1$ with dashed lines indicating information flow across timesteps.", "The figure uses linear algebra notation with bold lower case variables being vectors, whereas bold upper case variables represent matrices.", "The vector of state 0 values is ${\\bf s_0}$ , corresponding to the scalar variable S0 in the program above.", "The input weight matrices ${\\bf U_1}$ and ${\\bf U_2}$ correspond to the weight mappings indicated by lc1 and lc2 respectively.", "The recurrent weight matrix ${\\bf W_0}$ is hollow, that is has zeroes along its diagonal to exclude “self” from OtherPeepsLC.", "The vector ${\\bf a_0}$ contains so-called auxiliary weights and is multiplied element-wise (Hadamard product) as indicated by the operator $\\odot $ .", "It is not really needed here but corresponds to how the cons is translated by our compiler.", "The shaded boxes in the diagram implicitly contain weight vectors and matrices, which have been omitted in order not to clutter the diagram.", "The intermediate vector ${\\bf s_{0,0} }$ represents lc2 InputsLC + SelfPeep0 and ${\\bf s_{0,1} }$ represents lc1( cons( lc0 OtherPeepsLC, InputsLC ) ).", "When we introduce new intermediate variables, we use a second subscript as above to differentiate them In our actual C++ code, all input weight matrices are stacked in one matrix and the same goes for all recurrent weight matrices in order to reduce the number of matrix-vector multiplications." ], [ "Choosing activation functions", "ADATE needs to be given a set of predefined functions to use for program synthesis.", "Obviously, we include addition, subtraction, multiplication and division.", "It would be possible for ADATE to use these together with if-expressions to synthesize its own activation functions, for example ReLU [11] or good approximations of the hyperbolic ones, but after some preliminary experimentation we decided against using if-expressions altogether.", "Instead, we chose to include three predefined activation functions, which turn out to be very useful for synthesizing other and novel activation functions.", "First, we chose to include tanh but not sigmoid since ADATE easily can express it as a scaled version of tanh.", "Second, the relu function was included and after that we also chose to add a linear approximation of tanh which we called saturated ReLU (srelu).", "The latter is defined as being the identity function for values between -1 and 1, just 1 for values greater than 1 and -1 for values less than -1.", "It could be implemented using two ReLU functions, but we chose to relieve ADATE of that extra burden.", "Our intuition was that elimination of the more curved parts of tanh may lead to smaller problems with vanishing gradients since relu also is linear and seems to have this advantage.", "However, we still wanted a function more suitable for gating than relu and therefore defined srelu.", "Thus, our set of predefined activation functions is tanh, relu and srelu." ], [ "Experiments", "In this section, we will first explain our neural net architecture and hyperparameter optimization, then how to very quickly evaluate neuron candidates and then introduce the datasets that we tested ARN on.", "Finally, we will present the experimental results along with statistical analyses.", "In general, we always split datasets into three subsets, with 50% for training, 25% for validation and 25% for testing.", "The test set is used only once at the very end to prevent subtle forms of information leaks and the overfitting that may result." ], [ "Architecture and hyperparameter optimization", "Before starting an ADATE run for a new dataset, we optimized the architecture and the hyperparameters for LSTM and used these also for the neurons discovered by ADATE.", "Assume that we have a dataset with $n_o$ outputs and that there are $l$ nodes in the LSTM layer.", "We used the simple net architecture below.", "An LSTM layer with $l$ nodes.", "A hyperbolic tangent layer with $n_o$ nodes.", "A linear layer with $n_o$ nodes.", "We use Glorot initialization for forward weights and orthogonal initialization for recurrent weights with a scaling factor of 0.1 in both cases.", "Also, we use an initial extra bias of 1 for the forget gates as recommended by [18] and others.", "The loss function was mean squared error (MSE) for regression and softmax plus cross entropy for classification.", "We tried a variety of other backends such as one or more ReLU layers instead of the hyperbolic tangent layer, but found no significant gains in validation losses.", "Before doing any other optimization, we found the optimal number $l$ of LSTM nodes under the assumption that $l = 2^i$ for $i=1,2, \\ldots , 7$ .", "The reason that we could use a rather small upper bound for the number of nodes is that we did not use datasets such as those from NLP that require a big memorization ability.", "The next topic is hyperparameter optimization.", "Choi et.", "al.", "[9] found that comparisons of different weight optimizers such as ADAM [19] and NADAM [10] often is skewed by inadequate hyperparameter optimization and recommend that all parameters in the algorithms should be optimized.", "We used their methodology with the initial parameter ranges that they give in Appendix D.8 for every single dataset.", "As they recommend, we performed a random search within these ranges but used many more evaluations, namely 512 random parameter sets for each dataset and picked the best one for each dataset using the validation data.", "Thus, we optimized all parameters in the ADAM algorithm and not just the learning rate.", "We tried with both ADAM and NADAM, but the difference between them was negligible.", "ADAM was used in all subsequent experiments.", "We also used the same learning rate schedule as Choi et.", "al.", "[9], namely the one from Shallue et.", "al.", "[26], and optimized the initial learning rate, the duration of the decay and the amount of decay together with the ADAM hyperparameters.", "Our batch size is so small that no warm-up is needed.", "The amount of training is often measured in the number of epochs, but we will instead use the total number of training examples including duplicates, that is the number of epochs times the training set cardinality.", "Each training session consisted of running 320 000 training examples with a batch size of four, that is a total of 80 000 weight updates.", "After preliminary experiments, we abandoned L1 and L2 regularization and dropout but chose to employ model checkpointing as follows.", "For every 20 000 training examples, we ran the net with the current weights on the validation data and updated the best weights found if there was a validation improvement." ], [ "Experiment design and implementation", "A key challenge with ADATE as for many other evolutionary algorithms is that very many evaluations may be required, where evaluation means to compute an evaluation value, sometimes called “fitness”, for a solution candidate, which here is an ARN.", "ADATE first screens candidates using a quite limited and very quick evaluation.", "The candidates that pass the screening, typically less than 1%, then move on to a second stage evaluation that in most cases use at least 100 times more time per candidate.", "Finally, less than 1% of the ones who pass the second stage are then passed to a third and final stage.", "The first two stages use a smaller number of training steps and a smaller number of recurrent nodes as follows.", "First stage.", "The goal for this stage was to have an overall evaluation time for a new neuron that is less than 100 ms. For some CPUs and datasets, we achieved 50 ms and for other combinations it is up to 200 ms. Of course, the evaluation results will be very bad in this stage, but still enough to filter away really bad programs.", "The number of nodes was always four and we used only 5000 training examples.", "Additionally, we restricted the training to run on only the five last timesteps in each time series.", "Second stage.", "Here, we use 40 000 training examples, all timesteps and $l/4$ nodes.", "In comparison with the first stage, the run time is increased by a factor given by the following expression where $n_t$ is the number of time steps.", "$8 \\cdot (l/16)^2 \\cdot n_t / 5$ For example, with $l=64$ and 100 time steps, the increase is 2560 times.", "Third stage.", "This stage uses the full 320 000 training examples and $l$ nodes.", "Thus, its expected run time is $8 \\cdot 4^2$ , that is 128, times longer than the second stage.", "For say $l=64$ and 100 timesteps, the total speed amplification is about $2560 \\cdot $$ 128 /$ 3, that is about $10^5$ times, since the implementaion actually tries to spend equally long run time on each stage.", "Our compiler translates a functional program that represents a neuron into byte code, which then is run on-the-fly by a byte code interpreter implemented in C++.", "Since matrix operations require almost all the execution time, it does not matter that neuron evaluation is somewhat slower than optimized machine code.", "All of the neural net is also implemented in C++ and uses the automatic differentiation library AADC that kindly was made available by Matlogica Ltd, who also provided absolutely outstanding support for how to use it in the most efficient way, especially for matrix operations.", "The resulting vectorized machine code, that is AVX256 or optionally AVX512, appears to run several times faster than Tensorflow, when the latter is restricted to a single CPU core.", "We evaluated the C interfaces of all the leading neural net packages but did not find anything that could compete with the speed, stability and ease of use that characterizes AADC.", "To check the correctness of our implementation, we ran the same datasets with LSTM using both our C++ code and Tensorflow and compared the results.", "Additionally, we implemented one of the new neurons in Tensorflow and compared with the C++ code without finding any statistically significant differences.", "All comparisons and other experiments used 64-bit floating point numbers since a 32-bit AADC version was not yet available from Matlogica.", "An ADATE run used between 500 and 1000 CPU cores depending on what was available.", "The number of evaluated candidate neurons was on the order of one billion for each dataset and around three days of run time were typically needed to find the best results." ], [ "Datasets", "Since we need to evaluate new neurons rather quickly, we chose to focus on datasets that can be effectively run with 128 LSTM neurons or less, that is rather small nets.", "For this reason, we avoided speech and NLP datasets that typically require bigger nets as well as attention [4], [15] to get the best results.", "Many of the datasets come from the excellent online repository for time series datasets that is curated by Bagnall et.", "al.", "[3] and their time series classification (TSC) research group.", "Most of the datasets are time series from various physical sensors.", "The only preprocessing that we employed was either centering and scaling or one hot encoding, depending on whether a predictor or an output was ordinal or nominal.", "Note that the goal of the experiments is a reliable comparison between LSTM and new neurons and not to get state-of-the-art results for any dataset.", "For this reason, we did not do any feature engineering or data augmentation.", "Thus, due to not doing any more advanced preprocessing, our results may for some datasets be far from the best that is possible.", "We will first give a brief description of each dataset and then list the number of examples, that is time series, and the number of predictors for each one.", "3W This is a predictive maintenance dataset that was donated to the UCI machine learning repository by Petrobras [28] and consists of actual sensor values and events in oil wells supplemented by simulated and handcrafted values and events.", "The task is to predict undesirable events.", "We preprocessed the dataset into non-overlapping time windows and used only sensor readings, for example temperature and pressure, as predictors.", "Crop This dataset is from TSC but was originally collected by Tan et.", "al.", "[27].", "The goal is to classify the type of crop on a patch of land from the time series of images produced by the Sentinel-2A satellite, which has plant growth monitoring as one of its primary missions.", "Double pendulum The task of the RNN is to learn to simulate a double pendulum.", "In general, RNNs can be applied to modeling of chemical and physical processes and are, for example, used quite successfully for this purpose by the Borregaard biorefinery since the beginning of 2022.", "Inspired by this, we generated the Double Pendulum dataset from the simulation provided with the SimBody physics engine.", "The predictors are the center of gravity coordinates of the first as well as the second pendulum at a given timestep and the response is the corresponding coordinates for the next timestep.", "Each time series is generated from randomly chosen initial positions and angular velocities.", "We deliberately lowered the sampling frequency so that black box modeling from measurement data alone becomes challenging.", "ECG5000 This dataset is a 20-hour long ECG and was originally published by Goldberger et.", "al.", "[13], who preprocessed the signals to make each heartbeat equally long using interpolation.", "We downloaded it from TSC.", "FordB This dataset was originally used in a competition at WCCI 2008 [1] and the task is to determine if there is something wrong with an internal combustion engine based on recordings of the engine sound.", "This dataset was also taken from TSC.", "Insect wingbeat This dataset [7] contains spectrograms from the sounds generated by the wings of insects, in this case mostly mosquitoes and flies.", "The goal is to classify the species of an insect based on a sound recording.", "We downloaded it from TSC.", "LSST This is yet another TSC dataset but originally comes from a 2018 Kaggle competition [2].", "It contains simulated data from the Vera C. Rubin Observatory, also known as the Large Synoptic Survey Telescope (LSST), that are measurements of how the brightness of an astronomical object varies with time.", "The goal is to determine which kind of object that was observed.", "WISDM The task is to predict the activity of a cell phone user based on the time series of accelerometer values [21].", "We used the most difficult version of the WISDM dataset which is version 1.1 without overlapping time windows and downloaded it from the UCI machine learning repository.", "As can be seen in Table REF , most of the datasets have a small or medium number of examples.", "This is a use case which is just as important as big data since many datasets that arise in practice have sizes similar to those in the table.", "Indeed, most datasets gathered by TSC are even smaller, but we did not want to include these since the confidence intervals may become too big.", "Table: Number of examples, number of timesteps, number of inputs and number of outputs after one hot encoding." ], [ "Experimental results", "Table REF compares an LSTM net with an ARN net for the test data sets.", "We used categorical cross entropy (CCE) for the classification datasets and mean squared error (MSE) for the regression dataset.", "The p-value was calculated using McNemar's test for classification and the Wilcoxon signed-rank test for regression with continuity correction in both cases.", "As can be seen in Table REF , the p-values are better than five sigma for six of the eight datasets.", "The column “Factor better” shows how many times better the ARN was compared with LSTM with respect to CCE or MSE.", "Table: Test data cross entropies or MSE and accuracies for LSTM and ARN.Since we carefully optimized the hyperparameters for LSTM, the big improvements obtained for say 3W, Double pendulum, Ford B, LSST and WISDM, cannot be explained by the LSTMs being poorly tuned.", "Instead, the most likely explanation is that the new ARN neurons have a superior modeling and fitting ability.", "As can be seen in the table, the improvement that was obtained varies a lot between the datasets.", "For example, for the FordB dataset, we obtained a more than three-fold reduction of the test data CCE, whereas there was no improvement at all for the ECG5000 dataset.", "One possible explanation for the huge improvement for FordB is that this dataset has as many as 500 timesteps and that LSTM cannot effectively handle so long time dependencies, whereas ARN can do that using skip connections through time.", "By looking at the best program for FordB in Appendix A.4, it is easy to see that skip connections are indeed used, but otherwise this is a really hard neuron to analyze since it was not designed by human beings.", "Since the experiments optimized cross entropy instead of accuracy, it is likely that somewhat higher accuracies could be obtained if ADATE were to directly optimize the accuracy instead.", "We omitted the neuron for ECG5000 since it was no improvement, but the best ones for all the other datasets are listed in the appendices.", "As can be seen there, there is a huge variation from one dataset to the next, but both skip connections and the linear combinations of state 0 (OtherPeepsLC ) are frequently used.", "From the outset, we naively guessed that reading the memories of other neurons would give rise to problems akin to vanishing or exploding gradients, but this seems to be much less problematic than we thought.", "All of the best programs use novel activation functions that we have never seen before.", "For example, the best program for WISDM in Appendix A.7 uses srelu applied to quadratic forms in several places.", "Many of the other neurons also contain various quadratic expressions, but which roles they have is not clear.", "There are some obvious redundancies in the coding, for example applying relu twice.", "ADATE is able to remove such redundancies, but much more run time might be needed.", "In general, evolution in ADATE is so effective that neurons are likely to become exquisitely adapted to the special characteristics of the dataset.", "This means that neurons should not be reused from one class of datasets to the next, that is, a new ADATE run is needed for each new type of application.", "One way to avoid this special adaption would be to run on say hundreds of different datasets simultaneously using many more cores and possibly accelerators." ], [ "A closer look at some ARNs", "In this section, we will discuss the circuit diagrams for the best LSST neuron and the best FordB neuron.", "These were selected since they are among the simpler best neurons.", "In order to translate the neuron syntax from SML to diagrams, we first printed them as C expressions using our compiler, converted them to equations and then manually simplified these equations before using them to draw the neurons.", "Figure: A circuit diagram for the best neuron for the LSST dataset .Both the diagram and the equations for LSST are shown in Figure REF .", "All vector-vector multiplications are element-wise and all variables denoted by b or B are vectors or matrices manually derived from cons expressions in the SML code that after translation to C contains auxiliary weights, denoted by a variables.", "We do not show the full and lengthy manual translation from SML to simplified equations.", "It is a bit fascinating to see that ADATE actually has invented a small convolutional filter represented by the equation for ${\\bf y_0^{(t)}}$ .", "This filter combines the inputs from three consectutive timesteps using a linear combination.", "However, the output from the filter is combined with skip connections from previous outputs represented by ${\\bf y_1^{(t)}}$ .", "Thus, the convolution is in principle over inputs and some outputs and the result is fed to relu as can be seen in the equation for ${\\bf y^{(t)}}$ .", "Another interesting feature of this neuron is that it totally has abandoned tanh and srelu as activation functions.", "As can be seen in Figure REF , it only contains two relu functions.", "However, we are not able to theoretically understand the dynamic behaviour of RNNs that contain such neurons and can only claim that it produces superior accuracy for the LSST dataset.", "Figure: A circuit diagram for the best neuron for the FordB dataset .The best neuron for the FordB dataset is more complex and shown in Figure REF .", "We will now take a look at some elaborate skip connections in this neuron that use the output from time $t-4$ to calculate the output at time $t$ .", "Since the dataset contains 500 timesteps, we can speculate that it is especially important to curate information over long time periods and that special connections may have been evolved to handle that.", "Let $\\rightarrow $ mean that the left hand side is partially used to compute the right hand side.", "Then, the information flow from $y^{(t-4)}$ to $y^{(t)}$ is as follows as can be seen in the circuit diagram.", "$y^{(t-4)} \\rightarrow s_0^{(t-3)} \\rightarrow s_2^{(t-2)} \\rightarrow s_3^{(t-1)} \\rightarrow y^{(t)}$ Thus, the internal states are employed to implement information flow that appears to be custom designed to retain long term memory for the FordB dataset.", "As a curiosity, we can just mention that ADATE has discovered ARNs that retain long term memory much better than LSTM for thousands of timesteps when simulating complex Mealy machines." ], [ "Analysis of a Pareto front", "Recall that a Pareto front contains gradually bigger and better programs so that the smallest and worst program comes first whereas the last program is the biggest and the best.", "As an example, we will use the Pareto front for the Double pendulum dataset which consists of 24 programs.", "The first four of these are so small that they are not interesting but the last 20 are shown in Figure REF .", "For this dataset, an LSTM has a validation MSE of 0.146 as indicated by the horizontal line in Figure REF , which means that all programs below the line are better than LSTM for Double pendulum on validation data.", "By using Pareto fronts, ADATE tries to both minimize MSE and program size.", "This has been so successful that all programs in the front are smaller, at least according to the ADATE syntactic complexity, than the LSTM program, which has a size of 324 bits.", "Thus, ADATE has found a plethora of programs that are both smaller and better than LSTM for Double pendulum.", "We will take a closer look at the two programs that are indicated by the blue squares in Figure REF .", "The smallest one, corresponding to the leftmost square, has an MSE of 0.176 and is the following simple RNN, where no internal neuron states are used.", "Therefore, we have also manually edited the program to remove them relu( lc2( cons( lc1( OtherOutputsLC ), InputsLC ) ) ) This is easy to understand and just a linear combination of the inputs and the outputs from other neurons, albeit formulated as a linear combination of a linear combination.", "Apparently, relu is more suitable than the other activation functions.", "The next square square in Figure REF is the program in Figure REF that has an MSE of 0.139, which is better than for the LSTM.", "Even if this neuron is quite simple, it is already becoming hard to understand exactly why it is better, with analysis being impeded by the use of internal states from other neurons.", "A partial hypothesis may be that using quadratic base functions may give more suitable regression modeling of a double pendulum.", "The best neuron for Double Pendulum, which is the green and last square in Figure REF , is given in Appendix A.3.", "It is much better and has an MSE of 0.094 on validation data.", "It is also much more difficult to understand.", "Also, note that the test data MSE for this neuron is 0.11 and that using validation data for both hyperparameter optimization, ADATE runs, and model checkpointing leads to some overfitting on validation data.", "In spite of this, the neuron is significantly better than LSTM also on test data with a p-value around $10^{-19}.$ The syntactic complexity makes a big jump with little gain in MSE from the second last to the best program in the front.", "Due to Ockhams's razor [24], overfitting can be suspected when this happens." ], [ "Conclusions and future work", "We tested the ADATE recurrent neurons (ARNs) on eight datasets.", "For the four best datasets, the ARNs were a factor of 3.6, 1.7, 1.7 and 1.4 better respectively as measured by cross entropy or mean squared error on test data.", "The p-values for these four were below about $10^{-18}$ , which is clearly better than five sigma.", "For the other four datasets, there were significant improvements for three but failure for one.", "The reason it did not work for one dataset seems to be overfitting to the validation data caused by a combination of hyperparameter tuning, model check pointing and using validation data to calculate evaluation values for programs generated by ADATE and selecting the best one.", "The results are so good that ARNs should be tried in practice whenever recurrent nets are used.", "The reasons for the big improvements that were found for at least half of the datasets are as follows.", "ADATE is quite capable when it comes to automatically adapting the code of a neuron to a specific class of datasets.", "The ARNs contain several somewhat unusual ingredients such as Up to four internal states per neuron.", "Sometimes, these are apparently used by ADATE to construct intricate skip connections through time.", "Linear combinations of the states of other neurons in addition to the usual combinations of outputs.", "Custom made activation functions that sometimes contain quadratic forms.", "Differentiation between “self” and “all others”.", "The technology is ready to use right away in C++ with an option to manually port to Tensorflow.", "We encourage people interested in using it to contact the corresponding author of this paper.", "However, a cluster with between 500 and 1000 cores is needed to run on a subset of data with up to about 50 000 training examples for about three days.", "Thus, the number of datasets that can be processed may be somewhat limited by available computational resources.", "Here are some possibilities for future work.", "Let an ARN layer be an ensemble of different neuron types and use ADATE to evolve all the different types, either one type at a time or all types simultaneously.", "Use many stacked ARN layers with different neuron types in each layer.", "Find a way to efficiently optimize hyperparameters as a part of the evolution instead of doing it only once at the beginning.", "Run on hundreds of datasets during the same evaluation in order to find neurons that are generally better.", "However, we expect that evolving neurons for specific classes of datasets often will yield much better results.", "Study other neural architectures than RNNs.", "For example, the batch normalization and ResNet blocks in a CNN could be viewed as the function f to improve and the time dimension in an RNN could be replaced by a layer index dimension, where say four states per block are curated from one layer to the next by f. Another possibility would be define differentiable functional programs and neural nets that represent them.", "These could use differentiable datastructures and handle recursion using a predefined set of differentiable schemas that are generalizations of say catamorphisms, anamorphisms and hylomorphisms.", "It may also be possible to employ the automatic programming presented in this paper for a variety of other current and future neural net and differentiable programming technologies." ], [ "Appendix A. The best neurons according to validation data", "This appendix lists the best programs according to validation data for the seven datasets out of eight for which improvements were found.", "They are presented exactly as they were printed by the ADATE system and could typically be somewhat simplified manually.", "Thus, they may look more complex than they really are.", "The best neuron for WISDM has also been implemented by one of us as a custom layer in Tensorflow in addition to the automatically generated C++ code that our compiler generates based on the SML programs.", "Requests for neuron synthesis and C++ or Tensorflow implementations for other datasets are very welcome.", "fun f       (         SelfPeep0,         SelfPeep1,         SelfPeep2,         SelfPeep3,         SelfOutput,         OtherPeepsLC,         OtherOutputsLC,         InputsLC         ) =   case cons( SelfOutput, cons( SelfOutput, InputsLC ) ) of     V21902689 =>   case     (       SelfOutput,       tanh(         srelu(           (             (               lc3( cons( lc1( V21902689 ), V21902689 ) ) *               srelu( SelfPeep3 )               ) -             (               tanh(                 lc0(                   cons(                     SelfPeep3,                     cons( tanh( SelfOutput ), InputsLC )                     )                   )                 ) +               srelu(                 relu(                   tanh(                     lc2(                       cons(                         ~0.14531347527330391E~1,                         cons(                           SelfPeep3,                           cons(                             srelu( srelu( SelfPeep2 ) ),                             cons(                               SelfOutput,                               cons( SelfPeep0, InputsLC )                               )                             )                           )                         )                       )                     )                   )                 )               )             )           )         )       )   of     ( V21915A57, V21915A58 ) =>       (         V21915A58,         SelfPeep0,         srelu( relu( lc2( OtherOutputsLC ) ) ),         SelfPeep0,         srelu(           (             V21915A57 -             relu(               tanh(                 tanh(                   srelu(                     srelu(                       case ( SelfPeep1, V21915A58 ) of                         ( V21915A59, V21915A5A ) =>                           ( V21915A5A - V21915A59 )                       )                     )                   )                 )               )             )           )         ) fun f       (         SelfPeep0,         SelfPeep1,         SelfPeep2,         SelfPeep3,         SelfOutput,         OtherPeepsLC,         OtherOutputsLC,         InputsLC         ) =   case     (       SelfOutput,       (         ( lc3( OtherPeepsLC ) * lc0( InputsLC ) ) -         (           tanh(             srelu(               lc0(                 cons(                   SelfPeep1,                   cons(                     SelfPeep3,                     cons( lc4( OtherPeepsLC ), bias )                     )                   )                 )               )             ) +           lc3( InputsLC )           )         )       )   of     ( VDBC64A, VDBC64B ) =>   case srelu( srelu( ( VDBC64B - SelfPeep1 ) ) ) of     V1C427005 =>       (         VDBC64B,         SelfPeep0,         lc0( cons( SelfPeep0, OtherPeepsLC ) ),         SelfPeep2,         srelu(           case ( V1C427005, V1C427005 ) of             ( V1C481BD5, V1C481BD6 ) =>           let             fun g1CDF7C46 V1CDF7C47 =               ( VDBC64A - V1CDF7C47 )           in             g1CDF7C46( ( V1C481BD6 * V1C481BD5 ) )           end           )         ) fun f       (         SelfPeep0,         SelfPeep1,         SelfPeep2,         SelfPeep3,         SelfOutput,         OtherPeepsLC,         OtherOutputsLC,         InputsLC         ) =   (     (       relu( relu( lc1( cons( lc1( OtherPeepsLC ), InputsLC ) ) ) ) -       relu(         lc3(           cons(             lc4( OtherPeepsLC ),             cons( 0.22174599383632232, InputsLC )             )           )         )       ),     SelfPeep1,     SelfPeep2,     SelfPeep3,     (       (         (           lc4(             cons(               lc4( OtherPeepsLC ),               cons(                 lc0(                   cons(                     SelfPeep2,                     cons(                       ( ~0.6961519103176064 + SelfPeep1 ),                       InputsLC                       )                     )                   ),                 InputsLC                 )               )             ) *           lc0( OtherPeepsLC )           ) +         SelfPeep0         ) *       (         SelfPeep1 -         (           tanh(             case               cons(                 lc3(                   cons(                     SelfPeep2,                     cons( ~0.6961519103176064, InputsLC )                     )                   ),                 InputsLC                 )             of               V1662138D =>                 (                   srelu(                     lc0(                       cons(                         SelfPeep0,                         cons( lc0( V1662138D ), InputsLC )                         )                       )                     ) *                   srelu(                     lc0(                       cons(                         relu(                           lc3(                             cons(                               SelfPeep0,                               cons(                                 lc0(                                   cons( lc0( V1662138D ), V1662138D )                                   ),                                 InputsLC                                 )                               )                             )                           ),                         InputsLC                         )                       )                     )                   )             ) +           ~0.45284110969509517           )         )       )     ) fun f       (         SelfPeep0,         SelfPeep1,         SelfPeep2,         SelfPeep3,         SelfOutput,         OtherPeepsLC,         OtherOutputsLC,         InputsLC         ) =   case srelu( srelu( relu( relu( SelfOutput ) ) ) ) of     V281AF3E7 =>   case     (       V281AF3E7,       (         SelfPeep2 -         (           tanh( lc1( OtherPeepsLC ) ) +           lc0(             cons(               SelfPeep3,               cons( V281AF3E7, cons( SelfPeep2, InputsLC ) )               )             )           )         )       )   of     ( VDBC64A, VDBC64B ) =>       (         VDBC64B,         tanh( srelu( srelu( VDBC64B ) ) ),         lc0( cons( tanh( SelfPeep0 ), InputsLC ) ),         SelfPeep2,         srelu(           (             VDBC64A - (             case               case                 let                   fun g1C57E5DF V1C57E5E0 =                     (                       V1C57E5E0,                       (                         ~0.12690104588539253E~1 - (                         case ( VDBC64B, SelfPeep1 ) of                           ( V1C57EDC3, V1C57EDC4 ) =>                             srelu(                               (                                 ( V1C57EDC3 - SelfPeep0 ) +                                 ( V1C57EDC3 - V1C57EDC4 )                                 )                               ) )                         )                       )                 in                   g1C57E5DF(                     case g1C57E5DF 0.1029527350644156 of                       ( A, B) => A + B                     )                 end                of                 ( V5FAAD0B, V5FAAD0C ) =>                   ( V5FAAD0B, srelu( V5FAAD0C ) )             of               ( V1D190879, V1D19087A ) =>                 tanh( tanh( ( V1D19087A * V1D190879 ) ) ) )             )           )         ) fun f       (         SelfPeep0,         SelfPeep1,         SelfPeep2,         SelfPeep3,         SelfOutput,         OtherPeepsLC,         OtherOutputsLC,         InputsLC         ) =   case     (       SelfOutput,       (         (           srelu(             tanh(               tanh(                 lc3(                   cons(                     0.10022791851659179E1,                     cons( lc4( OtherOutputsLC ), InputsLC )                     )                   )                 )               )             ) *           srelu( tanh( lc0( InputsLC ) ) )           ) -         (           tanh( srelu( ~0.6873893912995532E~2 ) ) +           srelu( relu( SelfOutput ) )           )         )       )   of     ( VDBC64A, VDBC64B ) =>   case tanh( srelu( srelu( tanh( relu( VDBC64B ) ) ) ) ) of     V1C9C2AC6 =>   case ( V1C9C2AC6, V1C9C2AC6 ) of     ( V1C9C2AD2, V1C9C2AD3 ) =>       (         VDBC64B,         ( tanh( SelfPeep2 ) * SelfPeep3 ),         relu( tanh( relu( V1C9C2AD3 ) ) ),         SelfPeep2,         srelu( ( VDBC64A - tanh( V1C9C2AD2 ) ) )         ) fun f       (         SelfPeep0,         SelfPeep1,         SelfPeep2,         SelfPeep3,         SelfOutput,         OtherPeepsLC,         OtherOutputsLC,         InputsLC         ) =   (     relu( lc1( OtherOutputsLC ) ),     SelfOutput,     lc0( InputsLC ),     SelfPeep2,     (       relu(         lc1(           cons(             srelu( ~0.1267263484282487 ),             cons(               SelfPeep3,               case                 cons(                   SelfPeep0,                   cons(                     SelfPeep2,                     cons(                       lc2( cons( lc0( OtherOutputsLC ), InputsLC ) ),                       InputsLC                       )                     )                   )               of                 V259F26F2 => cons( lc0( V259F26F2 ), V259F26F2 )               )             )           )         ) +       SelfOutput       )     ) fun f       (         SelfPeep0,         SelfPeep1,         SelfPeep2,         SelfPeep3,         SelfOutput,         OtherPeepsLC,         OtherOutputsLC,         InputsLC         ) =   case     (       SelfOutput,       (         lc0( bias ) -         (           tanh(             srelu(               srelu(                 lc1(                   cons(                     SelfPeep1,                     cons(                       SelfPeep3,                       cons(                         (                           lc4( InputsLC ) *                           lc2( cons( SelfPeep0, InputsLC ) )                           ),                         cons(                           srelu(                             relu(                               lc2(                                 cons(                                   lc0( OtherOutputsLC ),                                   OtherPeepsLC                                   )                                 )                               )                             ),                           case                             cons( lc1( OtherPeepsLC ), InputsLC )                           of                             V271482D0 =>                               cons( lc2( V271482D0 ), V271482D0 )                           )                         )                       )                     )                   )                 )               )             ) +           srelu(             relu( lc2( cons( lc4( OtherPeepsLC ), InputsLC ) ) )             )           )         )       )   of     ( VDBC64A, VDBC64B ) =>       (         VDBC64B,         SelfPeep0,         srelu( srelu( srelu( lc0( OtherPeepsLC ) ) ) ),         SelfPeep2,         srelu(           (             VDBC64A -             (               ( VDBC64B - SelfPeep1 ) *               ( VDBC64B - SelfPeep1 )               )             )           )         )" ] ]
2207.03577
[ [ "Teaching Rotational Physics with Bivectors" ], [ "Abstract Angular momentum is traditionally taught as a (pseudo)vector quantity, tied closely to the cross product: familiar ideas to experts but full of challenges and subtleties for students.", "Here, we present an alternative approach to teaching the subject in which angular momentum is instead described using bivectors, which can be visualized as \"tiles\" with area and orientation whose components form an antisymmetric matrix.", "Bivectors have historically been considered mostly in specialized contexts like spacetime classification or geometric algebra, but they are no more complicated to understand than cross products.", "Teaching rotational physics in this language is ultimately more fundamental, and opens the door to understanding the rotations in relativity and extra dimensions." ], [ "Introduction", "Rotational physics is full of pitfalls.", "Novice students struggle with right-hand rules and multiple definitions of the cross product, each with non-intuitive features.", "More advanced students may still stumble over the subtle trap of a left-handed coordinate system or the unexpected behavior of pseudovectors under coordinate reflections.", "And even experts may find themselves at a loss if asked how to describe angular momentum in relativity or a theory with extra dimensions of space, since the cross product is only defined in three dimensions.", "All of these issues are resolved when we recognize that angular momentum and angular velocity are fundamentally bivector quantities.", "Bivectors can be visualized in terms of oriented “tiles” whose areas represent their magnitudes, as shown in Figure REF .", "Because the tile directly shows the plane and direction of rotation, no right hand rules are necessary.", "Coordinate reflections affect the oriented tile in the natural way: there is no extra minus sign as is required for cross products and pseudovectors.", "The components of a bivector are specified by two coordinate labels (e.g.", "$\\omega _{yz}$ ) rather than one (they form an antisymmetric matrix), and because their order specifies the orientation there is no need for the convention of a right-handed coordinate system.", "And in higher dimensions where there is no concept of an axis of rotation, the idea of a plane of rotation remains meaningful.", "Figure: Angular velocity ω -0.3ex[0ex][0ex][3]↔\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega } is fundamentally a bivector quantity, represented as a tile whose attitude and orientation match the object's plane and direction of rotation and whose area gives the magnitude.", "(The shape of the tile is unimportant.)", "Here we see angular velocity bivectors for a clock's second hand and a merry-go-round, and the angular momentum bivector (proportional to angular velocity) for a ball on a string.", "This rightmost tile is constructed as the “wedge product” r →∧p →\\vec{r} \\wedge \\vec{p}.Formally, bivectors are rank-2 antisymmetric tensors (just as vectors are rank-1 tensors): they are closely related to differential 2-forms (in the same way that vectors are related to 1-forms).", "But that mathematical description may make them sound more complicated than they are: in practice, understanding bivectors is no more complicated than understanding cross products.", "And even for those who prefer to focus on the more familiar treatment of rotational physics, parts of the bivector story may give students a helpful “bridge” to those cross product results.", "Although bivectors are unfamiliar to most physicists, they do appear in the literature.", "For instance, they play a role in the Petrov classification of spacetimes,[1], [2] and they are an important component of geometric algebra.", "[3], [4] In fact, there are a number of good pedagogical introductions to bivectors in the geometric algebra context, but most of those make no effort to separate the fundamental properties of bivectors from their specific role in the geometric algebra formalism.As of this writing, readers of the Wikipedia entry for bivectors (https://en.wikipedia.org/wiki/Bivector) could be forgiven for thinking that the bivector concept is only defined as a part of geometric algebra.", "And the website bivector.net is actually a geometric algebra site.", "But the only bivector that most physicists ever encounter is the electromagnetic field tensor, and it is rarely labeled as one.", "The idea of describing rotational physics in terms of bivectors (or 2-forms) is not at all new, but it has mostly appeared only in specialized work where it is unavoidable (general relativity with torsion, for example[6]) rather than as a pedagogical approach to the subject.", "The plan of this paper is as follows.", "In Section  we give some initial motivation for rethinking such a fundamental topic and comment on pedagogical advantages and disadvantages.", "In Section , we solve a conservation of angular momentum problem with the help of core bivector concepts and visualizations.", "In Section , we solve a gyroscopic precession problem by adding bivectors geometrically both as a sum of using normal vectors and as a sum of tiles.", "In Section  we compute the linear velocity vector of a point on the Earth's surface by multiplying a vector and a bivector.", "We state general conclusions in Section .", "Finally, Appendix  briefly discusses bivector angular momentum in more sophisticated contexts, including the inertia tensor, quantum commutation relations, relativity, and extra dimensions.", "Appendix  sketches the proof of a claim about bivectors as sums of orthogonal tiles.", "And Appendix  presents reference formulas and derivations relating products involving pseudovectors to their bivector equivalents." ], [ "Why teach bivectors?", "To experts with long experience studying rotational physics, the bivector description presented below may at first seem unfamiliar and complicated.", "But novice learners find cross products and the vector description of rotational motion unfamiliar and complicated as well.", "For instance, two thirds of second semester physics students tested on cross products get less than half the questions right, whether the context is electromagnetism or pure math.", "[7] Cross products of vectors that point out of the plane of the page or where the right hand rule requires awkward arm positions are especially challenging.", "[8] And even after studying rotational motion, the majority of first semester students still believe that a particle's angular velocity vector points in the plane of motion.", "[9] (Kustusch gives a thorough literature review.", "[8]) With these challenges in mind, the bivector formalism offers some clear pedagogical advantages.", "Representing angular quantities as oriented tiles immediately eliminates the tendency to conflate the vector directions of linear and angular velocity.", "It has a direct, visible relationship to the rotation of the system, as opposed to the traditional visualization as an axis of rotation which is always one step more abstract.", "The greatly reduced emphasis on the right hand rule reduces a significant stumbling block for new learners, and largely eliminates the need to worry about whether the coordinate system is right handed.", "Moreover, calculating the bivector angular momentum of a point particle as a wedge product in components (Section ) is more intuitive than the cross product procedure: for example, the expression $x p_y - y p_x$ corresponds to the $\\ell _{xy}$ component, with the index order matching the positive term (rather than to the vector component $L_z$ , which requires students to keep track of cyclic coordinate order).", "The tile is constructed as a parallelogram, whose area makes the magnitude formula clear.", "And in more advanced rotational physics (Appendix ) the bivector form removes a significant barrier for students studying its generalizations in relativity or extra dimensions.", "Balancing out those advantages, there are three main points of concern: two pedagogical and one social.", "First, as Section  shows, bivector tile addition can be genuinely far more complicated than vector arrow addition tip-to-tail.", "This is a strong argument in favor of a hybrid approach, where students learn the bivector language but practice converting bivectors to pseudovectors when addition is necessary.", "And second, where a vector's components can be presented as a single column of numbers, a bivector's components are naturally presented as a matrix: many students have little to no experience with matrix math, and it could feel intimidating.", "In practice, though, both of these difficulties can be largely avoided in typical teaching and problem solving.", "Especially at the introductory level, the vast majority of examples of rotational systems and static equilibrium problems lie in a single plane, so only one component of angular momentum or torque is relevant.", "In those, the bivector and cross product approaches are essentially indistinguishable: only one component is nonzero so angular momentum is a single signed quantity, and the subtleties of addition in three dimensions don't arise.", "And even in these simple cases, the bivector language can make things clearer: when considering a system in the $xy$ -plane, the angular momentum is correspondingly labeled $\\ell _{xy}$ rather than $L_z$ , and the order of coordinates in the subscript conveniently specifies the positive direction.", "Finally, the most significant concern may be a social one: we have an obligation to teach our students how to engage with the larger physics community and the existing literature, where rotations are almost always described with cross product and pseudovectors.", "We will address this in full after we have established the bivector formalism.", "In brief, this change in approach could never happen all at once: there are natural ways for individual instructors to gradually incorporate pieces of the bivector description into their teaching that may even help students to better understand the traditional approach." ], [ "Angular momentum and the wedge product", "To illustrate the bivector formalism in a pedagogical context, consider the scenario shown in Figure REF .", "A frictionless wheel of diameter 1 and mass 3 (essentially all at the rim) is initially rotating clockwise at 1.2.", "A stone of mass 0.3 falls straight down and (inelastically) hits the wheel with speed 20 at a horizontal position 0.4 left of center.", "The stone comes momentarily to rest (with velocity zero) before falling aside.", "What is the final angular speed of the wheel?", "We can solve this problem using conservation of angular momentum about the wheel's axle.", "Just as we illustrate vectors as arrows whose length represents the vector's magnitude, in Figure REF we have illustrated the initial bivector angular momentum of the wheel $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }_i$ as an oriented “tile” whose attitude in space shows the plane of rotation, whose orientation (clockwise or counterclockwise) within the plane specifies the direction of rotation, and whose area represents the bivector's magnitude.", "(The shape of the tile is unimportant, and we will see that it is sometimes helpful to change from one shape to another.)", "Our calculation of the magnitude of $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }_i$ proceeds in very much the usual way.", "The moment of inertia of the wheel is $I = M R^2 = {0.75}{}$ .", "The initial angular velocity is clockwise in the $xy$ -plane: as a shorthand, we might denote it as $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega }_i = {1.2}{}~{\\fbox{$\\circlearrowright $}}$ , where the symbol ${\\fbox{$\\circlearrowright $}}$ represents a clockwise tile in the plane of the page just as $\\otimes $ represents a vector into the page.", "Then for this rigid object, the initial angular momentum of the wheel is $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }_i = I \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega }_i= {0.9}{}~{\\fbox{$\\circlearrowright $}}\\quad .$ To find the contribution of the falling stone, we model it as a point particle.", "Its (linear) momentum is ${6}{}$ pointing down, and it strikes the wheel at a location 0.5 from the axle at an angle $\\sin ^{-1}(\\tfrac{0.4}{0.5}) \\approx {53.1}$ from vertical.", "(Formally, $\\vec{p} = -{6}{} \\,\\hat{y}$ and $\\vec{r} = -{0.4}{}\\, \\hat{x} + {0.3}{}\\, \\hat{y}$ .)", "We find the bivector angular momentum of the stone as the wedge product (or exterior product) of position and momentum (rather than the familiar cross product).", "The wedge product is illustrated geometrically in two different ways at the top right of Figure REF : the two vectors define a parallelogram with a fixed attitude in space, and the order of the vectors determines its orientation (counterclockwise) as described in the caption.", "The bivector's magnitude corresponds to the parallelogram's area: $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell } &= \\vec{r} \\wedge \\vec{p}= |\\vec{r}| |\\vec{p}| \\sin \\theta ~{\\fbox{$\\circlearrowleft $}}\\\\\\nonumber &= h\\, |\\vec{p}|~{\\fbox{$\\circlearrowleft $}}= {2.4}{}~{\\fbox{$\\circlearrowleft $}}\\quad ,$ where $\\theta $ is the angle between the two vectors and $h=|\\vec{r}| \\sin \\theta ={0.4}{}$ is the horizontal component of the position.", "(This bivector area formula is the meaning behind the usual cross product magnitude.)", "Reversing the order of the vectors in the product would reverse the orientation while leaving the area and attitude unchanged.", "We interpret this as reversing the sign of the bivector, so the wedge product is antisymmetric: $\\vec{p} \\wedge \\vec{r} = -\\vec{r} \\wedge \\vec{p}$ .", "Thus, the total angular momentum of the system is $\\nonumber \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }_\\text{total} &= {-0.9}{}~{\\fbox{$\\circlearrowleft $}}+ {2.4}{}~{\\fbox{$\\circlearrowleft $}}\\\\&= {1.5}{}~{\\fbox{$\\circlearrowleft $}}\\quad ,$ where the first term is negative in the counterclockwise direction.", "Since this is all carried by the wheel, its final angular velocity is $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega }_f = \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }_f/I = {2}{}~{\\fbox{$\\circlearrowleft $}}$ .", "Along with this geometric definition of the wedge product, we can also write the bivector explicitly in components.", "One natural way to do this is in terms of coordinate unit vectors: for our falling stone, $\\nonumber \\vec{r} \\wedge \\vec{p}&= (-{0.4}{}\\, \\hat{x} + {0.3}{}\\, \\hat{y}) \\wedge (-{6}{} \\,\\hat{y}) \\\\\\nonumber &= {2.4}{}~\\hat{x} \\wedge \\hat{y}- {1.8}{}~\\hat{y} \\wedge \\hat{y} \\\\&= {2.4}{}~\\hat{x} \\wedge \\hat{y}\\quad ,$ since $\\hat{y} \\wedge \\hat{y} = 0$ by antisymmetry.", "We can think of $\\hat{x} \\wedge \\hat{y}$ as a “coordinate unit bivector” with the orientation that rotates $\\hat{x}$ toward $\\hat{y}$ .", "This is counterclockwise in the $xy$ -plane, in agreement with our earlier result.", "As shown in Figure REF , these coordinate unit bivectors (labeled with their coordinate sides) can be an effective way to represent bivector orientations on the page.", "Figure: Labeled representations of coordinate unit bivectors based on the three dimensional coordinate system shown.Our full calculation (where the wheel's initial motion is in the orientation that rotates $\\hat{y}$ toward $\\hat{x}$ ) would then be $\\nonumber \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }_\\text{total} &= \\phantom{-{}}{0.9}{}~\\hat{y} \\wedge \\hat{x} + {2.4}{}~\\hat{x} \\wedge \\hat{y}\\\\\\nonumber &= -{0.9}{}~\\hat{x} \\wedge \\hat{y} + {2.4}{}~\\hat{x} \\wedge \\hat{y}\\\\&= {1.5}{}~\\hat{x} \\wedge \\hat{y}\\quad .$ We can also label components with subscripts.", "For $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell } = \\vec{r} \\wedge \\vec{p}$ (and similarly for any wedge product), $\\ell _{ij} = r_i p_j - r_j p_i \\quad ,$ where the indices $i$ and $j$ run over the coordinates $x,y,z$ .", "(Note that this implies the shorthand $r_x = x$ , $r_y = y$ , etc.)", "These components form an antisymmetric matrix: $\\nonumber \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell } &= \\begin{pmatrix}0 & \\ell _{xy} & \\ell _{xz} \\\\\\ell _{yx} & 0 & \\ell _{yz} \\\\\\ell _{zx} & \\ell _{zy} & 0\\end{pmatrix}= \\begin{pmatrix}0 & \\ell _{xy} & -\\ell _{zx} \\\\-\\ell _{xy} & 0 & \\ell _{yz} \\\\\\ell _{zx} & -\\ell _{yz} & 0\\end{pmatrix} \\\\&= \\begin{pmatrix}0 & x p_y - y p_x & x p_z - z p_x \\\\y p_x - x p_y & 0 & y p_z - z p_y \\\\z p_x - x p_z & z p_y - y p_z & 0\\end{pmatrix} \\\\\\nonumber &= \\begin{pmatrix}0 & {2.4}{} & 0 \\\\-{2.4}{} & 0 & 0 \\\\0 & 0 & 0\\end{pmatrix}\\quad ,$ where the last line shows our specific example.", "It is worth emphasizing that while this matrix language is natural, students who are unfamiliar with matrices can easily just consider individual components: e.g.", "$\\ell _{xy} = {2.4}{}$ .", "The antisymmetry of the bivector components is easy to see from these expressions, and the indices show exactly which components to multiply and in what order.", "Moreover, these formulas are valid whether or not the coordinate system is right-handed.", "Figure: The tile representing ℓ -0.3ex[0ex][0ex][3]↔\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell } is projected onto each of the three coordinate planes.", "Comparing the orientation of each projection to the plane's orientation (shown here in cyclic coordinate order), in this example we can see that ℓ xy \\ell _{xy} and ℓ yz \\ell _{yz} are positive while ℓ zx \\ell _{zx} is negative.", "(If we instead considered the other orientation for the xzxz-plane, we would say ℓ xz \\ell _{xz} was positive.", ")Also shown is the normal vector L →\\vec{L} to the tile, which is equivalent to the usual cross product: its direction relates to the orientation of ℓ -0.3ex[0ex][0ex][3]↔\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell } by the right hand rule.These components $\\ell _{ij}$ have a direct geometric meaning: they are the projections of the tile's area onto each corresponding coordinate plane, as shown in Figure REF .", "The order of indices specifies an orientation of the coordinate plane: the one that rotates the first coordinate into the second.", "Then the sign of each component indicates whether the projected orientation matches that orientation.", "Because the Pythagorean theorem applies to coordinate plane area projections of a flat tile just as it does to lengths,[10] the magnitude of a bivector is $|\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }| = \\ell _{xy}^2 + \\ell _{yz}^2 + \\ell _{zx}^2= \\frac{1}{2} \\sum _{i,j} \\ell _{ij}^2\\quad ,$ where the factor of $\\frac{1}{2}$ corrects for double counting when we sum over all possible index pairs (in both orders).", "(This is a special case of the “double dot product” of bivectors.)", "Although it is not a necessary part of studying bivectors, Figure REF also illustrates that (in three dimensions) every bivector tile has a unique normal vector whose “length” equals the tile's “area” and whose direction is related to the tile's orientation by a right hand rule: curl your fingers around with the orientation of the tile, and your thumb points along the vector.", "The components of this normal vector match specific components of the bivector: $\\vec{L} = \\begin{pmatrix} \\ell _{yz} \\\\ \\ell _{zx} \\\\ \\ell _{xy} \\end{pmatrix}= \\begin{pmatrix} y p_z - z p_y \\\\ z p_x - x p_z \\\\ x p_y - y p_x \\end{pmatrix}\\quad .$ The first equality applies to any bivector with known components.", "But for bivectors constructed as a wedge product $\\vec{r} \\wedge \\vec{p}$ , this can serve as the definition of the cross product $\\vec{r} \\times \\vec{p}$ .", "This definition supplies a helpful reminder of the sometimes baffling cross product component formulas: the bivector component indices show exactly which vector components to multiply and in which order.", "(In this context, a right handed coordinate system becomes necessary.)", "And the right hand rule is especially meaningful as used here: of all the right hand rules in physics, this one (corresponding to the loop rule in magnetism) may be the simplest to understand, and it is explicitly connected to the intuitive rotation direction of the bivector tile.", "To be clear, the quantity $\\vec{L}$ constructed from a bivector in this way is a pseudovector (or axial vector), as distinguished from normal (or polar) vectors.", "It behaves like a vector under rotations, but under a reflection like $y \\rightarrow -y$ its overall direction reverses (in addition to the reflection) as shown in Figure REF : $(L_x, L_y, L_z) \\rightarrow (-L_x, L_y, -L_z)$ .", "Figure: Left: The bivector tile r →∧p →\\vec{r} \\wedge \\vec{p} together with the cross product r →×p →\\vec{r} \\times \\vec{p}.", "Right: The same vectors after reflection y→-yy \\rightarrow -y, now labeled r → ' \\vec{r}\\,^{\\prime } and p → ' \\vec{p}\\,^{\\prime }.", "The bivector tile r → ' ∧p → ' \\vec{r}\\,^{\\prime } \\wedge \\vec{p}\\,^{\\prime } and its orientation have reflected across the xzxz-plane in exactly the natural way.", "But the cross product r → ' ×p → ' \\vec{r}\\,^{\\prime } \\times \\vec{p}\\,^{\\prime } is not simply its reflection (shown lightly dashed): instead, its direction must also be reversed to remain in agreement with the right hand rule.", "This is the defining behavior of a pseudovector.This awkward behavior of pseudovectors becomes entirely natural in bivector components, where the components whose sign changes are precisely the ones involving $y$ : $(\\ell _{yz}, \\ell _{zx}, \\ell _{xy}) \\rightarrow (-\\ell _{yz}, \\ell _{zx}, -\\ell _{xy})$ .", "This corresponds to reversing the signs of the $y$ -row and $y$ -column of $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }$ (exactly following the standard tensor transformation law)." ], [ "Precession and bivector addition", "It is more complicated to apply the definitions and operations above in an intrinsically three dimensional context.", "As a specific example, consider the precession of a gyroscope as illustrated in Figure REF .", "Figure: Gyroscopic precession.", "Figure (a) shows the torque due to gravity as a tile: the plane of rotation if there were no initial angular momentum.", "(b) shows the initial angular momentum.", "(c) shows the gyroscope precessing in the horizontal plane, but it is not yet clear how to understand this.The gyroscope disk has mass 0.2 and moment of inertia 0.001 and rotates about its axis at 117.", "It is supported by a horizontal massless rod with its center 12 from the frictionless pivot.", "What is its precession frequency?", "As shown in the figure, the initial torque due to gravity about the pivot point can be visualized as a tile oriented with the natural plane of rotation if the gyroscope were not spinning and dropped from rest: a bivector with orientation $\\hat{z}\\wedge \\hat{x}$ .", "In particular, since the force of gravity is $\\vec{F} = m \\vec{g} = -{1.96}{}\\, \\hat{z}$ acting at position $\\vec{r} = {0.12}{}\\, \\hat{x}$ , the torque is $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\tau } = \\vec{r} \\wedge \\vec{F} = {0.2352}{}~\\hat{z}\\wedge \\hat{x}$ .", "Meanwhile, the initial angular momentum tile of the gyroscope matches its natural plane of rotation: its orientation is $\\hat{z}\\wedge \\hat{y}$ , with a specific value of $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }_\\text{initial} = I\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega } = {0.117}{}~\\hat{z}\\wedge \\hat{y}$ .", "Torque is defined as $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\tau } \\equiv d\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }/dt$ .", "Our goal is to use the relationship $d\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell } = \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\tau } dt$ to see how the angle of the gyroscope changes from $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }_\\text{initial}$ to $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }_\\text{final} = \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }_\\text{initial} + d\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }$ in time $dt$ : we will find the angle $d\\phi $ between the two planes.", "It is possible to compute this sum in components and solve the problem purely algebraically, but to actually understand what is going on we need to understand bivector addition geometrically.", "The simplest way to do this is to convert to traditional vector addition.", "Find each bivector tile's pseudovector equivalent, add those vectors tip-to-tail in the usual way, and then convert back from the result of that sum to a tile again.", "(The new tile is normal to the result of the sum and has area equal to its magnitude; its orientation is determined by the right hand rule.)", "This is illustrated alongside the first and last steps in Figure REF : the resulting argument would be identical to traditional vector-based discussions of precession.", "But if we want to avoid using pseudovectors entirely and to work purely in the bivector language, there is also a natural geometric interpretation of addition for bivector tiles themselves.", "The general process for adding arbitrary bivectors $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{a}$ and $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}$ as tiles is illustrated in Fig.", "REF , and proceeds as follows: Add/subtract areas if parallel: If the tiles are parallel, work as in Section : add their areas (magnitudes) if the orientation is the same or subtract them if the orientation is opposite.", "Otherwise: Line of intersection: Let the tiles overlap, and find the line where the two tiles intersect.", "(In three dimensions, non-parallel planes always intersect.)", "Reshape to rectangles: Reshape each tile into a rectangle with the line of intersection as one edge.", "The other edges should have appropriate length and directions to keep the area and orientation the same.", "Shared edge cancels: Line up the two tiles so the shared edges overlap with opposite orientation, so that the two edges “cancel out.” That is, if you follow the orientation direction of each tile around its boundary, the two tiles should go in opposite directions along that shared edge.", "Stretch it tight: With the overlapping edges “cancelled out,” there is now just one long bent tile (whose edges have a consistent orientation around the bent surface).", "Flatten out the tile to remove the bend, so it extends between the two outside edges.", "There's no denying it: this is process of adding tiles is subtle, and for addition it's usually easier to just work in components or with the pseudovector equivalents.", "But it is still good to know that there is a meaningful geometrical interpretation.", "The benefit, once this tile addition process is understood, is that we can now directly visualize what it means to “add” two different planes of rotation (without the extra conceptual step of changing to the perspective of axes of rotation and back).", "And this finally allows us to solve our precession problem.", "Figure REF shows the same torque and angular momentum tiles that were introduced in Figure REF , with the torque multiplied by a short time $dt$ .", "Figure: Analyzing the precession of the gyroscope from Figure  as a sum of orthogonal tiles.The torque is multiplied by a small time dtdt, resulting in an area much smaller than the initial angular momentum.", "Adding the two requires reshaping the ℓ -0.3ex[0ex][0ex][3]↔ initial \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }_\\text{initial} tile to perform the sum: the resulting final angular momentum's orientation has rotated horizontally by angle dφd\\phi as the gyroscope's plane of rotation precesses.Since the tiles have been reshaped to all have the same height, each bivector's magnitude is proportional to its tile's length in the horizontal plane.", "Thus, the angle $d\\phi = \\sin ^{-1}( |\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\tau }| dt/|\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }|)$ .", "Since the time $dt$ is very short, we can use the small angle approximation to find $d\\phi \\approx |\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\tau }| dt/|\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }|$ , giving a precession frequency of $\\omega _P \\equiv \\frac{d\\phi }{dt} = |\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\tau }|/|\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }| \\approx {2.0}{}$ .", "For a student who has become comfortable with adding tiles, this explanation of precession based on adding planes of rotation may be conceptually easier to grasp than the usual explanation based on axes of rotation.", "Translating the visible rotational motion and the torque into vector language, formally adding those vectors, and then translating the result back to a new visible plane of rotation often comes across as very indirect and complicated.", "But the process of adding tiles is undeniably complicated as well: we are trading one set of challenges for another." ], [ "Rigid rotations and the matrix product", "Rigid body rotations give us the opportunity to consider one more core part of bivector math.", "Consider the following question, illustrated in Figure REF : The city of Chicago is located 42 north of the equator.", "Taking the Earth to be a sphere of radius 6400, what is Chicago's linear velocity relative to the center of the Earth?", "Figure: Left: The r →\\vec{r} vector points from the center of the Earth to the city of Chicago.", "The city's instantaneous linear velocity is v →\\vec{v}: parallel to the plane of the equator going east.Right: Finding the (dot) product of a vector r →\\vec{r} with a bivector ω -0.3ex[0ex][0ex][3]↔\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega } is a two-step process.", "First, project the vector into the plane of the bivector, and then rotate it 90 in the tile's orientation direction.", "The resulting magnitude is |r →||ω -0.3ex[0ex][0ex][3]↔|cosθ|\\vec{r}|\\,|\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega }|\\,\\cos \\theta , where θ\\theta is the angle between the vector and the bivector plane, and the cosθ\\cos \\theta comes from the projection.", "(This matches the formula for cross product magnitude, |ω →×r →|=|ω →||r →|sin(90-θ)|\\vec{\\omega }\\times \\vec{r}|=|\\vec{\\omega }|\\,|\\vec{r}|\\,\\sin ({90} - \\theta ).", ")Introductory discussions of rigid rotating objects generally include the relationship between angular speed and the linear speed of a given point on the object: in bivector language, $|\\vec{v}| = r |\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega }|$ for a particle a distance $r$ from the axis of rotation.", "This is the magnitude part of a full vector equation traditionally written as a cross product: $\\vec{v} = \\vec{\\omega } \\times \\vec{r}$ .", "Geometrically, in bivector language this can't correspond to a wedge product because we start with a bivector and end with a vector.", "Similarly, in components, $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega }$ is a a square matrix, while the result $\\vec{v}$ is a vector.", "Instead, for the reasons explained below, we express this relationship as a dot product (matrix product) of the vector and the bivector to yield a vector: $\\vec{v} = \\vec{\\omega } \\times \\vec{r}\\quad \\longrightarrow \\quad \\vec{v} = \\vec{r} \\cdot \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega }\\quad .$ The order of terms is opposite that in the traditional cross product equation, and order does matter: because of the antisymmetry of the bivector, $\\vec{r} \\cdot \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega } = -\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega } \\cdot \\vec{r}$ .", "The geometric interpretation of the product $\\vec{r} \\cdot \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega }$ consists of two steps, as shown in Figure REF .", "First, project the vector $\\vec{r}$ into the plane of the bivector.", "In our case, the result is a vector of magnitude $|\\vec{r}| \\cos \\theta = {6400}{} \\cos {42} \\approx {4756}{}$ pointing horizontally away from the Earth's axis of rotation.", "(The factor of $\\cos \\theta $ comes from the projection, exactly as it does for the vector dot product.)", "And second, multiply that vector by $|\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega }|$ and rotate the result 90 in the direction of the bivector's orientation (so the result is perpendicular to $\\vec{r}$ ).Intuitively, the idea behind this projection-rotation process is that the bivector is an area spanned by one vector sweeping around to a second vector.", "By taking a dot product with an additional vector from the left, we replace the first vector with a scalar, rotating the result toward the second.", "This can be made formal by reshaping the tile as a wedge product of a unit vector along the additional vector's projection and a second vector perpendicular to it.", "(A dot product from the right would rotate the resulting vector 90 opposite the bivector's orientation, reversing the sign.)", "In our case, $|\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega }| = 2\\pi \\,{}/{24}{} \\approx {7.27e-5}{}$ .", "Then the result has magnitude $|\\vec{v}|=|\\vec{r}|\\,|\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega }|\\,\\cos \\theta \\approx {346}{}$ , and the direction is due east.", "In component language, we interpret $\\vec{r} \\cdot \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega }$ as the matrix product of a row vector times a square matrix.", "Because it is more familiar to multiply a square matrix times a column vector, we will often reverse the order with a minus sign: $\\vec{v} = -\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega } \\cdot \\vec{r}\\,$ .Related to this, a rotation matrix can be constructed as the matrix exponential $\\mathsf {R}(t) = e^{-\\protect \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega } t}$ .", "Then for a point with initial position $\\vec{r}_0$ , the matrix product $\\vec{r}(t) = \\mathsf {R}(t) \\cdot \\vec{r}_0$ gives the rotated position as a function of time.", "Then in general, $\\begin{pmatrix} v_x \\\\ v_y \\\\ v_z \\end{pmatrix}&= -\\begin{pmatrix}0 & \\omega _{xy} & \\omega _{xz} \\\\\\omega _{yx} & 0 & \\omega _{yz} \\\\\\omega _{zx} & \\omega _{zy} & 0\\end{pmatrix}\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} \\\\\\nonumber &= -\\begin{pmatrix}\\omega _{xy} y + \\omega _{xz} z \\\\\\omega _{yx} x + \\omega _{yz} z \\\\\\omega _{zx} x + \\omega _{zy} y\\end{pmatrix}= \\begin{pmatrix}-\\omega _{xy} y + \\omega _{zx} z \\\\\\omega _{xy} x - \\omega _{yz} z \\\\-\\omega _{zx} x + \\omega _{yz} y\\end{pmatrix}\\;.$ For our particular case the Earth's rotation is in the $\\hat{x}\\wedge \\hat{y}$ orientation, and if we choose the prime meridian as the $+x$ direction then the position of Chicago is $\\vec{r} \\approx \\Bigl (\\!", "{\\begin{matrix}\\phantom{-{}}{166}{}\\\\ -{4750}{}\\\\ \\phantom{-{}}{4280}{}\\end{matrix}}\\!\\Bigr )$ , since its longitude is 88 W. Thus $\\begin{pmatrix} v_x \\\\ v_y \\\\ v_z \\end{pmatrix}&\\approx -\\begin{pmatrix}0 & 7.27 & 0 \\\\-7.27 & 0 & 0 \\\\0 & 0 & 0\\end{pmatrix}\\begin{pmatrix} \\phantom{-{}}166 \\\\ -4750 \\\\ \\phantom{-{}}4280 \\end{pmatrix}\\times 10^{-5} \\,{}\\;,$ with the result $\\vec{v} = \\Bigl (\\!", "{\\begin{matrix}{346}{}\\\\ {12}{}\\\\0\\end{matrix}}\\!\\Bigr )$ : east from Chicago.", "All of these results match the traditional cross product answers.", "Geometrically, the cross product of the normal vector to $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega }$ with $\\vec{r}$ has the correct direction by the right hand rule, and the magnitude formula also matches: the angle between those two vectors is $({90}-\\theta )$ , and $|\\vec{r}|\\,|\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega }|\\,\\cos (\\theta ) = |\\vec{\\omega }|\\,|\\vec{r}|\\sin ({90}-\\theta )$ .", "And the component expressions in Eq.", "(REF ) also match the traditional cross product components, using the relationship between bivector and pseudovector components from Eq.", "(REF )." ], [ "Conclusions", "It is more than a little audacious to propose a radical change in how we think and teach about a topic as fundamental as rotational motion.", "The existing formalism is very effective and very familiar.", "And on a practical and social level, there will always be a need to teach students the cross product and the pseudovector formalism: even if the bivector approach presented here were universally adopted, students would still need to engage with cross products throughout the literature.", "It is easy to argue that for those very real social reasons, making a change like this would be both impractical and unnecessary.", "Nevertheless, it is the perspective of this work that cross products in physics always hide some underlying bivector phenomenon, and that there is genuine value in finding a way to teach our students in a way that doesn't obscure that truth.", "The bivector description of rotational physics is approachable enough to be taught at a wide range of levels, and even for experts it may give fresh insight into this familiar topic.", "(The same may also be true of the magnetic field, another familiar quantity with an underlying bivector description.", "[13]) At the same time, a substantial change in approach doesn't happen all at once.", "Rather than completely rewriting lesson plans and textbooks, we could begin by planting seeds of the bivector description during traditional cross product instruction.", "That might mean showing students how to construct an oriented tile as an intermediate step to finding the cross product vector, both to justify its magnitude and to offer an easier right hand rule.", "It might mean setting up a problem in the $xy$ -plane and labeling the torque “$\\tau _{xy}$ ” rather than “$\\tau _z$ ” or “$\\tau $ ”.", "Or it might mean taking a few minutes to comment on the existence of the bivector description and some of its advantages during an upper level class, even if only so the students recognize the concept if they encounter it again.", "Describing rotational quantities as bivectors can be helpful to students while also providing a deeper understanding of the topic itself.", "It is our hope that this perspective will eventually become one small part of how the field as a whole thinks about rotational physics.", "Thanks to Brian Hancock and May Lee for very helpful comments on earlier drafts of this work, which have led to significant improvements in clarity and accessibility." ], [ "More advanced applications", "The bivector approach to rotational mechanics described above continues to be viable for more sophisticated topics, and can in fact make some relationships look arguably more natural.", "Here, we will sketch just a few of those applications as results without proof: in most cases, they can be deduced from the traditional forms using the methods of Appendix ." ], [ "The inertia tensor", "In the traditional formulation, the general relationship between angular velocity $\\vec{\\omega }$ and angular momentum $\\vec{L}$ is given by a symmetric rank-2 tensor $I_{ij}$ (which can be interpreted as a linear map from vectors to vectors).", "In index notation, this relationship is $L_i = \\sum _j I_{ij} \\omega _j$ .", "The equivalent relationship in bivector language is $\\ell _{ij} = \\frac{1}{2} \\sum _{k,l} I_{ijkl} \\, \\omega _{kl}\\quad .$ The factor of $\\frac{1}{2}$ is the usual correction for double-counting when summing over entries of an antisymmetric matrix, just as in Eq.", "(REF ).", "The rank-4 inertia tensor $I_{ijkl}$ can be interpreted as a linear map from bivectors to bivectors.", "This version of the inertia tensor can be calculated asIndex notation has an elegant shorthand for this result: $I_{ijkl} = 4 dm \\, x_{[j} \\delta _{i][k} x_{l]}$ , where square brackets denote antisymmetrization: $M_{[ij]}=\\frac{1}{2} (M_{ij} - M_{ji})$ .", "$\\nonumber I_{ijkl} &= \\int dm \\, \\left( \\delta _{ik} x_j x_l - \\delta _{il} x_j x_k- \\delta _{jk} x_i x_l + \\delta _{jl} x_i x_k \\right) \\\\&= \\int dm \\, \\bigl ( \\delta _{ik} x_j x_l + (\\text{symmetries}) \\bigr )\\quad .$ (All four terms are essentially the same: they just use the bivector antisymmetry to sum over the different ways that an $\\ell _{ij}$ index could match an $\\omega _{kl}$ index.)", "Thus, for example, $I_{xyxy} = \\int dm \\, (x^2 + y^2)$ and $I_{yzzx} = \\int dm\\, (-yx)$ .", "It is straightforward to see that this tensor has some simplifying symmetries: $I_{ijkl} = -I_{jikl} = -I_{ijlk} = I_{klij} \\\\I_{ijkl} + I_{iklj} + I_{iljk} = 0\\;.$ The antisymmetry of the initial and final pairs matches that of the bivectors, and the symmetry under exchange of pairs matches the symmetry of the inertia tensor in the usual formulation.Curiously, these are exactly the algebraic symmetries of the Riemann curvature tensor.", "In this language, rather than finding eigenvectors of $I_{ij}$ to identify principal axes of rotation, we find eigenbivectors of $I_{ijkl}$ to identify principal planes of rotation.", "In brief, as in the Petrov classification of spacetimes,[1], [2] to do this we treat the space of bivectors as an abstract vector space and create a matrix $I_{AB}$ showing the action of $I_{ijkl}$ on the basis bivectors $\\hat{y}\\wedge \\hat{z}$ , etc.", "(As will be discussed in Section REF , in $d$ dimensions these bivector space indices range from $A=1,\\ldots ,\\frac{d(d-1)}{2}$ .)", "In practice, the natural way to do this (in three dimensions) is to map bivector index pairs to pseudovector indices so that $I_{AB}$ becomes just the traditional inertia tensor $I_{ij}$ : e.g.", "$I_{xy} = I_{yzzx}$ ." ], [ "Quantum commutation relations", "In quantum mechanics, the angular momentum operator $\\hat{\\vec{J}}$ is the generator of rotations: the rotation operator for angle $\\phi $ about (unit vector) axis $\\vec{n}$ is $\\hat{R}(\\vec{n},\\phi ) = e^{-i\\phi \\vec{n} \\cdot \\hat{\\vec{J}}/\\hbar }$ .", "The essential properties of the angular momentum operator are encoded in its commutation relations: $[\\hat{J}_i, \\hat{J}_j] = i \\hbar \\sum _k \\epsilon _{ijk} \\hat{J}_k$ .", "All of this can be translated into bivector language in a very straightforward way, with essentially no change other than notation.", "Angular momentum is a bivector operator $\\hat{\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{J}}$ , and a plane of rotation is specified by a unit bivector $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{n}$ .", "The dot product in the exponent is replaced by a “double dot product” $\\frac{1}{2}\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{n} : \\hat{\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{J}} = \\frac{1}{2} \\sum _{i,j=1}^3 n_{ij} J_{ij}$ as defined in Eq.", "(REF ).", "The rotation operator is then $\\hat{R}(\\vec{n},\\phi ) = e^{-i\\phi \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{n} : \\hat{\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{J}}/2\\hbar }\\quad .$ The commutation relations become $\\nonumber [\\hat{J}_{ij}, \\hat{J}_{kl}] &= i \\hbar \\left(\\delta _{ik} \\hat{J}_{jl} - \\delta _{il} \\hat{J}_{jk} - \\delta _{jk} \\hat{J}_{il}+ \\delta _{jl} \\hat{J}_{ik}\\right) \\\\&= i \\hbar \\delta _{ik} \\hat{J}_{jl} + (\\text{symmetries})\\quad .$ At most one of these four terms will be non-zero: as with Eq.", "(REF ), the four terms just reflect the different ways that an index from the first operator could match one from the second.", "This expression exactly matches the commutation relations of the Lorentz group for spacelike indices (see, e.g., Weinberg, Eq.", "(2.4.12)[16]), as it must.", "Finally, because all vectors (and pseudovectors) transform in the same way under rotations, the traditional commutation relations of angular momentum with any vector operator $\\hat{\\vec{V}}$ match those for $\\hat{J}_i$ with itself: $[\\hat{J}_i, \\hat{V}_j] = i \\hbar \\epsilon _{ijk} \\hat{V}_k$ .", "In terms of the bivector angular momentum, this becomes $[\\hat{J}_{ij}, \\hat{V}_k]&= i \\hbar \\left( \\delta _{ik} \\hat{V}_j - \\delta _{jk} \\hat{V}_i \\right) = i \\hbar \\delta _{ik} \\hat{V}_j - (\\text{symmetry}) \\;.$ As before, the two terms reflect the antisymmetry of $\\hat{J}_{ij}$ ." ], [ "Relativistic angular momentum", "In special relativity, vector quantities are understood as the spatial parts of some corresponding four-vector in $(3+1)$ dimensions.", "Spacetime position is combined with time as $\\mathsf {X} = (ct, x, y, z)$ , and momentum $\\vec{p}$ is the spatial part of the four-momentum, whose time component is the energy: $\\mathsf {P} = (E/c, p_x, p_y, p_z) = \\gamma m \\,(c, v_x, v_y, v_z)$ .", "But this doesn't work for pseudovectors: not only is the cross product undefined in higher dimensions, but the relativistic transformation law for $\\vec{L}$ is entirely different than for vectors like $\\vec{p}$ or $\\vec{r}$ .", "The relativistic angular momentum of a point particle must be expressed as a four-bivector $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\mathsf {M}} = \\mathsf {X} \\wedge \\mathsf {P}$ , with (contravariant) components[17] $\\nonumber \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\mathsf {M}} &= \\begin{pmatrix}0 & -c N_x & -c N_y & -c N_z \\\\c N_x & 0 & \\ell _{xy} & -\\ell _{zx} \\\\c N_y & -\\ell _{xy} & 0 & \\ell _{yz} \\\\c N_z & \\ell _{zx} & -\\ell _{yz} & 0\\end{pmatrix}\\quad .$ The spatial components are the angular momentum bivector $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }$ , and the time-space components form a (true) vector $\\vec{N}$ in three dimensions.", "The standard Lorentz transformations of this rank-2 tensor lead to exactly the correct transformation laws for angular momentum, which mix the components of $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }$ with those of $\\vec{N}$ .", "To understand $\\vec{N}$ , we can write out the wedge product definition: $N_i = x_i \\frac{E}{c^2} - t p_i = \\gamma m (x_i - v_i t)$ .", "When summed over a collection of particles, this is effectively total energy times “the apparent location of the center of mass at $t=0$ ,” extrapolated in time to $t=0$ using the current center of mass velocity.", "Unfamiliar as this is, in the absence of external forces the center of mass velocity doesn't change, so this “$t=0$ position” is a conserved quantity, just like the (spatial) angular momentum.", "It may feel odd to see a specific time ($t=0$ ) as a defining aspect of a conserved quantity, but we have measured angular momentum about $\\vec{r}=0$ as well: all components of relativistic angular momentum depend on our choice of spacetime origin." ], [ "Rotations in extra dimensions", "String theory and other ideas in modern physics invite us to study physics in systems with more than three dimensions of space, and most basic laws of physics extend in an obvious way: for example, if $d=4$ then the momentum vector would have components $\\vec{p} = (p_x, p_y, p_z, p_w)$ .", "But just as with spacetime in relativity, the pseudovector description of rotational physics is not even well-defined in that case: there is no concept of a unique axis of rotation.", "As mentioned briefly in section REF , in $d$ dimensions of space there are $\\binom{d}{2} = \\frac{d(d-1)}{2}$ independent planes of rotation, each with its own bivector component.", "For example, when $d=4$ we can write the angular velocity as $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega } = \\begin{pmatrix}0 & \\omega _{xy} & \\omega _{xz} & \\omega _{xw} \\\\-\\omega _{xy} & 0 & \\omega _{yz} & \\omega _{yw} \\\\-\\omega _{xz} & -\\omega _{yz} & 0 & \\omega _{zw} \\\\-\\omega _{xw} & -\\omega _{yw} & -\\omega _{zw} & 0\\end{pmatrix} \\quad .$ Because there are more independent components than the number of dimensions, there is no way to represent angular velocity as a vector.", "(This is why $d=3$ is the unique special case where a cross product can be defined.)", "The greater number of planes allows rotations to be substantially more complicated than what we are used to in three dimensions.", "One aspect of this is that only rare bivectors in four or more dimensions can be represented as a single tile.", "The addition rules in section  give a single tile result for the case of two parallel tiles (such as $\\hat{x}\\wedge \\hat{y} + 3\\hat{x}\\wedge \\hat{y}$ ) and for two tiles that overlap along a line (such as $\\hat{x}\\wedge \\hat{y} + \\hat{z}\\wedge \\hat{x}$ , which overlap along a line parallel to the $x$ -axis and yield the bivector $\\hat{x}\\wedge (\\hat{y}-\\hat{z})$ ).", "But most tiles in four dimensions are entirely separate and overlap only at the single point at the origin (such as $\\hat{x}\\wedge \\hat{y} + \\hat{z}\\wedge \\hat{w}$ ) and those sums cannot be simplified at all.", "(A simple bivector is one that can be written in the form $\\vec{u}\\wedge \\vec{v}$ ; this is sometimes called a blade.)", "In general, any bivector in $d$ dimensions can be written as a sum of $d/2$ vector wedge products (rounded down), each represented by a tile.", "So in three dimensions it is always possible to represent a bivector as a single tile, but in four or five dimensions most require a sum of two tiles, and so on.", "It is even possible to require that all of the vectors in those wedge products be mutually orthogonal (so that each tile is orthogonal to all the others).", "Unless two tiles have the same magnitude, that orthogonal sum is unique.", "(A sketch of a proof is given in Appendix .)", "A concrete example of momentum conservation may help clarify all this.", "Consider a (hyper)spherical rock of mass 3.7e4 and moment of inertia 1 floating in space with its initial angular velocity (about the center of mass) in a single plane: $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega }_i = \\hat{y} \\wedge (10 \\hat{x} + 16 \\hat{z}) \\,{}$ .", "(About three rotations per second.)", "A pebble of mass 1 with velocity $\\vec{v} = (0,2,-1,0)\\,{}$ collides with the rock at position $\\vec{r} = (4,0,8,-1)\\,{}$ relative to the rock's center of mass.", "We want to know the rock's final angular velocity once the pebble is embedded in it.", "The rock's initial angular momentum is just $I \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega }_i$ : the numerical components match those of its angular velocity.", "The pebble's angular momentum is the wedge product $\\vec{r}\\wedge \\vec{p}$ , where $\\vec{p} = m\\vec{v} = (0,2000,-1000,0)\\,{}$ .", "Then $\\vec{r}\\wedge \\vec{p} &=\\left(\\!", "{\\begin{matrix}0 & 8& -4 & 0 \\\\-8 & 0 & -16 & 2 \\\\4 & 16 & 0 & -1 \\\\0 & -2 & 1 & 0\\end{matrix}}\\!\\right) {}\\quad .$ This means that the final total angular momentum is $\\nonumber \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\ell }_f &=\\left[\\left(\\!", "{\\begin{matrix}0 & -10 & 0 & 0 \\\\10 & 0 & 16 & 0 \\\\0 & -16 & 0 & 0 \\\\0 & 0 & 0 & 0\\end{matrix}}\\!\\right) +\\left(\\!", "{\\begin{matrix}0 & 8& -4 & 0 \\\\-8 & 0 & -16 & 2 \\\\4 & 16 & 0 & -1 \\\\0 & -2 & 1 & 0\\end{matrix}}\\!\\right)\\right] {} \\\\&=\\left(\\!", "{\\begin{matrix}0 & -2 & -4 & 0 \\\\2 & 0 & 0 & 2 \\\\4 & 0 & 0 & -1 \\\\0 & -2 & 1 & 0\\end{matrix}}\\!\\right) {}\\quad .$ After dividing by the moment of inertia (which is essentially unchanged), we can write the final angular velocity as $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{\\omega }_f = \\bigl [ -2\\hat{x}\\wedge (\\hat{y}+2\\hat{z})+ (2\\hat{y}-\\hat{z})\\wedge \\hat{w} \\bigr ] {}$ .", "This can be visualized as a sum of two rectangular tiles that are entirely orthogonal to each other: a simultaneous rotation in two different planes, one twice as fast as the other.If we do not demand that the two planes be mutually orthogonal, there are many other ways to write this final angular velocity as a sum of two wedge products.", "For example, $\\bigl [ (2\\hat{x}+\\hat{z})\\wedge (\\hat{y}+2\\hat{z}) + (2\\hat{y}-\\hat{z})\\wedge (\\hat{y}-\\hat{w} ) \\bigr ] {}$ .", "The math of bivectors has interesting implications for the nature of rotations in four (or more) dimensions.", "In four dimensions, rotations can be categorized as simple, double, or isoclinic.", "Simple rotations have angular velocities given by simple bivectors: they are rotations parallel to a single plane, just as in three dimensions.", "Double rotations, by far the most common, have angular velocities that can only be written as the sum of two tiles: at any point but the origin, all four coordinates are changing at once under simultaneous rotations in the two tile planes.", "And isoclinic rotations are the special case of double rotations where the two tiles have the same angular speed: in this case, there are infinitely many ways of dividing the motion into two orthogonal planes of rotation.", "One application of this is solar system formation, when a cloud of randomly moving dust collapses to form a star and structures in orbit around it.", "In three dimensions every bivector is simple, so the total angular momentum of any system lies in a single plane.", "Collisions between particles will eventually cancel out most motion orthogonal to that plane, concentrating the matter into a dense rotating disk where planets can form.", "In four dimensions angular momentum is usually not a simple bivector, so there is no preferred plane where the matter will collect and it will remain in a diffuse cloud where planet formation is unlikely.", "[19] (This difficulty is in addition to the radial instability of circular orbits in higher dimensions.)" ], [ "Bivectors as sums of orthogonal tiles", "Any bivector $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}$ can be expressed as a sum of $d/2$ vector wedge products (rounded down) with all of the vectors mutually orthogonal, each corresponding to a tile.", "Unless two of the tiles have the same magnitude, this orthogonal sum is unique.", "What follows is a sketch of the argument.", "[20] The matrix product of $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}$ with itself ($\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}\\cdot \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}$ ) is a symmetric matrix, so its eigenvectors form a basis.", "Consider one eigenvector $(\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}\\cdot \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b})\\cdot \\vec{u} = \\lambda \\vec{u}$ with $\\lambda \\ne 0$ , scaled for convenience so $|\\vec{u}|=1$ , and define $\\vec{v} \\equiv \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}\\cdot \\vec{u}$ .", "(This implies that $\\vec{v}$ is a second eigenvector with eigenvalue $\\lambda $ , and since $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}$ is antisymmetric, $\\vec{u}\\cdot \\vec{v}=\\vec{u}\\cdot \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}\\cdot \\vec{u}=0$ .)", "The antisymmetry of $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}$ also implies $\\vec{u}\\cdot \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}\\cdot \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}\\cdot \\vec{u}=-\\vec{v}\\cdot \\vec{v}=-|\\vec{v}|^2$ , which combines with $\\vec{u}\\cdot \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}\\cdot \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}\\cdot \\vec{u}=\\vec{u}\\cdot (\\lambda \\vec{u}) = \\lambda $ to show $\\lambda = -|\\vec{v}|^2$ .", "Given all this, we can write in components $b_{ij} \\equiv v_i u_j - u_i v_j + a_{ij}$ , where $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{a}$ is a new bivector that satisfies $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{a}\\cdot \\vec{u} = \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{a}\\cdot \\vec{v} = 0$ but otherwise has the same eigenvectors and eigenvalues as $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}$ .", "Repeat the whole process for $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{a}$ , choosing only vectors orthogonal to $\\vec{u}$ and $\\vec{v}$ : in the end, you'll get $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b} = \\vec{v} \\wedge \\vec{u} + \\cdots $ as desired.", "(Eigenvectors with different eigenvalues are orthogonal, so the only way this will fail to be unique is if more than one pair of vectors has the same $\\lambda $ .", "Then, any decomposition of that eigenspace into orthogonal tiles will work.)" ], [ "Supplementary reference: Bivector equivalents of pseudovector products", "We collect here a number of formal results relating pseudovector product equations to their bivector equivalents.", "The results are presented in index notation and involve the totally antisymmetric Levi-Civita symbol $\\epsilon _{ijk}$ , where $\\epsilon _{xyz}=+1$ , $\\epsilon _{yxz}=-1$ , etc.", "As we will see, the final bivector forms do not involve $\\epsilon _{ijk}$ at all.", "Although the derivations here are entirely in Cartesian three dimensional space for clarity, the final bivector results in index notation generalize directly to curved space in any dimension (where bivectors are antisymmetric rank-2 contravariant tensors).", "For clarity, in this appendix we will use the symbols $\\vec{U}$ , $\\vec{V}$ , and $\\vec{W}$ to refer to generic ordinary vectors (“polar vectors”), while the symbols $\\vec{B}$ , $\\vec{C}$ , and $\\vec{D}$ will refer to generic pseudovectors (“axial vectors”) whose bivector equivalents are $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}$ , $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{c}$ , and $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{d}$ , respectively.", "In most cases, the equivalences can be derived using the identity $\\sum _{i=1}^3 \\epsilon _{ijk} \\epsilon _{imn} = \\delta _{jm} \\delta _{kn} - \\delta _{jn} \\delta _{km} \\quad ,$ though at times a more general identity is needed: $\\nonumber \\epsilon _{ijk} \\epsilon _{lmn}= {}&\\delta _{il} \\delta _{jm} \\delta _{kn}+ \\delta _{im} \\delta _{jn} \\delta _{kl}+ \\delta _{in} \\delta _{jl} \\delta _{km} \\\\& - \\delta _{il} \\delta _{jn} \\delta _{km}- \\delta _{im} \\delta _{jl} \\delta _{kn}- \\delta _{in} \\delta _{jm} \\delta _{kl} \\;.$ For any pseudovector $\\vec{B}$ (in three dimensions), we can define the bivector $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}$ by the matrix components $b_{ij}$ : $b_{ij} \\equiv \\sum _{k=1}^3 \\epsilon _{ijk} B_k\\equiv \\epsilon _{ijk} B_k \\quad .$ This second form illustrates the use of Einstein summation notation (used implicitly through the rest of this section) where repeated indices within a single term are assumed to be summed over all coordinates.", "Explicitly in terms of components, this reads $b_{ij}= \\begin{pmatrix}0 & b_{xy} & -b_{zx} \\\\-b_{xy} & 0 & b_{yz} \\\\b_{zx} & -b_{yz} & 0\\end{pmatrix} = \\begin{pmatrix}0 & B_z & -B_y \\\\-B_z & 0 & B_x \\\\B_y & -B_x & 0\\end{pmatrix} \\quad .$ We can see that $b_{ji} = -b_{ij}$ , because of the antisymmetry of $\\epsilon _{ijk}$ .", "The relationship also works in the other direction, with a factor of a half to compensate for double counting (since each independent bivector component appears twice in the matrix): $B_i = \\frac{1}{2} \\epsilon _{ijk} b_{jk}= \\begin{pmatrix} b_{yz} \\\\ b_{zx} \\\\ b_{xy} \\end{pmatrix}\\quad .$ We begin with cross products, since that is where pseudovectors arise.", "The cross product of two ordinary vectors $\\vec{B}= \\vec{U}\\times \\vec{V}$ is expressed in index notation as $B_i = \\epsilon _{ijk} U_j V_k \\quad .$ Using the definition in Eq.", "(REF ) (after suitably relabeling summed indices), the symmetries of $\\epsilon _{ijk}$ , and the identity (REF ), we find $b_{ij} &= \\epsilon _{ijk} B_k= \\epsilon _{ijk}\\left( \\epsilon _{kmn} U_m V_n \\right)= \\left( \\epsilon _{kij} \\epsilon _{kmn} \\right)U_m V_n\\nonumber \\\\&= \\left( \\delta _{im} \\delta _{jn} - \\delta _{in} \\delta _{jm} \\right)U_m V_n= U_i V_j - U_j V_i \\quad .$ This is the definition of the wedge product: $\\boxed{\\vec{B}= \\vec{U}\\times \\vec{V}\\qquad \\longleftrightarrow \\qquad \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}= \\vec{U}\\wedge \\vec{V}}\\quad .$ This can be visualized as explained in Figure REF .", "Figure: In the wedge product b -0.3ex[0ex][0ex][3]↔=U →∧V →\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}= \\vec{U}\\wedge \\vec{V}, the two vectors define a parallelogram of area |b -0.3ex[0ex][0ex][3]↔|=|U →|h=|U →||V →|sinθ|\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}| = |\\vec{U}| h = |\\vec{U}| |\\vec{V}| \\sin \\theta with a fixed attitude in space.", "This figure illustrates two ways to visualize its orientation: either as the direction that rotates the first vector toward the second tail-to-tail, or by following the first vector tip-to-tail with the second around the edges.Meanwhile, the cross product of a pseudovector and an ordinary vector is an ordinary vector, $\\vec{V}= \\vec{B}\\times \\vec{U}$ , and can be expressed in index notation as $V_i= \\epsilon _{ijk} B_j U_k = U_k \\left(\\epsilon _{kij} B_j \\right)= U_k b_{ki} \\quad .$ We can immediately recognize this as matrix multiplication of a row vector times a square matrix, which we will represent symbolically as a dot product: $\\boxed{\\vec{V}= \\vec{B}\\times \\vec{U}\\qquad \\longleftrightarrow \\qquad \\vec{V}= \\vec{U}\\cdot \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}}\\quad .$ This can be visualized as explained in Figure REF .", "(Note that the order of the two terms has reversed.)", "Figure: Finding the (dot) product of a vector U →\\vec{U} with a bivector b -0.3ex[0ex][0ex][3]↔\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b} is a two-step process.", "First, project the vector into the plane of the bivector, and then rotate it 90 in the tile's orientation direction.", "The resulting magnitude is |U →||b -0.3ex[0ex][0ex][3]↔|cosθ|\\vec{U}|\\,|\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}|\\,\\cos \\theta , where θ\\theta is the angle between the vector and the bivector plane, and the cosθ\\cos \\theta comes from the projection.", "(This matches the formula for cross product magnitude, |B →×U →|=|B →||U →|sin(90-θ)|\\vec{B}\\times \\vec{U}|=|\\vec{B}|\\,|\\vec{U}|\\,\\sin ({90} - \\theta ).", ")Given the antisymmetry of $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}$ , if we reinterpret $\\vec{U}$ as a column vector we can instead write the matrix product in the opposite order: $\\vec{V}= -\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}\\cdot \\vec{U}$ .", "Performing the matrix multiplication often feels more familiar in this order.", "(In the remainder of this section, we will omit the derivations of each bivector expression: the methods are similar to those above, but the details can get tedious.)", "Although it is less common in physical formulas, for completeness we should also consider the cross product of two pseudovectors, whose result is another pseudovector: $\\vec{D}= \\vec{B}\\times \\vec{C}$ .", "Converting this to bivector language is messy: it starts like Eq.", "(REF ) and uses Eqs.", "(REF ) and (REF ), with the result $d_{ij} = \\left( c_{ik} b_{kj} - b_{ik} c_{kj} \\right) \\quad .$ Figure: The commutator [c -0.3ex[0ex][0ex][3]↔,b -0.3ex[0ex][0ex][3]↔][\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{c},\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}] of two tiles is non-zero if their planes intersect along a line (with angle θ\\theta between them).", "Reshape the tiles as rectangles with unit length along the shared line, and form a “pipe” out of two copies of each as shown.", "The cross section of the pipe (e.g.", "either “end cap”) is the tile representing the commutator: a parallelogram of area |c -0.3ex[0ex][0ex][3]↔||b -0.3ex[0ex][0ex][3]↔|sinθ|\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{c}|\\,|\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}| \\sin \\theta .", "As for orientation, find an edge of the “pipe” where the overlapping edges have the same orientation: the orientation of the result matches the edge from the first tile that points into the shared edge and the edge of the second that points out of it.", "Imagining that shared edges with opposite orientation “cancel out,” you can follow the arrows around the pipe for a full cycle.", "(Reversing the order of the commutator changes which path around the pipe you follow.", ")The result is recognizable as the matrix product of the two bivectors, antisymmetrized: $\\boxed{\\vec{D}= \\vec{B}\\times \\vec{C}\\quad \\longleftrightarrow \\quad \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{d}= \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{c}\\cdot \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}- \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}\\cdot \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{c}\\equiv [\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{c},\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}]}\\quad .$ The final square bracket notation reflects the fact that this is the commutator of the two matrices.", "For “simple” bivectors that can each be represented by a single tile, this can be visualized as described in Figure REF .", "Having established the formulas for cross products involving pseudovectors, we next consider dot products.", "The dot product of two pseudovectors $\\alpha = \\vec{B}\\cdot \\vec{C}$ is a scalar and can be written as $\\alpha = B_i C_i = \\tfrac{1}{2} b_{jk} c_{jk} \\quad .$ As in previous equations, the factor of $\\frac{1}{2}$ compensates for the duplication of entries in the two antisymmetric matrices.", "This sum over products of all matrix components is sometimes called the “double dot product” and is denoted by a colon: $\\boxed{\\alpha = \\vec{B}\\cdot \\vec{C}\\qquad \\longleftrightarrow \\qquad \\alpha = \\tfrac{1}{2} \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}\\!", ":\\!\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{c}}\\quad .$ As with the vector dot product, we can interpret this as a measure of whether two bivectors are oriented in the same direction: orthogonal tiles have double dot product zero.", "We can use this to define the magnitude of a bivector: $|\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}|^2 \\equiv \\frac{1}{2} \\sum _{i,j=1}^3 b_{ij}^2$ .", "If we use the antisymmetry of the bivector matrices, we can also interpret this in terms of the trace of the matrix product: $\\alpha = -\\frac{1}{2} \\operatorname{tr}(\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}\\cdot \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{c})$ .", "And if we express $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{c}= \\vec{U}\\wedge \\vec{V}$ , then $\\alpha = \\vec{U}\\cdot \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}\\cdot \\vec{V}$ .", "Geometrically, this is a measure of the degree to which the two planes have the same attitude and orientation in space.", "For two tiles whose planes overlap along a line (which includes any two tiles in three dimensions) the double dot product matches the dot product of their normal vectors: $\\frac{1}{2} \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}\\!", ":\\!\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{c}= |\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}|\\, | \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{c}| \\cos \\theta $ , where $\\theta $ is the angle between the planes.", "More generally in higher dimensions (where two planes may only overlap at a point), the double dot product equals zero if any vector in one tile is perpendicular to the other tile.", "The precise geometric meaning in general seems complicated: considering the form $\\vec{U}\\cdot \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}\\cdot \\vec{V}$ in light of Figure REF , we are projecting $\\vec{U}$ into the plane of $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}$ , multiplying it by the bivector magnitude, rotating it 90, and then taking the dot product with $\\vec{V}$ .", "Finally, the dot product of an ordinary vector and a pseudovector $\\phi = \\vec{U}\\cdot \\vec{B}$ is more subtle to understand, because the result $\\phi $ is a “pseudoscalar” rather than an ordinary scalar: $\\phi &= U_i B_i= \\tfrac{1}{2} \\epsilon _{ijk} U_i b_{jk}\\nonumber \\\\&= U_x b_{yz} + U_y b_{zx} + U_z b_{xy} \\quad .$ Every term contains all three spatial coordinates and no free indices: this does indeed look like a scalar quantity.", "But like a pseudovector, under a reflection like $x \\rightarrow -x$ , all three terms change sign, which an ordinary scalar would not.", "The appropriate interpretation (which correctly generalizes to higher dimensions) is not a scalar, but the totally antisymmetric wedge product of $U_i$ with $b_{ij}$ : $\\boxed{\\phi = \\vec{U}\\cdot \\vec{B}\\qquad \\longleftrightarrow \\qquad \\Phi _{(3)} = \\vec{U}\\wedge \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}} \\\\\\Phi _{ijk} = U_i b_{jk} + U_j b_{ki} + U_k b_{ij}\\quad .$ This result is a “trivector”: a totally antisymmetric rank-3 tensor.", "(This product is symmetric: $\\vec{U}\\wedge \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}= \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}\\wedge \\vec{U}$ .)", "Intuitively, this can be visualized as an (oriented) parallelepiped region of 3D space, with the bivector tile $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b}$ as its base and the vector $\\vec{U}$ showing how that base “extrudes” into the third dimension.", "(The sign of the 3D orientation is positive if the bivector's orientation looks counterclockwise when viewed from inside the parallelepiped.)", "This is the natural interpretation of the “triple product” of vectors, which in our language we would write as $\\vec{U}\\wedge \\vec{V}\\wedge \\vec{W}$ .", "Because this expression is totally antisymmetric in $i$ , $j$ , and $k$ , it is zero unless the three indices are different.", "In three dimensions, this means that Eq.", "(REF ) is the only independent term: that's why it looks like a scalar.", "But in $d$ dimensions, there are $\\binom{d}{3} = \\frac{1}{3!}", "d (d-1)(d-2)$ independent ways of choosing three coordinate labels out of $d$ , so that is the number of components in this relationship.", "Replacing pseudovectors with bivectors opens one more possible type of product that is not relevant in three dimensions: the wedge product of two bivectors $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b} \\wedge \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{c}$ .", "This is a totally antisymmetric rank-4 tensor, and with four antisymmetrized coordinate indices it is guaranteed to be zero in three dimensions.", "The explicit coordinate form can be written: $\\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{b} \\wedge \\overset{\\raisebox {-0.3ex}[0ex][0ex]{[3]\\mathbf {\\leftrightarrow }}}{c} \\rightarrow b_{ij} c_{k\\ell } + b_{ik} c_{\\ell j} + b_{i\\ell } c_{jk} +b_{jk} c_{i\\ell } + b_{j\\ell } c_{ki} + b_{k\\ell } c_{ij}\\;.$ This could be imagined as an oriented chunk of 4D space." ] ]
2207.03560
[ [ "On the greatest common divisor of $n$ and the $n$th Fibonacci number, II" ], [ "Abstract Let $\\mathcal{A}$ be the set of all integers of the form $\\gcd(n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$th Fibonacci number.", "Leonetti and Sanna proved that $\\mathcal{A}$ has natural density equal to zero, and asked for a more precise upper bound.", "We prove that \\begin{equation*} \\#\\big(\\mathcal{A} \\cap [1, x]\\big) \\ll \\frac{x \\log \\log \\log x}{\\log \\log x} \\end{equation*} for all sufficiently large $x$." ], [ "Introduction", "Let $(u_n)$ be a nondegenerate linear recurrence with integral values.", "Arithmetic relations between $n$ and $u_n$ have been studied by several authors.", "For example, the set of positive integers such that $n$ divides $u_n$ has been studied by Alba González, Luca, Pomerance, and Shparlinski [2], assuming that the characteristic polynomial of $(u_n)$ is separable, and by André-Jeannin [3], Luca and Tron [11], Sanna [17], and Somer [21], when $(u_n)$ is a Lucas sequence.", "Furthermore, Sanna [19] showed that the set of natural numbers $n$ such that $\\gcd (n, u_n) = 1$ has a natural density (see [13] for a generalization).", "Mastrostefano and Sanna [12], [18] studied the moments of $\\log \\!\\big (\\!\\gcd (n, u_n)\\big )$ and $\\gcd (n, u_n)$ when $(u_n)$ is a Lucas sequence, and Jha and Nath [7] performed a similar study over shifted primes.", "(See also the survey of Tron [23] on greatest common divisors of terms of linear recurrences.)", "Let $(F_n)$ be the linear recurrence of Fibonacci numbers, which is defined by $F_1 = F_2 = 1$ and $F_{n + 2} = F_{n + 1} + F_n$ for every positive integer $n$ .", "Sanna and Tron [20] proved that, for each positive integer $k$ , the set of positive integers $n$ such that $\\gcd (n, F_n) = k$ has a natural density, which is given by an infinite series.", "Kim [9] and Jha [6] obtained formally analogous results in cases of elliptic divisibility sequences and orbits of polynomial maps, respectively.", "Let $\\mathcal {A}$ be the set of numbers of the form $\\gcd (n, F_n)$ , for some positive integer $n$ .", "Leonetti and Sanna [10] provided an effective method to enumerate the elements of $\\mathcal {A}$ in increasing order.", "In particular, the first elements of $\\mathcal {A}$ are $1, \\quad 2, \\quad 5, \\quad 7, \\quad 10, \\quad 12, \\quad 13, \\quad 17, \\quad 24, \\quad 25, \\quad 26, \\quad 29, \\quad 34, \\quad 35, \\quad 36, \\quad \\dots $ see [1] for more terms.", "Then they proved that $\\#\\mathcal {A}(x) \\gg \\frac{x}{\\log x}$ for all $x \\ge 2$ .", "Their approach relied on a result of Cubre and Rouse [4], which in turn follows from Galois theory and the Chebotarev density theorem.", "Later, Jha and Sanna [8] obtained an elementary proof as an application of related arithmetic problem over shifted primes.", "Leonetti and Sanna [10] also gave the upper bound $\\#\\mathcal {A}(x) = o(x)$ as $x \\rightarrow +\\infty $ ; and asked for a more precise estimate.", "We prove the following upper bound on $\\#\\mathcal {A}(x)$ .", "Theorem 1.1 We have $\\#\\mathcal {A}(x) \\ll \\frac{x \\log \\log \\log x}{\\log \\log x}$ for all sufficiently large $x$ .", "In light of the gap between the upper bound of Theorem REF and the lower bound (REF ) it is natural to wonder which is the true order of $\\#\\mathcal {A}(x)$ .", "By performing some numerical experiments (see Section  later), we found that $\\#\\mathcal {A}(x)$ appears to be asymptotic to $x / (\\log x)^c$ , as $x \\rightarrow +\\infty $, for some constant $c \\approx 0.63$ , see Figure REF .", "Of course, these kind of experiments has to be taken with a grain of salt, since they cannot reveal slow-growing factors like $\\log \\log x$ .", "Figure: A plot of #𝒜(x)/(x/(logx) c )\\#\\mathcal {A}(x) / (x / (\\log x)^c) for xx up to 10 6 10^6." ], [ "Notation", "For every set of positive integers $\\mathcal {S}$ and for every $x > 0$ , we define $\\mathcal {S}(x) := \\mathcal {S} \\cap [1, x]$ .", "We employ the Landau–Bachmann “Big Oh” and “little oh” notation $O$ and $o$ , as well as the associated Vinogradov symbols $\\ll $ and $\\gg $ .", "In particular, all of the implied constants are intended to be absolute.", "We let $\\operatorname{Li}(x) := \\int _2^x (\\log t)^{-1}\\,\\mathrm {d}t$ denote the integral logarithm." ], [ "Preliminaries", "For each positive integer $n$ , let $z(n)$ be the rank of appearance of $n$ , that is, $z(n)$ is the smallest positive integer $k$ such that $n$ divides $F_k$ .", "It is well known that $z(n)$ exists.", "Moreover, put $\\ell (n) := \\operatorname{lcm}\\!\\big (n, z(n)\\big )$ and $g(n) := \\gcd \\!\\big (n, F_{n}\\big )$ .", "The next lemma collects some elementary properties of $z$ , $\\ell $ , and $g$ .", "Lemma 2.1 For all positive integer $m,n$ and all prime numbers $p$ , we have: $z(m) \\mid z(n)$ whenever $m \\mid n$ .", "$n \\mid g(m)$ if and only if $\\ell (n) \\mid m$ .", "$n \\in \\mathcal {A}$ if and only if $n = g\\big (\\ell (n)\\big )$ .", "$m \\mid n$ whenever $\\ell (m) \\mid \\ell (n)$ and $n \\in \\mathcal {A}$ .", "$z(p)\\mid p - (p/5)$ where $(p/5)$ is a Legendre symbol.", "$z(p^n)=p^{\\max (n-e(p)\\,,\\,0)}\\,z(p)$ , where $e(p) := \\nu _p(F_{z(p)})\\ge 1$ and $\\nu _p$ is the usual p-adic valuation.", "$\\ell (p^n)=p^n z(p)$ if $p\\ne 5$ , and $\\ell (5^n)=5^n$ .", "For REF , REF .", "and REF , see [10].", "Fact REF follows easily from REF and REF .", "Facts REF and  REF are well known (cf. [11]).", "Fact REF follows quickly from REF and REF .", "For each positive integer $d$ , let $\\mathcal {P}_d$ be the set of prime numbers $p$ such that $d$ divides $z(p)$ .", "Cubre and Rouse [4] proved that $\\#\\mathcal {P}_d(x) \\sim \\delta (d) \\operatorname{Li}(x)$ , as $x \\rightarrow +\\infty $ , where $\\delta (d) := \\frac{1}{d}\\prod _{p \\,\\mid \\, d} \\left(1 - \\frac{1}{p^2}\\right)^{-1} {\\left\\lbrace \\begin{array}{ll} 1 & \\text{ if } 10 \\nmid d; \\\\ 5/4 & \\text{ if } d \\equiv 10 \\!\\!\\!\\pmod {20}; \\\\ 1/2 & \\text{ if } 20 \\mid d .", "\\end{array}\\right.", "}$ Sanna [16] extended this result to Lucas sequences (under some mild restrictions) and provided also an error term.", "In particular, as a consequence of [16], we have the following asymptotic formula.", "Lemma 2.2 There exists an absolute constant $B > 0$ such that $\\#\\mathcal {P}_d(x) = \\delta (d) \\operatorname{Li}(x) + O\\!\\left(\\frac{x}{(\\log x)^{12/11}}\\right) ,$ for all odd positive integers $d$ and for all $x \\ge \\exp (B d^{40})$ .", "From [16] we have that there exists an absolute constant $B > 0$ such that $\\#\\mathcal {P}_d(x) = \\delta (d) \\operatorname{Li}(x) + O\\!\\left(\\frac{d}{\\varphi (d)} \\cdot \\frac{x \\, (\\log \\log x)^{\\omega (d)}}{(\\log x)^{9/8}}\\right) ,$ for all odd positive integers $d$ and for all $x \\ge \\exp (B d^{40})$ , where $\\varphi (d)$ and $\\omega (d)$ are the Euler totient function and the number of prime factors of $d$ , respectively.", "Note that we can assume that $B$ (and consequently $x$ ) is sufficiently large.", "In particular, we have that $d \\le (\\log x)^{1/40}$ .", "Put $\\varepsilon := 1/330$ .", "By the classic lower bound for $\\varphi (d)$ (see, e.g., [22]) we have that $\\frac{d}{\\varphi (d)} \\ll \\log \\log d \\ll \\log \\log \\log x \\le (\\log x)^{\\varepsilon } .$ Recall that $\\omega (d) \\le \\big (1 + o(1)\\big ) \\log d / \\log \\log d$ as $d \\rightarrow +\\infty $ (see, e.g., [22]).", "Therefore, there exists an absolute constant $C > 0$ such that if $d > C$ then $\\omega (d) \\le (1 + \\varepsilon ) \\frac{\\log d}{\\log \\log d} \\le \\left(\\frac{1}{40} + 2\\varepsilon \\right) \\frac{\\log \\log x}{\\log \\log \\log x} ,$ and consequently $(\\log \\log x)^{\\omega (d)} \\le (\\log x)^{\\frac{1}{40} + 2\\varepsilon }$ .", "Also, if $d \\le C$ then $(\\log \\log x)^{\\omega (d)} \\le (\\log x)^\\varepsilon $ .", "The claim follows.", "Remark 2.1 In Lemma REF the exponent $12/11$ can be replaced by $11/10 + \\varepsilon $ , for every $\\varepsilon > 0$ , assuming that $x$ is sufficiently large depending on $\\varepsilon $ .", "We also need an upper bound for $\\#\\mathcal {P}_d(x)$ .", "Lemma 2.3 We have $\\#\\mathcal {P}_d(x) \\ll \\frac{x}{\\varphi (d) \\log (x / d)}$ for all positive integers $d$ and for all $x > d$ .", "By Lemma  REFREF , we have that $\\#\\mathcal {P}_d(x) \\le 1 + \\#\\big \\lbrace p \\le x : p \\equiv \\pm 1 \\!\\!\\!\\pmod {d} \\big \\rbrace \\ll \\frac{x}{\\varphi (d) \\log (x / d)} ,$ where we applied the Brun–Titchmarsh inequality [22].", "Now we give an upper bound for the sum of reciprocals of primes in $\\mathcal {P}_d$ .", "Lemma 2.4 We have $\\sum _{p \\,\\in \\, \\mathcal {P}_d(x)} \\frac{1}{p} = \\delta (d)\\log \\log x + O\\!\\left(\\frac{\\log (2d)}{\\varphi (d)}\\right)$ for all odd positive integers $d$ and for all $x \\ge 3$ .", "First, suppose that $x < \\exp (Bd^{40})$ , where $B$ is the constant of Lemma REF .", "Hence, we have that $\\delta (d)\\log \\log x \\ll \\frac{\\log \\log x}{d} \\ll \\frac{\\log (2d)}{d} .$ Moreover, by [15], we have that $\\sum _{\\begin{array}{c}p \\,\\le \\, x \\\\[1pt] p \\,\\equiv \\, \\pm 1 \\!\\!\\!\\!\\pmod {d}\\end{array}} \\frac{1}{p} = \\frac{2\\log \\log x}{\\varphi (d)} + O\\!\\left(\\frac{\\log (2d)}{\\varphi (d)}\\right) .$ This together with Lemma  REFREF yields that $\\sum _{p \\,\\in \\, \\mathcal {P}_d(x)} \\frac{1}{p} \\le \\frac{1}{d} + \\sum _{\\begin{array}{c}p \\,\\le \\, x \\\\[1pt] p \\,\\equiv \\, \\pm 1 \\!\\!\\!\\!\\pmod {d}\\end{array}} \\frac{1}{p} \\ll \\frac{1}{d} + \\frac{\\log (2d)}{\\varphi (d)} .$ Hence, the claim follows.", "Now suppose that $x \\ge \\exp (Bd^{40})$ .", "By partial summation, we have that $\\sum _{p \\,\\in \\, \\mathcal {P}_d(x)} \\frac{1}{p} = \\frac{\\#\\mathcal {P}_d(x)}{x} + \\int _1^x \\frac{\\#\\mathcal {P}_d(t)}{t^2} \\,\\mathrm {d} t .$ Obviously, $\\#\\mathcal {P}_d(x) / x \\ll 1/d$ by the trivial inequality.", "Thus it remains to bound the integral.", "By Lemma  REFREF , we have that $\\int _1^{2d} \\frac{\\#\\mathcal {P}_d(t)}{t^2} \\,\\mathrm {d} t \\le \\frac{1}{d^{2}}\\int _1^{2d-2} 5 \\,\\mathrm {d} t \\ll \\frac{1}{d},$ after noticing that $\\#\\mathcal {P}_d(t)>0$ only if $t\\ge d-1$ .", "By Lemma REF , we have that $\\int _{2d}^{\\exp (Bd^{40})} \\frac{\\#\\mathcal {P}_d(t)}{t^2} \\,\\mathrm {d} t\\ll \\int _{2d}^{\\exp (Bd^{40})} \\frac{\\mathrm {d} t}{\\varphi (d) \\,t \\log (t / d)} = \\left[ \\frac{\\log \\log (t/d)}{\\varphi (d)}\\right]_{t \\,=\\, 2d}^{\\exp (Bd^{40})} \\ll \\frac{\\log d}{\\varphi (d)}.$ By Lemma REF , we have that $\\int _{\\exp (Bd^{40})}^x \\frac{\\#\\mathcal {P}_d(t)}{t^2} \\,\\mathrm {d} t&= \\int _{\\exp (Bd^{40})}^x \\frac{\\delta (d) \\operatorname{Li}(t)}{t^2} \\,\\mathrm {d} t + O\\!\\left(\\int _{\\exp (Bd^{40})}^x \\frac{\\mathrm {d}t}{t (\\log t)^{12/11}} \\right) \\\\&= \\delta (d) \\left[\\log \\log t - \\frac{\\operatorname{Li}(t)}{t} \\right]_{t \\,=\\, \\exp (Bd^{40})}^x + O\\!\\left( \\frac{1}{d^{40/11}} \\right) \\\\&= \\delta (d) \\left(\\log \\log x + O(\\log d) \\right) + O\\!\\left( \\frac{1}{d^{40/11}} \\right) \\\\&= \\delta (d) \\log \\log x + O\\!\\left(\\frac{\\log d}{d} \\right) .$ Putting these together, the claim follows.", "The following sieve result is a special case of [5] (cf. [14]).", "Lemma 2.5 We have $\\#\\lbrace n\\le x : p\\mid n \\Rightarrow p \\notin \\mathcal {P}\\rbrace \\ll x \\prod _{p \\,\\in \\, \\mathcal {P}(x)} \\left(1 - \\frac{1}{p}\\right) ,$ for all $x \\ge 2$ and for all sets of prime numbers $\\mathcal {P}$ ." ], [ "Proof of Theorem ", "Suppose that $x > 0$ is sufficiently large, and put $k := \\left\\lfloor \\frac{1}{\\log 5} \\log \\!\\left(\\frac{25}{24\\log 5} \\cdot \\frac{\\log \\log x}{\\log \\log \\log x}\\right)\\right\\rfloor $ and $d := 5^k$ .", "Note that $\\delta (d) = 5^{-k} \\cdot 25 / 24$ .", "Hence, we get that $\\log \\!\\left(\\frac{\\log d}{\\delta (d)}\\right) = k \\log 5 + \\log k + \\log \\!\\left(\\frac{24 \\log 5}{25}\\right) \\le \\log \\log \\log x .$ Therefore, we have that $(\\log d) / \\delta (d) \\le \\log \\log x$ and $(\\log x)^{\\delta (d)} \\ge d \\gg \\frac{\\log \\log x}{\\log \\log \\log x} .$ We split $\\mathcal {A}$ into two subsets: $\\mathcal {A}_1$ is the subset of $\\mathcal {A}$ consisting of integers without prime factors in $\\mathcal {P}_d$ , and $\\mathcal {A}_2 := \\mathcal {A} \\setminus \\mathcal {A}_1$ .", "First, we give an upper bound on $\\#\\mathcal {A}_1(x)$ .", "By Lemma REF and Lemma REF , we get that $\\#\\mathcal {A}_1(x) \\ll x \\prod _{p \\,\\in \\, \\mathcal {P}_d(x)} \\left(1-\\frac{1}{p}\\right)\\ll x \\exp \\!\\left(-\\!\\!\\sum _{p \\,\\in \\, \\mathcal {P}_d(x)} \\frac{1}{p}\\right)\\ll \\frac{x}{(\\log x)^{\\delta (d)}} ,$ where we also used the inequality $1 - x \\le \\exp (-x)$ , which holds for $x \\ge 0$ .", "Now we give an upper bound on $\\#\\mathcal {A}_2(x)$ .", "If $n \\in \\mathcal {A}_2$ then $n$ has a prime factor $p \\in \\mathcal {P}_d$ .", "Hence, we have that $p \\mid n$ and $d \\mid z(p)$ .", "Thus, by Lemma  REFREF , we get that $z(p) \\mid z(n)$ and so $d \\mid \\ell (n)$ .", "Recalling that $d = 5^k$ , by Lemma  REFREF we have that $\\ell (d) = d$ .", "Hence, we get that $\\ell (d) \\mid \\ell (n)$ and, by Lemma  REFREF , it follows that $d \\mid n$ .", "Thus all the elements of $\\mathcal {A}_2$ are multiples of $d$ .", "Consequently, we have that $\\#\\mathcal {A}_2(x) \\le \\frac{x}{d} .$ Therefore, putting together (REF ) and (REF ), and using (REF ), we obtain that $\\#\\mathcal {A}(x) = \\#\\mathcal {A}_1(x) + \\#\\mathcal {A}_2(x) \\ll \\frac{x}{(\\log x)^{\\delta (d)}} + \\frac{x}{d} \\ll \\frac{x \\log \\log \\log x}{\\log \\log x} ,$ as desired.", "The proof is complete." ], [ "Numerical computations", "We computed the elements of $\\mathcal {A} \\cap [1, 10^6]$ by using a program written in C that employs Lemma  REFREF .", "Note that computing $g\\big (\\ell (n)\\big )$ directly as $\\gcd \\!\\big (\\ell (n), F_{\\ell (n)}\\big )$ would be prohibitive, in light of the exponential grown of Fibonacci numbers.", "Instead, we used the fact that $g\\big (\\ell (n)\\big ) = \\gcd \\!\\big (\\ell (n), F_{\\ell (n)} \\bmod \\ell (n)\\big ),$ and we computed Fibonacci numbers modulo an integer by efficient matrix exponentiation." ] ]
2207.03521
[ [ "Finding Fallen Objects Via Asynchronous Audio-Visual Integration" ], [ "Abstract The way an object looks and sounds provide complementary reflections of its physical properties.", "In many settings cues from vision and audition arrive asynchronously but must be integrated, as when we hear an object dropped on the floor and then must find it.", "In this paper, we introduce a setting in which to study multi-modal object localization in 3D virtual environments.", "An object is dropped somewhere in a room.", "An embodied robot agent, equipped with a camera and microphone, must determine what object has been dropped -- and where -- by combining audio and visual signals with knowledge of the underlying physics.", "To study this problem, we have generated a large-scale dataset -- the Fallen Objects dataset -- that includes 8000 instances of 30 physical object categories in 64 rooms.", "The dataset uses the ThreeDWorld platform which can simulate physics-based impact sounds and complex physical interactions between objects in a photorealistic setting.", "As a first step toward addressing this challenge, we develop a set of embodied agent baselines, based on imitation learning, reinforcement learning, and modular planning, and perform an in-depth analysis of the challenge of this new task." ], [ "Visual perception networks.", "We train a Mask-RCNN using Feature Pyramid Networks with ResNet-50 backbone for detection and instance segmentation.", "We use a dataset of 60,000 images of 30 categories to train this model.", "We set batch size as 12 and use SGD as an optimizer.", "The learning rate is 0.001.", "The model converges after 70K iterations." ], [ "Auditory perception networks.", "We use ResNet-18 pre-trained on ImageNet-1K as a backbone which takes as input log Mel-scaled spectrograms extracted from waveforms.", "Following the backbone, we use three linear layers with a hidden dimension of 1024 to serve as a classifier.", "We update all parameters of the model during training.", "We set batch size as 128 and use Adam as an optimizer.", "The learning rate is 0.0003.", "The model converges within 200 epochs." ] ]
2207.03483
[ [ "Energy Correlators of Hadronically Decaying Electroweak Bosons" ], [ "Abstract Energy correlators are field-theoretically clean and phenomenologically valuable probes of QCD dynamics.", "We explore the possibility of using the information encoded in the energy correlators of a hadronically decaying electroweak vector boson in order to extract its full decay density matrix.", "The kinematics of the one- and two-point energy correlators can indeed discriminate between longitudinal and transverse modes and reveal the interference pattern between different vector polarizations.", "Such observables improve the sensitivity to microscopic new physics affecting the production rate of the different helicities.", "We assess the impact on higher dimensional EFT operators in simple scenarios." ], [ "warn luatex=false colorlinks, citecolor=black, linkcolor=black, urlcolor=bluscuro 1.1 justification=raggedright,singlelinecheck=false Energy Correlators of Hadronically Decaying Electroweak Bosons Lorenzo Ricci0000-0001-8704-3545 Theoretical Particle Physics Laboratory (LPTP), Institute of Physics, EPFL, Lausanne, Switzerland Marc Riembau0000-0002-9842-2425 Theoretical Particle Physics Laboratory (LPTP), Institute of Physics, EPFL, Lausanne, Switzerland Energy correlators are field-theoretically clean and phenomenologically valuable probes of QCD dynamics.", "We explore the possibility of using the information encoded in the energy correlators of a hadronically decaying electroweak vector boson in order to extract its full decay density matrix.", "The kinematics of the one- and two-point energy correlators can indeed discriminate between longitudinal and transverse modes and reveal the interference pattern between different vector polarizations.", "Such observables improve the sensitivity to microscopic new physics affecting the production rate of the different helicities.", "We assess the impact on higher dimensional EFT operators in simple scenarios.", "Introduction Scattering at high energies reveals the true nature of massive Standard Model (SM) particles: they are a mixture of fields with different dynamics.", "This difference is commonly exacerbated in presence of some putative microscopic physics, out of which the SM emerges.", "For instance, such dynamics might have a role in the EW symmetry breaking mechanism and couple predominantly to longitudinal vector bosons.", "It is therefore of utmost importance to develop a program centered on observables capable to discriminate the various helicity configurations.", "Observables of this kind were proposed long ago for $WW$ -production at LEP [1], [2], [3], or for $t\\bar{t}$ -production at Tevatron [4], [5], [6], and are a substantial part of the BSM precision program at the LHC [7], [8], [9], [10], [11], [12], [13].", "Advantages of using such observables to constrain BSM physics are particularly clear in the context of Effective Field Theories (EFT), where microscopic physics is encoded in higher dimensional operators.", "At high energy, a given operator modifies only certain helicity amplitude, which is often suppressed in the SM [14], [15].", "As a consequence, the interference between the SM and EFT amplitudes gets suppressed, which translates to a poor sensitivity and that might compromise their interpretation within a consistent EFT framework [16], [17].", "Recovering the sensitivity requires the detailed analysis of the differential kinematic distributions of the final states.", "For instance, the strategy proposed in [18] to measure the interference between BSM and SM was later used in [19], improving the linear constraints by an order of magnitude.", "In the particular case of diboson production, while many studies exist for leptonic decay products of vector bosons, the case of hadronic decays is less explored and substantially more challenging.", "Existing results rely on jet substructure and, recently, on machine learning in order to discriminate among longitudinal and transversely polarized vector bosons [20], [21], [22], [23], [24], [25].", "In this Letter, instead, we explore the possibility of using energy correlators (EC) in hadronically decaying $W$ and $Z$ bosons in order to extract information about their helicity.", "This means not only a better discrimination of signal versus background in searches for specific EFT deformations, but also to design observables sensitive to interferences among different helicity states.", "We follow this idea through two simple paths.", "First, we show how one-point EC can disentangle purely longitudinal and transverse decays.", "Then we prove that two-point EC can reveal the interference pattern among different helicities.", "In particular this enhances the sensitivity to EFT deformation whenever SM and BSM amplitudes are produced in different helicity configurations.", "We remark that most of our analysis will be theoretical and illustrative.", "We will explore relevant scenarios with a simplified analysis to grasp the advantages of using polarization-sensitive observables and comment on how to export our results to the LHC framework.", "Energy correlators & decay density matrix In general, the amplitude for the production and decay of a vector boson is described, under the narrow-width approximation, by the product of the production and decay amplitudes $\\mathcal {A}^{hard,V}_{h}$ and $\\mathcal {A}_{h,X}$ that describe the production and decay of a vector boson $V$ with helicity $h$ .", "The cross section involves then a double sum over the helicities of the vector in the amplitude and its conjugate.", "Therefore, it is convenient to write the cross section as the trace of the product of two matrices $d\\sigma = d\\rho ^{hard,V}_{{h^\\prime }h} d\\rho ^V_{{h^\\prime }h},$ where $d\\rho ^{hard,V}_{{h^\\prime }h}$ is the so-called density matrix for the “hard” production of the vector $V=W^\\pm ,Z$ and $d\\rho ^V_{{h^\\prime }h}$ is the –fully differential– density matrix for its decay.", "The indexes $h,{h^\\prime }=+,-,0$ run over the vector boson helicities.", "Common observables targeting to measure new physics in $d \\rho ^{hard,V}$ , like bump searches or many “high-$p_T$ ” EFT probes, ignore the full information carried by the decay products, integrating over their kinematic variables.", "For instance, since $d\\rho ^V_{{h^\\prime }h}\\propto e^{i(h-{h^\\prime })\\phi }$ the decay matrix becomes diagonal if the observable is inclusive in the azimuthal angle $\\phi $ and one becomes sensitive only to the diagonal entries of $d\\rho ^{hard,V}_{{h^\\prime }h}$ , i.e.", "one loses the information on the interference between different polarizations.", "This information can be in principle recovered from the kinematic distribution of the decay density matrix $d\\rho ^V_{{h^\\prime }h}$ .", "At leading order in the weak coupling constant, the vector boson decays into a $q\\bar{q}$ pair, which then interact, radiate and evolve according to QCD dynamics, resulting in a jet of hadrons.", "It is our goal to find sensible observables that can probe the decay density matrix $d\\rho ^V_{{h^\\prime }h}$ into hadrons in order to dissect the nature of the microscopic physics mediating the hard process.", "An approach to define such observables consists in studying jet substructure in terms of correlation functions of light-ray operators [26], [27], [28] ${\\mathcal {E}}_i =\\text{lim}_{r \\rightarrow \\infty } \\int _0^{+\\infty } dt\\, r^2 n^i T_{0i} (t,r \\vec{n})\\,,$ where $T_{\\mu \\nu }$ is the stress-energy tensor.", "In particular, we focus on the so-called light-ray density matrix [29], defined as the vacuum expectation value $\\begin{aligned}d\\rho ^V_{h {h^\\prime }}[\\left\\lbrace {\\mathcal {E}}_i\\right\\rbrace ] = \\mathcal {N} \\int d^4 x e^{i q \\cdot x} \\langle \\mathcal {O}_h(x) {\\mathcal {E}}_1\\ldots {\\mathcal {E}}_N \\mathcal {O}^{\\dagger }_{h^{\\prime }}(0)\\rangle \\end{aligned}\\,,$ where $\\mathcal {O}_h$ is some operator exciting the QCD vacuum and $\\mathcal {N}$ is just a normalization constant.", "Physically, $d\\rho ^V_{{h^\\prime }h} [{\\mathcal {E}}_i\\ldots {\\mathcal {E}}_N]$ measures the energy deposited in $N$ calorimeters placed at infinity in some directions $\\vec{n}_i$ in presence of a source.", "In our case, the source $\\mathcal {O}_h$ is a boosted on-shell vector of helicity $h$ and four-momentum $q$ in the lab.", "frame, “injecting” a $q\\bar{q}$ current in the correlator of eq.", "(REF ) $\\mathcal {O}_h(x) = \\left( \\bar{q}\\gamma _{\\mu } (g_LP_L+g_RP_R) q \\right) (x) \\, \\varepsilon ^{\\mu }_h \\,,$ Being Infrared and Collinear Safe, energy correlators are theoretically clean and can be safely computed with the usual rules of perturbative QCD.", "Moreover, they possess a series of phenomenologically interesting features.", "They can be computed and measured on tracks, improving the angular resolution, and the insensitivity to soft radiation mitigates the need of grooming [28], [30].", "In the recent years, there has been renewed interest due to new theoretical insights, allowing a simpler characterization of jet dynamics [31], [32], [33], [34].", "The $d\\rho ^V_{{h^\\prime }h}[\\left\\lbrace {\\mathcal {E}}_i\\right\\rbrace ]$ is in general independent of the microscopic dynamics.", "However, in the following, we will discuss how the detailed study of energy correlators allows to characterize the diagonal and off-diagonal entries of the decay density matrix, improving sensitivity to new physics encoded in the hard process.", "Diagonal entries: Longitudinal and transverse.— Figure: One and two point energy correlator of a hadronically decaying electroweak boson.", "Dots denote energy measurements from a calorimeter.", "The one point measures the energy deposit in a calorimeter at a distance z 〈ℰ〉 z_{\\langle {\\mathcal {E}}\\rangle } from the jet axis and azimuthal angle φ 〈ℰ〉 \\phi _{\\langle {\\mathcal {E}}\\rangle } with respect the plane formed with the beam.", "The two point correlator depends on similar quantities, but we integrate the distance and orientation of the calorimeters within the jet.The density matrix for the one point energy correlator, that we denote shortly by $\\langle {\\mathcal {E}}\\rangle $ , is given by the correlator of eq.", "(REF ) with only one insertion of the light-ray operator ${\\mathcal {E}}_{\\vec{n}}$ , pointing toward the direction $\\vec{n}$ .", "For clarity it useful to relate the energy correlator to the more “phenomenologically friendly” objects of the weighted cross-section (see for instance [35]).", "In particular we have $\\langle {\\mathcal {E}}_{\\vec{n}} \\rangle =\\frac{1}{2 m_V \\Gamma _V} _X \\mathcal {A}_{h,X} \\mathcal {A}_{h^{\\prime },X}^* \\sum _{i\\in X}\\frac{E_i}{E} \\delta \\left( \\vec{n}_i -\\vec{n}\\right)\\,,$ where $X$ is a generic multiparticle state to which the vector $V$ , of energy $E$ , decays.", "The factor $\\frac{E_i}{E}$ weights the energies of the partons in the decay.", "Notice that eq.", "(REF ), or equivalently eq.", "(REF ) with one ${\\mathcal {E}}$ insertion, depends at most by the two variables parametrizing the direction of $\\vec{n}$ .", "The natural choice are the two angles $0\\le \\theta < \\pi $ and $0 \\le \\phi < 2 \\pi $ , where $\\theta $ is the angle respect to jet momentum $q$ .", "The angle $\\phi $ is the angle between the line that connects the center of the jet and the calorimeter, and the plane that goes through the jet axis and the beam, see Fig.", "REF .", "Diagonal entries have a trivial $\\phi $ dependence and so they are functions only of $\\theta $ .", "At leading order, the density matrix is determined by the different decay amplitudes $\\mathcal {A}_h$ ($\\equiv \\mathcal {A}_{h,q\\bar{q}} $ ).", "In the vector rest frame the latter are simply proportional to the Wigner-d matrices, $\\mathcal {A}_\\pm = g_Vm_Ve^{\\pm i\\phi }\\frac{1\\pm \\cos \\theta _*}{2}\\,,\\quad \\mathcal {A}_0 = -g_Vm_V \\frac{\\sin \\theta _*}{\\sqrt{2}},$ where $\\theta _*$ is the angle between the direction of flight of the vector and the helicity-plus fermion.", "In the case of the $W$ , the coupling is left-handed and $g_W=g$ .", "The $Z$ boson, instead, couples to both left and right currents, with strength $g_Z=\\sqrt{2}(gc_\\text{W}T^3_f-g^\\prime s_\\text{W} Y_f)$ .", "In the lab frame, the vector boson is boosted up to an energy $E$ , which sets a characteristic angular scale $z_\\star \\equiv m_V^2/E^2$ that controls the kinematics.", "In this frame, we define $z$ to be the separation between a parton and the direction of the vector boson, $z=\\frac{1-\\cos \\theta }{2}\\simeq \\frac{1}{4}\\theta ^2$ , see Fig REF .", "At leading order in $z_\\star $ , the energy of a quark is in one-to-one with its separation $z_q$ , $E_q\\,=\\, E\\,\\left(1+\\frac{4z_q}{z_\\star }\\right)^{-1}\\,\\equiv \\, E \\,x(z),$ where it is convenient to define the quark energy fraction $x(z_q)\\equiv x$ .", "The rest frame angle is related to $z_q$ by $\\cos \\theta _*=\\frac{1-4z_q/z_\\star }{1+4z_q/z_\\star }$ .", "The position of the two quarks obeys $z_qz_{\\bar{q}}=(\\frac{z_\\star }{4})^2$ , since $z_\\star /4$ corresponds to $\\theta _*=\\frac{\\pi }{2}$ , so at parton level each event has two energy depositions at each side of $z_\\star /4$ .", "For $z\\ll z_\\star /4$ , the energy of the vector boson is entirely deposited in a single quark since the other quark's energy is redshifted to zero.", "For $z\\sim z_\\star /4$ , the energy is shared among quarks.", "In terms of the energy fraction of the positive-helicity quark, the amplitudes in the lab frame at leading order in $z_\\star $ are given by $\\begin{aligned}\\mathcal {A}_+ &\\simeq g_Vm_Ve^{i\\phi }x\\,,\\quad \\mathcal {A}_- \\simeq g_Vm_Ve^{-i\\phi }(1-x)\\,,\\\\&\\mathcal {A}_0 \\simeq -g_Vm_V \\sqrt{2}\\sqrt{x(1-x)}.\\end{aligned}$ Notice that the different helicities have a different behavior, given by the degree of overlap with the $\\cos \\theta _*\\sim 1$ region of the decay amplitude.", "When measuring the energy correlator, though, we average the position of the calorimeter.", "Placing it on the minus-helicity fermion is equivalent to $x\\rightarrow 1-x$ and $\\phi \\rightarrow \\pi +\\phi $ .", "The quark helicity is not directly measurable, so the energy correlator has the redundancy $\\mathcal {A}_h\\rightarrow (-1)^{h}\\mathcal {A}_{-h}$ .", "Notice also that at leading order in $z_\\star $ the kinematics only depend on the quantity $z/z_\\star $ .", "This will be of crucial importance, later on, for our discussion on the LHC.", "Figure: One-point energy correlator of a WW jet.", "The coordinate zz is the angular distance between the calorimeter and the center of the jet.The transverse and longitudinal terms in the diagonal entries of the density matrix behave differently, since $d\\rho ^V_{00}\\sim x(1-x)$ while $d\\rho ^V_{TT}\\equiv d\\rho ^V_{++}+d\\rho ^V_{--}\\sim x^2+(1-x)^2$ .", "In Fig.", "REF we show the spectrum of the one-point energy correlator of $d\\rho ^V_{00}$ (in cyan) and $d\\rho ^V_{TT}$ (in orange) as a function of $z$ with $\\phi $ integrated over, where we fix the $W$ boson energy at $1.5\\,\\mathrm {TeV}$ .", "The horizontal lines show the data points with the statistical Monte Carlo error, clearly negligible and in dashed we show the interpolation.", "The longitudinal spectrum peaks at $z=\\frac{1}{6}z_\\star $ .", "The transverse spectrum is actually the sum of two contributions.", "The one that dominates by far has the calorimeter placed at the position of the fermion with the same-sign helicity as the $W$ boson helicity, which has a peak at $z=\\frac{1}{16}z_\\star $ .", "The subdominant contribution has it placed in the opposite-sign helicity quark, which has a much smaller peak at larger $z=\\frac{3}{8}z_\\star $ and pushes the total transverse peak of Fig.", "REF to $z\\simeq \\frac{11}{144}z_\\star $ .", "In Fig.", "REF we compare the analytic LO result with the numerical simulation for a monoenergetic ($E=1.5$ TeV) hadronic $W$ jet.", "Concretely, we simulated through MadGraph [36] the semileptonic $W^+ W^-$ production from lepton-lepton scattering.", "The simulation is further processed via Pythia 8 [37], [38] to include the effects of showering and hadronization.", "The minimum $p_T$ for the $W$ is fixed at 300 GeV.", "Each event gives a distribution for the correlator, obtained by measuring the energies of the hadrons hitting the detector.", "In the Figure we show the sum of such distributions, corresponding to the average energy measured by a calorimeter placed at $z$ .", "The LO predictions are normalized to 1, while the Pythia 8 ones are fixed to have the same-height peak.", "The LO calculation gives already a very good prediction for the distributions from Pythia 8, which can be further improved using the higher order corrections.Actually, the transverse distribution can be directly imported from [39], [40], by considering their result in a center of mass $\\sqrt{\\hat{s}}=m_W$ , and boosting it by $e^\\eta \\simeq 2E/m_W$ .", "Figure: 95%CL contours of different observables of WWWW production.", "In the left (right), we assume anomalous production of longitudinal (transverse) WW-bosons due to non-vanishing δg 1Z (λ γ )\\delta g_{1Z}\\,(\\lambda _{\\gamma }), indicated by the star.In order to have an estimate of how the differences in the longitudinal and transverse distributions allow to discriminate them, we define the simple but intuitive asymmetry ${A}_{LT} =({A}_+-{A}_-)/({A}_++{A}_-)$ , where ${A}_\\pm $ is the energy accumulated with $z$ greater/smaller than the angle $z^{\\text{peak}}$ at which the transverse distribution has a maximum.", "Approximately, ${A}_{LT}\\simeq 0$ for a purely transverse sample, while ${A}_{LT}\\sim 0.5$ for a purely longitudinal one.", "At parton level, in the vector boson rest frame, $z^{\\text{peak}}$ corresponds to an angle $\\theta _*\\simeq 58^\\circ $ .", "In a typical observable where one compares the number of events at each side of a cut, the maximal sensitivity of an excess of longitudinal events over a purely transverse sample corresponds to a cut at $\\theta _*\\simeq 56^\\circ $ .We remark that the one-point energy correlator measures the average energy density deposited at $z$ in a sample of jet events, and has not to be thought as a probability distribution.", "However, since $z^{cut}\\ll z_\\star /4$ , at the left of the cut most of the energy is deposited in a single quark and therefore the asymmetry is roughly an event-level counting.", "Let us investigate the impact of the asymmetry in a simple but relevant scenario.", "In the left of Fig.", "REF we consider semileptonic $W^+W^-$ production at $\\sqrt{s}=3\\,\\text{TeV}$ with $p_T(W)>1.2$ TeV and $5/\\text{ab}$ of integrated luminosity.The choice for the $p_T$ cut is to reduce the large forward SM background improving the EFT sensitivity.", "The production of longitudinal modes is considered to be modified by an anomalous triple gauge coupling $\\mathcal {L}\\supset igc_{\\theta _W} \\delta g_{1Z}(W^+_{\\mu \\nu }W^-_\\mu -W^-_{\\mu \\nu }W^+_\\mu )Z_\\nu $ with $\\delta g_{1Z}=2\\times 10^{-3}$ [41], [42].", "We interpret the cross section measurement in the 2$d$ parameter space defined by $\\delta g_{1Z}$ and $\\lambda _\\gamma $ , the coefficient of $\\mathcal {L}\\supset i\\frac{e}{m_W^2}\\lambda _\\gamma W^+_{\\mu \\nu }W^-_{\\nu \\rho }A_{\\rho \\mu }$ , which instead controls the production of the transverse modes.", "The cross section measurement gives the gray contour, and cannot determine the origin of the excess.", "The measurement of the asymmetry ${A}_{LT}$ , in cyan, confirms that the excess comes from an anomalous production of longitudinal modes.", "We will comment the rest of the plot later, when discussing the interference terms.", "We now turn to the two point correlator $\\langle {\\mathcal {E}}_1 {\\mathcal {E}}_2 \\rangle $ , defined in a similar way inserting two light-ray operators into eq.", "(REF ).", "Potentially, $\\langle {\\mathcal {E}}_1 {\\mathcal {E}}_2 \\rangle $ has a non trivial dependence on four angles, parametrizing the position of $\\vec{n}_1$ and $\\vec{n}_2$ .", "In the following, we only consider two angles, being the angular distance $z=\\frac{1}{2}(1-\\vec{n}_1\\cdot \\vec{n}_2)$ between the two calorimeters and the azimuthal angle $\\phi $ between their line of separation and the scattering plane.With an abuse of notation, we use $z$ and $\\phi $ in both one-point and two-point correlators.", "see Fig REF .", "Should be clear from context which case we refer to.", "The two other coordinates will be integrated over.", "The amplitudes at leading order are given by Eq.", "(REF ), now with the energy fraction $x$ related to the angular separation $z$ between the quarks by $x(z)\\,=\\, \\frac{1}{2}\\left(1\\pm \\sqrt{1-z_\\star /z}\\right)$ , where the sign is determined by whether $\\cos \\theta _*$ is positive or negative.", "At this order in the perturbative expansion, there is a minimal separation between quarks given by $z_\\star $ , equal for all vector polarizations.", "The product of the quark energies is given by $E_qE_{\\bar{q}}/E^2=x(1-x)=\\frac{z_\\star }{4z}$ , so the spectrum peaks at $z\\sim z_\\star $ for all entries of the density matrix.", "This is shown in Fig.", "REF for the diagonal ones.", "Since the longitudinal entry also behaves as $\\propto x(1-x)$ , the longitudinal peak is more pronounced than the one of the transverse distribution.", "Two effects resolve the peak and give a contribution for lower $z$ .", "First, the finite decay width of the vector gives a small width of order $\\sim \\Gamma _V/m_V$ to the peak in the energy correlator.", "However, this effect is subleading with respect the one given by the QCD radiation from the quarks.", "Indeed, the width of the peak corresponds to the angular scale where the correlator enters in the scaling regime.", "Such scaling regime has received recent attention [34], [29] since it can be described as an operator product expansion of null-ray operators [31], [43].", "For our purposes, it is sufficient to say that this regime is purely controlled by QCD, the $z$ dependence does not depend on the quark helicity, and apparently it does not offer any information that can be used to disentangle the different contributions of the diagonal entries of the decay density matrix.", "Below the $\\sim (\\text{few}\\,\\mathrm {GeV})^2/E^2$ scale, one can observe deviations due to hadronization [44], [45], [46].", "Figure: Two-point energy correlator of a WW-boson jet.", "Both transverse and longitudinal distributions show a peak at the angular scale z ☆ =m V 2 /E 2 z_\\star =m_V^2/E^2.", "For lower zz, one explores the scaling regime of the individual quark sub-jets.", "At very small angular scales z∼ GeV 2 /E 2 z\\sim \\,\\mathrm {GeV}^2/E^2, hadronization enters and cuts off the correlator.Off-diagonal entries: Interference.— Both one- and two-point correlators of the decay density matrix depend on the azimuthal angles $\\phi _{\\langle {\\mathcal {E}}\\rangle }$ and $\\phi _{\\langle {\\mathcal {E}}{\\mathcal {E}}\\rangle }$ , see Fig REF , which we will denote generically $\\phi $ .", "The dependence is fixed to be $d\\rho ^V_{{h^\\prime }h} \\propto e^{i(h-{h^\\prime })\\phi }$ .", "If we use observables that are inclusive in $\\phi $ , like the invariant mass of the process or the $p_T$ of the vectors, only the diagonal terms with $h={h^\\prime }$ contribute while the off-diagonal ones with $h\\ne {h^\\prime }$ integrate to zero, giving the usual factorized picture of production rate times decay rate.", "Having control on the azimuthal angle $\\phi $ allows to be sensitive to the interference between the production of different vector polarizations.", "While this is inherently interesting, from a BSM perspective the attractiveness of being sensitive to interference terms is twofold.", "First, in the case of diboson production, while SM amplitudes are dominated by opposite-helicity transverse vectors, dimension six operators affect the production of either same-helicity vectors of longitudinally-polarized vectors.", "Therefore, interference terms between the dominant SM amplitudes and the BSM contribution have non-trivial $\\phi $ dependence.", "Second, inclusive observables with quadratic sensitivity to EFT coefficients tend to be dominated by the high-energy tail of the distribution, which might induce problems with the EFT interpretation.", "Interference is linearly sensitive to higher dimension operators, allowing to interpret the constraints in a broader class of theories [16].", "There are two types of interference depending on $\\Delta h\\equiv h-{h^\\prime }$ .", "Interference between the different transverse polarizations has $|\\Delta h|=2$ , while longitudinal-transverse interference has $|\\Delta h|=1$ .", "As mentioned previously, ignorance on whether the calorimeter is placed on the helicity-plus or helicity-minus quark amounts to a $x\\rightarrow 1-x$ and $\\phi \\rightarrow \\pi +\\phi $ redundancy.", "The effect of this redundancy on the $|\\Delta h|=1$ interference will be discussed below.", "The effect on $|\\Delta h|=2$ terms is simpler, since the phase $e^{i\\Delta h\\phi }$ is left invariant.", "As an example, we study diboson production as a probe of such interference.", "At high energy, the SM produces mainly opposite-helicity vectors.", "Therefore $|\\Delta h|=2$ interference of one vector should be studied along a $|\\Delta h|=2$ interference of the other vector, making clear that interference is a phenomenon affecting the whole amplitude, and each process should be studied individually.", "However, in presence of the operator $\\mathcal {L}\\supset c_{WWW}\\frac{v^2}{\\Lambda ^2}\\frac{g^3}{m_W^2}\\epsilon ^{ABC}W^A_{\\mu \\nu }W^B_{\\nu \\rho }W^C_{\\rho \\mu }$ , the triple gauge coupling discussed above is generated with $\\lambda _\\gamma =-6g^2c_{WWW}\\frac{v^2}{\\Lambda ^2}$ , giving an amplitude that produces same-sign-helicity $W$ -bosons and grows at high energy.", "Therefore, a $|\\Delta h|=2$ interference between the SM amplitude and the one driven by the EFT operator can be observed with a single vector while being inclusive on the kinematics of the other vector.", "This is shown in Fig.", "REF .", "We simulated the semileptonic decay of $e^+e^-\\rightarrow W^+ W^-$ at $\\sqrt{s}=2\\,\\mathrm {TeV}$ in the SM (cyan) and in presence of $\\lambda _\\gamma =0.01$ (magenta).", "The EFT operator induces a $\\cos 2\\phi $ interfering pattern in both one- and two-point energy correlator of the $W$ -boson jet.", "For the two point correlator, we impose the requirement $z_\\star /2<z<2z_\\star $ in order to avoid contributions from calorimeters separated a distance $z\\ll z_\\star $ .", "The study of azimuthal dependence of the energy correlators at such small separations only reveals interference between different quark helicities, which is suppressed by the mass and of limited interest for the density matrix of a vector decay.Interference inside QCD jets has received some attention in the last years, see [47], [48], [49].", "We impose no constraint on $z$ in the one-point correlator.", "Figure: Azimuthal dependence of the one- and two-point energy correlators of a 1 TeV \\,\\mathrm {TeV} WW-boson.A contribution with λ γ =0.01\\lambda _\\gamma =0.01 shows a cos2φ\\cos 2\\phi behaviour from the interference.Figure: Impact of the asymmetry A int {A}_\\text{int} on the projected 95%CL sensitivity to λ γ \\lambda _\\gamma .Figure: One-point (left) and two-point (right) p T p_T-correlators as a function of the boost invariant distance ΔR 2 /R ☆ 2 \\Delta R^2/R_\\star ^2 at the LHC, for the transverse and longitudinal WW polarizations and a quark and gluon jet.In order to have an estimate on how well the interference can be measured given a number of events, we design a simple observable, considering the accumulated energy in different quadrants of $\\phi $ .", "From a distribution $f(\\phi )$ for the one- or two-point energy correlator, we define ${N}_+=f(\\pi /4\\le |\\phi |<3\\pi /4)$ and ${N}_-=f(0\\le |\\phi |<\\pi /4)+f(3\\pi /4\\le |\\phi |<\\pi )$ .", "Then the asymmetry ${A}_\\text{int}=({N}_+-{N}_-)/({N}_++{N}_-)$ is sensitive to the interference between different $W$ -boson helicity production amplitudes.Notice that ${A}_\\text{int}$ , contrary to ${A}_\\text{LT}$ , is in one-to-one correspondence to the one that can be defined at parton level.", "For illustrative purposes we show in Fig.", "REF the bounds on $\\lambda _\\gamma $ as a function of the number of events at $\\sqrt{s}=2\\,\\mathrm {TeV}$ with $p_T(W)>300\\,\\mathrm {GeV}$ .", "In red we show the bounds from an inclusive measurement of the cross section assuming 10% (solid) and 3% (dashed) of systematic uncertainty.", "In cyan, we add the asymmetry ${A}_\\text{int}$ assuming negligible systematic uncertainty.", "For low number of events, the inclusive measurement saturates the sensitivity and the $\\phi $ distribution gives no further discrimination, while for large number of events where precision measurements can be made, there is a clear advantage of using the asymmetry.", "Moreover, the region where the asymmetry dominates scales faster with the number of events than the region dominated by the cross section measurement, since the interference term is linearly sensitive to $\\lambda _\\gamma $ .", "We've checked that if we only consider the terms up to $1/\\Lambda ^2$ in the cross section, adding the interference improves the bound by more than an order of magnitude, in accordance to [18], [19].", "A detailed study of the full kinematics of the process allows to further improve the constraints, as done at parton level in [50].", "We can now turn to the rest of Fig.", "REF .", "In the right plot, we consider the same process and bin studied before in the left plot, but instead with $\\lambda _\\gamma =10^{-3}$ and $\\delta g_{1z}=0$ .", "Now it is the asymmetry ${A}_\\text{int}$ the one confirming that the excess comes indeed from an anomalous production of transverse modes.", "Moreover, due to the linear sensitivity, it can discriminate between both signs of $\\lambda _\\gamma $ , contrary to the cross section measurement.", "We stress that the conclusions above can be imported to LHC analyses: bins with large data samples will benefit the most from including observables sensitive to interference effects, since the linear sensitivity scales better with more statistics and allows to extend the regime of validity of the EFT.", "We finish by commenting on the more challenging and, arguably, interesting $|\\Delta h|=1$ interferences.", "As mentioned, the $x\\rightarrow 1-x$ and $\\phi \\rightarrow \\pi +\\phi $ redundancy erases the interference pattern in $\\phi $ if we proceed like for the $|\\Delta h|=2$ interference.", "However, the interference can be observed by considering the dependence of both $\\phi $ and $z$ , since the $x\\rightarrow 1-x$ changes the behaviour in $z$ .", "This is equivalent as the necessity to observe two angles in the leptonic case as studied in [18], [12].", "Therefore, the very interesting case of longitudinal-transverse interference is necessarily at least a two-dimensional problem.", "In practice, it is even more complex.", "In the case of diboson production, the SM production is dominated by opposite-sign transverse vectors while the typical BSM effects reside in the longitudinal-longitudinal production.", "Therefore, interference requires to observe a $|\\Delta h|=1$ interference on both vector bosons, requiring a global study of all kinematics of the process.", "Such multidimensional problem is better suited for modern machine learning approaches [51], [52], [53], [54], [55], [12].", "Towards the LHC.— So far we have studied the energy correlators for a monoenergetic source of electroweak bosons.", "In a hadronic environment, as the LHC, one has a spectrum of energies.", "However, the previous theoretical predictions can be immediately imported due to two effects.", "First, it is possible to show that the energy ratio $E_i/E_J$ between a hadron and jet is related to the ratio of their transverse momenta, $E_i/E_J=p_{T,i}/p_{T,J}+\\mathcal {O}(z_\\star ^{1/2})$ .", "Moreover, the angular distance $z$ is related to the boost invariant distance $\\Delta R^2=(\\Delta \\eta )^2+(\\Delta \\phi )^2$ by $z/z_\\star =\\Delta R^2/R_\\star ^2+\\mathcal {O}(z_\\star ^{1/2})$ , where we have defined $R_\\star =4m_V^2/p_{T,J}^2$ .", "Second, at leading order in $z_\\star $ , the LO predictions only depend on the quantity $z/z_\\star $ .", "This way, by measuring $z_\\star $ for each event, the obtained distribution from an ensamble of events in terms of $z/z_\\star $ will look like a distribution from a monoenergetic source.", "Therefore, the boost invariant $p_T$ -correlator, where we weight by the $p_T$ of the hadrons as a function of the distance $(\\Delta R)^2/R_{\\star }^2$ , has the same distribution as the energy correlator up to $\\mathcal {O}(z_\\star ^{1/2})$ corrections.", "In Fig.", "REF we show the one- and two-point $p_T$ -correlator as a function of $(\\Delta R)^2/R_{\\star }^2$ for a longitudinal and transverse $W$ -boson, and for reference we also for a quark and gluon jet.", "The jets are reconstructed using the anti-$k_t$ algorithm [56] with $R=1$ .", "The distributions for the vector coincide with the predictions for the EC of a monoenergetic $W$ beam, at least up to $\\mathcal {O}(z_\\star ^{1/2})$ corrections.", "The quark and gluon distributions for the one-point peak at lower $(\\Delta R)^2/R_{\\star }^2$ as expected.", "Moreover, notice that the one-point correlator requires the definition of a jet axis, which can be sensitive to soft recoil.", "It would be interesting to study the one-point correlator with a stable definition for the jet axis [57], [58].", "The two-point shows a completely different behaviour between EW and QCD jets.", "While the vector distributions peak at $z_\\star $ , the quark and gluon jets show the scaling behaviour.", "In fact, the tail at low $(\\Delta R)^2/R_{\\star }^2$ of the vector distributions coincides with the quark-jet distribution, as expected.", "We conclude that the previous monoenergetic results can be imported for the LHC distributions for the diagonal entries of the density matrix.", "The azimuthal distribution of the off-diagonal entries receives, however, modifications due to boosts along the beam axis.", "While we expect mild effects, the assessment of the sensitivity requires further study.", "Figure: Impact on the one-point correlator of the selection cuts typically used to discriminate EW from QCD jets.An important question is the impact of the QCD background and how the spectrum of the energy correlators is deformed after applying the selection criteria typically used to discriminate EW and QCD jets [21], [22], [23], [59], [60].", "In particular, the invariant mass of the jet $m_J$ and the $D_2$ observable [23] are two of the main drivers that discern between EW and QCD jets.", "In order to assess the impact of such cuts in a clean environment, we consider a monoenergetic beam of $W$ bosons of $E_W=1.5\\,\\mathrm {TeV}$ in a lepton collider.", "The effect of the cuts on the one-point correlator is shown in Fig.", "REF , where we compare the spectrum without cuts, after the requirement $65\\,\\mathrm {GeV}<m_J<95\\,\\mathrm {GeV}$ , and after further requiring $D_2<5$ , 2 and 1.", "All curves are normalized to 1.", "We observe that the cut on $m_J$ has only mild effects on the spectrum.", "However, a selection on $D_2$ can have a large impact.", "The $D_2$ observable is defined as $D_2 = \\frac{\\frac{1}{E_J^3}\\sum _{i<j<k}E_iE_jE_k\\,z_{ij}z_{jk}z_{ki}}{\\left(\\frac{1}{E_J^2}\\sum _{i<j}E_iE_j\\,z_{ij}\\right)^3},$ where the sums run over the jet constituents.", "Low values of $D_2$ correspond to jets that look like two-prong while larger values correspond to one-prong jets, in this way cut selections can be designed to reject gluonic jets.", "In Fig.", "REF we see, however, that such selections have an inherent bias towards rejecting the transverse vectors that are kinematically distinct with respect the longitudinals.", "This is because, as explained in the discussion of the one-point correlator, low $z$ is in one-to-one with having the energy deposition dominated in a single quark, which translates to larger $D_2$ values.", "Such kinematical configuration is more frequent in the transverse polarizations.", "The $D_2$ cut imposes a requirement on the kinematical configuration within the jet and homogenizes the transverse and longitudinal distributions.", "We cannot end the discussion without remarking that $D_2$ is defined as a ratio of fully integrated energy correlators, and therefore it is suggestive to think that the exploration of fully-differential observables might allow a discrimination of EW and QCD jets that retains the different character of longitudinal and transverse vectors.", "Conclusions and outlook In this Letter we have explored the possibility of using the information encoded in the energy correlators of a hadronically decaying electroweak vector boson in order to probe its full decay density matrix.", "We observe that the dependence on the angular separation $z$ of the one-point energy correlator can indeed discriminate between longitudinal and transverse modes.", "The azimuthal dependence $\\phi $ alone of both one- and two-point energy correlators shows the interference pattern of the $|\\Delta h|=2$ interference terms.", "The $|\\Delta h|=1$ interference terms require the study of the distribution in both $\\phi $ and $z$ .", "The distributions are independent of the $W$ boost when expressed as a function of $z/z_\\star $ , which allows to predict the distributions observed at hadron colliders, like the LHC, in terms of boost invariant quantities.", "Kinematics of the decay allows to dissect the dynamics that produce an electroweak boson, enhancing the sensitivity to EFT operators and potentially to resonance searches.", "We presented how the energy correlators allow to identify the microscopic origin of an excess of events, see Fig.", "REF , and how the sensitivity to EFT operators is enhanced by unveiling the interference pattern, see Fig.", "REF .", "An important future avenue to pursue is the detailed analysis of the correlators in the hadronic environment of the LHC.", "The impact of QCD jets and selection criteria on the distributions is a crucial element to explore further.", "We observe that the usual $D_2$ selection cut has a bias towards eliminating vectors with transverse-like kinematics, forcing the exploration of other criteria that may retain the kinematical difference between the longitudinal and transverse distributions.", "This, together with the assessment of the different theoretical and experimental uncertainties is postponed for future work.", "While we studied the one-dimensional distributions of the energy correlator, extra information can be gained from the multi-dimensional distributions.", "For instance, the full 4-dimensional two-point and 6-dimensional three-point correlator would constitute a generalization of the $D_2$ variable used to discriminate QCD and EW jets, which might be interesting to investigate.", "From a BSM perspective, a better control of hadronic decays and the kinematical characterization of the decaying EW vectors might open the door to study rare multiboson processes that carry important information about the microscopic dynamics in the EW sector [61].", "Energy correlators are at the crossroads of conformal field theory, QCD, experimental physics and data analysis.", "As shown, they can also be used to enhance sensitivity to BSM physics.", "This makes them an extremely appealing and fruitful arena that aims to push precision physics at the LHC to new levels.", "We thank Alfredo Glioti, Ian Moult, Francesco Riva, Steven Schramm and Andrea Wulzer for valuable discussions, suggestions and comments.", "The work of L.R.", "and M.R.", "is supported by The Swiss National Science Foundation under contract 200021-178999 and 200020-188671." ] ]
2207.03511
[ [ "Radiative corrections to the forward-backward asymmetry in\n $e^+e^-\\to\\pi^+\\pi^-$" ], [ "Abstract We present a calculation of the $C$-odd radiative corrections to $e^+e^-\\to \\pi^+\\pi^-$ in a dispersive formalism, concentrating on the leading pion-pole contribution in the virtual box diagrams.", "In particular, we show how the effect of a general pion vector form factor in the loop integral can be incorporated in a model-independent way and how the cancellation of infrared singularities proceeds in this case.", "The numerical results, dominated by the infrared enhanced contributions, indicate significant corrections beyond scalar QED, essentially confirming recent findings in generalized vector-meson-dominance models." ], [ "Introduction", "The data-driven determination of hadronic vacuum polarization (HVP) is dominated by the $2\\pi $ channel, which gives about $70\\%$ of the HVP contribution to the anomalous magnetic moment of the muon [1].", "The consensus number [2], [3], [4], [5], [6], [7] $a_\\mu ^\\text{HVP, LO}\\big |_{e^+e^-}=693.1(4.0)\\times 10^{-10}$ for this quantity relies on a set of measurements from SND [8], [9], CMD-2 [10], [11], [12], [13], BESIII [14], and CLEO [15] (and, when dispersive constraints are included [4], [16], [6], [5], [17], also on space-like data [18], [19]), but is dominated by the precision data sets from BaBar [20], [21] and KLOE [22], [23], [24], [25].", "Since Ref.", "[1] new data have become available from SND [26], but resolving the tension between BaBar and KLOE requires new measurements at a similar level of precision, as are expected from CMD-3 [27], BaBar [28], BESIII [29], and Belle II [30].", "This question has become increasingly urgent in view of recent lattice-QCD results [31], [32], [33] challenging the data-driven value (REF ) at least for the intermediate window [34], and critical for the interpretation [35], [36], [37], [38], [39] of the $4.2\\sigma $ tension between experiment [40], [41], [42], [43], [44] and Standard-Model theory [1], [45], [46], [47], [48], [2], [3], [4], [5], [6], [7], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62].", "A potential weak point in the analysis concerns the treatment of radiative correction [28], [63], [64], which are implemented in Monte-Carlo generators relying on scalar QED supplemented by the pion vector form factor (VFF) wherever possible [65], to capture the dominant corrections from the structure of the pion.", "In the case of final-state radiation (FSR), this approach does capture the dominant infrared (IR) enhanced effects [66], [67], [68], [69], but it is not guaranteed that corrections can be neglected in all kinematic configurations relevant for precision measurements of $e^+e^-\\rightarrow \\pi ^+\\pi ^-$ .", "A possible test case concerns the forward–backward asymmetry in the same process, in which case radiative corrections arise from the interference of initial-state-radiation (ISR) and FSR diagrams [70] combined with box diagrams, see Fig.", "REF .", "While these corrections cancel when integrated over the entire phase space, they can be probed in the forward–backward asymmetry.", "Writing the Born cross section as $\\frac{d\\sigma _0}{dz}=\\frac{\\pi \\alpha ^2\\beta ^3}{4s}(1-z^2)\\big |F_\\pi ^V(s)\\big |^2,\\qquad \\beta =\\sqrt{1-\\frac{4M_\\pi ^2}{s}},\\qquad z=\\cos \\theta ,\\qquad \\alpha =\\frac{e^2}{4\\pi },$ with the pion VFF $F_\\pi ^V(s)$ defined by the matrix element of the electromagnetic current $j_\\text{em}^\\mu =(2\\bar{u}\\gamma ^\\mu u-\\bar{d}\\gamma ^\\mu d-\\bar{s} \\gamma ^\\mu s)/3$ , $\\langle \\pi ^\\pm (p^{\\prime }) | j_\\mathrm {em}^\\mu (0) | \\pi ^\\pm (p) \\rangle =\\pm (p^{\\prime }+p)^\\mu F_\\pi ^V((p^{\\prime }-p)^2),$ it is clear that the forward–backward asymmetry $A_\\text{FB}(z)=\\frac{\\frac{d\\sigma }{dz}(z)-\\frac{d\\sigma }{dz}(-z)}{\\frac{d\\sigma }{dz}(z)+\\frac{d\\sigma }{dz}(-z)}$ vanishes at tree-level, but radiative corrections give rise to odd terms in $z$ .", "In Ref.", "[71] these corrections were calculated in scalar QED, multiplied in the end globally with $|F_\\pi ^V|^2$ to account for structure corrections.", "However, in Ref.", "[72] it was observed that this approximation was insufficient to describe preliminary data from the CMD-3 experiment, proposing an improved approach using generalized vector-meson-dominance (GVMD) models in the loop integral.", "Figure: Representative diagrams contributing to A FB A_\\text{FB} (solid, dashed, and wiggly lines denote electrons, pions, and photons, respectively).", "The interference of ISR (a)(a) and FSR (b)(b) produces real terms odd in zz, while box diagrams (c)(c) give the virtual corrections.", "Only the sum of real and virtual contributions is IR finite.", "The gray blob denotes the pion VFF, which for (b)(b) can be justified because we only keep the soft limit.", "The short-dashed line in (c)(c) indicates that we only consider the pion-pole singularities, not general hadronic intermediate states that could contribute to the hadronic side of the box diagram.In this paper, we present a model-independent approach that captures the effect of the pion-pole singularities in the virtual diagrams, and show that the dominant corrections still arise from the IR enhanced effects.", "For the real contribution, due to the interference of the ISR and FSR diagrams in Fig.", "REF , we obtain the correction factor $\\delta _\\text{soft}&=\\frac{2\\alpha }{\\pi }\\Bigg \\lbrace \\log \\frac{\\lambda ^2}{4\\Delta ^2}\\log \\frac{1+\\beta z}{1-\\beta z}+\\log (1-\\beta ^2)\\log \\frac{1+\\beta z}{1-\\beta z}+\\log ^2(1-\\beta z)-\\log ^2(1+\\beta z)\\\\&+\\text{Li}_2\\bigg (\\frac{(z-1)\\beta }{1-\\beta }\\bigg )+\\text{Li}_2\\bigg (\\frac{(1+z)\\beta }{1+\\beta }\\bigg )-\\text{Li}_2\\bigg (\\frac{(z+1)\\beta }{\\beta -1}\\bigg )-\\text{Li}_2\\bigg (\\frac{(1-z)\\beta }{1+\\beta }\\bigg )\\Bigg \\rbrace ,$ which agrees with similar representations in Refs.", "[71], [72] and is defined relative to Eq.", "(REF ) $\\frac{d\\sigma }{dz}\\bigg |_{C\\text{-odd}}=\\frac{d\\sigma _0}{dz}\\Big [\\delta _\\text{soft}(\\lambda ^2,\\Delta )+\\delta _\\text{virt}\\big (\\lambda ^2\\big )\\Big ]+\\frac{d\\sigma }{dz}\\bigg |_\\text{hard}(\\Delta ).$ The parameter $\\lambda =m_\\gamma $ regularizes the IR divergence via a small photon mass, and $\\Delta $ is a cutoff in the photon energy (we do not consider the hard contribution with photon energy above $\\Delta $ any further).", "As we will show below, already the cancellation of the IR singularity becomes quite subtle for a general pion VFF.", "After discussing the analytic structure in Sec.", ", we derive our result for the virtual contribution in terms of the standard scalar loop functions in Sec.", ", before turning to the numerical analysis in Sec. .", "Our conclusions are summarized in Sec.", "." ], [ "Dispersion relations and cut structure", "We define the kinematic variables according to $e^+(p_2)e^-(p_1)\\rightarrow \\pi ^+(l_2)\\pi ^-(l_1),$ with Mandelstam variables $s&=(p_1+p_2)^2=(l_1+l_2)^2,\\\\t&=(p_1-l_1)^2=(p_2-l_2)^2=\\frac{1}{2}\\Big (2M_\\pi ^2+2m_e^2-s+z \\sigma _e\\sigma _\\pi s\\Big ),\\\\u&=(p_1-l_2)^2=(p_2-l_1)^2=\\frac{1}{2}\\Big (2M_\\pi ^2+2m_e^2-s-z \\sigma _e\\sigma _\\pi s\\Big ),$ where $\\sigma _\\pi \\equiv \\beta =\\sqrt{1-\\frac{4M_\\pi ^2}{s}},\\qquad \\sigma _e=\\sqrt{1-\\frac{4m_e^2}{s}},\\qquad z=\\frac{t-u}{s\\sigma _\\pi \\sigma _e},$ and $s+t+u=2M_\\pi ^2+2m_e^2$ .", "In the following, we will always put $m_e=0$ unless required to regularize collinear singularities.", "To isolate the contribution from the pion pole in the hadronic part of the loop diagram, one starts from fixed-$s$ dispersion relations, which for scalar particles would immediately allow one to identify the scalar loop integral $D_0$  [73] in terms of its double-spectral function.", "This procedure has been performed in detail for the pion boxes in hadronic light-by-light scattering [74], demonstrating that the non-box contributions that arise in a diagrammatic scalar-QED calculation are simply required by gauge invariance, in such a way that the final result can be obtained by multiplying the scalar-QED amplitude by the appropriate pion VFFs for the external photons.", "In the present case a similar argument applies if a dispersive representation is used for the pion VFF in the $e^+e^-\\rightarrow \\pi ^+\\pi ^-$ subamplitudes, i.e., if we replace $\\frac{F_\\pi ^V(s)}{s}=\\frac{1}{s}+\\frac{1}{\\pi }\\int _{4M_\\pi ^2}^\\infty d s^{\\prime }\\frac{\\text{Im}\\,F_\\pi ^V(s^{\\prime })}{s^{\\prime }(s^{\\prime }-s)}\\rightarrow \\frac{1}{s-\\lambda ^2}-\\frac{1}{\\pi }\\int _{4M_\\pi ^2}^\\infty d s^{\\prime }\\frac{\\text{Im}\\,F_\\pi ^V(s^{\\prime })}{s^{\\prime }} \\frac{1}{s-s^{\\prime }}$ for each of the two photon propagators.", "In Eq.", "(REF ) we have introduced the IR regulator in the pole terms, which are the ones that produce the IR divergence.", "The resulting representation for the box diagram then consists of pole times pole, mixed pole and dispersive, and dispersive times dispersive contributions.", "In the next section, we will express each of them in terms of standard loop functions." ], [ "Formulation in terms of scalar loop functions", "The tensor decomposition is easiest to derive directly for the spin sum of the interference of the box diagram with the tree-level amplitude.", "This gives the decomposition of the virtual correction $\\delta _\\text{virt}&=\\bar{\\delta }_\\text{virt}\\big (\\lambda ^2,\\lambda ^2\\big )-\\frac{1}{\\pi }\\int _{4M_\\pi ^2}^\\infty ds^{\\prime }\\frac{\\text{Im}\\,F_\\pi ^V(s^{\\prime })}{s^{\\prime }} \\big [\\bar{\\delta }_\\text{virt}\\big (s^{\\prime },\\lambda ^2\\big )+\\bar{\\delta }_\\text{virt}\\big (\\lambda ^2,s^{\\prime }\\big )\\big ]\\\\&+\\frac{1}{\\pi }\\int _{4M_\\pi ^2}^\\infty ds^{\\prime }\\frac{\\text{Im}\\,F_\\pi ^V(s^{\\prime })}{s^{\\prime }} \\frac{1}{\\pi }\\int _{4M_\\pi ^2}^\\infty ds^{\\prime \\prime }\\frac{\\text{Im}\\,F_\\pi ^V(s^{\\prime \\prime })}{s^{\\prime \\prime }}\\bar{\\delta }_\\text{virt}(s^{\\prime },s^{\\prime \\prime }),$ with $\\bar{\\delta }_\\text{virt}(s^{\\prime },s^{\\prime \\prime })&=-\\frac{\\text{Re}\\,F_\\pi ^V(s)}{2\\beta ^2 s(1-z^2)|F_\\pi ^V(s)|^2}\\frac{\\alpha }{\\pi }\\\\&\\times \\text{Re}\\,\\bigg [4t\\big (M_\\pi ^2-t\\big )\\Big (C_0\\big (m_e^2,t,M_\\pi ^2,s^{\\prime },m_e^2,M_\\pi ^2\\big )+C_0\\big (m_e^2,t,M_\\pi ^2,s^{\\prime \\prime },m_e^2,M_\\pi ^2\\big )\\Big )\\\\&\\quad -4u\\big (M_\\pi ^2-u\\big )\\Big (C_0\\big (m_e^2,u,M_\\pi ^2,s^{\\prime },m_e^2,M_\\pi ^2\\big )+C_0\\big (m_e^2,u,M_\\pi ^2,s^{\\prime \\prime },m_e^2,M_\\pi ^2\\big )\\Big )\\\\&\\quad -4s(t-u)C_0\\big (m_e^2,s,m_e^2,m_e^2,s^{\\prime },s^{\\prime \\prime }\\big )+4\\big (t^2-u^2\\big )C_0\\big (M_\\pi ^2,s,M_\\pi ^2,M_\\pi ^2,s^{\\prime },s^{\\prime \\prime }\\big )\\\\&\\quad +4\\big (M_\\pi ^2-t\\big )\\Big (\\big (M_\\pi ^2-t)^2+M_\\pi ^4+t(s^{\\prime }+s^{\\prime \\prime }-u)\\Big )\\\\&\\quad \\qquad \\times D_0\\big (m_e^2,m_e^2,M_\\pi ^2,M_\\pi ^2,s,t,s^{\\prime },m_e^2,s^{\\prime \\prime },M_\\pi ^2\\big )\\\\&\\quad -4\\big (M_\\pi ^2-u\\big )\\Big (\\big (M_\\pi ^2-u)^2+M_\\pi ^4+u(s^{\\prime }+s^{\\prime \\prime }-t) \\Big )\\\\&\\quad \\qquad \\times D_0\\big (m_e^2,m_e^2,M_\\pi ^2,M_\\pi ^2,s,u,s^{\\prime },m_e^2,s^{\\prime \\prime },M_\\pi ^2\\big )\\bigg ] + (\\text{Re}\\,\\rightarrow \\text{Im}\\,),$ and loop functions $C_0$ , $D_0$ in the conventions of Ref.", "[73]." ], [ "Pole–pole", "We first consider the case $s^{\\prime }=s^{\\prime \\prime }=\\lambda ^2$ .", "The IR divergences are easiest to extract using dispersive representations of the loop integrals, leading to the expressions given in App.", "REF .", "In combination, we find the following expression for the pole–pole contribution $\\delta _\\text{virt}^\\text{pole--pole}&=\\frac{\\alpha }{\\pi }\\frac{\\text{Re}\\,F_\\pi ^V(s)}{|F_\\pi ^V(s)|^2}\\Bigg \\lbrace 2\\log \\frac{\\lambda ^2}{s}\\log \\frac{1-\\beta z}{1+\\beta z} +\\frac{z}{(1-z^2)\\beta }\\bigg [\\frac{(1-\\beta )^2}{\\beta }\\frac{\\pi ^2}{6}+2\\log ^2 2\\\\&\\quad -\\log ^2(1-\\beta ^2)+\\frac{1+\\beta ^2}{2\\beta }\\bigg (4\\text{Li}_2\\bigg (\\frac{\\beta -1}{1+\\beta }\\bigg )+\\log ^2\\frac{1-\\beta }{1+\\beta }\\bigg )\\bigg ]\\\\&\\quad +\\frac{1+2\\beta z+\\beta ^2}{(1-z^2)\\beta ^2}\\bigg [\\frac{1}{2}\\log ^2(1+\\beta z)+\\log \\beta _+\\log \\frac{1+2\\beta z+\\beta ^2}{1+\\beta z}+\\text{Li}_2\\big (\\beta _+\\big )\\bigg ]\\\\&\\quad -\\frac{1-2\\beta z+\\beta ^2}{(1-z^2)\\beta ^2}\\bigg [\\frac{1}{2}\\log ^2(1-\\beta z)+\\log \\beta _-\\log \\frac{1-2\\beta z+\\beta ^2}{1-\\beta z}+\\text{Li}_2\\big (\\beta _-\\big )\\bigg ]\\Bigg \\rbrace \\\\&+\\frac{\\alpha }{\\pi }\\frac{\\text{Im}\\,F_\\pi ^V(s)}{|F_\\pi ^V(s)|^2}\\frac{\\pi }{(1-z^2)\\beta ^2}\\bigg [(1+\\beta ^2)z\\log \\frac{1-\\beta }{1+\\beta }+2\\beta z\\log \\frac{1-\\beta ^2 z^2}{1-\\beta ^2}\\\\&\\quad -\\Big (1+\\beta ^2(2z^2-1)\\Big )\\log \\frac{1-\\beta z}{1+\\beta z}\\bigg ],\\qquad \\beta _\\pm = \\frac{1-\\beta ^2}{2(1\\pm \\beta z)}.$ The functional form of the contribution proportional to $\\text{Re}\\,F_\\pi ^V/|F_\\pi ^V|^2$ agrees with Ref.", "[71], i.e., as expected, the point-like limit is recovered from the pole–pole contribution upon setting $F_\\pi ^V=1$ .", "In particular, we confirm that all collinear singularities cancel." ], [ "Pole–dispersive", "The loop integrals required for the mixed pole and dispersive contributions are provided in App.", "REF .", "As a first step, we consider the IR divergence: for the real part, we find $\\delta _\\text{virt}^\\text{pole--disp}\\big |_\\text{Re}^\\text{IR}&=\\frac{\\alpha }{\\pi }\\frac{\\text{Re}\\,F_\\pi ^V(s)}{|F_\\pi ^V(s)|^2} \\bigg [2\\log \\frac{\\lambda ^2}{s}\\log \\frac{1-\\beta z}{1+\\beta z} \\text{Re}\\,\\frac{s}{\\pi }\\int _{4M_\\pi ^2}^\\infty ds^{\\prime }\\frac{\\text{Im}\\,F_\\pi ^V(s^{\\prime })}{s^{\\prime }(s^{\\prime }-s)}\\bigg ]\\\\&=\\frac{\\alpha }{\\pi }\\frac{\\text{Re}\\,F_\\pi ^V(s)}{|F_\\pi ^V(s)|^2} \\bigg [2\\log \\frac{\\lambda ^2}{s}\\log \\frac{1-\\beta z}{1+\\beta z} \\big (\\text{Re}\\,F_\\pi ^V(s) -1\\big )\\bigg ],$ which, together with Eq.", "(REF ), combines to a prefactor $(\\text{Re}\\,F_\\pi ^V)^2/|F_\\pi ^V|^2$ .", "The IR divergence in the imaginary part is more subtle.", "It arises from the imaginary part generated by the $D_0$ function for $s^{\\prime }<s$ , since the integration over $s^{\\prime }$ displays an end-point singularity at $s^{\\prime }=s$ .", "To extract this singularity, we write $\\frac{1}{\\pi }\\int _{4M_\\pi ^2}^{s}ds^{\\prime } \\frac{\\text{Im}\\,F_\\pi ^V(s^{\\prime })}{s^{\\prime }} \\text{Im}\\,\\delta ^\\text{pole--disp}(s^{\\prime })&=\\frac{1}{\\pi }\\int _{4M_\\pi ^2}^{s}ds^{\\prime } \\frac{\\text{Im}\\,F_\\pi ^V(s^{\\prime })-\\text{Im}\\,F_\\pi ^V(s)}{s^{\\prime }} \\text{Im}\\,\\delta ^\\text{pole--disp}(s^{\\prime })\\\\&+\\frac{\\text{Im}\\,F_\\pi ^V(s)}{\\pi }\\int _{4M_\\pi ^2}^{s}ds^{\\prime } \\frac{\\text{Im}\\,\\delta ^\\text{pole--disp}(s^{\\prime })}{s^{\\prime }},$ with integrand $\\text{Im}\\,\\delta ^\\text{pole--disp}(s^{\\prime })&=\\frac{2\\pi \\big (s^{\\prime }-s+(s+s^{\\prime }-2s z^2)\\beta ^2\\big )\\log \\frac{1-\\beta z}{1+\\beta z}}{(s^{\\prime }-s)(1-z^2)\\beta ^2}\\\\&-\\frac{2\\pi z\\Big ((1+\\beta ^2)\\log \\frac{1-\\beta }{1+\\beta }+2\\beta \\log \\frac{1-\\beta ^2z^2}{1-\\beta ^2}\\Big )}{(1-z^2)\\beta ^2}.$ The second integral in Eq.", "(REF ), however, needs to be evaluated including the dominant effect of the IR regulator, which changes the imaginary part of the $D_0$ function to $\\text{Im}\\,D_0(s,t,s^{\\prime },\\lambda ^2)&=\\frac{2\\pi \\theta \\big ((\\sqrt{s}-\\lambda )^2-s^{\\prime }\\big )}{\\sqrt{(s^{\\prime }-s)^2(M_\\pi ^2-t)^2-2\\lambda ^2\\big ((s+s^{\\prime })(M_\\pi ^2-t)^2+2s s^{\\prime } t\\big )}}\\\\&\\times \\bigg (\\log \\frac{M_\\pi ^2-t}{s}+\\log \\frac{s}{m_e M_\\pi }\\bigg ).$ The second term in the last line does not give rise to an IR divergence, and cancels with the other collinear singularities.", "For simplicity, we treat the regular terms in the same way as the singular ones, which slightly simplifies the expressions and allows us to write the full contribution from the imaginary part as $\\delta _\\text{virt}^\\text{pole--disp}\\big |_\\text{Im}&=\\frac{\\alpha }{\\pi }\\frac{\\text{Im}\\,F_\\pi ^V(s)}{|F_\\pi ^V(s)|^2}\\bigg [\\frac{1}{\\pi }\\int _{4M_\\pi ^2}^s ds^{\\prime }\\frac{\\text{Im}\\,F_\\pi ^V(s^{\\prime })-\\text{Im}\\,F_\\pi ^V(s)}{s^{\\prime }} \\text{Im}\\,\\delta ^\\text{pole--disp}(s^{\\prime })\\bigg ]\\\\&+\\frac{\\alpha }{\\pi }\\frac{\\big (\\text{Im}\\,F_\\pi ^V(s)\\big )^2}{|F_\\pi ^V(s)|^2}\\Bigg \\lbrace 2\\log \\frac{4\\lambda ^2}{s}\\log \\frac{1-\\beta z}{1+\\beta z}\\\\&\\quad +4\\log \\frac{1-\\beta z}{2}\\log \\big [1-\\beta z+\\sqrt{1-2\\beta z+\\beta ^2}\\big ]-4\\log ^2(1-\\beta z)\\\\&\\quad -4\\log \\frac{1+\\beta z}{2}\\log \\big [1+\\beta z+\\sqrt{1+2\\beta z+\\beta ^2}\\big ]+4\\log ^2(1+\\beta z)\\\\&\\quad +\\frac{2\\log (1-\\beta ^2)}{(1-z^2)\\beta ^2}\\bigg (z(1+\\beta ^2)\\log \\frac{1-\\beta }{1+\\beta }-\\log \\frac{1-\\beta z}{1+\\beta z}+2\\beta z \\log \\frac{1-\\beta ^2z^2}{1-\\beta ^2}\\bigg )\\\\&\\quad -\\frac{2\\log \\frac{1-\\beta z}{1+\\beta z}}{1-z^2}\\bigg ((2z^2-1)\\log \\frac{1-\\beta ^2}{\\beta ^2}+2\\log \\beta \\bigg )\\Bigg \\rbrace .$ Summing up the IR divergences in Eqs.", "(REF ), (REF ), and (REF ), we finally obtain $\\delta _\\text{virt}^\\text{IR}=2\\frac{\\alpha }{\\pi } \\log \\frac{\\lambda ^2}{s}\\log \\frac{1-\\beta z}{1+\\beta z},$ which cancels against the real emission (REF ), as expected.", "However, we observe that for the virtual contribution the factor $|F_\\pi ^V|^2$ originates from the subtle interplay of three different contributions, which no longer applies for the finite terms.", "Ultimately, this is the reason why multiplying the point-like result by $|F_\\pi ^V|^2$ is a poor approximation.", "For completeness, we also give the full expression for the real part: $\\delta _\\text{virt}^\\text{pole--disp}\\big |_\\text{Re}&=\\frac{\\alpha }{\\pi }\\frac{\\text{Re}\\,F_\\pi ^V(s)}{|F_\\pi ^V(s)|^2}\\frac{1}{\\pi }\\int _{4M_\\pi ^2}^\\infty ds^{\\prime }\\frac{\\text{Im}\\,F_\\pi ^V(s^{\\prime })}{s^{\\prime }} \\text{Re}\\,\\delta ^\\text{pole--disp}(s^{\\prime }),\\\\\\text{Re}\\,\\delta ^\\text{pole--disp}&=\\frac{2s}{s^{\\prime }-s}\\bigg [\\log \\frac{\\lambda ^2 s^{\\prime }}{(s^{\\prime }-s)^2}\\log \\frac{1-\\beta z}{1+\\beta z}+\\text{Li}_2\\bigg (1-\\frac{s^{\\prime }(1-\\beta ^{\\prime })}{s(1-\\beta z)}\\bigg )\\\\&\\quad +\\text{Li}_2\\bigg (1-\\frac{s^{\\prime }(1+\\beta ^{\\prime })}{s(1-\\beta z)}\\bigg )-\\text{Li}_2\\bigg (1-\\frac{s^{\\prime }(1-\\beta ^{\\prime })}{s(1+\\beta z)}\\bigg )-\\text{Li}_2\\bigg (1-\\frac{s^{\\prime }(1+\\beta ^{\\prime })}{s(1+\\beta z)}\\bigg )\\bigg ]\\\\&+\\frac{2(1+\\beta ^2)}{(1-z^2)\\beta ^2}\\log \\frac{1-\\beta z}{1+\\beta z}\\log \\frac{2s}{(1-\\beta ^2)|s-s^{\\prime }|}\\\\&-\\frac{4z}{(1-z^2)\\beta }\\bigg [s \\text{Re}\\,\\bar{C}_0\\big (s,s^{\\prime },m_e^2\\big )+\\frac{s}{2}(1+\\beta ^2) \\text{Re}\\,C_0\\big (s,s^{\\prime },M_\\pi ^2\\big )+\\log ^2 2\\\\&\\quad +\\log ^2(1-\\beta ^{\\prime })+\\log ^2(1+\\beta ^{\\prime })-\\log ^2(1-\\beta ^2)-\\log \\frac{1-\\beta ^2}{2}\\log \\big (1-\\beta ^2z^2\\big )\\\\&\\quad +\\log \\frac{1-\\beta ^2 z^2}{(1-\\beta ^2)^2}\\log \\frac{s}{s^{\\prime }}-\\log \\frac{s^{\\prime }(1-\\beta ^2)}{s(1-\\beta ^2 z^2)}\\log \\frac{s^{\\prime }}{|s-s^{\\prime }]}\\bigg ]\\\\&+\\frac{2(1-2\\beta z+\\beta ^2)}{(1-z^2)\\beta ^2}\\bigg [\\text{Li}_2\\bigg (1-\\frac{s^{\\prime }(1-\\beta ^{\\prime })}{s(1-\\beta z)}\\bigg )+\\text{Li}_2\\bigg (1-\\frac{s^{\\prime }(1+\\beta ^{\\prime })}{s(1-\\beta z)}\\bigg )+\\text{Li}_2\\big (\\beta _-\\big )\\\\&\\quad +\\log \\big (1-2\\beta z +\\beta ^2\\big )\\log \\beta _-+\\frac{3}{2}\\log ^2(1-\\beta z)\\bigg ]\\\\&-\\frac{2(1+2\\beta z+\\beta ^2)}{(1-z^2)\\beta ^2}\\bigg [\\text{Li}_2\\bigg (1-\\frac{s^{\\prime }(1-\\beta ^{\\prime })}{s(1+\\beta z)}\\bigg )+\\text{Li}_2\\bigg (1-\\frac{s^{\\prime }(1+\\beta ^{\\prime })}{s(1+\\beta z)}\\bigg )+\\text{Li}_2\\big (\\beta _+\\big )\\\\&\\quad +\\log \\big (1+2\\beta z +\\beta ^2\\big )\\log \\beta _++\\frac{3}{2}\\log ^2(1+\\beta z)\\bigg ],$ where $\\bar{C}_0\\big (s,s^{\\prime },m_e^2)=\\bigg [C_0\\big (s,s^{\\prime },m_e^2)+\\frac{\\log \\frac{m_e^2}{s^{\\prime }}\\log \\frac{|s^{\\prime }-s|}{s^{\\prime }}}{s}\\bigg ]_{m_e\\rightarrow 0}$ is the finite part of the massless $C_0(s,s^{\\prime },m_e^2)$ loop function and again all the collinear singularities from $m_e\\rightarrow 0$ cancel.", "The singularity at $s^{\\prime }=s$ is to be interpreted as the principal value, but in contrast to the imaginary part no end-point singularity arises." ], [ "Dispersive–dispersive", "The purely dispersive correction can be expressed as $\\delta _\\text{virt}^\\text{disp--disp}&=\\frac{\\alpha }{\\pi }\\frac{\\text{Re}\\,F_\\pi ^V(s)}{|F_\\pi ^V(s)|^2}\\frac{1}{\\pi }\\int _{4M_\\pi ^2}^\\infty ds^{\\prime }\\frac{\\text{Im}\\,F_\\pi ^V(s^{\\prime })}{s^{\\prime }}\\frac{1}{\\pi }\\int _{4M_\\pi ^2}^\\infty ds^{\\prime \\prime }\\frac{\\text{Im}\\,F_\\pi ^V(s^{\\prime \\prime })}{s^{\\prime \\prime }}\\text{Re}\\,\\delta ^\\text{disp--disp}(s^{\\prime },s^{\\prime \\prime })\\\\&+\\frac{\\alpha }{\\pi }\\frac{\\text{Im}\\,F_\\pi ^V(s)}{|F_\\pi ^V(s)|^2}\\frac{1}{\\pi }\\int _{4M_\\pi ^2}^\\infty ds^{\\prime }\\frac{\\text{Im}\\,F_\\pi ^V(s^{\\prime })}{s^{\\prime }}\\frac{1}{\\pi }\\int _{4M_\\pi ^2}^\\infty ds^{\\prime \\prime }\\frac{\\text{Im}\\,F_\\pi ^V(s^{\\prime \\prime })}{s^{\\prime \\prime }}\\text{Im}\\,\\delta ^\\text{disp--disp}(s^{\\prime },s^{\\prime \\prime }),$ where the integrands follow directly from Eq.", "(REF ).", "All loop functions that contribute in this case, explicit expressions for which are provided in App.", "REF , are IR finite and free of collinear singularities." ], [ "Numerical analysis", "For the numerical analysis we use the pion VFF from Ref.", "[4] for $s\\le s_\\text{cut}$ , $s_\\text{cut}=1\\,\\text{GeV}^2$ .", "Since the corrections $\\delta $ are defined relative to the tree-level result (REF ), which itself depends on $F_\\pi ^V$ , they need to be determined in a self-consistent way.", "Accordingly, we restrict the analysis here to the energy below $s_\\text{cut}$ .", "However, for the dispersive integrals we also need to provide $\\text{Im}\\,F_\\pi ^V$ above, in particular, in writing Eq.", "(REF ) we have implicitly assumed the sum rule $\\frac{1}{\\pi }\\int _{4M_\\pi ^2}^\\infty ds^{\\prime } \\frac{\\text{Im}\\,F_\\pi ^V(s^{\\prime })}{s^{\\prime }}=1.$ In practice, this sum rule is easiest to fulfill by including excited $\\rho ^{\\prime }$ , $\\rho ^{\\prime \\prime }$ resonances in the $\\pi \\pi $ phase shift, which is one of the variants for the asymptotic continuation studied in Ref.", "[4] (using the implementation from Ref.", "[75], based on the data from Ref. [76]).", "In the following, we use the corresponding input for $F_\\pi ^V$ .", "Figure: Correction factors δ\\delta as a function of s\\sqrt{s} for fixed z=cos(1)z=\\cos (1) (left) and as a function of zz for fixed s=0.75GeV\\sqrt{s}=0.75\\,\\text{GeV} (right).", "The black lines denote the point-like result (dashed: real, dot-dashed: virtual, solid: sum of real and virtual), where in all cases the logarithmic terms including the IR divergence are not shown.", "In the same convention, we show our results for the pole–pole (blue), pole–dispersive (red), and dispersive–dispersive (green) contributions, as well as their sum minus the point-like virtual correction (maroon).", "The same quantity in the GVMD model of Ref.", "with a single Breit–Wigner is given for comparison (orange).Our numerical results for the corrections $\\delta $ are shown in Figs.", "REF and REF , where in all cases the terms proportional to $\\log \\frac{\\lambda ^2}{4\\Delta ^2}$ and $\\log \\frac{\\lambda ^2}{s}$ , respectively, are dropped.", "In addition to the separate curves for the pole–pole, pole–dispersive, and dispersive–dispersive contributions, we also show the quantity $\\delta _\\text{FF}$ , which was defined in Ref.", "[72] as the total minus the point-like virtual correction.", "We also include the GVMD model from the same reference (in the variant using a single Breit–Wigner function for the $\\rho (770)$ ), and, in Fig.", "REF , choose the same fixed parameters ($z=\\cos (1)$ and $\\sqrt{s}=0.75\\,\\text{GeV}$ , respectively) to facilitate the comparison.", "Figure REF shows our results for $\\delta _\\text{FF}$ as a function of both $\\sqrt{s}$ and $z$ .", "Figure: Structure-dependent correction δ FF \\delta _\\text{FF} in the dispersive approach (sum of pole–pole, pole–dispersive, and dispersive–dispersive minus point-like virtual), as a function of s\\sqrt{s} and zz.", "The contour lines give increments of 0.010.01.", "A data file is attached as supplemental material.The main observation is that we confirm significant departures from the point-like approximation, which, as remarked in the previous section, are a remnant of the intricate manner how IR singularities cancel in the sum of real and virtual contributions for a general VFF.", "In particular, while the pole–pole piece remains small and actually stays close to the point-like result, the pole–dispersive correction receives a large enhancement in the vicinity of the $\\rho $ resonance.", "The exact shape does differ when compared to the GVMD approximation, but the overall size of the effect seems to be captured correctly by the model.", "Finally, we find that the dispersive–dispersive contributions are negligible below $1\\,\\text{GeV}$ , supporting the expectation that the most important radiative corrections are the ones that display some form of IR enhancement." ], [ "Conclusions", "In this paper we presented a calculation of the radiative corrections to the forward–backward asymmetry in $e^+e^-\\rightarrow \\pi ^+\\pi ^-$ that includes the full effect of the pion vector form factor in the loop integral by means of a dispersive representation and thus captures the leading hadronic intermediate state.", "In particular, we studied how the cancellation of infrared divergences proceeds in the case of a general form factor, and found that an intricate interplay between a point-like pion-pole contribution and the real and imaginary parts of a mixed pole and dispersive correction becomes necessary.", "Overall, the numerical results support recent findings in a generalized vector-meson-dominance model [72], indicating significant deviations from the point-like approximation, but the dispersive analysis puts this conclusion on a more solid foundation and allows one to trace back the origin of the large correction to a remnant of the infrared singularities.", "This implies that the usual assumption that the most important radiative corrections for processes involving hadrons should be the ones that display some form of infrared enhancement actually proves correct, while the problem in the scalar-QED calculation multiplied by the pion vector form factor was that these pieces were not correctly identified.", "These insights should prove valuable for reassessing the role of radiative corrections in precision measurements of $e^+e^-\\rightarrow \\pi ^+\\pi ^-$ .", "We thank Peter Stoffer for comments on the manuscript.", "Financial support by the SNSF (Project Nos.", "200020_175791, PCEFP2_181117, and PZ00P2_174228) and by the Ramón y Cajal program (RYC2019-027605-I) of the Spanish MINECO is gratefully acknowledged." ], [ "Pole–pole", "For the pole–pole contribution we need the loop functions for $s^{\\prime }=s^{\\prime \\prime }=\\lambda ^2$ : $C_0\\big (t,\\lambda ^2\\big )&=\\frac{1}{M_\\pi ^2-t}\\bigg [\\log \\frac{\\lambda ^2}{M_\\pi ^2}\\log \\frac{M_\\pi ^2-t}{M_\\pi m_e}+\\log ^2\\frac{m_e}{M_\\pi }-\\log ^2\\frac{M_\\pi ^2-t}{M_\\pi ^2}-\\text{Li}_2\\bigg (\\frac{t}{M_\\pi ^2}\\bigg )\\bigg ],\\\\C_0\\big (s,\\lambda ^2,\\lambda ^2,M_\\pi ^2\\big )&=\\frac{1}{s\\beta }\\bigg [\\frac{\\pi ^2}{6}+\\frac{1}{2}\\log ^2\\frac{1-\\beta }{1+\\beta }+2\\text{Li}_2\\bigg (\\frac{\\beta -1}{1+\\beta }\\bigg )+i\\pi \\log \\frac{1-\\beta }{1+\\beta }\\bigg ],\\\\C_0\\big (s,\\lambda ^2,\\lambda ^2,m_e^2\\big )&= \\frac{\\pi ^2+3\\log ^2\\frac{m_e^2}{s}+6i\\pi \\log \\frac{m_e^2}{s}}{6s},\\\\D_0\\big (s,t,\\lambda ^2,\\lambda ^2\\big )&=\\frac{2}{s}\\bigg (\\log \\frac{\\lambda ^2}{s}+i\\pi \\bigg )\\frac{\\log \\frac{M_\\pi ^2-t}{m_eM_\\pi }}{M_\\pi ^2-t},$ where we suppressed the other arguments and only kept $m_e$ to regularize collinear singularities at intermediate steps of the calculation." ], [ "Pole–dispersive", "For the case $s^{\\prime }\\ge 4M_\\pi ^2$ , $s^{\\prime \\prime }=\\lambda ^2$ we need the additional loop functions $C_0(t,s^{\\prime })&=-\\frac{1}{M_\\pi ^2-t}\\bigg [\\text{Li}_2\\bigg (\\frac{t}{M_\\pi ^2}\\bigg )+\\text{Li}_2\\bigg (1-\\frac{s^{\\prime }(1-\\beta ^{\\prime })}{2(M_\\pi ^2-t)}\\bigg )+\\text{Li}_2\\bigg (1-\\frac{s^{\\prime }(1+\\beta ^{\\prime })}{2(M_\\pi ^2-t)}\\bigg )\\\\&\\quad +\\frac{\\pi ^2}{6}+\\frac{1}{2}\\log ^2(1-\\beta ^{\\prime })+\\frac{1}{2}\\log ^2(1+\\beta ^{\\prime })-\\log \\frac{2M_\\pi ^2}{M_\\pi ^2-t}\\log \\frac{2(M_\\pi ^2-t)}{s^{\\prime }}\\bigg ],\\\\C_0^{>}(s,s^{\\prime },M_\\pi ^2)&=\\frac{1}{s\\beta }\\bigg [\\text{Li}_2\\bigg (\\frac{1-\\beta }{1-\\beta ^{\\prime }}\\bigg )+\\text{Li}_2\\bigg (\\frac{1+\\beta ^{\\prime }}{1+\\beta }\\bigg )+\\text{Li}_2\\bigg (\\frac{1-\\beta }{1+\\beta ^{\\prime }}\\bigg )+\\text{Li}_2\\bigg (\\frac{1-\\beta ^{\\prime }}{1+\\beta }\\bigg )\\\\&\\quad +\\frac{1}{2}\\bigg (\\log ^2\\frac{1-\\beta }{1+\\beta }+\\log ^2\\frac{1+\\beta }{1+\\beta ^{\\prime }}+\\log ^2\\frac{1+\\beta }{1-\\beta ^{\\prime }}+2\\log \\frac{1+\\beta ^{\\prime }}{1-\\beta ^{\\prime }}\\log \\frac{\\beta -\\beta ^{\\prime }}{\\beta +\\beta ^{\\prime }}\\bigg )\\\\&\\quad -\\frac{\\pi ^2}{2}+2\\text{Li}_2\\bigg (\\frac{\\beta -1}{1+\\beta }\\bigg )+i\\pi \\log \\frac{1-\\beta }{1+\\beta }\\bigg ],\\\\C_0^{>}(s,s^{\\prime },m_e^2)&= \\frac{-\\frac{\\pi ^2}{3}+\\frac{1}{2}\\log ^2\\frac{s}{s^{\\prime }}-\\log \\frac{m_e^2}{s^{\\prime }}\\log \\frac{s-s^{\\prime }}{s^{\\prime }}+\\text{Li}_2\\big (\\frac{s^{\\prime }}{s}\\big )+i\\pi \\log \\frac{m_e^2}{s}}{s},\\\\C_0^{<}(s,s^{\\prime },M_\\pi ^2)&=\\frac{1}{s\\beta }\\bigg [-\\text{Li}_2\\bigg (\\frac{1-\\beta ^{\\prime }}{1-\\beta }\\bigg )-\\text{Li}_2\\bigg (\\frac{1+\\beta }{1+\\beta ^{\\prime }}\\bigg )+\\text{Li}_2\\bigg (\\frac{1-\\beta }{1+\\beta ^{\\prime }}\\bigg )+\\text{Li}_2\\bigg (\\frac{1-\\beta ^{\\prime }}{1+\\beta }\\bigg )\\\\&\\quad +\\frac{1}{2}\\bigg (\\log ^2\\frac{1-\\beta }{1+\\beta }-\\log ^2\\frac{1-\\beta }{1-\\beta ^{\\prime }}+\\log ^2\\frac{1+\\beta }{1-\\beta ^{\\prime }}+2\\log \\frac{1+\\beta ^{\\prime }}{1-\\beta ^{\\prime }}\\log \\frac{\\beta ^{\\prime }-\\beta }{\\beta +\\beta ^{\\prime }}\\bigg )\\\\&\\quad +\\frac{\\pi ^2}{6}+2\\text{Li}_2\\bigg (\\frac{\\beta -1}{1+\\beta }\\bigg )\\bigg ],\\\\C_0^{<}(s,s^{\\prime },m_e^2)&= \\frac{-\\log \\frac{m_e^2}{s^{\\prime }}\\log \\frac{s^{\\prime }-s}{s^{\\prime }}-\\text{Li}_2\\big (\\frac{s}{s^{\\prime }}\\big )}{s},\\\\D_0\\big (s,t,s^{\\prime },\\lambda ^2\\big )&=-\\frac{\\log \\frac{M_\\pi ^2-t}{m_eM_\\pi }\\Big [\\log \\frac{\\lambda ^2}{M_\\pi ^2}-\\log \\frac{m_e}{M_\\pi }+\\log \\frac{s^{\\prime 2}}{(s^{\\prime }-s)^2}-\\log \\frac{M_\\pi ^2-t}{M_\\pi ^2}+2\\pi i\\theta \\big (s-s^{\\prime }\\big )\\Big ]}{(M_\\pi ^2-t)(s^{\\prime }-s)}\\\\&-\\frac{1}{(M_\\pi ^2-t)(s^{\\prime }-s)}\\bigg [\\frac{\\pi ^2}{6}+\\text{Li}_2\\bigg (1-\\frac{s^{\\prime }(1-\\beta ^{\\prime })}{2(M_\\pi ^2-t)}\\bigg )+\\text{Li}_2\\bigg (1-\\frac{s^{\\prime }(1+\\beta ^{\\prime })}{2(M_\\pi ^2-t)}\\bigg )\\\\&\\quad +\\frac{1}{2}\\log ^2(1-\\beta ^{\\prime })+\\frac{1}{2}\\log ^2(1+\\beta ^{\\prime })-\\log \\frac{2M_\\pi ^2}{M_\\pi ^2-t}\\log \\frac{2(M_\\pi ^2-t)}{s^{\\prime }}\\bigg ],$ where $\\beta ^{\\prime }=\\sqrt{1-4M_\\pi ^2/s^{\\prime }}$ and $\\gtrless $ indicates that the expression applies to $s\\gtrless s^{\\prime }$ ." ], [ "Dispersive–dispersive", "For general $s^{\\prime },s^{\\prime \\prime }$ we write the additional loop functions in the form $C_0(s,s^{\\prime },s^{\\prime \\prime },M_\\pi ^2)&=\\frac{1}{\\pi }\\int _{(\\sqrt{s^{\\prime }}+\\sqrt{s^{\\prime \\prime }})^2}^\\infty dx \\frac{\\text{Im}\\,C_0(x,s^{\\prime },s^{\\prime \\prime },M_\\pi ^2)}{x-s},\\\\\\text{Im}\\,C_0(s,s^{\\prime },s^{\\prime \\prime },M_\\pi ^2)&=\\frac{\\pi \\theta \\big (s-\\big (\\sqrt{s^{\\prime }}+\\sqrt{s^{\\prime \\prime }}\\big )^2\\big )}{s \\beta }\\log \\frac{s-s^{\\prime }-s^{\\prime \\prime }-\\beta \\lambda ^{1/2}_s}{s-s^{\\prime }-s^{\\prime \\prime }+\\beta \\lambda ^{1/2}_s},\\\\D_0\\big (s,t,s^{\\prime },s^{\\prime \\prime }\\big )&=\\frac{1}{\\pi }\\int _{M_\\pi ^2}^\\infty dx \\frac{\\text{Im}_t\\,D_0(s,x,s^{\\prime },s^{\\prime \\prime })}{x-t}=\\frac{1}{\\pi }\\int _{(\\sqrt{s^{\\prime }}+\\sqrt{s^{\\prime \\prime }})^2}^\\infty dx \\frac{\\text{Im}_s\\,D_0(x,t,s^{\\prime },s^{\\prime \\prime })}{x-s},\\\\\\text{Im}_t\\,D_0^>(s,t,s^{\\prime },s^{\\prime \\prime })&=\\frac{\\pi \\theta \\big (t-M_\\pi ^2\\big )}{\\sqrt{\\Delta (s,t,s^{\\prime },s^{\\prime \\prime })}}\\bigg [2\\pi i \\theta \\big (s-(\\sqrt{s^{\\prime }}+\\sqrt{s^{\\prime \\prime }})^2\\big )\\\\&+\\log \\frac{(s^{\\prime }+s^{\\prime \\prime }-s)(M_\\pi ^2-t)^2+2t s^{\\prime }s^{\\prime \\prime }+(t-M_\\pi ^2)\\sqrt{\\Delta (s,t,s^{\\prime },s^{\\prime \\prime })}}{(s^{\\prime }+s^{\\prime \\prime }-s)(M_\\pi ^2-t)^2+2t s^{\\prime }s^{\\prime \\prime }-(t-M_\\pi ^2)\\sqrt{\\Delta (s,t,s^{\\prime },s^{\\prime \\prime })}}\\bigg ],\\\\\\text{Im}_t\\,D_0^<(s,t,s^{\\prime },s^{\\prime \\prime })&=\\frac{2\\pi \\theta \\big (t-M_\\pi ^2\\big )}{\\sqrt{-\\Delta (s,t,s^{\\prime },s^{\\prime \\prime })}}\\bigg [\\arctan \\frac{(t-M_\\pi ^2)\\sqrt{-\\Delta (s,t,s^{\\prime },s^{\\prime \\prime })}}{(s^{\\prime }+s^{\\prime \\prime }-s)(M_\\pi ^2-t)^2+2ts^{\\prime }s^{\\prime \\prime }}\\\\&\\quad +\\pi \\theta \\Big (\\big (s-s^{\\prime }-s^{\\prime \\prime }\\big )\\big (M_\\pi ^2-t\\big )^2-2ts^{\\prime } s^{\\prime \\prime }\\Big )\\bigg ],\\\\\\text{Im}_s\\,D_0(s,t,s^{\\prime },s^{\\prime \\prime })&=\\frac{\\pi \\theta \\big (s-(\\sqrt{s^{\\prime }}+\\sqrt{s^{\\prime \\prime }})^2\\big )}{\\sqrt{\\Delta (s,t,s^{\\prime },s^{\\prime \\prime })}}\\log \\frac{2s s^{\\prime } s^{\\prime \\prime }+\\lambda _s(M_\\pi ^2-t)+\\sqrt{\\lambda _s \\Delta (s,t,s^{\\prime },s^{\\prime \\prime })}}{2s s^{\\prime } s^{\\prime \\prime }+\\lambda _s(M_\\pi ^2-t)-\\sqrt{\\lambda _s \\Delta (s,t,s^{\\prime },s^{\\prime \\prime })}},$ where $\\Delta (s,t,s^{\\prime },s^{\\prime \\prime })=\\lambda _s\\big (M_\\pi ^2-t\\big )^2-4ts s^{\\prime }s^{\\prime \\prime },\\qquad \\lambda _s=\\lambda (s,s^{\\prime },s^{\\prime \\prime }),$ $\\lambda (a,b,c)=a^2+b^2+c^2-2(a b+a c+b c)$ , and the expressions are valid in the kinematic region of interest ($s^{\\prime }\\ge 4M_\\pi ^2$ , $s^{\\prime \\prime }\\ge 4M_\\pi ^2$ , $s\\ge 4M_\\pi ^2$ , $t\\le 0$ , $m_e= 0$ ).", "$\\gtrless $ indicates that the expression applies for $\\Delta (s,t,s^{\\prime },s^{\\prime \\prime })\\gtrless 0$ ." ] ]
2207.03495
[ [ "Nonparametric Estimation of the Potential Impact Fraction and Population\n Attributable Fraction with Individual-Level and Aggregated Data" ], [ "Abstract The estimation of the potential impact fraction (including the population attributable fraction) with continuous exposure data frequently relies on strong distributional assumptions.", "However, these assumptions are often violated if the underlying exposure distribution is unknown or if the same distribution is assumed across time or space.", "Nonparametric methods to estimate the potential impact fraction are available for cohort data, but no alternatives exist for cross-sectional data.", "In this article, we discuss the impact of distributional assumptions in the estimation of the population impact fraction, showing that under an infinite set of possibilities, distributional violations lead to biased estimates.", "We propose nonparametric methods to estimate the potential impact fraction for aggregated (mean and standard deviation) or individual data (e.g.", "observations from a cross-sectional population survey), and develop simulation scenarios to compare their performance against standard parametric procedures.", "We illustrate our methodology on an application of sugar-sweetened beverage consumption on incidence of type 2 diabetes.", "We also present an R package pifpaf to implement these methods." ], [ "Introduction", "The potential impact fraction (PIF), also known as the generalized impact fraction, quantifies the contribution of a risk factor to morbidity or mortality, by estimating the difference in disease burden resulting from a change in the exposure distribution to a counterfactual scenario [20], [24], [36], [33].", "A special case of the PIF is the population attributable fraction (PAF), often referred to as the population attributable risk, where the counterfactual exposure equals the baseline level, which in many cases is no exposure, for all individuals.", "While the selection of counterfactuals has focused on public health scenarios, little has been discussed about the methodological implications of parametric distributional assumptions in the PIF estimation [24].", "In what we call the “standard method”, the PIF is estimated by using exposure parameters (e.g.", "mean and variance) obtained from national surveys [12], [7], [23], [18], [11], [35], with meta-analytical relative risks [24], [10].", "Here, it is assumed that the exposure follows a certain distribution, usually without empirical verification, or that the same distribution applies to different settings and countries, such as in the case of the Global Burden of Disease project [10].", "The implications of these distributional assumptions are rarely discussed, despite previous evidence showing that the PIF estimates can be biased if the exposure distribution is misspecified [17].", "The problem of model misspecification to estimate the PIF is not new.", "Both semiparametric and nonparametric methods have been proposed to avoid making distributional assumptions in cases where both the relative risk and the exposure distribution are obtained from the same cohort data [6], [29], [32].", "However, international efforts to estimate the burden of disease in different countries, such as those conducted by the World Health Organization and the Institute for Health Metrics and Evaluation, are usually based on relative risks from meta-analyses and population-level point estimates from survey data, for which nonparametric methods are not available [10].", "Even in cases where datasets with disaggregated individual level exposure information are available, the lack of methods to estimate the PIF by combining these datasets with meta-analytical relative risks has forced researchers to aggregate the data and follow the standard method [35], [12].", "In this paper, we analyze the problems that arise when the PIF is estimated via an arbitrary selection of the exposure distribution.", "We then present two nonparametric alternatives that adequately estimate the PIF using meta-analytical relative risks, the first of which uses individual level exposure data and the second of which only uses the mean and variance of the exposure data.", "Finally, we conduct numerical simulations of our methods and illustrate them on an application of sugar sweetened beverages on incidence of type 2 diabetes in Mexico.", "Let ${X} = (X_1, X_2, \\dots , X_k)^T$ with $k$ components be the exposure, which takes values over $\\mathcal {X}$ , the set of all possible ${X}$ .", "The estimation of the PIF requires a relative risk function, $RR$ , that depends on the exposure ${X}$ , and ${\\beta } = (\\beta _0, \\beta _1, \\dots , \\beta _{k})^T$ , the regression coefficients corresponding to the exposure ${X}$ that usually obtained from previous study or meta-analysis.", "Here, we assume that ${\\beta }$ is a causal parameter if all confounders were adjusted for in the regression model.", "Examples of relative risk functions include the exponential function, $RR({X};{\\beta }) = \\textrm {exp}({\\beta }^T {X})$ , for logistic, Poisson, or Cox regression models, and linear function, $RR({X};{\\beta }) = 1 + {\\beta }^T {X}$ , for linear regression models.", "The PIF contrasts the differences in the burden of disease between an observed and a counterfactual exposure distribution.", "When the exposure is categorical, the PIF is defined as $\\textrm {PIF} =\\dfrac{\\sum _{{X} \\in \\mathcal {X}} p_{\\textrm {obs}}({X}) \\cdot RR({X};{\\beta }) - \\sum _{{X} \\in \\mathcal {X}} p_{\\textrm {cft}}({X}) \\cdot RR({X};{\\beta })}{\\sum _{{X} \\in \\mathcal {X}} p_{\\textrm {obs}}({X}) \\cdot RR({X};{\\beta })},$ where $p_{\\textrm {obs}}({X})$ is the observed probability mass function of the exposure ${X}$ in the population, and $p_{\\textrm {cft}}({X})$ is the probability mass function in the counterfactual intervention scenario.", "Alternatively, if ${X}$ is continuous with exposure distribution $f_{\\textrm {obs}}$ , the PIF is given by $\\textrm {PIF} = \\dfrac{\\int _{\\mathcal {X}} RR({X};{\\beta })f_{\\textrm {obs}}({X})d{X} - \\int _{\\mathcal {X}} RR\\big ({X};{\\beta } \\big )f_{\\textrm {cft}}({X})d{X}}{\\int _{\\mathcal {X}} RR({X};{\\beta })f_{\\textrm {obs}}({X})d{X}},$ where $f_{\\textrm {obs}}$ and $f_{\\textrm {cft}}$ represent probability density functions of the observed exposure and the counterfactual continuous exposure, respectively [24], [33].", "In general, unifying (REF ) and (REF ) to allow for both discrete and continuous exposures, the PIF can be written as: $\\textrm {PIF} = \\dfrac{ E^{\\textrm {obs}}_{{X}} \\Big [ RR\\big ({X};{\\beta } \\big )\\Big ] - E^{\\textrm {cft}}_{{X}} \\Big [ RR\\big ({X};{\\beta }\\big )\\Big ]}{E^{\\textrm {obs}}_{{X}} \\Big [ RR\\big ({X};{\\beta } \\big )\\Big ]},$ where $E^{\\textrm {obs}}_{{X}} \\left[ RR({X};{\\beta })\\right]$ represents the expected value of the relative risk under the observed exposure distribution in a given population and $E^{\\textrm {cft}}_{{X}} \\left[ RR({X};{\\beta })\\right]$ is the expected value of the relative risk under a counterfactual distribution of the exposure [32], [38].", "Often, the counterfactual exposure distribution under the intervention can be represented as a transformation $g$ on the exposure ${X}$ .", "For example, $g({X}) = 0.6 \\cdot {X}$ might represent a $40 \\%$ reduction in the exposure, or $g({X}) = {X} - 2$ , an overall decrease of 2 units of the exposure.", "Then, the PIF can be written as $\\textrm {PIF} = \\dfrac{ E^{\\textrm {obs}}_{{X}} \\Big [ RR\\big ({X};{\\beta } \\big )\\Big ] - E^{\\textrm {obs}}_{{X}} \\Big [ RR\\big (g({X});{\\beta }\\big )\\Big ]}{E^{\\textrm {obs}}_{{X}} \\Big [ RR\\big ({X};{\\beta } \\big )\\Big ]}.$ The PAF is defined as a special case of the $\\textrm {PIF}$ when the counterfactual is the baseline exposure for all individuals (i.e., $RR\\big (g({X});{\\beta }\\big )=1)$ .", "Thus, the expected value of the relative risk under the counterfactual scenario equals 1 [33], yielding $\\textrm {PAF}= 1 - \\dfrac{1}{E^{\\textrm {obs}}_{{X}} \\Big [ RR\\big ({X};{\\beta } \\big )\\Big ]}.$ The PIF and PAF can be thought of as causal estimands if the following assumptions hold.", "First, ${\\beta }$ is a causal parameter, which holds if the model to estimate the relative risk is adjusted for all known confounders.", "Second, ${\\beta }$ is also transportable to the counterfactual population if the confounder distribution is similar in the observed population and the counterfactual confounder distribution.", "Third, we also assume that there are no effect modifiers [22]." ], [ "Problems in the standard approach", "When the exposure is categorical, the PIF can be easily estimated from expression (REF ) [30].", "PIF estimation is more challenging for continuous exposures.", "The standard approach assumes a distribution for the exposure ${X}$ and fits its parameters through method of moments estimation, matching the mean and variance of the empirical exposure data.", "The PIF is then estimated using equation (REF ) through analytic integration or by Monte Carlo integration.", "This method depends heavily on the choice of the exposure distribution.", "The exposure is often assumed to follow a normal distribution, a log-normal distribution, a Weibull distribution, or other distributions.", "For example, studies estimating the $\\textrm {PAF}$ of obesity-related diseases assumed that the exposure variable was log-normally distributed [1], [35].", "However, the true distribution is actually unknown in practice.", "An incorrect distribution that misrepresents the data could yield a result that diverges substantially from the actual fraction, introducing a subjective bias.", "Consequently, approaches to estimate the PIF should avoid untested distributional assumptions about ${X}$ .", "Table REF shows how the bias of the PAF of the standard method changes as a function of the true exposure distribution (Gamma(1.15, 1.29), Normal(1.48, 1.38$^2$ ), or Weibull(1.08, 1.53)) when other distributions are assumed (Gamma, Lognormal, Normal, Weibull) and the relative risk function is exponential $RR({X};{\\beta }) = \\textrm {exp}({\\beta }^T {X})$ .", "The parameters of the true exposure distributions are taken from our illustrative example.", "Table: Relative bias percentage of PAF under different distributional assumptions for the standard method.When the assumed distribution is log-normal and the relative risk function is exponential, the $\\textrm {PAF}$ equals 1 for positive ${\\beta }$ , as shown in Table REF , because the denominator in (REF ) equals the log-normal moment generating function, which is infinite [5].", "The problem lies on the combination of a heavy-tailed distribution with an exponential relative risk.", "A random variable $X$ is said to have a heavy tail if the tail probabilities $P(X>t)$ decay more slowly than tails of any exponential distribution, that is, $\\displaystyle \\lim _{x\\rightarrow \\infty } e^{cx}P(X>x)=\\infty $ for all positive $c$ .", "In addition to the log-normal distribution, the Pareto, Cauchy, and Weibull (with shape parameter less than 1) distributions are also heavy-tailed [8].", "The PIF could be undefined with an exponential relative risk for a heavy-tailed distribution; hence additional constraints are required to guarantee the existence of the PIF.", "This problem has been pointed out previously without much mathematical detail by [17].", "As a potential solution to this problem, [17] truncated the assumed exposure distribution by providing an upper bound $M$ , thereby avoiding large exposure values and the infinite expected values.", "In addition, the zero and non-zero values of the exposure data are first separated, and the parameters of the positive values of the exposure are estimated using maximum likelihood estimation.", "This can be written as $\\textrm {PAF} = 1 - \\frac{1}{p_0 RR_0 + (1- p_0) \\int _0^M RR(x;\\beta ) f(x)\\text{d}x},$ where $M$ is the truncation bound, $p_0$ is the proportion of zero values in the exposure, and $RR_0$ is the relative risk under no exposure.", "Nevertheless, additional problems arise because the estimated PAF value now depends on the upper bound.", "For example, consider Figure REF which shows the PAF as a function of the exposure's upper bound $M$ .", "The figure shows that if an upper bound of $M = 20$ is selected, the resulting PAF approximates $50\\%$ ; truncating at $M = 40$ results in a PAF of $80\\%$ .", "By changing the truncation bound, $M$ , we can obtain PAF estimates ranging from anywhere between $0\\%$ to $100\\%$ .", "Here, we assume ${X}$ to be log-normally distributed with parameters $\\log \\mu = 0.05,\\log \\sigma = 0.98$ and an exponential relative risk function $RR({X};{\\beta }) = e^{\\log (1.27) {X}}$ .", "The relative risk function and fitted parameters are taken from the exposure of our illustrative example, discussed later.", "Figure: PIF and PAF as a function of the truncation limit MM considering X{X} to be log-normally distributed with parameters logμ=0.05,logσ=0.98\\log \\mu =0.05,\\log \\sigma =0.98 and an exponential relative risk function RR(X;β)=e βX RR({X};{\\beta }) = e^{\\beta X}, where β=log(1.27)\\beta = \\log (1.27).", "Parameters are taken from the illustrative example in Section ." ], [ "Methods", "In order to resolve these issues and improve the estimation of the PIF, we propose two nonparametric alternatives: one that requires individual-level exposure data, which we call the “empirical method”, and one that only uses the mean and variance of the exposure data, which we call the “approximate method”.", "Both methods are implemented in an R package pifpaf, available on Github (github.com/colleenchan/pifpaf)." ], [ "Empirical method", "Let ${X}_1, {X}_2, \\dots , {X}_n$ be a random sample of $n$ individuals.", "Denote ${\\mu }^{\\textrm {obs}}({\\beta })$ the mean of the relative risk, conditional on ${\\beta }$ .", "It can be estimated by $\\widehat{\\mu }_n^{\\textrm {obs}}({\\beta }) = \\dfrac{1}{n} \\sum \\limits _{i=1}^{n} RR\\big ( {X}_i; {\\beta } \\big ).$ Let $\\mu ^{\\textrm {cft}}({\\beta })$ denote the conditional mean under the counterfactual scenario.", "If the counterfactual exposure can be written as a function of the original exposure, $g({X_i})$ , the counterfactual conditional mean is estimated by: $\\widehat{\\mu }_n^{\\textrm {cft}}({\\beta }) = \\dfrac{1}{n} \\sum _{i=1}^n RR\\big ( g({X}_i); {\\beta } \\big ).$ Let $\\widehat{{\\beta }}$ be an estimate of ${\\beta }$ from a previous study or a meta analysis.. We define the empirical estimators of PAF and PIF as: $\\widehat{\\textrm {PAF}} := 1 - \\dfrac{1}{\\widehat{\\mu }_n^{\\textrm {obs}}(\\widehat{{\\beta }})}, \\qquad \\textrm {and} \\qquad \\widehat{\\textrm {PIF}} := 1 - \\dfrac{\\widehat{\\mu }_n^{\\textrm {cft}}(\\widehat{{\\beta }})}{\\widehat{\\mu }_n^{\\textrm {obs}}(\\widehat{{\\beta }})}.$ The asymptotic properties of the estimators are presented in the following theorem.", "Suppose that $\\widehat{{\\beta }}$ is a consistent and asymptotically normal estimator from an independent study.", "That is, $\\widehat{{\\beta }}\\overset{p}{\\longrightarrow }{\\beta }$ and $\\sqrt{m} (\\widehat{{\\beta }}-{\\beta })$ is asymptotically mean-zero multivariate normal with covariance matrix $\\Sigma _{\\beta }$ , where $m$ is the sample size of the independent study estimating ${\\beta }$ .", "Assume $RR({X};{\\beta })$ is a continuous function.", "Then $\\widehat{\\textrm {PAF}}$ converges in probability to $\\textrm {PAF}$ , and $\\sqrt{n} (\\widehat{\\textrm {PAF}}-\\textrm {PAF})$ converges toward a mean-zero normal distribution when both $m$ and $n$ approach infinity and $n/m\\longrightarrow \\lambda (<\\infty )$ .", "Furthermore, suppose $g({X})$ is continuous, then $\\widehat{\\textrm {PIF}}$ converges in probability to $\\textrm {PAF}$ , and $\\sqrt{n} (\\widehat{\\textrm {PIF}}-\\textrm {PIF})$ converges toward a mean-zero normal distribution when both $m$ and $n$ approach infinity and $n/m\\longrightarrow \\lambda (<\\infty )$ .", "The detailed proof is provided in the Appendix.", "We now derive the estimate of confidence intervals for $\\widehat{\\textrm {PAF}}$ .", "We have proved the asymptotic normality $\\hat{\\mu }_n^{\\textrm {obs}}(\\hat{{\\beta }})$ in (REF ).", "The variance of $\\hat{\\mu }_n^{\\textrm {obs}}(\\hat{{\\beta }})$ can be estimated by $\\begin{aligned}\\widehat{Var}(\\hat{\\mu }_n^{\\textrm {obs}}(\\hat{{\\beta }}))&= \\frac{1}{n}\\widehat{Var}(RR(X;\\hat{\\beta }))+\\mathbb {E}(\\nabla _{\\beta }RR(X;{\\beta }))\\widehat{Var}(\\hat{\\beta })\\mathbb {E}(\\nabla _{\\beta }RR(X;{\\beta }))^T\\nonumber \\\\&\\approx \\frac{1}{n} \\left(\\frac{1}{n}\\sum _{i=1}^n(RR(X_i;\\hat{\\beta }))^2-\\left(\\hat{\\mu }_n^{\\textrm {obs}}(\\hat{{\\beta }})\\right)^2\\right) + \\\\& \\qquad \\left(\\frac{1}{n}\\sum _{i=1}^n\\nabla _{\\beta }RR(X_i;{\\beta })\\Big |_{\\beta =\\hat{\\beta }}\\right)\\widehat{Var}(\\hat{\\beta }) \\left(\\frac{1}{n}\\sum _{i=1}^n\\nabla _{\\beta }RR(X_i;{\\beta })\\Big |_{\\beta =\\hat{\\beta }}\\right)^T .", "\\end{aligned}$ By the delta method, the variance of $\\widehat{\\textrm {PAF}}$ can be estimated by $\\widehat{Var}(\\widehat{\\textrm {PAF}})\\approx \\frac{\\widehat{Var}(\\hat{\\mu }_n^{\\textrm {obs}}(\\hat{{\\beta }}))}{(\\hat{\\mu }_n^{\\textrm {obs}}(\\hat{{\\beta }}))^4}$ Then the $(1-\\alpha )$ % confidence interval for $\\widehat{\\textrm {PAF}}$ is estimated as $\\widehat{\\textrm {PAF}}\\pm z_{1-\\frac{\\alpha }{2}} \\sqrt{\\widehat{Var}(\\widehat{\\textrm {PAF}})}$ with $z_{1-\\frac{\\alpha }{2}}$ being the $1-\\frac{\\alpha }{2}$ quantile of the standard normal distribution.", "Similarly, the confidence intervals can be constructed using the estimate of the variance of $\\widehat{\\textrm {PIF}}$ provided in the Appendix." ], [ "Approximate Method", "Often, such as encountered by the Global Burden of Disease group, data on the exposure and covariates distribution is not available [10].", "Rather, estimates $\\bar{{X}} = (\\bar{X}_1, \\bar{X}_2, \\dots , \\bar{X}_k)^T$ for the mean and estimators $\\hat{\\sigma }_{i,j}$ for the covariance between $X_i$ and $X_j$ , (where $X_i$ , $X_j$ represent the exposures or covariates of the random variable ${X} = (X_1, X_2, \\dots , X_k)^T$ ) are available.", "When the relative risk function $RR({X}; {\\beta })$ is twice differentiable in ${X}$ , (such would be the case of linear and exponential relative risk functions), by expanding the Taylor series to the second order, we can approximate $\\hat{\\mu }^{\\textrm {obs}}(\\hat{{\\beta }})$ by $\\hat{\\mu }^{\\textrm {obs}}(\\hat{{\\beta }}) \\approx RR(\\bar{{X}};\\hat{{\\beta }})+\\dfrac{1}{2} \\sum \\limits _{i,j} \\hat{\\sigma }_{i,j} \\frac{\\partial ^2 RR\\left({X};\\hat{{\\beta }}\\right)}{\\partial X_i \\partial X_j}\\big |_{{X} = \\bar{{X}}},$ leading to the estimator of the PAF $\\widehat{\\textrm {PAF}}= 1-\\dfrac{1}{RR(\\bar{{X}};\\hat{{\\beta }})+\\frac{1}{2} \\sum _{i,j} \\hat{\\sigma }_{i,j} \\frac{\\partial ^2 RR\\left({X};\\hat{{\\beta }}\\right)}{\\partial X_i \\partial X_j}\\big |_{{X} = \\bar{{X}}}}.$ The detailed derivation can be found in the Appendix.", "Similarly, if the counterfactual function $g({X})$ is a twice differentiable function of ${{X}}$ , then $\\widehat{\\textrm {PIF}}= 1-\\dfrac{RR\\big (g(\\bar{{X}} ),\\hat{{\\beta }} \\big ) + \\frac{1}{2}\\sum _{i,j} \\hat{\\sigma }_{i,j}\\frac{\\partial ^2 RR\\left(g({X}),\\hat{{\\beta }} \\right)}{\\partial X_i \\partial X_j}\\big |_{{X} = \\bar{{X}}}}{RR(\\bar{{X}};\\hat{{\\beta }})+\\frac{1}{2} \\sum _{i,j} \\hat{\\sigma }_{i,j} \\frac{\\partial ^2 RR\\left({X},\\hat{{\\beta }}\\right)}{\\partial X_i \\partial X_j}\\big |_{{X} = \\bar{{X}}}}.$ For an exponential relative risk that takes the form $RR( {X}, \\hat{\\beta }) = \\exp (\\hat{\\beta }{X})$ with $k=1$ , Eq.", "(REF ) and Eq.", "(REF ) simplify to $\\widehat{\\text{PAF}}= 1-\\dfrac{1}{ \\exp (\\hat{\\beta }\\bar{X}) \\left(1 +\\frac{1}{2} \\hat{\\beta }^2 \\sqrt{\\widehat{Var}(X)} \\right) },$ $\\widehat{\\text{PIF}}= 1-\\dfrac{\\exp (\\hat{\\beta }g(\\bar{X})) \\left(1 +\\frac{1}{2}\\sqrt{\\widehat{Var}(X)} \\left(\\hat{\\beta }^2 (g^{\\prime }(\\bar{X}))^2 + \\hat{\\beta }g^{\\prime \\prime }(\\bar{X}) \\right) \\right)}{ \\exp (\\hat{\\beta }\\bar{X}) \\left(1 +\\frac{1}{2} \\hat{\\beta }^2 \\sqrt{\\widehat{Var}(X)} \\right) },$ respectively.", "We approximate their variance using the multivariate delta method, which is shown in the Appendix.", "The corresponding confidence intervals are constructed similarly to the empirical method in the previous section." ], [ "Illustrative Example", "We illustrate the use of our methodology in an analysis of sugar sweetened beverage (SSB) consumption on incidence of type 2 diabetes, and compare the results of our method with the standard approach and the mixture approach [17].", "The mixture method separates out the zero values of the exposure from the positive values of the exposure and estimates the fitted parameters of the positive values of the exposure through maximum likelihood estimation.", "SSB's are drinks with added sugar including soft drinks, flavored juice drinks, sports drinks, or sweetened tea and coffee.", "SSB consumption has risen in many countries, most noticeably in developing countries, in recent decades, and comprises the largest source of added sugar in the U.S. diet [25].", "Consumption of SSB's has been linked to increased risks of incidences of obesity, diabetes, and heart disease, leading to calls for their reduced consumption from groups such as the American Heart Association [21], [14], [16], [34].", "The data on SSB consumption comes from ENSANUT 2016, a probabilistic national health and nutrition survey of the Mexican population gathered between May and October of 2016 [9].", "The average consumption in this data ($n = 7762$ ) is 1.48 with standard deviation 1.38; about 5% of the sample had zero consumption.", "The relative risk of an additional serving of SSB (336 ml) on incidence of type 2 diabetes in Mexico is taken from recent literature, which is 1.27 with 95% CI (1.16, 1.38) [31].", "In the standard approach, a parametric distribution $f(x)$ must first be chosen and its parameters are fit using method of moments.", "Then, the PAF can be estimated via formula (REF ).", "In the mixture method, we consider a mixture distribution where $p_0$ is the proportion of zeros in the data and the distribution $f(x)$ of the remaining $1-p_0$ non-zero values is fitted using maximum likelihood estimation, as in Equation REF .", "We assume the baseline relative risk $RR_0 = 1$ .", "For heavy-tailed distributions, the PAF is 1 since the second term in the denominator converges to infinity; choosing a truncation bound sidesteps this issue.", "We estimate the PAF using the standard approach and the mixture approach without and with a truncation bound, where we set $M$ to be the maximum value of the data (11.855 servings).", "We fit a number of parametric distributions (Gamma, log normal, normal, and Weibull) to the non-zero exposure values using maximum likelihood estimation.", "For the two-parameter Gamma and Weibull distribution, closed form solutions using the maximum likelihood estimation are not available so the optimization of the log-likelihood was implemented using the BFGS method [3].", "Figure REF shows the SSB consumption distribution and the fitted distributions.", "We use Gauss-Kronrod quadrature to compute the integrations.", "Table REF shows the estimated PAF's using the standard method, the mixture method, and proposed empirical and approximate methods.", "Figure: Distribution of SSB consumption in ENSANUT 2016.Gamma, Normal, Lognormal, and Weibull distributions with parameters fit using the standard method (method of moments) are superimposed.Table: PAF estimate of type II diabetes due to SSB consumption under different distributional assumptions.There is no method to determine whether a distribution under the standard or mixture approach will yield appropriate results.", "However, we note that the Weibull distribution seems to fit the SSB consumption data the best, and thus, seems to be closest to the ground truth PAF value in this application.", "Thus, we would hope that the PAF estimate from the empirical and approximate methods to be close to that of the Weibull estimate of approximately 0.34.", "This is indeed the case (0.345 for the empirical method and 0.325 for the approximate method).", "We observe that standard method and mixture method on an assumed lognormal distribution and the standard method on the normal distribution perform poorly.", "Setting a truncation bound as in the mixture approach mitigates the problem although the correct exposure distribution and truncation bound still must be properly chosen." ], [ "Simulation Studies", "In this section, we investigate the finite sample performance of the empirical and approximate methods based on $B=10,000$ simulations, varying the sample size $n = 100, 1000, 10000$ and the proportion of 0 values $p_0 = 0, 0.05, 0.25, 0.5, 0.75$ .", "We assume an exponential relative risk $RR(X;\\beta ) = \\exp (\\beta X)$ , where $\\beta \\sim Normal(\\beta _0, \\sigma ^2)$ with $\\beta _0 = \\log (1.27)$ and $\\sigma ^2 = 70000 \\cdot 0.0443^2 / (7 n) $ .", "The distribution is from the estimated relative risk in our illustrative example in Section , which had sample size 72,667.", "Additionally, we vary the true exposure $f(x)$ to be a truncated log normal, truncated normal, and truncated Weibull with best fit parameters also taken from our illustrative example, all truncated at $M=12$ with probability $1-p_0$ and 0 otherwise.", "For each simulation, we generate the true distribution of the exposure $f(x)$ from a mixture distribution, where we first generate $A_i, \\ldots , A_n \\sim \\text{Bernoulli}(p_0)$ .", "Then, we generate $X_1, \\ldots , X_n \\sim f(x)$ truncated at $M=12$ for $A_i > 0$ and 0 for $A_i = 0$ .", "We also simulate $\\beta \\sim Normal(\\beta _0, \\sigma ^2(\\beta _0) )$ , and use an exponential relative risk function $ RR({X}) = \\exp ( \\beta {X})$ , where $\\beta = \\log (1.27)$ from our illustrative example.", "For each simulation, we estimate the PAF and the corresponding 95% confidence interval using the empirical and approximate method, where we use the sample average and variance of the $X$ when required.", "We report the coverage and average relative bias percentage in Table over the $B$ simulations for $f(x)$ distributed truncated Normal, truncated Lognormal, and truncated Weibull, respectively.", "[!htbp] We vary the true underlying distribution $f(x)$ , the proportion of 0 values $p_0$ , and sample size $n$ .", "$f(x)$ is truncated at 12 for all distributions.", "The parameters of each parametric distribution are taken from the illustrative example.", "We report the average PAF estimate, average relative bias of the PAF estimates, average PAF standard error, standard deviation of the PAF estimates, and 95% confidence interval coverage probabilities over 10,000 simulations for each scenario.", "Table: NO_CAPTIONBoth the approximate and empirical method perform very well, achieving minimal relative bias and excellent coverage rates.", "In the simulations we considered, the empirical method maintained less than 5% relative bias with 95% confidence intervals achieving 93% coverage rates when $N \\ge 1000$ in all scenarios and less than 1% relative bias with 94% coverage rates in all cases when $ N \\ge 10000$ .", "For the empirical method estimates, the relative bias converges to 0 and the 95% coverage probability converges to 95% as sample size increases.", "this result is expected given our result from Theorem REF .", "The approximate achieves comparable coverage rate performance to the empirical method, also achieving less bias and converging to a 95% coverage rate as sample size increases.", "For both methods, the average standard error of the PAF estimate and standard deviation over the $B$ simulations are very similar across all scenarios considered." ], [ "Discussion", "The PIF is a critical epidemiological indicator and a primary input for disease prioritization, resource allocation and policy development.", "Currently, researchers rely on distributional assumptions to estimate the PIF using cross-sectional data and meta-analytic risk estimates from the literature [[10]].", "However, estimation methods are ill-prepared to deal with these data without making strong distributional assumptions.", "Moreover, no methods are available to produce PIF estimates with individual-level data using cross-sectional surveys.", "Here we characterize the implications of distributional assumptions in the estimation of the PIF and present two nonparametric approaches that overcome the observed limitations of parametric methods.", "The standard approach for estimating the PIF is widely used to quantify the burden of disease in different countries [26], [10]; however, we have found at least two reasons to be cautious when implementing it.", "First, different distributions of exposure can lead to quite different PIF and PAF estimates, including a possible undefined result when the distribution is heavy-tailed.", "The problems of arbitrary selection of a parametric distribution for statistical inference have been widely discussed in the literature [39].", "Second, truncation can also bias significantly the PIF.", "Only in the rare case where the underlying distribution matches the selected distribution will the standard method produce an unbiased result.", "However, correct specification of the distribution cannot be verified.", "[17] compared three distributions (Weibull, Gamma and Lognormal) to fit alcohol consumption data from various countries, and concluded that Weibull and Gamma were good empirical fits for alcohol consumption.", "[17] recommended using Gamma due to its flexibility.", "Yet, it is unclear if the distribution of alcohol consumption is similar across countries, and there is no reason to believe that a Gamma distribution would be an adequate representation in other settings.", "This subjective decision-making process and the aforementioned limitations can be prevented using the nonparametric approach to estimate the PIF.", "The branches of robust and nonparametric statistics deal with distribution selection problems.", "Nonparametric methods to estimate attributable fractions in cohort and case-control studies have been proposed [38], [13], [6], [29], [32].", "In particular, [28], [32] have proposed doubly robust nonparametric estimates for the PIF and PAF; however, these methods are designed for longitudinal data, where the exposure and outcome are available from the same population.", "When longitudinal data is available, using these methods will produce the best estimates.", "However, longitudinal data is frequently unavailable, particularly in low and middle income countries; thus, methods capable of handling exposure survey data and meta-analytical risks are needed.", "Our methods fill this gap, allowing researchers to combine different data sources avoiding strong distributional assumptions for the exposure.", "Moreover, to our knowledge, no alternatives have been developed for cross-sectional exposure data, which is the basis of most burden of disease estimates.", "We developed two nonparametric alternatives to estimate the PIF based on cross-sectional data: a consistent empirical method for individual-level data and an approximate method for aggregated data.", "Both methods require less assumptions and perform better than the standard approach when the underlying distribution is unknown.", "Despite not requiring a specific distribution, our method is bounded by known epidemiological assumptions such as: transportable and unbiased relative risks, and no reverse causation [27], [2], [40], [4].", "In addition, mathematical hypotheses over the relative risk function and its parameter estimator $\\hat{{\\beta }}$ are needed.", "For consistency, the empirical method needs either a finite population with a Fisher consistent estimator; or a convex, concave or Lipschitz relative risk function with an asymptotically consistent estimator.", "The approximate method additionally requires that the relative risk function is twice differentiable on ${\\beta }$ and that the exposure distribution is subgaussian.", "These technical conditions are fulfilled by most relative risk functions and exposure distributions encountered in practice.", "While the empirical method builds from the work of [32], the approximate is based on the delta method which has been used to estimate confidence intervals for the PAF [37].", "Both methods require less assumptions than the standard approach and fare significantly better even when only the mean and variance are available, as demonstrated by our simulation studies.", "We have implemented these methods in the pifpaf package for the statistical software R. The empirical framework can be easily extended to accommodate other methods of statistical estimation.", "For example, to account for outliers, robust mean estimators can be used to estimate PIF instead of the proposed $\\hat{\\mu }_n^{\\textrm {obs}}$ [15].", "Nonparametric Bayesian inference is also possible, providing a compromise between an epidemiologist's conception of the data and the sample [19].", "Additional work is needed to advance these methods." ], [ "Acknowledgements", "This work was supported by a grant from Bloomberg Philanthropies and the National Institute of Public Health of Mexico.", "TBG received support from Harvard University through the Lown Scholar's program.", "DS was supported by a grant from the National Institutes of Health DP1ES025459." ], [ "Proof of Theorem 1", "We first prove the consistency.", "We have, $\\hat{\\mu }_n^{\\textrm {obs}}(\\hat{{\\beta }})-\\mathbb {E}(RR(X;{\\beta })) = [\\hat{\\mu }_n^{\\textrm {obs}}(\\hat{{\\beta }})-\\hat{\\mu }_n^{\\textrm {obs}}({{\\beta }})] + [\\hat{\\mu }_n^{\\textrm {obs}}({{\\beta }})-\\mathbb {E}(RR(X;{\\beta }))]$ The first term converges to zero in probability due to the consistency of $\\hat{{\\beta }}$ and continuous mapping theorem, and the second term converges to zero in probability due to the law of large numbers.", "So $\\hat{\\mu }_n^{\\textrm {obs}}(\\hat{{\\beta }})\\overset{p}{\\longrightarrow }\\mathbb {E}(RR(X;{\\beta }))$ .", "Similarly, $\\hat{\\mu }_n^{\\textrm {cft}}(\\hat{{\\beta }})\\overset{p}{\\longrightarrow }\\mathbb {E}(RR(g(X);{\\beta }))$ .", "The consistency of $\\widehat{\\textrm {PAF}}$ and $\\widehat{\\textrm {PIF}}$ follows directly by the continuous mapping theorem.", "We then prove the asymptotic normality.", "By the Central Limit Theorem, conditional on $\\hat{\\beta }$ $\\sqrt{n}\\left[\\left(\\begin{array}{c}\\hat{\\mu }_n^{\\textrm {obs}}(\\hat{{\\beta }}) \\\\\\hat{\\mu }_n^{\\textrm {cft}}(\\hat{{\\beta }})\\end{array}\\right)-\\left(\\begin{array}{c}\\mathbb {E}(RR(X;\\hat{\\beta })) \\\\\\mathbb {E}(RR(g(X);\\hat{\\beta }))\\end{array}\\right)\\right]\\overset{\\mathcal {D}}{\\longrightarrow }N({0},\\Sigma _1(\\hat{\\beta })),$ where the covariance matrix ${\\Sigma }_1(\\cdot )$ is equal to ${\\Sigma }_1(\\beta ) = \\left(\\begin{array}{cc}Var(RR(X;{\\beta })) & Cov(RR(X;{\\beta }),RR(g(X);{\\beta })) \\\\Cov(RR(X;{\\beta }),RR(g(X);{\\beta })) & Var(RR(g(X);{\\beta }))\\end{array}\\right)$ By the consistency of $\\hat{\\beta }$ , we have ${\\Sigma }_1(\\hat{\\beta })\\overset{p}{\\longrightarrow }{\\Sigma }_1({\\beta })$ .", "Then we have, $ \\sqrt{n}\\left[\\left(\\begin{array}{c}\\hat{\\mu }_n^{\\textrm {obs}}(\\hat{{\\beta }}) \\\\\\hat{\\mu }_n^{\\textrm {cft}}(\\hat{{\\beta }})\\end{array}\\right)-\\left(\\begin{array}{c}\\mathbb {E}(RR(X;\\hat{\\beta })) \\\\\\mathbb {E}(RR(g(X);\\hat{\\beta }))\\end{array}\\right)\\right]\\overset{\\mathcal {D}}{\\longrightarrow }N({0},\\Sigma _1(\\beta ))$ By the Delta method, $ \\sqrt{m}\\left[\\left(\\begin{array}{c}\\mathbb {E}(RR(X;\\hat{\\beta })) \\\\\\mathbb {E}(RR(g(X);\\hat{\\beta }))\\end{array}\\right)-\\left(\\begin{array}{c}\\mathbb {E}(RR(X;{\\beta })) \\\\\\mathbb {E}(RR(g(X);{\\beta }))\\end{array}\\right)\\right]\\overset{\\mathcal {D}}{\\longrightarrow }N({0},\\Sigma _2),$ where the covariance matrix ${\\Sigma }_2$ is equal to $\\Sigma _2 =\\left(\\begin{array}{c}{\\mathbb {E}(\\nabla _{\\beta }RR(X;{\\beta }))} \\\\{\\mathbb {E}(\\nabla _{\\beta }RR(g(X);{\\beta }))}\\end{array}\\right)\\Sigma _{\\beta }\\left(\\begin{array}{c}{\\mathbb {E}(\\nabla _{\\beta }RR(X;{\\beta }))} \\\\{\\mathbb {E}(\\nabla _{\\beta }RR(g(X);{\\beta }))}\\end{array}\\right)^T.$ Notice that $&&\\left(\\begin{array}{c}\\hat{\\mu }_n^{\\textrm {obs}}(\\hat{{\\beta }}) \\\\\\hat{\\mu }_n^{\\textrm {cft}}(\\hat{{\\beta }})\\end{array}\\right)-\\left(\\begin{array}{c}\\mathbb {E}(RR(X;{\\beta })) \\\\\\mathbb {E}(RR(g(X);{\\beta }))\\end{array}\\right) \\\\&=&\\underbrace{\\left(\\begin{array}{c}\\hat{\\mu }_n^{\\textrm {obs}}(\\hat{{\\beta }}) \\\\\\hat{\\mu }_n^{\\textrm {cft}}(\\hat{{\\beta }})\\end{array}\\right)-\\left(\\begin{array}{c}\\mathbb {E}(RR(X;\\hat{\\beta })) \\\\\\mathbb {E}(RR(g(X);\\hat{\\beta }))\\end{array}\\right)}_{\\textrm {first term}}+\\underbrace{\\left(\\begin{array}{c}\\mathbb {E}(RR(X;\\hat{\\beta })) \\\\\\mathbb {E}(RR(g(X);\\hat{\\beta }))\\end{array}\\right)-\\left(\\begin{array}{c}\\mathbb {E}(RR(X;{\\beta })) \\\\\\mathbb {E}(RR(g(X);{\\beta }))\\end{array}\\right)}_{\\textrm {second term}}.$ The two terms are asymptotically unrelated since, by a double expectation argument, $&&\\mathbb {E}\\left[\\left\\lbrace \\left(\\begin{array}{c}\\hat{\\mu }_n^{\\textrm {obs}}(\\hat{{\\beta }}) \\\\\\hat{\\mu }_n^{\\textrm {cft}}(\\hat{{\\beta }})\\end{array}\\right)-\\left(\\begin{array}{c}\\mathbb {E}(RR(X;\\hat{\\beta })) \\\\\\mathbb {E}(RR(g(X);\\hat{\\beta }))\\end{array}\\right)\\right\\rbrace ^T\\left\\lbrace \\left(\\begin{array}{c}\\mathbb {E}(RR(X;\\hat{\\beta })) \\\\\\mathbb {E}(RR(g(X);\\hat{\\beta }))\\end{array}\\right)-\\left(\\begin{array}{c}\\mathbb {E}(RR(X;{\\beta })) \\\\\\mathbb {E}(RR(g(X);{\\beta }))\\end{array}\\right)\\right\\rbrace \\right] \\\\&=&\\mathbb {E}\\left[\\mathbb {E}\\left\\lbrace \\left(\\begin{array}{c}\\hat{\\mu }_n^{\\textrm {obs}}(\\hat{{\\beta }}) \\\\\\hat{\\mu }_n^{\\textrm {cft}}(\\hat{{\\beta }})\\end{array}\\right)-\\left(\\begin{array}{c}\\mathbb {E}(RR(X;\\hat{\\beta })) \\\\\\mathbb {E}(RR(g(X);\\hat{\\beta }))\\end{array}\\right)\\Big |\\hat{\\beta }\\right\\rbrace ^T\\left\\lbrace \\left(\\begin{array}{c}\\mathbb {E}(RR(X;\\hat{\\beta })) \\\\\\mathbb {E}(RR(g(X);\\hat{\\beta }))\\end{array}\\right)-\\left(\\begin{array}{c}\\mathbb {E}(RR(X;{\\beta })) \\\\\\mathbb {E}(RR(g(X);{\\beta }))\\end{array}\\right)\\right\\rbrace \\right]\\\\&\\overset{p}{\\longrightarrow }&0.$ Then we have, $ \\sqrt{n}\\left[\\left(\\begin{array}{c}\\hat{\\mu }_n^{\\textrm {obs}}(\\hat{{\\beta }}) \\\\\\hat{\\mu }_n^{\\textrm {cft}}(\\hat{{\\beta }})\\end{array}\\right)-\\left(\\begin{array}{c}\\mathbb {E}(RR(X;{\\beta })) \\\\\\mathbb {E}(RR(g(X);{\\beta }))\\end{array}\\right)\\right]\\overset{\\mathcal {D}}{\\longrightarrow }N({0},\\Sigma ),$ where $\\Sigma =\\Sigma _1(\\beta )+\\lambda \\Sigma _2$ .", "The asymptotic normality is derived by the delta method.", "That is, $\\sqrt{n}(\\widehat{\\textrm {PAF}}-\\textrm {PAF}) \\overset{D}{\\longrightarrow }N(0,\\sigma _{PAF}^2),$ where $\\sigma _{PAF}^2=\\Sigma _{11}/\\mathbb {E}(RR(X;{\\beta }))^4$ and $\\Sigma _{11}$ is the first diagonal entry of $\\Sigma $ , $\\Sigma _{11} = Var(RR(X;{\\beta }))+\\lambda \\mathbb {E}(\\nabla _{\\beta }RR(X;{\\beta }))\\Sigma _{\\beta }\\mathbb {E}(\\nabla _{\\beta }RR(X;{\\beta }))^T,$ and $\\sqrt{n}(\\widehat{\\textrm {PIF}}-\\textrm {PIF}) \\overset{D}{\\longrightarrow }N(0,\\sigma _{PIF}^2),$ where the asymptotic variance $\\sigma _{PIF}^2 =\\left(\\begin{array}{cc}\\frac{\\mathbb {E}(RR(g(X);{\\beta }))}{\\mathbb {E}(RR(X;{\\beta }))^2} & -\\frac{1}{\\mathbb {E}(RR(X;{\\beta }))}\\end{array}\\right)\\Sigma \\left(\\begin{array}{cc}\\frac{\\mathbb {E}(RR(g(X);{\\beta }))}{\\mathbb {E}(RR(X;{\\beta }))^2} & -\\frac{1}{\\mathbb {E}(RR(X;{\\beta }))}\\end{array}\\right)^T.$" ], [ "Approximate Method Estimator", "The first and second moments of ${X}$ are $\\mu _{{X}} = \\mathbb {E}({X}) \\quad \\textrm { and } \\quad {\\Sigma }_{{X}} = Var({X}),$ and their estimates are $\\widehat{\\mu }_{{X}} = \\frac{1}{n}\\sum _{i=1}^n{X}_i = \\bar{{X}} \\quad \\textrm { and } \\quad \\widehat{\\Sigma }_{{X}} = \\frac{1}{n}\\sum _{i=1}^n({X}_i-\\widehat{\\mu }_{{X}})({X}_i-\\widehat{\\mu }_{{X}})^T.$ Consider a general function $h({X})$ , which is twice differentiable.", "Let ${D}h({X}) = \\frac{\\partial h({X})}{\\partial {X}} \\quad \\textrm { and } \\quad {H}h({X}) = \\frac{\\partial ^2 h({X})}{\\partial {X} \\partial {X}^T}.$ The second-order Taylor polynomial for $h({X})$ is $h({X}) & \\approx & h(\\widehat{\\mu }_{{X}}) + {D}h(\\widehat{\\mu }_{{X}})({X}-\\widehat{\\mu }_{{X}})+\\frac{1}{2}({X}-\\widehat{\\mu }_{{X}})^T{H}h(\\widehat{\\mu }_{{X}})({X}-\\widehat{\\mu }_{{X}}) \\\\& = & h(\\widehat{\\mu }_{{X}}) + {D}h(\\widehat{\\mu }_{{X}})({X}-\\widehat{\\mu }_{{X}})+\\frac{1}{2}tr\\left[({X}-\\widehat{\\mu }_{{X}})({X}-\\widehat{\\mu }_{{X}})^T{H}h(\\widehat{\\mu }_{{X}})\\right].$ So applying the approximation to all subjects ${X}_1$ , ${X}_2$ , ..., ${X}_n$ , we have, $\\frac{1}{n}\\sum _{i=1}^nh({X}_i) \\approx h(\\widehat{\\mu }_{{X}}) + \\frac{1}{2}tr\\left[\\widehat{\\Sigma }_{{X}}{H}h(\\widehat{\\mu }_{{X}})\\right].$ Using this, we can approximate $\\hat{\\mu }_n^{\\textrm {obs}}(\\hat{{\\beta }})$ and $\\hat{\\mu }_n^{\\textrm {cft}}(\\hat{{\\beta }})$ as $\\hat{\\mu }^{\\textrm {obs}}(\\hat{{\\beta }}) \\approx RR(\\bar{{X}};\\hat{{\\beta }})+\\dfrac{1}{2} \\sum \\limits _{i,j} \\hat{\\sigma }_{i,j} \\frac{\\partial ^2 RR\\left({X};\\hat{{\\beta }}\\right)}{\\partial X_i \\partial X_j}\\big |_{{X} = \\bar{{X}}},$ $\\hat{\\mu }^{\\textrm {cft}}(\\hat{{\\beta }}) \\approx RR\\big (g(\\bar{{X}} ),\\hat{{\\beta }} \\big ) + \\frac{1}{2}\\sum _{i,j} \\hat{\\sigma }_{i,j}\\frac{\\partial ^2 RR\\left(g({X}),\\hat{{\\beta }} \\right)}{\\partial X_i \\partial X_j}\\big |_{{X} = \\bar{{X}}}.$ The approximate method PIF and PAF estimators follow directly by substituting these quantities into Eq.", "(REF )." ], [ "Variance of the Approximate Method Estimator", "Note that the PAF estimator in Eq.", "(REF ) can be expressed as a function of three parameters: $ \\widehat{ \\text{PAF}} = h( Z) = h(\\bar{X}, \\widehat{\\text{Var}}(X), \\hat{\\beta }) $ If $Z$ is a consistent estimator for $\\zeta $ , then we can use multivariate delta method to obtain asymptotic normality $\\sqrt{n} (h( Z) - h( \\zeta )) \\rightarrow \\mathcal {N}(0, \\nabla h( Z)^T \\cdot \\Sigma \\cdot \\nabla h(Z))$ where $ \\nabla h({Z})^T =\\begin{pmatrix}\\frac{ \\hat{\\beta }}{\\exp (\\beta \\bar{X}) \\left(1 + \\frac{1}{2} \\hat{\\beta }^2 \\sqrt{\\text{Var}(X) } \\right) } &\\frac{ \\hat{\\beta }^2 \\exp (\\hat{\\beta }\\bar{X}) \\left( 1 + \\frac{1}{2} \\hat{\\beta }^2 \\sqrt{\\text{Var}(X) } \\right)^2}{ 4 \\sqrt{\\text{Var}(X)} } &\\frac{ \\bar{X} + \\hat{\\beta }(1 + \\frac{1}{2} \\hat{\\beta }) \\sqrt{\\text{Var}(X)} }{ \\exp ( \\hat{\\beta }\\bar{X}) \\left( 1 + \\frac{1}{2} \\hat{\\beta }^2 \\sqrt{\\text{Var}(X)} \\right)^2 }\\end{pmatrix},$ and $\\begin{aligned}\\Sigma &=\\begin{pmatrix}\\text{Var}(\\bar{X}) & \\text{Cov}(\\bar{X}, \\widehat{\\text{Var}}(X)) & \\text{Cov}(\\bar{X}, \\hat{\\beta }) \\\\\\text{Cov}(\\bar{X}, \\widehat{\\text{Var}}(X)) & \\text{Var}( \\widehat{\\text{Var}}(X)) & \\text{Cov}(\\widehat{\\text{Var}}(X), \\hat{\\beta }) \\\\\\text{Cov}(\\bar{X}, \\hat{\\beta }) & \\text{Cov}(\\widehat{\\text{Var}}(X), \\hat{\\beta }) & \\text{Var}( \\hat{\\beta })\\end{pmatrix} \\\\&\\approx \\begin{pmatrix}\\frac{\\widehat{\\text{Var}}(X)}{n} & 0 & 0 \\\\0 & \\frac{3\\widehat{\\text{Var}}(X)^2}{n} - \\frac{(n-3)\\widehat{\\text{Var}}(X)^{3/2}}{n(n-1)} & 0 \\\\0 & 0 & \\text{Var}( \\hat{\\beta })\\end{pmatrix}.\\end{aligned}$ The covariance terms $\\text{Cov}(\\bar{X}, \\hat{\\beta })$ and $\\text{Cov}(\\widehat{\\text{Var}}(X), \\hat{\\beta })$ are 0 since they are taken from independent studies.", "If $X$ is normally distributed, $\\text{Cov}(\\bar{X}, \\widehat{\\text{Var}}(X)) = 0$ and $\\text{Var}( \\widehat{\\text{Var}(X)})= \\frac{ 3 \\text{Var}(X)^2 }{n} - \\frac{ \\text{Var}(X)^{3/2} (n-3)}{n(n-1)}$ .", "So we approximate $\\text{Cov}(\\bar{X}, \\widehat{\\text{Var}}(X)) \\approx 0$ and $\\text{Var}( \\widehat{\\text{Var}}(X))\\approx \\frac{ 3 \\widehat{Var}(X)^2 }{n} - \\frac{ \\widehat{Var}(X)^{3/2} (n-3)}{n(n-1)}$ .", "$\\widehat{\\text{Var}}(\\text{PAF}) = \\nabla h( Z)^T \\cdot \\Sigma \\cdot \\nabla h(Z)$ is used as the estimate of variance of PAF for the approximate method.", "Similarly, we can derive the variance of the PIF for the approximate method." ] ]
2207.03597
[ [ "An Embedding-Dynamic Approach to Self-supervised Learning" ], [ "Abstract A number of recent self-supervised learning methods have shown impressive performance on image classification and other tasks.", "A somewhat bewildering variety of techniques have been used, not always with a clear understanding of the reasons for their benefits, especially when used in combination.", "Here we treat the embeddings of images as point particles and consider model optimization as a dynamic process on this system of particles.", "Our dynamic model combines an attractive force for similar images, a locally dispersive force to avoid local collapse, and a global dispersive force to achieve a globally-homogeneous distribution of particles.", "The dynamic perspective highlights the advantage of using a delayed-parameter image embedding (a la BYOL) together with multiple views of the same image.", "It also uses a purely-dynamic local dispersive force (Brownian motion) that shows improved performance over other methods and does not require knowledge of other particle coordinates.", "The method is called MSBReg which stands for (i) a Multiview centroid loss, which applies an attractive force to pull different image view embeddings toward their centroid, (ii) a Singular value loss, which pushes the particle system toward spatially homogeneous density, (iii) a Brownian diffusive loss.", "We evaluate downstream classification performance of MSBReg on ImageNet as well as transfer learning tasks including fine-grained classification, multi-class object classification, object detection, and instance segmentation.", "In addition, we also show that applying our regularization term to other methods further improves their performance and stabilize the training by preventing a mode collapse." ], [ "Introduction", "A good representation should include useful features (those that facilitate downstream prediction tasks) while ignoring “nuisance\" features [3].", "Among the best known self-supervised methods, contrastive methods combine an attractive term between similar images (typically different perturbations of the same image) with an explicit repulsive term between distinct pairs.", "Recently, BYOL [16] utilized siamese neural networks (referred to as the online and target) with lagged (moving averaged) parameters in the target network, and simply minimized distance between online and target network embeddings.", "While there was no explicit repulsive term in BYOL, it was later shown to be highly dependent on the use of BatchNorm layers.", "The activation normalization in BatchNorm and other methods can be viewed as a global, dimension-wise dispersion of the set of embeddings, a desirable feature of a representation.", "However, other normalization methods such as LayerNorm were shown to be much less effective in BYOL suggesting the story is more complicated than normalization and global dispersion [1], [28], [11], [30].", "Inspection of the gradients in BatchNorm reveals that they have a strong stochastic component (beyond the global “normalizing\" component) that depends on differences between image activations and their batch centroid (i.e.", "on whatever other images happen to be in the same minibatch).", "From our perspective, these forces provide a local (stochastic) dispersive force between embeddings.", "Thus Batchnorm implements two of the desirable features of a good representation (local and global dispersion of embeddings), but in a suboptimal way.", "Here we define separate loss terms for local and global dispersion and apply them to the embedding layer only (as opposed to other intermediate network layers).", "By moving dispersion to loss layers, we allow network normalization layers (which ideally impact network training and stability but not losses) to be independently designed.", "We can also separately define and optimize the local and global dispersion losses.", "If we assume that the optimization method used to train the network is either stateless or sufficiently “fast” (e.g.", "the optimizer uses momentum=$0.9$ for an effective time constant of 10 steps), then the embeddings are part of a second-order dynamical system.", "The embeddings are defined by the parameters of the two networks (online and target), together with corresponding input images.", "The moving parameter average implemented on the target network, together with a fast optimizer which functions as an integrator of loss gradients, defines a second-order dynamical system.", "We exploit the dynamics of this system in two ways: by using “fast-slow” optimization for attractive and dispersive forces, and by showing that stochastic energy injected into the system should “stretch” attractive links with equal potential energy on average - so weaker attractive links will be stretched further.", "Multiview contrastive training, where more than two augmentations are compared, works very effectively with a lagged-arm network.", "Dispersive forces act in the network to situate embeddings with globally uniform density.", "While loss-gradients act as forces applied to the online network, online embeddings experience a strong viscous “drag” from their corresponding target network embeddings which they are attracted to.", "So embeddings move globally at the time constant of the lagged network, which is typically thousands to tens of thousands of time steps.", "On the other hand, embeddings within the same group, i.e.", "embeddings of views of the same image, experience no “drag” relative to their centroid.", "They collapse and are maintained close together at the time constant of the online network.", "Given a lagged-arm, siamese architecture inspired by BYOL, we explore Multiview, Singular value, Brownian Diffusive regularizations.", "These three loss terms address respectively, (i) fast-slow attractive/dispersive optimization, (ii) global, uniform, dispersion of embeddings (iii) local dispersive force.", "We evaluate our approach on visual benchmarks including ImageNet-100 [8], STL-10 [7], and ImageNet [8].", "We show that our model significantly outperforms prior work in the image classification task.", "Also, we show that joint local/global dispersive forces lead to a larger dissimilarity of negative pairs compared to other approaches.", "We summarize our contributions as follows: We analyze and optimize self-supervised learning using a dynamic model of embeddings.", "To optimize the placement of embeddings, we propose a MSBReg loss that consists of 1) Multiview centroid loss and 3) Singular value loss 3) Brownian diffusive loss and MSBReg outperforms other baselines by a significant margin.", "Figure: The architecture overview of MSBReg .", "This architecture is inspired by BYOL's architecture.", "Each model takes K augmented views as its inputs.", "MSBReg minimizes (1) multiview centroid loss, (2) singular value loss, (3) Brownian diffusive loss and .", "The first makes the online network predict the target network's representation of the centroid of K views.", "The second loss favors a spatially uniform (spherical) distribution.", "The last one induces noise into embedding space and makes the embeddings repulse each other on average, preventing the model from converging to collapsed solutions.Figure: (A) Multiview centroid loss applies attractive force to embeddings generated by online network.", "The solid shapes are embeddings generated by online network and the shapes with dashed line are the geometric centroids of the embeddings generated by target network.", "We can model such system as spring-mass system.", "(B) Brownian diffusive loss induces the random walk of embeddings, preventing the model from admitting collapsed solutions." ], [ "Related Work", "Self Supervised Learning.", "Recent works suggest that a state-of-the-art image representation can be successfully and efficiently learned by a discriminative approach with self supervised learning.", "These methods originally relied on a contrastive loss that encourages representation of different views of the same image (i.e.", "positive pairs) to be close in the embedding space, and representations of views from different images (i.e.", "negative pairs) to be pushed away from each other.", "Contrastive methods often require a careful treatment of negative pairs, which need a large memory overhead as they need to be sampled from a memory bank [35], [17] or from the current mini-batch of data [5].", "The contrastive approach is also unsatisfying from a modeling perspective - the fact that images are distinct does not imply that they are different - but the contrastive approad applies large repulsive gradients to distinct, close image pairs.", "Motivated by a desire to overcome the difficulties of contrastive approaches, recent works [16], [6] use two neural networks (referred to as online and target networks) are trained to minimize the distance between their embeddings of the same image.", "Some works use a moving average on parameters of one arm [16] while others use the same parameters [17], [6].", "These methods have been effective but their success is somewhat mysterious since there is no obvious force to prevent collapse of embeddings since forces are only attractive.", "It turn out that as batch normalization [20] was an important element of the success of BYOL.", "In contrast we employ explicit local and global dispersive losses in addition to attractive forces on groups of multiple image views.", "Regularizing Consistency of Singular Value.", "Whitening is the most similar approach to regularizing consistency of singular values.", "Recently, whitening output embeddings has received attention as a method to avoid a mode collapse.", "Whitening removes redundant information in input and prevents all dimensions of embeddings from being encoded with the same representation.", "Whitening features induces the contrastive effect of the embeddings by scattering them.", "This [9] performs a explicit whitening via Cholesky decomposition.", "Performing Cholesky decomposition, which requires the computation of inverse of the sample covariance matrix is not only computationally expensive but unstable.", "This method  [38] computes cross-correlation matrix and makes it close to identity matrix in Frobenius norm.", "This paper [2] suggested a similar approach.", "Unlike the methods mentioned above, which compute the covariance matrix with the only positive pairs, singular value loss term in MSBReg computes the covariance matrix along the batch dimension (with the negative pairs) and helps global dispersion of embeddings by making the embeddings be distributed isotropically.", "To emphasize this aspect, we coinage our loss as singular value loss, which regularizes the consistency of singular values of the empirical covariance matrix.", "Multiview Loss.", "In supervised learning settings, batch repetition method [19] is proposed to improve the image classification performance as well as training efficiency.", "Recent self-supervised learning based on contrastive learning usually uses two views of the same image as positive pairs.", "And it is trained to minimize the distance or maximize similarity of embeddings of those two views.", "Recent work [4] suggested multi-crop method which maximizes the similarity between views more than 2.", "To reduce computational cost, it generates 2 views with high resolution ($224 \\times 224$ for ImageNet) and several other views with low resolution ($96 \\times 96$ for ImageNet).", "This method [9] generates multiple positive views to perform whitening among them.", "In contrast, our method uses multiple views of the positive views of the same images to compute the centroid and distance between embeddings and the centroid.", "We discuss the relationship between batch repetition method and multiview centroid loss in the appendix.", "Uniformity of Embeddings.", "Our work is aligned with [32] in that our method also seeks to distribute the embeddings as uniformly as possible on the embedding space.", "That work claims that the contrastive learning is to make embeddings be distributed uniformly on the hypersphere.", "Similarly, our work also tries to distribute the embeddings uniformly on the embedding space.", "That method reformulates contrastive loss as the sum of alignment loss and uniformity loss.", "The first term, alignment loss, aligns positive views.", "The second term, uniformity loss, makes the embedding distribution $\\textit {uniform}$ on the surface of unit sphere.", "The uniformity loss is defined by Gaussian kernel.", "The difference to our method is that 1) [32] is based on constrastive method and 2) [32] hypothesizes the embedding space is hypersphere.", "However, our method seeks more general embedding space with the dynamical system modeling.", "The advantage of modeling dynamical system is that we can study the motion of embeddings and control them with this model.", "We further study the effect of our loss terms to uniformity and alignment trade-off in depth in the appendix." ], [ "BYOL Architecture", "We follow the recent BYOL architecture [16] that learns a joint embedding of an image $x \\in \\mathcal {X}$ with two networks – consists of two neural networks referred to as the online (or fast learner) and target (or slow learner) network.", "For completeness, we summarize some of the key details of the BYOL architecture.", "As shown in Figure REF , the online network is trained to predict the target network's representation of the augmented view of the same image.", "This online network is parameterized by a set of learnable weights $\\theta $ and consists of three consecutive components: a backbone $f_\\theta $ , a projection head $g_\\theta $ , and a prediction head $h_\\theta $ .", "The target network is parameterized by a set of weights $\\xi $ and consists of two components: a backbone $f_\\xi $ and a projection head $g_\\xi $ .", "The parameter $\\xi $ is updated by the bias-corrected exponentially weighted moving average of the online network's parameter $\\theta $ at each training step, i.e.", "$\\xi _{t+1} = \\tau _t \\xi _t + (1-\\tau _t) \\theta _t$ where $\\tau _t \\in [0, 1]$ is a target decay rate.", "Two augmented views $v\\triangleq t(x)$ and $v^{\\prime }\\triangleq t^{\\prime }(x)$ are generated by applying image augmentations $t\\sim \\mathcal {T}$ and $t^{\\prime }\\sim \\mathcal {T^{\\prime }}$ given two distributions of image augmentations $\\mathcal {T}$ and $\\mathcal {T}^{\\prime }$ .", "The online network outputs $z\\triangleq g_\\theta (f_\\theta (v))$ from the first augmented view $v$ , while the target network produces $z^{\\prime }\\triangleq g_\\xi (f_\\xi (v^{\\prime }))$ from the second augmented view $v^{\\prime }$ .", "A prediction from the online network $p\\triangleq h_\\theta (z)$ is then $l_2$ -normalized to compute the cosine similarity loss $\\mathcal {L}_{\\textnormal {byol}}$ by measuring mean squared error between the normalized prediction $p$ and the normalized target predictions $z^{\\prime }$ : $\\mathcal {L}_{\\text{byol}} (\\theta , \\xi ; \\mathcal {X}) := \\left\\Vert \\hat{p}-\\hat{z}^{\\prime }\\right\\Vert _2^{2} = 2-2\\frac{\\langle p, z^{\\prime } \\rangle }{||p||_{2} \\cdot ||z^{\\prime }||_{2}}$ where $\\hat{p}={p}/{\\left\\Vert p\\right\\Vert _2}$ and $\\hat{z}^{\\prime }={z^{\\prime }}/{\\left\\Vert z^{\\prime }\\right\\Vert _2}$ .", "Note that the loss $\\mathcal {L}_{\\text{byol}}$ is optimized with respect to $\\theta $ only, but not $\\xi $ .", "The gradient does not back-propagate through the target network as depicted by stop-gradient in Figure REF .", "After training, both the prediction head $h_\\theta $ and the projection head $g_\\theta $ are discarded and the representations $z$ of the online network are used for downstream tasks." ], [ "MSBReg ", "Built upon BYOL architecture, we use the following loss $\\mathcal {L}(\\theta , \\xi ; \\mathcal {X})$ (instead of using $\\mathcal {L}_{\\text{byol}}$ ) that consists of the following three loss terms: (i) multiview centroid loss $\\mathcal {L}_{c}$ , (ii) singular value loss $\\mathcal {L}_{s}$ , and (iii) Brownian diffusion loss $\\mathcal {L}_{b}$ .", "The overall loss is defined as follows: $\\mathcal {L}(\\theta , \\xi ; \\mathcal {X}) &= \\mathcal {L}_{c}(\\theta , \\xi ; \\mathcal {X})+\\lambda _{s} \\mathcal {L}_{s}(\\theta ; \\mathcal {X})+\\lambda _{b} \\mathcal {L}_{b}(\\theta ; \\mathcal {X}) $ Multiview Centroid Loss.", "As opposed to BYOL, we train the online network to predict the target network's centroid representation of differently augmented multi-views of the same image.", "Given an image $x\\in \\mathcal {X}$ , we generate $K$ differently augmented views (i.e.", "multi-view): $v_j \\triangleq t_j(x)$ and $v_l \\triangleq t_l(x)$ for $j,~l\\in \\lbrace 1, 2, \\dots , K\\rbrace $ by applying stochastic image augmentations $t_j, t_l \\sim \\mathcal {T}$ .", "Given $K$ outputs from the target network, $z^{\\prime }_l=g_\\xi (f_\\xi (v^{\\prime }_l))$ , we use the geometric center of these $K$ outputs as the centroid representation, i.e.", "$\\frac{1}{K}\\sum _{l=1}^K \\hat{z}^{\\prime }_{l}$ where $\\hat{z}^{\\prime }_l={z^{\\prime }_l}/{\\left\\Vert z^{\\prime }_l\\right\\Vert _2}$ .", "Lastly, we compute the sum of $L_2$ loss between the target network's centroid representation of embeddings of $K$ different views $\\lbrace \\hat{z}^{\\prime }_{l}\\rbrace _{1}^{K}$ and the online network's representation – thus, this loss applies an attractive force to pull together multiple augmented representations of the same image (positive pairs) into the geometric centroid as the pivot to cluster embeddings.", "Ultimately, we the following multiview centroid loss $\\mathcal {L}_{c}$ : $\\mathcal {L}_{c}(\\theta , \\xi ; \\mathcal {X}) &= \\frac{1}{K} \\sum _{j=1}^K \\left\\Vert \\hat{p}_j-\\frac{1}{K} \\sum _{l=1}^K \\hat{z}^{\\prime }_{l}\\right\\Vert _2^2 $ where $\\hat{p}_j={p_j}/{\\left\\Vert p_j\\right\\Vert _2}$ is $l_2$ -normalized predictions from the online network for the augmented view of the same input, i.e.", "$p_j=h_\\theta (g_\\theta (f_\\theta (v_j)))$ .", "Note that, minimizing Eq.", "REF is mathematically identical to minimizing pairwise distance between $\\lbrace \\hat{p}^{\\prime }_{j}\\rbrace _{1}^{K}$ and $\\lbrace \\hat{z}^{\\prime }_{l}\\rbrace _{1}^{K}$ .", "Therefore, this loss generates a stronger attractive force that aggregates the embeddings of the same image that BYOL loss in Eq.", "REF .", "Brownian Diffusion Loss.", "We use a dispersive loss, called Brownian diffusion loss, that induces a Brownian motion (or a random walk) of the online network's representation $p_j$ of $j$ -th augmented view of an input.", "A $d$ -dimensional random vector $n\\in \\mathbb {R}^d$ is sampled from unit normal distribution, i.e.", "$n \\sim \\mathcal {N}(0, I_d)$ with an identity matrix $I_d$ .", "Our Brownian diffusion loss is defined as follows: $\\mathcal {L}_{b}(\\theta ; \\mathcal {X}) &= \\frac{1}{K} \\sum _{j=1}^K \\langle \\hat{n}, \\hat{p}_j \\rangle $ where $\\hat{n} = n/\\left\\Vert n\\right\\Vert _2$ .", "The noise vector $\\hat{n}$ drives a diffusive motion by pushing particles in the embedding space in radial direction, which is uniformly sampled on the unit hyper-sphere.", "Importantly, we use the same random vector $\\hat{n}$ for the all augmented embeddings of the given image.", "This implies that the positive pairs which share the similar semantics are not spread apart.", "In contrast, the views from the different image moves to the different direction and the direction is likely to be orthogonal to other images' moving direction.", "I.e.", "Brownian diffusion loss disperses the embeddings locally, which gives implicit contrastive effect between embeddings of different images.", "We observe that our Brownian diffusion loss is critical to prevent mode-collapse [16].", "As the target network's parameter is updated by the exponentially weighted moving average of the online network's parameter at each training step (given a high target decay rate), the change of the target network's representation is relatively slower than that of the online network (effectively $\\frac{1}{1-\\tau }$ times slower).", "Such an imbalance may cause a mode collapse as the online network's representation can quickly collapse into a single point without any repulsive force between them.", "Singular Value Loss.", "Lastly, we use the singular value loss $\\mathcal {L}_w$ that decorrelates the different feature dimensions of the projections $\\hat{p}$ to prevent these dimensions from conveying the same information, thus avoid a dimension collapse.", "We minimize the following Euclidean distance between the empirical covariance matrix of the embeddings and the identity matrix $I_d$ – thus, we penalize the off-diagonal coefficients of the covariance matrix and make the distribution ball-shaped.", "Let the $p_{ij}$ be i-th batch and j-th augmented embeddings.", "Then the empirical covariance matrix of j-th augmented embeddings $S_j$ is: $S_j = \\frac{1}{n-1} \\sum _{i=1}^n (p_{ij} - \\bar{p}_j) (p_{ij} - \\bar{p}_j)^T$ where $n$ is the number of batches and $\\bar{p_j} = \\frac{1}{n} \\sum _{i=1}^n p_{ij}$ .", "Then we define singular value loss as: $\\mathcal {L}_{s}(\\theta ; \\mathcal {X}) &= \\frac{1}{K} \\sum _{j=1}^K {\\left\\Vert S_j - I_d\\right\\Vert _\\text{F}^2} \\\\&= \\frac{1}{K} \\sum _{j=1}^K \\sum _{i=1}^d (\\sigma _{ij} - 1)^2$ where $\\sigma _{ij}$ is singular values of the covariance matrix, $S_j$ .", "We found that this loss improves when corporated with Brownian diffusion loss.", "Some prior works [2], [9], [38] justify whitening loss as removing correlations between different embeddings.", "In our method however, we treat singular value loss as a dispersive force that encourages uniformity of the embedding distribution.", "Even though Brownian diffusion loss addresses local dispersion in the embedding space, singular value loss exerts the force to regularize the shape of embedding distribution to be globally spherical at large scale." ], [ "Evaluation of Representations with MSBReg ", "Evaluation on ImageNet-100 and STL-10.", "Following the linear evaluation protocol, we train a simple linear classifier with the frozen representations from our encoder, which is pre-trained with our MSBReg .", "We first evaluate the performance of the encoder on a small-size ImageNet-100 [29] and a mid-size STL-10 datasets.", "We observe in Table REF that the performance of MSBReg generally outperforms other state-of-the-art approaches on both datasets, especially we observe a large gain on the STL-10 dataset.", "The performance gain is more apparent that the following three approaches, MoCo, SimCLR, and Wang and Isola, use a more expressive ResNet-50-based backbone than our ResNet-18-based backbone.", "We also observe that the quality of the learned representation improves as the number of views $K$ increases (compare bottom two rows).", "Evaluation on ImageNet.", "We further evaluate the representations obtained after self-supervised pre-training with MSBReg on the large-scale ImageNet dataset with two evaluation metrics: 1) linear evaluation protocol and 2) semi-supervised learning with the subsets of ImageNet and 3) kNN classification.", "For linear evaluation protocol, likewise to Table REF , dashed-line means that the original paper did not report the corresponding value.", "We observe in Table REF that the performance of MSBReg outperforms other approaches and gets the result compatible to SwAV with multi-crop, which may confirm that the effectiveness of MSBReg for learning the better visual representation.", "Especially, compared to other baselines which are trained for 400 epochs, our method is only trained for 300 epochs.", "This implies that it has enough room to improve a lot.", "Again, note that ours uses a smaller batch size than alternatives except MoCo-v2 (i.e.", "512 vs. 4096 or 1024), but shows the matched or better performance.", "To evaluate semi-supervised leaning ability of our method, we report top-1 and top-5 accuracy over $1\\%$ and $10\\%$ of ImageNet subsets.", "The experiment results are in Table REF .", "For both $1\\%$ and $10\\%$ subsets, our method outperforms baselines, when we compare methods with top-1 accuracy.", "Especially, in the fine-tuning result with $1\\%$ subset of ImageNet dataset (see 1st column in Table REF ), our method surpasses with the large margin.", "For top-5 accuracy, our method gets matched performance with  [13] and outperforms other methods.", "kNN evaluation results are in Table REF .", "We report the accuracy of 20-NN and 200-NN classification results.", "Our method outperforms baselines.", "Table: Evaluation of the representations pretrained with MSBReg on various downstream tasks: 1) the performance linear classifier on top of frozen ResNet-50 backbone and 2) object detection with fine-tuning.", "For the linear probing, we report mAP for VOC07  benchmark, Top-1 accuracy (%\\%) for Places  and iNaturalist2018  benchmarks.", "For the object detection task, we report AP 50 \\text{AP}^{50}, AP 75 \\text{AP}^{75}, and AP all \\text{AP}^{\\text{all}} for VOC07+12 benchmark.Table: Comparison of the quality of representations between BYOL  and ours on the STL-10 dataset .", "The Top-1 classification accuracy is reported with different types of normalization techniques: a batch normalization (BN)  and a layer norm (LN) .", "To see the effect of our proposed Brownian Diffusive Loss, ℒ b \\mathcal {L}_b, we also report scores of BYOL with ℒ b \\mathcal {L}_b (4th row)." ], [ "Transfer Learning on Various Downstream Tasks", "We further evaluate the transferability of the features trained with MSBReg on ImageNet via transferring the features to various downstream tasks.", "In Table REF , we compare the performance of MSBReg with baselines.", "We first report the linear classification result on VOC07 [10], Places205 [39] and iNaturalist [31] visual benchmarks.", "Each of benchmarks is to evaluate 1) multi-label classification 2) scenic scenario and 3) fine-grained classifcation.", "We evalute the performance of linear classifier on top of the frozen ResNet-50 encoder pretrained with MSBReg method.", "We report mAP for VOC07 dataset and top-1 accuracy ($\\%$ ) for other benchmarks.", "We observe that our method shows generally matched results compared with alternatives.", "For object detection task, we finetune pre-trained ResNet-50 backbone with the PASCAL VOC07+12 object detection benchmark [10].", "We use Faster R-CNN [27] with C4 backbone as our baseline model.", "We report $\\text{AP}^{50}$ , $\\text{AP}^{75}$ , and $\\text{AP}^{\\text{all}}$ .", "We observe in Table REF that our model shows generally matched results compared with alternatives except for PixPro [36], which is proposed for .", "For $\\text{AP}_{50}$ , our method performs better than the baselines, while our method shows matched or slightly lower performance than other approaches.", "We report instance segmentation result on COCO dataset in the appendix." ], [ "Brownian Diffusive Loss against Mode Collapse", "BYOL [16] successfully uses only pairs of positives, but the reason why the online and target networks can avoid a so-called mode collapse, i.e.", "representations of all the examples are mapped to the same point in the embedding space, is not yet clearly explained.", "Existing work [11], [6], [29], [28] discussed that the use of the Batch Norm (BN) implicitly contributes to avoiding generating a collapsed representation.", "Especially, the original authors of [16] show that BYOL works without BN [28].", "However, those methods are impractical in terms of restricting the network architecture design and this fact implies that these approaches are suboptimal.", "In our work, we propose to use Brownian diffusive loss, $\\mathcal {L}_b$ , which pushes embeddings into the radial direction to be uniformly sampled on the unit hyper-sphere.", "This helps to avoid collapsed representations without the need of using the Batch Norm (BN).", "We further discuss this in the appendix.", "In Table REF , we empirically observe that BYOL suffers from a mode collapse when we replace the Batch Norm (in the prediction and projection heads) with another normalization technique, a Layer Norm (compare 1st vs. 3rd row).", "The top-1 classification accuracy is largely degraded from 89.5% to 10.6%, i.e.", "mode collapsed.", "Ours with the Brownian diffusive loss $\\mathcal {L}_b$ was not the case (compare 2nd vs. 6th row).", "Though we observe a slight degradation in the top-1 classification accuracy, ours sufficiently avoid collapsed representations.", "Further, we evaluate the BYOL with our Brownian diffusive loss to demonstrate its effectiveness against a mode collapse.", "We observe that our Brownian diffusive loss helps avoid collapsed representations (compare 3rd vs. 4th rows).", "We also observe that the quality of representations depends on the strength of the hyperparameter $\\lambda _b$ where we obtain the best performance with $\\lambda _b=5\\times 10^{-4}$ .", "We observe a tension as we see a smaller or larger $\\lambda _b$ slightly degrades the quality of representations." ], [ "Comparison with Multi-Crop Method", "We further compare Multiview centroid loss with the multi-crop method.", "The main difference between multiview centroid loss and multi-crop in SwAV is that our method uses the same resolution across all views while multi-crop uses low resolutions.", "We observe in Table REF that a BYOL model with the multi-crop method shows a degradation (compare 1st vs. 2nd row), while MSBReg improves the performance of BYOL with a large margin (compare 1st vs. 3rd).", "This fact is also reported in [2].", "Here, we describe the details of experiment.", "For a fair comparison, we implement the multi-crop method in the BYOL framework.", "The multi-crop method generates 2 views with full resolution ($224 \\times 224$ for ImageNet) and $V$ views with low resolution ($96 \\times 96$ for ImageNet).", "Cropping small parts of an image is used to generate low-resolution images.", "We choose $V = 6$ following [4].", "To apply the multi-crop method to BYOL, we reformulate BYOL loss (i.e.", "Eq.", "REF ) as follows: $\\mathcal {L}_{\\text{mc-byol}} (\\theta , \\xi ; \\mathcal {X}) &:= \\sum _{i, j}^{V+2}\\left\\Vert \\hat{p_i}-\\hat{z_j}^{\\prime }\\right\\Vert _2^{2} \\mathbb {1}(i\\ne j) \\\\&= \\sum _{i, j}^{V+2} \\bigg ( 2-2\\frac{\\langle p_i, z_j^{\\prime } \\rangle }{||p_i||_{2} \\cdot ||z_j^{\\prime }||_{2}} \\bigg ) \\mathbb {1}(i\\ne j)$ which shows that the multi-crop method minimizes the distance between pairs of embeddings, while our Multiview centroid loss minimizes the distance between each view and the geometric centroid of multi-views.", "Table: Comparison accuracy of downstream image classification task on ImageNet between multiview loss and multi-crop .", "We apply multi-crop method to BYOL." ], [ "Ablation Studies", "Table REF shows our ablation study to see the effect of our proposed three regularizations: (1) Multiview centroid loss $\\mathcal {L}_c$ , (2) Brownian diffusive loss $\\mathcal {L}_b$ , and (3) Singular value loss $\\mathcal {L}_s$ .", "Given the BYOL model as a baseline, we apply different combinations of our regularizations and measure the quality of representations following the linear evaluation protocol.", "We report scores on the ImageNet-100 dataset.", "We use ✓ and ✗ to indicate with and without, respectively.", "Note that we set $\\lambda _b$ and $\\lambda _w$ by default as 0.5 and $4.0\\times 10^{-3}$ , respectively.", "We first observe that a significant performance gain is obtained with our Multiview centroid loss $\\mathcal {L}_c$ (compare 1st vs. 5th and 9th).", "The quality of the learned representations consistently improves as the number of views $K$ increases.", "Since BYOL uses 2 views ($K=2$ ) for training, doubling the number of views provides more than $6\\%$ performance gain.", "The other two regularizations, Brownian diffusive loss $\\mathcal {L}_b$ and Singular value loss $\\mathcal {L}_s$ , also consistently improve the overall classification accuracy.", "For example, the classification performance improves $0.92\\%$ with the Brownian diffusive loss (compare 1st vs. 3rd) and the Singular value loss (compare 1st vs. 2nd).", "Such performance gain becomes more apparent with the Multiview centroid loss where we obtain a larger gain: $1.44\\%$ with the Singular value loss and $1.5\\%$ with the Brownian diffusive loss.", "Concretely, applying all our proposed regularizations together shows the best performance.", "We further study the sensitivity of the tuning of the two new hyperparameters $\\lambda _s$ and $\\lambda _b$ .", "We report the result in the supplementary.", "Table: Ablation study to study the effect of our proposed three regularizations: (1) Multiview centroid loss ℒ c \\mathcal {L}_c, (2) Brownian diffusive loss ℒ b \\mathcal {L}_b, and (3) singular value loss ℒ w \\mathcal {L}_w.", "Note that we compare the top-1 classification accuracy (in %) of a linear classifier on the ImageNet-100 dataset.In this work, we have explored multiview, singular value regularization and Brownian diffusion methods for self-supervised learning.", "Each method implicitly induces contrastive effect, which stabilizes the the training of self-supervised learning.", "Our method achieves a good downstream task performance for instance classification as well as various transfer learning such as object detection, semantic segmentation." ], [ "Content", "This supplementary material provides implementation details (Section ) including training strategy, architectures, and image augmentations.", "We also provide evaluation details (Section ) including linear evaluation protocol, semi-supervised learning setting, k-NN classification, and transfer learning on various downstream tasks.", "We also report supplemental experimental results of instance segmentation on COCO dataset (Section ).", "In addition, this supplementary presents ablation study results.", "Lastly, we compare our method with  [19] and  [32] in detail." ], [ "Implementation Details", "We first provide implementation details of our method.", "We would emphasize that our code will be made publicly available upon publication.", "In Section REF and , we explain details of our training strategy and architectures.", "Next, in Section REF , we explain details of the stochastic image data augmentation used in our experiment." ], [ "Training Strategy.", "We utilize the Layer-wise Adaptive Rate Scaling (LARS) [37] optimizer that is known to effectively overcome large-batch training difficulties.", "We also use the learning rate scheduler that applies a cosine decay function [23] without restarts to an optimizer step.", "As suggested by [14], we apply a learning rate warm-up for the first 10 epochs where we start training with a small safe learning rate, which is slowly increased to the max learning rate linearly.", "The max learning rate is $\\texttt {base\\_lr} \\times \\frac{\\texttt {batch size}}{256} \\times K$  [14].", "We set the base learning rate to 0.4 for ImageNet-100, 0.5 for STL-10 dataset and 0.15 for ImageNet datasets.", "Unless otherwise stated, we set the batch size to 512.", "The weight decay parameter is set to $1\\times 10^{-5}$ .", "We exclude biases and parameters in batch normalization layer following BYOL [16].", "We train the model for 320 epochs for ImageNet-100 and STL-10 benchmarks and 300 epochs for ImageNet with 8 V100 16GB GPUs." ], [ "Architectures", "For a fair comparison, we use ResNet-18 [18] as a backbone network architecture for STL-10 and ImageNet-100 datasets and ResNet-50 as a backbone for ImageNet dataset, which are widely experimented with conventional approaches for the self-supervised representation learning task.", "Following BYOL [16], the projection heads (i.e.", "$f_\\theta $ and $g_\\xi $ in Figure 2 in the main paper) and the prediction head of the online network (i.e.", "$h_\\theta $ ) use a 2-layer fully connected network with ReLU [25] as an activation function.", "We tune the size of hidden layers and output layers of projection and prediction heads, when the backbone network is ResNet-18.", "We use 512 hidden layer size and 128 output layer size instead of 2048 hidden units and 256 output size, which are used in BYOL.", "We apply batch normalization layer [20].", "Also, we experiment various normalization layers including weight standardization [26] and layer normalization [1] to show that our method does not suffer from mode collapse without batch normalization.", "Table: Image augmentation parameters" ], [ "Image Augmentations", "We use a stochastic data augmentation operator that is sampled from the family of augmentations $\\mathcal {T}$ and results in a randomly augmented view of any given data example.", "Following SimCLR [5], our data augmentation module sequentially applies the following four augmentations: (1) random cropping followed by resizing back to the original size, (2) aspect-ratio changes, (3) random flipping in the horizontal direction, (4) random color distortion (i.e.", "jitter and lighting).", "Detailed augmentation parameters are in Table REF ." ], [ "Evaluation Details", "In this section, we provide relevant information for evaluation of our method." ], [ "Linear Evaluation Protocol", "We use the linear evaluation protocol [21], which is the standard practice to evaluate the quality of the learned image representations.", "Using the trained encoder as the feature extractor, we train a linear classifier as a post-hoc manner, i.e.", "a simple image classifier given a set of features.", "Then, we measure its classification accuracy on the test set as a proxy of the quality of the learned representations.", "Note that the encoder is frozen during the evaluation phase.", "We use the following three standard image classification benchmarks: (1) STL-10 [7], (2) ImageNet-100 [29], and (3) ImageNet [8].", "Note that ImageNet-100 contains only 100-class examples that are randomly sampled from ImageNet" ], [ "Semi-Supervised Learning", "We also evaluate the semi-supervised learning ability of our method with subset of ImageNet training set.", "We finetune ResNet-50 encoder pretrained with our algorithm and the classifier on top of the encoder using $1\\%$ and $10\\%$ of ImageNet.", "These ImageNet splits can be found from the official implementation of  [5].", "We mainly follow the semi-supervised learning protocol of  [16].", "We use SGD with momentum of 0.9 and Nesterov, batch size of 1024.", "We use the separate learning rates for the classifier and the encoder.", "For fine-tuning task with $1\\%$ ImageNet subset, we set learning rate of the classifier 2.0 and freeze the encoder.", "For fine-tuning task with $10\\%$ ImageNet subset, we use the 0.25 as the learning rate of the classifier and $2.5\\times 10^{-4}$ as the learning rate of the encoder." ], [ "k-NN Classification", "We closely follow the existing work [35], [40] to evaluate the quality of representations learned by our model.", "We first collect representations from training and validation images with the frozen encoder.", "Then, we compute the classification accuracy of 20/200-nearest neighbor classifier." ], [ "Transferring to Downstream Tasks", "To test the transferability of representations trained with our method on ImageNet, we perform transfer learning to various datasets: Places205, iNaturalist2018, Pascal VOC, and COCO.", "Image classification.", "We train the linear classifier layer on top of the frozen ResNet-50 backbone pretrained with MSBReg .", "For VOC 07, we train a linear support vector machine (SVM).", "For other image classification benchmarks, iNaturalist 2018 and Places 205, we train the linear classifier with SGD with momentum of 0.9 and weight decay of $10^{-4}$ .", "The batch size is 256 and learning rate is 0.2 and we reduce the learning rate by factor of 10 two times with equally spaced intervals.", "For Places205, the training epoch is 28 and for iNaturalist 2018, the training epoch is 84.", "Object detection.", "Following previous works [15], [4], [2], [13], we finetune the network on VOC07+12 [10] dataset using Faster-RCNN [27].", "We report three metrics of the object detection, $\\text{AP}_{\\text{all}}$ , $\\text{AP}_{\\text{75}}$ and $\\text{AP}_{\\text{50}}$ .", "We use Detectron2 [33] to transfer our model to the object detection task.", "We set the initial learning rate 0.02.", "Other hyperparameters such as learning rate scheduling, warm-up steps are exactly same as [17].", "Instance Segmentation.", "For instance segmentation task, we evaluate our model with COCO dataset.", "We closely follow [17], [38], [2].", "We use Mask R-CNN FPN backbone.", "The backbone is initialized with our pretrained ResNet-50 backbone.", "We train the network for 90K iterations with a batch size of 16.", "A learning rate is 0.05 and reduced by a factor of 10 after 60K and 80K iterations.", "We linearly warm up the learning rate for 50 iterations." ], [ "Results on COCO Instance Segmentation", "We also evaluate the learned representation on COCO insstance segmentation task.", "We observe in Table REF that our method shows competitive performance with other methods.", "Our method is better than BYOL [16] (3rd row), which is our main baseline.", "SwAV [4] (5th row) shows similar performance to ours.", "Note that this method uses more augmentations than ours.", "Table: Performance comparison for transfer learning on instance segmentation task on COCO dataset.", "We use train2017 as training data and report the box detection AP (AP bb \\text{AP}^{\\text{bb}}) and instance segmentation AP (AP mk \\text{AP}^{\\text{mk}}) scores on val2017 dataset." ], [ "Ablation Studies", "We perform ablation experiments to study the trade off between major hyperparameters in MSBReg , $\\lambda _s$ and $\\lambda _b$ .", "In table REF , our experiment reports the top-1 classification accuracy on ImageNet-100.", "We train ResNet-18 with MSBReg for 300 epochs with various combinations of $K \\in \\lbrace 2, 4, 8\\rbrace $ , $\\lambda _s \\in \\lbrace 0, 0.002, 0.004, 0.01\\rbrace $ , and $\\lambda _b \\in \\lbrace 0, 0.25, 0.5, 1.0, 2.0\\rbrace $ .", "Note that the case of $K=2$ is the same as BYOL setting.", "Then, we train the linear classifier on top of frozen ResNet-18 backbone pretrained with MSBReg .", "Our study shows that the classification accuracy increases until $\\lambda _s$ =0.004$, \\lambda _b=0.5$ for the cases of $K=4$ and $K=8$ .", "Interestingly, both singular value loss and Brownian loss improve the performance for the case of BYOL ($K=2$ ).", "Table: Ablation studies to investigate the trade-off between losses in MSBReg ." ], [ "Related Works", "In this section, we supplement Section 2.", "We compare our work with batch repetition method [19], uniformity loss [32], and BYOL without BN [28] in detail.", "Batch Repetition.", "In the Section 2, we mention batch repetition method [19].", "Similar to this method, our multiview centroid loss partially benefits from the fact that simply seeing the same image with different augmentations at each iteration, stabilizes and accelerates training in self-supervised settings.", "However, the main difference between [19] and multiview centroid loss, is that multiview centroid loss considers the interactions between embeddings of the positive pairs.", "Uniformity of Embeddings.", "In this section, we report uniformity score of MSBReg and other baselines in Table REF .", "We train BYOL, BYOL with uniformity loss, BYOL+$\\mathcal {L}_b+\\mathcal {L}_s$ and MSBReg with ImageNet-100 with ResNet-18 backbone.", "Then, we evaluate each model with three metrics: 1) linear classifier accuracy 2) alignment loss and 3) uniformity loss.", "Here, both alignment loss and uniformity loss are introduced in [32].", "Alignment loss, $\\mathcal {L}_{\\text{align}}$ is defined as mean squared error between positive pairs and uniformity loss, $\\mathcal {L}_{\\text{uniform}}$ , is defined as the logarithm of the average pairwise Gaussian potential between negative pairs.", "In Table REF , uniformity loss improves the performance of BYOL (1st row vs 2nd row), by decreasing uniformity loss.", "Ours shows lower uniformity loss, higher alignment loss and the better performance than other baselines.", "This strengthen the argument of [32] and ours, which argues that the optimal distribution trained with self-supervised method is uniformly on the embedding manifold.", "Table: Evaluating methods with ℒ align \\mathcal {L}_{\\text{align}} and ℒ uniform \\mathcal {L}_{\\text{uniform}}BYOL without Batch Normalization Layer.", "The widely known fact about BYOL [16] is that this method falls into the mode collapse [11] without batch normalization layer.", "The authors of [16] performed studies that BYOL works even without BN layer [28].", "In this paper, authors showed that BYOL without BN gets matched performance using various normalization techniques including weight standardization [26] or the deliberately handled initialization.", "But still, BYOL fails to converge optimal solution with such deliberately tuned training techniques.", "In this section, we show that MSBReg also work with layer normalization [1] without any other techniques in Table REF .", "In Table REF , the top-1 classification accuracy is largely degraded from 89.5% to 10.6%, i.e.", "mode collapsed.", "Ours with the Brownian diffusive loss $\\mathcal {L}_b$ was not the case (compare 2nd vs. 6th row).", "Though we observe a slight degradation in the top-1 classification accuracy, ours sufficiently avoid collapsed representations.", "Further, we evaluate the BYOL with our Brownian diffusive loss to demonstrate its effectiveness against a mode collapse.", "We observe that our Brownian diffusive loss helps avoid collapsed representations (compare 3rd vs. 4th rows).", "We also observe that the quality of representations depends on the strength of the hyperparameter $\\lambda _b$ where we obtain the best performance with $\\lambda _b=5\\times 10^{-4}$ .", "We observe a tension as we see a smaller or larger $\\lambda _b$ slightly degrades the quality of representations.", "Table: Comparison of the quality of representations between BYOL  and ours on the STL-10 dataset .", "The Top-1 classification accuracy is reported with different types of normalization techniques: a batch normalization (BN)  and a layer norm (LN) .", "To see the effect of our proposed Brownian Diffusive Loss, ℒ b \\mathcal {L}_b, we also report scores of BYOL with ℒ b \\mathcal {L}_b (4th row)." ] ]
2207.03552
[ [ "The Spiderweb proto-cluster is being magnetized by its central radio jet" ], [ "Abstract We present deep broadband radio polarization observations of the Spiderweb radio galaxy (J1140-2629) in a galaxy proto-cluster at $z=2.16$.", "These yield the most detailed polarimetric maps yet made of a high redshift radio galaxy.", "The intrinsic polarization angles and Faraday Rotation Measures (RMs) reveal coherent magnetic fields spanning the $\\sim60$ kpc length of the jets, while $\\sim50$% fractional polarizations indicate these fields are well-ordered.", "Source-frame absolute RM values of $\\sim1,000$ rad/m/m are typical, and values up to $\\sim11,100$ rad/m/m are observed.", "The Faraday-rotating gas cannot be well-mixed with the synchrotron-emitting gas, or stronger-than-observed depolarization would occur.", "Nevertheless, an observed spatial coincidence between a localized absolute RM enhancement of $\\sim1,100$ rad/m/m, a bright knot of Ly$\\alpha$ emission, and a deviation of the radio jet provide direct evidence for vigorous jet-gas interaction.", "We detect a large-scale RM gradient totaling $\\sim1,000$s rad/m/m across the width of the jet, suggesting a net clockwise (as viewed from the AGN) toroidal magnetic field component exists at 10s-of-kpc-scales, which we speculate may be associated with the operation of a Poynting-Robertson cosmic battery.", "We conclude the RMs are mainly generated in a sheath of hot gas around the radio jet, rather than the ambient foreground proto-cluster gas.", "The estimated magnetic field strength decreases by successive orders-of-magnitude going from the jet hotspots ($\\sim90$ $\\mu$G) to the jet sheath ($\\sim10$ $\\mu$G) to the ambient intracluster medium ($\\sim1$ $\\mu$G).", "Synthesizing our results, we propose that the Spiderweb radio galaxy is actively magnetizing its surrounding proto-cluster environment, with possible implications for theories of the origin and evolution of cosmic magnetic fields." ], [ "Introduction", "Supermassive black holes (SMBH) in active galactic nuclei (AGN) interact with the larger cosmos via production of magnetized, radio-emitting synchrotron jets and lobesAnd also through AGN `winds', but this feedback channel is not relevant to the work presented here.", "(e.g.", ", ).", "This can help drive cosmic ecology and evolution (e.g. )", "by controling the flow of SMBH-bound gas on scales ranging from megaparsecs (i.e.", "in galaxy cluster gaseous halos; and references therein) down to milliparsecs (i.e.", "in SMBH accretion disks; e.g.", ", ), by heating galactic and intergalactic gas to limit star-formation and the growth of massive galaxies (e.g.", ", , ), or by conversely generating localized star formation via compression of gas clouds around the host galaxy (e.g.", ", , , ), and by enriching the universe in metals , chemical compounds (e.g.", ", ), magnetic fields (e.g.", ", ), and cosmic rays (e.g.", ", , , ).", "Such interactions may have had their greatest impact on galactic evolution in the approximate redshift range $2<z<4$ (e.g.", ", , , ), the nearside of which sees galactic star formation rates beginning to decline after `cosmic noon' , and a nascent red sequence of quenched galaxies already established , .", "However, the precise roles that radio jets play in driving high$-z$ galactic evolution remain unclear , and directly probing the jet-gas interaction regions at these redshifts remains challenging.", "Moreover, better-studied low-$z$ radio galaxies are probably poor analogs for their high-$z$ counterparts, since they operate in very different cosmic environments.", "For example, the mean density of cold gas in high-$z$ galaxies is several times higher than in the present-day universe (e.g.", ").", "The energy density of the cosmic microwave background scales as $(1+z)^4$ (meaning that inverse-Compton cooling of radio lobes may scale similarly — e.g.", ", , ).", "The star formation rate is also an order of magnitude greater than present-day levels , galaxy clusters are still in the process of forming (e.g.", ", ), and magnetic fields may not yet fully permeate the intergalactic gas , which can affect its viscosity, pressure support, and thermal conductivity.", "Therefore, to better understand how super-massive black holes have shaped the cosmic ecology, it is desirable to observe radio galaxies in nascent galaxy clusters at high redshift, using techniques that illuminate the locus of interaction between the jet and ambient gas, and probe the physics occurring therein.", "Broadband radio spectropolarimetry represents a singular set of such techniques.", "Exploiting analysis of Faraday rotation and depolarization, it provides exquisite probes of magnetic field strength and structure around radio-emitting plasma, and by extension, the physical processes operating therein (e.g.", ", , , , , , , , , , , , , , , ).", "Consider that the linear polarization state of radio emission can be described by a complex vector $P$ , related to the Stokes parameters $Q$ and $U$ , the polarization angle $\\psi $ , the fractional polarization $p$ and the total intensity $I$ as $P = Q + iU = pIe^{2i\\psi }$ In traveling from source to observer, linearly polarized radiation will be Faraday rotated by magnetized thermal plasma along the line of sight (LOS) to an observer by an amount equal to $\\Delta \\psi = \\text{RM}\\lambda ^2$ where $\\lambda $ is the observing wavelength, and RM is the Faraday Rotation Measure, which is related to the thermal electron density $n_e$ [cm$^{-3}$ ] and magnetic field $B$ [$\\mu $ G] along the LOS as $\\text{RM} = 0.812 \\int _{z_s}^{0} \\frac{n_e(z)B_{||}(z)}{(1+z)^2}\\frac{dl}{dz}\\text{dz}~~~\\text{rad m}^{-2}~,$ where $z_s$ is the redshift of the radio source, and the comoving path increment per unit redshift, dl/dz, is in parsecs.", "The RM thus provides a means to detect magnetized thermal plasma, and to help deduce its properties.", "In terms of high-$z$ proto-cluster/radio galaxy systems to target, there are none more compelling nor better-studied than J1140-2629 (`The Spiderweb Galaxy') at $z = 2.16$ , .", "The radio source is characterized by two powerful jets oriented roughly east-west, extending about 60 kpc from the AGN , .", "The jets appear to be interacting with gas in the system over a range of scales in multi-faceted ways: The proto-cluster contains a dense agglomeration of galaxies extending to radii $>100$ kpc, whose projected long axis appears to align with that of the radio jets, and whose chain and tadpole morphologies may indicate the radio galaxy has impacted their star formation history (e.g.", ", ).", "Observations of CO emission with the Jansky Very Large Array (VLA) and Australia Telescope Compact Array (ATCA) has revealed a remarkable molecular gas halo, which is also aligned with the radio jet, and which extends to a projected radius of 70 kpc .", "Atacama Large Millimeter Array (ALMA) observations of H$_2$ O, CI, and CO emission suggest that the passage of the radio jet can induce condensation of cold molecular gas clouds, intrinsically linking the action of the radio jet with star formation activity in the proto-cluster halo .", "The entire proto-cluster is enveloped in a spectacular Ly$\\alpha $ halo out to a projected radius of $>200$ kpc, which is again broadly aligned with the radio jet , , and which is strongly displaced by a newly-detected eastern radio lobe .", "An alignment between the radio jet and bright X-ray emission in the system, first described by , has been confirmed and delineated in great detail with a new $\\sim 700$ ksec Chandra observation .", "The new images show a remarkably close correlation between the radio and the extended X-ray emission from the jet.", "The data are consistent with the extended X-ray emission arising mostly from inverse-Compton up-scattering of the local photon field by the radio emitting relativistic electrons, with a minor thermal contribution from hot gas.", "High resolution, broadband, full-polarisation radio observations dis not previously exist, but represent a singular probe of magnetised gas physics in and around the radio jet, and may thereby shed new light on jet-gas interactions in this archetypal protocluster system.", "We therefore undertook such observations using the Jansky Very Large Array (VLA), which possess sub-arcsecond resolution and span 340 MHz to 36 GHz, as well as a 700 ksec Chandra exposure, as part of a definitive new study of the Spiderweb system , .", "In this paper, we present an analysis of polarization and Faraday rotation in the Spiderweb system, in order to better understand the properties of the radio jet, the surrounding proto-cluster medium, and interactions between the two, at an epoch where such interactions may peak in their cosmological importance.", "The paper is laid out as follows: Section describes our observations and their calibration, and Section describes our polarimetric imaging and analysis, as well as details of ancillary data sets.", "Section describes the results derived from our radio observations, and Section describes further results obtained via analysis of ancillary data sets.", "We discuss our findings and conclude in Section .", "We adopt the current Planck cosmology for our calculations, such that one arcsecond on the sky corresponds to 8.49 kpc at the $z=2.16$ redshift of the source." ] ]
2207.03498
[ [ "Online Trajectory Prediction for Metropolitan Scale Mobility Digital\n Twin" ], [ "Abstract Knowing \"what is happening\" and \"what will happen\" of the mobility in a city is the building block of a data-driven smart city system.", "In recent years, mobility digital twin that makes a virtual replication of human mobility and predicting or simulating the fine-grained movements of the subjects in a virtual space at a metropolitan scale in near real-time has shown its great potential in modern urban intelligent systems.", "However, few studies have provided practical solutions.", "The main difficulties are four-folds.", "1) The daily variation of human mobility is hard to model and predict; 2) the transportation network enforces a complex constraints on human mobility; 3) generating a rational fine-grained human trajectory is challenging for existing machine learning models; and 4) making a fine-grained prediction incurs high computational costs, which is challenging for an online system.", "Bearing these difficulties in mind, in this paper we propose a two-stage human mobility predictor that stratifies the coarse and fine-grained level predictions.", "In the first stage, to encode the daily variation of human mobility at a metropolitan level, we automatically extract citywide mobility trends as crowd contexts and predict long-term and long-distance movements at a coarse level.", "In the second stage, the coarse predictions are resolved to a fine-grained level via a probabilistic trajectory retrieval method, which offloads most of the heavy computations to the offline phase.", "We tested our method using a real-world mobile phone GPS dataset in the Kanto area in Japan, and achieved good prediction accuracy and a time efficiency of about 2 min in predicting future 1h movements of about 220K mobile phone users on a single machine to support more higher-level analysis of mobility prediction." ], [ "Introduction", "For crowd surveillance or traffic regulation systems, an accurate fine-grained prediction of human movements can help people make informed decisions and governments take timely countermeasures.", "Most existing studies predict human mobility at a coarse level, predicting either just the number of aggregated population or traffic volume or only the next-move/destination represented by the grid cell or coordinate, rather than a complete trajectory that is matched to the transportation network.", "Such coarse predictions are insufficient to support higher-level transportation predictive analysis that transportation bureaus or companies' are concerned with.", "Some examples of higher-level analysis are listed as: What will the traffic volume on a particular road/railway segment be in 1h?", "How many people will get on/off and transfer in a specific station using a particular line (e.g., Tokyo station and Yamanote line)?", "Find out the potential sources/sinks of a gathering or dispersing pattern from the massive predicted human trajectories.", "How the users will move alternatively if the line stopped.", "To this end, instead of modeling the mobility in aggregation, we need a more comprehensive virtual replica of the mobility in the physical world and make an accurate and efficient prediction of the future movements of the agents in the virtual space under the current urban status or simulated conditions.", "Recent progresses in digital twin, especially mobility digital twin [5], [27], have shed light on a potential solution to this problem that senses, predicts, simulates and visualizes the mobility of a city in real-time.", "However, few studies have proposed practical solutions of making a fine-grained prediction of the massive agents in the virtual space accurately and efficiently.", "In this study, by collecting the raw GPS stream that makes a more lossless replication for the mobility of the city, we aim to make more informative predictions by refining our predictions from aggregated population/traffic to individual, from a grid-cell or coordinates represented by trajectory to transportation network level, and from predicting the next move or destination to a complete future trajectory.", "As shown in Fig.", "REF , our fine-grained prediction can support further high-level analysis such as aggregation at the road network level to predict 1h ahead future traffic volume on a specific road segment, or aggregation at a station level to predict the future transfer behavior after people arrive at an interchange like the Tokyo station.", "The main challenges come from four aspects: 1) The predictor should be capable of capturing the differences of human mobility from day to day and adjust itself by perceiving the current crowd context.", "For example, a user goes to drinking places with a higher probability if we observe a higher population going to drinking places.", "Thus, we need to model both the sequential pattern of user trajectories and how the sequential pattern varies considering other trajectories within the crowd context.", "2) The multimodal transportation network, especially in the urban areas, enforces complex constraints on human mobility.", "Considering road networks as an example, the road network in the Kanto area includes 2,902,380 road segments of 9 road types with different properties (access type, tolled or free, speed limits, etc.)", "and 2,035,726 road nodes.", "Existing prediction algorithms can hardly address so many complex constraints on human mobility in a feasible manner.", "3) Learning to generate a rational human trajectory, which is a long sequence ($>$ 100) with a large vocabulary ($>$ 1K), is regarded as a challenging task in the machine learning community.", "Generating a 1h predicted trajectory at the transportation network level is an even more daunting problem because of the longer sequence length (as much as hundreds of road segments in 1h by car) and larger vocabulary (millions of road nodes or segments) size.", "4) Inferring fine-grained trajectories from raw GPS trajectories requires heavy computation, especially to disambiguate the localization uncertainty and search for a reasonable route considering trajectory uncertainty, multimodality and different road properties.", "To address these difficulties, we propose a two-stage human mobility predictor that stratifies coarse and fine-grained level movements, as shown in Fig.", "REF .", "In the first stage, we encode the current global state of the city mobility as crowd context automatically, and predict the cluster-represented destination distribution at a coarse level.", "Note that, in this stage, we focus on addressing large-scale, long-term, and inter-user dependencies, while not considering local transportation network structure.", "This is because our local transportation choice is largely determined by long-term destination.", "In the second stage, we develop a retrieval-based method to generate our predicted trajectories via a weighted sampling from the fine-grained historical trajectory database, where the weight is determined by the coarse level prediction in the first stage.", "Thus, we offload heavy computation to an offline phase; moreover our system is efficient and may practically generate timely, fine-grained future trajectories.", "The entire pipeline is shown in Fig.", "REF We summarize our contributions as follows: To make efficient prediction of the virtual replica of the mobility in physical world rather than aggregated data (e.g., traffic volume, population density), We propose a novel two-stage human mobility prediction framework that predicts metropolitan-scale human mobility at a fine-grained level in near real-time.", "To better encode the urban status, we propose a novel predictive model that perceives the citywide mobility as crowd context and adjusts the predictor in a meta-learning paradigm.", "We propose a novel probabilistic retrieval-based prediction approach that bridges cluster-level prediction with fine-grained future trajectory generation for the prediction or simulation in the mobility digital twin." ], [ "Preliminaries", "In this section, we define the terms and concepts frequently used throughout this paper.", "[Raw GPS data] The raw GPS data can be formally described as follows: $X=\\left\\lbrace \\left( user\\_id,\\,time,\\,latitude,\\,longitude \\right)\\right\\rbrace $ Thus, the trajectory of each user $Tr^{u}$ can be defined as $Tr^{u} = x^u_{0}, x^u_{1}, \\,\\dots \\, \\quad x^u_{i} \\in X$ where $x^u_{i}$ is the $i$ -th record (sorted by time) of user $u$ .", "To distinguish the online data stream and offline historical data, we use symbol prime t̂o denote those variables in or related to the historical trajectory database (e.g., the historical raw GPS data $\\hat{X}$ and user ID $\\hat{u}$ in the historical database).", "[Cluster-level trajectory] We represent each trajectory on day $d$ as a cluster-level trajectory by dividing the continuous time and location representation into time slots $t$ and location cluster ID $C^u_{t}$ .", "$Trc^{u}_{d} = \\left[\\, C^{u}_{d, \\,0},\\, C^u_{d, \\,1},\\, \\dots ,\\, C^u_{d, \\,T - 1}\\, \\right]$ where $T$ is the number of time slices for one day.", "Location is represented by the index of cluster, and cluster-level trajectories are obtained in an online manner.", "In addition, we denote the collection of the cluster-level trajectories for all users as $TRC_{d} = \\lbrace Trc^u_d \\,\\big |\\, u \\in U \\rbrace $ , where $U$ is the set of user IDs.", "[Node-level Trajectory] We incorporate the information obtained from a digital map to match the raw GPS trajectory with the transportation network.", "Thus, we get the node-level trajectory as $Trn^{u} = (t_0, \\, node_0), \\, link_{0 -> 1}, \\, (t_1, \\, node_1), \\, link_{1 -> 2}, \\, \\dots $ where $time_{i}$ and $node_{i}$ are the $i$ -th time stamp and transportation node ID of the node-level trajectory, respectively.", "$link_{i - 1, \\,i}$ is the link between $node_{i - 1}$ and $node_{i}$ , which usually represents a segment of road or railway.", "[Crowd Context] We define the crowd context $\\Phi _{t}$ at time $t$ to simplify the interdependence of the user trajectories $TRC[t - \\Delta T: \\, t]$ at time $t$ .", "The crowd context encodes the current collective mobility patterns into a vector, and configures the cluster-level predictor to be more adaptive to the current mobility trend." ], [ "Cluster-level trajectory pre-processing", "To model long-distance and long-term dependencies from user trajectories, we need to transform raw GPS trajectory to cluster-level trajectory, as shown in Fig.", "REF .", "We obtain a cluster-level trajectory from raw GPS log data using a four-step process: Figure: Our proposed collective cluster-level mobility predictor.", "Red part shows how crowd context is used to enhance our prediction.", "Forwarding-fill the trajectory to obtain a uniform-sampling trajectory.", "Note that forward-filling is suitable for an online system because no future information is required.", "Index the trajectory location using Uber's Hexagonal Hierarchical Spatial Index (H3 index)https://uber.github.io/h3/ with the 8th resolution (average hexagon area $0.74 km^2$ ).", "Count the visit frequency of the hexagons in the training data, and create two sets of hexagons: i) top 1000 most frequently visited hexagons, as shown in Fig.", "REF (left); and ii) sampled 2600 hexagons from the remaining hexagons with the probability proportional to their visit frequencies, as shown in Fig.", "REF (right).", "Simplify the H3-indexed trajectory by approximating the H3-indices that do not belong to Type i) using the nearest Type ii) hexagon.", "Thus, we use 3600 indices to represent all the locations in the dataset.", "Therefore, we regularize the raw GPS trajectory from both spatial and temporal aspects.", "From the temporal aspect, we re-sample the non-uniform sampling trajectory to obtain a uniform sampling trajectory, which simplifies spatiotemporal features and accelerates the processing via batching.", "From the spatial aspect, we prefer discrete location representation, rather than continuous values (coordinates), because complex transportation networks make spatial dependency highly non-linear, especially if locations along the highways or railways can be regarded as singularities and can hardly be modeled by continuous functions.", "In this study, we conduct a simplification of the H3-index with respect to the visit frequency.", "Many previous studies represent location by grid cell only.", "However, to model human mobility at a metropolitan scale, we need a more flexible resolution.", "Note that we apply a sampling strategy, rather than directly using the top 3600 hexagons for a compromise of spatial resolution.", "Compared with the geographical distribution of top 1001-3600 hexagons (middle) and sampled hexagons (right), we find that the top 1001-3600 is much more spatially concentrated; thus, the spatial resolution is relatively low compared with sampled hexagons concerning the fourth process step listed above." ], [ "Cluster-level predictor", "A basic model for cluster-level predictor can be described as $p \\left(Trc[t + \\Delta T] \\, \\bigg | \\,Trc[t - \\Delta T\\,:\\, t]\\right) = F\\left(Trc[t - \\Delta T\\,:\\, t]\\right)$ where $F$ is the predictive function that considers the most recent $\\Delta T$ records and predicts the probability of $\\Delta T$ -ahead future movement $Trc[t + \\Delta T]$ .", "A typical choice of $F$ is to utilize a gated recurrent unit (GRU) to model the sequential pattern of the most recent trajectory and a $SoftMax$ layer to transform the prediction into a probability distribution.", "This can be described as $\\begin{array}{cl}h_0 =& \\textbf {0}\\\\h_{\\tau }=& GRU \\left( \\left[ EL\\left(Trc[t - \\Delta T + \\tau ]\\right), ET\\left( t \\right) \\right],\\, h_{\\tau - 1} \\right)\\\\o_T=& SoftMax\\left( MLP\\left(h_{\\Delta T})\\right) \\right)\\end{array}$ where $\\tau = 1, \\dots , \\Delta T$ , $h$ is the hidden state of the GRU at each time step, and $o$ the output vector representing the distribution of $Trc_{t + \\Delta T}$ .", "We use an embedding layer $EL$ with the vocabulary size of the number of clusters to map the cluster ID to an $N_{EL}$ -dimensional vector.", "Similarly, the time-of-day is mapped to a $N_{ET}$ using the embedding layer $ET$ .", "Figure: Visualization of the 4th and 58th dimensions of crowd context changing with timeHowever, the basic model does not consider the interdependence of the user trajectories.", "For example, when we observe some trajectories gathering at the central office area, it will increase our belief that it is a regular weekday, and that other users will also have a higher probability of going to their workplace.", "Consequently, we add flexibility to the predictive function $F$ that can adjust itself according to the current crowd context $\\Phi _{t}$ , characterizing the spatiotemporal features from all contemporary user trajectories $TRC[t - \\Delta T:\\, t] = \\left\\lbrace Trc^{u}[t - \\Delta T\\,:\\, t] \\,\\bigg |\\, u \\in U\\right\\rbrace $ .", "Thus, following a meta-learning paradigm, Equation REF can be rewritten as $p \\left(Trc[t + \\Delta T] \\, \\bigg | \\,Trc[t - \\Delta T\\,:\\, t], \\,\\Phi _{t}\\right) =\\mathcal {F}\\bigg ( \\Phi _{t} \\bigg ) \\bigg (Trc[t - \\Delta T\\,:\\, t]\\bigg )$ where $\\mathcal {F}$ utilizes the crowd context $\\Phi _{t}$ to characterize the current crowd trend, and generates the adaptive predictor $\\mathcal {F}\\bigg ( \\Phi _{t} \\bigg )$ , which is equivalent to a flexible $F$ in Equation REF .", "That is, we use the adaptive predictor $\\mathcal {F}\\bigg ( \\Phi _{t} \\bigg )$ that changes with the crowd context $\\Phi _{t}$ to replace the fixed predictor $F$ in our proposed method.", "Thus, our proposed predictor can exploit the periodic patterns we learned from the training data, can learn how to cope with the fluctuations in human mobility from the training data, and thus, is more robust to irregular human mobility.", "Note that the crowd context $\\Phi _{t}$ is calculated from $TRC[t - \\Delta T:\\, t]$ , which is an unordered set of instances (trajectories); therefore, we need to choose a permutation-invariant function (pooling function) for estimating the crowd context by aggregating over all the instances.", "$Mean$ and $Max$ are the two most widely used and time-efficient permutation-invariant functions.", "$Max$ calculates the maximum value of each dimension of the feature vectors, and $Mean$ averages all these feature vectors.", "We will show in our later experiments (Fig.", "REF ) that $Mean$ pooling function outperforms $Max$ .", "This is because $Mean$ is more capable of reflecting the crowd proportion, while $Max$ is better at testing the existence of the set.", "Let us assume that a feature describes whether a user is heading for his/her workplace.", "$Mean$ pooling can easily distinguish between 1% and 20% population, while $Max$ pooling can only test whether there is someone who is headed to work.", "Note that, from Equation REF , the crowd context should be computed from the entire user set.", "However, it is ineffective to conduct mean pooling over the entire user set literally.", "In practice, we increase the size of the mini-batch (we set it to 4096 in our experiments) in our training phase to use the average of the mini-batch to approximate the crowd context; meanwhile, in the testing phase, we pre-compute and apply mean pooling to the crowd context over all the users.", "To encode the sequential pattern of crowd context, we use another GRU network (red part in Fig.", "REF ) and merge with the hidden state from the GRU (blue part in Fig.", "REF ) defined in Equation REF .", "The network structure of our proposed cluster-level predictor is shown in Fig.", "REF .", "Fig.", "REF shows how the crowd context encodes the periodic and irregular facets of human mobility.", "We select two dimensions (4th and 58th) from the crowd context, and observe how the value of these two dimensions change during the testing phase.", "In the upper figure, the 4th dimension of the crowd context is more sensitive to irregular human mobility, especially as it clearly distinguishes the Obon festival week from regular weekdays.", "By contrast, the 58th dimension of the crowd context (lower) is less sensitive to irregular aspects of human mobility, while characterizing more periodic mobility patterns, even demonstrating some differences between the Obon festival week and a regular week.", "Therefore, every dimension of our crowd context encodes a different facet of large-scale human mobility, which is complementary with each other and supports our cluster-level predictors to simultaneously cope with both periodic and unexpected patterns underlying human mobility.", "Trajectory interpolation and map matching is the key process bridging GPS coordinates and transportation nodes and links.", "However, this procedure is very time-consuming and relies heavily on future information to disambiguate the uncertainties, as shown in Fig.", "REF .", "We can hardly determine whether the user is on the railway (blue) or road (black) until we obtain the information of his/her next move (blue cross).", "Thus, it is difficult to interpolate and map-match the raw GPS trajectory in real-time for an online system.", "Therefore, we propose a retrieval-based prediction approach that offloads the heavy computation task to the offline phase.", "In the rest of this subsection, we will describe how to acquire a node-level trajectory and how to build a fine-grained historical trajectory database for online querying." ], [ "Node-level trajectory", "Based on the methods introduced in [32], we conduct a three-step process to get a node-level trajectory as shown in Fig.", "REF : 1) Trip segmentation: stay points (with a maximum moving distance of 300m and minimum stay period of 15 min) are extracted from time-series GPS data based on spatiotemporal features, and moving state (MOVE or STAY).", "The transportation mode (WALK, BICYCLE, CAR, TRAIN, OTHER) is determined and partitioned for each trip based on the methods in [32].", "2) Interpolation between segmented trips: owing to the sparse characteristics of raw GPS trajectories, there are blank gaps between segmented trips.", "Considering the moving distance (threshold = 200m), gap length (threshold = 1h), and the stay/move states at the two ends of the gap, we fill the blank following the rules.", "If a long moving distance is presented, we insert a move state in the mid-term.", "If a state transition is presented, we determine the exact transition time estimated from the trajectories at the moving state.", "We merge consecutive STAY->STAY trips or MOVE->MOVE trips with the same transportation modes.", "3) Route estimation: in order to snap the GPS points to transportation network nodes and estimate a more accurate travel time, we conduct a route search for each trip with the MOVE state depending on the transportation mode and road/railway type (highway, national road, railway of local/express train, etc.).", "This above procedure is difficult for an online system because future information is critical to disambiguate the uncertainty, especially for sparse GPS data with low and non-uniform sampling rates and positioning errors.", "Moreover, the route estimation step is a non-deterministic problem and thus requires a large computational resources.", "Thus, we aim at obtaining good quality fine-grained historical trajectories in the offline module.", "Furthermore, in the following parts of this paper, we will show how it can be used in an online system in our retrieval-based prediction approach." ], [ "Creating trajectory database", "By obtaining the fine-grained node-level trajectories, we create the key-value store historical trajectory database as: $DB\\left[{C}\\right] = \\left\\lbrace \\left( \\hat{u}, \\,\\hat{d},\\, \\hat{t} \\right)\\, : \\, Trn^{\\hat{u}}_{\\hat{d}} \\left[\\hat{t}: \\hat{t} + \\Delta T \\, \\right] \\:\\bigg |\\: Trc^{\\hat{u}}_{\\hat{d}}[\\hat{t} + \\Delta T] = C\\right\\rbrace $ where node-level trajectory $Trn^{\\hat{u}}_{\\hat{d}}$ of user $\\hat{u}$ on day $\\hat{d}$ is partitioned into slices with time interval $\\Delta T$ .", "Each slice has key $\\left( \\hat{u}, \\,\\hat{d},\\, \\hat{t} \\right)$ .", "The database is partitioned by the destination cluster $Trc^{\\hat{u}}_{\\hat{d}}[\\hat{t} + \\Delta T]$ , corresponding to our cluster-level prediction.", "For an efficient query of historical trajectories, we map the historical trajectory to a Euclidean metric space considering two principles: 1) trajectory spatiotemporal continuity, meaning that the predicted trajectory should start close to where the observed trajectory ends; and 2) periodicity, meaning that the trajectories at the same time-of-day should share similar patterns.", "Consequently, we define our spatiotemporal vector representation $V$ as $V \\left( \\hat{u}, \\,\\hat{d},\\, \\hat{t} \\right) \\, = \\, \\left(lat^{\\hat{u}}_{\\hat{d},\\,\\hat{t}},\\, lon^{\\hat{u}}_{\\hat{d},\\,\\hat{t}},\\, \\alpha \\cdot cos \\frac{2\\pi \\hat{t}}{T},\\, \\alpha \\cdot cos \\frac{2\\pi \\hat{t}}{T} \\right)$ where $\\alpha $ is the coefficient controlling the relative weights between continuity and periodicity.", "A $k$ -d tree is built in the offline phase to prepare for the online searching of the $N$ -nearest neighbors as candidates." ], [ "Online Prediction", "The online prediction problem can be formulated as: $p\\left(Trn^u[t:\\,t + \\Delta T] \\, \\bigg |\\, TR[t - \\Delta T:\\,t] \\right) =\\\\\\sum _{C} p\\left( Trn^u[t:\\,t + \\Delta T] \\, \\bigg |\\, C,\\, Tr^u[t]\\,\\right) p\\left(\\, C \\, \\bigg |\\, TRC[t - \\Delta T:\\,t] \\right)$ where the raw trajectories set from all user $TR[t - \\Delta T:\\, t] = \\left\\lbrace Tr^{u}[t - \\Delta T\\,:\\, t] \\,\\bigg |\\, u \\in U\\right\\rbrace $ , and $TR$ can be transformed to $TRC$ in a deterministic manner.", "$C$ denotes the $\\Delta T$ -ahead destination cluster $Trc^u_{t + \\Delta T}$ .", "Note that the two terms represent our two-stage prediction, respectively.", "1) $p\\left(\\, C \\, \\bigg |\\, TRC[t - \\Delta T:\\,t] \\right)$ represents the probability distribution predicted by the cluster-level predictor.", "This stage is simple, and we have introduced the pre-computing strategy in Section REF .", "2) Because the intrinsic structure of $Trn^u[t:\\,t + \\Delta T]$ is very hard to model, we approximate the distribution $p\\left( Trn^u[t:\\,t + \\Delta T] \\, \\bigg |\\, C,\\, Tr^u[t]\\,\\right)$ via a prediction-by-retrieval approach.", "This can be done using a four-step process in a Monte Carlo Markov chain.", "Determine the shard of historical trajectory database $DB[C]$ by sampling destination cluster prediction $C$ .", "Calculate the query vector $V$ similar to Equation REF from $Tr^u[t]$ , and search the $K$ nearest neighbors $\\left\\lbrace (Trn_k, \\,\\epsilon _k) \\,\\bigg |\\, Trn_k \\in DB[C] \\right\\rbrace $ where $k = 1,\\dots ,K$ with distance $\\epsilon _k$ .", "Weighting the probability of candidates $\\left\\lbrace Trn_k \\right\\rbrace $ by their distances: $p\\left(Trn_k\\right) = \\frac{exp(-\\beta \\epsilon _k)}{\\sum _j exp(-\\beta \\epsilon _j)} $ , where $\\beta $ is the parameter controlling the sampling temperature.", "Candidate is drawn from $p\\left(Trn_k\\right)$ .", "Minor changes such as changing the user ID to $u$ and updating the time to the current time are applied before exporting our fine-grained prediction results.", "Note that, because all the trajectories in the historical database are fine-grained trajectories, our predicted trajectories, which are retrieved from the database as the prototypes, are also fine-grained trajectories.", "In other words, we avoid the heavy computation and complex modeling asscociated with generating a fine-grained trajectory by retrieving fine-grained trajectories in the historical database, where the sampling weight is determined by the cluster-level prediction.", "Figure: Evaluation on cluster-level prediction on different days" ], [ "Data", "In this paper, we use a dataset \"Konzatsu-Tokei (R)\" data.", "“Konzatsu-Tokei (R)\" Data refers to people flows data collected by individual location data sent from mobile phone under users' consent, through applications provided by NTT DOCOMO, INC. Those data are processed collectively and statistically in order to conceal the private information.", "Original location data is GPS data (latitude, longitude) sent in about every a minimum period of 5 minutes and does not include the information to specify individual.", "Some applications such as “docomo map navi\" service (map navi and local guide).", "We use two months of data (from 2012.6.1 to 2012.07.31) for training/validating the cluster-level predictor and building the fine-grained historical trajectory database, and one month data (from 2012.08.01 to 2012.08.31) for evaluation.", "We cropped the data in Kanto area (covering Tokyo metropolitan area), with an average number of about 220K user IDs.", "For the sake of protecting user's privacy as well as efficiency, we do not store users' ID for long-observation and thus we are more focused on the movement distribution at the large scale while a long dependency of the user's mobility [9] is not taken into consideration." ], [ "Evaluation on Cluster-level Prediction", "We choose five baseline methods to evaluate our cluster-level prediction.", "GRU is the predictor that is defined in Equation REF ; Conditional is an extension to GRU, which uses day-of-week as auxiliary data to distinguish between weekdays and weekends; Ensemble is based on the method in [7], which trains an independent predictor for each single day in the training dataset, and use an gating function to fuse these predictors in an adaptive way.", "In this experiment, we pre-train 14 predictors from Jun 1 to Jun 14; Context (Mean) excludes the top GRU layer for crowd context.", "Thus, the ability of modeling sequential pattern of the crowd context is weakened.", "Context (Max) shares the same network structure with “Context (Mean)\", but uses $Max$ as a pooling function.", "We use cross entropy (a lower value that indicates a better performance) for evaluation.", "As is shown in Fig.", "REF , our cluster-level predictor achieves the best performance in our test data.", "The non-adaptive predictor GRU performs worst because it completely ignores the different mobility patterns of users on different days.", "“Conditional\" is capable of distinguishing between weekdays and holidays, but fails to capture the subtle difference between different “weekdays\" or “holidays\" from crowd context.", "“Mean\" pooling functions better than “Max\" pooling, because the average operation is better at preserving the crowd proportion information than maxing out.", "“Ensemble\" is limited by the number of pre-trained predictors.", "We observed a decrease of cross entropy if we add more predictors; however the speed drops significantly for several predictors (about 5 times slower than our proposed method if we use 14 predictors), making it less practical for an online prediction system." ], [ "Evaluation on Fine-grained Prediction", "As shown in Fig.", "REF , we visualize our prediction results at time 07:30 on a weekday morning (2012.08.01 Wed) to qualitatively evaluate how well the gathering pattern in the morning rush hour for Tokyo station is predicted.", "1h-ahead fine-grained trajectories are predicted and compared with ground truth trajectories for those trajectories passing Tokyo station during 08:15-08:30.", "We take a snapshot of the trajectories every 20 minutes.", "Consequently, it is observed that our proposed method can predict how people will gather at Tokyo station (e.g., the source distribution of the gathering people and which route they will take) well.", "In Fig.", "REF , we selected four types of aggregated analysis: 1) number of transfer, 2) traffic volume on a particular road segment, 3) number of passengers using the station on a particular line, and 4) the total number of passengers for all lines in an interchange.", "We compared our 1h-ahead prediction with the most widely used time-series prediction methods, namely ARIMA[2] and prophet[24] from 2012.08.01 to 2012.08.08.", "We can see from Fig.", "REF that our fine-grained prediction can predict the sharp peaks of rush hours accurately and switch between the weekday and weekend state via crowd context automatically, which are difficult for traditional time series methods.", "A more comprehensive quantitative evaluation is given in Table REF .", "In general, our proposed method achieves a higher accuracy, especially for \"all lines\" which is less affected by random noise caused by few observations.", "Note that our predictor can predict more accurately, and is more informative fo the fine-grained trajectories compared with baseline models.", "Table: Evaluation on Fine-grained PredictionFigure: Prediction of the users that are affected by the stop of the Musashino line (left) and simulation of the alternative routes without using Musashino line (right)." ], [ "Simulation", "Note that our proposed prediction system for mobility digital twins can not only predict the future movements of the users based on the current urban status, but also predicting the trajectories responding to different conditions by filtering or augmenting the historical database with respect to specific simulation tasks.", "For example, we can easily identify those users that are potentially affected by the suspension of the Musashino line (the Tokyo unclosed outer ring line) from our fine-grained trajectory prediction results, as shown on the left of Fig.", "REF .", "Assuming the users will not change their moving destinations, we can easily generate the simulated future movements of those affected users if the Musashino line is out-of-service by filtering out all those candidates trajectories in the historical trajectory database that travels with Musashino line.", "As shown on the right of Figure REF , alternative routes are found from the historical trajectory database with the consideration of the frequencies, the current location, and the destination region.", "Such analysis and simulation are not well-supported by most of existing aggregated level or destination-only coarse mobility prediction methods." ], [ "System Implementation and Time Efficiency", "We deployed our algorithm on a deep learning workstation with Intel Xeon E5-2690v4, 2 x TitanX Pascal 12GB GDDR5X, 128GB and 1.2TB Intel® NVMe SSD DC P3600 Series.", "The algorithm was implemented in Python, except the trajectory interpolation and map-matching part were implemented in Java.", "We utilized the deep learning framework PyTorch 1.3.1 https://pytorch.org/ to construct the cluster-level predictor, and key-value storage library levelDB 1.20 https://github.com/google/leveldb for building and online querying the fine-grained history trajectory database.", "We measure the time efficiency of the three steps in our online prediction phase.", "From Table REF , we can see that fine-grained 1h-ahead trajectory prediction for about 220K users at a metropolitan scale takes about 2 min, which is a reasonable time latency for a practical system.", "Note that, except computing the crowd context, cluster-level and node-level prediction are conducted independently on each user.", "Thus a parallel acceleration strategy can be expected to achieve a desirable acceleration in our future work.", "Table: Time efficiencyDigital twin [23], which makes a replication of the physical world in the digital world, has drawn increasing attention in recent years, especially in the research of smart city [5], [27], [4], [20].", "However, most studies on digital twin for smart city focus more on the real-time sensing and interactive visualization, while the real-time trajectory prediction algorithm that is suitable for the mobility replica in the digital twin is relatively under-explored.", "Most existing studies predict human mobility at a small scale [22], [1], [31], aggregated level [16], [17], [3], [29], [14], [30], [8], [15] or coarse-level [7], [21], [9], [31].", "Compared with our fine-grained predictor, such coarse predictions are insufficient to support higher-level transportation predictive analysis, such as transportation transfer prediction (e.g., transfer between railway lines or to other transportation modes).", "To predict the fine-grained level human mobility, we need to generate a realistic trajectory, which is a long timestamped sequence, on the transportation network.", "[12] applied the seq2seq model to human trajectories and predicted future movements at the coarse level (only a few steps with continuous location representation), without considering the transportation network.", "[19] predicted the rest-of-the-day trajectory of the user in a predicting-by-retrieving paradigm, which is similar to the node-level prediction phase in our work.", "However, only the user's historical trajectories are considered, and only the dynamic time warping distances are calculated for measuring the similarity between the most recent trajectory and historical trajectories.", "Thus, our work is more suitable for predicting human mobility at a metropolitan scale.", "Moreover, the various nodes/links in the network makes the SoftMax layer very difficult to train.", "[11] uses clustering on the classes based on the frequency, and can accept a much larger number of classes.", "However, a transportation network within a metropolitan area has a much larger number of unique nodes/links (e.g., in our application we have $10^6$ unique nodes/links in the Greater Tokyo area), making it hard to model.", "An alternative that circumvents the above problem is to use ranking models [13], which well preserves the structure information.", "If our expected output text is most probably included, or can be approximated from the database, ranking models will significantly outperform generative models in terms of the quality of generated sequences.", "In this study, we utilize the second method, but approximate distribution over historical trajectories in a feasible manner to avoid bias in the aggregated traffic flow.", "Many existing predictors for individual users emphasize predicting the regularities of human mobility [10], [9], [31], [18].", "This may lead to a drift when the crowd trend changes.", "Some studies have explored methods considering crowd conformity: [6] predicted the user attendance of an event by leveraging both local and global historical data from all users; [28] proposed a flashback on hidden states of recurrent neural network to model the periodicity of spatiotemporal contexts of location-based service user's sparse trajectory.", "[26] modeled the regularity and conformity in human mobility, where conformity is the pattern in which some users follow others.", "In this study, conformity is modeled via crowd context implicitly.", "[7] trained an independent component predictor on each day in the training set, and performed online adaptive learning to leverage the crowd trend information.", "This method can make a good prediction for both regular and irregular mobility; however, numerous component predictors make the the prediction very inefficient (time and memory).", "Social LSTM [1] designs a social pooling that is capable of integrating the mobility of a crowd.", "This idea is the most relevant to our cluster-level predictor with crowd context; however, this study aims to solve the human mobility prediction for a surveillance scene.", "Inspired by Social LSTM and some following studies [25], we design a batchwise mean pooling method to capture the crowd context at each time step, and a context GRU to model the sequential pattern in the crowd context, which is more suitable for large-scale mobility prediction." ], [ "Conclusion", "This study proposes a novel two-stage fine-grained human mobility prediction method for predicting the mobility replica for the mobilty digital twin.", "Crowd context, that describes the current state of the mobility replica, is considered to make the predictor more flexible to perceive the current crowd trend information, and a predict-by-retrieval method is proposed to predict fine-grained trajectories in a practical time.", "We also note some limitations of the current work.", "1) Our crowd context is described in a global scale, while local irregularity can be ignored by our global crowd contexts if it is less significant to be aware at the metropolitan scale.", "Thus, a fusion of global and local crowd context is considered as a promising future direction.", "2) We currently implement our online prediction system on a single machine in which little optimization of time efficiency is considered.", "As we can see from Table REF , the time efficiency bottleneck of current system is within node-level prediction, which takes up about 84% of the total elapsed time.", "The interdependence of different users are mainly modeled in the crowd context computing phase, which takes only about 0.3% of the total elapsed time, while cluster-level and node-level prediction is conducted on each user independently.", "Thus, parallel computing is very promising in accelerating our prediction significantly." ], [ "Appendix", "In this appendix, we give more details on the implementation details of our system, and another experiment on DiDi Chengdu (a publicly available dataset) for reproducibility." ], [ "Experimental Setups", "In our previous experiments, we set the time interval to be 5 min for trajectory representation, and used the most recent 1h (12 steps) trajectories to make a 1h-ahead prediction every 15 minutes.", "The cluster and time embedding dimensions were set to 128 and 64 respectively.", "Moreover, a two-layered GRU with a hidden size of 256 was utilized to model the sequential pattern for individual trajectories and a single-layered GRU with a hidden size of 64 was utilized to model the sequential pattern of the crowd context.", "A Multi-class MLP layer was implemented as a two-layered network, with a 256 dimension latent layer." ], [ "Data Details", "As shown in Fig.", "REF , we show one-week number of unique user IDs, number of rows (every time our node-level trajectories passing a transportation node, we write one row describing the timestamp and the node in the database), and the distribution of the trips with respect to different transportation model.", "We determined the home location each user in the dataset, which was compared with census data on 1km grid sections, and estimated the linear relationship as: $N_{GPS} \\,=\\, 0.0063\\, *\\, N_{census}\\, +\\, 0.74, \\: R^2\\, =\\, 0.79$ where $N_{GPS}$ is the population estimated from GPS dataset, $N_{census}$ is the population given by the national census data, and $R^2$ is the coefficient of determination.", "Because the dataset used in this paper cannot be published due to privacy concerns, to help researchers to reproduce this paper, we conduct additional experiment on DiDi Chengdu City Second Ring Road Regional Trajectory Data Set in Oct and Nov 2016 https://gaia.didichuxing.com.", "We have open-sourced the experiment scripts with detailed descriptions and all baseline models (in the cluster-level prediction) we used in this paper https://github.com/fanzipei/crowd-context-prediction/tree/master.", "The code has been modified to work with DiDi Chengdu dataset, so the researchers can reproduce our system easily.", "Figure: DiDi taxi trajectory 6 min / 12 min ahead prediction.", "A significant increase of taxi trajectories in the yellow region can be seen from the upper row, and our predictor can successfully predict the increase of the taxi trajectories.In this experiment, we split the DiDi Chengdu dataset into training set (from 2016.10.01 to 2016.11.14) and testing set (from 2016.11.15 to 2016.11.30).", "Considering the target region of this dataset is smaller and the taxi trips do not have a continuous observation of the user, we cropped a shorter period (all days from 7 a.m. to 11 a.m.) with a smaller interval (1 min).", "We take the order IDs as the pseudo \"user IDs\" use the latest 12 time steps observations of trajectories to predict their future movements (12 min ahead).", "Table REF shows the performance of our cluster-level prediction on DiDi dataset.", "Our proposed predictor outperforms all the baseline models.", "Same to our previous experiments, Context (Mean) is the second-best predictor, still performs slightly better than the Context (Max).", "Conditional and Ensemble predictors do not achieve a good prediction results in this experiment.", "One probable reason is the training dataset include too many irregular patterns.", "Note that the whole first week in Oct is the national holidays in China, and thus we labeled them all as holidays.", "Conditional and Ensemble do not have sufficient ability to distinguish these holidays from normal holidays, which leads to a worse prediction performance.", "In the fine-grained prediction stage, we skipped the stay-move detection and transportation mode classification steps since all the trajectories are all moving cars.", "The original GPS trajectories can be directly taken as fine-grained trajectories because it has a very short time interval (3-4 seconds) and has already been snapped to the road network.", "Our proposed two-stage fine-grained mobility prediction algorithm can also handle this type of data without minor modifications.", "Figure REF shows our fine-grained prediction results.", "Notably, our two-stage fine-grained prediction can not only predict complete trajectories for each individuals at a low cost (on average it takes about 7 seconds on each prediction for over 40K trajectories), but also successfully predict the aggregated trajectory patterns, such as the gathering pattern at a office area in Chengdu as shown on the right column in Figure REF ." ] ]
2207.03575
[ [ "Hypercritical accretion during common envelopes in triples leading to\n binary black holes in the pair-instability-supernova mass gap" ], [ "Abstract Hydrodynamic studies of stellar-mass compact objects (COs) in a common envelope (CE)have shown that the accretion rate onto the CO is a few orders of magnitude below the Bondi-Hoyle-Lyttleton (BHL) estimate.", "This is several orders of magnitude above the Eddington limit and above the limit for neutrino-cooled accretion (i.e., hypercritical accretion, or HCA).", "Considering that a binary system inside the CE of a third star accretes material at nearly the same rate as a single object of the same total mass, we propose stellar-evolution channels which form binary black hole (BBH) systems with its component masses within the pair-instability supernova (PISN) mass gap.", "Our model is based on HCA onto the BBH system engulfed into the CE of a massive tertiary star.", "Furthermore, we propose a mass transfer mode which allows to store mass lost by the binary onto a third star.", "Through the use of population synthesis simulations for the evolution of BBHs and standard binary-evolution principles for the interaction with a tertiary star, we are able to produce BBHs masses consistent with those estimated for GW190521.", "We also discuss the massive binary system Mk34 as a possible progenitor of BBHs in the PISN gap, as well as the spin distribution of the observed mergers in the gravitational-wave catalog." ], [ "Introduction", "Stellar evolution predicts that low-metallicity stars with initial masses $M_{\\rm ZAMS} \\lesssim 133 \\; \\mbox{M$_\\odot $}$ produce black holes (BHs) with $M_{\\rm BH} \\lesssim 50 \\; \\mbox{M$_\\odot $}$ [27], [31], [3].", "If a star has $M_{\\rm ZAMS} \\gtrsim 140 \\; \\mbox{M$_\\odot $}$ , it may experience a pair-instability supernova [6], [31], [32].", "PISN occur when a massive oxygen core is sufficiently hot and dense to efficiently create electron-positron pairs.", "These pairs convert internal gas energy into rest mass, abruptly decreasing the radiation pressure and prompting a rapid contraction of the core.", "This contraction ignites oxygen, leading to a runaway thermonuclear explosion that leaves no remnant [31].", "The PISN results in a predicted mass gap for non-interacting massive stars with masses between $50 \\; \\mbox{M$_\\odot $}\\lesssim M_{\\rm gap} \\lesssim 130 \\; \\mbox{M$_\\odot $}$ [88].", "The binary black hole (BBH) system progenitor of the GW source GW190521 consisted of two black holes (BHs) with masses $M_{\\rm BH, 1} = 85^{+21}_{-14}\\mbox{M$_\\odot $}$ and $M_{\\rm BH, 2} = 66 ^{+17}_{-18}\\mbox{M$_\\odot $}$ [2] which fall in the theoretically predicted PISN mass gap.", "The merger of the BBH radiated about $7.6^{+2.2}_{-1.9} \\; \\mbox{M$_\\odot $}$ in GWs, and resulted in a single 142$^{+28}_{-16}\\mbox{M$_\\odot $}$ BH [2].", "The inferred Kerr spin parameters of the merging BHs was $0.1 \\lesssim a_{\\star } \\lesssim 0.9$ , where $a_\\star \\equiv Jc^2/GM_{\\rm BH}^2$ , being $J$ the angular momentum of the BH, and the spin parameter of the remnant BH was $a_\\star \\simeq 0.72^{+0.09}_{-0.12}$ .", "Several mechanisms have been proposed to produce BBH in the PISN gap.", "These include: a) multiple generations BBH mergers either in clusters [26], [46], [89], in AGN disks [51], [7], or in the field [98]; b) accretion in proto-clusters [71], [72], isolated binaries [95], halos [74], or onto primordial BHs [22]; c) creation of massive BBHs that form from isolated binaries [86], the merger of two stars below the PISN forming a star with a small core and an oversized envelope [67], massive BHs from population III stars [39], [101]; d) mechanisms beyond the standard models of stellar evolution [75], [90], [106].", "The common envelope (CE) phase is a short lived stage during the evolution of a binary system in which the envelope of one of the stellar components engulfs the other component [64], [35].", "Due to mass transfer and orbital decay the orbital separation decreases up to a point in which highly energetic and transitory phenomena may be produced, for example: type Ia supernovae [18], [66], short GRBs [97], long GRBs [14], [57], and GWs [1].", "During the CE phase, which may occur in several occasions during the evolution of the binary system, compact objects may be the engulfed component [37] and the accretion onto the secondary component may play an important role [16].", "Most studies (e.g., those in the third paragraph) assume that accretion is limited by the Eddington luminosity.", "However, accretion onto a single star or a binary system may be super-Eddington or hypercriticalIn super-Eddington accretion ($\\dot{M}_{\\rm Edd} < \\dot{M} \\lesssim 3000\\; \\dot{M}_{\\rm Edd}$ ) the energy excess is lost through photons, while in hypercritical accretion (HCA) the thermal energy is mainly lost through neutrinos, given that for $\\dot{M} \\gtrsim 3000 \\dot{M}_{\\rm Edd}$ and the accreting material has large density and temperature [17], [12].", "[63], [76], [91], [44].", "This is especially true for a compact object (CO) accretor.", "Radiation-hydrodynamics simulations have shown that steady accretion happens once the accretion rate is $\\dot{M} \\gtrsim 3000\\; \\dot{M}_{\\rm Edd}$ , with $\\dot{M}_{\\rm Edd}$ being the Eddington-limited mass-accretion rate [36].", "In addition, HCA can also produce the masses and spins observed in some BH binaries in high-mass, X-ray binary systems [58], [59].", "In this paper, we propose that HCA can play an important role in the formation of massive stellar BHs.", "We show that a BBH system with HCA during a CE phase may produce BHs with masses within the PISN gap.", "We discuss two possible scenarios: a) the evolution of a hierarchical triple system, or b) the outcome of a BBH capturing a massive star.", "We propose a mechanism which allows the system to store material, initially part of the more massive binary stars, in the tertiary star.", "Both scenarios produce BBH with component masses within the PISN mass gap through HCA during the CE phase of the massive star.", "The paper is organized as follows.", "In Section  we discuss the conditions for HCA onto a BBH during the CE phase.", "In Section   we discuss mass-transfer and storage in triple systems.", "In Section  we describe possible evolutionary paths of triple-systems which produce hyper accreting BHs within the CE of a red giant or red super giant.", "In Section  and   we discuss our methods and results and drive our conclusions." ], [ "Hypercritical accretion onto a BBH system during the CE phase", "In this section we describe how during HCA onto a BBH system which is within the CE of a massive star a non-negligible fraction of the massive star may be accreted by the BBH.", "We also discuss how accretion and the ejection process are the key competing mechanisms to determine the amount of hydrogen-envelope accreted onto the inner BBH." ], [ "Accretion onto a BBH", "Previous studies have shown that accretion onto a star or compact object orbiting inside the CE of a red giant (RG) or red super-giant (RSG) occurs at a fraction ($\\epsilon \\approx 0.01-0.1$ ) of the Bondi-Hoyle-Littleton rate [68], [50], [60], [16], [48], [49].", "The accretion rate onto the companion ($\\dot{M}_{\\rm a}$ ) is: $\\dot{M}_{\\rm a} &\\equiv & \\epsilon \\dot{M}_{\\rm BHL} =\\epsilon \\frac{4 \\pi \\rho G^2 M^2}{(v^2+c_s^2)^{3/2}} \\nonumber \\\\&\\simeq & 2.8\\times 10^{-5} \\ \\epsilon _{0.1} \\ \\rho _{-6} \\ M_{50}^2\\ v_7^{-3} \\ \\mbox{M$_\\odot $}\\; {\\rm s}^{-1}$ where $G$ is the Gravitational constant, $\\rho _{-6}$ is the density of the environment in units of $10^{-6}$ g cm$^{-3}$ , $M_{50}$ the mass of the accretor in units of 50 $\\mbox{M$_\\odot $}$ , $v_{7}$ the velocity of the accretor in units of $10^{7}$ cm s$^{-1}$ , and $\\epsilon _{0.1}$ the accretion efficiency normalized to $\\epsilon =0.1$ (which depends on the structure of the envelope, in particular on the presence of density and velocity gradients).", "Since the sound speed within the envelope of a RG or RSG is $c_s \\sim 10\\; (T/10^4 {\\rm K})^{1/2}$ km s$^{-1}$ , and the velocity of the accreting material is of order of the keplerian velocity ($v \\sim v_{k}\\gtrsim 100$ km s$^{-1}$ ), we have $(v^2+c_s^2)^{3/2}\\sim v^3$ .", "If the secondary is engulfed by the CE in a sudden plunge, the orbital period of the accretor as it enters the envelope of the CE may be completely asynchronised with respect to the rotation period of the donor.", "This scenario leads, at first order approximation, to BHL accretion.", "The case of a gradual plunge occurs when the donor and the accretor are fully synchronized, and thus, the relative velocity is zero.", "On the other hand, if the secondary is engulfed gradually by the CE, then approximately, Bondi accretion takes place (case in which $v\\lesssim c_s$ in equation REF ).", "Density and velocity gradients, magnetic fields, convection flows, Coriolis, centrifugal and Euler forces, and the presence of a preexisting disk, to name some, modify the simple picture of BHL or Bondi accretion, and thus it is uncertain, in any case, whether a disk may or may not form around the accretor.", "Meanwhile, accretion onto a BBH system depends on the orbital separation $a$ between the BHs and on the accretion radius ($r_a = 2 G M/v^2 \\sim 3 \\times 10^{14}~M_{100}~v_7^{-2}~{\\rm cm}$ ; see, e.g., [24]).", "If the BBH forms part of a triple system, a fraction of the envelope of the tertiary can be accreted during a CE phase by the CE-engulfed BBH (see Section REF ).", "Also, in the case of a triple system, it must be considered that the Keplerian velocity of the binary will never be zero, regardless of whether synchronization is achieved or not with the envelope of the tertiary.", "In order to model the effect of accretion of a binary system inside a CE of a third star, we follow [19], [20].", "These studies present numerical simulations of accretion onto binaries for a wide range of orbital periods and inclinations.", "They find that for small orbital separations ($a \\ll r_a$ ) the accretion takes place onto a point mass (located at the center of mass of the BBH) with a mass given by the sum of the two individual masses ($M=M_{\\rm BH_1} + M_{\\rm BH_2}$ ).", "When $a \\gg r_a$ , each BH accretes as a single object.", "The intermediate case ($a \\sim r_a$ ) is complex and remains to be fully studied.", "In the following we will focus on the case in which $a\\gg r_a$ .", "From equation REF it is clear that the mass of the black holes ($M_{\\rm BH,i}$ , with $i=1,2$ ) changes as $\\dot{M}_{\\rm BH,i} = k M^2_{\\rm BH,i}$ (where $k= 4 \\pi \\epsilon \\rho G^2/v^3$ ), which by direct integration gives: $M_{\\rm BH,i} = \\frac{M_{{\\rm BH,{i_0}}}}{1- k M_{{\\rm BH,{i_0}}}t}\\;,$ where $M_{{\\rm BH,{i_0}}}$ is the initial mass of the black hole $i$ .", "The black hole $i$ achieves a mass $M_{\\rm BH,i}$ in a time: $t_{\\rm f,i} = \\left( 1-\\frac{M_{{\\rm BH,{i_0}}}}{M_{\\rm BH,i}}\\right) \\frac{1}{k M_{{\\rm BH,{i_0}}}}\\;.$ Since $t_{\\rm f,1} = t_{\\rm f,2}$ , then, the following relationship between the ratio of the initial and the final masses of the black holes ($q_0=M_{{\\rm BH,{2_0}}}/M_{{\\rm BH,{1_0}}}$ and $q=M_{\\rm BH,2}/M_{\\rm BH,1}$ respectively) is obtained: $q_0 = \\frac{q M_{\\rm BH,1}/M_{{\\rm BH,{1_0}}}}{1+q( M_{\\rm BH,1}/M_{{\\rm BH,{1_0}}}-1)}\\;.$ Using equations 2, 3, and 4 the initial masses of the BHs and the accretion time $t_f$ (for the BHs to reach their final masses), are obtained as a function of the final masses.", "In Figure REF we plot the evolution of the BBH during HCA when inside the CE of a third star.", "Specifically, we plot the range of possible initial masses of the two merging BHs (with $M_{\\rm BH,1} > M_{\\rm BH,2}$ ), the amount of mass accreted by the binary system, and the time required for the BHs to reach the masses of GW190521 (85$\\mbox{M$_\\odot $}$ and 66$\\mbox{M$_\\odot $}$ ).", "For the latter, we assume that the accretion rate onto the BBH during the CE phase is $10\\%$ of the BHL accretion rate.", "The accreted mass will be a substantial fraction of the envelope of the tertiary star in which the BBH is immerse.", "If the initial mass of both BHs is large, the accretion time required to reach masses similar to those of GW190521 is considerably smaller (a few days to over a month) and the required mass transferred and accreted from the donor star drops to a few tens of solar masses.", "For example, if the initial mass of the secondary BH is $M_{\\rm BH,2}\\approx 40\\; \\mbox{M$_\\odot $}$ , then the mass of the primary BH must be $M_{\\rm BH,1}\\approx 45\\; \\mbox{M$_\\odot $}$ (from the solid black line) and the accretion time for each of the BHs to reach $65\\; \\mbox{M$_\\odot $}$ and $85\\; \\mbox{M$_\\odot $}$ (respectively, and assuming 10% of the BHL accretion rate) is $\\simeq 10$ d (red solid line).", "The total mass transferred and accreted from the envelope of the tertiary star needs to be $\\Delta M_{\\rm CE}$ $\\gtrsim 65\\; \\mbox{M$_\\odot $}$ (dashed black line).", "Assuming a maximum available envelope mass of 100 $\\mbox{M$_\\odot $}$ then $\\gtrsim 65\\%$ of the envelope must be accreted (for further details see Section ).", "Figure: The solid black line indicates the initial mass of the primary BH (M BH ,1 M_{\\rm BH,1}) as a function of the initial mass of the secondary BH (M BH ,2 M_{\\rm BH,2}) in the binary system.", "The dashed black line represents the mass accreted by the BBH system.", "The red solid line and the right yy-axis show the accretion time needed for the primary and secondary BHs to end up with 85 M ⊙ \\mbox{M$_\\odot $} and 65 M ⊙ \\mbox{M$_\\odot $} respectively during the CE phase (assuming an accretion rate given by 10% of the BHL rate).", "The gray shaded area is excluded as the initial mass of the primary BH is larger than ≳\\gtrsim 55 M ⊙ \\mbox{M$_\\odot $}.", "Larger BH masses are excluded by single stellar evolution models (see Section )." ], [ "Limits on the accretion onto the BBH", "The accretion onto a BBH system may be limited by several effects: a) the radiation pressure of the accreting material, b) the orbital timescale, and c) the amount of material ejected from the CE.", "In this section, we discuss these effects and argue that they do not represent a strong limiting factor to the accretion onto the BBH.", "Firstly, we discuss the feedback due to radiation pressure and neutrino effects.", "The accretion rate given by equation ($\\ref {eq:BHL}$ ) is $\\sim $ 10 orders of magnitude above the Eddington mass accretion rate given by: $\\dot{M}_{\\rm Edd} = \\frac{L_{\\rm Edd}}{c^2} \\simeq 3.5\\times 10^{-15} M_{50}\\ \\mbox{M$_\\odot $}\\; {\\rm s}^{-1}\\;,$ where $L_{\\rm Edd}$ is the Eddington luminosity and $c$ is the speed of light.", "Nevertheless, the Eddington limit assumes spherical symmetry, i.e., Bondi accretion.", "BHL accretion breaks spherical symmetry, leading to radiation ejected through polar jets and material accreted via the equatorial plane.", "Additionally, velocity and density gradients, as well as Coriolis and centrifugal forces, may further break the cylindrical symmetry of the BHL accretion [60], [48], [49].", "Thus, the BHL accretion rate onto a BH which is within the CE of a massive RG or RSG is HCA.", "In this case, as the accretion rate is $\\dot{M}_{\\rm a}\\gtrsim 3\\times 10^3 \\; \\dot{M}_{\\rm Edd}$ , neutrinos are produced [33], [12].", "Since the density and temperature of the material in the accretion flows around the BH are well below those in core-collapse supernovae, close to where the Eddington limit for neutrinos lies, these neutrinos will not interact with the accretion flow.", "Therefore, most of the released binding energy will be radiated away by neutrinos which leave the CE without exerting pressure on the accreting material and allowing for extremely large accretion rates.", "Hence, the radiation pressure and neutrino emissivity will not have effect on our results.", "Second, we consider the limit imposed on accretion due to the orbital timescale.", "With $\\dot{M}_{\\rm a}\\gtrsim 10^{9} \\dot{M}_{\\rm Edd}$ (see equations 1 and 5), and most of the binding energy being radiated away by neutrinos, the BBH could increase its mass from $M_{\\rm BBH,i} \\simeq 55 \\;\\mbox{M$_\\odot $}+ 45\\; \\mbox{M$_\\odot $}= 100\\; \\mbox{M$_\\odot $}$ to $M_{\\rm BBH,f} \\simeq 85\\; \\mbox{M$_\\odot $}+ 65\\;\\mbox{M$_\\odot $}= 150\\; \\mbox{M$_\\odot $}$ within a few weeks of CE (see Figure REF ).", "However, if the initial orbital separation in the triple system is of order $a_{\\rm tri,i} \\gtrsim 10^{14}$ cm (considering a $M_\\star = 150 \\mbox{M$_\\odot $}$ donor star as well as the BBH system), the initial orbital period will be: $T = \\frac{2\\pi R}{v_k} = 2\\pi \\sqrt{\\frac{R^3}{GM}} \\sim 100~a_{\\rm tri,14}^{3/2}~M_{100}^{-1/2}~{\\rm d}\\;.$ As the BBH can not accrete an amount of mass larger than the stellar material available during their orbit around the companion star, the timescale for the accretion onto the BBH system will be the largest between $T$ and $\\tau _{\\rm BHL}\\sim 30 \\; (\\epsilon /0.1)^{-1}$ d. Since the BHL accretion timescale ($\\tau _{\\rm {BHL}}$ ) is smaller than the orbital timescale, accretion is limited by $T$ .", "As the orbital separation decreases, so does the orbital period, and it continues to decrease until $T\\sim \\tau _{\\rm {BHL}}$ .", "Last, the ejection of stellar material from the CE could limit accretion onto the BBH (as the material ejected from the CE and that lost by the triple system could unbinding the envelope).", "The BBH massHere, by accretion on the BBH we refer to the accretion onto the system which could include an accretion disk around the BBH.", "The actual accretion onto the BBH will be then delayed by the viscous time.", "increases as $\\dot{M}_a = \\epsilon \\dot{M}_{\\rm BHL}$ (equation REF ), while the envelope mass drops by the same amount, i.e.", "$\\dot{M}_{\\rm env} = - \\dot{M}_a$ .", "We limit the amount of accreted mass as the minimum between the BHL accreted one and the mass available at radius larger than the orbital separation ($M_\\star (r>a)$ ), that is: $M_{\\rm BBH}(t) &=& M_{\\rm BBH, 0} + \\min \\left(M_*(r>a),\\int _0^t \\dot{M}_a(t^{\\prime }) dt^{\\prime }\\right) \\;, \\\\M_{\\rm env}(t) &=& M_{\\rm env, 0} - (M_{\\rm BBH}(t) - M_{\\rm BBH, 0})\\:,$ where $M_{\\rm BBH, 0}$ , $M_{\\rm env, 0}$ are the initial BBH and envelope masses.", "The inspiral of the BBH will be driven by the the gravitational drag due to the envelope material accreting onto the BBH [47], which produces a change in the BBH orbital energy (around the massive star) as: $\\frac{dE_{\\rm orb}}{dt} = - \\dot{M}_a v_k^2\\;,$ where $\\dot{M}_a$ is the mass accretion rate (given by equation REF ), $v_k$ is the Keplerian velocity ($v_k=\\sqrt{G M_\\star (r \\le a)/a}$ , with $M_\\star (r \\le a)$ the enclosed mass within the orbital separation), and the orbital energy is $E_{\\rm orb} = -0.5\\;G M_{\\rm BBH} M_\\star (r \\le a)/a$ .", "Assuming that a fraction $\\alpha $ of the orbital energy is deposited (as thermal energy) into the envelope, thus, the energy deposited ($E_d$ ) is given by $\\frac{dE_d}{dt} = \\alpha \\dot{M}_a v_k^2\\;,$ Taking the $\\alpha \\lambda $ energy formalism of [94] in which the envelope will be ejected once the energy deposited is of order of the binding energy [102], [62], and using equations REF , REF (as well as the binding energy of the envelope, $E_b$ ), then the condition $E_b\\sim E_d$ reads $\\frac{G M(a)M(r>a_f)}{\\alpha \\lambda a}\\sim \\int _0^t \\dot{M}_a v_k^2 dt\\;,$ where $\\lambda $ is a numerical factor which depends on the density structure of the star [23].", "By numerically integrating this set of equations (REF -REF ) (and considering a stellar core with a mass $35\\ \\mbox{M$_\\odot $}$ , an envelope density profile, between $2\\times 10^{11}$ cm and $2 \\times 10^{13}$ cm, $\\rho \\propto r^{-n}$ (with $n=2.5-2.8$ ) and a total envelope mass of 65 $\\mbox{M$_\\odot $}$ , the BBH inspiral time is $\\sim 100-170$  days, and most the material can be accreted by the BBH (since $E_b\\gtrsim 10 E_d$ at all times) for $\\alpha \\le 1$ .", "Thus, the loss of material during the accretion phase is minor (at least, in the framework of this simple analytical method), and the system is stable." ], [ "Jets as an extra energy injection source.", "If an accretion disk does not form around the accreting component, HCA may only proceed through neutrino-radiation cooling.", "[17], [12].", "In this case, there is no extra energy injection into the CE other than the orbital energy, parametrised by the $\\alpha \\lambda $ formalism.", "On the other hand, when an accretion disk forms around an accreting BH, a collimated and relativistic jet may be launched [5], [82].", "In this case, as shown by [60], the energy deposition into the CE will dominate over the $\\alpha \\lambda $ and other energy sources [37].", "Switching on this mechanism can bring the system into the so-called grazing envelope phase [73], [83], [78], [79] and may even terminate the CE phase in extreme cases by removing the envelope [84].", "Nonetheless, as the energy deposited by the jet will drop the accretion rate, the jet power will also drop, then alternating periods of accretion with period of quiescence.", "The “self-regulation” of the jet guarantees that a significant percentage of the CE available material will be accreted (although it does not guarantee that the orbital separation will decrease).", "Given the complexity of the problem, future numerical works are needed to understand the full impact of jets or outflows in the CE evolution (hereafter, CEE).", "One of the main processes determining the evolution of a multiple-stellar system is mass transfer between the components.", "In this Section, we describe our assumptions and the uncertainties on mass transfer mechanism, particularly in the context of a CEE from a tertiary donor with an inner BBH accretor.", "We provide a qualitative description of each mechanism and we estimate the physical quantities (in particular, orbital separations and periods) characterizing the system.", "In the Roche-lobe overflow (RLOF) mechanism the mass is transferred to the companion once the radius of the envelope of the donor exceeds the volume-averaged Roche-lobe (RL) radius $r_{\\rm RL}$ [25]: $\\frac{r_{\\rm RL}}{a} = \\frac{0.49 q^{2/3}}{0.6 q^{2/3} + \\ln (1 + q^{1/3})} \\; ,$ where $q$ is the mass ratio of the binary components ($q=M_{\\rm {d}}/M_{\\rm {a}}$ , with $M_{\\rm {d}}$ the donor and $M_{\\rm {a}}$ the accretor).", "During RLOF two important processes may allow a binary system to transfer mass [80], these are: a) when the donor star is more massive than the accretor the orbital separation decreases (as well as the RL of the donor); b) the reaction of the donor star to the loss of its outer layers.", "In the latter if the envelope is radiative it will contract as a response to mass loss, while if it is convective it will expand [81].", "The envelopes of RG stars is usually radiative, however, RSG stars have convective envelopes [40].", "If the star expands, it is likely that the star will continue to fill its RL (or that it will fill it up again, if the orbital separation increased); this process occurs in a Kelvin-Helmholtz (or thermal) timescale [80].", "In order to maintain a stable RLOF phase, at least one of these runaway processes must be at play [81].", "Another important mass-transfer mechanism in the evolution of multiple stellar systems is the wind RLOF (wRLOF), in which the wind escaping the donor star is swept by the accreting object.", "For close binary systems, [55] found that mass transfer through wRLOF can be highly efficient (with up to $\\sim 70\\%$ of the wind being captured and accreted by the companion), when the orbital velocity of the binary is comparable to the wind velocity.", "A compact orbit is a necessary condition for a highly efficient wRLOF mass-transfer, because it allows for a large orbital velocity as well as a stellar wind which has not been fully accelerated.", "Stellar winds, particularly in massive stars, require $\\sim 10-15\\ \\mbox{R$_\\odot $}$ to fully accelerate, hence the winds in this scenario have not reached escape velocity and can be substantially funneled through the first Lagrange point (L1) and focused towards the accretor [99].", "This mechanism is an efficient mass transfer mechanism in the evolutionary stages of the massive stellar systems [100].", "An additional mass-transfer mechanism may occur in triple systems, specifically, when the inner binary system is in a CE phase and the tertiary is far orbiting around the binary system.", "In this stage, a substantial amount of material from the outer layers of a CE may be inflated without being fully unbound from the binary system; in fact, this material may even form a circumbinary disk [38].", "This material may slowly move to a large orbit around the binary (and may even fall back towards the binary if its velocity is below the escape velocity).", "So, if a tertiary star orbits the binary it may be able to capture a large fraction of this material, see Section REF and Section REF for further details.", "This mass-transfer mechanism can be important in triple systems and we term it common-envelope Roche-lobe overflow (ceRLOF).", "In principle, ceRLOF allows for an effective mechanism to transfer mass in a conservative way (hereafter, we refer to conservative mass-transfer as a process in which the mass lost from the binary system is accreted by the tertiary star) and store a substantial fraction of the envelopes of both stars in the inner binary in the tertiary star.", "During the evolution of our triple-stellar systems, mass is transferred back and forth between the stellar components at different stages, particularly during CE stages from the inner binary (ceRLOF).", "The captured mass will later be returned to the inner BBH after the tertiary, and last-to-evolve star, becomes a giant which fills its RL and engulfs the BBH in a third CE stage.", "In this scenario, a large-mass BBH can be formed successfully (see Section ) without requiring extremely massive stars ($M_\\star > 150 \\mbox{M$_\\odot $}$ ) or systems with more than three starsWe assume that the two SNe forming heavy BHs lose only a small fraction of the mass of binary [28].." ], [ "CE-BBH scenarios", "In this section, we discuss two possible scenarios which may produce a BBH system with masses close to those in the PISN mass gap which have been inferred for GW190521 by the Ligo-Virgo-Kagra collaboration.", "First, we describe the evolution of a binary system, composed by two massive stars, which produces a BBH (Section REF ).", "Then, we consider the case of ceRLOF from a tertiary star in the case when the tertiary is pre-existent (Section REF ), and the case in which a tertiary star has been captured by the BBH system (Section REF )." ], [ "Evolution of the inner massive binary in a hierarchical triple stellar system", "In order to look for a binary a binary system with component masses close to the lower edge of the pair-instability black-hole mass gap ($M_{\\rm BH,1} \\sim M_{\\rm BH,2} \\sim 45\\mbox{M$_\\odot $}$ ) we use the population synthesis code COMPAS [69].We adopt the default choices of COMPAS parameters for binary evolution which follows the classic CE formation scenario of BBH mergers [9], [87], [96].", "The evolution of such binary is shown in Figure REF and discussed next (see also Table REF ): Figure: Evolutionary pathway for a binary system with a primary star with M 1 =119.2M ⊙ M_1=119.2\\ \\mbox{M$_\\odot $}, secondary with M 2 =79.7M ⊙ M_2=79.7\\ \\mbox{M$_\\odot $} (both with Z=0.0001Z=0.0001), and an orbital separation of a=464.6R ⊙ a=464.6\\ \\mbox{R$_\\odot $}.", "The upper panel shows the evolution of the primary, secondary and the total masses.", "The bottom panel shows the evolution of the radii of the two stars (R 1 R_1 and R 2 R_2), the correspondent Roche lobe radii of each of the stars (R RL ,1 R_{\\rm RL,1} and R RL ,2 R_{\\rm RL,2}), and the orbital separation of the binary (semi-major axis aa and periastron a p a_{\\rm p}).Table: Principal stages (and main values) of the evolution of the binary shown in Figure .", "*Note: 40M ⊙ 40~\\mbox{M$_\\odot $} are accreted by the most massive BH (M 1 M_1) and 26M ⊙ 26~\\mbox{M$_\\odot $} by the least massive BH (M 2 M_2).", "The binary system begins at the zero-age main sequence (ZAMS) with a primary of 119.2 $\\mbox{M$_\\odot $}$ and a companion of 79.7 $\\mbox{M$_\\odot $}$ (both with a metallicity of $Z=0.0001$ ) in a circular orbit with a semi-major axis of 464.6 $\\mbox{R$_\\odot $}$ .", "At $\\approx $ 3.4 Myr, the post-main-sequence primary fills its RL and begins a semi-conservative episode of mass transfer with the main-sequence companion.", "In this first episode, the primary donates $\\approx $ 67.1 $\\mbox{M$_\\odot $}$ , of which, $\\approx $ 33.0 $\\mbox{M$_\\odot $}$ are accreted by the main-sequence companion, and the remaining $\\approx $ 34.1 $\\mbox{M$_\\odot $}$ are lost from the (inner) binary.", "This semi-conservative episode where mass is lost via isotropic re-emission widens the (inner) orbit to $\\approx $ 737.1 $\\mbox{R$_\\odot $}$ .", "Shortly after, at $\\approx $ 3.6 Myr, the primary collapses into a 45.8 $\\mbox{M$_\\odot $}$ BH.", "The secondary star finishes the main sequence and enters the Hertzsprung gap.", "At $\\approx $ 4.6 Myr it begins a dynamically unstable mass transfer episode.", "This mass transfer episode results in a common-envelope episode, a process where $\\approx $ 68 $\\mbox{M$_\\odot $}$ of envelope can be removed from the (inner) binary.", "This process results in an $\\approx $ 44.1 $\\mbox{M$_\\odot $}$ stripped star in a close binary with orbital separation of several solar radii.", "Finally, the secondary collapses at $\\approx $ 4.9 Myr to form an $\\approx $ 39.6 $\\mbox{M$_\\odot $}$ BH.", "We choose this low-metallicity binary system in order to neglect the effect of stellar winds in our calculations.", "Moreover, lower metallicity environments are predicted to lead to more massive BBH remnants [61] and compact BBHs [98]." ], [ "Hierarchical triple system", "The hierarchical-triple-system (HTS) channel which we propose involves a massive inner stellar binary (described in Section REF ) and a dynamically-stable tertiary star orbiting around it [34].", "The tertiary can capture and store a large fraction of the mass which is ejected from the inner binary system as it becomes a BBH system.", "Once the BBH is produced the tertiary may begin its RLOF phase and with this transfer mass back onto the BBH.", "Such a system may form in a massive stellar-forming regions via gravitational capture, or as a triple system in the field [54].", "The condition for dynamical stability of a HTS is that the orbit of the tertiary star ($a_{\\rm out}$ ) is always larger than three times the orbit of the binary ($a_{\\rm in}$ ), this is, $a_{\\rm out} \\gtrsim 3 a_{\\rm in}$ [42], [45], [77].", "To simplify our model, we assume that mass is transferred from the inner binary and onto the tertiary star during the ceRLOF stage (and not during the RLOF stage when the primary star expands; nonetheless this early stage could also transfer up to 33.4$\\mbox{M$_\\odot $}$ onto the tertiary).", "If this was not the case, the initial orbital separation of the tertiary star, as well as the subsequent stages, would need to be larger to prevent the HTS from becoming unstable.", "Using the results from COMPAS for the binary system (see Section REF ), and for the tertiary star evolution either conservative mass transfer during RLOF, CEE, or ceRLOF stages, as well as isotropic mass loss during SNe, we broadly calculate the evolution of the HTS (see Table REF for details).", "As mentioned in Section REF the maximum orbital expansion of the inner binary ($a_{\\rm in}$ ), which occurs after the SN explosion of the primary, is $a_{\\rm in} = 766.5 \\mbox{R$_\\odot $}$ .", "With the latter, $a_{\\rm out} = 2300 \\mbox{R$_\\odot $}$ and the evolution of the HTS can be modeled.", "The main stages are next listed: At ZAMS, the tertiary star, of mass $M_{3} \\simeq 35 \\mbox{M$_\\odot $}$ and metallicity $z = 0.0001$ , is located in a circular orbit at $a_{\\rm out} \\sim 1817~\\mbox{R$_\\odot $}$ .", "Approximately 3.4 Myr after ZAMS, the primary star reaches RLOF and transfers $\\Delta M \\sim 67.1 \\mbox{M$_\\odot $}$ , the secondary accretes $\\sim 33.4\\ \\mbox{M$_\\odot $}$ (the rest is lost).", "At this stage the tertiary could accrete a fraction of the mass which is lost (if its orbital velocity is comparable to the ejected mass velocity), but our model does not account for this.", "In response to the mass loss the binary orbit expands to $a_{\\rm in} \\simeq 737.1\\ \\mbox{R$_\\odot $}$ , and the orbit of the tertiary star to $a_{\\rm out} \\simeq 2216\\ \\mbox{R$_\\odot $}$ .", "At 3.6 Myr, the primary star collapses onto a $45.8\\ \\mbox{M$_\\odot $}$ BH (losing $6.1\\ \\mbox{M$_\\odot $}$ ).", "The inner binary orbit expands to $a_{\\rm in} = 766\\ \\mbox{R$_\\odot $}$ , and the orbit of the tertiary star reaches the minimum stable orbit for the HTS with $a_{\\rm out} = 2300\\ \\mbox{R$_\\odot $}$ .", "At 4.6 Myr, the secondary star expands and produces a CE with the first-born BH.", "We assume that no accretion onto the BH occurs (although a few $\\mbox{M$_\\odot $}$ could be accreted); instead, $68\\ \\mbox{M$_\\odot $}$ are ejected (not necessarily unbound) from the secondary star, which gets stripped down to $44.1\\ \\mbox{M$_\\odot $}$ .", "The ejected material, has low velocity and thus a large fraction of it may be captured by the tertiary star, i.e., it undergoes ceRLOF mass transfer.", "During the CE phase of the binary system, the orbit of the inner binary decays to $a_{\\rm in}=6.4\\ \\mbox{R$_\\odot $}$ , while the tertiary star captures up to $68\\ \\mbox{M$_\\odot $}$ of the ejected mass.", "Assuming conservative mass transfer during this ceRLOF phase, the mass of the tertiary star grows to $M_{3} \\simeq 103 \\mbox{M$_\\odot $}$ which brings the mass ratio of the tertiary to the inner binary close to one ($q_3=M_{3}/(M_{1}+M_{2})=103/(45.8+44.1)=1.15$ ), thus reducing its orbit to $a_{\\rm out} \\simeq 818\\ \\mbox{R$_\\odot $}$ .", "At 4.9 Myr, the secondary star collapses to a $39.6\\ \\mbox{M$_\\odot $}$ BH, losing $4.4\\ \\mbox{M$_\\odot $}$ in the process.", "Both orbits expand a little after the mass loss to $a_{\\rm in} \\simeq 6.7\\ \\mbox{R$_\\odot $}$ and $a_{\\rm out} \\simeq 836\\ \\mbox{R$_\\odot $}$ .", "The RL radius of the tertiary star reaches $R_{\\rm RL,3} \\simeq 330\\ \\mbox{R$_\\odot $}$ .", "A few hundred kyrs later, the tertiary star ends its MS stage, expands, fills its $R_{\\rm RL,3}$ and produces a CEE with the BBH.", "In this CEE, the BHs accrete several tens of solar masses, ending at $M_{\\rm BH1} \\simeq 86\\ \\mbox{M$_\\odot $}$ and $M_{\\rm BH2} \\simeq 66\\ \\mbox{M$_\\odot $}$ .", "The CEE further decreases the inner orbital separation.", "While, the loss of mass from the tertiary onto the binary decreases the mass ratio and expands the orbit of the tertiary to $a_{\\rm out} > 2074\\ \\mbox{R$_\\odot $}$ .", "Finally, the BBH will merge via GW emission in $\\sim $ Myr.", "This timescale is similar to that when the tertiary star will become a BH, but a sequential BBH merger will not occur within the age of the Universe.", "Depending on what happens first, and the magnitude and direction of the recoil kick of the BBH product, the system might be disrupted or remain bound, now as a binary." ], [ "Star-forming region capture and CEE - BBH captures a massive star", "We consider the binary system described in Section REF .", "If it captures a tertiary massive star companion ($M_3 > 35\\ \\mbox{M$_\\odot $}$ ) before the CE phase, then we can obtain a HTS system which can also produce the desired BBH in the PISN.", "The tertiary star could capture $68~\\mbox{M$_\\odot $}$ of the binary system in a ceRLOF and evolve similarly to the HTS in Section REF .", "Given the CEE stage of the inner binary, the tertiary star could be captured into a much smaller orbital separation ($a_{\\rm out} \\gtrsim 30~\\mbox{R$_\\odot $}$ ) without disturbing the evolution of the BBH.", "If this is the case, the CEE in which mass is transferred onto the BBH may occur in a shorter timescale than in our HTS scenario (as the tertiary star will RLOF at an earlier time).", "Alternatively, the tertiary could be captured at a later stage (e.g.", "once the BBH has formed).", "In this scenario the captured star must be more massive ($M_3 > 100 \\mbox{M$_\\odot $}$ ) given that it will not necessarily be able to capture mass during the ceRLOF of the binary.", "A caveat here would be that the ejected mass (the $68~\\mbox{M$_\\odot $}$ ) may not have escape velocity and fall back onto the binary (and be accreted by the BBH), and another part may form a circumbinary disk around the tertiary star.", "Eventually the captured tertiary would star evolve, expand, fills its RL and produce a CEE with the BBH where the BHs reach their final masses before merging ($M_1 > 86~\\mbox{M$_\\odot $}$ and $M_2 > 66~\\mbox{M$_\\odot $}$ ).", "The capture of a wandering star, or of a star ejected from another binary/tertiary system may happen at different evolutionary phases of the binary.", "At early stages of BBH formation (e.g., stage i in Section REF ), the binary increases its mass (to $\\approx 200~\\mbox{M$_\\odot $}$ ) and has a large cross section; however, the evolutionary lifetime as a stellar binary is relatively short ($\\sim $ Myr).", "If the tertiary is captured during the next 200 kyr (stages ii and iii), given the decrease in total mass ($\\sim 160~\\mbox{M$_\\odot $}$ ) and the increase in orbital separation ($\\lesssim 766~\\mbox{R$_\\odot $}$ ), the BBH will be more prone to disruption.", "Finally, after the CE phase (stage iv) the binary system will have a much smaller cross section ($a_{\\rm in} \\lesssim 7~\\mbox{R$_\\odot $}$ ), less mass ($M_1 \\gtrsim 46~\\mbox{M$_\\odot $}$ and $M_2 \\gtrsim 40~\\mbox{M$_\\odot $}$ ), but will have a large amount of time available to capture a tertiary before merging ($\\tau _{\\rm GW} \\sim 7$ Gyr).", "The expected capture rate is larger during the ZAMS phase, with twice the mass and 70 times larger orbital separation (so that the the cross section is $\\sim 10^4$ larger) with respect to the BBH phase even though the timescale is 2000 times smaller.", "Also, the binary will be closer to the star formation region where it was formed, thus the stellar density will be larger.", "Table: PISN-BH candidates from .", "The columns correspond to the GW event, the mass of the BH remnant (M rem M_{\\rm rem}), the masses of the merging BHs (M BH ,1 M_{\\rm BH,1} and M BH ,2 M_{\\rm BH,2}), and the effective spin (χ eff \\chi _{\\rm eff})." ], [ "Channels to produce BHs with masses in the PISN mass gap", "In this paper we propose two evolutionary channels in which, through HCA onto a BBH during the CE phase, BHs with masses falling in the theoretically predicted PISN mass gap may be produced.", "Thus, the BBH system progenitor of the GW source GW190521 may have been formed via such scenarios.", "Our results are based on two key processes: i) a ceRLOF, where a fraction of the material lost by the stellar progenitors of the BBH system is stored into a tertiary, massive star, and ii) HCA, which allows a large amount of mass to be accreted by the BBH, growing one of the BHs (or the two BHs) well above the PISN mass gap.", "We stress that, while the Eddington limit applies for spherical symmetry and assumes electron scattering for the momentum deposition by the photons in the accreting material, during HCA most of the energy is lost through neutrinos (whose mean free path is many orders of magnitude larger than that of photons).", "In the first channel, a binary system with $\\sim 120~\\mbox{M$_\\odot $}$ and $\\sim 80~\\mbox{M$_\\odot $}$ ZAMS stars evolves inside a HTS producing a BBH system of $\\sim 45~\\mbox{M$_\\odot $}$ and $\\sim 40~\\mbox{M$_\\odot $}$ .", "In this scenario, most of the mass lost by the binary system is stored in a tertiary star which later returns it to the BBH enhancing both BH masses.", "In the second scenario, a tertiary star is captured by the binary system (which in turn evolves in a similar path as the first proposed evolutionary channel).", "In both scenarios, the tertiary star evolves out of the MS stage after the BBH with BH masses of $M_1 \\simeq {45}~\\mbox{M$_\\odot $}$ and $M_2 \\simeq {40}~\\mbox{M$_\\odot $}$ has formed.", "The tertiary expands and engulfs the BBH in a CE, transferring through HCA in the CE phase $\\sim 40 \\mbox{M$_\\odot $}$ to the $45-\\mbox{M$_\\odot $}$ BH, and $\\sim 25 \\mbox{M$_\\odot $}$ to the $40-\\mbox{M$_\\odot $}$ BH, leaving behind two BHs with masses of $M_1 \\simeq {85}~\\mbox{M$_\\odot $}$ and $M_2 \\simeq {66}~\\mbox{M$_\\odot $}$ .", "Then, a BBH system with an orbital separation $a_{\\rm BS}\\lesssim 10\\; \\mbox{R$_\\odot $}$ will merge in a timescale of $\\tau _{\\rm GW} \\lesssim T_H/2$ , where $T_H$ is the Hubble timescaleUsing $z = 0.82\\rm $ in [15] one obtains a light travel time $\\lesssim 28/[1+\\rm (1+z)^2]\\; {\\rm Gyr} = 6.5$ Gyr (with which the maximum time for the BBH to merge is $\\approx 7.3$ Gyr).", "From [53] the maximum orbital separation is: $a &=& \\left(\\frac{256 G^3 \\eta M^3 \\tau _{GW}}{5 c^5}\\right)^{1/4} \\nonumber \\\\&\\simeq & 55 \\ \\mbox{R$_\\odot $}\\left(\\frac{\\tau _{GW}}{7 {\\rm Gyr}}\\right)^{1/4} \\left(\\frac{M_{\\rm BBH}}{150\\mbox{M$_\\odot $}}\\right)^{3/4} \\left(\\frac{\\eta }{0.25}\\right)^{1/4}$ with $M_{\\rm BBH} = M_{\\rm BH,1} + M_{\\rm BH,2} = 85\\ \\mbox{M$_\\odot $}+ 65\\ \\mbox{M$_\\odot $}= 150\\ \\mbox{M$_\\odot $}$ , $\\eta = (M_{\\rm BH,1} M_{\\rm BH,2})/M_{\\rm BBH}^2 \\simeq 0.25$ .", "This corresponds to an orbital period of 3.88 d. in a galaxy which is $5.3$ Gpc ($z = 0.82$ ) from us." ], [ "Possible progenitors", "MelnicK 34 (Mk34) is a spectroscopic binary, X-ray colliding-wind system, with two hydrogen rich WR stars [21], [30].", "Mk34 has a $155.1 \\pm 1$ d orbital period, and a projected distance of 2 pc from R136 (the stellar cluster with the most massive stars in 30 Doradus, [65]).", "Its current age is $\\sim 0.6\\pm 0.3$ Myr, and the masses of the WRs are $139^{+21}_{-18}~\\mbox{M$_\\odot $}$ for the primary and $127^{+17}_{-17}~\\mbox{M$_\\odot $}$ for the secondary, corresponding to ZAMS masses of $144^{+22}_{-17}~\\mbox{M$_\\odot $}$ and $131^{+18}_{-16}~\\mbox{M$_\\odot $}$ respectively [93].", "Thus, this system represents a possible progenitor of the BBH studied in this paper.", "Modeling the future evolution of Mk34, the primary star will have a maximum radius of $R_{\\rm max} \\sim 78-126~\\mbox{R$_\\odot $}$ during MS, with a periastron orbital separation of $254~\\mbox{R$_\\odot $}$ [93].", "Also, the primary star in Mk34 will fill its Roche lobe (with $400 - 500~\\mbox{R$_\\odot $}$ ) and begin a mass transfer phase before it forms a BBH system (with masses $<45~\\mbox{M$_\\odot $}$ ) in 2 to 2.5 Myr [105].", "A more recent study using COMPAS finds that the future evolution of Mk34 is uncertain [11], allowing for a system like the one predicted in Section REF .", "Thus, it is possible that even at metallicities as large as those of LMC these kind of BBH mergers may occur.", "If Mk34 was accompanied by a $35~\\mbox{M$_\\odot $}$ tertiary or if the binary system captures a tertiary star (probably another runaway star from R136) within the mass range $50~\\mbox{M$_\\odot $}\\lesssim M_3 \\lesssim 100~\\mbox{M$_\\odot $}$ and orbital separation as large as $\\sim $ 10 AU, then the system would follow a path similar to that described in Section .", "HCA in the CE phase could lead to BBH system with masses very close to those from GW190521.", "GW190521 is also the first BBH merger which may be associated to an electromagnetic (EM) counterpart.", "The optical flare (with $E\\sim 10^{51}$ erg) could be interpreted as evidence of interaction between the BBH and the accretion disk of an active galactic nucleus [29].", "However, GW190521 was at a redshift $\\simeq 0.82^{+0.28}_{-0.34}$ [3], and the EM event was detected $\\sim 34$ d after the GW signal.", "Thus, the association between GW190521 and the EM radiation is weak and remains to be confirmed.", "Nonetheless, we must note that, in our scenarios, if the BBH were to merge during the the CE of the tertiary star, it is likely that an energetic EM counterpart (GRB-like, in terms of energy) would be observed at the time of the merger [85].", "This would be particularly the case if the BBH merger occurs in the core of the star.", "The BBH merger, unlike a binary neutron star merger, produces its jets from accretion disks formed from external material, whereas binary neutron star may produce their own disk from neutron star material [43], [70], [52].", "The amount of angular momentum available for the infalling material is high, so, most likely a pair of extremely energetic jets would form and produce an energetic electromagnetic transient counterpart to the GW signal.", "Fallback from the exploded star could also help to produce a late EM signal (a hypernova-like transient).", "CEE of a BBH in a tertiary star could result in mergers within the CE if the drag is strong.", "This can be prevented if the BHs accrete at BHL rates.", "However, a merger of the BBH inside the core of the triple star can also account for a GW signal accompanied by a GRB." ], [ "BH spins", "Table REF shows a list of GW events which have at least one of the merging BHs in the PISN mass gap [4].", "These BHs could be produced by the HCA onto the BBH system during the CE of the tertiary star (see Section  for further details).", "Apart from the formation channels which may produce massive BHs, we analyse the spin that such BHs would have.", "Assuming that during the formation of the BH (or BHs) in the PISN mass gap there are weak SN kicks, mass transfer stages, and tidal synchronization episodes, the expectation is that massive BHs will be formed with low spins which would tend to be aligned with the orbital angular momentum (see, e.g., discussion in [56], [57]; unless something such as SN kicks may alter them [92]).", "We highlight the fact that the effective spins $\\chi _{\\rm eff}$ are distributed over a large range of values (from $-0.29$ to $0.70$ ) and that there is no strong trend.", "Values of $\\chi _{\\rm eff}\\approx 0.70$ are expected in BBHs which experience multiple sequential mergers.", "Focusing in our triple system evolutionary channels, we enumerate a few mechanisms that will likely increase and/or misalign the spins of the BHs (and which may be taken into account when predicting the characteristics of the resulting BHs since massive stars are preferentially in multiple stellar systems, see e.g.", "[54]): Strong SN kicks can misalign as well as spin up BHs.", "HCA during CE phases, especially when the BH mass doubles, can turn a spinless BH to a Kerr BH (see Figure 6 in [13]) and align the BH spin to the orbital angular momentum of the binary system.", "HCA in a CE of the tertiary star where the orbits are misaligned could produce almost anything, from aligned small spins to completely misaligned large spins.", "Tertiary-star orbital plane misaligned with binary orbital plane (particularly likely in captures) can misalign the BH spins (Kozai-Lidov)." ], [ "Other considerations", "The amount of mass accreted during the CE stage, estimated in Section  should be taken as an upper limit.", "The error bars of the BH masses in GW190521 are $\\sim 15-20$ % at $1\\sigma $ [2].", "So, it is possible that the BHs may have lower masses, thus reducing the amount of mass accreted.", "The PISN mass gap for BHs ($M_{\\rm GAP}$ ) is limited between $55 \\lesssim M_{\\rm GAP}/\\mbox{M$_\\odot $}\\lesssim 135$ [27] but the lower limit may be higher.", "The latter may increase to $65\\ \\mbox{M$_\\odot $}$ considering low mass-loss during the evolution of massive ZAMS stars [103], [104].", "Other studies propose that it could be of order 80 or 90 $\\mbox{M$_\\odot $}$ [10], [8].", "Thus, by taking an upper BH mass limit $\\sim 65\\ \\mbox{M$_\\odot $}$ , our models can form a BBH with M$_{\\rm BBH,1} \\simeq 65\\ \\mbox{M$_\\odot $}$ and M$_{\\rm BBH,2} \\simeq 60\\ \\mbox{M$_\\odot $}$ , hence reducing the mass accreted by $\\sim $ a half (to $\\sim 25$ M$_\\odot $ ) in order to form the observed BBH in GW190521.", "Stellar models which attempt to form BHs with masses within the PISN mass gap, assume that the hydrogen envelope collapses onto the BHs [67], and drives them directly into the PISN; after that, they still need to find another BH which goes through a similar process to merge with.", "In our model we get BHs with masses above the PISN mass gap via the ceRLOF mechanism by transferring the hydrogen envelope onto the tertiary star and returning it to the BBH at a later stage.", "Another mechanism which could lead a stellar binary to evolve into a GW190521-like system is when a star with $M_\\star \\gtrsim 100~\\mbox{M$_\\odot $}$ passes close to the BBH and is tidally disrupted (or a Mk34-like system with $\\sim 50-\\mbox{M$_\\odot $}$ BHs is produced).", "In this case the BHs accrete several tens of solar masses of disrupted-star material, bringing them into the mass range of GW190521 as well as closer together.", "Mass loss in pulsational PISNe may also lead to mass transfer onto the tertiary star, especially for material which does not acquire escape velocity or with a velocity $v \\sim v_{\\rm orb}$ .", "This material is usually lost during single stellar evolution, but a fraction of it can be recovered in binary- or triple-system evolution.", "The presence of strong jets and/or large $\\alpha \\lambda $ values could prevent efficient mass transfer during ceRLOF.", "The formation of jets/winds/outflows from accretion disks around the accreting BHs can remove the envelope of a massive star, thus they can limit the mass growth of the BHs.", "A small orbital separation between the BBH components can limit the formation of large accretion disks.", "Thus the orbital separation of the BBH could be an important factor for the amount of mass accreted by the BHs.", "Another relevant factor is that the ore massive the BHs the larger the BHL radius, thus it is likely that substantial mass growth by HCA during a CE will also depend on the initial mass of the BHs as they undergo CEE.", "[41] argue that unless the CEE occurs when the envelopes are mostly convective, these cannot be ejected.", "Then, assuming that accretion is negligible, the CE mass which is not ejected produces an orbital drag, and the binary merges.", "In our case, the mass which is not ejected ends up being rapidly accreted by the BHs, thus, preventing a merger.", "Also, in a triple system, mass can be removed from the inner binary by reaching the Roche lobe, without having to reach the escape velocity.", "It is then transferred onto the (tertiary) companion.", "Hence, the closer the tertiary star is, the more efficient the ceRLOF phase." ], [ "Conclusion", "In this paper we propose an evolutionary channel in which a BBH system, via HCA during the CE phase, produces BHs with masses within the PISN mass gap (e.g., the BBH system which produced the GW190521 event).", "We have considered two main scenarios.", "In the first scenario, the evolution of a HTS with an inner binary system with two very-massive stars.", "During the evolution of these two stars, part of the material is shed away and it is stored in a tertiary companion (our ceRLOF mass transfer mode as described in Section ).", "The massive binary produces a BBH system with masses below the PISN gap limit (e.g., $\\sim $ 45 M$_{}$ and $\\sim $ 40 M$_{}$ ).", "Later, the tertiary star leaves the MS, expands and engulfs the BBH in its envelope, the BBH accretes back the previously lost mass and results in a BBH with masses in the PISN mass gap.", "In the second scenario, a massive binary captures a tertiary star.", "If the star is captured during the MS of the binary, it may capture by ceRLOF mass from the inner binary and proceed similarly to the first model.", "Alternatively, the captured star may be a very-massive star, captured after the BBH has formed.", "As the tertiary evolves out of its MS stage it produces a CE with the BBH and transfers mass onto it, leaving, again a BBH with masses in the PISN mass gap.", "The main bottleneck for these scenarios may come from whether the BBHs are faster at accreting the CE material than the CE at absorbing energy from the system and unbounding it.", "If these processes are nearly equally efficient, the accretion during a CE stage of a star with $M_\\star \\gtrsim 100 \\mbox{M$_\\odot $}$ by the BBH could account for the masses of the GW190521 event.", "We also discuss possible progenitors of BBH systems with massive components.", "Mk34 represents a possible progenitor of the BBH studied in this paper.", "BBHs which are born in multiple systems will produces massive BHs with a disperse distribution of $\\chi _{\\rm eff}$ values.", "Our paper takes simulations of binaries from a population-synthesis code (COMPAS) and includes them in the simplified analytical evolution of a HTS.", "Future work should study the hydrodynamic evolution of the ceRLOF, the role of HCA in BBHs, and incorporate them into a triple-star population-synthesis software.", "Such population study would determine rates and configurations of BBHs in the PISN mass gap via the ceRLOF+HCA formation mechanism described in this manuscript.", "The rates and configurations could be then compared with the current GW catalog and the forthcoming detections of current and future GW observatories." ], [ "Acknowledgements", "F.D.C and D.L.C.", "acknowledge support from the UNAM-PAPIIT grant AG100820.", "D.L.C.", "is supported by Cátedras CONACyT at the Instituto de Astronomía (UNAM).", "A.V.G.", "acknowledges support by the Danish National Research Foundation (DNRF132).", "This research has made use of NASA’s Astrophysics Data System as well as arXiv." ] ]
2207.03514
[ [ "Real-time evolution of SU(3) hadrons on a quantum computer" ], [ "Abstract The theory of quarks and gluons - quantum chromodynamics - has been known for decades.", "Yet it is not fully understood.", "A recent example is the discovery of tetraquarks that led to a new research field.", "To address the many unsolved questions of the standard model, nonperturbative calculations are crucial.", "Quantum computers could simulate problems for which traditional QCD methods are inapplicable, such as real-time evolutions.", "We take a key step in exploring this possibility by performing a real-time evolution of tetraquark physics in one-dimensional SU(3) gauge theory on a superconducting quantum computer.", "Our experiment represents a first quantum computation involving quarks with three colour degrees of freedom, i.e.", "with the gauge group of QCD." ], [ "Introduction", "Quantum chromodynamics (QCD) provides the fundamental understanding of the strong nuclear force.", "It describes a vast range of hadrons and their properties in terms of just the quark masses and a gauge coupling.", "The recent discoveries [1] of several tetraquark candidates are reminders of the richness still remaining to be understood within QCD.", "Lattice gauge theory is the first-principles nonperturbative theoretical tool for studying QCD.", "Emerging quantum computers will allow lattice studies to access new topics within QCD, such as real-time evolution [2], [3].", "In this work, we use real-time evolution to present the first study of a tetraquark on a quantum computer.", "The calculations use SU(3) gauge theory with a single fermion flavor.", "The theory is written in one spatial dimension and, to match the available quantum hardware, our computations focus on the basic building block (i.e.", "minimal lattice length).", "Previous quantum computations within U(1) gauge theory [4], [5], [6], [7], [8], [9], [10], [11] showed electron-positron pair production.", "Moving from this Abelian case to a non-Abelian theory reveals qualitatively new phenomena.", "For example, in addition to quark-antiquark pair production (and the existence of a meson), there is also a gauge-singlet particle having valence quarks without valence antiquarks (i.e.", "the baryon).", "A recent paper [12] presented the first quantum computation of a baryon mass in a SU(2) gauge theory.", "In the present work we consider SU(3) [13], which is the gauge group of QCD itself, and study the tetraquark.", "Specifically, we identify the state possessing two quarks in a color antitriplet plus two antiquarks in a color triplet.", "The mixing of this state with the meson and with other quantum states is extracted from the quantum computation of time evolution.", "This simulation uses a resource-efficient and error-mitigation-augmented Trotter protocol performed on superconducting quantum computers.", "Very recent quantum simulations of SU(2) and SU(3) gauge theories for particle physics [14], [15], [12], [13], [16], [17], [18] have succeeded in accessing increasingly complex model systems on the route to QCD.", "Our experiment contributes to this quest by taking the crucial step to arrive at the simulation of quarks using the SU(3) gauge group relevant for hadron physics experiments.", "All these experiments appear in a timely manner to benchmark and max out the current noisy intermediate scale quantum devices to eventually address the open questions in high energy and particle physics.", "In fact, upon completion of our manuscript an independent but related work has appeared only forty-eight hours earlier [19], underlining the ongoing demand for techniques that allow quantum simulations of lattice gauge theories.", "Figure: Gauge theory on the lattice.", "In order to study the SU(3) theory in one dimension, we use the lattice as shown in panel a, where space is discretised into unit cells that each hold up to three quarks (red, green, blue) represented by filled circles and up to three antiquarks (antired, antigreen, antiblue) represented by stripped circles.", "Each particle is mapped to a qubit according to the staggerisation table in panel b.", "As example, panel c shows a pictorial representation of the tetraquark state for an elementary cell in the strong coupling limit along with its spin formulation (see text for details)." ], [ "Our calculations use the Hamiltonian approach where time is not discretized, and the lattice is purely spatial.", "We consider a one-dimensional (1D) lattice with open boundary conditions, where each site $n$ hosts a fermionic field with three color components, $\\hat{\\phi }_{n}=\\left(\\hat{\\phi }_{n}^{1}, \\hat{\\phi }_{n}^{2}, \\hat{\\phi }_{n}^{3}\\right)^{T} $ .", "We choose to work with staggered fermions [20] with the convention that odd sites host antimatter while even sites host matter.", "The gauge fields $\\hat{U}_{n}$ are defined on the link between sites $n$ and $n+1$ , and mediate the interaction between color degrees of freedom.", "The gauge-invariant lattice Hamiltonian in natural units ($\\hbar =c=1$ ) reads $\\hat{H}_l = & \\frac{1}{2a} \\sum _{n=1}^{N-1} \\left( \\hat{\\phi }_{n}^{\\dagger } \\hat{U}_{n} \\hat{\\phi }_{n+1} + \\operatorname{H.c.}\\right) \\\\&+ m \\sum _{n=1}^{N} (-1)^{n} \\hat{\\phi }_{n}^{\\dagger } \\hat{\\phi }_{n} + \\frac{a g^{2}}{2} \\sum _{n=1}^{N-1} \\mathbf {\\hat{L}}_{n}^{2} ,$ where $\\operatorname{H.c.}$ denotes the Hermitian conjugate, $N$ is the number of lattice sites with spacing $a$ , $m$ is the bare mass, and $g$ is the bare coupling.", "The alternating sign appearing in the mass term is the signature of the staggered formulation.", "The first term in the Hamiltonian describes the creation of particle-antiparticle pairs.", "The last term encodes the color electric energy of the system and is expressed in terms of the left electric field $\\mathbf {L}_{n}$ on the link $n$ .", "Furthermore, it is convenient to introduce the non-Abelian charges at site $n$ , $\\hat{Q}_{n}^{a}=\\sum _{i,j=1}^{3}\\hat{\\phi }_{n}^{i\\dagger }(T^{a})_{ij}\\hat{\\phi }_{n}^{j}$ where $T^{a}=\\lambda ^{a}/2$ , and $\\lambda ^{a}, a=1,\\dots ,8$ , are the Gell-Mann matrices [21].", "These charges appear in the non-Abelian version of the Gauss law that physical states must satisfy [22].", "We work in the sector with zero external charges and zero total non-Abelian charge, i.e.", "a colour singlet state must satisfy $\\hat{Q}_{tot}^{a}{\\Psi }\\equiv \\sum _{n}\\hat{Q}_{n}^{a}{\\Psi }=0$ .", "Besides the eight non-Abelian charges, the Hamiltonian also conserves the baryon number, $B$ , which measures the matter-antimatter imbalance (see Methods).", "Targeting tetraquark physics in our quantum simulations, we are especially interested in the $B=0$ subsector where all states contain an equal number of quarks and antiquarks." ], [ "To simulate and study the rich physics of the SU(3) theory, we encode Eq.", "(REF ) in a Hamiltonian suitable for quantum simulations.", "In a first step, a gauge transformation is applied that eliminates the gauge degrees of freedom from the Hamiltonian and allows us to express the Hamiltonian in terms of fermions only [23].", "This first step saves resources (as gauge fields are not stored explicitly in the qubit register) at the expense of introducing long-range interactions.", "In a second step, a Jordan-Wigner transformation [24] translates fermionic matter degrees of freedom into spin 1/2, i.e.", "qubit degrees of freedom (see Fig.", "REF ).", "After rescaling by the lattice spacing $a$ , the resulting Hamiltonian reads $\\hat{\\tilde{{H}}}=\\hat{H}_{kin}+\\tilde{m}\\hat{H}_{m}+\\frac{1}{2x}\\hat{H}_{e},$ where $\\tilde{m}=am$ and $x=1/g^{2}a^2$ are the dimensionless mass and coupling constant respectively.", "In the spin formulation, the kinetic term is given by $\\hat{H}_{kin}=\\frac{1}{2}\\sum _{n=1}^{N-1}(-1)^n& \\left(\\hat{\\sigma }_{3n-2}^{+}\\hat{\\sigma }_{3n-1}^{z}\\hat{\\sigma }_{3n}^{z} \\hat{\\sigma }_{3n+1}^{-}\\right.", "\\\\&-\\left.", "\\hat{\\sigma }_{3n-1}^{+}\\hat{\\sigma }_{3n}^{z}\\hat{\\sigma }_{3n+1}^{z} \\hat{\\sigma }_{3n+2}^{-}\\right.", "\\\\& +\\left.", "\\hat{\\sigma }_{3n}^{+}\\hat{\\sigma }_{3n+1}^{z}\\hat{\\sigma }_{3n+2}^{z} \\hat{\\sigma }_{3n+3}^{-} +\\operatorname{H.c.}\\right), $ and the mass term reads $\\hat{H}_{m}=\\frac{1}{2}\\sum _{n=1}^{N}\\left[(-1)^{n}\\left(\\hat{\\sigma }_{3n-2}^{z}+\\hat{\\sigma }_{3n-1}^{z}+\\hat{\\sigma }_{3n}^{z}\\right)+3\\right].", "$ The electric field Hamiltonian takes the form $\\hat{H}_{e}=\\sum _{n=1}^{N-1}\\left( \\sum _{m\\le n}\\mathbf {\\hat{Q}}_{m}\\right)^2, $ where $\\hat{\\mathbf {Q}}_{m}$ is a vector with eight components given by the non-Abelian charges.", "The exact expression of the non-Abelian charges in terms of qubits can be found in the Methods section.", "In Eq.", "(REF )-(REF ), the operators $\\hat{\\sigma }^x=(\\hat{\\sigma }^- +\\hat{\\sigma }^+)$ , $\\hat{\\sigma }^y=i(\\hat{\\sigma }^- -\\hat{\\sigma }^+)$ and $\\hat{\\sigma }^{z}$ are the usual Pauli matrices.", "Figure: Strong coupling states.", "The energy eigenstates in the strong coupling limit form a convenient basis for N=2N=2 discretised lattice sites.", "For the basic building block of the lattice (see Fig.", "1), these are given by the bare vacuum |vac〉|vac\\rangle , meson |m〉|m\\rangle , tetraquark |𝕋〉|\\mathbb {T}\\rangle , and baryon-antibaryon (baryonium) |B ¯B〉|\\bar{B}B\\rangle .", "We resort to a two column representation for the states in the fermion occupation number basis, where the first and second column indicates the state of the antimatter and matter respectively.Figure: Trotter time evolution.", "We performed experiments for the Hamiltonian parameters m ˜=1.2\\tilde{m}=1.2, x=0.8x=0.8 (panels a,b,c) and m ˜=0.45\\tilde{m}=0.45, x=0.8x=0.8 (panels d,e,f).", "The energy spectra on the left are shown to scale and reflect the proportion to which the strong coupling states in Fig.", "2 contribute to the individual energy eigenstates.", "The data obtained on three different IBM quantum computers [panel b ibm_peekskill, panel e ibm_geneva for N T =4N_\\mathrm {T}=4 and ibmq_lima for N T =8N_\\mathrm {T}=8] is shown in the middle column, where the different markers denote different Trotter steps and hence different circuit lengths.", "The circuit is heavily optimized (see Methods) and contains N T ·10N_\\mathrm {T}\\cdot 10 CNOT gates and a total circuit depth of N T ·25+1N_\\mathrm {T}\\cdot 25 + 1 after transpilation to the employed native gates.", "For the data points shown in the figure, the error mitigation has already been applied (see Supplemental Information).", "The dashed lines mark the exact Trotter evolution obtained via a numerical exponentiation.", "We further composed an expected graph for the evolution obtained from the Trotter protocol, which is plotted in a solid black line.", "The error bars shown here originate via bootstrapping of the error mitigation method .", "Error bars corresponding to the quantum projection noise are small due to the 2048 performed shots and would be hidden by the size of the markers and hence not shown.", "To obtain the energy differences indicated by the arrows in the left column (panel a and d), we resort to Bayesian inference.", "The results are shown in panels c and f respectively, where the solid lines denote the mean of 5000 samples drawn from the posterior predictive distribution (see Methods).", "From these samples we also compute the highest density interval (HDI) equivalent, i.e.", "the grey area marks the interval between the 2.52.5 and 97.597.5 percentiles.", "The point estimate for the energy gap between |B ¯B〉|\\bar{B}B\\rangle and |𝕋〉|\\mathbb {T}\\rangle [panel a] is given by ω 0 =(2π)·0.181/[m ˜t]\\omega _0 = (2\\pi )\\cdot 0.181/[\\tilde{m}t] with an HDI of (2π)·[0.153,0.204]/[m ˜t])(2\\pi )\\cdot [0.153,\\,0.204]/[\\tilde{m}t]).", "For the second case in panel d, we find that ω 1 =(2π)·0.482/[m ˜t]\\omega _1 = (2 \\pi ) \\cdot 0.482 / [\\tilde{m} t] with HDI =(2π)·[0.409,0.545]/[m ˜t]\\mathrm {HDI} = (2 \\pi ) \\cdot [0.409,\\, 0.545]/ [\\tilde{m} t] and ω 2 =(2π)·0.427/[m ˜t]\\omega _2 = (2 \\pi ) \\cdot 0.427/ [\\tilde{m} t] where HDI =(2π)·[0.297,0.502]/[m ˜t]\\mathrm {HDI} = (2 \\pi ) \\cdot [0.297,\\, 0.502]/ [\\tilde{m} t].", "Values for other parameters in our probabilistic model can be found in the Supplemental Information.In the following, we consider and study the basic building block consisting of $N=2$ discretized lattice sites.", "A convenient basis is the strong coupling one given by $\\tilde{m}\\rightarrow \\infty $ and $x\\rightarrow 0$ (i.e.", "in the limit in which $\\hat{H}_m$ and $\\hat{H}_{e}$ dominate over the pair-creation term $\\hat{H}_{kin}$ ).", "The gauge-invariant basis states in that limit can be constructed by successively applying the kinetic term to the vacuum state.", "The different basis states obtained are depicted in Fig.", "REF in the fermion occupation picture, where the first and second column of the ket describe the antimatter and matter content of the state respectively.", "While all of them possess the same quantum baryon number $B=0$ , the number of particle-antiparticle pairs contained in them differ.", "These states are all eigenstates of the mass Hamiltonian in Eq.", "(REF ), which counts the number of particles and antiparticles in a state, with eigenvalues $0,2,4$ and 6 for the vacuum, meson, tetraquark and baryon-antibaryon states respectively.", "The meson state consists of a color-singlet superposition of particle-antiparticle pairs.", "By contrast the tetraquark state is a color superposition of diquark-antidiquark pairs.", "Like a single antiquark, the diquark is a color antitriplet state.", "At finite coupling $x$ and mass $\\tilde{m}$ , the eigenstates of the Hamiltonian are given by a superposition of the strong coupling basis states.", "By studying time evolution under the Hamiltonian in Eq.", "(REF ), we can probe the transitions between the different eigenstates.", "In particular, by chosing the initial state as the strong coupling limit baryon-antibaryon state (containing six particles and antiparticles in total) and in the regime where $x/\\tilde{m}\\le 1$ , we can probe a single transition between the baryon-antibaryon and the tetraquark state.", "When the parameters are chosen outside of this regime, more than one transition becomes involved in the time evolution, which makes the dynamics richer and more complex.", "The time evolution $e^{-i\\hat{\\tilde{H}}t}$ is obtained from a Trotter decomposition [26] that we optimize for minimal gate depth.", "Fig.", "REF in the Methods shows the Trotter circuit for a basic building block ($N=2$ ).", "While this minimal lattice is described by six spins (compare Fig.", "REF ), we can simulate the $B=0$ sector using only three qubits, due to a particle-antiparticle symmetry in this case (see Methods).", "We are interested in tracking the particle number $\\langle \\hat{n}(t)\\rangle ={\\Psi _{0}}e^{it\\hat{H}^{(3)}}\\hat{H}^{(3)}_{m}e^{-it\\hat{H}^{(3)}}{\\Psi _{0}}, $ as we evolve the system in time starting from an initial state ${\\Psi _{0}}$ with $\\hat{H}^{(3)}$ the three qubits Hamiltonian derived in Methods.", "We focus here on ${\\Psi _{0}}={\\Psi _{\\bar{B}B}}$ , the baryon-antibaryon state in the strong coupling limit (see Fig.", "REF ).", "In terms of spins, the strong coupling baryon-antibaryon state is given by ${\\Psi _{0}}={\\downarrow \\downarrow \\downarrow }{\\uparrow \\uparrow \\uparrow }$ where the first ket refers to antiquarks and the second to quarks (note that only the first ket is implemented in the quantum simulation and the second is implied)." ], [ "We are using self-mitigation as introduced in Ref. [17].", "The basic idea is to use our quantum circuit in two ways.", "A “physics run” applies the desired number of Trotter steps, $N_{\\mathrm {T}}$ , forward in time to reach the final time of interest.", "A “mitigation run” applies $N_{\\mathrm {T}}/2$ steps forward in time followed by $N_{\\mathrm {T}}/2$ steps backward in time, which results in a noisy experimental determination of the known initial state.", "Randomized compiling is used to surround the CNOT gates with Pauli gates that turn coherent errors into incoherent errors.", "We find that the number of physics and mitigation runs can be as low as 40 each, and up to 560 depending on the quantum computing device chosen.", "Throughout this work we always collect 2048 shots from a single circuit execution.", "As described in Ref.", "[17], the information gleaned from the mitigation runs provides an excellent error mitigation for the physics runs.", "As in [12], each separate calculation is further accompanied by a set of $2^3$ calibration circuits to estimate the transfer map mixing the true outcome probabilities into the observed ones." ], [ "We perform two Trotter time evolution experiments on a universal superconducting quantum computer [27] using up to eight Trotter time steps.", "In both experiments, the system is initialized in the strong coupling baryon-antibaryon state $|\\bar{B}B\\rangle $ and evolved in time under the gauge-invariant three-qubit Hamiltonian in the Methods.", "Since the Hamiltonian preserves the baryon number of the initial time, the observed evolution remains within the $B=0$ sector.", "The first experiment is carried out for the Hamiltonian parameters $x=0.8$ and $\\tilde{m}=1.2$ [see Eq.", "(REF )].", "This quark mass is large enough to organize the hadron spectrum in an intuitive way: the vacuum, meson, tetraquark, and baryon-antibaryon states have energies near 0, $2\\tilde{m}$ , $4\\tilde{m}$ and $6\\tilde{m}$ respectively.", "For reference, Fig.", "REF (a) shows the spectrum and the composition of physical hadron states in terms of strong coupling states.", "Since we choose the strong-coupling baryon-antibaryon as initial state, quark-antiquark annihilation provides a direct connection to the tetraquark state, and indeed the collected data allows to observe this oscillation to dominate the time evolution as shown in Fig.", "REF (b).", "The experiment has been performed on the ibm_peekskill device, where for each data point we run 280 (140, 140) repetitions in the case of $N_{\\mathrm {T}}=4$ ($N_{\\mathrm {T}}=2$ , $N_{\\mathrm {T}}=6$ ) for physics and mitigation runs respectively, where $N_\\mathrm {T}$ is the number of Trotter steps.", "We perform a Bayesian analysis [28] to extract the frequency of the oscillation and thus calculate the mass gap between the baryon-antibaryon state and the tetraquark state (see Methods for more details).", "We identify one frequency, where the point estimate is given by $\\omega _{0} = (2\\pi )\\cdot 0.181/[\\tilde{m}t]$ with the highest density interval (HDI) of $(2\\pi )\\cdot [0.153,\\,0.204]/[\\tilde{m}t])$ , which corresponds to the energy gap between $|\\bar{B}B\\rangle $ and $|\\mathbb {T}\\rangle $ .", "The HDI, is the interval where we find 95% of the values during the sampling procedure.", "In Fig.", "REF (c) we plot 5000 samples from the posterior predictive distribution which within the 2.5 and 97.5 percent quantile agrees well with the collected data.", "The second experiment is carried out on ibm_geneva ($N_{\\mathrm {T}}=4$ , 560 repetitions) and ibmq_lima ($N_{\\mathrm {T}}$ =8, 40 repetitions), employing a smaller quark mass $\\tilde{m}=0.45$ but the same coupling constant $x=0.8$ .", "In this case, the hadron spectrum is not ordered simply by counting the number of quarks in each state.", "Instead, our quantum calculation of the real-time dynamics is able to reveal two dominant energies that produce a beat frequency between them which is easily visible in Fig.", "REF .", "Applying the same Bayesian inference techniques, we extract two frequency components $\\omega _1 = (2 \\pi ) \\cdot 0.482 / [\\tilde{m} t]$ with $\\mathrm {HDI} = (2 \\pi ) \\cdot [0.409,\\, 0.545]/ [\\tilde{m} t]$ and $\\omega _2 = (2 \\pi ) \\cdot 0.427/ [\\tilde{m} t]$ where $\\mathrm {HDI} = (2 \\pi ) \\cdot [0.297,\\, 0.502]/ [\\tilde{m} t]$ , which confirms the underlying physics.", "In particular, the strong-coupling $|\\bar{B} B\\rangle $ initial state is once again mixing with the strong-coupling tetraquark, but this tetraquark is now a significant percentage of two physical eigenstates, as shown in Fig.", "3." ], [ "Discussion", "Recent observations of tetraquarks and other hadrons beyond the traditional mesons and baryons have sparked a great deal of theoretical activity [1], with lattice gauge theory playing a central role in the determination of static properties.", "To access time-dependent dynamics, we turn to the Hamiltonian approach on quantum computers.", "Building on previous proposals [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40] and demonstrations in simpler gauge theories[14], [15], [12], [17], [16], [18], we have constructed a one-dimensional lattice gauge theory for QCD itself, with one flavor of dynamical matter coupled to SU(3) gauge fields.", "Real-time oscillations of the tetraquark with other hadron states are observed by running on IBM Quantum hardware [27].", "Specifically, we begin from a strong-coupling baryon-antibaryon state at time $t=0$ and see how the tetraquark emerges.", "The success of these simulations required the use of recent advances in error mitigation [17].", "Our approach is based on an elimination of the gauge degrees of freedom, with that physics being re-expressed as nonlocal interactions among matter fields.", "Future work will extend this methodology to two (and ultimately three) spatial dimensions using the methods developed in [41].", "Another important route for generalisations is the extension of wider classes of simulated time evolutions to extract truly dynamical quantities.", "Interesting applications include time correlation functions and the ongoing quest to simulate particle collisions with quantum computers.", "Our simulation of SU(3) hadrons on a quantum computer accomplishes a key step on the path toward accessing increasingly relevant quantum computations for QCD.", "We are immensely grateful to Thomas Fernholz and Christian Sommer for sharing their important scientific insights and invaluable input to our project.", "We thank John Watrous for his support, and we are grateful for the IBM Quantum Researchers Program Access Award enabling the use of IBM Quantum services for this work.", "The views expressed are those of the authors, and do not reflect the official policy or position of IBM or the IBM Quantum team.", "This work has been supported by Transformative Quantum Technologies Program (CFREF), NSERC and the New Frontiers in Research Fund.", "CM acknowledges the Alfred P. Sloan foundation for a Sloan Research Fellowship.", "Methods Gauge elimination and qubit formulation.", "Due to gauge invariance, the Hamiltonian in Eq.", "(REF ) commutes with the Gauss' law operators (which generate the local gauge transformations) $\\hat{G}_{n}^{a}\\equiv \\hat{L}_{n}^{a}-\\hat{R}_{n-1}^{a}-\\hat{Q}_{n}^{a}$ , where $\\hat{L}_{n}^{a}$ and $\\hat{R}_{n-1}^{a}$ are the $a$ -component (with $a=1,\\dots ,8$ ) of the left and right electric field living on the link $n$ respectively.", "For a non-Abelian gauge group, the right and left color electric field are related via the adjoint representation $\\hat{R}_{n}^{a}=(\\hat{U}_n^\\text{adj})_{ab}\\hat{L}_{n}^{b}$ , with $(\\hat{U}_{n}^\\text{adj})_{ab}=2\\mathrm {Tr}\\left[ \\hat{U}_{n}T^{a}\\hat{U}_{n}^{\\dagger }T^{b}\\right]$ , where $T^{a}=\\lambda ^{a}/2$ , and $\\lambda ^{a}\\ (a=1,\\dots ,8)$ are generators of the SU(3) Lie algebra and are given by the Gell-Mann matrices [21].", "The Hamiltonian also commutes with the redness operator $\\hat{\\mathcal {R}} =\\sum _{n=1}^{N} \\hat{\\phi }_{n}^{1 \\dagger }\\hat{\\phi }_{n}^{1}-N/2$ , the greenness $\\hat{\\mathcal {G}} =\\sum _{n=1}^{N} \\hat{\\phi }_{n}^{2 \\dagger }\\hat{\\phi }_{n}^{2}-N/2$ and blueness operator $\\hat{\\mathcal {B}} =\\sum _{n=1}^{N} \\hat{\\phi }_{n}^{3 \\dagger }\\hat{\\phi }_{n}^{3}-N/2$ which measures the matter-antimatter imbalance of a specific color.", "It is however more convenient to combine these three operators into a single one which measure the matter-antimatter imbalance irrespective of the color.", "We therefore define the baryon number operator as $\\hat{B}=\\frac{1}{3}\\left(\\hat{\\mathcal {R}}+\\hat{\\mathcal {G}}+\\hat{\\mathcal {B}}\\right).", "$ In order to simulate time dynamics in the baryon sector $B=0$ on a quantum computer, we first transform the fermionic Hamiltonian in Eq.", "(REF ) into one involving only qubits degrees of freedom.", "The transformation is achieved in two steps.", "We first eliminate the gauge fields by following the methods developed in [12], the dimensionless Hamiltonian reads $\\hat{H} = & \\frac{1}{2} \\sum _{n=1}^{N-1} \\sum _{i=1}^{3}\\left( \\hat{\\phi }_{n}^{i \\dagger }\\hat{\\phi }_{n+1}^{i} + \\operatorname{H.c.}\\right) \\\\&+ \\tilde{m} \\sum _{n=1}^{N}\\sum _{i=1}^{3} (-1)^{n} \\hat{\\phi }_{n}^{i \\dagger } \\hat{\\phi }_{n}^{i} + \\frac{1}{2x} \\sum _{n=1}^{N-1}\\left( \\sum _{m\\le n}\\hat{{Q}}_{m}\\right)^{2} ,$ where $\\tilde{m}=am$ and $x=1/g^{2}a^2$ .", "The last term represents the color electric energy of the system and is expressed in terms of the non-Abelian charges at site $n$ $\\hat{Q}_{n}^{a}=\\sum _{i,j=1}^{3}\\hat{\\phi }_{n}^{i\\dagger }(T^{a})_{ij}\\hat{\\phi }_{n}^{j}.", "$ In a second step, we triple the size of the lattice to define $3N$ new sites and distribute the color components of the fermionic field among them by defining the single component fields $\\hat{\\phi }_{n}^{i}=\\hat{\\psi }_{3n-3+i}$ with $n=1,2,\\dots ,N$ and $i=1,2,3$ (see Fig.", "REF ).", "We then perform a generalised Jordan-Wigner transformation on the single component fermionic field $\\hat{\\psi }_{n}$ [24] $\\hat{\\psi }_{n}=\\left(\\prod _{l<n} s_{l} \\hat{\\sigma }_{l}^{z}\\right)\\hat{\\sigma }_{n}^{-}, \\quad \\hat{\\psi }_{n}^{\\dagger }=\\hat{\\sigma }_{n}^{+}\\left(\\prod _{l<n} s_{l} \\hat{\\sigma }_{l}^{z}\\right)$ where $s_{l}$ are phase factors that we choose equal to $+1$ on antimatter cells and $-1$ on matter cells.", "This choice is convenient because it matches the standard colour notation, such as $(\\bar{r}r+\\bar{g}g+\\bar{b}b)/\\sqrt{3}$ .", "After the Jordan-Wigner transformation, the kinetic Hamiltonian in terms of qubits is given by Eq.", "(REF ) while the mass Hamiltonian is given by Eq.", "(REF ).", "In the qubit formulation, the non-Abelian charges defined in Eq.", "(REF ) are given by $\\hat{Q}_{n}^{1}&=\\frac{(-1)^{n}}{2}\\left( \\hat{\\sigma }_{3n-2}^{+}\\hat{\\sigma }_{3n-1}^{-}+\\operatorname{H.c.}\\right), \\\\\\hat{Q}_{n}^{2}&=\\frac{i(-1)^{n}}{2}\\left( \\hat{\\sigma }_{3n-1}^{+}\\hat{\\sigma }_{3n-2}^{-}-\\operatorname{H.c.}\\right),\\\\\\hat{Q}_{n}^{3}&=\\frac{1}{4}\\left( \\hat{\\sigma }_{3n-2}^{z}-\\hat{\\sigma }_{3n-1}^{z}\\right),\\\\\\hat{Q}_{n}^{4}&=-\\frac{1}{2}\\left( \\hat{\\sigma }_{3n-2}^{+}\\hat{\\sigma }_{3n-1}^{z}\\hat{\\sigma }_{3n-2}^{-}+\\operatorname{H.c.}\\right), \\\\\\hat{Q}_{n}^{5}&=\\frac{i}{2}\\left( \\hat{\\sigma }_{3n-2}^{+}\\hat{\\sigma }_{3n-1}^{z}\\hat{\\sigma }_{3n}^{-}-\\operatorname{H.c.}\\right), \\\\\\hat{Q}_{n}^{6}&=\\frac{(-1)^n}{2}\\left( \\hat{\\sigma }_{3n-1}^{+}\\hat{\\sigma }_{3n}^{-}+\\operatorname{H.c.}\\right), \\\\\\hat{Q}_{n}^{7}&=\\frac{i(-1)^n}{2}\\left( \\hat{\\sigma }_{3n}^{+}\\hat{\\sigma }_{3n-1}^{-}-\\operatorname{H.c.}\\right),\\\\\\hat{Q}_{n}^{8}&=\\frac{1}{4\\sqrt{3}}\\left( \\hat{\\sigma }_{3n-2}^{z}+\\hat{\\sigma }_{3n-1}^{z}-2\\hat{\\sigma }_{3n}^{z}\\right).$ The electric field Hamiltonian can be obtained by injecting the expressions of the non-Abelian charges (REF )-() in the color electric term in Eq.", "(REF ).", "We obtain $\\hat{H}_{e}=&\\frac{1}{3}\\sum _{n=1}^{N-1}\\left( 3-\\hat{\\sigma }_{3n-2}^{z}\\hat{\\sigma }_{3n-1}^{z}-\\hat{\\sigma }_{3n-2}^{z}\\hat{\\sigma }_{3n}^{z}-\\hat{\\sigma }_{3n-1}^{z}\\hat{\\sigma }_{3n}^{z}\\right)\\\\&+\\sum _{n=1}^{N-2}\\sum _{m=n+1}^{N-1}\\left[ (N-m)\\left( \\hat{\\sigma }_{3n-2}^{+}\\hat{\\sigma }_{3n-1}^{-}\\hat{\\sigma }_{3m-1}^{+}\\hat{\\sigma }_{3m-2}^{-}\\right.", "\\right.", "\\\\& +\\left.", "\\hat{\\sigma }_{3n-1}^{+}\\hat{\\sigma }_{3n}^{-}\\hat{\\sigma }_{3m-1}^{-}\\hat{\\sigma }_{3m}^{+}+\\operatorname{H.c.}\\right)(-1)^{n+m} \\\\& +(N-m)\\left(\\hat{\\sigma }_{3n-2}^{+}\\hat{\\sigma }_{3n-1}^{z}\\hat{\\sigma }_{3n}^{-}\\hat{\\sigma }_{3m-2}^{-}\\hat{\\sigma }_{3m-1}^{z}\\hat{\\sigma }_{3m}^{+}+\\operatorname{H.c.}\\right) \\\\&-\\frac{1}{12}(N-m) \\hat{\\sigma }_{3m-2}^{z}(\\hat{\\sigma }_{3n-1}^{z}+\\hat{\\sigma }_{3n}^{z}-2\\hat{\\sigma }_{3n-2}^{z}) \\\\&-\\frac{1}{12}(N-m) \\hat{\\sigma }_{3m-1}^{z}(\\hat{\\sigma }_{3n}^{z}+\\hat{\\sigma }_{3n-2}^{z}-2\\hat{\\sigma }_{3n-1}^{z})\\\\& \\left.", "-\\frac{1}{12}(N-m) \\hat{\\sigma }_{3m}^{z}(\\hat{\\sigma }_{3n-2}^{z}+\\hat{\\sigma }_{3n-1}^{z}-2\\hat{\\sigma }_{3n}^{z}) \\right], $ which exhibits long range spin-spin interaction as a direct consequence of the gauge elimination.", "The baryon number operator is proportional to the total magnetization of the system in the qubit encoding $\\hat{B}=\\frac{1}{6}\\sum _{n=1}^{3N}\\hat{\\sigma }_{n}^{z}.", "$ We are interested in a basic building block consisting of $N=2$ discretized lattice sites.", "The model is then described by a chain with 6 qubits and the terms in the Hamiltonian read $\\hat{H}_{kin}&=-\\frac{1}{2}\\left( \\hat{\\sigma }_{1}^{+}\\hat{\\sigma }_{2}^{z}\\hat{\\sigma }_{3}^{z} \\hat{\\sigma }_{4}^{-}-\\hat{\\sigma }_{2}^{+}\\hat{\\sigma }_{3}^{z}\\hat{\\sigma }_{4}^{z} \\hat{\\sigma }_{5}^{-}+\\hat{\\sigma }_{3}^{+}\\hat{\\sigma }_{4}^{z}\\hat{\\sigma }_{5}^{z} \\hat{\\sigma }_{6}^{-}+\\operatorname{H.c.}\\right), \\\\\\hat{H}_{m}&=\\frac{1}{2}\\left(6- \\hat{\\sigma }_{1}^{z}-\\hat{\\sigma }_{2}^{z}-\\hat{\\sigma }_{3}^{z}+\\hat{\\sigma }_{4}^{z}+\\hat{\\sigma }_{5}^{z}+\\hat{\\sigma }_{6}^{z} \\right),\\\\\\hat{H}_{e}&=\\frac{1}{3}\\left(3-\\hat{\\sigma }_{1}^{z}\\hat{\\sigma }_{2}^{z}-\\hat{\\sigma }_{1}^{z}\\hat{\\sigma }_{3}^{z} -\\hat{\\sigma }_{2}^{z}\\hat{\\sigma }_{3}^{z}\\right).$ In the sector with baryon number $B=0$ (i.e.", "as much matter as antimatter), the three terms composing the Hamiltonian commute with the following operator $\\hat{CP}=\\prod _{n=1}^{3}\\hat{\\sigma }_{n}^{x}\\hat{\\sigma }_{7-n}^{x}\\hat{W}_{n,7-n}$ where $\\hat{W}_{n,n^{\\prime }}$ is the SWAP unitary operator between qubit $n$ and $n^{\\prime }$ .", "This symmetry corresponds to the composition of a spatial reflection with respect to the middle of the chain ($\\hat{P}$ ) followed by a charge conjugation operation ($\\hat{C}$ ) which flips the spins.", "Local spin operators $\\hat{\\sigma }_{n}^{a}$ transforms as $(\\hat{CP})^{\\dagger }\\hat{\\sigma }_{n}^{a}\\hat{CP}=(\\hat{\\sigma }^{x}\\hat{\\sigma }^{a}\\hat{\\sigma }^{x})_{7-n}$ under the $\\hat{CP}$ operation with $a=x,y,z$ and $n=1,2,\\dots ,6$ .", "It is thus clear that a convenient basis is the one spanned by states of the form $|\\Psi \\rangle = \\sum c_{ijk} |i\\rangle _1|j\\rangle _2 |k\\rangle _3 \\otimes \\hat{\\sigma }_{4}^x \\hat{\\sigma }_{5}^x\\hat{\\sigma }_{6}^x|i\\rangle _4|j\\rangle _5|k\\rangle _6$ , which are invariant under the $\\hat{CP}$ operation and have $B=0$ .", "When one works with this basis, the state of the last three qubits is completely determined by the state of the first three.", "As a direct consequence, we can encode the states by using only the first 3 qubits (i.e.", "the state of the antimatter) rather than 6.", "The reduced three-qubit Hamiltonian reads $\\hat{H}_{kin}^{(3)}&=-\\frac{1}{2}\\left( \\hat{\\sigma }_{1}^{x}\\hat{\\sigma }_{2}^{z}\\hat{\\sigma }_{3}^{z}+\\hat{\\sigma }_{1}^{z}\\hat{\\sigma }_{2}^{x}\\hat{\\sigma }_{3}^{z}+\\hat{\\sigma }_{1}^{z}\\hat{\\sigma }_{2}^{z}\\hat{\\sigma }_{3}^{x}\\right),\\\\\\hat{H}_{m}^{(3)}&=3-\\hat{\\sigma }_{1}^{z}-\\hat{\\sigma }_{2}^{z}-\\hat{\\sigma }_{3}^{z},\\\\\\hat{H}_{e}^{(3)}&=\\frac{1}{3}\\left(3-\\hat{\\sigma }_{1}^{z}\\hat{\\sigma }_{2}^{z}-\\hat{\\sigma }_{1}^{z}\\hat{\\sigma }_{3}^{z} -\\hat{\\sigma }_{2}^{z}\\hat{\\sigma }_{3}^{z}\\right), $ and the time evolution is obtained using the Hamiltonian $\\hat{H}^{(3)}=\\hat{H}_{kin}^{(3)}+\\tilde{m}\\hat{H}_{m}^{(3)}+\\frac{1}{2x}\\hat{H}_{e}^{(3)}$ Trotter evolution Although the Hamiltonian of Eqs.", "(REF -) is expressed in terms of Pauli X and Z gates, a simple rotation to Y and Z gates allows for more cancellations among CNOT gates.", "This is especially valuable on hardware that does not provide all-to-all connectivity among the qubits.", "The first half of our first-order Trotter step is displayed in Fig.", "REF and, to match the available hardware, it does not use any entangling gates directly between $q_1$ and $q_3$ .", "The second half of the Trotter step is the same except for a relabeling of $q_1 \\leftrightarrow q_3$ .", "Figure: The first half of a Trotter step is shown in this figure.The second half is identical except for the interchange of two qubits: q 1 ↔q 3 q_1 \\leftrightarrow q_3.Note that a pair of CNOT gates can be canceled where the two halves meet.", "Bayesian inference analysis.", "We model the obtained data $D$ as draws from a normal distribution with variance $\\sigma $ , where the mean $S_K$ is given by a cosine series $S_K = \\sum _i^K A_i \\cos \\left(\\omega _i t + \\phi _i\\right) + \\xi ,$ with uniform priors on the frequencies $\\omega _i$ and the amplitudes $A_i$ , while the priors for the phases $\\phi _i$ and the offset $\\xi $ are normal distributions located at zero.", "The uniform priors are adjusted to include the frequencies that are visible from the data by eye.", "The prior on $\\sigma $ is given by the maximum of $0.3$ and the value of the largest errorbar in the dataset.", "Note that each dataset stemming from different $N_\\mathrm {T}$ is modeled as a separate likelihood while the parameters are shared.", "After obtaining a representation of the posterior distribution for the parameters $\\theta = \\lbrace A_i, \\omega _i, \\phi _i, \\xi , \\sigma \\rbrace $ , $P\\left(\\theta \\vert D\\right)$ , via Monte Carlo sampling from Bayes' rule, we sample the posterior predictive distribution $P(D^\\prime \\vert D) = \\int _\\Theta P(\\Theta =\\theta \\vert D)P(D^\\prime \\vert \\Theta = \\theta )$ , where $\\Theta $ denotes the collective random variables for all parameters." ], [ "Due to gauge invariance, the Hamiltonian in Eq.", "(REF ) commutes with the Gauss' law operators (which generate the local gauge transformations) $\\hat{G}_{n}^{a}\\equiv \\hat{L}_{n}^{a}-\\hat{R}_{n-1}^{a}-\\hat{Q}_{n}^{a}$ , where $\\hat{L}_{n}^{a}$ and $\\hat{R}_{n-1}^{a}$ are the $a$ -component (with $a=1,\\dots ,8$ ) of the left and right electric field living on the link $n$ respectively.", "For a non-Abelian gauge group, the right and left color electric field are related via the adjoint representation $\\hat{R}_{n}^{a}=(\\hat{U}_n^\\text{adj})_{ab}\\hat{L}_{n}^{b}$ , with $(\\hat{U}_{n}^\\text{adj})_{ab}=2\\mathrm {Tr}\\left[ \\hat{U}_{n}T^{a}\\hat{U}_{n}^{\\dagger }T^{b}\\right]$ , where $T^{a}=\\lambda ^{a}/2$ , and $\\lambda ^{a}\\ (a=1,\\dots ,8)$ are generators of the SU(3) Lie algebra and are given by the Gell-Mann matrices [21].", "The Hamiltonian also commutes with the redness operator $\\hat{\\mathcal {R}} =\\sum _{n=1}^{N} \\hat{\\phi }_{n}^{1 \\dagger }\\hat{\\phi }_{n}^{1}-N/2$ , the greenness $\\hat{\\mathcal {G}} =\\sum _{n=1}^{N} \\hat{\\phi }_{n}^{2 \\dagger }\\hat{\\phi }_{n}^{2}-N/2$ and blueness operator $\\hat{\\mathcal {B}} =\\sum _{n=1}^{N} \\hat{\\phi }_{n}^{3 \\dagger }\\hat{\\phi }_{n}^{3}-N/2$ which measures the matter-antimatter imbalance of a specific color.", "It is however more convenient to combine these three operators into a single one which measure the matter-antimatter imbalance irrespective of the color.", "We therefore define the baryon number operator as $\\hat{B}=\\frac{1}{3}\\left(\\hat{\\mathcal {R}}+\\hat{\\mathcal {G}}+\\hat{\\mathcal {B}}\\right).", "$ In order to simulate time dynamics in the baryon sector $B=0$ on a quantum computer, we first transform the fermionic Hamiltonian in Eq.", "(REF ) into one involving only qubits degrees of freedom.", "The transformation is achieved in two steps.", "We first eliminate the gauge fields by following the methods developed in [12], the dimensionless Hamiltonian reads $\\hat{H} = & \\frac{1}{2} \\sum _{n=1}^{N-1} \\sum _{i=1}^{3}\\left( \\hat{\\phi }_{n}^{i \\dagger }\\hat{\\phi }_{n+1}^{i} + \\operatorname{H.c.}\\right) \\\\&+ \\tilde{m} \\sum _{n=1}^{N}\\sum _{i=1}^{3} (-1)^{n} \\hat{\\phi }_{n}^{i \\dagger } \\hat{\\phi }_{n}^{i} + \\frac{1}{2x} \\sum _{n=1}^{N-1}\\left( \\sum _{m\\le n}\\hat{{Q}}_{m}\\right)^{2} ,$ where $\\tilde{m}=am$ and $x=1/g^{2}a^2$ .", "The last term represents the color electric energy of the system and is expressed in terms of the non-Abelian charges at site $n$ $\\hat{Q}_{n}^{a}=\\sum _{i,j=1}^{3}\\hat{\\phi }_{n}^{i\\dagger }(T^{a})_{ij}\\hat{\\phi }_{n}^{j}.", "$ In a second step, we triple the size of the lattice to define $3N$ new sites and distribute the color components of the fermionic field among them by defining the single component fields $\\hat{\\phi }_{n}^{i}=\\hat{\\psi }_{3n-3+i}$ with $n=1,2,\\dots ,N$ and $i=1,2,3$ (see Fig.", "REF ).", "We then perform a generalised Jordan-Wigner transformation on the single component fermionic field $\\hat{\\psi }_{n}$ [24] $\\hat{\\psi }_{n}=\\left(\\prod _{l<n} s_{l} \\hat{\\sigma }_{l}^{z}\\right)\\hat{\\sigma }_{n}^{-}, \\quad \\hat{\\psi }_{n}^{\\dagger }=\\hat{\\sigma }_{n}^{+}\\left(\\prod _{l<n} s_{l} \\hat{\\sigma }_{l}^{z}\\right)$ where $s_{l}$ are phase factors that we choose equal to $+1$ on antimatter cells and $-1$ on matter cells.", "This choice is convenient because it matches the standard colour notation, such as $(\\bar{r}r+\\bar{g}g+\\bar{b}b)/\\sqrt{3}$ .", "After the Jordan-Wigner transformation, the kinetic Hamiltonian in terms of qubits is given by Eq.", "(REF ) while the mass Hamiltonian is given by Eq.", "(REF ).", "In the qubit formulation, the non-Abelian charges defined in Eq.", "(REF ) are given by $\\hat{Q}_{n}^{1}&=\\frac{(-1)^{n}}{2}\\left( \\hat{\\sigma }_{3n-2}^{+}\\hat{\\sigma }_{3n-1}^{-}+\\operatorname{H.c.}\\right), \\\\\\hat{Q}_{n}^{2}&=\\frac{i(-1)^{n}}{2}\\left( \\hat{\\sigma }_{3n-1}^{+}\\hat{\\sigma }_{3n-2}^{-}-\\operatorname{H.c.}\\right),\\\\\\hat{Q}_{n}^{3}&=\\frac{1}{4}\\left( \\hat{\\sigma }_{3n-2}^{z}-\\hat{\\sigma }_{3n-1}^{z}\\right),\\\\\\hat{Q}_{n}^{4}&=-\\frac{1}{2}\\left( \\hat{\\sigma }_{3n-2}^{+}\\hat{\\sigma }_{3n-1}^{z}\\hat{\\sigma }_{3n-2}^{-}+\\operatorname{H.c.}\\right), \\\\\\hat{Q}_{n}^{5}&=\\frac{i}{2}\\left( \\hat{\\sigma }_{3n-2}^{+}\\hat{\\sigma }_{3n-1}^{z}\\hat{\\sigma }_{3n}^{-}-\\operatorname{H.c.}\\right), \\\\\\hat{Q}_{n}^{6}&=\\frac{(-1)^n}{2}\\left( \\hat{\\sigma }_{3n-1}^{+}\\hat{\\sigma }_{3n}^{-}+\\operatorname{H.c.}\\right), \\\\\\hat{Q}_{n}^{7}&=\\frac{i(-1)^n}{2}\\left( \\hat{\\sigma }_{3n}^{+}\\hat{\\sigma }_{3n-1}^{-}-\\operatorname{H.c.}\\right),\\\\\\hat{Q}_{n}^{8}&=\\frac{1}{4\\sqrt{3}}\\left( \\hat{\\sigma }_{3n-2}^{z}+\\hat{\\sigma }_{3n-1}^{z}-2\\hat{\\sigma }_{3n}^{z}\\right).$ The electric field Hamiltonian can be obtained by injecting the expressions of the non-Abelian charges (REF )-() in the color electric term in Eq.", "(REF ).", "We obtain $\\hat{H}_{e}=&\\frac{1}{3}\\sum _{n=1}^{N-1}\\left( 3-\\hat{\\sigma }_{3n-2}^{z}\\hat{\\sigma }_{3n-1}^{z}-\\hat{\\sigma }_{3n-2}^{z}\\hat{\\sigma }_{3n}^{z}-\\hat{\\sigma }_{3n-1}^{z}\\hat{\\sigma }_{3n}^{z}\\right)\\\\&+\\sum _{n=1}^{N-2}\\sum _{m=n+1}^{N-1}\\left[ (N-m)\\left( \\hat{\\sigma }_{3n-2}^{+}\\hat{\\sigma }_{3n-1}^{-}\\hat{\\sigma }_{3m-1}^{+}\\hat{\\sigma }_{3m-2}^{-}\\right.", "\\right.", "\\\\& +\\left.", "\\hat{\\sigma }_{3n-1}^{+}\\hat{\\sigma }_{3n}^{-}\\hat{\\sigma }_{3m-1}^{-}\\hat{\\sigma }_{3m}^{+}+\\operatorname{H.c.}\\right)(-1)^{n+m} \\\\& +(N-m)\\left(\\hat{\\sigma }_{3n-2}^{+}\\hat{\\sigma }_{3n-1}^{z}\\hat{\\sigma }_{3n}^{-}\\hat{\\sigma }_{3m-2}^{-}\\hat{\\sigma }_{3m-1}^{z}\\hat{\\sigma }_{3m}^{+}+\\operatorname{H.c.}\\right) \\\\&-\\frac{1}{12}(N-m) \\hat{\\sigma }_{3m-2}^{z}(\\hat{\\sigma }_{3n-1}^{z}+\\hat{\\sigma }_{3n}^{z}-2\\hat{\\sigma }_{3n-2}^{z}) \\\\&-\\frac{1}{12}(N-m) \\hat{\\sigma }_{3m-1}^{z}(\\hat{\\sigma }_{3n}^{z}+\\hat{\\sigma }_{3n-2}^{z}-2\\hat{\\sigma }_{3n-1}^{z})\\\\& \\left.", "-\\frac{1}{12}(N-m) \\hat{\\sigma }_{3m}^{z}(\\hat{\\sigma }_{3n-2}^{z}+\\hat{\\sigma }_{3n-1}^{z}-2\\hat{\\sigma }_{3n}^{z}) \\right], $ which exhibits long range spin-spin interaction as a direct consequence of the gauge elimination.", "The baryon number operator is proportional to the total magnetization of the system in the qubit encoding $\\hat{B}=\\frac{1}{6}\\sum _{n=1}^{3N}\\hat{\\sigma }_{n}^{z}.", "$ We are interested in a basic building block consisting of $N=2$ discretized lattice sites.", "The model is then described by a chain with 6 qubits and the terms in the Hamiltonian read $\\hat{H}_{kin}&=-\\frac{1}{2}\\left( \\hat{\\sigma }_{1}^{+}\\hat{\\sigma }_{2}^{z}\\hat{\\sigma }_{3}^{z} \\hat{\\sigma }_{4}^{-}-\\hat{\\sigma }_{2}^{+}\\hat{\\sigma }_{3}^{z}\\hat{\\sigma }_{4}^{z} \\hat{\\sigma }_{5}^{-}+\\hat{\\sigma }_{3}^{+}\\hat{\\sigma }_{4}^{z}\\hat{\\sigma }_{5}^{z} \\hat{\\sigma }_{6}^{-}+\\operatorname{H.c.}\\right), \\\\\\hat{H}_{m}&=\\frac{1}{2}\\left(6- \\hat{\\sigma }_{1}^{z}-\\hat{\\sigma }_{2}^{z}-\\hat{\\sigma }_{3}^{z}+\\hat{\\sigma }_{4}^{z}+\\hat{\\sigma }_{5}^{z}+\\hat{\\sigma }_{6}^{z} \\right),\\\\\\hat{H}_{e}&=\\frac{1}{3}\\left(3-\\hat{\\sigma }_{1}^{z}\\hat{\\sigma }_{2}^{z}-\\hat{\\sigma }_{1}^{z}\\hat{\\sigma }_{3}^{z} -\\hat{\\sigma }_{2}^{z}\\hat{\\sigma }_{3}^{z}\\right).$ In the sector with baryon number $B=0$ (i.e.", "as much matter as antimatter), the three terms composing the Hamiltonian commute with the following operator $\\hat{CP}=\\prod _{n=1}^{3}\\hat{\\sigma }_{n}^{x}\\hat{\\sigma }_{7-n}^{x}\\hat{W}_{n,7-n}$ where $\\hat{W}_{n,n^{\\prime }}$ is the SWAP unitary operator between qubit $n$ and $n^{\\prime }$ .", "This symmetry corresponds to the composition of a spatial reflection with respect to the middle of the chain ($\\hat{P}$ ) followed by a charge conjugation operation ($\\hat{C}$ ) which flips the spins.", "Local spin operators $\\hat{\\sigma }_{n}^{a}$ transforms as $(\\hat{CP})^{\\dagger }\\hat{\\sigma }_{n}^{a}\\hat{CP}=(\\hat{\\sigma }^{x}\\hat{\\sigma }^{a}\\hat{\\sigma }^{x})_{7-n}$ under the $\\hat{CP}$ operation with $a=x,y,z$ and $n=1,2,\\dots ,6$ .", "It is thus clear that a convenient basis is the one spanned by states of the form $|\\Psi \\rangle = \\sum c_{ijk} |i\\rangle _1|j\\rangle _2 |k\\rangle _3 \\otimes \\hat{\\sigma }_{4}^x \\hat{\\sigma }_{5}^x\\hat{\\sigma }_{6}^x|i\\rangle _4|j\\rangle _5|k\\rangle _6$ , which are invariant under the $\\hat{CP}$ operation and have $B=0$ .", "When one works with this basis, the state of the last three qubits is completely determined by the state of the first three.", "As a direct consequence, we can encode the states by using only the first 3 qubits (i.e.", "the state of the antimatter) rather than 6.", "The reduced three-qubit Hamiltonian reads $\\hat{H}_{kin}^{(3)}&=-\\frac{1}{2}\\left( \\hat{\\sigma }_{1}^{x}\\hat{\\sigma }_{2}^{z}\\hat{\\sigma }_{3}^{z}+\\hat{\\sigma }_{1}^{z}\\hat{\\sigma }_{2}^{x}\\hat{\\sigma }_{3}^{z}+\\hat{\\sigma }_{1}^{z}\\hat{\\sigma }_{2}^{z}\\hat{\\sigma }_{3}^{x}\\right),\\\\\\hat{H}_{m}^{(3)}&=3-\\hat{\\sigma }_{1}^{z}-\\hat{\\sigma }_{2}^{z}-\\hat{\\sigma }_{3}^{z},\\\\\\hat{H}_{e}^{(3)}&=\\frac{1}{3}\\left(3-\\hat{\\sigma }_{1}^{z}\\hat{\\sigma }_{2}^{z}-\\hat{\\sigma }_{1}^{z}\\hat{\\sigma }_{3}^{z} -\\hat{\\sigma }_{2}^{z}\\hat{\\sigma }_{3}^{z}\\right), $ and the time evolution is obtained using the Hamiltonian $\\hat{H}^{(3)}=\\hat{H}_{kin}^{(3)}+\\tilde{m}\\hat{H}_{m}^{(3)}+\\frac{1}{2x}\\hat{H}_{e}^{(3)}$" ], [ "Although the Hamiltonian of Eqs.", "(REF -) is expressed in terms of Pauli X and Z gates, a simple rotation to Y and Z gates allows for more cancellations among CNOT gates.", "This is especially valuable on hardware that does not provide all-to-all connectivity among the qubits.", "The first half of our first-order Trotter step is displayed in Fig.", "REF and, to match the available hardware, it does not use any entangling gates directly between $q_1$ and $q_3$ .", "The second half of the Trotter step is the same except for a relabeling of $q_1 \\leftrightarrow q_3$ .", "Figure: The first half of a Trotter step is shown in this figure.The second half is identical except for the interchange of two qubits: q 1 ↔q 3 q_1 \\leftrightarrow q_3.Note that a pair of CNOT gates can be canceled where the two halves meet." ], [ "We model the obtained data $D$ as draws from a normal distribution with variance $\\sigma $ , where the mean $S_K$ is given by a cosine series $S_K = \\sum _i^K A_i \\cos \\left(\\omega _i t + \\phi _i\\right) + \\xi ,$ with uniform priors on the frequencies $\\omega _i$ and the amplitudes $A_i$ , while the priors for the phases $\\phi _i$ and the offset $\\xi $ are normal distributions located at zero.", "The uniform priors are adjusted to include the frequencies that are visible from the data by eye.", "The prior on $\\sigma $ is given by the maximum of $0.3$ and the value of the largest errorbar in the dataset.", "Note that each dataset stemming from different $N_\\mathrm {T}$ is modeled as a separate likelihood while the parameters are shared.", "After obtaining a representation of the posterior distribution for the parameters $\\theta = \\lbrace A_i, \\omega _i, \\phi _i, \\xi , \\sigma \\rbrace $ , $P\\left(\\theta \\vert D\\right)$ , via Monte Carlo sampling from Bayes' rule, we sample the posterior predictive distribution $P(D^\\prime \\vert D) = \\int _\\Theta P(\\Theta =\\theta \\vert D)P(D^\\prime \\vert \\Theta = \\theta )$ , where $\\Theta $ denotes the collective random variables for all parameters." ] ]
2207.03473
[ [ "Eureka!: An End-to-End Pipeline for JWST Time-Series Observations" ], [ "Abstract $\\texttt{Eureka!", "}$ is a data reduction and analysis pipeline for exoplanet time-series observations, with a particular focus on James Webb Space Telescope (JWST) data.", "The goal of $\\texttt{Eureka!", "}$ is to provide an end-to-end pipeline that starts with raw, uncalibrated FITS files and ultimately yields precise exoplanet transmission and/or emission spectra.", "The pipeline has a modular structure with six stages, and each stage uses a \"Eureka!", "Control File\" (ECF; these files use the .ecf file extension) to allow for easy control of the pipeline's behavior.", "Stage 5 also uses a \"Eureka!", "Parameter File\" (EPF; these files use the .epf file extension) to control the fitted parameters.", "We provide template ECFs for the MIRI, NIRCam, NIRISS, and NIRSpec instruments on JWST and the WFC3 instrument on the Hubble Space Telescope (HST).", "These templates give users a good starting point for their analyses, but $\\texttt{Eureka!", "}$ is not intended to be used as a black-box tool, and users should expect to fine-tune some settings for each observation in order to achieve optimal results.", "At each stage, the pipeline creates intermediate figures and outputs that allow users to compare $\\texttt{Eureka!", "}$'s performance using different parameter settings or to compare $\\texttt{Eureka!", "}$ with an independent pipeline.", "The ECF used to run each stage is also copied into the output folder from each stage to enhance reproducibility.", "Finally, while $\\texttt{Eureka!", "}$ has been optimized for exoplanet observations (especially the latter stages of the code), much of the core functionality could also be repurposed for JWST time-series observations in other research domains thanks to $\\texttt{Eureka!", "}$'s modularity." ], [ "0pt" ], [ "0pt" ], [ "0pt *" ], [ " Figure: NO_CAPTIONFigure: NO_CAPTION Bell et al.", "(2022).", "Eureka!", ": An End-to-End Pipeline for JWST Time-Series Observations.", "Journal of Open Source Software, 0(0), ¿PAGE?", "https://doi.org/10.xxxxxx/draft.Bell et al.", "(2022).", "Eureka!", ": An End-to-End Pipeline for JWST Time-Series Observations.", "Journal of Open Source Software, 0(0), ¿PAGE?", "https://doi.org/10.xxxxxx/draft.", "[L]4.5cm [L]4.5cm repobox colback=red, colframe=red!75!black, boxrule=0.5pt, arc=2pt, left=6pt, right=6pt, top=3pt, bottom=3pt centerflushleft centerflushleft 1em 1=0 Ligatures=TeX,Scale=MatchLowercase same [main,import]american 0 1]Taylor J.", "Bell 0000-0003-4177-2149  2]Eva-Maria Ahrer 0000-0003-0973-8426  3]Jonathan Brande 0000-0002-2072-6541  4]Aarynn L. Carter 0000-0001-5365-4815  5]Adina D. Feinstein 0000-0002-9464-8101  6]Giannina Guzman Caloca 0000-0001-6340-8220  7,8]Megan Mansfield 0000-0003-4241-7413  9]Sebastian Zieba 0000-0003-0562-6750  10]Caroline Piaulet 0000-0002-2875-917X  10]Björn Benneke 0000-0001-5578-1498  11]Joseph Filippazzo 0000-0002-0201-8306  12]Erin M. May 0000-0002-2739-1465  10]Pierre-Alexis Roy 0000-0001-6809-3520  9]Laura Kreidberg 0000-0003-0514-1147  12]Kevin B. Stevenson 0000-0002-7352-7941  [1]BAER Institute, NASA Ames Research Center, Moffet Field, CA 94035, USA [2]Department of Physics, University of Warwick, Gibbet Hill Road, CV4 7AL Coventry, UK [3]Department of Physics and Astronomy, University of Kansas, 1082 Malott, 1251 Wescoe Hall Dr., Lawrence, KS 66045, USA [4]Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA [5]Department of Astronomy & Astrophysics, University of Chicago, 5640 S. Ellis Avenue, Chicago, IL 60637, USA [6]Department of Astronomy, University of Maryland, College Park, MD USA [7]Steward Observatory, University of Arizona, Tucson, AZ 85719, USA [8]NHFP Sagan Fellow [9]Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany [10]Department of Physics and Institute for Research on Exoplanets, Université de Montréal, Montreal, QC, Canada [11]Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA [12]Johns Hopkins APL, 11100 Johns Hopkins Road, Laurel, MD 20723, USA DOI: 10.xxxxxx/draft Software Review [x=1.2ex, y=1.2ex, baseline=-0.05ex][x=1ex, y=1ex] (-0.1,-0.1) –++ (-0, 1.2) –++ (0.6, 0) –++ (0, -0.6) –++ (0.6, 0) –++ (0, -1); [draw, line width = 0.5, rounded corners=0.5] (0,0) rectangle (1,1); [draw, line width = 0.5] (0.5, 0.5) – (1, 1); [draw, line width = 0.5] (0.6, 1) – (1, 1) – (1, 0.6); Repository [x=1.2ex, y=1.2ex, baseline=-0.05ex][x=1ex, y=1ex] (-0.1,-0.1) –++ (-0, 1.2) –++ (0.6, 0) –++ (0, -0.6) –++ (0.6, 0) –++ (0, -1); [draw, line width = 0.5, rounded corners=0.5] (0,0) rectangle (1,1); [draw, line width = 0.5] (0.5, 0.5) – (1, 1); [draw, line width = 0.5] (0.6, 1) – (1, 1) – (1, 0.6); Editor: @dfm [x=1.2ex, y=1.2ex, baseline=-0.05ex][x=1ex, y=1ex] (-0.1,-0.1) –++ (-0, 1.2) –++ (0.6, 0) –++ (0, -0.6) –++ (0.6, 0) –++ (0, -1); [draw, line width = 0.5, rounded corners=0.5] (0,0) rectangle (1,1); [draw, line width = 0.5] (0.5, 0.5) – (1, 1); [draw, line width = 0.5] (0.6, 1) – (1, 1) – (1, 0.6); Reviewers: @catrionamurray @christinahedges Submitted: 03 June 2022Published: unpublished LicenseAuthors of papers retain copyright and release the work under a Creative Commons Attribution 4.0 International License (CC BY 4.0)." ], [ "Summary", "Eureka!", "is a data reduction and analysis pipeline for exoplanet time-series observations, with a particular focus on James Webb Space Telescope (JWST) data.", "The goal of Eureka!", "is to provide an end-to-end pipeline that starts with raw, uncalibrated FITS files and ultimately yields precise exoplanet transmission and/or emission spectra.", "The pipeline has a modular structure with six stages, and each stage uses a “Eureka!", "Control File” (ECF; these files use the .ecf file extension) to allow for easy control of the pipeline’s behavior.", "Stage 5 also uses a “Eureka!", "Parameter File” (EPF; these files use the .epf file extension) to control the fitted parameters.", "We provide template ECFs for the MIRI, NIRCam, NIRISS, and NIRSpec instruments on JWST and the WFC3 instrument on the Hubble Space Telescope (HST).", "These templates give users a good starting point for their analyses, but Eureka!", "is not intended to be used as a black-box tool, and users should expect to fine-tune some settings for each observation in order to achieve optimal results.", "At each stage, the pipeline creates intermediate figures and outputs that allow users to compare Eureka!’s performance using different parameter settings or to compare Eureka!", "with an independent pipeline.", "The ECF used to run each stage is also copied into the output folder from each stage to enhance reproducibility.", "Finally, while Eureka!", "has been optimized for exoplanet observations (especially the latter stages of the code), much of the core functionality could also be repurposed for JWST time-series observations in other research domains thanks to Eureka!’s modularity." ], [ "Outline of ", "Eureka!", "is broken down into six stages, which are as follows (also summarized in fig:overview): Stage 1: An optional step that calibrates raw data (converts ramps to slopes for JWST observations).", "This step can be skipped within Eureka!", "if you would rather use the Stage 1 outputs from the jwst pipeline.", "Stage 2: An optional step that further calibrates Stage 1 data (performs flat-fielding, unit conversion, etc.", "for JWST observations).", "This step can be skipped within Eureka!", "if you would rather use the Stage 2 outputs from the jwst pipeline.", "Stage 3: Using Stage 2 outputs, performs background subtraction and optimal spectral extraction.", "For spectroscopic observations, this stage generates a time series of 1D spectra.", "For photometric observations, this stage generates a single light curve of flux versus time.", "Stage 4: Using Stage 3 outputs, generates spectroscopic light curves by binning the time series of 1D spectra along the wavelength axis.", "Optionally removes drift/jitter along the dispersion direction and/or sigma clips outliers.", "Stage 5: Fits the light curves with noise and astrophysical models using different optimization or sampling algorithms.", "Stage 6: Displays the planet spectrum in figure and table form using results from the Stage 5 fits.", "Figure: An overview flowchart showing the processing done at each stagein Eureka!.", "The outputs of each stage are used as the inputs tothe subsequent stage along with the relevant settings file(s)." ], [ "Statement of Need", "The calibration, reduction, and fitting of exoplanet time-series observations is a challenging problem with many tunable parameters across many stages, many of which will significantly impact the final results.", "Typically, the default calibration pipeline from astronomical observatories is insufficiently tailored for exoplanet time-series observations as the pipeline is more optimized for other science use cases.", "As such, it is common practice to develop a custom data analysis pipeline that starts from the original, uncalibrated images.", "Historically, data analysis pipelines have often been proprietary, so each new user of an instrument or telescope has had to develop their own pipeline.", "Also, clearly specifying the analysis procedure can be challenging, especially with proprietary code, which erodes reproducibility.", "Eureka!", "seeks to be a next-generation data analysis pipeline for next-generation observations from JWST with open-source and well-documented code for easier adoption; modular code for easier customization while maintaining a consistent framework; and easy-to-use but powerful inputs and outputs for increased automation, increased reproducibility, and more thorough intercomparisons.", "By also allowing for analyses of HST observations within the same framework, users will be able to combine new and old observations to develop a more complete understanding of individual targets or even entire populations." ], [ "Documentation", "Documentation for Eureka!", "is available at https://eurekadocs.readthedocs.io/en/latest/." ], [ "Similar Tools", "jwst (JWST calibration pipeline developers, 2022), exoplanet (Foreman-Mackey et al., 2021), juliet (Espinoza et al., 2019), POET (Cubillos et al., 2013; Stevenson et al., 2012), WFC3 (Stevenson et al., 2014), PACMAN (Kreidberg et al., 2014; Zieba & Kreidberg, 2022), SPCA (Bell et al., 2021; Dang et al., 2018)" ], [ "Acknowledgements", "Eureka!", "allows for some variations upon the STScI’s jwst pipeline (JWST calibration pipeline developers, 2022) for Stages 1 and 2, but presently these stages mostly act as wrappers around the jwst pipeline.", "This allows Eureka!", "to run the jwst pipeline in the same manner as Eureka!’s latter stages.", "Eureka!", "then uses its own custom code for additional calibration steps, spectral or photometric extraction, and light curve fitting.", "Several parts of the spectroscopy-focused code in Stages 3 and 4 of Eureka!", "were inspired by, or were initially written for, the WFC3 (Stevenson et al., 2014) pipeline.", "Other parts of the spectroscopy code and several parts of the photometry focused code in Stage 3 were inspired by, or were initially written for, the POET pipeline (Cubillos et al., 2013; Stevenson et al., 2012).", "Some of the Stage 5 code comes from Kreidberg et al.", "(2014) and PACMAN (Zieba & Kreidberg, 2022).", "Small pieces of the SPCA (Bell et al., 2021; Dang et al., 2018) and Bell_EBM (Bell & Cowan, 2018) repositories have also been reused.", "ALC is supported by a grant from STScI (JWST-ERS-01386) under NASA contract NAS5-03127.", "ADF acknowledges support by the National Science Foundation Graduate Research Fellowship Program under Grant No. (DGE-1746045).", "CP acknowledges financial support by the Fonds de Recherche Québécois—Nature et Technologie (FRQNT; Québec), the Technologies for Exo-Planetary Science (TEPS) Trainee Program and the Natural Sciences and Engineering Research Council (NSERC) Vanier Scholarship.", "JB acknowledges support from the NASA Interdisciplinary Consortia for Astrobiology Research (ICAR).", "KBS is supported by JWST-ERS-01366.", "MM acknowledges support through the NASA Hubble Fellowship grant HST-HF2-51485.001-A awarded by STScI, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555.", "We also thank Ivelina Momcheva for useful discussions.", "Support for this work was provided in part by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127.", "In addition, we would like to thank the Transiting Exoplanet Community Early Release Science program for organizing meetings that contributed to the writing of Eureka!." ], [ "References", "tocsectionReferences preBell, T. J., & Cowan, N. B.", "(2018).", "Increased Heat Transport in Ultra-hot Jupiter Atmospheres through H$_{2}$ Dissociation and Recombination.", "ApJL, 857(2), L20.", "https://doi.org/10.3847/2041-8213/aabcc8 preBell, T. J., Dang, L., Cowan, N. B., Bean, J., Désert, J.-M., Fortney, J. J., Keating, D., Kempton, E., Kreidberg, L., Line, M. R., Mansfield, M., Parmentier, V., Stevenson, K. B., Swain, M., & Zellem, R. T. (2021).", "A comprehensive reanalysis of Spitzer’s 4.5 $\\mu $ m phase curves, and the phase variations of the ultra-hot Jupiters MASCARA-1b and KELT-16b.", "MNRAS, 504(3), 3316–3337.", "https://doi.org/10.1093/mnras/stab1027 preCubillos, P., Harrington, J., Madhusudhan, N., Stevenson, K. B., Hardy, R. A., Blecic, J., Anderson, D. R., Hardin, M., & Campo, C. J.", "(2013).", "WASP-8b: Characterization of a Cool and Eccentric Exoplanet with Spitzer.", "ApJ, 768(1), 42. https://doi.org/10.1088/0004-637X/768/1/42 preDang, L., Cowan, N. B., Schwartz, J. C., Rauscher, E., Zhang, M., Knutson, H. A., Line, M., Dobbs-Dixon, I., Deming, D., Sundararajan, S., Fortney, J. J., & Zhao, M. (2018).", "Detection of a westward hotspot offset in the atmosphere of hot gas giant CoRoT-2b.", "Nature Astronomy, 2, 220–227.", "https://doi.org/10.1038/s41550-017-0351-6 preEspinoza, N., Kossakowski, D., & Brahm, R. (2019).", "juliet: a versatile modelling tool for transiting and non-transiting exoplanetary systems.", "MNRAS, 490(2), 2262–2283.", "https://doi.org/10.1093/mnras/stz2688 preForeman-Mackey, D., Luger, R., Agol, E., Barclay, T., Bouma, L. G., Brandt, T. D., Czekala, I., David, T. J., Dong, J., Gilbert, E. A., Gordon, T. A., Hedges, C., Hey, D. R., Morris, B. M., Price-Whelan, A. M., & Savel, A.", "B.", "(2021).", "exoplanet: Gradient-based probabilistic inference for exoplanet data & other astronomical time series (Version 0.5.1).", "Zenodo; Zenodo.", "https://doi.org/10.5281/zenodo.1998447 preJWST calibration pipeline developers.", "(2022).", "jwst: Python library for science observations from the james webb space telescope.", "In GitHub repository.", "GitHub.", "https://github.com/spacetelescope/jwst preKreidberg, L., Bean, J. L., Désert, J.-M., Benneke, B., Deming, D., Stevenson, K. B., Seager, S., Berta-Thompson, Z., Seifahrt, A., & Homeier, D. (2014).", "Clouds in the atmosphere of the super-Earth exoplanet GJ 1214b.", "Nature, 505(7481), 69–72.", "https://doi.org/10.1038/nature12888 preStevenson, K. B., Bean, J. L., Seifahrt, A., Désert, J.-M., Madhusudhan, N., Bergmann, M., Kreidberg, L., & Homeier, D. (2014).", "Transmission Spectroscopy of the Hot Jupiter WASP-12b from 0.7 to 5 $\\mu $ m. AJ, 147(6), 161. https://doi.org/10.1088/0004-6256/147/6/161 preStevenson, K. B., Harrington, J., Fortney, J. J., Loredo, T. J., Hardy, R. A., Nymeyer, S., Bowman, W. C., Cubillos, P., Bowman, M. O., & Hardin, M. (2012).", "Transit and Eclipse Analyses of the Exoplanet HD 149026b Using BLISS Mapping.", "ApJ, 754(2), 136. https://doi.org/10.1088/0004-637X/754/2/136 preZieba, S., & Kreidberg, L. (2022).", "PACMAN.", "In GitHub repository.", "GitHub.", "https://github.com/sebastian-zieba/PACMAN" ] ]
2207.03585
[ [ "Towards Optimal Integrated Planning of Electricity and Hydrogen\n Infrastructure for Large-Scale Renewable Energy Transport" ], [ "Abstract The imminent advent of large-scale green hydrogen (H2) production raises the central question of which of the two options, transporting \"green\" molecules, or transporting \"green\" electrons, is the most cost-effective one.", "This paper proposes a first-of-its-kind mathematical framework for the optimal integrated planning of electricity and H2 infrastructure for transporting large-scale variable renewable energy (VRE).", "In contrast to most existing works, this work incorporates essential nonlinearities such as voltage drops due to losses in high-voltage alternating current (HVAC) and high-voltage direct current (HVDC) transmission lines, losses in HVDC converter stations, reactive power flow, pressure drops in pipelines, and linepack, all of which play an important role in determining the optimal infrastructure investment decision.", "Capturing these nonlinearities requires casting the problem as a nonconvex mixed-integer nonlinear program (MINLP), whose complexity is further exacerbated by its large size due to the relatively high temporal resolution of RES forecasts.", "This work then leverages recent advancements in convex relaxations to instead solve a tractable alternative in the form of a mixed-integer quadratically constrained programming (MIQCP) problem.", "The impact of other fundamental factors such as transmission distance and RES capacity is also thoroughly analysed on a canonical two-node system.", "The integrated planning model is then demonstrated on a real-world case study involving renewable energy zones in Australia." ], [ "Introduction", "Due to the variability of renewable energy sources (RES), maximising their utilisation is arguably one of the biggest challenges facing energy system operators in Australia and around the world.", "Maximising this utilisation requires energy storage, which, in the case of large energy volumes, may be problematic as large-scale battery storage alone is too costly and pumped-hydro storage is limited for geographical reasons .", "A promising long-term solution for maximising the integration of VRE consists of building a new infrastructure for transporting VRE in the form of electricity and/or H$_2$ .", "Large-scale renewable energy hubs coupled to H$_2$ production hubs might unlock substantial economies of scale predicated on building a cost-effective VRE transport infrastructure.", "Designing a cost-effective infrastructure will need to address the challenging questions of (i) whether VRE hubs and electrolysers should be co-located, (ii) whether to transport VRE as molecules in H$_2$ pipelines or as electrons in electricity transmission lines, and (iii) the drivers and conditions that favour one investment option over another.", "Answering the above questions is a massive undertaking that requires an integrated electricity and H$_2$ system (IEHS) modelling framework to assess costs and benefits of different investment options.", "As many of the challenges identified here are relatively new, existing knowledge and modelling tools are inadequate for performing such a large-scale optimal integrated infrastructure design exercise.", "In particular, existing state-of-the-art literature is either limited in scope to H$_2$ supply chain only , , , , i.e., disregarding electricity infrastructure options, or is limited in the variety of considered infrastructure technologies , , , .", "Considering all the relevant transport and storage technologies in an integrated framework can unlock superior design solutions.", "This is especially true when considering the specific features associated with RES, and in particular when they are clustered in large-scale renewable energy hubs where wind and solar farms may be located far from the location of H$_2$ utilisation.", "Other essential aspects that are ignored in the literature include voltage drops due to losses in transmission lines, pressure drops in pipelines, linepack,The linepack is the amount of pressurised gas stored in a pipeline network.", "compressor sizing, water availability for electrolysers, and reactive power compensation, all of which play an important role in determining the optimal infrastructure investment decision.", "The modelling of the linepack is instrumental in quantifying the VRE storage capacity of the H$_2$ pipeline network, which can in turn influence the sizing of H$_2$ pipelines and compressors.", "In fact, most (if not all) existing works use steady-state gas flow models, which are generally inadequate in gas transmission networks where H$_2$ injections from the VRE introduce time-varying accumulation rates.", "More importantly, with the exception of , , the majority of existing works, including , , , , only examine transport options between just two nodes, as opposed to over a network with a general topology (which may include loops and parallel links).", "In light of the knowledge gaps identified above, this paper introduces a novel mathematical optimisation model aiming at finding the optimal integrated infrastructure planning for transporting large-scale VRE as either electricity lines and/or H$_2$ pipelines.", "Specifically, the model not only considers all relevant infrastructure technologies such as HVDC, HVAC, reactive power plants, and H$_2$ pipelines and compressors, but also incorporates all the essential nonlinearities that directly influence the optimal infrastructure investment decision, such as voltage drops due to losses in HVAC and HVDC transmission lines, losses in HVDC converter stations, reactive power flow, pressure drops in pipelines, and linepack.", "Additionally, the model adopts a relatively high temporal resolution to fully capture the variability of RES and its impact on the optimal investment decision.", "Instead of directly solving the resulting large-scale nonconvex mixed-integer nonlinear programming (MINLP) problem, which is computationally intractable, the paper introduces a tractable alternative in the form of a mixed-integer quadratically constrained programming (MIQCP) relaxation.", "This novel MIQCP model is demonstrated on a set of studies that rigorously analyse the impact of the two fundamental factors, distance and RES capacity, on the optimal planning decision.", "The MIQCP model is also demonstrated on a real-world case study involving actual renewable energy zones in Australia.", "The paper is organised as follows.", "Section  introduces the optimal integrated VRE transport infrastructure design model and Section  describes how to derive a strong MIQCP relaxation of the problem.", "Section  numerically evaluates the proposed MIQCP model on a canonical 2-node system as well as on a real-world case study involving renewable energy zones in Australia.", "The paper concludes in Section ." ], [ "Mathematical modelling", "A prototype integrated VRE transport infrastructure design model is shown in Figure REF , where electricity transmission line options include both HVDC and HVAC, as well as their associated control equipment such as transformers, reactive power compensation, and converters.", "The H$_2$ pipeline options also include compressors and pressure regulators.", "Figure: Illustration of a prototype integrated VRE transport infrastructure design model where the energy from multiple VRE hubs can be transported to an H 2 _2 demand point using electricity lines and/or H 2 _2 pipelines over a network." ], [ "Electrolyser station model", "Electrolysers use electricity to split water into H$_2$ and oxygen (O$_2$ ) in a process called electrolysis.", "Since the output pressure of a typical proton exchange membrane (PEM) electrolyser is around 3.5 , an electrolyser station in this work is assumed to include a gas compressor to boost the pressure to transmission levels (up to 10), as shown in Figure REF .", "Figure: Model of an electrolyser station.The decision to install an electrolyser station at a certain location can be captured by a binary variable $z_{im}^{\\rm ptg}$ which takes a value of 1 if the electrolyser station is installed and 0 otherwise.", "In constraint form this can be written as $& z_{im}^{\\rm ptg} \\in \\left\\lbrace 0,1 \\right\\rbrace , & im \\in \\mathcal {E}, $ where $\\mathcal {E}$ is the set of all candidate electrolyser stations in the network.", "The process of converting electrical energy to chemical energy can be mathematically written as $& \\phi _{mty}^{\\rm ptg} = \\frac{p_{ity}^{\\rm ptg} \\eta _{im}^{\\rm ptg}}{HHV}, \\qquad \\qquad im \\in \\mathcal {E}, \\ ty \\in \\mathcal {T} \\times \\mathcal {Y} \\\\& w_{ity}^{\\rm ptg} = w_{i(t-1)y}^{\\rm ptg} - 10\\phi _{mty}^{\\rm ptg} \\rho \\Delta \\tau , \\ im \\in \\mathcal {E}, \\ ty \\in \\mathcal {T} \\times \\mathcal {Y} $ where $p_{ity}^{\\rm ptg}$ () is the input electrical power to the electrolyser station, $\\phi _{mty}^{\\rm ptg}$ () is the aggregated output H$_2$ volumetric flow rate of the electrolyser modules, and $w_{ity}^{\\rm ptg}$ () is the input water consumed by the electrolyser station over a period of $\\Delta \\tau $ ().", "In (REF ), $\\eta _{im}^{\\rm ptg} = 70\\%$ is the efficiency of each electrolyser module, $HHV = {12.1948}{}$ is the higher heating value of H$_2$ , and $\\rho = {0.086}{}$ is the density of H$_2$ .", "The efficiency of the electrolyser station $\\eta _{im}^{\\rm ptg}$ includes rectifiers and transformers (including transformer cooling and gas cooling).", "Constraint () is founded on the fact that producing 1 of H$_2$ requires 10 of water.", "Each candidate electrolyser station location in the network is associated with a predetermined initial amount of water $w_{i01}^{\\rm ptg}$ (at $t=0$ and $y=1$ ).", "The input electrical power is constrained by a maximum predetermined upper limit on the size of the station, $\\overline{p}_{i}^{\\rm ptg}$ , through $& 0 \\le p_{ity}^{\\rm ptg} \\le \\overline{p}_{i}^{\\rm ptg} z_{im}^{\\rm ptg}.", "& im \\in \\mathcal {E}, \\ ty \\in \\mathcal {T} \\times \\mathcal {Y} $ The size of the compressor can be determined from the required horsepower $p_{mty}^{\\rm cp}$ () $p_{mty}^{{\\rm cp}} = \\frac{K T Z_{m}^{\\rm cp} \\gamma \\phi _{mty}^{\\rm ptg}}{(\\gamma - 1)\\eta _{im}^{\\rm cp}}\\left( \\left( \\frac{\\overline{\\wp }_{m}}{\\wp _{mty}^{\\rm ptg}} \\right)^{\\frac{\\gamma - 1}{\\gamma }} - 1 \\right), \\nonumber \\\\im \\in \\mathcal {E}, \\ ty \\in \\mathcal {T} \\times \\mathcal {Y}$ where $\\gamma = 1.296$ is the isentropic exponent (dimensionless), $K=0.351121 \\times 10^{-3}$ (), and $\\eta _{im}^{\\rm cp}$ (dimensionless) is the overall efficiency of the compressor.", "The maximum output pressure of the compressor can be set to the maximum operating pressure of the H$_2$ network $\\overline{\\wp }_{m}= {10}{}$ , which, combined with a fixed output pressure of the electrolyser modules $\\wp _{mty}^{\\rm ptg} = {3.5}{}$ , makes (REF ) linear in $p_{mty}^{{\\rm cp}}$ and $\\phi _{mty}^{\\rm ptg}$ .", "Note that at the demand point a compressor is not needed and therefore $p_{mty}^{{\\rm cp}} = 0$ .", "Finally, the compressibility factor in (REF ) is obtained from the Soave-Redlich-Kwong (SRK) equation of state with $T = {288.15}{}$ as the H$_2$ gas temperature at standard conditions.", "The investment cost of electrolyser stations is given by $I^{\\rm ptg} = \\sum _{im \\in \\mathcal {E}} \\left( c_{i,0}^{\\rm ptg}z_{im}^{\\rm ptg} + c_{i,1}^{\\rm ptg} \\left\\Vert \\left( p_{ity}^{\\rm ptg} \\right)_{ty \\in \\mathcal {T} \\times \\mathcal {Y}} \\right\\Vert _{\\infty } \\right.", "\\\\\\left.", "+ c_{m}^{\\rm cp} \\left\\Vert \\left( p_{mty}^{\\rm cp} \\right)_{ty \\in \\mathcal {T} \\times \\mathcal {Y}} \\right\\Vert _{\\infty } \\right),$ where $c_{i,0}^{\\rm ptg}$ ($) is the base installation cost of an electrolyser station, $c_{i,1}^{\\rm ptg}$ ($) is the unit cost of an electrolyser station, $c_{m}^{\\rm cp}$ ($) is the unit cost of the compressor.", "The unit cost of an electrolyser station $c_{i,1}^{\\rm ptg}$ includes the cost of the step-down transformer and rectifier." ], [ "H$_2$ pipeline model", "Each gas transmission corridor $mn \\in \\mathcal {P}$ between junctions $m$ and $n$ is associated with a predetermined set of candidate H$_2$ pipeline link options $\\mathcal {O}^{\\rm p}$ .", "A model of an H$_2$ pipeline link over gas transmission corridor $mn$ is shown in Figure REF .", "Figure: Model of an H 2 _2 pipeline link.Different pipeline link options are distinguished by different pipeline diameters including 0.5, 0.9, and 1.2.", "The gas transmission capacity $\\phi _{mnty}^{o}$ () of a pipeline increases with the diameter $D_{mn}^{o}$ ().", "The decision of choosing a certain pipeline option can be captured by a binary variable $z_{mn}^{{\\rm p},o}$ which takes a value of 1 if option $o$ is installed and 0 otherwise.", "In constraint form this can be written as $& z_{mn}^{{\\rm p},o} \\in \\left\\lbrace 0,1 \\right\\rbrace , & mn \\in \\mathcal {P}, \\ o \\in \\mathcal {O}^{\\rm p} $ where $\\mathcal {P}$ is the set of all tentative pipeline corridors where H$_2$ gas is flowing from junction $m$ towards junction $n$ .", "In this work, the direction of gas flow is known in advance owing to the predetermined locations of RES and H$_2$ demand (off-take) locations.", "The average gas volume flow rate $\\phi _{mnty}^{o}$ () across pipeline option $o$ over transmission corridor $mn$ can be obtained from the discretised equation of motion along the full length of the pipe $& \\left( \\phi _{mnty}^{o} \\right)^2 = \\Phi _{mn}^{o} \\left((\\wp _{mty}^{o})^2 - (\\wp _{nty}^{o})^2 \\right), & mn \\in \\mathcal {P}, $ for all $ o \\in \\mathcal {O}^{\\rm p}$ , $ ty \\in \\mathcal {T} \\times \\mathcal {Y}$ , where $\\Phi _{mn}^{o} = \\frac{\\eta _{mn}^{o}\\pi ^2 (D_{mn}^{o})^5}{16 \\rho ^2 Z_{mn}^{o} R T L_{mn}^{o} f_{mn}^{o}},$ and $f_{mn}^{o} = 4\\left(20.621 (D_{mn}^{o})^{1/6} \\right)^{-2}$ defines the Weymouth friction factor , and $\\eta _{mn}^{o}$ is the pipe efficiency.", "The compressibility factor $Z_{mn}^{o}$ is computed from the SRK equation of state .", "In (REF ), the pressures $\\wp _{nty}^{o}$ () are related to the junction pressures $\\wp _{nty}$ () through $& \\underline{\\wp }_{m}z_{mn}^{{\\rm p},o} \\le \\wp _{mty}^{o} \\le \\overline{\\wp }_{m}z_{mn}^{{\\rm p},o}, & \\\\& \\underline{\\wp }_{n}z_{mn}^{{\\rm p},o} \\le \\wp _{nty}^{o} \\le \\overline{\\wp }_{n}z_{mn}^{{\\rm p},o}, & \\\\& \\underline{\\wp }_{m}\\left( 1 - z_{mn}^{{\\rm p},o} \\right) \\le \\wp _{mty} - \\wp _{mty}^{o} \\le \\overline{\\wp }_{m}\\left( 1 - z_{mn}^{{\\rm p},o} \\right), & \\\\& \\underline{\\wp }_{n}\\left( 1 - z_{mn}^{{\\rm p},o} \\right) \\le \\wp _{nty} - \\wp _{nty}^{o} \\le \\overline{\\wp }_{n}\\left( 1 - z_{mn}^{{\\rm p},o} \\right), & $ for all $mn \\in \\mathcal {P}$ , $ o \\in \\mathcal {O}^{\\rm p}$ , $ ty \\in \\mathcal {T} \\times \\mathcal {Y}$ and $& \\underline{\\wp }_{m} \\le \\wp _{mty} \\le \\overline{\\wp }_{m}, & m \\in \\mathcal {J}, \\ ty \\in \\mathcal {T} \\times \\mathcal {Y} $ where $\\mathcal {J}$ is the set of all gas junctions in the network.", "The volumetric gas flows entering and leaving the pipe are related to the average volumetric flow rate across the pipe through $& \\phi _{mnty}^{o} = 0.5 \\left( \\phi _{mnty}^{{\\rm in},o} + \\phi _{mnty}^{{\\rm out},o} \\right), & mn \\in \\mathcal {P} \\\\& 0 \\le \\phi _{mnty}^{o},\\phi _{mnty}^{{\\rm in},o},\\phi _{mnty}^{{\\rm out},o} \\le \\overline{\\phi }_{mnty}^{o}, & mn \\in \\mathcal {P} $ for all $ o \\in \\mathcal {O}^{\\rm p}$ , $ ty \\in \\mathcal {T} \\times \\mathcal {Y}$ , and the average pressure across a pipe is defined as $\\hspace{-8.5359pt} \\wp _{mnty}^{o} = \\frac{2}{3} \\left(\\wp _{mty}^{o} + \\wp _{nty}^{o} - \\frac{\\wp _{mty}^{o}\\wp _{nty}^{o}}{\\wp _{mty}^{o} + \\wp _{nty}^{o}} \\right), \\ mn \\in \\mathcal {P}$ for all $ o \\in \\mathcal {O}^{\\rm p}$ , $ ty \\in \\mathcal {T} \\times \\mathcal {Y}$ .", "The linepack in the pipeline can now be captured by $& \\ell _{mnty}^{o}=\\Psi _{mn}^{o} \\wp _{mnty}^{o}, & \\\\& \\ell _{mnty}^{o}=\\ell _{mn(t-1)y}^{o} + \\Delta \\tau \\left( \\phi _{mnty}^{{\\rm in},o} - \\phi _{mnty}^{{\\rm out},o} \\right), & $ for all $mn \\in \\mathcal {P}$ , $ o \\in \\mathcal {O}^{\\rm p}$ , $ ty \\in \\mathcal {T} \\times \\mathcal {Y}$ , where () is the discretised continuity equation over the full length of the pipe and $\\Psi _{mn}^{o} = \\frac{\\pi (D_{mn}^{o})^2 L_{mn}^{o}}{4\\rho Z_{mn}^{o} R T}.$ To ensure fairness, the initial value of the linepack (at $t=0$ and $y=1$ ) for all the pipeline options is set to its minimum value, i.e., $& \\ell _{mn01}^{o}=\\Psi _{mn}^{o} \\underline{\\wp }_{n} z_{mn}^{{\\rm p},o},$ for all $mn \\in \\mathcal {P}$ , $ o \\in \\mathcal {O}^{\\rm p}$ , $ ty \\in \\mathcal {T} \\times \\mathcal {Y}$ .", "Finally, the gas balance equations at each junction of the gas network can now be written as $\\left( \\phi _{mty}^{\\rm ptg} - \\mu p_{mty}^{\\rm cp} \\right) = \\\\\\sum _{o \\in \\mathcal {O}^{\\rm p}} \\left( \\sum _{mn \\in \\mathcal {P}} \\phi _{mnty}^{{\\rm in},o} - \\sum _{nm \\in \\mathcal {P}} \\phi _{nmty}^{{\\rm out},o} \\right) + \\phi _{mty}^{\\rm H_2,d},$ for all $m \\in \\mathcal {J}$ , $ ty \\in \\mathcal {T} \\times \\mathcal {Y}$ , where $\\phi _{mty}^{\\rm H_2,d}$ is the H$_2$ volumetric flow rate demand at each junction and $\\mu = {0.2624}{}$ is the gas turbine fuel rate coefficient of a centrifugal H$_2$ compressor.", "The product $\\mu p_{mty}^{\\rm cp}$ delineates the amount of gas consumed by the compressor in the electrolyser station (see Figure REF ) during the pressure boosting process.", "Figure: Model of an HVDC link consisting of two converter stations and a transmission line.The investment cost of the pipeline link is given by $I^{\\rm pipe} = \\sum _{mn \\in \\mathcal {P}} \\sum _{o \\in \\mathcal {O}^{\\rm p}} c_{mn}^{{\\rm p},o} z_{mn}^{{\\rm p},o},$ where $c_{mn}^{{\\rm p},o}$ ($) is the installation cost of a pipeline link of option $o$ over corridor $mn$ ." ], [ "HVDC link model", "An HVDC link over transmission corridor $ij$ consists of an HVDC transmission line connecting two converter stations, one at the sending end (rectifier) and one at the receiving end (inverter) of the link as shown in Figure REF .", "Each HVDC transmission corridor $ij \\in \\mathcal {L}^{\\rm dc}_{\\rm f}$ between buses $i$ and $j$ is associated with a predetermined set of candidate HVDC link options $\\mathcal {O}^{\\rm dc}$ and a maximum number of parallel links $c \\in \\mathcal {C}^{{\\rm dc},o} = \\left\\lbrace 1, \\ldots , \\overline{c}_{ij}^{{\\rm dc},o} \\right\\rbrace $ for each option $o \\in \\mathcal {O}^{\\rm dc}$ .", "Examples of an HVDC transmission link option include a 500 bipole at 1, 2, or 3 rated capacity.", "The decision to install a certain option and number of parallel links can be captured by a binary variable $z_{ij}^{{\\rm dc},oc}$ which takes a value of 1 if option $o$ and $c$ th link are installed and 0 otherwise.", "It therefore follows that $& z_{ij}^{{\\rm dc},oc} \\in \\left\\lbrace 0,1 \\right\\rbrace , & \\hspace{-28.45274pt} ij \\in \\mathcal {L}^{\\rm dc}_{\\rm f}, \\ o \\in \\mathcal {O}^{\\rm dc}, \\ c \\in \\mathcal {C}^{{\\rm dc},o} \\\\& \\sum _{c \\in \\mathcal {C}^{{\\rm dc},o}} z_{ij}^{{\\rm dc},oc} \\le \\overline{c}_{ij}^{{\\rm dc},o}, & ij \\in \\mathcal {L}^{\\rm dc}_{\\rm f}, \\ o \\in \\mathcal {O}^{\\rm dc} \\\\& z_{ij}^{{\\rm dc},oc} \\le z_{ij}^{{\\rm dc},o(c-1)}, & ij \\in \\mathcal {L}^{\\rm dc}_{\\rm f}, \\ c \\in \\mathcal {C}^{{\\rm dc},o} \\setminus \\lbrace 1\\rbrace $ where constraints () and () enforce the sequential installation of links in each option and transmission corridor.", "In this work, HVDC converter stations are assumed to be of the voltage-source (VSC) type, which can control active and reactive power independently.", "The main reason for this assumption is that, unlike other converter types such as line commutated converters (LCC), VSC HVDC incorporates self-commutating switching elements that can control active and reactive power independently without additional compensation equipment .", "This therefore makes VSC HVDC more suitable for transporting renewable energy over long distances, as is the case in this paper.", "The active, reactive, and apparent power of a VSC are bounded by its ratings and MVA capacity ($\\overline{S}^{{\\rm cv},o}$ ) as follows $& \\underline{p}^{{\\rm cv},o}z_{ij}^{{\\rm dc},oc} \\le p_{ijty}^{{\\rm cv}_{i},oc} \\le \\overline{p}^{{\\rm cv},o}z_{ij}^{{\\rm dc},oc}, & \\\\& \\underline{q}^{{\\rm cv},o}z_{ij}^{{\\rm dc},oc} \\le q_{ijty}^{{\\rm cv}_{i},oc} \\le \\overline{q}^{{\\rm cv},o}z_{ij}^{{\\rm dc},oc}, & \\\\& \\sqrt{ \\left( p_{ijty}^{{\\rm cv}_{i},oc} \\right)^2 + \\left( q_{ijty}^{{\\rm cv}_{i},oc} \\right)^2 } \\le \\overline{S}^{{\\rm cv},o} z_{ij}^{{\\rm dc},oc}, &$ for all $ij \\in \\mathcal {L}^{\\rm dc}_{\\rm f} \\cup \\mathcal {L}^{\\rm dc}_{\\rm t}$ , $ o \\in \\mathcal {O}^{\\rm dc}$ , $ c \\in \\mathcal {C}^{{\\rm dc},o}$ , $ ty \\in \\mathcal {T} \\times \\mathcal {Y}$ , where $\\mathcal {L}^{\\rm dc}_{\\rm t}$ is the set of HVDC corridors such that $j$ is the sending-end bus and $i$ is the receiving-end bus.", "A power electronic converter is an active device whose losses can be obtained from the following parametric equation $& p_{ijty}^{{\\rm loss}_{i},oc} = \\alpha ^{{\\rm dc},o} z_{ij}^{{\\rm dc},oc} + \\beta ^{{\\rm dc},o} i_{ijty}^{{\\rm cv}_{i},o} + \\gamma ^{{\\rm dc},o} \\left( i_{ijty}^{{\\rm cv}_{i},o} \\right)^2, $ for all $ij \\in \\mathcal {L}^{\\rm dc}_{\\rm f} \\cup \\mathcal {L}^{\\rm dc}_{\\rm t}$ , $ o \\in \\mathcal {O}^{\\rm dc}$ , $ c \\in \\mathcal {C}^{{\\rm dc},o}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ , where $i_{ijty}^{{\\rm cv}_{i},o}$ is the magnitude of the AC-side current, $\\alpha ^{{\\rm dc},o}$ captures the no-load losses of transformers and averaged auxiliary equipment losses, $\\beta ^{{\\rm dc},o}$ captures the switching losses of valves and freewheeling diodes as well as some conduction losses due to the series voltage drop, and $\\gamma ^{{\\rm dc},o}$ captures the conduction losses of transformers, switches, and inductors in the VSC.", "Typical values for the loss parameters are $\\alpha ^{{\\rm dc},o} = {6.62}{}$ , $\\beta ^{{\\rm dc},o} = {1800}{}$ , and $\\gamma ^{{\\rm dc},o} ={1.98}{}$ .", "The converter current $i_{ijty}^{{\\rm cv}_{i},o}$ is in turn bounded by $& 0 \\le i_{ijty}^{{\\rm cv}_{i},o} \\le \\overline{i}^{{\\rm cv}_{i},o}z_{ij}^{{\\rm dc},oc},& $ for all $ij \\in \\mathcal {L}^{\\rm dc}_{\\rm f} \\cup \\mathcal {L}^{\\rm dc}_{\\rm t}$ , $o \\in \\mathcal {O}^{\\rm dc}$ , $c \\in \\mathcal {C}^{{\\rm dc},o}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ .", "The AC and DC sides of the converter are related through $& p_{ijty}^{{\\rm cv}_{i},oc} + p_{jity}^{{\\rm cv}_{i},oc} = p_{ijty}^{{\\rm loss}_{i},oc}, &$ for all $ij \\in \\mathcal {L}^{\\rm dc}_{\\rm f} \\cup \\mathcal {L}^{\\rm dc}_{\\rm t}$ , $o \\in \\mathcal {O}^{\\rm dc}$ , $c \\in \\mathcal {C}^{{\\rm dc},o}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ , and the DC-side power of the converter is linked to the power flowing through the HVDC transmission line, $p_{ijty}^{{\\rm dc},oc}$ , through $& p_{jity}^{{\\rm cv}_{i},oc} + p_{ijty}^{{\\rm dc},oc} = 0, &$ for all $ij \\in \\mathcal {L}^{\\rm dc}_{\\rm f} \\cup \\mathcal {L}^{\\rm dc}_{\\rm t}$ , $o \\in \\mathcal {O}^{\\rm dc}$ , $c \\in \\mathcal {C}^{{\\rm dc},o}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ .", "Additionally, the AC-side current and voltage are related to the active and reactive power injections of the converter through $& \\left( p_{ijty}^{{\\rm cv}_{i},oc} \\right)^2 + \\left( q_{ijty}^{{\\rm cv}_{i},oc} \\right)^2 = \\left( v_{ity} \\vphantom{i_{ijty}^{{\\rm cv}_{i},o}} \\right)^2 \\left( i_{ijty}^{{\\rm cv}_{i},o} \\right)^2, &$ for all $ij \\in \\mathcal {L}^{\\rm dc}_{\\rm f} \\cup \\mathcal {L}^{\\rm dc}_{\\rm t}$ , $o \\in \\mathcal {O}^{\\rm dc}$ , $c \\in \\mathcal {C}^{{\\rm dc},o}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ .", "Finally, the power flowing through the HVDC transmission line, $p_{ijty}^{{\\rm dc},oc}$ , is defined by $& p_{ijty}^{{\\rm dc},oc} = \\frac{\\left( v_{i}^{{\\rm dc},oc} \\right)^2 - v_{i}^{{\\rm dc},oc}v_{j}^{{\\rm dc},o}}{r_{ij}^{oc}}, &$ for all $ij \\in \\mathcal {L}^{\\rm dc}_{\\rm f} \\cup \\mathcal {L}^{\\rm dc}_{\\rm t}$ , $o \\in \\mathcal {O}^{\\rm dc}$ , $c \\in \\mathcal {C}^{{\\rm dc},o}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ , where $r_{ij}^{o}$ is the equivalent resistance of the HVDC bipole transmission line and the DC voltage $v_{i}^{{\\rm dc},oc}$ is bounded by $& \\underline{v}_{i}^{{\\rm dc},oc}z_{ij}^{{\\rm dc},oc} \\le v_{i}^{{\\rm dc},oc} \\le \\overline{v}_{i}^{{\\rm dc},oc}z_{ij}^{{\\rm dc},oc}, & $ for all $ij \\in \\mathcal {L}^{\\rm dc}_{\\rm f} \\cup \\mathcal {L}^{\\rm dc}_{\\rm t}$ , $o \\in \\mathcal {O}^{\\rm dc}$ , $c \\in \\mathcal {C}^{{\\rm dc},o}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ .", "The investment cost of the HVDC transmission link is given by $& I^{\\rm HVDC} = \\sum _{ij \\in \\mathcal {L}^{\\rm dc}_{\\rm f}} \\sum _{o \\in \\mathcal {O}^{\\rm dc}} \\sum _{c \\in \\mathcal {C}^{{\\rm dc},o}} \\left( c_{ij}^{{\\rm dc},o} z_{ij}^{{\\rm dc},oc} \\right), &$ where $c_{ij}^{{\\rm dc},o}$ ($) is the investment cost of HVDC link option $o$ over corridor $ij$ .", "The investment cost $c_{ij}^{{\\rm dc},o}$ includes the cost of HVDC transmission lines as well as the cost of the two converter stations at the sending-end and receiving-end of the line." ], [ "HVAC link model", "An HVAC link over transmission corridor $ij$ consists of an HVAC transmission line connecting two ideal transformers, one at the sending end and one at the receiving end of the link as shown in Figure REF .", "Each HVAC transmission corridor $ij \\in \\mathcal {L}^{\\rm ac}_{\\rm f}$ between buses $i$ and $j$ is associated with a predetermined set of candidate HVAC line options $\\mathcal {O}^{\\rm ac}$ and a maximum number of parallel links $c \\in \\mathcal {C}^{{\\rm ac},o} = \\left\\lbrace 1, \\ldots , \\overline{c}_{ij}^{{\\rm ac},o} \\right\\rbrace $ for each option $o \\in \\mathcal {O}^{\\rm ac}$ .", "Figure: Model of an HVAC link consisting of two transformer substations and a transmission line.Examples of an HVAC transmission link option include 345 at 0.75 rated capacity, 500 at 1.5 rated capacity, and 765 at 1.5 rated capacity in both single and double circuit arrangements.", "The decision of choosing a certain option and number of parallel links (single or double circuits) can be captured by a binary variable $z_{ij}^{{\\rm ac},oc}$ which takes a value of 1 if option $o$ and $c$ th link are installed and 0 otherwise.", "It therefore follows that $& z_{ij}^{{\\rm ac},oc} \\in \\left\\lbrace 0,1 \\right\\rbrace , & \\hspace{-28.45274pt} ij \\in \\mathcal {L}^{\\rm ac}_{\\rm f}, \\ o \\in \\mathcal {O}^{\\rm ac}, \\ c \\in \\mathcal {C}^{{\\rm ac},o} \\\\& \\sum _{c \\in \\mathcal {C}^{{\\rm ac},o}} z_{ij}^{{\\rm ac},oc} \\le \\overline{c}_{ij}^{{\\rm ac},o}, & ij \\in \\mathcal {L}^{\\rm ac}_{\\rm f}, \\ o \\in \\mathcal {O}^{\\rm ac} \\\\& z_{ij}^{{\\rm ac},oc} \\le z_{ij}^{{\\rm ac},o(c-1)}, & ij \\in \\mathcal {L}^{\\rm ac}_{\\rm f}, \\ c \\in \\mathcal {C}^{{\\rm ac},o} \\setminus \\lbrace 1\\rbrace $ where constraints () and () enforce the sequential installation of links in each option and transmission corridor.", "The complex voltage $\\tilde{v}_{i}$ (pu) at bus $i$ can be expressed as $\\tilde{v}_{i}=v_{i} \\mathrm {e}^{\\mathrm {i} \\theta _{i}}=v_{i} \\angle \\theta _{i} = v_{i}\\cos \\left(\\theta _{i}\\right) + \\mathrm {i} v_{i}\\sin \\left(\\theta _{i}\\right)$ in polar form, where $\\mathrm {i} = \\sqrt{-1}$ .", "HVAC transmission lines and phase-shifting transformers are represented by their $\\pi $ -model equivalents, in which the admittance is defined as $\\tilde{y}_{ij}=1/(r_{ij} + \\mathrm {i}x_{ij}) = g_{ij} + \\mathrm {i} b_{ij}$ , where $g_{ij}$ and $b_{ij}$ are the conductance (pu) and susceptance (pu), respectively.", "Additionally, the charging susceptance in the $\\pi $ -model of branch $ij$ is denoted by $b^{\\rm ch}_{ij}$ (pu).", "By defining $w_{i} & =\\left|\\tilde{v}_{i}\\right|^2 = v^{2}_{i}, & i \\in \\mathcal {B} \\\\w_{ij}^{\\rm r} & =\\Re \\left\\lbrace \\tilde{v}_{i} \\tilde{v}_{j}^*\\right\\rbrace =v_{i}v_{j}\\cos \\left(\\theta _{i}-\\theta _{j}\\right), & ij \\in \\mathcal {L} \\\\w_{ij}^{\\rm i} & =\\Im \\left\\lbrace \\tilde{v}_{i} \\tilde{v}_{j}^*\\right\\rbrace =v_{i}v_{j}\\sin \\left(\\theta _{i}-\\theta _{j}\\right), & ij \\in \\mathcal {L}$ the active and reactive power flows over link $c$ of option $o$ in corridor $ij$ can be written as $& p_{ijty}^{{\\rm ac},oc} = g^{{\\rm c},o}_{ij} w_{ity}^{{\\rm ac},oc} - g_{ij}^{o} w_{ijty}^{{\\rm r},oc} + b_{ij}^{o} w_{ijty}^{{\\rm r},oc}, & \\\\& q_{ijty}^{{\\rm ac},oc} = b^{{\\rm c},o}_{ij} w_{ity}^{{\\rm ac},oc} - b_{ij}^{o} w_{ijty}^{{\\rm r},oc} - g_{ij}^{o} w_{ijty}^{{\\rm r},oc}, & $ for all $ij \\in \\mathcal {L}^{\\rm ac}_{\\rm f} \\cup \\mathcal {L}^{\\rm ac}_{\\rm t}$ , $oc \\in \\mathcal {O}^{\\rm ac} \\times \\mathcal {C}^{{\\rm ac},o}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ , where $g^{{\\rm c},o}_{ij}=\\Re \\lbrace (\\tilde{y}_{ij}^{o})^*-0.5\\mathrm {i}b^{{\\rm ch},o}_{ij} \\rbrace = g_{ij}^{o}$ , $b^{{\\rm c},o}_{ij}=\\Im \\lbrace (\\tilde{y}_{ij}^{o})^*-0.5\\mathrm {i}b^{{\\rm ch},o}_{ij} \\rbrace = -b_{ij}^{o} - 0.5b^{{\\rm ch},o}_{ij}$ , and $\\mathcal {L}^{\\rm ac}_{\\rm t}$ is the set of HVAC corridors such that $j$ is the sending-end bus and $i$ is the receiving-end bus.Note that $w_{ijty}^{{\\rm r},oc} = -w_{jity}^{{\\rm r},oc}$ .", "In (REF ), $w_{ity}^{{\\rm ac},oc}$ , $w_{ijty}^{{\\rm r},oc}$ , and $w_{ijty}^{{\\rm i},oc}$ are set to zero if link $c$ of option $o$ in corridor $ij$ is not installed through $& \\underline{v}_{i}^{2}z_{ij}^{{\\rm ac},oc} \\le w_{ity}^{{\\rm ac},oc} \\le \\overline{v}_{i}^{2} z_{ij}^{{\\rm ac},oc}, & \\\\& \\underline{v}_{j}^{2}z_{ij}^{{\\rm ac},oc} \\le w_{jty}^{{\\rm ac},oc} \\le \\overline{v}_{j}^{2} z_{ij}^{{\\rm ac},oc}, &$ for all $ij \\in \\mathcal {L}^{\\rm ac}_{\\rm f}$ , $oc \\in \\mathcal {O}^{\\rm ac} \\times \\mathcal {C}^{{\\rm ac},o}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ , and $& \\underline{v}_{i}\\underline{v}_{j}\\cos \\left( \\overline{\\theta }_{ij} \\right)z_{ij}^{{\\rm ac},oc} \\le w_{ijty}^{{\\rm r},oc} \\le \\overline{v}_{i}\\overline{v}_{j}z_{ij}^{{\\rm ac},oc}, & \\\\& \\overline{v}_{i}\\overline{v}_{j}\\sin \\left( \\underline{\\theta }_{ij} \\right)z_{ij}^{{\\rm ac},oc} \\le w_{ijty}^{{\\rm i},oc} \\le \\overline{v}_{i}\\overline{v}_{j}\\sin \\left( \\overline{\\theta }_{ij} \\right)z_{ij}^{{\\rm ac},oc}, & $ for all $ij \\in \\mathcal {L}^{\\rm ac}_{\\rm f}$ , $oc \\in \\mathcal {O}^{\\rm ac} \\times \\mathcal {C}^{{\\rm ac},o}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ .", "Conversely, if link $c$ of option $o$ in corridor $ij$ is installed, $w_{ity}^{{\\rm ac},oc}$ is set equal to $w_{ity}$ (which is another design variable) through $& \\underline{v}_{i}^{2} \\left( 1 - z_{ij}^{{\\rm ac},oc} \\right) \\le w_{ity} - w_{ity}^{{\\rm ac},oc} \\le \\overline{v}_{i}^{2} \\left( 1 - z_{ij}^{{\\rm ac},oc} \\right), & \\\\& \\underline{v}_{j}^{2} \\left( 1 - z_{ij}^{{\\rm ac},oc} \\right) \\le w_{jty} - w_{jty}^{{\\rm ac},oc} \\le \\overline{v}_{j}^{2} \\left( 1 - z_{ij}^{{\\rm ac},oc} \\right),$ for all $ij \\in \\mathcal {L}^{\\rm ac}_{\\rm f}$ , $oc \\in \\mathcal {O}^{\\rm ac} \\times \\mathcal {C}^{{\\rm ac},o}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ , and $& \\underline{v}_{i}^{2} \\le w_{ity} \\le \\overline{v}_{i}^{2}.", "& i \\in \\mathcal {B}, \\ ty \\in \\mathcal {T} \\times \\mathcal {Y} $ Constraints eqwijzij,eqwrwi,eqwijwij ensure that $p_{ijty}^{{\\rm ac},oc}$ and $q_{ijty}^{{\\rm ac},oc}$ in (REF ) are set to zero when the $c$ th link of option $o$ in corridor $ij$ is not installed, by setting the corresponding $z_{ij}^{{\\rm ac},oc} = 0$ and therefore $w_{ity}^{{\\rm ac},oc}$ , $w_{ijty}^{{\\rm r},oc}$ , and $w_{ijty}^{{\\rm i},oc}$ all to zero.", "Additionally, links installed in parallel in option $o$ along corridor $ij$ should have their $w_{ijty}^{{\\rm r},oc}$ and $w_{ijty}^{{\\rm i},oc}$ equal to $w_{ijty}^{{\\rm r},o1}$ and $w_{ijty}^{{\\rm i},o1}$ , respectively, as follows $& 0 \\le w_{ijty}^{{\\rm r},o1} - w_{ijty}^{{\\rm r},oc} \\le \\overline{v}_{i}\\overline{v}_{j} \\left( 1 - z_{ij}^{{\\rm ac},oc} \\right), & \\\\& \\overline{v}_{i}\\overline{v}_{j}\\sin \\left( \\underline{\\theta }_{ij} \\right) \\left( 1 - z_{ij}^{{\\rm ac},oc} \\right) \\le w_{ijty}^{{\\rm i},o1} - w_{ijty}^{{\\rm i},oc}, & \\\\& w_{ijty}^{{\\rm i},o1} - w_{ijty}^{{\\rm i},oc} \\le \\overline{v}_{i}\\overline{v}_{j}\\sin \\left( \\overline{\\theta }_{ij} \\right) \\left( 1 - z_{ij}^{{\\rm ac},oc} \\right), & $ for all $ij \\in \\mathcal {L}^{\\rm ac}_{\\rm f}$ , $o \\in \\mathcal {O}^{\\rm ac}$ , $c \\in \\mathcal {C}^{{\\rm ac},o} \\setminus \\lbrace 1\\rbrace $ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ .", "Since all installed parallel links are constrained by eqwijwij and eqwrc12 to all have the same values for $w_{ity}^{{\\rm ac},oc}$ , $w_{ijty}^{{\\rm r},oc}$ , and $w_{ijty}^{{\\rm i},oc}$ as $w_{ity}^{o1}$ , $w_{ijty}^{{\\rm r},o1}$ and $w_{ijty}^{{\\rm i},o1}$ , respectively, it suffices to enforce the (nonconvex) rotated second-order cone constraints $& w_{ity}^{o1}w_{jty}^{o1} = \\left( w_{ijty}^{{\\rm r},o1} \\vphantom{w_{ijty}^{{\\rm i},o1}} \\right)^2 + \\left( w_{ijty}^{{\\rm i},o1} \\right)^2 , & $ for all $ij \\in \\mathcal {L}^{\\rm ac}_{\\rm f}$ , $o \\in \\mathcal {O}^{\\rm ac}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ , the angle difference constraints $& \\tan \\left( \\underline{\\theta }_{ij} \\right) w_{ijty}^{{\\rm r},o1} \\le w_{ijty}^{{\\rm i},o1} \\le \\tan \\left( \\overline{\\theta }_{ij} \\right) w_{ijty}^{{\\rm r},o1}, & $ for all $ij \\in \\mathcal {L}^{\\rm ac}_{\\rm f}$ , $o \\in \\mathcal {O}^{\\rm ac}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ , and the apparent power limit constraints $& \\sqrt{\\left( p_{ijty}^{{\\rm ac},o1}\\right)^2 + \\left( q_{ijty}^{{\\rm ac},o1} \\right)^2 } \\le \\overline{S}^{{{\\rm ac},o1}}z_{ij}^{{\\rm ac},o1} & $ for all $ij \\in \\mathcal {L}^{\\rm ac}_{\\rm f} \\cup \\mathcal {L}^{\\rm ac}_{\\rm t}$ , $o \\in \\mathcal {O}^{\\rm ac}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ , on the first circuit only.", "The angle difference constraints in (REF ) are necessary to ensure angular displacement is below 45 to maintain transient stability of the HVAC power system.", "As identified in , enforcing constraints eqrsoc1,eqanglediff1,eqMVA1 on all links can further improve the lower bound at the root node of a MIQCP branch-and-bound algorithm and the effect on computing time can be favorable only if the subsequent reduction in the number of solved continuous problems at each node of the branch-and-bound tree is not offset by the increase in their size.", "Note that since there are no cycles (loops) in this expansion planning setting the (nonconvex) cycle constraints would be redundant and can therefore be ignored without affecting the feasibility of the solution (refer to for more detail).", "The nodal active power balance constraints can now be written as $& p^{\\rm res}_{ity} = \\sum _{j \\in \\mathcal {B}_{i}} \\left( \\sum _{oc \\in \\mathcal {O}^{\\rm ac} \\times \\mathcal {C}^{{\\rm ac},o}} p_{ijty}^{{\\rm ac},oc} \\right.", "+ \\nonumber \\\\& \\left.", "\\sum _{oc \\in \\mathcal {O}^{\\rm dc} \\times \\mathcal {C}^{{\\rm dc},o}} p_{ijty}^{{\\rm cv}_{i},oc} \\right) + p_{ity}^{\\rm ptg}, \\ i \\in \\mathcal {B}, \\ ty \\in \\mathcal {T} \\times \\mathcal {Y} $ where $\\mathcal {B}_{i}$ is the set of HVAC buses adjacent to bus $i$ , and $p^{\\rm res}_{ity}$ is the power injection from wind and solar assets connected to bus $i$ during time slot $t$ .", "This RES power injection is in turn bounded by $0 \\le p^{\\rm res}_{ity} \\le \\overline{p}^{\\rm res}_{ity}, \\ i \\in \\mathcal {B}, \\qquad ty \\in \\mathcal {T} \\times \\mathcal {Y}$ where $\\overline{p}^{\\rm res}_{ity}$ is the total generated VRE (wind and solar) at bus $i$ during time slot $t$ .", "The HVAC planning options typically include reactive power compensation at some HVAC buses, which can improve line loadability by maintaining voltages near rated values and angular displacement below 45.", "A shunt reactor, or more generally a static VAr compensator (SVC), at bus $i$ can be modelled as $& \\underline{q}_{i}^{\\rm var} z_{i}^{\\rm var} \\le q_{ity}^{\\rm var} \\le \\overline{q}_{i}^{\\rm var} z_{i}^{\\rm var}, \\ i \\in \\mathcal {B}, & \\ ty \\in \\mathcal {T} \\times \\mathcal {Y} $ where $q_{ity}^{\\rm var}$ denotes the reactive power output of a VAr plant at bus $i$ at time $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ and $z_{i}^{\\rm var}$ is a binary variable that takes a value of 1 when the shunt reactor is installed and 0 otherwise.", "The nodal reactive power balance constraints can now be written as $& - q_{ity}^{\\rm var}= \\sum _{j \\in \\mathcal {B}_{i}} \\left( \\sum _{oc \\in \\mathcal {O}^{\\rm ac} \\times \\mathcal {C}^{{\\rm ac},o}} q_{ijty}^{{\\rm ac},oc} \\right.", "+ \\nonumber \\\\&\\left.", "\\sum _{oc \\in \\mathcal {O}^{\\rm dc} \\times \\mathcal {C}^{{\\rm dc},o}} q_{ijty}^{{\\rm cv}_{i},oc} \\right).", "\\ i \\in \\mathcal {B}, \\ ty \\in \\mathcal {T} \\times \\mathcal {Y} $ Finally, the investment cost of the combined HVAC transmission links and VAr plants is given by $I^{\\rm HVAC} = \\sum _{ij \\in \\mathcal {L}^{\\rm ac}_{\\rm f}} \\sum _{o \\in \\mathcal {O}^{\\rm ac}} \\sum _{c \\in \\mathcal {C}^{{\\rm ac},o}} \\left( c_{ij}^{{\\rm ac},o} z_{ij}^{{\\rm ac},oc} \\right) + \\\\\\sum _{i \\in \\mathcal {B}} \\left( c_{i,0}^{\\rm var} z_{i}^{\\rm var} + c_{i,1}^{\\rm var} \\left\\Vert \\left( q_{ity}^{\\rm var} \\right)_{ty \\in \\mathcal {T} \\times \\mathcal {Y}} \\right\\Vert _{\\infty } \\right),$ where $c_{ij}^{{\\rm ac},o}$ ($) is the investment cost of HVAC link option $o$ over corridor $ij$ , $c_{i,0}^{\\rm var}$ ($) is the installation cost the VAr device (SVC) at bus $i$ , and $c_{i,1}^{\\rm var}$ ($) is the unit cost of reactive power output from the VAr device at bus $i$ .", "The investment cost $c_{ij}^{{\\rm ac},o}$ includes the cost of HVAC transmission lines as well as the cost of step-up and step-down transformer substations tx$_{i}$ and tx$_{j}$ at the sending-end and receiving-end of the line." ], [ "Optimal integrated infrastructure planning", "Mathematically, the objective of the integrated infrastructure planning problem is to simultaneously minimise the total investment cost and maximise the H$_2$ sale over the whole planning horizon $ \\mathcal {Y} = \\lbrace 1, \\ldots , Y \\rbrace $ as $& \\underset{\\begin{array}{c}x\\end{array}}{\\mbox{minimise}} \\quad I^{\\rm ptg} + I^{\\rm pipe} + I^{\\rm HVDC} + I^{\\rm HVAC} - \\nonumber \\\\& \\qquad \\qquad \\quad \\sum _{y \\in \\mathcal {Y}} \\sum _{t \\in \\mathcal {T}} \\sum _{m \\in \\mathcal {J}} \\frac{c_{m}^{\\rm H_2} \\phi _{mty}^{\\rm H_2,d} \\Delta \\tau }{(1+\\iota )^{y}} \\\\& \\text{subject to {eq_electrolyserbinary,eq_powertogas,eq_powerlimits,eq_ptgcompressorpower,eq_pipelinebinary,eq_motion,eq_pressurelimitsofpipem,eq_pressurelimitsofpipen,eq_pressurecouplingofpipem,eq_pressurecouplingofpipen,eq_pressurelimits,eq_flowinout,eq_flowlimits,eq_averagepressure,eq_linepack,eq_linepack0,eq_gasbalance,eq_dcbinary,eq_dcsequential1,eq_dcsequential2,eq_hvdcconverter,eq_converterlosses,eq_convertercurrentlimits,eq_converterACDC,eq_converterDCDC,eq_converterpowerdefinition,eq_DClinepower,eq_DClinevoltagelimits,eq_acbinary,eq_acsequential1,eq_acsequential2,eq_pijqij,eq_wijzij,eq_wrwi,eq_wijwij,eq_voltagelimits,eq_wrc12,eq_rsoc1,eq_anglediff1,eq_MVA1,eq_RESbounds,eq_pbalance,eq_qvar,eq_qbalance}},$ where $c_{m}^{\\rm H_2}$ ($) is the selling price (profit) of H$_2$ , $\\iota $ is the discount rate, and $x$ is a vector that concatenates all the variables of the problem.The original selling price is typically in $ but is then converted to $ for consistency of dimensions." ], [ "Mixed-integer convex relaxation", "Due to the nonconvex nonlinear constraints in (REF ), (REF ), (REF ), (REF ), (REF ) and (REF ), Problem REF belongs to the class of mixed-integer nonlinear programming (MINLP) problems that have a nonconvex continuous relaxation, thus making it extremely difficult to solve to global, or even local, optimality.", "To make matters worse, Problem REF requires disjunctive constraints (to encode “or” statements) associated different design variables that correspond to the optimal choice of transport option.", "These disjunctive constraints require a Big-M reformulation to transform them into MILP constraints such as the ones in eqpressurelimitsofpipem,eqpressurelimitsofpipen,eqpressurecouplingofpipem,eqpressurecouplingofpipen, eqwijzij, eqwijwij, and eqwrc12.", "Unfortunately, Big-M reformulations are notorious for having weak root node relaxations in general.", "As a result, this class of MINLPs in particular is intractable even for small scale problems.", "Luckily, there exists a tractable alternative to Problem REF in the form of a strong mixed-integer quadratically constrained programming (MIQCP) problem.", "It is straightforward to check that the nonconvex second-order cone (SOC) constraint in (REF ) becomes convex when it is relaxed into an inequality constraint of the form $& \\left( \\phi _{mnty}^{o} \\right)^2 \\le \\Phi _{mn}^{o} \\left((\\wp _{mty}^{o})^2 - (\\wp _{nty}^{o})^2 \\right), & mn \\in \\mathcal {P} $ for all $ o \\in \\mathcal {O}^{\\rm p}$ , $ ty \\in \\mathcal {T} \\times \\mathcal {Y}$ .", "Moreover, because the nodal pressures are nonnegative, constraint eqaveragepressure becomes convex when it is relaxed into an inequality constraint of the form $\\hspace{-7.11317pt} \\wp _{mnty}^{o} \\ge \\frac{2}{3} \\left(\\wp _{mty}^{o} + \\wp _{nty}^{o} - \\frac{\\wp _{mty}^{o}\\wp _{nty}^{o}}{\\wp _{mty}^{o} + \\wp _{nty}^{o}} \\right), \\ mn \\in \\mathcal {P}$ for all $ o \\in \\mathcal {O}^{\\rm p}$ , $ ty \\in \\mathcal {T} \\times \\mathcal {Y}$ .The proof can be found in .", "However, although convex, constraint (REF ) cannot be directly handled by state-of-the-art MIQCP solvers such as Gurobi .", "For this reason, constraint (REF ) can instead be replaced by a tight polyhedral envelope $\\hspace{-7.11317pt} \\wp _{mnty}^{o} \\ge \\frac{2}{3} \\left(\\wp _{mty}^{o} + \\wp _{nty}^{o} + {\\rm conv} \\left( - \\frac{\\wp _{mty}^{o}\\wp _{nty}^{o}}{\\wp _{mty}^{o} + \\wp _{nty}^{o}} \\right) \\right)$ for all $mn \\in \\mathcal {P}$ , $ o \\in \\mathcal {O}^{\\rm p}$ , $ ty \\in \\mathcal {T} \\times \\mathcal {Y}$ , as detailed in .", "The first step towards conferring a strong convex relaxation property to constraints (REF ), (REF ), and (REF ) is to define new variables to substitute the square of the voltage and the square of the current terms, i.e., $w := v^2$ and $l:=i^2$ , respectively.", "Constraints (REF ), (REF ), and (REF ) can now be equivalently rewritten as $& p_{ijty}^{{\\rm loss}_{i},oc} = \\alpha ^{{\\rm dc},o} z_{ij}^{{\\rm dc},oc} + \\beta ^{{\\rm dc},o} i_{ijty}^{{\\rm cv}_{i},o} + \\gamma ^{{\\rm dc},o} l_{ijty}^{{\\rm cv}_{i},o}, \\\\& l_{ijty}^{{\\rm cv}_{i},o} = \\left( i_{ijty}^{{\\rm cv}_{i},o} \\right)^2 , \\\\& \\left( p_{ijty}^{{\\rm cv}_{i},oc} \\right)^2 + \\left( q_{ijty}^{{\\rm cv}_{i},oc} \\right)^2 = w_{ity} l_{ijty}^{{\\rm cv}_{i},o} , & \\\\& p_{ijty}^{{\\rm dc},oc} = \\frac{ w_{i}^{{\\rm dc},oc} - w_{ij}^{{\\rm dc},o}}{r_{ij}^{oc}}, &$ for all $ij \\in \\mathcal {L}^{\\rm dc}_{\\rm f} \\cup \\mathcal {L}^{\\rm dc}_{\\rm t}$ , $o \\in \\mathcal {O}^{\\rm dc}$ , $c \\in \\mathcal {C}^{{\\rm dc},o}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ , and $& w_{i}^{{\\rm dc},oc} w_{j}^{{\\rm dc},oc} = \\left( w_{ij}^{{\\rm dc},o} \\right)^2, &$ for all $ij \\in \\mathcal {L}^{\\rm dc}_{\\rm f}$ , $o \\in \\mathcal {O}^{\\rm dc}$ , $c \\in \\mathcal {C}^{{\\rm dc},o}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ .", "It is now straightforward to verify that the nonconvex quadratic constraint in () and the nonconvex rotated SOC constraints in (), and (REF ) become convex when they are relaxed into inequality constraints of the form $& l_{ijty}^{{\\rm cv}_{i},o} \\ge \\left( i_{ijty}^{{\\rm cv}_{i},o} \\right)^2 , \\\\& \\left( p_{ijty}^{{\\rm cv}_{i},oc} \\right)^2 + \\left( q_{ijty}^{{\\rm cv}_{i},oc} \\right)^2 \\le w_{ity} l_{ijty}^{{\\rm cv}_{i},o} , & $ for all $ij \\in \\mathcal {L}^{\\rm dc}_{\\rm f} \\cup \\mathcal {L}^{\\rm dc}_{\\rm t}$ , $o \\in \\mathcal {O}^{\\rm dc}$ , $c \\in \\mathcal {C}^{{\\rm dc},o}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ and $& w_{i}^{{\\rm dc},oc} w_{j}^{{\\rm dc},oc} \\ge \\left( w_{ij}^{{\\rm dc},o} \\right)^2, &$ for all $ij \\in \\mathcal {L}^{\\rm dc}_{\\rm f}$ , $o \\in \\mathcal {O}^{\\rm dc}$ , $c \\in \\mathcal {C}^{{\\rm dc},o}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ .", "Finally, the nonconvex rotated SOC constraint (REF ) can also be similarly relaxed into a convex constraint of the form $& w_{ity}^{o1}w_{jty}^{o1} \\ge \\left( w_{ijty}^{{\\rm r},o1} \\vphantom{w_{ijty}^{{\\rm i},o1}} \\right)^2 + \\left( w_{ijty}^{{\\rm i},o1} \\right)^2 , & $ for all $ij \\in \\mathcal {L}^{\\rm ac}_{\\rm f}$ , $o \\in \\mathcal {O}^{\\rm ac}$ , $ty \\in \\mathcal {T} \\times \\mathcal {Y}$ .", "The MIQCP relaxation of Problem REF can now be written as $& \\underset{\\begin{array}{c}x\\end{array}}{\\mbox{minimise}} \\quad I^{\\rm ptg} + I^{\\rm pipe} + I^{\\rm HVDC} + I^{\\rm HVAC} - \\nonumber \\\\& \\qquad \\qquad \\quad \\sum _{y \\in \\mathcal {Y}} \\sum _{t \\in \\mathcal {T}} \\sum _{m \\in \\mathcal {J}} \\frac{c_{m}^{\\rm H_2} \\phi _{mty}^{\\rm H_2,d} \\Delta \\tau }{(1+\\iota )^{y}} \\\\& \\text{subject to {eq_electrolyserbinary,eq_powertogas,eq_powerlimits,eq_ptgcompressorpower,eq_pipelinebinary},{eq_pressurelimitsofpipem,eq_pressurelimitsofpipen,eq_pressurecouplingofpipem,eq_pressurecouplingofpipen,eq_pressurelimits,eq_flowinout,eq_flowlimits},{eq_linepack,eq_linepack0,eq_gasbalance,eq_dcbinary,eq_dcsequential1,eq_dcsequential2,eq_hvdcconverter},{eq_convertercurrentlimits,eq_converterACDC,eq_converterDCDC}}, \\nonumber \\\\& \\text{{eq_DClinevoltagelimits,eq_acbinary,eq_acsequential1,eq_acsequential2,eq_pijqij,eq_wijzij,eq_wrwi,eq_wijwij,eq_voltagelimits,eq_wrc12},{eq_anglediff1,eq_MVA1,eq_RESbounds,eq_pbalance,eq_qvar,eq_qbalance},{eq_motion_r},{eq_averagepressure_r_poly},{eq_converterlosses_li},{eq_DClinepower_ww},{eq_isquared_r,eq_converterpowerdefinition_wi_r,eq_wiwj_wij_r,eq_rsoc1_r}}.$ Although Problem  is a relaxation of Problem REF , which entails that its solution will most likely be infeasible in the original space of Problem REF , blackthis relaxation is expected to provide a high-quality lower bound on the globally optimal solution of Problem REF .", "This is especially true when the solution of this MIQCP relaxation is compared to solutions from MILP approximations , , whose solution quality is arbitrary as their feasible region does not contain the original feasible region defined by Problem REF ." ], [ "Numerical evaluation", "The integrated modelling in this work adopts a half-hourly resolution $\\Delta t = {0.5}{}$ , which matches the granularity of the RES forecasts from the Australian Energy Market Operator (AEMO) .", "The lifespan of the project is assumed to be 20 years and the discount factor is assumed to be 6% ($\\iota = 0.06$ ).", "However, instead of also considering a planning horizon of $\\left| \\mathcal {Y} \\right| = 20$ years, which would translate to millions of variables and constraints that would make Problem  intractable, only one representative year is carefully chosen in the planning horizon, i.e., $\\left| \\mathcal {Y} \\right| = 1$ .", "In a similar attempt to reduce the size of the problem while still capturing the necessary variability in RES, only 4 representative weeks, one in each season, are chosen in this representative year.", "This translates to $ (24/\\Delta t) \\times 7 \\times 4 = 1344$ time steps, i.e., $t \\in \\mathcal {T}=\\lbrace 1,2,\\ldots , 1344 \\rbrace $ , which still results in large-scale problems as will be shown in the case studies below.", "As a result, the hydrogen sales in the subsequent 19 years are assumed to be constant cash flows.", "The corresponding term in the objective function therefore becomes $C^{\\rm H_2} = \\sum _{t \\in \\mathcal {T}} \\sum _{m \\in \\mathcal {J}} \\frac{13.04 c_{m}^{\\rm H_2} \\phi _{mt}^{\\rm H_2,d} \\Delta \\tau }{\\iota }\\left( 1 - \\frac{1}{(1+\\iota )^{20}}\\right)$ which is effectively the net present value of the hydrogen sales.", "Since 28 representative days are considered in the representative year, the selling price is multiplied by $365.25/28=13.04$ to cover a whole year's worth of H$_2$ sales.", "The hydrogen selling price is taken from as 3.23$, i.e., $c_{m}^{\\rm H_2} = 3.23 \\rho = {0.277}{\\$}$ .", "The minimum and maximum operating pressures in the H$_2$ network are assumed to be 3.5 and 10, respectively.All the costs in this paper are in US dollars.", "The cost and parameter assumptions of different transport technologies are shown in Tables REF -REF .", "Table: Cost and parameter assumptions of electrolysers , and compressors in electrolyser stations (see Figure ).Table: Cost and parameter assumptions of H 2 _2 pipelines .Table: Cost and parameter assumptions of VSC HVDC systems , , .Table: Cost and parameter assumptions of HVAC systems , , , .At the time of writing this paper, the largest commissioned 500 VSC HVDC project has a capacity of 700.", "However, 1, 2, and 3 500 VSC HVDC technology is assumed to be available in the near future.", "The costs for those are assumed to be the same as existing LCC HVDC technology of the same capacity.", "Since the base installation costs of electrolyser stations and SVC are negligible compared to total station cost, both $c_{i,0}^{\\rm ptg}$ and $c_{i,0}^{\\rm var}$ are assumed to be zero, which obviates the need for binary variables, i.e., $z_{im}^{\\rm ptg}$ and $z_{i}^{\\rm var}$ are no longer needed.", "It should be emphasised that the cost and parameter assumptions in Tables REF -REF are for the sole purpose of demonstrating the novel integrated modelling in Problem .", "Therefore, in the context of these specific costs and parameter assumptions, the findings in this paper should be considered solely for illustration and demonstration purposes rather than real guidelines for energy infrastructure planners and stakeholders, for which specific studies based on agreed input data and assumptions should be performed.", "In this implementation setup, Julia v1.7.2 is used as a programming language along with JuMP v0.22.3 as a mathematical modelling layer for all the optimisation problems.", "All simulations are conducted on a computing platform with an Intel Core i7-6820HK CPU at 2.7GHz, 64-bit operating system, and 32GB RAM.", "The MIQCP problem in () is solved using Gurobi v9.5.0 with the branch-and-bound algorithm which solves continuous QCP relaxations at each node.", "In contrast, the linearised outer-approximation approach performed poorly on this problem.", "The proposed MIQCP formulation in Problem  is demonstrated on two case studies, one consisting of a canonical two-node system, and one involving actual renewable energy zones (REZ) in Queensland, Australia.", "In both case studies the profiles of RES forecasts are obtained from AEMO's Integrated System Plan (ISP) “Central” scenario ." ], [ "Case study 1: Canonical 2-node system", "This case study is intended to thoroughly analyse the two fundamental drivers, namely distance and RES capacity, affecting the investment decision between two nodes.", "In particular, the RES capacity is varied from 1 and 10 and the distance is varied from 200 to 1000, and the results are shown in Figure REF .", "The RES capacity is shared equally between wind and solar and a single solar profile and a single wind profile are used, with capacity factors of 0.2731 and 0.4041, respectively.", "The immediate inference that can be drawn from Figure REF is that H$_2$ pipelines tend to be preferred for higher RES capacities (7500 and above) transmitted over medium to long distances (400 and above), whereas lower capacities (5000 and below) are dominated by electricity options.", "In particular, HVAC systems are the preferred option for short distances (200 and below) across all RES capacities and HVDC systems are the preferred option for medium to long distances (600 and above) and medium RES capacities (2500 and 5000).", "Despite lower losses in HVDC systems, the high cost of converter stations places them at a disadvantage compared to HVAC systems for short to medium distances and an RES capacity of 2500 to 5000.", "However, the larger cost of cables of HVAC tips the scale in favour of HVDC systems for medium to long distances.", "Additionally, angle displacement constraints on HVAC systems require SVCs to absorb the large reactive power induced by inductive and capacitive effects of long distance AC transmission, which further increases the cost of HVAC links.", "These results are congruent with HVAC vs HVDC comparisons in existing literature, which identify a break-even distance of around 600, beyond which HVDC becomes more competitive .", "At smaller RES capacities (1000), where the capacity factors of 0.2731 and 0.4041 for solar and wind, respectively, translate to an average available RES power of 338, a 345 750 HVAC with an SVC at node 1 is more cost-effective than a 1 500 VSC HVDC system for transmission distances below 800.", "On the other hand, the high operating pressure range (3.5 to 10) of H$_2$ pipelines translates to much smaller transmission losses and large linepack capacities that together give H$_2$ pipelines an edge over electricity options for higher RES capacities (7500 and above) transmitted over medium to long distances (400 and above).", "The linepack can be thought of as a large storage element that can smooth out the variability of RES.", "Each one of the MIQCP problems in Figure REF has more than 400,000 continuous variables, 28 binary variables, and more than 900,000 constraints including more than 69,000 (convex) quadratic constraints.", "It takes Gurobi between 1 and 3 hours to solve each one." ], [ "Case study 2: REZ in Queensland, Australia", "AEMO's ISP identifies potential renewable energy zones (REZ) across the national electricity market (NEM) .", "In more detail, this case study considers four REZ in Queensland and one H$_2$ demand point (off-take), as shown in Figure REF .", "Figure REF also shows the RES capacity forecast for 2040.", "The solution to this integrated planning problem, shown in Figure REF , is a hybrid system consisting of a 2 VSC HVDC link between REZ Q1 and Q4 (570), a 3 500 double circuit HVAC link between REZ Q8 and the demand point (200), and a 0.5 diameter H$_2$ pipeline between REZ Q4 and Q6 (300) and Q6 and the demand point (500).", "No VAr plants (SVCs) were needed in this case as the HVAC link is installed over the relatively short distance of 200 where both the voltage drop and angle difference (and also power losses) are small.", "These results are in congruence with the 2-node results in the previous section.", "Finally, the profiles of available (forecast) VRE, accommodated VRE ($p^{\\rm res}_{ity}$ ), and demand ($\\phi _{mty}^{\\rm H_2,d} HHV$ ) at the optimal solution of Problem  are shown in Figure REF .", "It can be seen from Figure REF that the demand profile varies over a smaller range compared to the profile of available (forecast) VRE input and this is due to the effect of the linepack in the two H$_2$ pipelines installed between REZ Q4 and Q6 (300) and Q6 and the demand point (500).", "This linepack profile is shown in Figure REF .", "Figure REF also shows that the optimal infrastructure investment planning in Figure REF has an energy transmission factor of 0.9901, which means that 99.01% of the total generated VRE is accommodated by the installed infrastructure.", "Recall that the energy transmission factor is defined as $TF = \\frac{ \\displaystyle \\underset{t \\in \\mathcal {T}}{\\sum } \\underset{y \\in \\mathcal {Y}}{\\sum } \\underset{i \\in \\mathcal {B}}{\\sum } p^{\\rm res}_{ity}}{ \\displaystyle \\underset{t \\in \\mathcal {T}}{\\sum } \\underset{y \\in \\mathcal {Y}}{\\sum } \\underset{i \\in \\mathcal {B}}{\\sum } \\overline{p}^{\\rm res}_{ity}} .$ Figure: RES capacity in 2040 according to AEMO's REZ .Figure: Optimal solution to the infrastructure planning problem in Figure .Figure: Profiles of available (forecast) VRE (p ¯ ity res \\overline{p}^{\\rm res}_{ity}), accommodated VRE (p ity res p^{\\rm res}_{ity}), and demand (φ mty H 2 ,d HHV\\phi _{mty}^{\\rm H_2,d} HHV) at the optimal solution of Problem .Figure: Linepack profile in the two H 2 _2 pipelines installed between REZ Q4 and Q6 (300) and Q6 and the demand point (500).The MIQCP problem in this case study has around than 560,000 continuous variables, 44 binary variables, and more than 1,160,000 constraints including more than 107,000 (convex) quadratic constraints.", "It takes Gurobi around 4 days to solve it." ], [ "Conclusion", "To address the challenging question of whether to transport large-scale VRE as molecules in H$_2$ pipelines or as electrons in electricity transmission lines, this paper introduced a first-of-its-kind mathematical framework for finding the optimal integrated planning of electricity and H$_2$ infrastructure.", "The model fills the gap in existing state-of-the-art literature by (i) considering all relevant infrastructure technologies such as HVDC, HVAC, SVC, and H$_2$ pipelines and compressors, and by (ii) incorporating essential nonlinearities such as voltage drops due to losses in HVAC and HVDC transmission lines, losses in HVDC converter stations, reactive power flow, pressure drops in pipelines, and linepack, all of which play an important role in determining the optimal infrastructure investment decision.", "The high temporal resolution of the RES forecasts makes this model a large-scale nonconvex MINLP problem that is intractable if solved directly using MINLP solvers.", "The paper therefore proposes a tractable alternative in the form of an MIQCP relaxation that is demonstrated on a canonical two-node system as well as on a real-world case study involving actual renewable energy zones in Australia." ], [ "Acknowledgment", "This work is supported by Future Fuels Cooperative Research Centre as part of the RP1.1-02B: “Transport and Storage Options for Future Fuels” project.", "The cash and in-kind support from the industry participants is gratefully acknowledged." ] ]
2207.03567
[ [ "ALMA-IMF IV -- A comparative study of the main hot cores in W43-MM1:\n detection, temperature and molecular composition" ], [ "Abstract W43-MM1 is a young region, very rich in terms of high-mass star formation.", "We aim to systematically identify the massive cores which contain a hot core and compare their molecular composition.", "We used ALMA high-spatial resolution (2500 au) data of W43-MM1 to identify line-rich protostellar cores and make a comparative study of their temperature and molecular composition.", "The identification of hot cores is based on both the spatial distribution of the complex organic molecules and the contribution of molecular lines relative to the continuum intensity.", "We rely on the analysis of CH3CN and CH3CCH to estimate the temperatures of the selected cores.", "Finally, we rescale the spectra of the different hot cores based on their CH3OCHO line intensities to directly compare the detections and line intensities of the other species.", "W43-MM1 turns out to be a region rich in massive hot cores with at least 1 less massive and 7 massive hot cores.", "The excitation temperature of CH3CN is of the same order for all of them (120-160 K).", "There is a factor of up to 30 difference in the intensity of the complex organic molecules (COMs) lines.", "However the molecular emission of the hot cores appears to be the same within a factor 2-3.", "This points towards both a similar chemical composition and excitation of most of the COMs over these massive cores, which span about an order of magnitude in core mass.", "In contrast, CH3CCH emission is found to preferentially trace more the envelope, with a temperature ranging from 50 K to 90 K. Core 1, the most massive hot core of W43-MM1, shows a richer line spectrum than the other cores.", "In core 2, the emission of O-bearing molecules does not peak at the dust continuum core center; the blue and red shifted emission correspond to the outflow lobes, suggesting a formation via the sublimation of the ice mantles through shocks or UV irradiation on the walls of the cavity." ], [ "Introduction", "W43-MM1 is a massive star formation region 5.5 kpc away [45] and located at the tip of the Galactic bar [36].", "Among the 131 cores with 2000 au typical sizes, identified by [32] (hereafter mentioned as motte18) using ALMA data, 18 cores have masses $>$ 10 M$_\\odot $ .", "The large number of massive cores detected in W43-MM1 make it an ideal laboratory for understanding the physical processes and chemical evolution involved in the formation of massive stars.", "The identification of massive cores and the characterisation of their environment that we obtained in W43-MM1 results from many years of sustained effort involving observations and state-of-the-art simulations and models.", "[33] found such a high star-formation rate and efficiency in W43 that they deemed this region “mini-starburst”, reminiscent of the galaxies thus called.", "Herschel and ground-based observations revealed a complex structure of molecular filaments hosting dense cores exposed to the radiation from neighbouring massive stars [2], [10], [8], [36], [35], [34], [5].", "[22], [21] studied the water emission from the W43-MM1 dense filament; they found a very turbulent and infalling medium, and inferred a high accretion luminosity.", "[26] used NOEMA observations of the dust continuum emission in W43-MM1 with a typical angular resolution of 3$$ to highlight a linear correlation between the star formation efficiencies and the density in different layers of the filament.", "In [25], they used the same dataset together with grids of models provided by the Paris-Durham shock model [19], [18] to characterise the ongoing shock processes.", "They showed that the filament was probably the result of a cloud-cloud collision, and revealed the presence of numerous bipolar outflows.", "In the meantime, [41] had confirmed the presence of dense cores with the SMA and revealed local variations of the magnetic field.", "[9] studied the magnetic field structure at $\\sim $  0.5$$ resolution.", "ALMA observations revolutionised our understanding of this region: motte18 identified and characterised 131 pre- and proto-stellar cores in the region at $\\sim 0.5$ resolution, measuring their mass, temperature, size and density, and obtaining an unexpected core mass function (CMF) with an excess of massive cores.", "[37] studied the physical structure of the remarkably massive ($\\sim $ 55 M$_\\odot $ in 1300 au radius) pre-stellar core candidate (core #6) found in the region, while its chemistry was studied by [29].", "[38] investigated the ejection/accretion link by studying the characteristics of 46 molecular outflow lobes identified with ALMA and found evidence for time variable ejection processes with a timescale of $\\sim $  500 yr.", "The large program ALMA-IMF: ALMA transforms our view of the origin of stellar masses (project # 2017.1.01355.L) [17], [31], [40] extends the work by motte18 that found the first \"top-heavy\" core mass function in the W43-MM1 protocluster.", "It consists of the observation of 15 massive protoclusters to investigate the distribution of the 0.5-200 M$_\\odot $ cores at a $\\sim $ 2000 au scale and thus characterise the core mass function evolution and to determine the pre-stellar, protostellar, or UCHII region nature of the cores.", "In this paper we focus on the identification and characterisation of the hot cores in W43-MM1.", "By analogy with the Orion hot core [30], a hot core is usually defined as a hot (T $\\ge $ 100 K), dense (density $\\ge $ 10$^{6}$ cm$^{-3}$ ) and compact (diameter <0.1 pc) region where a large number of molecular lines from complex organic molecules (COMs) are detected [6], [20], [7].", "The study of the chemistry of star-forming regions can provide us with precious information on the physical evolution of protostars [23].", "The article is organised as follows.", "We present the data in Sect. .", "In Sect.", ", we identify eight hot cores with two methods: one using the spatial distribution of molecules and the other one using the line densities compared to the continuum level in the continuum cores, and we confirm the nature of these cores using methyl formate and methyl cyanide maps.", "In Sect.", "we determine the temperature of the hot cores from the CH$_3$ CN and CH$_3$ CCH emission.", "In Sect.", "we compare the molecular composition of the hot cores from their spectra normalised to the methyl formate lines intensities.", "We discuss the molecular similarity of the hot cores using scaled spectra and correlation plots in Sect. .", "Our conclusions are presented in Sect.", "." ], [ "Observations", "We use band 3 and band 6 ALMA observations of W43-MM1, carried out between 2014 and 2018.", "The 1.3 mm observations (216 to 234 GHz) are from ALMA Cycle 2 (project #2013.1.01365.S) and Cycle 3 (#2015.1.01273.S) and were previously presented in [29], together with our continuum subtraction method.", "The 1.3 mm dataset is composed of nine bands of bandwidths between 0.1 and 1.9 GHz, with a spatial resolution of $\\sim $ 0.45$$ , a spectral resolution ranging from 0.2 to 1.3 km s$^{-1}$ and an rms between 0.1 and 0.5 K (see table:specwin3mm).", "The observations are 2.1 pc $\\times $ 1.4 pc mosaics taken with the ALMA 12 m and ACA 7 m arrays.", "The gridding was performed with Briggs' weighting using a robustness parameter of 0.5, and the cleaning used the multiscale option excluding the borders of the mosaic to avoid divergence problems.", "Hence, 120 to 129 out of the 131 cores of motte18 are in the cleaned field, depending on the band.", "The 3 mm observations (91.7 – 105.4 GHz) are from ALMA Cycle 5 and are part of the large program ALMA-IMF.", "This program covered the W43-MM1 region at 3 mm at a comparable resolution to the previous 1 mm data.", "In this paper we present a preliminary data reduction and analysis of these 3 mm data.", "We use the same cleaning and continuum subtraction method (based on the distribution of channel intensities) as presented in [29], using CASA https://casa.nrao.edu.", "Cunningham et al.", "(in prep) present the standardised data reduction methods applied by the ALMA-IMF consortium to homogenise the analysis of all the regions observed.", "On average the observational parameters for the 4 selected bands have a spatial resolution of 0.46$$ (2500 au), a spectral resolution of 1.5 km s$^{-1}$ (0.5 MHz) and an rms of 0.6 K per channel; the detailed parameters for each band are given in table:specwin3mm.", "These bands include several lines of CH$_3$ CN (spw1), CH$_3$ CCH (spw2), and CH$_3$ OH (spw3).", "We have also made 1.3 mm maps with a higher spatial resolution, applying a uniform weighting to visibilities in the gridding, in order to study the distribution of the molecules in Sect.", "REF .", "We reached a resolution 1.5 times better (0.3$$ or 1600 au) at the expense of a lower sensitivity.", "Figure: Left: spectrum with molecular line emission in the Band 6 spw7 band.", "Right: distribution of the intensity channels (in black).", "An exponentially modified Gaussian (in red) is adjusted to fit the Gaussian part due to the noise, whose peak is taken as the continuum value, and the tail associated with the molecular emission.The conversion from flux density to brightness temperature was made using the formula https://science.nrao.edu/facilities/vla/proposing/TBconv: ${T = 1.222 \\times 10^3 \\frac{I}{\\nu ^{2}\\theta _{\\rm maj}\\theta _{\\rm min}}}$ where $T$ is the brightness temperature, $I$ the flux in mJy beam$^{-1}$ , $\\nu $ the frequency in GHz and $\\theta _{\\rm maj}$ and $\\theta _{\\rm min}$ the half-power beam widths along the major and minor axes, respectively." ], [ "Identification of hot cores", "To identify hot cores, we used two different approaches, one which uses molecules known to trace hot cores (see Sect.", "REF ) and another one which does not need any line identification and which relies on the richness of the hot core spectra in COM lines.", "Two methods based on this later approach are presented in Sect.", "REF and Sect.", "REF .", "The first one is based on the sum of the brightness temperature of all lines over the Band 6 spw7 band; it is computed for each pixel of a 2500 au spatial resolution map, and requires no a priori knowledge of the region.", "The selected band is particularly rich in COM lines.", "The second one is based on the sum over a band of the line contribution, averaged over the area of a continuum core and compared to the continuum emission.", "We use the different bands from table:specwin3mm (bandwidth between 0.1 and 2 GHz).", "We study first the brightness temperature of the line emission to estimate the \"contamination\" of the continuum emission by line emission, then just the number of lines detected to get the \"density in lines\", which is the fraction of the band showing detected lines.", "This method requires as a first step the identification of the continuum cores.", "Note that, as indicated in Sect.", ", up to 11 of the 131 cores from motte18 are out of the bounds of our data-cubes.", "Specifically, this is the case of cores #15, 39, 44, 59 and 67 located in W43-MM1 SW, all of them associated with molecular outflows [38].", "Among these cores, motte18 indicated that only core #15 has detectable molecular lines.", "None of these are included in the analysis that follows." ], [ "Continuum level ", "We have separated the continuum and the molecular emission in each pixel of the image by the method presented in [29].", "The continuum level is estimated from the spectrum intensity channel distribution, after fitting it with an exponentially modified Gaussian to adjust both the Gaussian distribution of the noise and the asymmetric distribution of the lines intensities (see fig:cont)." ], [ "Indicators of richness in lines ", "To quantify the line richness, we take the line brightness temperature at a pixel or average it over a given spatial region $R$ , and sum it on individual channels over a given frequency range.", "We call $I_{\\rm Lines}^{pix}$ and $I_{\\rm Lines}^R$ , respectively, the integrated line emission.", "$I_{\\rm Lines}^{\\rm pix} = \\sum _i^{n_{\\rm chan}} T^{\\rm pix}_{{\\rm Lines}, \\,i} \\Delta \\nu \\,\\, \\mbox{and} \\,\\,I_{\\rm Lines}^R = \\sum _i^{n_{\\rm chan}} <T_{{\\rm Lines}, \\,i}>_R \\Delta \\nu $ Where $T_{{\\rm Lines}, \\,i}$ is the brightness temperature due to the lines in channel $i$ ; it is computed using the total brightness temperature $T_{{\\rm Total}, \\,i}$ and the continuum value $T_{\\rm Cont}$ determined previously: $T_{{\\rm Lines}, \\,i} = T_{{\\rm Total}, \\,i}-T_{\\rm Cont}$ $<>_R$ indicates the average of the line brightness temperature over the spatial region $R$ .", "$n_{\\rm chan}$ is the number of channels.", "$\\Delta \\nu $ is the channel width.", "Figure: Lines, total and continuum emission maps obtained from the spw7 band at 233 GHz.", "Contours represent 3, 5, 7, 10, 20, 30, 50, 70 and 100 σ\\sigma , with σ=0.22\\sigma =0{.", "}22 K, the rms in a channel of resolution Δν\\Delta \\nu = 0.122 MHz.", "The hot cores are marked by a star symbol.", "The total emission is averaged over the 1.9 GHz band.", "For the line map, the first two contours are added in red to reveal the fainter hot cores.Figure: Separation of continuum and line contributions for cores #5 and 11 as examples.", "The spectra are spatially averaged over the source size.", "The grey horizontal line across the observed spectra (in black) is the continuum level obtained by the method of .", "Below this line, the grey area represents the continuum integrated flux.", "Above this line, the blue area is the continuum brightness of the lines.", "It is estimated in each channel depending on the noise, whose 1σ\\sigma and 2σ\\sigma values are represented by horizontal lines." ], [ "Hot core identification from the spatial distribution of COMs", "To highlight the presence of hot cores, we focus on the analysis of the “continuum” band (spw7) at 233 GHz, because this band offers a large band width ($\\sim $ 2 GHz) not contaminated by strong emission lines coming from the simplest molecules (like H$_2$ CO, CO or SiO) but mainly by COMs.", "Moreover, motte18 and [29] have already studied the 233 GHz band towards W43-MM1.", "For each pixel we compute the integrated line emission $I_{\\rm Lines}^{\\rm pix}$ and divide by the number of channels ($T_{\\rm Lines}^{\\rm pix}=I_{\\rm Lines}^{\\rm pix}/n_{\\rm chan}$ ) to produce the mean line brightness temperature map in Figure REF .", "Similarly using the total channel intensity and the continuum contribution we obtain maps of $T_{\\rm Total}^{\\rm pix}$ and $T_{\\rm Cont}^{\\rm pix}$ which are also shown in the same figure.", "We have assumed that the continuum intensity does not vary significantly over the spectral band ($\\Delta \\nu =1.9$  GHz), the expected difference being only 2% in this range of frequency.", "Because the continuum emission is bright at 1.3 mm, the structures in the total integrated map are very similar to the ones found in the continuum map.", "To reveal the presence of hot cores, we look on the map for places where line emission $T_{\\rm Lines}^{\\rm pix}$ is greater than the noise (first 3 sigma contour in fig:continuum).", "With this method, seven structures are highlighted, among which the largest one previously identified as N1a, detected at 5$$ x 3$$ resolution with NOEMA [26], and separated in two substructures at 2400 au resolution with ALMA [32], [38].", "The center of these eight structures is in agreement with the position of eight of the brightest continuum cores identified by motte18 and marked on the continuum map: cores #1, 2, 3, 4, 5, 9, 10 and 11, with structures 1 and 4 contributing to one (possibly double) hot core candidate in the line map.", "All of them are high-mass cores [16$-$ 102 M$_\\odot $ , see][]motte18 located on the main filament, except core #11 (2 M$_\\odot $ ).", "Figure: Relative line contribution to the total flux (left) and fraction of channels that contains molecular emission (right) as a function of the continuum level obtained for the 1.3 mm bands.", "Red dots represent high mass cores with M >> 10 M ⊙ _\\odot , gray dots represent the other cores.", "The blue areas represent the results for spectra without many lines above the 1σ1\\sigma , 2σ2\\sigma and 3σ3\\sigma levels, where σ\\sigma is the rms noise level in one channel of the corresponding band.", "Cores identified as containing a hot core following the criterion described in Sect.", "are marked by their core number.The spectra of the other cores identified by motte18 do not show any lines in this band or the lines are not bright enough for the comparative line methods discussed in Sects.", "and .", "The detected lines are individually not significant enough for a clear constraint of line parameters (shape, center and width) and thus the uncertainties on physical parameters such as temperature and column densities are too high.", "Nonetheless, a detailed previous study showed that we can analyse the same lines in the 1.3 mm line faintest sources of W43-MM1, such as in core #6, which is a high-mass prestellar or very young protostellar core.", "The complete analysis of the spectra from cores #3 and #6 can be found in [29].", "A direct analysis for cores #1 and 4 is difficult because they appear to be in interaction and affected by significant spectral confusion." ], [ "Hot core identification from the line densities in continuum cores", "As motte18 have already identified the continuum cores in the W43-MM1 region and defined their sizes and locations, we propose another method to highlight the ones that contain a hot core.", "Unlike the pixel-based analysis in Sect.", "REF , we look for the relative contributions of continuum and molecular line emission in the spectra of the continuum cores, spatially integrated over the source size given by motte18.", "For each spectrum, we distinguish channels with molecular emission from line free (or continuum) channels if the core total intensity $T_{{\\rm Total}, \\,i}^{\\rm core}$ of the channel $i$ , once the continuum is subtracted, is above or below $1 \\sigma $ (the rms measured in a line free part of the spectra) (see fig:spectres).", "After tests on synthetic spectra, we find that the contribution of the lines is better taken into account if we consider all the line signal for a channel with an intensity greater than $2 \\sigma $ , and only signal above $1 \\sigma $ for a channel with an intensity between $1 \\sigma $ and $2 \\sigma $ .", "Thus, for a core spectrum, we define the line contribution $C_{\\rm Lines}^{\\rm core}$ to the total brightness as: $C_{\\rm Lines}^{\\rm core}=I_{\\rm Lines}^{\\rm core}/I_{\\rm Total}^{\\rm core}$ And using eq:Ilinesmap: $C_{\\rm Lines}^{\\rm core} = \\sum ^{n_{\\rm chan}} _{i} \\frac{ 1}{ T_{{\\rm Total}, \\,i}^{\\rm core} } \\times {\\left\\lbrace \\begin{array}{ll}0 & \\mbox{if } T_{{\\rm Lines}, \\,i}^{\\rm core} \\le 1 \\sigma \\\\(T_{{\\rm Lines}, \\,i}^{\\rm core}-\\sigma ) & \\mbox{if } 1 \\sigma < T_{{\\rm Lines}, \\,i}^{\\rm core} \\le 2 \\sigma \\\\T_{{\\rm Lines}, \\,i}^{\\rm core} & \\mbox{if } T_{{\\rm Lines}, \\,i}^{\\rm core} > 2 \\sigma \\end{array}\\right.", "}$ The left panel of fig:contamination1 presents, for the nine 1.3 mm ALMA bands, the relative contribution $C_{\\rm Lines}^{\\rm core}$ of lines to the total flux versus $T_{\\rm Cont}^{\\rm core}$ for all cores.", "Note that the rapid increase of the relative line contribution at low continuum values corresponds to the cores fainter in lines, for which the method is biased by the noise (blue areas).", "However, as in Sect.", "REF , cores #1, 2, 3, 4, 5, 9, 10 and 11 clearly stand out.", "The results are consistent between the bands but are more obvious for the bands with no strong line (e.g., at 216 GHz), as the strong lines come from molecules which are widespread; the results are also better in bands with a large frequency range (e.g., the 3 bands from 231 to 234 GHz).", "To avoid the confusion observed for the weak continuum values, another approach of this method is to focus on the fraction $F_{\\rm Lines}^{\\rm core}$ of channels considered as containing lines detected at a $2\\sigma $ level, defined as: $F_{\\rm Lines}^{\\rm core} = \\frac{ 1}{ n_{\\rm chan}} \\times \\sum ^{n_{\\rm chan}} _{i}{\\left\\lbrace \\begin{array}{ll}0 & \\mbox{if } T_{{\\rm Lines}, \\,i}^{\\rm core} \\le 2 \\sigma \\\\1 & \\mbox{if } T_{{\\rm Lines}, \\,i}^{\\rm core} > 2 \\sigma \\end{array}\\right.", "}$ The results are presented in the right panel of fig:contamination1.", "The relative comparison highlights the same eight cores and we also observe a relation between the number of channels with molecular emission and the continuum level; this correlation is clearer than in the left panel for all bands, even for the \"CO\" 230.30$-$ 230.76 GHz band.", "Figure: Top: methyl formate intensity map integrated over the velocity (bandwidth of 25 km s -1 ^{-1} around the 216.21 GHz transitions).", "The contours correspond to 3, 6, 9, 12, 24, 50, 100, 200 σ \\sigma , where the rms noise σ\\sigma = 2.6 K km s -1 ^{-1}.", "Bottom: methyl cyanide intensity map integrated over the velocity (bandwidth of 120 km s -1 ^{-1} around the 91.97 GHz transitions).", "The contours correspond to 5, 10, 20, 30, 50, 100, 200 σ \\sigma , where the rms noise σ\\sigma = 83 K km s -1 ^{-1}.", "The red stars indicate the position of the hot cores identified on Fig.", "." ], [ "Identification from the spatial distribution of CH$_3$ OCHO and CH{{formula:6e0c0bda-a3c4-4b28-9821-bdd0bee47fd3}} CN", "Methyl formate (CH$_3$ OCHO) and methyl cyanide (CH$_3$ CN) are two abundant complex organic molecules and are thus often used to trace hot cores [4], [44].", "We used eq:Ilinesmap to integrate the line emission at each pixel and Fig.", "REF presents maps of the two molecules using the CH$_3$ OCHO doublet at 216.21 GHz (see Sect. )", "and the CH$_3$ CN transitions from 91.958 to 91.987 GHz (see Sect. ).", "The methyl formate map highlights the eight hot cores identified on Fig.", "REF .", "These hot cores also stand out in the methyl cyanide map but they are surrounded by more widespread extended emission.", "Such an extended CH$_3$ CN emission was also observed in DR21(OH) by [12] who proposed that it traces warm gas associated with the low-velocity shocks due to converging flows coinciding with velocity shears.", "In W43-MM1 the extended CH$_3$ CN emission follows the spatial distribution of the narrow linewidth component of the SiO emission which originates from low-velocity shocks [25].", "These shocks are also likely associated with the ridge formation through colliding flows or cloud-cloud collision." ], [ "Comparison of the hot cores identification methods", "The two methods described in Sect.", "REF and REF succeed in identifying the same hot cores.", "The interest of using only the spatial distribution of COMs is that it allows to identify potential hot cores independently of the identification of the continuum cores.", "However one needs a spectral band with lines mainly coming from COMs, like the Band 6 spw7 band, and the sensitivity will be limited by the width of the spectral band and the number of strong COM lines therein.", "The second set of methods based on the relative contribution of lines $C_{\\rm Lines}^{\\rm core}$ with respect to the continuum emission or the fraction $F_{\\rm Lines}^{\\rm core}$ of channels with detected line emission, needs first to identify the continuum cores and the catalog of continuum cores will depend on the software package used for extraction [40].", "motte18 also identified potential hot cores from the line contamination in the emission of the continuum cores.", "They compared the fluxes measured in a 1.9 GHz “continuum” band and in a selection of line-free channels summing up to 65 MHz and they found the same eight cores as in Sect.", "REF .", "They detected however two more cores: core #15 which is not included in our analysis and core #30 which does not display any COMs lines when looking at the spectra.", "The method used in Sect.", "REF directly uses hot core tracers.", "However it requires first to identify the lines and to be sure that these lines are not blended with other species, which is often the case in hot cores.", "Furthermore one needs to determine the velocity of the cores to center the map on the emission line.", "When the field of view is large, there is a velocity gradient which makes more difficult to make a map: one can make a \"composite\" map adapting the velocity throughout the field, or one can take a large velocity window to integrate the emission but the sensibility will be less and the risk of blending lines will be higher.", "In the case of methyl cyanide, we have also noted that an extended emission is also present which makes more difficult the identification of the faintest hot cores." ], [ "Core temperatures", "Methyl cyanide (CH$_3$ CN) and methyl acetylene (CH$_3$ CCH) are considered as two good thermometers, as long as the lines are optically thin, because their emission K-ladder lines are close in frequency and cover a large enough range of upper level energies $E_{\\rm u}$ [16]." ], [ "CH$_3$ CN", "There are five lines of CH$_3$ CN ($J=5-4$ ) between 91.95 and 91.99 GHz, with upper level energies $E_{\\rm u}$ ranging from 13 to 128 K. CH$_3$ CN (5$_{0}$ –4$_{0}$ ) and CH$_3$ CN (5$_{1}$ –4$_{1}$ ) are only separated by 2.3 MHz, and because of the average linewidth of 5 km s$^{-1}$ , these two lines are blended.", "In the same ALMA spectral window, there are also five lines of the isotopologue CH$_3^{13}$ CN ($J=5-4$ ) with the same $E_{\\rm u}$ ranging from 13 to 128 K, as well as ten CH$_3$ CN ($J=5-4$ ) $_8$ =1 lines with $E_{\\rm u}$ ranging from 532 to 706 K. The spectroscopic parameters of the lines are given in table:MFlines.", "The CH$_3$ CN and CH$_3^{13}$ CN spectra averaged over the beam for the 8 cores are plotted on fig:CH3CN.", "The pattern of the CH$_3$ CN lines is similar for all the cores, except for cores #1, 2, 3 and 4 where the five lines are almost equally intense because of line opacity.", "Nonetheless, the relative intensities of the optically thin lines of the isotopologue CH$_3^{13}$ CN of these four cores are the same as the optically thin CH$_3$ CN pattern of the other cores.", "The similarity of the pattern with lines of different $E_{\\rm u}$ suggests an equivalent temperature for all the cores.", "The CH$_3$ CN $_8$ =1 lines are only detected towards cores #1, 2, 3, 4 and 11 and it is marginally detected towards core #5 (see fig:CH3CNv8).", "Figure: CH 3 13 _3^{13}CN (from 91.91 to 91.94 GHz) and CH 3 _3CN (91.95 to 91.99 GHz) synthetic spectra (in red for cores #10 and 11, in black for the others) overlaid on the observed spectra.", "The parameters used for the synthetic spectra are listed in table:NT.Figure: CH 3 _3CN 8 _8=1 synthetic spectra (in red for core #11, in black for the others) overlaid on the observed spectra.", "The parameters used for the synthetic spectra are listed in table:NT.", "The cores #3, 5 and 11 spectra are smoothed to a velocity resolution of 6.35 km s -1 ^{-1}.In Fig.", "REF we have overlaid for each core a synthetic spectrum considering the temperatures, column densities, linewidths and a source size indicated in table:NT.", "Due to the high average H$_2$ density of these cores (3 –76 $\\times $ 10$^8$  cm$^{-3}$ , see motte18), we consider that all lines are thermalised.", "The values are obtained with the Monte-Carlo Markov Chain algorithm and the LTE model of the CASSIS software http://cassis.irap.omp.eu [43].", "For cores #1, 2, 3, 4, 5 and 11, the temperatures, column densities and linewidths are first derived from a fit to the optically thin CH$_3^{13}$ CN and CH$_3$ CN $_8$ =1 lines assuming a source size equal to the beam (0.49$$ ).", "Then the CH$_3$ CN lines are taken into account to derive the source size.", "For cores #9 and 10, the parameters are derived from a fit to the optically thin CH$_3$ CN and CH$_3^{13}$ CN lines assuming a source size equal to the beam.", "We find an isotopic ratio of about 42, consistent with the 40–50 value at 5.5 kpc [27].", "We assume here a simple model with a uniform source and the emission of CH$_3$ CN, CH$_3^{13}$ CN and CH$_3$ CN $_8$ =1 coming from the same region.", "A more realistic source model will be used in a forthcoming paper.", "The temperatures are similar for all the cores, ranging from 120 K to 160 K with uncertainties of $\\pm $ 20 K. The broader linewidths for cores #2 and 5 can be due to multiple velocity components as seen in other COM lines (see Sect.", "REF and fig:multiples).", "The CH$_3$ CN transitions are also detected towards the possibly younger core #6 studied by [29] and the derived temperature is 60 $\\pm $ 20 K in agreement with the determinations in that paper.", "This temperature is notably different from the temperatures T$_{\\rm ex} \\sim $ 150 K we find here towards the hot cores.", "Figure: CH 3 _3CCH synthetic spectra (in red for cores #10 and 11, in black for the others) overlaid on the observed spectra.", "The parameters used for the synthetic spectra are listed in table:NT.", "The CH 3 _3CCH lines are contaminated by an ethanol line at 102.534 GHz in cores #1, 2, 3 and 4, an ethylene glycol line at 102.539 GHz mainly in core #1, and acetone lines at 102.547 GHz in cores #1, 2 and 4.Table: CH 3 _3CN and CH 3 _3CCH column densities and temperatures towards the hot cores.Table: Spectroscopic parameters of the CH 3 _3CN, CH 3 _3CCH and CH 3 _3OCHO lines studied." ], [ "CH$_3$ CCH", "We have selected five CH$_3$ CCH ($J=6-5$ ) lines between 102.51 and 102.55 GHz, with upper level energies $E_{\\rm u}$ ranging from 17 to 132 K. CH$_3$ CCH (6$_{5}$ –5$_{5}$ ) is not studied here, because the line is too weak towards all the cores.", "As for CH$_3$ CN, the CH$_3$ CCH (6$_{0}$ –5$_{0}$ ) and CH$_3$ CCH (6$_{1}$ –5$_{1}$ ) lines are blended.", "They are also contaminated by acetone (CH$_3$ COCH$_3$ ) lines at 102.547 GHz in cores #1, 2 and 4.", "Furthermore the CH$_3$ CCH (6$_{2}$ –5$_{2}$ ) line at 102.540 GHz is contaminated by the ethylene glycol, (CH$_2$ OH)$_2$ , (9$_{2,7}$ –8$_{2,6}$ ) line at 102.539 GHz mainly in core #1.", "The CH$_3$ CCH spectra averaged over the beam are presented in Fig.", "REF .", "The emission from this molecule appears to be optically thin in all the cores.", "We have overlaid synthetic spectra which parameters are indicated in table:NT.", "In the figure, we note the detection of an ethanol (C$_2$ H$_5$ OH) line at 102.534 GHz in cores #1, 2, 3 and 4, exhibiting a varying intensity from one core to the other.", "Twelve CH$_3$ CCH $_{10}$ =1 transitions with $E_{\\rm u}$ ranging from 487 to 741 K are included in the frequency range of the observations (between 102.74 and 102.94 GHz), but the intensities estimated from the parameters in table:NT are significantly below the noise level for a detection in any of the cores.", "The difference of temperatures between CH$_3$ CN (120-160 K) and CH$_3$ CCH (50-90 K) suggests that CH$_3$ CCH traces the outer envelope whereas CH$_3$ CN traces the inner part.", "Furthermore the linewidth of the CH$_3$ CCH lines compared to the CH$_3$ CN lines is also smaller for each core.", "The observations and a gas-grain chemical modelling suggest that the CH$_3$ CN emission in IRAS 16293-2422 also arises from a warmer and inner region of the envelope than the CH$_3$ CCH emission [1]." ], [ "Similarity of the normalised spectra", "To compare the molecular composition of the selected cores, we have first made a superposition of their spectra.", "Because some molecular cores have much more intense lines than the others, we normalised the spectra using three bright methyl formate (CH$_3$ OCHO, hereafter MF) doublets for this purpose; their spectroscopic parameters are listed in table:MFlines.", "The MF doublets transitions have similar $E_{\\rm u}$ levels (99 to 109 K) and are therefore most probably tracing the same volume of gas.", "This strategy has the following advantages: MF lines are common in all hot core spectra, easy identification of lines, lower optically thickness than for CH$_3$ OH lines, transitions of the two torsional A– and E–species close in frequency, very low contamination level of these doublets.", "Table: Mass, dust temperature, velocity and intensity scale factors of the hot cores.Figure: Superposition of the methyl formate doublets spectra of the hot cores.", "The spectra are normalised with respect to the six strongest and lightly contaminated methyl formate lines.", "The DCO + ^+ (3-2) emission at 216.1126 GHz is weak with respect to the methyl formate emission towards the hot cores.Figure: Comparison of the spectra of the hot cores.", "The spectra are aligned in velocity and multiplied by a factor in order to normalise to a peak value of 1 for the methyl formate lines of the 216200 and 218200 MHz bands.The spectra of each core have been aligned in velocity and multiplied by a factor so that the MF intensity of these three doublets is the same, taking core #4 as a reference.", "The individual velocity of the cores and the derived intensity ratio are displayed in table:normvalues.", "By applying these corrections, we obtain the superpositions shown in fig:MFlines for the six selected MF lines.", "The first doublet is slightly contaminated by the DCO$^+$ (3-2) line at 216.1126 GHz.", "We have mapped the spatial distribution of DCO$^+$ emission, it is located in the high-density regions but avoids the hot cores [28].", "Nonetheless, the line stands out on spectra for cores #5, 9 and 10 because the MF lines are much fainter than in the other hot cores.", "The superposition result for the Band 6 spw7 is shown in fig:compaspectexte and the entire spectral bands superposition results are shown in Appendix .", "After normalisation by the MF lines, the spectra of the eight hot cores are relatively similar.", "The upper level energy of the transitions are different, with a large $E_{\\rm u}$ coverage for some molecules (e.g., CH$_3$ OH, CH$_3$ OCHO, C$_2$ H$_5$ CN).", "The fact that the intensity factor between the cores is the same for low-$E_{\\rm u}$ and high-$E_{\\rm u}$ transitions implies that the excitation temperatures of the cores are of the same order, which is in agreement with the results from the CH$_3$ CN analysis (see Sect. ).", "The strongest lines come from the simplest molecules; they are spotted by vertical dotted lines in fig:compaspec.", "They are associated to the following molecules: CO (and C$^{18}$ O), SO, SiO, DCN, H$_2$ CO (and H$_2^{13}$ CO), HC$_3$ N, OCS (and $^{33}$ S and $^{13}$ C isotopologues), $^{13}$ CS and H$_2$ C$^{34}$ S. The study of the distribution of these molecules for core #3 showed that they are mainly not peaking at the core, except for H$_2$ C$^{34}$ S, $^{13}$ CS, OCS and its isotopologues [29], [28].", "For the molecules not centred on cores, the lines are generally wider and we can see line wings associated to the outflows.", "A detailed study of the CO(2-1) and SiO(5-4) outflows in W43-MM1 can be found in [38].", "A broad and bright high-velocity component for core #9, especially visible on the HC$_3$ N, CO and SO lines, is due to the presence in the projection plane of an outflow knot close to the core center [38].", "Lines are relatively intense for cores #5, 9, 10, 11.", "This is probably because they are less optically thick and avoid self-absorption.", "A high-velocity component is visible on OCS and $^{13}$ CS lines for core #9.", "The effect of optical thickness is visible on the line profiles of the OCS line at 231.061 GHz.", "If we consider that the OCS/MF ratio is the same in all the cores, the relative thickness of the OCS line for each core compared to the other is directly observable in fig:compaspec.", "Core #11 is the less optically thick in OCS, while cores #1 and 4 are the most.", "Furthermore, the dip at the center of the line for these two cores confirms their strong opacity.", "Likewise lines of CH$_3$ OH and its isotopologue $^{13}$ CH$_3$ OH in core #11 are more intense than in the other cores as they are optically thin in this core.", "In the continuum spw7 band at 233 GHz, some lines are clearly more intense in the three cores #1, 4 and 11.", "They are all associated to NH$_2$ CHO transitions.", "As $E_u$ for these lines ranges from 94 K to 258 K, this is not an effect of temperature but can come from a larger relative abundance or a difference in the NH$_2$ CHO emission size.", "We note also that core #1 is the richest core in molecules, for example with transitions of (CH$_2$ OH)$_2$ , H$^{13}$ CONH$_2$ and NH$_2$ CN which are not present in the spectra of the other cores.", "A c-C$_3$ H$_2$ transition is detected in the core #10 spectrum at 216.278 GHz, but the spatial distribution map seems to indicate that it is mainly associated to the outflows of core #2 [28].", "Figure: Two velocity components are visible in COM lines for cores #2 and #5.", "For CH 3 _3OCHO and OC 33 ^{33}S, the red component is at 99.2 km s -1 ^{-1} and the blue component at 94.8 km s -1 ^{-1}, with a at half-power width of 4.2 km s -1 ^{-1}.", "For 13 ^{13}CH 3 _3OH, the red component is at 99.2 km s -1 ^{-1} and the blue component at 94.7 km s -1 ^{-1}, with a half-power width of 6.0 km s -1 ^{-1}.Figure: The emission inside the dotted circle is the 218 GHz methyl formate doublet integrated in velocity (blue: 90–97 km s -1 ^{-1}, red: 97–104 km s -1 ^{-1}) towards core #2.", "The emission outside is the SiO (blue: 82–88 km s -1 ^{-1}, red: 108–119 km s -1 ^{-1}).", "Contours are 50 and 80% of maximum.", "The black ellipse is the methyl formate 0.28 ×\\times 0.20 beam.", "The black cross marks the center of the continuum core and the red and blue crosses the maximum of the methyl formate emission." ], [ "Methyl formate as a tracer of hot cores", "The current understanding is that COMs, like methyl formate, mainly form through ice chemistry on grains and are then released when dust temperatures become high enough for ices to sublimate [46], [42].", "The hot core is the inner part where the temperature is higher than $\\sim $ 100 K and a question is to know whether methyl formate can be a good tracer of the heating due to the luminosity of protostellar objects.", "All COMs do not originate from gas with the same physical conditions.", "Chemical differentiation (CN versus O-bearing molecules) has been largely observed in a lot of sources [11].", "Towards G328.2551–0.5321 [11] find that several O-bearing COMs peak at the proposed accretion shocks rather than the radiatively heated core whereas CN-bearing molecules peak towards the central protostar.", "In W43-MM1, at a $\\sim $ 2500 au spatial resolution, the emission of the COMs are spatially centred on the hot cores.", "However a clear second component of methyl formate (shifted by $\\sim $ 4 km s$^{-1}$ ) is visible for cores #2 and 5 (fig:multiples).", "This component is also spatially centred on the continuum core and does not come from a nearby source.", "A second faint component may be present as well for cores #9 and 10 and a faint third component for core #5 (shifted by $\\sim $ 7 km s$^{-1}$ ).", "These components are visible in other O-bearing molecules like CH$_3^{18}$ OH and OCS and its isotopologues and for optically thin lines.", "Figure REF presents the spatial distribution, on a higher resolution (0.24$$ or $\\sim $ 1300 au), of the blue and red components of the methyl formate lines for core #2.", "These components do not peak at the continuum center and their positions coincide with the blue and red parts of the outflows as traced by the CO and SiO emission [38].", "It suggests that the methyl formate, as well as the methanol and OCS, emission is related to the outflows and that they could have been released from the ice mantles via sublimation through shocks or UV irradiation by the protostar on the walls of the outflow cavity.", "The enhancement of COMs in regions of outflows is commonly observed towards high-mass [15], [39] and low-mass [14], [24], [3], [13] protostars.", "If the protostellar luminosity increases, the phenomena of accretion, ejection and shocks will be enhanced and the methyl formate emission will increase as well and can be used to trace the hot cores and the thermal heating from their embedded protostellar objects (Bonfand et al.", "in prep.).", "Figure: For cores #1, 2, 4, 5, 9, 10 and 11, plot of the intensity of a frequency channel versus the intensity of the same frequency channel of core #3 for the spectra of the spw7 band after correction of the velocity.", "The circles mark the peaks of the spectra.", "The line is a linear fit to the point distribution." ], [ "A general similarity of spectra", "The simple superposition of the spectra of the different cores (fig:compaspec) suggests a general similarity.", "If confirmed, this would point towards both a similar chemical composition and excitation of most of the COMs.", "A full modelling of each source in terms of physical structure and chemical composition would be the ideal approach but it is a lengthy process.", "We propose here a preliminary simpler approach to start quantifying the similarity of the cores based on correlation plots of each source spectrum with that of a reference source.", "Figure: Slope of the linear fit of fig:correlation versus the methyl formate scaling factor of fig:compaspectexte.The plots are presented in fig:correlation, taking core #3 as a reference.", "The intensity of each channel in the spectra of the different cores are plotted with respect to the intensity of the same velocity channel of core #3, after shifting the spectra with respect to the core velocities.", "Here we have selected core #3 as a reference source, as the lines are intense but less optically thick than in core #4, which avoids biases (see Sect.", "REF ).", "Larger circles indicate peaks in the core #3 spectrum, defined as channels above their two closest neighbours and stronger than 1 K (to remain well above the noise).", "The plots show a general correlation but with some dispersion, and a tendency in some cases towards a curved rather than a linear relation.", "The slopes of the linear fits (renormalised to 1 for core #4 for comparison) are given in table:normvalues and they are plotted versus the methyl formate scaling factors in fig:correlation-plot.", "It appears that the two methods to compare the line spectra give coherent results.", "Hereafter we first discuss various causes of possible spectra dissimilarity and how they affect these spectrum versus spectrum plots, then we discuss the observed similarity in a more quantitative way and find a factor 2–3 agreement.", "Figure: For cores #1, 2, 4, 5, 9, 10 and 11, plot of the logarithm of the normalised ratio R i =T i S /(a 1 ×T i S 3 )R_i = T_i^S/(a_1 \\times T_i^{S_3}) versus T i S 3 T_i^{S_3}, the channel intensity of the reference source core #3 (see Sect. ).", "Horizontal red and green lines indicate a departure from R i =1R_i=1 by a factor 2 and 3 respectively.", "Circles indicatepeaks in the core #3 spectrum.", "Most points right of T i S 3 =2KT_i^{S_3} = 2~K remain with a factor 2 of the linear fit shown in fig:correlation (slope a 1 a_1), which indicates a good general similarity in the spectra." ], [ "Reasons for dissimilarity", "We consider that all lines are thermalised (see Sect.", "REF ).", "If the molecular emission of two cores is identical, the spectrum-spectrum plot should be linear, only affected by the observational noise (5 sigma $\\sim $ 1 K).", "Possible reasons of dissimilarity are listed below and illustrated using simulated spectra in Appendix : Velocities: if the emission lines of the molecules are not centred at the same velocity in two cores, each line would appear as a loop in the diagram due to the Doppler shift.", "To remove this effect, we have realigned each core spectrum with respect to the reference core spectrum, by varying the relative velocity shift to minimise the dispersion in the spectrum-spectrum plot.", "Linewidths: a difference in linewidth would increase the dispersion, but here all sources have similar linewidths, except core #5 (see table:NT).", "Masses: if the cores have a similar structure, but one is more massive, the relation between optically thin lines will be linear, but with a slope different from 1.", "The stronger lines will not be affected by optical thickness in the same way, and the line profiles will differ.", "Temperatures: if the temperature is different in the two cores, each individual optically thin lines will still lead to linear relations but with different slopes for different energy levels.", "Abundances: if the relative abundance of molecules is not exactly the same, thin lines will present a linear relation but with different slopes for each species.", "From an astrophysical point of view, sources in our sample differ by a factor 6 in mass for the massive cores (even up to $\\sim $ 50 if core #11 is include), leading to much higher column densities and opacities in some of them.", "Moreover, some cores might include unresolved multiple sources and some might have a noticeable proportion of molecules released by shocks (e.g.", "linked to bipolar flows) in addition to thermal desorption and ice sublimation.", "In those cases, the kinematics and the composition of the gas could be affected to some extent.", "However, observationally, as shown below, the molecular emission spectra of the cores are quite similar." ], [ "Similarity from ratio plots", "To get a more quantitave indication of the similarity of the spectra, we have plotted in fig:correlation-2 the logarithm of the ratio of one spectrum to the linear fit (see fig:correlation).", "More precisely, we define the quantity $R_i$ for each frequency channel $i$ as $R_i = T_i^S/(a_1 \\times T_i^{S_{\\rm ref}})$ where $T_i^S$ and $T_i^{S_{\\rm ref}}$ are the channel intensities respectively for core S and a core S$_{\\rm ref}$ taken as reference, and $a_{\\rm 1}$ is the slope of the linear fit (with no constant term) of $T_i^S$ versus $T_i^{S_{\\rm ref}}$ .", "As mentioned before, the spectrum of S has been first realigned in velocity with respect to the spectrum of S$_{\\rm ref}$ .", "fig:correlation-2 presents plots of log $(R_i)$ versus $T_i^{S_{\\rm ref}}$ .", "For all sources $R_i$ remains close to 1 within a factor 3, and in many cases within a factor 2, for most channels $i$ where the core #3 spectrum is above 2 K (core #3 maximum intensity being about 12.9 K); these limits are indicated respectively by the green and red horizontal lines.", "One notes a slight decrease of the ratio $R_i$ with $T_i^{S_3}$ which is due to opacity effects in the strongest lines.", "Core #11, which is the least massive, appears the least well correlated to core #3.", "The rather limited dispersion in the plots indicates a limited role of the potential dissimilarity factors listed above.", "The 2–3 factor of agreement between the spectra suggests a similar molecular composition of the cores, which can be due to the formation of the molecules.", "The lines in the spw7 band are principally those of COMs which are mainly formed on similar ices in the filament and desorbed by shocks and/or by the high temperature." ], [ "Conclusion", "In this article we have studied with ALMA at high spatial resolution (0.5$$ ) the molecular composition of the rich high-mass star forming region W43-MM1.", "This study proposes analysis tools and lays the groundwork for future comparisons to similar systems.", "To identify the molecular hot cores we have developed different methods.", "The first one relies on the continuum versus line emission separation method developed in [29], applied to a 2 GHz band around 233 GHz rich in Complex Organic Molecules lines and not contaminated by strong lines of simpler species.", "Hot cores are then identified in the map of the continuum-subtracted brightness temperature averaged over the band, the peaks of which highlight intense COMs emission.", "A second hot core identification method uses the relative contribution of lines and continuum, but in spectra spatially averaged on previously identified continuum cores.", "In this case all the nine bands we observed are used - narrow as well as broad ones.", "The criterion relies either on the summed line intensities or on the number of line channels.", "The results are in general agreement with the first method but some bands with strong lines of simple species are definitely less sensitive.", "We made methyl formate (CH$_3$ OCHO) and methyl cyanide (CH$_3$ CN) maps which highlight the same hot cores as previously determined and confirmed their nature.", "We also note an extended methyl cyanide emission which may trace the warm gas associated with the low-velocity shocks also observed in SiO.", "Seven hot cores with 16 to 100 M$_\\odot $ in mass and one less massive 2 M$_\\odot $ core were identified.", "For each identified core we have determined mean temperatures using the classical \"thermometer molecules\" methyl cyanide (CH$_3$ CN) and methyl acetylene (CH$_3$ CCH) lines at 3 mm.", "The CH$_3^{13}$ CN isotopologue lines allowed to circumvent optical thickness of the lines in the strongest sources (#1-4).", "CH$_3$ CN temperatures are consistently all around 150 K whereas CH$_3$ CCH leads to lower value in the 50-90 K range.", "This is interpreted as due to a distribution extending further in the envelope beyond the hot core region where ice mantles have been sublimated.", "The previous studied core #6 is confirmed as atypical with a CH$_3$ CN lower temperature of $\\sim $ 60 K. The chemical composition of the cores is compared using two methods.", "First a direct superposition of the COMs-rich $\\sim $  2 GHz wide spectra around 233 GHz is made after a scaling in intensity based on methyl formate lines.", "Secondly correlation diagrams of the brightness temperature in each channel are plotted.", "No line identification is required.", "We conclude to a general good agreement within a factor 2–3 of the mean in the chemical composition of the various hot cores, which cover an order of magnitude in mass.", "Core #1 is especially rich in molecular lines, including ethylene glycol (CH$_2$ OH)$_2$ , formamide isotopologue H$^{13}$ CONH$_2$ , and cyanamide NH$_2$ CN which are not detected in other cores.", "Simpler species like SiO, DCN, H$_2$ CO, CO, SO do not have an emission concentrated in the cores; H$_2$ C$^{34}$ S,$^{13}$ CS and OCS (and isotopologues) however do.", "In core #2 we find a spatial association between the blue and red velocity components of methyl formate and the outflow lobes.", "We plan to develop these studies in the frame of forthcoming analyses of the sources in the W43-MM2 and W43-MM3 ALMA-IMF regions.", "We thank the anonymous referee for his helpful comments improving the manuscript.", "This paper makes use of the following ALMA data: ADS/JAO.ALMA#2013.1.01365.S, ADS/JAO.ALMA#2015.1.01273.S, ADS/JAO.ALMA#2017.1.01355.L.", "ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile.", "The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.", "This work is based on an analysis carried out with the CASSIS software and the CDMS and JPL spectroscopic databases.", "CASSIS has been developed by IRAP-UPS/CNRS (http://cassis.irap.omp.eu).", "This work was supported by the Programme National de Physique Stellaire and Physique et Chimie du Milieu Interstellaire (PNPS and PCMI) of CNRS/INSU (with INC/INP/IN2P3).", "The project leading to this publication has received support from ORP, that is funded by the European Union's Horizon 2020 research and innovation programme under grant agreement No 101004719 [ORP].", "This project has received funding from the European Research Council (ERC) via the ERC Synergy Grant ECOGAL (grant 855130), from the French Agence Nationale de la Recherche (ANR) through the project COSMHIC (ANR-20-CE31-0009).", "AG acknowledges support from NSF grant AST 2008101.", "P.S.", "was partially supported by a Grant-in-Aid for Scientific Research (KAKENHI Number 18H01259) of the Japan Society for the Promotion of Science (JSPS).", "G.B.", "acknowledges support from the PID2020-117710GB-I00 grant funded by MCIN/AEI/10.13039/501100011033 and from the \"Unit of Excellence María de Maeztu 2020-2023'\" award to the Institute of Cosmos Sciences (CEX2019-00918-M).", "AS gratefully acknowledges funding support through Fondecyt Regular (project codes 1180350 and 1220610), from the ANID BASAL project FB210003, and from the Chilean Centro de Excelencia en Astrofísica y Tecnologías Afines (CATA) BASAL grant AFB-170002.", "LB gratefully acknowledges support by the ANID BASAL projects ACE210002 and FB210003.", "RGM and TN acknowledge support from UNAM-PAPIIT project IN108822." ], [ "Sources of dissimilarity in the correlation plots. ", "To help visualizing the effect of various sources of dissimilarity discussed in Sect.", "REF on the \"correlation\" plots of Fig.", "REF , we provide in Fig REF a few examples of such plots drawn with simple synthetic spectra.", "These spectra are presented in the left column and the corresponding correlation plots in the right column for each case.", "The reference spectrum is plotted in black, and the compared spectrum in red.", "For optically thin lines the profile is gaussian.", "The following effects are shown: (a) a shift in frequency by 1 MHz (corresponding for spw7 band to about 1.3 km s$^{-1}$ ), (b) a larger line width (9 km s$^{-1}$ compared to 5 km s$^{-1}$ in the reference spectrum), (c) spectral confusion: a line present only in the second spectrum merges partially into one of the three lines (otherwise similar in both spectra), (d) the lines in the second spectrum are more intense and optically thick, (e) the relative intensity of the three lines is different in both spectra.", "This mimics either a difference in abundances if the lines are from different species, or a difference in rotational temperatures if the lines are from the same species but have different upper level energies.", "(f) a combination of the two previous effects (d) and (e).", "A perfect proportionality of the two spectra would of course lead to a single line in the correlation plots.", "The first effect should not be present for single component sources, as the spectra have been realigned on purpose.", "All other effects would widen and/or deform the correlation.", "The correlation plots in Sect.", "REF obtained with the observed spectra show that the combined effect of all these sources of dissimilarity remains limited.", "Figure: Effects of dissimilarity in the correlation plots." ] ]
2207.03537
[ [ "On the Metric Properties of IR Evaluation Measures Based on Ranking\n Axioms" ], [ "Abstract The axiomatic analysis of IR evaluation metrics has contributed to a better understanding of their properties.", "Some works have modelled the effectiveness of retrieval measures with axioms that capture desirable properties on the set of ranked lists of documents.", "Recently, it has been shown that three of these axioms lead to some orderings.", "This work formally explores the metric properties of the set of rankings, endowed with these orderings.", "Based on lattice theory, the possible metrics and pseudo-metrics, defined on these structures, are determined.", "It is found that, when the relevant documents are prioritized, precision, recall, RBP and DCG are metrics on the set of rankings, however they are pseudo-metrics when the swapping of documents is considered." ], [ "Introduction", "In the Cranfield paradigm [2], relevance judgements are assigned to a ranked list of documents for a topic.", "Then, an IR evaluation metric can be seen as a mapping that relates the set of possible lists of judged documents (empirical domain) with the real numbers (scores).", "They quantify, in numeric terms, the ability of systems to leave aside the irrelevant documents from the relevant ones.", "There have been proposed a very large number of measures which evaluate different aspects of IR systems [3], [4], [5], [6].", "In designing these measures, much of the effort has gone into capture the effectiveness of retrieval systems.", "Sometimes, these attempts have led to pay less attention to the three postulates of a metricHere, the mathematical term of “metric” collides with the commonly used term of IR evaluation “metric”.", "To solve this issue, in this paper, the term “metric” will have its mathematical sense.", "The term “measure” will design any mapping, from the set of rankings to the real numbers.", "Thus, IR evaluation metrics will be referred as IR evaluation measures., in the mathematical sense: identity of indiscernible, symmetry and triangular inequality.", "This can have inadequate consequences, such as drawn counter-intuitive conclusions.", "The metric properties of a mapping are directly related to the properties of the set where it is defined [7], [8]: if there exists a mapping or function that verifies the three postulates, then the set where it is defined is a metric space and the function that verifies these properties is a metric.", "The importance of the empirical domain can also be observed on the IR evaluation measures, for example, in the statement “an evaluation measure is an ordinal scale”, the evaluation measure is preserving the existing ordering between rankings of documents.", "Some works [9], [10], [11], [12] have presented desirable properties that any reasonable IR evaluation measure should satisfy, for instance, “it is preferred to retrieve relevant documents in the top ranking positions”.", "These properties (axioms, constraints or heuristics) lead to different orderings in the set of ranked lists of documents [1].", "Then, some obvious questions arise: Is the set of rankings, endowed with this orderings, a metric space?", "Which are the corresponding metrics?", "Are the values assigned by the IR evaluation measures consistent with the orderings derived from the desirable properties?", "Here, the consistency can be understood as meaning that the scores assigned by the IR evaluation measures do not contradict the orderings derived from the desirable properties.", "This paper determines if the set of ranked lists of documents, endowed with the orderings derived in [1], is a metric or pseudo-metric space, i.e., if there exists at least one metric or pseudo-metric defined on it.", "Then, in the set of rankings where it is feasible, all the possible metrics/pseudo-metrics are inferred.", "To this end, it is presented the concept of valuation, which is the ground of the concepts of measure and probability in measurement theory.", "There are shown explicit expressions of the mappings, in terms of a remarkable subset of rankings presented in [1]: the join-irreducible elements.", "Then, it is checked if some IR evaluation measures are metrics or pseudo-metrics.", "It is found that precision, recall, $RBP$ and $DCG$ are metrics on the set of rankings where the relevant documents are prioritized, however they are pseudo-metrics when the swapping of documents is considered.", "The rest of the paper is organized as follows: In Section , some related works are reported.", "In Section , we briefly introduce notations and some basic results of lattice theory.", "In Section , the problem is contextualized in the IR setting.", "Then, in Sections and , it is determined when precision, recall, $RBP$ or $DCG$ are metrics for the set-based and rank-based retrieval respectively.", "Finally, in Section some conclusions are drawn." ], [ "Lattice Theory", "In this section we recall some basic notions of lattice theory, for additional background, we refer the reader to the textbooks [13], [14].", "A partially ordered set or poset is a non-empty set, $X$ , together with a binary, reflexive, antisymmetric and transitive relation, $\\preceq $ , defined on $X$ .", "A poset $(X, \\preceq )$ is called a totally ordered set or a chain if any two elements, $x$ , $y \\in X$ , are comparable, i.e., $x \\preceq y$ or $y \\succeq x$ .", "A (closed) interval is a subset of $X$ defined as $[x, y] = \\lbrace z \\in X : x \\preceq z \\preceq y\\rbrace $ .", "It is said that $y$ covers $x$ in $X$ , denoted by $x \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }y$ , if there does not exist $z \\in X$ such that $x \\ne z \\ne y$ and $x \\preceq z \\preceq y$ .", "An interval $[x, y]$ is called prime, if $x \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }y$ , i.e., $[x, y] = \\lbrace x, y\\rbrace $ .", "Thus, $(X, \\preceq )$ has associated a graph, called its Hasse diagram, where nodes are labelled with the elements of $X$ and the edges indicate the covering relation.", "Nodes are represented in different levels, if $y$ covers $x$ , then $x$ is below $y$ in the diagram.", "A poset $X$ is called graded if there exists an integer-valued function, $\\rho $ , defined on $X$ , such that, if $x \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }y$ , then $\\rho (y) = \\rho (x) + 1$ .", "The function $\\rho $ is called the rank function.", "An upper bound for a subset, $S \\subseteq X$ , is an element, $x \\in X$ , for which $s \\preceq x$ , $\\forall s \\in S$ .", "A lower bound for a subset of a poset is defined analogously.", "The greatest lower bound or meet of two elements $x, y \\in X$ is denoted $x \\wedge y$ .", "The least upper bound or join of two elements $x, y \\in X$ is denoted $x \\vee y$ .", "It is said that $X$ has a top element, denoted by $1 \\in X$ , if $x \\preceq 1$ , $\\forall x \\in X$ ; dually, $0 \\in X$ is the bottom element, if $0 \\preceq x$ , $\\forall x \\in X$ .", "Considering the join and the meet as operators, it is said that $(X, \\wedge , \\vee , \\preceq )$ is a lattice if $(X, \\preceq )$ is a poset and every two elements have a meet and a join.", "A lattice is modular if $x \\preceq y$ implies $x \\vee (z \\wedge y) = (x \\vee z) \\wedge y$ , where $x$ , $y$ and $z$ are arbitrary elements.", "A lattice is distributive if its meet operator distributes over its join operator, i.e., $x \\wedge (y \\vee z)$ $=$ $(x \\wedge y) \\vee (x \\wedge y)$ , $\\forall x$ , $y$ , $z \\in X$ .", "The distributive property is a stronger condition than the modular property; thus, any distributive lattice is a modular lattice [13].", "An example of distributive lattices are the chains [14].", "There are some remarkable elements in a lattice, given $j \\in X$ (with $j \\ne 0$ ), it is said to be a join-irreducible element, if $x \\vee y = j$ implies $x = j$ or $y = j$ .", "Thus, $j \\in X$ is join-irreducible if it cannot be expressed as the join of two elements that are strictly smaller than $j$ .", "The set of the join-irreducible elements of $X$ is denoted by $J_{X}$The join-irreducible elements are those that have only one descendant in the Hasse diagram of $X$ [14].", "Let $x$ be an element of $X$ , it could be useful to consider the subset of join-irreducible elements smaller than $x$ , which will be denoted by $J_{x} = \\lbrace j \\in J_{X} : j \\preceq x\\rbrace $ .", "The following result highlights the importance of the join-irreducible elements of a finite distributive lattice.", "Theorem 1 .S.", "Blyth [15]] If $X$ is a finite distributive lattice, then every element of $X \\setminus \\lbrace 0\\rbrace $ can be expressed uniquely as an irredundant join of join-irreducible elements.", "A real-valued mapping defined on a lattice, $v: X \\rightarrow \\mathbb {R}$ , is a valuation, if it satisfies the following property: $v(x) + v(y) = v(x \\vee y) + v(x \\wedge y) $ , $\\forall x$ , $y \\in X$ .", "A valuation, $v: X \\rightarrow \\mathbb {R}$ is called isotone or monotone (resp.", "positive), if $x \\preceq y \\Rightarrow v(x) \\le v(y)$ , where $\\le $ denotes the ordering on $\\mathbb {R}$ (resp.", "$x \\preceq y$ and $x \\ne y$ $\\Rightarrow v(x) < v(y)$ ).", "The next result shows a characterization of the valuations in terms of the join-irreducible elements.", "Theorem 2 (G.C.", "Rota [16]) A valuation, defined on a finite distributive lattice, $X$ , is uniquely determined by the values it takes on the set of join-irreducible elements of $X$ , and these values can be arbitrarily assigned.", "In [17], is given an explicit expression of these functions.", "Considering an assignment or mapping from the set of join-irreducible elements to the positive real numbers, $w:J_{X} \\longrightarrow \\mathbb {R}^{+}$ , any isotone valuation, $v$ , on a finite distributive lattice can be obtained as: $v(x) = \\sum _{j \\in J_{x}} w(j)$ Theorem 3 (G. Birkhoff [13]) In any lattice, $L$ , with an isotone valuation, the distance function, $d_v (x,y) = v(x \\vee y) - v(x \\wedge y)$ , satisfies for all $x$ , $y$ and $z \\in X$ : (i) $d_v(x,x) = 0$ , $d_v(x, y) \\ge 0$ and $d_v(x,y) = d_v(y,x)$ ; (ii) $d_v(x,z) \\le d_v(x, y) + d_v(y,z)$ ; and (iii) $d_v(z \\vee x, z\\vee y) + d_v(z \\wedge x, z \\wedge y) \\le d_v(x,y)$ .", "This result enables to define a pseudo-metric or semi-metric lattice as a lattice with an isotone valuation.", "A pseudo-metric lattice is so named because the distance, $d_v(x, y)$ , verifies the properties of a pseudo-metric: symmetry, $d_v(x,y) = d_v(y, x)$ , and triangular inequality, $d_v(x,y) \\le d_v(x,z) + d_v(z,y)$ .", "In addition, if $d_v(x,y)=0 \\Leftrightarrow x=y$ , then $d_v$ verifies the three postulates of a metric and $(X, \\wedge , \\vee , \\preceq )$ is called a metric lattice.", "As $v$ is an isotone valuation, this condition is equivalent to $x \\wedge y \\preceq x \\vee y \\Rightarrow v(x \\wedge y) < v(x \\vee y)$ [13].", "Hence, a pseudo-metric lattice yields a metric lattice if and only if the valuation, which defines, is positive.", "Theorem 4 (G. Birkhoff [13]) Any metric lattice is modular.", "The converse of this theorem holds for finite modular lattices [18].", "From this result, any finite distributive lattice is a metric lattice since distributive implies modular.", "Indeed, every isotone valuation, in a finite distributive lattice, is positive since, by Theorem REF and Equation REF , if $x \\preceq y$ and $x\\ne y$ , then $J_x \\subset J_y$ .", "Let $v$ be any isotone valuation on a finite distributive lattice generated by a strictly positive real-assignment, $w$ , of the join-irreducible elements, when each edge, from $x$ to $y$ , of the covering graph is weighted by $\\big \\vert v(y) - v(x) \\big \\vert $ , then the minimum path length metric coincides with $d_v$ and it verifies: $d_{v}(x,y) = v(x) + v(y) - 2 \\cdot v(x \\wedge y) = \\sum _{j \\in J_{x} \\triangle J_{y}} w(j)$ where $\\triangle $ is the symmetric difference operator [19], [20].", "In addition, if $w$ is defined by $w(j) = k$ for all $j \\in J_X$ , $k \\in \\mathbb {R}^{+}$ and $X$ is a distributive lattice, then $\\big \\vert v(x) - v(y) \\big \\vert = k$ for every edge in the Hasse diagram of $X$ .", "Thus, the valuation can be expressed as $v(x) = k \\cdot \\big \\vert J_x \\big \\vert $ and $d_v$ is the natural distance on the covering graph, $d_{v}(x, y) = k \\cdot \\big \\vert J_{x} \\triangle J_{y} \\big \\vert $ .", "Finally, fixing one of the arguments to the bottom element, 0, and choosing $k=1$ , this valuation coincides with the rank function: $\\rho (x) = d_v(0,x) = \\big \\vert J_x \\big \\vert $ ." ], [ "IR Setting", "In this section, some notation needed throughout the paper is introduced.", "The formalism of [21] is adopted.", "Consider a finite set of documents that are retrieved from query terms provided by a user.", "An (ordered or not) collection of $N$ retrieved documents will be called a system run of length $N$ for a topic.", "Once documents have been retrieved, they can be classified by their relevance degree.", "Let $(REL, \\preccurlyeq )$ be a finite and totally ordered set of relevance degrees, where $REL = \\lbrace \\mathcal {a}_0, \\ldots , \\mathcal {a}_{c}\\rbrace $ with $\\mathcal {a}_{i} \\prec \\mathcal {a}_{i+1}$ , for each $i \\in \\lbrace 1, \\ldots , c-1\\rbrace $ .", "These relevance degrees can be categorical labels.", "To handle numerical values, a gain function, $g: REL \\rightarrow \\mathbb {R}^{+}$ , is considered as a map that assigns a positive real number to any relevance degree.", "In this paper, it can be assumed that $g(\\mathcal {a}_0) = 0$ and $g$ is a strictly increasing function.", "A particular case of such a map is the indicator function, defined as $\\delta _{\\mathcal {a}}(\\mathcal {a}_j) = j$ , for each $j = 0, \\ldots c$ .", "When every retrieved document of a system run is classified with a relevance degree, a judged run is obtained, denoted by $\\hat{r}$ .", "In the set-based retrieval, each judged run is an unordered set of $N$ relevance degrees, which may contain the same element several times.", "For instance, for $N=4$ and 3 relevance degrees, a judged run is given by $\\hat{r} = \\lbrace \\mathcal {a}_2, \\mathcal {a}_1, \\mathcal {a}_1, \\mathcal {a}_0\\rbrace $ .", "For the sake of clarity, the convention is used to represent $\\hat{r}$ as $\\lbrace \\hat{r}_1, \\ldots , \\hat{r}_N\\rbrace $ , where $\\hat{r}_i \\succcurlyeq \\hat{r}_{i + 1}$ , for any $i \\in \\lbrace 1, \\ldots N-1\\rbrace $ , i.e., the relevance degrees are listed in decreasing order.", "This process does not affect the results obtained.", "The $j$ -th element of the set $\\hat{r}$ is denoted by $\\hat{r}_j$ .", "In the rank-based retrieval, each judged run is an ordered list of $N$ relevance degrees.", "For instance, for $N=4$ and 3 relevance degrees, a judged run is given by $\\hat{r} = (\\mathcal {a}_1, \\mathcal {a}_0, \\mathcal {a}_2, \\mathcal {a}_1)$ .", "In this case, the $j$ -th element of $\\hat{r}$ is denoted by $\\hat{r}[j]$ .", "The set of all the possible judged runs of length $N$ is a finite set, which will be denoted by $R(N)$ .", "In [1], three commonly accepted operations about the set where an IR evaluation measure is defined are stated: Replacement: “Replacing a document for another one with a higher relevance degree is a preferred option” [12], [9], [22].", "Projection: “Removing all documents that are not considered in the highest or more relevant part of a list is a preferred option” [11], [22].", "Swapping: “Swapping a less relevant document in a higher rank position with a more relevant one in a lower rank position is a preferred option” [10], [11], [12].", "Combining these properties, five orderings are obtained: two for the set-based retrieval (Projection+Replacement and Replacement) and three for the rank-based retrieval (Projection+Replacement, Replacement and Swapping+Replacement).", "$R(N)$ endowed with any of these orderings is a poset and considering the meet and join operations, $(R(N), \\wedge , \\vee , \\preceq )$ can be considered as a lattice [1].", "An IR evaluation measure can be seen as a mapping from $R(N)$ to the set of real numbers.", "In the set based retrieval some remarkable evaluation measures are as follows.", "Generalized Precision ($gP$ ) [21]: $gP(\\hat{r}) = \\frac{1}{N} \\cdot \\sum _{i=1}^{N} \\frac{g(\\hat{r}_i)}{g(\\mathcal {a}_{c})} \\ .$ This measure represent a user model which always reads from rank 1 down to $k$ , and then stops.", "This score represents the average rate at which a user accures relevance.", "Generalized Recall ($gR$ ) [21] $gR(\\hat{r}) = \\frac{1}{RB} \\cdot \\sum _{i=1}^{N} \\frac{g(\\hat{r}_i)}{g(\\mathcal {a}_{c})} \\ ,$ where $RB$ is the recall base (total number of relevant documents).", "The user model of this measure is similar to $gP$ , but the score represents the average rate that a relevant document will be retrieved.", "These evaluation measures can also be considered in the rank-based retrieval, in addition to the following.", "Graded Rank-Biased Precision ($gRBP$ ) [23], [24], [21]: $gRBP(\\hat{r}) = \\frac{(1-p)}{g(\\mathcal {a}_c)} \\cdot \\sum _{i=1}^N p^{i-1} \\cdot g(\\hat{r}[i]) \\ ,$ where $p$ reflects the persistence scanning a system output run.", "This score is numerically equal to the expected relevance of the last document inspected.", "Discounted Cummulated Gain ($DCG$ ) [25]: $DCG_{b}(\\hat{r}) = \\sum _{i=1}^N \\frac{g(\\hat{r}[i])}{\\max \\lbrace 1, \\log _{b}i \\rbrace } \\ ,$ where the parameter $b$ typically ranges over 2 and 10.", "The user is assumed to inspect an unbounded number of items.", "It models a user that scans down a ranked list, growing less interested with each successive rank; the gain function models the utility the user derives from each document.", "The valuations on $(R(N), \\wedge , \\vee , \\preceq )$ are real-valued mappings, $M: R(N) \\longrightarrow \\mathbb {R}$ , satisfying $M(\\hat{r} \\vee \\hat{s}) = M(\\hat{r}) + M(\\hat{s}) - M(\\hat{r} \\wedge \\hat{s})$ , $\\forall \\hat{r}$ , $\\hat{s} \\in R(N)$ .", "They are isotone or monotone (resp.", "positive), if $\\hat{r} \\preceq \\hat{s} \\Rightarrow M(\\hat{r}) \\le M(\\hat{s})$ , where $\\le $ denotes the ordering on $\\mathbb {R}$ (resp.", "$\\hat{r} \\preceq \\hat{s}$ and $\\hat{r} \\ne \\hat{s}$ $\\Rightarrow M(\\hat{r}) < M(\\hat{s})$ ).", "Remark 1 An IR evaluation measure, $M:R(N)\\longrightarrow \\mathbb {R}$ , is an ordinal scale or order preserving, if $\\hat{r} \\preceq \\hat{s} \\Leftrightarrow M(\\hat{r}) \\le M(\\hat{s})$ .", "The difference with the definition of an isotone mapping is the implication from right to left, thus not every isotone mapping is an ordinal scale, however every ordinal scale is an isotone mapping.", "One of the aims of this paper is to determine all the isotone valuations (i.e., pseudo-metrics) defined on $R(N)$ , thus the obtained results generalize the ordinal scales or even higher order scale types, which are valuations.", "To classify the metric properties of the empirical domain of IR evaluation measures, the following results of Section are considered: (i) $(R(N), \\wedge , \\vee , \\preceq )$ is a pseudo-metric lattice if and only if there exists an isotone valuation defined on $R(N)$ ; and (ii) $(R(N), \\wedge , \\vee , \\preceq )$ is a metric lattice if and only if there exists a positive valuation defined on $R(N)$ .", "If $(R(N), \\wedge , \\vee , \\preceq )$ is a finite distributive lattice and $w:J_{R(N)} \\longrightarrow \\mathbb {R}^{+}$ is an assignment to its join-irreducible elements, then any isotone valuation is as follows (see Section ).", "$M(\\hat{r}) = \\sum _{\\hat{j} \\in J_{\\hat{r}}} w(\\hat{j})$ These isotone valuations, in a finite distributive lattice, are really positive valuations; thus, they generate a metric, $d_M$ , which coincides with the minimum path metric on the covering graph and is given by $d_{M}(\\hat{r}, \\hat{s}) = M(\\hat{r}) + M(\\hat{s}) - 2 \\cdot M(\\hat{r} \\wedge \\hat{s})$ (see Section ).", "If $w$ is a constant assignment, with value $k \\in \\mathbb {R}^{+}$ , then $d_{M}(\\hat{r}, \\hat{s}) = k \\cdot \\big \\vert J_{\\hat{r}} \\triangle J_{\\hat{s}} \\big \\vert $ is the natural distance on the Hasse diagram.", "In the particular case of $k=1$ , this valuation is the rank function and it is given by $\\rho (\\hat{r}) = \\big \\vert J_{\\hat{r}} \\big \\vert $ .", "In the following sections, it will be considered the five orderings derived in [1].", "The key point is to determine if the empirical domain is a metric or pseudo-metric lattice.", "Then, the possible mappings on this set can be analysed.", "If the set of rankings, endowed with one of these orderings, is not a metric lattice, then there can be no positive valuations.", "However, it may exists isotone valuations.", "If there exists a pseudo-metric, then the set of rankings would be a pseudo-metric lattice.", "If the set of rankings, endowed with one of these orderings, is a metric space, then there is at least one mapping that is a positive valuation, which could be an ordinal scale.", "In addition, if the join-reducible assignment is constant, then it represents the natural distance.", "However, in a metric lattice there may also exist other valuations that need not be isotone or positive.", "Some IR evaluations measures may exhibit this situation, for instance, in a metric lattice, it could be an IR evaluation measure which is an isotone valuation, but not positive, i.e., it would be a pseudo-metric." ], [ "Set-based Retrieval", "In this Section, the set-based retrieval is considered and the two orderings which were deduced in [1] are analysed." ], [ "Projection+Replacement [set-based]", "Let $\\hat{r}, \\hat{s} \\in R(N)$ such that $\\hat{r} \\ne \\hat{s}$ , and let $k$ be the largest relevance degree at which the two runs differ for the first time, i.e., $k = max \\big \\lbrace j \\le c : \\big \\vert \\lbrace i : \\hat{r}_i = \\mathcal {a}_j\\rbrace \\big \\vert \\ne \\big \\vert \\lbrace i : \\hat{s}_i = \\mathcal {a}_j\\rbrace \\big \\vert \\big \\rbrace $ ; then, they are ordered by: $\\hat{r} \\preceq \\hat{s} \\Longleftrightarrow \\big \\vert \\lbrace i : \\hat{r}_i = \\mathcal {a}_k\\rbrace \\big \\vert \\le \\big \\vert \\lbrace i : \\hat{s}_i = \\mathcal {a}_k\\rbrace \\big \\vert $ A system run is preferred to another if for the highest relevance degree at which the two system runs differ, the first has more documents of this relevance degree.", "As $(R(N), \\preceq )$ is a totally ordered set [1], then every element is join-irreducible, except $\\hat{0}$ , i.e., $J_{R(N)} = R(N) \\setminus \\lbrace \\hat{0}\\rbrace $ .", "Thus, $J_{\\hat{r}}$ is the set of all the elements smaller than $\\hat{r}$ , except $\\hat{0}$ .", "$(R(N), \\wedge , \\vee , \\preceq )$ is a metric lattice since it is a chain (see Section ).", "In the following, it will be determined all the possible isotoneThey are really positive since $R(N)$ is a finite distributive lattice.", "valuations on $R(N)$ .", "Consider an assignment, $w:R(N) \\setminus \\lbrace \\hat{0}\\rbrace \\longrightarrow \\mathbb {R}^{+}$ and a judged run, $\\hat{r} \\in R(N) \\setminus \\lbrace \\hat{0}\\rbrace $ , it can be assumed that $\\hat{0} =\\hat{r}^{0} \\preceq \\hat{r}^{1} \\preceq \\ldots \\preceq \\hat{r}^{k} = \\hat{r}$ .", "As every totally ordered set is a distributive lattice, then, by Equation REF , any isotone valuation can be obtained as: $M(\\hat{r}) = \\sum _{i = 1}^{k} w(\\hat{r}^{i})$ Considering $\\hat{r}$ , $\\hat{s} \\in R(N)$ , each of these isotone valuations, $M$ , generates a metric, $d_M$ , which coincides with the minimum path metric and is given by $d_{M}(\\hat{r}, \\hat{s}) = M(\\hat{r}) + M(\\hat{s}) - 2 \\cdot M(\\hat{r} \\wedge \\hat{s})$ .", "As $R(N)$ , endowed with the projection+replacement [set-based] ordering, is a totally ordered set; then, it can be assumed that $\\hat{r} \\preceq \\hat{s}$ , i.e., $\\hat{r} \\wedge \\hat{s} = \\hat{r}$ .", "Therefore, $d_{M}(\\hat{r}, \\hat{s}) = M(\\hat{s}) - M(\\hat{r})$ .", "If $w(\\hat{r}) = k$ , $\\forall \\hat{r} \\in R(N) \\setminus \\lbrace \\hat{0}\\rbrace $ , with $k \\in \\mathbb {R}^{+}$ , then these valuations are expressed as $M(\\hat{r}) = k \\cdot \\big \\vert J_{\\hat{r}} \\big \\vert $ and the natural distance on the covering graph is $d_{M}(\\hat{r}, \\hat{s}) = k \\cdot \\big \\vert J_{\\hat{s}} \\triangle J_{\\hat{r}} \\big \\vert $ .", "Thus, the isotone valuations, $M(\\hat{r})$ , with constant assignment to the join-irreducible elements count the number of elements from $\\hat{0}$ to $\\hat{r}$ .", "It can be given an alternative expression of $M(\\hat{r})$ .", "Considering that $M(\\hat{0}) = 0$ , the Projection+Replacement [set-based] ordering is the following: $\\lbrace \\mathcal {a}_0, \\mathcal {a}_0, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0\\rbrace \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }\\lbrace \\mathcal {a}_1, \\mathcal {a}_0, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0\\rbrace \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }\\lbrace \\mathcal {a}_1, \\mathcal {a}_1, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0\\rbrace \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }\\cdots \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }\\lbrace \\mathcal {a}_1, \\mathcal {a}_1, \\mathcal {a}_1, \\ldots , \\mathcal {a}_1\\rbrace \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }\\lbrace \\mathcal {a}_2, \\mathcal {a}_0, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0\\rbrace \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }\\lbrace \\mathcal {a}_2, \\mathcal {a}_1, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0\\rbrace \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }\\lbrace \\mathcal {a}_2, \\mathcal {a}_1, \\mathcal {a}_1, \\ldots , \\mathcal {a}_0\\rbrace \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }\\cdots $ .", "For example, if $\\hat{r}= \\lbrace \\mathcal {a}_k, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0\\rbrace $ , the number of elements from $\\hat{0}$ to $\\hat{r}$ are the combinations of $N$ elements taken $\\delta _{\\mathcal {a}}(\\mathcal {a}_k)= k$ at a time, with repetitions, i.e., $\\binom{\\delta _{\\mathcal {a}}(\\hat{r}_0) + N - 1}{N}$ .", "Now, if $\\hat{s}= \\lbrace \\mathcal {a}_k, \\mathcal {a}_m, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0\\rbrace $ , to count the number of elements from $\\hat{r}$ to $\\hat{s}$ , it should be considered subsets of $N-1$ elements, by dropping the first common element $\\mathcal {a}_k$ .", "Then, the number of elements from $\\hat{r}$ to $\\hat{s}$ is the number of elements from $\\hat{0}$ to $\\lbrace \\mathcal {a}_m, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0\\rbrace $ .", "They are the combinations of $N-1$ elements taken $\\delta _{\\mathcal {a}}(\\mathcal {a}_m) - 1= m -1$ at a time, with repetitions, i.e., $\\binom{\\delta _{\\mathcal {a}}(\\hat{r}_1) + N - 1}{N}$ .", "In general, the positive valuations with constant assignment to the join-irreducible elements can also be expressed as: $M(\\hat{r}) = k \\cdot \\sum _{j=1}^N \\binom{\\delta _{\\mathcal {a}}(\\hat{r}_j) + N - j}{N - j + 1}$ Once, it has been seen that $(R(N), \\wedge , \\vee , \\preceq )$ is a metric lattice and it has been determined all the possible metrics on $R(N)$ , it can be explored the metric properties of the IR evaluation measures defined on it.", "As $(R(N), \\preceq )$ is a totally ordered set, every IR evaluation measure is a valuation on this ordering.", "However, the generalized precision and recall are not isotone on this partial order.", "For instance, for three relevance degrees ($c=2$ ), when $N=4$ , $\\hat{r} = \\lbrace \\mathcal {a}_1, \\mathcal {a}_1, \\mathcal {a}_1, \\mathcal {a}_0 \\rbrace \\preceq \\lbrace \\mathcal {a}_2, \\mathcal {a}_0, \\mathcal {a}_0, \\mathcal {a}_0\\rbrace = \\hat{s}$ .", "However, $gP(\\hat{s}) = 0.25 < 0.375 = gP(\\hat{r})$ .", "The same result is valid for the generalized recall since it reaches the same values than the generalized precision.", "Thus, the generalized precision and recall are not pseudo-metrics on the projection+replacement [set-based] ordering, thus they cannot represent the natural distance.", "A simple inspection of the analytical expressions of $gP$ and $gR$ it is sufficient to see that they are not similar to Equation REF ." ], [ "Replacement [set-based]", "Any pair of system runs, $\\hat{r}, \\hat{s} \\in R(N)$ , such that $\\hat{r} \\ne \\hat{s}$ , is ordered by: $\\hat{r} \\preceq \\hat{s} \\Longleftrightarrow \\big \\vert \\lbrace i : \\hat{r}_i \\succcurlyeq \\mathcal {a}_j\\rbrace \\big \\vert \\le \\big \\vert \\lbrace i : \\hat{s}_i \\succcurlyeq \\mathcal {a}_j\\rbrace \\big \\vert , \\forall j \\in \\lbrace 0, \\ldots , c\\rbrace $ This approach considers a run greater than another if, fixing any relevance degree, it has more documents above that relevance degree than does the other.", "In [1], it is shown that this is a partial order and $(R(N), \\wedge , \\vee , \\preceq )$ is a distributive lattice.", "The following result determines the number of join-irreducible elements smaller than a judged run.", "Proposition 1 Considering $(R(N), \\wedge , \\vee , \\preceq )$ , where $\\preceq $ is the replacement [set-based] ordering and the indicator function on $REL$ , $\\delta _{\\mathcal {a}}$ , then, $\\big \\vert J_{\\hat{r}} \\big \\vert = \\sum _{i=1}^N \\delta _{\\mathcal {a}}(\\hat{r}_i)$ $(R(N), \\wedge , \\vee , \\preceq )$ is a metric lattice since it is distributive (see Section ).", "In the following, it will be determined all the possible isotoneThey are really positive since $R(N)$ is a finite distributive lattice.", "valuations on $R(N)$ .", "Considering an assignment, $w:J_{R(N)} \\longrightarrow \\mathbb {R}^{+}$ and the Equation REF ; then, any isotone valuation can be expressed as $M(\\hat{r})=\\sum _{\\hat{j} \\in J_{\\hat{r}}} w(\\hat{j})$ .", "Each of these isotone valuations, $M$ , generates a metric, $d_M$ , which coincides with the minimum path metric and is given by $d_{M}(\\hat{r}, \\hat{s}) = M(\\hat{r}) + M(\\hat{s}) - 2 \\cdot M(\\hat{r} \\wedge \\hat{s})$ , $\\forall \\hat{r}$ , $\\hat{s} \\in R(N)$ .", "If $w(\\hat{j}) = k$ , $\\forall \\hat{j} \\in J_{R(N)}$ , with $k \\in \\mathbb {R}^{+}$ , then the natural distance on $(R(N), \\preceq )$ is $d_{M}(\\hat{r}, \\hat{s}) = k \\cdot \\big \\vert J_{\\hat{s}} \\triangle J_{\\hat{r}} \\big \\vert $ and, by Proposition REF , the isotone valuations are expressed as: $M(\\hat{r}) = k \\cdot \\sum _{i=1}^N \\delta _{\\mathcal {a}}(\\hat{r}_i)$ Once, it has been seen that $(R(N), \\wedge , \\vee , \\preceq )$ is a metric lattice and it has been determined all the possible metrics on $R(N)$ , it can be explored the metric properties of the IR evaluation measures defined on this partial order.", "In [1], it was shown that the generalized precision and recall are valuations on the replacement [rank-based] ordering, the following result shows that they are positive.", "Proposition 2 The generalized precision and recall are positive valuations, when they are defined in $(R(N), \\preceq )$ with the replacement [set-based] ordering.", "As the the generalized precision, $gP$ , and recall, $gR$ , are positive valuations, then they are metrics on this partial order.", "In addition, they have a constant assignment to the join-irreducible elements since they can be obtained from Equation REF , when $k= \\frac{1}{N \\cdot \\mathcal {a}_c}$ and $k= \\frac{1}{RB \\cdot \\mathcal {a}_c}$ respectively.", "Thus, they are the natural distance metric on the replacement [set-based] partial order." ], [ "Rank-based Retrieval", "In this Section, the rank-based retrieval is considered and the three orderings which were deduced in [1] are analysed." ], [ "Projection+Replacement [rank-based]", "Let $\\hat{r}$ , $\\hat{s} \\in R(N)$ such that $\\hat{r} \\ne \\hat{s}$ , then there exists $k = min \\lbrace j \\le N : \\hat{r}[j] \\ne \\hat{s}[j] \\rbrace $ .", "Any pair of distinct system runs is ordered by: $\\hat{r} \\preceq \\hat{s} \\Longleftrightarrow \\hat{r}[k] \\preccurlyeq \\hat{s}[k]$ A system run is preferred to another if it has a higher relevance degree in the highest rank position at which the two system runs differ.", "$(R(N), \\preceq )$ is a totally ordered set [1].", "As $(R(N), \\preceq )$ is a totally ordered set [1], then every element is join-irreducible, except $\\hat{0}$ , i.e., $J_{R(N)} = R(N) \\setminus \\lbrace \\hat{0}\\rbrace $ .", "Thus, $J_{\\hat{r}}$ is the set of all the elements smaller than $\\hat{r}$ , except $\\hat{0}$ .", "$(R(N), \\wedge , \\vee , \\preceq )$ is a metric lattice since it is a chain (see Section ).", "In the following, it will be determined all the possible isotoneThey are really positive since $R(N)$ is a finite distributive lattice.", "valuations on $R(N)$ .", "Consider an assignment, $w:R(N) \\setminus \\lbrace \\hat{0}\\rbrace \\longrightarrow \\mathbb {R}^{+}$ and a judged run $\\hat{r} \\in R(N) \\setminus \\lbrace \\hat{0}\\rbrace $ , it can be assumed that $\\hat{0} =\\hat{r}^{0} \\preceq \\hat{r}^{1} \\preceq \\ldots \\preceq \\hat{r}^{k} = \\hat{r}$ .", "As every totally ordered set is a distributive lattice, then, by Equation REF , any isotone valuation can be obtained as: $M(\\hat{r}) = \\sum _{i = 1}^{k} w(\\hat{r}^{i})$ Considering $\\hat{r}$ , $\\hat{s} \\in R(N)$ , each of these isotone valuations, $M$ , generates a metric, $d_M$ , which coincides with the minimum path metric and is given by $d_{M}(\\hat{r}, \\hat{s}) = M(\\hat{r}) + M(\\hat{s}) - 2 \\cdot M(\\hat{r} \\wedge \\hat{s})$ .", "As $R(N)$ , endowed with the projection+replacement [rank-based] ordering, is a totally ordered set; then, it can be assumed that $\\hat{r} \\preceq \\hat{s}$ , i.e., $\\hat{r} \\wedge \\hat{s} = \\hat{r}$ .", "Therefore, $d_{M}(\\hat{r}, \\hat{s}) = M(\\hat{s}) - M(\\hat{r})$ .", "If $w(\\hat{r}) = k$ , $\\forall \\hat{r} \\in R(N) \\setminus \\lbrace \\hat{0}\\rbrace $ , with $k \\in \\mathbb {R}^{+}$ , then these valuations are expressed as $M(\\hat{r}) = k \\cdot \\big \\vert J_{\\hat{r}} \\big \\vert $ and the natural distance on the covering graph is $d_{M}(\\hat{r}, \\hat{s}) = k \\cdot \\big \\vert J_{\\hat{s}} \\triangle J_{\\hat{r}} \\big \\vert $ .", "Thus, the isotone valuations, $M(\\hat{r})$ , with constant assignment to the join-irreducible elements count the number of elements from $\\hat{0}$ to $\\hat{r}$ .", "It can be given an alternative expression of $M(\\hat{r})$ .", "Considering the binary case ($c=1$ ), the projection+replacement [rank-based] ordering is the following: $(\\mathcal {a}_0, \\mathcal {a}_0, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0) \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }(\\mathcal {a}_1, \\mathcal {a}_0, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0) \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }(\\mathcal {a}_0, \\mathcal {a}_1, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0) \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }(\\mathcal {a}_1, \\mathcal {a}_1, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0) \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }(\\mathcal {a}_0, \\mathcal {a}_0, \\mathcal {a}_1, \\ldots , \\mathcal {a}_0) \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }(\\mathcal {a}_1, \\mathcal {a}_0, \\mathcal {a}_1, \\ldots , \\mathcal {a}_0) \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }(\\mathcal {a}_1, \\mathcal {a}_1, \\mathcal {a}_1, \\ldots , \\mathcal {a}_0) \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }\\cdots $ .", "That is, they are the sequence of the ordinal numbers expressed in base 2.", "In general, the system runs of this ordering can be represented by the conversion in base 10 of the numbers in base $c+1$ identified by $\\delta _{\\mathcal {a}}(\\hat{r})$ , and the ordering among system runs, $\\preceq $ , corresponds to the ordering, $\\le $ , among numbers in base $c+1$ .", "Thus, the positive valuations with constant assignment to the join-irreducible elements can also be expressed as: $M(\\hat{r}) = k \\cdot \\sum _{j=1}^N \\delta _{\\mathcal {a}}(\\hat{r}[j]) \\cdot (c+1)^{N-j}$ Once, it has been seen that $(R(N), \\wedge , \\vee , \\preceq )$ is a metric lattice and it has been determined all the possible metrics on $R(N)$ , it can be explored the metric properties of the IR evaluation measures defined on this partial order.", "As $R(N)$ is a totally ordered set, every IR evaluation measure is a valuation on this ordering.", "However, the generalized precision and recall are not isotone, for instance, in the binary case ($c=1$ ), for $N=4$ , it can be easily checked that $\\hat{r} = (\\mathcal {a}_0, \\mathcal {a}_0, \\mathcal {a}_1, \\mathcal {a}_1) \\preceq (\\mathcal {a}_1, \\mathcal {a}_0, \\mathcal {a}_0, \\mathcal {a}_0) = \\hat{s}$ .", "However, $gP(\\hat{s}) = 0.25 < 0.5 = gP(\\hat{r})$ .", "The same result is valid for the generalized recall, since, in the binary case, it has the same values than the generalized precision.", "On the other hand, the average precision is not an isotone valuation on the projection+replacement [rank-based] ordering, considering the same example of the previous paragraph, it can be checked that $AP(\\hat{s}) = 0.125 < 0.208 = AP(\\hat{r})$ .", "The following result shows an interesting property of the $RBP$ evaluation measure.", "Proposition 3 (M. Ferrante [21]) Considering the ordering projection+replacement [rank-based], $RBP$ is an isotone valuation if and only if $p \\le G/(G + 1)$ , where $G= \\min _{j \\in \\lbrace 1, \\ldots , c\\rbrace } (g(\\mathcal {a}_j) - g(\\mathcal {a}_{j-1})) / g(\\mathcal {a}_c)$ is the normalized smallest gap between the gain of two consecutive relevance degrees.", "Finally, $DCG_b$ is not an isotone valuation with this partial ordering.", "For instance, in the binary case ($c=1$ ), when $N=4$ ; then, $\\hat{r} = (\\mathcal {a}_0, \\mathcal {a}_0, \\mathcal {a}_1, \\mathcal {a}_1) \\preceq (\\mathcal {a}_0, \\mathcal {a}_1, \\mathcal {a}_0, \\mathcal {a}_0) = \\hat{s}$ .", "However, $DCG_2(\\hat{s}) = 1.0 < 1.131 = DCG_2(\\hat{r})$ .", "In all the cases where the IR evaluation measures are not isotone valuations, it follows that they are not pseudo-metrics on the projection+replacement [rank-based] ordering, thus they cannot represent the natural distance.", "The isotone valuation $RBP$ with the above mentioned restrictions is a pseudo-metric, which is not the natural distance, since it can not be expressed in terms of the Equation REF ." ], [ "Replacement [rank-based]", "Any pair of system runs, $\\hat{r}, \\hat{s} \\in R(N)$ , such that $\\hat{r} \\ne \\hat{s}$ , is ordered by: $\\hat{r} \\preceq \\hat{s} \\Longleftrightarrow \\hat{r}[k] \\preccurlyeq \\hat{s}[k], \\ \\ \\forall k \\in \\lbrace 1, \\ldots , N\\rbrace $ A run is greater than another if, for each rank position, it has higher relevance degrees.", "In [1], it is shown that this is a partial order and $(R(N), \\wedge , \\vee , \\preceq )$ is a distributive lattice.", "The following result determines the number of join-irreducible elements smaller than a judged run.", "Proposition 4 Considering $(R(N), \\wedge , \\vee , \\preceq )$ , where $\\preceq $ is the replacement [rank-based] ordering and the indicator function on $REL$ , $\\delta _{\\mathcal {a}}$ , then, $\\big \\vert J_{\\hat{r}} \\big \\vert = \\sum _{i=1}^N \\delta _{\\mathcal {a}}(\\hat{r}[i])$ $(R(N), \\wedge , \\vee , \\preceq )$ is a metric lattice since it is distributive (see Section ).", "In the following, it will be determined all the possible isotoneThey are really positive since $R(N)$ is a finite distributive lattice.", "valuations on $R(N)$ .", "Considering an assignment, $w:J_{R(N)} \\longrightarrow \\mathbb {R}^{+}$ and the Equation REF , any positive valuation can be expressed as $M(\\hat{r})=\\sum _{\\hat{j} \\in J_{\\hat{r}}} w(\\hat{j})$ .", "Each of these isotone valuations, $M$ , generates a metric, $d_M$ , which coincides with the minimum path metric and is given by $d_{M}(\\hat{r}, \\hat{s}) = M(\\hat{r}) + M(\\hat{s}) - 2 \\cdot M(\\hat{r} \\wedge \\hat{s})$ , $\\forall \\hat{r}$ , $\\hat{s} \\in R(N)$ .", "If $w(\\hat{j}) = k$ , $\\forall \\hat{j} \\in J_{R(N)}$ , with $k \\in \\mathbb {R}^{+}$ , then the natural distance on $(R(N), \\preceq )$ is $d_{M}(\\hat{r}, \\hat{s}) = k \\cdot \\big \\vert J_{\\hat{s}} \\triangle J_{\\hat{r}} \\big \\vert $ and, by Proposition REF , the isotone valuations are expressed as: $M(\\hat{r}) = k \\cdot \\sum _{i=1}^N \\delta _{\\mathcal {a}}(\\hat{r}_i)$ Once, it has been seen that $(R(N), \\wedge , \\vee , \\preceq )$ is a metric lattice and it has been determined all the possible metrics on $R(N)$ , it can be explored the metric properties of the IR evaluation measures defined on this partial order.", "In [1], it was shown that the generalized precision and recall are valuations on the replacement [rank-based] ordering, the following result shows that they are positive.", "Proposition 5 The generalized precision and recall are positive valuations, when they are defined in $(R(N), \\preceq )$ with the replacement [rank-based] ordering.", "As the the generalized precision, $gP$ , and recall, $gR$ , are positive valuations, then they are metrics on this partial order.", "In addition, they have a constant assignment to the join-irreducible elements since they can be obtained from Equation REF , when $k= \\frac{1}{N \\cdot \\mathcal {a}_c}$ and $k= \\frac{1}{RB \\cdot \\mathcal {a}_c}$ respectively.", "Thus, they are the natural distance metric on the replacement [rank-based] partial order.", "Similar results are obtained for $RBP$ and $DCG_b$ .", "In [1], it was shown that these IR evaluation measures are valuations on the replacement [rank-based] ordering, the following result shows that they are positive.", "Proposition 6 $RBP$ and $DCG_b$ are positive valuations, when they are defined in $(R(N), \\preceq )$ with the Replacement [rank-based] ordering.", "As $RBP$ and $DCG_b$ are positive valuations, then they are metrics on this partial order.", "However, they have not a constant assignment to the join-irreducible elements.", "Thus, they are not the natural distance metric on the replacement [rank-based] partial order." ], [ "Swapping+Replacement [rank-based]", "Given two system runs $\\hat{r}$ , $\\hat{s} \\in R(N)$ , then $\\hat{r} &\\preceq \\hat{s} & &\\Longleftrightarrow & \\big \\vert \\lbrace i \\le k : \\hat{r}[i] \\succcurlyeq \\mathcal {a}_j\\rbrace \\big \\vert &\\le \\big \\vert \\lbrace i \\le k : \\hat{s}[i] \\succcurlyeq \\mathcal {a}_j\\rbrace \\big \\vert & \\\\& & & & \\forall j \\in \\lbrace 0, \\ldots , c\\rbrace , & \\forall k \\in \\lbrace 1, \\ldots , N\\rbrace &$ A run is considered larger than another one when, for each rank position, it has more relevant documents than the other up to that rank for every relevance degree.", "$(R(N), \\wedge , \\vee , \\preceq )$ is a finite lattice, which contains a $N_5$ sublattice [1], i.e., it is not a modular lattice [13].", "By the results of Section , $(R(N), \\wedge , \\vee , \\preceq )$ is not a metric lattice.", "In this case, there cannot exists a positive valuation defined on $R(N)$ .", "As it was noted in [1], there does not exist the natural distance on the covering graph of $R(N)$ .", "However, there may exist isotone valuations defined on $R(N)$ , which are weaker valuations.", "In these case, the isotone valuation generates a pseudo-metric, which satisfy the same postulates of the metrics, except $d_M(\\hat{r},\\hat{s})=0 \\Leftrightarrow \\hat{r}=\\hat{s}$ .", "Once, it has been determined the metric properties of $(R(N), \\wedge , \\vee , \\preceq )$ , it can be determined the metric properties of the IR evaluation measures defined on this partial order.", "The generalized precision is not a positive valuation, for instance, in the binary case for $N=3$ , $\\hat{r} = (\\mathcal {a}_1, \\mathcal {a}_0, \\mathcal {a}_1) \\preceq (\\mathcal {a}_1, \\mathcal {a}_1, \\mathcal {a}_0) = \\hat{s}$ and $\\hat{r} \\ne \\hat{s}$ , when considering the swapping property at the two last positions.", "However, $gP(\\hat{r}) = 0.22 = gP(\\hat{s})$ .", "The same result is valid for the generalized recall since it reaches the same values than the generalized precision.", "However, the following results show that the IR evaluation measures are isotone valuations on $(R(N), \\wedge , \\vee , \\preceq )$ .", "Proposition 7 The generalized precision and recall are isotone valuations, when they are defined on $(R(N), \\wedge , \\vee , \\preceq )$ with the Swapping+Replacement [rank-based] ordering.", "Proposition 8 $RBP$ and $DCG_b$ are isotone valuations, when they are defined on $(R(N), \\wedge , \\vee , \\preceq )$ with the Swapping+Replacement [rank-based] ordering.", "These results indicate that $(R(N), \\wedge , \\vee , \\preceq )$ is a pseudo-metric lattice, where the generalized precision and recall, $RBP$ or $DCG_b$ are pseudo-metrics." ], [ "Related Work", "The formal analysis of IR evaluation metrics has contributed to a better understanding of their properties.", "One of the first attempts is given in [26], where the effectiveness and the efficiency of IR evaluation measures are analysed in terms of a 2-by-2 contingency table of pertinence and retrieval.", "Later, van Rijsbergen [27], [28], tackle the issue of the foundations of measurement in IR through a conjoint (additive) structure based on precision and recall; then, he examines the properties of a measure on this prec-recall structure.", "In [29], a similar conjoint structure is defined, but on the contingency table of the binary retrieval; then, they study the properties of the proposed MZ-metric.", "In [30], user judgements on documents are formally described by a weak order, then, a measure of system performance is presented, whose appropriateness is demonstrated through an axiomatic approach.", "In [31], a framework for the theoretical comparison of IR models, based on situation theory, is presented.", "It allows an inference mechanism with the axiomatised concept of aboutness.", "In [32], [33] an axiomatic definition of IR effectiveness metric is provided, in a unifying framework for ranking, classification and clustering.", "In [34], it is discussed what properties should enjoy an evaluation measure for quantification.", "In the same formal research line, some works have analysed desirable properties that any suitable IR evaluation measure should satisfy.", "The three metric postulates, studied in this paper, are implicitly considered in some of these properties or axioms.", "In [35], the axioms that an evaluation measure for classification should satisfy are discussed, then the K evaluation metric is proposed.", "In [10], the evaluation measures are characterized by seven numeric properties, such as convergence, top-weightedness and localization, which provides a framework to classify the measures according to their effectiveness.", "In [6], [11], [22], the properties of priority, deepness, closeness threshold, confidence, etc., are stated to derive evaluation measures, which can be applied to several information access tasks.", "Recently, in [1], three of these ranking axioms are considered; then, the possible orderings in the set of rankings of documents are derived.", "One of the closest works to the present is [9], where two axioms are considered: the monotonicity, also known as the homogeneity law in measurement theory and the Archimedian, then they determine explicit expressions of the evaluation measures in the binary retrieval, by supporting the results on measurement theory.", "However, this paper considers ranking axioms, which are directly related to preferences on retrieval measures.", "In addition, it is grounded on lattice theory and the results are generalized to multi-grade relevance.", "The other close work is [21], where a theory of IR evaluation measures, based on the representational theory of measurements, is developed to determine whether and when IR measures are interval scales.", "Some orderings on the set of rankings are derived from two operations: replacement and swap; then, the results are proven by a common generic procedure, using the notion of interval and differences.", "However, this work is based on ranking orderings derived from three operations, which introduces a new ordering.", "In addition, the concepts of metric and pseudo-metric are less restrictive than the concepts of ordinal and interval scales, thus the results of this paper generalize some results of [21]." ], [ "Conclusions", "The properties of IR evaluation measures do not depend only on its analytical expression.", "The empirical domain where they are defined play an important role.", "This set can be characterized by desirable properties that any reasonable IR evaluation measure should satisfy.", "These properties lead to different orderings in the set of ranked lists of documents [1].", "This paper checks if the set of rankings endowed with these orderings are metric or pseudo-metric spaces and determine the possible metrics and pseudo-metrics defined on it.", "There are shown explicit expressions of these mappings, in terms of a remarkable subset of system runs: the join-irreducible elements.", "It is found that, when the relevant documents are prioritised, precision, recall, $RBP$ and $DCG$ are metrics on the set of rankings, however they are pseudo-metrics when the swapping of documents is considered.", "Thus, it has no sense to state that “a specific IR evaluation measure is a metric or pseudo-metric”, since it has to be considered the empirical domain where it is defined.", "The use of evaluation measures is a common practice to evaluate, benchmark and track the effectiveness of IR systems.", "Then, a natural question arises: which measure should be chosen?", "A (partial) answer to this question is given by the formal analysis of their metric properties, more specifically, by their axiomatic characterization.", "Some works have characterised when IR evaluation measures are ordinal scales ($a \\ge b \\Leftrightarrow f(a) \\ge f(b)$ ) [12] or interval scales [21], in the orderings of [1].", "Here, these results have been generalised, by determining the pseudo-metrics, or equivalently, the valuations that verify: $a \\ge b \\Rightarrow f(a) \\ge f(b)$ ." ], [ "Appendix", "Proposition REF : Considering $(R(N), \\wedge , \\vee , \\preceq )$ , where $\\preceq $ is the replacement [set-based] ordering and the indicator function on $REL$ , $\\delta _{\\mathcal {a}}$ , then, $ \\big \\vert J_{\\hat{r}} \\big \\vert = \\sum _{i=1}^N \\delta _{\\mathcal {a}}(\\hat{r}_i) $ Consider the convention to represent $\\hat{r}$ as noted in Section ; then, it can be assumed that $\\hat{r}=\\lbrace \\mathcal {a}_h, \\mathcal {a}_j, \\mathcal {a}_k, \\ldots \\rbrace $ , where $0 \\le \\cdots \\le k \\le j \\le h \\le c$ .", "In [1], it is shown that the joint-irreducible elements of $R(N)$ with the replacement [set-based] partial ordering, are: $ \\begin{matrix}\\lbrace \\mathcal {a}_1, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0 \\rbrace , & \\lbrace \\mathcal {a}_1, \\mathcal {a}_1, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0 \\rbrace , & \\ldots , & \\lbrace \\mathcal {a}_1, \\ldots , \\mathcal {a}_1 \\rbrace , \\\\\\lbrace \\mathcal {a}_2, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0 \\rbrace , & \\lbrace \\mathcal {a}_2, \\mathcal {a}_2, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0 \\rbrace , & \\ldots , & \\lbrace \\mathcal {a}_2, \\ldots , \\mathcal {a}_2 \\rbrace , \\\\\\vdots & \\vdots & \\vdots & \\vdots \\\\\\lbrace \\mathcal {a}_c, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0 \\rbrace , & \\lbrace \\mathcal {a}_c, \\mathcal {a}_c, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0 \\rbrace , & \\ldots , & \\lbrace \\mathcal {a}_c, \\ldots , \\mathcal {a}_c \\rbrace \\end{matrix}$ On the other hand, the subset of join-irreducible elements smaller that $\\hat{r}$ , i.e., $J_{\\hat{r}}$ , can be seen as the union of the following subsets of join-irreducible elements: (i) there are $h = \\delta _{\\mathcal {a}}(\\hat{r}_1)$ join-irreducible elements corresponding to the first judged document, $\\mathcal {a}_h$ , of $\\hat{r}$ : $\\lbrace \\mathcal {a}_1, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0 \\rbrace , \\lbrace \\mathcal {a}_2, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0 \\rbrace , \\ldots , \\lbrace \\mathcal {a}_h, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0 \\rbrace $ .", "Note that by definition of the partial order, $\\preceq $ , these elements are smaller than $\\hat{r}$ ; (ii) there are $j = \\delta _{\\mathcal {a}}(\\hat{r}_2)$ join-irreducible elements corresponding to the second judged document, $\\mathcal {a}_j$ , of $\\hat{r}$ : $\\lbrace \\mathcal {a}_1, \\mathcal {a}_1, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0 \\rbrace , \\lbrace \\mathcal {a}_2, \\mathcal {a}_2, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0 \\rbrace , \\ldots , \\lbrace \\mathcal {a}_j, \\mathcal {a}_j, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0 \\rbrace $ .", "Note that by definition of the partial order, $\\preceq $ , these elements are smaller than $\\hat{r}$ ; (iii) and so on.", "Now, suppose that there exists a join-irreducible element smaller than $\\hat{r}$ and different from the elements listed in the previous paragraph; then, inspecting the form of the possible join-irreducible elements in Equation REF , there exists a join-irreducible, $\\hat{m} = \\lbrace \\mathcal {a}_m, \\ldots , \\mathcal {a}_m, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0\\rbrace $ , such that $\\hat{m} \\preceq \\hat{r}$ and $p$ is the largest position of $\\hat{m}$ where the judged document is classified as $\\mathcal {a}_m$ .", "It can be assumed that $\\hat{r} = \\lbrace \\mathcal {a}_h, \\mathcal {a}_j, \\mathcal {a}_k, \\ldots , \\mathcal {a}_n, \\ldots \\rbrace $ , where $\\mathcal {a}_n$ is placed at position $p$ , $\\mathcal {a}_n \\preccurlyeq \\mathcal {a}_m$ and $\\mathcal {a}_n \\ne \\mathcal {a}_m$ .", "By Theorem REF , in a finite distributive lattice, every element can be expressed in a unique manner as a join of the join-irreducible elements.", "Note that these join-irreducible elements are smaller than the element since it is join of them.", "As $(R(N), \\wedge , \\vee , \\preceq )$ is a distributive lattice with the replacement [set-based] partial order, thus, $\\hat{r} = \\bigvee J_{\\hat{r}} \\vee \\hat{m}$ .", "In addition, in [1], it is shown that the join operation in the replacement [set-based] partial order is the maximum component wise.", "However, this contradicts the last equality since the judged document of $\\hat{r}$ at position $p$ is $\\mathcal {a}_n$ and the judged document of $\\bigvee J_{\\hat{r}} \\vee \\hat{m}$ at position $p$ is $\\mathcal {a}_m$ .", "Therefore, the elements listed bellow, as subsets of $J_{\\hat{r}}$ , are the join-irreducible elements smaller than $\\hat{r}$ and it holds: $ \\big \\vert J_{\\hat{r}} \\big \\vert = \\sum _{i=1}^N \\delta _{\\mathcal {a}}(\\hat{r}_i) $ Proposition REF : The generalized precision and recall are positive valuations, when they are defined in $(R(N), \\preceq )$ with the replacement [set-based] ordering.", "The generalized precision will be a positive valuation if the implication: $\\hat{r} \\preceq \\hat{s}$ and $\\hat{r} \\ne \\hat{s} \\Rightarrow gP(\\hat{r}) < gP(\\hat{s})$ holds true.", "In [1], it is shown that the covering relation of the replacement [set-based] ordering is as follows: given a pair of system runs, $\\hat{r}, \\hat{s} \\in R(N)$ with $\\hat{r} \\ne \\hat{s}$ , then $\\hat{r} \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }\\hat{s}$ $\\Leftrightarrow $ $\\exists k : \\hat{r}_k = \\mathcal {a}_i, \\hat{s}_k = \\mathcal {a}_{i+1}$ and $\\hat{r}_j = \\hat{s}_j, \\forall j \\ne k$ .", "Thus, if $\\hat{r} \\preceq \\hat{s}$ and $\\hat{r} \\ne \\hat{s}$ , then there exists at least $k$ , such that $\\hat{r}_k \\prec \\hat{s}_k$ and $\\hat{r}_j \\preccurlyeq \\hat{s}_j, \\forall j \\ne k$ .", "Considering that $g$ is an strictly increasing function, it follows: $\\hat{r} \\preceq \\hat{s},\\ \\ \\hat{r} \\ne \\hat{s} \\Longrightarrow \\sum _{i=1}^N g(\\hat{r}_i) < \\sum _{i=1}^N g(\\hat{s}_i) \\ .$ As $N$ and $g(\\mathcal {a}_c)$ are constants, $gP(\\hat{r}) = \\frac{1}{N} \\cdot \\sum _{i=1}^{N} \\frac{g(\\hat{r}_i)}{g(\\mathcal {a}_{c})} < \\frac{1}{N} \\cdot \\sum _{i=1}^{N} \\frac{g(\\hat{s}_i)}{g(\\mathcal {a}_{c})} = gP(\\hat{s}) \\ .$ Similarly, as $RB$ is constant, $gR(\\hat{r}) = \\frac{1}{RB} \\cdot \\sum _{i=1}^{N} \\frac{g(\\hat{r}_i)}{g(\\mathcal {a}_{c})} < \\frac{1}{RB} \\cdot \\sum _{i=1}^{N} \\frac{g(\\hat{s}_i)}{g(\\mathcal {a}_{c})} = gR(\\hat{s})$ Proposition REF : Considering $(R(N), \\wedge , \\vee , \\preceq )$ , where $\\preceq $ is the replacement [rank-based] ordering and the indicator function on $REL$ , $\\delta _{\\mathcal {a}}$ , then, $ \\big \\vert J_{\\hat{r}} \\big \\vert = \\sum _{i=1}^N \\delta _{\\mathcal {a}}(\\hat{r}[i]) $ It can be assumed that $\\hat{r}=(\\mathcal {a}_h, \\mathcal {a}_j, \\mathcal {a}_k, \\ldots )$ , where $\\mathcal {a}_h$ , $\\mathcal {a}_j$ , $\\mathcal {a}_k$ , $\\ldots \\in REL$ .", "In [1], it is shown that the joint-irreducible elements of $R(N)$ with the replacement [rank-based] partial ordering, are: $ \\begin{matrix}(\\mathcal {a}_1, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0 ), & (\\mathcal {a}_0, \\mathcal {a}_1, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0 ), & \\ldots , & (\\mathcal {a}_0, \\mathcal {a}_0, \\ldots , \\mathcal {a}_1), \\\\(\\mathcal {a}_2, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0 ), & (\\mathcal {a}_0, \\mathcal {a}_2, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0 ), & \\ldots , & (\\mathcal {a}_0, \\mathcal {a}_0, \\ldots , \\mathcal {a}_2 ), \\\\\\vdots & \\vdots & \\vdots & \\vdots \\\\(\\mathcal {a}_c, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0 ), & (\\mathcal {a}_0, \\mathcal {a}_c, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0), & \\ldots , & (\\mathcal {a}_0, \\mathcal {a}_0, \\ldots , \\mathcal {a}_c)\\end{matrix}$ On the other hand, the subset of join-irreducible elements smaller that $\\hat{r}$ , i.e., $J_{\\hat{r}}$ , can be seen as the union of the following subsets of join-irreducible elements: (i) there are $h = \\delta _{\\mathcal {a}}(\\hat{r}[1])$ join-irreducible elements corresponding to the first judged document, $\\mathcal {a}_h$ , of $\\hat{r}$ : $(\\mathcal {a}_1, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0 ), (\\mathcal {a}_2, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0), \\ldots , (\\mathcal {a}_h, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0)$ .", "Note that by definition of the partial order, $\\preceq $ , these elements are smaller than $\\hat{r}$ ; (ii) there are $j = \\delta _{\\mathcal {a}}(\\hat{r}[2])$ join-irreducible elements corresponding to the second judged document, $\\mathcal {a}_j$ , of $\\hat{r}$ : $(\\mathcal {a}_0, \\mathcal {a}_1, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0), (\\mathcal {a}_0, \\mathcal {a}_2, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0), \\ldots , (\\mathcal {a}_0, \\mathcal {a}_j, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0)$ .", "Note that by definition of the partial order, $\\preceq $ , these elements are smaller than $\\hat{r}$ ; (iii) and so on.", "Now, suppose that there exists a join-irreducible element smaller than $\\hat{r}$ and different from the elements listed in the previous paragraph; then, inspecting the form of the possible join-irreducible elements in Equation REF , there exists a join-irreducible, $\\hat{m} = (\\mathcal {a}_0, \\ldots ,\\mathcal {a}_0, \\mathcal {a}_m, \\mathcal {a}_0, \\ldots , \\mathcal {a}_0)$ , such that $\\hat{m} \\preceq \\hat{r}$ and $p$ is the position of $\\mathcal {a}_m$ .", "It can be assumed that $\\hat{r} = (\\mathcal {a}_h, \\mathcal {a}_j, \\mathcal {a}_k, \\ldots , \\mathcal {a}_n, \\ldots )$ , where $\\mathcal {a}_n$ is placed at position $p$ , $\\mathcal {a}_n \\preccurlyeq \\mathcal {a}_m$ and $\\mathcal {a}_n \\ne \\mathcal {a}_m$ .", "By Theorem REF , in a finite distributive lattice, every element can be expressed in a unique manner as a join of the join-irreducible elements.", "Note that these join-irreducible elements are smaller than the element since it is join of them.", "As $(R(N), \\wedge , \\vee , \\preceq )$ is a distributive lattice with the replacement [rank-based] partial order, thus, $\\hat{r} = \\bigvee J_{\\hat{r}} \\vee \\hat{m}$ .", "In addition, in [1], it is shown that the join operation in the replacement [rank-based] partial order is the maximum coordinate wise.", "However, this contradicts the last equality since the judged document of $\\hat{r}$ at position $p$ is $\\mathcal {a}_n$ and the judged document of $\\bigvee J_{\\hat{r}} \\vee \\hat{m}$ at position $p$ is $\\mathcal {a}_m$ .", "Therefore, the elements listed bellow, as subsets of $J_{\\hat{r}}$ , are the join-irreducible elements smaller than $\\hat{r}$ and it holds: $ \\big \\vert J_{\\hat{r}} \\big \\vert = \\sum _{i=1}^N \\delta _{\\mathcal {a}}(\\hat{r}_i) $ Proposition REF : The generalized precision and recall are positive valuations, when they are defined in $(R(N), \\preceq )$ with the replacement [rank-based] ordering.", "The generalized precision will be a positive valuation if the implication: $\\hat{r} \\preceq \\hat{s}$ and $\\hat{r} \\ne \\hat{s} \\Rightarrow gP(\\hat{r}) < gP(\\hat{s})$ holds true.", "In [1], it is shown that the covering relation of the replacement [rank-based] ordering is as follows: given a pair of system runs, $\\hat{r}, \\hat{s} \\in R(N)$ with $\\hat{r} \\ne \\hat{s}$ , then $\\hat{r} \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }\\hat{s} \\Leftrightarrow \\exists k : \\hat{r}[k] = \\mathcal {a}_i, \\hat{s}[k] = \\mathcal {a}_{i+1}$ and $\\hat{r}[j] = \\hat{s}[j], \\forall j \\ne k$ .", "Thus, if $\\hat{r} \\preceq \\hat{s}$ and $\\hat{r} \\ne \\hat{s}$ , then there exists at least $k$ , such that $\\hat{r}[k] \\prec \\hat{s}[k]$ and $\\hat{r}[j] \\preccurlyeq \\hat{s}[j], \\forall j \\ne k$ .", "Considering that $g$ is an strictly increasing function, it follows: $\\hat{r} \\preceq \\hat{s},\\ \\ \\hat{r} \\ne \\hat{s} \\Longrightarrow \\sum _{i=1}^N g(\\hat{r}[i]) < \\sum _{i=1}^N g(\\hat{s}[i]) \\ .$ As $N$ and $g(\\mathcal {a}_c)$ are constants, $gP(\\hat{r}) = \\frac{1}{N} \\cdot \\sum _{i=1}^{N} \\frac{g(\\hat{r}[i])}{g(\\mathcal {a}_{c})} < \\frac{1}{N} \\cdot \\sum _{i=1}^{N} \\frac{g(\\hat{s}[i])}{g(\\mathcal {a}_{c})} = gP(\\hat{s}) \\ .$ Similarly, since $RB$ is constant, $gR(\\hat{r}) = \\frac{1}{RB} \\cdot \\sum _{i=1}^{N} \\frac{g(\\hat{r}[i])}{g(\\mathcal {a}_{c})} < \\frac{1}{RB} \\cdot \\sum _{i=1}^{N} \\frac{g(\\hat{s}[i])}{g(\\mathcal {a}_{c})} = gR(\\hat{s})$ Proposition REF : $RBP$ and $DCG_b$ are positive valuations, when they are defined in $(R(N), \\preceq )$ with the Replacement [rank-based] ordering.", "$RBP$ will be a positive valuation if the implication: $\\hat{r} \\preceq \\hat{s}$ and $\\hat{r} \\ne \\hat{s} \\Rightarrow gP(\\hat{r}) < gP(\\hat{s})$ holds true.", "In [1], it is shown that the covering relation of the replacement [rank-based] ordering is as follows: given a pair of system runs, $\\hat{r}, \\hat{s} \\in R(N)$ with $\\hat{r} \\ne \\hat{s}$ , then $\\hat{r} \\prec \\mathrel {\\hspace{-2.77771pt}}\\mathrel {\\cdot }\\hat{s} \\Leftrightarrow \\exists k : \\hat{r}[k] = \\mathcal {a}_i, \\hat{s}[k] = \\mathcal {a}_{i+1}$ and $\\hat{r}[j] = \\hat{s}[j], \\forall j \\ne k$ .", "Thus, if $\\hat{r} \\preceq \\hat{s}$ and $\\hat{r} \\ne \\hat{s}$ , then there exists at least $k$ , such that $\\hat{r}[k] \\prec \\hat{s}[k]$ and $\\hat{r}[j] \\preccurlyeq \\hat{s}[j], \\forall j \\ne k$ .", "Considering that $g$ is an strictly increasing function, it follows: $\\hat{r} \\preceq \\hat{s},\\ \\ \\hat{r} \\ne \\hat{s} \\Longrightarrow \\sum _{i=1}^N g(\\hat{r}[i]) < \\sum _{i=1}^N g(\\hat{s}[i]) \\ .$ Note that if $g(\\hat{r}[i]) < g(\\hat{s}[i])$ , then $p^{i-1} \\cdot g(\\hat{r}[i]) < p^{i-1} \\cdot g(\\hat{s}[i])$ , $\\forall i \\in \\lbrace 1, \\ldots , N\\rbrace $ , then, $gRBP(\\hat{r}) = \\frac{(1-p)}{g(\\mathcal {a}_c)} \\cdot \\sum _{i=1}^N p^{i-1} \\cdot g(\\hat{r}[i]) < \\frac{(1-p)}{g(\\mathcal {a}_c)} \\cdot \\sum _{i=1}^N p^{i-1} \\cdot g(\\hat{s}[i]) = gRBP(\\hat{s}) \\ .$ And similarly, if $g(\\hat{r}[i]) < g(\\hat{s}[i])$ , then $\\frac{g(\\hat{r[i]})}{\\max \\lbrace 1, \\log _{b}i \\rbrace } < \\frac{g(\\hat{s[i]})}{\\max \\lbrace 1, \\log _{b}i \\rbrace }$ , $\\forall i \\in \\lbrace 1, \\ldots , N\\rbrace $ , then $DCG_{b}(\\hat{r}) = \\sum _{i=1}^N \\frac{g(\\hat{r[i]})}{\\max \\lbrace 1, \\log _{b}i \\rbrace } < \\sum _{i=1}^N \\frac{g(\\hat{s[i]})}{\\max \\lbrace 1, \\log _{b}i \\rbrace } = DCG_{b}(\\hat{s})$ Proposition REF : The generalized precision and recall are isotone valuations, when they are defined on $(R(N), \\wedge , \\vee , \\preceq )$ with the Swapping+Replacement [rank-based] ordering.", "The generalized precision will be an isotone valuation if the implication: $\\hat{r} \\preceq \\hat{s} \\Rightarrow gP(\\hat{r}) \\le gP(\\hat{s})$ holds true.", "Let $\\hat{r}, \\hat{s} \\in R(N)$ such that $\\hat{r} \\ne \\hat{s}$ , by definition of the Swapping+Replacement [rank-based] ordering and considering that $g$ is an strictly increasing function: $\\hat{r} &\\preceq \\hat{s} & &\\Longleftrightarrow & \\big \\vert \\lbrace i \\le k : \\hat{r}[i] \\succcurlyeq \\mathcal {a}_j\\rbrace \\big \\vert &\\le \\big \\vert \\lbrace i \\le k : \\hat{s}[i] \\succcurlyeq \\mathcal {a}_j\\rbrace \\big \\vert & &\\Longleftrightarrow & \\sum _{i=1}^k g(\\hat{r}[i]) &\\le \\sum _{i=1}^k g(\\hat{s}[i]), \\\\& & & & \\forall j \\in \\lbrace 0, \\ldots , c\\rbrace , & \\forall k \\in \\lbrace 1, \\ldots , N\\rbrace & & & \\forall \\ k \\in & \\lbrace 1, \\ldots , N\\rbrace $ In particular, when $k=N$ , the last expression is $\\sum _{i=1}^N g(\\hat{r}[i]) \\le \\sum _{i=1}^N g(\\hat{s}[i])$ .", "Considering that $g(\\mathcal {a}_c)$ and $N$ are constants, $gP(\\hat{r}) = \\frac{1}{N} \\cdot \\sum _{i=1}^{N} \\frac{g(\\hat{r}[i])}{g(\\mathcal {a}_{c})} \\le \\frac{1}{N} \\cdot \\sum _{i=1}^{N} \\frac{g(\\hat{s}[i])}{g(\\mathcal {a}_{c})} = gP(\\hat{s}) \\ .$ Similarly, as $RB$ is constant, $gR(\\hat{r}) = \\frac{1}{RB} \\cdot \\sum _{i=1}^{N} \\frac{g(\\hat{r}[i])}{g(\\mathcal {a}_{c})} \\le \\frac{1}{RB} \\cdot \\sum _{i=1}^{N} \\frac{g(\\hat{s}[i])}{g(\\mathcal {a}_{c})} = gR(\\hat{s})$ Proposition REF : $RBP$ and $DCG_b$ are isotone valuations, when they are defined on $(R(N), \\wedge , \\vee , \\preceq )$ with the Swapping+Replacement [rank-based] ordering.", "$RBP$ will be an isotone valuation if the implication: $\\hat{r} \\preceq \\hat{s} \\Rightarrow gRBP(\\hat{r}) \\le gRBP(\\hat{s})$ holds true.", "Let $\\hat{r}, \\hat{s} \\in R(N)$ such that $\\hat{r} \\ne \\hat{s}$ , by definition of the Swapping+Replacement [rank-based] ordering and considering that $g$ is an strictly increasing function: $\\hat{r} &\\preceq \\hat{s} & &\\Longleftrightarrow & \\big \\vert \\lbrace i \\le k : \\hat{r}[i] \\succcurlyeq \\mathcal {a}_j\\rbrace \\big \\vert &\\le \\big \\vert \\lbrace i \\le k : \\hat{s}[i] \\succcurlyeq \\mathcal {a}_j\\rbrace \\big \\vert & &\\Longleftrightarrow & \\sum _{i=1}^k g(\\hat{r}[i]) &\\le \\sum _{i=1}^k g(\\hat{s}[i]), \\\\& & & & \\forall j \\in \\lbrace 0, \\ldots , c\\rbrace , & \\forall k \\in \\lbrace 1, \\ldots , N\\rbrace & & & \\forall \\ k \\in & \\lbrace 1, \\ldots , N\\rbrace $ In particular, when $k=N$ , the last expression is $\\sum _{i=1}^N g(\\hat{r}[i]) \\le \\sum _{i=1}^N g(\\hat{s}[i])$ .", "Notice that if $g(\\hat{r}[i]) \\le g(\\hat{s}[i])$ , then $p^{i-1} \\cdot g(\\hat{r}[i]) \\le p^{i-1} \\cdot g(\\hat{s}[i])$ , $\\forall i \\in \\lbrace 1, \\ldots , N\\rbrace $ , thus $gRBP(\\hat{r}) = \\frac{(1-p)}{g(\\mathcal {a}_c)} \\cdot \\sum _{i=1}^N p^{i-1} \\cdot g(\\hat{r}[i]) \\le \\frac{(1-p)}{g(\\mathcal {a}_c)} \\cdot \\sum _{i=1}^N p^{i-1} \\cdot g(\\hat{s}[i]) = gRBP(\\hat{s}) \\ .$ And similarly, if $g(\\hat{r}[i]) \\le g(\\hat{s}[i])$ , then $\\frac{g(\\hat{r[i]})}{\\max \\lbrace 1, \\log _{b}i \\rbrace } \\le \\frac{g(\\hat{s[i]})}{\\max \\lbrace 1, \\log _{b}i \\rbrace }$ , $\\forall i \\in \\lbrace 1, \\ldots , N\\rbrace $ , thus $DCG_{b}(\\hat{r}) = \\sum _{i=1}^N \\frac{g(\\hat{r[i]})}{\\max \\lbrace 1, \\log _{b}i \\rbrace } \\le \\sum _{i=1}^N \\frac{g(\\hat{s[i]})}{\\max \\lbrace 1, \\log _{b}i \\rbrace } = DCG_{b}(\\hat{s})$" ] ]
2207.03588
[ [ "Oxygen vacancy dynamics in Pt/TiOx/TaOy/Pt memristors: exchange with the\n environment and internal electromigration" ], [ "Abstract Memristors are expected to be one of the key building blocks for the development of new bio-inspired nanoelectronics.", "Memristive effects in transition metal oxides are usually linked to the electromigration at the nanoscale of charged oxygen vacancies (OV).", "In this paper we address, for Pt/TiOx/TaOy/Pt devices, the exchange of OV between the device and the environment upon the application of electrical stress.", "From a combination of experiments and theoretical simulations we determine that both TiOx and TaOy layers oxidize, via environmental oxygen uptake, during the electroforming process.", "Once the memristive effect is stabilized (post-forming behavior) our results suggest that oxygen exchange with the environment is suppressed and the OV dynamics that drives the memristive behavior is restricted to an internal electromigration between TiOx and TaOy layers.", "Our work provides relevant information for the design of reliable binary oxide memristive devices." ], [ "Introduction", "Memristive systems -defined as metal/insulator/metal structures able to switch between different resistive states upon the application of external electrical stimuli [1], [2]- are expected to be one of the key building blocks for the development of new neuromorphic hardware [3], intended to outperform current software-based machine learning algorithms running on computers with the Von Neumann architecture [4] .", "Memristive mechanisms strongly rely on the presence and electromigration of defects [1], [5]; in the case of oxides, these defects are usually the ubiquitous charged OV [6].", "Typically, the electromigration of OV can lead into the formation and disruption of conducting nanofilaments or to the modulation of the resistance of Schottky metal/oxide interfaces [1].", "It has been reported that both mechanisms could coexist for single devices, being possible to select one or the other by controlling external stimuli parameters such as the compliance current programmed during the transition from high to low resistance states [7] or other device operation conditions [8].", "Among single oxides, TaO$_y$ presents a high potential to be implemented in memristive systems with neuromorphic behavior.", "This is based on its CMOS compatibility -which would ease the integration with standard electronics-, analog response - in order to mimic the adaptable synaptic weights of biological synapses- [9], [10], [11], very high endurance (from 10$^{10}$ to 10$^{12}$ cycles) [12], [13], ultrafast switching time ($\\approx $ 10 ps) [14], large ON-OFF ratio ($\\approx $ 10$^6$ ) [15] and ultra-low power operation ($\\approx $ 60 fJ/bit) [15].", "The already reported memristive mechanisms in TaO$_y$ -based devices include the formation of conducting nanofilaments [9], [16], [17], [18], the modulation of energy barriers present at metal-oxide interfaces [19], [10] or the OV exchange between TaO$_y$ /TaO$_h$ bilayers (with different degree of oxidation) [20], [13], [21].", "In the latter case, it is usually assumed that the more reduced layer acts as OV source/sink that eases the reduction/oxidation of the more oxidized one that drives the resistance changes -usually Ta$_2$ O$_5$ - [13].", "Many mechanisms proposed to describe the memristive behavior of oxide-based devices have usually assumed that OV dynamics takes place internally between different layers or zones of the device [22], [23]; in other words, the total amount of OV present in the device is considered as a constant.", "More recently, it has been experimentally shown for different oxide-based memristors that molecular oxygen transfer across oxide-metal interfaces [24], [25], which could eventually lead to oxygen exchange with the environment, could be a non-negligible effect and must be taken into account to properly describe the memristive effect.", "Advanced characterization tools such as in-operando transmission electron microscopy [26] or secondary mass ion spectrometry [27] were used to get evidence on this.", "It has been proposed that moisture seems to play a key role in providing oxygen to the device oxidation [28], [29], [30].", "We also notice that the incorporation of protons to the device was also proposed to affect the device electrical behavior in the case of cationic resistive switches [31].", "The influence of ambient conditions on the memristive response of the device is therefore not a trivial issue for the technological applications of these systems, and it needs to be fully understood and controlled in order to develop strategies -such as, for example, a proper encapsulation of the device if necessary- to warrant a reliable memristive behavior.", "In this paper, we address, from a combination of experiments and theoretical simulations, the memristive response of Pt/TiO$_x$ /TaO$_y$ /Pt heterostructures, making focus on the presence of oxygen exchange with the environment.", "Our findings indicate that the electroforming process is accompanied by a strong oxygen uptake from the ambient -which oxidizes both TiO$_x$ and TaO$_y$ layers-, but after the memristive cycling is stabilized oxygen exchange with the environment is spontaneously suppressed and OV dynamics is restricted to an internal exchange between TiO$_x$ and TaO$_y$ layers." ], [ "Methods", "We have grown by pulsed laser deposition TiO$_x$ /TaO$_y$ bilayers on top of platinized silicon substrates.", "The depositions were made at room temperature and at oxygen pressures of 0.01 and 0.1 mbar, respectively.", "Top Pt electrodes were microfabricated by a combination of sputtering and optical lithography.", "Electrical characterization was performed with a source measure unit Keithley 2612B hooked to a commercial probe station.", "High resolution Scanning Transmission Electron Microscopy with a High Angular Annular Dark Field Detector (STEM-HAADF) was performed using a FEI Titan G2 microscope with a probe corrector (60–300 keV).", "In situ chemical analysis was performed by Energy Dispersive Spectroscopy (EDS).", "Samples for TEM were prepared by Focused Ion Beam (FIB) in a Helios 650 dual beam equipment." ], [ "Device electroforming", "Fig.", "1(a) shows a STEM-HAADF cross-section corresponding to a virgin Pt/TiO$_x$ /TaO$_y$ /Pt heterostructure, before the application of any voltage stress.", "The STEM-HAADF image suggests that TiO$_x$ and TaO$_y$ thicknesses are 81.5 nm and 10.5 nm, respectively.", "EDS linescans, shown in Fig.", "1(b), indicate composition gradients for both oxide layers: the Ti oxide chemistry goes from TiO$_{1.22}$ in the region close to the top Pt electrode to TiO$_{2.24}$ close to the interface with Ta oxide.", "On the other hand, the Ta oxide layer displays a stochiometry ranging from TaO$_{2.33}$ close the interface with the Ti oxide to TaO$_{3}$ close to the bottom Pt electrode.", "We notice that memristors with graded chemical composition and reliable behavior were reported in the literature [32], [33].", "EDS line scans also show that the TiO$_x$ /TaO$_y$ interface is not sharp but it displays a zone ($\\approx $ 10 nm) of Ti and Ta intermixing.", "Fast Fourier Transforms (FFT) performed in both oxide layers (not shown here) show the absence of diffraction poles, evidencing their amorphous character.", "In addition, the top Pt electrode shows the presence of columns and grain boundaries, which could behave eventually as fast paths for oxygen migration in and out of the device [34].", "Figure: (a) STEM-HAADF cross-section corresponding to a pristine Pt/TiO x _x/TaO y _y/Pt device; (b) EDS line-scans corresponding to the cross-section shown in (a), as shown with a dashed white line.", "Ti, Ta, O and Pt species are quantified.", "The scans start at the Pt top electrode and end at the bottom Pt electrode; (c) STEM-HAADF cross-section corresponding to a electroformed Pt/TiO x _x/TaO y _y/Pt device; (d) EDS line-scans corresponding to the formed device, shown in (c).", "The scans, shown with a dashed white line in (c), start at the Pt top electrode and end at the bottom Pt electrode; (e) FFT corresponding to the TiO x _x layer of the formed device.", "The appearance of faint diffraction poles -estimated interplanar distances are shown in the figure- indicates the formation of nanocrystallites but does not allow a precise identification of the crystalline phase.The forming process, shown in Fig.", "2 for a $\\approx 3x10^3$ $\\mu $ m$^2$ device, started with the application of an initial pulsed ramp consisting in voltage pulses (10ms wide) of increasing amplitudes from 0 V to V$_{FO}$ $\\approx $ -5.5 V (see Fig.", "2(b)).", "After each pulse, we measured the remanent resistance by applying a small voltage of 100 mV.", "This initial stimulus produced a spike-like change in the device resistance, as it is observed in Fig.", "2(c), from a virgin state of $\\approx $ 100$ \\Omega $ to $\\approx $ 80 k$\\Omega $ and then to $\\approx $ 8 k$\\Omega $ .", "Afterwards, symmetric pulsed ramps with V$_{MIN}$ = -2 and -2.5 V and V$_{MAX}$ = 2 and 2.5 V, respectively, were applied -notice that -V$_{MIN}$ , V$_{MAX}$ < -V$_{FO}$ -, which produced a progressive resistance recovery until a stable resistive switching effect between $\\approx $ 10 k$\\Omega $ and $\\approx $ 14 k$\\Omega $ was found, as it is displayed in Fig.", "2(d) .", "Figs.", "1(c) and 1(d) show a STEM-HAADF cross-section of a formed device and the corresponding EDS linescans, respectively.", "It is found that, upon forming, both Ti and Ta oxide layers become more oxidized in relation to the virgin device: the Ti oxide layer displays an uniform TiO$_{2.3}$ stochiometry while the Ta oxide layer displays a TaO$_{3.4}$ stochiometry, also uniform in thickness.", "We notice that both layers display higher oxygen content than the standard (and stable) TiO$_{2}$ and Ta$_{2}$ O$_{5}$ phases, as has been reported for TaO$_{y}$ [35] and related to the absorption of environmental water molecules and the formation of Ta-O-O-H bonds by means of a protonation reaction [36].", "Also, FFT performed from the STEM-HAADF cross section show, for the formed device, the appearance of faint diffraction poles, indicating the formation of nanocrystallites of Ti and Ta oxides (see Fig.", "1(e) for the case of Ti oxide).", "This crystallization process is likely related to the presence of thermal effects -via Joule heating- [37], [38] during the initial stage of the electroforming process, and we associate it to the resistance spike we described before.", "The experiments described above show that environmental oxygen -and eventually protons- are incorporated to the device upon forming.", "However, it is unclear if this interaction with the ambient is maintained once the resistive switching effect becomes stable.", "In order to tackle this issue, we have performed numerical simulations using the Voltage Enhance OV drift model (VEOV) [23], [39], adapted here to describe the oxidation process during forming and the subsequent remanent resistance vs. voltage cycles in the Pt/TiO$_x$ /TaO$_y$ /Pt system." ], [ "Simulating the electroforming process", "The VEOV model simulates the migration of OV, ubiquitous in transition metal oxides, due to an applied external electrical stimulus and has been extensively employed to unveil the memristive response of several oxide based devices, even ferroelectrics and topotactic manganites [40], [41], [21], [42], [43], [44].", "The key ingredients of the model are i) the dependence of the resistivity of an oxide on its local oxygen stoichiometry and ii) the strong electric fields that develop close to the electrode(s)/oxide interface(s) -usually forming Schottky barriers- and/or at the interface between different oxides composing the device.", "Under an external voltage, OV electromigrate back and forth depending on the polarity of the applied stimulus along nanoscale regions close to where strong electric fields develop, with the concomitant change in the device resistance.", "In particular, TiO$_x$ and TaO$_y$ behave as an n-type semiconductors in which OV are electron donors.", "To have further insight into the model details, Fig.", "2(a) shows a sketch of the present device where we have defined the memristive active regions relevant for the simulations.", "The left (L) region comprises the interface Pt/TiO$_{x}$ while the right (R) one represents the interface TaO$_y$ /Pt.", "The central (C) region mainly comprises the interface TiO$_{x}$ /TaO$_y$ , where our TEM experiments evidenced some Ti and Ta intermixing (recall Fig.", "1).", "Due to the largest thickness of the TiO$_{x}$ layer in our devices, we assume that the L region is larger than the C and R respectively, the latter two including all the TaO$_{y}$ layer.", "On the other hand, the remaining TiO$_{x}$ at the right of the L zone represents an inert bus zone for the transfer of OV that, as we will show below, essentially does not participate in the RS effect.", "For the simulations we define a 1D chain of $N= NL+NC+NR+ NB$ total sites, where the first $NL$ sites correspond to the L layer, $NC$ sites to the central C layer and $NR$ sites to the R layer, respectively.", "In addition $NB$ sites are assigned to the bus region in the TiO$_x$ .", "Taking into account the previous device description (recall the STEM-HAADF cross-section displayed in Fig.", "1), we consider $NB> NL > NC>NR$ (see Table 1 for values of the parameters employed in the simulations).", "Each site $i$ represents a domain of (sub)nanoscopic dimensions characterized by its resisitivity $\\rho _{i}=\\rho _{0} (1 - A_{i} \\delta _{i})$ [39] that decreases with $\\delta _{i}$ , the local density of OV.", "We define ${\\rho _{0}}$ as the residual resistivity for negligible OV concentration and, following the reported resistivities for TiO$_{x}$ and TaO$_{y}$ [45], [46] we take different values of ${\\rho _{0}}$ in each oxide, accordingly (see also Table 1).", "The coefficients $A_{i}$ characterize the different interfaces and can be taken either smoothly dependent on the site position or as constants (as we do for simplicity), without affecting the qualitative behaviour of the simulated results.", "In all the cases we have $ A_{i} \\delta _{i} < 1 \\; \\forall i$ .", "Table: Parameters employed in the numerical simulations of OV dynamics.The activation voltages V α V_\\alpha are in units of the thermal energy K B TK_B T.The reservoir´s activation voltage was taken V RE =0.16V_{RE}=0.16.", "The bus zone B hasN B =297N_B=297 sites and same parameters than the L zone.The total resistivity of the system can be computed as $\\rho = \\sum _{i=1}^{N}{\\rho _{i}}$ and, as we are considering a 1D model, the resistance $R$ can be trivially computed from $\\rho $ through a length scaling factor.", "To account for the absorption of oxygen observed experimentally during the forming process (see Fig.", "1(d)), we assume that the sample can exchange OV with an external reservoir and thus the total number of OV in the sample is not conserved during the application of electrical stress.", "This is a new key ingredient that settles a difference with previous studies [39], [21], [42], in which any possible exchange of OV between the sample and the ambient was neglected in the VEOV model.", "Following this line of reasoning, a net decrease in the sample OV content will be interpreted as an oxidation process (a net uptake of oxygen).", "Although this might be a oversimplified assumption, as the net uptake of oxygen could be concomitant with other effects such as proton incorporation, as we mentioned before, it allows capturing non-trivial characteristics of the experimental forming process, as we will describe below.", "We simulate the external reservoir as a region in contact with the L zone that can allocate an arbitrarily large number of OV.", "Notice that as reported in Figs.", "1(c) and 1(d), the uptake of oxygen is more favoured at the oxide layer close to the Pt top electrode and thus in the simulations we consider that the interchange of OV is through the L zone and the reservoir.", "We emphasize that the electroforming is a complex out-of-equilibrium process under which structural changes, like crystallization, can additionally contribute to the change of the device resistance.", "We assume that: i) the crystallization process is concomitant with the resistance spike we observed at the first stage of the electroforming process, and it is finished afterwards ii) after crystallization, the device still presents a large number of OV which are progressively filled during the next electroforming steps.", "Our simulations start at ii) and describe the resistance evolution assuming that no further structural changes occur.", "In Fig.", "2(b) we show the experimental voltage protocol $V(t)$ during the forming process together with the $V_s (t)$ employed for the simulation, which follows quite well the experimental one.", "Given a value of $V_s (t)$ , the OV density at each site i is updated for each simulation step according to the rate probability $p_{ij} = \\delta _i (1-\\delta _j) \\exp (-V_{\\alpha } + \\Delta V_{i})$ [23] for a transfer from site i to a nearest neighbor j= i $ \\pm 1$ .", "Notice that $p_{ij}$ is proportional to the OV density at site i and to the available OV density at the neighbour site j.", "In the Arrhenius factor, $\\exp (-V_{\\alpha } + \\Delta V_{i})$ , $\\Delta V_i$ is the local potential drop at site i defined as ${\\Delta V}_i (t) = V_{i}(t) - V_{i-1}(t)$ with $V_i(t) = V_s (t) \\rho _{i} / \\rho $ and $V_\\alpha $ the activation energy for vacancy diffusion in the absence of external stimulus.", "The values of $V_\\alpha = V_{L}, V_{B}, V_{C}$ and $V_R$ for the L, B, C and R layers are given in Table 1.", "In all the calculations, the energy scales are taken in units of the thermal energy $k_{B}T$ .", "As we mentioned, the reservoir is modelled by a region external to the sample which can allocate a large amount of OV.", "Thus following the usual statistical assumption, once OV are injected into the reservoir they tend to remain in it.", "To accomplish this, we consider for the reservoir a transfer rate $p_{ij}$ with an activation energy $V_\\alpha = V_{RE}\\gg V_L$ .", "In addition we do not include the external voltage in the reservoir.", "At each simulation time step $t_k$ , we compute the local voltage profile $V_i(t_k)$ , the local voltage drops ${\\Delta V}_i (t_k)$ and employing the probability rates $p_{ij}$ we obtain the transfers between nearest neighboring sites.", "Afterwards, the values $\\delta _i(t_k)$ are updated to a new set of densities $\\delta _i(t_{k+1})$ , with which we compute at time $t_{k+1}$ , the local resistivities $\\rho _i(t_{k+1})$ , the local voltage drops under the applied voltage $V_s (t_{k+1})$ , and finally the total resistivity $\\rho (t_{k+1})$ , to start the next simulation step at $t_{k+1}$ .", "Figure: (a) Scheme of the device assumed for the simulations.", "The different active zones -L,R and C- are indicated.", "The red arrows sketch the exchange of OV with the ambient during the electroforming process; (b) Simulated (green dashed) and experimental (red circles) electroforming voltage protocols.", "The inset shows a blow-up of the main panel for times between 0 and 100 s; (c) Left axis: resistance evolution with time during the electroforming process of a ≈3x10 3 \\approx 3x10^3 μ\\mu m 2 ^2 Pt/TiO x _x/TaO y _y/Pt device.", "The red (green) curve corresponds to the experimental (simulated) remanent resistance values.", "Right axis: simulated OV percentage remaining in the TiO x _x/TaO y _y bilayer during the electroforming process, as a function of time; (d) Stable remanent resistance vs. voltage cycle measured experimentally (red) and simulated (green), for the same device.We consider an initial OV configuration consistent with the value of $\\approx $ 8k$\\Omega $ attained in the experiment after the resistance spike observed in the first stage of the electroforming process.", "Therefore we start the simulations at time $0^{+}$ , immediately after the application of a post-resistance spike short positive pulse (that we consider instantaneous for the simulation purposes).", "Fig.", "2(c) shows the time evolution of the simulated resistance of the sample during the application of the electroforming protocol, $V_s(t)$ , shown in Fig.", "2(b).", "The agreement between the simulated and the experimental curve is remarkable.", "After a transient, in which the device resistance fluctuates for a time scale of the order of 100 s, the resistance finally stabilizes in the remanent resistance vs. voltage loop shown in Fig.", "2(d).", "Notice that the simulations perfectly capture the time scale of this process with an attained stable resistance loop that reproduces most of the characteristics of the experimental one.", "Fig.", "2(c) additionally shows the time evolution of the relative fraction (percentage) of OV remaining in the sample $\\delta _0\\equiv N_{VO}(t)/N_{V0}(0^+) \\%$ (that is the percentage ratio between the total number of OV at time t and at the initial time).", "A saturation close to 75$\\%$ , once the stable resistant loop is attained, is clearly observed.", "This might correspond to an equilibrium state between the device and the environment with no subsequent OV exchange during the stable memristive cycling, as we address in the next section." ], [ "Stable memristive behavior: experiments and simulations", "We have measured and simulated the stable memristive response of another device with a larger area ($\\approx 1.2x10^4$ $\\mu $ m$^2$ ) than the previous one, which was electroformed in a similar way than described before and stimulated with different writing voltage cycles, characterized by their maximum (minimum) excursions V$_{MAX}>0$ (-V$_{MIN}>0$ ).", "Fig.", "3(a) displays the case of V$_{MAX}$ < -V$_{MIN}$ , with a remanent resistance loop vs. voltage that presents a clockwise (CW) evolution, while Fig.", "3(b) displays the case of V$_{MAX}$ > -V$_{MIN}$ , characterized by a loop with a counter-clockwise (CCW) evolution.", "Figure: (a) CW remanent resistance vs. voltage loops for a ≈1.2x10 4 \\approx 1.2x10^4 μ\\mu m 2 ^2 Pt/TiO x _x/TaO y _y/Pt device.", "Experimental resistance values are shown in red and simulated ones are shown in green; (b) CCW loops for the same device.", "The transition between CW and CCW loops is attained by tuning the maximum and minimum writing voltage excursions; (c) OV profiles correspondent to the HR 1 _1 state of CW (pink) and HR 2 _2 of CCW (orange) simulated loops; (d) OV profiles correspondent to the LR 1 _1 state of CW (dark blue) and LR 2 _2 of CCW (light blue) simulated loops.", "The OV density per site (δ ˜ i )\\widetilde{\\delta }_i) is normalized in terms of δ=0.0039\\delta = 0.0039, the initial (uniform) OV density considered.We notice that both remanent resistance vs. voltage loops share quite the same low resistance value (LR$_{1}$ $\\sim $ LR$_{2} \\approx $ 1.2 k$\\Omega $ ) but they differ in their high resistance state ($HR_1 \\approx $ 1.5 k$\\Omega $ for the CW loop and $HR_2 \\approx $ 2.4 k$\\Omega $ for the CCW loop).", "We remark that the resistance levels of the CW loop (Fig.", "3(a)) are around one order of magnitude lower than those found for the device described previously, with a similar evolution (recall Fig.", "2(d)).", "This indicates that the resistance levels increase as the device area is decreased, consistently with a non-filamentary, area distributed memristive effect, as previously reported for other simple oxides-based memristive systems [19], [10], [13], [21].", "In addition, our results shows the possibility of tuning the evolution (circulation) of the remanent resistance loop in a reversible way, as it was previously found in Pt/TaO$_y$ /TaO$_h$ /Pt devices and linked to the control at the nanoscale of the OV dynamics through asymmetric electrical stimuli, allowing the selective activation/deactivation of both oxide/Pt interfaces [21].", "In Figs.", "3(a) and 3(b) are shown the simulations of both CW and CCW resistance loops (dashed lines), displaying an excellent agreement with the experimental ones (dotted lines).", "The distinctive feature of these simulations is that they are performed conserving the total amount of OV present in the system.", "In other words, our results confirm that the interaction between the device and the environment is restricted to the electroforming process, but later on the memristive effect relies exclusively on the internal redistribution of OV between different zones of the device.", "This result is at odds with Ref.", "[47], where it was suggested that oxygen exchange takes place during the stable memristive cycling for ohmic top metal/oxide interfaces.", "The difference in our case might rely on the existence of an energy barrier (i.e.", "Schottky type) at the top Pt/TiO$_x$ interface [48].", "The OV distribution along the device is shown for the HR$_1$ , HR$_2$ and LR$_1 \\sim $ LR$_2$ states, pointed out in the correspondent resistance loops displayed in Fig.", "3.", "For the low resistance states, in both CW and CCW loops, OV accumulate at the central Ti and Ta intermixing (C) zone, reducing its resistance and driving the overall drop of the total device resistance.", "Notice that the residual penetration of OV in the R zone in the case of the CW loop is responsible for the tiny difference between the LR$_1$ and LR$_2$ values.", "Additionally for the CW(CCW) loop the HR$_1$ (HR$_2$ ) state corresponds to OV located mainly at the L(R) interface, while in both cases the C region, being depleted from OV, is responsible for the (high) resistance value of the device.", "We notice that in order to properly simulate the experimental electrical behavior, it should be assumed that the C zone has the highest residual resistivity of the device.", "This implies that the RS effect is dominated by the resistivity changes of zone C, driven by OV electromigration between this zone and the two metal/oxide interfaces, depending on the polarity and the asymmetry of the applied stimulus.", "Starting the simulations from an OV distribution compatible with the post forming HR$_1$ state, the SET transition to the LR$_1$ state in the CW loop takes place when the OV, initially located at the L interface, have been driven to the C zone under the positive SET stimulus.", "For the CCW loop, the LR$_2$ state is attained when the OV, former located at the R interface, are drifted to the C zone under the negative SET voltage.", "The latter analysis can be complemented by fitting the dynamic current-voltage (I-V) curves, recorded simultaneously with the CW and CCW remanent resistance loops (see Ref.", "[21] for further experimental details), after proposing an equivalent circuit.", "Figs.", "4(a) and (b) display the dynamic I-V curves related to the remanent resistance loops with CW and CCW circulations, respectively.", "It is seen that the I-V curves display an inverse circulation in relation to the remanent resistance loops, as expected.", "They show a non-linear evolution indicating the presence of non-ohmic transport mechanisms, as usually found in capacitor-like structures with memristive non-filamentary behavior [49], [41], [21], [43].", "To perform the fittings we considered the $\\gamma $ = dLn(I)/dLn(V) parameter representation, firstly introduced in Ref.", "[50] and which was proved as a suitable way for undisclosing multiple conduction mechanisms, usually found in oxide based memristors [41], [21], [43].", "Figure: Dynamic I-V curves recorded simultaneously with CW (a) and CCW (b) remanent resistance loops.", "The different I-V circulations -mirrored with respect to the remanent resistance loops- are attained by changing the maximum and minimum voltage excursions (see text for details).", "The insets show the minimal equivalent circuits necessary to fit the experimental data; (c), (d) γ\\gamma vs. V 1/2 V^{1/2} representation of the Dynamic Low and High Resistance states (DHR and DLR, respectively), obtained from the I-V curves displayed in (a) and (b).", "Symbols correspond to the experimental data, while the fittings (see the text for details) are shown in solid lines.Figs.", "4(c) and (d) display the $\\gamma $ vs. V$^{1/2}$ behavior derived from the aforementioned I-V curves, both for low and high resistance branches, which we name as Dynamic Low (High) Resistance state or DLR (DHR).", "We notice that only voltages with absolute values lower that the SET/RESET ones are considered.", "First, we stress the existence of ohmic conduction for low voltages ($\\gamma $ $\\approx $ 1); for higher voltages, a non-linear conduction mechanism, which we identify as Space Charge Limited Current (SCLC) conduction with traps, prevails, as $\\gamma $ increases smoothly with V and reaches values $\\ge $ 2 [51].", "The simplest circuit representations consistent with the evolution of the $\\gamma $ parameter with voltage are displayed as insets in Figs.", "4(a) and 4(b).", "For the case of the I-V displayed in Fig.", "4(a) -corresponding to a CW remanent resistance loop-, the equivalent circuit comprises a parallel combination of a resistor $R_{1}$ and a SCLC channel ($SCLC_{1}$ ), in series with a resistor $R_{2}$ (see the inset of Fig.", "4(a)).", "The current flowing through the device can be described by I = $I_{R_{1}}$ + I$_{SCLC_{1}}$ = (V - I$R_{2}$ ) / R$_{1}$ + $A_{1}$ (V-I$R_{2}$ )$^{n_{1}}$ , where $I_{R_{1}}$ and $I_{SCLC_{1}}$ are the currents through the $R_{1}$ resistor and the SCLC element, respectively.", "$A_{1}$ is related to the mobility, the dielectric constant and the width of the transport channel and $n_{1}$ is an exponent $\\ge $ 2.", "This implicit equation was solved numerically in order to fit the experimental $\\gamma $ vs. V$^{1/2}$ curves, both for DLR and DHR, by determining in each case the fitting parameters R$_{1}$ , A$_{1}$ , $n_{1}$ and R$_{2}$ .", "The results of the fittings can be observed in Fig.", "4(c), showing a very good agreement with the experimental data We also tested the possible contributions of Schottky diodes present at both metal-oxide interfaces, finding that they don't significantly contribute to the electronic transport in the range of (small) voltages used to perform the fittings.", "They would only contribute with a small part of the conduction in the range of higher voltages, where some deviations between the fits and the experimental values can be observed..", "It is found that the transition from DLR to DHR is driven by changes in the non-linear SCLC$_{1}$ element (A$_{1}$ and $n_{1}$ are $\\approx $ 64% and $\\approx $ 17% lower for the DHR state, respectively) and its parallel leakage channel R$_{1}$ .", "A similar analysis can be made for the case of the I-V curve displayed in Fig.", "4(b) -corresponding to a CCW remanent resistance loop-, where the equivalent circuit in this case corresponds to a resistor R$_{1}$ in series with the parallel combination of a SCLC element (SCLC$_{2}$ ) and a resistor R$_{2}$ (see the inset of Fig.", "4(b)).", "The corresponding $\\gamma $ vs. V$^{1/2}$ fittings for both DLW and DHR states are shown in Fig.", "4(d), again with a good agreement between the experimental and calculated curves.", "In this case, the memristive effect is dominated by changes in the element SCLC$_{2}$ (A$_{2}$ and $n_{2}$ are $\\approx $ 25% and $\\approx $ 7% lower for the DHR state, respectively) and its leakage channel R$_{2}$ .", "From the analysis of the two presented cases (I-V curves with opposite circulations) it is found that both the transport mechanism and memristive effect are strongly dependant on a SCLC channel.", "We recall that SCLC is a bulk conduction mechanism, which supports our previous statement that the resistance change is not dominated by the oxide/metal interfaces but by a bulk zone of the device in between both metal/oxide interfaces, including the intermixed TiO$_x$ /TaO$_y$ interface." ], [ "Concluding remarks", "From a combination of electrical measurements, analytic characterization and modelling, we have unveiled the role of OV exchange between Pt/TiO$_x$ /TaO$_y$ /Pt memristive devices and the environment.", "Our microscopy experiments show a clear oxidation process of both TiO$_x$ and TaO$_y$ layers during the electroforming process, validated by our numerical simulations based on the VEOV model.", "It is reasonable to assume that the top Pt electrode, with a microstructure of columns and grain boundaries, behaves as a permeable layer that allows oxygen transport between the environment and the oxide bilayer trough grain boundaries, as has been previously suggested in Ref.", "[34].", "In addition to this oxidation process, electroforming also shows the formation of oxide nanograins -likely at the first stages of the forming process-, reflecting the presence of strong thermal effects that trigger the partial crystallization of the oxide bilayer.", "Once the electroforming process is complete, the system is able to switch between stable low and high resistance states in two different ways: if the writing voltages are such that V$_{MAX}$ > -V$_{MIN}$ the remanent resistance shows a CCW evolution, while if V$_{MAX}$ < -V$_{MIN}$ the loops display a CW evolution.", "For both cases, the remanent resistance loops can be numerically reproduced by assuming that the system maintains a constant number of OV, indicating that it is in equilibrium with the environment.", "This difference with respect to the electroforming process can be related to the fact that V$_{MAX}$ and -V$_{MIN}$ are lower that the forming maximum voltage -V$_{FO}$ .", "However, other features such as the nature of the top metal/oxide interface might also play a significant role [47].", "Based on our numerical simulations, it was established that the OV dynamics for the stable CW (CCW) loop is constrained to the OV exchange between TiO$_x$ (TaO$_y$ ) layer and the central device zone comprising the TiO$_x$ /TaO$_y$ interface, where Ti and Ta interdiffusion was observed.", "This central zone was also shown to dominate the electrical transport and to control the resistive changes of the device for both cases.", "Our work provides relevant information for the design of reliable binary oxides memristive systems, which are strong candidates for the implementation of neuromorphic computing devices such as physical neural networks [53]." ], [ "Acknowlegments", "We acknowledge support from UNCuyo (06/C591), ANPCyT (PICT2017-1836, PICT2019-02781, PICT2019-0654 and PICT2020A-00415) and EU-H2020-RISE project \"MELON\" (Grant No.", "872631).", "We also acknowledge the LMA-Universidad de Zaragoza for offering access to the microscopy instruments.", "MJS acknowledges the hospitality of the LPMC, Université of Picardie Jules Verne.", "The data that support the findings of this study are available from the corresponding author upon reasonable request" ] ]
2207.03538
[ [ "Dynamic Community Detection via Adversarial Temporal Graph\n Representation Learning" ], [ "Abstract Dynamic community detection has been prospered as a powerful tool for quantifying changes in dynamic brain network connectivity patterns by identifying strongly connected sets of nodes.", "However, as the network science problems and network data to be processed become gradually more sophisticated, it awaits a better method to efficiently learn low dimensional representation from dynamic network data and reveal its latent function that changes over time in the brain network.", "In this work, an adversarial temporal graph representation learning (ATGRL) framework is proposed to detect dynamic communities from a small sample of brain network data.", "It adopts a novel temporal graph attention network as an encoder to capture more efficient spatio-temporal features by attention mechanism in both spatial and temporal dimensions.", "In addition, the framework employs adversarial training to guide the learning of temporal graph representation and optimize the measurable modularity loss to maximize the modularity of community.", "Experiments on the real-world brain networks datasets are demonstrated to show the effectiveness of this new method." ], [ "Introduction", "Neuroscience is emerging into a generation marked by a large amount of complex neural data obtained from large-scale neural systems [1].", "The majority of these extensive data are in the form of data from networks that cover the relationships or interconnections of elements within different types of large-scale neurobiological systems.", "Significantly, these data often span multiple scales (neurons, circuits, systems, whole brain) or involve different data types in neurobiology (e.g., structural networks expressing anatomical connectivity of nerves, functional networks representing connectivity of distributed brain regions associated with neural activity).", "The brain network consists of anatomical structures segmenting different brain regions and connecting them by functional networks showing their complex neuronal communication and signaling patterns.", "Attributed to advancements in current imaging techniques and advanced methods of medical image processing [2] [3] [4] [5], this sophisticated pattern of neural signals may be studied using functional imaging, in which neuronal activity is associated with a variety of behaviors and cognitive functions as well as brain diseases [6] [7] [8] [9] [10].", "At the same time, network science is the study of complex network representation through theories and techniques of computer science and mathematics.", "With the convergence of two significant scientific developments in recent years, new techniques and analytical methods in the network science field are emerging for evaluating real-world biological networks [11].", "Subgraphs, network modules, and communities have been extensively studied in the context of network structures, and in particular, community detection [12] [13] methods have been widely used in network neuroscience [14].", "Network structure identification or community detection(see schematics in bottom right of Figure.", "REF ) is the partition of nodes in a network into groups in which nodes in the communities are tightly connected, and nodes in different communities are sparsely connected.", "The organizational principles and operational functions of complex network systems can be revealed and understood through mining the network structure.", "Furthermore, the creation of comprehensive network maps of neural circuits and systems has resulted from the development of new techniques for mapping the structure and functional connectivity of the brain.", "A wide range of graph-theoretic tools can be used to examine and analyze the structure of these brain networks.", "Therefore, methods for detecting modules or network communities in brain networks are of specialized application, and they reveal tightly connected primary building elements or substructures, which frequently relate to particular functional components.", "Data-driven models have gotten much attention for a long time, and when combined with machine learning techniques, it has led to great success in building pattern recognition models within the field of medical image computing [15] [16].", "The models have the potential to achieve high accuracy at a low computational cost.", "Deep learning is currently widely perceived as one of the most significant developments in machine learning [17], [18], [19].", "The method of deep learning is a new approach for dealing with high-dimensional biological data and learning low-dimensional representations of medical image [20].", "The approach based on the generative adversarial methodse [21] and the graph neural network are good instances.", "Generative adversarial network(GAN) [22], [23], [24], which can bee seen as variational-inference [25] based generative model, is commonly employed in medical image representational analysis [26] [27] [28].", "The utilization of GAN for community detection is inspired by the fact that GANs are often supervised in training, and the newly generated data (in principle) has the same distributioun as real data, allowing for robust, complex data analysis [29] [30] [31].", "Convolutional neural network (CNN) [32] approach reduces the dimensions of medical imaging data by pooling and convolution , enabling it to successfully recognize pattern in biomedical task [33] [34].", "Graph convolutional network (GCN) is developed to extract community features since it derives the CNN capabilities and directly processes on network structured data.", "However, existing methods to process dynamic network data to obtain temporal graph representations for community detection remain challenging, especially for small sample network datasets.", "To end these issues, we developed a novel adversarial temporal graph representation learning (ATGRL) to complete the clustering of brain nodes in dynamic brain networks, detect different communities containing similar brain regions in dynamic brain networks and their evolution, and improve the robustness of the model while handling with small sample network data by employing generative adversarial approaches.", "The proposed temporal graph attention encoder is efficient to graph representation learning, and more helpful graph embeddings are obtained to complete the clustering to detect more accurate dynamic communities.", "The detected communities with sound classification effects can be used as biological markers.", "Figure: Proposed adversarial temporal graph representation learning framework for detecting brain communities." ], [ "Method", "Our ATGRL includes two core parts:1) a temporal graph auto-encoder consists of a temporal graph attention encoder and a decoder, and 2) an adversarial regularizer including a discriminator.The architecture are illustrated in Fig.", "REF .", "In the autoencoder, the encoder ($E$ ) adopts temporal graph attention networks to transform the time series of brain regions ($\\left\\lbrace X^t\\right\\rbrace ^T_{t=1}$ ) and brain functional connections ($A$ ) into the embeddings ($\\left\\lbrace Z^t\\right\\rbrace ^T_{t=1}$ ) .", "Moreover, in the adversarial regularizer, a min-max adversarial game is led between the encoder, which regards as the generator ($G$ ), and the discriminator ($D$ ) to learn better embeddings.", "In order to detect communities, the measurable soft modularity loss is employed which optimizes the community assignment matrix $P$ .", "Therefore, the encoder is trained with triple objectives: a classic reconstruction loss of autoencoder, an adversarial training loss from the discriminator and the measurable modularity loss for detecting community." ], [ "Temporal Graph Autoencoder", "The temporal graph autoencoder aims to embed the dynamic brain network attributes in a low-dimensional latent space.", "First, we use two different network blocks to construct the encoder: the topological attention block and temporal attention block.", "Each block is formed by several stacked layers of the corresponding layer.", "They both employ self-attention mechanisms to obtain an efficient temporal graph representation from its neighboring and historical context information.", "The initial input for this layer is a set of brain network attributes $\\left\\lbrace x_i\\in \\mathbb {R}^d,\\forall i\\in \\mathcal {V}\\right\\rbrace $ where $d$ is the dimension of time series.", "The output is a set of brain region representations $\\left\\lbrace h_i\\in \\mathbb {R}^f,\\forall i\\in \\mathcal {V}\\right\\rbrace $ where $f$ is the dimension of captured topological properties.", "Similar to graph attention networks(GAT) [35],our topological attention layer is concerned with the near neighbors of the brain region $i$ by calculating attention weight from input brain region representations: $\\begin{aligned}h_{i}&=\\sigma \\left(\\sum _{u \\in \\mathcal {N}_{i}} \\alpha _{i j} W x_{j}\\right), \\\\ \\alpha _{i j}&=\\frac{\\exp \\left(\\sigma \\left(A_{i j} \\cdot \\left[W x_{j} \\Vert W x_{i}\\right]\\right)\\right)}{\\sum _{w \\in \\mathcal {N}_{i}} \\exp \\left(\\sigma \\left(A_{w i} \\cdot \\left[W x_{w} \\Vert W x_{i}\\right]\\right)\\right)}\\end{aligned}$ Here $\\mathcal {N}_i=\\left\\lbrace j\\in \\mathcal {V}:(i,j)\\in \\mathcal {E}\\right\\rbrace $ is the set of near neighbor of region $i$ which are linked by functional connection $A$ ; $W\\in \\mathbb {R}^{d\\times f}$ is a weight transformation matrix for each region representations; $\\sigma (\\cdot )$ is sigmoid activation function and $\\Vert $ is the concatenation operation.", "The learnt coefficients $\\alpha _{ij}$ , which is computed by performing softmax on each neighbors, indicates the significance of brain region $i$ to region $j$ .Note that topological attention layer applies on brain region representation at a single timestamp, and multiple topological attention layer can calculate the entire time sequence in parallel." ], [ "Temporal Attention Layer.", "Dynamically capturing constant changing patterns of brain networks is essential for dynamic community detection.", "When extracting the local timestamp features, it is critical to consider the influence of the global temporal context.", "The key question is how to capture the temporal alterations in brain networks structure throughout variety of time steps.", "Temporal attention layer is designed for tackling this issue with the help of the scaled dot-product attention [36].", "Its queries, keys,and values are being used to represent the attributes of input brain regions.", "We define $H_s=\\left\\lbrace h^1_s,h^2_s,...,h^T_s\\right\\rbrace $ , a representation sequence of a brain region $s$ at continuous timestamps as input, where $T$ is the number of time steps.", "And the output of the layer is $Z_s=\\left\\lbrace z^1_s,z^2_s,...,z^T_s\\right\\rbrace $ , a new brain network representation sequence for region $s$ at different timestamp.", "Using $h^t_s$ as the query, temporal attention layer evaluate its historical representations, inquiring the temporal context of the neighborhood around region $s$ .", "Hence, temporal self-attention allows the discovery of relationships between time-varying representations of a brain region across several time steps.", "Formally, the temporal attention layer is computed as: $\\begin{aligned}Z_{s}&=\\beta _{s}\\left(H_{s} W_{s}\\right),\\\\\\beta _{s}^{i j}=\\frac{\\exp \\left(e_{s}^{i j}\\right)}{\\sum _{k=1}^{T} \\exp \\left(e_{s}^{i k}\\right)}, \\quad e_{s}^{i j}&=\\left(\\frac{\\left(\\left(H_{s} W_{q}\\right)\\left(H_{s} W_{k}\\right)^{T}\\right)_{i j}}{\\sqrt{F^{\\prime }}}\\right)\\\\\\end{aligned}$ where $\\beta _s\\in \\mathbb {R}^{T\\times T}$ is the attention coefficient matrix computed by the query-key dot product attention operation;$W_q\\in \\mathbb {R}^{d\\times f}$ ,$W_k\\in \\mathbb {R}^{d\\times f}$ and $W_v\\in \\mathbb {R}^{d\\times f}$ are linear projections matrices which transform representations into a particular space.", "The two attention blocks are calculated in sequence to obtain the final temporal representation, $i,e.$ , the output embeddings $Z$ .", "And It is utilized to reconstruct the brain network topology in the decoder: $\\hat{A}=\\sigma (ZZ^T)$ $\\hat{A}$ is the reconstructed brain functional connection and $\\sigma (\\cdot )$ is still sigmoid function.", "The classic reconstruction loss is defined by the form of cross entropy: $\\mathcal {L}_{RE}=\\sum \\mathbb {E}\\left[A_{i j} \\log \\hat{A}_{i j}+\\left(1-A_{i j}\\right) \\log \\left(1-\\hat{A}_{i j}\\right)\\right]$ In this adversarial model, the main objective is enforcing brain network embeddings $Z$ to match the prior distribution.", "Other naive regularizers push the learned embeddings to conform to the Gaussian distribution rather than capture semantic diversity.", "As a result, conventional techniques to network embedding cannot effectively profit from adversarial learning.", "Therefore, we derive the previous distribution of communities by counting different kinds of modules in the functional brain network that have been confirmed by neuroscience.", "The adversarial model serves as a discriminator by using a three-layer fully connected network to identify whether a latent code drawn from the prior distribution $p_{{z}^{\\prime }}$ (positive samples) or embeddings $z$ from the temporal graph encoder $E$ (negative samples).", "The regularizer will eventually enhance the embedding during the minimax competition between the encoder and the discriminator in the training phase.", "The loss of the encoder(generator) $\\mathcal {L}_G$ and discriminator $\\mathcal {L}_D$ in the adversarial model, defined as follows: $\\mathcal {L}_{G}=-\\mathbb {E}_{x \\sim p_{\\text{data }}}\\log \\mathcal {D}_\\phi \\left(E_\\psi \\left(X,A\\right)\\right)$ $\\begin{aligned}\\mathcal {L}_{D}=&-\\mathbb {E}_{z^{\\prime } \\sim p_{z^{\\prime }}}\\log \\mathcal {D}_\\phi ({z}^{\\prime }) \\\\&-\\mathbb {E}_{x \\sim p_{\\text{data }}}\\log \\left(1-\\mathcal {D}_\\phi \\left(E_\\psi \\left(X,A\\right)\\right)\\right)\\end{aligned}$ in this expression, ${z}^{\\prime }$ is a latent code sampled from the prior distribution $ p_{z^{\\prime }}$ of empirically confirmed brain communities; $\\mathcal {D}_\\phi (\\cdot )$ and $E_\\psi (\\cdot )$ is the above-mentioned discriminator and encoder.", "Formally, the objective of this adversarial learning model can be indicated as a minmax criterion: $\\begin{aligned}\\mathcal {L}_{\\mathcal {AL}} &=\\min _{G} \\max _{D} \\Theta \\left(D, G\\right) \\\\&=\\min _{G} \\max _{D}\\left( \\mathbb {E}_{z^{\\prime } \\sim p_{z^{\\prime }}}\\log \\mathcal {D}({z}^{\\prime };\\phi )\\right.", "\\\\&+\\left.\\mathbb {E}_{x \\sim p_{\\text{data }}}\\log \\left(1-\\mathcal {D} \\left(E\\left(X,A\\right);\\phi \\right)\\right)\\right)\\end{aligned}$" ], [ "Measurable Modularity Loss", "Modularity maximization is a technique for community discovery that is commonly used in the detection of brain modules.", "A partition is regarded high quality (and so has a higher $Q$ score [37]) conceptually if the communities it forms are more dense internally than would be predicted by chance.", "Thus, the partition that gets the maximum value of $Q$ is considered to be a good estimation of the community structure of a brain network.", "This intuition may be expressed as follows: $Q=\\frac{1}{2 m} \\sum _{i j}\\left[A_{i j}-c_{i j}\\right] \\delta \\left(\\omega _{i}, \\omega _{j}\\right)$ here $a_{ij}$ indicates the number of functional connection between region $i$ and $j$ ; $c_{ij}=\\frac{k_ik_j}{2m}$ denotes the estimated number of connections based on a null model where $k_i=\\sum _j A_{ij}$ is a degree of the region $i$ and $2m=\\sum _{ij}A_{ij}$ is overall amount of connections in the brain networks; $\\delta (\\omega _i,\\omega _j)=1$ if $\\omega _i=\\omega _j$ , which means reigon $i$ and reigon $j$ are in the same community and 0 otherwise.", "Inspired by [38], to develop a differentiable objective for optimizing the community assignment matrix $P=softmax(Z)\\in \\mathbb {R}^{N\\times C}$ which represents a matrix of probabilities of brain region attribution to communities, the measurable modularity loss employed by our framework is defined as: $\\mathcal {L}_{\\mathrm {MM}}=\\underbrace{-\\frac{1}{\\sum |A_{ij}|} \\operatorname{tr}\\left(\\mathbf {P}^{\\top } \\mathbf {B P}\\right)}_{\\text{measurable modularity }}+\\lambda \\underbrace{\\left(\\sum _{i}^{C}\\left(\\sum _{j}^{N} P_{i j}-\\frac{1}{C}\\right)^{2}\\right)}_{\\text{regularization }}$ where the modularity matrix $B=A-\\frac{dd^T}{2m}$ ; $C$ is the amount of communities and $N$ is the number of regions in the brain networks.", "The regularization ensures that the model can identify communities of the predicted size.", "Thus, the total loss for the encoder optimization in the train process to obtain better embeddings is sum of the above three loss terms, expressed as follows: $\\mathcal {L}_{total}=\\mathcal {L}_{RE}+\\mathcal {L}_{G}+\\mathcal {L}_{MM}$ In this part, we assess the performance of ATGRL in terms of both dynamic community detection and graph representation learning." ], [ "Dataset Preparation.", "We obtained the dynamic brain network dataset required for the experiment by preprocessing long-term functional MRI images of experimental rats.", "The first preprocessing was carried out in MATLAB utilizing the Statistical Parametric Mapping 8 (SPM8) tool.", "To adjust for head motion, functional signals were aligned and unwrapped, and the mean motion-corrected image was coregistered with the high-resolution anatomical T2 image.", "Following that, the functional data were smoothed using a $3mm$ full-width at half-maximum (FWHM) isotropic Gaussian kernel.", "On the basis of the Wister rat brain atlas, 150 functional network areas were outlined.", "We used magnitude-squared coherence to assess the spectral relationship between regional time series, resulting in a $150\\times 150$ functional connection matrix for each time step, whose members showed the intensity of functional connectivity between all pairs of areas." ], [ "Implementation Details.", "ATGRL was implemented using pytorch backend.", "The training of the network was accelerated by one Nvidia GeForce RTX 2080 Ti.", "The training epoch was set at 500, while the learning rate was set to 0.001 during training.", "To minimize overfitting, Adam [39] was utilized as an optimizer with a weight decay of 0.01.", "We trained the encoder with 2 topological attention layers and 2 temporal attention layers.", "We repeat all trials ten times and average the findings.", "For all datasets and approaches, we set the regularization value to 0.5 and the number of communities at 15." ], [ "Baseline.", "Our approach was compared against the following two kinds of baselines:" ], [ "GAE.", " [40]is recently the most common autoencoder-based unsupervised framework for graph data, in which the encoder is composed of two-layer graph convolutional networks to leverage topological information." ], [ "ARGA.", " [41]is an adversarially regularized autoencoder method that employs graph autoencoder to learn the representations, regularizes the latent codes, and forces the latent codes to match a prior distribution; differing from ours, it used simple Gaussian distribution as the prior distribution." ], [ "Metrics.", "For graph-level metrics, we report average community conductance$\\mathcal {C}$ and modularity$\\mathcal {Q}$ .", "For ground-truth label correlation analysis, we report normalized mutual information (NMI) between the community assignments and labels and pairwise F-1 score between all node pairs and their associated community pairs.", "Table: Community detection performance on rat brain networks dataset by graph conductance 𝒞\\mathcal {C}, modularity 𝒬\\mathcal {Q}, NMI, and pairwise F1 measure." ], [ "Ablation Study.", "As indicated in Table.REF , we conducted ablation research on community detection to evaluate the effectiveness of our proposed encoder and adversarial learning, and three significant outcomes were achieved: 1) In the comparison of graph-level metrics, the k-means based method showed impressive performance on community conductance and the modularity loss based method did better than it on community modularity.", "This is due to the fact that the two algorithms are fundamentally different in terms of optimization; modularity loss originates with the goal of maximizing modularity.", "2) The approach with adversarial regularizer is generally performed well; it represents that adversarial learning does play its role as an auxiliary to graph representation learning.", "3) Our algorithm that replaced the proposed encoder with a two-layer graph convolution encoder performs worse; it shows in some way that our proposed encoder may learn better embeddings to make it perform well.", "Figure: Visualization of dynamic community detection performance within four time steps." ], [ "Visualization of Dynamic Community Detection.", "We illustrated our result of dynamic community detection by Fig.REF .", "It shows the changes in the positional distribution of the three major brain communities detected by our approach with increasing time steps.", "We can see that there is no significant change in the distribution of brain communities at time steps 1 to 2, but there is a more remarkable change at time steps 2 to 3.", "It is because the rats in the original dataset did change their brain network properties and topology due to experimental factors.", "Therefore, the outcomes of the experiment are in line with the neuroscientific truth.", "Figure: Classification result of the proposed graph representaion learning and the competing method on baseline." ], [ "Graph Representation Learning Performance", "We grouped the rat data collected before and after the severe change into two groups and verified whether the model learned efficient graph representations by competing with the state-of-the-art graph representation learning model on classification performance." ], [ "Competing Methods.", "DGI [42] highlights the importance of cluster and representation learning in combination.", "We learn unsupervised graph representation with DGI and two algorithms both run SVM on the final representations as the classifier." ], [ "Metrics.", "Evaluation of diagnostic performance is based on quantitative measures in five key areas: To summarize: 1) accuracy (ACC); 2) area under receiver operating characteristic curve (AUC); 3) Precision (PRE); 4) Recall (REC); and 5) balanced accuracy (BAC).", "Our suggested technique is being evaluated using leave-oneout cross-validation (LOOCV), since we only have a small quantity of data.", "One of the $N$ individuals is omitted from the testing process, and the $N-1$ subjects that remain are used for training purposes only.", "It is the greedy search that sets the hyperparameters in each technique to the optimum values." ], [ "Prediction Results.", "As demonstrated in Fig.REF , our approach achieved generally better results on classification performance.", "In a respect, it verifies that the representations obtained by our method are more strong in the unsupervised learning process." ], [ "Conclusion", "In this research, we propose a novel framework called Adversarial Temporal Graph Representation Learning (ATGRL) for introducing community detection into a deep graph representation learning process directed by an adversarial regularizer.", "In addition to using temporal graph attention encoder to merge input spatial topology features and temporal contextual representation to represent latent variables, adversarial training with a neuroscientific prior is used to deconstruct the embedding space in the ATGRL framework.", "Our method outperformed two unsupervised deep embedding and community identification approaches in dynamic brain network datasets.", "And we obtained better results than the comparison method when using the obtained graph representations for classification, indicating that there is an advantage in graph representation learning that may yield better graph embeddings in the latent space.", "Detailed model discussions were conducted to investigate the proposed ATGRL and the superiority of the encoder and the adversarial regularizer." ] ]
2207.03580
[ [ "A Learn-and-Control Strategy for Jet-Based Additive Manufacturing" ], [ "Abstract In this paper, we develop a predictive geometry control framework for jet-based additive manufacturing (AM) based on a physics-guided recurrent neural network (RNN) model.", "Because of its physically interpretable architecture, the model's parameters are obtained by training the network through back propagation using input-output data from a small number of layers.", "Moreover, we demonstrate that the model can be dually expressed such that the layer droplet input pattern for (each layer of) the part to be fabricated now becomes the network parameter to be learned by back-propagation.", "This approach is applied for feedforward predictive control in which the network parameters are learned offline from previous data and the control input pattern for all layers to be printed is synthesized.", "Sufficient conditions for the predictive controller's stability are then shown.", "Furthermore, we design an algorithm for efficiently implementing feedback predictive control in which the network parameters and input patterns (for the receding horizon) are learned online with no added lead time for computation.", "The feedforward control scheme is shown experimentally to improve the RMS reference tracking error by more than 30% over the state of the art.", "We also experimentally demonstrate that process uncertainties are compensated by the online learning and feedback control." ], [ "Introduction", " Jet-based AM refers to manufacturing techniques in which droplets of a material to be fabricated are deposited onto a substrate based on a input pattern.", "These droplets then solidify (either by polymerization or solidification), creating a solid layer.", "As this is carried out layer after layer, the 3D part develops.", "Like other AM techniques, the additive nature of jet-based AM enables the efficient manufacture of intricate and miniature parts which are otherwise difficult to fabricate.", "Hence, jet-based AM techniques find application in the manufacture of electronics, medical models, and compliant features for robots and synthetic tissues [1], [2], [3], [4].", "A major objective in these applications is to ensure that the droplet deposition results in parts that conform to desired geometry.", "Several studies have focused on heuristically tuning fabrication parameters such as the rate of deposition, spacing between droplets, and temperature to find the suitable process parameters acceptable [5].", "However, these lack a formal relationship between the deposited droplets and the ensuing height profile, which is necessary to optimally control the finished geometry.", "Moreover, solely heuristic tuning precludes the ability to compensate for droplet or layer height uncertainties during the fabrication process.", "The earliest work on 3D part geometry control for jet-based AM was reported in [6] where the authors implemented a so-called greedy geometry feedback scheme on parts made from paraffin wax.", "A clear drawback of the scheme is its heuristic control law and lack of a droplet deposition model.", "Subsequent approaches in geometry-level control have incorporated a model into the control scheme.", "In [7], a stochastic greedy-type control algorithm was employed based on an empirical model.", "However, this model is not generalizable and suffers from poor scalability.", "Other researchers have used simplified linear models for control.", "In [8], the deposited droplets are modelled as a Gaussian distribution and a spatial iterative learning control (SILC) scheme was proposed to minimize geometry tracking error.", "By design, the SILC is limited to feedforward, or feedback control only if the reference profiles for each layer are identical [9].", "In [10] the droplets were assumed to be hemispherical and a predictive feedback controller was proposed and validated in simulation.", "Practical implementation of geometry control in high resolution AM requires the process model to be fairly accurate for feedforward control and requires the layer control input patterns to be synthesized in a timely manner for feedback control [11].", "[12] experimentally demonstrated model predictive control (MPC) using the linear graph-based model presented in [13].", "Though the graph-based model is a substantial simplification of the actual height evolution dynamics, the work showed that by implementing feedback control in a layer-wise fashion, geometries can be accurately tracked, although with increased lead time for computations.", "In this work, we use the physics-guided data-driven model presented in [14].", "The model is a convolutional RNN (convRNN) with a physically interpretable architecture.", "This model accurately predicts the height evolution of parts under various scenarios using sparse data.", "Because of the model's accuracy, we deploy it for feedforward control and show significantly improved feedforward control performance over the state-of-the-art in [12].", "We develop an adaptive control framework in which the model parameters may be updated online while the process is simultaneously controlled in a closed loop fashion.", "This is made possible due to the low data requirement of the convRNN.", "The online learning and control frame work follows a strategy presented in [15].", "In simulation, the model was learned after each layer and the control input pattern for the next layer was computed.", "By using the model developed in [14], we allow the learning to occur after an arbitrary number of layers and generalize a nonlinear MPC for an arbitrary number of prediction layers.", "Moreover, the algorithm is designed efficiently such that no computational lead time, other than that required for measurement, is added to the process and thus may be implemented in practice.", "The paper is organized as follows.", "Table REF summarizes relevant notations.", "In Sec.", "we describe the control problem.", "The following section briefly reviews the state of the art in geometry control in droplet based AM methods.", "In Sec.", "we develop a predictive controller for the process.", "The dynamical stability of the predictive controller is analyzed in Sec.", ".", "Sec.", "discusses a strategy for implementing predictive control in an efficient manner.", "In Sec.", ", we demonstrate both feedforward and feedback (with online learning) control.", "Sec.", "concludes the paper and previews future work.", "Table: Table of relevant notations" ], [ "Problem Description", "Fig.", "REF shows the basic scheme of the jet-based AM system considered in this work, a closed-loop drop-on-demand inkjet 3D printing process.", "A desired geometry $R$ is to be fabricated.", "The geometry is resolved in the horizontal plane into an $n_x$ by $n_y$ grid space to obtained a discretized height distribution (or profile).", "It is additionally sliced horizontally into layers.", "Assume we have just printed layer $L$ and the current height profile is $Y^{(L)} \\in \\mathbb {R}^{n_x \\times n_y}$ .", "Now supposing the desired height profile of the reference geometry at layer $L+Z$ ($Z > 0$ ) is $R^{(L+Z)}\\in \\mathbb {R}^{n_x \\times n_y}$ , we aim to determine the sequence of future input patterns $\\mathcal {U}_{(L)}^{(L+Z)} = \\lbrace U^{(L)}, U^{(L+1)},\\hdots , U^{(L+Z-1)} \\rbrace $ , for $U^{({L+{k}})} \\in \\mathbb {R}^{n_x \\times n_y}, {k} \\in [\\!", "[0, Z-1]\\!]", "$ , that achieve the desired height profile.", "Moreover, we desire to compensate for process uncertainties or change in the evolution of the height profile.", "This calls for a feedback control framework in which the system's model may be adaptively updated in real-time.", "To design the control framework, we use the data-driven dynamical model presented in [14] as it lends itself to in-process learning.", "Consider the function $\\Phi $ of [14] parameterized on $\\theta $ such that $\\hat{Y} ^{(L+1)} = \\Phi (\\theta ,{Y} ^{(L)},U^{(L)}) \\in \\mathbb {R}^{n_x \\times n_y}$ .", "The optimal set of parameters $\\theta ^{*}$ is obtained by minimizing the error ${\\Vert {Y}^{(\\ell _f)}-\\hat{Y} ^{(\\ell _f)}(\\theta )\\Vert }_2^2$ using stored data pairs of $\\mathcal {U}_{(\\ell _i)}^{(\\ell _f)} $ and $\\lbrace {Y}^{(\\ell _i)}, {Y}^{(\\ell _f)}\\rbrace $ .", "${Y}^{(\\ell _i)}$ is the measured initial height profile at layer $\\ell _i$ , upon which the input sequence $\\mathcal {U}_{(\\ell _i)}^{(\\ell _f)} $ produces ${Y}^{(\\ell _f)}$ at layer $\\ell _f$ .", "Once $\\theta ^*$ is obtained, $\\Phi $ may be written such that the control input becomes the optimization variable $\\hat{Y} ^{(L+1)} = \\Phi ^*(\\hat{Y} ^{(L)},U^{(L)})$ .", "The optimal control input sequence for layers $L$ to $L+Z$ , $\\mathcal {U}_{(L)}^{(L+Z)*}$ , may now be obtained by minimizing ${\\Vert R^{(L+Z)}-\\hat{Y} ^{(L+Z)}\\Vert }_2^2$ using the pair ($Y^{(L)}$ , $R^{(L+Z)}$ ).", "To compensate for uncertainties after printing some layer $L+Z_u$ ($Z_u \\le Z$ ), feedback information (the height profile ${Y} ^{(L+Z_u)}$ ) may be collected and the control input for the next $Z$ layers may be recomputed.", "To alleviate plant-model mismatch, the set of parameters $\\theta $ is updated using data pairs obtained of $\\lbrace {Y}^{(\\ell _i)}, {Y}^{(\\ell _f)}\\rbrace $ and $\\mathcal {U}_{(\\ell _i)}^{(\\ell _f)} $ during the printing session.", "Subsequently, our two-fold objectives are: (1) to develop a procedure for finding optimal control sequence $\\mathcal {U}_{(L)}^{(L+Z)*}$ ; and (2) to generate an efficient strategy for an online update of set $\\theta ^*$ and implementing feedback control.", "Figure: 3D inkjet printing scheme.", "For each layer, a reference profile based on the desired part geometry is fed to a controller which generates a suitable input sequence." ], [ "Related Work", "In Sec.", ", we note that several strategies have been employed for geometry-level feedback control in jet-based AM processes.", "In this section, we summarily discuss these control strategies and assess the drawbacks that springboard the control framework presented in this work.", "Greedy Feedback Control: Early attempts to implement geometry-level control involved using greedy algorithms to determine when and where to deposit droplets.", "In [6], the next location to deposit a droplet was chosen based on a heuristic score assigned to each location.", "The algorithm is quite limited as it does not consider droplet interaction post-deposition or droplet size/scale.", "In [7] a `stochastic greedy-type' search algorithm is employed to minimize tracking error $\\Vert R-\\hat{Y}^{(L)}\\Vert _2^2$ and surface roughness.", "This algorithm empirically models the effect of neighboring droplets when the edges of the part shrink.", "Consequently, it is not applicable in scenarios where the edges are elevated [14].", "Further, due to the `greedy' search approach employed, the algorithm increases in expense proportionally to the size of the grid resolution.", "Linear Time-Invariant MPC: In [12], following the lifted model of [16], the geometry control objective is formulated as an MPC problem, where a height-tracking cost-function $J$ is to be minimized over a finite receding horizon of $Z$ layers: $\\begin{aligned}& \\underset{U^{(L)}}{\\text{min}}& & J\\left(U^{L}\\right) \\\\& \\text{s.t.", "}& & \\hat{y}^{(L+k+1)} = \\mathcal {A} \\hat{y}^{(L+k)} + \\mathcal {B} u^{(L+k)}, {k} \\in [\\!", "[0, Z-1]\\!", "], \\\\& & & U_{\\text{low}} \\le {U^{L}}\\le U_{\\text{high}},\\end{aligned}$ where $U^{L}=[u^{(L)^T}~...~u^{(L+Z+1)^T}]^T \\in \\mathbb {R}^{nZ}$ , $u^{(L+k)}$ is the $i^{th}$ layer control input in the receding horizon, and $\\hat{y}^{(L)}$ is the current height.", "$(\\mathcal {A}, {\\mathcal {B}})$ models the height evolution from layer to layer: $\\mathcal {A}\\in \\mathbb {R}^{n \\times n}$ is the state matrix that captures the dynamics of the height evolution over the entire layer; ${\\mathcal {B}}\\in \\mathbb {R}^{n \\times n}$ is the input matrix accounting for the height distribution of each droplet on deposition.", "The cost function is designed to penalize tracking error.", "$U_{\\text{low}}$ and $U_{\\text{high}}$ define the upper and lower constraints on the input.", "The optimization is performed each layer, and $u^{(L)^\\star }$ is applied.", "The model $({\\mathcal {A}}, {\\mathcal {B}})$ does not adequately capture nonlinear fluid behavior when droplets overlap.", "Hence, the control approach requires layer-to-layer feedback to adequately compensate for the plant-model mismatch, in addition to compensating for uncertainties.", "Further, if operating conditions change, the model would require offline re-identification.", "Iterative Learning Control: In [8], a spatial iterative learning (SILC) scheme is proposed with the following learning law: $u^{(L+1)} = \\Gamma _uu^{(L)} + \\Gamma _ee^{(L)}$ Here, $\\Gamma _u, \\Gamma _e \\in \\mathbb {R}^{n \\times n}$ are the input and error learning matrices.", "$e^{(L)}$ is the error between the desired and model output: $e^{(L)} ={y}^{(L)} - \\hat{y}^{(L)}$ .", "$\\hat{y}^{(L)}$ is evaluated in a similar way as in (REF ), but with $\\mathcal {A} = I$ .", "As with the linear MPC strategy, $\\Gamma _u, \\Gamma _e$ are evaluated to minimize a height tracking cost function.", "The SILC is limited to feedforward, or feedback control if the reference profiles for each layer are identical [9], [17].", "Though promising, typical demonstrations of the SILC scheme have been limited to simulations.", "The above approaches have been shown to achieve improvement in geometry output [18], [12].", "However, the model performance is limited as the linear models do not well capture the nonlinear fluid behavior when droplets interact.", "Furthermore, it is difficult to propagate any plant-model mismatch (disturbance) since the reference geometry for each layer is unique.", "Hence, it is important to use a model that: (1) more accurately captures the fluid behavior, (2) can be refined online if necessary, and (3) can be used to design a control strategy.", "Next, we employ the model presented in [14] and design a predictive controller based on the model." ], [ "Predictive Control of the Printing Process", "In the physics-guided convRNN model proposed in [14], the model (network) parameters are time invariant, while the input pattern is fed to the network as a time series.", "In this section, we re-express the model such that the input is now time-invariant.", "This re-expression allows us determine the (sub)optimal control input pattern by leveraging the similar gradient expressions as were used for the model identification." ], [ "ConvRNN Model Reformulated for Predictive Control", "In [14], for a given layer $L$ , the time-step to time-step evolution of the height evolution was modeled as: $\\begin{split}&h_{t+1} = \\phi _1(h_{t})+ \\text{vec}~(b * U_{t}), \\hspace{14.22636pt} t \\in [\\!", "[0, N_L-1]\\!", "], \\\\&\\hat{y}^{(L+1)} = \\phi _2(\\phi _1(h_{N_L})), \\hspace{14.22636pt} h_0 = \\hat{y}^{(L)},\\end{split}$ where $h_t \\in \\mathbb {R}^{n}$ , $ n = n_x \\times n_y$ , is the network's internal state (or height).", "The function $\\phi _1 (h_{t})$ is defined as: $\\phi _1 (h_{t}) = h_{t}-D\\sigma (\\kappa D^Th_{t}),$ and denoting $h^{(L)} \\phi _1(h_{N_{L-1}})$ , $\\phi _2 (h^{(L)})$ is defined elementwise as: $\\phi _2(h^{(L)}(i)) = \\log (\\gamma + \\exp {h^{(L)}(i)+v_0}).$ In (REF ), $D \\in \\mathbb {R}^{n\\times n_l}$ is an incidence matrix that transforms the height profile vector into height differences across links where $n_l$ is the number of links.", "These differences are then weighted by a flowability constant $\\kappa $ such that $\\kappa D^Th_t$ is the effective flow across links at time $t$ due height to differences across each link, $D^Th_t$ .", "The activation function $\\sigma $ thresholds the effective height difference, $\\alpha $ , across a link that would cause flow at the time $t$ .", "The soft-thresholding is applied to capture surface tension effect.", "(The reader is referred to [14] for additional details).", "Meanwhile, $\\phi _2$ in (REF ) is a generic softplus function, parameterized on $\\gamma $ , that is applied to the element-wise sum of the internal state $h^{(L)}$ and a negative scalar $v_0$ .", "The softplus function accounts for the curing effect and ensures the output profile $\\hat{y}^{(L)}$ is non-negative.", "In (REF ), $b \\in \\mathbb {R}^{p \\times p}$ is a convolution kernel that represents the effect of a droplet deposition on the height profile.", "$U_t \\in \\mathbb {R}^{n_x \\times n_y}$ , $t \\in [0, N_L-1] $ are the admissible inputs at time step $t$ such that $\\sum _{t=0}^{N_L-1} U_t = U^{(L)}$ .", "The 2D convolution can be expressed as $\\text{vec}(b * U_t) = W_uu_t$ , where $W_u$ is a Toeplitz matrix corresponding to kernel $b$ and $u_t \\in \\mathbb {R}^n$ is $U_t$ vectorized.", "Thus, we can now rewrite (REF ) as: $h_{t+1} = \\phi _1(h_t)+ W_{u,t}u^{(L)}, t \\in [0, N_L-1]$ Here, $u^{(L)}$ is the input vector for the entire layer and $W_{u,t} = W_uI_t$ .", "$I_t$ is a sparse matrix holding a one at each position corresponding to where a deposition may take place: $I_tu^{(L)} = u_t$ .", "Once the model parameters in (REF ) are identified and given a reference profile, we can find a (sub)optimal input for layer $L$ , $u^{(L)*}$ by gradient-based means." ], [ "Predictive Control Using the Reformulated Model", "In [14], the optimal values of the model parameters were determined from input-output data.", "These values were determined via gradient descent direction using back-propagation through time.", "Now, we follow an analogous approach to determine (sub)optimal input, given knowledge of the model parameters and desired output.", "Assume layer $L$ has been printed and we are concerned with the output of the next $Z$ layers.", "We define cost function $J({U^{L}}) \\mathrel {{[1pt]{=}{\\scriptstyle \\Delta }}}\\sum _{i=0}^{Z} J^{(L+i)}$ where: $J^{(L+i)} = {\\left\\lbrace \\begin{array}{ll} {P (r^{(L+Z)}-\\hat{y}^{(L+Z)})}_{2}^{2} & {i = Z}, \\\\ { Q (r^{(L+i)}-\\hat{y}^{(L+i)})}_{2}^{2} + { G u^{(L+i)}}_{2}^{2} &\\textrm {else.}\\end{array}\\right.", "}\\nonumber $ The control optimization problem may be written as: ${U^{L^*}} & = \\underset{{U^{L}}}{\\text{argmin }}J({U^{L}}) \\\\\\text{s.t. }", "h_{t_k+1} &= \\phi _1(h_{t_k})+ W_{u,t_k}u^{(L)}, t_k \\in [\\!", "[t_{0_k}, N_{L+k}-1]\\!", "],\\\\\\hat{y}^{(L+k+1)} & = \\phi _2(\\phi _1(h_{N_{L+k}})), h_{t_{0_k}} = \\hat{y}^{(L+k)}\\\\0_n & \\leqslant {u^{(L+k)}}\\leqslant 1_nu_{max} ~\\forall k \\in [\\!", "[0, Z-1]\\!", "]\\,$ where $P$ , $Q$ , $G$ $\\in \\mathbb {R}^{n\\times n}$ are weighting matrices, and $u_{max}$ is an upper bound to the values in ${U^{L*}}$ .", "The total derivative of $J^{(L+i)}$ with respect to $U^{L}$ is: $ &\\dfrac{dJ^{(L+i)}}{{dU^{L}}}= {J^{(L+i)}}{\\hat{y}^{(L+i)}} {\\hat{y}^{(L+i)}}{{U^{L}}} + {J^{(L+i)}}{{U^{L}}}\\nonumber \\\\&={J^{(i)}}{\\hat{y}^{(L+i)}} \\sum \\limits _{j=0}^{i}\\sum \\limits _{t_{j}=t_{0_{j}} }^{N_{j}}{\\hat{y}^{(L+i)}}{h_{t_j}} {h_{t_j}}{{U^{L}}} + {J^{(L+i)}}{{U^{L}}}.", "\\quad $ The terms in (REF ) may be expressed as: ${J^{(L+i)}}{\\hat{y}^{(L+i)}} &= {\\left\\lbrace \\begin{array}{ll} 2({r^{(L+Z)}}-\\hat{y}^{(L+Z)})^{T}P ~\\textrm {for } i = Z,\\\\2({r^{(L+i)}}-\\hat{y}^{(L+i)})^{T}Q ~\\textrm {otherwise },\\end{array}\\right.", "}\\\\{\\hat{y}^{(L+i)}}{h_{t_j}} &= {\\hat{y}^{(L+i)}}{h_{N_{L+i}}} {h_{N_{L+i}}}{\\hat{y}^{(L+i-1)}} \\cdots {\\hat{y}^{(L+j)}}{h_{N_{L+j}}} {h_{N_{L+j}}}{h_{t_j}},\\\\{h_{t_j}}{{U^{L}}} &= W_{t_j} = \\begin{bmatrix}0_{n\\times n}^{(0)} \\cdots 0_{n\\times n}^{(i-1)} W_{u,t_i} \\cdots 0_{n\\times n}^{(Z-1)}\\end{bmatrix},\\\\{J^{(L+i)}}{{U^{L}}} &= {\\left\\lbrace \\begin{array}{ll} 0 &~\\textrm {for } i = Z, \\\\u^{(i)^T}G &~\\textrm {else; }\\end{array}\\right.", "}$ where: ${\\hat{y}^{({L+j})}}{h_{N_{L+j}}} &= diag(1/(1+\\gamma \\exp {1v_0 - {h}_{N_{L+j}}})),\\\\{h_{N_{L+j}}}{h_{t_j}} &= \\prod _{N_{L+j}-1>k\\geqslant {t_j}}\\bigg ( I - Ddiag (\\sigma ^{\\prime }(l_{k}))KD^T\\bigg ).$ Then we can write that ${\\hat{y}^{(L+i)}}{{U^{L}}} = {\\hat{y}^{(L+i)}}{h_{N_{L+i}}} {h_{N_{L+i}}}{{U^{L}}}$ where: $\\hspace{-427.0pt}{h_{N_{L+i}}}{{U^{L}}} = {\\left\\lbrace \\begin{array}{ll} 0 ~ \\textrm {for } i = 0, \\\\\\sum \\limits _{t_{1}=t_{0_{i_1}} }^{N_{L+1}} {h_{N_{L+1}}}{h_{t_{1}}} W_{t_{1}} ~ \\textrm {for } i = 1,\\\\\\sum \\limits _{t_{i}=t_{0_{i}} }^{N_{L+i}} {h_{N_{L+i}}}{h_{t_{i}}} W_{t_{i}} + {h_{N_{L+i}}}{\\hat{y}^{(L+i-1)}} {\\hat{y}^{(L+i-1)}}{{U^{L}}} ~ \\textrm {else};\\end{array}\\right.", "}\\hspace{1000.0pt}$ Note that since the gradients are expressed analytically rather than as finite differences, the computation to find ${U^{L^*}}$ is expedited.", "We use the sequential quadratic programming method in which the Hessian is estimated using the Broyden–Fletcher–Goldfarb–Shanno algorithm and a penalty function is used to enforce the constraint [19], [20].", "The approach is implemented using MATLAB's fmincon function." ], [ "Predictive Control Stability", "In this section, we analyze the stability of the MPC problem.", "For this analysis, we neglect the shrinkage effect in the output equation ($y = h_{N_L +1}$ ).", "We follow the Lyapunov's direct method [21], where the cost function of the finite horizon optimization problem is used to establish stability [22], [23], [24].", "The height evolution from time step to time step within layer $L$ can be written as: $&{h_{t+1} = \\phi _1 (h_{t}) + B_t{u_{t}}, \\hspace{14.22636pt} t \\in [\\!", "[0, N_L-1]\\!", "],} \\\\&{\\hat{y}^{(L+1)} = \\phi _1 (h_{N_L}), h_0 = \\hat{y}^{(L)},}$ where $B_t = (1/ {u_{t}})W_uu_t \\in \\mathbb {R}^{n}$ , $\\phi _1 (h_{t}) = (I -DK_{\\delta }(h_t) D^T)h_{t} = A(h_{t})h_{t}$ and $K_{\\delta } (h_t) = diag(k_{\\delta }) \\in \\mathbb {R}^{n_l \\times n_l}$ .", "Each diagonal element is defined as: $k_{\\delta }(i) \\mathrel {{[1pt]{=}{\\scriptstyle \\Delta }}}{\\left\\lbrace \\begin{array}{ll}\\kappa \\big (1 - (1-\\delta )\\alpha /l_t(i)\\big ) & \\text{if } l_t(i) > {\\alpha },\\\\\\delta \\kappa & \\text{if } {-\\alpha } \\leqslant {l_t(i)} \\leqslant {\\alpha },\\\\\\kappa \\big (1 + (1-\\delta )\\alpha /l_t(i)\\big ) & \\text{if } l_t(i) < {-\\alpha },\\end{array}\\right.", "}$ with $l_t = \\kappa D^Th_t \\in \\mathbb {R}^{n_l}$ and $\\underline{\\kappa } < \\kappa < \\bar{\\kappa }$ ($\\underline{\\kappa }$ and $\\bar{\\kappa }$ are lower and upper bounds on the flowability constant).", "We can lift the time-step height evolution of (REF ) for each layer to yield a layer-to-layer height evolution model to obtain $\\hat{y}^{(L+1)} = \\mathcal {A}_L\\hat{y}^{(L)}+ \\mathcal {B}_L u^{(L)}$ where $\\mathcal {A}_L \\in \\mathbb {R}^{n\\times {n}}$ is $\\prod _{i=N_L}^{0}A(h_{i})$ , $\\mathcal {B}_L \\in \\mathbb {R}^{n\\times {N_L}}$ is $\\Big [\\begin{matrix}(\\prod _{i=N_L}^{1}A(h_i))B_0 & (\\prod _{i=N_L}^{2}A(h_i))B_1 &\\cdots \\end{matrix} \\nonumber \\\\\\begin{matrix}(\\prod _{i=N_L}^{t+1}A(h_i))B_t & (\\prod _{i=N_L}^{t+2}A(h_i))B_{t+1}\\end{matrix} \\cdots & A_{N_L}B_{N_L-1} \\Big ], \\nonumber $ and we denote $\\begin{bmatrix}{u_0} & {u_1} & \\cdots & {u_t} & \\cdots & {u_{N_L -1}}\\end{bmatrix}^T \\equiv u^{(L)}$ .", "Note that $\\mathcal {A}_L$ and $\\mathcal {B}_L$ are functions of $\\hat{y}^{(L)}$ and $u^{(L)}$ , and may be explicitly written as $\\mathcal {A}(\\hat{y}^{(L)},u^{(L)})$ and $\\mathcal {B}(\\hat{y}^{(L)},u^{(L)})$ respectively.", "For brevity, we retain earlier notations.", "Let error $e^{(L)} = r^{(L)}-\\hat{y}^{(L)}$ , where $r^{(L)}$ is the reference height profile for layer $L$ .", "We assume there exists an ideal control input $u^{*(L)}$ such that $r^{(L+1)} = \\mathcal {A}_Lr^{(L)} + \\mathcal {B}_Lu^{*(L)}$ .", "The closed-loop MPC system is now defined as $e^{(L+1)} = \\mathcal {A}_Le^{(L)} + \\mathcal {B}_Lw^{(L)}$ , where $w^{(L)} = u^{*(L)} - u^{(L)}$ .", "The control objective is to minimize the following cost over the next $Z$ layers: $\\begin{split}& J({W_{L}}) \\mathrel {{[1pt]{=}{\\scriptstyle \\Delta }}}(e^{({L+Z})})^T P e^{({L+Z})}\\\\& \\hspace{10.0pt} + \\sum _{k=0}^{Z-1} (e^{({L+i})})^T Q e^{({L+i})} + (w^{({L+i})})^TG w^{({L+i})}.\\end{split}$ Stability Lemma: The closed-loop MPC system $e^{(L+1)} = \\mathcal {A}_Le^{(L)} + \\mathcal {B}_Lf(e^{(L)})$ , where $f(e^{(L)})$ is the receding horizon control law that associates the optimal input $w^{*(L)}$ to the current state $e^{(L)}$ is stable at the point $e^{(L)} = 0$ if: $P = cP_D$ where $c$ is some constant and $P_D$ is a diagonal matrix such that $0 \\preceq P_D \\preceq I$ , and $-P+Q+c\\bar{A}^T\\bar{A} \\preceq 0$ , where $Q {\\succeq 0}$ and $\\bar{A}=(I-\\delta {\\kappa } DD^T)$ .", "Proof: Define the Lyapunov function: $V(e^{(L)})=\\underset{W_L}{\\textrm {min}}J(W_L)$ .", "Suppose the optimal input sequence is: $\\begin{split}W_L^{\\star }(e^{(L)})&=\\textrm {arg}\\min _{W_L}J(W_L) \\\\ &=\\lbrace w^{*({L})},w^{*({L+1})},\\cdots ,w^{*({L+Z-1})}\\rbrace .\\end{split}$ The following shifted input sequence at layer $L+1$ is: $\\begin{split}\\tilde{W}_{{L+1}}(e^{({L+1})})=&\\lbrace w^{*({L+1})},\\cdots , F_{{L+Z}}e^{({L+Z})}\\rbrace ,\\end{split}$ where $F_{L+Z}$ is some state feedback controller (gain) at $L+Z$ .", "Note $\\tilde{W}_{L+1}$ is not necessarily the optimal input at layer $L+1$ for $e^{(L+1)}$ .", "Let $\\tilde{V}(e_{L+1})=J(\\tilde{W}_{L+1})$ .", "We have: $ {}\\begin{split}&\\tilde{V}(e^{(L+1)})-V(e^{(L)})=-(e^{(L)})^TQe^{(L)}-(u^{(L)})^TGu^{(L)} \\\\&+(e^{(L+Z)})^T \\Big (-P+Q+(F_{L+Z})^TG(F_{L+Z}) \\\\&(\\mathcal {A}_{L+Z}+\\mathcal {B}_{L+Z}F_{L+Z})^TP(\\mathcal {A}_{L+Z}+\\mathcal {B}_{L+Z}F_{L+Z})\\Big )e^{(L+Z)},\\end{split}$ The first two terms on the right-hand-side of (REF ) are non-positive.", "For the case where $F_{L+Z} = 0$ , for any $L+Z$ , we want to show that the third term is also non-positive.", "From Lemma Condition 1, it can be proved that $\\mathcal {A}_{L+Z}^TP\\mathcal {A}_{L+Z} \\preceq c\\bar{A}^T\\bar{A} ~ \\forall ~{L+Z}.$ Since $-P+Q+c\\bar{A}^T\\bar{A} \\preceq 0$ (Lemma Condition 2), we have: $-P+Q+\\mathcal {A}_{L+Z}^TP\\mathcal {A}_{L+Z} \\preceq -P+Q+c\\bar{A}^T\\bar{A} \\preceq 0 ~ \\forall {L+Z}.", "\\nonumber $ Thus, $\\tilde{V}(e^{(L+1)})-V(e^{(L)}) \\le 0 ~ \\forall ~ e_L \\ne 0$ .", "Now, noting that $\\tilde{W}_{L+1}(e^{L+1})$ (for the case $F_{L+Z} = 0$ ) is not necessarily optimal, it follows that: ${V}(e^{(L+1)})-V(e^{(L)}) \\le \\tilde{V}(e^{(L+1)})-V(e^{(L)}) \\le 0 ~ \\forall ~ e_L.$ Proof of Equation (REF ): We want to show that $ \\mathcal {A}_{L}^TP\\mathcal {A}_{L} \\preceq \\bar{A}^T\\bar{A}$ $\\forall L$ .", "Let ${A}_t= I -DK_t D^T$ , where $K_t = K_\\delta (h_t)$ as already defined.", "Recall that $P=cP_D$ , where $P_D \\preceq I$ (Lemma Condition 1), and $\\bar{A} = (I-\\delta {\\kappa } DD^T)$ .", "The inequality may then be written as: ${A_{N_L}}\\cdots {A_0}P_D{A_0}\\cdots {A_{N_L}} \\preceq \\bar{A}^T\\bar{A}$ Let $\\mathcal {S}$ denote the set of all positive definite matrices with spectral radii less or equal to 1, $\\mathcal {S}=\\lbrace S : 0 \\preceq S \\preceq I; S =S^T\\rbrace $ .", "To prove (REF ), we need to show that for any element $M \\in \\mathcal {S}$ , $A_t^TMA_t \\in \\mathcal {S} $ $\\forall ~ t$ .", "[14] establishes that the spectral radius of the Laplacian $DD^T$ , $\\rho (DD^T) \\leqslant 12$ .", "Further, the lower and upper bounds on the flowability constant $\\kappa $ ($\\underline{\\kappa }$ and $\\bar{\\kappa }$ ) were given as 0 and $1/6$ respectively.", "Since $K_t$ is diagonal, $2(K_t - \\delta \\underline{\\kappa } I)^{-1} \\succeq 12I \\succeq D^TD.$ The second inequality follows from $\\rho (DD^T) \\leqslant 12$ .", "Hence: $2(K_t - \\delta \\underline{\\kappa } I) \\succeq (K_t - \\delta \\underline{\\kappa } I)D^TD(K_t - \\delta \\underline{\\kappa } I),$ which evaluates to: $I - 2\\delta \\underline{\\kappa } DD^T & + (\\delta \\underline{\\kappa })^2 DD^T DD^T \\succeq \\nonumber \\\\ & I - 2DK_tD^T + DK_tD^TDK_tD^T.$ Thus, $\\bar{A}^T\\bar{A} \\succeq A_t^TA_t$ for any $t$ .", "By definition, $M \\preceq I$ , therefore: $A_t^TMA_t \\preceq A_t^TA_t\\preceq \\bar{A}^T\\bar{A} \\preceq I ~~ \\forall t.$ Furthermore, for any $M \\in \\mathcal {S}$ , $M \\succeq 0$ and so $A_t^TMA_t \\succeq 0$ ; thus (REF ) can be expanded as: $0 \\preceq A_t^TMA_t \\preceq A_t^TA_t\\preceq \\bar{A}^T\\bar{A} \\preceq I ~~ \\forall t.$ Finally, (REF ) implies that $A_t^TMA_t \\in \\mathcal {S}$ $\\forall t$ .", "This result indicates that to guarantee stability, $P_D$ should be chosen such that $P_D\\succeq \\bar{A}^T\\bar{A}$ , and then, $0\\preceq Q \\preceq P-c\\bar{A}^T\\bar{A}$ ." ], [ "Given an accurate model of the height evolution, the optimal control scheme of Section REF can provide good tracking performance.", "As the number of layers increases however, uncertainty in the printing process may begin to substantially impact the height profile evolution.", "In addition, given that the MPC depends on a data-driven model, we may want to update the model online utilizing data from the current print session (especially if geometry or printing conditions change).", "The computational expense of updating (training) the model and computing new control input may lead to substantial lead times in the fabrication process.", "Therefore, in this section, we discuss a strategy for efficient adaptive feedback control.", "We propose a semi-feedback approach that allows printing and computation to occur simultaneously.", "This strategy considers that the actual process is relatively slow and that necessary computations will be made in parallel with the printing of one or more layers.", "We begin by implementing the MPC similar to the traditional fashion, that is, we calculate input for $Z$ layers in the horizon, implement $Z_u$ layer(s) and obtain feedback; but while recomputing control input for the next $Z$ layers, we proceed to print $Z_d$ layer(s).", "The algorithm is demonstrated in Fig.", "REF for a 6-layer part and proceeds as follows: Given the total number of layers to be printed $T_L$ and control horizon $Z$ , set the data size to be used for training as $\\Delta \\ell $ .", "Also select the number of layers $Z_u$ that will be implemented before obtaining the next feedback measurement, such that $Z_u+ Z_d\\le Z$ and $Z_u> Z_d$ .", "Let $L = 0$ and define $T_s \\mathrel {{[1pt]{=}{\\scriptstyle \\Delta }}}\\left\\lfloor (T_L - (Z_d+Z_u))/Z_u)\\right\\rfloor $ .", "Use the last $\\Delta \\ell $ input layer pairs in the data base to identify the set of model parameters.", "If $\\Delta \\ell = 0$ , assume the parameters are based on linear superposition.", "Calculate the control input for $Z$ layers into the future.", "For $i = {0}\\cdots {T_s}$ : Implement control input for $Z_u$ layers into the future.", "Get feedback measurement of the height profile $L = L + Z_u$ and add to database.", "Proceed to print $Z_d$ more layers.", "Simultaneously, update model parameters using the $\\Delta \\ell $ last input layer pairs in the augmented data base and calculate the control input for the next $Z$ layers, fixing $\\big [u^{(L)^T} \\cdots u^{(L+Z_d-1)^T}\\big ]^T$ already being implemented.", "Set $L \\leftarrow L+Z_d$ .", "If $i = 1$ , set $~Z_u \\leftarrow Z_u - Z_d$ ." ], [ "Experimental Results", "Fig.", "REF shows the experimental setup for control execution.", "Initially, the 3D model of a part is sliced horizontally into layers and an associated motion path for the nozzle is generated, along with droplet deposition locations.", "Motion stages move the build substrate while the nozzle deposits droplets according to the input pattern.", "Once all depositions for a layer are complete, the part is cured under UV light.", "Then a laser sensor measures the layer height profile for feedback control.", "The process is repeated until all layers are printed.", "Table: Printing process parameters" ], [ "Feedforward Control Implementation", "In this subsection, we experimentally implement the feedforward MPC in Sec.", "REF .", "The driving signal applied to the piezoelectric nozzle (see [25] for details), the ink type, substrate, and the grid dimensions used for the experiment are given in Table REF .", "We print four layers of a cross shaped part.", "The convRNN model parameters are learned from this print using the single data pair, $\\lbrace {Y}^{(0)}, {Y}^{(4)}\\rbrace $ and $\\mathcal {U}_{(0)}^{(3)}$ (Fig.", "REF ).", "The identified parameters are given in Fig.", "REF .", "First, we attempt to find a feedforward control input for a similar cross shape.", "We find the input pattern for each layer as specified in Sec REF , solving for $U^0 \\in \\mathbb {R}^{4n}$ with $u_{max} = 2$ .", "Because the system is limited to discrete droplets, the solution is quantized by rounding up to the nearest integer and implemented.", "For comparison, a second cross-shape part with the same reference is printed based on the linear superposition model described in [16].", "The cross-shape parts printed based on the linear superposition and the convRNN models are meshed in Fig.", "REF .", "Note that the sensor measurement gives a finer resolution ($36 \\times 36 $ ) than the above grid size.", "The convRNN-based control yields a $34\\%$ improvement in RMS tracking error over that of the linear superposition.", "Longitudinal sections through the parts (Fig.", "REF ) accentuate the convRNN-based control compensation for elevation of the cross edges.", "We then carry out similar feedforward control for a 4-layered T-shape part (Fig.", "REF ) using only the original identified model parameters $\\theta ^*$ from the cross shape.", "Similar RMS tracking error performance improvement is observed.", "Fig.", "REF shows longitudinal sections through the T-shape part.", "We observe that the convRNN-based controller not only compensates for elevation of the surface edges, but it also rectifies mismatches in the side walls.", "An improvement of the reference tracking for all layers (not shown) is also noted.", "Figure: Longitudinal sections of open and closed loop (online learning & control) profiles after every other layer of an N-shaped part.", "At the top of each subplot is displayed the RMS error of each control approach with respect to the reference.Figure: Comparison of feedforward control performance of the convRNN with the learn & control algorithm for an N-shape part.", "RMS error is based on the reference profile (not shown)." ], [ "Online Learning and Control Implementation", "We demonstrate the learn & control strategy in Sec.", "using an `N' shape printed in the same fashion as the parts in Sec.", "REF .", "We first print 6 layers of the part in open-loop based on the linear superposition model.", "We print 6 additional layers of the part using a feedforward control input that relies on the identified convRNN model $\\theta ^*$ used in Sec.", "REF .", "Fig.", "REF summarizes the results.", "The figure presents longitudinal sections through the printed part every other layer.", "Although the convRNN-based feedforward result outperforms that of the superposition at lower layers, as the number of layers grows, bias in the learned $\\theta ^*$ begin to dominate and convRNN-based feedforward is no longer advantageous.", "Recall that the convRNN feedforward input is based on merely one data pair (the cross-shaped part in Sec.", "REF ).", "Not only does the cross-shape part have fewer layers than the N-shape part, the shrinkage observed in printing this N-shaped part is more significant than for the former.", "Hence, it is imperative to populate the dataset and learn from it as printing proceeds.", "Thus, we now print the same `N' shape with learn & control strategy developed in Sec.", "as illustrated in Fig.", "REF .", "The following parameters are used: $Z = 3,~ Z_u = 2, ~Z_d = 1$ .", "Results for the online learning and feedback control strategy are superimposed on Fig.", "REF (purple line).", "Observe that the feedforward output based on the linear superposition model exhibits the largest RMS error at the 2nd layer (Fig.", "REF ).", "This is because this model only accounts for droplet deposition and does not capture any further dynamics.", "The feedforward convRNN, with a control horizon of $Z=6$ , has the best performance at this layer.", "(Note that the learn & control strategy is solving a different MPC problem with $Z=3$ ).", "The learn & control algorithm takes the lowest RMS error at the 4th layer (Fig.", "REF ) because of feedback.", "Although the feedforward convRNN profile is better laterally aligned with the reference than the linear superposition model, the volume of material being deposited is inadequate.", "By the final layer (Fig.", "REF ), this volume inadequacy results in the greatest deviation from the reference.", "On the other hand, the learn & control feedback algorithm improves the RMS error over the convRNN-based feedforward profile by over $50\\%$ (Fig.", "REF ) and the superposition-based feedforward profile by $45\\%$ .", "Using a 4.1 GHz Intel Core i7 16GB RAM computer, the computational time required for online model update and control calculations is about a half minute or less.", "Meanwhile, the typical print time for a layer is about 4 minutes.", "Since the computations and printing occur simultaneously, no additional lead time is required.", "Discussion: We have demonstrated that we can achieve significant improvement in reference geometry tracking in the inkjet 3D printing process using a feedback control scheme with minimal downtime.", "The height profile need not be measured frequently and the computations required for model training and control learning, though expensive, may be carried out simultaneously with the printing process.", "These benefits are made possible through a physics-guided data-driven model that requires little data for training and is a good predictor of the process dynamics.", "For larger grid sizes, the computational time will grow exponentially while the actual printing time grows linearly [12].", "Hence, future work will address decentralization of the MPC scheme to scale it almost linearly with grid size." ], [ "Conclusions", "In this paper we proposed a novel predictive control method to improve geometry accuracy in jet-based AM.", "Our method improves upon existing linear MPC and iterative learning control methods by using a physics-guided data-driven model that captures the nonlinear fluid behavior of interacting droplets.", "We showed how the nonlinear predictive controller may be synthesized using backpropagation gradients.", "We also established conditions for stability of the controlled system.", "We implemented the feedforward nonlinear MPC scheme and showed it to outperform the state-of-the-art open-loop control for inkjet 3D printing.", "We further developed an efficient online learning and control algorithm that allows for feedback control in real time without adding substantial lead time to the fabrication process.", "The algorithm was also implemented on an inkjet 3D printing system and shown to substantially improve the reference geometry tracking over open-loop printing.", "Future work will aim to distribute the MPC optimization for faster computation and implement the feedback control strategy in multi-material printing." ] ]
2207.03556
[ [ "Gradients of O-information: low-order descriptors of high-order\n dependencies" ], [ "Abstract O-information is an information-theoretic metric that captures the overall balance between redundant and synergistic information shared by groups of three or more variables.", "To complement the global assessment provided by this metric, here we propose the gradients of the O-information as low-order descriptors that can characterise how high-order effects are localised across a system of interest.", "We illustrate the capabilities of the proposed framework by revealing the role of specific spins in Ising models with frustration, and on practical data analysis on US macroeconomic data.", "Our theoretical and empirical analyses demonstrate the potential of these gradients to highlight the contribution of variables in forming high-order informational circuits" ], [ "Bounds for first order gradients of the O-information", "Here we present the proof of the bounds in equation (3) in the main manuscript.", "Let us consider $n$ random variables $\\mathbf {X}^n = (X_1,X_2,\\ldots ,X_n)$ .", "Two popular extensions of the mutual information are the total correlation $\\text{TC}(\\mathbf {X}^n)$ and the dual total correlation $\\text{DTC}(\\mathbf {X}^n)$ $\\text{TC}(\\mathbf {X}^n) = \\sum _{i=1}^n H(X_i) - H(\\mathbf {X}^n); \\qquad \\qquad \\text{DTC}(\\mathbf {X}^n) = H(\\mathbf {X}^n) - \\sum _{i=1}^n H( X_i \\mid \\mathbf {X}^n_{-i})$ where $H(\\cdot )$ is the Shannon entropy.", "The O-information $\\Omega $ of the system is given by the difference: $\\Omega (\\mathbf {X}^n) &\\equiv \\text{TC}(\\mathbf {X}^n) - \\text{DTC}(\\mathbf {X}^n) = (n-2)H(\\mathbf {X}^n) + \\sum _{k=1}^n \\Big [ H(X_j) - H(\\mathbf {X}^n_{-j}) \\Big ].$ The gradient of the O-information (see the main manuscript) is given by: $ \\partial _i\\Omega (\\mathbf {X}^n) &= \\Omega (\\mathbf {X}^n) - \\Omega (\\mathbf {X}^n_{-i}) = (2-n)I(X_i;\\mathbf {X}^n_{-i}) + \\sum _{k=1}^{n-1} I(X_k;\\mathbf {X}^n_{-ik}).$ Analogously we can consider the gradient of the total correlation $\\partial _i \\text{TC}&= \\text{TC}(\\mathbf {X}^n) - \\text{TC}(\\mathbf {X}^n_{-i}) = I(X_i; \\mathbf {X}_{-i}^n) \\ge 0,$ which, being equivalent to a mutual information, satisfies $0 \\le \\partial _i\\text{TC}\\le \\log |\\mathcal {X}|$ , where $|\\mathcal {X}|$ is the cardinality of the largest alphabet in $\\mathbf {X}^n$ .", "On the other hand the gradient of the dual total correlation can be written as a sum of conditional mutual information terms: $\\partial _i \\text{DTC}(\\mathbf {X}^n)&= \\text{DTC}(\\mathbf {X}^n) - \\text{DTC}(\\mathbf {X}^n_{-i}) \\nonumber \\\\&=\\sum _{k=1}^{n-1}\\Big [ H(\\mathbf {X}_{-i}^n) -H(\\mathbf {X}^n_{-ki}) -H(\\mathbf {X}^n) + H(\\mathbf {X}_{-k}^n) \\Big ] \\nonumber \\\\&= \\sum _{k=1}^{n-1} \\Big [ I(X_i; \\mathbf {X}_{-i}^n) - I(X_i; \\mathbf {X}_{-ki}^n) \\Big ] \\nonumber \\\\&= \\sum _{k=1}^{n-1} \\Big [ H(X_k\\vert \\mathbf {X}^n_{-jk}) - H(X_k\\vert \\mathbf {X}^n_{-k}) \\Big ] \\\\&= \\sum _{k=1}^{n-1} I(X_k; X_j \\vert \\mathbf {X}^n_{-jk}) \\ge 0,$ thus implying that $0 \\le \\partial _i\\text{DTC}(\\mathbf {X}^n)\\le (n-1)\\log |\\mathcal {X}|$ .", "Putting these results together one can find that $-(n-2)\\log |\\mathcal {X}| \\le \\partial _i \\Omega (\\mathbf {X}^n) \\le \\log |\\mathcal {X}|.$ The reminding of the proof demonstrates these bounds and their tightness.", "The upper bound is trivial, indeed we have already shown that $\\partial _i\\text{DTC}(\\mathbf {X}^n) \\ge 0$ , hence: $\\partial _i \\Omega (\\mathbf {X}^n) = \\partial _i \\text{TC}(\\mathbf {X}^n) - \\partial _i \\text{DTC}(\\mathbf {X}^n) \\le \\partial _i \\text{TC}(\\mathbf {X}) \\le \\log \\vert \\mathcal {X}\\vert .$ The tightness of the upper bound can be proven by showing that is achieved by the $n$ -COPY gate, specifically by taking $X_1$ as a Bernoulli variable with $p = 1/2$ and $X_1 = X_2 = \\dots = X_n$ .", "Since we have that $I(X_i; \\mathbf {X}^n_{-i}) = 1$ and $I(X_i; \\mathbf {X}^n_{-ik}) = 1$ for $i=1,2,\\ldots ,n$ , using Equation (REF ) it follows that $\\partial _i\\Omega (\\mathbf {X}) = (2-n) + (n-1) = 1 ,\\qquad i = 1,2,\\ldots ,n.$ This covers the case of binary random variables, but the result can be readily generalized to $\\vert \\mathcal {X}\\vert > 2$ .", "To prove the lower bound is a little more tricky: we start noting that $\\partial _i\\Omega (\\mathbf {X}^n)$ can be written as a sum of conditional interaction information terms.", "Indeed it has been shown in [10] that O-information can be decomposed as a sum of interaction information terms $\\Omega (\\mathbf {X}^n) = \\sum _{k = 2}^{n-1} I(X_k; \\mathbf {Y}^{k-1}_1; \\mathbf {Y}_{k+1}^n),$ where we used the notation $\\mathbf {Y}^{q}_k = (X_k, \\dots , X_{q})$ .", "For definiteness we fix $i = 1$ , obtaining $\\nonumber \\partial _1\\Omega (\\mathbf {X}^n) &= \\Omega (\\mathbf {X}^n) - \\Omega (\\mathbf {X}^n_{-1}) = \\sum _{k = 2}^{n-1} I(X_k; \\mathbf {Y}^{k-1}_1; \\mathbf {Y}_{k+1}^n) - \\sum _{k = 3}^{n-1} I(X_k; \\mathbf {Y}^{k-1}_2; \\mathbf {Y}_{k+1}^n) \\\\\\nonumber &= I(X_2; X_1; \\mathbf {Y}_3^n) + \\sum _{k = 3}^{n-1} \\left[I(X_k; \\mathbf {Y}^{k-1}_1; \\mathbf {Y}_{k+1}^n) - I(X_k; \\mathbf {Y}^{k-1}_2; \\mathbf {Y}_{k+1}^n)\\right] \\\\&= \\sum _{k = 2}^{n-1} I(X_k; X_1; \\mathbf {Y}_{k+1}^n | \\mathbf {Y}_2^{k-1}).", "$ Now we notice that each term in the sum can be written as a difference of two conditional mutual information terms (bounded between 0 and $\\log \\vert \\mathcal {X}\\vert $ ), hence each term has the following bounds $-\\log \\vert \\mathcal {X}\\vert \\le I(X_k; X_1; \\mathbf {Y}_{k+1}^n | \\mathbf {Y}_2^{k-1}) \\le \\log \\vert \\mathcal {X}\\vert .$ This implies that $-(n-2)\\log _2\\vert \\mathcal {X}\\vert \\le \\partial _1\\Omega (\\mathbf {X}^n),$ thus proving the lower bound.", "Finally, to prove that the lower bound is tight, we consider the $n$ -XOR gate, that is $X_1 \\dots X_{n-1}$ as Bernoulli random variables with $p = 1/2$ and $X_n = (\\sum _{j = 1}^{n-1} X_j ) \\ \\text{mod} \\ 2$ .", "Using Equation (REF ) we have $I(X_i; \\mathbf {X}_{-i}^n) = 1$ and $I(X_k; \\mathbf {X}_{-ik}^n) = 0$ then $\\partial _i\\Omega = (2-n), \\qquad i = 1,2,\\ldots ,n$ Then the lower bound is tight, since is achieved by the variables composing a $n$ -XOR gate." ], [ "O-information of US macroeconomic indicators: triplets and quadruplets ", "Concerning the US economic data set, we report here the conventional O-information analysis, taking into account triplets and quadruplets of variables.", "We first obtain all the triplets which are significantly synergistic and those which are significantly redundant.", "Then, for each variable, we sum $\\Omega $ over all the redundant and significant triplets which contain that variable, obtaining $R_\\Omega $ which is an index of redundancy of that variable.", "The same is done summing over synergistic triplets, thus leading to an index of synergy of that variable $R_\\Omega $ : the results are shown in figure REF top-left, where these indexes are compared with the first order gradient as found by the proposed approach.", "Analogously, for each pair of variables we sum over the triplets containing that pair, and obtain a synergy index and a redundancy index for all pairs of variables, depicted in figureREF top-middle and compared with second order gradients.", "In figureREF top-right the distribution of $\\Omega $ for all the significant triplets.", "In the second row of figureREF the same quantities are calculated using quadruplets: we stress that no synergistic quadruplet is found to be statistically significant.", "These results show that, as far as the redundancy is concerned, the proposed approach leads to a pruning of redundant pairs of variables w.r.t.", "the index $R_\\Omega $ , as shown by the vertical cloud of red points around $\\partial _{ij}^2 \\Omega =0$ .", "Moreover, the redundant pattern of $R_\\Omega $ is quite stable going from triplets to quadruplets.", "On the other hand, as far as the synergy is concerned, $\\partial _{i} \\Omega $ seems unrelated to $S_\\Omega $ calculated on triplets.", "Figure: The first row depicts the redundancy index R Ω R_\\Omega and the synergy index S Ω S_\\Omega in the univariate (left) and pairwise (middle) case, see the text, plotted as a function the first order gradients and second order gradients, respectively, of the corresponding variable or pair of variables; in the right we depict the distribution of the O-information values of all the significant triplets.", "In the second row the same analysis has been shown for the significant O-information quadruplets.", "Red and blue dots indicates R Ω R_\\Omega and S Ω S_\\Omega , respectively." ] ]
2207.03581
[ [ "Interlayer excitonic insulator in two-dimensional double-layer\n semiconductor junctions: An explicitly solvable model" ], [ "Abstract Excitonic insulators conduct neither electrons nor holes but bound electron-hole pairs, excitons.", "Unfortunately, it is not possible to inject and detect the electron and hole currents independently within a single semiconducting layer.", "However, interlayer excitonic insulators provide a spatial separation of electrons and holes enabling exciton current measurements.", "The problem is that the spatial separation weakens electron-hole pairing and may lead to interlayer exciton disassociation.", "Here we develop an explicitly solvable model to determine an interlayer separation that is strong enough to prevent electron and hole hopping across the layers but still allows for electron-hole pairing sufficient for transition into an interlayer excitonic insulator state.", "An ideal junction to realize such a state would comprise a pair of identical narrow-gap two-dimensional semiconductors separated by a wide-gap dielectric layer with low dielectric permittivity.", "The present study quantifies parameters of such a junction by taking into account interlayer coherence effects." ], [ "Introduction", "The concept of excitonic insulator (XI) dates back to the 60's when the normal insulating ground state was found to be unstable against the formation of electron-hole bound states (excitons) in semiconductors with a narrow bandgap [1], [2], [3].", "The instability emerges as soon as the exciton binding energy exceeds the semiconducting bandgap.", "The resulting state remains insulating for holes and electrons separately but turns out to be able to conduct excitons.", "Unfortunately, low exciton binding energy and lack of separate control over electron and hole populations spoil manifestations of the XI state in bulk semiconductors [4], [5], [6], [7], [8].", "However, the recent advent of two-dimensional (2D) materials has revived the field and led to the interlayer excitonic insulator (IXI) concept [9], [10], [11], [12], [13], [14], [15], [16].", "Figure: (a) Interlayer excitonic insulator in a drag-counterflow geometry with the current densities 𝐣\\mathbf {j} and 𝐣 d \\mathbf {j}_d shown.The interlayer spacer thickness, dd, must be chosen within a certain value range.", "(b) If dd is too large, then electrons and holes are not paired.", "(c) If dd is too small, then the spacer is not able to prevent charge hopping across the layers.", "In this case, the Coulomb interactionmay lead to transition into the conventional single-layer XI phase .The idea is to make use of a double-layer semiconductor structure with a dielectric spacer that prevents the interlayer electron-hole pairs from recombination but allows for strong Coulomb pairing [15], [16], [9], [10], [11], [12].", "Besides higher exciton binding energies in 2D semiconductors, the double-layer configuration makes it possible to realise a drag-counterflow setup [17], [19] with two pairs of contacts for independent control of electron and hole transport, see Fig.", "REF (a).", "However, the requirements for a suitable dielectric spacer are somewhat contradictory.", "On the one hand, the electron-hole attraction across the junction must be much weaker than the intralayer confinement to avoid interlayer charge hopping.", "On the other hand, the electron-hole interactions must be sufficiently strong to ensure stability of the IXI phase state.", "To achieve better interlayer electrical isolation one could increase the spacer thickness $d$ , see Fig.", "REF (b).", "However, increasing $d$ leads to strong reduction of the bare Coulomb 2D Fourier transform, $V_q=2\\pi /q$ , by the form-factor $\\mathrm {e}^{-qd}/\\epsilon $ , where $\\epsilon $ is the dielectric permittivity of the interlayer media.", "The reduction is especially strong for larger in-plane wave vectors $\\mathbf {q}$ relevant for tightly bound excitons [20].", "To achieve stronger electron-hole pairing one could decrease $d$ , see Fig.", "REF (c).", "This increases the interlayer charge hoping probability and gradually reduces the double-layer structure to a bilayer material hosting conventional excitons.", "Hence, even if the IXI state exists at all, it remains stable only within a certain interval of values $d$ limited from below and above by material parameters.", "The quantum mechanical effects add even more interesting physics into the IXI problem.", "The eigenstate of a charge carrier in the symmetric double-layer structure does not obviously coincide with that of a separated layer.", "Once an electron (or a hole) is created in an eigenstate of a given layer its further evolution is governed by the double-layer Hamiltonian.", "The resulting probability density oscillates between two layers, and at certain time points its maximum occurs on the opposite side.", "Hence, the electrons and holes injected into the eigenstates of the separated layers can hop between the layers when evolving in time.", "The phenomenon could be seen as an interlayer coherence that can be suppressed by either the double-layer asymmetry or disorder.", "The main question addressed in the present paper is whether the interlayer coherence between electrons and holes is beneficial for bringing them into the IXI state.", "To answer this question we maximize the coherence effect by employing a perfectly symmetric double-layer structure modeled by a double-delta-shaped out-of-plane confinement.", "We reveal two competing mechanisms: (i) electron-hole pairing with larger in-plane wave vectors making interlayer excitons tightly bound, and (ii) interlayer hopping that hampers formation of the IXI state.", "We find the set of parameters at which the mechanism (i) dominates in symmetric double-layer structures and makes transition into the IXI state possible.", "The paper is organized as follows.", "Section introduces the one-particle framework, Section adds a mean-field treatment of electron-hole pairing, Section provides discussion of asymmetric double-layer structures with different relative permittivities of the dielectric spacer, and Section concludes with a recipe for the IXI using existing 2D materials." ], [ "Single-particle prerequisites", "Let us first describe the double-layer junction at a single-particle level.", "The junction comprises two identical 2D semiconductors separated by a dielectric layer of thickness $d$ , see Fig.", "REF (a,b).", "Each semiconducting layer is described by the effective low-energy 2D Hamiltonian neglecting spin-orbit interactions [21].", "As the semiconductors are 2D, the out-of-plane confinement must be very narrow for each layer.", "Such ultimately narrow confinements are conveniently described by $\\delta $ -shaped potentials.", "The junction must be described as a whole, rather than as a stack of weakly coupled layers, to take into account quantum mechanical effects properly.", "Thus, the model Hamiltonian is written as $H=\\left(\\begin{array}{cc}\\frac{\\hbar ^2 k_z^2}{2m_0}+U(z) +\\frac{\\Delta _\\infty }{2}& \\hbar v_0 (k_x-ik_y)\\\\\\hbar v_0 (k_x+ik_y) & \\frac{\\hbar ^2 k_z^2}{2m_0}+U(z) -\\frac{\\Delta _\\infty }{2}\\end{array}\\right),$ with $U(z)$ given by $U(z)= - u_0\\left[ \\delta \\left(z-\\frac{d}{2}\\right)+ \\delta \\left(z+\\frac{d}{2}\\right)\\right],$ where $\\hbar k_{x,y,z}$ are the components of the electron momentum $\\hbar \\mathbf {k}$ with $\\hbar $ being the Planck constant, $m_0$ is the electron mass, $v_0$ is the band parameter (effective velocity), $u_0$ is the potential depth parameter, and $\\Delta _\\infty $ is the bandgap at $d\\rightarrow \\infty $ .", "The eigenstates of $H$ can be written explicitly as $\\Psi _{1,2}^\\pm =\\psi _{1,2}(z)\\chi ^\\pm (x,y)$ .", "Here, the indices “1,2” stand respectively for the even and odd states, see Fig.", "REF (c,d), “$\\pm $ ” refers to the conduction and valence bands, see Fig.", "REF (e), and the factorized functions read [22], [23] $\\psi _{1,2}(z)=\\frac{B_{1,2}}{\\sqrt{2}}\\left\\lbrace \\begin{array}{ll}(1\\pm \\mathrm {e}^{\\kappa _{1,2}d})\\mathrm {e}^{\\kappa _{1,2}z}, & z\\le -\\frac{d}{2};\\\\\\mathrm {e}^{\\kappa _{1,2}z} \\pm \\mathrm {e}^{-\\kappa _{1,2}z}, & -\\frac{d}{2}< z <\\frac{d}{2};\\\\\\pm (1\\pm \\mathrm {e}^{\\kappa _{1,2}d})\\mathrm {e}^{-\\kappa _{1,2}z}, & z\\ge \\frac{d}{2};\\end{array}\\right.$ where $B_{1,2}=\\sqrt{\\kappa _{1,2}/(\\mathrm {e}^{\\kappa _{1,2}d}\\pm \\kappa _{1,2}d\\pm 1)}$ , $\\kappa _{1,2} = \\frac{m_0u_0}{\\hbar ^2}+\\frac{1}{d}W_0\\left(\\pm \\frac{m_0u_0d}{\\hbar ^2}\\mathrm {e}^{-\\frac{m_0u_0d}{\\hbar ^2}}\\right),$ with $W_0$ being the Lambert function (ProductLog in Wolfram's Mathematica), and $\\chi ^+(x,y)=\\frac{1}{L}\\mathrm {e}^{ik_x x+ik_y y}\\left(\\begin{array}{c}\\cos \\frac{\\gamma }{2}\\\\\\sin \\frac{\\gamma }{2}\\mathrm {e}^{i\\phi }\\end{array}\\right),$ $\\chi ^-(x,y)=\\frac{1}{L}\\mathrm {e}^{ik_x x+ik_y y}\\left(\\begin{array}{c}\\sin \\frac{\\gamma }{2}\\\\-\\cos \\frac{\\gamma }{2}\\mathrm {e}^{i\\phi }\\end{array}\\right),$ where $\\tan \\gamma =2\\hbar v k/\\Delta _\\infty $ , $\\tan \\phi =k_y/k_x$ , and $L$ is the layer size.", "The corresponding eigenvalues are given by $E_{1,2}^\\pm =E_{1,2}(d)\\pm \\varepsilon _k$ , where $\\varepsilon _k=\\sqrt{(\\hbar v k)^2 +(\\Delta _\\infty /2)^2},$ and $E_{1,2}(d) = E_0 \\left[1+\\frac{\\hbar ^2}{m_0 u_0 d}W_0\\left(\\pm \\frac{m_0u_0d}{\\hbar ^2}\\mathrm {e}^{-\\frac{m_0u_0d}{\\hbar ^2}}\\right) \\right],$ with $E_0=-m_0u_0^2/(2\\hbar ^2)$ .", "The two energy branches, $E_{1,2}(d)$ , merge to $E_0$ in the limit of $d\\rightarrow \\infty $ , see Fig.", "REF (d).", "As $d$ descreases, the bands shift in opposite directions forming a type II junction that potentially can host interlayer excitons [24].", "Note, however, that we consider a homojunction, not a heterostructure [24].", "The interlayer bandgap reads $\\Delta _d = \\Delta _\\infty - E_2(d)+E_1(d)$ , see Fig.", "REF (f).", "It naturally reduces when the layers get closer to each other.", "If the Fermi level is fixed, then the left semiconducting layer becomes n-doped, whereas the right one acquires p-doping.", "Note, that such a doping-by-proximity effect is intrinsic for our model.", "The single-particle model is able to indicate the mechanisms potentially hampering IXI formation.", "First of all, the electrons and holes should sit deeply in the respective layers to prevent recombination caused by their mutual attraction.", "Neglecting dependence on $d$ , we can estimate the critical $u_0$ as $\\sim e^2/\\epsilon $ that results in the desirable depth $E_0 \\ll -m_0 e^4/(2\\epsilon ^2 \\hbar ^2)$ .", "This is not a strong criterion in the presence of a dielectric spacer with $\\epsilon \\gg 1$ , see the red line in Fig.", "REF (g).", "It is instructive to consider the quantum mechanical effects leading to electron-hole interlayer hopping.", "The effect of quantum mechanical superposition is especially obvious when the two layers are perfectly identical.", "In this case, an electron (or a hole) is not localised in either layer.", "The position probability density $|\\Psi _{1,2}^\\pm |^2$ is symmetric with respect to $z=0$ for both states 1 and 2, meaning that a position measurement would reveal an electron (or a hole) with the same probability in either layer.", "An electron (or a hole) state created at time $t=0$ in a given layer involves a superposition between $\\psi _1(z)$ and $\\psi _2(z)$ .", "To be specific, consider an electron localised in the left layer and a hole localised in the right layer described, respectively, by the wave functions $\\Psi _L^e(x,y,z) &= &\\frac{1}{\\sqrt{2}}\\left[\\psi _1(z)-\\psi _2(z)\\right]\\chi ^+(x,y),\\\\\\Psi _R^h(x,y,z) &= &\\frac{1}{\\sqrt{2}}\\left[\\psi _1(z)+\\psi _2(z)\\right]\\chi ^-(x,y),$ see Fig.", "REF (c) for $\\psi _{1,2}(z)$ profiles.", "The states are not stationary, and they evolve in accordance with the standard solutions of the time-dependent Schrödinger equation written as $&& \\Psi _L^e(x,y,z,t) = \\\\\\nonumber && \\frac{1}{\\sqrt{2}}\\left[\\psi _1(z){\\mathrm {e}}^{-iE_1^+t/\\hbar }-\\psi _2(z) {\\mathrm {e}}^{-iE_2^+t/\\hbar }\\right]\\chi ^+(x,y),\\\\&& \\Psi _R^h(x,y,z,t) = \\\\\\nonumber && \\frac{1}{\\sqrt{2}}\\left[\\psi _1(z){\\mathrm {e}}^{-iE_1^-t/\\hbar }+\\psi _2(z){\\mathrm {e}}^{-iE_2^-t/\\hbar }\\right]\\chi ^-(x,y).$ Obviously, the probability density $|\\Psi _{L,R}^{e,h}(x,y,z,t)|^2$ oscillates with the period given by $\\tau (d)=\\frac{2\\pi \\hbar }{E_2(d)-E_1(d)}.$ The interlayer probability density oscillation period could also be seen as an interlayer coherence time.", "Within the period $\\tau (d)$ , the electron (or hole) probability density maximum hops back and forth between the layers.", "Hence, an electron (or a hole) injected into one of the two layers in a symmetric double-layer structure can be found in any layer at the a random time point $t\\gg \\tau (d)$ .", "Microscopically, the quantum mechanical “measurement” takes place each time when a charge carrier is trapped by a defect.", "The process is usually associated with non-radiative exciton recombination and occurs in 2D semiconductors [25], [26] at the time scale $\\tau _\\mathrm {nr} \\sim $ $10^{-12}$ – $10^{-10}$ s. Obviously, if $\\tau _\\mathrm {nr}\\gg \\tau (d)$ , then electrons and holes have already hoped between the layers many times before recombining.", "This is the case when the interlayer spacer is thin, see Fig.", "REF (g).", "If XI state can form at all in such conditions, then it should be seen as a single-layer XI, where drag-counterflow measurements are impossible, see Fig.", "REF .", "In contrast, if $\\tau _\\mathrm {nr}\\ll \\tau (d)$ , then electrons and holes remain in the respective layers within the excitonic lifetime, and an XI state, if formed, could be detected in drag-counterflow measurements." ], [ "Many-body model", "We are now ready to write a mean-field Hamiltonian describing the many-body IXI state.", "To do that, we reduce the four-band model shown in Fig.", "REF (e) to a two-band one shown in Fig.", "REF (a).", "The effective model involves one conduction and one valence band hosting electrons and holes in the single-particle states $\\Psi _1^+$ and $\\Psi _2^-$ , respectively.", "We symmetrize the bands placing the zero-energy level in the middle of the interlayer bandgap.", "The mean-field Hamiltonian can be then written as [27] $H_\\mathrm {MF}=\\sum \\limits _\\mathbf {k}\\left(a_\\mathbf {k}^\\dagger b_\\mathbf {k}^\\dagger \\right)\\left(\\begin{array}{cc}\\xi _\\mathbf {k} & -\\Delta _\\mathbf {k}^\\dagger \\\\-\\Delta _\\mathbf {k} & -\\xi _\\mathbf {k}\\end{array}\\right)\\left(\\begin{array}{c}a_\\mathbf {k} \\\\b_\\mathbf {k}\\end{array}\\right),$ where $a_\\mathbf {k}^\\dagger $ ($b_\\mathbf {k}^\\dagger $ ) are the electron creation operators in the conduction (valence) band, $a_\\mathbf {k}$ ($b_\\mathbf {k}$ ) are the respective hole creation operators, $\\xi _\\mathbf {k}=\\varepsilon _k+(E_1(d)-E_2(d))/2$ , and the mean-field parameter reads [27] $\\Delta _\\mathbf {k} = \\sum \\limits _{\\mathbf {k}^{\\prime }}|V_{\\mathbf {k}\\mathbf {k}^{\\prime }}|\\langle a_{\\mathbf {k}^{\\prime }}^\\dagger b_{\\mathbf {k}^{\\prime }} \\rangle .$ Using the Bogolubov transformation $a_\\mathbf {k} & = & c_{1\\mathbf {k}} \\cos \\frac{\\zeta _\\mathbf {k}}{2} - c_{2\\mathbf {k}} \\sin \\frac{\\zeta _\\mathbf {k}}{2}, \\\\b_\\mathbf {k} & = & -c_{1\\mathbf {k}} \\sin \\frac{\\zeta _\\mathbf {k}}{2} - c_{2\\mathbf {k}} \\cos \\frac{\\zeta _\\mathbf {k}}{2},$ with $\\tan \\zeta _\\mathbf {k} =\\Delta _\\mathbf {k}/\\zeta _\\mathbf {k}$ , we arrive at the canonical form of the mean-field Hamiltonian given by $H_\\mathrm {MF}=\\sum \\limits _\\mathbf {k}\\sqrt{\\xi _\\mathbf {k}^2 + \\Delta _\\mathbf {k}^2}\\left( c_{1\\mathbf {k}}^\\dagger c_{1\\mathbf {k}} - c_{2\\mathbf {k}}^\\dagger c_{2\\mathbf {k}}\\right).$ In the low-temperature limit, the mean-field order parameter reads $\\Delta _\\mathbf {k} = \\frac{1}{2}\\sum \\limits _{\\mathbf {k}^{\\prime }} |V_{\\mathbf {k}\\mathbf {k}^{\\prime }}|\\frac{\\Delta _{\\mathbf {k}^{\\prime }}}{\\sqrt{\\xi _{\\mathbf {k}^{\\prime }}^2 + \\Delta _{\\mathbf {k}^{\\prime }}^2}}.$ Equation (REF ) is formally equivalent to the gap equation derived in the seminal paper [3].", "However, $V_{\\mathbf {k}\\mathbf {k}^{\\prime }}$ and $\\xi _{\\mathbf {k}}$ both depend on $d$ in our case.", "Finally, we assume that the order parameter does not depend on $\\mathbf {k}$ and represents the mean-field bandgap, $\\Delta _\\mathrm {MF}$ , which can be found from the gap equation written as $\\frac{1}{2}\\sum \\limits _{\\mathbf {q}}\\frac{V_q}{\\sqrt{\\xi _{\\mathbf {q}}^2 + \\Delta _\\mathrm {MF}^2}}=1.$ Evaluation of $V_q$ must take into account the wave function overlap between the single particle states $\\Psi _1^+$ and $\\Psi _2^-$ .", "The two-particle wave function can be written as an antisymmetric combination of the single-particle states given by $\\nonumber \\Psi ({\\mathbf {k}_1,\\mathbf {r}_1;\\mathbf {k}_2,\\mathbf {r}_2}) &= & \\frac{1}{\\sqrt{2}}\\left[\\Psi _1^+(\\mathbf {k}_1,\\mathbf {r}_1)\\Psi _2^-(\\mathbf {k}_2,\\mathbf {r}_2) \\right.", "\\\\&& \\left.", "-\\Psi _1^+(\\mathbf {k}_2,\\mathbf {r}_2)\\Psi _2^-(\\mathbf {k}_1,\\mathbf {r}_1) \\right],$ where $\\mathbf {r}_{1,2}=(x_{1,2},y_{1,2},z_{1,2})$ are the coordinates of particles 1 and 2, and $\\mathbf {k}_{1,2}$ are their in-plane wave vectors.", "Transition from the state with $\\mathbf {k}_1$ , $\\mathbf {k}_2$ to the state with $\\mathbf {p}_1$ , $\\mathbf {p}_2$ is described by the following matrix element $\\nonumber V_{\\mathbf {p}_1\\mathbf {p}_2\\mathbf {k}_1\\mathbf {k}_2} &&= \\int d\\mathbf {r}_1^3 \\int d\\mathbf {r}_2^3\\Psi ^*({\\mathbf {p}_1,\\mathbf {r}_1;\\mathbf {p}_2,\\mathbf {r}_2})\\\\&& \\times V(\\mathbf {r}_1,\\mathbf {r}_2)\\Psi ({\\mathbf {k}_1,\\mathbf {r}_1;\\mathbf {k}_2,\\mathbf {r}_2}),$ where $V(\\mathbf {r}_1,\\mathbf {r}_2)=e^2/(\\epsilon |\\mathbf {r}_2-\\mathbf {r}_1|)$ .", "The integrand in Eq.", "(REF ) contains four terms, but we have $\\gamma \\sim 0$ for low-energy electrons and holes ($2\\hbar vk/\\Delta _\\infty \\ll 1$ ), and the terms containing spinor products between $\\chi ^+(x_{1,2},y_{1,2})$ and $\\chi ^-(x_{1,2},y_{1,2})$ become negligible, see Eqs.", "(REF ) and (REF ).", "The remaining two terms are equal.", "Neglecting unimportant phase factors we have $V_{\\mathbf {p}_1\\mathbf {p}_2\\mathbf {k}_1\\mathbf {k}_2} \\approx (2\\pi )^2\\delta \\left(\\mathbf {q}-\\mathbf {s}\\right) V_q,$ where $\\mathbf {q}=\\mathbf {p}_1-\\mathbf {k}_1$ , $\\mathbf {s}=\\mathbf {k}_2-\\mathbf {p}_2$ , and $V_q=2\\pi e^2 F_q/(\\epsilon q)$ with $F_q$ given by $F_q= \\int \\limits _{-\\infty }^\\infty d z_1\\int \\limits _{-\\infty }^\\infty d z_2{\\mathrm {e}}^{-q|z_2-z_1|}|\\psi _1(z_1)|^2|\\psi _2(z_2)|^2.$ In the conventional limit of the states localized in the respective layers we can approximate $|\\psi _{1,2}(z_{1,2})|^2=\\delta (z_{1,2}\\pm d/2)$ , and $F_q={\\mathrm {e}}^{-qd}$ .", "If the wave functions $\\psi _{1,2}(z)$ overlap, then $F_q$ differs strongly from ${\\mathrm {e}}^{-qd}$ , as demonstrated in Fig.", "REF (b).", "Introducing $\\nu _q=\\hbar v q$ we rewrite the gap equation (REF ) as $\\int \\limits _0^\\infty d\\nu _q\\frac{F_{\\nu _q/(\\hbar v)}}{\\sqrt{\\Delta _\\mathrm {MF}^2 +\\left(\\sqrt{\\nu _q^2+\\frac{\\Delta _\\infty ^2}{4}}+\\frac{E_1(d)-E_2(d)}{2}\\right)^2}}=\\frac{2}{r_s},$ where $r_s=e^2/(\\epsilon \\hbar v)$ .", "Figure REF (c) shows solutions of Eq.", "(REF ) for different semiconductor bandgaps, with $\\Delta _\\infty =1.6$ eV being relevant for 2D WSe$_2$ [28].", "The velocity $v\\sim 10^7$ cm/s is typical for 2D transition metal dichalcogenides [21], and the spacer is assumed to be made of $h$ -BN with the relative dielectric permittivity of $\\epsilon =6.9$ [29].", "The energy depth is taken to be $E_0= -2$ eV but it could be even deeper up to $E_0\\sim -3$ eV (one-half of the $h$ -BN monolayer bandgap) [28], [30].", "Figure REF (c) demonstrates clearly that the wave function overlap in $F_q$ is crucial for creating an IXI state at a reasonable $d$ .", "If the overlap is neglected, then the solutions of Eq.", "(REF ) shown in Fig.", "REF (c) by dashed curves exist only at $d<0.5$ nm, i.e.", "the critical $d$ is smaller than the monolayer thickness.", "The local maximum of $\\Delta _\\mathrm {MF}(d)$ occurs at the point when $\\Delta _\\infty \\sim E_2(d)-E_1(d)$ .", "It shifts to even smaller $d$ when $\\Delta _\\infty $ increases.", "If the overlap is taken into account, then the critical $d$ shifts towards larger values reaching 2 nm for $\\Delta _\\infty =1$ eV, see solid curves in Fig.", "REF (c).", "Figure REF (d) combines the data shown in Figs.", "REF (c) and REF (g).", "The lower right corner of the phase diagram is the region where IXI state is expected.", "In that region, Eq.", "(REF ) allows for a solution with respect to $\\Delta _\\mathrm {MF}$ , and, at the same time, the interlayer coherence time, $\\tau (d)$ , is much larger than the typical exciton life-time in 2D semiconductors.", "In simple terms, the electrons are holes interact strongly but are well separated." ], [ "Discussion", "Having established the crucial role of the interlayer coherence in the IXI state we now focus on the effects of the double-layer asymmetry, relative dielectric permittivity of the interlayer media, and the size of a semiconductor bandgap.", "The double-layer asymmetry influences the IXI state in two different ways.", "On the one hand, the asymmetry reduces the wave function overlap that results in $F_q\\rightarrow \\mathrm {e}^{-qd}$ making the IXI state harder to reach.", "On the other hand, the asymmetry triggers the collapse of electron and hole wave functions into their respective layers precluding the interlayer coherence effects [22] and improving electron-hole separation beneficial for the IXI phase.", "Note that Eq.", "(REF ) estimating $\\tau (d)$ makes sense for a symmetric double-layer only.", "To quantify the effect of the double-layer asymmetry we introduce $E_{0L,R}=-m_0u_{0L,R}^2/(2\\hbar ^2)$ , and instead of Eq.", "(REF ) we have $U(z)= - u_{0R} \\delta \\left(z-\\frac{d}{2}\\right)- u_{0L} \\delta \\left(z+\\frac{d}{2}\\right).$ The eigenstates of $H$ can be still written as $\\Psi _{1,2}^\\pm =\\psi _{1,2}(z)\\chi ^\\pm (x,y)$ with $\\psi _{1,2}(z)$ given by $\\psi _{1,2}(z)=\\left\\lbrace \\begin{array}{ll}A_{1,2L}\\mathrm {e}^{\\kappa _{1,2}z}, & z\\le -\\frac{d}{2};\\\\B_{1,2}\\mathrm {e}^{\\kappa _{1,2}z} + C_{1,2} \\mathrm {e}^{-\\kappa _{1,2}z}, & -\\frac{d}{2}< z <\\frac{d}{2};\\\\A_{1,2R}\\mathrm {e}^{-\\kappa _{1,2}z}, & z\\ge \\frac{d}{2};\\end{array}\\right.$ where $\\kappa _{1,2}$ are the two solutions of the eigenvalue equation given by $\\nonumber && \\frac{m_0^2 u_{0L} u_{0R}}{\\hbar ^4}-\\left(\\kappa - \\frac{m_0 u_{0R}}{\\hbar ^2}\\right)\\left(\\kappa - \\frac{m_0 u_{0L}}{\\hbar ^2}\\right){\\mathrm {e}}^{2\\kappa d}=0.\\\\$ In contrast to the symmetric case, Eq.", "(REF ) does not allow for an explicit solution but it can be solved numerically [22].", "Figure REF (a) shows the states $\\psi _{1,2}(z)$ in a slightly asymmetric double-layer.", "It is clear that $\\psi _1(z)$ tends to collapse into the left layer whereas $\\psi _2(z)$ does the same into the right one.", "As $\\psi _1(z)$ and $\\psi _2(z)$ describe, respectively, electron and hole states in our many-body model, the carriers having opposite charges turn out to be well separated in space.", "Hence, the IXI state should exist even at smaller separations.", "Figure REF (a) demonstrates, however, that $F_q$ drops significantly even for a slightly asymmetric double-layer and rapidly approaches the ${\\mathrm {e}}^{-qd}$ form.", "As a consequence, the many-body gap equation has a solution only at a very small $d$ , in the region enclosed by the dashed line in Fig.", "REF (d).", "Such a small separation does not have physical sense because it is smaller than the monolayer thickness.", "Hence, an asymmetric double-layer structure made of 2D semiconductors with $\\Delta _\\infty >1$ eV separated by $h$ -BN is not suitable for creating the IXI state.", "A very recent paper [13], however, offers a way to suppress interlayer tunneling in semiconducting bilayers without dielectric spacer.", "There are two obvious options to remedy the situation.", "First, we could reduce $\\Delta _\\infty $ by using a narrow gap 2D semiconductor, such as one of X-enes [28], [31], [32] or $1T$ -TiS$_2$ [33], [34].", "Second, we could substitute $h$ -BN by a dielectric material with a smaller relative dielectric permittivity, such as silicon dioxide ($\\epsilon =3.9$ ) or polyethylene ($\\epsilon =2.25$ ).", "Figures REF (c,d) compares the IXI phase diagrams for the symmetric, Fig.", "REF (c), and asymmetric, Fig.", "REF (d), double-layer junctions.", "The IXI state is supposed to be stable as long as solution of Eq.", "(REF ) exists.", "The interlayer distance is shown in the respective panels.", "The temperature is assumed to be zero.", "Note, however, that decreasing semiconductor bandgap might induce transition to the XI state in each layer separately that would make the desired drag-counterflow setup impossible to implement.", "This transition cannot be described by the simplified two-band many-body model employed above." ], [ "Outlook", "The results discussed above suggest that the interlayer overlap between electron and hole wave functions potentially facilitates a transition into the IXI state.", "Figure REF (d) is among the main results of the present paper offering a recipe for the IXI state using known 2D materials.", "The ingredients are one insulating and two semiconducting layers.", "The semiconducting layers must be identical and possess a direct bandgap of about 1 eV.", "A much larger bandgap would require unrealistically strong interaction or small interlayer separation to bring the double-layer into the IXI state, whereas a much smaller bandgap would convert each layer into the XI state separately.", "At the moment, the best choice seems to be 2D TiS$_3$ with the direct bandgap very close to $\\Delta _\\infty \\sim 1$ eV taken in Fig.", "REF (d).", "The bandgap size has been predicted by means of ab-initio calculations [35] and confirmed experimentally [36].", "There is strong anisotropy in electronic structure [37] but it is not able to spoil the qualitative applicability of the model proposed.", "Moreover, 2D TiS$_3$ can be assembled into heterostructures with other 2D materials, [38] including $h$ -BN [39] employed in Fig.", "REF (d) as a spacer.", "As 2D TiS$_3$ and $h$ -BN have similar work functions of about 5 eV, and a $h$ -BN monolayer has a bandgap of about 6 eV, the resulting band diagram should be similar to that shown in Fig.", "REF (e) with $E_0\\sim -3$ eV.", "Having two or three monolayers of $h$ -BN as spacer would bring the electron-hole system into the desired lower-right corner of the phase diagram in Fig.", "REF (d).", "The black curve separating the single-layer XI from the true IXI state also applies to TiS$_3$ , demonstrating subpicosecond exciton lifetime [40].", "This research is supported by the Ministry of Education, Singapore, under its Research Centre of Excellence award to the Institute for Functional Intelligent Materials (I-FIM, Project No.", "EDUNC-33-18-279-V12)." ] ]
2207.03501
[ [ "Accurate Hellmann-Feynman forces from density functional calculations\n with augmented Gaussian basis sets" ], [ "Abstract The Hellmann-Feynman (HF) theorem provides a way to compute forces directly from the electron density, enabling efficient force calculations for large systems through machine learning (ML) models for the electron density.", "The main issue holding back the general acceptance of the HF approach for atom-centered basis sets is the well-known Pulay force which, if naively discarded, typically constitutes an error upwards of 10 eV/Ang in forces.", "In this work, we demonstrate that if a suitably augmented Gaussian basis set is used for density functional calculations, the Pulay force can be suppressed and HF forces can be computed as accurately as analytical forces with state-of-the-art basis sets, allowing geometry optimization and molecular dynamics to be reliably performed with HF forces.", "Our results pave a clear path forwards for the accurate and efficient simulation of large systems using ML densities and the HF theorem." ], [ "Introduction ", "A pressing issue in contemporary materials simulations is the accurate and efficient first principles calculation of atomic forces for materials with computational unit cells surpassing tens of thousands of atoms.", "A representative class of such materials are protein molecules—some containing beyond 100,000 atoms—where accurate forces would allow for advancements in the understanding of processes such as protein folding [3], [4], [5], [6], [7], [8], [9].", "Regarding periodic systems, accurate simulation of the correlated electronic structure of van der Waals materials like twisted bilayer graphene would require force calculations on superlattices surpassing 40,000 atoms [10], [11], [12], [13].", "Even with adaptations for increased efficiency [14], [15], [16], traditional first principles methods, that is, density functional theory (DFT) [17], [18] is unable to efficiently and accurately compute forces for such large-scale systems.", "Machine learning (ML) has recently emerged as an effective solution to the seemingly intractable problem of accurate computation of properties of large systems.", "The effectiveness of ML methods stems from their ability to extrapolate solutions from simple training data to more complex use cases: traditional first principles methods are only needed for generating the training data for the ML model, while the training configurations are typically orders of magnitude smaller than the systems to which the ML techniques are eventually applied.", "ML models are able to predict properties such as hopping parameters [10] and potential energy surfaces and forces [19], [20], [21] for large molecules and complicated solids.", "While ML models have been trained to compute forces directly [19], the Hellmann–Feynman (HF) theorem presents a promising alternative approach.", "Namely, under the Born–Oppenheimer approximation, the analytic expression for the force acting on nucleus $I$ is $\\vec{F}^E_{I} = - \\frac{\\partial E}{\\partial \\vec{R}_I} = -\\frac{\\partial \\langle \\Psi (\\lbrace \\vec{R}\\rbrace ) | \\hat{H} | \\Psi (\\lbrace \\vec{R}\\rbrace ) \\rangle }{\\partial \\vec{R}_I} = \\vec{F}_I^\\text{HF} + \\vec{F}_I^\\text{Pulay}, $ where the HF term is $\\vec{F}^\\text{HF}_{I} = -\\left\\langle \\Psi (\\lbrace \\vec{R}\\rbrace ) \\left| \\frac{\\partial \\hat{H}}{\\partial \\vec{R}_I} \\right| \\Psi (\\lbrace \\vec{R}\\rbrace ) \\right\\rangle $ and the Pulay term originating from the geometry dependence of the wave function [22] is $\\vec{F}^\\text{Pulay}_{I} = -2 \\left\\langle \\Psi (\\lbrace \\vec{R}\\rbrace ) \\Bigg | \\hat{H} \\Bigg | \\frac{\\partial \\Psi (\\lbrace \\vec{R}\\rbrace )}{\\partial \\vec{R}_I} \\right\\rangle , $ where $\\lbrace \\vec{R} \\rbrace $ is the set of all nuclear positions, $\\hat{H}$ is the electronic Hamiltonian, and $\\Psi $ is the electronic wave function.", "The only terms in the electronic Hamiltonian that depend on the nuclear coordinates $\\vec{R}_I$ are nuclear-nuclear repulsion $\\hat{H}^\\text{nuc-nuc} = \\sum _{IJ} \\frac{Z_{I} Z_{J}}{|\\vec{R}_{I}-\\vec{R}_{J}|} $ and nuclear-electron attraction $\\hat{H}^\\text{nuc-el} =- \\sum _{I} Z_{I} \\int \\frac{ \\rho (\\vec{r})}{|\\vec{r}-\\vec{R}_{I}|} {\\rm d}^3 r, $ where $Z_I$ is the atomic number of nucleus $I$ .", "As these are the only terms that contribute to hft, it is remarkable that the HF force can be computed based solely on the knowledge of the electron density $\\rho $ —in line with the Hohenberg–Kohn theorems [17] of DFT—regardless of the level of sophistication of the employed many-electron wave function.", "Importantly, this also means that if the Pulay force can be suppressed sufficiently, accurate forces on large systems can be computed from an accurate model of the electron density, only.", "Accurate ML density prediction is also simpler than accurate ML force prediction: constructing the training data for ML force prediction requires calculating analytical forces from first principles via analytic, while the HF approach only requires calculations of the electron density to evaluate hft which is considerably less expensive [23], [24].", "Pioneering work by [25] has demonstrated that specially optimized basis sets can suppress the Pulay force.", "In their proof of principle work, [25] showed that atom-centered Slater-type orbital basis sets constructed to be flexible enough to describe the derivative of the wave function $|\\partial \\Psi / \\partial \\vec{R}_{I} \\rangle $ afford noticeably attenuated Pulay forces.", "[25] were able to decrease the Pulay forces to an extent where HF forces became accurate for small molecules at the Hartree–Fock level of theory.", "In this work, we follow [25] and demonstrate that specially optimized atom-centered Gaussian basis sets yield HF forces with an accuracy comparable to analytic forces computed with state-of-the-art Gaussian basis sets.", "In hfbasis, we construct a series of basis sets dubbed $\\sigma $ NZHF (N = Single, Double, Triple) that are optimized to reduce the Pulay term in analytic.", "The performance of these basis sets are then analyzed in results.", "In waterclusters, we carry out an in-depth analysis of total energies, analytical gradients, and HF forces over large cluster configurations of water at room temperature using DFT.", "Our results show that the HF forces computed using the $\\sigma $ DZHF and $\\sigma $ TZHF basis sets yield similar accuracy to analytical forces computed using the size-matched $\\sigma $ NZ [26], cc-pVNZ [1], and pcseg-N [2] basis sets.", "With this error analysis in hand, in dnafragments we move towards a system of practical interest—DNA fragments—where we find that HF forces computed with the $\\sigma $ DHZF basis again yield comparable accuracy to analytic gradient forces with the pcseg-2 and aug-pcseg-2 basis sets.", "We finish in conclusions with a short summary and conclusions.", "The $\\sigma $ NZHF basis sets H, C, N, O, F, P, S and Cl are published in the Supplementary Material and on the Basis Set Exchange [27], [28]." ], [ "Hellmann–Feynman Optimized Basis Set ", "We follow the strategy of [25] in constructing the HF optimized basis sets.", "This approach relies on the observation that the Pulay force can be reduced with a basis that accurately reproduces its own spatial derivatives with respect to nuclear coordinates [29], [30].", "[25] provided a technique where an initial basis set is improved iteratively to reduce the error $\\Delta _\\lambda $ in the basis set projection of the nuclear gradients of the basis functions, see Eq.", "6 of ref. rico2007generation.", "The basis set is extended until the error is suitably small as determined by a parameter $\\epsilon $ , $\\Delta _\\lambda < \\epsilon $ .", "As a result of this procedure, the Pulay forces are reduced to the order of $\\epsilon $ .", "Below we provide a sufficient overview of the basis set construction procedure for the reader, with additional implementation details in the Supplementary Material.", "We start from the family-style sigma basis sets of [26] ($\\sigma $ NZ, N = Single, Double, Triple).", "The $\\sigma $ NZ basis sets have been optimized following the general lines of the procedure of [1] by minimizing the CISD energy of isolated atoms [26].", "The composition of the $\\sigma $ NZ basis sets is shown in tabl.", "The $\\sigma $ NZ basis sets were chosen as the starting point of this work, as the $\\sigma $ NZ basis sets consist of contractions of primitive Gaussians (pGTO) whose exponents are shared by several angular momenta: if a given primitive appears multiplied by a spherical harmonic of quantum number $l = L$ , analogous functions with the same exponent are also present for $0 \\le l < L$ .", "The use of shared exponents results in the inclusion of a great deal of the gradient space in the basis, simplifying the efforts of this work.", "Table: Number of primitive and contracted basis functions for the σ\\sigma NZ and σ\\sigma NZHF (this work) basis sets for the considered first row and second row atoms.N exp N_\\text{exp} denotes the number of unique exponents in the basis set.To develop a basis with a high degree of fulfillment of the HF theorem, we extend the $\\sigma $ NZ basis sets with functions corresponding to the occupied basis functions' derivatives.", "As shown in Eq.", "16 of the Supplementary Material, the derivative of a pGTO with a given $l > 0$ yields three functions with the same exponent as in the pGTO: one function corresponding to $l+1$ , another to $l-1$ , and one function to $l-1$ but bearing an additional factor $r^2$ .", "As the present basis sets employ spherical functions only, the radial factors included in the pGTOs for angular momentum $l$ is $r^l$ .", "Because of this, we remove the additional functions with $r^2$ from consideration, yielding what we call a reduced set of primitives that span the reduced space.", "Our goal is to iteratively improve the reduced space so that the distance between it and the reference space—the space spanned by contractions of $\\sigma $ NZ and their derivatives—becomes smaller than the used threshold $\\epsilon = 10^{-3}$ .", "The procedure proceeds as follows.", "Choose the initial basis set, yielding a reduced space and a reference space.", "Compute $\\Delta _\\lambda $ with the given reduced space and reference space.", "If $\\Delta _\\lambda < \\epsilon $ , stop.", "Otherwise, add additional primitive functions to the reduced set and go back to computedelta.", "Details on computing $\\Delta _\\lambda $ for Gaussian basis functions as well as the decision criteria for exponents and angular momenta of the added pGTOs can be found in Sections II–V of the Supplementary Material.", "Once the improved reduced set of primitives has been formed, we carry out an expansion of the functions of the reference set in an orthogonal basis of the reduced set of primitives to form the final contracted $\\sigma $ NZHF basis sets.", "We find that projecting into the full reduced set of primitives is generally unnecessary.", "Rather, by choosing a suitable subspace of the reduced set of primitives to project into, we can reduce the number of final contracted basis functions in $\\sigma $ NZHF up to 20% for heavier atoms.", "Details of the iterative approach for finding a suitable subspace can be found in Section V of the Supplementary Material.", "The $\\sigma $ NZHF basis sets resulting from this procedure are the final HF basis sets we use in this work.", "The compositions of the $\\sigma $ NZHF basis sets are shown in tabl in terms of primitive and contracted functions; the compositions of the original $\\sigma $ NZ basis sets are also included for comparison.", "It should be noted that additional pGTOs (namely, $s$ and $p$ primitives) were only required for the $\\sigma $ TZHF basis sets.", "The $\\sigma $ NZHF basis sets for H, C, N, O, F, P, S and Cl are provided in the Supplementary Material and can be found on the Basis Set Exchange [27], [28]." ], [ "Water Clusters ", "To determine the efficacy of the new basis sets, we computed HF forces, analytic forces, and total energies using DFT with the PBE0 functional [31], [32] with a custom version of the Psi4 package [33] available at https://github.com/JoshRackers/psi4.", "All DFT calculations are carried out using a (75, 302) quadrature grid and density functionals from Libxc [34].", "Density fitting was employed in the DFT calculations with the def2-universal-jkfit auxiliary basis of [35].", "The aug-cc-pV5Z basis set [1], [36] was used as the reference point of comparison for the accuracy of all other energy and force calculations.", "To understand the effect of the added basis functions for fulfillment of the HF theorem, in addition to the $\\sigma $ NZHF basis sets of this work we also computed energies and forces using the $\\sigma $ NZ basis sets that were the starting point for the $\\sigma $ NZHF basis sets.", "Furthermore, to make a fair comparison to the state-of-the-art in basis set development, we also computed energies and forces using size-matched cc-pVNZ basis sets [1] that are the recommended choice for total energy calculations with post-Hartree–Fock methods, as well as size-matched pcseg-N basis sets [2] that are optimized for accurate self-consistent field total energies.", "tabl2 presents the comparison of basis set sizes for the $\\sigma $ NZHF, cc-pVNZ and pcseg-N basis sets.", "Table: Comparison of basis set sizes for a single water molecule.", "The σ\\sigma NZHF series of this work are contrasted to two commonly used basis sets: pcseg-N and cc-pVNZ.The \"Small,\" \"Medium,\" and \"Large\" categories are used in this work for comparison between various basis sets of similar size.Comparison to the σ\\sigma NZ basis set size is presented in tabl.The comparison between forces was carried out over configurations of water molecules sampled from a classical molecular dynamics (MD) simulation.", "Snapshots were taken from an equilibrated simulation with the HIPPO force field [37].", "The simulation was run using the Tinker molecular dynamics package in the NPT ensemble, at 298 K and 1 atm.", "with a Bussi thermostat and Monte Carlo barostat, a time step of 1 fs, and the RESPA integrator [38].", "2000 snapshots of a 512 molecule periodic box simulation were collected and used to sample cluster geometries for force comparison.", "A total of 50 configurations of 10 molecule clusters were generated from the MD snapshots.", "The procedure to generate the clusters was the following.", "First, a snapshot was chosen uniformly at random from the 2000 available snapshots.", "Second, a molecule in that snapshot was then selected uniformly at random from the 512 available molecules.", "This molecule is the central molecule of the cluster, and the cluster is formed from the central molecule and its 9 nearest neighbors.", "Figure: Error in the computed a) HF forces and b) analytic forces over water clusters using the σ\\sigma NZHF and size-matched energy optimized σ\\sigma NZ, cc-pVNZ and pcseg-N basis sets.The sizes of the basis sets are presented in tabl2.The reference force F ref F^{\\text{ref}} was computed using the aug-cc-pV5Z basis with the analytical gradient technique with a root mean square (RMS) magnitude of 0.71 eV/Å.We begin by comparing the HF force computed using the $\\sigma $ NZ, $\\sigma $ NZHF, cc-pVNZ, and pcseg-N basis sets over water clusters in forcea).", "The errors in the force components $|\\vec{F}_\\text{basis}^\\text{HF} - \\vec{F}_\\text{aug-cc-pV5Z}^{\\text{ref}}|$ are determined against analytic forces computed in the aug-cc-pV5Z basis.", "The distribution of errors is shown with standard box plots against the size of the basis set $N_\\text{bas}$ , where the box indicates the first to third quantiles, the whiskers denote 90% confidence interval, and the center line denotes the median error.", "We find that the $\\sigma $ DZHF and $\\sigma $ TZHF basis sets yield HF forces with errors that are nearly two orders of magnitude smaller than those of the similarly sized $\\sigma $ NZ, cc-pVNZ and pcseg-N basis sets, demonstrating the efficacy of the optimized $\\sigma $ NZHF sets in reducing the Pulay forces.", "It should be noted that this increased accuracy does not appear for the $\\sigma $ SZHF basis — this error is addressed in the following paragraph.", "Analytic gradients are compared in forceb).", "Similarly to the HF force comparison, the smallest basis set $\\sigma $ SZHF does not afford accurate gradients, indicating that the basis set is too small for quantitative electronic structure calculations.", "However, the larger $\\sigma $ DZHF and $\\sigma $ TZHF basis sets afford a 5-fold to 10-fold reduction in the force error relative to the $\\sigma $ NZ and cc-pVNZ basis sets, which are optimized for CISD energies.", "The pcseg-N basis sets have been designed for DFT calculations and perform better than the $\\sigma $ NZ and cc-pVNZ sets, but are nonetheless matched in accuracy by the $\\sigma $ DZHF and $\\sigma $ TZHF basis sets.", "These results indicate that the $\\sigma $ NZHF basis sets are useful not only for HF force calculations, but can also be used for accurate calculations of analytical forces.", "A final important point is the ability of the $\\sigma $ NZHF basis sets in computing the total energy.", "We present these results in energy.", "The $\\sigma $ SZHF basis set is again seen to result in large errors, confirming the analysis made above for the forces.", "However, the $\\sigma $ NZHF basis sets yield fast convergence to the basis set limit with an accuracy similar to or better than that of the pcseg-N basis sets that have been optimized for DFT calculations.", "Both the largest $\\sigma $ NZHF and pcseg-N basis sets afford errors 1–2 orders of magnitudes smaller than the similarly sized $\\sigma $ NZ and cc-pVNZ basis sets that have not been optimized for DFT calculations.", "Figure: Error in the total energy over water clusters using the σ\\sigma NZHF and size-matched σ\\sigma NZ, cc-pVNZ and pcseg-N basis sets.The reference energy E ref E^{\\text{ref}} was computed using the aug-cc-pV5Z basis with mean and standard deviation 20787.02 ±\\pm 0.31 eV." ], [ "DNA Fragments ", "As a demonstration of the power of the HF basis sets on another type of system, we computed the HF forces and analytical gradients on small DNA fragments with the same PBE0 functional, code version and basis sets as in the water cluster study.", "DNA fragments are formed of more elements than water, containing C, N, and P atoms in addition to H and O.", "Therefore, the DNA calculations also allow us to determine the accuracy of the $\\sigma $ NZHF basis sets for a more diverse set of first and second row atoms.", "DNA fragment geometries were obtained from MD simulations performed in a previous study [39] using the Amber 20 package [40] and the BSC1 force field [41].", "DNA was modeled as 12 base pair strands (“12-mers\") in the canonical B [42] form, the most common structural form of DNA.", "To ensure adequate sampling of DNA's conformational space, we included four different 12-mer base sequences.", "Solvent molecules were modeled explicitly as TIP3P water [43] with a Mg2+ and Cl- counterion concentration of about 100 mmol/L.", "Initial DNA structures were first minimized and then allowed to heat up from 0 K to 300 K for 40 ps.", "After heating, 50 ns production runs were performed in the NPT ensemble at 1 atm and 300 K. We used the Langevin thermostat with a collision frequency of 1 ps$^{-1}$ , the Berendsen barostat with a relaxation time of 2 ps, and a time step of 2 fs.", "For each DNA structure, three simulations were performed with different starting trajectories for a combined simulation time of 150 ns.", "Fragment geometries were constructed from production run snapshots by extracting the central two base pairs of the 12-mer sequence and stripping away the rest of the molecule.", "To ensure a representative sampling of DNA structures, we analyzed ten configurations of six types of DNA fragments for a total of 60 structures.", "The six fragments include the four nucleotides (base + sugar + phosphate) with each of the bases (A, C, G, and T), and the two base pair structures (A-T and C-G), each fragment containing around 35 atoms.", "Note that the nucleotides are negatively charged due to the phosphate group whereas the base pair structures are neutral.", "A comparison of the HF forces with the $\\sigma $ DZHF basis and analytical forces in the $\\sigma $ DZHF, pcseg-2 and aug-pcseg-2 basis sets against reference analytical gradients in the aug-pcseg-3 basis set is shown in dna.", "The box plot notation is the same as in waterclusters.", "Figure: Error in the computed forces for the DNA structures, with the σ\\sigma DZHF, pcseg-2 and aug-pcseg-2 basis sets.The reference force F ref F^{\\text{ref}} was computed using the aug-pcseg-3 basis with the analytic gradient technique with RMS magnitudes 0.27 eV/Å, 1.78 eV/Å, 1.85 eV/Å, 1.47 eV/Å and 5.42 eV/Å for the H, C, N, O and P atoms respectively.We begin by noting that there is little difference between the errors in the pcseg-2 and aug-pcseg-2 analytical gradients.", "The lack of difference is likely due to the non-equilibrium nature of the DNA MD samples, which had 26 meV additional energy per mode, washing out the importance of diffuse functions in describing the electronic structure.", "In contrast to our results, equilibrium calculations on DNA have indicated the importance of diffuse functions particularly in describing the ionic phosphate group [44], [45].", "Due to the negligible difference between the pcseg-2 and aug-pcseg-2 forces on our non-equilibrium configurations, we limit the discussion of force errors to the $\\sigma $ DZHF and pcseg-2 basis sets.", "We find excellent agreement for forces computed with the $\\sigma $ DZHF and pcseg-2 bases across all atomic species.", "Analytic gradients computed with $\\sigma $ DZHF have comparable error to those computed with pcseg-2, while $\\sigma $ DZHF HF forces have slightly larger errors than pcseg-2 analytic gradients.", "To quantify the larger errors present in the HF force calculations, we compute the increase in the median errors shown in dna relative to the pcseg-2 analytic gradients.", "For the H, C, N, O and P atoms, we find the median errors for the $\\sigma $ DZHF HF forces are larger by a multiplicative factor of 1.5, 1.8, 1.6, 1.1 and 1.9 relative to the pcseg-2 analytic gradients, all below a factor of 2.", "As the quality of the $\\sigma $ NZHF basis set is contingent on the quality of the starting basis set, systematic improvements to the HF error can be pursued by beginning with a higher quality starting basis set than the $\\sigma $ NZ bases used in this work.", "As a final point, we note the error distribution for oxygen is considerably wider than for the other atoms.", "This reflects the wider range of environments for O in the DNA structures.", "For example, there are neutral oxygen atoms in the A, C, G and T bases while the phosphate group houses the negatively charged oxygen anion [46].", "This can be contrasted with the H, C, N and P atoms, which are only found in their electrically neutral forms, resulting in a smaller error distribution [46]." ], [ "Summary and Conclusions ", "In this work, we construct specially optimized basis sets named $\\sigma $ NZHF by systematically improving the family-style basis set $\\sigma $ NZ [26], and demonstrate that $\\sigma $ NZHF basis sets can be used to suppress the Pulay force in Gaussian basis set calculations.", "The specially optimized $\\sigma $ NZHF basis sets of this work afford HF forces of comparable accuracy to analytical gradients for state-of-the-art Gaussian basis sets for both water clusters and DNA fragments.", "We note that the accuracy of the $\\sigma $ NZHF basis set in this work is contingent on the quality of the starting point basis set, and that systematic reductions in HF force errors can be pursued by beginning with a higher quality basis set than the $\\sigma $ NZ set used in this work.", "By demonstrating that accurate forces can be computed just from an accurate electronic density in line with the Hohenberg–Kohn theorem [17], our work shines light on an interesting path for first principles ML force calculations.", "Instead of focusing on ML models that directly predict the force, one could train an ML density model on first principles densities computed with an appropriately optimized basis set like $\\sigma $ DZHF or $\\sigma $ TZHF.", "Accurate forces can then be computed directly from the predicted ML density using the HF term, hft.", "Indeed, there are already many ML models which can accurately learn densities [47], [48], [49]; however, these models typically predict the density on an auxiliary basis set.", "As such, the last major hurdle to integrating our accurate basis set to ML HF forces is the construction of optimized auxiliary basis sets according to the ordinary basis sets constructed in this work.", "With these optimized auxiliary basis sets in hand—which might be generated automatically [50]—a promising pipeline would emerge for the accurate computation of forces for large-scale systems built on an ML model for the electronic density.", "We hope to follow up with work of this nature in the near future.", "J.A.R., S.P., and A.J.L.", "were supported by the Harry S. Truman Fellowship, and the Laboratory Directed Research and Development and Academic Alliance Programs of Sandia National Laboratories.", "Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.", "We thank LDRD ACORN under OSP number A21-0245 and the Sandia Academic Alliance for supporting this work.", "We thank Sandia National Laboratories and the UNM Center for Advanced Research Computing, supported in part by the National Science Foundation, for providing the high performance computing and large-scale storage resources used in this work.", "This paper describes objective technical results and analysis.", "Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.", "Supplementary Material: Accurate Hellmann–Feynman forces with optimized atom-centered Gaussian basis sets Shivesh Pathak and Joshua Rackers Center for Computing Research, Sandia National Laboratories Ignacio Ema López and Rafael López Fernández Departamento de Química Física Aplicada, Universidad Autónoma de Madrid Alex J. Lee and William P. Bricker Department of Chemical and Biological Engineering, University of New Mexico Susi Lehtola Molecular Sciences Software Institue (Dated: 2023/01/09 06:06:45)" ], [ "Definitions", "Gaussian primitive functions (pGTO) are defined as the product of a Gaussian radial factor, $e^{-\\xi \\, r^2}$ , and an angular factor that can be either a product of powers of Cartesian coordinates, or a regular spherical harmonic.", "In this work, we will use normalized spherical pGTO, $g_{lm}$ , defined as: $g_{lm}(\\xi ,{\\bf r}) = {\\mathcal {N}}_{l,\\xi }^{r} \\; {\\mathcal {N}}_{lm}^\\Omega \\;e^{-\\xi \\, r^2} \\; z_l^m({\\bf r})$ where $z_l^m ({\\bf r})$ are unnormalized regular solid harmonics: $z_l ({\\bf r}) = r^l \\; (-1) \\; P_l^{|m|}(\\cos \\theta )\\;\\left\\lbrace \\begin{array}{lr}\\cos m \\phi & \\hspace*{14.22636pt} (m \\ge 0) \\\\\\sin |m| \\phi & \\hspace*{14.22636pt} (m <0)\\end{array}\\right.$ and ${\\mathcal {N}}_{\\xi _i}$ and ${\\mathcal {N}}_{LM}^\\Omega $ are the radial and angular normalization constants given by: ${\\mathcal {N}}_{l,\\xi }^{r} = 2^{l+1} \\; \\sqrt{\\frac{\\xi ^l}{(2l+1)!!}}", "\\; \\;\\left[\\frac{(2 \\, \\xi )^3}{\\pi }\\right]^{1/4}$ ${\\mathcal {N}}_{lm}^\\Omega = \\left[\\frac{(2l+1) \\; (l-|m|)!", "}{2 \\, (1+\\delta _{m0}) \\;\\pi \\; (l+|m|)!", "}\\right]^{1/2}$ Usual Gaussian basis sets consist of contractions (linear combinations) of Gaussian primitives.", "They are known as contracted Gaussian functions (GTO), $G_{LM}$ , defined as: $G_{lm}({\\bf r}) = \\sum _{i=1}^{N_G} c_i \\; g_{lm,i}({\\bf r})$ where $i$ labels the primitives in the set, and the coefficients $c_i$ are determined by different procedures subject to the normalization of the GTO.", "The number of coefficients and their values depend on the recipe used in the construction of a particular basis set.", "Sigma basis sets ($\\sigma $ BS) are a particular type of GTO BS in which if a given primitive appears multiplied by a spherical harmonic of quantum number $l = L$ , then functions with the same exponent are also present for $0 \\le l < L$ .", "This is an interesting property for the development of atomic BS with a high degree of fulfillment of the Hellmann–Feynman theorem (BSHF), as will be shown later." ], [ "Distance between basis sets", "In the comparison of basis sets, it is very useful to consider the distance between the subspaces generated by them.", "Let $\\mathcal {V}_1$ and $\\mathcal {V}_2$ be two subspaces of the real Hilbert space generated by two GTO basis sets, $\\lbrace G^{(1)}_{i}\\rbrace _{i=1}^{N_G^{(1)}}$ and $\\lbrace G^{(2)}_{i}\\rbrace _{i=1}^{N_G^{(2)}}$ .", "It has been proved that there exist orthonormal basis sets $\\lbrace {\\widetilde{G}}^{(1)}_{i}\\rbrace _{i=1}^{N_G^{(1)}}$ and $\\lbrace {\\widetilde{G}}^{(2)}_{i}\\rbrace _{i=1}^{N_G^{(2)}}$ , such that: ${\\langle \\, }{\\widetilde{G}}^{(1)}_{i} {\\, | \\,}{\\widetilde{G}}^{(1)}_{j} {\\, \\rangle }& = & \\delta _{ij} \\hspace*{28.45274pt}{\\langle \\, }{\\widetilde{G}}^{(2)}_{i} {\\, | \\,}{\\widetilde{G}}^{(2)}_{j} {\\, \\rangle }= \\delta _{ij}\\nonumber \\\\{\\langle \\, }{\\widetilde{G}}^{(1)}_{i} {\\, | \\,}{\\widetilde{G}}^{(2)}_{j} {\\, \\rangle }& = & \\lambda _i \\; \\delta _{ij}$ The orthonormal BS, ${\\widetilde{G}}$ , are related to the original ones, $G$ , by simple linear transformations: ${\\widetilde{G}}^{(1)} = {\\mathcal {D}}^{(1)} \\cdot G^{(1)} \\hspace*{28.45274pt}{\\widetilde{G}}^{(2)} = {\\mathcal {D}}^{(2)} \\cdot G^{(2)}$ The pairs of functions ${\\widetilde{G}}^{(1)}_{i}$ , ${\\widetilde{G}}^{(2)}_{i}$ are partners, and the distance between them can be obtained as: $d_{ii} = {\\, | \\,}{\\widetilde{G}}^{(1)}_{i} - {\\widetilde{G}}^{(2)}_{i} {\\, | \\,}= \\sqrt{2 \\; (1-\\lambda _i)}$ These distances can be used to define the distance, $\\Delta $ , between the subspaces.", "Among the possible definitions for this distance, we choose: $\\Delta ^2 = \\sum _i d_{ii}^2$ From eqs (REF ) and (REF ), it is clear that the closer the $\\lambda _i$ are to 1, the smaller the distance between the subspaces $\\mathcal {V}_1$ and $\\mathcal {V}_2$ .", "Thus, if both BS span the same subspace, all the $\\lambda _i$ are equal to 1 and the distance between them is zero." ], [ "Development of basis sets using a distance criterion", "Let us consider a GTO BS, $G^{(1)}$ , that we will call the reference, and the problem of developing an alternative BS, $G^{(2)}$ , such that the distance between them is minimum.", "This can be useful, for instance, when $G^{(2)}$ is smaller than the reference $G^{(1)}$ , but one wants to reproduce as much as possible the subspace spanned by $G^{(1)}$ .", "It has been proved that this can be done by defining the projector $\\hat{\\mathcal {P}}^{(1)}$ onto $\\mathcal {V}_1$ , whose matrix representation on the $\\mathcal {V}_2$ subspace reads: ${\\bf P}^{(1)} = ( [ \\, {\\bf S}^{(2)} \\, ]^{-1/2} \\, {\\bf M}\\, \\mathbf {C}) \\; ( \\mathbf {C}^\\dagger \\,{\\bf M}^\\dagger \\,[ \\, {\\bf S}^{(2)} \\, ]^{-1/2} )\\equiv {{\\mathcal {M}}} \\; {{\\mathcal {M}}}^\\dagger $ where ${\\bf S}^{(2)}$ is the overlap matrix in the subspace $\\mathcal {V}_2$ generated by $G^{(2)}$ and ${\\bf M}$ is the mixed overlap matrix between elements of $\\mathcal {V}_2$ and $\\mathcal {V}_1$ , whose elements are given respectively by: $S_{ij} = {\\langle \\, }G^{(2)}_{i} {\\, | \\,}G^{(2)}_{j} {\\, \\rangle }\\hspace*{14.22636pt} \\mbox{and} \\hspace*{14.22636pt}M_{ij} = {\\langle \\, }G^{(2)}_{i} {\\, | \\,}G^{(1)}_{j} {\\, \\rangle }$ and $\\mathbf {C}$ is the coefficient matrix that defines an orthonormal set of functions, $G^{ON(1)}$ , in $\\mathcal {V}_1$ : $G^{ON(1)}_i = \\sum _r G^{(1)}_r \\; C_{ri}$ It should be noticed that the matrix ${\\bf M}$ is not symmetric, even it can be non square.", "However, the ${\\bf P}^{(1)}$ is a symmetric square matrix.", "Diagonalizing ${\\bf P}^{(1)}$ by a suitable unitary transformation, ${\\bf R}$ , one obtains the $\\lambda _i$ as the square root of the eigenvalues: ${\\bf R}^\\dagger \\, {\\bf P}^{(1)} \\, {\\bf R}= {\\Lambda }_d^2$ with $\\lambda _i = {\\Lambda }_{ii}$ being the elements of the diagonal matrix ${\\Lambda }_d$ .", "The partner basis of the spaces are given by: ${\\widetilde{G}}^{(2)}_{i} = \\sum _{r=1}^{N_G^{(2)}} G^{(2)}_{r} \\; ( \\, [ \\, {\\bf S}^{(2)} \\, ]^{-1/2}\\, {\\bf R})_{ri}$ ${\\widetilde{G}}^{(1)}_{i} = \\lambda _i \\; \\sum _{r=1}^{N_G^{(1)}} G^{(1)}_{r} \\;( \\mathbf {C}\\, {{\\mathcal {M}}}^\\dagger \\, {\\bf R})_{ri}$" ], [ "Overlap between Gaussian functions and their derivatives.", "In the application of the procedure described in sec to the development of atomic BSHF, the reference BS is taken as a $\\sigma $ BS  plus the derivatives of its functions with respect to the space coordinates.", "Therefore, overlap involving GTO and their derivatives are required.", "As GTO are linear combinations of primitive pGTO, the integrals involving the former can be obtained as linear combinations of integrals among the latter.", "Due to the orthogonality of the pGTO with different $M$ quantum numbers and given the symmetry in atoms, it is sufficient to consider the first derivatives with respect to $Z$ , the remaining integrals involving derivatives to $X$ or to $Y$ being equal to them.", "Let us consider first the derivative of a pGTO with respect to $Z$ Cartesian coordinate, given by: $\\frac{\\partial g_{lm}(\\xi ,{\\bf r})}{\\partial Z} = {\\mathcal {N}}_{l,\\xi }^{r} \\; {\\mathcal {N}}_{lm}^\\Omega \\; \\left[(l+|m|) \\; z_{l-1} ({\\bf r}) \\; \\left( 1 - \\frac{2 \\, \\xi \\, r^2}{2l+1} \\right)-\\frac{(l+1-|m|)}{2l+1} \\; 2 \\xi \\; z_{l+1} ({\\bf r})\\right] \\; e^{-\\xi r^2}$ As it can be seen, this derivative yields functions with $l-1$ and $l+1$ and the same exponent as the original pGTO.", "This fact makes $\\sigma $ BS  specially useful for the development of Hellmann-Feynman BS ($\\sigma $ BSHF), as the introduction of the derivatives does not increase the number of pGTO in the basis set as much as with usual GTO BS.", "It is also important to note that the functions with $l-1$ contain an extra $r^2$ factor that it is not included in the functions of the $\\sigma $ BSHF.", "The different types of overlap integrals required to build matrices ${\\bf S}$ and ${\\bf M}$ and to orthogonalize the $G^{(1)}$ functions, can be obtained with the aid of: $\\int _0^\\infty dr \\; r^{2l+2} \\; e^{-(\\alpha +\\beta ) \\, r^2} =\\frac{(l+1/2)!", "}{2 \\; (\\alpha +\\beta )^{l+3/2}}$ and $\\int _0^\\pi d\\theta \\; \\sin (\\theta ) \\int _0^{2\\pi } d\\phi \\;z_l ({\\bf r}) \\; z_{l^{\\prime }}^{m^{\\prime }}({\\bf r})=\\delta _{ll^{\\prime }} \\; \\delta _{mm^{\\prime }} \\; r^{2l} \\;\\frac{2 \\; (1+\\delta _{m0}) \\; \\pi \\; (l+|m|)!", "}{(2l+1) \\;(l-|m|)!", "}$ Thus, the overlap integral of a pair of pGTO is given by: ${\\langle \\, }g_{lm}(\\xi ) {\\, | \\,}g_{l^{\\prime }m^{\\prime }}(\\xi ^{\\prime }) {\\, \\rangle }={\\mathcal {N}}_{l,\\xi }^{r} \\; {\\mathcal {N}}_{l^{\\prime },\\xi ^{\\prime }}^{r} \\; {\\mathcal {N}}_{lm}^\\Omega \\; {\\mathcal {N}}_{l^{\\prime }m^{\\prime }}^\\Omega \\;\\delta _{ll^{\\prime }} \\; \\delta _{mm^{\\prime }} \\;\\frac{2 \\; (1+\\delta _{m0}) \\; \\pi \\; (l+|m|)!", "}{(2l+1) \\; (l-|m|)!}\\;\\frac{(l+1/2)!", "}{2 \\; (\\xi +\\xi ^{\\prime })^{l+3/2}}$ the integral involving a pGTO and the derivative of a pGTO with respect to $Z$ , by $\\left\\langle \\frac{\\partial }{\\partial Z} g_{lm}(\\xi ) \\Bigl | g_{l^{\\prime }m^{\\prime }}(\\xi ^{\\prime }) \\right\\rangle & = &{\\mathcal {N}}_{l,\\xi }^{r} \\; {\\mathcal {N}}_{l^{\\prime },\\xi ^{\\prime }}^{r} \\; {\\mathcal {N}}_{lm}^\\Omega \\; {\\mathcal {N}}_{l^{\\prime }m^{\\prime }}^\\Omega \\;\\delta _{mm^{\\prime }} \\;\\frac{2 \\; (1+\\delta _{m0}) \\; \\pi \\; (l^{\\prime }+|m|)!", "}{(2l^{\\prime }+1) \\;(l^{\\prime }-|m|)!}", "\\;\\nonumber \\\\& \\times &\\frac{(l^{\\prime }+1/2)!", "}{2 \\; (\\xi +\\xi ^{\\prime })^{l^{\\prime }+3/2}} \\;\\left\\lbrace \\delta _{l-1,l^{\\prime }} \\; (l+|m|) \\; \\frac{\\xi ^{\\prime }}{\\xi +\\xi ^{\\prime }}-\\delta _{l+1,l^{\\prime }} \\; 2 \\xi \\; \\frac{(l+1-|m|)}{2l+1}\\right\\rbrace $ and the integrals involving the derivatives with respect to $Z$ of two pGTO, by: $\\left\\langle \\frac{\\partial }{\\partial Z} g_{lm}(\\xi ) \\Bigl | \\frac{\\partial }{\\partial Z} g_{l^{\\prime }m^{\\prime }}(\\xi ^{\\prime }) \\right\\rangle & = &{\\mathcal {N}}_{l,\\xi }^{r} \\; {\\mathcal {N}}_{l^{\\prime },\\xi ^{\\prime }}^{r} \\; {\\mathcal {N}}_{lm}^\\Omega \\; {\\mathcal {N}}_{l^{\\prime }m^{\\prime }}^\\Omega \\;\\delta _{mm^{\\prime }} \\;2 \\; (1+\\delta _{m0}) \\; \\pi \\nonumber \\\\& \\times &\\left\\lbrace \\frac{(l+|m|)!", "}{(2l-1) \\;(l-1-|m|)!}", "\\; \\frac{(l-1/2)!", "}{2 \\; (\\xi +\\xi ^{\\prime })^{l+3/2}}\\right.\\nonumber \\\\& \\times &\\left[\\delta _{ll^{\\prime }} \\; (1 - \\delta _{l \\, |m|}) \\; (l+|m|) \\; \\frac{2l+3}{2l+1} \\;\\frac{\\xi \\; \\xi ^{\\prime }}{\\xi +\\xi ^{\\prime }}-\\delta _{l-1 \\, l^{\\prime }+1} \\; \\frac{(l^{\\prime }+1-|m|) \\;2 \\xi ^{\\prime 2}}{(2l^{\\prime }+1)}\\right]\\\\ \\nonumber & + &\\frac{(l+1+|m|)!", "}{(2l+3) \\;(l-|m|)!}", "\\; \\frac{(l+3/2)!", "}{2 \\; (\\xi +\\xi ^{\\prime })^{l+5/2}}\\\\ \\nonumber & \\times &\\left.\\left[-\\delta _{l+1 \\, l^{\\prime }-1} \\; \\frac{(l^{\\prime }+|m|) \\; 2 \\xi ^2}{(2l+1) \\; (\\xi +\\xi ^{\\prime })}+ \\delta _{ll^{\\prime }} \\; \\frac{(l+1-|m|) \\; 4 \\xi \\; \\xi ^{\\prime }}{(2l+1)^2}\\right]\\right\\rbrace $" ], [ "Construction of Hellmann-Feynman basis sets", "To develop a BS with a high degree of fulfillment of the Hellmann-Feynman theorem, we start choosing the set of pGTO corresponding to a $\\sigma $ BS  of a given quality ($\\sigma $ NZ), and extend it with the functions corresponding to their derivatives.", "As eq (REF ) shows, the derivative of a pGTO with a given $l > 0$ yields three functions with the same exponent as in the pGTO: one function corresponds to $l+1$ , and the other two to $l-1$ , one of them bearing a factor $r^2$ .", "As in usual GTO BS no power of $r$ is included in the primitives, we remove the functions with $r^2$ from the set, yielding what we call a reduced set of primitives, which will span the reduced space.", "Next, we consider the contractions of the $\\sigma $ BS  and their derivatives, which form the reference set, and try to reproduce the space spanned by them, the reference space, in the reduced space.", "For this purpose, we compute the distance between the reference and reduced spaces, as indicated in section .", "If the distance is lower than a given threshold, that we take as $10^{-3}$ , the reduced space is considered sufficient to expand the contractions and their derivatives for practical purposes.", "If it is higher than this threshold, more primitives are added to the reduced set of primitives until the test on distance is fulfilled.", "It is worth noticing that the extra primitives, when necessary, are required to remedy the lack of functions with the $r^2$ factor, and this provides a hint for their selection.", "The process for choosing the extra primitives when required is essentially heuristic.", "Based on the fact that the $r^2$ factor in the derivatives is divided by $2l+1$ , we guess that primitive functions with lower $l$ values will be more effective.", "Likewise, as the $r^2$ factor shifts the function towards higher values of $r$ than the pure exponential, we also guess that small exponents will be also advantageous in this respect.", "With these guidelines, we analyzed the $\\sigma $ TZHF  of oxygen in which extra primitives are necessary.", "A first attempt was made by adding a single $s$ function followed by exponent optimization for reducing the distance.", "As this attempt and the addition of a second $s$ function failed, a pair $s$ , $p$ was tried.", "In this case, after exponents optimization (without restrictions) this combination was successful.", "The process was repeated with other atoms and we found that, in the atoms treated so far (see Table REF ) the addition of this pair of functions, with exponents optimized for each atom, was sufficient to get a distance below the selected threshold.", "Once the reduced set has been completed, we carry out an expansion of the functions of the reference set in an orthogonal basis of the reduced set, that we will call reduced contractions.", "The process could end here, but we have noticed that choosing suitable orthogonal functions in the reduced set, the number of contractions necessary to satisfactorily expand the functions of the reference set can be further reduced.", "The selection of these contractions is made by computing the distance between the spaces spanned by subsets of reduced contractions and the reference space.", "This is accomplished in an iterative way, until a reduced subset fulfills the distance criterion.", "As it can be seen in Table REF , in case the $\\sigma $ TZHF  of C, N, O, and F atoms, the GTO contractions plus their derivatives give 108 functions, whereas 87 functions are sufficient in the reduced subset to achieve the desired accuracy (distance).", "In case of P, S and Cl, the corresponding numbers for the $\\sigma $ TZHF  are 122 and 100.", "Finally, it is worth mentioning that the contractions added to the original $\\sigma $ BS  to give the $\\sigma $ BSHF  also affect the value of the energy (which will be lowered) and the wavefunction.", "This would lead to and endless optimization loop.", "Fortunately, the degree of fulfillment of the Hellmann-Feynman theorem achieved in the first iteration of the loop is sufficient for practical purposes, making it unnecessary to continue the process.", "Table: Number of primitives and contracted basis functions" ] ]
2207.03587
[ [ "Convolution Neural Network based Mode Decomposition for Degenerated\n Modes via Multiple Images from Polarizers" ], [ "Abstract In this paper, a mode decomposition (MD) method for degenerated modes has been studied.", "Convolution neural network (CNN) has been applied for image training and predicting the mode coefficients.", "Four-fold degenerated $LP_{11}$ series has been the target to be decomposed.", "Multiple images are regarded as an input to decompose the degenerate modes.", "Total of seven different images, including the full original near-field image, and images after linear polarizers of four directions (0$^\\circ$, 45$^\\circ$, 90$^\\circ$, and 135$^\\circ$), and images after two circular polarizers (right-handed and left-handed) has been considered for training, validation, and test.", "The output label of the model has been chosen as the real and imaginary components of the mode coefficient, and the loss function has been selected to be the root-mean-square (RMS) of the labels.", "The RMS and mean-absolute-error (MAE) of the label, intensity, phase, and field correlation between the actual and predicted values have been selected to be the metrics to evaluate the CNN model.", "The CNN model has been trained with 100,000 three-dimensional images with depths of three, four, and seven.", "The performance of the trained model was evaluated via 10,000 test samples with four sets of images - images after three linear polarizers (0$^\\circ$, 45$^\\circ$, 90$^\\circ$) and image after right-handed circular polarizer - showed 0.0634 of label RMS, 0.0292 of intensity RMS, 0.1867 rad of phase MAE, and 0.9978 of average field correlation.", "The performance of 4 image sets showed at least 50.68\\% of performance enhancement compared to models considering only images after linear polarizers." ], [ "Introduction", "Because of the boost in data traffic and intensity of high-power lasers, the demands for multimode optical waveguides have been increasing [1], [2], [3].", "multimode waveguides have an advantage not only in communications and high-power transmissions, but also in various application fields such as imaging, nonlinear optics, and quantum physics [4], [5], [6].", "In specific, multimode waveguide is capable to generate or transmit radially polarized light, which is suitable for optical trapping, machining, and longitudinal focusing [7], [8], [9], [10], [11], [12], [13].", "To understand the ratio of different modes, modal decomposition (MD) technique is inevitable.", "MD has been studied intensively, and achieved in various methods, such as cross-correlated imaging, spatially and spectrally resolved imaging, low-coherence interferometry, multi-variable optimization algorithm, and correlation filter [14], [15], [16], [17], [18].", "Thanks to various algorithms and increase in computing power, artificial intelligence has been explosively studied and applied in various fields, including the optics research field [19], [20].", "Especially, convolution neural network (CNN) technique which is proper for image process has been applied in various fields, including MD[21], [22], [23].", "[cnn-md1] has studied MD for one polarization state from near-field image, and [cnn-md2] has applied far-field images as additional input data.", "However, these existing studies have not considered degenerated modes.", "$LP_{01}$ mode is two-fold degenerated, and $LP_{11}$ mode is four-fold degenerated, which are $TM_{01}, TE_{01}$ , and two $HE_{21}$ modes.", "To obtain the specific mode ratio such as radially polarized mode, the coefficient of each mode is required.", "These degenerated modes show exactly identical near-field profiles, therefore it is difficult to exactly decompose the degenerated modes by a single near-field image.", "Recently, [cnn-md3] has studied MD for degenerated modes using polarization filters.", "The results were remarkable which showed 99.1% of field correlations.", "The study used three polarized images, and the $LP_{11}$ mode is a four-fold degenerated state, therefore three polarized images are the minimum input to examine the weights of four modes.", "However, as the image shows only the intensity of the complex field, images after LPs are unable to distinguish the rotating direction of circular or elliptical polarized lights.", "In this paper, images after linear polarizers (LPs) and circular polarizers (CPs) are applied to compare the degenerated modes.", "Four $LP_{11}$ groups are considered, and six additional near-field images after LPs and CPs are considered.", "Four images after LPs of direction with $x \\:(0^\\circ )$ , $x+y\\: (45^\\circ )$ , $y\\: (90^\\circ )$ , and $x-y \\:(135^\\circ )$ , and two images after right-handed CP (RHCP) and left-handed CP (LHCP) are considered.", "Among the seven images, three, four, and seven sets of images are chosen to train the CNN model.", "Eigenmodes have been obtained and reproduced based on numerical simulations, and sample images are also calculated based on eigenmodes from numerical results.", "The images are stacked to form a three-dimensional tensor with a depth of three, four, and seven and are regarded as an input of CNN, and the model is trained by generated images.", "The electric field of waveguide can be decomposed by linear combination of its eigenmodes, such as following equation.", "$\\vec{E}_{net}(r,\\theta )= \\sum _{n=1}^{N} C_n \\vec{E}_n(r,\\theta ) ,$ where $C_n$ is the complex coefficient of the eigenmode $\\vec{E}_n$ .", "The complex coefficient $C_n$ can be expressed as both intensity-phase form and real-imaginary form, such as following equation.", "$C_n=\\rho _n exp(j \\phi _n) = x_n + j y_n$ The modes of interest in this paper are $LP_{11}$ series, which are $TE_{01}, TM_{01}$ , and two $HE_{21}$ modes.", "Each mode can be expressed as the following equations.", "$\\begin{split}&\\vec{E}_{TE_{01}}(r,\\theta )=B(r)(-sin\\theta a_x+cos\\theta a_y)\\\\&\\vec{E}_{TM_{01}}(r,\\theta )=B(r)(cos\\theta a_x+sin\\theta a_y)\\\\&\\vec{E}_{HE_{21o}}(r,\\theta )=B(r)(-sin\\theta a_x-cos\\theta a_y)\\\\&\\vec{E}_{HE_{21e}}(r,\\theta )=B(r)(-cos\\theta a_x+sin\\theta a_y)\\end{split}$ where $B(r)$ is the radial distribution of $LP_{11}$ , which is generally a Bessel function [24].", "The mode profile of $LP_{11}$ series has been calculated via finite element method via COMSOL.", "The multimode waveguide has been selected to be a conventional multimode fiber, FG025LJA (Thorlabs), a step-index multimode fiber with 0.1 of NA, and a core diameter of 25 um.", "The operation wavelength has been assumed to be 1064 nm, which is a typical value for Yb doped lasers [25].", "Hereinafter, the four modes, $TE_{01}, TM_{01}, HE_{21o}$ , and $HE_{21e}$ are indexed as mode 1, 2, 3, and 4, respectively.", "The degenerated $LP_{11}$ series can be expressed by four imaginary coefficients $C_1, C_2, C_3$ , and $C_4$ .", "The net intensity $|\\vec{E}^2|$ is not phase-sensitive, so one could tune one coefficient to be real, or phase to be 0.", "Hereinafter, the phase is tuned as $C_1$ to be a positive real number, which indicates $\\phi _1=0$ , $x_n>0$ , and $y_1=0$ .", "Now to generate a degenerated sample, four random coefficients are generated.", "The coefficients are weighed to each eigenmode and the final near-field image is generated.", "Not only the image without the polarizer, but images after four LPs – $x \\:(0^\\circ )$ , $x+y\\: (45^\\circ )$ , $y\\: (90^\\circ )$ , and $x-y \\:(135^\\circ )$ – and two CPs with different rotation directions - RHCP and LHCP - are also calculated and generated.", "The pixel number of the generated image has been chosen to be 121$\\times $ 121, where the core region has been divided into 100 pieces, and 10% margin on both sides was regarded as the area of interest.", "However, as discussed previously, the left half of the image has an identical tendency to the right half, so taking half of the image is enough for comparison on degenerated $LP_{11}$ series MD.", "Therefore, the tenor size of the final input for training becomes 61$\\times $ 121$\\times n$ , where $n$ is the number of images selected in a single case.", "Generated seven images for four eigenmodes are shown in Fig.", "REF .", "The electric vector field is shown within the unpolarized full image.", "Figure: Seven images of LP 11 LP_{11} series eigenmodes.", "The electric vector field is depicted within the full image.From Fig.", "REF , it seems that the modes are available to be compared only by using three images after three LPs.", "As the degenerated mode has four unknown coefficients $C_1 \\sim C_4$ , and the problem is to find the ratio between four variables, three equations are enough to solve the problem.", "However, the image only shows the absolute value not the exact complex value, three images are not enough for some case.", "Fig.", "REF shows two different mixed modes.", "The first and second mixed mode is chosen to be sum of $TE_{01}$ mode and $TM_{01}$ mode with different phase constant, which can be expressed as $\\vec{E_1}-i\\vec{E_2}$ , respectively.", "As the electric field direction of $TE_{01}$ mode and $TM_{01}$ mode are perpendicular in all the regions, the electric field of mixed mode becomes circularly polarized in every positions.", "Therefore, it is unavailable to distinguish two modes only via LP.", "In other words, the MD system using LP cannot compare two perpendicular modes have leading phase or retarding phase.", "It is shown in Fig.", "REF that two different modes show exactly the same images after all directions of LPs.", "In this paper, four sets of images and seven sets of images are considered to be the input.", "Four images are selected to be the images after LPs of angle with $x \\:(0^\\circ )$ , $x+y\\: (45^\\circ )$ and $y\\: (90^\\circ )$ , and an additional image after RHCP.", "Seven sets are chosen to be all the images depicted in Fig.", "REF and Fig.", "REF .", "For comparison, case of three LPs are also tested.", "Figure: Seven images of twp mixed LP 11 LP_{11} series, E 1 →+iE 2 →\\vec{E_1}+i\\vec{E_2} and E 1 →-iE 2 →\\vec{E_1}-i\\vec{E_2}." ], [ "CNN Model", "The CNN model has been constructed for MD.", "The size of the input is selected to be three, four, and seven groups of 61$\\times $ 121 images, resulting in 61$\\times $ 121$\\times n$ images where $n=$ 3,4, and 7.", "The input image goes through a CNN network, which is composed of four convolution rectified linear unit (ReLU) layers, three max pooling layers, a flatten layer, two fully connected ReLu layers, three dropout layers, and the output target of 7 tanh layer.", "To avoid overfitting, all layers have $10^{-8}$ L2 regulation, and the dropout ratio of 5% has been applied.", "The full system is shown in Fig.", "REF .", "Note that the numbers at the convolution block of Fig.", "REF represent the filter size of the convolution tensor, and the size of the block represents the shape of the output after the layer.", "The loss of the model was selected to be root-mean-square (RMS), or mean-squared-error (MSE).", "In terms of loss function, [cnn-md2] has used field correlation function.", "However, the degenerated case shows similar correlation functions, so in this paper, the loss function is assumed to be RMS between the values.", "It is notable that the phase and intensity are different units, so if coefficient $\\rho $ and $\\phi $ are used to measure loss such as [cnn-md1], one must carefully determine the weight of intensity and phase.", "It is reasonable to assume the loss between the actual coefficient $C_{n,a}$ and the predicted coefficient $C_{n,p}$ to be the distance between the two imaginary values.", "The distance is available to be expressed as $\\sqrt{(x_{n,a}-x_{n,p})^2+(y_{n,a}-y_{n,p})^2}$ , so if the output label is determined to be $x_1, x_2, y_2, x_3, y_3, x_4$ , and $y_4$ , the RMS or MSE will represent the distance between the answer and predicted value.", "The labels are normalized, the largest value to become 1.", "Labels are denoted as $z_n$ , where $z_{1,2,4,6}=x_{1,2,3,4}$ and $y_{2,3,4}=z_{3,5,7}$ .", "Note that the first mode is tuned to be real, so $y_1$ is always 0, which can be removed from the output label.", "The RMS loss of real and complex components, will be referred to hereinafter as label RMS.", "The label RMS is shown in the following equation.", "$Label \\:RMS = \\sqrt{\\frac{\\sum _{n=1}^{4} (x_{n,a}^2-x_{n,p}^2) + \\sum _{n=2}^{4} (y_{n,a}^2-y_{n,p}^2) }{7}}=\\sqrt{\\overline{\\sum _{n=1}^{7} (z_{n,a}^2-z_{n,p}^2) }}=\\sqrt{\\overline{\\Delta z^2}}$ The optimizer of model has been chosen to be “ADAM”, and the batch size was selected as 128.", "300 epoch of training was performed with 100,000 training sets and 3,000 validation sets.", "After training, the mode intensity ratio $\\rho $ , phase $\\phi $ and reconstructed field images have been calculated from the predicted labels.", "The metric of the model was evaluated in terms of RMS, mean-absolute-error (MAE) and the field correlation between the actual and seven reconstructed near-field images, which are the original full image and six images after the polarizers.", "Figure: The schematic of the full CNN system.", "3D tensors with depth of three, four, and seven are inserted into the CNN architecture and 7 labels, the real and image value of each mode coefficients, are obtained as an output." ], [ "Results", "The model is trained with 100,000 generated 61$\\times $ 121$\\times n$ training images with 3000 validation sets, and the RMS loss in terms of epoch is shown in Fig.", "REF for three cases, $n=$ 3, 4, and 7.", "RMS of the training set and validation set are depicted, respectively.", "It is clear that the case of $n=3$ , which only considers images after LPs, converges slower, and shows higher loss compared to the cases which consider image after CP.", "Both case of $n=$ 4 and 7, which consider images after CP, shows similar converge speed and losses.", "Figure: The RMS loss history while training, in terms of epochs for case of n=n=3, 4, and 7.", "Both RMS losses of the training set and validation set are depicted.To evaluate the performance of the model, the model predicted 10,000 new test sets.", "From the predicted seven labels, obtain four weight coefficients ($\\rho _1 \\sim \\rho _4$ ) and three relative phases($\\phi _2 \\sim \\phi _4$ ), and the field correlation of seven images are inversely calculated.", "The label MAE ($\\overline{|\\Delta z|}$ ), label RMS ($\\sqrt{\\overline{\\Delta z^2}}$ ), MAE and RMS of weight coefficient ($\\overline{|\\Delta \\rho |}$ and $\\sqrt{\\overline{\\Delta \\rho ^2}}$ ), normalized average difference (MAE) of phase ($\\overline{|\\Delta \\phi |}/2\\pi $ ), the field correlation of the full unpolarized image ($Corr_{full}$ ), and the average of seven field correlations ($\\overline{Corr}$ ) have been calculated and shown in Table .", "Note the angle difference angle $\\Delta \\rho $ has been calculated from the angle of $C_{n,a}/C_{n,p}$ .", "It is shown that all the cases of $n=$ 4 and 7 which considered images after CP show better performance compared to the case of $n=3$ which only considered images after LP.", "Especially the average correlation of $n=3$ is less than 99%.", "Because the case of $n=3$ is unable to compare the direction of circular or Elliptical polarized light, the field correlations of images after RHCP and LHCP are relatively low, which are calculated as 0.9855 and 0.9859.", "The performance enhancement between the case of $n=$ 3 and 4 is also calculated and shown in the last row in Table .", "Note that the loss of correlation has been regarded as $1-Corr$ .", "It is shown that all the metrics showed at least 50% of reduced loss, which indicates the case with the CP predicts the degenerated mode with higher accuracy.", "Table: NO_CAPTION tableCalculated metrics for $n=$ 3, 4 and 7.", "The metrics – RMS of weight coefficient ($\\sqrt{\\overline{\\Delta \\rho ^2}}$ ), MAE of normalized phase ($\\overline{|\\Delta \\phi |}/2\\pi $ ), and field correlation of the full image ($Corr_{full}$ ) – sorted by the label RMS ($\\sqrt{\\overline{\\Delta z^2}}$ ), are depicted in Fig.", "REF for the case of $n=$ 3, 4, and 7.", "It is shown that low label loss tends to show high field correlation, and low mode weight MAE and phase MAE.", "However, the metrics are not exactly correlated to the label MAE, where some of the outliers are observed where low label MAE loss has high intensity/phase loss or low field correlation.", "The fact indicates the field correlation is not the best metric to be optimized while training, where similar mode images can be generated from different combinations.", "One may use the seven field correlations as a metric, however, it would add complexity to the calculation.", "Comparing the three cases, it is obvious that the case of $n=3$ shows the highest loss.", "The label RMS (green solid line) has a higher value compared to other cases ($n=4,7$ ).", "The intensity weight and phase difference of $n=3$ also show more outliers.", "For the case of $n=4$ and 7, most of intensity and phase losses (red and black dots) are positioned at the bottom line and the field correlation (blue dots) is mostly shown at the top line.", "From the results, it is clear that the cases which considered the image after CP shows higher performance.", "The case of $n=$ 4 and 7 shows similar performance, so the case of four sets of images is assumed to be proper which requires low memory and calculation time.", "Figure: The intensity RMS loss (red dots), normalized phase RMS loss (black dots), and field correlation (blue dots) of 10,000 test sets for (a) n=3n=3, (b) n=4n=4, and (c) n=7n=7.", "The metrics are sorted by the label RMS loss (green solid line).The comparison of modal weight $\\rho $ , phase $\\phi $ , and the seven images for the case of $n=4$ are shown in Fig.", "REF .", "The samples were chosen as the lowest quartile, median, and highest quartile in terms of the label MAE loss – which has $2500^{th}, 5000^{th}$ , and 7500th smallest label MAE loss.", "It is clear that the coefficients and the images show relatively good matching between the actual value/image and predicted ones.", "Figure: Comparison between intensity ρ\\rho and phase φ\\phi of mode coefficient and their seven images for (a) lowest quartile, (b) median, and (c) highest quartile." ], [ "Discussions", "The MD study based on CNN has not been thoroughly studied.", "The metrics, loss function, and method to verify the model are not standardized.", "For in case of loss function, the results show that field correlation is not a proper metric to understand the exact ratio of each mode.", "Using both intensity and phase, the weight function on different dimensions is another hyper-parameter to be optimized.", "Therefore, the paper proposes the real and imaginary component of the modal coefficient is the best choice to be the label, and MSE or RMS will simply be the loss to train and predict the model.", "Most of the MD studies used a small number of test sets.", "Samples for comparisons on coefficients and images are selectively chosen, without any standards or metrics.", "In this paper, 10,000 test samples evaluated the performance of the trained model, and quartiles based on loss have been selected as an example to be shown.", "As the labels are not discrete but continuous, numerous samples are required to exactly predict the MD system.", "For example, if each label is divided into five levels, - which leads to under  20% of loss - samples of $5^n$ will be required, where $n$ is the number of labels.", "Four modes need seven labels, which requires 78125 samples.", "However, as the mode theory is studied intensively, it is available to generate training sets.", "If the number of modes increases, the required training set for model prediction will exponentially increase.", "Assuming one linear polarization and using $LP$ series will reduce the label by half.", "However, the $LP$ based model will fail to examine the ratio of cylindrical vector modes such as $TM_{01}$ mode or $TE_{01}$ mode.", "In addition, if one desires to use two-dimensional input image, connecting multiple images horizontally (or vertically) - which generates 61$\\times $ (121*n)$\\times $ 1 image in the case of the paper - will perform similar results.", "The depth of the CNN layers must be reduced, as the 3D CNN layer calculates the third dimension of the tensors at once." ], [ "Conclusion", "The paper has proposed CNN model for MD on degenerated modes based on four or seven sets of input images from LPs and CPs.", "The CPs detect the direction of circularly polarized light and enhances the performance of the CNN model.", "$LP_{11}$ series has been decomposed from 7 images including the full image and images from four LPs of 0$^\\circ $ , 45$^\\circ $ , 90$^\\circ $ , and 135$^\\circ $ and two CPs, RHCP and LHCP.", "Output labels of the model have been chosen to be the real and imaginary parts of mode coefficients.", "The model was trained via 100,000 training sets, and 10,000 test sets were evaluated.", "The trained model showed 0.0634 of label RMS, 0.0292 of intensity RMS, 0.1867 rad of phase MAE, and 0.9978 of average field correlation for case of 4 image sets.", "The performance of 4 image sets showed at least 50.68% of performance enhancement compared to models considering only images after LPs.", "Quartiles in terms of label RMS have been selected to visualize the performance.", "The predicted values showed relatively reasonable match compared to the actual value.", "The study is believed to be applied to obtain the coefficient of cylindrical vector mode.", "In addition, the proposed methods to decompose degenerate modes, select the loss function, and standards to select examples are expected to assist further CNN based MD studies or related works." ] ]
2207.03489
[ [ "ACCESS: Confirmation of a Clear Atmosphere for WASP-96b and a Comparison\n of Light Curve Detrending Techniques" ], [ "Abstract One of the strongest ${\\rm Na~I}$ features was observed in WASP-96b.", "To confirm this novel detection, we provide a new 475-825nm transmission spectrum obtained with Magellan/IMACS, which indeed confirms the presence of a broad sodium absorption feature.", "We find the same result when reanalyzing the 400-825nm VLT/FORS2 data.", "We also utilize synthetic data to test the effectiveness of two common detrending techniques: (1) a Gaussian processes (GP) routine, and (2) common-mode correction followed by polynomial correction (CMC+Poly).", "We find that both methods poorly reproduce the absolute transit depths but maintain their true spectral shape.", "This emphasizes the importance of fitting for offsets when combining spectra from different sources or epochs.", "Additionally, we find that for our datasets both methods give consistent results, but CMC+Poly is more accurate and precise.", "We combine the Magellan/IMACS and VLT/FORS2 spectra with literature 800-1644nm HST/WFC3 spectra, yielding a global spectrum from 400-1644nm.", "We used the PLATON and Exoretrievals retrieval codes to interpret this spectrum, and find that both yield relatively deeper pressures where the atmosphere is optically thick at log-pressures between $1.3^{+1.0}_{-1.1}$ and 0.29$^{+1.86}_{-2.02}$ bars, respectively.", "Exoretrievals finds a solar to super-solar ${\\rm Na~I}$ and ${\\rm H_2O}$ log-mixing ratios of $-5.4^{+2.0}_{-1.9}$ and $-4.5^{+2.0}_{-2.0}$, respectively, while PLATON finds an overall metallicity of $log_{10}(Z/Z_{\\odot}) = -0.49^{+1.0}_{-0.37}$dex.", "Therefore, our findings are in agreement with literature and support the inference that the terminator of WASP-96b has few aerosols obscuring prominent features in the optical to near-infrared (near-IR) spectrum." ], [ "Introduction", "In-depth studies of exoplanetary atmospheres (exo-atmospheres) is a key pathway to obtaining more detailed insights about the formation and evolution of planetary systems.", "Many of these planets are in extreme environments not found in the solar system and understanding how their atmospheres are sculpted by such unique environments gives us detailed insights on the complex chemistry and physics at play.", "Examples include the cause of observed temperature inversions in hot Jupiters [8], [31], the cause and composition of high altitude aerosols (clouds and/or hazes) [61], [102], [32], and observed super sonic wind speeds [29].", "Observing exo-atmospheres also can improve our understanding of the formation and evolutionary processes that exoplanets undergo, e.g., host disk dissipation rate [74] and hot Jupiter migration timescales [60], [74].", "To improve our understanding of exoplanets, observations have had to push the performance of instruments, which were not designed with exo-atmosphere studies in mind, to their limits.", "This has also been the case for data analysis techniques aimed at removing both instrumental and astrophysical systematics both from space-based observatories [73], [88] and ground-based ones [35], [108].", "As expected in any developing field, there have been a number of cases with disagreeing results (e.g., [89] and [17]; [85] and [22]; and [86] and [37], [36], and [58]).", "These discrepancies are attributed to imperfect understanding of systematics, whether it be instrumental, observational, or astrophysical in nature.", "This highlights the importance of confirming features of interest via independent analyses in order to isolate spurious signals and instill more confidence in agreeing detections.", "This is particularly important in attempts to find correlations between atmospheric features and other system parameters, such as the cause of high-altitude aerosol formation in gas giants.", "There have been several studies attempting to find such a correlation [87], [40], [91], [30], [96], [19], with no clear answer yet [3].", "WASP-96b [39], is one of few exoplanets observed to-date to have little evidence supporting the presence of high-altitude aerosols in both optical [64] and near-infrared [109] transmission spectra.", "An exoplanet with a transmission spectrum that can be modeled excluding high-altitude aerosols is also called a “clear” atmosphere, which does not mean the planet has no aerosols but little to no aerosols obscure the summed terminator's spectrum.", "Other planetary atmosphere that can be explained without including high-altitude aerosols are WASP-17b [87], WASP-39b [26], [63], [101], [48], WASP-62b [4], and WASP-94Ab [1].", "Finding planets like these is essential for understanding the evolutionary, chemical, and physical processes underway in this class of planets because aerosols mute the features needed to probe exo-atmospheres.", "This is particularly challenging given that it is estimated only ${\\sim } 7\\%$ of hot Jupiters have clear atmospheres [102].", "As such, it is vital to find and thoroughly study clear planets like WASP-96b.", "The optical transmission spectrum of WASP-96b was first observed by [64] with VLT/FORS2 and recently [109] combined the VLT/FORS2 spectrum with a near-IR spectrum derived using HST/WFC3 observations from GO program 15469 (PI: Nikolay Nikolov) to obtain a better picture of the planet's clear nature.", "These studies report strong ${\\rm Na~I}$ and ${\\rm H_2O}$ features with abundances of $-3.88^{+1.05}_{-0.82}$ and $-3.65^{+0.90}_{-0.94}$ , respectively.", "In this paper, we present a new optical transmission spectrum of WASP-96b derived from new observations obtained as part of ACCESSThe Atmospheric Characterization Collaboration for Exoplanet Spectroscopic Studies (ACCESS) survey on the Magellan Telescopes [45], [75], [12], [22], [104], [58], [105], [49].", "These observations are described in Section .", "We then combined the ACCESS observations with an independent re-analysis of the original VLT/FORS observations and the available near-IR observations to derive a new 400–1644 nm optical to near-infrared transmission spectrum for this planet and re-inspect its atmospheric properties via retrieval models.", "To understand individual and relative performances of commonly used detrending methods, we also conduct a detailed comparison of two often-used techniques in ground-based transmission spectroscopy: (1) a Gaussian processes routine [35], [104], [108], [58], [105], and (2) common-mode correction followed by polynomial correction [34], [63], [37], [64], [14].", "We discuss these detrending techniques and their performance comparison in Section .", "The remainder of this paper is structured as follows.", "Section  describes our methods for constructing the transmission spectra, and Section  outlines how we consider stellar activity affecting the transmission spectra.", "The spectra are then interpreted using retrievals in Section .", "In Section , we present and discuss the retrieval analysis results, and contextualize WASP-96b within the exoplanet population in Section .", "We conclude in Section .", "We used two transit observations of WASP-96b on the nights of UT170729 and UT170822 (UTYYMMDD) with the FOcal Reducer and Spectrograph (FORS2) FORS instrument website mounted on the 8.2 m Very Large Telescope (VLT) in the European Southern Observatory on Cerro Paranal, Chile.", "These observations were originally collected, reduced, and published by [64].", "For our re-analysis of this data, described in Section , we used the same extracted spectra produced in [64], where additional detail of the data collection and extraction process can be found.", "Both observations were taken with the multi-object spectroscopy (MOS) mode and the same two-slit mask.", "Each slit in the mask was ${22}{}$ by ${120}{}$ and centered on the target and comparison star.", "Given the wide slits, the spectral resolution of the observations were determined by the seeing, with an average resolving power of R $\\sim $ 600.", "The first transit was observed using the bluer dispersive element, GRIS600B (600B), which had approximate spectral coverage of 360–620 nm.", "Contrary to [64], we excluded the spectrum from 360–400 nm to avoid systematics caused by the low counts, where that region had over 85% fewer counts than the rest of the spectrum.", "The second night used the redder GRIS600RI (600RI) grism and the GG435 filter, producing an approximate spectral coverage of 520–835 nm." ], [ "Magellan/IMACS Transits", "We observed two transits of WASP-96b as part of the ACCESS Survey ACCESS generally acquires $\\ge $ 3 transits of a target to reduce systematics and increase precision, but determined two sufficient when considering the already observed VLT/FORS2 transits.", "on the nights of UT170804 and UT171108 with the Inamori-Magellan Areal Camera & Spectrograph [18] on the 6.5-m Baade Magellan Telescope at Las Campanas Observatory in Chile.", "Both transits were observed using the 8K $\\times $  8K CCD mosaic camera at the f/2 focus, which provides a 27.4 field of view (FoV).", "We used a 300 line/mm grating at blaze angle of 17.5$^{\\circ }$ , yielding a spectral coverage of 0.44–0.97 $\\mu $ m (without a filter).", "To reduce readout time and improve the duty cycle of the observations, we used 2$\\times $ 2 binning.", "Both observations used the FAST readout mode, which had a 29 s readout time and a 3 s overhead.", "The first observation had no filter, but we applied the GG495 filter (coverage 0.49–0.97 $\\mu $ m) for the second night to prevent second-order light contamination.", "The effect of second-order contamination on the first observation is discussed in Appendix .", "For each observation we used the MOS mode and designed a custom science mask with ${10}{}$ by ${90}{}$ slits for the target and a number of comparison stars in the field.", "Identical masks, but with slit widths of ${0.5}{}$ instead of ${10}{}$ , were also designed for wavelength calibrations.", "The average seeing-limited resolving power of the observations were R $\\sim $ 900 The comparison stars were selected using the same prescription described in [75].", "We consider a nearby star as a suitable comparison if it has a color difference with WASP-96 of $D < 1$ , where $D=\\sqrt{[(B-V)_c - (B-V)_t]^2 + [(J-K)_c - (J-K)_t]^2}$ The uppercase letters in the equation correspond to the Johnson-Cousin apparent magnitudes of the stars, and the subscripts $t$ and $c$ indicate the target and comparison, respectively.", "The comparisons used in both the ACCESS and VLT/FORS2 observations are summarized in Table REF .", "cccccccc[htb] Target and comparison star magnitudes and coordinates from the UCAC4 catalog [111].", "We used comparisons 14 and 15 for the analysis of both IMACS transits and comparison 1 for both FORS2 transits.", "Comparison 3 was in the IMACS slit but was over saturated and not used.", "Star RA Dec B V J K D WASP-96 00:04:11.12 -47:21:38.25 13.25 12.51 11.27 10.91 0 COMP1 00:04:18.87 -47:16:31.05 13.52 12.88 11.76 11.41 0.1 COMP3 00:04:02.03 -47:14:49.87 12.19 11.28 9.1 9.67 0.94 COMP14 00:05:12.82 -47:03:37.02 13.87 12.98 11.40 10.85 0.24 COMP15 00:04:25.01 -47:00:42.96 13.86 13.17 11.93 11.44 0.14 We limited observations of each transit to airmass below 2 during the full transit window to minimize differential atmospheric refraction effects, so we ended up with two full transits with additional out-of-transit baselines of 1.29 and 1.35 hours for the UT170804 and UT171108 transits, respectively.", "To maintain count levels of 25,000–35,000 ADUThis is well within the linearity limits of the IMACS CCDs [12]., given the average seeing conditions at each night, we held exposure times constant to 60 seconds for transit UT170804 (average seeing of ${1.55}{}$ ) and 45 seconds for transit UT171108 (average seeing of ${0.71}{}$ ).", "A summary of each transit's observing conditions are given in Table REF .", "Finally, we collected bias frames, quartz lamp flats, HeNeAr calibrations (using the ${0.5}{}$ by ${90}{}$ masks), each with the same binning as the science observations.", "ccccccccc[htb] Observing log for WASP-96b data from Magellan/IMACS.", "Night Obs.", "Obs.", "Start/ Airmass Frames min/max Start (UTC) End (UTC) seeing [\"] 2017 Aug. 04 02:55/06:38 1.93–1.09 148 1.4/1.71 2017 Nov. 08 00:01/03:49 1.11–1.18 180 0.68/0.8" ], [ "ACCESS Reduction Pipeline", "We reduced the data using the ACCESS custom pipeline introduced in [45] and described in detail in [20].", "We first subtracted the bias level from each frame using the median of the overscan region.", "The pipeline also has the option of flat-fielding each frame using a master flat, but as with previous ACCESS datasets [75], [12], [58], [105], [49], we found that flat-fielding worsens the photometric precision of the transit light curves, so we opted to not flat-field the data.", "The ineffectiveness of flat-fielding is likely due to the lack of sufficient flats needed for the high precision analysis.", "We collected 10 to 25 flats per night with 1 to 5 second exposures.", "We then performed a bad-pixel correction using a bad-pixel map obtained from the flats.", "Any pixel with 10 times higher or lower photon noise relative to neighboring pixels was added to the map.", "The correction was done to the science images by replacing the value of each flagged pixel with a value interpolated from the neighboring pixels in the dispersion direction of the detector.", "We traced each spectrum in the images by first using a second-order polynomial to identify each resolution element's centroid (relative to both spatial and dispersion directions).", "Then a fourth-order polynomial was fit on the centroids in order to ensure the smoothness of the trace.", "The sky background was subtracted using the median of the spectral counts outside of the central apertures for a given wavelength element.", "The appropriate aperture sizes were empirically determined to be ${3.6}{}$ (9 pixels) in radius for UT170804 and ${3.2}{}$ (8 pixels) for UT171108.", "The extracted spectrum was produced by summing the spectral profile within the central aperture in the spatial direction and subtracting the sky background.", "A Lorentzian profile was used to fit each spectral line in the HeNeAr calibration spectra.", "The wavelengths of each line as a function of pixel position was then fit using sixth-order polynomials.", "The pipeline iteratively excluded deviant datapoints until the root mean square error value of the fit was less than 2 km s$^{-1}$ ($\\sim $ 0.05 Å).", "This wavelength map was applied to each extracted spectrum.", "We identified and removed cosmic rays by first creating a global median spectrum using the median of all the normalized spectra for each exposure.", "This normalized, median spectrum was used to compare against each individual normalized spectrum, and any counts in a spectrum more than $10\\sigma $ greater than the global median spectrum (on a corresponding wavelength) were replaced by interpolating the counts at the appropriate wavelength from the preceding and following spectra in the time series.", "It should also be noted that the pipeline is capable of doing optimal extraction [57].", "This could potentially be used in place of bad-pixel and cosmic-ray removal, as it effectively highlights and removes deviant counts (Allen et.", "al., in prep.).", "However, we tested this for both WASP-96 observations, and found very little difference between resulting spectra.", "Figure REF shows the median extracted spectrum for both nights.", "The final wavelength range used was 475–825 nm for UT170804 and 525–825 nm for UT171108 (see Fig.", "REF ).", "Given the diminishing throughput on the edge of the spectra, the bluest wavelengths had too few counts to be useful.", "We also excluded data from 759.4–767.2 nm, a region of strong tellurics.", "Though the sky subtraction and dividing by the comparison(s) should in principle negate the telluric features, that region still had a drastic decrease in counts (and likely residual telluric-induced systematics).", "This made it difficult to properly fit a light-curve at that wavelength range.", "Unfortunately, this coincides with the ${\\rm K~I}$ doublet (centered at 768.15 nm).", "We also excluded wavelengths longer than 825 nm due to the diminishing throughput and increase in tellurics (see Fig.", "REF ).", "We could have attempted to obtain useful information from that range, but given that the available HST/WFC3/IR/G102 data [109] is not afflicted by any of these effects, we instead omit that wavelength range from the ACCESS analysis.", "When using the f/2 mode, each stellar spectrum is dispersed on two CCD chips, causing a gap in the spectra (see Figure REF ).", "Fortunately, all utilized spectra of the target and COMP14 fell on one chip.", "For COMP15, the gap was approximately between 5580-5675Å; however, that region still had COMP14 for systematic corrections.", "Figure: Median extracted spectra of WASP-96 and comparisons, excluding the oversaturated COMP 3.", "The top row shows IMACS data (UT170804 and UT171108) and the bottom shows FORS2 data from (UT170729 and UT170822).", "Each spectroscopic bin used is shaded in light gray and demarked by dotted lines.", "Telluric regions with less than 0.1 transmission are shaded in light red.", "The only gap in the binning scheme is the strong telluric region from 7594.0–7672.0Å." ], [ "HST WFC3 Transits", "The HST data were obtained directly from [109] (GO program 15469, P.I.", "N. Nikolov).", "It consisted of two transits, one on UT181218 with the G102 grism (0.8–1.1 $\\mu $ m) and another on UT181228 G141 (1.1–1.7 $\\mu $ m) grism.", "The raw spatially scanned spectra were reduced with Iraclis [95].", "The quadratic limb-darkening parameters were obtained using the models of [16] and the inclination (i = 1.486 radians), semimajor axis relative to stellar radius ($a/R_s$  = 8.84), and orbital period (P = 3.4252602 days) were all held fixed to parameters determined from [39] and [64]." ], [ "Photometric Monitoring", "We compiled and analyzed available time-series photometry of the host star WASP-96 from the Transiting Exoplanet Survey Satellite [80] and the All-Sky Automated Survey for Supernovae [50] to constrain the presence of star spots that could contaminate the transmission spectrum of WASP-96b.", "TESS observed WASP-96 in 2-minute-cadence mode over 27 days during Sector 2 (between 23 August and 20 September 2018) and over 24 days in Sector 29 (between 26 August and 20 September 2020).", "To model its photometric modulation, we used the Pre-search Data Conditioning Simple Aperture Photometry [43] light curve obtained from the Barbara A. Mikulski Archive for Space Telescopes (MAST) portal.", "We masked the transit data using a period of 3.42526 days, initial mid-transit time (t0) of 2458354.319946 BJD, and a transit window of 3.04 hours (25% buffer in duration) where the transit duration of 2.445 hours was calculated using the system parameters of [109].", "The resulting out-of-transit light curves have a median absolute deviation (MAD) of 2.76 parts-per-thousand (ppt) and a peak-to-peak difference of 25.5 ppt for Sector 2 (17676 data points), and a MAD of 2.9 ppt and peak-to-peak difference of 37.3 ppt for Sector 29 (14252 data points).", "For our analysis of the TESS data, we binned every 100 observations together.", "ASAS-SN observed WASP-96 in two filters.", "$V$ filter observations were collected between April 2014 and September 2018.", "$g$ filter observations were collected between September 2017 and February 2020.", "To remove deviating observations, we sigma-clipped points that deviated by more than $3\\sigma $ from the mean.", "This led to excluding 11 of 921 and 27 of 1626 observations for the V and g filters, respectively.", "Because of the lower cadence of the ASAS-SN observations ($\\sim $ 3 per day) compared to the TESS observations, their larger variance, and since we do not expect significant stellar modulation in a day, we weighted averaged observations taken on the same night.", "We then removed observations with uncertainties larger than 3 times the mean uncertainty, which led to six binned observations removed in V band, and seven in g band.", "The final light curves included 351 & 520 datapoints with MADs of 10.4 & 9.5 ppt and peak-to-peak differences of 83.5 & 96.5 ppt, respectively." ], [ "Light-curve Analysis and Comparison of Detrending Techniques", "All the wavelengths used in each extracted spectrum obtained in Section REF were summed for every exposure to construct a photometric white-light curve.", "Furthermore, binned-light curves were constructed by partitioning the spectra into distinct spectro-photometric bins that were used to determine the change in transit depth over wavelength (i.e., the transmission spectrum).", "The binning scheme was designed by considering spectro-photometric precision, the overlap of spectral bands from different observations, high telluric absorption regions, and the desire to properly probe for atmospheric features, such as a scattering slope, sodium, and potassium lines.", "The bins used to construct the transmission spectrum are shaded grey in Figure REF and their wavelength coverages are listed in the first column of Table .", "An overview of the white-light and binned light curve detrending technique implemented for our final light curves is provided in sections REF & REF .", "However a more detailed description of our detrending routines are given in Appendix .", "This section ends (REF ) with an explanation of how we determined the best detrending methods for our data." ], [ "White Light Curve Fitting", "Our first step in detrending the VLT/FORS2 and Magellan/IMACS white-light curves was a sigma clipping of the raw light curves (target divided by mean of comparisons).", "This was done by individually evaluating each data point in the light curve with a moving average of 11 points centered around the point of interest.", "If the value of the specific point deviated by more than $3\\sigma $ from the mean it was removed.", "We opted for this method because it does not depend on an initial transit model fit, which is often used as a sigma-clipping criteria.", "It resulted in zero, two, three, and two points being removed from each transit in chronological order.", "We then used the PCA+GP routine (see Appendix REF ), to fit the sigma-clipped white-light curves.", "The transit parameters used in the fits were quadratic limb darkening (LD) coefficients, $q_1, q_2$ , planet orbital period, $P$ , semi-major axis relative to the stellar radius, $a/R_s$ , the planet-to-star radius ratio, $R_p/R_s$ , impact parameter, $b$ , mid-transit time, $t_0$ , eccentricity, $e$ , and the argument of periastron, $\\omega $ .", "$q_1$ and $q_2$ are parameterizations of the more commonly used $u_1$ and $u_2$ quadratic LD parameters, where $q_1=(u_1 + u_2)^2$ and $q_2=u_1/2(u_1 + u_2)$ [47].", "For our analysis we had to express $b$ in terms of inclination, $i$ , as $i=\\cos ^{-1}\\left(\\frac{b}{a}*\\frac{(1 + e*\\sin (\\omega ))}{1 - e^2}\\right).$ We adopted a circular orbit for WASP-96b, based on [39], and used a quadratic LD law as in [64].", "We also held $P$ (3.4252602 days), $a/R_s$ (8.84), and $i$ (1.486 radians) fixed to the values used by [109] in order to later combine the optical and HST spectra.", "The quadratic LD parameters were set to be uniform from 0 to 1, and $R_p/R_s$ had a normal prior with mean value of 0.115 and a standard deviation of 0.02.", "Here priors were based off of the $R_p/R_s$ values obtained by [64].", "For $t_0$ we used normal priors with uncertainties of 0.005 days (7.2 minutes) and mean values of $t_0\\,-\\,2,450,000$ =7963.33672[MJD], 7970.69261[BJD], 7987.31195[MJD], and 8066.59828[BJD] days for the transits in chronological order small priors were used for the FORS2 data because they were based off of the $t_0$ values found by [64].", "In order to use consistent priors for all transits, we fit the IMACS data first with wider priors (0.5 days, 12 hours), then with the 0.005 day priors after using the initial fit to obtain estimates on $t_0$ ..", "The best-fit white light curves and their corresponding orbital parameters are shown in Figure REF and Table REF .", "We note that the difference in depth for the last transit (UT171108) is because the orbital parameters were held fixed.", "When allowed to be free, more consistent depths are obtained, and the other system parameter still agree within their uncertainty ranges.", "However, the absolute depth is not as relevant, as will be discussed in Section REF .", "CcRRRR[htb] Fitted white light curve values.", "The period (P=3.4252602), semi-major axis (relative to stellar radius, $a/R_s$ =8.84), and inclination (i=85.14) were all held fixed to parameters used by [109].", "The mid-transit times are in terms of MJD for the FORS2 observations (UT170729 and UT170822) and BJD for the IMACS observations (UT170804 and UT171108).", "Note: the difference in depth for the last transit is because the orbital parameters were held fixed, when allowed to be free more consistent depths are obtained.", "parameter definition UT170729 UT170804 UT170822 UT171108 Rp/Rs planet radius/star radius 0.1148+0.0029-0.0033 0.1168+0.0046-0.0048 0.1188+0.0015-0.0018 0.1018+0.0049-0.0051 t0-2.45e6 mid-transit (JD) 7963.33662+0.00050-0.00048 7970.69093+0.00049-0.00046 7987.31216$\\pm $ 0.00024 8066.59703+0.00069-0.00071 q1 LD coeff 1 0.37+0.10-0.12 0.23+0.11-0.09 0.31+0.06-0.05 0.32+0.18-0.15 q2 LD coeff 2 0.29+0.24-0.15 0.33+0.34-0.23 0.63+0.18-0.22 0.31+0.32-0.21 Figure: Top: White light curves (LC) composed from dividing the target by the sum of comparison stars, i.e., the raw LC.", "The first two columns are for the VLT/FORS2 transits, showing the exact data of .", "The last two columns are new Magellan/IMACS observations directly from ACCESS’s custom pipeline.", "The data points marked with an x were identified as outliers and are not included in the rest of the analysis.", "Middle: The detrended white LC (blue points) produced utilizing our PCA+GP routine, the corresponding best-fit transit LC (solid black line) with orbital parameters shown in Table , and the residuals (red points) produce by subtracting the detrended data from the model.", "The value of σ\\sigma given on each residual panel is the standard deviation of the residuals in ppm.", "Bottom: The common mode correction term, which was produced by dividing the top LC by the best fit transit model (the black LC in the middle panel, see Appendix  for detailed discussions).", "This term was used to correct for common systematics in the binned light curves." ], [ "Binned light curve fitting", "The next step on the path to produce the transmission spectrum is fitting the binned light curves to obtain values of $R_p/R_s$ as a function of wavelength.", "With the uncorrected binned-light curves in hand, we first used the same 3$\\sigma $ clipping process discussed in Section REF for each bin.", "Then the bins were detrended with a combination of common mode correction and polynomial fitting (CMC+Poly), where the CMC term (shown in Figure REF ) was obtained from a PCA+GP fit of the white-light curve (detail in Appendix REF ).", "We also fixed all parameters, aside from $q_1$ , $q_2$ , and $R_p/R_s$ , to values obtained by the PCA+GP white-light fit obtained in Section REF $P$ , $a/R_s$ , and $i$ were already fixed to the values used by [109].", "The bins used and their corresponding radii, along with plots of the raw (target/comparisons) and detrended light curves can all be found in Appendix .", "We determined the CMC+Poly routine was best to detrend the binned light curves for this dataset, by testing its effectiveness against synthetic data, which is discussed in the following section (Sec.", "REF )." ], [ "Comparing the performance of GP and CMC+Poly with Synthetic data", "A number of techniques have been used to detrend the light curves of exoplanet transits (i.e.", "PCA e.g.", "[45], GP e.g.", "[35], CMC e.g.", "[34]), but there are few instances in the literature that compare the effectiveness of one method over another [34], [69].", "Furthermore, using two separate analysis techniques could produce widely different results (e.g.", "[86] and [37]).", "Thus, it is important to ensure the method utilized for a particular dataset yields as accurate results as possible and provides consistent uncertainty measurements.", "To instill confidence in our analysis procedures of the binned light curves, we synthesize transmission spectra to compare the precisions and accuracies of obtained transit depths to synthesized values.", "We test the synthetic data against two common transit reduction techniques: 1) a Gaussian Processes (GP) routine, and 2) Common-Mode-Correction followed by polynomial correction (CMC+Poly).", "To test both methods, we created 500 synthetic light curves similar to the VLT/FORS2 observations.", "A detailed description of how the synthetic data were created can be found in Appendix , but in general we simulated 50 flat (constant $R_p/R_s$ ) transmission spectra out of 10 bins each.", "All of the 10 bins had approximately the same shot noise levels ($\\sim $  400 ppm), which was assigned to produce a white-light curve noise level of about 125 ppm, consistent to the VLT/FORS2 white-light photon noise limits ($\\sim $ 63 and 162 for transit UT170729 and UT170822, respectively).", "All bins in a given spectrum had the same overarching systematic generated by a random draw in a GP distribution, where the GP was constructed to be correlated to a few of the UT170729 observation's auxiliary parameters.", "Then each individual bin had additional systematics generated from up to a second-order polynomial fit using other auxiliary parameters with random polynomial coefficients.", "Each bin also had their own quadratic limb darkening coefficients.", "Because the VLT/FORS2 observations only had one comparison star, we also only created one comparison star in our synthetic data.", "As outlined in Appendix REF , when there is only one comparison star, PCA cannot be done and the PCA+GP routine becomes a GP routine with the comparison star used as a linear regressor.", "This is why in the synthetic analysis we refer to this method as a GP routine, but when using the same routine with the Magellan/IMACS data, which has two comparison stars, we refer to it as PCA+GP.", "Images of each step in the synthetic data production process are shown in Appendix .", "After the synthetic spectra were produced, we fit all 50 white light curves using the GP method in the same way described in Appendix REF .", "We used the results of each white light curve as priors for the analysis of the binned light curves (see Appendix ).", "With those white-light parameters we produced the transmission spectra following first the GP process to detrend the bins, and then with the CMC+Poly process, which are both discussed in Appendix .", "Both methods used to detrend the binned-light curves utilized the parameters determined from the GP detrended white-light data, and the CMC+Poly method used that white-light curve model to produce the CMC term.", "This allowed us to compare the effectiveness and accuracy of both methods given that each true depth is known.", "Additionally, because all 50 spectra were flat with the same inputted depth, we could collectively compare the results from every reduced bin.", "To first understand the accuracies of both methods, we plot a histogram of the difference of each bin's obtained depth relative to the true depth ($R_p/R_s=0.1157$ ), shown in the first column of Figure REF .", "From this we see that though the true depth is on average obtained using the GP method (first row), neither method consistently reproduced the true depths, where only 46.32$\\pm $ 2.82%uncertainties obtained through bootstrapping.", "of the bins detrended using the GP method obtained a depth 1$\\sigma $ from its average uncertainties levels.", "The mean uncertainty of obtained $R_p/R_s$ with the GP method was 0.00774.", "This is significantly worse using the CMC+Poly method, which only has 12.54$\\pm $ 2.00% of the bins within its 1$\\sigma $ level, where the mean uncertainty with the CMC+Poly method was 0.00127.", "This suggests that if one were to fit any given transmission spectrum with the GP routine there is only a 46.32$\\pm $ 2.82% likelihood of those obtained depths being consistent with the physical depths.", "This is only worse using the CMC+Poly method.", "One likely strong contribution for this is that the bins are already biased by the white-light fit, given that the bins reduction is dependent on the white-light's.", "This is especially the case for the CMC method, because the common mode term, which drives the binned light curves correction model, can only be determined from the white-light fit.", "To support this we compared the obtained binned depths relative to their corresponding white-light fits' depths and find much more consistency of the depths with 41.79$\\pm $ 2.82% and 57.28$\\pm $ 2.88% within the 1$\\sigma $ average uncertainty for the CMC+Poly and GP methods, respectively.", "Still, neither method can consistently re-obtain the true white-light depths.", "However, when comparing each bin to a corresponding mean depth determined by averaging all 10 bins for a given transmission spectrum (column two of Figure REF ), we find 78.21$\\pm $ 2.42% and 70.72$\\pm $ 2.67% are within 1$\\sigma $ for the CMC+Poly and GP methods, respectively.", "This implies that though the absolute depth is not consistently obtained, a relative depth is for both detrending methods.", "If that is the case, it provides justification for the required offsets often needed when combining transmission spectra from different nights and/or different instruments [58], [105], [109].", "These results would also explain why the inconsistency in white-light depth found for transit UT171108 (Table REF and Fig.", "REF ) does not imply that the transmission spectrum of that night is incorrect.", "The likely scenario for that dataset is the GP routine misestimated the white-light depth (likely because of the fixed parameters), but this is common for the majority of the synthetic data as well.", "However, the relative binned depths can still be preserved using this white-light depth for the CMC correction, because the difference in depths from one bin to another is still maintained.", "To further highlight that the structure of the spectrum is preserved we plot the standard deviation of all bins in a given spectrum.", "Since, each synthesized spectrum is flat, each bin (of a given spectrum) should not significantly vary from one another.", "This is exactly what we find, where every spectrum has a bin standard deviation significantly lower than that spectrum's average $R_p/R_s$ uncertainty width (see column three of Figure REF ).", "Furthermore, the third column shows that the standard deviation is higher for the GP method, implying that the CMC+Poly method is more consistent.", "Additionally, because the CMC+Poly method inherently produces lower uncertainties (average $R_p/R_s$ uncertainty of 0.00127 for CMC+Poly compared to 0.00774 for GP), for this set of data, the CMC+Poly method is consistently more accurate and precise than the GP method.", "As such we elect to use the CMC+Poly method to detrend all binned light curves.", "It is important to understand that these finding are only for this specific dataset, which is constructed to mimic the particular VLT/FORS2 observations of WASP-96b; i.e.", "the synthetic data has a relatively low shot noise level ($\\sim $ 400 for bins), the bulk of the systematics are dominated by the white-light systematics (see Appendix ), and there was only one comparison star used.", "Therefore, this should not be extrapolated to every dataset.", "For example, we reduced the VLT/FORS2 data with just a CMC correction and found a similar fit compared to using CMC+Poly, outlining how little chromatic systematics persists in the given VLT/FORS2 data.", "If chromatic systematics are more dominant in a dataset or there are multiple comparison stars, then it could be possible that the data would be better reduced with a method like PCA+GP, which is less heavily dependent on the initial white-light fit.", "For this reason, we still ran the ACCESS data through both the CMC+Poly and PCA+GP routines as described below to confirm CMC+Poly still performed better for those data.", "In summary, one should explore the best detrending method for specific data before assigning one.", "Figure: Histograms used to outline the biases and precisions of both the GP (top row) and the CMC+GP (bottom row) routines.", "Left column: We fit each of the 500 total synthetic binned light curves using the GP and CMC+Poly routines; then take the difference of each fitted binned depth from the true depth (R p /R s R_p/R_s=0.1157) used to produce all synthetic data.", "That difference was divided by the uncertainty of the fitted depth, in order to obtain the bias from the input depth, relative to the uncertainties.", "The distribution of the relative differences are plotted.", "Middle column: This is similar to the left column, but instead is the difference of each fitted binned depth from the mean depth of all 10 bins in its corresponding transmission spectra.", "This provides a relative bias on obtained transit depths.", "In this case `relative' means relative to the transmission spectrum's mean depth.", "For both the left and middle columns, the black dashed line represents the average 16 and 84 percentiles of the histograms, the middle solid black line is the 50 percentile, and the red dotted line is 0.", "This corresponds to the true value of the depth, thus when the dashed black line and dotted red line do not overlap the routine has an inherent bias on that obtained depth.", "Right column: This is the standard deviation of each of the 50 transmission spectra.", "It shows the measured scatter of the transmission spectra, where the true scatter is 0, because the spectra were made to be flat.", "Here the black dashed line is the mean width of all bins uncertainties.", "Note that though both routines produce scatter well under their respective mean uncertainty widths, because the mean uncertainty width is over 6 times larger for the GP routine, the CMC+Poly routine is much more precise." ], [ "Optical Transmission Spectrum from the VLT/FOR2 and ACCESS Data", "We produced the transmission spectrum by plotting the $R_p/R_s$ found from each detrended binned-light curve against that wavelength interval (bin).", "Three optical transmission spectra were created.", "One from combining the two FORS2 transits where we weighted averaged all overlapping bins, another from combining the two IMACS transits, and the third from combining all four transits (global optical spectrum).", "These are plotted in Figure REF .", "As justified in Section REF , we fit an offset when combining each of the spectra.", "For a given combined spectrum a white-light depth was determined for each individual spectrum using a weighted mean of only overlapping bins.", "The weighted mean of these white-light depths was then used as the central depth for which each individual spectra was offset to.", "The average precision of each bin for the combined IMACS, FORS2, and global optical spectra are 0.00129, 0.00094, and 0.00076 $R_p/R_s$ , respectively.", "The IMACS data has more scatter and lower precision than the FORS2 data (see Fig.", "REF ), even though both use two transits.", "One explanations to this is that the size of Magellan Baade (6.5 m) is smaller than VLT (8.2 m), meaning less collecting area and more shot noise.", "Additionally, the difference in comparisons used could cause the IMACS data to have more systematics.", "The comparison [64] used (D=0.1) was more similar to the target than either of ours (D=0.24 and 0.14; see Table REF ).", "The importance of the comparisons is emphasized by [1], where they were able to detect a strong sodium signal and a super-Rayleigh scattering slope in the atmosphere of WASP 94Ab with just one comparison and one transit, partially attributed to the comparison being nearly identical in spectral type and location in the sky.", "Furthermore, for both IMACS transits the baselines, which is needed to properly correct for systematics, were relatively short (seen in Fig REF ), where ideally the baseline should be the same length as the transit duration.", "Likely the largest cause of deviation between the two datasets is that the IMACS data has more chromatic systematics.", "This can be seen when looking at the binned light curves in Appendix .", "Why the chromatic systematics are stronger for the IMACS observations is unclear.", "It might be due to chromatic differences in comparisons, or some other issue.", "Regardless, the CMC method relies on the assumption that the bulk of the systematics found in the white-light curve can be applied to each of the binned light curves.", "If each bin's systematics are more unique the initial CMC is not as effective, and might even introduce more spurious signals that the polynomial fit has to correct for.", "Concern over this led us to analyze the IMACS data with both CMC+Poly and PCA+GP routines.", "In doing so we found that the two transmission spectra were consistent to one another, with each bin deviating from one another by 0.562$\\sigma $ on average.", "However, the uncertainties of the PCA+GP method were nearly 5 times larger than the CMC+Poly's.", "Given both methods produced similar features, the accuracy of the CMC+Poly method is supported by Section REF , and the desire to use the same routine for all datasets we choose to continue using the CMC+Poly routine for the IMACS data.", "Figure: The transmission spectra using only the two Magellan/IMACS transits (top), only the two VLT/FORS2 transits (middle), and combining all four transits (bottom).", "In all three plots the black points correspond to the final spectrum produced by taking the weighted average of all overlapping bins.", "When combining each spectrum an offset is applied, so the means of each individual spectra are consistent, before the bins are averaged together.", "The center of the Na I{\\rm Na~I} and KI{\\rm K~I} absorption are plotted as dotted gray and yellow lines, respectively." ], [ "Stellar Activity", "Activity in the host star can introduce signals into the planetary transmission spectrum [72], [11], [67], [75], [2], [100], [77].", "Therefore, in order to prevent misinterpretation of WASP-96b’s transmission spectrum, we parameterize the host star’s level of activity.", "The three proxies for stellar activity we explore are 1) rotational period [68], 2) Ca ii lines[97], [59], [65], and 3) photometric modulation [46], [104]." ], [ "Rotational Period", "We infer the stellar rotational period by combining the radius of the star with the projected stellar rotational velocity, $\\hbox{$v$\\,sin\\,$i$}$ , as well as fitting a periodic signal to the photometric monitoring data.", "The $\\hbox{$v$\\,sin\\,$i$}$ for WASP-96 was determined by [39] using the Euler/CORALIE spectrograph.", "It was not well constrained but was found to be 1.5$\\pm $ 1.3 km s-1.", "This provides a 3-$\\sigma $ upper limit on $\\hbox{$v$\\,sin\\,$i$}$ of 5.4 km s-1.", "Combining this with the stellar radius of 1.05$\\pm $ 0.05 R$\\odot $ [39] we estimate a 3-$\\sigma $ lower limit on rotation period of about 9.8 days.", "Thus, we scanned the photometric data for periodic peaks within a range of 9.8 to 300 days using Lomb-Scargle periodogram analysis [55].", "With the binned combined TESS data the highest periodic peaks were 35.9, 37.7, and 31.2, and with the combined ASAS-SN data they were 28.3, 28.8, and 11.2.", "However, for all peaks the False Alarm Probability (FAP) were greater than 10-1, thus no significant peaks were found with the periodogram analysis.", "This is likely because the photometric modulation is in general very small (see Sec.", "REF or Sec.", "REF ).", "We then jointly fit the ASAS-SN and binned TESS data using the Juliet package [21] with a semi-periodic kernel.", "In this joint fit, only the period and timescale terms (see equation 9 of [21]) are set common for all photometric campaigns.", "All other parameters were specific to the combined TESS (sector 2 and 29), the V band ASAS-SN, or the g band ASAS-SN data.", "When doing this we assume the periodicicity is due to stellar inhomogeneities Even with a relatively quiet star like WASP-96, we are assuming that smaller spots/faculae can be used to measure modulation and rotational period.", "coming in and out of view as the star rotates, which is often done [41], [86], [62].", "We use wide uniform priors on the period from 9.8 to 50 days, which is consistent with what the periodograms and $\\hbox{$v$\\,sin\\,$i$}$ weakly suggest.", "The resulting period was found to be 31.3$^{+0.3}_{-3.4}$ days.", "Given the observed correlation of rotational period to activity levels [68], [78], this also implies that WASP-96 is a relatively quiet star.", "Figure REF shows the photometric monitoring campaigns along with the Juliet best fits.", "Figure: Photometeric monitoring of WASP-96: The top row is TESS monitoring.", "There the red dashed lines are the Juliet semi-periodic best fit.", "The grey dots are the original 30-minute cadence TESS observations and the green dots are the data at 100 (∼\\sim 3.34 hours) binning, with their associated error bars in blue.", "The right figure is sector 29 data and the left is from sector 2, where both sectors were combined to act as one dataset for the Juliet fit (the binned data was used for the fitting).", "The bottom row is ASAS-SN monitoring, where the blue dashed lines are the Juliet semi-periodic best fit.", "On the right, the green hollow circles with associated error bars correspond to the g band data.", "On the left, the red hollow circles with associated error bars correspond to the V band data.", "The V and g band data were used as separated datasets in the Juliet fit.", "For all Juliet fits (blue and red dotted lines), there is a gradient of shaded gray regions representing the 1,2,& 3 sigma confidence intervals.", "For the TESS data, the confidence intervals are about the size of the dotted line." ], [ "Ca ", "The Ca ii H & K lines were measured using two R = 48000 spectra collected with the MPG 2.2-m/FEROS spectrograph on 19 December 2016.", "Each spectrum has an average SNR of 27.", "The reduced data was acquired from ESO’s online archive.", "Figure REF shows the Ca ii H & K lines and we see no emission in the core of either of the lines, implying that WASP-96 is a relatively inactive star.", "Figure: High-resolution FEROS spectrum of WASP-96: Both Ca ii H & K lines (top), where the dashed red lines correspond to the K line core centered at 393.366 nm and the H line core centred at 396.847 nm.", "The left bottom panel is zoomed in on the K line and the bottom right is zoomed in on the H line, with the central cores highlighted with red dotted lines in both.", "There is no emission seen in either of the cores." ], [ "Photometric Modulation", "The analysis in the above two subsections suggest that WASP-96 is a quiet G-type star.", "However, we use the amplitude of the photometric modulation of the star to quantify its level of activity in terms of spot covering fraction and spot temperature, which are the parameters needed for the retrievals (see Sec. ).", "This is done by using equation 2 and Table 2 of [76] and twice the TESS MAD of 0.0058 (see Sec.", "REF ), assuming that the MAD is approximately the amplitude of sinusoidal variation.", "We use the TESS data because the variability in the ASAS-SN data seems to be driven by more than just the star (likely low precision), which is apparent when comparing the MAD of the TESS and ASAS-SN monitoring campaigns of the same target.", "Following that procedure, we estimate a spot covering fraction of WASP-96 1.35$\\pm $ 0.97% assuming only spots are present and 1.40$\\pm $ 1.09% assuming spots and faculae are both present.", "To capture these limitations in the retrieval analysis, we constrained the heterogeneity covering fraction, $f$ , to have normal priors with a mean of 0.014 and standard deviation of 0.009.", "For a G8 star like WASP-96, we should expect the heterogeneity temperature contrast, $\\Delta $  T, to be roughly 1600 K [10], [76].", "We used wider uniform priors from -3000 to 3000 K on $\\Delta $  T, see Appendix ." ], [ "Retrieval Analysis", "We combined each of the three optical spectra discussed in Section (IMACS only, FORS2 only, and global spectra, see Fig.", "REF ) with the HST/WFC3 data.", "We ran each of these three optical to near-IR spectra against two retrievals: PLATON [112] and Exoretrievals [22].", "We used both PLATON and Exoretrievals, because their differing approaches of modeling exo-atmospheres provides different insights about the observed transmission spectra.", "The key differences between PLATON and Exoretrievals are: (1) PLATON includes collision induced absorption, where Exoretrievals does not, (2) Exoretrievals models the transmission spectrum using a semi-analytical formalism with an isothermal, isobaric, atmosphere and non-equilibrium chemistry, but PLATON uses an isothermal atmosphere and imposes equilibrium chemistryThough equilibrium chemistry, might be an inaccurate assumption for observed transmission spectral features [98], [51], [83], using it still provides useful insights that could not be obtained without this assumption., and (3) the bulk of the line list used by PLATON are from HITRAN [82], whereas the majority of what Exoretrievals uses is from HITEMP [81] and ExoMol [110], [92].", "Additionally, testing a transmission spectrum against multiple retrievals provides a robustness against assumptions that are unique to each retrieval [58], [49].", "Both retrievals used nested sampling to explore their parameter space (dynesty, [90] [90], for PLATON and PyMultiNest, [13] [13], for Exoretrievals), as such we used the differences in log Bayesian evidences, $\\Delta \\ln Z$ , to test which specific model was favored over another.", "Following the same prescription as [58] [58] [94], [9], we interpreted the $\\Delta \\ln Z$ values in a frequentist significance as: |$\\Delta \\ln Z$ | of 0 to 2.5 is inconclusive with < 2.7$\\sigma $ support for the higher evidence model, |$\\Delta \\ln Z$ | of 2.5 to 5 corresponds to a moderately significant detection of 2.7$\\sigma $ to 3.6$\\sigma $ , and |$\\Delta \\ln Z$ | $\\ge $ 5 corresponds to strong support for one model over the other.", "With Exoretrievals, we tested the spectra against having either water, potassium, sodium, water and sodium, and all three species in the transmission spectra.", "Along with these molecular and atomic species, we tested if the spectra warranted high altitude scattering agents, stellar activity, and a combination of scatters and activity.", "Lastly, a model with no features (flat spectrum) and only activity features was tested.", "In total, we tested 22 different combinations of models.", "For all models, aside from the flat spectrum, a reference radius (parameterized with $f$ ) and reference pressure (P$_0$ ) were fit.", "These are the pressure and corresponding radius where the atmosphere is optically thick in all wavelengths.", "With PLATON we tested a model with scattering agents, stellar activity, models with both scattering agents and stellar activity, and a model without either (clear).", "All models fit for a reference radius (R$_0$ ) corresponding to the radius of the planet at an arbitrary reference pressure, which was set to 1 bar.", "The models also fit for a pressure (P$_{cloud}$ ) where the atmosphere is optically thick.", "The priors did not change for each spectrum we ran against the retrievals and we set the priors between PLATON and Exoretrievals as consistent to one another as possible.", "For all three spectra and both retrieval models we fit an offset between the optical and near-IR dataPLATON v3 could not fit an offset between different dataset, as such, we modified the code to do so., which is justified given that the detrending method used for the optical data does not preserve absolute depth (see Sec.", "REF )." ], [ "IMACS & FORS2 Transmission Spectra", "Tables of $\\Delta \\ln Z$ for both datasets against both retrievals are shown in Tables REF & REF .", "In the tables the $\\Delta \\ln Z$ values for each model was determined by comparing the clear model, for PLATON, or the clear model and flat spectrum, for Exoretrievals, to the other models.", "One can also compare any given model from another by examining the difference of $\\Delta \\ln Z$ 's between one another.", "Using Exoretrievals, we find from both the IMACS and FORS2 spectra, independently, that the model with the highest $\\Delta \\ln Z$ is one with water and sodium (no potassium).", "We find that models with additional stellar activity or scatters do not make a significant difference in the $\\Delta \\ln Z$ , aside for the IMACS dataset that has a significant decrease in $\\Delta \\ln Z$ when including scatters ($\\Delta \\ln Z$ decreases by 4, compared to a model with only water and sodium).", "The feature blue-ward of 500 nm is likely what the retrievals with scatters and activity are attempting to fit.", "However, because the feature is not extreme, relative to the uncertainties (especially for the IMACS dataset), there is not sufficient support for the extra complexity of these models.", "As such we can only claim a tentative detection of a slight blue-ward slope either attributed from the star or a possible Rayleigh scattering slope.", "For the IMACS data the best fit model that included stellar activity found spot parameters of $\\Delta T$ =-150$^{+1300}_{-1700}$  K and $f$ =0.0114$^{+0.0085}_{-0.0072}$ .", "For FORS2 those activity parameters were $\\Delta T$ =-1060$^{+1100}_{-1200}$  K and $f$ =0.0124$^{+0.0083}_{-0.0080}$ .", "The best model which included a haze scattering slope for the IMACS data obtained a haze power law of $\\gamma $ =-9.3$^{+7.8}_{-3.4}$ , and $\\gamma $ =-9.7$^{+5.1}_{-3.0}$ was obtained for the FORS2 best scatters model.", "With Exoretrievals $\\gamma $ of -4 corresponds to a Rayleigh slope (see Appendix D of [22]), implying that even though the retrieved slopes are unconstrained, they are consistent with Rayleigh scattering.", "The detections of $H_2O$ and Na were highly significant ($\\Delta \\ln Z >$ 11) for both datasets.", "The highest-evidence retrieval model parameters for this data subset and the others can be seen in Table REF |l|C|C|C|C|C|C|C|C|C|[h!]", "$\\Delta $ ln Z for various Exoretrievals (left) and PLATON (right) models relative to a clear (and flat for Exoretrievals's case) spectrum with the subset of data that included only the Magellan/IMACS and HST/WFC3 data.", "The retrievals with water and sodium were heavily supported by Exoretrievals.", "Models that included scattering were the most supported with PLATON.", "6|c|Exoretrievals 2|c|PLATON Model: flat $H_2O$ $Na$ $K$ $H_2O +Na$ $H_2O + K +Na$ Model: clear 0.0 6.99 -0.39 4.07 12.78 10.94 clear 0.0 scatterers — 5.5 -0.89 3.49 8.78 8.39 scattering 4.54 activity 0.35 5.97 -1.23 4.04 10.65 10.1 activity 0.6 Both — 4.91 -1.92 3.34 9.63 -4.05 Both 4.14 Interestingly, with PLATON there is less consistency amongst the two datasets.", "All PLATON models with the FORS2 dataset were indistinguishable from one another, which is in agreement with what Exoretrievals found for the FORS2 dataset.", "That is the slope blue-ward of 500 nm could be explained by either activity or a scattering slope, but neither is required for the data.", "For the FORS2 dataset the best fit model including activity found $\\Delta T$ =-1040$^{+1200}_{-1000}$ and $f$ =0.0143$^{+0.0083}_{-0.0089}$ ; the best fit model including scatterers found a scattering slope, $\\alpha $ , of 5.7$^{+5.3}_{-6.0}$ ($\\alpha $ =4 is Rayleigh); and the highest evidence model (clear model) obtained a metallicity, $\\log _{10}(Z/Z_{\\odot })$ , of 0.26$^{+0.75}_{-0.78}$  dex, C/O of 1.11$^{+0.55}_{-0.39}$ , $log_{10}(P_0)$ of 0.3$^{+1.7}_{-1.3}$  bars, and T$_p$ of 987$^{+92}_{-52 }$ .", "Contrarily to PLATON, the IMACS data obtained a significantly higher evidence ($\\Delta \\ln Z >$ 4) for the models that included scatterers.", "The retrieved values with IMACS differed from the fit with FORS2 likely because the Na feature was not as prominent in the IMACS data.", "Thus, the retrieval increases the metallicity ($log_{10}(Z/Z_{\\odot })$ =0.51$^{+0.53}_{-0.75}$  dex) and decreases the temperature (T$_p$ of 752$^{+62}_{-94}$  K) to mute the Na feature.", "In turn, the best retrieved spectrum is overall different from the FORS2 spectrum.", "Still the retrieved scattering slope of both datasets was consistent with a Rayleigh scattering slope, though not well constrained.", "|l|C|C|C|C|C|C|C|C|C|[h!]", "Same as Table REF but with the subset of data that included only the VLT/FORS2 and HST/WFC3 data.", "The retrievals with water and sodium were heavily supported by Exoretrievals.", "There was no model that had high enough evidence to favor it over another with PLATON.", "7|c|Exoretrievals 2|c|PLATON Model: flat $H_2O$ $Na$ $K$ $H_2O +Na$ $H_2O + K +Na$ Model: clear 0.0 4.94 0.13 5.81 11.32 10.65 clear 0.0 scatterers — 3.54 -1.41 3.73 9.48 8.97 scattering -1.18 activity 0.06 4.13 -1.43 4.18 11.38 11.11 activity -0.18 Both — 3.11 -2.12 3.11 9.13 -4.07 Both -1.25 |l|C|C|C|l|C|C|C|[h!]", "Parameters obtained by the best-fit retrievals for each data subsets (IMACS with WFC3, FORS2 with WFC3, and combined optical with WFC3).", "The model which only included water and sodium was the highest-evidence model with Exoretrievals for each data subset (the evidence for the model with and without activity was nearly identical in the FORS2 subset).", "With PLATON, the model including scattering was favored with the combined dataset and the IMACS one, but for the FORS2 dataset a featureless model was marginally preferred.", "Here Tp, P0, offset, $\\alpha $ , Z/Z$_{\\odot }$ , C/O, and $\\log _{10}(H_2O)$ , $\\log _{10}(Na)$ correspond to planet terminator temperature [K], reference pressure at which the atmosphere is optically thick [bar], offset in transit depth [Rp/Rs], scattering slope ($\\alpha $ of 4 is Rayleigh), metallicity of the star relative to solar, carbon-to-oxygen abundance ratio, and log mixing ratios of water and sodium, respectively.", "4|c|Exoretrievals 4|c|PLATON IMACS+WFC3 FORS2+WFC3 combined data IMACS+WFC3 FORS2+WFC3 combined data Tp 730$^{+180}_{-140}$ 790$^{+220}_{-160}$ 830$^{+160}_{-140}$ Tp 752$^{+62}_{-94}$ 987$^{+92}_{-52}$ 877$\\pm 40$ $\\log _{10}(P\\textsubscript {0})$ 0.64$^{+1.63}_{-1.93}$ 0.4$^{+1.81}_{-2.17}$ 0.29$^{+1.86}_{-2.02}$ $\\log _{10}(P\\textsubscript {0})$ 1.58$^{+0.84}_{-0.97}$ 0.3$^{+1.7}_{-1.3}$ 1.3$^{+1.0}_{-1.1}$ offset 0.01201$\\pm $ 0.0004 0.00238$^{+0.00041}_{-0.00038}$ 0.00547$^{+0.00031}_{-0.00032}$ offset 0.00967$^{+0.00034}_{-0.00031}$ 0.00199$^{+0.00044}_{-0.00043}$ 0.00414$^{+0.00036}_{-0.00035}$ $\\log _{10}(H_2O)$ -3.9$^{+1.8}_{-2.0}$ -4.3$^{+1.9}_{-2.1}$ -4.5$\\pm 2.0$ $\\log _{10}(Z/Z_{\\odot }$ ) 0.51$^{+0.53}_{-0.75}$ 0.26$^{+0.75}_{-0.78}$ -0.49$^{+1.00}_{-0.37}$ $\\log _{10}(Na)$ -5.5$^{+1.8}_{-1.9}$ -5.0$^{+2.0}_{-2.1}$ -5.4$^{+2.0}_{-1.9}$ C/O 0.69$^{+0.72}_{-0.41}$ 1.11$^{+0.55}_{-0.39}$ 0.97$^{+0.65}_{-0.50}$ $\\alpha $ 3.7$^{+6.3}_{-5.0}$ — 10.4$^{+2.5}_{-4.5}$" ], [ "Combined Transmission Spectrum", "The $\\Delta \\ln Z$ of all models run using the global data against both retrievals is shown in Table REF .", "When running the retrievals against the combined data, we find a similar trend as that for the individual datasets (aside for IMACS with PLATON).", "That is, a major detection of water and sodium, and tentative signs of a blue-ward slope attributed to stellar activity or a scattering slope.", "For Exoretrievals the evidence, $\\Delta \\ln Z$ = 19.45, is even stronger than the individual datasets, showing that combining the data does improve the overall detection of the species.", "The corner plot of the highest-evidence model retrieved by PLATON (one with scatters) and Exoretrievals (one without activity and scatters but including H$_2$ O and Na) is shown in Figures REF and REF .", "Figures REF and REF show the global data with over-plotted models that either include stellar activity, a scattering slope, or none (flat and clear) for both retrievals.", "We elect to use the global transmission spectra which included all optical data to interpret the retrievals and atmosphere of WASP-96b, because the maximum relative retrieved evidence and transmission spectrum precision are higher when including both optical data.", "|l|C|C|C|C|C|C|C|C|C|[h!]", "Same as Table REF but with the dataset that included the combined IMACS, FORS2, and HST data.", "The retrievals with water and sodium were heavily supported by Exoretrievals and no model was prefered with PLATON.", "7|c|Exoretrievals 2|c|PLATON Model: flat $H_2O$ $Na$ $K$ $H_2O+Na$ $H_2O+K+Na$ Model: clear 0.0 5.73 -0.56 12.52 19.45 19.11 clear 0.0 scatterers — 6.87 -0.53 11.63 18.35 17.46 scattering 0.8 activity 0.1 5.06 -1.45 12.36 18.33 18.14 activity -0.28 Both — 4.95 -1.92 10.79 17.62 -4.16 Both 0.55 Figure: Corner plot of the PLATON best fit retrieval model with, which is run against the combined Magellan/IMACS, VLT/FORS2, and HST/WFC3 data.", "Its corresponding transmission spectrum is shown in Figure (red model).", "Vertical dashed lines mark the 16% and 84% quantiles.Figure: Same as Figure , but for Exoretrievals, where its corresponding transmission spectrum is shown in Figure (blue colored model).Figure: The final transmission spectra of WASP-96b, which include the combined FORS2 and IMACS data (purple) and the G102 (green) and G141 (pink) data both from .", "Best fit PLATON retrieval models that include stellar activity (grey), scatters (gold), and a clear atmosphere (red) are also plotted, with there 1-σ\\sigma confidence interval shaded in the same colors.", "Because for each of the three models, the optical offsets were slightly different, what is shown here is the mean depth where the true offsets were 0.00476 -.00039 +.00040 ^{+.00040}_{-.00039}, 0.00414 -.00035 +.00036 ^{+.00036}_{-.00035}, and 0.00467 -.00034 +.00039 ^{+.00039}_{-.00034} for the activity, scatters, and clear models, respectively.", "As can be seen in Table the model with the highest evidence is one with scatters; however, no model has an evidence strong enough to favor it over another.Figure: Same as Figure , but with Exoretrievals.", "In this figure the optical (combined FORS2 and IMACS), G102, and G141 observations are cyan, white, and red, respectively.", "Again, the evidence of each plotted model (activity - gray, scatters - green, and clear - purple) are indistinguishable from each other, but the clear atmosphere has the highest evidence.", "The offset with each fit were 0.00554±\\pm .00031, 0.00558±\\pm .00034, and 0.00547 -.00032 +.00031 ^{+.00031}_{-.00032} for the activity, scatters, and clear models, respectively." ], [ "Atomic and Molecular Features", "The retrieved mixing ratios of sodium ($\\log _{10}(Na)=-5.4^{+2.0}_{-1.9}$ ) and water ($\\log _{10}(H_2O)=-4.5\\pm 2.0$ ) are in agreement both with what was obtained by [109] [109] ($\\log _{10}(Na)=-3.88^{+1.05}_{-0.82}$ and $\\log _{10}(H_2O)=-3.65^{+0.90}_{-0.94}$ ) and what was obtained by [64] [64] ($\\log _{10}(Na)=-5.1^{+0.6}_{-0.4}$ ).", "The Na mixing ratios obtained here are consistent with solar abundance ($log_{10}(Na)=-5.78\\pm 0.03$ ; [6] [6]) and WASP-96's stellar abundance [64].", "The water abundance is also consistent with water abundances of Jupiter constrained by Juno ($\\log _{10}(H_2O)=-2.6^{+0.27}_{-0.44}$ ; [54]).", "This implies that the formation process for WASP-96b might have been similar to our own Jovians, but that conclusion is limited by the uncertainties in the measured mixing ratios.", "Theoretical predictions for clear atmosphere planets predict absorption signatures from the optical alkali features of Na and K [84].", "However, our observations of WASP-96b show no observable evidence of K absorption, even though strong ${\\rm Na~I}$ and ${\\rm H_2O}$ features imply WASP-96b has a clear transmission spectrum.", "There are multiple possibilities as to why potassium was not significantly detected.", "The most obvious hindrance in detecting K was the gap in the transmission spectrum (759.4–767.2 nm) that was nearly aligned with the center of the most prominent K doublet feature (768.15 nm).", "Essentially, this only allowed K to be constrained by its wings, which might not be enough even if K is present.", "The possibility of K being present in the atmosphere is hinted in Figure REF , where PLATON's retrievals by default include K because of imposed equilibrium chemistry and relative abundances determined by metallicity.", "In those models we see that the transmission spectrum is somewhat consistent with the K doublets' features, where the blue-ward bin in the K wing is within the model's 1-$\\sigma $ interval and the red-ward bin is not.", "The fact that the evidences of the Exoretrievals models including K are lower than those excluding it is likely only because the data could be explained with or without K, given the gap in the central doublet band.", "Another possibility is that the potassium is locked away in KCl, which has been suggested to have a highly efficient formation rate in this temperature regime [32].", "Further space-based observations, or ground based high resolution observations will be needed to determine if abundant K absorption is truly present in the atmosphere of WASP-96b." ], [ "Activity and Optical Slope", "For the models that included activity, retrieved $\\Delta T$ temperatures from both retrievals and all datasets were consistent ($<$ 1-$\\sigma $ ) with no active regions ($\\Delta T$ = 0 K), implying a non-detection of stellar activity.", "Therefore, stellar photometry (see sec.", "REF & REF ), stellar spectroscopy (see sec.", "REF ), and the transmission spectrum retrieval analysis all support the idea that WASP-96 has very little activity.", "H$_2$ Rayleigh scattering is expected blue-wards of $\\sim $ 450–550 nm in a hydrogen dominated atmosphere without high altitude clouds [84], [53].", "The fact that we were unable to significantly identify one is likely due to limitations of the data, given that we only have two and a half broader bins in this wavelength range.", "The slope could be better constrained with HST/STIS G430L has coverage approximately from 0.29–0.57 $\\mu $ m https://www.stsci.edu/hst/instrumentation/stis or HST/WFC3/UVIS WFC3/UVIS G280 covers 0.2-0.8 $\\mu $ m and has better throughput in the blue than STIS [103] observations.", "This would likely be sufficient to distinguish from stellar activity, because the star is relatively quiet.", "Therefore, when imposing these constraints, as was done in Section REF , the only viable activity features produced are very shallow slopes.", "This is outlined in Figures REF and REF where the models with activity fit are shown in gray for both figures.", "Furthermore, the activity of quiet stars, like this one, is dominated by faculae [79], [77], which produces the opposite signal of what is expected of a Rayleigh slope, when the activity regions are unocculted.", "Meaning that large cold unocculted spots, needed to mimic a Rayleigh slope, would be in direct contradiction to all other observations of WASP-96.", "Thus, it is clear that if a blue-ward scattering slope is found with additional observations, the most viable explanation would be attributing the feature to the planet.", "Though the data is not sufficient enough to significantly distinguish between a model with and without a scattering slope, all retrieved scattering slopes were found to be consistent ($<$ 2-$\\sigma $ ) with a Rayleigh scattering slope.", "If this can be confirmed the Rayleigh like slope would best be attributed to H$_2$ scattering." ], [ "Aerosols", "The higher reference pressures obtained by PLATON and Exoretrievals (a few to tens of bars) both agree with one another and supports the idea that if WASP-96b hosts thick aerosol layers, they are confined to beneath the top of the atmosphere.", "The strong water features and broad ${\\rm Na~I}$ absorption wings observed also support this conclusion.", "Additionally, when fitting for aerosols with Exoretrievals the cloud cross section, $\\sigma _{cloud}$ (see [22] Appendix D) was initially set to be very wide, with log-uniform priors from -80 to 80 for $\\sigma _{cloud}$ .", "In doing so, we found that across all 3 datasets (i.e.", "optical data from IMACS, FORS2, and combined) and all iterations of models that include scattering (i.e.", "no mater which molecules were included or if including activity) the means of $\\sigma _{cloud}$ were nearly the same at $\\sim $ -55.", "This value is so low it effectively means that the retrievals find no evidence of clouds affecting the spectra, which is consistent with the strong water and sodium features, and the larger reference pressures.", "Aerosols are likely present in the atmosphere of all planets, but when aerosols are not warranted by the retrievals that implies that the aerosols are not thick enough at the high altitudes in which transmission spectroscopy probes to significantly mute the spectral features in the observed wavelengths, and/or they are not significantly present in the terminator.", "Also, though the retrievals, suggests that data can be explained without thick aerosols, color dependent scatters may still affect the transmission spectra.", "In particular, the possible Rayleigh scattering slope in the optical cannot be confidently refuted in the combined optical data." ], [ "Temperature and C/O ratio", "The terminator temperatures found with PLATON (T$_p$ =877$\\pm $ 40 K) and Exoretrievals (T$_p$ =830$^{+160}_{-140}$  K) are within agreement of one another and with what [109] retrieved (T$_p$ =954$^{+198}_{-195}$  K), given their uncertainties.", "However, these values are vastly different from the temperature retrieved by [64] (T$_p$ =1710$^{+150}_{-200}$  K), who used the ATMO models [5], [38] to perform retrieval analysis on their data.", "Our retrieved temperatures are also far from the 0 albedo equilibrium temperature of 1285$\\pm 40$  K [39].", "A lower retrieved terminator temperature is often found in the literature and is expected when applying a 1D model to probe a 3D atmosphere.", "1D models are useful for fitting features found in transmission spectra substantially more quickly than 3D models.", "Unfortunately, they can artificially shift the retrieved temperature up to hundreds of Kelvin cooler than the true average temperature [56], [71].", "Other physical assumptions prescribed in the retrieval frameworks could also contribute to a discrepancy in retrieved temperatures [106].", "The only feature in the observed spectrum that could directly constrain C or O was H$_2$ O, thus, the quoted C/O ratios should be taken lightly, given that it is only using the water features and chemical equilibrium constraints to obtain the ratio.", "Including the two Spitzer/IRAC points centered at 3.6 and 4.5 $\\mu $ m (program ID 14255), as was done by [109], would include data near carbon bearing features (i.e.", "CO, CO$_2$ , and CH4).", "However, we elected to exclude these points because the two points have larger uncertainties (in wavelength and depth), which would not significantly constrain any of these carbon species, as can be seen in [109].", "Given the lack of a constrained C/O ratio, it is hard to ascertain the formation region of WASP-96b, but JWST observations would be able to refine the C/O ratio with higher precision and higher spectral resolution observations in the near- to mid-IR wavelengths.", "The observed lack of aerosols obscuring features in the optical to near-IR spectrum of WASP-96b makes it an ideal target for such observations, which has the potential to provide key insight of where hot Jupiters formed and when/if they migrated." ], [ "Mass-Metallicity Trend", "Our metallicity with Platon is consistent with the constraints from [64] ($\\log _{10}(Z/Z_{\\odot })=0.4^{+0.7}_{-0.5}$ ).", "Furthermore, it is within 3 $\\sigma $ agrement to the solar system mass-metallicity trend [101], [3], [99], shown in Figure REF .", "In this figure we plot the metallicites and masses of the solar system giants and a linear fit of that data in log-log space.", "We also show planets with metallicities derived from molecular and atomic abundance of one to a few different species.", "Looking at all exoplanets in the figure, the trend found for the solar system does not persist among the exoplanet sample.", "However, given that high-altitude aerosols make it more difficult to accurately determine molecular abundances, we also highlighted the five planets which are thought to have little high-altitude aerosols obscuring their spectra.", "When only including these planets, four of the five planets (WASP-17b, WASP-62b, WASP-94b, and WASP-96b) are less than 3 $\\sigma $ from the predicted values.", "Though most of the clear atmospheres prove to be consistent with the solar system mass-metallicity trend, it is still hard to claim that we can interpret this as most exoplanets undergo the same formation mechanisms as our solar system for multiple reasons.", "For one, all three of the consistent metallicities are in similar mass regimes, therefore, giving perspective only on a small fraction of exoplanets.", "Additionally, most of these metallicities were obtained assuming the abundances of one to a few species can be directly translated to the bulk metallicity of the atmosphere.", "How far this assumption is skewed from reality is unknown, since interactions among transport, chemistry, and phase changes may mean the elements are not well-mixed in the atmosphere [113].", "Furthermore, most of the planets in Figure REF are hot jupiters/neptunes, which are outliers that are not representative to exoplanets in general.", "For these reasons, and others, there is much headway required in order to obtain a more complete grasp of exoplanet formation mechanisms and trends.", "Observing more planets like WASP-96b, with relatively clear atmospheres, in many wavelengths and improving our analysis processes around these observations is the best path forward.", "Interestingly, though the work from this paper and others [3], [99] find no clear trend amongst atmospheric composition and mass, there has been a found mass-metallicity trend in terms of planet bulk composition [93].", "Still, much work is needed in order to determine if there truly is or is not an atmospheric metallicity-mass trend before the root of this potential discrepancy can be explored further.", "Figure: Observed mass–metallicity trend for transiting exoplanets.", "The bulk of the data was acquired from https://stellarplanet.org/science/mass-metallicity/, but we added HAT-P-32b , WASP-17b (PLATON run with data from ), WASP-62b (using Na abundance from as a proxy), WASP-94b , and WASP-96 (this work, magenta star).", "The dashed black line corresponds to a linear fit in log–log space to the solar system points (black dots).", "We highlight the clear planets with magenta and all other planets are gray." ], [ "Correlations to aerosol formation rates", "Understanding why WASP-96b is one of the few planets that has an observed transmission spectrum unobscured by aerosols is vital for advances in exoplanetology.", "This would allow astronomers to a priori determine what key characteristics make an exoplanet transmission spectrum clear and allow for more targeted exo-atmosphere surveys.", "Figure REF puts WASP-96b in a broader context, it was initially used by [91], who proposed a temperature-gravity trend in the formation of high-altitude clouds.", "They used the H$_{2}$ O-J index as a proxy for the cloud levels, which inherently assumes a similar absolute water abundance of all hot Jupiters.", "HAT-P-11b is a prime example of the limitations of using this index as a proxy for cloud formation, where its water feature is extremely strong (2.499$\\pm $ 0.505), but its optical spectra revealed it to have a cloudy atmosphere [15].", "Nonetheless, the H$_{2}$ O-J index provides an easy index to parameterize cloud levels.", "Multiple other planets have been added to the original figure [3], [105], and make it clear that the trend does not strictly exist.", "As seen in our Figure REF , WASP-96b (marked as a star) highlights the lack of an obvious trend even further, where it is near the fitted temperature-log(g) boundary line but is in fact one of the most clear atmospheres observed to-date.", "Thus, there is still extended work needed to isolate what causes high-altitude aerosol formation.", "Figure: The H2O-J indices (color coded squares) defined by relative to the planetary surface gravities and equilibrium temperatures.", "Other planets with surface gravities and effective temperatures obtained from TEPCat are plotted as green circles for context.", "We also plot the dividing line in T equ _{equ}-log 10 _{10}(g) phase space that proposed delineates between cloudy and clear planets.", "Where the authors say H2O-J indices greater than 2 is likely completely clear, and indices less than 1 are greatly obscured by clouds.", "The only planets with indices greater than 2 are HAT-P-11b (2.5±\\pm 0.5), WASP-96 (2.4±\\pm 0.7), and WASP-17b (2.3±\\pm 0.9).", "WASP-96 (star) is shown to be one of the clearest observed planets, both through this index and in the optical, yet it sits very close to the dividing line.", "The data in this table was produced in ." ], [ "Summary & Conclusion", "We observed two transmission spectra of WASP-96b with Magellan/IMACS as part of the ACCESS survey.", "In the process of analyzing the data, we tested the precisions and accuracies of two commonly used spectroscopic light curve detrending techniques: A) Common mode correction followed by a polynomial fitting (CMC+Poly) and B) a Gaussian processes (GP) routine.", "Both routines were tested against simulated data, where we find that for data without substantial chromatic systemics the CMC+Poly procedure produces more accurate depths with higher precision.", "Additionally, we find that neither method (worse for CMC+Poly) was able to consistently reproduce absolute depths.", "This provided justification of fitting for offsets amongst transmission spectra from different nights and instruments.", "The transmission spectrum from IMACS was then added with reanalyzed transmission spectra from FORS2, both of which were reduced using the CMC+Poly routine.", "The optical data was combined with a previously published HST/WFC3 (G102 and G141) transmission spectrum [109], collectively producing a nearly continuous coverage transmission spectrum from 400–1237 nm.", "This spectrum was run against two retrievals: PLATON and Exoretrievals.", "Both retrievals found that the terminator of WASP-96b was shrouded by little or no aerosols, as such it is still one of the most clear exo-atmospheres observed.", "More specifically PLATON found a metallicity, C/O ratio, reference pressure, and terminator temperature of $\\log _{10}(Z/Z_{\\odot }) = -0.49^{1.0}_{-0.37}$  dex, C/O = 0.97$^{+0.65}_{-0.50}$ , $\\log _{10}(P_0)$ = 1.3$^{+1.0}_{-1.1}$  bars, and T$_p$ = 877$\\pm $ 40 K, respectively.", "Exoretrievals found a terminator temperature, reference pressure, and water and sodium mixing ratios of T$_p$ = 830$^{+160}_{-140}$  K, $\\log _{10}(P_0)$ = 0.29$^{+1.86}_{-2.02}$  bars, $\\log _{10}(H_2O)$ = -4.5$\\pm $ 2.0, and $\\log _{10}(Na)$ = -5.4$^{+2.0}_{-1.9}$ , respectively.", "With the constraints on stellar activity imposed from photometric monitoring, neither retrieval had strong support for stellar activity explaining the data.", "Though there is a hint of a slight slope blue-ward of 550 nm, there was not substantial evidence supporting the need for a scattering slope.", "However, bluer, space-borne observations of the planet could discern if the hint of a blue-ward slope is indeed a true feature.", "Finally, in an attempt to put WASP-96b in a broader context, we found that it along with three other clear planets (WASP-62b, WASP-94b, and WASP-17b) are consistent with the mass-metallicity trend observed in the solar system.", "However this sample is not complete enough to make overarching claims about the extrasolar giants' formation mechanisms compared to giant planet formation pathways in the solar system.", "We also find no clear correlation to what causes the aerosol formation rate in exo-atmospheres, and recommend more in-depth analysis of a higher sample of planets in order to find such a correlation.", "The results reported herein benefited from support, collaborations and information exchange within NASA's Nexus for Exoplanet System Science (NExSS), a research coordination network sponsored by NASA's Science Mission Directorate.", "This material is partly based upon work supported by the National Aeronautics and Space Administration under Agreement No.", "80NSSC21K0593 for the program “Alien Earths”.", "This paper includes data gathered with the 6.5 meter Magellan Telescopes located at Las Campanas Observatory, Chile.", "We thank the staff at the Magellan Telescopes and Las Campanas Observatory for their ongoing input and support to make the ACCESS observations presented in this work possible.", "This work uses observations collected at the European Organization for Astronomical Research in the Southern Hemisphere under European Southern Observatory programme 199.C-0467(H).", "We also appreciate the support from the NSF Graduate Research Fellowship (GRFP), grant number DGE1745303.", "The computations in this paper were conducted on the Smithsonian High Performance Cluster (SI/HPC), Smithsonian Institution.", "https://doi.org/10.25572/SIHPC.", "B.V.R.", "thanks the Heising-Simons Foundation for support.", "C.M.", "thanks Neale Gibson for conversations on marginalization techniques.", "A.J.", "acknowledges support from ANID – Millennium Science Initiative – ICN12_009 and from FONDECYT project 1210718.", "Astropy [7], corner [27], Matplotlib [42], NumPy [66], Multinest [25], PyMultiNest [13], SciPy [44], batman [52], george [28] dynesty [90], PLATON [112], Juliet [21] Magellan:Baade, Smithsonian Institution High Performance Cluster (SI/HPC)" ], [ "Second Order Contamination", "As mentioned in Section REF , we did not use a blocking filter during the UT170804 transit, but we added the GG495 blocking filter in the UT171108 observations after we learned from other ACCESS observations that second order light introduces contamination in some cases.", "To check for possible contamination in the UT170804 observation, we modeled its effect by convolving the unfiltered stellar spectra, which inherently is a convolution of the instrument's throughput and WASP-96’s stellar spectra, with the G495 filter’s throughput.", "This gave us the spectral profile of the UT170804 observation, if the G495 filter had been added, but with any second order light still present.", "Then we modeled the normalized continuum of this light curve against the normalized continuum of the UT171108 observations light curve.", "Because we assumed that the only difference between both continuum structures should be from the 2nd order light, we used the ratio of the two continua as a correction term for the second order light.", "This process is illustrated in Figure REF .", "We applied this correction method to each comparison star and WASP-96 spectrum observed on UT170804 (the exact correction term is unique per spectrum) to determine the effect of the second order light.", "We then used both the corrected and uncorrected spectra to produce a white-light curve and binned light curves, which are constructed by integrating either all wavelengths of light (white-light) or specific band passes (binned) in a given spectrum, and plotting each integrated spectral counts relative to time.", "Next, we compared the final white-light and binned transits with and without the correction, and found an insignificant difference between the two.", "This is likely, because the relative effect of the second order light is negated when dividing by the comparison stars (or using a PCA correction with the comparison stars).", "Given that the correction had minimal effects, in our final analysis we used the uncorrected spectra.", "Figure: The process used to correct for second order light.", "The left panel shows the normalized IMACS observations taken on UT170804, which had no filter (red); UT170804 convolved with the GG495 transmission profile (green); and UT171108, which had the GG495 filter (blue).", "The middle panel shows the normalized second order light correction, determined by dividing the filter convolved first night (green) by the second night (blue).", "The right panel shows normalized UT170804 (red), UT170804 convolved with the GG495 profile and corrected for second order light (green), and UT171108 (blue).", "Dotted lines of each spectra’s modeled continua is overlayed in the same respective colors as their spectra.", "These steps are also applied to each comparison so every spectrum is corrected from second order light." ], [ "Principle Component Analysis (PCA) and Gaussian Processes (GP)", "The PCA+GP routine we used has been implemented often in recent years [104], [108], [58], [105], but we outline it here.", "We first model out commmon systematics found in all the comparison stars by performing singular value decomposition on a matrix composed of comparison light curves ($L_k(t)$ ) in the form: $L_k(t)=\\sum _{i=1}^{K}A_{k,i}s_i(t),$ where $s_i(t)$ is a set of signals representing the systematics affecting a given light curve and $A_{k,i}$ is the weight for each of those signals.", "This allows us to identify the principal components (i.e.", "PCA) of the light curve, which is driven by systematics.", "This is used rather than just dividing the target by the sum or mean light curve of the comparison stars, because when doing that you combine systematics that are unique to a specific star, instead of isolating which systematics persist amongst all stars.", "Therefore, this method is less likely to incorrectly divide out systematics in the comparison star that were not present in the target star.", "To model out systemtics unique to the target's light curve we used a Gaussian process (GP) regression with a joint probability distribution of the form $\\mathcal {N}[0,\\Sigma ]$ , where the covariance function ($\\Sigma $ ) is defined as $K_{SE}(x_i,x_j) +\\sigma ^2_w\\delta _{i,j}$ .", "Here, $\\sigma ^2_w$ and $\\delta _{i,j}$ are a jitter term and the Kroenecker delta function, respectively.", "$K_{SE}(x_i,x_j)$ is a multidimensional squared-exponential kernel of the form: $K_{SE}(x_i, x_j)=\\sigma ^2_{GP} \\exp \\Bigg (-\\sum _{d=1}^{D} \\alpha _d(x_{d,i}-x_{d,j})^2\\Bigg ),$ where $\\sigma ^2_{GP}$ is the amplitude of the GP and $\\alpha _d$ are the inverse (squared) length-scales of each components of the GP.", "The priors on the jitter term and $\\sigma ^2_{GP}$ were both log-uniform, where the jitter term ranged from 0.01 to 100 ppm and $\\sigma ^2_{GP}$ from 0.01 to 100 mmag.", "The prior on each $\\alpha _d$ was an exponential function, similar to what was done by [37] and [24].", "In the above equations index i denotes each time-stamp and d corresponds to a set of different time-dependent external (auxiliary) parameters used.", "For the Magellan/IMACS transits these auxiliary parameters were variation of air mass, full-width at half-maximum (FWHM) of the spectra, mean sky flux, position of the central pixel trace (perpendicular to the dispersion axis), and drift of the wavelength solution.", "For the VLT/FORS2 transits they were dispersion and cross-dispersion drift, variation of the FWHM of the spectra, air mass, mean sky flux, and change in rotator angle.", "The PCA and GP components were combined using the following equation: $M_k(t)=c_k + \\sum _{i=1}^{N_k} A_{k,i} s_i(t) - 2.51*log_{10}T(t|\\phi ) + \\epsilon ,$ where M$_k(t)$ is the (mean-subtracted) magnitude of the target star in the kth model, $c_k$ is a magnitude offset, $N_k$ is the number of PCA signals $s_i(t)$ , $A_{i,k}$ is the weight for each signal in each models, $T(t|\\phi )$ is the transit model with parameters $\\phi $We used the Python package batman [52] to produce the analytic transit model., and $\\epsilon $ our GP component We used george [28] to evaluate the GP marginalized likelihoods.", "We used [13]'s Nested sampling routine, PyMultiNest (a python wrapper of MultiNest, [25]), to explore the posteriors of our PCA+GP models.", "Because Nested sampling provides Bayesian evidences of models, we used those evidences as weights to combine posterior distributions of each M$_k(t)$ model in a technique called Bayesian Model Averaging [33].", "In the limiting case where there is only one comparison star, the PCA portion of the routine cannot be performed.", "In that case equation REF becomes $M(t)=c_0 + A*m_c(t) - 2.51*log_{10}T(t|\\phi ) + \\epsilon ,$ where M(t), $T(t|\\phi )$ , $c_0$ , and $\\epsilon $ are the same as M$_k(t)$ , $T(t|\\phi )$ , $c_k$ , and $\\epsilon $ from equation REF , but without summing over different $N_k$ PCA signals.", "A is now a weight for the mean-subtracted comparison star magnitude, m$_c$ .", "This equation is what was used by [108], because they only had one comparison star.", "When this detrending routine can be used to its fullest (i.e.", "with multiple comparison stars) it is referred as “PCA+GP.” In the case when there is only one comparison star, like in the FORS2 data, it is referred to as a “GP” routine.", "As discussed in Section REF and Appendix , the synthetic data only had one comparison star, so we in fact are comparing the GP and CMC+Poly routines in that analysis.", "This general procedure can be used to detrend both the white-light curves and the binned-light curves.", "When producing the binned-light curves solely with PCA+GP we first used this method on the white-light data, then fixed all parameters based on the PCA+GP white-light fit (aside from $q_1$ , $q_2$ , and $R_p/R_s$ ), then ran the PCA+GP routine against the binned-light curves.", "When producing the binned-light curves with the CMC+Poly routine we again used the PCA+GP method on the white-light data, used that white-light fit to obtain the CMC term, fixed all binned-light curve parameters based on the PCA+GP white-light fit (aside from $q_1$ , $q_2$ , and $R_p/R_s$ ), and ran the CMC+Poly routine (see sec.", "REF )." ], [ "Common Mode Correction (CMC)", "The general process in CMC is to first produce a partially detrended white-light curve, where the larger systematics common to the target and the comparison(s) are removed.", "Two common ways to do this is either by dividing the light curve of the target by the light curve of the star(s) or with PCA.", "Next, a best fit transit model for the light curve is found using a routine that fits for both a transit and additional systematics (e.g.", "GPs).", "This transit model is divided by the uncorrected white-light curve to produce a constant systematic correction term (common mode).", "The correction term is then used for each individual bin, because the systematics are expected to be relatively consistent throughout each spectral band pass.", "After the common systematics are removed, each bin has an additional detrending routine (e.g.", "GPs or polynomial correction) to correct for any residual chromatic systematics.", "Doing this technique has proven to produce relatively high precision (though accuracy was not tested) for retrieved transit depths of each band pass [34], [63], [37], [64], [14].", "Our best fit white-light curve transit model (used for the common mode term) is obtained with the same steps of Appendix REF .", "After the common mode correction is applied we fit the remaining systematics using 1st and 2nd order polynomial fits of all external parameters.", "We individually explore the posterior space of all possible combinations of systematic corrections.", "This means for each bin, including a fit without external parameters (just a single constant coefficient), 729 and 243 models were fit for the IMACS and FORS2 data, respectively.", "The posterior space was explored in three steps: first, we fit for just the polynomial systematic coefficients on the out-of-transit data using scipy.optimize.minimize [44] with the 'Powell' method, coefficient bounds of -5 to 5, and initial points of 0.", "Then we use the found coefficients as the initial start points of another scipy.optimize.minimize run which includes the in transit data and a transit model fit.", "Lastly, we use PyMultiNest as our final posterior exploration where the transit parameters found with scipy were used as priors on the mean value, while maintaining the prior bounds (see Sec.", "REF ), and the found polynomial coefficients were used as the mean values for a normal prior distributions with a standard deviations of 0.05.", "Again, we used BMA to best combine posteriors from all explored posterior spaces.", "For the fits we held $t_0$ fixed to what the white-light curve fit found, thus, the only transit parameters fit were the LD parameters and $R_p/R_s$ .", "There are differences in how this technique is implemented compared to [64]'s method; however, it produces a nearly identical transmission spectrum, as shown in Figure REF .", "Figure: The original transmission spectrum of WASP-96b analyzed by (black), and a re-reduction of the same data (VLT/FORS2) with the same binning scheme using our CMC+Poly analysis discussed in section .", "The two analyses produce similar mean R p _p/R s _s precisions (0.00108 for the original and 0.0079 our reanalysis) and have an average difference of depth of only 0.56-σ\\sigma , suggesting strong agreement of the two data.", "An offset is applied so the means of both datasets are the same." ], [ "Light Curves", "For each bin we estimated the theoretical photon noise precision $\\sigma _w$ .", "This was calculated by taking the mean counts (per star) over each exposure, summing all the counts (N) in a given bin, and using $\\sqrt{N}$ to estimate the shot noise for each star.", "The error of each star was propagated where the final (uncorrected) transit light curve was produced by dividing WASP-96's flux by the sum of the comparison stars' flux.", "We also estimate the noise of the corrected binned light curves.", "This was done with two approaches.", "First we calculate the root mean squared (r.m.s.)", "of the residuals.", "To also include an estimate that is not dependent on the best fit transit model, we take the standard deviation of the out-of-transit data.", "Both methods produced similar precisions and we use the higher of the two for our estimates of the the corrected light curve precision.", "We quantify the red noise by taking the ratio of the detrended light curve's noise level and the theoretical photon noise.", "We call this ratio $\\beta $ , as it is similar to the $\\beta $ term used by [107] (though derived differently).", "The average $\\beta $ is 1.122, 1.671, 1.089, and 1.528 for each night in chronological order.", "Their corresponding $R_p/R_s$ uncertainties are 0.001262, 0.002284, 0.000935, and 0.001485, correlated to the total light curve precision (red and white noise) but overall determined by how well the posterior space can be constrained, given the data.", "Figure: Left: Raw binned light curves (LC) of VLT/FORS2 transit UT170729 produced by dividing the comparison star LC (or the sum of the comparison star LC, in the case of the IMACS data) by the target's.", "The wavelength range of each bin is printed in the bottom left corner.", "Middle: LC produced by dividing the raw LC by the Common Mode Correction (CMC) term shown in Figure (final row).", "The number of sigma clipped points is printed in the bottom left and those points are marked with a black x, the theoretical photon noise precision is also printed as σ w \\sigma _w.", "The additional polynomial systematic correction specifically fit for each bin is shown as a magenta line.", "Right: The final CMC and polynomial corrected light curve, with the best fit transit model in magenta and the residuals of the detrended data from the best-fit transit model on the bottom.", "The standard deviation (σ r \\sigma _r) of the residuals in ppm is printed in the bottom left corner and the ratio of σ r \\sigma _r and σ w \\sigma _w (β\\beta ) is printed.Figure: Same as Figure , but for Magellan/IMACS transit UT170804.Figure: Same as Figure , but for VLT/FORS2 transit UT170822.", "Though there are a few cases (8th, 9th, and 12th bins) where β\\beta is slightly less than one.", "We do not interpret this as the light curve was over corrected, but given that they are nearly one, we see this as we achieved photon noise precision for those bins.Figure: Same as Figure , but for Magellan/IMACS transit UT171108.|C|C|C|C|[htb] Combined Magellan/IMACS (UT170804 and UT171108) optical transmission spectrum (Rp/Rs), combined VLT/FORS2 (UT170729 and UT170822) optical transmission spectrum, and global combined optical spectrum (all four transits).", "The data were produced implementing the reduction and detrending processes discussed in Section .", "These depths are offset so the mean depth is 0.1158.", "Wavelength () Magellan/IMACS VLT/FORS2 Global 4000.0-4500.0 —- 0.1173+0.0014-0.0013 0.1177+0.0014-0.0013 4500.0-4750.0 —- 0.1161+0.0011-0.0013 0.1165+0.0011-0.0013 4750.0-5025.0 0.1165+0.0036-0.0043 0.1153+0.0011-0.0013 0.1158+0.0013-0.001 5025.0-5255.0 0.1143+0.0014-0.0015 0.1149+0.0011-0.0013 0.11470.0009 5255.0-5435.0 0.1138+0.0011-0.001 0.11530.0006 0.1149+0.0006-0.0005 5435.0-5615.0 0.1207+0.0011-0.0012 0.11580.0007 0.11730.0006 5615.0-5800.5 0.1185+0.0023-0.002 0.1169+0.0007-0.0008 0.11740.0007 5800.5-5985.5 0.11840.0009 0.1185+0.0007-0.0009 0.1183+0.0005-0.0006 5985.5-6175.0 0.1186+0.0011-0.001 0.11730.0008 0.11770.0006 6175.0-6360.0 0.11390.0009 0.1162+0.0008-0.0007 0.1151+0.0005-0.0006 6360.0-6545.0 0.11390.0009 0.1156+0.001-0.0008 0.1145+0.0006-0.0007 6545.0-6730.0 0.1135+0.001-0.0009 0.1151+0.0012-0.0011 0.11390.0007 6730.0-6930.0 0.11410.0009 0.1147+0.001-0.0009 0.1142+0.0006-0.0007 6930.0-7110.0 0.1155+0.0015-0.0018 0.115+0.0007-0.0008 0.11540.0007 7110.0-7290.0 0.1167+0.0014-0.0015 0.115+0.001-0.0009 0.1156+0.0007-0.0009 7290.0-7470.0 0.1073+0.0012-0.001 0.1154+0.001-0.0009 0.11160.0007 7470.0-7594.0 0.1168+0.0012-0.0011 0.1161+0.0009-0.001 0.1165+0.0008-0.0007 7672.0-7836.0 0.1134+0.0008-0.0009 0.1152+0.0009-0.0008 0.11420.0006 7836.0-8000.0 0.1160.0009 0.1163+0.001-0.0009 0.1159+0.0006-0.0007 8000.0-8250.0 0.12250.0008 0.114+0.001-0.0011 0.1187+0.0007-0.0006 |C|C|c|C|C|[htb] Near-IR transmission spectrum obtained from [109].", "The G102 grism is from 8000–11250  Å and the G141 grism is from 11372–12180 Å. Wavelength () Rp/Rs Wavelength () Rp/Rs 8000.00 - 8250.00 0.116327 0.000772 12370.00 - 12559.00 0.117175 0.000650 8250.00 - 8500.00 0.117499 0.001021 12559.00 - 12751.00 0.117512 0.000790 8500.00 - 8750.00 0.117439 0.000940 12751.00 - 12944.00 0.119013 0.000780 8750.00 - 9000.00 0.117597 0.001338 12944.00 - 13132.00 0.117490 0.000816 9000.00 - 9250.00 0.117473 0.001331 13132.00 - 13320.00 0.118453 0.000971 9250.00 - 9500.00 0.118305 0.000801 13320.00 - 13509.00 0.118013 0.000765 9500.00 - 9750.00 0.116508 0.000883 13509.00 - 13701.00 0.118878 0.000717 9750.00 - 10000.00 0.117490 0.000919 13701.00 - 13900.00 0.120121 0.000995 10000.00 - 10250.00 0.117427 0.000850 13900.00 - 14100.00 0.120768 0.001002 10250.00 - 10500.00 0.118545 0.000915 14100.00 - 14303.00 0.119470 0.000865 10500.00 - 10750.00 0.117456 0.001000 14303.00 - 14509.00 0.119771 0.000800 10750.00 - 11000.00 0.118174 0.000594 14509.00 - 14721.00 0.117461 0.000970 11000.00 - 11250.00 0.119214 0.000466 14721.00 - 14941.00 0.118351 0.000776 11153.00 - 11372.00 0.117894 0.000689 14941.00 - 15165.00 0.118482 0.000924 11372.00 - 11583.00 0.118786 0.000845 15165.00 - 15395.00 0.117843 0.000809 11583.00 - 11789.00 0.118811 0.000887 15395.00 - 15636.00 0.116880 0.000661 11789.00 - 11987.00 0.117792 0.000771 15636.00 - 15889.00 0.118089 0.000722 11987.00 - 12180.00 0.116893 0.000932 15889.00 - 16153.00 0.117456 0.001005 12180.00 - 12370.00 0.117520 0.000897 16153.00 - 16436.00 0.117435 0.000928" ], [ "Synthetic Spectra", "In our initial analysis, we reduced the VLT/FORS2 transmission spectrum using the PCA+GP routine (just `GP', given that there is only one comparison), discussed in appendix REF , to reduce the white light curves and the binned light curves.", "When comparing the VLT/FORS2 transmission spectrum produced with this method to the original spectrum [64], shown in Figure REF .", "We found that the uncertainties from the GP method were over 3 times larger than when [64] used their CMC+Poly method on the same data.", "This was independent of if the squared exponential kernel (see Appendix REF ) or the george Matern32Kernel was used.", "Naturally, this is concerning, because though both methods produce similar overall structures, Figure REF implies that either CMC+Poly underestimates the uncertainties or GP overestimates it.", "As such, we test the accuracy and precision of both methods using synthetic data.", "Figure: The transmission spectra of VLT/FORS2 data.", "The black diamonds with uncertainties is the analysis done by , and the cyan dots with uncertainties is our initial analysis of the VLT/FORS2 data with the same wavelength bins and using the GP detrending method.", "An offset is applied so their mean depths are equal to one another.", "Each spectrum is a weighted average of observations taken with the B filter (350–617.3 nm) on the night of UT170729 and the R filter (529.3–801.3 nm) on the night of UT170822.", "The average deviation of each point relative to their uncertainties is 0.28σ\\sigma , suggesting that the spectral structures are the same.The first step to produce our synthetic data was to generate a transit light curve with parameters similar to what produces the transit light curve of WASP-96b.", "For generating all light curves we used batman [52] with the same time stamps as FORS2 transit UT170729 (i.e.", "90 points covering $\\sim $ 5 hours).", "The exact light curve parameters used are shown in Table .", "Next, we generated a realization of a Gaussian Process, GP(t, instrumental_systematics), for the common systematics that affect all spectrophotometric bands.", "Here the 'instrumental_systematics' were the FORS2 transit UT170729 auxiliary parameters of airmass, full-width at half-maximum (FWHM), and rotational angle.", "Using george [28], we combined three squared exponential kernels each with an inverse natural-log length scale of -2 and a constant term of -5, -12, and -12 for the GP inputs of airmass, FWHM, and rotational angle, respectively.", "The values of the inverse length scale and constant terms were empirically deduced in order to produce systematics with amplitudes and structure similar to what was seen in the FORS2 transit UT170729 light curves.", "An example of the white light curve and its GP systematics is shown in Figure REF .", "Our third step, was to produce the binned light curves.", "This was done by generating 10 light curves each containing the same GP systematic realization as above, but with quadratic limb darkening (LD) coefficients determined using PyLDTk [70] with a stellar effective temperature of 5500 K, stellar log$_{10}$ (g) of 4.42, metallicity (log$_{10}$ (Z)/log$_{10}$ (Z$_{\\odot }$ )) of 0.25 dex, and assuming each bin was 150Å wide starting from 3800 to 5300Å.", "Each binned light curve had a unique realization of shot noise with a 400 ppm standard deviation.", "In addition, each binned light curve had an unique systematic added by generating a first order polynomial on dispersion drift, a second order polynomial on cross-dispersion drift, and a first order polynomial on rotator angle.", "The coefficients of each polynomial on each bin were randomly drawn from a uniform distribution from -0.1 to 0.1.", "Each polynomial function was standardized [23], [22], [49] and their product was used as the chromatic systematics.", "An example of binned light curves and their polynomial systematics can be seen in Figure REF .", "The final step in producing the synthetic data was to decompose the light curves into two components, the target light curves and the comparison star light curves.", "When making the comparison star light curve we take a constant light curve with normalized value of 1 and add a separate draw from the above GP systematics, i.e.", "using the same auxiliary parameters and kernels, but a different random draw from the GP distribution.", "Because it is assumed that the systematics affecting the comparison star's light curve should also affect the targets, we multiple this GP draw by the original target light curve.", "Thus, when the target is divided by the comparison, the effect of the comparison's systematics are completely divided out.", "This might not happen in real datasets, because the comparision could add additional systematics, but for our synthetic data, we assume any such systematics is included in the initial construction of the light curve (i.e.", "the first GP draw).", "The first column of Figure REF shows examples of synthesized binned target light curves and comparison light curves.", "The white light curve can then be constructed by combining all 10 bins, and assuming each bin had the same number of counts.", "When producing all synthetic data we followed these steps 50 times, where each white light curve was produced with a different draw from the GP distribution, unique realizations of shot noise, and unique coefficients for the polynomial systematics on each bin.", "Figure: An example synthetic white light curve (black dots) constructed by adding all 10 binned light curves in Figure .", "Given that all of the individual chromatic systematics are smaller relative to the white-light systematics and the transit, we do not see their effect in this light curve.", "This is exactly what we would see in a true light curve (e.g.", "see Fig.", ".", "The red line is the systematics produced by a random draw from a GP distribution created using airmass, full-width at half-maximum (FWHM), and rotational angle as inputs.", "σ w \\sigma _w is the residuals of the out-of-transit data from the GP systematic model, which would be 0 if there was no white noise added.Figure: The specific binned light curve used to construct the WLC in Figure .", "The left column shows the binned target light curves (blue) and the binned comparison light curves (purple).", "The target light curve is composed of the transit, systematics common to all bins and the comparions, systematics common to all bins and just the target, and systematics unique to each bin.", "The LD coefficients used for each bin are printed below the light curves, and were determined with PyLDTk.", "The right column shows the 'raw' (target/comparison) light curve in black dots.", "The larger amplitude GP systematics (constant for each bin) is shown as a yellow dashed line, the smaller amplitude polynomial systematics (unique per bin) is shown in red and the combined GP and polynomial systematics is shown as green dashes.", "The residuals of the out-of-transit data from the green dashed line is printed in the bottom left corner (σ w \\sigma _w).Cccc[htb] The initial transit parameters to construct the synthetic data (column 2), the white-light priors used when fitting the data (column 3), and the binned priors (column 4).", "The period (P) of 3.42526 days and mid transit time (t0) of 2457963.336499 days were held fixed for all fits.", "Here 'b' is the impact parameter, which can be transformed to inclination (i) via equation REF using $a/R_s$ , eccentricity (fixed at 0), and $\\omega $ (fixed at 90$^{\\circ }$ ).", "The limb-darkening parameters were determined using PyLDTk and were different for each bin, what is shown in the table below is the average.", "parameter value white-light binned $R_p/R_s$ 0.1157 uniform: 0.1–0.14 normal: m=(WL fit), $\\sigma _n$ = 0.05 <q1> 0.8208 uniform: 0–1 uniform: 0–1 <q2> -0.0201 uniform: 0–1 uniform: 0–1 b 0.7456 uniform: 0.5–1 Fix to WL fit $a/R_s$ 9.0 uniform: 8–10 Fix to WL fit" ], [ "Retrieval Modeling Priors", "|C|C|C|C|C|C|[htb] The priors for Exoretrievals and PLATON.", "These priors were set to allow for a wide parameter space to be surveyed, but contained within physical regimes.", "Not all parameters were included in each model fit (see Tab.", "REF ).", "We used 5000 live points for all runs.", "For further description of the parameters of Exoretrievals, please refer to the Appendix D of [22].", "The log cloud absorbing ($\\sigma _{cloud}$ ) parameter was fixed because across all datasets and all models which included aerosols, this retrieved parameter was near -55.", "As such, we decided to reduce the dimensionality of the explored posterior space by fixing it to this value, effectively turning off clouds.", "3|c|Exoretrievals 3|c|PLATON parameter function bounds parameter function bounds reference pressure (P0, bars) log-uniform -8 to 3 reference pressure (Pclouds, Pa) log-uniform -3.99 to 7.99 planetary atmospheric uniform 500 to 1600K planetary atmospheric uniform 500 to 1600K temperature (Tp) temperature (Tp) stellar temperature uniform 5300 to 5780K stellar temperature gaussian $\\mu $ =5540K, $\\sigma $ =150K (Tocc) (Tstar) stellar heterogeneities uniform 2540 to 8540K stellar heterogeneities uniform 2540 to 8540K temperature (Thet) temperature (Tspot) heterogeneity covering gaussian $\\mu $ =0.014, $\\sigma $ =0.009 heterogeneity covering gaussian $\\mu $ =0.014, $\\sigma $ =0.009 fraction (fhet) fraction (fspot) offset (depth) gaussian $\\mu $ =0, $\\sigma $ =2000ppm offset (depth) uniform -6000 to 6000ppm haze amplitude ($a$ ) log-uniform -1 to 10 scattering factor log-uniform -10 to 10 haze power law ($\\gamma $ ) $^*$ uniform -14 to 4 scattering slope ($\\alpha $ ) $^{\\ddagger }$ uniform -4 to 14 log cloud absorbing Fixed -55 metallicity (Z/Z$_{\\odot }$ ) log-uniform -1 to 3 cross-section ($\\sigma $cloud) trace molecules' log-uniform -30 to 0 C/O uniform 0.05 to 2 mixing ratios reference radius factor ($f$ ) $^{\\diamond }$ uniform 0.8 to 1.2 1 bar, reference radius (R0) uniform 1 to 1.4Rj *This is the exponent of the scattering slope power law, where $-4$ is a Rayleigh scattering slope.", "This is the wavelength dependence of scattering, with 4 being Rayleigh.", "This is a factor multiplied by the inputted planetary radius to produce the reference radius, i.e.", "R$_0=f$ R$_p$" ] ]
2207.03479
[ [ "Night Sky Brightness Measurement, Quality Assessment and Monitoring" ], [ "Abstract Ground-based optical astronomy necessarily involves sensing the light of astronomical objects along with the contributions of many natural sources ranging from the Earth's atmosphere to cosmological light.", "In addition, astronomers have long contended with artificial light pollution that further adds to the 'background' against which astronomical objects are seen.", "Understanding the brightness of the night sky is therefore fundamental to astronomy.", "The last comprehensive review of this subject was nearly a half-century ago, and we have learned much about both the natural and artificial night sky since.", "This Review considers which influences determine the total optical brightness of the night sky; the means by which that brightness is measured; and how night sky quality is assessed and monitored in the long term." ], [ "Introduction", "Environmental pollution caused by artificial light at night, commonly known as “light pollution,” is a source of significant known and suspected hazards [1], [2], [3].", "Light pollution now touches every continent except Antarctica [4] and yields steadily increasing environmental pressure.", "[5], [6] Of the world population, more than 80% of all people and more than 99% of people in the U.S. and Europe live in places where the brightness of the night sky is elevated due to light pollution.", "[4] Both the extent to which the indication of artificial light appears in satellite remote sensing data and the quantity of emitted light have increased globally on average by about two percent per year in recent years.", "[7] The spatial variance of anthropogenic light is large, [8] and both lit area and quantity of light are stable or decreasing in only a handful of countries.", "[7] Light pollution manifests itself both as a presence on the ground and in the night sky.", "On the ground, we perceive its effects directly in forms such as glare and light trespass, [9] and indirectly in threats to human and wildlife biology, [10], [11], [12] public safety, [13], [14] and energy security.", "[15], [16] In the night sky, light pollution yields skyglow, a condition in which artificial light directed upward is scattered back to the ground where it obscures our view of the stars.", "[17] The brightness of the night sky, relative to its assumed `pristine' state absent anthropogenic light pollution, is related to the amount of artificial light emitted on the ground, [18], [19] so night sky quality is often taken as a proxy for the conditions affecting natural darkness on Earth.", "[20], [21] Managing the resource of natural nighttime darkness involves understanding its nature and the influences that threaten its integrity, which in turn require knowing the initial state of the resource and how that state evolves with time.", "[22] Ground-based optical astronomy depends crucially on this knowledge in order to extract the maximum information value from the cosmic light that our telescopes and instruments collect.", "[3] Understanding the sources of light controlling the brightness of the night sky and how those components change on various timescales is therefore a fundamental concern to astronomy.", "The factors that determine night sky brightness were last comprehensively reviewed in the literature almost 50 years ago.", "[23] In the time since, research combining elements of astronomy, atmospheric science and space physics has revealed a more complete picture of those factors.", "The deeper understanding of this phenomenon that results has implications for both how modern ground-based astronomical observatories are designed and operated as well as the outdoor lighting policies and practices that best support observatory site protection.", "This Review considers which factors determine the brightness of the night sky (§ ); how sky brightness is quantified (§ ); relevant units of measurement (§ ); and how night sky quality is classed, compared and ranked (§ ).", "Throughout the Review we refer to two acronyms frequently, mirroring their use in the literature: “night sky brightness” (NSB), and “artificial light at night” (ALAN).", "From sunset to the onset of astronomical darkness, the brightness of the sky at the zenith decreases by a factor of about $4\\times 10^{5}$ (Figure REF a).", "The time required to complete this transition varies with season and latitude; in the tropics it can take place in as little as 72 minutes.", "After sunset, the Sun continues to directly illuminate the atmosphere for some time.", "This period is called twilight, during which sky brightness is dominated by the scattering of sunlight illuminating the atmosphere at progressively higher altitudes.", "Figure: Natural light in the sky from day to night.", "(a) Natural outdoor illumination levels adapted from .", "Horizontal illuminance is shown on the ordinate, while the abscissa shows the altitudes of the Sun (heavy solid line) and Moon (dotted lines).", "SS = sunset, CT = civil twilight, NT = nautical twilight, AT = astronomical twilight.", "Figure courtesy of T. Longcore.", "(b) The anti-twilight arch appears just above the Earth’s shadow at dusk.", "Photo by G. Donatiello.Since the amount of scattered sunlight depends on the number of scatterers along the direction of travel of light rays, and the density of scatterers decreases with height above the surface of the Earth, the brightness of the twilight sky drops rapidly as the Sun descends further below the horizon.", "First-order scattering of light dominates sky brightness from sunset to until about the end of civil twilight, when the solar altitude reaches $-6^{\\circ }$ .", "As the angle increases, second- and higher-order scattering become important.", "[25] The color of the twilight sky is generally the same blue as the daytime sky except for contributions resulting from a combination of Rayleigh scattering and absorption by ozone (O$_{3}$ ) molecules whose concentration in the stratosphere peaks at altitudes of $30-35$ kilometers.", "[26] When there are unusual quantities of particulates in the atmosphere, such as following significant wildfire events and volcanic eruptions, scattered sunlight is subject to additional `de-blueing', resulting in vivid colors.", "The same circumstances are known to cause significant variations in atmospheric extinction, affecting astronomical photometry.", "[27] Opposite the setting Sun in the sky, the anti-twilight arch rises (Figure REF b).", "Its warm tones represent the enhancement of backscattered sunset light de-blued by its passage through the atmosphere at near-grazing angles.", "It reaches a maximum width of between $3-6^{\\circ }$ above the anti-solar point; below the arch, a deep blue tinged with purple indicates the projection of the Earth's shadow onto the twilight sky.", "Direct illumination of the atmosphere reaches the local zenith when the altitude of the Sun is approximately $-6^{\\circ }$ and second-order scattering begins to become important.", "Around this time the lower boundary of direct solar irradiation reaches 35 km altitude where it excites a layer of neutral sodium atoms to the $^{2}P^{0}$ state, causing them to emit in the 589.0/589.6-nm “D” lines of Na i.", "By the time solar angle reaches $-6.5^{\\circ }$ , irradiation is reduced to the point where the intensity of the emission equals that of the corresponding absorption lines in the solar spectrum.", "The episode is so relatively short in duration that it has been described as the “sodium flash.” [28] Figure: Night airglow phenomena seen from Earth orbit.", "(a) Emitting layers of neutral sodium (yellow; 80-100 km) and oxygen (green, 100 km; red, 200-300 km) atoms seen in an oblique view of the Earth’s limb from low- earth orbit near the terminator.", "NASA photo ISS042-E-037847 taken by European Space Agency astronaut Samantha Cristoforetti.", "(b) Structured airglow observed above thunderstorms in west Texas, U.S., originating along a dry line on 15 April 2012.", "(c) Airglow waves emanating from the site of the eruption of the Calbuco volcano in southern Chile on 22 April 2015.", "Images by Jesse Allen (NASA Earth Observatory).As twilight progresses, a similar “oxygen flash” happens as neutral oxygen atoms at altitudes from $200-300$ km are directly irradiated by the Sun, yielding emission in the $^{1}S{\\rightarrow }^{3}P$ 630/636-nm lines of O i.", "This is thought to contribute to the purplish hues of the sky during late twilight along with molecular scattering of sunlight at high altitudes and de-blued illumination of the stratosphere and the troposphere.", "[29] Since emission in the O i doublet is radiatively excited, the lines largely fade out after the onset of astronomical darkness.", "However, a chemical process excites the same transitions on the night side of Earth, particularly at low magnetic latitudes, resulting in red oxygen “airglow” that varies in intensity and sky distribution through the night in the tropics.", "[30], [31] Oxygen atoms at lower altitudes are collisionally excited and emit characteristically in the forbidden $^{1}S{\\rightarrow }^{1}D$ transition at 577 nm.", "Resulting O i emission persists throughout the night.", "[32] Figure REF a shows the stratification of sodium and oxygen airglow toward the limb of the Earth as seen from low-earth orbit.", "This phenomenon is discussed further in Section REF .", "Direct illumination of the entire atmosphere above an observer continues until the Sun reaches an altitude of $-12^{\\circ }$ (nautical dusk).", "From here to a solar altitude of $-18^{\\circ }$ , only the uppermost parts of the atmosphere seen in the direction of the horizon are still directly illuminated by the Sun.", "Once the Sun descends below $-18^{\\circ }$ , direct illumination ceases and astronomical darkness begins.", "The sequence of events described here unfolds in reverse on the morning side of night, from the point when the Sun again reaches an altitude of $-18^{\\circ }$ until sunrise." ], [ "Sources of natural light in the night sky", "The total brightness at any point on the night sky is the sum of contributions from both natural and artificial sources, each of which is a function of azimuth ($\\alpha $ ) and altitude ($\\gamma $ ).", "Some sources are also a function of time ($t$ ), [33] so in general: $B_{sky}(\\alpha , \\gamma , t) = \\sum _{i=1}^{N} B_i(\\alpha , \\gamma , t)$ The natural, optical-light components of NSB have been reviewed in several publications, [23], [34], [20] and are summarized in Tables REF and REF .", "These components are divided into two sources: those that originate in or near the Earth's atmosphere and those that originate in the space beyond the Earth's atmosphere.", "Because the boundary between the atmosphere and outer space is not well defined, we include the Earth's magnetic environment as part of `atmosphere' such that, e.g., aurorae are not considered an astronomical phenomenon.", "Kocifaj et al.", "[35] recently showed that artificial objects orbiting the Earth are a non-trivial contributor of diffuse light to the night sky as seen from the ground.", "We have included a value for this effect in Table REF .", "On the other hand, distinctly terrestrial sources of natural light at night, such as wildfire, lightning and bioluminescence, are typically neglected in NSB calculations because they are regarded as insignificant contributors except on highly local spatial scales.", "Sources of natural optical light in the night sky originating in and near the Earth's atmosphere.", "For each component (column 1), the angular extent of the light in the sky (column 2) is given along with its average surface brightness (column 3), the factors that influence its extent and/or brightness (column 4), and its physical cause (column 5).", "Table: NO_CAPTION [a]S$_{10}$ (vis) is a linear unit equal to the surface brightness of a star whose visual magnitude is +10 and whose light is distributed over one square degree.", "In SI units, 1 S$_{10}$ (vis) $\\approx $ $1.04\\times 10^{-6}$ cd m$^{-2}$ .", "[b][36] [c][23] [d]The large range in auroral brightness is rated from zero to four on a base-ten logarithmic scale (the International Brightness Coefficient, or IBC;  [37], [38]).", "[e]All auroral brightnesses are drawn from [39].", "[f][40], [41], [42].", "[g]The ADL reaches a maximum at very large zenith angles ($\\sim 90^{\\circ }$ ).", "[43] [h]Defined in [35] to include both satellites and space debris.", "[i]In addition to an overall diffuse glow, NSB contributions from space objects may be higher along the paths of certain orbital shells across the sky.", "See, e.g., [44].", "[j]Estimated value as of mid-2019.", "Sources of natural optical light in the night sky originating beyond the Earth's atmosphere.", "The order and contents of the columns are the same as in Table REF .", "Table: NO_CAPTION [a]All zodiacal light brightnesses are as reported in [34].", "[b][45], [23], [46], [47], [48].", "[c][49].", "[d][50].", "[e]At 4250 Å.", "[51], [52].", "[f][53].", "[g]`Diffuse Cosmic Optical Background'; [54].", "[h]Lauer et al.", "[55] report “a flux component of unknown origin” of 8.06$\\pm $ 1.92 nW m$^{-2}$ sr$^{-1}$ in New Horizons Long-range Reconnaissance Imager (LORRI) measurements at $\\lambda =0.608$ $\\mu $ m after subtracting the estimated contribution from the integrated light of external galaxies.", "Natural sources of light in the night sky range over several orders of magnitude in both surface brightness and wavelength.", "[34] The total natural NSB, generally quoted for the zenith, is the sum of the astronomical sources with an allowance for quiescent, (pseudo-)continuum airglow, but not including other transient terrestrial sources like aurora.", "This averages about 205 S$_{10}$ (vis) or $\\sim 22.0$ m$_{V}$ mag arcsec$^{-2}$ ($\\sim 2\\times 10^{-4}$ cd m$^{-2}$ ).Assuming the ecliptic pole were viewed at the zenith and with no contribution from aurora or airglow.", "See the further discussion of the minimum brightness of the night sky in Section REF .", "This value is itself proposed as a unit of measurement, called one Night Sky Unit (NSU) or one “sky”.", "[56] Note that the brightnesses of all natural light phenomena are quoted for “clear air” conditions at typical atmospheric optical depth ($\\tau $ ) values near mean sea level.", "NSB values $<1$ NSU reported in the literature result from light losses due to, e.g., turbidity in the lower troposphere that absorbs and scatters light out of the incoming beam.", "Sources of light in the sky yield corresponding horizontal illuminances that span many orders of magnitude.", "To the extent that total NSB is related to the total ALAN emission on the ground, it also serves as a predictor of surface illuminance, which has ecological implications.", "Hänel et al.", "[57] tabulated literature values for conditions ranging from full daylight to naturally dark nighttime conditions, which we reproduce here in an abbreviated form in Table REF .", "Table: Typical horizontal illuminances (E V,H E_{V,H}) and corresponding zenith luminances (L V, zenith L_{V,\\textrm {zenith}}) from the literature, adapted from Table 2 in .", "Lighting sources are noted in column 1.", "Base-ten logarithms of E V,H E_{V,H} and L V, zenith L_{V,\\textrm {zenith}} are given in columns 2 and 3, respectively.", "Where no measurement is available or a given entry has no physical meaning, an em dash (—) is shown." ], [ "Airglow and aurorae", "In directions away from the ecliptic, these two energetic processes in the Earth's upper atmosphere are the most significant contributors to the total brightness of the natural night sky.", "The two mechanisms are distinguished primarily by the source of excitation: airglow derives from photoionization followed by radiative recombination, photochemical reactions, or ambient collisional excitation of atoms, whereas aurorae result from the collisional excitation of atoms by solar charged particles spiraling down the Earth's magnetic field lines.", "Airglow dominates at virtually all latitudes, while aurorae are most important at high magnetic latitudes.", "Both light sources are temporally variable in terms of their distribution on the sky, and their intensities vary on timescales ranging from seconds to hours.", "In addition to the polar aurorae, an adjunct phenomenon known as a mid-latitude stable auroral arc (MSAR) yields a lower-intensity glow seen across hundreds to thousands of kilometers on the ground.", "It is thought to result from impacts of magnetospheric electrons accelerated downward along the Earth's magnetic field.", "Unlike the polar aurorae, MSAR excitation favors the $^{1}D$ transition of O i and emission in the 630 nm doublet.", "[58] Another transient subauroral arc only recently identified is the Strong Thermal Emission Velocity Enhancement (STEVE), which seems to be related to ion drift in the ionosphere.", "[59] As observed and photographed, the airglow often displays a periodic structure attributable to gravity waves in the atmosphere.", "[60] These waves can be generated by phenomena ranging from volcanic eruptions to turbulence in the stratosphere induced by the action of supercell thunderstorms (Figure REF b and c).", "Shepherd and Cho [61] presented orbital O i $\\lambda $ 577 nm observations indicating airglow enhancements up to a full order of magnitude over quiescent conditions when multiple zonal components of gravity waves come into phase at the same longitude.", "The strength of the airglow contribution depends on the phase and amplitude of the solar cycle.", "Its time average can be approximated in S$_{10}$ (vis) units according to $B_{airglow} = 145 + 108(S-0.8)$ where $S$ is the solar 10.7-cm flux in units of MJy.", "[34] As $S$ is observed to vary roughly sinusoidally between 0.8 and 2.0 during the solar cycle, the quiescent airglow contribution ranges between $145-270$ S$_{10}$ (vis).", "Excluding aerosol extinction in the lower atmosphere, natural NSB can therefore vary by a factor of nearly two both within one night and from night to night.", "NSB is also observed to vary seasonally, [33] and certain episodes seem to correlate with solar activity even near solar minimum [62].", "At least some of the seasonal variability is attributable specifically to changes in the strength of the airglow lines.", "[63] There are further indications of natural NSB enhancements that correlate in time at observing stations separated by thousands of kilometers.", "[64] These observations highlight the importance of temporal sampling and the ongoing need to develop a deeper understanding of airglow physics.", "After accounting for line sources of radiation, there is an observed residual airglow “continuum” that shows no spectroscopic structure.", "[65] It is spectrally flat over the visible wavelengths and adds no more than about 50 S$_{10}$ (vis) to the brightness of the night sky.", "[36] The source of this pseudo-continuum is not entirely clear, in part because it is exceedingly difficult to fully distinguish it from zodiacal light and integrated starlight.", "Roach and Gordon concluded that the source of this light is a combination of “a real upper-atmosphere phenomenon” and sunlight multiply scattered far into the Earth's night-side atmosphere.", "Kenner and Ogryzlo [66] reviewed various proposed chemical reactions involving compounds of oxygen and nitrogen, along with their own laboratory data, to explain this emission.", "Bates [67] argued that the continuum is formed by a multitude of products resulting from collisions between metastable oxygen atoms and air molecules, “thereby forming complexes that dissociate by allowed radiative transitions.” The consensus among researchers now is that the airglow continuum is composed of many hundreds of individual atomic and molecular spectral lines whose sharp features overlap to create a broad emission continuum." ], [ "Anthropogenic skyglow", "Light from human-caused sources comprises the balance of NSB when added to the spatially and temporally variable contributions of natural sources.", "The visible manifestation of this light, commonly referred to as “skyglow”, forms in the lower atmosphere as a result of both small- and large-particle scattering.", "ALAN emitted on the ground influences skyglow through its spectral power distribution, angular emission function and the total lumen output of contributing light sources.", "[68] Skyglow is observed to vary in brightness relative to the assumed zenith brightness of an unpolluted night sky by up to a factor of about 6,000, [69] at which point only a handful of the brightest stars are visible to the unaided human eye.", "In most urban contexts, skyglow dominates the light of the night sky (Figure REF ).", "Figure: The color and brightness of the light-polluted night sky.", "Uncalibrated (left) and luminance-calibrated (right) images of the sky on UT 23 May 2017 as seen from Mission San Xavier del Bac near Tucson, U.S.", "The zenith is at the center of both images and the horizon runs around the edge; north is at top and east at left.", "Warm tones in the uncalibrated image indicate the color of skyglow then dominated by high-pressure sodium lighting emissions.", "False colors at right correspond to luminance in units of m V _{V} arcsec -2 ^{-2} according to the color bar at right.", "Images by the author.The presence of ice and snow on the ground intensifies skyglow due to their high reflectivities, enhancing upward-directed emissions from cities; models of skyglow formation over cities show an almost linear relationship between ground reflectance and artificial NSB.", "[70] Measurements of the effect show an up to three-fold increase in NSB in cities due to snow cover on the ground, [71] and snow cover further amplifies skyglow itself due to reflections of the sky from the ground.", "[72], [73] Skyglow is also sensitive to the presence of very fine particles in the air, which may be increased by certain kinds of air pollution.", "[74], [75] Cloudy nights make the problem even worse; [76] overcast conditions over cities increase horizontal illuminance at ground level by a factor of up to ten.", "[77] On the other hand, the comparative absence of ALAN in rural places means that cloud cover tends to darken the nighttime sky and landscape.", "On an overcast night far from sources of anthropogenic light, the measured NSB can be up to about a factor of six lower than a clear night at the same location.", "[78] Skyglow is now well understood as a matter of radiative transfer, and models have steadily become more sophisticated and better representative of real-world conditions.", "The total light output of a city is the strongest predictor of NSB in the urban environment, and of the brightness of `light domes' over cities as seen from remote locations.", "[79] Modelers have also attempted to infer the so-called city emission function (CEF) as a means of describing the distribution of the anthropogenic light over a city causing skyglow.", "[80], [81], [82] Understanding the CEF is crucial for predicting the appearance of skyglow both within and outside of the city; for example, light rays emitted at very shallow upward angles yield the greatest impact on NSB as seen at distances from dozens to hundreds of kilometers.", "[83] Shadowing of city light emissions by topographic features is also known to influence the CEF.", "[84], [70]" ], [ "The scattering of light in the Earth's atmosphere", "The behavior of light during its flight through the atmosphere is governed by the frequency-dependent radiative transfer equation: $\\frac{dI_{\\nu }}{d\\tau _\\nu } = -I_{\\nu } + S_{\\nu }$ where $I_{\\nu }$ is the light intensity, $\\tau _{\\nu }$ is the optical depth, and $S_{\\nu }$ is the so-called “source function” defined by the ratio of emission and absorption coefficients.", "[85] In the simple case of a plane-parallel atmosphere, or otherwise in conditions where the atmosphere is horizontally homogeneous, it has the general solution.", "[86] $I_{\\nu }(\\tau _{\\nu }) = I_{\\nu }(0) e^{{-\\tau _{\\nu }}} + \\int _{0}^{\\tau _{\\nu }} S_{\\nu } e^{-({\\tau _{\\nu }-{\\tau ^{\\prime }_{\\nu }}})} S_{\\nu }(\\tau ^{\\prime }_{\\nu }) d\\tau ^{\\prime }_{\\nu }.$ This formula governs the frequency-dependent intensity of light as it traverses a medium with optical depth $\\tau _{\\nu }$ .", "In short, it holds that the change in intensity during the traverse is the sum of light added to the beam less the sum of light removed from the beam.", "Light can be added through direct emission from the medium and removed via absorption and scattering.", "The remainder of this section focuses on the scattering of light by atmospheric constituents, as this process dominates the radiative transfer process in determining NSB.", "We assume local thermodynamic equilibrium, which is a reasonable approximation for clear sky/air conditions, and for the moment we neglect both complete absorption of light and direct emission from the atmosphere through, e.g., airglow.", "Two modes of scattering control the behavior of light as it transits the atmosphere: Rayleigh scattering and Mie scattering.", "These modes depend largely on the size of the scattering particles, which are composed of two principal atmospheric constituents: molecules and `aerosols' (a suspension of fine solid particles or liquid droplets in air).", "Rayleigh scattering is important when the wavelengths of light involved are much smaller than the size of the scattering particles.", "The scattering strength is strongly dependent on wavelength ($I \\propto I_{0}\\lambda ^{-4}$ ).", "Mie scattering applies exclusively to homogeneous, spherical particles and shows almost no scattering strength dependence on wavelength; in comparison, the particles in the Earth's atmosphere are distinctly inhomogeneous and irregular in shape.", "To the extent that atmospheric particles are comparable in size to the wavelength of light, rather than much smaller or much larger, circumstances are reasonably approximated by Mie scattering.", "The distinction between Rayleigh and Mie scattering is obvious in everyday experience: the blue color of the daytime sky results from strong Rayleigh scattering of short-wavelength visible light by diatomic nitrogen molecules comprising the majority of the lower atmosphere, while clouds are white or gray depending on whether they reflect or transmit (attenuated) sunlight.", "Short-wavelength light is efficiently Rayleigh-scattered even along short optical paths, yielding blue-rich spectra near light sources and progressively `de-blued' spectra at large distances.", "Rayleigh scattering dominates NSB in both cases.", "Mie scattering becomes an important influence in and near cities, especially where particulate pollution from vehicle exhaust and industrial activity are common.", "[83], [70] Furthermore, multiple-order scattering is often important in real-world situations.", "Sophisticated radiative transfer codes account for this in skyglow models, although they tend to become processor-intensive as the number of scattering orders increases.", "Some light exits the atmosphere completely, whether directly or after one or more scattering events, and can be detected from orbit.", "This forms the basis of remote sensing of upward radiance, which is discussed in Section REF ." ], [ "Existing sensing and monitoring capabilities", "The measurement of NSB from ground-based platforms has been recently reviewed by Hänel et al.", "[57] We summarize the relevant points here." ], [ "Sensing", "There are two basic approaches to measure and monitor NSB: look upward from the ground or look down from orbit.", "The former mode involves direct sensing of NSB, while the latter mode predicts NSB seen from the ground by sensing upward-directed radiance and applying a model of how light propagates through the atmosphere.", "Raw ground-based measurements are model-independent but typically limited geographically and temporally.", "We focus here largely on the ground-based approach, but briefly comment on new capabilities for remotely sensing NSB in Section REF .", "Direct measurements of NSB from the ground involve sensors that integrate the flux of light through a known solid angle, within some wavelength range, and over some length of time.", "These divide into two types: single-channel devices and multichannel devices.", "Single-channel devices are patterned on photoelectric photometers used by astronomers for almost a century.", "These devices, such as the popular Sky Quality Meter (SQM; [87], [88]), rely on simple and well-understood physics, require little electric current to operate, and are usually small enough to be easily portable.", "They typically employ light-to-frequency (LTF) converters whose output is a signal pulse stream, the frequency of which is linearly proportional to received light intensity.", "Their light response is determined in the laboratory, with on-board lookup tables relating measured frequency to light intensity tied to calibrated light sources.", "Since the response of LTF converters is also sensitive to ambient operating temperature, sensing of the air temperature is required to properly correct the measured frequency.", "This is usually done on board the measurement device.", "Most commercially available devices have their own photometric passbands modeled on Johnson-Cousins $V$ .", "[89] Researchers have experimented with other filters, but $V$ was chosen to match the bulk of existing literature data and the human visual response to light under photopic conditions.", "Infrared blocking filters are often used in combination with the quantum efficiency profile of the semiconductor material of the LTF to achieve the desired effective passband.", "Optics may be used to constrain the opening angle defining the device's angular field of view.", "Although single-channel device measurements indicate only the brightness of the night sky averaged across a fairly large acceptance angle, some authors report creating crude two-dimensional maps of NSB by interpolating spot measurements from these devices.", "[90] Single-channel devices have a number of advantages, including ease of use; portability; a physically simple sensing mechanism; temperature compensation; good repeatability; rapid capture and display of data; and a relatively long historical record of use.", "However, there are certain drawbacks to these devices.", "In order to sense a sufficient amount of light to yield a measurable signal, they must integrate it over a relatively large solid angle.", "They offer little meaningful spatial resolution in most applications, making them generally unsuitable for monitoring the behavior of light domes near the horizon.", "Lastly, there are differences among commercially available devices in terms of photometric passbands that complicate comparison of results among different device types." ], [ "Multichannel devices", "Multichannel detectors consist of arrays of light-sensitive elements whose output is multiplexed through one or more signal amplifiers.", "The ideal example is an imaging spectroradiometer, which provides a complete set of information about the wavelength-dependent brightness of the night sky in any given direction.", "However, the current generation of such devices is too slow for capturing time-resolved NSB data, and they tend to be prohibitively expensive.", "One more often encounters cameras capturing two-dimensional images, particularly commercial digital single-lens reflex (DSLR) cameras and mirrorless interchangeable lens cameras (MILC); see, e.g.,  [71].", "Some are operated with photometric filters to yield a particular effective passband, while others use Bayer filter mosaics to capture native (pseudo-)true-color images through the combination of broadband red-, green- and blue-filtered data.", "(Figure REF a) Figure: Typical broadband digital imaging passbands and night sky spectra.", "(a) Representative spectral sensitivity curves of some commercial digital cameras (blue, green and red lines corresponding to broad RGB bands) and the astronomical V-band response (black line).", "Figure 2 in .", "(b) The night sky spectrum over Zselic International Dark Sky Park, Hungary (yellow) and its decomposition and fit (other colors).", "See main text for a description of the fit components.", "Figure 1 in .The main advantage these cameras have over single-channel devices is the ability to produce two-dimensional images with some amount of both angular and spectral resolution.", "They are often paired with very wide-angle lenses to capture views with solid angles as large as 2$\\pi $ steradians (180$^{\\circ }$ ) in a single exposure, [92] while others build up multiple-image mosaics with angular offsets between exposures so that the results can later be “stitched” together in software.", "[93] As a result, these devices provide significantly more spatial information about the distribution of NSB than do single-channel devices.", "Depending on the pixel scale of the detector, star images may be sufficiently sampled that flux calibration can be performed using spectrophotometric standard stars; other imaging systems make use of lab calibrations from reference light sources and employ integrating spheres for illumination of the camera and lens.", "Spatial distortion information for particular lens and camera combinations can be used to correct lens aberrations after the fact in software.", "[94], [95] Multichannel devices have their own drawbacks.", "Due to sensor size and pixel scale, they generally have limited angular resolution.", "When imagers are used with fisheye lenses to capture all-sky data in single exposures, significant angular distortions are induced near the horizon.", "Their multi-spectral functionality is usually limited to a few broad passbands.", "And, lastly, there is as yet no standard, SI-traceable reporting unit for NSB measurements.", "This issue is discussed further in Section ." ], [ "Color considerations", "An concern adjunct to characterizing NSB is the spectral power distribution (SPD) of sky light.", "As the preceding discussion suggests, the sensed NSB is the result of integrating, with respect to wavelength, the convolution of the SPD of the night sky with the spectral bandpass of the measuring device.", "The SPD of the night sky is a complex function of the various physical processes from which it results (see Secton ); it is further modulated by wavelength-dependent scattering during the transit of night sky light through the Earth's atmosphere.", "Measurements of NSB in both radiometric and photometric units are therefore strongly dependent on the night sky's spectrum.", "[96] Because most devices used to sense NSB have relatively large spectral bandpasses, the responses of those instruments interact with the night sky SPD in complex ways and call for careful consideration when interpreting measurements.", "[97] Some authors report the use of metrics such as the correlated color temperature (CCT) of the night sky as a means of characterizing its spectral qualities.", "[78], [69] While CCT relates to the spectra of thermal sources, its utility is diminished as the SPDs of sources become increasingly non-thermal.", "Since many NSB components, such as airglow and aurorae, have decidedly non-thermal SPDs, the use of CCT alone is unlikely to give reliable color information about the night sky." ], [ "Data modeling", "Modeling of NSB observations can assist with their analysis and interpretation.", "For example, Duriscoe [20] reported successfully recovering the anthropogenic component of NSB from mosaicked all-sky image data by subtracting 2-D models of natural sources of light.", "To the extent that construction and application of such models can be automated, they hold the promise of rapidly disentangling natural sources of light in the night sky from artificial sources for the purposes of modeling the angular and temporal evolution of skyglow.", "For spectrally resolved measurements, it is possible to model the natural components of NSB in wavelength space to subtract and remove them, leaving behind only the spectrum of artificial light sources.", "Figure REF b shows the decomposition of a night sky spectrum containing both natural light sources and anthropogenic skyglow.", "The “Continuous” component (light blue) is the sum of contributions from natural light sources other than airglow line emission.", "“Emission” (dark blue) consists of modeled line airglow contributions.", "“ALAN” (red) is the sum of several common lamp spectra.", "The sum of all components is the “Fit” line (gray).", "From this decomposition it was determined that the `continuous' component of the natural sky (zodiacal light, scattered starlight and airglow pseudo-continuum) is nearly constant at all visible wavelengths and has a spectral radiance of $\\sim 2$ nW m$^{-2}$ sr$^{-1}$ nm$^{-1}$ .", "[91] There are a handful of additional approaches to the modeling NSB that add other inputs to the direct sensing of light.", "For example, Kolláth and Kolláth [98] used raw backscatter data from a laser ceilometer to provide inputs to Monte Carlo simulations of sky radiances measured simultaneously from the ground using calibrated cameras.", "The authors applied this technique to infer the vertical structure of the radiance distribution of the night sky." ], [ "Remote sensing of NSB", "NSB is now routinely measured by remotely sensing upward-directed radiance using a variety of platforms, including Earth-orbiting satellites, [99] the International Space Station [100], airplanes, [101], [102] drones [103] and balloons.", "[104], [105] The use of remote sensing to infer NSB in this manner offers a number of attractive qualities.", "Chief among these is the ability to collect information about NSB from essentially anywhere on Earth, which decouples NSB measurement and monitoring from the deployment of ground-based sensors.", "Falchi and coworkers provided such a global data product most recently in 2016.", "[4] They calibrated the radiance-NSB relationship using many thousands of ground-based NSB measurements, but their predictions are sometimes inaccurate.", "This may be the result of models assuming a flat Earth, and which therefore do not take into account the screening effect of regional topography, or due to the fact that locally variable atmospheric turbidity can induce time-dependent scattering effects.", "In particular, at astronomical observatories, which tend to be located in comparatively dry, high-altitude sites, the aerosol content is probably overestimated as therefore also is the computed scattering.", "Yet this map remains our only truly global view of light pollution.", "Diffuse light seen around cities in remote sensing imagery from Earth orbit was long thought to result from a combination of sensing artifacts and low spatial resolution, [106], [107] but it is now recognized as a real signal corresponding to light scattered in the atmosphere.", "Kocifaj and Bará [108] showed that certain aerosol properties, such as the particle size number distribution, can be successfully retrieved from orbital radiometry of the angular radiance distribution of the scattered light near cities.", "Sánchez de Miguel et al.", "[109] recently found a strong correlation between the zenith NSB measured on the ground and orbital radiance measurements at both low and high resolution.", "They suggested that creating accurate, regional or even global NSB maps based on radiance measurements from the newest generation of orbital radiometers should be possible.", "However, there are other problems with existing satellite remote sensing platforms.", "For example, the Day-Night Band of the Visible Infrared Imaging Radiometer Suite (VIIRS-DNB), deployed on board the Suomi NPP and NOAA-20 satellites, has no spectral sensitivity shortward of 500 nm.", "The instrument is therefore effectively blind to the strong peak in white LED light emissions near 450 nm.", "This limits what can be reliably inferred concerning short-wavelength light sources within the data set.", "[110]" ], [ "Monitoring", "In context of this Review, “monitoring” of NSB refers to its repeated measurement to look for trends on timescales ranging from minutes to years.", "Monitors, like sensing devices, fall into two general categories: those that function autonomously, and those whose operation requires human attendants.", "Autonomous monitors are sensing devices fitted into weatherproof housings with their own electric power supplies and, optionally, network connections.", "Some of them save their measurements to on-board memory, while others relay them to another location for storage via a local network or the Internet.", "At present, autonomous monitors tend to be single-channel devices with few requirements for field calibration.", "These monitors are subject to regular insolation during the daytime, which appears to contribute to photometric zeropoint drift, possibly through the deterioration of optical and/or electronic components due to solar ultraviolet light irradiation.", "[111] Some authors report attempts to calibrate these long-term secular trends using luminance sources like the twilight sky. [112].", "Attempts to construct autonomous all-sky imagers have tended to leverage existing facilities marketed to amateur astronomers as cloud sensors; other, purpose-built devices, such as the ASTMON system [92], are intended as fully robotic instruments whose data acquisition and reduction are automated and which function as permanent monitors.", "Attended monitors may function automatically, but they require a human operator for setup and maintenance.", "This is usually because the monitoring device is not permanently installed and lacks equipment to make it durable in the natural environment.", "The operator may also direct details of the data collection protocol such as manually switching slides in a rotating filter wheel.", "An example of this is the Road Runner system, in which a single-channel sensing device is mounted to the roof of an automobile and collects NSB data continuously while the vehicle is driven.", "[113] Another example is the U.S. National Park Service Night Sky Team (NPS NST) imaging method.", "[93] Its camera, situated on an automatic `go-to' mount, executes an imaging program automatically, but it must be transported to each measurement site and set up by NPS NST staff.", "There is also considerable human effort required to reduce, analyze, and report the resulting data.", "Monitoring entails the concerns of data handling, transmission and storage, as well as reduction and analysis.", "Some autonomous monitors log NSB data to on-board storage media, which must be periodically retrieved and offloaded.", "Other systems, such as the Telescope Encoder and Sky Sensor-WiFi (TESS-W), [114] make use of wireless networking and transmission of measurements to a central storage location via the Internet, leaving them vulnerable to network interruptions.", "There are also concerns about data reporting formats, although some effort has been put into designing and promoting a standard protocol for recording NSB data.", "[115]" ], [ "Temporal sampling frequency", "Other monitoring considerations involve the frequency of data collection, both in the temporal and spatial senses.", "Given the timescales on which the natural NSB varies, sampling frequency is important so as to fully understand the brightness range of the natural nighttime environment; the same applies to skyglow, which tends to vary in slower and more predictable ways.", "The presence of skyglow can `stabilize' NSB if it significantly exceeds the radiance of natural sources of light in the night sky, as in many bigger cities.", "In such cases, only weak apparent variations exist from night to night.", "NSB monitors therefore typically perform best in urban environments while potentially giving ambiguous information in naturally dark locations.", "Various approaches to visualizing NSB time-series data are suggested in the literature.", "Perhaps the most common method is the NSB densitogram, commonly referred to as a `jellyfish plot'.", "In this representation, the NSB is plotted against the local time, and each pixel is color-coded to represent the number of observations in a time series that fall into that particular (time, NSB) bin.", "It is a convenient way to both compress a lengthy time series into a single plot as well as to quickly discern between typical and atypical NSB conditions.", "Figure: NSB histogram made by integrating a time series of measurements obtained over one year at Paramos, Spain.", "The upper plot demonstrates the method schematically, with the result shown in the lower plot.", "‘Arms’ on either side of the plot (“A”) represent the rapid sky brightness change during twilight.", "The brightest nights (“B”) are those during which moonlight directly illuminates the detector.", "“C” represents nights during which clouds scatter moonlight and reflect city lights.", "Nights “D” correspond to moonless nights when clouds amplify city skyglow.", "The darkest nights (“E”) correspond to clear conditions with no moonlight contribution.", "Figure 2 in .This kind of data visualization helps inform efforts to characterize night sky quality at a given location and follow its evolution in time.", "For example, Bará et al.", "[110] suggest that a well-sampled jellyfish plot can be used to extract meaningful sky quality metrics.", "In Figure REF , a jellyfish plot is collapsed to a 2-D distribution in frequency versus zenith NSB.", "Peaks corresponding to the various brightness regimes are evident in the result.", "From this, the authors conclude that no single value of the NSB fully represents the variety of conditions at any particular observing site.", "Some limited efforts have been made to apply, e.g., Fourier analysis techniques to time-domain measurements of NSB.", "For instance, Puschnig, Wallner and Posch [116] used fast Fourier transform frequency analysis of nightly mean NSB measurements made using a network of Sky Quality Meters in Austria.", "From this analysis they concluded that the circalunar periodicity of NSB, of biological importance to a number of nocturnal species, essentially disappears for maintained zenith brightnesses higher than about 16.5 mag arcsec$^{-2}$ ($\\sim 32$ mcd m$^{-2}$ ).", "Bará et al.", "[110] further considered whether the NSB sampling rate on a timescale of minutes influences average sky quality indicators using measurement collected in long (e.g., yearly) time periods, concluding that it does not.", "Resampling a series of zenith brightnesses obtained with Sky Quality Meters in one-minute readings to sampling intervals of five and ten minutes, they found that the the maximum absolute difference of the full width at half-maximum (FWHM) of the darkest peak in a histogram of time-series NSB values was $<0.0009$ mag arcsec$^{-2}$ for a five-minute sampling interval and $<0.0017$ mag arcsec$^{-2}$ for a ten-minute sampling interval.", "These values are well below the measured precision of the SQM ($\\lesssim 5$ %).", "However, the question of which temporal NSB sampling frequencies are sufficient to yield a sense of the typical night sky quality at a given location is not well formed because there is yet no general agreement as to what we mean by `typical'.", "If this were clearly and definitively decided, a simple analysis would easily reveal the optimal sampling parameters to yield the desired metric.", "An example of how this approach may be applied to NSB data is discussed in Section REF ." ], [ "Spatial sampling frequency", "Characterizing the typical NSB across a large geographic area demands consideration of the proper spatial sampling frequency in order to ensure uniform results, especially with respect to acceptable measurement uncertainties.", "To date there is one published study on this subject by Bará [117], based on data from Falchi et al.", "[4] Bará found that a useful rule of thumb is that one measurement per square kilometer is sufficient to constrain the zenith NSB at any point in a sampled region to a precision of $\\pm 0.1$ $V$ mag arcsec$^{-2}$ rms.", "However, the author notes that “exact reconstruction of the zenithal night sky brightness maps from samples taken at the Nyquist rate seems to be considerably more demanding.”" ], [ "Measurement units", "NSB measurements found in the literature are reported in several different, and sometimes confusing, units.", "Although one occasionally finds illuminances reported in SI units like microlux, the majority of measurements are surface brightness terms.", "As a further complication, measurements can be either radiometric or photometric depending on whether they refer broadly to the entire visible spectrum or instead are weighted by the spectral response of the human eye, respectively.", "Some units characterizing NSB in surface brightness terms are as follows: Candela per square meter (cd m$^{-2}$ ), a linear, SI unit informally called the “nit”.", "The unit is based on the SI units of luminous intensity (candela) and area (meter).", "The CGS equivalent is the stilb.", "1 stilb = $1.04\\times 10^{2}$ cd m$^{-2}$ .", "Lambert (L), a linear, non-SI unit defined as $\\pi ^{-1}$ cd cm$^{-2}$ ($\\approx $ 3183 cd m$^{-2}$ ).", "S$_{10}$ (vis), a linear, non-SI unit defined as the surface brightness of a $m_{V}=+10$ star whose light is distributed over one square degree.", "1 S$_{10}$ (vis) $\\approx $ $1.04\\times 10^{-6}$ cd m$^{-2}$ .)", "Magnitude per square arcsecond (mag arcsec$^{-2}$ , or mpsa), a logarithmic, non-SI unit defined such that if an area on the sky contained only exactly one magnitude $N$ star in each square arcsecond, the sky brightness would be $N$ mag arcsec$^{-2}$ .", "Night Sky Unit (NSU), a linear, non-SI unit introduced in Section 5.2 as the average zenith NSB away from the ecliptic assuming quiescent airglow conditions and the absence of skyglow ($\\sim 0.2$ mcd m$^{-2}$ in the $V$ band).", "It is sometimes called a “Natural Sky Unit” or a “sky”.", "Of these, the magnitude per square arcsecond is most often encountered, being the native reporting unit of, among other devices, the popular Sky Quality Meter.", "Transformations between magnitudes per square arcsecond and SI luminance units have been derived so that astronomical brightnesses in, e.g., $V$ magnitudes can be approximately transformed to photometric values.", "Noting that the relationship between these quantities depends on the spectral power distribution of the source, transformation equations have been derived for scotopic [96] and mesopic [118] viewing conditions and calibrated using zero-point luminances determined from a variety of night sky spectra.", "Kolláth et al.", "[91] lately discussed the problem of different effective photometric passbands among both single-channel and multichannel devices used to measure NSB, as well as the lack of standardized, SI-traceable reporting units.", "Since the range of the band-averaged spectral radiance of a device is independent of the selected passband for spectrally flat or constant sky radiance, the measured band-averaged spectral radiance is of the same order and takes the SI unit of W m$^{-2}$ sr$^{-1}$ m$^{-1}$ .", "Given typical NSB values and the wavelengths of light involved, a more natural unit is nW m$^{-2}$ sr$^{-1}$ nm$^{-1}$ , which the authors propose as one Dark Sky Unit (DSU).", "In this unit, the band-averaged radiance of the natural night sky under clear-air and quiescent-airglow conditions is approximately 1–2, while a cloudy sky yields a value of about 1.", "These numbers are applicable for any passband defined in the visible spectrum.", "On nights when airglow is particularly active and its spectrum is dominated by line emission rather than the pseudo-continuum, it can increase the band-averaged radiance at the zenith by almost a factor of two as compared to nights when airglow is relatively inactive." ], [ "Classifying and ranking night sky quality", "Measurement and monitoring of NSB are usually conducted to meet one or more objectives.", "These may involve gathering a baseline of nighttime conditions to initially characterize the quality of the night sky before monitoring begins, with an eye toward assessing the impact of skyglow on an area.", "This supports site selection for new ground-based astronomical observatories.", "Long-term monitoring helps site managers identify potential threats to night sky quality and assess the efficacy of various mitigations.", "While NSB can be quantified, there is no fully objective quality determination or ranking system for the night sky in part because the experience is distinctly human-focused.", "Any attempt to compare or rank sky quality must admit the inability to completely specify what for many people is an emotional, psychological and sensory experience.", "Furthermore, we now know that the natural night sky varies considerably in brightness on timescales ranging from minutes (aurorae) to years (atmospheric extinction from events like volcanic eruptions).", "As a result, relative measurements that look for trends in NSB at a single location are often more reliable than meta-analyses that aim to quantitatively compare two or more locations.", "The aerosol content of the lower atmosphere is an important factor controlling the perceived quality of the night sky for two reasons.", "One is that aerosols are responsible for the direct attenuation of light from astronomical objects, making them appear to be less bright than they would be above the Earth's atmosphere.", "The other reason is that aerosols scatter light – whether natural or artificial – and raise the level of the sky background.", "These effects jointly act to reduce the contrast between astronomical objects and the background, making them difficult to see.", "[119] In cities, the scattered light of skyglow overwhelmingly dominates attenuation of starlight as the cause of this contrast reduction, but in rural areas direct attenuation competes with scattering.", "The importance of aerosols in otherwise naturally dark places may well explain measurements of NSB apparently falling below the naturally imposed `floor' discussed in Section REF .", "But with these unnaturally dark skies should come a diminished ability to see faint stars whose dim light is substantially or completely extinguished during its passage through the atmosphere.", "This suggests that night sky quality is ultimately determined by some combination of objective sky brightness measurements and subjective impressions of the visibility of faint astronomical objects.", "Both subjective (observer-dependent) and objective (device-dependent) quality metrics have been proposed and are discussed below.", "Several of these metrics are inter-compared in Figure REF a.", "Figure: A comparison of subjective and objective night sky quality metrics.", "(a) Various metrics indicating night sky quality combined into a nomogram, a type of figure in which any horizontal line displays the same quantity on multiple scales.", "In addition to objective measures of NSB, several subjective scales (Bortle, Milky Way visibility, and Sky Quality Index) are shown.", "Nomogram by Henk Spoelstra, used with kind permission.", "(b) An empirical calibration between naked-eye limiting magnitude and approximate Johnson-Cousins BB-band NSB.", "Approximate conversion formulae to and from each variable are indicated on the plot.", "Data and plot courtesy of Anthony Tekatch (Unihedron)." ], [ "Naked-Eye Limiting Magnitude", "Subjective metrics tend to rely on the human visual system for sensing NSB.", "Some approaches use estimates of the visual or naked-eye limiting magnitude (NELM) as an indicator of NSB, given empirical relationships between the two (Figure REF b).", "NELM estimates are the basis for citizen science efforts such as Globe At Night, whose data have been shown in aggregate to correctly approximate the NSB as measured through direct sensing.", "[120]" ], [ "Bortle Scale", "Other subjective metrics are more impressionistic, such as the Bortle Scale, [121] which ranks night sky quality on a scale ranging from one to nine.", "While the Bortle Scale is intuitive, anecdotally some users report sufficient ambiguity in the descriptions of the Bortle classes that estimates are often not reliable to better than one full step in the scheme of classes.", "The apparent brightness and degree of visual structure in naked-eye observations of the Milky Way is sometimes suggested as an alternative indicator of night sky quality.", "Crumey [119] argued that the fundamental visibility of the Milky Way is an indicator of what we would call a “dark” sky, but notes the useful limits of this idea." ], [ "Anthropogenic Light Ratio", "The U.S. National Park Service (NPS) has proposed the Anthropogenic Light Ratio (ALR) as the most basic means of representing the relative amount of skyglow visible from a given site.", "[122] It is the ratio of anthropogenic to natural NSB averaged over the entire sky.", "The latter quantity is usually taken as the “night sky unit” described here in Section REF .", "For purposes of computing ALR, NPS assumes an unpolluted night sky to have a zenith luminance of 78 nL ($\\sim 0.25$ mcd m$^{-2}$ , or 21.79 mag arcsec$^{-2}$ ).", "ALR is linear and unitless, and it can be equivalently expressed as a percentage.", "As a consequence, comparisons between sites in terms of ALR are easily made.", "[122] ALR is in fairly wide use as a NSB metric.", "For example, Falchi et al.", "[4] used ALR as the basis for the maps presented in the New World Atlas of Artificial Night Sky Brightness.", "Duriscoe et al.", "[123] presented a method for estimating ALR with high confidence over large regions using cloud-free, composite satellite images and a simplified spatial model.", "However, as an all-sky average, ALR fails to adequately characterize the distribution of light near the horizon where the `light domes' of cities appear.", "A potential adaptation of ALR that would make it more robust is to specify it as a function of altitude and azimuth." ], [ "Sky Quality Index", "NPS has also devised and promoted the Sky Quality Index (SQI), a metric derived from the distribution of sky luminance values in all-sky image data modeled using the method of Duriscoe [20], described here in Section REF , to remove the light of the natural sky and leave only the anthropogenic contribution.", "SQI can take any value from 0 to 100, where 100 is defined as a night sky entirely devoid of skyglow." ], [ "Illuminance metrics", "Duriscoe [20] also proposed a number of quantities derived from calibrated all-sky imagery that could be used to objectively characterize NSB at a given site.", "These include the average all-sky luminance (both total observed and that from skyglow alone) and the maximum horizontal and vertical illuminances.", "Patat [124] suggested characterizing a night sky by sampling the darkest area of the sky relative to the distance between that point and the ecliptic and the galactic equator in order to minimize contributions from the zodiacal light and diffuse galactic light, respectively.", "Duriscoe criticized this approach because it does not account for the influence of the (spatially averaged) airglow, which increases with zenith distance by a factor of five between the zenith and the horizon." ], [ "NSB histograms", "Bertolo et al.", "[125] recently analyzed NSB data obtained by the Veneto Sky Quality Meter Network, comparing the night sky quality at seven sites in northeastern Italy.", "They characterized the distribution of time-series SQM measurements by the FWHM of the darkest peak of the histogram.", "Bará, Lima and Zamorano [110] elaborated on this idea, proposing as a metric $m_{\\textrm {\\tiny {FWHM}}}$ , which is the average value of the readings within the FWHM of the darkest peak of the histogram under No-Sun No-Moon (NSNM) conditions.", "They contrasted $m_{\\textrm {\\tiny {FWHM}}}$ against $m_{1/3}$ , defined as the average of the upper tertile of NSB values obtained under the same NSNM conditions.", "While $m_{1/3}$ is advantageous in bright, generally urban contexts and suitable for long-term site monitoring, $m_{\\textrm {\\tiny {FWHM}}}$ is superior to $m_{1/3}$ in naturally dark locations.", "In the latter case, $m_{1/3}$ can be strongly influenced by very dark NSB measurements due to effects such as overcast or foggy conditions and snow cover obscuring the detector field of view." ], [ "Sky brightness percentiles", "Hung [21] examined over 1,500 NPS Night Sky Team imaging datasets collected along with Sky Quality Meter measurements and visual observations from hundreds of U.S. national parks and monuments over nearly two decades, finding strong correlations between various commonly used night sky quality metrics.", "She performed a principal component analysis on 53 metrics derived from 1,391 complete datasets and concluded that only five principal components are required to explain 99% of variations among the metrics.", "Hung concluded that zenith brightness and five brightness percentiles (50, 95, 99, 99.995, and 99.999) represent a minimum set that provides non-redundant characteristics of night sky quality." ], [ "Declarations", "The author declares the following competing interests.", "Financial competing interests: The author is self-employed as a consultant in the field that is the subject of this Review.", "Non-financial competing interests: The author is an unpaid member of committees of the American Astronomical Society and International Astronomical Union that advocate or lobby for interests that are the subject of this Review." ] ]
2207.03551
[ [ "Bridging the Gap between Object and Image-level Representations for\n Open-Vocabulary Detection" ], [ "Abstract Existing open-vocabulary object detectors typically enlarge their vocabulary sizes by leveraging different forms of weak supervision.", "This helps generalize to novel objects at inference.", "Two popular forms of weak-supervision used in open-vocabulary detection (OVD) include pretrained CLIP model and image-level supervision.", "We note that both these modes of supervision are not optimally aligned for the detection task: CLIP is trained with image-text pairs and lacks precise localization of objects while the image-level supervision has been used with heuristics that do not accurately specify local object regions.", "In this work, we propose to address this problem by performing object-centric alignment of the language embeddings from the CLIP model.", "Furthermore, we visually ground the objects with only image-level supervision using a pseudo-labeling process that provides high-quality object proposals and helps expand the vocabulary during training.", "We establish a bridge between the above two object-alignment strategies via a novel weight transfer function that aggregates their complimentary strengths.", "In essence, the proposed model seeks to minimize the gap between object and image-centric representations in the OVD setting.", "On the COCO benchmark, our proposed approach achieves 36.6 AP50 on novel classes, an absolute 8.2 gain over the previous best performance.", "For LVIS, we surpass the state-of-the-art ViLD model by 5.0 mask AP for rare categories and 3.4 overall.", "Code: https://github.com/hanoonaR/object-centric-ovd." ], [ "0pt", "8pt plus 4pt minus 2pt0pt plus 2pt minus 2pt" ], [ "0pt", "8pt plus 4pt minus 2pt0pt plus 2pt minus 2pt" ], [ "0pt", "8pt plus 4pt minus 2pt0pt plus 2pt minus 2pt 2pt plus 2pt minus 2pt Existing open-vocabulary object detectors typically enlarge their vocabulary sizes by leveraging different forms of weak supervision.", "This helps generalize to novel objects at inference.", "Two popular forms of weak-supervision used in open-vocabulary detection (OVD) include pretrained CLIP model and image-level supervision.", "We note that both these modes of supervision are not optimally aligned for the detection task: CLIP is trained with image-text pairs and lacks precise localization of objects while the image-level supervision has been used with heuristics that do not accurately specify local object regions.", "In this work, we propose to address this problem by performing object-centric alignment of the language embeddings from the CLIP model.", "Furthermore, we visually ground the objects with only image-level supervision using a pseudo-labeling process that provides high-quality object proposals and helps expand the vocabulary during training.", "We establish a bridge between the above two object-alignment strategies via a novel weight transfer function that aggregates their complimentary strengths.", "In essence, the proposed model seeks to minimize the gap between object and image-centric representations in the OVD setting.", "On the COCO benchmark, our proposed approach achieves 40.3 AP$_{50}$ on novel classes, an absolute 11.9 gain over the previous best performance.", "For LVIS, we surpass the state-of-the-art ViLD model by 5.0 mask AP for rare categories and 3.4 overall.", "Code: https://bit.ly/3byZoQp.", "[0]*Equal contribution" ], [ "Introduction", "Open-vocabulary detection (OVD) aims to generalize beyond the limited number of base classes labeled during the training phase.", "The goal is to detect novel classes defined by an unbounded (open) vocabulary at inference.", "Owing to the challenging nature of the OVD task, different forms of weak-supervision for novel categories are typically used, e.g., extra image-caption pairs to enlarge the vocabulary [1], image-level labels on classification datasets [2] and pretrained open-vocabulary classification models like CLIP [3].", "The use of weak-supervision to enlarge the vocabulary is intuitive as the cost of annotating large-category detection datasets is monumental while the image-text/label pairs are readily available via large classification datasets [4] or internet sources [3], [5].", "One of the major challenges with enlarging vocabulary via image-level supervision (ILS) or pretrained models learned using ILS is the inherent mis-match between region and image-level cues.", "For instance, pretrained CLIP embeddings used in the existing OVD models [6], [2] do not perform well in locating object regions [7] since the CLIP model is trained with full scale images.", "Similarly, weak supervision on images using caption descriptions or image-level labels does not convey the precise object-centric information.", "For label grounding in images, the recent literature explores expensive pretraining with auxiliary objectives [1] or use heuristics such as, the max-score or max-size boxes [2].", "In this paper, we set out to bridge the gap between object and image-centric representations within the OVD pipeline.", "To this end, we propose to utilize high-quality class-agnostic and class-specific object proposals via the pretrained multi-modal vision transformer (ViT) [8].", "The class-agnostic object proposals are then used to distill region-specific information in the CLIP visual embeddings, making them suitable for local objects.", "Furthermore, the class-specific proposal set allows us to visually ground a larger vocabulary, thereby aiding in generalization to novel categories.", "Next, the final and important question is how to make visual-language (VL) mapping amenable to local object-centric information.", "For this purpose, we introduce a region-conditioned weight transfer process which closely ties together image and region VL mapping.", "In a nut-shell, the proposed approach connects the image, region and language representations to generalize better to novel open-vocabulary objects.", "The major contributions of this work include: We propose region-based knowledge distillation to adapt image-centric CLIP embeddings for local regions, thereby improving alignment between region and language embeddings.", "We show that the resulting well-aligned representations aid in improving the overall performance of our text driven OVD pipeline.", "In order to visually ground weak image labels, our approach performs pseudo-labeling using the high-quality object proposals from pretrained multi-modal ViTs.", "This helps in enlarging the class vocabulary and therefore generalizes better to new object classes.", "The above contributions mainly target the visual domain.", "In order to preserve the benefits of object-centric alignment in the language domain, we also propose to explicitly condition the (pseudo-labeled) image-level VL mapping on the region-level VL mapping via a novel weight transfer function.", "In this manner, we are the first to simultaneously integrate object-centric visual and language alignment within a single architecture for OVD.", "Our extensive experiments demonstrate the improved OVD capability of the proposed approach.", "On COCO and LVIS benchmarks, our method achieves absolute gains of 11.9 and 5.0 AP on novel and rare classes over the current SOTA methods.", "Further generalizability is demonstrated by our cross-dataset evaluations performed on COCO, OpenImages and Objects365, leading to consistent improvements compared to existing methods." ], [ "Related Work", "Zero-shot Object Detection (ZSD): This setting involves detecting novel class objects at inference, for which no visual examples are available during training.", "Zhu et al.", "[9] use semantic information with visual features to get proposals for both seen and unseen classes.", "Bensal et al.", "[10] show that learning a good separation between background and foreground is critical in ZSD and propose to use multiple latent classes for modeling background during training.", "Rahman et al.", "[11] propose a polarity loss to solve the ambiguity between background and unseen classes.", "DELO [12] focuses on generating good proposals for unseen classes by synthesizing visual features for unseen objects using a generative model.", "Gupta et al.", "[13] benefits from the contemporary cues in semantic and visual space ensuring better class separation for ZSD.", "Other works use additional learning signals, including unlabeled images from target domain [14] and raw textual descriptions from the internet [15].", "Although significant progress has been made on this topic [14], [15], [13], the inherent complexity of the task makes it challenging for the ZSD models to generalize well to unseen object classes.", "Weakly-supervised Object Detection (WSOD): In this setting, only image-level labels are used to approach object detection [16], [17], [18], [19], [20], or are used alongside the detection dataset to enlarge the detector vocabulary [21], [22], [23].", "Bilen et al.", "[24] proposed a weakly-supervised deep detection network (WSDNN) that uses off-the-shelf region proposals [25], [26] and computes objectness and recognition scores for each proposal using separate subnetworks.", "Cap2Det [27] operates in a similar setting and uses raw text captions to generate pseudo-labels to guide image-level supervision.", "Li et al.", "[28] uses segmentation-detection collaborative network (SDCN) for accurate detection under weakly-supervised setting using only image labels.", "PCL [29] proposes to cluster the spatially adjacent proposals and then assign image labels to each cluster.", "CASD [30] argues that the detectors trained only with image-level labels are prone to detect boxes around salient objects and propose feature attention along with self-distillation to address the issue.", "YOLO9000 [31] and DLWL [32] augments the detection training by assigning image-level labels to the max-score proposal.", "Detic [2] shows that using max-size proposal is an optimal choice for assigning image-level labels as it does not rely on the predictions of the network being optimized and provides better signals for the novel classes.", "We also operate in a similar WSOD setting and use high-quality object proposals from pretrained multi-modal ViT [8] to enlarge detector vocabulary and generalize towards novel object categories.", "Open-vocabulary Object Detection (OVD): In OVD, the objective is to detect target class objects not present in the training/base class vocabulary.", "A typical solution of the problem is to replace the classifier weights with text embeddings of the target vocabulary (e.g., GloVe [33], BERT [34], CLIP [3]).", "OVR-RCNN [1] uses BERT embeddings as classifier weights and proposes to use open-vocabulary captions to learn the vision-to-language mapping.", "It surpasses the ZSD approaches by a large margin.", "ViLD [6] uses pretrained CLIP [3] to distill knowledge into a two-stage object detector [35] and replaces the classifier weights with CLIP text embeddings obtained by ensembling multiple text prompts (e.g., a {category}, a photo of a {category}).", "Gao et al.", "[36] generate pseudo bounding-box labels using pretrained VL models for training open-vocabulary detector.", "All these methods use carefully designed manual prompts for generating text embeddings.", "DetPro [37] and PromptDet [38] replace these manual prompts with learnable tokens and achieve competitive results on novel/rare categories.", "However, in our work, we use fixed manual prompts and instead focus on improving the object-centric representations for open-vocabulary object detection." ], [ "Object-centric Open-Vocabulary Detection", "Here, we first present a brief overview of the proposed open-vocabulary detection (OVD) framework.", "As discussed earlier, existing OVD methods use different forms of weak supervision that employ image-centric representations, making them less suited for the end detection task.", "Our proposed method aims to bridge the gap between image and object-centric visual-language (VL) representations.", "We summarize the architectural overview of our method in Fig.", "REF .", "The proposed design has three main elements.", "1) Our region-based knowledge distillation (refer Sec.", "REF ) adapts image-centric language representations to be object-centric.", "A VL mapping learns to align the local region representations of the detector to the language representations by distilling the detector's region representations with region representations from a VL model (CLIP).", "2) Given weak image-level supervision, we use pseudo-labeling from pretrained multi-modal ViTs (refer Sec.", "REF ) to improve generalization of the detector to novel classes.", "3) For efficient combination of the above two proposed components, we condition the VL mapping learned during the weak supervision on the VL mapping learned with region-based distillation via a novel weight transfer function (refer Sec.", "REF ).", "Specifically, we follow a stage-wise learning strategy to first align the region and language embeddings using RKD, and use this distilled VL mapping for object-centric visual and language alignment in the subsequent stage.", "Figure: An overview of our proposed object-centric framework for OVD.", "We pair a two-stage object detector with fixed language embeddings from a pretrained visual-language (VL) model, CLIP .", "Our proposed pseudo-labeling strategy 𝒬 pseudo \\mathcal {Q}_{\\text{pseudo}} uses pretrained multi-modal ViTs to obtain high-quality class-agnostic and class-specific proposals.", "The overall pipeline follows a stage-wise learning strategy.", "First, we introduce region-based knowledge distillation (RKD) to adapt image-centric CLIP embeddings for local regions.", "Using the pretrained VL image encoder as a teacher model, we train the detector to induce point-wise and inter-embedding relationship alignment with our region embeddings using class-agnostic proposals from 𝒬 pseudo \\mathcal {Q}_{\\text{pseudo}}.", "Next, we utilize a weakly-supervised learning framework by combining instance-level labels from detection dataset and image-level labels from classification dataset which are visually grounded using 𝒬 pseudo \\mathcal {Q}_{\\text{pseudo}}.", "This weak-supervision helps in enlarging the class vocabulary and generalizes the detector to novel classes.", "To preserve the benefits of object-centric alignment in the language domain learned via RKD, we explicitly condition the image-level VL mapping W P W_P, on the learned region-level VL mapping W D W_D via a novel weight transfer function." ], [ "Detection Pipeline: Preliminaries", "In the open-vocabulary detection problem, we have access to an object detection dataset where the training set, $\\mathcal {D}_{\\text{det}}$ , comprises of samples from the set of base object categories, $\\mathcal {C}_{\\text{B}}$ .", "The images of $\\mathcal {D}_{\\text{det}}$ are exhaustively annotated with bounding-box labels and corresponding class labels $y_r \\in {\\mathcal {C}_{\\text{B}}}$ , for the different objects in the image.", "Given an image $I \\in \\mathbb {R}^{H \\times W \\times 3}$ , we design an open-vocabulary object detector to solve two subsequent problems: (1) effectively localize all objects in the image, (2) classify the detected region into one of the class label of $\\mathcal {C}_{\\text{test}}$ , which is provided by the user at test time.", "The categories during test time also include novel categories $\\mathcal {C}_{\\text{N}}$ beyond the closed set of base categories seen during the training phase, i.e., $\\mathcal {C}_{\\text{test}} = \\mathcal {C}_{\\text{B}} \\cup \\mathcal {C}_{\\text{N}}$ .", "We convert a generic two-stage object detector [35] to an open-vocabulary detector by replacing the learnable classifier head with fixed language embeddings, $\\mathcal {T}$ corresponding to the category names of $\\mathcal {C}_{\\text{test}}$ , that are obtained using a large-scale pretrained VL model.", "Following [6], we use the text embeddings from CLIP text encoder [3] for classification, where only the embeddings of $\\mathcal {C}_{\\text{B}}$ categories, $\\mathcal {T}_{\\mathcal {C}_{\\text{B}}}$ are used during training.", "Specifically, we generate the text embeddings offline, by processing the prompts corresponding to each category with a template of ‘a photo of {category}’ through the CLIP text encoder.", "The RoI [35] head computes pooled feature representations $\\phi (r)$ of the proposals $r$ generated by the region proposal network (RPN).", "These feature embeddings are projected to a common feature space shared by the text embedding $\\mathcal {T}$ using a linear layer $f(\\cdot )$ , which we represent as region embeddings, $\\mathcal {R} = f(\\phi (r)) \\in \\mathbb {R}^{D}$ .", "For classification, we compute the cosine similarity between the region embeddings and text embeddings to find the matching pairs.", "During training, the regions that do not match with any of the ground-truths are assigned to the background category represented by a fixed all zero embedding.", "We compute the cosine similarity by comparing each region to each base class, $\\mathcal {V} = {sim}(r, b) = \\cos \\big (\\mathcal {R}(r), \\mathcal {T}_b \\big ) \\ \\forall \\ b \\in \\mathcal {C}_{\\text{B}}$ .", "The classification loss is a softmax cross-entropy (CE) where the logits are the cosine similarity scores, $\\mathcal {L}_{cls} = \\frac{1}{N} \\sum _{r} \\mathcal {L}_{CE} \\left({{\\texttt {softmax}}}\\Big (\\frac{\\mathcal {V}}{\\tau } \\Big ), y_r\\right), \\ \\ y_r \\in \\mathcal {C}_{\\text{B}}.$ where $\\tau $ is the temperature, $N$ is the total number of proposals per image, and $r$ represents a single proposal with the ground-truth label $y_r$ ." ], [ "Region-based Knowledge Distillation", "In the OVD setting, we assume that $f(\\cdot )$ learns a VL mapping and aligns the output region embeddings of the detector with the corresponding CLIP text embeddings.", "However, the performance on novel categories is not comparable to what CLIP encoded embeddings would provide (refer Appendix  for details).", "We hypothesize that this performance gap is mainly due to two reasons, 1) the data that has been used for training CLIP model consists of scene-centric images, making it less suitable for region classification, e.g., in our case where object-centric tightly bounded proposals are used, 2) the zero-shot generalization ability of the pair-wise trained CLIP image and text embeddings cannot be fully utilized due to the mismatch between regions representations from CLIP image encoder and our detector.", "Based on these insights, we propose a region-based knowledge distillation (RKD).", "The proposed RKD uses distillation in the detection pipeline by distilling region embeddings from high-quality class-agnostic proposals ($\\tilde{r}$ ) obtained from a pretrained multi-modal ViT (MViT) [8].", "Note that we obtain both class-agnostic (used in RKD) and class-specific (refer Sec.", "REF ) object proposals using this pseudo-labeling process, which we refer to as $\\mathcal {Q}_{\\text{pseudo}}$ .", "This is possible via using intuitive text queries to interact with the MViT model that can locate generic objects and provides the corresponding set of candidate proposals.", "The queries can be generic or targeted, based on the task, e.g., ‘all objects’ to generate class-agnostic proposals, or ‘every dog’ for a specific class.", "For RKD, we compute class agnostic proposals on $\\mathcal {D}_{\\text{det}}$ using simple text query, ‘all objects’ and select top-K proposals (Fig.", "REF b).", "CLIP embeddings $\\mathcal {I}(\\tilde{r})$ are then computed offline using the CLIP image encoder $\\mathcal {I}(\\cdot )$ .", "With the detector region embeddings and the corresponding CLIP region representations, we propose to use two types of distillation losses to improve the alignment.", "(1) Point-wise embedding matching loss: The $\\mathcal {L}_1$ loss matches the individual region embeddings $\\mathcal {\\tilde{R}} = f(\\phi (\\tilde{r}))$ with the CLIP region representations $\\mathcal {I}(\\tilde{r})$ , $\\mathcal {L}_1 = \\frac{1}{K} \\sum _{\\tilde{r}} \\parallel \\mathcal {\\tilde{R}} - \\mathcal {I}(\\tilde{r}) \\parallel _{1}.$ Using this criteria, our visual encoder, along with the VL projection layer $f(\\cdot )$ , approximates the CLIP image encoder and consequently aligns our region embeddings with the CLIP text embeddings.", "(2) Inter-embedding relationship matching loss (IRM): It is a knowledge distillation based loss $\\mathcal {L}_{irm}$ that instills inter-embedding relationships within our region representations to be consistent to the CLIP region representations [39].", "Instilling such inter-embedding relations would be beneficial as we know that the teacher model $\\mathcal {I}(\\cdot )$ , and the student model (our detector), are different in nature with respect to their training methods (Fig.", "REF ).", "The IRM loss is defined on pairwise similarity matrices of the two different set of embeddings.", "Specifically, with the top-K proposals computed from $\\mathcal {Q}_{\\text{pseudo}}$ , we compose $K \\times K$ similarity matrices for $\\mathcal {I}(\\tilde{r})$ and $\\tilde{\\mathcal {R}}$ denoted by ${S_I}$ and ${S_R}$ respectively.", "Notably, these matrices are normalized by L2 norm applied row-wise.", "The IRM loss is a Frobenius norm $\\parallel \\cdot \\parallel _F$ , over the mean element-wise squared difference between ${S_{\\mathcal {I}}}$ and ${S_R}$ , ${S_R} = \\frac{\\tilde{\\mathcal {R}} \\cdot \\tilde{\\mathcal {R}}^{T}}{\\parallel \\tilde{\\mathcal {R}} \\cdot \\tilde{\\mathcal {R}}^{T} \\parallel _2}, \\ \\ \\ {S_{\\mathcal {I}}}= \\frac{\\mathcal {I}(\\tilde{r}) \\cdot \\mathcal {I}(\\tilde{r})^{T}}{\\parallel {\\mathcal {I}(\\tilde{r}) \\cdot \\mathcal {I}(\\tilde{r})^{T}} \\parallel _2}, \\nonumber \\\\\\mathcal {L}_{{irm}} = \\frac{1}{K^2} \\parallel {S_R} - {S_{\\mathcal {I}}} \\parallel _F^2.$ We weight the $\\mathcal {L}_{1}$ and $\\mathcal {L}_{irm}$ losses by factors $\\beta _1$ and $\\beta _2$ , respectively.", "Together with the standard two-stage detector losses; RPN loss ($\\mathcal {L}_{rpn}$ ), regression loss ($\\mathcal {L}_{reg}$ ) and classification loss ($\\mathcal {L}_{cls}$ ) [35], [40]; the overall training objective with RKD can be expressed as, $\\mathcal {L}_{RKD} = \\mathcal {L}_{rpn} + \\mathcal {L}_{reg} + \\mathcal {L}_{cls} + \\beta _1 \\ \\mathcal {L}_{1} + \\beta _2 \\ \\mathcal {L}_{irm}.$ Figure: Top-row: Similarity matrices computed on the CLIP (S I {S_I}) and detector (S R {S_R}) region embeddings for COCO novel classes.", "A subset of 100 randomly selected samples per category form a batch represented by a column are grouped together.", "Our region-based distillation enforces the similarity patterns in the RKD model to be closer to the teacher model, CLIP, indicated by the bright colors along diagonals.", "Bottom-row: t-SNE plots of CLIP and detector region embeddings on novel COCO categories.", "The CLIP aligned RKD and weight transferred detector embeddings shows improved separability among novel class features as compared to the supervised detector region embeddings (figure best viewed in-zoom)." ], [ "Image-level Supervision with Pseudo Box Labels", "In the open-vocabulary setting, a fundamental challenge is to generalize the detector to novel classes.", "However, due to the daunting task of densely locating all objects in natural scenes, the existing detection datasets are of relatively smaller magnitude compared to the classification datasets, which are easier to annotate.", "To this end, Zhou et al.", "[2] proposed to take advantage of large-scale image classification dataset during training to expand the detector's vocabulary.", "However, an important question is how to effectively associate the region proposals of novel objects with the corresponding labels.", "We note that the existing approach uses heuristics such as, selecting the whole image as a single box, or just the maximum sized box from the RPN, which can ignore potential objects (Fig.", "REF a).", "We propose a weakly-supervised method to generalize the detector to novel categories by using pseudo-box labels from pretrained MViT [8].", "We follow [2] to train the detector with a combination of detection and classification dataset.", "A batch of data is prepared by combining data from the detection dataset $\\mathcal {D}_{\\text{det}}$ that are exhaustively annotated with bounding-box and class labels, with data from a classification dataset $\\mathcal {D}_{\\text{cls}}$ that only contains image-level labels.", "With $\\mathcal {Q}_{\\text{pseudo}}$ , we obtain the pseudo-box labels on this classification dataset, which we use for image-level supervision (ILS).", "Specifically, consider a sample image $I \\in \\mathcal {D}_{\\text{cls}}$ , which has a total of $N$ ground-truth class labels, we generate object proposals offline with the use of MViT corresponding to these weak labels.", "Specifically, we construct $N$ class-specific text queries $\\lbrace t_n\\rbrace _{n=1}^N$ with template ‘every {category}’, and obtain $K$ proposals $\\lbrace \\tilde{r}_k\\rbrace _{k=1}^{K}$ and corresponding confidence scores $\\lbrace \\tilde{s}_k\\rbrace _{k=1}^{K} $ for each query, $[(\\tilde{r_1}, \\tilde{s_1}), (\\tilde{r_2}, \\tilde{s_2}), \\cdots (\\tilde{r_K}, \\tilde{s_K})] = \\mathcal {Q}_{\\text{pseudo}}(I, t_n); \\ \\ I\\in \\mathcal {D}_{\\text{cls}}, n \\in N .\\nonumber $ We select the top-1 proposal with the highest confidence score, as the pseudo-box label for a particular category.", "This gives us $N$ high-quality pseudo-box labels for each image, corresponding to its $N$ image-level category labels (Fig.", "REF a).", "We compute the region embeddings $\\mathcal {\\tilde{R}}$ for proposals $\\tilde{r}$ as, $\\mathcal {\\tilde{R}}_n = f(\\phi ({\\tilde{r}_{\\hat{k}}})), \\ \\ \\hat{k}= \\text{argmax}_k (\\tilde{s_k}).", "\\nonumber $ In the case of $\\mathcal {D}_{\\text{det}}$ , the training follows the standard two-stage RCNN training recipe.", "However, for $\\mathcal {D}_{\\text{cls}}$ , only the classification loss is updated.", "We call this pseudo-max score, $\\mathcal {L}_{pms}$ loss.", "$\\mathcal {L}_{pms} = \\frac{1}{N} \\sum _{n} BCE(\\mathcal {V}, y_{\\tilde{r}}), \\text{ where } \\mathcal {V} = \\cos \\big (\\mathcal {\\tilde{R}}_n, \\mathcal {T} \\big ).$ We weight $\\mathcal {L}_{pms}$ by a factor $\\alpha $ and the overall training objective with our ILS can be expressed as, $\\mathcal {L}_{ILS} = \\left\\lbrace \\begin{array}{lcr}\\mathcal {L}_{rpn} + \\mathcal {L}_{reg} + \\mathcal {L}_{cls}, & \\mbox{if} & I \\in \\mathcal {D}_{\\text{det}} \\\\\\alpha \\ \\mathcal {L}_{pms}, & \\mbox{if} & I \\in \\mathcal {D}_{\\text{cls}}.\\\\\\end{array}\\right.$ Figure: (a) Class-specific Proposals: A visual comparison of heuristic methods (left) used for visual grounding in image-level supervision , with our proposed method (right).", "Using heuristic approaches like selecting maximum sized box from the RPN can ignore local objects in the scene.", "In our method, we design class-specific text queries with known class labels for pseudo-labeling potential objects.", "(b) Class-agnostic Proposals: In region-based knowledge distillation (RKD), we induce better region-level alignment with fewer high-quality proposals from a generalized class-agnostic proposal generator .", "We compare top-K RPN proposals (left) with top-K multi-modal ViTs proposals used in a class-agnostic manner (right)." ], [ "Weight Transfer Function", "To combine the alignment from region-based distillation (Sec.", "REF ) with the benefits from weak supervision with pseudo-box labels (Sec.", "REF ), a naive approach would be to train the detector with a combination of losses: $\\mathcal {L}_{1}$ (REF ), $\\mathcal {L}_{irm}$ (REF ) and $\\mathcal {L}_{pms}$ (REF ).", "However, we demonstrate that a simple combination of the two approaches does not lead to complimentary benefits, instead they compete with each other (Table REF ).", "The additional supervision from pseudo-labels improves the generalization of the detector, while the region-based distillation works towards object-centric alignment in the language domain, thereby improving the overall performance of the detector.", "Our aim is to incorporate the benefits from the two approaches and preserve the object-centric alignment in the language domain.", "To this end, we use a weight transfer mechanism [41] from VL projection used in region-based distillation to the weak supervision by learning a weight transfer function, $\\mathcal {W_T(\\cdot )}$ .", "In other words, the VL projection function $f(\\cdot )$ used during the weak image-level supervision is explicitly conditioned on the mapping function used for alignment in the distillation process.", "This way, both the transformations are tied together to reinforce mutual representation capability and avoid any conflict in the learned function mapping.", "Let the weights of the projection layer in RKD and weak image-level supervision be represented as $W_D$ and $W_P$ respectively.", "The weight transfer operation is given by, $W_P = \\mathcal {W_T}(W_D) = \\Big ( W_{\\theta _2} \\ \\rho (W_{\\theta _1} \\ W_{D}) \\Big ) &; \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mathcal {W_T}\\colon \\ W_D \\rightarrow W_P.$ Here, $W_D$ is kept frozen and we design $\\mathcal {W_T}$ as a 2-layer MLP, $W_{\\theta _1}$ followed by $W_{\\theta _2}$ a with LeakyReLU ($\\rho $ ) activation with a negative slope of 0.1.", "Further, we use a skip connection across $W_P$ by projecting the original representations using a separate 2-layer MLP (Fig.", "REF ).", "The total loss here is a combination of $\\mathcal {L}_{RKD}$  (Eq.", "REF ) and $\\mathcal {L}_{ILS}$  (Eq.", "REF ) loss, given by, $\\mathcal {L} = \\mathcal {L}_{rpn} + \\mathcal {L}_{reg} + \\mathcal {L}_{cls} + \\beta _1 \\ \\mathcal {L}_{1} + \\beta _2 \\ \\mathcal {L}_{irm} + \\alpha \\ \\mathcal {L}_{pms}.$" ], [ "Datasets", "We conduct our experiments on COCO [42] and LVIS v1.0 [43] under OVD setting.", "For evaluation, we use the generalized ZSD setting where the classifier contains both base and novel categories.", "Table REF summarizes all the datasets used in our work.", "Following [2], [1], we use a subset of ImageNet-21K having 997 overlapping LVIS categories and COCO captions dataset for ILS in LVIS and COCO experiments respectively (refer Appendix.", "for more details).", "For the pseudo-labeling process $\\mathcal {Q}_{\\text{pseudo}}$ , we use the MViT pretrained on a Large-scale Modulated Detection (LMDet) dataset [8].", "We ensure that MViT pretraining dataset has no overlap with any of the evaluation datasets in our work.", "COCO OVD: We use COCO-2017 dataset for training and validation.", "We follow the ZS splits proposed in [10], in which 48 categories are selected as base and 17 are selected as novel classes.", "LVIS OVD: LVIS contains 1203 categories which are further split into frequent, common and rare categories.", "Inline with [6], [2], we combine the frequent and common categories to form base classes and keep all rare classes as novel, resulting in 866 base and 337 rare classes.", "Cross-transfer Datasets: To validate the adaptability of our method, we evaluate and compare results of our LVIS trained model on OpenImages[44] and Objects365 [45] and COCO [42] datasets." ], [ "Implementation details", "We conduct COCO experiments using Faster R-CNN [35] with ResNet-50 backbone.", "We train the supervised-base model on 48 base classes ($\\mathcal {C}_{\\text{B}}$ ) for 1x schedule ($\\sim $ 12 COCO epochs) and report box AP$_{50}$ .", "For RKD, we finetune this model for another 1x schedule using box-labels from $\\mathcal {C}_{\\text{B}}$ and class-agnostic proposals from the pretrained MViT [8].", "This model is further finetuned for 1x schedule with ILS and the associated weight transfer function using class labels from COCO captions and corresponding class-specific proposals from MViT.", "This sums to an overall 3x training schedule.", "For LVIS experiments, we use Mask R-CNN [40] with federated loss [46] and sigmoid cross-entropy, and report mask AP.", "For RKD and weight transfer, we use the same training schedules as of COCO and report the average over three runs.", "For comparison with Detic [2], we apply our proposed method on their strong CenterNetV2 [46] baseline under the same settings.", "It uses ImangeNet21K pretrained backbone with 4x schedule using large scale jittering (LSJ) [47] augmentations.", "All of our models are trained using 8 A100 GPUs with an approximate training time of 9 and 6 hours for 1x schedule of COCO and LVIS respectively.", "In our experiments, we use SGD optimizer with a weight decay of $1e^{-4}$ and a momentum of 0.9.", "We train for 1x schedule with batch size of 16 and initial learning rate of 0.02 which drops by a factor of 10 at the 8th and 11th epoch.", "We set temperature $\\tau $ to 50.", "Our longer schedules experiments use 100-1280 LSJ [47].", "We use $\\alpha $ of 0.1 to weight $\\mathcal {L}_{pms}$ .", "For computing CLIP embeddings we use the CLIP model ViT-B/32 [3], with input size of 224$\\times $ 224.", "We use the query ‘a photo of a {category}’ for to compute the text embeddings for the classifier.", "For distillation, we use top 5 proposals from the pretrained MViT [8] evaluated with generic query, `all objects', generating class-agnostic proposals.", "We refer to Appendix  for additional details of the approach we use to generate class-agnostic and class-specific proposals from MViT.", "In COCO experiments, we set weights $\\beta _1$ and $\\beta _2$ to 0.15.", "In LVIS, we set $\\beta _1$ to 0.15 and $\\beta _2$ to 0.25.", "We choose these values using a randomized hyperparameter-search on the corresponding held-out datasets.", "The 2-layer MLP in our weight transfer function has a hidden dim of 512, and a hidden dim of 1024 is used in the MLP skip connection across $W_P$ in Fig.", "REF (refer Appendix for more details)." ], [ "Our Approach: Main results", "Table REF shows the contribution of individual components in our proposed approach.", "Building on top of the supervised-base model, RKD shows an absolute gain of 19.9 and 1.2 AP for COCO novel and base classes respectively, indicating the adaptability of image-centric CLIP embeddings for local regions.", "With ILS, novel class AP improves by 32.5, demonstrating generalization to novel classes and thus enlarging the detector's vocabulary.", "Naively combining the two approaches shows improvement, but however struggles to maintain the gains from the individual components.", "In contrast, our weight transfer method suitably combines the complimentary benefits of both components (Fig.", "REF ), achieving 40.3 AP on novel classes while maintaining performance on base classes.", "[][h!]", "Table: NO_CAPTION Effect of individual components in our method.", "Our weight transfer method provides complimentary gains from RKD and ILS, achieving superior results as compared to naively adding both components.", "Open-vocabulary Detection - COCO: We compare our OVD results with previously established methods in Table REF .", "OVR-CNN learns a vision-to-language mapping with expensive pretraining.", "Detic uses ILS to improve detection on novel classes.", "We use a novel weight transfer function to perform object-centric VL alignment and achieve 54.1 AP on the base classes, surpassing OVR-CNN and Detic by 8.1 AP and 0.3 AP respectively.", "On novel classes our method achieves 40.3 AP, the highest novel AP achieved over all methods.", "In comparison with ViLD, which trains for 8x schedule (${\\sim }$ 96 epochs), our method with same schedule provides 56.7 base AP, lagging by 2.8.", "Table: OVD results on COCO.", "Here 𝒞 B \\mathcal {C}_{\\text{B}} and 𝒞 N \\mathcal {C}_{\\text{N}} represents the base and novel classes respectively.", "§§The results quoted from .", "†ViLD and our methods are trained for longer 8x schedule (shown in gray).", "‡We train detic for another 1x for a fair comparison with our method.", "For ViLD, we use their unified model that trains ViLD-text and ViLD-Image together.", "For Detic, we report their best model.On novel classes, we achieve 40.5 AP surpassing ViLD by a gain of 12.9.", "In contrast to ViLD design, our weight transfer function allows both RKD and ILS to provide complimentary gains without any competition among the two methods [6].", "Open-vocabulary Detection - LVIS: Table REF (left) compares our results with ViLD [6] on LVIS benchmark.", "With 3x training schedule ($\\sim $ 36 epochs) we perform reasonably well compared to ViLD 32x schedule ($\\sim $ 384 epochs), already surpassing the rare AP by 1.1 while having slightly lower performance on frequent class.", "Extending our model to 8x schedule fills the gap, surpassing ViLD by 0.8 in frequent and 5.0 AP in rare classes respectively.", "In Table REF (right), we compare our method with Detic using their strong LVIS baseline that uses CenterNetV2 network.", "Following similar settings, we finetune their box-supervised model using our weight transfer method and show improvements.", "Table: OVD results on LVIS.", "(Left): Comparison with prior work ViLD, using their unified model (ViLD-text + ViLD-Image), show improvement across novel and base categories.", "(Right): We show the comparison with Detic, by building on their strong LVIS baseline using CenterNetV2 detector Table: Cross-dataset evaluation.", "†The results evaluated using official implementation.Cross-dataset evaluation performance: We provide cross-dataset evaluation of our model in Table REF and compare with prior OVD works.", "ViLD-text[6] and Detic-base[2] are box-supervised baseline models for ViLD and Detic respectively.", "Our method builds on top of Detic-base and shows favourable results when directly transferred to cross-datasets without any dataset-specific finetuning.", "We use our method trained on LVIS and report AP$_{50}$ on COCO [42], OpenImages [44] and Objects365 [45].", "We refer to Appendix for qualitative results." ], [ "Analysis of RKD and ILS", "Effect of Region-based Knowledge Distillation (RKD): Table: Analysis on our region-based KD.We ablate the effect of $\\mathcal {L}_{1}$ (Eq.", "REF ) and $\\mathcal {L}_{irm}$ (Eq.", "REF ) RKD approach on COCO (Table REF ).", "The results show the importance of both loss functions, where using $\\mathcal {L}_{1}$ loss over base model with top-5 proposals from MViT [8] improves the base and novel class by 1.9 and 15.0 AP (row-1 vs 3).", "Using $\\mathcal {L}_{irm}$ in row-4 further improves the overall and novel class AP.", "To show the importance of using quality proposals in RKD, we compare the model trained with $\\mathcal {L}_1$ loss using top-5 RPN vs MViT proposals (row-2 vs 3).", "All the models in rows 2-4 are finetuned on the base model.", "Effect of Weak Image-level Supervision (ILS): Table: Analysis on our weak IL supervision.We compare different choices of ILS in Table REF .", "Our $\\mathcal {L}_{pms}$ loss (Eq.", "REF ) is compared with previously adopted ILS approaches [31], [32], [2] (rows 2-3).", "In row-4, we generate class-agnostic object proposals using ‘all objects' text query with multi-modal ViTs (MViTs) [8] and select max-size proposal for ILS.", "In row-5, our proposed ILS approach uses target specific ‘every {category}' text query with MViT and selects top-1 proposal for each ILS category.", "Our method (row-5) shows better performance compared to other alternatives.", "Additionally, we present all ablations on LVIS dataset in Appendix ." ], [ "Conclusion", "This paper develops a novel framework to leverage the representation and generalization capability of pre-trained multi-modal models towards improved open-vocabulary detection (OVD).", "Specifically, we note that the existing OVD methods use weak supervision modes that are more image-centric, rather than object-centric for the end detection task.", "We proposed a novel knowledge distillation approach together with object-level pseudo-labeling to promote region-wise alignment between visual and language representations.", "Our weight transfer module provide an integration mechanism to combine the benefits of knowledge distillation and object-level pseudo-labeling.", "We demonstrate encouraging results on a four popular OVD benchmarks, demonstrating sound generalization ability.", "Supplemental Material In this section, we provide additional information regarding, Implementation details (Appendix ) Qualitative Results (Appendix ) Zero-shot Region Classification (Appendix ) Additional Ablation Experiments (Appendix ) Pseudo-labeling using Multi-modal ViTs (Appendix ) Limitations (Appendix ) Potential Negative Social Impacts (Appendix ) Ethical Considerations (Appendix ) Datasets License Details (Appendix )" ], [ "Implementation Details", "We provide additional implementation details for our approach and datasets used in this work.", "We use standard Faster R-CNN [35] with ResNet-50 C4 backbone and Mask R-CNN [40] with ResNet-50 FPN backbone for COCO and LVIS experiments respectively.", "We use L2 normalization on the region and text embeddings before computing the RKD loss and final classification scores.", "We note that this normalization is helpful to stabilize the training.", "For ILS, we sample images from detection and classification datasets with a ratio of 1:4.", "Specifically, we use a batch size of 16 and 64 for detection and classification datasets, respectively.", "We will release our codes and pretrained models publicly to ensure reproducibility of our results.", "Datasets for ILS: We use COCO captions and ImageNet-21k [4] datasets for our proposed Image Level supervision (ILS) on COCO and LVIS datasets respectively.", "COCO captions dataset uses images from COCO detection dataset and provides five captions for each image.", "The words in a caption are compared heuristically, with every category name in the list of categories in COCO (base + novel).", "Using this method, we generate a list of positive categories for each image which is used as labels for ILS.", "We use ImageNet-21k [48] for LVIS experiments which is a large scale classification dataset containing approximately 14M images and 21K classes.", "We use categories from ImageNet-21k which overlaps with LVIS categories, resulting in a subset containing 997 categories.", "Cross-dataset evaluation: We provide cross-dataset evaluation of our LVIS trained model in Table REF .", "Following [2], [6], we use validation sets of OpenImages V5 containing $\\sim $ 41K images and Objects365 V2 containing $\\sim $ 80K images for evaluation.", "We report AP$_{50}$ for cross-data evaluation." ], [ "Zero-shot Region Classification", "We compare the zero-shot classification performance of open-vocabulary detector with pretrained CLIP [3] model on COCO validation dataset.", "Table REF shows the results where the top-1 classification accuracy is evaluated using the ground-truth object bounding boxes from COCO.", "The CLIP pretrained model shows better results for novel classes as compared to supervised-base model, indicating the strong generalization of the CLIP (row-1 vs 2).", "However the base class accuracy is higher for the supervised-base model as it is trained using COCO base classes.", "Further, using our region-based knowledge distillation (RKD) and novel weight transfer function improves the base and novel class performance, indicating the object-centric alignment in latent space.", "Table: Classification results on novel and base classes with boxes cropped from COCO validation dataset using ground truth annotations.", "The pretrained CLIP shows competitive novel class accuracy.", "Our proposed RKD and weight transfer approach further improve the performance." ], [ "Ablation Experiments on LVIS", "Effect of individual components: Table REF shows the contribution of individual components in our proposed approach on LVIS dataset.", "The baseline Mask-RCNN model (row-1) is trained on LVIS frequent and common classes using only the box-level supervision along with the zero-shot CLIP [3] classifier.", "The results indicate the effectiveness of our region-based distillation (RKD) which explicitly aligns the image-centric CLIP embeddings to the object-centric region embeddings.", "Our image-level supervision (ILS) which uses class-specific pseudo-labels from the pretrained multi-modal ViT [8], effectively enlarges the detector's vocabulary indicated by an increase of 4.8 AP over the base model for rare categories.", "Further, our proposed weight transfer scheme combines the strengths of the two methods and achieves better results on the common and frequent categories, while performing on par for the rare classes compared to naively combining the two approaches (row-4 vs 5).", "Table: Effect of individual components in our method on LVIS dataset.", "Using RKD provides improvements over the baseline in all metrics (row-1 vs 2).", "Using ILS mainly helps in improving rare class performance (row-1 vs 3).", "Simply combining two methods shows improvements over the baseline but struggles to retain the individual performances especially for common and frequent categories (row-4).", "Our weight transfer approach provides complimentary gains from RKD and ILS, achieving good results as compared to simply adding both components (row-4 vs 5).Table: Analysis on our RKD method on LVIS.Effect of RKD: Table REF shows the effect of different loss functions ($\\mathcal {L}_{1}$ and $\\mathcal {L}_{irm}$ in Eq.", "REF and Eq.", "REF respectively) used in our region-based knowledge distillation (RKD) on LVIS dataset.", "It shows the effectiveness of using proposals from multi-modal ViT (MViT) [8] as compared to RPN for region-level alignment (row-2 vs 3).", "Using high-quality MViT proposals provides significant gains compared to using RPN proposals.", "Further, using our inter-embedding relationship matching (IRM) loss along with $\\mathcal {L}_{1}$ loss provides an overall good trade-off between rare, common and frequent class AP.", "Table: Analysis on our weak ILS on LVIS.Effect of ILS: Table REF compares the different heuristics based approaches opted for image-level supervision (ILS) versus our method that utilizes class-specific proposals from the pretrained MViT on LVIS dataset.", "Selecting top-1 proposal from MViT using target specific specific queries such as ‘every {category}' provides optimal performance for rare classes." ], [ "Initialization for RKD Training", "We note that it is important to properly initialize the RKD training to gain its full advantages.", "Table REF shows that training RKD from scratch (row-2) results in lower base class AP.", "However, initializing the RKD training from the Supervised base model recovers this loss and provides improvements over the base model.", "This indicates that region-based alignment is sensitive to the distribution of the features and requires mature features for effectively distilling knowledge from pretrained CLIP model.", "This observation is same as in [49] where the contrastive clustering is enabled only on the mature features after a few training epochs for open-world object detection." ], [ "Additional Ablation Experiment", "Table REF shows the ablation on using a MLP skip connection across $\\mathcal {W_P}$ in Fig.", "REF .", "We add this skip connection to form a direct path for region classification using CLIP in ILS.", "This allows the weight transfer function to specifically focus on the residual signal in the ILS pathway.", "It improves the convergence and helps to attain better results in most cases on LVIS/COCO datasets.", "Table: The ablation on using MLP skip connection in Fig.", "." ], [ "Pseudo Labeling using Multi-modal ViTs", "In this section, we describe the process of generating class-agnostic and class-specific proposals using multi-modal ViTs (MViTs) [8], [50].", "We name this process as pseudo labeling $\\mathcal {Q}_{\\text{pseudo}}$.", "The MViT model is trained using aligned image text pairs and is capable of locating novel and base class objects using relevant human-intuitive text queries.", "For example, targeted text queries such as ‘every person' and ‘every elephant' can be used to locate all persons and all elephants in an image respectively (Fig.", "REF b).", "Maaz et al.", "[8] show that the MViTs encode the object-centric concepts using aligned image-caption pairs and are excellent class-agnostic object detectors.", "The authors designed text queries such as ‘all objects' and ‘all entities' and demonstrated state-of-the-art class-agnostic object detection results on multiple datasets across different domains.", "We use these MViTs to generate class-agnostic and class-specific object proposals for region-based knowledge distillation (RKD) and weak image-level supervision (ILS), respectively.", "Figure: (a) Class-agnostic Proposals: The figure shows the top 5 class-agnostic proposals obtained from the MViT  using ‘all objects' text query.", "As illustrated, these high-quality tightly bound object proposals provide rich local-semantics for RKD in our proposed pipeline.", "(b) Class-specific Proposals: The figure shows the class-specific proposals obtained from the MViT using ‘every <category name>' text queries.", "The left image in each pair shows all proposals while the corresponding right image shows the selected top 1 proposal per category for ILS.Class-agnostic proposals for RKD: We generate class-agnostic object proposals from the MViT [8] using ‘all objects' text query.", "The generated proposals are ranked using predicted objectness scores and the top 5 proposals per image are selected for RKD as shown in Fig.", "REF a.", "Next, the CLIP [3] image-encoder and our OVD detector is used to generate embeddings corresponding to these proposals which are then used for calculating the RKD loss in Eq.", "REF .", "To save the computation load and increase the training efficiency, we compute the class-agnostic proposals and the corresponding CLIP region embeddings offline and load them during training.", "Further for LVIS experiments, we use images from a subset of ImageNet-21K (consisting of 997 overlapping LVIS categories) for RKD as well.", "Class-specific proposals for ILS: We generate class-specific proposals from the MViT [8] using ‘every <category name>' text query.", "Given the $N$ category names present in an image, we use $N$ queries of format ‘every <category name>' to generate class-specific proposals followed by selecting top 1 proposal for each category.", "This provides us $N$ high-quality box proposals per image corresponding to $N$ categories present in the image.", "These proposals are used to effectively enhance the detector's vocabulary using ILS during training.", "Further, to maintain the training efficiency of our experiments, we compute these class-specific proposals offline and load them during training." ], [ "Limitations", "Our proposed OVD method encourages object centric visual-language (VL) alignment using a novel weight transfer method which combines benefits from RKD and ILS.", "Irrespective of the state-of-the-art results on novel/rare classes, there is still a significant gap between base and novel class performances (e.g.", "56.7 and 40.5 AP for COCO base and novel categories in Table REF , 29.1 and 21.1 Mask AP for LVIS frequent and rare categories in Table REF ).", "Further, the open-vocabulary capabilities of our model largely depends or are limited to the vocabulary of the pretrained CLIP [3] model, which is used as a teacher in our RKD pipeline." ], [ "Potential Negative Social Impacts", "The results of cross-dataset transfer evaluations show that the vocabulary of our detector is highly flexible and can be expanded to any number of categories, based on the downstream tasks and datasets.", "This poses a risk on how our OVD detector with a large vocabulary can be used in inappropriate ways in the community such as for large scale illegal video surveillance.", "Furthermore, OVD capabilities can be modulated for targeted detections instead of generic detections by tuning the classifier weights using specialized prompts.", "This could add biases in the detector and can lead to unfair predictions." ], [ "Ethical Considerations", "The OVD response to recognize object categories strongly depends on the image-text pretraining datasets used for the training of VL model (CLIP in our case).", "Thus, the source of these datasets can pose ethical issues.", "For example, datasets extracted from internet can contain racial and unethical bias and can modulate the ethical behaviour of the detector as well.", "Thus, before applying our OVD detector in a practical scenario, such biases of the pretraining/training datasets should be removed to have fairness and ethically correct results of the detector.", "Moreover, the detector vocabulary is flexible and it can be tuned to show racial biasness while detecting humans.", "For example, weights of the zero-shot classifier generated with specialized biased prompts could lead to biased and unethically targeted human detections (e.g., black vs white) which must be taken into consideration." ], [ "License Details", "Here we provide license details of the datasets used in our work, summarized in Table REF .", "COCO is available for non-commercial use under the Creative Commons Attribution 4.0 (CC BY 4.0) license.", "LVIS is based on the COCO dataset, and it is licensed under both CC BY 4.0 and the COCO license.", "ImageNet-21k is a publically available dataset available for research and non-commercial use.", "It is licensed under Creative Commons (CC), and its type is \"CC BY-NC\".", "We use a pretrained MViT model for proposal generation, which is trained on LMDet (Large scale Modulated Detection dataset).", "It uses Flicker30k, Visual Genome, and GQA datasets.", "The license type of Flicker30k is CC BY-NC.", "Visual Genome and GQA both have the same license type CC BY 4.0.", "For cross-datasets evaluation, Objects365 and OpenImages are used, which are licensed under Creative Commons Attribution 4.0.", "Annotations of OpenImages are licensed by Google LLC under Creative Commons Attribution 2.0." ] ]
2207.03482
[ [ "Flow Synthesis Based Visual Servoing Frameworks for Monocular Obstacle\n Avoidance Amidst High-Rises" ], [ "Abstract We propose a novel flow synthesis based visual servoing framework enabling long-range obstacle avoidance for Micro Air Vehicles (MAV) flying amongst tall skyscrapers.", "Recent deep learning based frameworks use optical flow to do high-precision visual servoing.", "In this paper, we explore the question: can we design a surrogate flow for these high-precision visual-servoing methods, which leads to obstacle avoidance?", "We revisit the concept of saliency for identifying high-rise structures in/close to the line of attack amongst other competing skyscrapers and buildings as a collision obstacle.", "A synthesised flow is used to displace the salient object segmentation mask.", "This flow is so computed that the visual servoing controller maneuvers the MAV safely around the obstacle.", "In this approach, we use a multi-step Cross-Entropy Method (CEM) based servo control to achieve flow convergence, resulting in obstacle avoidance.", "We use this novel pipeline to successfully and persistently maneuver high-rises and reach the goal in simulated and photo-realistic real-world scenes.", "We conduct extensive experimentation and compare our approach with optical flow and short-range depth-based obstacle avoidance methods to demonstrate the proposed framework's merit.", "Additional Visualisation can be found at https://sites.google.com/view/monocular-obstacle/home" ], [ "Introduction", "As UAVs gain popularity, interest has grown in the development of robust navigation methods.", "For the urban setting, navigation through high-rises has become a problem of imminent interest.", "Low-cost off-the-shelf RGBD sensors are noisy [1] and detect obstacles in the typical stereo depth range, not efficient for long-range obstacle detection and maneuvering.", "In this context, we propose a novel flow synthesis based deep visual servoing framework for monocular obstacle avoidance, wherein by obstacles, we indicate urban high-rises/skyscrapers.", "Independently and otherwise, flow-based approaches to monocular avoidance [2], [3] have been popular in literature; nonetheless, their outcomes have tended to be unreliable due to often inaccurate flow estimates and controllers based on such flow.", "Instead, by synthesizing a desired flow, we preempt the challenges associated with the noisy flow, and the multi-step CEM-based control achieves superior goal convergence than a single-step reactive control based on flow.", "Recent Visual Servoing frameworks [4], [5], [6] have leveraged deep optical flow to do high precision visual servoing.", "In this paper, we ask the question: If these frameworks can utilise optical flow to reach a goal image with high precision, can we design a surrogate flow for the obstacle that can push the obstacle out of the scene?", "Given an urban high-rise scene as a monocular image input, our novel pipeline segments the Building of Concern (BoC) through a modified saliency segmentation network.", "This network segments the BoC amongst other competing high-rises.", "The BoC is typically the building that will collide first with the MAV if it continues along its current trajectory.", "The pixels contained in the segmentation mask of the BoC are subject to a flow that quintessentially is the pixel error that needs to be minimized by the multi-step visual servoing based CEM controller.", "Fig.", "REF showcases a high-level overview of our idea.", "Figure: Our algorithm explicitly segments the building to generate a surrogate desired flow for the high precision Flow-Based visual servoing algorithms.", "The 'pink' obstacle flow pushes the building leftward, and hence a rightward drone trajectory is generated by the servoing framework and vice-versa for 'blue' flow, which pushes the building rightward.To the best of the authors' knowledge, this approach is novel yet uncomplicated.", "It parts ways from prior flow-based methods that synthesized a control corresponding to the observed flow of an obstacle.", "Rather, in this case, a flow error is synthesized that is devoid of any noise and easily reduced by a controller.", "Moreover, unlike data-driven frameworks that directly learn and map pixels to control often through black-box neural networks, the proposed CEM controller offers sufficient interpretability of what it strives to accomplish.", "The paper contributes in the following ways: To the best of the authors' knowledge, this is the first approach to tackling the problem of navigation amongst urban high-rises with a single monocular camera as the essential sensing modality.", "We revisit the concept of salient object detection as collision obstacle detection.", "This deep model segments only the Building of Concern (BoC), thereby ensuring the relevance of our servoing based avoidance mechanism for an identified obstacle.", "In contrast to previous methods of synthesizing control from estimated flow, which is often noisy, we propose noiseless surrogate flow synthesis and demonstrate its efficacy for monocular obstacle avoidance.", "The paper benchmarks tangible performance gain in comparison with popular flow-based obstacle avoidance methods [3], and short-range depth-based obstacle avoidance pipelines [7] on both photorealistic and real-world scenes imported into the AirSim Simulation Environment [8]." ], [ "Related Work", "Monocular Obstacle Avoidance: While the literature is sufficiently populated with methods relating to obstacle avoidance with depth sensors such as RGBD cameras [7] or stereoscopy [9], [10], prior art relating to monocular obstacle avoidance has been sparse.", "In [11] a Monocular Direct SLAM framework popularly called LSD-SLAM is used to construct occupancy maps over which navigation and exploration is performed.", "However, monocular SLAM based occupancy maps are typically noisy that entail uncertainty-based obstacle avoidance formulation along the lines of [1] as an overhead.", "Moreover, as the authors mentioned, maps often need to be regenerated by intermittent hovering maneuvers that can prove costly as UAVs and MAVs work with limited onboard power and compute budget.", "In [12] a similar SLAM-based approach using feature-based ORB SLAM is presented.", "The sparse map is interpreted in terms of planes, and an RRT-based trajectory planner is presented.", "Once again, such sparse point cloud methods are subject to uncertainty-based overheads and need extra modules beyond SLAM to reinterpret the sparse point clouds.", "Optical Flow and Feature Based Avoidance Methods: In contrast to methods that explicitly build a map, there exists a class of methods that make use of optical flow cues to compute obstacle avoidance behavior.", "Flow balancing is a popular method of obstacle avoidance where the difference in flow between the horizontal and vertical sections of the image are balanced by inducing an appropriate yaw and height changes in the vehicle [2], [3].", "Elsewhere flow is used to compute depth or time to collision that guides the control performance [13].", "In [13] correlation-based motion detectors were also tried out as a mechanism for obstacle detection taking cues from biology.", "Whereas in [14] growth in feature sizes were the cue to detect the presence of an obstacle, and heuristic controls were used for avoidance.", "A primary challenge with optical flow-based control lies in the flow estimates being noisy.", "Often successive flow estimates can cause conflicting control actions such as back and forth motion primarily due to inaccuracies or noisy flow estimates.", "Figure: Here we summarize our obstacle avoidance pipeline.", "We use the saliency-based detection module BASNet to get the obstacle mask ℳ(t)\\mathcal {M}(t).", "We then combine it with the radial flow ℛ\\mathcal {R} to get the desired flow ℱ * (t)\\mathcal {F}^*(t).", "We use the synthesized desired flow ℱ * (t)\\mathcal {F}^*(t) along with a flow-based Visual Servoing pipeline to generate a control command using Cross Entropy Method.Data Driven Methods: Data-driven methods for monocular obstacle avoidance have also been popular.", "Some paradigms use black-box neural nets to learn disparity [15] followed by a traditional controller to avoid the perceived obstacles over the depth maps; other methods directly map and infer pixels to controls [16], [17].", "Depth maps learned from monocular vision are noisy and need further retraining in new scenes and entail that the controller resorts to uncertainty-based formulations for obstacle avoidance such as in [1].", "Visual Servoing: This is an active camera control technique that derives a control command to achieve the pose for a target view, with respect to the current viewpoint [18].", "Literature abounds with a variety of servoing frameworks [19], [20] while recently data-driven methods have gained in popularity [4], [5], [6].", "This paper uses a multi-step CEM-based servoing controller that outputs control and minimizes synthesized flow errors.", "Contrast with Previous Methods: Unlike the mapping formulations, the present approach operates directly on images than on the estimates of the 3D world from the images, thereby circumventing problems associated with noisy maps and the sparseness of the reconstruction.", "Synthesizing the control corresponding to flow estimated over images has been the prevalent practice, such as in optical flow-based and depth-based avoidance methods [2], [3], [15].", "However, the present method synthesizes the flow and generates control that achieves this synthesized flow.", "Since the synthesized flow is noiseless as obtained by construction, the CEM Servoing Controller that minimizes this synthesized flow error is untroubled largely by uncertainty in the flow estimates.", "Further, unlike the reactive approaches to flow-based obstacle avoidance [2] the DeepMPC controller introduced in [5] is a controller over a time horizon that provides increased robustness.", "We show vastly superior performance over reactive optical flow-based avoidance controller under the duress of flow noise.", "Further, this paper is believably the first attempt to explore the role of visual servoing in an obstacle avoidance setting, whereas all previous approaches have used servoing to achieve a goal-reaching behavior (reaching the pose corresponding to the desired image)" ], [ "Method", "Given a camera and a GPS sensor, our aim is to generate the optimal velocity control commands to reach a desired goal GPS location $g^*$ in a collision-free manner.", "Modern Deep Learning-based servoing methods [4], [5] use optical flow for high precision visual servoing.", "We ask the question: if these methods can use optical flow to reach a precise goal image, can we design a surrogate flow for obstacles that can push them out of the image?", "Referring to Fig.REF , we first utilise the saliency-based architecture described in [21] to identify the obstacle or building in the line of attack of the MAV.", "We then introduce the Radial Flow in Sec.", "REF , which can be combined with the obstacle mask obtained using BASNet to provide a desired flow for the servoing frameworks introduced in [4], [5].", "We use MPC+CEM [5] to optimise for velocity commands; however, this only serves as the design choice." ], [ "Saliency Based Obstacle Avoidance", "Saliency detection methods [22] aim to mimic the human visual attention mechanism to identify objects more attentive than the surroundings.", "Saliency detection has a range of applications, such as foreground-background segmentation, object tracking, identification, reidentification, detection, especially in cluttered environments.", "A relevant yet sparsely explored application is collision obstacle detection during navigation.", "A MAV traversing in an urban environment must navigate many structurally diverse high-rises at different times during the flight.", "To identify any such 'Building of Concern', we leverage the power of Salient Object Detection models and find them to perform well in detecting obstacles even at long distances.", "Further, our downstream avoidance task utilises the generated saliency map to synthesise flow and generate velocity commands that can steer the drone clear from a collision.", "It can also efficiently work with imperfect object segmentation masks; hence, salient object detection methods justify our case for detecting objects of collision.", "The salient object detection method we are using is BASNet[21], a U-Net[23] like supervised encoder-decoder based method with an additional residual refinement module which improves the coarse prediction, extending it to the object boundary.", "We train BASNet to identify obstacles that appear in the line of sight of collision.", "Typical semantic segmentation methods [24] would segment all competing object instances in the entire field of view without any criteria such as the object's location.", "Depth-based methods like [7] are not reliable beyond stereo camera ranges and would detect large obstacles like buildings very late as compared to when they are visible.", "Notably, Salient object detection methods handle the task well for long-range collision obstacle detection from a single RGB image.", "We train BASNet on a curated dataset of MAV FPV images of skyscrapers from around the world and model buildings in UrbanScene3D and AirSim (building 99 environment).", "We perform data augmentation to increase the dataset size to 4000, with transformations such as ColorJitter, MotionBlur, RandomBrightness, RandomRotate and follow the training methodology of the original authors, using the same hyperparameters." ], [ "Radial Flow", "Recent visual servoing methods [4], [5] calculate the flow between the current image and desired image to do high precision visual servoing.", "However, during the monocular obstacle avoidance, the desired image is unavailable.", "Here we explain how we can design a surrogate flow, which can help us robustly avoid obstacles.", "Consider an image of dimension $(H \\times W)$ .", "For any pixel coordinate $(i, j)$ , we want to assign a flow value that pushes the pixel out of the image.", "This turns out to be a flow in a radially outward direction from the image center.", "This “radial\" flow $R$ and for any pixel coordinate can be described using : $\\textstyle R[i, j] &= [i - \\frac{H}{2}, \\Lambda *(j - \\frac{W}{2})]$ Where $\\Lambda $ is a parameter and is set to 10.", "$\\Lambda $ remains same for each scene in our benchmark.", "We normalize the Radial Flow as: $\\textstyle N[i, j] &= \\sqrt{(i - \\frac{H}{2})^2 + (j - \\frac{W}{2})^2)} \\\\\\textstyle R[i, j] &= \\frac{R[i, j]}{N[i,j]}$ We use the mask $M(t)$ obtained using our saliency based obstacle avoidance module to get the obstacle mask.", "This mask $M(t)$ is then multiplied with Normalised Radial Flow $R$ to get a desired flow for servoing as described in the next section.", "$\\textstyle \\mathcal {F}^*(t) &= R \\cdot M(t)$ The motivation behind the design of the radial flow can be explained using Fig.REF .", "We want the desired flow as calculated in Eq.", "REF to be biased to take the MAV away from the obstacle while also taking the MAV forward.", "However, we cannot assign the pixel-wise optical flow in a radially outward direction since it is the same as when the drone moves forward.", "Hence, we scale the component along the (left, right) direction, forcing the optimisation to produce a velocity that takes it away from the building while also moving forward.", "Additionally, if the center patch of the image is covered with an obstacle, we damp the forward velocity by a factor $\\mu $ .", "This provides a robust surrogate desired flow $\\mathcal {F}^*(t)=R*\\mathcal {M}(t)$ which can be used with a flow-based servoing algorithm to reliably generate optimal velocity commands as explained in section REF .", "Figure: Visualisation of flow vectors according to how the drone moves : (a), (b) show the flow obtained when the drone moves forward and right, respectively.", "(c) shows the flow when the drone moves forward and right simultaneously.", "(d) shows a visualisation of the synthesized radial flow using Eq.. We leverage the fact that when (d) is combined with the obstacle mask ℳ(t)\\mathcal {M}(t), it forces the CEM optimisation to generate a velocity that moves the obstacle out of the scene.", "So if the mask shows that the building is dominantly in the right direction, the servoing generates the velocity, which takes it in (forward+left) direction, towards the side of the building, and vice-versa if the mask is dominantly on the left side." ], [ "Avoidance Using Visual Servoing Framework", "We now explain our algorithmic pipeline, which combines the saliency mask predicted using BASNet and the radial flow to give a velocity control command to avoid the obstacles.", "The salient obstacle mask is obtained using the BASNet module.", "The mask at time $t$ is combined with Radial Flow $R$ to give a desired flow $F^*(t)$ for the current scene using Eq.", "REF .", "We then calculate the 2-View Flowdepth $Z_T$ introduced in [4] which considers the optical flow between time step $t-1$ and $t$ as a proxy estimate for depth.", "We use the flow network FlowNet2[25] to construct the interaction matrix $L(Z_t)$ .", "$L(Z_t) = \\begin{bmatrix*}[r]\\frac{-1}{Z_t} & 0 & \\frac{x}{Z_t} & xy & -(1+{x}^2) & y \\\\0 & \\frac{-1}{Z_t} & \\frac{y}{Z_t} & 1+{y}^2 & -xy & -x\\end{bmatrix*}.$ The interaction matrix $L(Z_t)$ maps the drone velocity to the velocity of pixels (or optical flow) on the image plane.", "We generate the flow predictions using the interaction matrix $L(Z_t)$ .", "$\\begin{split}\\widehat{\\mathcal {F}}(\\mathbf {v}_{t+1:t+T}) = \\sum _{k=1}^{T}[L(Z_{t})\\mathbf {v}_{t+k}]\\end{split}$ We can then optimize the velocity for the desired flow: $\\mathcal {L}_{flow} = \\Vert \\widehat{\\mathcal {F}}(\\widehat{\\mathbf {v}_{t}}) - \\mathcal {F}^{*}\\Vert = \\Vert [L(Z_{t})\\widehat{\\mathbf {v}}]- \\mathcal {F}^*\\Vert $ Optimising the loss function in Eq.", "REF gives the desired velocity for the avoidance of obstacle.", "We use Cross-Entropy Method (CEM) to solve for the desired velocity.", "At each time step we sample 'N' velocities from a Gaussian distribution ${{g}(\\mu , \\sigma ^{2})}$ and calculate the loss (Eq.", "REF ) for each of them.", "The losses are then sorted, and the top 'K' velocities (with least losses) are used to update the parameters of Gaussian distribution ${{g}(\\mu , \\sigma ^{2})}$ .", "We run CEM for several iterations before convergence is achieved.", "This is analogous to the MPC+CEM described in [5]." ], [ "Overall Pipeline and Implementation Details", "We now describe our overall goal-reaching pipeline.", "We divide our algorithm into two parts, a) Goal Reaching mode: Here, we use GPS location to orient and move towards the goal location $g^*$ .", "b) Obstacle avoidance mode: We give the velocity commands obtained (as explained in Section REF ) to the drone.", "We use the obstacle mask $\\mathcal {M}(t)$ obtained using our Saliency Based Obstacle detection algorithm to decide whether to follow Obstacle avoidance mode or not.", "If the center patch of the image is covered with an obstacle, we damp the forward velocity by a factor $\\mu $ .", "Our method can be used to do 6-DOF obstacle avoidance, however, we used a 4-DoF ($x, y, z, yaw$ ) while presenting the results in this paper." ], [ "Flow Balancing with Radial flow", "In this section, we describe how we can use the desired radial flow $\\mathcal {F}$ obtained using Eq.", "REF to improve the performance of the Flow Balancing Method [2].", "Flow balancing has been classically used to avoid obstacles using a monocular camera.", "It uses the flow between the current image and the previous image to get a yaw angle for the drone.", "The yaw rate is given using the equation : $\\mathcal {\\dot{\\theta }} =(\\frac{\\omega _L - \\omega _R}{\\omega _L + \\omega _R}) \\times k$ Here $\\omega _L$ and $\\omega _R$ represent the sum of magnitudes in the left and right half of the Optical flow between the current image $I_t$ and previous image $I_{t-1}$ and k is a constant which is an adjustable parameter.", "The reasoning behind the approach is that closer obstacles have higher flows, so we should move to the direction that has less flow.", "However, this method fails to work on our benchmark because, in long-range avoidance, the optical flow fails to provide enough information to give an optimal velocity command.", "To have a fair comparison against flow-balancing, we use the formulated radial flow $F^*(t)$ described in (Eq.", "REF ) to get the yaw rate.", "Our Radial Flow-based Flow balancing gives much better results when compared to naive flow balancing.", "Figure: Qualitative results on the benchmark for selected scenes: Our method successfully avoids the obstacles on all the 10 scenes in the building 99 environment and 5 of 6 in the UrbanScene3D environments.", "Here we show images for 5 intermittent poses captured during the obstacle avoidance for selected scenes in the simulation benchmark.", "In Building 1, we are able to segment and avoid the building even though there are several buildings behind it.", "Building 4 covers a large part of the image, but our algorithm is able to move in the correct direction.", "In Building 10, the MAV can navigate the narrow path between the two buildings successfully.", "We also present results on certain challenging configurations from the real-world dataset UrbanScene3D.The lines in the trajectory column indicate the following: (RED) –>> Our Method, (BLUE) –>> Flow Balancing, (GREEN) –>> Flow Balancing with Radial Flow.", "The goal and start positions are marked with a red star and a yellow circle, respectively." ], [ "Simulation Benchmark", "Our simulation benchmark consists of 10 photo-realistic scenes from the building 99 environment and 6 real-world reconstruction scenes from the UrbanScene3D dataset [26].", "These scenes span across a varying difficulty level depending on the number of buildings on the path to the goal location, the amount of free space available to the MAV while navigating between buildings, and the angle at which it approaches the building.", "The urban scene 3D dataset contains real-world urban environments for cities like Shanghai, New York, etc., with realistic textures.", "This benchmark further tests the generalization of our methods by putting them in near-realistic scenes.", "We have chosen to conduct our experiments on the real scenes: Sci-Art, Residence, and Campus.", "Our method is able to navigate to the provided goal GPS location in all the scenes and is able to make intelligent choices required to navigate in scenes with space constraints.", "Fig.", "REF shows the obstacle avoidance sequence for several scenes on our benchmark.", "We show that our algorithm is able to navigate to the goal location in scenes across varying levels of difficulty.", "For Building 1, BASNet segments only the building in the line of attack amongst multiple competing buildings.", "For Building 4 and Building 10, the algorithm has to navigate in a narrow passage between two skyscrapers and make several key decisions to reach the respective desired goal locations.", "We are also able to navigate in scenes where a large part of the scene is covered with an obstacle.", "In Building 5, our visual servoing optimisation pipeline is able to find the correct direction to move in order to avoid the obstacle.", "On the photo-realistic UrbanScene3D dataset, our algorithm generalises and gives similar performance.", "Our algorithm is able to work on 15 out of 16 scenes.", "Compared to our proposed novel algorithm, naive Flow Balancing works on only 2 scenes out of 16 scenes from our simulation benchmark.", "It is clear that the optical flow between the current and previous image is not able to provide any meaningful direction for the MAV to follow.", "However, our radial-flow guided flow-balancing algorithm shows a stark improvement compared to naive flow balancing.", "Still, it fails to generalise and can successfully reach goal location only on 5 out of 16 scenes in our simulation benchmark.", "The success-rate results are summarised in Table REF , and it is clear that our method shows robust performance and is able to generalise across the Building-99 and UrbanScene3D datasets.", "Table REF shows the minimum distance of the camera from any building.", "Our algorithm consistently performs better than other algorithms and maintains the highest safe distance from the obstacles.", "This is also evident from Fig.", "REF where we plot the trace of trajectories obtained using different algorithms.", "Table: Success Rate Comparison: Our method generalises across different scenes in the Building-99 and UrbanScene3D dataset to give a consistent controller performance." ], [ "Salient Object Detection as Collision Obstacle Detection", "In this section, we present qualitative results of BASNet in detecting the BoC during flight.", "Fig.", "REF a)-d) show detection of obstacles in/close to the line of attack of the drone, except in Fig.", "REF d) owing to its small size and distance from the drone.", "However, it must be noted that this is not a collision scenario, and such behavior is actually desirable so the drone does not remain in a perpetual state of detecting and avoiding an obstacle and can continue moving on the original trajectory.", "Fig.", "REF a) and c) also show how imperfect/partial segmentation masks do not hinder downstream flow synthesis and subsequent avoidance, as the velocity computed from the desired flow still moves the drone away from the obstacle.", "Figure: Obstacle segmentation masks for scenes in the AirSim Building 99 environment computed from BASNetFigure: Evaluation of stereo depth avoidance for a large structure.", "The MAV is enclosed in a white bounding box, position highlighted with a green circle.", "a) shows the simulation setup in Gazebo consisting of a building with a 30mx30m front and an Iris drone approaching it from 30m distance and 10m altitude.", "b) shows that the detection of a building (voxel map) starts when the MAV is 9m away from the front and ends up being 1m close till the whole front is detected.", "The planner fails to find a trajectory to avoid the obstacle; subsequently, the MAV is halted." ], [ "Stereo Depth based Avoidance", "We compare the performance of our obstacle detection and avoidance method with a stereo depth based avoidance method.", "We use Fast-Planner[7] a kinodynamic path searching method using a 3D point cloud as perception information on a MAV in a simulated environment in Gazebo and approaching a building 30m wide and 60m tall.", "This method works on voxelised occupancy mapping for environment representation, making it agnostic to the scene/object texture.", "Fig.", "REF shows the experiment setup, obstacle occupancy map, and MAV trajectory.", "We highlight the result in terms of obstacle detection range and subsequent failure for avoidance.", "Stereo range based depth cameras work reliably in the range of 7-10m, which is essentially how close the MAV gets to the building before seeing it as an obstacle.", "At this point, for a typically wide building, it becomes unfeasible to view around the sides and plan any path for avoidance, resulting in the drone halting in its track with no further course of action other than the nondeductive exploration of a vast structure to find a safe passage.", "This becomes inefficient for MAVs by wasting limited flight endurance." ], [ "Conclusion", "This paper presented a robust monocular obstacle avoidance algorithm deployed for UAVs navigating amongst urban environments.", "Our framework leverages saliency-based segmentation to identify the Building of Concern.", "We then combine the obtained segmentation mask with the proposed radial flow to get the desired flow, which can be fed into the high precision flow-based visual servoing methods to avoid the obstacles successfully.", "We conduct extensive experiments to demonstrate the merit of our approach.", "Our work shows how a visual servoing framework and a saliency-based obstacle detection can be combined to avoid obstacles robustly.", "In future work, we aim to explore saliency and visual servoing based frameworks to avoid obstacles at a higher velocity." ] ]
2207.03557
[ [ "G2L: A Geometric Approach for Generating Pseudo-labels that Improve\n Transfer Learning" ], [ "Abstract Transfer learning is a deep-learning technique that ameliorates the problem of learning when human-annotated labels are expensive and limited.", "In place of such labels, it uses instead the previously trained weights from a well-chosen source model as the initial weights for the training of a base model for a new target dataset.", "We demonstrate a novel but general technique for automatically creating such source models.", "We generate pseudo-labels according to an efficient and extensible algorithm that is based on a classical result from the geometry of high dimensions, the Cayley-Menger determinant.", "This G2L (``geometry to label'') method incrementally builds up pseudo-labels using a greedy computation of hypervolume content.", "We demonstrate that the method is tunable with respect to expected accuracy, which can be forecast by an information-theoretic measure of dataset similarity (divergence) between source and target.", "The results of 280 experiments show that this mechanical technique generates base models that have similar or better transferability compared to a baseline of models trained on extensively human-annotated ImageNet1K labels, yielding an overall error decrease of 0.43\\%, and an error decrease in 4 out of 5 divergent datasets tested." ], [ "Introduction", "In the field of supervised deep learning, one often ends up with very little labeled training data.", "To alleviate this problem, a well-known technique used is Transfer Learning [36].", "It uses trained weights from a source model as the initial weights for training a target dataset.", "A well-chosen source with a large amount of labeled data leads to significant improvement in accuracy for the target.", "The task we address is the scenario of providing an efficient machine learning service for clients whose data, particularly image data, varies greatly from the “natural scenes” that typically populate the image databases on which off-the-shelf classifiers are trained.", "Typically, these clients will present a rather small dataset taken from specialized environments, often industrial, which have been captured under unforeseen circumstances and parameters.", "Our method creates pseudo-labels for this data by determining their geometric relationships to the feature space of existing labeled data.", "In this work, we therefore develop a content-aware labeling technique.", "First, we take data points such as images, and compute labels for them by calculating distances of these data points from a set of named anchor data points representing known and labeled categories, like $animal$ , $plant$ , $tool$ , etc.", "Second, pseudo-labels are then constructed for the incoming data points based on these distances, or, more accurately, based on “contents”, which is the high-dimensional generalization of distances, calculated using a geometric approach.", "Each pseudo-label then consists of a sequence of semantically descriptive names: for example, $\\langle tool, plant \\rangle $ , could be the pseudo-label for data from a previously unseen category like $rake$ .", "Finally, we train a source model using these automatically labeled labels.", "We show that when an incoming dataset has a high divergence from the data on which a classifier has been trained, this method helps to focus transfer learning on the most relevant aspects of the training data, and the resultant model is more accurate than the original “vanilla” classifier.", "Applying this G2L (“geometry to label”) method, we evaluate workloads from the Visual Decathlon  [38] and other labeled datasets, comparing how well our pseudo-labeling scheme performs in generating sources for transfer learning against ImageNet1K data  [31] using standard human annotated labels.", "We show that our purely mechanical approach wins in many circumstances and that we can specify those circumstances.", "Our contributions are: (1) a disciplined extensible algorithm to create semantically interpretable pseudo-labels, derived from methods in the geometry of high dimensions; (2) an analysis of the effects of algorithm hyperparameters on the amount and quality of these pseudo-labels; (3) a characterization of the relationship of source-to-target similarity (divergence) vs. pseudo-label variety, optimal transfer learning rate, and final accuracy; and (4) a discussion and visualizations of 280 experiments on a wide variety of transfer tasks." ], [ "Related Work", "Several well-established approaches attempt to assign labels to unlabeled images automatically.", "Some utilize feature clusters to predict labels [17] or augment image data with linguistic constraints from sources such as WordNet [2], [18].", "They address tasks by pretraining models using larger unlabeled data-sets.", "Pretraining approaches have also improved results when attempting a target task with a limited amount of accurately labeled training data [24], by using weakly labeled data, such as social media hashtags, which are very plentiful.", "Again, however, effectiveness only appears to grow as the log of the image count.", "Other approaches use generative models such as GANs [30] to explore and refine category boundaries between data clusters, which exploit the rich statistical structures of both real and generated examples, sometimes augmented with labels or linguistic constraints.", "These automatic approaches use the structures present in large unlabeled datasets to extend the expressivity of known labels and augment the raw size of training sets.", "More broadly, various approaches attempt to learn a representation of a class of data and later use that representation in service of a target task.", "For example, [16] clustered images in an embedding space and developed a meta-learner to find classifications which distinguished various clusters within this embedding.", "Other approaches to mapping the feature space have used autoencoders [5].", "Knowledge distillation techniques [26] teach a student an accurate but more compact feature space, similar in spirit to the work presented here.", "The above existing literature attempts to find hybrid approaches that find productive ways to leverage machine-learned distributions of examples to find new ways of characterizing unlabeled data.", "The current work presents a novel approach in this domain." ], [ "General Approach", "In this section we give an overview of the conceptual flow of our method, discuss the feature space of one application of it to a computer vision task, and suggest a real-world analogy that illustrates some of the semantic considerations behind its central geometry-based algorithm." ], [ "Labeling Method", "Generating rich pseudo-labels from models trained on distributionally similar data involves a trade off between an expressive, long label, and a generalizable, short label.", "Longer labels carry more information about similarity between previous models and the target image, and differences between the previously trained models could be critical for adequately labeling new examples.", "For example, an incoming set of data including pictures of household objects might be well described by combining the labels of “tool, fabric, furniture.” However, domains that possess substantial differences from previous data might be better defined by the magnitude and direction of such a difference.", "For example, a “flower” dataset would share some features with “plant,” but it is perhaps better defined by statements such as “flowers are very unlike furniture”.", "In other, ambiguous cases, negative features may be necessary to distinguish between overlapping cases: a suit of armor might have similarities with the body shapes of people but could be contrasted with these categories by its dissimilarity with “sport,” a category otherwise close to “person.”" ], [ "Labeling Example", "We illustrate our method using a specific case study involving images, and with source datasets created by vertically partitioning ImageNet22K [7] along these distinct subtrees: $animal$ , $building$ , $fabric$ , $food$ , $fruit$ , $fungus$ , $furniture$ , $garment$ , $music$ , $nature$ , $person$ , $plant$ , $sport$ , $tool$ , $tree$ , $weapon$ , illustrated later in Fig.", "REF .", "These subtrees vary in their number of images (from 103K images for $weapon$ to 2,783K images for $animal$ ) and in their number of classes (from 138 for $weapon$ to 4,040K for $plant$ ).", "These 16 subtrees were used since they were easy to partition from Imagenet22K, but our method could also be used with any other labeled data ontology.", "We represent each such source dataset by a single average feature vector.", "This study generates this vector from the second to last layer of a reference VGG16 [32] model trained on ImageNet1K, with the average taken over 25% of all the images in the dataset.", "To label a new image, we first calculate its own feature vector, then compute its Euclidean distance from each of the representatives of the datasets.", "Together with other geometric computations in this high dimensional space, these distance measures are then used for a full pseudo-labeling process described in Sec.", "." ], [ "An Analogy", "Our G2L approach for pseudo-labeling an image can be understood as being similar to the “Blind Men and the Elephant” parable, where blind men, who have never learned about an elephant, try to categorize an elephant just by touching it, then relating it to something that they already know.", "Their categorizations of Elephant then include Fan (ear), Rope (tail), Snake (trunk), Spear (tusk), Tree (leg), and Wall (flank).", "Basically, by touching and feeling an elephant, the blind men are measuring its closeness to things known by them.", "Our approach also measures the closeness of an unknown image, in feature space, to existing known categories and then generates a pseudo-label for it." ], [ "Analogy Extended", "However, additionally, our work extends this analogy in three critical ways.", "First, we also compare unknown images to the existing categories that are farthest from them: the elephant is “not Feather”.", "Second, we observe a strong predictive relationship between (a) the measurement of the similarity of unknown imagery to existing categories, and (b) the computation of the number of pseudo-labels necessary to derive good transfer performance: the elephant could also use “Curtain, Leather, Bark, Mud”.", "Third, we also observe a strong predictive relationship between measured similarity and optimal learning rate: the elephant is most easily described starting from “Manatee”.", "The significance and computational advantages of these extensions are detailed in Sec.", "and .", "In this section we present the main algorithm for generating semantically meaningful geometric pseudo-labels.", "We first start with five motivating principles and some necessary math background.", "Then we walk through the algorithm, and discuss some of its properties and results." ], [ "Motivation", "Pseudo-labels for a target dataset can be generated by using a large labeled dataset organized within a semantic hierarchy such as ImageNet22K, and an off-the-shelf robust classifier, such as VGG16 trained on ImageNet1K.", "(The robust classifier need not be trained on the same labeled dataset.)", "Our algorithm exploits these two tools in a way that promotes five desirable properties for pseudo-labels, to ensure that the pseudo-labels have an easily understandable meaning, and that their computation are reasonable efficient.", "First, the pseudo-labels should be easily interpretable to humans.", "As an example, we can start by partitioning ImageNet22K into 16 non-intersecting sets, each of which carries the name of an object category, such as in Sec. .", "Second, the pseudo-labels should therefore also follow a simple grammar.", "We refer to the 16 subsets that comprise the above partition as the source subsets.", "Then, a pseudo-label for an incoming target data item can be defined as the concatenation of some number of source subset names, such as the sequence $\\langle person, music, tool \\rangle $ .", "This produces an informative natural language description of the incoming target data item.", "Third, to make comparisons possible, these pseudo-labels should be geometrically interpretable within the space of feature vectors.", "Distances between pseudo-labels can be defined by various metrics: $L^1$ (city-block), $L^2$ (Euclidean), $\\sqrt{JS}$ (the square root of Jenson-Shannon divergence [12]), or others.", "This can be tricky; the feature vector spaces used in machine learning are difficult to visualize, and such high-dimensional spaces generate geometric paradoxes even at relatively low dimensions [14].", "For example, each feature vector of a dataset is very likely to be on the convex hull of that dataset's representation in that space [37].", "Nevertheless, by methods like that of barycentric coordinates [19], particular “anchor” vectors can be used to represent regions within these spaces.", "Fourth, the computation of pseudo-labels should leverage known efficient geometric algorithms that localize incoming data.", "Metrics defined over these spaces can be used to partition the space into cells that form equivalence classes of locations based on individual anchor points (“1st-order Voronoi diagram”).", "These locations are characterizable by geometric properties such as “the nearest point to this cell is $P$ ”; see Fig.", "REF (a).", "These cells can be efficiently determined [10].", "Further, these metrics can also partition the space into cells that form equivalence classes of locations based on sets of points (“$n^{th}$ -order Voronoi diagram”), characterizable by geometric properties such as “the $n$ -nearest points to this cell are $\\lbrace P_1, P_2, \\dots , P_n\\rbrace $ ; see Fig.", "REF (b).", "The extreme case for $N$ points is the $(N{-}1)^{th}$ -order partition (“farthest-point Voronoi diagram”); see Fig.", "REF (c).", "However, general farthest-point algorithms are provably hard, and the only efficient algorithms are approximate [29].", "Fifth, the concepts of lengths and distances in these spaces should be further generalized to all measures of higher-dimensional “content”, following the progression of polytopes [4], as point, length, area, volume, hypervolume, etc., to arbitrarily high dimension.", "Elegant algorithms exist for computing such content, in particular, the Cayley-Menger determinant [34].", "Figure: Voronoi tessellation regions in two-dimensions generated by the N=4N{=}4 points shown in red.", "Colored regions depict equivalence classes of points that share: (a) “closest point” (policy c̰c̰), (b) “top 2 closest points”, (c) “farthest point” (policy f̰f̰), (d) “closest and farthest points” (policy C̰C̰ or F̰F̰), the intersection of (a) and (c).", "Policies are explained in Sec.", "Therefore, pulling these observations together, we seek to devise a method that composes a small number of semantically-named anchor vectors derived from the source datasets, into a sequence that defines location descriptions for target data items, based on the generalization of closest and farthest (Voronoi) distances into minimal and maximal (Cayley-Menger) contents.", "These location descriptions then become the pseudo-labels for the incoming data." ], [ "Mathematical Foundation", "Some necessary mathematical preliminaries now follow.", "Cayley-Menger determinant.", "Our method depends on the generalization of the concept of a single distance between a target and a single source, to that of the content of a $d$ -dimensional simplex defined by the target and $d$ sources.", "The computation of content is a well-studied algorithm based on the Cayley-Menger determinant (“$CM$ ”).", "The determinant itself generalizes several earlier classic algorithms, including the familiar Heron formula for the area of a triangle, and the less familiar Piero formula for computing the volume of a tetrahedron.", "Cayley-Menger computation.", "For an $d$ -simplex, composed of $d{+}1$ anchors, the math to compute content $C_d$ proceeds in three steps.", "The derivation of these steps is tedious, and is explained in [3].", "First, it forms $\\mathbf {M}_d$ , a particular symmetric $(d{+}2){\\times }(d{+}2)$ matrix.", "It incorporates a symmetric submatrix that expresses the squares of all pair-wise distances, that is, $D_{i,j}{=}distance(i,j)^2$ .", "$\\mathbf {M}_d=\\begin{bmatrix}0 & 1 & 1 & 1 & 1 & \\cdots & 1 \\\\1 & 0 & D_{0,1} & D_{0,2} & D_{0,3} & \\cdots & D_{0,n} \\\\1 & D_{1,0} & 0 & D_{1,2} & D_{1,3} & \\cdots & D_{1,n} \\\\1 & D_{2,0} & D_{2,1} & 0 & D_{2,3} & \\cdots & D_{2,n} \\\\1 & D_{3,0} & D_{3,1} & D_{3,2} & 0 & \\cdots & D_{3,n} \\\\\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\1 & D_{n,0} & D_{n,1} & D_{n,2} & D_{n,3} & \\cdots & 0 \\\\\\end{bmatrix}$ Second, it computes the coefficient $a_d=(-1)^{d+1} 2^d (d!", ")^2$ , which records the effects that various matrix operations have had on the determinant of $\\mathbf {M}_d$ , during its simplification from more complex geometric volume computations into its above form.", "This coefficient also defines the integer sequence A055546 at [33], where it has an imaginative interpretation involving roller coasters.", "Third, it solves for the value of $C_d$ implicitly expressed by $a_d C^2_d=\\det {\\mathbf {M}_d}$ .", "The complexity of computing the determinant, by the usual and reasonably efficient method of LU decomposition, is $\\mathcal {O}(d^3)$ .", "No simpler approach involving the reuse of previously computed subdeterminants appears feasible, as the determinant has been proven to be irreducible for dimensions greater than 3 [11].", "Nevertheless, in the context of our overall machine learning problem, this cost has proven to be negligible with respect to training costs." ], [ "Pseudo-label Creation", "Now, we give an overview of the algorithm.", "An incoming data point is compared at each step against a collection of named category anchor points.", "The name of the anchor point that minimizes (or maximizes, depending on a policy) its distance to the incoming point is chosen as the first component of an evolving sequence of names.", "Thereafter, the process repeats, and at each step the sequence is extended with the name of the anchor point that best extremizes the content—the area, volume, hypervolume, etc.— of the evolving polytope formed by these selected points.", "After a stopping criteria, this sequence gives the pseudo-label.", "The full G2L algorithm is summarized in Algorithm REF .", "The algorithm requires a number of hyperparameters that are set by experiment.", "An example is shown for each of these choices, in the pseudocode of the precondition (“Require”) preamble.", "These examples use image classification as the domain, and record the exact configuration that is used in the experiments reported in Fig.", "REF .", "Most of the parameters (and hyperparameters) are straightforward.", "The indicator $Layer$ is the choice of a particular layer within the data representation of $f$ , usually but not necessarily the second-last.", "The function $Met$ is the choice of a distance function that has been derived from an inner product, as required for the derivation of $CM$ .", "The remaining three hyperparameters are more complex.", "Parameters needing explanation.", "The method $Aggr$ is the choice of an aggregation method that represents a set of $Layer$ vectors in a sparser form.", "This can be as trivial as using a single mean vector, or as more elaborate as using a set of representatives derived from clustering methods.", "For example, as Fig.", "REF suggests, the source $food$ is probably adequately represented by a single aggregate vector, but the source $fruit$ probably is better represented by a pair of aggregate vectors $fruit_{plant}$ and $fruit_{food}$ .", "Figure: The 16 sources, using 1NN under Euclidean metric to define “closest”.", "Arrow a→ba{\\rightarrow }b means “aa includes a bb subcluster”; thickness is subcluster weight.The integer $d_{max}$ determines the number of dimensions to be explored using $CM$ during the creation of the output pseudo-label name sequences.", "It also bounds the length of the pseudo-label name sequence $pls_i$ , by $d_{max} \\le \\vert pls_i \\vert \\le 2d_{max}$ .", "The exact length of $pls_i$ , which is constant over a given execution of the complete algorithm, is determined by $Pol$ .", "Extremizing policies.", "The extrema decision sequence $Pol$ , and its summarizing notation, are best explained by a walkthrough of the algorithm.", "At $d{=}1$ , the algorithm considers the length of the line (the 1-simplex) formed from the target data item $t_i$ , and a representative vector $sour_{j,k}$ from the source representation $Sour_j$ .", "(If $Aggr$ was a simple mean, then each $Sour_j$ will be a singleton set.)", "Each $sour_{j,k}$ is examined, and the content (here, the length), computed by $CM$ , is recorded in $cont_{i,j,k}$ .", "Now we can choose the first dimension's extremizing pseudo-label sequence $pls_1$ for $t_i$ , from one of four short sequences: (1) the source name of the closest vector, if $Pol$ starts with $\\langle c \\rangle $ , as shown in Fig.", "REF (a); or (2) the source name of farthest vector, if $Pol$ starts with $\\langle f \\rangle $ , as shown in Fig.", "REF (c); or (3) the source name of the closest vector followed by the source name of the farthest vector, if $Pol$ starts with $\\langle C \\rangle $ , as shown in Fig.", "REF (d); or (4) the source name of the farthest vector followed by the source name of the closest vector, if $Pol$ starts with $\\langle F \\rangle $ , as shown in Fig.", "REF (d) again (the repeat is expected in this case).", "For example, if $Pol{=}\\langle c \\rangle $ , one possible pseudo-label $pls_1$ for a particular $t_i$ could be the sequence $\\langle fruit_{food} \\rangle $ .", "Whereas, if $Pol{=}\\langle F \\rangle $ , it could be $\\langle fungus$ , $fruit_{food} \\rangle $ instead.", "The four choices of extremizing policy at any dimension are therefore captured by the quaternary alphabet $\\lbrace c, f, C, F\\rbrace $ .", "And in particular, the policy $\\langle C \\rangle $ is the special case already explored in prior work [8], which forms pseudo-labels consisting of the names of $\\langle closest, farthest \\rangle $ pairs.", "Proceeding to $d{=}2$ , the algorithm considers the areas, computed by $CM$ , of the triangle (2-simplex) formed by the target data item $t_i$ , a representative vector $sour_{j,k}$ , and a single prior extremizing vector, chosen according to the first dimension's policy.", "This single vector would be the length-minimizing vector if the policy had been $\\langle c \\rangle $ or $\\langle C \\rangle $ ; or the length-maximizing vector if the policy had been $\\langle f \\rangle $ or $\\langle F \\rangle $ .", "At this point, again we can efficiently choose one of four short sequences that capture the names of the area-extremizing sources for this dimension's pseudo-label, which we then append to the evolving sequence $pls_i$ .", "At two dimensions, there are therefore 16 total policies, ranging from $\\langle c, c \\rangle $ to $\\langle F, F \\rangle $ .", "These 16 policies create 4 different name sequences of length 2, 8 different name sequences of length 3, and 4 different name sequences of length 4.", "By establishing and solving straightforward recurrence relations that are similar to those describing Pascal's triangle, we find that the number of possible name sequences at dimension $d$ with length $l$ is given by $P(d,l){=}2^l \\cdot \\binom{d}{l{-}d}$ , and that the total possible sequences at dimension $d$ is given by $\\sum _{l} P(d,l){=}4^d$ .", "The algorithm proceeds likewise for each higher dimension, up to $d_{max}$ , by first building simplices that extend the prior dimension's simplex, and then selecting names according to this higher dimension's policy.", "[hbt!]", "G2L pseudo-label algorithm, in pseudocode [1] $Tar \\leftarrow $ target dataset of data items, $Tar{=}\\cup t_i$ $Full \\leftarrow $ semantically-partionable labeled dataset e.g.", "ImageNet22K $Part \\leftarrow $ partition of $Full$ , $Part{=}\\cup \\lbrace P_j\\rbrace $ $f \\leftarrow $ classifier e.g.", "VGG16 on Imagenet1K $Layer \\leftarrow $ feature vector layer e.g.", "second-last $Aggr \\leftarrow $ feature vector aggregator e.g.", "mean $Met \\leftarrow $ feature vector metric e.g.", "Euclidean $d_{max} \\leftarrow $ max simplex dimension e.g.", "4 $Pol \\leftarrow $ extrema decision sequence e.g.", "$$ pseudo-label sequence $pls_i$ for each $t_i$ INITIALIZATION each data item $t_i \\in Tar$ represent $t_i$ by $vert_i \\leftarrow $ $Layer$ vector of $t_i$ within $f$ each subset $P_j \\in Part$ represent $P_j$ by set $Sour_j{=}\\cup \\lbrace sour_{j,k}\\rbrace \\leftarrow $ aggregation of $Layer$ vectors of $P_j$ within $f$ , using $Aggr$ PROCESS each $t_i \\in Tar$ $pls_i \\leftarrow \\langle \\rangle $ $d{=}1$ to $d_{max}$ each $Sour_j$ each $sour_{j,k} \\in Sour_j$ $X_{i,j,k} \\leftarrow $ simplex, using $vert_i, pls_i, sour_{j,k}$ each vertex pair in $X_{i,j,k}$ compute edge distance, using $Met$ $cont_{i,j,k} \\leftarrow $ content of $X_{i,j,k}$ , using $CM$ $e_d \\leftarrow $ $argextreme_{j,k}$ of $cont_{i,j,k}$ , using $Pol$ $names_d \\leftarrow $ names of $sour_{j,k}$ , using $\\langle e_d \\rangle $ $pls_i \\leftarrow pls_i$ concat $names_d$" ], [ "Empirical Properties of Pseudo-labels", "In what follows, we have used VGG16 trained on Imagenet1K as classifier $f$ , and we have partitioned the $Full$ ImageNet22K dataset into a $Part$ collection of the 16 non-intersecting semantic subsets given in Sec.", "REF .", "We will now refer to policies without angle brackets or commas; $\\langle C, f, f, f \\rangle $ becomes simply $$ .", "Outliers and tractability.", "We note that a few of our 16 mean vectors, particularly $fungus$ , $sport$ , and $furniture$ , repeatedly show up as outlier vertices in the polytopes under construction, as suggested by their relations shown in Fig.", "REF .", "They therefore tend to occur early in the output pseudo-label sequences.", "We observe empirically, as suggested in [11], that the search for these maximal and minimal simplicies has to be done exhaustively.", "For example, through exhaustive search on our test dataset, we find that the 1-simplex with minimal content is $\\langle plant, tree\\rangle $ , yet the minimal 2-simplex is $\\langle fabric, garment, person \\rangle $ , and then the minimal 3-simplex is $\\langle fabric$ , $garment$ , $plant$ , $tree \\rangle $ .", "Impact of the first policy.", "We illustrate an important statistical property of the resulting pseudo-labelings in the heatmap of Fig.", "REF , which displays the entropy of pseudo-labels generated for $Part$ under the 256 possible policies of order $d{=}4$ .", "Figure: Heatmap showing the entropy of the pseudo-labels generated for d=4d{=}4 policies applied to downsampled ImageNet22K data.", "The diagram is fractal.What is visually apparent is that the variability depends primarily on the initial policy in the sequence.", "The left half of the diagram shows policies that begin with $c̰$ or $C̰$ (first policy decision is “closest”); the right half begins with $f̰$ .or $F̰$ (first policy decision is “farthest”).", "The top half shows policies that begin with $c̰$ or $f̰$ , which produce a single source name at $d{=}1$ ; the bottom half begins with $C̰$ or $F̰$ , which produce two source names at $d{=}1$ .", "The diagram shows that diversity of pseudo-labels increases roughly from upper left, $$ , to lower right, $$ .", "The general progression is $c̰, f̰, C̰, F̰$ , in the pattern of $\\begin{bmatrix} c̰ & f̰ \\\\ C̰ & F̰ \\end{bmatrix}$ .", "Closer examination shows that the diagram is fractal, and that it follows this pattern at all scales.", "The major anomaly is $$ , at position $(1,9)$ , colored strongly dark, which can be interpreted as “apply the 4-simplex consisting of the worst outlier source and its three nearest neighbors”—which for many different inputs is exactly the same, giving minimal entropy.", "In contrast, the rightmost column of the diagram, which traces policies from $$ at $(1,16)$ to $$ at $(16,16)$ , shows a monotonic and nearly linear increase in entropy to the global maximum.", "Growth and predictability.", "The G2L algorithm creates pseudo-label sequences that are combinations of source names.", "The number of potential sequences of any given length can be very large, for example, $\\binom{16}{4}{=}1,820$ , $\\binom{16}{6}{=}8,008$ and $\\binom{16}{8}{=}12,870$ .", "However, when the algorithm was applied to the ImageNet22K training dataset, the number of unique sequences were typically much less, as shown in Fig.", "REF .", "Even $$ of length 8 in the prolific extreme lower right corner produced only 2,643.", "This slower growth reflects the non-random correlated semantic clustering of the data." ], [ "Experimental Evaluation", "Using our geometric technique, we created a number of pseudo-labeled datasets for the images in ImageNet1K, as shown in Fig.", "REF .", "We then trained ResNet27 models using six pseudo-labeled datasets, creating base models for further transfer learning.", "These six were: $\\texttt {cccc}$ , $\\texttt {Cfff}$ , $\\texttt {Ffff}$ , $\\texttt {FCCC}$ , $\\texttt {cccccc}$ , $\\texttt {cfffff}$ .", "These six were chosen because they represent a broad spectrum of unique label counts, they explore policies starting with different initial extremizing decisions, and they show the effect of increased dimensions.", "We observe that the CFA policy introduced in [8] can be obtained from our approach; by our notational convention it would be, simply, policy $\\texttt {C}$ .", "Figure: (a) Base models and (b) Target datasetsWe also created a baseline model using the vanilla ImageNet1K dataset of images and human-annotated labels.", "This model attained a top-1 average accuracy of 66.6%, which is suitable for a ResNet27 model.", "The same hyperparameters and training setup was used for all the pseudo-labeling models.", "We chose ResNet27 because residual networks [15] are considered state of the art, and ResNet27 is easy to train, while still being large enough for our datasets.", "To evaluate the usefulness of these base models, we focused on eight target workloads taken from the Visual Domain Decathlon [38] and other fine-grained visual classification tasks.", "The choice of target datasets was made to have sufficient diversity in terms of number of labels, number of images, number of images per label, and divergence with respect to ImageNet1K.", "Divergence is here computed by first normalizing the representative vectors of each dataset so that their components (which are all non-negative) sum to 1, then applying the usual Kullback–Leibler divergence formula [22].", "Since we want to compare the performance of pseudo-labeling with respect to vanilla ImageNet1K, we selected only those datasets whose transfer learning accuracy under vanilla were not close to 1.", "This ensures that the comparison with vanilla is not trivial (otherwise, all policies also have accuracies very close to 1).", "The target workloads evaluated included Aircraft [25], CIFAR100 [21], Describable Textures (DTD) [6], Omniglot [23], Street View House Number (SVHN) [27], UCF101 [20], Oxford VGG Flowers [28], and Caltech-UCSD Birds (CUBS) [35].", "These span a range of divergence from ImageNet1K, and possess different labels and dataset sizes, as shown in Fig.", "REF .", "These target workloads were then learned from pseudo-labeled and human-annotated (vanilla ImageNet1K) source models over five different learning rates.", "The inner layers were set to learning rates ranging over 0.001, 0.005, 0.010, 0.015, and 0.020, and the last layer was set to a learning rate ten times that.", "Each source model was trained using Caffe1 and SGD for 900K iterations, with a step size of 300K iterations, an initial learning rate of 0.01, and weight decay of 0.1.", "The target models were trained with identical network architecture but with a training method with one-tenth of iterations (90K) and step size (30K).", "A fixed random seed was used throughout all training.", "Thus a total of 280 transfer learning experiments (8 targets $\\times $ 7 policies $\\times $ 5 learning rates), with same set of hyperparameters, were conducted and compared for top-1 accuracy." ], [ "Observations", "Overall Accuracy.", "Figs.", "REF and REF compare the transfer learning top-1 accuracy of vanilla with our pseudo-labeling approach.", "Fig.", "REF shows that the divergence measure closely tracks the accuracy of both vanilla and G2L, with a correlation coefficient of 0.7.", "For datasets whose divergence is above 0.6, G2L beats vanilla four times out of five.", "Our approach outperforms vanilla in four high divergent cases (Omniglot, SVHN, Oxford, and UCF101).", "For the other four cases (DTD, Cubs, CIFAR100, and Aircraft) where it underperforms, its performance was very similar to vanilla.", "Taken together, the average of all eight winners, compared to the average of just vanilla, decreases the overall error rate by 0.43%.", "Figure: (a) Divergence versus accuracy for 8 targets.", "(b) All 280 experiments.Divergence vs.", "Number of Labels.", "As divergence from Imagenet1K increased for the targets, pseudo-labeling schemes with a lesser number of unique labels performed better; see Fig.", "REF (a).", "In cases where pseudo-labeling schemes did better than the vanilla ImagNet1K labels, the label count was 40 to 160, in contrast to 1,000 in the vanilla.", "Divergence vs. Learning Rates.", "As divergence from Imagenet1k increased for the targets, higher learning rates were better for the pseudo-labeling schemes; see Fig.", "REF (b).", "In the cases where pseudo-labeling schemes did better than vanilla ImageNet1K, learning rates of 0.015 did best.", "Even for the vanilla ImageNet1K labels, a higher learning rate was generally better for high divergent workloads.", "Number of Labels vs. Learning Rates.", "Pseudo-labeling schemes with fewer unique labels performed better at higher learning rates; see Fig.", "REF (c).", "In cases where pseudo-labeling schemes did better than vanilla, high learning rates like 0.015 were best.", "Figure: (a) Divergence vs.", "Number of Labels.", "(b) Divergence vs. Learning Rate.", "(c) Number of Labels vs. Learning Rate." ], [ "Limitations and Future Work", "Theory Limitations.", "The initial partitioning of the ImageNet22K dataset into 16 categories is currently heuristic.", "It known that the use of the second-last vector as a representative is not helpful if incoming data is significantly different on the signal level [13], and that vectors of other layers are.", "The aggregation method to aggregate the representative vectors of the sources relies on a simple mean.", "An efficient method for selecting the probable best policy is currently unexplored.", "Theory Future Work.", "Initial partitioning could be cast as an (approximate) optimization problem, as can the selection of representative processing levels.", "Simple aggregation by a mean vector can be augmented by a multi-vector clustering technique.", "Inter-cluster distance measures like “energy distance” [1] would then be used instead of Euclidean.", "More powerful intermediate $n^{th}$ order Voronoi methods should be explored; see Fig.", "REF (b).", "The space of policies over many problems should be examined in order to determine heuristics about policy behaviors, particularly those regarding the likelihood of accuracy improvement.", "The G2L methods needs to be applied to data modalities other than vision.", "Practice Limitations.", "How well a given initial partition spans the available high-dimensional feature space has not been quantified.", "Learning rates for each layer in these experiments were identical.", "Policies were applied according to a fixed script.", "The value of $d_{max}$ is a completely free meta-parameter.", "Practice Future Work.", "A method for optimizing the initial partition according to some figure of merit would be useful, particularly since human-annotated labels are quite sensitive to fine details such as color, texture, shape, etc.", "A more appropriate selection of learning rates for different layers of network can significantly improve accuracy during fine-tuning [9]; a thorough exploration, coordinated with knowledge of specific policy strengths and weaknesses, should be attempted.", "Search techniques other than greedy should be explored, that more intelligently select the best policy to execute next, and that stop without requiring a value for $d_{max}$ .", "These G2L approach, particularly since it generates descriptive pseudo-labels, can and should be applied to the problem of data augmentation, and to human label error correction." ], [ "Conclusion", "We have demonstrated a novel but general technique for automatically creating descriptive content-aware pseudo-labels for transfer learning, based on a classical result from the geometry of high dimensions.", "It computes hypervolume content as a heuristic for label quality.", "The expected accuracy is approximately forecast by the divergence between the representative vectors of sources and targets.", "Our 280 experiments show that this mechanical technique generates base models that have similar or better transferability, compared to usual methods.", "This G2L approach overall increases classification accuracy, particularly on incoming datasets that are unusually divergent from the human-labeled training set.", "Our novel geometry-based pseudo-labeling method can be extended in several theoretic and practical ways, and can be applied to other data modalities." ] ]
2207.03554
[ [ "Emergent Higher-Symmetry Protected Topological Orders in the Confined\n Phase of $U(1)$ Gauge Theory" ], [ "Abstract We consider compact $U^\\kappa(1)$ gauge theory in 3+1D with the $2\\pi$-quantized topological term ${\\sum_{I, J =1}^\\kappa\\frac{K_{IJ}}{4\\pi}\\int_{M^4}F^I\\wedge F^J}$.", "At energies below the gauge charges' gaps but above the monopoles' gaps, this field theory has an emergent ${\\mathbb{Z}_{k_1}^{(1)}\\times\\mathbb{Z}_{k_2}^{(1)}\\times\\cdots}$ 1-symmetry, where $k_i$ are the diagonal elements of the Smith normal form of $K$ and $\\mathbb{Z}_{0}^{(1)}$ is regarded as $U(1)^{(1)}$.", "In the $U^\\kappa(1)$ confined phase, the boundary's IR properties are described by Chern-Simons field theory and has a ${\\mathbb{Z}_{k_1}^{(1)}\\times\\mathbb{Z}_{k_2}^{(1)}\\times\\cdots}$ 1-symmetry that can be anomalous.", "To show these results, we develop a bosonic lattice model whose IR properties are described by this field theory, thus acting as its UV completion.", "The lattice model in the aforementioned limit has an exact ${\\mathbb{Z}_{k_1}^{(1)}\\times\\mathbb{Z}_{k_2}^{(1)}\\times\\cdots}$ 1-symmetry.", "We find that a gapped phase of the lattice model, corresponding to the confined phase of the $U^\\kappa(1)$ gauge theory, is a symmetry protected topological (SPT) phase for the ${\\mathbb{Z}_{k_1}^{(1)}\\times\\mathbb{Z}_{k_2}^{(1)}\\times\\cdots}$ 1-symmetry, whose SPT invariant is ${e^{i\\pi\\sum_{I, J}K_{IJ}\\int B_I\\smile B_J+B_I\\underset{1}{\\smile} d B_J}e^{i\\pi\\sum_{I< J}K_{IJ}\\int d B_I\\underset{2}{\\smile}d B_J}}$.", "Here, the background 2-cochains $B_I$ satisfy ${d B_I=\\sum_I B_{I}K_{IJ} = 0}$ mod $1$ and describe the symmetry twist of the ${\\mathbb{Z}_{k_1}^{(1)}\\times\\mathbb{Z}_{k_2}^{(1)}\\times\\cdots}$ 1-symmetry.", "We apply this general result to a few examples with simple $K$ matrices.", "We find the non-trivial SPT order in the confined phases of these models and discuss its classifications using the fourth cohomology group of the corresponding 2-group." ], [ "Introduction", "Symmetry protected topological (SPT) phases of quantum matter are short-range entangled gapped phases whose ground states cannot be adiabatically connected to a trivial product state due to the presence of a symmetry , , , .", "While the bulk of an SPT phase is trivial, its boundary is nevertheless nontrivial because on the boundary the symmetry is anomalous.", "In fact, such a boundary theory can only exist as the boundary of an invertible phase and cannot be a standalone bulk theory in one lower dimension.", "Furthermore, from the contemporary prospective of anomaly inflow, the SPT order in the bulk provides a unique characterization of the 't Hooft anomaly on the boundary.", "Since their discovery, there have been numerous interesting generalizations of SPT phases.", "This includes SPT phases with intrinsic topological order in the bulk , , , , SPT phases with a gapless bulk , , , and higher-order SPTs where edge states exist only on a subregion of the boundary , .", "In this paper, we consider the generalization where the SPT order is protected by a higher-symmetry , , , , , , .", "Global symmetries, called 0-symmetries, are symmetries whose transformation acts on a codimension-1 submanifold of spacetime (e.g., all of space), and the charged operators correspond to 0-dimensional objects (i.e., particles).", "Higher-symmetry is a generalization that includes $p$ -symmetry, where now the symmetry transformation acts on a codimension-${(p+1)}$ submanifold of spacetime and the charged operators are $p$ -dimensional closed objects , , .", "A $p$ -symmetry is mathematically described by a $(p+1)$ -group , .", "Just like global symmetries, higher-symmetries can be spontaneously broken , can be anomalous , and can be gauged .", "Generic lattice Hamiltonians do not commute with closed string, membrane, operators and thus do not have exact higher-symmetries.", "Instead, the lattice models with exact higher-symmetries are quite special.", "For instance, the Hamiltonians of many exactly soluble models , , , , , with topological orders , , have exact higher-symmetries.", "While higher-symmetries are typically not exact UV symmetries, they can nevertheless be emergent symmetries occurring in the IR.", "Intuitively, this is because at high energies the charged $p$ -dimensional closed objects can become open, and ${(p-1)}$ -dimensional excitations living on their boundary explicitly breaks the $p$ -symmetry.", "From this perspective, the aforementioned models with topological order have exact higher-symmetries because they are in a limit without dynamics.", "Thus even if the aforementioned ${(p-1)}$ -dimensional excitations are excited, they are frozen and do not break the symmetry .", "So, at energies much smaller than the gap of the excitations that explicitly break a higher-symmetry, the system can have emergent higher-symmetries at low energies.", "This gives rise to an interesting scenario where some low-energy excitations condense, while the higher-symmetry breaking excitations remain to have a large energy gap.", "Then, near the critical point of the phase transition, the system will still have the emergent higher-symmetry (which may be spontaneously broken).", "If the condensed phase happens be a short-range entangled state  with the emergent higher-symmetry, it can be an SPT phase protected by the emergent higher symmetry .", "We denote such a higher SPT phase as an $n$ -SPT phase if it is protected by an $n$ -symmetry.", "In this paper, we extend the work presented in W181202517 and further investigate this mechanism for creating SPT phases protected by emergent higher-symmetries.", "In particular, we consider abelian gauge theory in 3+1D which at energies smaller than the gauge charge's and monopole's gap has two emergent $U(1)$ 1-symmetries (which we denote as $U(1)^{(1)}$ ) commonly denoted as the electric and magnetic symmetries.", "In the strong coupling limit, the gauge theory is in a short-range entangled gapped confined phase where the monopoles condense and the magnetic $U(1)^{(1)}$ symmetry is explicitly broken.", "However, at energies below the gauge charge gap, the electric symmetry is still present in the confined phase.", "This gives rise to the aforementioned scenario and the possibility that near the confinement transition, the confined phase has nontrivial 1-SPT order protected by the electric symmetry.", "In fact, here we will show that with $2\\pi $ -quantized topological terms, the confined phase of abelian gauge theory near the transition has nontrivial emergent 1-SPT orders.", "Usually, a topological term can affect the dynamics of the strong coupling limit in a very non-trivial way, and can make it impossible to calculate the physical properties (such as energy gap) in this limit.", "However, a $2\\pi $ -quantized topological terms are much easier to handle , , , , and we can still determine the strong coupling limit to be a short-range entangled gapped confined phase.", "The remaining of this paper is organized as follows.", "In section  we introduce the notations used in this paper.", "In section , we present a simple example of a nontrivial 1-SPT order in the confined phase of 3+1D $_2$ gauge theory.", "In doing so, we review the cochain lattice field theory formalism and techniques which we use throughout the rest of the paper.", "Then, in section  we consider the same scenario but in 3+1D abelian gauge theory with $\\kappa $ -types of $U(1)$ gauge fields and $2\\pi $ -quantized topological terms.", "Using the bosonic lattice model we develop, we find that the total emergent electric symmetry ${U(1)^{(1)}\\times U(1)^{(1)}\\times \\cdots }$ below the gauge charges' gaps and the monopoles' gaps is reduced to ${_{k_1}^{(1)} \\times _{k_2}^{(1)} \\times \\cdots }$ at energies above the monopoles' gaps (but still blow the gauge charges' gaps).", "Subsequently, we find that the confined phase of $U^(1)$ gauge theory with $2\\pi $ -quantized topological terms has nontrivial ${_{k_1}^{(1)} \\times _{k_2}^{(1)} \\times \\cdots }$ 1-SPT order.", "We construct the associated 1-SPT invariant and consider some examples." ], [ "Notations and conventions", "In this paper, we will use the notion of cochain, cocycle, and coboundary, as well as their higher cup product $\\underset{{\\scriptscriptstyle k}}{\\smile }$ and Steenrod square $\\mathbb {Sq}^k$ .", "A brief introduction of chains and cochains can be found in the Appendix of TW190802613.", "From its definition, we note that $\\underset{{\\scriptscriptstyle 0}}{\\smile } =\\smile $ , the cup product.", "We will abbreviate the cup product $a\\smile b$ as $ab$ by dropping $\\smile $ .", "We will also use $\\overset{\\scriptscriptstyle n}{=}$ to mean equal up to a multiple of $n$ , and use $\\overset{\\scriptscriptstyle }{=}$ to mean equal up to $f$ (up to a coboundary).", "An important identity which we will repeatedly use is that for cochains $f_m, h_n$ , ( fm k hn)= fm k hn + (-)m fm k hn+ (-)m+n-k fm k-1 hn + (-)mn+m+n hn k-1 fm .", "Furthermore, in this paper we will deal with many $_n$ -valued quantities.", "We will denote them as, for example, $a^{_n}$ .", "However, we will always lift the $_n$ -value to $$ -value, so the value of $a^{_n}$ has a range from $ -\\lfloor \\frac{n}{2} \\rceil $ to $ \\lfloor \\frac{n}{2} \\rceil $ , where $\\lfloor x \\rceil $ denotes the integer that is closest to $x$ (if two integers have the same distance to $x$ , we will choose the smaller one, $\\lfloor \\frac{1}{2} \\rceil =0$ ).", "In this case, the expression like $a^{_n}a^{_m}$ makes sense." ], [ "$_2^{(1)}$ 1-SPT order in 3+1D theories", "In this section, we review one of the simplest ways to realize $_2^{(1)}$ 1-SPT order in 3+1D , .", "Our purpose of doing so is to introduce the formalism we use and warm-up in a simple context before beginning section , where things get more involved.", "We start by considering a twisted $_2$ 2-gauge theory.", "By considering its confined phase, we then construct a model with $_2^{(1)}$ ${\\text{1-SPT}}$ order by “ungauging” the twisted $_2$ 2-gauge theory.", "The $_2^{(1)}$ 1-SPT order is exact in this model, but survives else where in the confined phase diagram as an emergent $_2^{(1)}$ 1-SPT, existing at energies much smaller than the $_2$ gauge charge gap." ], [ "Twisted $_2$ 2-gauge theory", "To construct a 3+1D bosonic model that realizes $_2^{(1)}$ 1-SPT order, we first construct a local bosonic model with topological order described by a $_2$ ${\\text{2-gauge}}$ theory.", "We triangulate spacetime $^4$ and, working in the Euclidean signature, consider the lattice path integral of cochain fields .", "The bosonic degrees of freedom for the $_2$ ${\\text{2-gauge}}$ theory are described by a $_2$ -valued 2-cochain field $b^{_2}$ .", "As a 2-cochain, $b^{_2}$ is a map from 2-chains to $_2$ , as opposed to ${\\text{1-gauge}}$ theory which is described by a 1-cochain field.", "Consider the local bosonic model: $Z(^4, g) = \\sum _{b^{_2}} ^{-\\frac{1}{2g} \\sum _{ijkl}\\left( b^{_2}_{ijkl} - 2\\lfloor \\frac{1}{2} b^{_2}_{ijkl} \\rceil \\right)},$ where ${\\sum _{ijkl}}$ sums over all spacetime 3-simplices for a fixed triangulation, and $\\sum _{b^{_2}}$ sums over all the 2-cochain field corresponding to the path integral.", "In the exactly solvable limit ${g\\rightarrow 0}$ , the path integral becomes $Z = \\sum _{b^{_2}\\overset{\\scriptscriptstyle 2}{=}0} 1,$ where ${b^{_2}\\overset{\\scriptscriptstyle 2}{=}0}$ means ${b^{_2}=0~\\mod {~}2}$ .", "Now, $Z$ captures the topological order described by the deconfined phase of pure $_2$ 2-gauge theory.", "However, we note that in ${3+1}$ D, $_2$ 2-gauge theory is dual to $_2$ 1-gauge theoryIn $_2$ 2-gauge theory in 3+1D, loop excitations carry $_2$ gauge charge while particle excitations carry the $_2$ gauge flux.", "On the other hand, in $_2$ 1-gauge theory in 3+1D, particles carry the $_2$ gauge charge and loops carrying the $_2$ gauge flux.", "The duality between $_2$ 2-gauge theory and $_2$ 1-gauge theory in ${3+1}$ D simply switches which excitations are called gauge charges and which are called gauge fluxes..", "Thus, the topological order is also described by $_2$ 1-gauge theory, which is $_2$ topological order.", "In fact, Eq.", "(REF ) is the ${3+1}$ D toric code.", "We now consider the equivalent limit in a twisted $_2$ 2-gauge theory.", "This is described by a different bosonic model, which is Eq.", "(REF ) but with the 1 replaced with the action amplitude $^{\\pi \\int _{^4} (b^{_2})^2}$ .", "Indeed, the path integral is Z(4) = b22=0 4 (b2)2 , where we use the shorthand ${(b^{_2})^2 \\equiv b^{_2}\\smile b^{_2}}$ and $\\int _{^4}$ is a sum over all 4-simplicies of $^4$ .", "Note that this action amplitude is correctly invariant under the gauge transformation ${b^{_2} \\rightarrow b^{_2} + 2n^{}}$ , where $n^{}$ is an arbitrary $$ -valued 2-cochain.", "The twisted $_2$ 2-gauge theory realizes a twisted $_2$ topological order where the $_2$ charges are fermions ." ], [ "Lattice model with $_2$ 1-SPT order", "We now use the twisted $_2$ 2-gauge theory in Eq.", "(REF ) to obtain a local bosonic model that realizes a 1-SPT order protected by the $_2$ 2-group.", "The classifying space of the $_2$ 2-group is a topological space denoted by $B(_2,2)$ , which satisfies ${\\pi _2(B(_2,2))=_2}$ and ${\\pi _n(B(_2,2))=0}$ for $n\\ne 2$ .", "The associated symmetry is a $_2$ 1-symmetry, which we denote as $_2^{(1)}$ .", "The $_2$ 2-gauge theory can be “ungauged” by first parameterizing the dynamical 2-cochain field $b^{_2}$ as $b^{_2} = B^{_2} + a^{_2},$ where $a^{_2}$ is a $_2$ -valued 1-cochain field describing the pure 2-gauge fluctuations.", "However, we now reinterpret the meaning of $B^{_2}$ and $a^{_2}$ by treating $a^{_2}$ as the dynamical field and $B^{_2}$ as a $_2$ -valued 2-cocycle background field.", "This produces a new local bosonic model whose resulting path integral is obtained from the twisted $_2$ 2-gauge theory Eq.", "(REF ): $Z (^4, B^{_2}) = \\sum _{a^{_2}}^{\\pi \\int _{^4} (B^{_2}+a^{_2})^2}.$ This path integral is invariant under the gauge transformation a2 a2 +2, B2 B2 - 2.", "$B^{_2}$ describes the symmetry-twist of the $_2^{(1)}$ 1-symmetry.", "Turning off the background symmetry-twist field, and hence ungauging the $_2^{(1)}$ 1-symmetry, the model becomes Z (4,0) = a2 4 (a2)2 , which has an exact $_2^{(1)}$ 1-symmetry is generated by $_2$ -valued 2-cocycles $^{_2}$ : a2 a2 +2, 2 2= 0.", "By construction, there is no obstruction to gauging the $_2^{(1)}$ 1-symmetry and therefore the $_2^{(1)}$ 1-symmetry is anomaly-free.", "This can further be seen from the fact that the path integral $Z (^4)$ is invariant under the $_2^{(1)}$ transformation even when $^4$ has boundary.", "Using that ${\\int _{^4} (a^{_2})^2 = \\int _{\\partial ^4} a^{_2}a^{_2}}$ , when spacetime $^4$ is closed (i.e., ${\\partial ^4 = \\emptyset }$ ) then ${\\int _{^4} (a^{_2})^2 = 0}$ .", "Therefore, the action amplitude ${^{\\pi \\int _{^4} (a^{_2})^2}=1}$ and so for a closed spacetime $Z (^4,0) = \\sum _{a^{_2}}^{\\pi \\int _{^4} (a^{_2})^2 } = 2^{N_e},$ where $N_e$ is the number of the edges in the spacetime complex $^4$ .", "According to a conjecture presented in KW1458, this implies that the ground state of the model Eq.", "(REF ) has no topological order (is short range entangled).", "Since the ground state has $_2^{(1)}$ 1-symmetry and no topological order, it may have a $_2^{(1)}$ 1-SPT order, which are classified by the fourth cohomology group ${H^4(B(_2,2),{\\mathbb {R}/\\mathbb {Z}}) =_4}$  , , , .", "To see which 1-SPT order is realized, we note that the SPT order is characterized by the volume-independent partition function $Z^{\\text{top}}(^4,B^{_2})= \\dfrac{Z (^4,B^{_2})}{Z (^4,0)},$ which is called the SPT-invariant , , , .", "We compute the 1-SPT invariant from Eq.", "Bda, by integrating out $a^{_2}$ for closed spacetime $^4$ and for $B^{_2}=0$ mod 2.", "Using Eq.", "(REF ) and the fact that $B^{_2}$ is a cocycle, we find that Ztop(4,B2) =4 (B2)2 =m4 24 Sq2(B2) |m=2.", "Here, the generalized Steenrod square $\\mathbb {Sq}^{k}$ is defined as Sqk(cl) cll-k cl + cll-k+1 cl, where $c_l$ is any $l$ -cochain.", "From the above 1-SPT invariant, we see that the model defined by Eq.", "(REF ) realizes a $_2^{(1)}$ 1-SPT order that corresponds to $2\\in _4$ in the confined phase." ], [ "Emergent $_2^{(1)}$ 1-SPT order in the confined phase of {{formula:59d62d80-8e9f-4cd9-b3b1-e1c4f107c1fb}} \ngauge theory", "The fact that the theory Eq.", "(REF ) has an exact $_2^{(1)}$ symmetry makes it rather special.", "Indeed, for a typical condensed matter model, the lattice theory would be more like Z[4,g,h] = a2 4 (a2)2 -hij (a2)ij -12g ijk ( a2)ijk -212 (a2)ijk , where $\\sum _{ijk}$ sums over all the triangles and $\\sum _{ij}$ sums over all the edges of $^4$ .", "This path integral does not have the $_2^{(1)}$ symmetry (REF ), it is explicitly broken by the $h$ term.", "Only when ${h=0}$ does Eq.", "(REF ) have the $_2^{(1)}$ symmetry.", "Thus, at first glance, when ${h\\ne 0}$ this generic model does not realize a $_2^{(1)}$ SPT state since it does not even have the symmetry.", "However, while the $_2^{(1)}$ symmetry is no longer a UV symmetry, for small $|h|$ the low-energy sectors of the Hilbert space enjoys an emergent $_2^{(1)}$ symmetry.", "Indeed, since only the motion of the $_2$ charge excitations can break the $_2^{(1)}$ 1-symmetry (i.e., the $h$ term), a $_2^{(1)}$ symmetry emerges at energies much smaller than the $_2$ charge excitation gap.", "Figure: The schematic phase diagram of the model described by Eq. ().", "Thereis a topologically ordered phase described by the deconfined phase of 2 _2gauge theory, and a gapped short-range entangled phase corresponding to theconfined phase of 2 _2 gauge theory.", "At h=0{h=0}, the model has an exact 2 (1) _2^{(1)} symmetry, while for h≠0{h\\ne 0} the 2 (1) _2^{(1)} symmetry is anemergent symmetry at energies below the 2 _2 gauge charge gap.", "Within theconfined phase, there is an SPT order protected by an emergent 2 (1) _2^{(1)}symmetry.", "The SPT invariant describing this 1-SPT order is given byEq.", "()When $|h|\\ll 1$ and $|g|\\ll 1$ , the model Eq.", "(REF ) realizes the $_2$ topological order described by the deconfined phase of $_2$ gauge theory.", "As we increase $g$ , it will undergo a phase transition into a confined phase with short-range entanglement.", "Let us assume this transition is continuous (if it is not, we can modify the model to make the confinement transition nearly continuous).", "Then, approaching the transition, the $_2$ -flux fluctuations are low energy excitations while the $_2$ charge excitations remain to have a large energy gap.", "This is exactly the scenario for the emergent $_2^{(1)}$ symmetry.", "So in the confined phase near the transition (when the $_2$ charge excitations energy gap remains large), the model realizes an 1-SPT order protected by the emergent $_1^{(1)}$ 1-symmetry, described by $2\\in _4$ .", "The region of $h$ where the emergent $_2^{(1)}$ SPT order exists gets smaller the larger $g$ becomes, while ${h=0}$ always as an exact 1-SPT order (see Fig.", "REF ).", "The low-energy effective theory describing the phase with emergent 1-SPT order is Eq.", "(REF ).", "Introducing the Poincaré dual of $a^{_2}$ , denoted as $\\hat{f}$ , the lattice action ${\\int _{^4} (a^{_2})^2 }$ is equal to the intersection number of $\\hat{f}$ : ${\\int _{^4} (a^{_2})^2 = \\#(\\hat{f}\\cdot \\hat{f})}$ .", "We note that $\\hat{f}$ corresponds to the world sheets of $_2$ flux loops so ${\\#(\\hat{f}\\cdot \\hat{f})}$ is the intersection number of $_2$ flux world sheets in spacetime.", "The topological term ${^{\\pi \\int _{^4}(a^{_2})^2 }}$ is therefore 4 (a2)2 =(-1)#(ff) .", "In general, the path integral of a $_2$ gauge theory may or may not contain the topological term ${(-1)^{\\#(\\hat{f}\\cdot \\hat{f})}}$ .", "When the topological term is included, then the confined phase of the $_2$ gauge theory near the confinement transition will be a 1-SPT state protected by the emergent $_2^{(1)}$ 1-symmetry.", "However, when the path integral does not include the topological term, then the confined phase of the $_2$ gauge theory near the confinement transition will be a trivial SPT state of the emergent $_2^{(1)}$ 1-symmetry.", "Therefore, in a 3+1D $_2$ gauge theory, by adjusting the presence or the absence of the topological term ${(-1)^{\\#(\\hat{f}\\cdot \\hat{f})}}$ (the intersection term for the $_2$ -flux world sheet), we can control the presence or the absence of the 1-SPT order protected by the emergent $_2^{(1)}$ 1-symmetry in the confined phase near the confinement transition." ], [ "Using confined phases of 3+1D $_n$ gauge theory to realize\n{{formula:3e89b0ec-b859-42e8-b088-f89bdd3cba0a}} 1-SPT orders for even {{formula:8e86b4fe-1912-4757-a176-ce2234c71c9d}}", "For simplicity, we've presented the above in the $_2$ case, but it can easily be generalize by replacing $_2$ with $_n$ , where $n$ is a positive even integer.", "Indeed, introducing the $_n$ -valued 1-cochain field $a^{_n}$ , the generalized generic latice model is Z = an mn 4 Sq2(an) -hij (an)ij -1ng ijk ( an)ijk -n1n (an)ijk , where the topological term is now proportional to ${\\int _{^4} \\mathbb {Sq}^2(a^{_n})}$ .", "In TW190802613, it was shown that ${\\mathbb {Sq}^2(a^{_n} +nb)\\overset{\\scriptscriptstyle 2n}{=} \\mathbb {Sq}^2(a^{_n})}$ and that ${^{\\pi \\frac{m}{n} \\int _{^4}\\mathbb {Sq}^2(a^{_n}) } = 1}$ when $^4$ is closed.", "Thus, the inclusion of the topological term ${^{\\pi \\frac{m}{n} \\int _{^4} \\mathbb {Sq}^2(a^{_n}) }}$ does not affect the local dynamics of the model, which is the level shift symmetry discussed in .", "As a result, when $|h|\\ll 1$ and $|g|\\ll 1$ , the model (REF ) realizes the $_n$ topological order described by the deconfined phase of a $_n$ gauge theory.", "As we increase $g$ , the model will undergo a confinement phase transition.", "Assuming the transition is continuous, near the transition the $_n$ -flux energy gap is much smaller than the gap for the $_n$ charge excitations.", "So near the transition, the model has an emergent $_n^{(1)}$ 1-symmetry, at energies much less than $_n$ charge energy gap.", "After the confinement transition, the confined phase has 1-SPT order protected by the emergent $_n^{(1)}$ 1-symmetry and described by $m\\in H^4(B(_n,2);{\\mathbb {R}/\\mathbb {Z}})=_{2n}$ for even $n$ .The lattice model (REF ) is well defined even for $m\\notin $ .", "However, when $m\\notin $ it is not clear if the model has a gapped confined phase when $|g|\\gg 1$ .", "Here $B(_n,2)$ is the classifying space of the $_n$ 2-group describing $_n^{(1)}$ 1-symmetry.", "It satisfies $\\pi _2(B(_n,2))=_n$ and $\\pi _i(B(_2,2))=0,\\ i\\ne 2$ .", "The physical consequence of $_n^{(1)}$ 1-SPT order, such as boundary states, was discussed in TW190802613.", "In last section, we saw how $_{2n}^{(1)}$ 1-SPT state can be realized in the confined phase near the confinement transition of 3+1D $_n$ gauge theory.", "Now we investigate more complicated 1-SPT orders which are protected by finite 1-symmetries.", "In this section, we will construct a 3+1D bosonic model, that corresponds to lattice $U^(1)$ “gauge theory” with a $2\\pi $ -quantized topological term.", "We will show that, due to the topological term, the model has a reduced $ _{k_1}^{(1)} \\times _{k_2}^{(1)} \\times \\cdots $ 1-symmetry.", "We will also show that the confined phase of the $U^(1)$ gauge theory can have a 1-SPT orders protected by the $_{k_1}^{(1)} \\times _{k_2}^{(1)} \\times \\cdots $ 1-symmetry." ], [ "3+1D $U^(1)$ pure gauge field theory and {{formula:11e17354-dfa1-4df1-bbac-e1639f1009d3}} -quantized topological term", "Before we consider the bosonic lattice model, we first consider the corresponding continuum theory.", "We do so in a timely, but non-rigorous, fashion to see how the results from the lattice theory which we present in the next sections are hinted towards in the continuum theory.", "It will set the stage for the lattice theory where the formal manipulations are much more involved than those in the field theory.", "We consider the theory described by the Euclidean action $S = \\frac{1}{2g^2}\\sum _{I}\\int _{M^4} f^{I} {\\mathchoice{\\,{\\scriptstyle \\wedge }\\,}{{\\scriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}}\\mathop {*}f^{I} + S_\\text{top},$ where $a^{I}$ with ${I = 1,\\ldots , \\kappa }$ are ${\\mathbb {R}/\\mathbb {Z}}$ -valued 1-form fieldsTypically the $U(1)$ connection is a map ${A:^4\\mapsto /2\\pi }$ .", "We define $a = A/2\\pi $ to absorb factors of $2\\pi $ and match the convention used in the bosonic lattice model.", "In terms of the coupling constant $g$ , the typical $U(1)$ coupling constant is $e = 2\\pi g$ ., the 2-form curvature $f^{I} = a^{I}$ , and ${k_{IJ}\\in }$ .", "The first term is Maxwell's kinetic term and the second term is the $2\\pi $ -quantized topological term (topological in the sense that it is independent of the metric) $S_{\\text{top}} = -2\\pi \\sum _{I\\le J}k_{IJ}\\int _{M^4} f^I {\\mathchoice{\\,{\\scriptstyle \\wedge }\\,}{{\\scriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}}f^J,$ Furthermore, the quantity ${k_{IJ}\\int f^I{\\mathchoice{\\,{\\scriptstyle \\wedge }\\,}{{\\scriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}}f^J}$ is quantized as an integer when $M^4$ is closed.", "Thus, for closed $M^4$ the action amplitude of the topological term is unity, but for $M^4$ with a boundary it can have a nontrivial effect.", "Since the action depends only on $f^{I}$ , it is left unchanged by $a^{I} \\rightarrow a^{I} + \\Gamma ^{I}, \\ \\ \\ \\ \\Gamma ^{I} = 0.$ This corresponds to a real symmetry transformation (not a gauge transformation) when ${\\oint \\Gamma ^{I} \\ne }$ .", "Since there are $\\kappa $ field $a^{I}$ , Eq.", "(REF ) is associated with $\\kappa $ different $U(1)^{(1)}$ 1-form symmetries: ${U(1)^{(1)}\\times U(1)^{(1)}\\times \\cdots }$ .", "The associated Noether current can be found by introducing a background symmetry twisted field $^I$ in Lorentzian signature and having ${a^I \\rightarrow a^I - ^I}$ .", "Noting that the conserved current $J^I$ minimally couples to $^I$ as ${\\int ^I ~{\\mathchoice{\\,{\\scriptstyle \\wedge }\\,}{{\\scriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}}\\mathop {*}J^I}$ , we find that for the transformation of the $I$ th field: $J^{I} = \\dfrac{1}{g^2}f^{I} + 2 \\pi \\sum _J K_{IJ}\\mathop {*}f^{J},$ where $K_{IJ}$ is given by KII=2kII, KIJ=KJI=kIJ, I < J.", "The fact that the current is conserved means that ${^\\dagger J^I = 0}$ , where ${^\\dagger = \\mathop {*}\\mathop {*}}$ in the adjoint of $$ .", "In the above analysis of the symmetry, we consider the field theory without $U(1)$ charges and $U(1)$ monopoles.", "This ${U(1)^{(1)}\\times U(1)^{(1)}\\times \\cdots }$ is really an emergent 1-form symmetry at energies below the $U(1)$ charge gaps and the monopole gaps.", "Indeed, at energies above the $U(1)$ gauge charge gaps, terms like ${\\int a^I {\\mathchoice{\\,{\\scriptstyle \\wedge }\\,}{{\\scriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}}\\mathop {*}j^I}$ will contribute to the action and this symmetry will be explicitly broken.", "Let's now introduce the 1-form ${j^{I}_m = \\mathop {*}f^{I}}$ , the Dirac monopole current density associated with the $I$ th field $a^{I}$ whose Poincaré dual is the world-line of the monopole.", "The continuity equation ${^\\dagger J^{I} =0}$ then implies that $\\dfrac{1}{g^2}^\\dagger f^{I} = 2\\pi K_{IJ}j_m^{J}.$ The effect of the nonzero righthand side is a generalized version of the Witten effect where $U(1)$ monopoles of the $J$ th field carries $K_{IJ}$ units of the $I$ th $U(1)$ gauge charge.", "The presence of magnetic monopoles complicates things.", "At energies below the $U(1)$ charge gaps but above the monopole gaps, due to the topological term, the monopoles fluctuations imply $U(1)$ charge fluctuations.", "This may break the ${U(1)^{(1)}\\times U(1)^{(1)}\\times \\cdots }$ 1-form symmetry to a smaller symmetry.", "In the continuum, monopole configurations can be easily considered by parametrizing the curvature as ${f^I = \\tilde{a}^I + G^I}$ .", "The 1-form fields $\\tilde{a}^I$ describe the smooth local fluctuations of $a^I$ and satisfy the Bianchi identity $(\\tilde{a}^I) = 0$ , while the 2-form fields $G^I$ capture the singular monopole configurations and satisfy ${j^I_m =\\star G^I}$ .", "At energies above the monopole gap, the field theory that describes the lattice model instead has the topological term $\\begin{aligned}S_\\text{top} &= -2\\pi \\sum _{I\\le J} k_{IJ} \\int _{M^4} \\tilde{a}^I {\\mathchoice{\\,{\\scriptstyle \\wedge }\\,}{{\\scriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}}\\tilde{a}^J +G^I {\\mathchoice{\\,{\\scriptstyle \\wedge }\\,}{{\\scriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}}G^J \\\\&\\hspace{70.0pt} -2\\pi \\sum _{I, J} K_{IJ}\\int _{M^4}\\tilde{a}^I {\\mathchoice{\\,{\\scriptstyle \\wedge }\\,}{{\\scriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}}\\mathop {*}j_m^J.\\end{aligned}$ This is equivalent to Eq.", "(REF ) up to a boundary term.", "For all practical purposes, we may treat the density in Eq.", "(REF ) as the definition of $f^I{\\mathchoice{\\,{\\scriptstyle \\wedge }\\,}{{\\scriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}}f^J$ for energies above the monopole gap.", "This distinction is important as the ${U(1)^{(1)}\\times U(1)^{(1)}\\times \\cdots }$ symmetry of Eq.", "(REF ) is broken down to a finite subgroup in Eq.", "(REF ), agreeing with the symmetries of the lattice model we study.", "Indeed, above the monopole energy gap, Eq.", "(REF ) is invariant under the transformation $\\tilde{a}^{I} \\rightarrow \\tilde{a}^{I} + \\Gamma ^{I}, \\ \\ \\ \\ \\sum _{I} K_{IJ} \\oint _{C^1}\\Gamma ^I \\in ,\\ \\ \\ \\ \\Gamma ^{I} = 0,$ for any closed 1-submanifold $C^1$ .", "The additional restriction ${\\sum _{I} K_{IJ} \\oint _{C^1}\\Gamma ^I \\in }$ ensures that the action amplitude ${^{2\\pi \\sum _{I, J} K_{IJ}\\int _{M^4}\\tilde{a}^I {\\mathchoice{\\,{\\scriptstyle \\wedge }\\,}{{\\scriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}}\\mathop {*}j_m^J}}$ is invariant since ${\\oint \\mathop {*}j^I_m \\in }$ .", "We note that this term in Eq.", "(REF ) also recovers the Gauss-Witten law Eq.", "(REF ).", "Thus, at a fixed point in spacetime, the values of allowed $\\Gamma ^I$ form a rational lattice $K^{-1}$ .", "So, above the monopole gap the theory has the 1-symmetries ${_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots }$ , where $k_i$ are the diagonal elements of the Smith normal form for $K$ .", "Below the monopole gap when $j^I_m$ vanishes, this constraint on $\\Gamma ^I$ does not apply so there is instead the aforementioned ${U(1)^{(1)}\\times U(1)^{(1)}\\times \\cdots }$ symmetries.", "Let's now turn on 2-form background fields $^I$ that are the flat connections describing the twist of the ${_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots }$ symmetry and satisfy the quantization conditions $\\sum _{I}K_{IJ} \\oint _{C^2} ^I \\in ,$ for any closed 2-submanifold $C^2$ .", "We'll work locally at the level of differential forms, ignoring topological subtitles and monopoles.", "The background fields minimally couple to the dynamical fields $a^I$ by replacing the curvature $a^I$ in the Euclidean action by ${a^I - ^I}$ .", "Making this replacement and taking the ${g\\rightarrow \\infty }$ limit, the action becomes $S = -2\\pi \\sum _{I\\le J}k_{IJ}\\int _{M^4} (a^I- ^I) {\\mathchoice{\\,{\\scriptstyle \\wedge }\\,}{{\\scriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}}(a^J- ^J).$ We can use Eq.", "(REF ) to find the continuum SPT invariant which describes the 1-SPT order in the confined phase.", "Indeed, let's consider spacetime $M^4$ to be closed.", "Then, since we ignore monopoles and because ${_I = 0}$ , integrating by parts we can rewrite the Euclidean action as $\\begin{aligned}S &= -2\\pi \\sum _{I\\le J}k_{IJ}\\int _{M^4} ^I{\\mathchoice{\\,{\\scriptstyle \\wedge }\\,}{{\\scriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}}^J,\\\\&= -\\pi \\sum _{I, J}K_{IJ}\\int _{M^4} ^I{\\mathchoice{\\,{\\scriptstyle \\wedge }\\,}{{\\scriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}}^J.\\end{aligned}$ Thus, the path integral $Z$ in this limit is $\\begin{aligned}Z[M^4,B^I] &= \\int D[a^I]~e^{\\pi \\sum _{I,J}K_{IJ}\\int _{M^4} ^I{\\mathchoice{\\,{\\scriptstyle \\wedge }\\,}{{\\scriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}}~^J},\\\\&= \\operatorname{Vol}^\\kappa ({\\mathbb {R}/\\mathbb {Z}})~e^{\\pi \\sum _{I,J}K_{IJ}\\int _{M^4} ^I{\\mathchoice{\\,{\\scriptstyle \\wedge }\\,}{{\\scriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}}~^J},\\end{aligned}$ where we've used that the action amplitude does not depend on the dynamical fields $a^I$ and introduced $\\operatorname{Vol}^\\kappa ({\\mathbb {R}/\\mathbb {Z}}) = \\int D[a^I].$ The SPT invariant is given by the volume-independent part of the path integral $Z^{\\text{top}}(M^4,B^I)= \\dfrac{Z(M^4,B^I)}{Z (M^4,0)}.$ Therefore, using Eq.", "(REF ) we find that in the continuum theory the 1-SPT invariant is $Z^{\\text{top}}(M^4,B^I) = e^{\\pi \\sum _{I,J}K_{IJ}\\int _{M^4} ^I{\\mathchoice{\\,{\\scriptstyle \\wedge }\\,}{{\\scriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}}~^J}.$ Thus, without much work we can characterize the 1-SPT order.", "However, in doing so we ignored nontrivial fibre bundles and magnetic monopoles.", "In the remainder of this section, we'll regulate this continuum theory by considering a bosonic lattice model whose IR properties are described by the field theory.", "Using this lattice model, we'll be able to recalculate the SPT invariant more rigorously (see Eq.", "(REF )), and find lattice-dependent terms in addition to one which captures Eq.", "(REF )." ], [ "Lattice Regularization of $U^(1)$ gauge theory with {{formula:b4fa26c8-171a-4463-9cb5-2056fa9bdeb4}} -quantized topological\nterm", "We now regulate the field theory discussed in the previous section by triangulating spacetime.", "The 1-form fields $a^I$ will be represented by ${\\mathbb {R}/\\mathbb {Z}}$ -valued 1-cochains $a_I^{\\mathbb {R}/\\mathbb {Z}}$ .", "There are three key properties that the $U^(1)$ gauge theory on a lattice must include: Letting $m_I^$ be an arbitrary $Z$ -valued 1-cochain, the action amplitude is invariant under ${a^{\\mathbb {R}/\\mathbb {Z}}_I \\rightarrow a^{\\mathbb {R}/\\mathbb {Z}}_I+m_I^}$ , even when spacetime $^4$ has a boundary; When $^4$ is closed, the action amplitude of the $2\\pi $ -quantized topological term becomes unity; In the smooth field limit (the low energy limit) when ${a^{{\\mathbb {R}/\\mathbb {Z}}}_J\\sim \\lfloor a^{{\\mathbb {R}/\\mathbb {Z}}}_J \\rceil }$ , which implies no monopoles), the action amplitude reduces to its continuum limit $ ^{2\\pi \\int _{M^4} \\sum _{I\\le J} k_{IJ}f^I {\\mathchoice{\\,{\\scriptstyle \\wedge }\\,}{{\\scriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}}f^J }$ .", "Regularizing the Maxwell term on the lattice is straight forward, but the $2\\pi $ -quantized topological term is highly non-trivial.", "Noting the relationship between the topological term and Chern-Simons theory in the continuum, this motivates us to define the $2\\pi $ -quantized topological term on the lattice as the derivative of the lattice Chern-Simons action.", "Indeed, we start with 2+1D $U^(1)$ Chern-Simons theory on spacetime lattice $^3$ obtained in DW190608270 $\\begin{aligned}&\\hspace{-5.0pt}Z_{\\text{CS}}=\\int D[a^{\\mathbb {R}/\\mathbb {Z}}_I]\\ ^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^3} \\hspace{-2.0pt} \\big (a^{/}_I(a^{/}_J-\\lfloor a^{/}_J \\rceil )\\big ) }\\\\&\\hspace{5.0pt}\\times \\hspace{-3.0pt}^{2 \\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^3} \\hspace{-2.0pt}a^{\\mathbb {R}/\\mathbb {Z}}_{I} (a^{\\mathbb {R}/\\mathbb {Z}}_{J} -\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_J \\rceil )-\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_I \\rceil a^{\\mathbb {R}/\\mathbb {Z}}_J}\\\\&\\hspace{5.0pt}\\times \\hspace{-3.0pt}^{-2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^3} \\hspace{-2.0pt} a^{{\\mathbb {R}/\\mathbb {Z}}}_J\\underset{{\\scriptscriptstyle 1}}{\\smile }\\lfloor a^{{\\mathbb {R}/\\mathbb {Z}}}_I \\rceil }\\hspace{-2.0pt}^{- \\hspace{-2.0pt}\\sum \\limits _{I}\\hspace{-2.0pt}\\int \\limits _{^3} \\hspace{-2.0pt} \\frac{|a^{/}_I - \\lfloor a^{/}_I \\rceil |^2}{g_3}}\\hspace{-4.0pt},\\end{aligned}$ where $a^{\\mathbb {R}/\\mathbb {Z}}_I$ are the aforementioned ${\\mathbb {R}/\\mathbb {Z}}$ -valued 1-cochain, the path-integral notation is shorthand for ${\\int D[a^{\\mathbb {R}/\\mathbb {Z}}_I]= \\prod _{ij,I}\\int _{-\\frac{1}{2}}^{\\frac{1}{2}}(a^{\\mathbb {R}/\\mathbb {Z}}_I)_{ij}}$ , and $k_{IJ}\\in $ .", "This lattice model is rather complicated as it captures the effects of magnetic monopoles.", "We note that DW190608270 found that Eq.", "CSlatt1 is invariant under the gauge transformation aR/ZI aR/ZI + mI for any $$ -valued 1-cochain $m_I^$ even when $^3$ has boundary.", "The path integral of the 3+1D bosonic model (for spacetime $^4$ with or without boundary) is then obtained from Eq.", "CSlatt1 by taking a derivative and setting ${g_3\\rightarrow \\infty }$ .", "Using the properties of the (higher) cup product, the first line of Eq.", "CSlatt1 vanishes since it is already the $$ of something, the second line of Eq.", "CSlatt1 becomes $\\begin{aligned}& ^{2\\pi \\int _{^4} k_{IJ}\\big ( a^{\\mathbb {R}/\\mathbb {Z}}_{I} (a^{\\mathbb {R}/\\mathbb {Z}}_{J} -\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_J \\rceil )-\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_I \\rceil a^{\\mathbb {R}/\\mathbb {Z}}_J \\big )}\\\\&\\hspace{15.0pt}=^{2\\pi k_{IJ} \\int _{^4} (a^{\\mathbb {R}/\\mathbb {Z}}_{I} -\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_I \\rceil ) (a^{\\mathbb {R}/\\mathbb {Z}}_{J} -\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_J \\rceil ) }\\times \\\\&\\hspace{50.0pt}^{2\\pi k_{IJ} \\int _{^4}a^{\\mathbb {R}/\\mathbb {Z}}_{I} \\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_J \\rceil - \\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_I \\rceil a^{\\mathbb {R}/\\mathbb {Z}}_{J}},\\end{aligned}$ and the third line of Eq.", "CSlatt1 becomes $\\begin{aligned}&^{-2\\pi k_{IJ} \\int _{^4} \\big ( a^{{\\mathbb {R}/\\mathbb {Z}}}_J\\underset{{\\scriptscriptstyle 1}}{\\smile }\\lfloor a^{{\\mathbb {R}/\\mathbb {Z}}}_I \\rceil \\big ) }\\\\&\\hspace{15.0pt}=^{-2\\pi k_{IJ} \\int _{^4} a^{{\\mathbb {R}/\\mathbb {Z}}}_J\\underset{{\\scriptscriptstyle 1}}{\\smile }\\lfloor a^{{\\mathbb {R}/\\mathbb {Z}}}_I \\rceil }\\times \\\\&\\hspace{45.0pt}^{2\\pi k_{IJ} \\int _{^4} a^{{\\mathbb {R}/\\mathbb {Z}}}_J\\lfloor a^{{\\mathbb {R}/\\mathbb {Z}}}_I \\rceil + \\lfloor a^{{\\mathbb {R}/\\mathbb {Z}}}_I \\rceil a^{{\\mathbb {R}/\\mathbb {Z}}}_J } .\\end{aligned}$ Putting this all together and including the lattice Maxwell term ${ ^{- \\sum _I \\int _{^4} \\frac{|a^{\\mathbb {R}/\\mathbb {Z}}_I-\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_I \\rceil |^2}{g} } }$ , we obtain a 3+1D bosonic model on spacetime lattice with a $2\\pi $ -quantized topological term $\\begin{aligned}&\\hspace{-5.0pt}Z = \\int D[a^{\\mathbb {R}/\\mathbb {Z}}_I] \\ ^{-2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} a^{{\\mathbb {R}/\\mathbb {Z}}}_J \\underset{{\\scriptscriptstyle 1}}{\\smile }\\lfloor a^{{\\mathbb {R}/\\mathbb {Z}}}_I \\rceil }\\\\&\\hspace{7.0pt}\\times ^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} (a^{\\mathbb {R}/\\mathbb {Z}}_I-\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_I \\rceil )(a^{\\mathbb {R}/\\mathbb {Z}}_J-\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_J \\rceil )}\\\\&\\hspace{7.0pt}\\times ^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{IJ}\\hspace{-1.0pt}K_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} a^{{\\mathbb {R}/\\mathbb {Z}}}_I \\lfloor a^{{\\mathbb {R}/\\mathbb {Z}}}_J \\rceil }^{- \\hspace{-2.0pt}\\sum \\limits _{I} \\hspace{-2.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} \\frac{|a^{\\mathbb {R}/\\mathbb {Z}}_I-\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_I \\rceil |^2}{g} }\\hspace{-2.0pt},\\end{aligned}$ where $K_{IJ}$ is given by KII=2kII, KIJ=KJI=kIJ, I < J.", "Because the lattice Chern-Simons path integral was invariant under the gauge transformation Eq.", "aZgauge even when $^4$ has boundary, by definition the path integral Eq.", "(REF ) is also invariant.", "Thus, requirement (1) from above is satisfied.", "Furthermore, since we defined the action as the derivative of something, requirement (2) is also automatically satisfied.", "Lastly, lets check that Eq.", "(REF ) satisfies requirement (3).", "In the ${g\\sim 0}$ limit, the Maxwell term enforces fluctuations ${a^{\\mathbb {R}/\\mathbb {Z}}_I-\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_I \\rceil \\sim }$ to be small.", "Therefore, using that $\\sim (a^{\\mathbb {R}/\\mathbb {Z}}_I-\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_I \\rceil )=\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_I \\rceil ,$ since $\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_I \\rceil \\overset{\\scriptscriptstyle 1}{=}0$ and $$ is small, this implies that $\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_I \\rceil = 0,$ and hence there are no monopoles.", "When $a^{\\mathbb {R}/\\mathbb {Z}}_J$ describes a monopole, it cannot be smooth and thus ${\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_J \\rceil \\ne 0}$ .", "In fact, $\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_J \\rceil $ is the Poincaré dual of the Dirac monopoles' worldsheets (the trajectory of the Dirac strings of the monopole in spacetime).", "Thus $\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_J \\rceil $ is the Poincaré dual of the boundary of the Dirac worldsheet, which is the worldline of the $U(1)$ monopoles.", "Therefore, the $g\\sim 0$ limit corresponds to the smooth field limit.", "In this limit, the action amplitude for the topological term in Eq.", "(REF ) becomes $^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} (a^{\\mathbb {R}/\\mathbb {Z}}_I-\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_I \\rceil )^2}.$ Relating the 1-form field $a^I$ to the 1-cochain $(a^{{\\mathbb {R}/\\mathbb {Z}}}_I)_{ij}$ by $\\int _i^j a^I = (a^{{\\mathbb {R}/\\mathbb {Z}}}_I)_{ij}$ and the 2-form curvature field ${f^I = a^I}$ by $\\int _{A(ijk)}\\hspace{-8.0pt}f^I = (a^{\\mathbb {R}/\\mathbb {Z}}_I -\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_I \\rceil )_{ijk},$ the action amplitude (REF ) becomes $^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} (a^{\\mathbb {R}/\\mathbb {Z}}_I-\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_I \\rceil )^2}\\approx ^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J}\\hspace{-2.0pt} k_{IJ} \\hspace{-4.0pt} \\int \\limits _{M^4} \\hspace{-2.0pt} f^I{\\mathchoice{\\,{\\scriptstyle \\wedge }\\,}{{\\scriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}{{\\scriptscriptstyle \\wedge }}}f^J}.$ Therefore, in the smooth field limit (the low energy limit) the $2\\pi $ -quantized term on the lattice is captured by the continuum field theory and requirement (3) is satisfied.", "In the absence of monopoles, Eq.", "(REF ) correctly becomes unity on a closed spacetime.", "For large $g$ , however, due to the presence of monopoles the lattice term ${^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} (a^{\\mathbb {R}/\\mathbb {Z}}_I-\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_I \\rceil )(a^{\\mathbb {R}/\\mathbb {Z}}_J-\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_J \\rceil )} }$ is no longer unity when $^4$ is closed and thus is neither $2\\pi $ -quantized nor topological.", "Therefore, for large $g$ the low-energy limit of the lattice model may not be described by the continuum topological term (REF ) since the lattice topological term must be described using all terms in Eq.", "(REF ).", "It's more likely that the low-energy physics of the lattice model for large $g$ , where the highly nontrivial terms in the first and third line of Eq.", "(REF ) are included, is better captured by the continuum topological term Eq.", "(REF )." ], [ "1-symmetries in 3+1D $U^(1)$ bosonic model", "Now that we've introduced the $U^(1)$ bosonic model, we now focus our attention on studying its symmetries and phase diagram.", "Firstly, let's review the case when $^4$ has no boundary and the topological term vanishes and Eq.", "K4Ddada becomes Maxwell's theory Z(4) = D[aR/ZI] -I 4|aR/ZI-aR/ZI |2g .", "When ${g\\sim 0}$ , the lattice curvature $a^{\\mathbb {R}/\\mathbb {Z}}_I$ fluctuate weakly and so the above model is in a deconfined phase of a compact $U^(1)$ gauge theory and has a gapless photon excitation.", "On the other hand, when ${g \\rightarrow \\infty }$ the model is in a gapped confined phase.", "Using that ${\\int _{-\\frac{1}{2}}^{\\frac{1}{2}} (a^{\\mathbb {R}/\\mathbb {Z}}_I)_{ij} =1}$ , the partition function is Z(4) = D[aR/ZI] =1, for any closed spacetime $^4$ .", "According a conjecture in KW1458, this implies that the gapped confined phase has a trivial topological order.", "In what follows, we now consider $^4$ with a boundary so the $2\\pi $ -topological term contributes to the path integral.", "We'll show that the gapped confined phase now has a 1-SPT order characterized by $k_{IJ}$ (see Fig.", "REF ).", "This is similar in spirit to section  where in order to get $_2^{(1)}$ SPT order we had to include the twist term Eq.", "(REF ).", "Regardless the value of $g$ and even on $^4$ with boundary, the path integral Eq.", "(REF ) is invariant under the transformation aR/ZI aR/ZI + Q/ZI, I Q/ZIKIJ , Q/ZI1=0.", "$^{{\\mathbb {Q}/\\mathbb {Z}}}_{I}$ are ${\\mathbb {Q}/\\mathbb {Z}}$ -valued 1-cocycles to ensure that the quantities ${a^{\\mathbb {R}/\\mathbb {Z}}_I-\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_I \\rceil }$ and ${\\lfloor a^{\\mathbb {R}/\\mathbb {Z}}_I \\rceil }$ are invariant under the transformation (REF ).", "If this were the only requirement, Eq.", "(REF ) would correspond to the $\\kappa $ different $U(1)^{(1)}$ 1-symmetries.", "However, the additional constraint that $\\sum _I^{{\\mathbb {Q}/\\mathbb {Z}}}_{I}K_{IJ}$ are $$ -valued cochains is required when there are magnetic monopoles to ensure the term $^{2\\pi \\int _{^4} \\sum _{IJ} a^{{\\mathbb {R}/\\mathbb {Z}}}_I K_{IJ} \\lfloor a^{{\\mathbb {R}/\\mathbb {Z}}}_J \\rceil } $ is invariant.", "Indeed, under the transformation Eq.", "1symm, this term changes by a phase factor $^{2\\pi \\int _{^4}\\sum _{IJ}^{{\\mathbb {Q}/\\mathbb {Z}}}_I K_{IJ} \\lfloor a^{{\\mathbb {R}/\\mathbb {Z}}}_J \\rceil },$ which is 1 provided $^{{\\mathbb {Q}/\\mathbb {Z}}}_{I}$ satisfy ${\\sum _I ^{{\\mathbb {Q}/\\mathbb {Z}}}_{I}K_{IJ} \\in }$ .", "As an integer matrix, $K$ has the following Smith normal form K = U k1 k2 k3 V , where $k_I$ are integers and $U,V$ are invertible integer matrices.", "Now the 1-symmetry can be written as aR/ZI aR/ZI + Q/ZI = aR/ZI + J Q/ZJ (U-1)JI, I Q/ZI UIJkJ = Q/ZJkJ , Q/ZI1=0.", "We see that the 1-symmetry is a $_{k_1}^{(1)}\\times _{k_2}^{(1)} \\times \\cdots $ 1-symmetry generated by the quantized $^{{\\mathbb {Q}/\\mathbb {Z}}}_{J}$ .", "When $k_I=0$ , $^{{\\mathbb {Q}/\\mathbb {Z}}}_{I}$ is not quantized and generates $U(1)^{(1)}$ 1-symmetry.", "The above result remains valid if we regard $_{0}^{(1)}$ as the $U(1)^{(1)}$ 1-symmetry.", "We note that, since the ${_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots }$ 1-symmetry is valid on the spacetime lattice with or without boundary, the 1-symmetry is anomaly-free.", "In addition to giving rise to a finite 1-symmetry, the term $^{2\\pi \\int _{^4} \\sum _{IJ} a^{{\\mathbb {R}/\\mathbb {Z}}}_IK_{IJ} \\lfloor a^{{\\mathbb {R}/\\mathbb {Z}}}_J \\rceil } $ also causes the $U(1)$ monopoles to be bounded with the $U(1)$ charges.", "In particular, the unit monopole of the $J^\\text{th}$ $U(1)$ field carries the $I^\\text{th}$ $U(1)$ charge $K_{IJ}$ .", "This is precisely the lattice version of the generalized Witten effect discussed in the continuum theory (see Eq.", "(REF )).", "Figure: The schematic phase diagram of the model described by Eq.", "() withthe additional term contributing to the action amplitude h∑ ij,I a I,ij ℝ/ℤ {^{h\\sum _{ij,I}a_{I,ij}^{\\mathbb {R}/\\mathbb {Z}}}}.", "When h=0{h=0}, there is no fluctuations of U(1)U(1) gaugecharge, and the model has an exact k 1 (1) × k 2 (1) ×⋯{_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots } 1-symmetry.", "For h≠0{h\\ne 0}, this becomes an emergent k 1 (1) × k 2 (1) ×⋯{_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots } 1-symmetry existing belowthe energy gaps of U(1)U(1) gauge charges.", "There is a deconfined phase describedby U ( 1)U^(1) gauge theory, and a gapped short-range entangled confined phasewhere the k 1 (1) × k 2 (1) ×⋯{_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots } 1-symmetry isnot spontaneously broken.", "In the confined phase there is an SPT orderprotected by the emergent k 1 (1) × k 2 (1) ×⋯{_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots }1-symmetry whose SPT invariant is given by Eq.", "().For large $g$ , these monopole-charge bound states condense which gives rise to a gapped oblique confined phase with $_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots $ 1-symmetry.", "We note that the 2+1D lattice $U^(1)$ Chern-Simons theory (REF ) also has the $_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots $ 1-symmetry, which can actually be anomalous .", "Since the 2+1D lattice $U^(1)$ Chern-Simons theory is the boundary of the $U^(1)$ model in the gapped confined phase, from the point of view of anomaly inflow , the gapped confined phase may have a non-trivial ${_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots }$ 1-SPT order.", "Indeed, in the next section we'll show that this confined phase is characterized by the $K$ -matrix and has a 1-SPT order protected by the ${_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots }$ 1-symmetry.", "Indeed, the 1-SPT invariant found in the next section is given by Eq. 1SPTinvk.", "Before concluding this subsection, we remark that the fact that the ${_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots }$ 1-symmetry is exact in the bosonic model is a special feature of the theory.", "A more generic lattice theory would also include the action amplitude ${^{h\\sum _{ij,I} a_{I,ij}^{\\mathbb {R}/\\mathbb {Z}}}}$ in the path integral, which explicitly breaks the ${_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots }$ 1-symmetry.", "However, like in the $_2$ gauge theory case discussed in section REF , for energies below the $U(1)$ gauge charge gaps, there is a region of $h\\ne 0$ where the ${_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots }$ 1-symmetry is an emergent symmetry.", "In this region, the corresponding 1-SPT order would also affect the low-energy physics and be protected by the emergent symmetry (see Fig.", "REF )." ], [ "Gauging the ${_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times ... }$ \n1-symmetry", "The fact that the boundary Chern-Simons theory has an anomalous ${_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots }$ 1-symmetry  means that the bulk theory has 1-SPT order in the large-$g$ confined phase protected by ${_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots }$ .", "In this section, we will characterize the 1-SPT theory in the bulk 3+1D theory by finding the SPT invariant obtained by gauging the 1-symmetry.", "This subsection contains mostly detailed calculations in order to derive the 1-SPT invariant given by Eq.", "(REF ).", "Before gauging the symmetry, it's convenient to first slowly turn on addition terms in the action which will not affect the 1-SPT order.", "In particularly, to the Euclidean lattice action we add S U ij,J (2I (aR/ZI)ij KIJ ).", "Note that in the ${U\\rightarrow \\infty }$ limit, this term makes $a^{\\mathbb {R}/\\mathbb {Z}}_I$ satisfy the quantization condition I aR/ZIKIJ 1= 0.", "Crucially, this preserves the ${_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots }$ 1-symmetry whose transformation is Eq.", "(REF ).", "Furthermore, when ${g\\rightarrow \\infty }$ the path integral for any closed spacetime changes smoothly as $U$ changes from 0 to $\\infty $ .", "This is because in this limit the only term in the action is Eq.", "(REF ) which is independently defined on each 1-simplex of the spacetime triangulation (i.e., non-interacting).", "Thus, the $U=0$ state and the ${U\\rightarrow \\infty }$ state belong to the same phase and so the ${(g,U)=(\\infty ,\\infty )}$ phase has the 1-SPT order as the ${(g,U)=(\\infty ,0)}$ phase.", "By considering the $U\\rightarrow \\infty $ state, the quantization condition turns the $U(1)$ cochain fields $a^{{\\mathbb {R}/\\mathbb {Z}}}_I$ into discrete cochain fields ${a^{\\mathbb {Q}/\\mathbb {Z}}_I}$${a^{\\mathbb {R}/\\mathbb {Z}}_I}$ is renamed as ${a^{\\mathbb {Q}/\\mathbb {Z}}_I}$ , since the quantized ${a^{\\mathbb {Q}/\\mathbb {Z}}_I}$ 's, ${\\sum _I a^{\\mathbb {Q}/\\mathbb {Z}}_{I} K_{IJ} \\overset{\\scriptscriptstyle 1}{=}0}$ , have values in ${\\mathbb {Q}/\\mathbb {Z}}$ .", "satisfying Eq.", "aq, which allows us to use results and techniques for discrete fields from section  to study the 1-SPT order in the $U(1)$ model.", "We now consider the ${U\\rightarrow \\infty }$ state, which in the strongly-interacting limit ${g\\rightarrow \\infty }$ the path integral Eq.", "K4Ddada becomes $\\begin{aligned}Z &=\\hspace{-15.0pt}\\sum _{\\sum _I a^{\\mathbb {Q}/\\mathbb {Z}}_{I} K_{IJ} \\overset{\\scriptscriptstyle 1}{=}0}\\hspace{-15.0pt}^{-2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt}a^{{\\mathbb {Q}/\\mathbb {Z}}}_J \\underset{{\\scriptscriptstyle 1}}{\\smile }\\lfloor a^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\rceil }\\times \\\\&\\hspace{5.0pt} ^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} (a^{\\mathbb {Q}/\\mathbb {Z}}_I-\\lfloor a^{\\mathbb {Q}/\\mathbb {Z}}_I \\rceil )(a^{\\mathbb {Q}/\\mathbb {Z}}_J-\\lfloor a^{\\mathbb {Q}/\\mathbb {Z}}_J \\rceil )},\\end{aligned}$ where we have use that for quantized $a^{\\mathbb {Q}/\\mathbb {Z}}_I$ satisfying Eq.", "(REF ), 24 IJ aR/ZI KIJ aR/ZJ =1.", "As mentioned, just like the original path integral Eq.", "(REF ), this path integral Eq.", "(REF ) also has the anomaly-free 1-symmetry aQ/ZI aQ/ZI + Q/ZI, I Q/ZIKIJ , Q/ZI1=0.", "Lastly, using that ${\\lfloor a^{{\\mathbb {R}/\\mathbb {Z}}}_J \\rceil \\underset{{\\scriptscriptstyle 1}}{\\smile }\\lfloor a^{{\\mathbb {R}/\\mathbb {Z}}}_I \\rceil }\\in $ and ${(a^{{\\mathbb {R}/\\mathbb {Z}}}_I) = 0}$ , we insert unity of the form $\\begin{aligned}1 &= ^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt}\\lfloor a^{{\\mathbb {R}/\\mathbb {Z}}}_J \\rceil \\underset{{\\scriptscriptstyle 1}}{\\smile }\\lfloor a^{{\\mathbb {R}/\\mathbb {Z}}}_I \\rceil }\\times \\\\&\\hspace{20.0pt}^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt}(a^{{\\mathbb {R}/\\mathbb {Z}}}_J-\\lfloor a^{{\\mathbb {R}/\\mathbb {Z}}}_J \\rceil )\\underset{{\\scriptscriptstyle 1}}{\\smile }(a^{{\\mathbb {R}/\\mathbb {Z}}}_I)}\\end{aligned}$ into Eq.", "(REF ) such that the path integral becomes $\\begin{aligned}Z &=\\hspace{-20.0pt}\\sum _{\\sum _I a^{\\mathbb {Q}/\\mathbb {Z}}_{I} K_{IJ} \\overset{\\scriptscriptstyle 1}{=}0}\\hspace{-20.0pt}^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt}( a^{{\\mathbb {Q}/\\mathbb {Z}}}_J -\\lfloor a^{{\\mathbb {Q}/\\mathbb {Z}}}_J \\rceil )\\underset{{\\scriptscriptstyle 1}}{\\smile }(a^{{\\mathbb {Q}/\\mathbb {Z}}}_I-\\lfloor a^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\rceil )}\\times \\\\&\\hspace{15.0pt} ^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} (a^{\\mathbb {Q}/\\mathbb {Z}}_I-\\lfloor a^{\\mathbb {Q}/\\mathbb {Z}}_I \\rceil )(a^{\\mathbb {Q}/\\mathbb {Z}}_J-\\lfloor a^{\\mathbb {Q}/\\mathbb {Z}}_J \\rceil )}.\\end{aligned}$ To determine the SPT order realized by the theory Eq.", "(REF ), we gauge the 1-symmetries by first replacing $a^{\\mathbb {Q}/\\mathbb {Z}}_I$ with ${a^{\\mathbb {Q}/\\mathbb {Z}}_I - B^{\\mathbb {Q}/\\mathbb {Z}}_I}$ which for convience we'll denote as bIQ/Z aQ/ZI - BQ/ZI, where $B^{\\mathbb {Q}/\\mathbb {Z}}_I$ is a background symmetry twist field satisfying $B^{\\mathbb {Q}/\\mathbb {Z}}_I \\overset{\\scriptscriptstyle 1}{=} 0, \\quad \\quad \\sum _{I} B^{\\mathbb {Q}/\\mathbb {Z}}_I K_{IJ} \\overset{\\scriptscriptstyle 1}{=} 0.$ We use “$\\overset{\\scriptscriptstyle 1}{=}$ ” instead of “$=$ ” here since shifting $B_I^{\\mathbb {Q}/\\mathbb {Z}}$ by a $$ -valued 2-cochain corresponds to performing a gauge transformation.", "After this, the path-integral Eq.", "K4DdadaD1 of course becomes $\\begin{aligned}Z &=\\hspace{-20.0pt}\\sum _{\\sum _I a^{\\mathbb {Q}/\\mathbb {Z}}_{I} K_{IJ} \\overset{\\scriptscriptstyle 1}{=}0}\\hspace{-20.0pt}^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt}( b^{{\\mathbb {Q}/\\mathbb {Z}}}_J -\\lfloor b^{{\\mathbb {Q}/\\mathbb {Z}}}_J \\rceil )\\underset{{\\scriptscriptstyle 1}}{\\smile }(b^{{\\mathbb {Q}/\\mathbb {Z}}}_I-\\lfloor b^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\rceil )}\\times \\\\&\\hspace{15.0pt} ^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} (b^{\\mathbb {Q}/\\mathbb {Z}}_I-\\lfloor b^{\\mathbb {Q}/\\mathbb {Z}}_I \\rceil )(b^{\\mathbb {Q}/\\mathbb {Z}}_J-\\lfloor b^{\\mathbb {Q}/\\mathbb {Z}}_J \\rceil )}.\\end{aligned}$ Note that for $^4$ with or without boundary, Eq.", "(REF ) is importantly invariant under the gauge transformations $\\begin{aligned}a_I^{\\mathbb {Q}/\\mathbb {Z}}&\\rightarrow a_I^{\\mathbb {Q}/\\mathbb {Z}}+ m^_I,\\\\b_I^{\\mathbb {Q}/\\mathbb {Z}}&\\rightarrow b_I^{\\mathbb {Q}/\\mathbb {Z}}+ n^_I,\\end{aligned}$ where ${m^_I\\overset{\\scriptscriptstyle 1}{=}0}$ and ${n^_I \\overset{\\scriptscriptstyle 1}{=}0}$ .", "If we had not inserted Eq.", "(REF ) into Eq.", "(REF ), the gauged theory would have not been gauge invariant.", "The path integral Eq.", "(REF ) is a bit cumbersome in its current form and it's hard to see how $^4$ being opened or closed changes the action amplitude.", "Thus, let's massage the action amplitude of Eq.", "(REF ) a bit to get it in a more enlightening form.", "First, we consider the first line of Eq.", "(REF ).", "Using that ${b_I^{{\\mathbb {Q}/\\mathbb {Z}}} \\overset{\\scriptscriptstyle 1}{=} 0}$ and rewriting $\\begin{aligned}&^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt}(b^{{\\mathbb {Q}/\\mathbb {Z}}}_J -\\lfloor b^{{\\mathbb {Q}/\\mathbb {Z}}}_J \\rceil )\\underset{{\\scriptscriptstyle 1}}{\\smile }(b^{{\\mathbb {Q}/\\mathbb {Z}}}_I - \\lfloor b^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\rceil )}\\\\&\\hspace{20.0pt}=^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt}b^{{\\mathbb {Q}/\\mathbb {Z}}}_J \\underset{{\\scriptscriptstyle 1}}{\\smile }(b^{{\\mathbb {Q}/\\mathbb {Z}}}_I - \\lfloor b^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\rceil )},\\end{aligned}$ we then can use Eq.", "() and once again ${b_I^{{\\mathbb {Q}/\\mathbb {Z}}} \\overset{\\scriptscriptstyle 1}{=} 0}$ to write this as $\\begin{aligned}&^{2\\pi \\sum \\limits _{I\\le J} k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt}b^{{\\mathbb {Q}/\\mathbb {Z}}}_J \\underset{{\\scriptscriptstyle 1}}{\\smile }(b^{{\\mathbb {Q}/\\mathbb {Z}}}_I - \\lfloor b^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\rceil )}\\\\&\\hspace{5.0pt}= ^{2\\pi \\sum \\limits _{I\\le J} k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt}\\big (b^{{\\mathbb {Q}/\\mathbb {Z}}}_J \\underset{{\\scriptscriptstyle 1}}{\\smile } (b^{{\\mathbb {Q}/\\mathbb {Z}}}_I -\\lfloor b^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\rceil ) \\big ) -b^{{\\mathbb {Q}/\\mathbb {Z}}}_J \\underset{{\\scriptscriptstyle 1}}{\\smile } b^{{\\mathbb {Q}/\\mathbb {Z}}}_I }\\times \\\\&\\hspace{25.0pt}^{2\\pi \\sum \\limits _{I\\le J} k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt}\\lfloor b^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\rceil b^{{\\mathbb {Q}/\\mathbb {Z}}}_J -b^{{\\mathbb {Q}/\\mathbb {Z}}}_J \\lfloor b^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\rceil } .\\end{aligned}$ Next, we consider the second line of Eq.", "(REF ).", "We can use the fact that since ${\\lfloor b_I^{\\mathbb {Q}/\\mathbb {Z}} \\rceil \\lfloor b_J^{\\mathbb {Q}/\\mathbb {Z}} \\rceil \\in }$ , then $\\begin{aligned}&^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} (b_I^{\\mathbb {Q}/\\mathbb {Z}}-\\lfloor b_I^{\\mathbb {Q}/\\mathbb {Z}} \\rceil )(b_J^{\\mathbb {Q}/\\mathbb {Z}}-\\lfloor b_J^{\\mathbb {Q}/\\mathbb {Z}} \\rceil ) }\\\\&\\hspace{20.0pt}= ^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} b_I^{\\mathbb {Q}/\\mathbb {Z}}b_J^{\\mathbb {Q}/\\mathbb {Z}}-b_I^{\\mathbb {Q}/\\mathbb {Z}}\\lfloor b_J^{\\mathbb {Q}/\\mathbb {Z}} \\rceil -\\lfloor b_I^{\\mathbb {Q}/\\mathbb {Z}} \\rceil b_J^{\\mathbb {Q}/\\mathbb {Z}}}.\\end{aligned}$ Using these simplifications, the gauged model (REF ) can be rewritten as Z = I aQ/ZI KIJ 1=0 2IJ kIJ 4 bJQ/ZbIQ/Z-bQ/ZJ 1 bQ/ZI 2IJ kIJ 4 (bQ/ZJ 1(bQ/ZI - bQ/ZI ) ) , where we have used that ${\\sum _{IJ} b^{\\mathbb {Q}/\\mathbb {Z}}_I K_{IJ} \\overset{\\scriptscriptstyle 1}{=} 0 }$ , 2IJ kIJ 4 -bIQ/ZbJQ/Z -bJQ/ZbIQ/Z =1.", "Therefore, starting from the $U^{\\kappa }(1)$ bosonic model and gauging the ${_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots }$ , Eq.", "(REF ) gives the path integral of the gauged model from which we can find the 1-SPT invariant.", "When the spacetime $^4$ has no boundary, the total derivative term in Eq.", "(REF ) vanishes and the path integral becomes $Z(B^{\\mathbb {Q}/\\mathbb {Z}}_I) =\\hspace{-16.99997pt}\\sum _{\\sum _I a^{\\mathbb {Q}/\\mathbb {Z}}_{I} K_{IJ} \\overset{\\scriptscriptstyle 1}{=}0}\\hspace{-16.99997pt}^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} b_J^{\\mathbb {Q}/\\mathbb {Z}}b_I^{\\mathbb {Q}/\\mathbb {Z}}-b^{{\\mathbb {Q}/\\mathbb {Z}}}_J \\underset{{\\scriptscriptstyle 1}}{\\smile } b^{{\\mathbb {Q}/\\mathbb {Z}}}_I } .$ We note that Eq.", "(REF ) is invariant under the following gauge transformation: bQ/ZI bQ/ZI +Q/ZI, I Q/ZI KIJ 1=0.", "It is straight forward to check that this is indeed the case.", "When ${^4 =\\emptyset }$ , the gauge transformation Eq.", "bgauge changes the term ${^{2\\pi \\sum _{I\\le J}k_{IJ} \\int _{^4} \\hspace{-2.0pt} b_J^{\\mathbb {Q}/\\mathbb {Z}}b_I^{\\mathbb {Q}/\\mathbb {Z}}} }$ by a factor $\\begin{aligned}&^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} \\omega _J^{\\mathbb {Q}/\\mathbb {Z}}b_I^{\\mathbb {Q}/\\mathbb {Z}}+ b_J^{\\mathbb {Q}/\\mathbb {Z}}\\omega _I^{\\mathbb {Q}/\\mathbb {Z}}+\\omega _J^{\\mathbb {Q}/\\mathbb {Z}}\\omega _I^{\\mathbb {Q}/\\mathbb {Z}}} \\\\&\\hspace{20.0pt}= ^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} \\omega _J^{\\mathbb {Q}/\\mathbb {Z}}b_I^{\\mathbb {Q}/\\mathbb {Z}}-b_J^{\\mathbb {Q}/\\mathbb {Z}}\\omega _I^{\\mathbb {Q}/\\mathbb {Z}}}.\\end{aligned}$ However, using that ${^{2\\pi \\sum _{I, J} K_{IJ} \\int _{^4} \\omega _J^{\\mathbb {Q}/\\mathbb {Z}}b_I^{\\mathbb {Q}/\\mathbb {Z}}} =1}$ from Eq.", "(REF ), and also Eq.", "cupkrel, we can rewrite Eq.", "(REF ) as $^{-2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt}\\omega _I^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 1}}{\\smile } b_J^{\\mathbb {Q}/\\mathbb {Z}}} .$ Furthermore, the gauge transformation (REF ) changes the term $ {^{2\\pi \\sum _{I\\le J} k_{IJ} \\int _{^4} b^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\underset{{\\scriptscriptstyle 1}}{\\smile } b^{{\\mathbb {Q}/\\mathbb {Z}}}_J } }$ by a factor 2IJ kIJ 4 Q/ZI 1 bQ/ZJ .", "Eq.", "(REF ) and (REF ) perfectly cancel each other and therrefor action amplitude in Eq.", "Zab0 is gauge invariant.", "Because the action amplitude is invariant under Eq.", "(REF ), it will not depend on the coboundaries $a_I^{\\mathbb {Q}/\\mathbb {Z}}$ and, therefore, we will be able to evaluate the path integral Eq.", "Zab when $^4=\\emptyset $ .", "Plugging in $b_I^{\\mathbb {Q}/\\mathbb {Z}}$ and integrating by parts using that $^4$ is closed, the path integral Eq.", "(REF ) becomes $\\begin{aligned}Z &=\\hspace{-16.99997pt}\\sum _{\\sum _I a^{\\mathbb {Q}/\\mathbb {Z}}_{I} K_{IJ} \\overset{\\scriptscriptstyle 1}{=}0}\\hspace{-16.99997pt}^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} -a_J^{\\mathbb {Q}/\\mathbb {Z}}B_I^{\\mathbb {Q}/\\mathbb {Z}}+ B_J^{\\mathbb {Q}/\\mathbb {Z}}a_I^{\\mathbb {Q}/\\mathbb {Z}}}\\times \\\\&^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_J^{\\mathbb {Q}/\\mathbb {Z}}B_I^{\\mathbb {Q}/\\mathbb {Z}}+ B_J^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 1}}{\\smile } a_I^{\\mathbb {Q}/\\mathbb {Z}}- B_J^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 1}}{\\smile } B_I^{\\mathbb {Q}/\\mathbb {Z}}}.\\end{aligned}$ Now, we can use Eq.", "() to rewrite the terms ${a_J^{\\mathbb {Q}/\\mathbb {Z}}B_I^{\\mathbb {Q}/\\mathbb {Z}}}$ and ${B_J^{\\mathbb {Q}/\\mathbb {Z}}a_I^{\\mathbb {Q}/\\mathbb {Z}}}$ such that $Z$ becomes $\\begin{aligned}Z &=\\hspace{-16.99997pt}\\sum _{\\sum _I a^{\\mathbb {Q}/\\mathbb {Z}}_{I} K_{IJ} \\overset{\\scriptscriptstyle 1}{=}0}\\hspace{-16.99997pt}^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_J^{\\mathbb {Q}/\\mathbb {Z}}B_I^{\\mathbb {Q}/\\mathbb {Z}}- B_J^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 1}}{\\smile } B_I^{\\mathbb {Q}/\\mathbb {Z}}}\\times \\\\&\\hspace{30.0pt}^{-2\\pi \\hspace{-2.0pt}\\sum \\limits _{I,J} \\hspace{-2.0pt}K_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} a_I^{\\mathbb {Q}/\\mathbb {Z}}B_J^{\\mathbb {Q}/\\mathbb {Z}}}.\\end{aligned}$ Because the path integral only sums over $a_I^{\\mathbb {Q}/\\mathbb {Z}}$ satisfying the quantization condition ${\\sum _I a^{\\mathbb {Q}/\\mathbb {Z}}_{I} K_{IJ} \\overset{\\scriptscriptstyle 1}{=}0}$ and that ${B_J^{\\mathbb {Q}/\\mathbb {Z}}\\overset{\\scriptscriptstyle 1}{=} 0}$ , the term in the second line of $Z$ becomes unity.", "Then, using Eq.", "() to rewrite ${B_J^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 1}}{\\smile } B_I^{\\mathbb {Q}/\\mathbb {Z}}}$ , the path integral becomes $\\begin{aligned}Z &=\\hspace{-16.99997pt}\\sum _{\\sum _I a^{\\mathbb {Q}/\\mathbb {Z}}_{I} K_{IJ} \\overset{\\scriptscriptstyle 1}{=}0}\\hspace{-16.99997pt}^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_J^{\\mathbb {Q}/\\mathbb {Z}}B_I^{\\mathbb {Q}/\\mathbb {Z}}+ B_I^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 1}}{\\smile } B_J^{\\mathbb {Q}/\\mathbb {Z}}}\\times \\\\&\\hspace{30.0pt}^{-2\\pi \\hspace{-2.0pt}\\sum \\limits _{I,J} \\hspace{-2.0pt}K_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_J^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 2}}{\\smile } B_J^{\\mathbb {Q}/\\mathbb {Z}}}.\\end{aligned}$ Firstly, note that the action amplitude on the second line is unity since ${B_J^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 2}}{\\smile } B_J^{\\mathbb {Q}/\\mathbb {Z}}\\in }$ .", "Additionally, the action amplitude no longer contains the cochains $a_I^{\\mathbb {Q}/\\mathbb {Z}}$ which the path integral is summing over.", "Thus, performing the sum we obtain $Z \\hspace{-1.0pt}= \\hspace{-1.0pt}|\\det (K)|^{N_e}~^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_J^{\\mathbb {Q}/\\mathbb {Z}}B_I^{\\mathbb {Q}/\\mathbb {Z}}+ B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J } \\hspace{-6.0pt}.$ where $N_e$ is the number of edges in the triangulated spacetime $^4$ ." ], [ "The 1-SPT invariant", "The SPT order is characterized by the volume-independent partition function $Z^{\\text{top}}(^4,B_I^{{\\mathbb {Q}/\\mathbb {Z}}})= \\dfrac{Z (^4,B_I^{{\\mathbb {Q}/\\mathbb {Z}}})}{Z (^4,0)}.$ From this, we find that 1-SPT invariant for the 1-SPT state is $Z^\\text{top}(^4, B^{\\mathbb {Q}/\\mathbb {Z}}_I) =^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_J^{\\mathbb {Q}/\\mathbb {Z}}B_I^{\\mathbb {Q}/\\mathbb {Z}}+ B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J } ,$ where as a reminder $B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\overset{\\scriptscriptstyle 1}{=} 0, \\ \\ \\ \\ \\ \\sum _I B^{\\mathbb {Q}/\\mathbb {Z}}_{I} K_{IJ} \\overset{\\scriptscriptstyle 1}{=}0.$ Such a non-trivial 1-SPT invariant Eq.", "(REF ) suggests that the 1-SPT order can be non-trivial.", "We note that, as confirmed in appendix section , this 1-SPT invariant is correctly gauge invariant.", "However, before going on consider some examples of non-trivial 1-SPT invariants (see section REF ), we want to show that any two matrices $K$ and $K$ related by ${K =U^\\top K U}$ with ${U\\in GL(,)}$ actually describes the same 1-SPT invariant.", "We will first try to express the 1-SPT invariant Eq.", "(REF ) in terms of only the $K$ -matrix instead of $k_{IJ}$ .", "In doing so, we'll also find a nice form for the 1-SPT invariant which we can use when considering examples in the next section.", "Consider the term ${B_J^{\\mathbb {Q}/\\mathbb {Z}}B_I^{\\mathbb {Q}/\\mathbb {Z}}}$ in the SPT invariant Eq.", "(REF ).", "We can first rewrite it as $\\begin{aligned}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_J^{\\mathbb {Q}/\\mathbb {Z}}B_I^{\\mathbb {Q}/\\mathbb {Z}}&=\\frac{1}{2} \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_J^{\\mathbb {Q}/\\mathbb {Z}}B_I^{\\mathbb {Q}/\\mathbb {Z}}- B_I^{\\mathbb {Q}/\\mathbb {Z}}B_J^{\\mathbb {Q}/\\mathbb {Z}}\\\\&\\hspace{15.0pt}+\\frac{1}{2} \\hspace{-2.0pt}\\sum \\limits _{I,J} \\hspace{-2.0pt}K_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_I^{\\mathbb {Q}/\\mathbb {Z}}B_J^{\\mathbb {Q}/\\mathbb {Z}}.\\\\\\end{aligned}$ Then using Eq.", "cupkrel and the fact that $^4$ is closed, this can become $\\begin{aligned}&\\frac{1}{2} \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_J^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 1}}{\\smile } B_I^{\\mathbb {Q}/\\mathbb {Z}}+ B_J^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 1}}{\\smile } B_I^{\\mathbb {Q}/\\mathbb {Z}}\\\\&\\ \\ \\ \\ +\\frac{1}{2} \\sum _{I,J} K_{IJ} \\int _{^4} B_I^{\\mathbb {Q}/\\mathbb {Z}}B_J^{\\mathbb {Q}/\\mathbb {Z}}, \\\\&=\\frac{1}{2} \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_J^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 1}}{\\smile } B_I^{\\mathbb {Q}/\\mathbb {Z}}- B_I^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 1}}{\\smile } B_J^{\\mathbb {Q}/\\mathbb {Z}}\\\\&\\ \\ \\ \\ +\\frac{1}{2} \\sum _{I,J} K_{IJ} \\int _{^4} B_I^{\\mathbb {Q}/\\mathbb {Z}}B_J^{\\mathbb {Q}/\\mathbb {Z}}+ B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J.\\end{aligned}$ Plugging this expression for ${B_J^{\\mathbb {Q}/\\mathbb {Z}}B_I^{\\mathbb {Q}/\\mathbb {Z}}}$ into the action in Eq.", "(REF ), we find $\\begin{aligned}&\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_J^{\\mathbb {Q}/\\mathbb {Z}}B_I^{\\mathbb {Q}/\\mathbb {Z}}+ B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J\\\\&=\\frac{1}{2} \\sum _{I,J} K_{IJ} \\int _{^4} B_I^{\\mathbb {Q}/\\mathbb {Z}}B_J^{\\mathbb {Q}/\\mathbb {Z}}+ B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J\\\\&\\ \\ \\ \\ +\\frac{1}{2} \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_J^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 1}}{\\smile } B_I^{\\mathbb {Q}/\\mathbb {Z}}+ B_I^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 1}}{\\smile } B_J^{\\mathbb {Q}/\\mathbb {Z}}.\\end{aligned}$ We can go further by again using Eq.", "cupkrel to rewrite second line of the right hand side and get $\\begin{aligned}&\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_J^{\\mathbb {Q}/\\mathbb {Z}}B_I^{\\mathbb {Q}/\\mathbb {Z}}+ B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J\\\\&= \\frac{1}{2} \\sum _{I,J} K_{IJ} \\int _{^4} B_I^{\\mathbb {Q}/\\mathbb {Z}}B_J^{\\mathbb {Q}/\\mathbb {Z}}+ B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J\\\\&\\ \\ \\ \\ -\\frac{1}{2} \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_J^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 2}}{\\smile } B_I^{\\mathbb {Q}/\\mathbb {Z}}.\\end{aligned}$ Therefore, the 1-SPT invariant that characterizes the 1-SPT order has the form $\\begin{aligned}\\hspace{-5.0pt}Z^\\text{top}(^4, B^{\\mathbb {Q}/\\mathbb {Z}}_I\\hspace{-1.0pt}) &\\hspace{-2.0pt}=\\hspace{-2.0pt}^{\\pi \\hspace{-2.0pt}\\sum \\limits _{I, J} \\hspace{-2.0pt}K_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_I^{\\mathbb {Q}/\\mathbb {Z}}B_J^{\\mathbb {Q}/\\mathbb {Z}}+ B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J} \\\\&\\ \\ \\ \\ \\times ^{\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_I^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 2}}{\\smile } B_J^{\\mathbb {Q}/\\mathbb {Z}}} .\\end{aligned}$ We can recast the relationship between $k_{IJ}$ and $K_{IJ}$ , given by Eq.", "(REF ), by treating $k_{IJ}$ as the elements of the upper triangular integer matrix $k$ that satisfies K=k+k.", "Then, using that ${B_J^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 2}}{\\smile } B_I^{\\mathbb {Q}/\\mathbb {Z}}\\overset{\\scriptscriptstyle }{=} -B_I^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 2}}{\\smile } B_J^{\\mathbb {Q}/\\mathbb {Z}}}$ from Eq.", "cupkrel, we can replace ${\\sum _{I\\le J}k_{IJ}}$ by ${\\sum _{I< J} K_{IJ}}$ in Eq.", "(REF ) to obtain $\\begin{aligned}\\hspace{-5.0pt}Z^\\text{top}(^4, B^{\\mathbb {Q}/\\mathbb {Z}}_I\\hspace{-1.0pt}) &\\hspace{-2.0pt}=\\hspace{-2.0pt}^{\\pi \\hspace{-2.0pt}\\sum \\limits _{I, J} \\hspace{-2.0pt}K_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_I^{\\mathbb {Q}/\\mathbb {Z}}B_J^{\\mathbb {Q}/\\mathbb {Z}}+ B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J} \\\\&\\ \\ \\ \\ \\times ^{\\pi \\hspace{-2.0pt}\\sum \\limits _{I<J} \\hspace{-2.0pt}K_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_I^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 2}}{\\smile } B_J^{\\mathbb {Q}/\\mathbb {Z}}} .\\end{aligned}$ Eq.", "(REF ) thus provides a form of the 1-SPT invariant in terms of only the $K$ -matrix.", "However, due to the sum on the second term only being over ${I<J}$ , it is not covariant.", "We see that, from Eq.", "(REF ), the 1-SPT invariant is characterized by a pair of integer matrices $(K,k)$ .", "At first glance, due to the $k$ dependence, or equivalently Eq.", "(REF ) not being covariant, it appears that the SPT invariant is changed by the transformation ${B^{/}_I \\rightarrow B^{/}_I}$ and ${K\\rightarrow K }$ where B/I = (U-1)IJ B/J, K = UK U , and ${U_{IJ} \\in GL(,)}$ .", "Therefore, it would appear that $K$ and $K$ do not describe the same 1-SPT invariant.", "However, it turns out that $K$ and $K$ actually do describe the same 1-SPT invariant.", "To show this, we first show that the 1-SPT invariant is left unchanged when $k$ is replaced by another integer matrix $k^{\\prime }$ (not necessarily upper triangular) such that ${K = k^{\\prime }+k^{\\prime \\top }}$ .", "The difference ${A=k-k^{\\prime }}$ is an antisymmetric integer matrix.", "The respective lattice Lagrangian densities of the 1-SPT invariant Eq.", "(REF ) for $k$ and $k^{\\prime }$ (after dividing by $2\\pi $ ) differ by $\\sum _{I, J} \\frac{A_{IJ}}{2} B_I^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 2}}{\\smile } B_J^{\\mathbb {Q}/\\mathbb {Z}}.$ Using that $A$ is antisymmetric, that integer multiples of $2\\pi $ can be added to the Lagrangian density without changing the path integral, and Eq.", "cupkrel, this can be rewritten as $\\begin{aligned}&\\sum _{I, J} \\frac{A_{IJ}}{2} B_I^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 2}}{\\smile } B_J^{\\mathbb {Q}/\\mathbb {Z}}\\\\&\\hspace{20.0pt}\\overset{\\scriptscriptstyle 1}{=}\\sum _{I< J} \\frac{A_{IJ}}{2} \\big (-B_I^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 3}}{\\smile } B_J^{\\mathbb {Q}/\\mathbb {Z}}\\big ),\\end{aligned}$ which vanishes on a closed manifold.", "Therefore, the two Lagrangian densities differ by only a coboundary term and give the same topological invariants for a closed $^4$ .", "The above result allows us to show that the SPT invariant is unchanged under the transformation Eq.", "(REF ).", "Indeed, we now only need to check the $k_{IJ}$ term in Eq.", "(REF ).", "Let $\\bar{k}$ be an integer matrix defined by $\\bar{k}_{IJ} = \\sum _{I^{\\prime }\\le J^{\\prime }} (U^\\top )_{II^{\\prime }} k_{I^{\\prime }J^{\\prime }} U_{J^{\\prime }J},$ such that ${K = \\bar{k} + \\bar{k}^\\top }$ .", "Using that, from Eq.", "(REF ), ${B^{/}_I = U_{IJ} B^{/}_J}$ and plugging it into the second line of Eq.", "(REF ), it becomes $^{\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_I^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 2}}{\\smile } B_J^{\\mathbb {Q}/\\mathbb {Z}}} \\hspace{-8.0pt}= ^{\\pi \\hspace{-2.0pt}\\sum \\limits _{I, J} \\hspace{-1.0pt} \\bar{k}_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt}B_I^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 2}}{\\smile } B_J^{\\mathbb {Q}/\\mathbb {Z}}} .$ Let's now introduce the upper triangular integer matrix $k$ such that ${K = k + k^\\top }$ .", "Using the above result for a closed $^4$ , the SPT invariant is unchanged by replacing $\\bar{k}$ with $k$ .", "Therefore, the SPT invariant Eq.", "(REF ) is unchanged under the transformation Eq.", "(REF ).", "The fact that $K$ and $K$ describes the same SPT invariant also allows us to find a convenient expression for the SPT invariant.", "Indeed, recall that the integer matrix $K$ has the Smith normal form given by Eq.", "(REF ).", "From the above discussion, we see that, without loosing generality, we may transform ${K\\rightarrow U^\\top K U}$ without changing the SPT invariant and thus may assume $K$ to have the following form K= V k1 k2 k3 = k1 k2 k3 V .", "The invertible integer matrix $V$ satisfies (V)IJ kJ = kJ VJI = kI VIJ or VIJVJI = kJkI.", "Using this expression for $K$ , the 1-SPT invariant in its original form given by Eq.", "(REF ) can be rewritten as $\\begin{aligned}&Z^\\text{top}(^4, B^{\\mathbb {Q}/\\mathbb {Z}}_I) = ^{\\pi \\sum \\limits _{I} k_{I}V_{II}\\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_I^{\\mathbb {Q}/\\mathbb {Z}}B_I^{\\mathbb {Q}/\\mathbb {Z}}+ B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_I } \\times \\\\&\\hspace{80.0pt} ^{2\\pi \\sum \\limits _{I< J} \\hspace{-1.0pt}k_{I}V_{IJ}\\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_J^{\\mathbb {Q}/\\mathbb {Z}}B_I^{\\mathbb {Q}/\\mathbb {Z}}+ B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J } .\\end{aligned}$ Furthermore, the quantization condition on the background cochain field then becomes ${\\sum _I B^{\\mathbb {Q}/\\mathbb {Z}}_{I} k_I V_{IJ} \\overset{\\scriptscriptstyle 1}{=}0}$ .", "This can be automatically satisfied if we let $B^{\\mathbb {Q}/\\mathbb {Z}}_{I}$ take the form BQ/ZI = kI-1 BkII, where $B^{_{k_I}}_I$ is a $_{k_I}$ -valued 2-cocycle and thus satisfies ${B^{_{k_I}}_I \\overset{\\scriptscriptstyle k_I}{=} 0 }$ .", "Using this, the 1-SPT invariant for the ${_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots }$ 1-symmetry becomes Ztop(4, BkII) = I VIIkI-1 4 BIkI BIkI + BkII 1 BkII         2I< J VIJkJ-1 4 BJkJ BIkI + BkII 1 BkJJ ." ], [ "Some Examples of SPT Invariants", "In the previous section, we found that $U^\\kappa (1)$ gauge theory with a $2\\pi $ topological term in the confined phase has a non-trivial 1-SPT invariant, Eq.", "(REF ).", "We then massaged the SPT invariant into other forms, such as Eq.", "(REF ) and Eq.", "(REF ).", "This suggests that generically there is a phase in the confined phase with non-trivial 1-SPT order which is protected by the ${_{k_1}^{(1)}\\times _{k_2}^{(1)}\\times \\cdots }$ symmetry discussed in section REF .", "Now we will consider same simple examples of different $K$ -matrices and the corresponding 1-SPT order.", "The first example will have ${\\kappa = 1}$ while the second and third will be ${\\kappa =2}$ .", "Example 1 Let's first consider the case where there is only one type of cochain field $a^{\\mathbb {R}/\\mathbb {Z}}$ so ${=1}$ and the $K$ -matrix would become $K=\\begin{pmatrix}2n\\end{pmatrix},$ with ${n\\in }$ .", "In this case, the 3+1D bosonic model on spacetime lattice Eq.", "(REF ) becomes Z = D[aR/Z] - 4 |aR/Z-aR/Z |2g 2n 4 (aR/Z-aR/Z )(aR/Z-aR/Z ) 4n 4 aR/Z aR/Z - aR/Z 1aR/Z .", "From our previous discusions, this theory has a $_{2n}^{(1)}$ symmetry.", "Let's see this explicitly.", "The path integral is invariant under the transformation ${a^{\\mathbb {R}/\\mathbb {Z}}\\rightarrow a^{\\mathbb {R}/\\mathbb {Z}}+ \\frac{1}{2n}\\beta ^}$ where $\\beta ^$ is an arbitrary $$ -valued 1-cochain satisfying ${\\beta ^\\overset{\\scriptscriptstyle 2n}{=} 0}$ .", "The physical part of $\\beta ^$ is defined modulo $2n$ because shifting $\\beta ^$ by $2n$ -valued 1-cochain corresponds to shifting $a^{\\mathbb {R}/\\mathbb {Z}}$ by an integer-valued 1-cochain, which is a gauge transformation.", "Therefore, this theory indeed has a $_{2n}^{(1)}$ symmetry.", "When $g\\ll 1$ , the above bosonic model at low-energies describes the deconfined phase of $U(1)$ gauge field theory.", "At energies much smaller than the energy gap of the $U(1)$ monopole, ${\\lfloor a^{\\mathbb {R}/\\mathbb {Z}} \\rceil = 0}$ and the $_{2n}^{(1)}$ symmetry is promoted to an emergent $U(1)^{(1)}$ symmetry.", "When $g\\gg 1$ , the above bosonic model is in a gapped phase with $_{2n}^{(1)}$ 1-symmetry, which corresponds to the confined phase of the $U(1)$ gauge theory.", "From our general discussion, this gapped phase is an SPT phase protected by the $_{2n}^{(1)}$ 1-symmetry.", "Indeed, using Eq.", "1SPTinvk, this SPT phase is characterized by the 1-SPT invariant $\\begin{aligned}Z^\\text{top}(^4, B^{_{2n}})&= ^{\\frac{2\\pi }{4n} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B^{_{2n}}B^{_{2n}} + B^{_{2n}} \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{_{2n}}},\\\\&= ^{\\frac{2\\pi }{4n} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} \\mathbb {Sq}^2(B^{_{2n}}) }.\\end{aligned}$ The 3+1D 1-SPT order for the $_{2n}^{(1)}$ 1-symmetry is classified by ${H^4(B(_{2n},2);{\\mathbb {R}/\\mathbb {Z}})=_{4n}}$ .", "From the SPT invariant, we find that the 1-SPT order realized by the confined phase is given by $1\\in _{4n}$ , and thus is the generator of the SPT orders classified by $H^4(B(_{2n},2);{\\mathbb {R}/\\mathbb {Z}})$ .", "Example 2 Let's now consider an example where there are two types of 1-cochain fields $a^{\\mathbb {R}/\\mathbb {Z}}_1$ and $a^{\\mathbb {R}/\\mathbb {Z}}_2$ , so ${\\kappa = 2}$ , and the $K$ -matrix is given by $K =\\begin{pmatrix}2 & 1\\\\1& 2\\\\\\end{pmatrix}.$ We'd like to find the SPT invariant for this $K$ matrix using Eq. 1SPTinvk.", "This requires us to first find the integers ${k_1,k_2}$ and the integer matrix $V$ from Eq.", "(REF ).", "The diagonal elements of the Smith normal form of $K$ are ${(k_1,k_2) = (3,1)}$ .", "Thus, by finding $k_I$ we can immediately conclude that the 1-symmetry is ${_3^{(1)}\\times _1^{(1)}\\equiv _3^{(1)}}$ .", "However, there does not exist an integer matrix $V$ which will work for this $K$ .", "To find the SPT invariant, we can instead consider the matrix $K =\\begin{pmatrix}6 & 3\\\\3 & 2\\\\\\end{pmatrix}.$ Since ${K = U K U^\\top }$ , where $U =\\begin{pmatrix}1 & 1\\\\1& 0\\\\\\end{pmatrix}\\in GL(2,),$ our results from section REF show that the SPT order of $K$ is equivalent to that of the $K$ matrix Eq.", "(REF ).", "Therefore, we now attempt to find the SPT invariant using the same approach but now with $K$ .", "First note that the diagonal elements of the Smith normal form of $K$ are still ${(k_1,k_2) = (3,1)}$ .", "The $K$ matrix can be written as K = 6 3 3 2 = 3 0 0 1 2 1 3 2 , and we find the integer matrix $V$ to be $V=\\begin{pmatrix}2 & 1\\\\3 & 2\\\\\\end{pmatrix}.$ From the $k_I$ and $V$ found for $K$ , we have that (VIJkJ-1) = 23 1 1 2 .", "Using Eq.", "(REF ), the corresponding 1-SPT invariant for the ${_3^{(1)}\\times _1^{(1)}\\equiv _3^{(1)}}$ 1-symmetry is given by $\\begin{aligned}&Z^\\text{top}(^4, B^{_{k_I}}_I) =^{\\frac{2\\pi }{3} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt}B_1^{_3} B_1^{_3} + B^{_3}_1 \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{_3}_1} \\times \\\\&\\hspace{90.0pt}^{2\\pi \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_2^{_1} B_1^{_3} + B^{_3}_1 \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{_1}_2}\\times \\\\&\\hspace{90.0pt}^{2\\pi \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt}B_2^{_1} B_2^{_1} + B^{_1}_2 \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{_1}_2}.\\end{aligned}$ We can now use the fact that the SPT invariant is invariant under the gauge transformation ${B_2^{_1} \\rightarrow B_2^{_1} + m^}$ , where $m^$ is a $$ -valued 2-cochain, to set ${B_2^{_1} = 0}$ .", "Doing so, the SPT invariant simplifies to $Z^\\text{top}(^4, B^{_{k_I}}_I) =^{\\frac{2\\pi }{3} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt}B_1^{_3} B_1^{_3} + B^{_3}_1 \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{_3}_1}.$ Therefore, the 1-SPT invariant of the K matrix Eq.", "(REF ) is given by Eq.", "(REF ).", "1-SPT order protected by the 1-symmetry $_3^{(1)}$ is classified by ${H^4(B(_3,2);{\\mathbb {R}/\\mathbb {Z}})=_3}$ .", "Therefore, from Eq.", "(REF ) the SPT order realized in the confined phase for the $K$ -matrix Eq.", "(REF ) is given by $1 \\in _3$ and is thus the generator for SPT orders classified by ${H^4(B(_3,2);{\\mathbb {R}/\\mathbb {Z}})}$ .", "Example 3 For our final example, let's again consider the scenario where there are ${\\kappa = 2}$ cochain fields $a^{\\mathbb {Q}/\\mathbb {Z}}_I$ , but now where the $K$ -matrix is $K=\\begin{pmatrix}0 & n\\\\n& 0\\\\\\end{pmatrix}, \\ \\ \\ \\ n\\in .$ The diagonal elements of the Smith normal form of this $K$ -matrix are ${(k_1,k_2) = (n,n)}$ .", "Therefore, the model with this $K$ matrix has a ${_n^{(1)}\\times _n^{(1)}}$ symmetry.", "Furthermore, this $K$ matrix can be written as K = 0 n n 0 = n 0 0 n 0 1 1 0 .", "Thus, unlike example 2, using the $K$ matrix we start with, there exists the integer matrix $V=\\begin{pmatrix}0 & 1\\\\1 & 0\\\\\\end{pmatrix}.$ From this matrix $V$ and from $k_I$ , we find that $(V_{IJ}k_J^{-1}) =\\begin{pmatrix}0 & \\frac{1}{n}\\\\\\frac{1}{n} & 0\\\\\\end{pmatrix} .$ Using Eq.", "(REF ), the corresponding 1-SPT invariant for the $_n^{(1)}\\times _n^{(1)}$ 1-symmetry is given by Ztop(4, BnI) = 2n 4 B2n B1n + Bn1 1 Bn2 .", "Thus, we find that the 1-SPT order in the confined phase of ${U(1)\\times U(1)}$ 3+1D gauge theory with $K$ matrix Eq.", "(REF ) is a mixed SPT order between the two $_n^{(1)}$ 1-symmetries.", "In other words, the boundary Chern-Simons theory has a mixed anomaly between two $_n^{(1)}$ 1-symmetries.", "This Chern-Simons theory describes 2+1D $_n$ topological order.", "Indeed, the loop operators charged under the two $_n^{(1)}$ 1-symmetries are the loop objects whose open ends correspond to $e$ and $m$ type anyons, respectively.", "Furthermore, the fact that the $e$ and $m$ anyons have nontrivial mutual statistics is a manifestation of the mixed anomaly between the two $_n^{(1)}$ 1-symmetries." ], [ "Conclusion", "In this paper, we have considered 3+1D compact $U^(1)$ gauge theory with $2\\pi $ -quantized topological terms.", "In section REF , we developed a bosonic lattice model acting as the UV regularization for the continuum theory.", "Working with this lattice model, we found that at energies much smaller than the gauge charges' gaps but larger than the monopoles' gaps, there is an emergent ${_{k_1}^{(1)} \\times _{k_2}^{(1)} \\times \\cdots }$ 1-symmetry.", "We found that the confined phase of the $U^(1)$ gauge theory (the symmetric gapped phase of the bosonic model) has non trivial symmetry protected topological (SPT) order which is protected by the emergent ${_{k_1}^{(1)} \\times _{k_2}^{(1)} \\times \\cdots }$ 1-symmetry.", "We then went on to gauge this emergent symmetry in section REF and found the corresponding SPT invariant in section REF .", "We gave some examples of different $K$ matricies where the confined phases of the $U^(1)$ gauge theories realizes a $_{2n}^{(1)}$ 1-SPT phase, a $_{3}^{(1)}$ 1-SPT phase, and a $_{n}^{(1)}\\times _{n}^{(1)}$ mixed 1-SPT phase.", "Note: after the completion of this paper, we noticed the independent work MF220607725 which studied the emergent 1-symmetry for the ${=1}$ case in a phase where monopoles were only partially condensed." ], [ "Acknowledgements", "SDP is thankful for helpful and fun discussions with Ethan Lake and Ho Tat Lam on higher-form symmetries and with Michael DeMarco on lattice Chern-Simons theory.", "SDP, additionally, acknowledges support from the Henry W. Kendall Fellowship.", "This work is partially supported by NSF DMR-2022428 and by the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation (651446, XGW)." ], [ "The Gauge Invariance of the 1-SPT Invariant", "In section REF of the main text, we found that the 1-SPT invariant for the $_{k_1}^{(1)} \\times _{k_2}^{(1)} \\times \\cdots $ 1-symmetry given by Eq.", "(REF ): $Z^\\text{top}(^4, B^{\\mathbb {Q}/\\mathbb {Z}}_I) =^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} B_J^{\\mathbb {Q}/\\mathbb {Z}}B_I^{\\mathbb {Q}/\\mathbb {Z}}+ B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J }.$ Here, $B^{{\\mathbb {Q}/\\mathbb {Z}}}_I$ , with ${I=1,\\ldots \\kappa }$ , are background symmetry twist 2-cochain fields satisfiying $B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\overset{\\scriptscriptstyle 1}{=} 0, \\ \\ \\ \\ \\ \\sum _I B^{\\mathbb {Q}/\\mathbb {Z}}_{I} K_{IJ} \\overset{\\scriptscriptstyle 1}{=}0.$ In this appendix section, we confirm a claim made in the main text that the above 1-SPT invariant for closed $^4$ is invariant under the gauge transformations BQ/ZI BQ/ZI + nI, BQ/ZI BQ/ZI + aIQ/Z, where $n_I$ are $$ -valued 2-cochains and $a_I^{\\mathbb {Q}/\\mathbb {Z}}$ are ${\\mathbb {Q}/\\mathbb {Z}}$ -valued 1-cochains satisfying the quantization conditions ${\\sum _I a^{\\mathbb {Q}/\\mathbb {Z}}_{I} K_{IJ}\\overset{\\scriptscriptstyle 1}{=}0}$ .", "First, we'll check the $$ -gauge transformation ${B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\rightarrow B^{{\\mathbb {Q}/\\mathbb {Z}}}_I + n_I}$ , which causes the 1-SPT invariant to change by a factor $^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} n_J B_I^{\\mathbb {Q}/\\mathbb {Z}}+B_J^{\\mathbb {Q}/\\mathbb {Z}}n_I }\\hspace{-2.0pt}^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} n_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J +B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\underset{{\\scriptscriptstyle 1}}{\\smile } n_J }\\hspace{-2.0pt}.$ Assuming that $^4 =\\emptyset $ and using (), this can be rewritten as unity: $\\begin{aligned}&^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} n_J B_I^{\\mathbb {Q}/\\mathbb {Z}}+B_J^{\\mathbb {Q}/\\mathbb {Z}}n_I }\\hspace{-2.0pt}^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} n_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J +B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\underset{{\\scriptscriptstyle 1}}{\\smile } n_J }\\hspace{-2.0pt}\\\\&=^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I, J} \\hspace{-2.0pt}K_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt}B_J^{\\mathbb {Q}/\\mathbb {Z}}n_I }^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} n_J B_I^{\\mathbb {Q}/\\mathbb {Z}}- B_I^{\\mathbb {Q}/\\mathbb {Z}}n_J }\\nonumber \\\\&\\ \\ \\ \\ \\ \\quad \\times ^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} n_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J +B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\underset{{\\scriptscriptstyle 1}}{\\smile } n_J }\\nonumber \\\\&=^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} - B_I^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 1}}{\\smile } n_J - B_I^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 1}}{\\smile } n_J }\\nonumber \\\\&\\ \\ \\ \\ \\quad \\times ^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} n_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J +B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\underset{{\\scriptscriptstyle 1}}{\\smile } n_J }\\nonumber \\\\&= ^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} n_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J - B_I^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 1}}{\\smile } n_J }\\nonumber \\\\&= ^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} n_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J + n_J \\underset{{\\scriptscriptstyle 1}}{\\smile } B_I^{\\mathbb {Q}/\\mathbb {Z}}+ n_J \\underset{{\\scriptscriptstyle 2}}{\\smile } B_I^{\\mathbb {Q}/\\mathbb {Z}}}\\nonumber \\\\&= ^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I, J} \\hspace{-2.0pt}K_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} n_I \\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J } = 1.\\end{aligned}$ Therefore, the SPT invariant is unchanged by the gauge transformation ${B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\rightarrow B^{{\\mathbb {Q}/\\mathbb {Z}}}_I + n_I}$ .", "Lastly, let's check the gauge transformation ${B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\rightarrow B^{{\\mathbb {Q}/\\mathbb {Z}}}_I + a_I^{\\mathbb {Q}/\\mathbb {Z}}}$ , which causes the 1-SPT invariant to change by a factor $^{2\\pi \\hspace{-2.0pt}\\sum \\limits _{I\\le J} \\hspace{-2.0pt}k_{IJ} \\hspace{-4.0pt}\\int \\limits _{^4} \\hspace{-2.0pt} a_J^{\\mathbb {Q}/\\mathbb {Z}}B_I^{\\mathbb {Q}/\\mathbb {Z}}+B_J^{\\mathbb {Q}/\\mathbb {Z}}a_I^{\\mathbb {Q}/\\mathbb {Z}}+ a_I^{\\mathbb {Q}/\\mathbb {Z}}\\underset{{\\scriptscriptstyle 1}}{\\smile } B^{{\\mathbb {Q}/\\mathbb {Z}}}_J }.$ Once again, assuming $^4 =\\emptyset $ and using (), we can show that this change is equal to unity: 2IJ kIJ 4 aJQ/ZBIQ/Z+BJQ/ZaIQ/Z+ aIQ/Z1 BQ/ZJ = 2IJ kIJ 4 aJQ/ZBIQ/Z+ aIQ/ZBJQ/Z- aIQ/Z1 BJQ/Z    2IJ kIJ 4 aIQ/Z1 BQ/ZJ =2I, J KIJ 4 aJQ/ZBIQ/Z =1 Therefore, the SPT invariant is also unchanged by the gauge transformation ${B^{{\\mathbb {Q}/\\mathbb {Z}}}_I \\rightarrow B^{{\\mathbb {Q}/\\mathbb {Z}}}_I + a_I^{\\mathbb {Q}/\\mathbb {Z}}}$ ." ] ]
2207.03544
[ [ "CMC Tori in the Generalised Berger Spheres and their Duals" ], [ "Abstract The study of minimal surfaces has a long history, due to the important applications.", "Given a fixed boundary, one wants to minimise the surface area: this can be used, for example, to minimise the area of the roof of a building.", "Similarly, looking for constant mean curvature (CMC) provides us with many interesting applications in physics: one of the easiest examples are soap bubbles.", "In this work however we occupy ourselves with minimal and constant mean curvature surfaces in the three-dimensional sphere $S^3$ and its dual space $\\Sigma^3$.", "In Chapter 1 we give a brief overview of the tools of Riemannian and Lorentzian geometry that we will use.", "We then take a closer look at $S^3$, computing its Levi-Civita connection and sectional curvatures: in Chapter 2 with respect to the Riemannian metric g and in Chapter 4 with respect to the Lorentzian metric h. Further, we determine some minimal and CMC tori inside ($S^3$, g) in Chapter 3 and in ($S^3$, h) in Chapter 5.", "We then proceed with the dual space $\\Sigma^3$ of $S^3$.", "In Chapter 6, we calculate the Levi-Civita connection and sectional curvatures with respect to g, and with respect to h in Chapter 8.", "Again we look for minimal and CMC tori of a certain family in ($\\Sigma^3$, g) in Chapter 7 and in ($\\Sigma^3$, h) in Chapter 9.", "In the appendix, the reader will find a Maple program.", "It was written to check the computations of the $S^3$ cases, but it can easily be adapted to$\\Sigma^3$." ], [ " CMC Tori in the Generalised Berger Spheres and their Duals Johanna Marie Gegenfurtner 20228 LUNFMA41352022 Abstract The study of minimal surfaces has a long history, due to the important applications.", "Given a fixed boundary, one wants to minimise the surface area: this can be used, for example, to minimise the area of the roof of a building.", "Similarly, looking for constant mean curvature (CMC) provides us with many interesting applications in physics – one of the easiest examples are soap bubbles.", "In this work however we occupy ourselves with minimal and constant mean curvature surfaces in the three-dimensional sphere $S^3$ and its dual space $\\Sigma ^3$ .", "In Chapter 1 we give a brief overview of the tools of Riemannian and Lorentzian geometry that we will use.", "We then take a closer look at $S^3,$ computing its Levi-Civita connection and sectional curvatures: in Chapter 2 with respect to the Riemannian metric $g$ and in Chapter 4 with respect to the Lorentzian metric $h$ .", "Further, we determine some minimal and CMC tori inside $(S^3,g)$ in Chapter 3 and in $(S^3,h)$ in Chapter 5.", "We then proceed with the dual space $\\Sigma ^3$ of $S^3.$ In Chapter 6, we calculate the Levi-Civita connection and sectional curvatures with respect to $g,$ and with respect to $h$ in Chapter 8.", "Again we look for minimal and CMC tori of a certain family in $(\\Sigma ^3,g)$ in Chapter 7 and in $(\\Sigma ^3,h)$ in Chapter 9.", "In the appendix, the reader will find a Maple program.", "It was written to check the computations of the $S^3$ cases, but it can easily be adapted to $\\Sigma ^3.$ Throughout this work it has been my firm intention to give reference to the stated results and credit to the work of others.", "All theorems, propositions, lemmas and examples left unmarked are either assumed to be well known, or are the fruits of my own efforts.", "Acknowledgments I'd like to thank my supervisor Sigmundur Gudmundsson for his help and guidance.", "I immensely appreciate the patience and confidence he showed me throughout this cooperation.", "Further, I'd like to express my deep gratitude towards my family, who has supported me during my studies with relentless encouragement.", "I'd like to thank my friends for all the beautiful distractions during the work on this project and the previous years.", "Lastly, I want to thank la mia amica geniale Lauren Tropeano for her unfailing warmth and wisdom.", "Johanna Marie Gegenfurtner m A main result in Lawson's paper from 1970 is that for any $g\\in \\mathbb {N},$ there exists an embedding of a compact minimal surface of genus $g$ into $S^3.$ Moreover, the embedding is not unique if $g$ is not prime.", "Lawson further conjectured that for $g=1,$ the only compact minimal surface embedded in $S^3$ is the flat Clifford torus given by $T=\\lbrace (x_1,x_2,x_3,x_4)\\in S^3\\subset \\mathbb {R}^4 \\ | \\ x_1^2+x_2^2=x_3^2+x_4^2=\\tfrac{1}{2}\\rbrace .$ This was finally proven by Brendle in 2013, see .", "It is important to speak of an embedding, since in Hsiang and Lawson constructed an infinite family of minimally immersed tori into $S^3.$ The Lawson conjecture however fails to hold in the Berger spheres.", "In , Torralbo considered the two-parametric Berger sphere $S^3_b (\\kappa ,\\tau )$ with the metric $g(A,B)=\\frac{4}{\\kappa }\\cdot (\\langle A,B \\rangle +(\\frac{4\\tau ^2}{\\kappa }-1)\\langle A,X \\rangle \\langle B,X \\rangle ).$ Here $X_{(z,w)}=(iz,iw)$ belongs to an orthogonal frame for the tangent bundle $TS^3$ of $S^3$ , $\\langle , \\rangle $ denotes the usual metric on the sphere and $\\kappa ,\\tau \\in \\mathbb {R}$ with $\\kappa >0$ and $\\tau \\ne 0.$ Note that $S^3_b(4,1)$ denotes the standard round sphere.", "Torralbo gave an example of another compact embedded minimal torus in the Berger sphere $S^3_b (4, 0.4),$ which is not the Clifford torus $T.$ It can also be interesting to study when the mean curvature of a manifold is constant.", "In , De Lima et al.", "considered a family of tori given by an embedding $\\Phi _{r,\\tau }:S^1\\times S^1\\mapsto S_b^3$ into the Berger sphere $S_b^3(4,\\tau ).$ Here the map $\\Phi _{r,\\tau }$ is defined by $(z,w)\\mapsto (z\\cdot r, \\sqrt{1-r^2}\\cdot w),$ where $r\\in (0,1).$ As it turns out each of those tori has constant mean curvature $H_r=\\frac{2r^2-1}{2r\\sqrt{1-r^2}}.$ In this work we will consider a family of tori parametrised analogously to the above, we however generalise the Berger metric further and obtain a three parameter family of metrics.", "Then we check under which conditions this matches the result of De Lima et al.", "in .", "Then we define a Lorentzian Berger metric and investigate the mean curvature.", "Additionally we will look for constant mean curvature tori of a certain family in $\\Sigma ^3=\\lbrace (z,w)\\ | \\ z,w\\in {\\mathbb {C}}^2,\\ |z|^2-|w|^2=1\\rbrace ,$ under both the Riemannian and Lorentzian Berger metrics.", "As it will be shown, the switch from the Riemannian to Lorentzian metrics does not impact the mean curvature H of the tori in question inside $S^3$ and $\\Sigma ^3$ respectively.", "Further we show that under certain conditions, for every real $C\\ge 0,$ there exist tori in $S^3$ with constant mean curvature $\\Vert H_g\\Vert \\equiv C.$ For the tori in $\\Sigma ^3,$ this can only be shown for $C>\\tfrac{1}{\\mu },$ where $\\mu $ is a parameter belonging to the generalised Berger metric.", "A similar statement, but using a different generalisation of the Berger metric, was given by Torralbo in .", "We also compute the Levi-Civita connection and sectional curvatures in all four cases.", "We assume that the reader is familiar with fundamental Riemannian geometry, however we first review some essential facts, that will be needed in our study of minimal and CMC submanifolds.", "This chapter is largely based on do Carmo's introductory textbook and Gudmundsson's lecture notes , whereas O'Neill's textbook was used for the parts on Lorentzian and semi-Riemannian geometry.", "A differentiable manifold $(M^m,\\mathcal {A})$ is defined to be a topological manifold $M$ of dimension $m$ together with a family of local charts $\\mathcal {A}.$ We require $M$ to be locally homeomorphic to $\\mathbb {R}^m,$ i.e.", "for every $p\\in M$ there exists a neighbourhood $U_p$ containing $p$ and a homeomorphism $x_p: U_p\\mapsto \\mathbb {R}^m.$ Further, the atlas $\\mathcal {A}=\\lbrace (U_\\alpha ,x_\\alpha ), \\alpha \\in \\mathcal {I}\\rbrace ,$ needs to cover the whole of $M$ and must be maximal, subject to the condition that the transition maps $x_\\beta \\circ x_\\alpha ^{-1}\\vert _{x_\\alpha (U_\\alpha \\cap U_\\beta )}:x_\\alpha (U_\\alpha \\cap U_\\beta )\\subset \\mathbb {R}^m\\rightarrow \\mathbb {R}^m, \\quad \\forall \\ \\alpha ,\\beta \\in \\mathcal {I}$ are differentiable of class $C^\\infty .$ Let $(M^m,\\mathcal {A}_M)$ be a differentiable manifold, where $\\mathcal {A}_M$ is an atlas on $M.$ Following Proposition 2.11 in , a subset $N$ of $M$ , is called a submanifold of $M$ if for each $p\\in N,$ $\\mathcal {A}_M$ contains some $(U_p,x_p),$ such that $p\\in U_p$ and $x_p(U_p\\cap N)=x_p(U_p)\\cap (\\mathbb {R}^n\\times \\lbrace 0\\rbrace ).$ Here $n\\in \\mathbb {N}$ such that $n\\le m$ denotes the dimension of $N,$ and $m-n$ is called the codimension of $N$ in $M.$ The atlas $\\mathcal {A}_M$ induces a structure on N, denoted $\\mathcal {A}_N,$ see Proposition 2.11 in .", "Let $\\lbrace e_1,\\dots ,e_n\\rbrace $ be an orthonormal basis of a semi-Euclidean space $V,$ and for a nondegenerate scalar product $\\langle , \\rangle ,$ set $\\epsilon _i=\\langle e_i,e_i \\rangle =\\pm 1.$ The number $v,$ where $0\\le v\\le n,$ of negative signs in the signature $(\\epsilon _1,\\dots ,\\epsilon _n)$ is called the index of $V.$ Note that the index is independent of the choice of basis, as shown in Lemma 26 in .", "Let $M$ be a smooth manifold.", "A metric tensor $h$ is a symmetric and nondegenerate tensor field of type (0,2) of constant index.", "That is, to every point $p\\in M,$ $h$ associates a scalar product $h_p: T_p M\\times T_p M\\mapsto \\mathbb {R},$ such that the index of $h_p$ is the same for all $p.$ The pair $(M,h)$ is a semi-Riemannian manifold.", "A metric on a submanifold $N\\subset M$ is obtained by restricting $h$ to $N.$ (Also see Definition 5.5 in .)", "A Riemannian metric has index 0 and is positive-definite.", "If $v=1$ and $\\dim M\\ge 2,$ $M$ is a so called Lorentz manifold.", "Particularly in the context of relativity we might want to describe the causal character of a tangent vector $W$ of a semi-Riemannian manifold $(M,h):$ If $h(W,W)<0,& &\\textrm {W is {\\it timelike,}}\\\\h(W,W)=0,& \\ \\textrm {and W\\ne 0,}& \\textrm {W is {\\it lightlike} }\\\\h(W,W)>0,& \\ \\textrm {or W=0},& \\textrm {W is {\\it spacelike}}.$ According to Lemma 25 in each vector $W\\in V$ can uniquely be written as $W=\\sum _{i=1}^{n} \\epsilon _i\\cdot h(W,e_i)\\cdot e_i.$ To distinguish the Riemannian from the Lorentzian case, we will write $g$ for a Riemannian metric and $h$ for its Lorentzian counterpart.", "For a differentiable manifold $M,$ and vector fields $X,Y\\in C^{\\infty }(TM),$ and $p\\in M,$ the Lie bracket $[X,Y]_p:C^{\\infty }(M)\\rightarrow \\mathbb {R}$ is defined by $X_p(Y(f))-Y_p(X(f)).$ We recall that the Levi-Civita connection $\\nabla : C^\\infty (TM)\\times C^\\infty (TM)\\mapsto C^\\infty (TM)$ on a Riemannian manifold $(M,g)$ is defined to be the unique affine connection that is symmetric and compatible with the Riemannian metric.", "Given an orthonormal frame $\\lbrace E_1,\\dots ,E_n\\rbrace $ of the tangent bundle, the Levi-Civita connection for $A,B\\in C^\\infty (TM)$ is given by $\\hbox{$\\nabla $\\hspace{-3.00003pt}$A$}\\hspace{-1.00006pt}{B}$ =i=1n g($\\nabla $$A$$B$ ,Ei)Ei.", "$The coefficients are given by the {\\it Koszul formula}: For $ , B, CC(TM),$\\begin{eqnarray}g(\\hbox{$\\nabla $\\hspace{-3.00003pt}$A$}\\hspace{-1.00006pt}{B}\\end{eqnarray},C)&=&\\frac{1}{2}\\cdot \\lbrace A(g(B,C))+B(g(C,A))-C(g(A,B))\\\\&&\\quad +g(C,[A,B])+g(B,[C,A])-g(A,[B,C])\\rbrace .\\nonumber $ We will later use the Levi-Civita connection to compute the sectional curvatures of $S^3$ and $\\Sigma ^3.$ Sectional curvatures describe the curvature of Riemannian manifold of dimension larger than 1.", "The Riemann curvature operator $R$ is given by $R(A,B)C&=&\\hbox{$\\nabla $\\hspace{-3.00003pt}$A$}\\hspace{-1.00006pt}{\\hbox{$\\nabla $\\hspace{-3.00003pt}$B$}\\hspace{-1.00006pt}{C}}$ -" ] ]
2207.03535
[ [ "Counting rotational subsets of the circle $\\mathbb{R}/\\mathbb{Z}$ under\n the angle multiplying map $t\\mapsto dt$" ], [ "Abstract A rotational set is a finite subset $A$ of the unit circle $\\mathbb{T}=\\mathbb{R}/ \\mathbb{Z}$ such that the angle-multiplying map $\\sigma_{d}:t\\mapsto dt$ restricted to $A$ is order-preserving and onto.", "Each rotational set has a geometric rotation number $p/q$.", "These sets were introduced by Lisa Goldberg to study the dynamics of complex polynomial maps.", "In this paper we provide a necessary and sufficient condition for a set to be $\\sigma_{d}$-rotational with rotation number $p/q$.", "As applications of our condition, we recover two classical results and enumerate $\\sigma_d$-rotational sets with rotation number $p/q$ that consist of a given number of orbits." ], [ "Introduction and preliminaries", "Let $\\mathbb {T}=\\mathbb {R}/\\mathbb {Z}$ be the unit circle parametrized by angle measured in revolutions.", "Impose a total order on $\\mathbb {T}$ by choosing representatives in $[0,1)\\subset \\mathbb {R}$ .", "That is, given $s,t\\in \\mathbb {T}$ we say that $s<t$ if and only if $s^{\\prime }-\\lfloor s^{\\prime }\\rfloor <t^{\\prime }-\\lfloor t^{\\prime }\\rfloor $ for any $s^{\\prime },t^{\\prime }\\in \\mathbb {R}$ corresponding to the equivalence classes $s,t$ .", "Likewise we impose a total order on $\\mathbb {Z}/q\\mathbb {Z}$ by choosing representatives in $\\lbrace 0,1,\\dots ,q-1$ }.", "Fix an integer $d\\ge 2$ and define the angle-multiplying map $\\sigma _{d}$ : $\\mathbb {T}\\rightarrow \\mathbb {T}$ with $\\sigma _{d}(t)=dt, \\forall t\\in \\mathbb {T}$ .", "Given $\\sigma _{d}$ , the orbit of an element $ t\\in \\mathbb {T}$ is the set $\\lbrace \\sigma _{d}^{k}(t):k=0,1,2,\\dots \\rbrace $ .", "Definition 1.1 Let $A=\\lbrace t_{0}<t_{1}<t_{2}<\\dots <t_{q-1}\\rbrace $ be a finite subset of $\\mathbb {T}$ indexed in increasing order, where indices are viewed as elements of $\\mathbb {Z}/ q\\mathbb {Z}$ .", "Then $A$ is $\\sigma _{d}$ -rotational if for some fixed $0\\ne p\\in \\mathbb {Z}/ q\\mathbb {Z}$ we have $\\sigma _{d}(t_{i})=t_{i+p}$ for every $t_{i}\\in A.$ Each rotation set is assigned a rotation number, which is the rational number $p/q$ whose expression may be further reducible.", "A set is rotational if it is $\\sigma _d$ -rotational for some $d\\ge 2$ .", "Rotational sets were introduced by Goldberg in [2] to study and classify polynomial maps $f:\\mathbb {C}\\rightarrow \\mathbb {C}$ in the setting of complex dynamics [3].", "Goldberg shows in [2] that rotational sets are uniquely determined by their rotation number and deployment sequence, which describes the placement of the set with respect to the fixed points of $\\mathbb {T}$ under $\\sigma _{d}$ .", "This allows for an enumeration of $\\sigma _{d}$ -rotational orbits with rotation number $p/q$ .", "(Rotational sets are rotational orbits or the union of rotational orbits.)", "This is generalized in [5] by Petersen and Zakeri, who enumerate periodic orbits under $\\sigma _{d}$ with any combinatorial type.", "That is, periodic orbits which are generally permuted under $\\sigma _{d}$ are considered.", "We continue progress on the combinatorics of rotational sets by introducing a new parametrization of rotational sets.", "Two results from [2] promptly follow: There are ${{d-2+q}\\atopwithdelims ()d-2}$ $\\sigma _{d}$ -rotational orbits with rotation number $p/q$ (in lowest terms).", "A $\\sigma _{d}$ -rotational set contains at most $d-1$ orbits.", "In Theorem REF we enumerate all $\\sigma _{d}$ -rotational sets with rotation number $p/q$ .", "The proof is constructive in the sense that it suggests an algorithm for generating the rotational sets.", "Example 1.1 The set $\\lbrace \\frac{1}{15}, \\frac{2}{15}, \\frac{4}{15}, \\frac{8}{15}\\rbrace $ is $\\sigma _{2}$ -rotational with rotation number $\\frac{1}{4}$ .", "The set $A=\\big \\lbrace \\frac{8}{26}, \\frac{17}{26}, \\frac{20}{26}, \\frac{23}{26}, \\frac{24}{26}, \\frac{25}{26} \\big \\rbrace $ is $\\sigma _{3}$ -rotational with rotation number $ \\frac{4}{6}=\\frac{2}{3}$ .", "Notice that $A$ is the union of two $\\sigma _d$ -rotational orbits $ \\lbrace \\frac{8}{26},\\frac{24}{26},\\frac{20}{26}\\rbrace $ and $ \\lbrace \\frac{17}{26},\\frac{25}{26},\\frac{23}{26}\\rbrace .$ Both smaller orbits also have rotation number $\\frac{2}{3}$ .", "A subset of $\\mathbb {Z}/q\\mathbb {Z}$ consists of consecutive elements if it has the form $\\lbrace i,i+1,i+2,\\dots \\rbrace $ for some $i\\in \\mathbb {Z}/q\\mathbb {Z}$ .", "A collection indexed by elements in $\\mathbb {Z}/q\\mathbb {Z}$ consists of consecutive elements if the indices are consecutive.", "Proposition 1.1 Let $A=\\lbrace t_{0}<t_{1}<t_{2}<\\dots <t_{q-1}\\rbrace $ be a $\\sigma _{d}$ -rotational set such that $\\sigma _d(t_i)=t_{i+p}$ .", "Then $A$ is the union of $n=\\text{gcd}(p,q)$ orbits.", "Each orbit in the union is $\\sigma _{d}$ -rotational and has rotation number $p/q$ .", "The orbits are interlaced, meaning, given any $n$ consecutive elements of $A$ no two elements come from the same orbit.", "Use the correspondence $t_{i}\\leftrightarrow i$ for $t_{i}\\in A$ and $i\\in \\mathbb {Z}/q\\mathbb {Z}$ .", "Then the orbit of $t_{i}$ corresponds to the coset $H+i$ of the subgroup $H$ generated by $p$ .", "Working over the integers, the least $m$ such that $mp\\equiv 0$ (mod $q$ ) is $m=q/n$ .", "Thus $|H|=q/n$ and the index of $H$ in $\\mathbb {Z}/q\\mathbb {Z}$ is $n$ .", "Equivalently, $A$ is the union of $n$ orbits of size $q/n$ .", "The consecutive elements $a,a+1,\\dots ,a+n-1\\in \\mathbb {Z}/q\\mathbb {Z}$ come from the consecutive cosets $H,H+1,\\dots ,H+n-1\\in (\\mathbb {Z}/q\\mathbb {Z})/H$ (perhaps not in the same order) which are pairwise disjoint; this proves the interlacing property.", "Relabel the elements of $A$ by the rule $t_j^{(k)}=t_{jn+k-1}$ .", "$A=\\lbrace t_0^{(1)}<t_0^{(2)}<\\dots <t_0^{(n)}<t_1^{(1)}<t_1^{(2)}<\\dots <t_1^{(n)}<\\cdots \\cdots <t_{q/n-1}^{(n)}\\rbrace $ Applying the interlacing property to the collections $t_0^{(1)},\\dots ,t_0^{(n)}$ and $t_0^{(2)},\\dots ,t_1^{(1)}$ we see that $t_0^{(1)}$ and $t_1^{(1)}$ come from the same orbit.", "More generally, $t_{i}^{(k)}$ and $t_{j}^{(k)}$ belong in the same orbit.", "Picking out an orbit of $\\mathcal {O}\\subset A$ , we have $\\mathcal {O}=\\lbrace t_0^{(k)}<t_1^{(k)}<\\dots <t_{q/n-1}^{(k)}\\rbrace $ and $\\sigma _d(t_j^{(k)})=\\sigma _d(t_{jn+k-1})=t_{jn+k-1+p}=t_{(j+p/n)n+k-1}=t_{j+p/n}^{(k)}.$ Therefore $\\mathcal {O}$ is $\\sigma _d$ -rotational with rotation number $(p/n)/(q/n)=p/q$ ." ], [ "Supporting lemmas", "Definition 2.1 Let the $q$ -tuple $(a_{0},a_{1},a_{2}, \\dots ,a_{q-1})$ , $a_i\\in \\lbrace 0,1,\\dots ,d-1\\rbrace $ , denote the element in $\\mathbb {T}$ with base $d$ representation $0.\\overline{a_{0}a_{1}a_{2}\\ldots a_{q-1}}.$ To make calculations easier, we view the indices as elements of $\\mathbb {Z}/q\\mathbb {Z}$ .", "Note that the lexicographical order on $q$ -tuples agrees with their order as points in $\\mathbb {T}$ .", "The $q$ -tuple representation has other nice properties.", "Notice that the map $\\sigma _d$ acts on the set of $q$ -tuples by $\\sigma _d(a_0,a_1,a_2,\\dots )=(a_1,a_2,a_3,\\dots )$ which leads to the equation $\\sigma _d^k(a_0,a_1,\\dots ,a_{q-1})=(a_k,a_{k+1},\\dots ,a_{k+q-1}), \\text{ $k\\in \\mathbb {Z}/q\\mathbb {Z}$}.$ The notation $\\sigma _d^k$ means $\\sigma _d^{k^{\\prime }}$ for any representative $k^{\\prime }\\in \\mathbb {Z}$ of $k$ .", "Suppose $A$ is a $\\sigma _{d}$ -rotational set with rotation number $p/q$ in lowest terms.", "Then $A$ is the union of orbits of size $q$ so that every point in $A$ has a $q$ -tuple representation.", "By assumption gcd$(p,q)=1$ , hence $p$ has a multiplicative inverse in $\\mathbb {Z}/q\\mathbb {Z}$ which we denote $p^*$ .", "Lemma 2.1 Let $\\mathcal {O}=\\lbrace t_{0}<t_{1}<t_{2}<\\dots < t_{q-1}\\rbrace $ be an orbit in $\\mathbb {T}.$ Then $\\mathcal {O}$ is $\\sigma _{d}$ -rotational with rotation number $p/q$ in lowest terms if and only if $\\sigma _{d}^{p^*}(t_{i})=t_{i+1}$ for every $t_{i}\\in \\mathcal {O}.$ Suppose $\\mathcal {O}$ is $\\sigma _{d}$ -rotational with rotation number $p/q$ in lowest terms.", "Then for every $t_{i}\\in \\mathcal {O}$ , we have $\\sigma _{d}(t_{i})=t_{i+p}$ which implies $\\sigma _{d}^{p^*}(t_{i})=t_{i+pp^*}=t_{i+1}$ .", "Conversely, if $\\sigma _{d}^{p^*}(t_{i})=t_{i+1}$ for every $t_{i}\\in \\mathcal {O}$ then $\\sigma _{d}(t_{i})=(\\sigma _{d}^{p^*})^{p}(t_{i})=t_{i+p}.$ The existence of $p^*$ implies that gcd$(p,q)=1$ so $p/q$ is in lowest terms.", "Define disjoint open intervals $I_{j}= ( \\frac{j}{d},\\frac{j+1}{d})\\subset \\mathbb {T}$ for $j=0,1,2$ ,..., $d-1$ , setting $(\\frac{d-1}{d},1) := (\\frac{d-1}{d},\\infty )$ when $j=d-1$ .", "Notice that the boundary points of these intervals are precisely the points in the $\\sigma _d$ -preimage of 0.", "Suppose the orbit of $t=(a_0,\\dots ,a_{q-1})$ is rotational.", "Since we exclude fixed points from the definition of rotational sets, $t\\ne 0$ .", "Moreover, $t\\notin \\sigma _d^{-1}(0)=\\mathbb {T}\\setminus (\\cup _j I_j)$ or equivalently $t\\in I_j$ for some $j$ .", "It follows that $t\\in I_j$ if and only if $j=a_0$ .", "Lemma 2.2 Suppose $s=(a_0,a_1,\\dots ,a_{q-1})$ and $t=(b_0,b_1,\\dots ,b_{q-1})$ are distinct points of $\\mathbb {T}$ such that their orbits are rotational sets.", "Then $a_0<b_0$ if and only if there exists $s<r<t$ such that $\\sigma _d(r)=0$ .", "Suppose $a_0<b_0$ .", "Since $s\\in I_{a_0}$ and $t\\in I_{b_0}$ there is a point in $r\\in \\sigma _d^{-1}(0)$ such that $s<r<t$ .", "Conversely, suppose $r\\in \\sigma _d^{-1}(0)$ such that $s<r<t$ .", "Since $s\\in I_{a_0}$ and $t\\in I_{b_0}$ we have $a_0<b_0$ .", "Lemma 2.3 Let $s=(a_0,a_1,\\dots ,a_{q-1})$ and $t=(b_0,b_1,\\dots ,b_{q-1})$ .", "Fix an index $m\\ne 0$ .", "If $a_i\\le b_i$ for $i\\ne m$ and $a_{m-1}<b_{m-1}$ then $s<t$ .", "Since $m\\ne 0$ , $m-1<m$ .", "Now we use the lexicographical ordering and compare digits, noticing that $a_0\\le b_0, a_1\\le b_1,\\dots ,a_{m-1}<b_{m-1}$ .", "Therefore $s<t$ .", "Definition 2.2 An $n$ -sequence of length $k$ is a nondecreasing sequence $a_{1}, a_{2}, a_{3},\\dots ,a_{k}$ with each $a_{i}$ chosen from $\\lbrace 0,1,2,\\dots ,n-1\\rbrace .$ A simple counting argument shows that there are ${{n-1+k} \\atopwithdelims (){n-1}}$ $n$ -sequences with length $k$ .", "Lemma 2.4 There is a bijection between the set $A$ of $n$ -sequences $s=a_{1},a_{2},a_{3},\\dots ,a_{k}$ with length $k$ such that $a_{i}<a_{i+1}$ for some fixed $1\\le i\\le k-1$ and the set $B$ of $(n-1)$ -sequences with length $k.$ Given a sequence in $A$ , subtract 1 from the terms $a_{i+1},a_{i+2},\\dots ,a_{n}$ to get a sequence in $B$ .", "Given a sequence in $B$ , add 1 to the terms $a_{i+1},a_{i+2},\\dots ,a_{n}$ to get a sequence in $A$ ." ], [ "Main results", "The proof of the following theorem was greatly simplified using a key idea from the proof of Theorem 2.8 in [1].", "Theorem 3.1 Let $A$ be the union of $n$ distinct orbits of $\\sigma _d$ having the $n$ least elements ${\\begin{array}{c}t_{1} =(a_{0}^{1}, a_{1}^{1}, a_{2}^{1},\\dots ,a_{q-1}^{1}),\\\\t_{2} =(a_{0}^{2}, a_{1}^{2}, a_{2}^{2},\\dots ,a_{q-1}^{2}),\\\\t_{3} =(a_{0}^{3},a_{1}^{3}, a_{2}^{3},\\dots ,a_{q-1}^{3}),\\\\\\vdots \\text{\\qquad \\qquad \\qquad \\qquad }\\\\t_{n} =(a_{0}^{n}, a_{1}^{n}, a_{2}^{n}, \\dots , a_{q-1}^{n}).\\end{array}}$ Then $A$ is $\\sigma _d$ -rotational with rotation number $p/q$ in lowest terms if and only if $a_{0}^{1},a_{0}^{2},a_{0}^{3},\\dots ,a_{0}^{n}, a_{p^*}^{1}, a_{p^*}^{2},a_{p^*}^{3},\\dots ,a_{p^*}^{n},a_{2p^*}^{1}, a_{2p^*}^{2},a_{2p^*}^{3},\\dots ,a_{2p^*}^{n},\\dots \\dots , a_{(q-1)p^*}^{n}$ is a $d$ -sequence with $a_{-(p+1)p^*}^{n}<a_{-pp^*}^{1}$ .", "Suppose $A$ is $\\sigma _{d}$ -rotational with rotation number $p/q$ in lowest terms.", "Index $A=\\lbrace t_{1},t_{2},t_{3},\\dots ,t_{nq}\\rbrace $ in increasing order.", "Then the sequence of leading digits $b_{1},b_{2},b_{3},\\dots ,b_{nq}$ where $t_{k}=(b_{k},\\dots )$ is a $d$ -sequence.", "It follows from Lemma REF that this sequence is (2).", "The points $(a_{(q-1)p^*}^n,...)$ and $(a_0^1,\\dots )$ in $A$ are consecutive in since the pair has the form $t_i,t_{i+1}$ .", "Note that $\\sigma _d$ and $\\sigma _d^{-1}$ preserve consecutiveness.", "We have $\\sigma _{d}(a_{(q-1)p^*-1}^{n},a_{(q-1)p^*}^{n},\\dots )=(a_{(q-1)p^*}^{n},\\dots ) \\quad \\text{ and }\\quad \\sigma _{d}(a_{q-1}^{1},a_{0}^{1},\\dots )=(a_{0}^{1},\\dots ).$ Since $(a_{(q-1)p^*}^{n},\\dots )>(a_{0}^{1},\\dots )$ there must be an element of the $\\sigma _{d}$ -preimage of 0 in the interval $[(a_{(q-1)p^*-1}^{n},\\dots ),(a_{q-1}^{1},\\dots )]$ .", "By Lemma REF we obtain $a_{-(p+1)p^*}^{n}=a_{(q-1)p^*-1}^{n}<a_{q-1}^{1}=a_{-pp^*}^{1}.$ Conversely, suppose (2) is a $d$ -sequence with $a_{-(p+1)p^*}^{n}<a_{-pp^*}^{1}$ .", "Observe that going down a column of (1) corresponds to going across a segment of (2).", "Thus $a_{i}^{j}\\le a_{i}^{j+1}$ for any $i\\in \\mathbb {Z}/q\\mathbb {Z}$ and $1\\le j\\le n-1$ .", "Then we have $t_{1}<t_{2}<\\dots <t_{n}$ since the $t_{j}$ are distinct.", "More generally, $\\sigma _{d}^{kp^*}(t_{1})<\\sigma _{d}^{kp^*}(t_{2})<\\dots <\\sigma _{d}^{kp^*}(t_{n})$ for any integer $k$ .", "We wish to additionally show that $\\sigma _{d}^{kp^*}(t_{n})<\\sigma _{d}^{(k+1)p^*}(t_{1})$ for $0\\le k\\le q-2$ so that $t_{1},t_{2},t_{3},\\dots ,t_{n},\\sigma _{d}^{p^*}(t_{1}),\\sigma _{d}^{p^*}(t_{2}),\\sigma _{d}^{p^*}(t_{3}),\\dots ,\\sigma _{d}^{p^*}(t_{n}),\\sigma _{d}^{2p^*}(t_{1}),\\dots \\dots ,\\sigma _{d}^{(q-1)p^*}(t_{n})$ is increasing.", "By Lemma REF this would imply that $A$ is $\\sigma _{d}$ -rotational with rotation number $p/q$ in lowest terms.", "To this end, notice that for $0\\le k\\le q-2$ $\\sigma _{d}^{kp^*}(t_{n}) &=(a_{kp^*}^{n},\\ a_{kp^*+1}^{n},\\ a_{kp^*+2}^{n},\\ \\dots ,\\ a_{kp^*+q-1}^{n}),\\\\\\sigma _{d}^{(k+1)p^*}(t_{1}) &=(a_{(k+1)p^*}^{1},\\ a_{(k+1)p^*+1}^{1},\\ a_{(k+1)p^*+2}^{1},\\ \\dots ,\\ a_{(k+1)p^*+q-1}^{1}).$ From (2) we have $a_{kp^*+i}^{n}=a_{(k+ip)p^*}^{n}\\le a_{(k+ip+1)p^*}^{1}=a_{(k+1)p^*+i}^{1}$ unless $k+ip=q-1$ .", "But $k+ip=q-1=-1$ implies that $a_{kp^*+i-1}^{n}=a_{(k+ip-p)p^*}^{n}=a_{-(p+1)p^*}^{n}<a_{-pp^*}^{1}=a_{(k+ip+1-p)p^*}^{1}=a_{[k+1+(i-1)p]p^*}^{1}=a_{(k+1)p^*+i-1}^{1}.$ By Lemma REF we see that $\\sigma _{d}^{kp^*}(t_{n})<\\sigma _{d}^{(k+1)p^*}(t_{0})$ except when $k+ip=q-1$ and $i=0$ , i.e.", "when $k=q-1.$ Corollary 3.1 Let $t=(a_0,a_1,a_2,\\dots ,a_{q-1})$ .", "The orbit $\\mathcal {O}$ of $t$ is a $\\sigma _d$ -rotational with rotation number $p/q$ in lowest terms if and only if $a_0\\le a_{p^*}\\le a_{2p^*}\\le \\dots \\le a_{-(p+1)p^*} < a_{-pp^*}\\le \\dots \\le a_{(q-1)p^*}.$ This is Theorem REF when $n=1$ .", "The sequence $(\\ref {ss})$ gives us a way of identifying every $\\sigma _{d}$ -rotational orbit for a given rotation number $p/q$ , assumed to be in lowest terms.", "We can make this simpler since, by Lemma REF , $(\\ref {ss})$ corresponds to the $(d-1)$ -sequence $a_{0},a_{p^*},\\dots ,a_{-(p+1)p^*},a_{-pp^*}-1,\\dots ,a_{(q-2)p^*}-1, a_{(q-1)p^*}-1$ which we call the representative sequence of $\\mathcal {O}.$ Let $A$ be the union of $n$ distinct rotational orbits.", "Using Theorem REF , we see that $A$ is a rotational set if and only if there is a labeling $\\mathcal {O}_{1}, \\mathcal {O}_{2},\\dots ,\\mathcal {O}_{n}$ of its orbits such that the corresponding representative sequences $s_{i}=b_{0}^{i},b_{1}^{i},b_{2}^{i},\\dots ,b_{q-1}^{i}$ can be interlaced, meaning $b_{0}^1,b_{0}^2,b_{0}^3,\\dots ,b_{0}^n,b_{1}^1,b_{1}^2,b_{1}^3,\\dots ,b_{1}^n,b_{2}^1,b_{2}^2,b_{2}^3,\\dots ,b_{2}^{n},\\dots \\dots ,b_{q-1}^{n}$ is a $(d-1)$ -sequence.", "McMullen in [4] gives an explicit algorithm for computing rotational orbits for a fixed rotation number $p/q$ .", "Example REF describes this same algorithm.", "In fact, Theorem REF gives us a something stronger: we can compute all rotational sets by checking if representative sequences can be interlaced then taking unions of rotational orbits.", "Figure: Graph representing σ 4 \\sigma _{4}-rotational sets with rotation number 1 4\\frac{1}{4} or 3 4\\frac{3}{4}Example 3.1 Let us construct a $\\sigma _{4}$ -rotational orbit with rotation number $\\frac{2}{5}$ .", "The multiplicative inverse of 2 in $\\mathbb {Z}/ 5\\mathbb {Z}$ is 3.", "Choose a representative sequence, say the 3-sequence $0,1,1,1,2.$ Add 1 to the last 2 terms to get $0,1,1,2,3$ .", "Substituting these numbers for $a_0,a_3,a_6=a_1,a_9=a_4,a_{12}=a_2$ in $(a_0,a_1,a_2,a_3,a_4)$ we obtain the 5-tuple $(0,1,3,1,2)$ which has fractional representation $\\frac{118}{1028}$ .", "This gives the $\\sigma _4$ -rotational orbit $\\lbrace \\frac{118}{1028},\\frac{391}{1028},\\frac{472}{1028},\\frac{541}{1028},\\frac{865}{1028}\\rbrace $ with rotation number $\\frac{2}{5}$ .", "Example 3.2 The undirected graph in Figure 1 shows all $\\sigma _{4}$ -rotational sets with rotation number $p/4$ for any $p\\in \\mathbb {Z}/ 4\\mathbb {Z}$ with gcd$(p,4)=1$ .", "The vertices represent $\\sigma _{4}$ -rotational orbits, labeled by their representative sequences.", "Two vertices are connected by an edge if and only if their representative sequences can be interlaced.", "It follows that any set of $n$ mutually adjacent vertices, or $n$ -clique, gives sequences which can be interlaced.", "Therefore each $n$ -clique in the graph corresponds to a $\\sigma _{4}$ -rotational set which is a union of $n$ orbits.", "There are 15 1-cliques, 30 2-cliques, and 16 3-cliques in the graph.", "Altogether there are 61 $\\sigma _{4}$ -rotational sets with rotation number $p/4.$ Corollary 3.2 There are ${{d-2+q}\\atopwithdelims (){d-2}}$ $\\sigma _{d}$ -rotational orbits with rotation number $p/q$ in lowest terms.", "There are ${{d-2+q}\\atopwithdelims (){d-2}}$ representative sequences each corresponding to a unique orbit.", "Corollary 3.3 A $\\sigma _{d}$ -rotational set contains at most $d-1$ orbits.", "Suppose that the union of $n$ orbits with rotation number $p/q$ is a $\\sigma _{d^{-}}$ rotational set.", "Then there exists a $(d-1)$ -sequence with length $nq$ which can be unlaced to give $n$ distinct $(d-1)$ -sequences with length $q$ .", "Let $s^{\\prime }=b_{0},b_{1},b_{2},\\dots ,b_{nq-1}$ be this $(d-1)$ -sequence and let $s_{i}=b_{i},b_{i+n},b_{i+2n},\\dots ,b_{i+(q-1)n}\\qquad \\text{ for } 0\\le i\\le n-1.$ Since the $s_{i}$ are distinct, adjacent sequences in $s_{0},s_{1},\\dots ,s_{n-1}$ must differ by at least one term.", "Then going across $s^{\\prime }$ we should find at least $n-1$ pairs of nonequal adjacent terms.", "But since $s^{\\prime }$ is a $(d-1)$ -sequence (there are only $d-1$ possibilities for the terms), there can be at most $d-2$ such pairs.", "This shows that $n-1 \\le d-2$ or $n\\le d-1$ .", "Theorem 3.2 Let $N_{k}$ be the number of $\\sigma _{d}$ -rotational sets with rotation number $p/q$ (in lowest terms) that contain precisely $k$ orbits.", "Each $N_{k}$ is given recursively by $\\begin{split}N_{1} &={{d-2+q}\\atopwithdelims (){d-2}},\\text{ }{\\it and}\\\\N_{k} &={{d-2+kq}\\atopwithdelims (){d-2}} - \\sum _{j=1}^{k-1} {{k-1}\\atopwithdelims (){j-1}} N_{j},\\text{ for } 2\\le k\\le d-1.\\end{split}$ We know $N_{1}={{d-2+q}\\atopwithdelims (){d-2}}$ from Corollary REF .", "Let $k\\in \\mathbb {Z}$ such that $2\\le k\\le d-1$ .", "Each rotational set counted by $N_{k}$ can be identified by a $(d-1)$ -sequence with length $kq$ obtained by interlacing $k$ distinct $(d-1)$ -sequences, each with length $q$ .", "By Lemma 2.4 the number of $(d-1)$ -sequences with length $kq$ is ${{d-2+kq}\\atopwithdelims (){d-2}}$ .", "To find $N_{k}$ , we want to subtract from ${{d-2+kq}\\atopwithdelims (){d-2}}$ the number of sequences obtained by interlacing nondistinct representative sequences.", "Let $s^{\\prime }=b_{0},b_{1},b_{2},\\dots ,b_{kq-1}$ be a $(d-1)$ -sequence with length $kq$ and let $s_{i}=b_{i},b_{i+k},b_{i+2k},\\dots ,b_{i+(q-1)k} \\qquad \\text{ for } 0\\le i\\le k-1.$ Now, because $s^{\\prime }$ is nondecreasing, identical sequences in $s_{0},s_{1},\\dots ,s_{k-1}$ group together with consecutive indices.", "Suppose there are precisely $j$ distinct sequences in $s_{0}, s_{1},\\dots ,s_{k-1}$ , where $1< j\\le k-1$ .", "There are $N_{j}$ possibilities for these distinct sequences.", "Furthermore, there are ${{k-1}\\atopwithdelims (){j-1}}$ possibilities for how they are grouped, given by a choice of $j-1$ dividers in between elements of the list $s_{0},s_{1},\\dots ,s_{k-1}$ .", "Thus, we subtract ${{k-1}\\atopwithdelims (){j-1}} N_{j}$ from ${{d-2+kq}\\atopwithdelims (){d-2}}$ for each $1\\le j\\le k-1$ .", "The application of a combinatorial inversion formula (Proposition REF ) gives us the following nonrecursive version of (REF ).", "$N_k = \\sum _{j=1}^k (-1)^{k+j} {{k-1}\\atopwithdelims (){j-1}}{{d-2+jq}\\atopwithdelims (){d-2}}$" ], [ "Appendix", "Proposition 4.1 Let $b=[b_1,\\dots ,b_m]^t\\in \\mathbb {R}^m$ be a column vector with real entries.", "The formulas $n_{1} =b_1,\\quad n_{k} =b_k - \\sum _{j=1}^{k-1} {{k-1}\\atopwithdelims (){j-1}} n_{j}\\text{ for } 2\\le k\\le m$ and $n_k = \\sum _{j=1}^k (-1)^{k+j} {{k-1}\\atopwithdelims (){j-1}}b_j$ define the same vector $n=[n_1,\\dots ,n_m]^t\\in \\mathbb {R}^m$ .", "Let $L=(a_{ij})$ and $L^{\\prime }=(a^{\\prime }_{ij})$ be the $m\\times m$ lower triangular matrices defined by $a_{ij}={{i-1}\\atopwithdelims (){j-1}}$ and $a^{\\prime }_{ij}=(-1)^{i+j}{{i-1}\\atopwithdelims (){j-1}}$ .", "For any $x\\in \\mathbb {R}$ the binomial theorem gives $L[1,x,\\dots ,x^{m-1}]^t &=[1,x+1,\\dots ,(x+1)^{m-1}]^t, \\\\L^{\\prime }[1,x,\\dots ,x^{m-1}]^t &=[1,x-1,\\dots ,(x-1)^{m-1}]^t.$ It follows that $L^{\\prime }=L^{-1}$ (by properties of the Vandermonde matrix).", "Formula (REF ) produces the solution $n$ of $Ln=b$ using forwards substitution, whereas (REF ) can be written as $n=L^{-1}b$ ." ], [ "Acknowledgements", "I would like to thank Dr. John Mayer and Dr. Debra Gladden for introducing me to rotation sets.", "I also thank Dr. Debra Gladden for her support and encouragement throughout my undergraduate years.", "I thank Dr. Saeed Zakeri for his helpful comments on a draft of this paper." ], [ "References", "  [1] Alexander Blokh, James Malaugh, John Mayer, Lex Oversteegen, and Daniel Parris.", "Rotational subsets of the circle under $z^{d}$ .", "Topology and its Applications, 153(1):1540-1570, 2006.", "[2] Lisa Goldberg.", "Fixed points of polynomial maps.", "I. Rotation subsets of the circles.", "Annales scientifiques de l'École Normale Supérieure, Ser.", "4, 25(6):679-685, 1992.", "[3] Lisa Goldberg and John Milnor.", "Fixed points of polynomial maps.", "Part II.", "Fixed pointportraits.", "Annales scientifiques de l'cole Normale Suprieure, 26(1):51-98, 1993.", "[4] Curtis McMullen.", "Dynamics on the unit disk: Short geodesics and simple cycles.", "Commentarii Mathematici Helvetici, 85:723-749, 2010.", "[5] Carsten L. Petersen and Saeed Zakeri.", "On combinatorial types of periodic orbits of the map x $\\mapsto kx$ (mod $\\mathbb {Z}$ ).", "Part I: Realization, arXiv:1712.04506, 2017." ] ]
2207.03594
[ [ "Central limit theorem for the principal eigenvalue and eigenvector of\n Chung-Lu random graphs" ], [ "Abstract A Chung-Lu random graph is an inhomogeneous Erd\\H{o}s-R\\'enyi random graph in which vertices are assigned average degrees, and pairs of vertices are connected by an edge with a probability that is proportional to the product of their average degrees, independently for different edges.", "We derive a central limit theorem for the principal eigenvalue and the components of the principal eigenvector of the adjacency matrix of a Chung-Lu random graph.", "Our derivation requires certain assumptions on the average degrees that guarantee connectivity, sparsity and bounded inhomogeneity of the graph." ], [ "Introduction", "The spectral properties of adjacency matrices play an important role in various areas of network science.", "In the present paper we consider an inhomogeneous version of the Erdős-Rényi random graph called the Chung-Lu random graph and we derive a central limit theorem for the principal eigenvalue and eigenvector of its adjacency matrix." ], [ "Setting", "Recall that the homogeneous Erdős-Rényi random graph has vertex set $[n] = \\lbrace 1,\\ldots ,n\\rbrace $ , and each edge is present with probability $p$ and absent with probability $1-p$ , independently for different edges, where $p \\in (0,1)$ may depend on $n$ (in what follows we often suppress the dependence on $n$ from the notation; the reader is however warned that most quantities depend on $n$ ).", "The average degree is the same for every vertex and equals $(n-1)p$ when self-loops are not allowed, and $np$ when self-loops are allowed (and are considered to contribute to the degrees of the vertices).", "In [15] the following generalisation of the Erdős-Rényi random graph is considered, called the Chung-Lu random graph, with the goal to accommodate general degrees.", "Given a sequence of degrees $\\vec{d}_n=(d_i)_{i \\in [n]}$ , consider the random graph $\\mathcal {G}_n(\\vec{d}_n)$ in which to each pair $i,j$ of vertices an edge is assigned independently with probability $p_{ij}=d_id_j/m_1$ , where $m_1=\\sum _{i=1}^n d_i$ (for computational simplicity we allow self-loops).", "Here, the degrees can act as vertex weights.", "Vertices with low weights are more likely to have less neighbours than vertices with high weights which act as hubs (see [33] for a general introduction to generalised random graphs).", "If $m_\\infty ^2 \\le m_1$ with $m_\\infty = \\max _{i\\in [n]} d_i$ , then $p_{ij} \\le 1$ for all $i,j \\in [n]$ , and the sequence $\\vec{d}_n$ is graphical.", "Note that in $\\mathcal {G}_n(\\vec{d}_n)$ the expected degree of vertex $i$ is $d_i$ .", "The classical Erdős-Rényi random graph (with self-loops) corresponds to $d_i=np$ for all $i \\in [n]$ ." ], [ "Principal eigenvalue and eigenvector", "The largest eigenvalue of the adjacency matrix $A$ and its corresponding eigenvector, written as $(\\lambda _1,v_1)$ , contain important information about the random graph.", "Several community detection techniques depend on a proper understanding of these quantities [32], [25], [1], which in turn play an important role for various measures of network centrality [26], [27] and for the properties of dynamical processes (such as the spread of an epidemic) taking place on networks [12], [28].", "For Erdős-Rényi random graphs, it was shown in [24] that with high probability (whp) $\\lambda _1$ scales like $\\lambda _1 \\sim \\max \\lbrace \\sqrt{D_\\infty }, np\\rbrace , \\qquad n \\rightarrow \\infty ,$ where $D_\\infty $ is the maximum degree.", "This result was partially extended to $\\mathcal {G}_n(\\vec{d}_n)$ in [16], and more recently to a class of inhomogeneous Erdős-Rényi random graphs in [5], [6].", "For a related discussion on the behaviour of $(\\lambda _1,v_1)$ in real-world networks, see [12], [28].", "In the present paper we analyse the fluctuations of $(\\lambda _1,v_1)$ .", "We will be interested specifically in the case where $\\lambda _1$ is detached from the bulk, which for Erdős-Rényi random graphs occurs when $\\lambda _1 \\sim np$ whp, and for Chung-Lu random graphs when $\\lambda _1 \\sim m_2/m_1$ , where $m_2=\\sum _{i\\in [n]} d_i^2$ .", "Note that the quotient $m_2/m_1$ arises from the fact that the average adjacency matrix is rank one and that its only non-zero eigenvalue is $m_2/m_1$ .", "Such rank-one perturbations of a symmetric matrix with independent entries became prominent after the work in [4].", "Later studies extended this work to finite-rank perturbations [3], [7], [11], [10], [20], [21].", "Erdős-Rényi random graphs differ, in the sense that perturbations live on a scale different from $\\sqrt{n}$ .", "For Chung-Lu random graphs we assume that $m_2/m_1\\rightarrow \\infty $ .", "In the setting of inhomogeneous Erdős-Rényi random graphs, finite-rank perturbations were studied in [13].", "In that paper the connection probability between between $i$ and $j$ is given by $p_{ij} = \\varepsilon _n f(i/n, j/n)$ , where $f\\colon \\,[0,1]^2\\rightarrow [0,1]$ is almost everywhere continuous and of finite rank, $\\varepsilon _n \\in [0,1]$ and $n\\varepsilon _n\\gg (\\log n)^{8}$ .", "However, for a Chung-Lu random graph with a given degree sequence it is not always possible to construct an almost everywhere continuous function $f$ independent of $n$ such that $\\varepsilon _n f(i/n, j/n) = d_i d_j/m_1$ .", "In the present paper we extend the analysis in [13] to Chung-Lu random graphs by focussing on $(\\lambda _1,v_1)$ .", "For Erdős-Rényi random graphs it was shown in [19], [18] that $\\lambda _1$ satisfies a central limit theorem (CLT) and that $v_1$ aligns with the unit vector.", "These papers extend the seminal work carried out in [22]." ], [ "Chung-Lu random graphs", "In the present paper, subject to mild assumptions on $\\vec{d}_n$ , we extend the CLT for $\\lambda _1$ from Erdős-Rényi random graphs to Chung-Lu random graphs, and derive a pointwise CLT for $v_1$ as well.", "It was shown in [16] that if $m_2/m_1\\gg \\sqrt{m_\\infty }\\,(\\log n)$ , then $\\lambda _1 \\sim m_2/m_1$ whp, while if $\\sqrt{m_\\infty } \\gg (m_2/m_1)( \\log n)^2$ , then $\\lambda _1 = m_\\infty $ whp.", "In fact, examples show that a result similar to (REF ) does not hold, and that $\\lambda _1$ does not scale like $\\max \\lbrace m_2/m_1, \\sqrt{m_{\\infty }}\\rbrace $ .", "These facts clearly show that the behaviour of $\\lambda _1$ is controlled by subtle assumptions on the degree sequence.", "In what follows we stick to a bounded inhomogeneity regime where $m_2/m_1\\asymp m_\\infty $ .", "The behaviour of $v_1$ is interesting and challenging, and is of major interest for applications.", "One of the crucial properties to look for in eigenvectors is the phenomenon of localization versus delocalization.", "An eigenvector is called localized when its mass concentrates on a small number of vertices, and delocalized when its mass is approximately uniformly distributed on the vertices.", "The complete delocalization picture for Erdős-Rényi random graphs was given in [19].", "In fact, it was proved that $\\lambda _1$ is close to the scaled unit vector in the $\\ell _\\infty $ -norm.", "In the present paper we do not study localization versus delocalization for Chung-Lu random graphs in detail, but we do show that in a certain regime there is strong evidence for delocalization because $v_1$ is close to the scaled unit vector.", "In [9] the authors found that the eigenvectors of a Wigner matrix with independent standard Gaussian entries are distributed according to a Haar measure on the orthogonal group, and the coordinates have Gaussian fluctuations after appropriate scaling.", "Our work shows that the coordinate-wise fluctuations hold as well for the principal eigenvector of the non-centered Chung-Lu adjacency matrix and that they are Gaussian after appropriate centering and scaling." ], [ "Outline", "In Section REF we define the Chung-Lu random graph, state our assumption on the degree sequence, and formulate two main theorems: a CLT for the largest eigenvalue and a CLT for its associated eigenvector.", "In Section REF we discuss these theorems and place them in their proper context.", "Section contains the proof of the CLT of the eigenvalue and Section studies the properties of the principal eigenvector." ], [ "Set-up", "Let $\\mathbb {G}_n$ be the set of simple graphs with $n$ vertices.", "Let $\\vec{d}_n=(d_i)_{i \\in [n]}$ be a sequence of degrees, such that $d_i\\in \\mathbb {N}$ for all $i\\in [n]$ and abbreviate $m_k = \\sum _{i \\in [n]} (d_i)^k, \\qquad m_\\infty = \\max _{i \\in [n]} d_i, \\qquad m_0= \\min _{i\\in [n]} d_i,$ Note that these numbers depend on $n$ , but in the sequel we will suppress this dependence.", "For each pair of vertices $i,j$ (not necessarily distinct), we add an edge independently with probability $p_{ij}=\\frac{d_id_j}{m_1}.$ The resulting random graph, which we denote by $\\mathcal {G}_n(\\vec{d}_n)$ , is referred to in the literature as the Chung-Lu random graph.", "In [15] it was assumed that $m_\\infty ^2 \\le m_1$ to ensure that $p_{ij} \\le 1$ .", "In the present paper we need sharper restrictions.", "Assumption 1.1 Throughout the paper we need two assumptions on $\\vec{d}_n$ as $n\\rightarrow \\infty $ : Connectivity and sparsity: There exists a $\\xi >2$ such that $(\\log n)^{2\\xi } \\ll m_\\infty \\ll n^{1/2}.$ Bounded inhomogeneity: $m_0 \\asymp m_\\infty $ .", "$\\spadesuit $ The lower bound in Assumption REF (D1) guarantees that the random graph is connected whp and that it is not too sparse.", "The upper bound is needed in order to have $m_\\infty = {\\protect {\\mathrm {o}}}(\\sqrt{m_1})$ , which implies that (REF ) is well defined.", "Assumption REF (D2) is a restriction on the inhomogeneity of the model and requires that the smallest and the largest degree are comparable.", "Remark 1.2 The lower bound on $m_\\infty $ in Assumption REF (D1) can be seen as an adaptation to our setting of the main condition in [16] for the asymptotics of $\\lambda _1$ .", "As mentioned in Section REF , under the assumption $\\frac{m_2}{m_1}\\gg \\sqrt{m_\\infty }\\,(\\log n)^\\xi ,$ [16] shows that $\\lambda _1= [1+{\\protect {\\mathrm {o}}}(1)]\\,m_2/m_1$ whp.", "It is easy to see that the above condition together with Assumption REF (D2) gives the lower bound in Assumption REF (D1).", "$\\spadesuit $ Remark 1.3 When $m_\\infty \\ll n^{1/6}$ , [33] implies that our results also hold for the Generalized Random Graph (GRG) model with the same average degrees.", "This model is defined by choosing connection probabilities of the form $p_{ij}=\\frac{d_id_j}{m_1+d_id_j},$ and arises in statistical physics as the canonical ensemble constrained on the expected degrees, which is also called the canonical configuration model.", "Note that in the above connection probability, $d_i$ plays the role of a hidden variable, or a Lagrange multiplier controlling the expected degree of vertex $i$ , but does not in general coincide with the expected degree itself.", "However, under the assumptions considered here, $d_i$ does coincide with the expected degree asymptotically.", "The reader can find more about GRG and their use in [33], and about their role in statistical physics in [31].", "In the corresponding microcanonical ensemble the degrees are not only fixed in their expectation but they take a precise deterministic value, which corresponds to the microcanonical configuration model.", "The two ensembles were found to be nonequivalent in the limit as $n\\rightarrow \\infty $ [30].", "This result was shown to imply a finite difference between the expected values of the largest eigenvalue $\\lambda _1$ in the two models [17] when the degree sequence was chosen to be constant ($d_i=d$ for all $i \\in [n]$ ).", "In this latter case the canonical ensemble reduces to the Erdős-Rényi random graph with $p=d/n$ , while the microcanonical ensemble reduces to the $d$ -regular random graph model.", "Although ensemble nonequivalence is not our main focus here, we will briefly relate some of our results to this phenomenon.", "$\\spadesuit $" ], [ "Notation", "Let $A$ be the adjacency matrix of $\\mathcal {G}_n(\\vec{d}_n)$ and $\\mathbb {E}[A]$ its expectation.", "The $(i,j)$ -th entry of $\\mathbb {E}[A]$ equals to $p_{ij}$ in (REF ).", "The $(i,j)$ -th entry of $A-\\mathbb {E}[A]$ is an independent centered Bernoulli random variable with parameter $p_{ij}$ .", "Let $\\lambda _1\\ge \\ldots \\ge \\lambda _n$ be the eigenvalues of $A$ and let $v_1, \\ldots , v_n$ be the corresponding eigenvectors.", "The vector $e$ will the $n\\times 1$ row vector given by $e=\\frac{1}{\\sqrt{m_1}}( d_1, \\cdots , d_n)^t,$ where $t$ stands for transpose.", "It is easy to see that $\\mathbb {E}[A]=ee^t$ .", "Definition 1.4 Following [19], we say that an event $\\mathcal {E}$ holds with high probability (whp) when there exist $\\xi >2$ and $\\nu >0$ such that $\\mathbb {P}(\\mathcal {E}) \\le \\mathrm {e}^{-\\nu (\\log n)^\\xi }.$ Note that this is different from the classical notion of whp, because it comes with a specific rate.", "We write $\\stackrel{w}{\\longrightarrow }$ to denote weak convergence as $n\\rightarrow \\infty $ , and use the symbols ${\\protect {\\mathrm {o}}},\\mathrm {O}$ to denote asymptotic order for sequences of real numbers." ], [ "CLT for the principal eigenvalue", "Our first theorem identifies two terms in the expectation of the largest eigenvalue, and shows that the largest eigenvalue follows a central limit theorem.", "Theorem 1.5 Under Assumption REF , the following hold: (I) $\\mathbb {E}[\\lambda _1] = \\frac{m_2}{m_1} + \\frac{m_1m_3}{m_2^2} +{\\protect {\\mathrm {o}}}(1), \\qquad n\\rightarrow \\infty .$ (II) $\\frac{m_2}{m_1}\\left(\\frac{\\lambda _1-\\mathbb {E}[\\lambda _1]}{\\sigma _1}\\right) \\stackrel{w}{\\longrightarrow }\\mathcal {N}(0,2), \\qquad n \\rightarrow \\infty ,$ where $\\sigma _1^2 = \\sum _{i,j} (p_{ij})^3(1-p_{ij}) \\sim \\frac{m_3^2}{m_1^3},\\qquad n \\rightarrow \\infty .$" ], [ "CLT for the principal eigenvector", "Our second theorem shows that the principal eigenvector is parallel to the normalised degree vector, and is close to this vector in $\\ell ^\\infty $ -norm.", "It also identifies the expected value of the components of the principal eigenvector, and shows that the components follow a central limit theorem.", "Theorem 1.6 Let $\\tilde{e} = e\\sqrt{m_1/m_2}$ be the $\\ell ^2$ -nomalized degree vector.", "Let $v_1$ be the eigenvector corresponding to $\\lambda _1$ and let $v_1(i)$ denote the $i$ -th coordinate of $v_1$ .", "Under Assumption REF , the following hold: (I) $\\langle v_1,\\tilde{e} \\rangle = 1+{\\protect {\\mathrm {o}}}(1)$ as $n\\rightarrow \\infty $ whp.", "(II) $\\Vert v_1- \\tilde{e}\\Vert _{\\infty } \\le \\mathrm {O}\\left(\\frac{(\\log n)^{\\xi }}{\\sqrt{n m_{\\infty }}}\\right)$ as $n\\rightarrow \\infty $ whp.", "(III) $\\mathbb {E}[v_1(i)]=\\frac{d_i}{\\sqrt{m_2}}+\\mathrm {O}\\left(\\frac{(\\log n)^{2\\xi }}{\\sqrt{m_2}}\\right)$ as $n\\rightarrow \\infty $ .", "Moreover, if the lower bound in Assumption REF (D1) is strengthened to $(\\log n)^{4\\xi } \\ll m_\\infty $ , then for all $i \\in [n]$ , (IV) $\\frac{m_2}{m_1}\\left(\\frac{v_1(i)-d_i/\\sqrt{m_2}}{s_1(i)}\\right)\\stackrel{w}{\\longrightarrow }\\mathcal {N}(0,1), \\qquad n \\rightarrow \\infty ,$ where $s_1^2(i)=\\sum _j d_j^2 p_{ij}(1-p_{ij}) \\sim d_i\\frac{m_3}{m_1},\\qquad n \\rightarrow \\infty .$" ], [ "Discussion", "We place the theorems in their proper context.", "1.", "Theorems REF –REF provide a CLT for $\\lambda _1,v_1$ .", "We note that $m_2/m_1$ is the leading order term in the expansion of $\\lambda _1$ , while $m_1m_3/m_2^2$ is a correction term.", "We observe that Theorem REF (I) does not follow from the results in [16], because the largest eigenvalue need not be uniformly integrable and also the second order expansion is not considered there.", "We also note that in Theorem REF (II) the centering of the largest eigenvalue, $\\mathbb {E}[\\lambda _1]$ , cannot be replaced by its asymptotic value as the error term is not compatible with the required variance.", "2.", "The lower bound in Assumption REF (D1) is needed to ensure that the random graph is connected, and is crucial because the largest eigenvalue is very sensitive to connectivity properties.", "Assumption REF (D2) is needed to control the inhomogeneity of the random graph.", "It plays a crucial role in deriving concentration bounds on the central moments $\\langle e, (A-\\mathbb {E}[A])^k e\\rangle $ , $k \\in \\mathbb {N}$ , with the help of a result from [19].", "Further refinements may come from different tools, such as the non-backtracking matrices used in [5], [6].", "While Assumption REF (D1) appears to be close to optimal, Assumption REF (D2) is far from optimal.", "It would be interesting to allow for empirical degree distributions that converge to a limiting degree distribrution with a power law tail.", "3.", "As already noted, if the expected degrees are all equal to each other, i.e., $d_i=d$ for all $i \\in [n]$ , then the Chung-Lu random graph, or canonical configuration model, reduces to the homogeneous Erdős-Rényi random graph with $p=d/n$ , while the corresponding microcanonical configuration model reduces to the homogeneous $d$ -regular random graph model (here, all models allow for self-loops).", "This implies that, for the homogeneous Erdős-Rényi random graph with connection probability $p \\gg (\\log n)^{2\\xi }/n$ , $\\xi >2$ , Theorem REF (I) reduces to $\\mathbb {E} [\\lambda _1] = np+1+{\\protect {\\mathrm {o}}}(1), \\qquad n\\rightarrow \\infty ,$ while Theorem REF (II) reduces to $\\frac{1}{\\sqrt{p}} \\left(\\lambda _1- \\mathbb {E}[\\lambda _1]\\right) \\stackrel{w}{\\longrightarrow }\\mathcal {N}(0, 2), \\qquad n \\rightarrow \\infty .$ Both these properties were derived in [18] for homogeneous Erdős-Rényi random graphs and also for rank-1 perturbations of Wigner matrices.", "In [17], the fact that $\\mathbb {E}[\\lambda _1]$ in the canonical ensemble differs by a finite amount from the corresponding expected value (here, $d=np$ ) in the microcanonical ensemble ($d$ -regular random graph) was shown to be a signature of ensemble nonequivalence.", "4.", "In case $d_i=d$ for all $i \\in [n]$ , Theorem REF (III) reduces to the following CLT, which was not covered by [18] and [17].", "Corollary 1.7 For the Erdős-Rényi random graph with $(\\log n)^{4\\xi }/n \\ll p \\ll n^{-1/2}$ for some $\\xi >2$ , $n\\sqrt{\\frac{p}{1-p}}\\left(v_1(i)-\\frac{1}{\\sqrt{n}}\\right)\\stackrel{w}{\\longrightarrow }\\mathcal {N}(0,1), \\qquad n \\rightarrow \\infty .$ Note that, in the corresponding microcanonical ensemble ($d$ -regular random graph), $v_1$ coincides with the constant vector where $v_1(i)=1/\\sqrt{n}$ for all $i \\in [n]$ .", "Therefore in the canonical ensemble each coordinate $v_1(i)$ has Gaussian fluctuations around the corresponding deterministic value for the microcanonical ensemble.", "This behaviour is similar to the degrees having, in the canonical configuration model, either Gaussian (in the dense setting) or Poisson (in the sparse setting) fluctuations around the corresponding deterministic degrees for the microcanonical configuration model [23].", "5.", "In [14] the empirical spectral distribution of $A$ was considered under the assumption that $(m_\\infty )^2/m_1 \\ll 1 \\ll n(m_\\infty )^2/m_1,$ which is weaker than Assumption REF .", "It was shown that if $\\mu _n \\stackrel{w}{\\longrightarrow }\\mu $ with $\\mu _n = n^{-1} \\sum _{i=1}^n \\delta _{d_i/m_\\infty }$ and $\\mu $ some probability distribution on $\\mathbb {R}$ , then $\\mathrm {ESD}\\left(\\frac{A}{\\sqrt{n(m_\\infty )^2/m_1}}\\right) \\stackrel{w}{\\longrightarrow }\\mu \\boxtimes \\mu _\\mathrm {sc}$ with $\\mu _{\\mathrm {sc}}$ the Wigner semicircle law and $\\boxtimes $ the free multiplicative convolution.", "Since $\\mu \\boxtimes \\mu _{\\mathrm {sc}}$ is compactly supported, this shows that the scaling for the largest eigenvalue and the spectral distribution are different." ], [ "Proof of Theorem ", "In what follows we use the well-known method of writing the largest eigenvalue of a matrix as a rank-1 perturbation of the centered matrix.", "This method was previously successfully employed in [22], [29], [19].", "Given the adjacency matrix $A$ of our graph $G$ , we can write $A=H+\\mathbb {E}[A]$ with $H=A-\\mathbb {E}[A]$ .", "Let $v_1$ be the eigenvector associated with the eigenvalue $\\lambda _1$ .", "Then $A v_1 =\\lambda _1 v_1, \\quad (H+\\mathbb {E}[ A])v_1 =\\lambda _1v_1, \\quad (\\lambda _1 I - H) v_1 = \\mathbb {E}[A] v_1.$ Using that $\\mathbb {E}[A]=ee^t$ , we have $(\\lambda _1 I - H) v_1= \\left\\langle e, v_1\\right\\rangle e$ , where $I$ is the $n \\times n$ identity matrix.", "It follows that if $\\lambda _1$ is not an eigenvalue of $H$ , then the matrix $(\\lambda _1 I - H)$ is invertible, and so $v_1= \\left\\langle e, v_1\\right\\rangle (\\lambda _1 I - H)^{-1} e.$ Eliminating the eigenvector $v_1$ from the above equation, we get $1 = \\left\\langle e, (\\lambda _1 I - H)^{-1} e\\right\\rangle ,$ where we use that $\\left\\langle e, v_1\\right\\rangle \\ne 0$ (since $\\lambda _1$ is not an eigenvalue of $H$ ).", "Note that this can be expressed as $\\lambda _1 = \\left\\langle e, \\left(I- \\frac{H}{\\lambda _1}\\right)^{-1} e\\right\\rangle = \\sum _{k=0}^{\\infty } \\left\\langle e, \\left(\\frac{H}{\\lambda _1}\\right)^{k}e\\right\\rangle \\quad whp,$ where the validity of the series expansion will be an immediate consequence of Lemma REF below.", "Section REF derives bounds on the spectral norm of $H$ .", "Section REF analyses the expansion in (REF ) and prove the scaling of $\\mathbb {E}[\\lambda _1]$ .", "Section REF is devoted to the proof of the CLT for $\\lambda _1$ , Section to the proof of the CLT for $v_1$ .", "In the expansion we distinguish three ranges: (i) $k=0,1,2$ ; (ii) $3 \\le k \\le L$ ; (iii) $L < k < \\infty $ , where $L=\\lfloor \\log n\\rfloor .$ We will show that (i) controls the mean and the variance in both CLTs, while (ii)-(iii) are negligible error terms." ], [ "The spectral norm", "In order to study $\\lambda _1$ , we need good bounds on the spectral norm of $H$ .", "The spectral norm of matrices with inhomogeneous entries has been studied in a series of papers [5], [6], [2] for different density regimes.", "An important role is played by $\\lambda _1(\\mathbb {E}[A])$ .", "In recent literature this quantity has been shown to play a prominent role in the so-called BBP-transition [4].", "Given our setting (REF ), it is easy to see that $\\lambda _1(\\mathbb {E}[A])=\\frac{m_2}{m_1},$ while all other eigenvalues of $\\mathbb {E}[A]$ are zero.", "Remark 2.1 Since $m_0\\le \\frac{m_2}{m_1}\\le m_\\infty $ , Assumption REF (D2) implies that $\\frac{m_2}{m_1}\\asymp m_\\infty .$ $\\spadesuit $ We start with the following lemma, which ensures concentration of $\\lambda _1$ and is a direct consequence of the results in [6] (which match Assumption REF ).", "Lemma 2.2 Under Assumption REF , whp $\\left|\\frac{\\lambda _1(A)-\\lambda _1(\\mathbb {E}[A])}{\\lambda _1(\\mathbb {E}[A])}\\right|= \\mathrm {O}\\left(\\frac{1}{\\sqrt{m_\\infty }}\\right), \\qquad n \\rightarrow \\infty ,$ and consequently $\\frac{\\lambda _1(A)}{\\lambda _1\\left(\\mathbb {E}[A]\\right)} \\overset{\\mathbb {P}}{\\rightarrow }1, \\qquad n \\rightarrow \\infty .$ In the proof it is understood that all statements hold whp in the sense of (REF ).", "Let $A=H+\\mathbb {E}[A]$ .", "Due to Weyl's inequality, we have that $\\lambda _1(\\mathbb {E}[A])-\\Vert H\\Vert \\le \\lambda _1(A)\\le \\lambda _1(\\mathbb {E}[A])+\\Vert H\\Vert .$ From [6] we know that there is a universal constant $C>0$ such that $\\mathbb {E}\\left[\\Vert H\\Vert \\right]\\le \\sqrt{m_\\infty }\\left(2+\\frac{C}{q}\\sqrt{\\frac{\\log n}{1\\vee \\log \\left(\\frac{\\sqrt{\\log n}}{q}\\right)}}\\right),$ where $q=\\sqrt{m_\\infty }\\wedge n^{1/10} \\kappa ^{-1/9}$ with $\\kappa $ defined by $\\kappa = \\max _{ij} \\frac{p_ij}{m_\\infty /n}=\\frac{n m_\\infty }{m_1}.$ Thanks to Assumption REF (D2), we have $\\kappa =\\mathrm {O}(1)$ .", "By Remark 3.1 of [6] (which gives us that $q=\\sqrt{m_\\infty }$ for $n$ large enough) and Assumption REF , we get that $\\mathbb {E}\\left[\\Vert H\\Vert \\right] \\le {\\left\\lbrace \\begin{array}{ll}\\sqrt{m_\\infty }\\left(2+\\frac{C\\sqrt{\\log n}}{\\sqrt{m_\\infty }}\\right), \\qquad (\\log n)^{2\\xi }\\le \\sqrt{m_\\infty }\\le n^{1/10} \\kappa ^{-1/9},\\\\[0.2cm]\\sqrt{m_\\infty }\\left(2+\\frac{C^{\\prime }\\sqrt{\\log n}}{n^{1/10}}\\right), \\qquad \\sqrt{m_\\infty }\\ge n^{1/10} \\kappa ^{-1/9}.\\end{array}\\right.", "}$ Using [8] or [6] (the Talagrand inequality), we know that there exists a universal constant $c>0$ such that $\\mathbb {P}\\left(\\left|\\Vert H\\Vert -\\mathbb {E}[\\Vert H\\Vert ]\\right|>t\\right)\\le 2\\mathrm {e}^{-ct^2}.$ For $t=\\sqrt{\\nu (\\log n)^\\xi }$ , $\\mathbb {E}[\\Vert H\\Vert ]-\\sqrt{\\nu } (\\log n)^{\\xi /2}\\le \\Vert H\\Vert \\le \\mathbb {E}[\\Vert H\\Vert ]+\\sqrt{\\nu } (\\log n)^{\\xi /2}.$ Thus, we have $\\left|\\lambda _1(A)-\\lambda _1(\\mathbb {E}[A])\\right|\\le \\Vert H\\Vert \\le \\sqrt{m_{\\infty }}(2+{\\protect {\\mathrm {o}}}(1))+ \\sqrt{\\nu } (\\log n)^{\\xi /2}.$ Using that $\\lambda _1(\\mathbb {E}[A])= m_2/m_1$ , we have that whp the following bound holds: $\\left|\\frac{\\lambda _1(A)-\\lambda _1(\\mathbb {E}[A])}{\\lambda _1({\\rm E}[A])}\\right|\\le \\frac{\\sqrt{m_\\infty }}{m_2/m_1}\\left(2+{\\protect {\\mathrm {o}}}(1)\\right)+\\frac{ \\sqrt{\\nu } (\\log n)^{\\xi /2}}{m_2/m_1}= \\mathrm {O}\\left(\\frac{\\sqrt{m_\\infty }}{m_2/m_1}\\right).$ Via Assumption REF and (REF ) the claim follows.", "Remark 2.3 $\\mbox{}$ (a) The proof of Lemma REF works well if we replace Assumption REF (D2) by a milder condition.", "Indeed, the former is directly linked to the parameter $\\kappa $ that appears in the proof of Lemma REF and in the proof of [6], which contains a more general condition on the inhomogeneity of the degrees.", "(b) Note that a consequence of proof of Lemma REF is that whp $\\frac{\\Vert H\\Vert }{\\lambda _1(A)}\\le 1-C_0$ for some $C_0\\in (0,1)$ .", "This allows us to claim that whp the inverse $\\left(I- \\frac{H}{\\lambda _1(A)}\\right)^{-1}$ exists.", "$\\spadesuit $ Lemma 2.4 Let $1\\le k\\le L$ .", "Then, under Assumption REF , whp $\\left| \\left\\langle e, H^k e\\right\\rangle - \\protect {\\mathbb {E}}\\left[\\left\\langle e, H^k e\\right\\rangle \\right]\\right| \\le C \\frac{m_2}{m_1}\\frac{ m_\\infty ^{\\frac{k}{2}}(\\log n)^{k\\xi }}{\\sqrt{n}},$ i.e., $\\max _{1 \\le k \\le L} \\mathbb {P}\\left( \\left|\\left\\langle e, H^k e\\right\\rangle - \\protect {\\mathbb {E}}\\left[\\left\\langle e, H^k e\\right\\rangle \\right]\\right|> \\frac{C (\\log n)^{k\\xi }m_\\infty ^{\\frac{k}{2}}}{\\sqrt{n}} \\frac{m_2}{m_1}\\right) \\le \\mathrm {e}^{-\\nu (\\log n)^\\xi },\\qquad n\\ge n_1(\\nu ,\\xi ).$ Lemma REF is a generalization to the inhomogeneous setting of [19].", "We skip the proof because it requires a straightforward modification of the arguments in [19].", "Lemma 2.5 Under Assumption REF , for $2\\le k\\le L$ , there exists a constant $C>0$ such that $\\mathbb {E}\\left[ \\left\\langle e, H^k e\\right\\rangle \\right] \\le \\frac{m_2}{m_1} (Cm_\\infty )^{k/2}.$ Let $\\mathcal {E}$ be the high probability event defined by (REF ), i.e., $\\Vert H\\Vert \\le {\\rm E}[\\Vert H\\Vert ] + \\sqrt{\\nu }(\\log n)^{\\xi /2}\\le m_\\infty \\left( 1+ \\mathrm {O}\\left( \\frac{(\\log n)^{\\xi /2}}{m_\\infty }\\right)\\right).$ Due to Assumption REF (D1) we can bound the right-hand side by $Cm_\\infty $ .", "Since $\\Vert e\\Vert _2^2= m_2/m_1$ , on this event we have $\\mathbb {E}\\left[ \\left( \\left\\langle e, H^k e\\right\\rangle \\right){\\bf 1}_{\\mathcal {E}}\\right]\\le \\Vert e\\Vert _2^2\\, \\mathbb {E}[\\Vert H\\Vert ^k{\\bf 1}_{\\mathcal {E}}]\\\\\\le \\frac{m_2}{m_1} (Cm_\\infty )^{k/2} .$ We show that the expectation when evaluated on the complementary event is negligible.", "Indeed, observe that $\\mathbb {E}\\left[ \\left\\langle e, H^k e\\right\\rangle \\right]&= \\mathbb {E}\\left(\\sum _{i_1, \\ldots , i_{k+1}=1}^n e_{i_1}e_{i_{k+1}} \\prod _{j=1}^k H(i_j, i_{j+1})\\right)^2\\\\&\\le \\left( \\frac{n^{k+1} m_\\infty ^2}{m_1}\\right)^2\\le C\\mathrm {e}^{(2k+2)\\log n} \\le \\mathrm {e}^{2(\\log n)^2},$ where in the last inequality we use that $m_\\infty ={\\protect {\\mathrm {o}}}(\\sqrt{m_1})$ .", "This, combined with the exponential decay of the event $\\mathcal {E}^c$ , gives $\\mathbb {E}\\left[ \\left\\langle e, H^k e\\right\\rangle {\\bf 1}_{A^c}\\right] \\le C\\mathrm {e}^{- \\nu (\\log n)^{\\xi }},$ and so the claim follows." ], [ "Expansion for the principal eigenvalue", "We denote the event in Lemma REF by $\\mathcal {E}$ , which has high probability.", "As noted in Remark REF (b), $I-\\frac{H}{\\lambda _1}$ is invertible on $\\mathcal {E}$ .", "Hence, expanding on $\\mathcal {E}$ , we get $\\lambda _1= \\sum _{k=0}^\\infty \\left\\langle e, \\frac{H^k}{\\lambda _1^k} e\\right\\rangle .$ We split the sum into two parts: $\\lambda _1 = \\sum _{k=0}^{L} \\frac{ \\left\\langle e, H^k e\\right\\rangle }{\\lambda _1^{k}} + \\sum _{k=L+1}^\\infty \\frac{\\left\\langle e, H^k e\\right\\rangle }{\\lambda _1^{k}}.$ First we show that we may ignore the second sum.", "To that end we observe that, by Assumption REF (D1), on the event $\\mathcal {E}$ we can estimate $\\left| \\sum _{k= L+1}^\\infty \\frac{ \\left\\langle e, H^k e\\right\\rangle }{\\lambda _1^{k}}\\right|&\\le \\sum _{k=L+1}^\\infty \\frac{\\Vert e\\Vert _2^2 \\Vert H\\Vert ^k}{\\lambda _1^k}\\le \\sum _{k=L+1}^\\infty \\frac{m_2}{m_1} \\frac{m_{\\infty }^{k/2}}{ (Cm_2/m_1)^k}\\nonumber \\\\&\\le \\sum _{k=L+1}^\\infty \\frac{C^{\\prime }}{ m_{\\infty }^{ k/2-1}}=\\mathrm {O}\\left(\\mathrm {e}^{-c\\log \\sqrt{n}}\\right).$ Because of (REF ) and the fact that $\\mathbb {E}(\\left\\langle e,H e\\right\\rangle )=0$ , (REF ) reduces to $\\lambda _1&=\\sum _{k=3}^{L} \\frac{\\mathbb {E}\\left[\\left\\langle e, H^k e\\right\\rangle \\right]}{\\lambda _1^{k}}+ \\sum _{k=3}^{L} \\frac{ \\left\\langle e, H^k e\\right\\rangle - \\mathbb {E}\\left[\\left\\langle e, H^k e\\right\\rangle \\right]}{\\lambda _1^{k}}\\\\&\\qquad +\\left\\langle e, e\\right\\rangle + \\frac{1}{\\lambda _1} \\left\\langle e, H e\\right\\rangle + \\frac{1}{\\lambda _1^2} \\left\\langle e, H^2 e\\right\\rangle +{\\protect {\\mathrm {o}}}(1).$ Next, we estimate the second sum in the above equation.", "Using Lemma REF , we get $&\\left|\\sum _{k=3}^{L} \\frac{ \\left\\langle e, H^k e\\right\\rangle - \\mathbb {E}\\left[\\left\\langle e, H^k e\\right\\rangle \\right]}{\\lambda _1^{k}}\\right|\\\\&\\le \\sum _{k=3}^{L} \\frac{C m_{\\infty }^{\\frac{k}{2}}(\\log n)^{k\\xi }}{\\sqrt{n}(m_2/m_1)^{k-1}}\\le \\sum _{k=3}^{L} \\frac{C(\\log n)^{k\\xi }}{\\sqrt{n}m_{\\infty }^{k/2-1}}\\le \\mathrm {O}\\left( \\frac{C(\\log n)^{\\xi +1}}{\\sqrt{nm_\\infty }}\\right) = {\\protect {\\mathrm {o}}}(1).$ From Lemma REF we have $\\sum _{k=3}^L \\frac{\\mathbb {E}\\left\\langle e, H^k e\\right\\rangle }{\\lambda _1^{k}}\\le \\sum _{k=3}^L \\frac{ \\frac{m_2}{m_1} (Cm_{\\infty })^{k/2} }{\\left(m_2/m_1\\right)^k}=\\mathrm {O}\\left( \\frac{1}{\\sqrt{m_{\\infty }}}\\right)= {\\protect {\\mathrm {o}}}(1),$ where the last estimate follows from Assumption REF (D1).", "Hence, on $\\mathcal {E}$ , $\\lambda _1= \\left\\langle e, e\\right\\rangle +\\frac{1}{\\lambda _1} \\left\\langle e, H e\\right\\rangle + \\frac{\\left\\langle e, H^2 e\\right\\rangle }{\\lambda _1^2}+ {\\protect {\\mathrm {o}}}(1).$ Iterating the expression for $\\lambda _1$ in the right-hand side, we get $\\lambda _1&= \\left\\langle e, e\\right\\rangle + \\left\\langle e, H e\\right\\rangle \\left( \\left\\langle e, e\\right\\rangle + \\frac{1}{\\lambda _1}\\left\\langle e, He\\right\\rangle +\\frac{1}{\\lambda _1^2} \\left\\langle e, H^2 e\\right\\rangle + {\\protect {\\mathrm {o}}}(1) \\right)^{-1} \\\\&\\quad + \\left\\langle e, H^2 e\\right\\rangle \\left( \\left\\langle e, e\\right\\rangle + \\frac{1}{\\lambda _1}\\left\\langle e, H e\\right\\rangle +\\frac{1}{\\lambda _1^2} \\left\\langle e, H^2 e\\right\\rangle + {\\protect {\\mathrm {o}}}(1) \\right)^{-2} + {\\protect {\\mathrm {o}}}(1),$ Expanding the second and third term we get, $\\lambda _1&= \\left\\langle e, e\\right\\rangle + \\frac{\\left\\langle e, H e\\right\\rangle }{\\left\\langle e, e\\right\\rangle }\\left( 1 - \\frac{ \\left\\langle e, H e\\right\\rangle }{\\lambda _1 \\left\\langle e, e\\right\\rangle }-\\frac{\\left\\langle e, H^2 e\\right\\rangle }{\\lambda _1^2 \\left\\langle e, e\\right\\rangle } + {\\protect {\\mathrm {o}}}(1) \\right) \\\\&\\quad + \\frac{\\left\\langle e, H^2 e\\right\\rangle }{(\\left\\langle e, e\\right\\rangle )^2}\\left( 1 - \\frac{2\\left\\langle e, H e\\right\\rangle }{\\lambda _1 \\left\\langle e, e\\right\\rangle }-\\frac{2\\left\\langle e, H^2 e\\right\\rangle }{\\lambda _1^2 \\left\\langle e, e\\right\\rangle } + {\\protect {\\mathrm {o}}}(1) \\right) + {\\protect {\\mathrm {o}}}(1),\\\\&= \\left\\langle e, e \\right\\rangle + \\frac{\\left\\langle e, H e\\right\\rangle }{ \\left\\langle e, e\\right\\rangle } - \\frac{\\left\\langle e, H e\\right\\rangle ^2}{\\lambda _1 \\left\\langle e, e\\right\\rangle ^2}+ \\frac{\\left\\langle e, H^2 e\\right\\rangle }{ \\left\\langle e, e\\right\\rangle ^2} + {\\protect {\\mathrm {o}}}(1).$ Here we use that $\\left\\langle e, e\\right\\rangle = m_2/m_1\\rightarrow \\infty $ , and we ignore several other terms because they are small whp, for example, $\\frac{\\left\\langle e, H e\\right\\rangle \\left\\langle e, H^2 e\\right\\rangle }{\\lambda _1^2\\left\\langle e, e\\right\\rangle ^2} = \\mathrm {O}\\left( \\frac{m_\\infty ^{3/2}}{(m_2/m_1)^4}\\right)={\\protect {\\mathrm {o}}}(1).$ One more iteration gives $\\lambda _1&= \\left\\langle e, e \\right\\rangle + \\frac{\\left\\langle e, H e\\right\\rangle }{\\left\\langle e, e \\right\\rangle } + \\frac{\\left\\langle e, H^2 e\\right\\rangle }{\\left\\langle e, e \\right\\rangle ^2} \\\\&\\quad - \\frac{\\left\\langle e, H e\\right\\rangle ^2}{\\left\\langle e, e \\right\\rangle ^2}\\left( \\left\\langle e, e \\right\\rangle + \\frac{1}{\\lambda _1}\\left\\langle e, H e\\right\\rangle +\\frac{1}{\\lambda _1^2} \\left\\langle e, H^2 e\\right\\rangle + {\\protect {\\mathrm {o}}}(1) \\right)^{-1} + {\\protect {\\mathrm {o}}}(1) \\\\&= \\left\\langle e, e \\right\\rangle + \\frac{\\left\\langle e, H e\\right\\rangle }{\\left\\langle e, e \\right\\rangle } + \\frac{\\left\\langle e, H^2 e\\right\\rangle }{ \\left\\langle e, e \\right\\rangle ^2}- \\frac{\\left\\langle e, He\\right\\rangle ^2}{\\left\\langle e, e \\right\\rangle ^3} + \\frac{\\left\\langle e, H^2 e\\right\\rangle ^2 \\left\\langle e, H e\\right\\rangle }{\\lambda _1 \\left\\langle e, e \\right\\rangle ^3}+ \\frac{\\left\\langle e, H^2 e\\right\\rangle ^3}{\\lambda _1^2 \\left\\langle e, e \\right\\rangle ^3} + {\\protect {\\mathrm {o}}}(1).$ Since the probability of $\\mathcal {E}^c$ decays exponentially with $n$ , taking the expectation of the above term and using that $\\mathbb {E}[e^\\prime W e]=0$ , we obtain $\\mathbb {E} [\\lambda _1] = \\left\\langle e, e \\right\\rangle + \\frac{\\mathbb {E}[\\left\\langle e, H^2 e\\right\\rangle ]}{ \\left\\langle e, e \\right\\rangle ^2}- \\frac{\\mathbb {E}[\\left\\langle e, He\\right\\rangle ^2]}{ \\left\\langle e, e \\right\\rangle ^3} + {\\protect {\\mathrm {o}}}(1)= \\frac{m_2}{m_1}+ \\frac{m_1m_3}{m_2^2}- \\frac{ m_3^2}{m_2^3}+ {\\protect {\\mathrm {o}}}(1).$ Note that $\\frac{ m_3^2}{m_2^2}\\le \\frac{m_{\\infty }^2}{n}={\\protect {\\mathrm {o}}}(1),\\qquad \\frac{m_1 m_3}{m_2^2}\\le \\left(\\frac{m_{\\infty }}{m_0}\\right)^4=\\mathrm {O}(1),$ and so we can write $\\mathbb {E} [\\lambda _1] = \\frac{m_2}{m_1}+\\frac{m_1m_3}{m_2^2}+ {\\protect {\\mathrm {o}}}(1).$" ], [ "CLT for the principal eigenvalue", "Again consider the high probability event on which (REF ) holds.", "Recall that from the series decomposition in (REF ) we have $\\lambda _1&= \\frac{\\left\\langle e, He\\right\\rangle }{\\lambda _1} + \\sum _{k=0}^{L} \\frac{\\mathbb {E}\\left\\langle e, H^ke\\right\\rangle }{\\lambda _1^k}+ \\sum _{k=2}^{L} \\frac{ \\left\\langle e, H^ke\\right\\rangle - \\mathbb {E}\\left\\langle e, H^k e\\right\\rangle }{\\lambda _1^k}+ \\sum _{k>L} \\frac{\\left\\langle e, H^k e\\right\\rangle }{\\lambda _1^k}.$ Lemma 2.6 The equation $x= \\sum _{k=0}^{L} \\frac{\\mathbb {E}\\left\\langle e, H^ke\\right\\rangle }{x^k}$ has a solution $x_0$ satisfying $\\lim _{n\\rightarrow \\infty } \\frac{x_0}{m_2/m_1} = 1.$ Define the function $h\\colon \\, (0,\\infty )\\rightarrow \\mathbb {R}$ by $h(x)= \\sum _{k=0}^{\\log n} \\frac{\\mathbb {E}\\left\\langle e, H^ke\\right\\rangle }{x^k}.$ Since $\\mathbb {E}[e^\\prime He]=0$ , we have $h\\left(\\frac{xm_2}{m_1}\\right)= \\frac{m_2}{m_1} +\\sum _{k=2}^{\\log n} \\frac{\\mathbb {E}\\left\\langle e, H^ke\\right\\rangle }{(xm_2/m_1)^k}.$ For $x>0$ , $\\left|\\sum _{k=2}^{\\log n} \\frac{\\mathbb {E}[\\left\\langle e, H^ke\\right\\rangle ]}{(xm_2/m_1)^k}\\right|&\\le \\sum _{k=2}^\\infty \\frac{1}{(xm_2/m_1)^k}\\frac{m_2}{m_1} (Cm_\\infty )^{k/2} \\\\&= {\\protect {\\mathrm {o}}}\\left( \\frac{m_2}{m_1} \\sum _{k=2}^\\infty \\frac{1}{x^k (\\log n)^{k\\xi }}\\right)={\\protect {\\mathrm {o}}}\\left( \\frac{m_2}{m_1} x^{-2}\\right).$ This shows that $\\lim _{n\\rightarrow \\infty } \\frac{1}{m_2/m_1} \\sum _{k=0}^{\\log n} \\frac{\\mathbb {E}\\left\\langle e, H^ke\\right\\rangle }{(xm_2/m_1)^k} = 1.$ Hence, for any $0<\\delta <1$ , $\\lim _{n\\rightarrow \\infty } \\frac{1}{m_2/m_1} \\left[ \\frac{m_2}{m_1}(1+\\delta )- h\\left((1+\\delta ) \\frac{m_2}{m_1}\\right)\\right] = \\delta .$ So, for large enough $n$ , $h\\left((1+\\delta ) \\frac{m_2}{m_1}\\right) < \\frac{m_2}{m_1} (1+\\delta ).$ Similarly, for any $0<\\delta <1$ , $h\\left((1-\\delta ) \\frac{m_2}{m_1}\\right) > \\frac{m_2}{m_1} (1-\\delta ).$ This shows that there is a solution for (REF ), which lies in the interval $[ \\frac{m_2}{m_1}(1-\\delta ), \\frac{m_2}{m_1}(1-\\delta )]$ .", "Lemma 2.7 Let $x_0$ be a solution for (REF ).", "Define $R_n=\\lambda _1 -x_0 -\\frac{\\left\\langle e, H e\\right\\rangle }{m_2/m_1}.$ Then $R_n={\\protect {\\mathrm {o}}}_\\mathbb {P}\\left(\\frac{m_3}{m_2\\sqrt{m_1}}\\right),\\qquad \\mathbb {E}\\left[|R_n|\\right]={\\protect {\\mathrm {o}}}\\left(\\frac{m_3}{m_2\\sqrt{m_1}}\\right).$ From the previous lemmas we have $\\lambda _1= x_0+ \\frac{\\left\\langle e, H e\\right\\rangle }{m_2/m_1}+ R_n.$ Therefore $\\mathbb {E}[\\lambda _1]= x_0+ \\mathbb {E}[R_n]$ and $\\lambda _1- \\mathbb {E}[\\lambda _1]= \\frac{\\left\\langle e, H e\\right\\rangle }{m_2/m_1}+ {\\protect {\\mathrm {o}}}\\left(\\frac{m_3}{m_2\\sqrt{m_1}}\\right).$ Hence $\\frac{m_2}{m_1}\\left(\\lambda _1- \\mathbb {E}[\\lambda _1]\\right)= \\left\\langle e, H, e\\right\\rangle + {\\protect {\\mathrm {o}}}\\left(\\frac{m_3}{m_1^{3/2}}\\right).$ Observe that $\\left\\langle e, H e\\right\\rangle = \\sum _{i,j=1}^N h_{i,j} \\frac{d_i d_j}{m_1}= 2\\sum _{i\\le j}h_{i,j} \\frac{d_i d_j}{m_1}$ Let $\\sigma _1^2= \\sum _{i\\le j} \\mathrm {Var}\\left(\\frac{2}{m_1}h_{i,j} d_i d_j\\right)=\\sum _{i\\le j}\\frac{4d_i^3d_j^3}{m_1^3}\\left(1- \\frac{d_id_j}{m_1}\\right)\\sim 2 \\frac{m_3^2}{m_1^3}\\left(1+\\mathrm {O}\\left(\\frac{m_\\infty ^2}{n}\\right)\\right),$ where we use the symmetry of the expression in the last equality.", "We can apply Lyapunov's central limit theorem, because $\\lbrace h_{i,j}\\colon \\, i\\le j\\rbrace $ is an independent collection of random variables and Lyapunov's condition is satisfied, i.e., $\\lim _{n \\rightarrow \\infty } \\frac{1}{\\sigma _n^3} \\sum _{i > j} \\protect {\\mathbb {E}}\\left[ \\left| H(i,j) d_i d_j \\right|^3\\right]\\le K\\lim _{n \\rightarrow \\infty } \\frac{m_1^{3/2}}{m_3^3}\\frac{m_4^2}{m_1} =0,$ where $K$ is a constant that does not depend on $n$ .", "Hence $\\frac{m_1^{3/2}\\left\\langle e, H e\\right\\rangle }{\\sqrt{2} m_3}\\overset{w}{\\longrightarrow }N(0,1).$ Returning to the eigenvalue equation in (REF ) and dividing by $\\sigma _1$ , we have $\\frac{\\sqrt{m_1}{m_2}}{m_3} \\left( \\lambda _1- {\\rm E}[ \\lambda _1]\\right)= \\frac{m_1^{3/2}\\left\\langle e, H e\\right\\rangle }{ m_3}+ {\\protect {\\mathrm {o}}}(1)\\overset{w}{\\longrightarrow }N(0, 2).$ We next prove Lemma REF , on which the proof of the central limit theorem relied.", "Note that by (REF ) and (REF ) we can write $\\lambda _1-x_0= \\frac{\\left\\langle e, H e\\right\\rangle }{\\lambda _1}+\\sum _{k=2}^{L} \\mathbb {E}\\left\\langle e, H^k e\\right\\rangle \\left( \\frac{1}{\\lambda _1^k}- \\frac{1}{x_0^k}\\right)+L_n,$ where $L_n=\\sum _{k=2}^L \\frac{ \\left\\langle e, H^k e\\right\\rangle - \\mathbb {E}\\left\\langle e, H^k e\\right\\rangle }{\\lambda _1^k}+\\sum _{k>L}\\frac{\\left\\langle e, H^k e\\right\\rangle }{\\lambda _1^k}.$ Thanks to Lemma REF , Lemma REF and (REF ) we have $L_n = \\mathrm {O}\\left(\\frac{m_\\infty (\\log n)^{2\\xi }}{\\sqrt{n}m_2/m_1}\\right).$ Note that $L_n= {\\protect {\\mathrm {o}}}( \\frac{m_3}{m_2 \\sqrt{m_1}})$ .", "Indeed, using $m_3\\ge n m_0^3$ and Assumption REF (D1), we get $\\frac{m_\\infty (\\log n)^{2\\xi } m_2\\sqrt{m_1}}{\\sqrt{n}(m_2/m_1) m_3}\\le \\frac{m_{\\infty }^{5/2} n^{3/2} (\\log n)^{2\\xi }}{\\sqrt{n} m_0 n m_0^3 (\\log n)^\\xi }=\\frac{ m_{\\infty }^{5/2}(\\log n)^{\\xi }}{m_0^4 }=\\mathrm {O}\\left(\\frac{(\\log n)^{\\xi }}{m_0^{3/2}}\\right).$ Observe that (REF ) can be rearranged as $(\\lambda _1-x_0)= \\frac{\\left\\langle e, He\\right\\rangle }{\\lambda _1}-\\sum _{k=2}^{L} (\\lambda _1-x_0)\\mathbb {E}\\left\\langle e, H^ke\\right\\rangle \\lambda _1^{-k} x_0^{-k} \\sum _{j=0}^{k-1} x_0^{k-1-j}+ L_n.$ Hence, bringing the second term from the right to the left, we have $(\\lambda _1-x_0)\\left[ 1+ \\sum _{k=2}^{L} \\mathbb {E}\\left\\langle e, H^ke\\right\\rangle \\lambda _1^{-k} x_0^{-k}\\sum _{j=0}^{k-1} x_0^{k-1-j}\\right]= \\frac{\\left\\langle e, H e\\right\\rangle }{\\lambda _1} + L_n.$ Using the bounds on $\\lambda _1$ and $x_0$ , we get $&\\left|\\sum _{k=2}^{L} \\mathbb {E}\\left\\langle e, H^ke\\right\\rangle \\lambda _1^{-k} x_0^{-k} \\sum _{j=0}^{k-1} x_0^{k-1-j}\\right|\\le \\sum _{k=2}^{L} \\frac{k}{(m_2/m_1)^{k+1}} \\mathbb {E}\\left\\langle e, H^ke\\right\\rangle \\\\&\\le \\sum _{k=2}^{L} \\frac{k}{(m_2/m_1)^{k+1}}\\frac{m_2}{m_1} (Cm_\\infty )^{k/2}= \\mathrm {O}\\left(\\frac{m_{\\infty }}{(m_2/m_1)^2(\\log n)^{2\\xi -1}}\\right)={\\protect {\\mathrm {o}}}(1).$ We can therefore write $\\lambda _1-x_0= \\frac{\\left\\langle e,H e\\right\\rangle }{\\lambda _1}+L_n,$ where $L_n= {\\protect {\\mathrm {o}}}_P( \\frac{m_3}{m_2 \\sqrt{m_1}})$ .", "Finally, to go to $R_n$ , note that $R_n= \\lambda _1-x_0- \\frac{\\left\\langle e, H e\\right\\rangle }{m_2/m_1} =\\left\\langle e, H e\\right\\rangle \\left( \\frac{1}{\\lambda _1}- \\frac{1}{m_2/m_1}\\right)+ L_n.$ To bound $R_n$ , it is enough to show that the first term on the right-hand side is whp bounded by $\\frac{m_3}{m_2 \\sqrt{m_1}}$ .", "Using Lemma REF (for $k=1$ ) and (REF ), we have whp $\\frac{\\left| \\left\\langle e, H e\\right\\rangle \\right| |\\lambda _1- m_2/m_1|}{\\lambda _1 m_2/m_1}\\le \\frac{\\sqrt{m_\\infty }(\\log n)^{\\xi }}{\\sqrt{n}}\\frac{\\sqrt{m_{\\infty }}}{(m_2/m_1)}.$ Using again Assumption REF (D1), $m_3\\ge n m_0^3$ , $m_1\\le n m_\\infty $ and $m_2\\le n m_{\\infty }^2$ , we get that $\\frac{m_\\infty (\\log n)^{\\xi }}{\\sqrt{n}(m_2/m_1)}\\frac{m_2\\sqrt{m_1}}{m_3}\\le \\left(\\frac{m_\\infty }{m_0}\\right)^3\\frac{c}{\\sqrt{m_{\\infty }}}={\\protect {\\mathrm {o}}}(1).$ This controls the right-hand side of (REF ), and hence $R_n={\\protect {\\mathrm {o}}}( \\frac{m_3}{m_2\\sqrt{m_1}})$ whp.", "We want to show that the latter is negligible both pointwise and in expectation.", "We already have that this is so whp on $R_n$ .", "We want to show that the same bound holds in expectation.", "Let $\\mathcal {A}$ be the high probability event of Lemma REF and REF , and write $\\mathbb {E}[|R_n|] = \\mathbb {E}[|R_n| \\textbf {1}_{\\mathcal {A}^c}] + \\mathbb {E}[|R_n|\\textbf {1}_{\\mathcal {A}}],$ where $\\textbf {1}_{\\mathcal {A}}$ is the indicator function of the event $\\mathcal {A}$ .", "Since all the bounds hold on the high probability event $\\mathcal {A}$ , it is immediate that $\\mathbb {E}[|R_n|\\textbf {1}_{\\mathcal {A}}]={\\protect {\\mathrm {o}}}\\left(\\frac{m_3}{\\sqrt{m_1} m_2}\\right).$ The remainder can be bounded via the Cauchy-Schwarz inequality, namely, $\\mathbb {E}[|R_n| \\textbf {1}_{\\mathcal {A}^c}] \\le \\left(\\mathbb {E}[|R_n|^2]\\mathbb {E}[\\textbf {1}_{\\mathcal {A}^c}]\\right)^{\\frac{1}{2}}\\le \\left(\\mathbb {E}\\left[|R_n|^2\\right]e^{-\\nu (\\log n)^\\xi }\\right)^{\\frac{1}{2}}.$ We see that if $\\mathbb {E}[|R_n|^2]={\\protect {\\mathrm {o}}}(\\mathrm {e}^{-\\nu (\\log n)^\\xi })$ , then we are done.", "Expanding, we see that $\\mathbb {E}[|R_n|^2] = \\mathbb {E}\\left[\\left|\\lambda _1 - x_0 - \\frac{\\left\\langle e, H e\\right\\rangle }{m_2/m_1} \\right|^2\\right] \\le n^C$ for some $C>0$ , where we use that $\\mathbb {E}[(\\lambda _1^2)] \\le \\mathbb {E}[A^2] = \\sum _{i,j=1}^N \\mathbb {E}[(A(i,j))^2] \\le m_\\infty n$ and the trivial bound $|\\left\\langle e, He\\right\\rangle |\\le n^{C_*}$ for some $C_*<C$ .", "Hence we have $\\left(\\mathbb {E}[|R_n|^2]\\mathbb {E}[\\textbf {1}_{A^c}]\\right)^{\\frac{1}{2}} \\le \\mathrm {e}^{-\\nu (\\log n)^{\\xi }}$ and $\\mathbb {E}[|R_n|] ={\\protect {\\mathrm {o}}}\\left(\\frac{m_3}{\\sqrt{m_1} m_2}\\right).$" ], [ "Proof of Theorem ", "In this section we study the properties of the principal eigenvector.", "Let $v_1$ be the normalized principal eigenvector, i.e., $\\Vert v_1\\Vert =1$ , and let $e$ be as defined in (REF ).", "Recall from (REF ) that $\\lambda _1 \\left(1-\\frac{H}{\\lambda _1}\\right) v_1= e\\langle e,v_1\\rangle ,$ and after inversion (which is possible on the high probability event) we have $v_1=\\frac{\\langle e,v_1\\rangle }{\\lambda _1}(1-H/\\lambda _1)^{-1} e.$ If $K$ denotes the normalization factor, then we can rewrite the above equation whp as the series $v_1=\\frac{K}{\\lambda _1}\\sum _{k=0}^\\infty \\frac{H^ke}{\\lambda _1^k}.$ Our first step is to determine the value of $K$ in (REF ).", "We adapt the results from [19] to derive a component-wise central limit theorem in the inhomogeneous setting described by (REF ) under Assumption REF .", "By the normalization of $v$ , $1 = \\langle v_1 , v_1\\rangle = \\frac{K^2}{\\lambda _1^2}\\left\\langle \\sum _{k=0}^\\infty \\frac{H^k}{\\lambda _1^k} e,\\sum _{\\ell =0}^\\infty \\frac{H^{\\ell }}{\\lambda _1^{\\ell }} e \\right\\rangle = \\frac{K^2}{\\lambda _1^2}\\sum _{k=0}^\\infty \\frac{(k+1)\\left\\langle e,{H^k}e\\right\\rangle }{\\lambda _1^k},$ where we use the symmetry of $H$ .", "The following lemma settles Theorem REF (I).", "Lemma 3.1 Under Assumption REF , and with $\\tilde{e}=e\\sqrt{\\frac{m_1}{m_2}}$ , whp $\\langle \\tilde{e},v_1\\rangle =1+{\\protect {\\mathrm {o}}}(1).$ Recall that $L=\\lfloor \\log n\\rfloor $ .", "We rewrite (REF ) as $\\begin{split}\\left(\\frac{\\lambda _1}{K}\\right)^2&= \\sum ^{L}_{k= 0}\\frac{(k+1)}{\\lambda _1^k}\\,\\mathbb {E}\\left[\\left\\langle e, H^k e\\right\\rangle \\right]+\\sum _{k=1}^{L} \\frac{(k+1)}{\\lambda _1^k}\\left|\\left\\langle e, H^k e\\right\\rangle -\\mathbb {E}\\left[\\left\\langle e,H^k e\\right\\rangle \\right]\\right| \\\\&\\qquad \\qquad +\\sum _{k=L+1}^\\infty \\frac{(k+1)}{\\lambda _1^k}\\left\\langle e, H^k e\\right\\rangle .\\end{split}$ We first show that the last two parts are negligible and then show that the main term of the first part is the term with $k=0$ , i.e., $\\langle e,e\\rangle =m_2/m_1$ .", "The last term in (REF ) is dealt with as follows.", "Using (REF ), we have whp $\\begin{split}\\sum _{k=L+1}^\\infty \\frac{(k+1)}{\\lambda ^k}\\left\\langle e,H^ke\\right\\rangle &\\le \\sum _{k=L+1}^\\infty (k+1) \\frac{\\Vert e\\Vert ^2\\Vert H\\Vert ^k}{(m_2/m_1)^k}\\le \\frac{m_2}{m_1}\\sum _{k=L+1}^\\infty (k+1) (1-C_0)^k\\\\&\\le \\frac{m_2}{m_1} (\\log n+2) \\mathrm {e}^{-c^{\\prime }\\log n}\\frac{1}{C_0^2}\\end{split}$ with $c^{\\prime }=-\\log (1-C_0)$ , where we use that $\\sum _{k=0}^\\infty (k+1)(1-c)^k=1/c^2$ for $|1-c|<1$ .", "We tackle the second sum in (REF ) by using Lemma REF .", "Indeed, whp we have $\\begin{split}\\sum _{k=1}^L \\frac{(k+1)}{\\lambda _1^k}\\left|\\left\\langle e, H^k e\\right\\rangle -\\mathbb {E}\\left[\\left\\langle e,H^ke\\right\\rangle \\right]\\right|&\\le \\sum _{k=1}^L (k+1) \\frac{Cm_\\infty ^{k/2}(\\log n)^{k\\xi }}{\\sqrt{n}}\\left(\\frac{m_2}{m_1}\\right)^{1-k}\\\\&\\le \\frac{C^{\\prime }\\sqrt{m_\\infty }(\\log n)^\\xi (\\log n +1)}{\\sqrt{n}}\\le \\frac{C^{\\prime }\\sqrt{m_\\infty }(\\log n)^{2\\xi }}{\\sqrt{n}},\\end{split}$ where the constant varies in each step.", "By Assumption REF (D1), the last term goes to zero.", "As to the first term, note that by (REF ) for $k\\ge 3$ we have $\\sum ^{L}_{k=3}\\frac{(k+1)}{\\lambda _1^k}\\mathbb {E}\\left[\\left\\langle e, H^k e\\right\\rangle \\right]&\\le \\sum _{k=3}^{L} (k+1)\\left(\\frac{m_2}{m_1}\\right)^{-k+1} (Cm_\\infty )^{k/2} \\\\&\\le \\sum _{k=3}^L \\frac{C m_{\\infty }^{k/2}}{(m_2/m_1)^{(k-1)}}=\\mathrm {O}\\left(\\frac{1}{\\sqrt{m_\\infty }}\\right).$ The term with $k=1$ is zero, while for $k=2$ we have $3\\frac{\\mathbb {E}\\langle e,H^2e\\rangle }{\\lambda ^2_1}\\le c \\frac{m_1m_3}{m_2^2}= \\mathrm {O}\\left(1\\right)$ for some constant $c$ .", "After substituting these results into (REF ), we find $\\left(\\frac{\\lambda _1}{K}\\right)^2= \\frac{m_2}{m_1}\\left(1+\\mathrm {O}\\left(\\frac{1}{m_2/m_1}\\right)\\right)$ and the proof follows by normalizing the vector $e$ and using (REF ).", "The following lemma is an immediate consequence of (REF ) and Lemma REF .", "Lemma 3.2 Under Assumptions REF , whp $v_1 = \\left(1+\\mathrm {O}\\left(\\frac{m_1}{m_2}\\right)\\right)\\sqrt{\\frac{m_1}{m_2}}\\sum _{k=0}^\\infty \\frac{H^k}{\\lambda _1^k}e.$ In order to estimate how the components of $v_1$ concentrate, we need the following lemma.", "Lemma 3.3 For $1 \\le k \\le L$ , whp $|H^k e(i)| =\\left| \\frac{1}{\\sqrt{m_1}}\\sum _{i_1,\\dots ,i_k}h_{i i_1}h_{i_1i_2}\\dots h_{i_{k-1}i_k}d_{i_k}\\right|\\le \\frac{m_\\infty }{\\sqrt{m_1}}\\left((\\log n)^{\\xi } \\sqrt{m_\\infty } \\right)^k.$ The proof of this lemma is a direct consequence of Lemma REF , is similar to [19] and therefore we skip it.", "An immediate corollary of the above estimate is the delocalized behaviour of the largest eigenvector stated in Theorem REF (II).", "Lemma 3.4 Let $v_1$ be the normalized principal eigenvector, and $\\tilde{e}=e\\sqrt{\\frac{m_1}{m_2}}$ .", "Then whp $\\Vert v_1- \\tilde{e}\\Vert _{\\infty } \\le \\mathrm {O}\\left(\\frac{(\\log n)^{\\xi }}{\\sqrt{n m_{\\infty }}}\\right).$ Recall from (REF ) that $v_1(i) =\\frac{K}{\\lambda _1} \\sum _{k=0}^\\infty \\frac{H^ke(i)}{\\lambda _1^k}= \\frac{K}{\\lambda _1} e(i) + \\frac{K}{\\lambda _1} \\sum _{k=1}^L \\frac{H^ke(i)}{\\lambda _1^k}+\\frac{K}{\\lambda _1} \\sum _{k=L+1}^\\infty \\frac{H^ke(i)}{\\lambda _1^k}.$ The last term is negligible whp, because it is the tail sum of a geometrically decreasing sequence.", "For the sum over $1 \\le k \\le L$ fwe can use Lemma REF and the fact that $K/\\lambda _1= \\sqrt{\\frac{m_1}{m_2}}+ {\\protect {\\mathrm {o}}}(1)$ whp.", "So we have $\\frac{K}{\\lambda _1} \\sum _{k=1}^L \\frac{H^ke(i)}{\\lambda _1^k}\\le \\frac{m_{\\infty }}{\\sqrt{n}m_0} \\frac{(\\log n)^{\\xi }}{\\sqrt{m_{\\infty }}}\\le \\mathrm {O}\\left(\\frac{(\\log n)^{\\xi }}{\\sqrt{nm_{\\infty }}}\\right).$ The first term whp is $\\frac{K}{\\lambda _1} e(i) = \\tilde{e}(i)+ {\\protect {\\mathrm {o}}}(1)$ and the error is uniform over all $i$ .", "Indeed, whp $\\left| \\frac{K}{\\lambda _1} e(i) - \\frac{K}{m_2/m_1} e(i)\\right|\\le \\frac{K d_i}{\\sqrt{m_1}} \\frac{\\left|\\lambda _1- m_2/m_1\\right|}{(m_2/m_1)^2}\\le \\sqrt{\\frac{m_2}{m_1}}\\frac{cm^{3/2}_\\infty }{\\sqrt{m_0n}} \\frac{c^{\\prime }}{m_{\\infty }^2} =\\mathrm {O}\\left(\\frac{1}{\\sqrt{n m_{\\infty }}}\\right),$ where we use Assumption REF , Remark REF and (REF ).", "Since the detailed computations are similar to the previous arguments, we skip the details.", "We next prove the central limit theorem for the components of the eigenvector stated in Theorem REF (IV).", "Theorem 3.5 Under Assumption REF , with the extra assumtion $m_{\\infty }\\gg (\\log n)^{4\\xi }$ , $\\sqrt{\\frac{m_2^3}{d_i m_3 m_1}} \\Big (v_1(i)-\\frac{d_i}{\\sqrt{m_2}}\\Big ) \\overset{w}{\\rightarrow }\\mathcal {N}(0,1).$ First we compute $\\mathbb {E}[v_1(i)]$ , and afterwards we show that the CLT holds componentwise.", "We use the law of total expectation.", "Conditioning on the high probability event $\\mathcal {E}$ in Lemma REF , we can write the expectation of the normalized eigenvector $v_1$ as $\\mathbb {E}[v_1(i)]=\\mathbb {E}[v_1(i)|\\mathcal {E}]\\,\\mathbb {P}(\\mathcal {E})+\\mathbb {E}[v_1(i)|\\mathcal {E}^c]\\,\\mathbb {P}(\\mathcal {E}^c).$ Because the components of a normalized $n$ -dimensional vector are bounded, we know that $\\mathbb {E}[v_1(i)]=\\mathbb {E}[v_1(i)|\\mathcal {E}]\\,\\mathbb {P}(\\mathcal {E})+\\mathrm {O}\\left(e^{-c_\\nu (\\log n)^\\xi }\\right)$ for some suitable constant $c_\\nu >0$ , dependent on $\\nu $ and on the the bound on $v_1(i)$ .", "On $\\mathcal {E}$ , we can expand $v_1$ as $v_1(i) = \\frac{K}{\\lambda _1} \\left(e(i)+ \\frac{(He)(i)}{\\lambda _1}+\\frac{(H^2e)(i)}{\\lambda _1^2}+ \\sum _{k=3}^\\infty \\frac{(H^ke)(i)}{\\lambda _1^k} \\right).$ Using the notation $\\mathbb {E}_{\\mathcal {E}}$ for the conditional expectation on the event $\\mathcal {E}$ , we have $&\\mathbb {E}_\\mathcal {E}[v_1(i)]=\\mathbb {E}_\\mathcal {E}\\left[\\frac{K}{\\lambda _1} e(i)\\right]+ \\mathbb {E}_\\mathcal {E}\\left[\\frac{K}{\\lambda _1} \\frac{(He)(i)}{\\lambda _1}\\right]++ \\mathbb {E}_\\mathcal {E}\\left[\\frac{K}{\\lambda _1} \\sum _{k=2}^\\infty \\frac{(H^ke)(i)}{\\lambda _1^k} \\right].$ For the first term we have, using (REF ), $\\mathbb {E}_{\\mathcal {E}}\\left[ \\frac{K}{\\lambda _1}e_i\\right]&= \\mathbb {E}_{\\mathcal {E}}\\left[ \\frac{1}{\\sqrt{m_2/m_1}}e_i\\right]+\\mathrm {O}\\left(\\frac{d_i}{\\sqrt{m_1} (m_2/m_1)^{3/2}}\\right)=\\frac{d_i}{\\sqrt{m_2}} +\\mathrm {O}\\left(\\frac{d_i}{\\sqrt{m_1} (m_2/m_1)^{3/2}}\\right).$ For the term corresponding to $k=1$ , we know that $\\mathbb {E}[(He)(i)]=0$ by construction on the whole space.", "However, under the event $\\mathcal {E}$ we can show that its contribution is exponentially negligible.", "We have $\\mathbb {E}_\\mathcal {E}\\left[\\frac{K}{\\lambda _1}\\frac{(He)(i)}{\\lambda _1}\\right]=\\mathbb {E}_\\mathcal {E}\\left[\\frac{K}{\\lambda _1}\\frac{\\sum _jh_{ij}d_j}{\\sqrt{m_1}\\lambda _1}\\right]=\\mathbb {E}_\\mathcal {E}\\left[\\frac{\\left(1+\\mathrm {O}\\left(\\frac{1}{m_2/m_1}\\right)\\right)}{\\sqrt{m_2/m_1}}\\left(\\frac{\\sum _j h_{ij}d_j}{\\sqrt{m_1}(m_2/m_1)}\\right.\\right.\\\\\\left.\\left.+\\frac{\\sum _j h_{ij}d_j}{\\sqrt{m_1}}\\frac{|\\lambda _1-(m_2/m_1)|}{(m_2/m_1)^2}\\right)\\right].$ Since $m_2/m_1\\rightarrow \\infty $ , there exists a constant $\\tilde{C}$ such that $\\frac{\\left(1+\\mathrm {O}\\left(1/(m_2/m_1)\\right)\\right)}{\\sqrt{m_2/m_1}}\\le \\tilde{C} \\frac{1}{\\sqrt{m_2/m_1}}.$ We can therefore write $\\mathbb {E}_\\mathcal {E}\\left[\\frac{K}{\\lambda _1}\\frac{\\sum _jh_{ij}d_j}{\\sqrt{m_1}\\lambda _1}\\right]&\\le \\tilde{C}\\frac{1}{\\sqrt{m_2/m_1}}\\mathbb {E}_\\mathcal {E}\\left[\\frac{\\sum _j h_{ij}d_j}{\\sqrt{m_1}(m_2/m_1)}+\\frac{\\sum _j h_{ij}d_j}{\\sqrt{m_1}}\\frac{|\\lambda _1-(m_2/m_1)|}{(m_2/m_1)^2}\\right]\\\\&\\le \\mathbb {E}_\\mathcal {E}\\left[\\frac{\\sum _j h_{ij}d_j}{\\sqrt{m_1}(m_2/m_1)}\\right]+\\mathbb {E}_\\mathcal {E}\\left[\\frac{\\sum _j h_{ij}d_j}{\\sqrt{m_1}}\\frac{|\\lambda _1-(m_2/m_1)|}{(m_2/m_1)^2}\\right]\\\\&\\le \\mathbb {E}_\\mathcal {E}\\left[\\sum _j h_{ij}d_j\\right]\\left(\\frac{1}{\\sqrt{m_1}(m_2/m_1)}+\\frac{\\sqrt{m_\\infty }}{\\sqrt{m_1}(m_2/m_1)}\\right).$ Here we use (REF ) to bound the difference $|\\lambda _1-(m_2/m_1)|$ .", "Next, write $0&=\\mathbb {E}\\left[\\sum _j h_{ij}d_j\\right]=\\mathbb {E}_\\mathcal {E}\\left[\\sum _j h_{ij}d_j\\right]\\mathbb {P}(\\mathcal {E})+\\mathbb {E}_{\\mathcal {E}^c}\\left[\\sum _j h_{ij}d_j\\right]\\mathbb {P}(\\mathcal {E}^c)\\\\&\\le \\mathbb {E}_\\mathcal {E}\\left[\\sum _j h_{ij}d_j\\right]\\mathbb {P}(\\mathcal {E})+m_1\\mathbb {P}(\\mathcal {E}^c)=\\mathbb {E}_\\mathcal {E}\\left[\\sum _j h_{ij}d_j\\right]\\mathbb {P}(\\mathcal {E})+\\mathrm {O}\\left(e^{-c_\\nu (\\log n)^\\xi }\\right),$ where $c_\\nu $ is a constant depending on $\\nu $ , and we use that $|h_{ij}|\\le 1$ and $m_1=\\mathrm {O}\\left(e^{3/2\\log n}\\right)$ .", "We can therefore conclude that $\\mathbb {E}_\\mathcal {E}\\left[\\frac{(He)(i)}{\\lambda _1}\\right]=\\mathrm {O}\\left(e^{-c^\\prime _\\nu (\\log n)^\\xi }\\right),$ where $c^\\prime _\\nu >0$ is a suitable constant depending on $\\nu $ , and possibly different from $c_\\nu $ .", "To bound the remaining expectation terms, we use Lemma REF , which gives a bound on $(H^ke)(i)$ on the event $\\mathcal {E}$ .", "As before, we break up the sum into two contributions: $\\mathbb {E}_\\mathcal {E}\\left[\\frac{K}{\\lambda _1} \\sum _{k=2}^\\infty \\frac{(H^ke)(i)}{\\lambda _1^k}\\right]=\\mathbb {E}_\\mathcal {E}\\left[\\frac{K}{\\lambda _1} \\sum _{k=2}^L \\frac{(H^ke)(i)}{\\lambda _1^k}\\right]+\\mathbb {E}_\\mathcal {E}\\left[\\frac{K}{\\lambda _1} \\sum _{k=L}^\\infty \\frac{(H^ke)(i)}{\\lambda _1^k}\\right].$ For the second term we have $\\begin{aligned}\\sum _{k=L+1}^\\infty \\frac{\\left(H^ke\\right)(i)}{\\lambda _1^k}\\le C \\sqrt{\\frac{m_2}{m_1}}\\,\\mathrm {e}^{-C_c\\log n},\\end{aligned}$ where we use (REF ) and $C_c=|\\log (1-C_0)|$ .", "The first term can be bounded via Lemma REF , which gives $\\begin{aligned}\\sum _{k=2}^L \\frac{(H^ke)(i)}{\\lambda _1^k}&\\le \\sum _{k=2}^L\\frac{m_\\infty \\left((\\log n)^\\xi \\sqrt{m_\\infty }\\right)^k}{\\sqrt{m_1}(m_2/m_1)^k}=\\mathrm {O}\\left(\\frac{(\\log n)^{2\\xi }}{\\sqrt{m_1}}\\right).\\end{aligned}$ Using the above bounds, taking expectations and using (REF ), we get $\\mathbb {E}_\\mathcal {E}\\left[\\frac{K}{\\lambda _1} \\sum _{k=2}^\\infty \\frac{(H^ke)(i)}{\\lambda _1^k}\\right]=\\mathrm {O}\\left(\\frac{(\\log n)^{2\\xi }}{\\sqrt{m_2}}\\right).$ Thus, we have obtained that $\\mathbb {E}[v_1(i)]=\\frac{d_i}{\\sqrt{m_2}}+\\mathrm {O}\\left(\\frac{(\\log n)^{2\\xi }}{\\sqrt{m_2}}\\right),$ which settles Theorem REF (III).", "We can write $v_1(i)-\\frac{d_i}{\\sqrt{m_2}}= \\frac{\\left(1+\\mathrm {O}\\left(\\frac{1}{m_2/m_1}\\right)\\right)e(i)}{\\sqrt{m_2/m_1}}-\\frac{d_i}{\\sqrt{m_2}}+ \\frac{K}{\\lambda _1} \\frac{(He)(i)}{\\lambda _1} +\\mathrm {O}\\left(\\frac{(\\log n)^{2\\xi }}{\\sqrt{m_2}}\\right),$ where we replace the last terms of the expansion of $v_1$ by the bounds derived above (note that these bounds are of the same order as the ones obtained for the same terms in expectation).", "The first term of the centered quantity $v_1(i)-d_i/\\sqrt{m_2}$ is given by $\\frac{\\left(1+\\mathrm {O}\\left(\\frac{1}{m_2/m_1}\\right)\\right)e(i)}{\\sqrt{m_2/m_1}}=\\mathrm {O}\\left(\\frac{d_i}{\\sqrt{m_1} (m_2/m_1)^{3/2}}\\right).$ This last error can be easily seen to be ${\\protect {\\mathrm {o}}}\\left(\\frac{(\\log n)^{2\\xi }}{\\sqrt{m_2}}\\right)$ .", "We can therefore write $v_1(i)-\\mathbb {E}[v_1(i)]= \\frac{K}{\\lambda _1} \\frac{(He)(i)}{\\lambda _1} +\\mathrm {O}\\left(\\frac{(\\log n)^{2\\xi }}{\\sqrt{m_2}}\\right).$ We proceed to show that the first term on the right-hand side of the above equality gives a CLT when the expression is rescaled by an appropriate quantity, and the error term goes to zero.", "It turns out that $s_n^2(i)=\\mathrm {Var}\\left(\\sum _j h_{ij} d_j\\right)= \\sum _j \\frac{d_id^3_j}{m_1}\\left(1+\\mathrm {O}\\left(\\frac{1}{m_0}\\right)\\right)\\sim \\frac{d_im_3}{m_1}.$ Multiplying by $\\sqrt{\\frac{m_2^3}{d_im_3m_1}}$ , we have $\\sqrt{\\frac{m_2^3}{d_im_3m_1}} \\Big (v_1(i)-\\langle \\tilde{e},v_1\\rangle \\tilde{e}(i)\\Big )= \\frac{1}{s_n}\\sum _j h_{ij}d_j +\\mathrm {O}\\left(\\sqrt{\\frac{m_2^2 (\\log n)^{4\\xi }}{d_im_3m_1}}\\right).$ The error term is $\\sqrt{\\frac{m_2^2 (\\log n)^{4\\xi }}{d_im_3m_1}}=\\mathrm {O}\\left(\\frac{(\\log n)^{2\\xi }}{\\sqrt{m_0}}\\right)={\\protect {\\mathrm {o}}}(1),$ where last inequality follows from the assumption that $m_0\\gg (\\log n)^{4\\xi }$ .", "We now apply Lindeberg's CLT to the term $\\frac{\\sum _j h_{ij}d_j}{s_n}$ .", "The Lindeberg condition for the CLT reads $\\lim _{n\\rightarrow \\infty }\\frac{1}{s_n^2(i)}\\sum _{j}^n\\mathbb {E}\\left[(h_{i j}d_j)^2 \\,{\\bf 1}_{\\lbrace |h_{i j}d_j|\\ge \\epsilon s_n(i) \\rbrace }\\right]=0.$ Defining $\\sigma ^2_j(i)={\\rm Var}(h_{ij}d_j)$ , we note that $\\lim _{n\\rightarrow \\infty }\\frac{\\sigma ^2_j(i)}{s^2_n(i)}=\\lim _{n\\rightarrow \\infty }\\frac{d_i d_j^3 m_1}{m_1 m_3 d_i}\\le \\lim _{n\\rightarrow \\infty }\\frac{m_\\infty ^3}{m_3}\\le \\lim _{n\\rightarrow \\infty }\\frac{m_\\infty ^3}{nm_0^3}=0.$ Let us finally examine the event $|h_{ij}d_j|\\ge \\epsilon s_n(i)=\\epsilon \\sqrt{\\frac{d_im_3}{m_1}}\\iff |h_{ij}|\\ge \\epsilon \\sqrt{\\frac{m_3}{m_1}\\frac{d_i}{d_j^2}}.$ By definition, $|h_{ij}|<1$ .", "If we show that $\\lim _{n\\rightarrow \\infty } \\sqrt{\\frac{m_3}{m_1}\\frac{d_i}{d_j^2}}=\\infty ,$ then for all $\\epsilon >0$ there exists $n_\\epsilon $ such that the event $\\epsilon \\sqrt{\\frac{m_3}{m_1}\\frac{d_i}{d_j^2}}>1>|h_{ij}|$ has probability 1.", "Indeed, $\\lim _{n\\rightarrow \\infty }\\epsilon \\sqrt{\\frac{m_3}{m_1}\\frac{d_i}{d_j^2}}> \\lim _{n\\rightarrow \\infty }\\epsilon \\sqrt{\\frac{nm_0^4}{nm_\\infty ^3}}\\ge \\lim _{n\\rightarrow \\infty }\\epsilon \\, C\\sqrt{m_0} = \\infty $ for a suitable constant $C$ .", "Thus, (REF ) holds." ] ]
2207.03531
[ [ "An Analysis of Uplink Success Probability in Multi-Cell Lora Networks\n Under Different Channel Models" ], [ "Abstract The development of the low power wide area network (LPWAN) for the internet of things (IoTs) is expected to grow widely, allowing remote monitoring of smart devices from a distance of up to several kilometers.", "This paper studies the performance and success probability of multi-cell LoRa networks.", "Using tools of stochastic geometry, the paper analyzes the important metric namely success probability in both Rayleigh and Rician channel models.", "The obtained analysis helps investigate and evaluate other quality criteria in the multi-cell LoRa network such as throughput, SNR and SIR requirements.", "Moreover, we provide numerical simulation results to corroborate the theoretical analysis and to verify how our analysis can characterize the given reliability target." ], [ "1..4em 1..4em 1..4em An Analysis of Uplink Success Probability in Multi-Cell Lora Networks Under Different Channel Models Tien Hoa Nguyen*    Tien Hoa Nguyen is with the school of electronic and electrical engineering (SEEE), Hanoi University of Science and Technology, Vietnam.", ", Van Dai Do    Van Dai Do is with the school of electronic and electrical engineering (SEEE), Hanoi University of Science and Technology, Vietnam.", "Corresponding author: Tien Hoa Nguyen (e-mail: [email protected]) Manuscript received December 6, 2021; revised December 31, xxxx; accepted March 07, xxxx.", "Digital Object Identifier 10.31130/jst-ud.xxxx.xxx ISSN 1859-1531 The development of the low power wide area network (LPWAN) for the internet of things (IoTs) is expected to grow widely, allowing remote monitoring of smart devices from a distance of up to several kilometers.", "This paper studies the performance and success probability of multi-cell LoRa networks.", "Using tools of stochastic geometry, the paper analyzes the important metric namely success probability in both Rayleigh and Rician channel models.", "The obtained analysis helps investigate and evaluate other quality criteria in the multi-cell LoRa network such as throughput, SNR and SIR requirements.", "Moreover, we provide numerical simulation results to corroborate the theoretical analysis and to verify how our analysis can characterize the given reliability target.", "LoRa, LoRaWAN, Probability, Rayleigh, Rice, SNR, SIR." ], [ "Introduction", "The positive impacts of IoT on society, the environment and the industry are expected to be significant, with billions of devices that are connected relating to IoT [1].", "There are plenty of reasons why IoT plays a key role in our lives; however, there are two main ones, which are the connectivity and affordability of wireless infrastructures [2], [3] LPWAN technologies have drawn much interest in this industry for its main features: interconnecting battery-powered devices with low-bandwidth, low bit rates over long ranges.", "Together with other LPWAN technologies (SigFox, Weightless-W, etc.", "), it proposes connectivity up to tens of kilometers for low data rate, low power and low throughput applications.", "The market for this technology is expected to be huge, mainly in smart cities and smart agricultural projects [4], [5].", "LoRa networks and their applications have been deployed in many countries; however, there is one aspect of LoRa that has not been studied adequately, which is the oversimplification of non-interactable and independent gateways [6].", "In addition, there have also not been enough comparisons to evaluate the effectiveness for different fading models.", "Those might lead to incorrect conclusions in papers and false optimize protocols [7], [8].", "In a context in smart cities with a large number of connected measuring devices, the exploitation of the LoRa system depends on the number of devices supported and its scalability [9].", "In addition, devices are often clustered at gateways, where the interference models such as co-cell and intra-cell of a multi-cell network must be defined [10].", "This stems from the imperfect orthogonality between the different SFs.", "In addition, transmission from different SFs does not eliminate interference with neighboring SFs, so LoRa systems need to require a certain level of Signal-to-interference (SIR) protection [11].", "In the presence of SIR, the performance of the LoRa system depends on specific signal to noise ratios (SNR) at the SFs.", "In this respect, SNR and SIR are important metrics of evaluating the LoRa system [12].", "In this paper, our main contribution is to focus on analyzing the success probability in two different fading models, namely Rayleigh and Rician, hence comparing their performances under the same configuration.", "Earlier works widely used Rayleigh fading model [13] while not considering Rician fading model, which are clearly stimulated and illustrated in our paper.", "Both models stimulated in our paper also do allow gateways to interact within their ranges, and clearly illustrate the traceability of them.", "In addition, we argue that elements in LoRa networks, which are the gateways and the end devices, can be approximately described by inhomogeneous Poisson point processes (PPPs).", "Then, we obtain closed form expressions of many network features.", "Finally, we compare the probability between the two models to find whether there are any significant differences between them under the same conditions.", "Notations: Vectors and Matrices are denoted by boldface small and big letters, correspondingly.", "The superscripts T and H stand for the transpose and conjugate transpose.", "$I_K$ is the $K \\times K$ identity matrix.", "The operator $E{.", "}$ is the expectation of a random variable.", "The notation $||.||$ for the Euclidean norm." ], [ "LoRa standardization and Motivation", "In this part, we briefly discuss the related standardization of LoRa and LoRaWAN.", "LoRa is known as a spread spectrum modulation technique, working based on chirp spectrum technology.", "LoRa operates in sub-gigahertz radio frequency bands.", "For each region, LoRa has different working ranges.", "In Europe, it works according to EU868 (863-870/873 MHz), while in Asia, it follows AS923 (915-928 MHz).", "LoRa systems include gateways (GWs), end-devices (EDs) and the NetServer.", "These elements form a star of stars topology with NetServer at the root, GWs at level one and EDs as the leaves.", "Tab.", "REF illustrates the main features of LoRa, specifically for a 25-byte message.", "Table: LoRa Characteristics of a 25-Byte message at BW = 125 kHzThe heart of the technology, chirp spread spectrum (CSS) modulation, ensures adaptive data rates and lets the system trade-off throughput for different properties, such as coverage range, or energy consumption, while maintaining the same bandwidth (BW).", "This process is often done by the Net Server, which regulates the BW and the SF.", "LoRa modulation has a total of 6 SF from SF7 to SF12, which controls the chirp symbol $T_s = {2^{SF}}{\\rm BW}$ .", "For that reason, the time-on-air of a transmission using LoRa increases exponentially with SF (see Tab.", "REF ).", "Table: SIR collision thresholds in dB between different SFsLoRaWAN is an open protocol letting devices use LoRa for communication.", "It makes use of the pseudo-orthogonality between SFs to serve more EDs.", "Tab.", "REF shows the SIR threshold needed to successfully send a packet between different SFs.", "While LoRa is in the physical layer, LoRaWAN is in the datalink and network layers.", "As mentioned in the introduction, LoRaWAN key attributes are long-range, low-power, and low data rates.", "The range of communication mostly depends on whether the Line-of-Sight (LoS) channel is available or not.", "In places where there are any objects that create non-Line-of-Sight (nLoS), such as the building blocks in the cities, the communication distance in fact can be much shorter than 10 km.", "LoRaWAN protocol follows the Aloha method, which lets devices communicate only when there is data ready to be sent.", "Since there is no need to synchronize, a device can go back to silent mode after sending a packet.", "This contributes to the low-power characteristic of LoRaWAN.", "The data rates also vary from region to region.", "In Europe, for example, the data rate range is from 250bps to 5.5kbps.", "This is considered low for daily activities, for instance, surfing the internet or watching movies.", "However, it meets the requirements for tiny amounts of data, from simple devices such as sensors.", "This is where LoRa and LoRaWAN really stand out from other means of wireless communications." ], [ "Spatial Distribution of GWs and EDs", "We suppose GWs and EDs are described by the homogeneous Poisson point processes GW and ED, with constant intensity functions $\\phi _{\\rm GW}$ and $\\phi _{\\rm ED}$ , respectively [5].", "Since the number of ED is much greater than the number of GW in our simulations, we have $\\delta _{\\rm GW} \\ll \\delta _{\\rm ED}$ .", "Each ED and GW is described as a point $x_i$ and $y_i$ in the PPPs, respectively.", "In addition, our coordinate system is supposed to have ED $i=0$ at the origin to simplify calculations.", "$d_{ij}= |x_i - y_j|$ is the Euclidean distance in km from $i^{th}$ ED to $j^{th}$ GW." ], [ "Rayleigh and Rician Fading Models", "In the state-of-the-art studies, where the authors in [13] just consider multi-gateway interactions of LoRa networks under Rayleigh channel fading.", "However, it is obvious that Rayleigh fading is suitable for rich scatter environments, it, as a result, is the best suit for the urban and/or indoor environments where LoS is generally hard to exist.", "The Rician fading model, on the other hand, is employed when a strong LoS path exists, thus, is typically used in the outskirts and/or rural areas.", "Additionally, it is noted that LoRa can be applied to either the urban or rural areas.", "As a consequence, we investigate the performance of LoRa networks under both fading distributions.", "In the present paper, two different fading distributions are taken into account, namely, the Rayleigh and Rician distributions.", "It is noted that Rayleigh distribution is employed in the urban and/or indoor environments where LoS hardly exists.", "Rician distribution, on the other hand, is typically used in rural areas where the received signal is dominated by a strong LoS path.", "In the former circumstance, we model the channel gain $h_{ij}^2$ between $i^{th}$ ED and $j^{th}$ GW by an exponential random variable of mean 1, assuming that our channel $h$ (quasi-static) here is modeled as an independent, circularly symmetric, zero-mean complex Gaussian random variable with unit variance.", "In the latter circumstance, the channel gain $h_{ij}^2$ is described as a Rician variable of mean 1, with its variance is also 1.", "We do not include the effects of log-normal shadowing in this paper, since such fluctuations are not expected to significantly change the results of our qualitative analysis.", "The white Gaussian noise (AWGN) in this model is assumed to have zero-mean and variance [14] $N({\\rm dBm}) = -174 + 10 \\log ({\\rm BW}) + {\\rm NF},$ where the first term is the thermal noise in 1 Hz of bandwidth and can only be affected by changing the temperature of the receiver.", "NF is the fixed receiver noise figure, here it is taken to be 6 dBm due to the model's hardware implementation.", "For simplicity, we assume that uplink uses a BW = 125kHz channel and a 25 Byte packet.", "In addition, all end-devices are assumed to transmit with a constant power $\\varepsilon = 19$ dBm." ], [ "Path Loss Attenuation", "Path loss is a major component in the analysis and design of a telecommunication system.", "In order to adjust the coverage and capacity of wireless networks, we pay attention to the behavior of the attenuation function.", "For simplicity, in this paper, we assume that the transmitted signal undergoes a path loss attenuation, described by the function $p(d_{ij})=\\left(\\frac{\\lambda }{4 \\pi d_{ij}} \\right)^2$ , derives from the Friis transmission equation, where $\\lambda = 34.5 cm$ is the carrier wavelength, and $\\eta \\ge 2$ is the path loss exponent." ], [ "Spreading Factor in different regions of Uplink Transmissions", "$SF$ in a LoRa network controls the speed of data transmission.", "Lower SF reduces the range of LoRa transmission, since the bit rate is increased, and processing gain is reduced.", "In this paper, we assume that each ED will transmit the $SF$ set by the distance between that ED and its nearest GW, according to Tab.1.", "For instance, if $d_{00}$ is in $d_2,d_3$ , then ED $i^{th}=0$ , it will transmit with $SF=9$ .", "This allows our network to effectively divide EDs eight regions orbiting around each GW.", "By doing so, we can easily assign $SF$ to each ED and make sure that every ED has only one $SF$ .", "It is important to note that $\\phi _{\\rm GW} << \\phi _{\\rm ED}$ , to guarantee that there is also ED with higher $SF$ in the analysis." ], [ "SNR and SIR requirements", "Usually, in wireless telecommunications models, signal-to-interference-plus-noise ratio (SINR) is used to represent path loss with distance and other factors, such as background noise, interfering strength of other transmission.", "Unlike other means of transmissions that use SINR to measure the effectiveness, LoRa has these two separate conditions of SNR and SIR, which we will further discuss below." ], [ "SNR requirement", "The first condition is concerned with whether the received SNR is below the SF specific threshold $q_{SF}$ (see Tab.", "REF ).", "The instantaneous SNR between ED ith and GW jth can be defined as ${\\rm SNR}_{ij} = \\frac{\\varepsilon |h_{ij}|^2}{Np(d_{ij})},$ where $\\varepsilon $ and $N$ are the transmit power and noise variance which are defined in Section 3.2; $p(d_{ij})$ is the path-loss and defined in Section 3.3, and $h_{ij}^2$ is the channel gain between $i^{th}$ ED and $j^{th}$ GW modeled as described in Section 3.2.", "In the case of Rayleigh channel, the channel $h_{ij}^2$ is modeled by an exponential random variable of mean 1, whereas in the case of the Rician channel, it is described as a Rician variable of mean 1 and unit variance.", "The first condition can now be formulated as the complement of the connection probability $H_1 = P[{\\rm SNR}_{ij} \\ge q_{SF}|d_{ij}],$ which captures the probability that at any instance of time, a received signal $s_1(t)$ from $i^{th}$ ED locating $d_{ij}$ km away from $j^{th}$ GW will not satisfy the SNR threshold $q_{SF}$ , a piece-wise function of the distance $d_{ij}$ ." ], [ "SIR requirement", "The second condition examines whether the signal-to-interference (SIR) between an ED and a GW is greater or equal than the SIR ratio threshold.", "As seen in Tab.", "2, there is a same SIR ratio threshold for every co-SF collision $\\tau = 1dB = 1.259$ , while for non-co-SF collisions, that number varies from -8 to -25 dB (which corresponds to 0.15 to 0.0032).", "We can now define the SIR between $i^{th}$ ED and $j^{th}$ GW as ${\\rm SIR}_{ij} = \\frac{\\varepsilon |h_{ij}|^2}{p(d_{ij})} \\frac{1}{I_j},$ where $I_j = \\Sigma \\frac{\\psi _{ik} \\varepsilon |h_{ij}|^2}{p(d_{ij})}$ where $k \\ne i$ is the total interference of co-SF collisions at $j^{th}$ .", "$\\psi _{ik}$ is set to 1 if $k^{th}$ ED is having the same SF while transmitting with $i^{th}$ ED, and zero otherwise.", "It is noted that the SNR and SIR in the LoRa networks are identical for both uplink and downlink transmissions and are different from the cellular network, where power control is compulsory in all uplink transmissions.", "The second condition can now be formulated as $H_2 = P[{\\rm SIR}_{ij} \\ge \\tau | d_{ij}].$" ], [ "Probability of successful uplink transmission", "With the given conditions, we formulate the probability of a successful uplink transmission by ED $i^{th}$ as $H(x_i) = P \\left[ \\cup \\left[ \\left( {\\rm SNR}_{ij} \\ge q_{SF_i} \\right) \\cap \\left({\\rm SNR}_{ij} \\ge \\tau \\right) \\right] \\right],$ where the events $H_{1ij}= \\left({\\rm SNR}_{ij} \\ge q_{SF_i} \\right)$ and $H_{2ij}= \\left({\\rm SIR}_{ij} \\ge \\tau \\right)$ are assumed to be independent to make the problem more mathematically tractable and more suitable for simulations.", "This probability is also used to evaluate the performance of Rayleigh channel and Rician channel." ], [ "Results and Discussions", "In this section the numerical simulation results to validate the theoretical analysis and verify our analysis will be presented.", "The simulation configurations are given in the Tab.", "REF as follows: Table: The simulation parameters" ], [ "SNR results", "SNR is the ratio of the useful signal power to the unwanted signal power, for example noise.", "To improve LoRa performance, this system often uses forward error correction (FEC) and spread factor ($SF$ ) that allows for significant SNR improvement.", "The $SF$ factor had the most significant impact on the LoRa system.", "Lower $SF$ reduces power consumption time, increases data rate.", "Figure: Linear plot of simulation results of the probability P[ SNR 00 ≥q 0 |d 00 ]P[{\\rm SNR}_{00}\\ge q_0|d_{00}] between ED i th =0{\\rm ED} i^{th} = 0 and its nearest GW j th =0{\\rm GW} j^{th} = 0 for (a) Rayleigh channel and (b) Rician channel.Fig.1a and Fig.1b illustrate Monte Carlo simulation results of the SNR condition of the two different channels.", "Overall, it is obvious that the probability of both cases decreases over time; however, the Rayleigh channel's probability decreases faster compared to Rician's.", "Over the range considered (8km), the probability of $H_1$ satisfying the Rician channel is higher than Reyleigh's.", "In contrast, EDs with lower $SF$ undergo different patterns for the two different channels.", "For the Rician channel, EDs having their closest GW within 1km obtain a success rate of more than 95% as observed in Fig.1b.", "However, that probability of Rayleigh channel drops to just more than 65%." ], [ "SIR results", "The signal-to-interference ratio (SIR) is used to evaluate the performance of our models here.", "The ratio is the quotient between the mean received modulated carrier power and the mean received co-channel interference power from other transmitters.", "Inter-SF interference resulting from transmissions from neighboring $SFs$ is not orthogonal, so in LoRa systems a certain degree of SIR protection is required in addition to evaluating the SNR performance.", "Fig.2 illustrates the simulation results of the SIR condition for the two channels.", "We observe a striking saw-tooth of the SIR condition, and the successful probability with respect to the transmission distance.", "In fact, this is a unique feature of LoRa and is the direct sequence of $q_{SF}$ .", "The main reason is that when the transmission distance belongs to different $SF$ regions, we have a novel threshold that generally decreases with the increase of the transmission distance, thus, the probability first boosts up and then goes down for each region.", "Figure: Numerical simulations of P[ SIR 00 ≥τ]P[{\\rm SIR}_{00}\\ge \\tau ] as a function of d 00 d_{00} for λ ED =5 ED \\lambda _{\\rm ED} =5{\\rm ED} per km and λ GW =0.05 GW \\lambda _{\\rm GW} = 0.05{\\rm GW} per km for (a) Rayleigh channel and (b) Rician channelLooking at Fig.2a, in the first kilometer, Rayleigh channel also witnesses a high probability of meeting the SIR requirement; however, for the rest of the distance range, it stays in between just under 60 per cent and 80%, which is considerably lower than Rician's.", "On the other hand, in Fig.2b, for the Rician channel, the rate of ED satisfying SIR requirement is high over the range considered.", "The rate maintains at nearly 100% in the first kilometer, then it stays more than 80% in the rest of the range." ], [ "Success transmission rate", "We now consider the success transmission rate, which is made by combining the SIR requirement and SNR requirement for two channels.", "As mentioned earlier in the explanation of Fig.2, the saw-tooth of the success transmission rate is a unique feature of LoRa.", "Overall, from Fig.3b, it is seen that the success rate of the Rician channel is higher than Rayleigh's.", "Figure: Comparisons of numerical simulations for the success probability H(x)H(x) between (a) Rayleigh channel and (b) Rician channel.Observing from Fig.3b, we can clearly see that the Rician channel maintains a success rate of more than 50% in the first 5 kilometers.", "On the other hand, in Fig.3a, Rayleigh's stats show that it can only keep that same number in the first 1.5 kilometers.", "The success rate decreases over distance for both channels.", "In both cases, EDs having $SF$ of 12 have a very low chance of successfully transmitting due to the vast number of co-SF interfering EDs.", "This is clearly understandable, since the further EDs are away from their nearest GW, the higher $SF$ they have, which leads to the fact that EDs are more likely to counter interference from other co-SF EDs." ], [ "Conclusion", "In this paper, we investigated the performance and success probability of multi-cell LoRa networks.", "In such a system, interference caused by simultaneous transmissions using the same SF as well as different SFs is a problem that needs to be analyzed.", "To this end, we use the tool of stochastic geometry to analyze the important metric namely success probability in both Rayleigh and Rician channel models, while considering the impact of interference among transmissions over the same $SF$ (co-SF) as well as different $SFs$ (inter-SF).", "Moreover, we derive the important quality criterias such as throughput, SNR and SIR requirements in the multi-cell LoRa network.", "In addition, we summarized the network performance under Rayleigh and Rician channel models.", "[Figure: NO_CAPTIONFigure: NO_CAPTION" ] ]
2207.03548
[ [ "Millisecond Pulsars in Dense Star Clusters: Evolution, Scaling\n Relations, and the Galactic-Center Gamma-ray Excess" ], [ "Abstract The number of millisecond pulsars (MSPs) observed in Milky Way globular clusters has increased explosively in recent years, but the underlying population is still uncertain due to observational biases.", "We use state-of-the-art $N$-body simulations to study the evolution of MSP populations in dense star clusters.", "These cluster models span a wide range in initial conditions, including different initial masses, metallicities, and virial radii, which nearly cover the full range of properties exhibited by the population of globular clusters in the Milky Way.", "We demonstrate how different initial cluster properties affect the number of MSPs, for which we provide scaling relations as a function of cluster age and mass.", "As an application, we use our formulae to estimate the number of MSPs delivered to the Galactic Center from inspiralling globular clusters to probe the origin of the Galactic-Center gamma-ray excess detected by Fermi.", "We predict about $400$ MSPs in the Galactic Center from disrupted globular clusters, which can potentially explain most of the observed gamma-ray excess." ], [ "Introduction", "Since the discovery of a highly-magnetized, fast-spinning, radio-pulsating neutron star (NS) around 50 years ago [42], our understanding of these stellar remnants has grown tremendously.", "They can now be observed across the electromagnetic spectrum, from the radio to X-rays and gamma-rays, and even in gravitational waves [56], [4].", "NSs are ubiquitous in the Universe, and have close connections to a number of current puzzles in astronomy, including the origins of the mysterious Fast Radio Bursts [72].", "NSs in the form of millisecond pulsars (MSPs) and X-ray binaries are especially abundant in globular clustershttp://www.naic.edu/~pfreire/GCpsr.html, where the MSP specific abundance is about an order of magnitude larger than the one in the Galactic field [20], [57], [73].", "It is now well understood that the frequent gravitational encounters dictated by the high stellar densities of globular clusters efficiently catalyze the dynamical formation of MSPs [12], [79], [53], [86].", "These past few years have seen a rapid increase in the number of detected cluster MSPs thanks to the advent of new radio telescopes, such as FAST and MeerKAT [70], [74].", "However, the true underlying pulsar population is still rather uncertain owing to selection biases, including the luminosity thresholds of surveys, the dispersion by the interstellar medium, the spreading of pulsar signals by Doppler shifting in binaries, and the beaming fraction [66].", "Many previous studies have estimated the pulsar population in star clusters using different methods, including pulsar luminosity functions [32], [64], [85], [50], [8], binary population synthesis with a cluster background [53], combining stellar encounter rates with observations (X-ray sources, [41]; gamma-ray luminosities, [6]), and Bayesian analysis [81].", "However, different methods have yielded very different results and no studies have explored systematically the population of cluster pulsars with self-consistent globular cluster simulations.", "In the past decade, MSPs from globular clusters have been connected to the Galactic-Center gamma-ray excess detected by the Fermi-Large Area Telescope [36].", "This gamma-ray excess is roughly spherical symmetric about the Galactic Center, and extends out to $\\sim 2~$ kpc, which cannot be explained by the cosmic ray interaction with interstellar medium or other known gamma-ray sources [69].", "The two main competing explanations for the excess are dark matter annihilation and/or an unresolved MSP population [44], [45], [1], [2], [69].", "While the former scenario is challenged by the non-detection of gamma-rays from dwarf spheroidal galaxies [69], the small number of low-mass X-ray binaries, which are believed to be the progenitors of MSPs [7], [12], detected in the Galactic Center [19], [37] argues against the in-situ MSP hypothesis (although see [33]).", "Instead, previous studies have suggested that MSPs delivered by globular clusters that inspiraled into the Galactic central region may explain the excess [14], [3], [28].", "In this study, we systematically explore a large number of cluster models simulated with the $N$ -body Monte Carlo code Cluster Monte Carlo (CMC; [61]) to calculate the number of MSPs in globular clusters with a wide range in initial conditions, including different initial masses, metallicities, and virial radii.", "We derive scaling relations as a function of the cluster mass and age, which we then apply to estimate the population of MSPs delivered to the Galactic Center by inspiraling globular clusters.", "Compared to previous studies that estimated the number of MSPs [32], [64], [85], [41], [53], [6], [50], [8], [81] or their gamma-ray luminosities in dense star clusters [14], [3], [28], our approach is more realistic since the catalog models include a self-consistent detailed treatment of single and binary pulsar evolution in a dynamical environment, and are representative of the Milky Way globular clusters.", "Our paper is organized as follows.", "In Section , we discuss how the number of MSPs in globular clusters depends on the clusters’ properties, and provide scaling relations to estimate their number as a function of cluster mass and age.", "In Section , we estimate the population of MSPs delivered to the Galactic Center from inspiralling globular clusters and compare their gamma-ray luminosity to the observed Galactic-Center gamma-ray excess.", "We discuss the implications of our finding and draw our conclusions in Section .", "We use the CMC cluster catalog models [61] to estimate the population of MSPs in globular clusters.", "These catalog models were run with the CMC $N$ -body dynamics code [76], and span a wide range of initial conditions, including different initial numbers of stars ($N=2\\times 10^5$ , $4\\times 10^5$ , $8\\times 10^5$ , $1.6\\times 10^6$ ), Galactocentric distances ($R_g/\\rm {kpc}=2, 8, 20$ ), virial radii ($R_v/\\rm {pc} = 0.5, 1, 2, 4$ ), and metallicities ($Z = 0.0002, 0.002, 0.02$ ).", "The stellar density distributions in the models follow a King profile [59] with a concentration parameter $W_0=5$ .", "The initial masses of the stars are drawn from a Kroupa initial mass function [62] between $0.08$ and $150~M_{\\odot }$ , and all models have a $5\\%$ initial binary fraction where the masses of the companion stars are drawn from a flat distribution in mass ratio to the primary star in the range $[0.1-1]$ [26].", "The initial binary separations are sampled from a log-uniform distribution from Roche lobe overflow to the hard/soft boundary, and the initial binary eccentricities follow a thermal distribution [40].", "Finally, the natal kicks of NSs formed in core-collapse supernovae are sampled from a Maxwellian distribution with a standard deviation $\\sigma _{ccsn}=265\\,\\rm {km \\,s^{-1}}$ [43], while NSs formed in electron-capture supernovae and accretion-induced collapses receive smaller natal kicks drawn from a Maxwellian distribution with a standard deviation $\\sigma _{ecsn} = 20\\,\\rm {km\\,s^{-1}}$ [58], [86].", "All models are evolved up to a Hubble time, and they reproduce well the observed properties of the population of Galactic globular clusters [61].", "In our cluster models, MSPs are modeled following [86].", "In short, NSs are all assumed to be born as young pulsars with large magnetic fields between $10^{11.5}-10^{13.8}~$ G and spin periods between 30 ms and 1000 ms. MSPs are formed during periods of stable mass transfer from their companion stars in binaries.", "As a result of Roche lobe overflow, the young pulsar or the old non-radiating NS will be spun up by the mass and angular momentum transferred from the inner edge of a Keplerian accretion disc [52], and its magnetic field will decay following $B = \\frac{B_0}{1+\\frac{\\Delta M}{10^{-6}\\,M_{\\odot }}}\\exp \\left(-\\frac{T-t_{\\rm acc}}{\\tau }\\right)+5\\times 10^7 \\,\\rm {G}\\,,$ where $B_0$ is the magnetic field at the beginning of the mass transfer period, $t_{\\rm acc}$ is the time spent on accretion, and $\\tau =3~$ Gyr is the decay timescale of the magnetic field of isolated pulsars.", "We assume a lower limit of $5\\times 10^7~$ G for the MSP magnetic fields.", "Note that our simple prescriptions allow us to reproduce closely the observed magnetic fields and spin periods of cluster pulsars, as shown in [86]." ], [ "Gamma-ray luminosity and Millisecond Pulsar Population", "The Fermi Gamma-ray Space Telescope has detected gamma-ray emission from a number of globular clusters, generally thought to originate from MSPs therein present.", "To compare our models against observations, we estimate the total gamma-ray luminosity of our cluster models from their population of MSPs.", "The gamma-ray luminosities of model MSPs are calculated using [47] $L_{\\gamma MSP} \\approx 4.8\\times 10^{33} \\left(\\frac{B}{10^{8.5}\\,\\rm {G}}\\right)^2\\left(\\frac{P}{3\\,\\rm {ms}}\\right)^{-4}\\left(\\frac{\\eta }{0.1}\\right)\\,\\rm {erg/s}\\,,$ where $B$ and $P$ are the magnetic field and spin period of a MSP, respectively, and $\\eta $ , which we fix to $0.1$ , is the gamma-ray luminosity emission efficiency.", "Figure REF shows the observed gamma-ray luminosities per unit mass from 25 globular clusters detected by Fermi [46], and model gamma-ray luminosities from 70 catalog models with a non-zero number of MSPs (within the last 2 Gyr).", "We find that most of our models well overlap with the region defined by the observed populationThe outliers where $L_{\\gamma }/M\\lesssim 10^{28}\\,\\rm {erg\\,s^{-1}}\\,M_{\\odot }^{-1}$ are from $\\sim 10~$ models with different initial conditions.", "All of them except one has only one MSP (one model has four MSPs) within the last 2 Gyr.", "Most of these MSPs have low gamma-ray luminosities because they were formed early in the evolution of their host clusters and evolved in isolation where their magnetic fields decayed to $\\lesssim 10^8~$ G and they spun down slightly, or they went through more mass transfer periods and their magnetic fields were further reduced.", "Therefore, the offsets from the observations are from a small number of MSPs combined with their low gamma-ray luminosities.. As already discussed, our models are consistent with the observed Galactic globular cluster properties such as their masses and half-light radii.", "Figure REF illustrates that our catalog models can also reproduce the observed gamma-ray luminosities of Galactic globular clusters quite well.", "Figure: Gamma-ray luminosity per cluster mass as a function of the cluster mass.", "We show gamma-ray luminosities from multiple time steps, within 2 Gyrs until either the time of disruption or a Hubble time.", "We only show model clusters that survived to the present day or dissolved after 77~Gyr, which is about the age of the known youngest globular clusters in the Milky Way , , , , Figure: Numbers of MSPs for models with different initial numbers of stars NN and virial radii R v R_v.", "All models have metallicity Z=0.002Z=0.002 and Galactocentric distance R g =8R_g = 8~kpc.", "From left to right the panels show the numbers of MSPs at 0.1, 1., 5., and 11.5 Gyr, respectively.To further break down the MSP number dependence on the properties of a globular cluster, we show in Figure REF the numbers of MSPs for models with metallicity $Z=0.002$ , Galactocentric distance $R_g=8~$ kpc, and various initial numbers of stars $N$ and viral radii $R_v$ .", "Models with different metallicities and Galactocentric distances are shown in Figure REF -REF in the Appendix.", "We find that the most massive and densest globular clusters with initial $N\\ge 8\\times 10^5$ and $R_v=0.5~$ pc contribute the largest number of MSPs (see Figures REF and REF -REF ).", "This is expected since the more massive and denser a globular cluster is, the higher is its dynamical interaction rate, which is the key process to form MSPs [12], [79], [53], [86]." ], [ "Scaling Relations", "We calculate the average number of MSPs for catalog models that survived to the present day [61], and express the averages as a function of the cluster age with a polynomial function.", "The polynomial parameters can be taken to be linear functions of the initial numbers of cluster stars or the initial cluster masses $M$ , where $N=M/0.6M_{\\odot }$ for a canonical Kroupa initial mass function.", "Our polynomial fits comparing to the model data are shown in Figure REF .", "The polynomial fits as a function of the cluster age and the initial cluster mass are as follows $\\begin{aligned}N_{MSP} &= A \\times t + B \\times t^2 +C \\times t^3 + D \\times t^4 + E,\\\\A &= 0.046 \\times M_5 - 0.060,\\\\B &= -0.011 \\times M_5+0.008,\\\\C &= 0.0011 \\times M_5-0.0003,\\\\D &= -0.00003 \\times M_5-0.00001,\\\\E &= 0.089 \\times M_5-0.104,\\end{aligned}$ where $M_5 = M/(0.6\\times 10^5 M_{\\odot })$ and $t$ is in unit of Gyr.", "Note that the number of MSPs for different $N$ (or $M$ ) could have large fluctuations (see Figures REF and REF -REF ) because of the different initial conditions and small number statistics.", "The standard deviations can be up to a factor of about 2 the average values for dense star cluster models with $N>2\\times 10^5$ .", "Figure: Top panel: The average number of MSPs for models with different initial number of stars, NN, as a function of the cluster age.", "The solid curves represent the model averages, while the black dashed curves show the fits from Eq. .", "Bottom panel: the markers and the black dashed lines show the polynomial parameters and their fits from Eq.", ", respectively.The initial cluster mass can be estimated from the present-day mass taking into account the fact that most catalog models that survived to the present day lost $\\sim 50-60\\%$ of their initial masses.", "As an example, we estimate the number of MSPs in the globular cluster 47 Tucanae, assuming its present-day mass to be $\\sim 10^6M_{\\odot }$ ([39]; [10], [88]), and that it lost about half of its mass in $\\sim 11$  Gyr.", "We use a factor of about 2 for the $1\\sigma $ upper limit and take into account that $\\sim 70\\%$ of its MSPs may come from binary formation through giant star collisions with NSs and tidal capture interactions between a NS and a main-sequence star [88].", "The estimated number of MSPs in 47 Tucanae at the present day from Eq.", "REF is $\\sim 50$ , consistent with the estimates from the previous targeted simulation of the cluster [88] and with the observations [41], [5].", "In this Section, we adopt the semi-analytical method described in [35] for building a Galactic potential, sampling an initial population of globular clusters, and inspiraling the clusters through dynamical friction into the Galactic Center (Section REF ).", "We then use the masses of the inspiraled globular clusters to estimate the number of MSPs delivered to the Galactic Center and their gamma-ray emission (Section REF ).", "This semi-analytical approach allows us to sample a large number of globular clusters without significant computational costs.", "For the purpose of estimating the number of MSPs in the Galactic Center from inspiraling globular cluster, we also calculate the average MSPs for all models with initial $R_g=2~$ kpc (including dissolved models).", "Here, we only consider models with $R_g=2~$ kpc because they can most closely represent the inspiraling globular cluster affected strongly by the Galactic potential.", "The polynomial fits as a function of the cluster age and the initial cluster mass are as follows $\\begin{aligned}N_{MSP} &= A_{rg2} \\times t + B_{rg2} \\times t^2 +C_{rg2} \\times t^3 + D_{rg2},\\\\A_{rg2} &= 0.018 \\times M_5 - 0.009,\\\\B_{rg2} &= -0.0030 \\times M_5+0.0015,\\\\C_{rg2} &= 0.0002 \\times M_5-0.0002,\\\\D_{rg2} &= 0.068 \\times M_5-0.157\\end{aligned}$ Figure: Similar to Figure , but only for models with a Galactocentric distance R g =2R_g=2~kpc.Figure REF compares the fits to the model data.", "When compared to the fits using all models as in Figure REF , we find that there could be larger deviations, especially for initial $N=8\\times 10^5$ in Figure REF (top panel).", "This is probably due to the fact that the number of models is smaller (48 models with $R_g=2~$ kpc compared to 119 total non-dissolved models).", "Note that we include both not-yet-dissolved and non-dissolved models in this calculation." ], [ "Semi-analytical Methods", "We briefly summarize the semi-analytical methods from [35].", "We assume that the Galaxy is composed of stars whose distribution follows a spherical Sérsic mass density profile [80], and a dark matter halo with an Navarro–Frenk–White profile [13].", "The Sérsic profile has a total mass $M_s=5\\times 10^{10}\\,M_{\\odot }$ , a concentration index $n_s=2.2$ , and an effective radius $R_s=4$ kpc.", "The Navarro–Frenk–White profile has a total mass $M_h=10^{12}\\,M_{\\odot }$ , and a scale radius $R_h=20$ kpc.", "In addition, we also include a central supermassive black hole with a mass $M_{SMBH} = 4\\times 10^6\\,M_{\\odot }$ .", "To initialize a population of globular clusters, we assume that they follow the mass distributions of the stars in the Galaxy, and their total mass is a fixed fraction, $1.2\\%$ , of the field stars [35].", "We also assume that clusters are on circular orbits, and sample their initial semi-major axis in the Galaxy between $0.1$ and 100 kpc.", "The individual masses of the globular clusters are drawn from a power-law distribution $\\frac{dN_{GC}}{dM_{GC}} \\propto M_{GC}^{-2},$ where $N_{GC}$ is the number of globular clusters at a certain mass, and $M_{GC}$ is the cluster mass, in the range $10^4\\,M_{\\odot }$ -$10^7\\,M_{\\odot }$ .", "Finally, we adopt the average density at the half-mass radius $\\rho _{\\mathrm {h}}=10^3\\min \\left\\lbrace 10^2,\\max \\left[1,\\left(\\frac{M}{10^5\\ M_{\\odot }}\\right)^2\\right]\\right\\rbrace \\,\\frac{M_{\\odot }}{\\mathrm {pc}^3}.$ Dynamical friction leads to the gradual inspiral of the globular clusters in the Galactic potential.", "We calculate the evolution of a cluster's distance to the Galactic center following [13], [35] $\\frac{dr^2}{dt} = -\\frac{r^2}{t_{df}},$ $t_{df} \\approx 0.23 \\left(\\frac{r}{\\rm {kpc}}\\right)^2 \\left(\\frac{M_{GC}}{10^5 M_{\\odot }}\\right)^{-1} \\left(\\frac{v_c}{\\rm {km\\,s^{-1}}}\\right)\\,\\rm {Gyr}\\,,$ where $v_c$ is the circular orbital velocity of a globular cluster.", "During inspiral, a globular cluster will lose mass from the stripping of stars by the Galactic tidal field, evaporation of stars through two-body relaxation, and stellar evolution.", "The timescale of mass loss from the stripping of stars by the Galactic tidal field can be estimated by [34], [35] $t_{tid} \\approx 10\\left(\\frac{M_{GC}}{2\\times 10^5M_{\\odot }}\\right)^{2/3}P(r)\\, \\rm {Gyr}\\,,$ where $P(r) = 41.4\\left(\\frac{r}{\\rm {kpc}}\\right)\\left(\\frac{v_c}{\\rm {km\\,s^{-1}}}\\right)^{-1}\\,.$ For isolated globular clusters, the evaporation time follows [35] $t_{iso} = 17\\left(\\frac{M_{GC}}{2\\times 10^5M_{\\odot }}\\right)\\,\\rm {Gyr}\\,.$ The mass loss rate of a globular cluster from these two effects is $\\frac{dM_{GC}}{dt} = -\\frac{M_{GC}}{\\min (t_{iso}, t_{tid})}.$ Typically, $t_{tid}<t_{iso}$ in the inner regions of the galaxy, which harbor the globular clusters that can potentially spiral into the Galactic Center.", "Finally, to incorporate mass loss from stellar evolution, we assume a broken-power-law initial mass function as in [62], with stellar masses between $0.08$ and $150\\,M_{\\odot }$ .", "We model the initial-to-final mass relation following [67], and the turn-off mass at a given time is approximated using $m_{\\rm TO}\\approx (t_{ms}/ 10\\,{\\rm Gyr})^{-2/5}$ [38].", "Note that the mass lost from the globular clusters is added to the field stellar mass at each time step.", "We evolve our population of globular clusters for $11.5$ Gyr assuming that all clusters formed from a burst of star formation at redshift $z=3$ [28].", "We classify a globular cluster as disrupted when the cluster mass densities at their half-mass radii are smaller than the surrounding Galactic field density, when $M_{GC} < 500 M_{\\odot }$ ([39]; [10], also see Figure REF below), or when its distance with respect to the Galactic Center is smaller than 1 pc." ], [ "Millisecond Pulsars and the Gamma-ray Excess in the Galactic Center", "We sample about 8700 globular clusters with an initial total mass of about $5.4\\times 10^8\\,M_{\\odot }$ .", "We find that after $11.5$  Gyr most of the globular clusters are disrupted, with only about 200 systems surviving to the present day, which is in nice agreement with the number of observed globular clusters in our Galaxy [39].", "Figure REF compares the number density distribution and mass distribution of the survived sample globular clusters to the observations.", "The results from our simple semi-analytical method are consistent with most of the features in the observed population of Galactic globular clusters.", "Figure: Top panel: Number density of globular clusters as a function of the Galactocentric distance in the Milky Way.", "The orange and purple dots show the observations from the Harris catalog and the Gaia EDR3 , repsectively.", "The black dots show the survived globular clusters at 11.5 Gyr from our cluster samples.", "Bottom panel: Mass distribution of globular clusters in the Milky Way.", "The orange histogram is from the Harris catalog assuming a mass-to-light ratio of 1.5.", "The purple histogram is from NN-body fits to the observed cluster surface brightness profiles and velocity dispersion profiles , .", "The mass distribution of the survived sample clusters are shown in the black histogram.We use Eq.", "REF to estimate the number of MSPs delivered to the Galactic Center by inspiraling globular clustersWe assume that all MSPs formed in a globular cluster that are not ejected stay close to the cluster center and do not get stripped during the gradual disruption of the outermost regions of their host globular clusters..", "In addition, a non-negligible number of MSPs are ejected from the globular clusters, where $\\sim 90\\%$ of them are ejected at $\\lesssim 100$  MyrThere are also $\\sim 15$ NS-white dwarf and NS-main sequence star binaries ejected per cluster, some of which may become MSPs in the future.", "For simplicity we do not consider these systems..", "The number of ejected MSPs scales with the initial number of cluster stars $N$ as $N_{\\rm MSP, ej}\\approx \\log _2(N/2\\times 10^5)$ .", "Therefore, for simplicity, we add an ejected number of MSPs at the initial position of a globular cluster in the sample.", "Figure: Top panel: Cumulative distribution of mass lost from the inspiraling globular clusters and mass from the remnants of the disrupted globular clusters.", "The orange markers show the observational constraints on the mass of the Milky Way nuclear star cluster , , .", "Bottom panel: Cumulative number of MSPs from the disrupted globular clusters, including MSPs that were ejected initially.", "The black curve is calculated from the averages of Eq. .", "The blue dashed curve is the 1σ1\\sigma upper limit, while the green dot-dashed curve shows an extreme upper limit assuming that 70%70\\% of the MSPs in a cluster are formed from giant star collisions with NSs and tidal capture interactions, which were not considered in the catalog models.", "The vertical line marks 2 kpc.The number of MSPs brought by inspiraled globular clusters to the Galactic Center is shown in Figure REF .", "Note that the stellar mass contributed by our sample of inspiraled globular clusters are within the limits of the observed masses of the Milky Way nuclear star cluster, whose mass is equally contributed to from local star formation and inspiralled star clusters ([65], [77], [78]; top panel).", "We find that the number of MSPs within 2 kpc of the Galactic Center is about 400, with a $1\\sigma $ upper limit of about 500.", "While taking into account binary formation through giant star collisions with NSs and tidal capture of main-sequence stars by NSs, and assuming that $70\\%$ of MSPs are formed from these channels [88], the number of MSPs can be up to $\\sim 1000$ .", "However, it is important to point out that this is an optimistic upper limit since the $70\\%$ used is for a cluster like 47 Tucanae, which is one of the most massive and densest globular clusters in the Milky Way, and is likely to have more dynamical interactions than typical globular clusters [88].", "Note that these numbers are consistent with the numbers predicted by gamma-ray luminosity functions [23].", "In Figure REF , we show the gamma-ray surface brightness from the population of MSPs from inspiraled globular clusters, compared to the detected Galactic Center gamma-ray excess [48], [16], [21], [49], [22].", "The gamma-ray luminosity of each MSP is drawn randomly from a list of MSPs from all catalog models with initial $R_g = 2~$ kpc, depending on when the host cluster is disrupted or when the MSP is ejected.", "The large fluctuation (the black star at $\\gtrsim 10^{-5}\\,\\rm {GeV}\\,cm^{-2}\\,s^{-1}\\,sr^{-1}$ ) is resulted from random draws of MSPs with large gamma-ray luminosities.", "The average luminosity of these MSPs is $\\sim 3\\times 10^{34}\\,\\rm {erg\\,s^{-1}}$ .", "For comparison, we also show in the figure the surface brightness profile when all MSPs have a gamma-ray luminosity of $4.8\\times 10^{33}\\,\\rm {erg\\,s^{-1}}$ (gray diamonds; see also Eq.", "REF ).", "We find that, for high gamma-ray luminosities of $\\sim 10^{34}\\,\\rm {erg\\,s^{-1}}$ per MSP, MSPs from inspiraled globular clusters can potentially explain most of the detected excess.", "However, if MSPs have lower luminosities of $\\sim 10^{33}\\,\\rm {erg\\,s^{-1}}$ , they may only contribute to $\\sim 10\\%$ of the detected gamma-ray excess.", "Figure: Gamma-ray surface brightness of MSPs from disrupted globular clusters and the gamma-ray excess detected around the Galactic Center as a function of the angular distance to the Galactic Center , , , , .", "Black stars show the MSP gamma-ray luminosities calculated using Eq.", ", while gray diamonds assume that all MSPs have a gamma-ray luminosity of 4.8×10 33 erg s -1 4.8\\times 10^{33}\\,\\rm {erg\\,s^{-1}} ." ], [ "Discussions and Conclusions", "To summarize, we have explored in this study how the initial conditions of dense star clusters affect the number of MSPs they produce using the CMC  cluster catalog models [61].", "We have presented scaling relations, which express the numbers of MSPs as a function of the cluster age for different initial cluster masses in Eq.s REF -REF .", "Our scaling relations allow quick estimates of the number of MSPs in dense star clusters.", "We have also demonstrated that our model MSPs can reproduce the gamma-ray luminosities observed from globular clusters.", "We have applied the scaling relation we have derived to estimate the number of MSPs delivered to the Galactic Center from inspiraling globular clusters.", "On average, about 400 MSPs are brought to the Galactic Center by inspiralling globular clusters, with an optimistic upper limit of $\\sim 1000$ .", "These MSPs have an average gamma-ray luminosity of $\\sim 3 \\times 10^{34}\\,\\rm {erg\\,s^{-1}}$ , and can potentially explain most of the detected gamma-ray excess from the Galactic Center.", "There are a few uncertainties on the predicted number of MSPs from the catalog models.", "We have briefly mentioned one uncertainty in Section REF ; that is, for massive and dense globular clusters, dynamical binary formation through NS-giant star collisions or tidal capture interactions may also contribute to a large portion of cluster MSPs [88].", "Furthermore, observations have shown that core-collapsed globular clusters contain more isolated MSPs, which could be formed from tidal disruption events between a NS and a main sequence star [60].", "These effects combined together could contribute to a factor of a few in the number of MSPs formed in dense globular clusters, most of which are core-collapsed.", "However, only around $20\\%$ of the present-day globular clusters are core-collapsed [39], and if the fraction was similar for all clusters ever formed in the Milky Way, the boost from these dynamical interactions may not be very significant.", "In addition, we have only studied models with an initial binary fraction of $5\\%$ , but the rates of binary-mediated dynamical encounters could be significantly affected by different initial binary fractions.", "Finally, the predicted number of MSPs in the Galactic Center is also limited by models that are only at the constant Galactocentric distance $R_g=2~$ kpc.", "In reality, orbits of globular clusters in a galaxy are not always circular [84], and the galactic potential they are subjected to are time-dependant during inspiral.", "We leave the detailed exploration of the previous uncertainties to future work [87].", "We thank Fred Rasio, Kyle Kremer, Carl Rodriguez for useful discussions.", "This work was supported by NSF Grants AST-1716762 and AST-2108624 at Northwestern University.", "G.F. acknowledges support from NASA Grant 80NSSC21K1722.", "This research was supported in part through the computational resources and staff contributions provided for the Quest high performance computing facility at Northwestern University, which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology.", "CMC [55], [54], [30], [31], [17], [82], [18], [71], [68], [75], [76], Fewbody [29], COSMIC [15], BSE [52], SSE [51] Similar to Figure REF , we show here the number of MSPs in the catalog models with all the combinations of different initial conditions.", "Figure: Same as Figure  but for models with a metallicity Z=0.0002Z=0.0002 and Galactocentric distance R g =2R_g = 2~kpc.Figure: Same as Figure  but for models with a metallicity Z=0.0002Z=0.0002 and Galactocentric distance R g =8R_g = 8~kpc.Figure: Same as Figure  but for models with a metallicity Z=0.0002Z=0.0002 and Galactocentric distance R g =20R_g = 20~kpc.Figure: Same as Figure  but for models with a metallicity Z=0.002Z=0.002 and Galactocentric distance R g =2R_g = 2~kpc.Figure: Same as Figure  but for models with a metallicity Z=0.002Z=0.002 and Galactocentric distance R g =20R_g = 20~kpc.Figure: Same as Figure  but for models with a metallicity Z=0.02Z=0.02 and Galactocentric distance R g =2R_g = 2~kpc.Figure: Same as Figure  but for models with a metallicity Z=0.02Z=0.02 and Galactocentric distance R g =8R_g = 8~kpc.Figure: Same as Figure  but for models with a metallicity Z=0.02Z=0.02 and Galactocentric distance R g =20R_g = 20~kpc." ] ]
2207.03504
[ [ "The ACII 2022 Affective Vocal Bursts Workshop & Competition:\n Understanding a critically understudied modality of emotional expression" ], [ "Abstract The ACII Affective Vocal Bursts Workshop & Competition is focused on understanding multiple affective dimensions of vocal bursts: laughs, gasps, cries, screams, and many other non-linguistic vocalizations central to the expression of emotion and to human communication more generally.", "This year's competition comprises four tracks using a large-scale and in-the-wild dataset of 59,299 vocalizations from 1,702 speakers.", "The first, the A-VB-High task, requires competition participants to perform a multi-label regression on a novel model for emotion, utilizing ten classes of richly annotated emotional expression intensities, including; Awe, Fear, and Surprise.", "The second, the A-VB-Two task, utilizes the more conventional 2-dimensional model for emotion, arousal, and valence.", "The third, the A-VB-Culture task, requires participants to explore the cultural aspects of the dataset, training native-country dependent models.", "Finally, for the fourth task, A-VB-Type, participants should recognize the type of vocal burst (e.g., laughter, cry, grunt) as an 8-class classification.", "This paper describes the four tracks and baseline systems, which use state-of-the-art machine learning methods.", "The baseline performance for each track is obtained by utilizing an end-to-end deep learning model and is as follows: for A-VB-High, a mean (over the 10-dimensions) Concordance Correlation Coefficient (CCC) of 0.5687 CCC is obtained; for A-VB-Two, a mean (over the 2-dimensions) CCC of 0.5084 is obtained; for A-VB-Culture, a mean CCC from the four cultures of 0.4401 is obtained; and for A-VB-Type, the baseline Unweighted Average Recall (UAR) from the 8-classes is 0.4172 UAR." ], [ "Introduction", "The Affective-Vocal Burst (A-VB ) competition is exploring the expression of affect and emotion in brief nonverbal vocalizations (e. g., vocal bursts such as laughs, sighs, and shouts).", "Within this competition, the organizers provide several emotion modeling strategies and aim to discuss each during the workshop held at the 2022 Affective Computing and Intelligent Interactions (ACII) Conference.", "Thus far, vocal bursts have been largely overlooked in machine learning, affective computing, and emotion science.", "Given the focus in these fields on facial expressions, the voice has been a relatively understudied medium for communicating emotion.", "To the extent that the voice has been studied as a modality of emotion expression, it has been chiefly understood from the perspective of speech prosody [1].", "But another way humans communicate emotion with the voice is with the brief sounds that occur in the absence of speech – laughs, cries, and shouts (to name a few).", "Recent studies have discussed the range of emotions conveyed by vocal bursts (known as affect bursts [2], [3]), with findings demonstrating that brief vocalizations reliably express over ten emotions and that the meanings of vocal bursts are generally preserved across diverse cultures [4], [5].", "The field of machine learning has recently seen increased interest in vocal burst modeling, with the Expressive Vocalizations (ExVo) competition at ICML in 2022 [6] being the first-of-its-kind competition to explore various modeling strategies to understand and generate vocal bursts.", "More broadly, computational speech-based emotion modeling has become a prevalent area of research since the success of computational paralinguistic methods [7] and general advances in machine and deep learning speech recognition strategies [8].", "Computational modeling of emotion promises to inform a wide range of wellbeing domains, with applications including diagnostic tools for psychiatric illnesses [9], and bio-markers for remote wellness monitoring [10].", "In the A-VB  competition, we extend on our recent works [6], with a more specific focus on comparing and contrasting the various strategies available for modeling emotion in vocal bursts.", "In particular, the A-VB  competition presents four sub-challenges utilizing a single dataset: [(1)] the high-dimensional emotion task (A-VB-High), in which participants must predict a high-dimensional (10 class) emotion space, as a multi-output regression task, the two-dimensional emotion task (A-VB-Two), where the two-dimensional emotion space based on the circumplex model of affect [11] (arousal and valance) is to be recognized, again as a multi-output regression task, the cross-cultural emotion task (A-VB-Culture), where participants will be challenged with predicting the intensity of 10 emotions associated with each vocal burst as a multi-output regression task, using a model or multiple models that generate predictions specific to each of the four cultures provided in the dataset (the U.S., China, Venezuela, or South Africa), and the expressive burst-type task (A-VB-Type), in which participants are challenged with classifying the type of expressive vocal burst from 8-classes; Cry, Gasp, Groan, Grunt, Laugh, Other, Pant, Scream.", "The dataset used within the A-VB  competition, the Hume Vocal Bursts dataset (Hume-VB), comprises 59,201 recordings totaling more than 36 hours of audio data from 1,702 speakers.", "First utilized in the A-VB  competition[6], to our knowledge, this dataset remains one of the largest available of vocal bursts.", "The recordings in Hume-VB are rich and diverse in several ways that present unique opportunities, with the labeling enabling an array of emotion characteristics to be explored from vocal bursts.", "A single vocal burst can combine classes such as gasps infused with a cry or a scream, ending with a laugh, and offers a vibrant testing bed for emotion understanding and modeling [5].", "Thus, the Hume-VB dataset enables distinct but complementary strategies: allowing participants to model continuous blends of utterances such as laughs, cries, and gasps as well as the specific meanings of different laughs (amusement, awkwardness, and triumph), cries (distress, horror, and sadness), gasps (awe, excitement, fear, and surprise), and more.", "In this paper, we include a description of the Hume-VB dataset in detail (sec:data), provide rules for the four competition tasks (sec:tasks), and present baseline results for each task (sec:baselines).", "We summarize our results in sec:results and conclude with a discussion of insights from baseline development in sec:conc.", "The A-VB Data The A-VB  competition relies on the Hume-VB dataset, a large-scale dataset of emotional non-linguistic vocalizations (vocal bursts).", "This dataset consists of 36 :47 :04 (HH :MM :SS) audio data from 1 702 speakers aged from 20 to 39 years.", "The data was gathered in 4 countries with broadly differing cultures: China, South Africa, the U.S., and Venezuela, and individuals are performing emotional mimicry of seed emotion examples.", "Furthermore, speakers' are recorded in their homes via their microphones.", "Each vocal burst has been labeled in terms of the intensity of 10 different expressed emotions, each on a [1 :100] scale, and these are averaged over an average of 85.2 raters' responses[1] Amusement Awe Awkwardness Distress Excitement Fear Horror Sadness Surprise and Triumph.", "As well as the distribution of arousal and valance, and cultural-based emotion dimensions of Hume-VB, in fig:tsne (left), a t-SNE representation of emotional expressions based on the human ratings across the training set is visualized.", "We can see that the expressions vary, with clearly defined regions corresponding to each expressed emotion and continuous gradients between emotions (e. g., amusement and excitement).", "Of note, fewer samples convey Triumph, so we expect this class to be more challenging to model.", "The intensity ratings for each emotion were scaled to [0 :1].", "For our baseline experiments, the audio files were normalized to -3 decibels and converted to 16 kHz, 16 bit, mono (we also provide participants with the raw unprocessed audio, captured at 48 kHz).", "The data was subsequently partitioned (see tab:splits,) into training, validation, and test splits, considering speaker independence and a balance across classes.", "Figure: t-SNE representation of the emotional expression (left), the distribution of arousal and valance (middle), and a t-SNE representation of the culture-based emotion labels (right), from the Hume-VB training set.Table: An overview of the Hume-VB data.", "Including (No.)", "Samples, Duration (HH: MM: SS), Speakers, country-of-origin, and vocalization type.", "The age range for speakers is 20.5:39.5 years.", "For the purposes of the competition, the test set is blinded.The Competition Tasks In the A-VB competition, we present four tasks of varying nature utilizing the Hume-VBdata.", "Each explores a different aspect of the affective samples, with our aim to understand more deeply the various strategies for modeling emotion in vocalizations – an ongoing area of research for machine learning.", "A-VB High In the High-Dimensional Emotion Sub-Challenge (A-VB-High), participants are challenged with predicting the intensity of 10 emotions (Awe, Awkwardness, Amusement, Distress, Excitement, Fear, Horror, Sadness, Surprise, and Triumph) associated with each vocal burst as a multi-output regression task.", "Participants will report the mean Concordance Correlation Coefficient (CCC) across all ten emotions.", "A-VB Two In the Two-Dimensional Sub-Challenge (A-VB-Two), participants predict values of arousal and valence (based on 1=unpleasant/subdued, 5=neutral, 9=pleasant/stimulated), derived from the circumplex model for affect [12] as a regression task.", "Participants will report the mean CCC across the two dimensions.", "A-VB Culture The Cross-Cultural High-Dimensional Emotion Sub-Challenge (A-VB-Culture) is a 10-dimensional, 4-country culture-specific emotion intensity regression task.", "In A-VB-Culture, participants are challenged with predicting the intensity of 40 emotions (10 from each culture) as a multi-output regression task.", "The label for each vocal burst consists of a culture-specific gold standard created from the average of annotations from the sample's culture.", "Participants will report the mean CCC across all 40 emotions.", "A-VB Type In the Expressive Burst-Type Sub-Challenge (A-VB-Type), participants are challenged with classifying the type of expressive vocal burst from 8 classes (Gasp, Laugh, Cry, Scream, Grunt, Groan, Pant, Other).", "Participants will report the Unweighted Average Recall (UAR) as a measure of accuracy.", "Table: Baseline scores for A-VB 2022.", "Reporting the mean Concordance Correlation Coefficient (CCC) for the three regression tasks and the Unweighted Average Recall (UAR) across the 8-classes (chance level .125) for A-VB-Type.", "For each task, the best score on the test set is emphasized as the official baseline.", "We report the best scores from 5 seeds.", "General Guidelines To participate in the A-VB  2022 competition, all participants are asked to provide a completed copy of the Hume-VB End-User License Agreement (EULA) (more details can be found on the competition homepagehttp://competitions.hume.ai/avb2022).", "Participants should submit a paper that meets the official ACII guidelines, describing their methods.", "(The A-VB  workshop also accepts contributions on related topics.)", "To obtain test scores, participants should submit their test set predictions to the competition organizers (each team can do this up to 5 times).", "Participants are free to compete in any or all tasks and are encouraged to explore combinations.", "Baseline Experiments For each sub-challenge of the A-VB competition, we provide a baseline system utilizing well-established methods known in audio-based emotion recognition modeling [13], [14], [15].", "We provide reproducible code supporting each baseline system on GitHubhttp://github.com/HumeAI/competitions/tree/main/A-VB2022.", "Feature-based Approach We extract two sets of features, each having precedence in related tasks [16], [17], [18].", "One feature vector is extracted per sample for each feature set.", "Using the openSMILE toolkit [19], we extracted the 6,373-dimensional ComParE set and the 88-dimensional eGeMAPS  set.", "The 2016 COMputational PARalinguistics ChallengE (ComParE) [20] set contains 6,373 static features computed based on functionals from low-level descriptors (LLDs) [21], [16].", "The extended Geneva Minimalistic Acoustic Parameter Set (eGeMAPS) [14], which is smaller in size (88-dimensions), was designed for affective-based computational paralinguistic tasks.", "We apply a standard neural network (NN) for these experiments.", "The NN consists of three fully-connected layers, with layer normalization between each and a leaky rectified linear unit (Leaky ReLU) as the activation function.", "For the regression experiments, sigmoid is applied to the output layer.", "The loss for each task is varied, with multi-label emotion experiments utilizing a combined Mean Square Error (MSE) loss and the classification tasks applying cross-entropy loss, including softmax on the output layer.", "From several experiments for each task, a global learning rate ($lr$ ) and batch size ($bs$ ) is chosen of $lr=10^{-3}$ and $bs=8$ .", "We also apply early stopping (patience of 5, maximum of 25 epochs) to avoid the effects of overfitting the model.", "End-to-End Approach For our end-to-end baseline, we use the multimodal profiling toolkit End2You [15].", "The baseline model comprises a convolutional neural network (CNN) that extracts features from each audio frame and a recurrent neural network (RNN) that extracts temporal features.", "We use the Emo-18 (CNN) network architecture [22], which consists of three cascade blocks of 1-D CNN layers, a Leaky ReLU activation function ($\\alpha =0.1$ ), and max-pooling operations.", "Both convolution and pooling operations are performed in the time domain, using the raw waveform as input.", "Before the final emotion prediction, we exploit temporal patterns in the signals using a 2-layer Long-Short Term Memory (LSTM) network.", "The input audio frame passed to the CNN is $0.1$  sec long, corresponding to a $1\\,600$ dimensional vector, corresponding to the audio sampling rate of 16 kHz.", "Audio signals with a length not divisible by the input length are padded with zeros.", "Our model is trained with the Adam optimization algorithm [23], a batch size of 8, and an initial learning rate of $10^{-4}$ .", "The network weights have been initialized with Kaiming uniform [24] initialization, and the biases are initially set to zero.", "The LSTM network comprises 256 hidden units and is trained with a gradient norm clipping of $5.0$ .", "Finally, we use the MSE loss function and the CCC evaluation metric for the regression tasks.", "For the classification task, we use the cross-entropy loss with UAR as the evaluation metric.", "Figure: Normalized confusion matrix for validation results of A-VB-Type, with eGeMAPS (left) and End2You (right) approaches.", "Discussion of Competition Baselines In tab:results, we provide the baseline results for each of the four sub-challenges of the A-VB competition.", "In all cases, the baseline score is set by the end-to-end approach End2You, with feature-based strategies falling short in all cases.", "For the A-VB-High task, a baseline on the test set of 0.5687 CCC is obtained utilizing the end-to-end, End2You method.", "Of interest here, we see the ComParE features closely following much better than eGeMAPS.", "This suggests that the prosodic- and spectral-based features included with the ComParE set may benefit this task.", "On the other hand, the limited samples available may also restrict the potential performance possible from the End2You method.", "We see similar results for A-VB-Two, with a baseline on the test set of 0.5084 CCC obtained for the mean across the two classes, arousal, and valance.", "Of interest, we find that the score for valance is higher than for arousal, 0.5701 and 0.4468 CCC, respectively.", "Typically, arousal would be easier than valance to model from speech [25].", "However, arousal tends to correlate highly with traits including speech-rate [26], and volume [27].", "With this in mind, this data is non-language based, and we consider that arousal may be more of a challenge in this context, as these samples are mainly single bursts, and volume may be less impacting on the perception of arousal given varied recording environments.", "As with A-VB-High and A-VB-Two, the baseline is set by the End2You approach for A-VB-Culture, with a CCC of 0.4401 CCC on the test set.", "Given the multi-cultural nature of this task, the overall CCC is lower than the others, as some cultures are more difficult to model.", "This deficiency is shown for Venezuela (a mean of 0.3888 CCC) and China (a mean of 0.3870 CCC), possibly due to the smaller sample size and the cultural difference in these samples.", "For the A-VB-Type task, we explore classification for the first time with this data, classifying 8-classes of vocalization types.", "Once again, the End2You approach is set as the baseline (0.4172 UAR on the test set), with a similar margin to the hand-crafted feature-based methods.", "In fig:cm, we can see the confusion matrix for the test results of the baseline system and the eGeMAPS approach.", "The most commonly confused class appears to be `Gasp' in both cases, possibly caused by the class imbalance, given that the `Gasp' class is the most dominant (7,104 samples vs. 4,940 for `Laugh', on the training set).", "Furthermore, we see that the hand-crafted features perform better for some classes, mainly `Screaming' in the case of eGeMAPS; this may indicate that the speech-based features are valuable for this task, supported by their strong performance across tasks.", "Concluding Remarks This contribution introduced the guidelines and baseline scores for the first ACII Affective Vocal Bursts (A-VB ) competition.", "The competition focuses on strategies for computationally modeling emotion in vocal bursts and utilizes a large-scale dataset, the Hume-VB corpus.", "In this year's A-VB , four tasks are presented: (1) A-VB-High, a multi-label regression task utilizing 10 dimensions of emotion, we report a baseline score of 0.5686 CCC for A-VB-High; (2) A-VB-Two, modeling two-dimensions of emotion (arousal and valance), we report a baseline score of CCC of 0.5084 for A-VB-Two; (3) A-VB-Culture in which participants should model 40 emotional dimensions, 10 for each culture, we report a baseline score of 0.4401 CCC for A-VB-Culture; and (4) A-VB-Type, an 8-class classification of vocalization type, we report a baselines score of 0.4172 UAR for A-VB-Type.", "Several aspects can be explored by participants of the A-VB  competition to improve on the provided baselines.", "Namely, exploring the advantages of jointly learning from the various labeling provided and knowledge-based approaches which harness the diversity present in the Hume-VB dataset." ], [ "The A-VB Data", "The A-VB  competition relies on the Hume-VB dataset, a large-scale dataset of emotional non-linguistic vocalizations (vocal bursts).", "This dataset consists of 36 :47 :04 (HH :MM :SS) audio data from 1 702 speakers aged from 20 to 39 years.", "The data was gathered in 4 countries with broadly differing cultures: China, South Africa, the U.S., and Venezuela, and individuals are performing emotional mimicry of seed emotion examples.", "Furthermore, speakers' are recorded in their homes via their microphones.", "Each vocal burst has been labeled in terms of the intensity of 10 different expressed emotions, each on a [1 :100] scale, and these are averaged over an average of 85.2 raters' responses[1] Amusement Awe Awkwardness Distress Excitement Fear Horror Sadness Surprise and Triumph.", "As well as the distribution of arousal and valance, and cultural-based emotion dimensions of Hume-VB, in fig:tsne (left), a t-SNE representation of emotional expressions based on the human ratings across the training set is visualized.", "We can see that the expressions vary, with clearly defined regions corresponding to each expressed emotion and continuous gradients between emotions (e. g., amusement and excitement).", "Of note, fewer samples convey Triumph, so we expect this class to be more challenging to model.", "The intensity ratings for each emotion were scaled to [0 :1].", "For our baseline experiments, the audio files were normalized to -3 decibels and converted to 16 kHz, 16 bit, mono (we also provide participants with the raw unprocessed audio, captured at 48 kHz).", "The data was subsequently partitioned (see tab:splits,) into training, validation, and test splits, considering speaker independence and a balance across classes.", "Figure: t-SNE representation of the emotional expression (left), the distribution of arousal and valance (middle), and a t-SNE representation of the culture-based emotion labels (right), from the Hume-VB training set.Table: An overview of the Hume-VB data.", "Including (No.)", "Samples, Duration (HH: MM: SS), Speakers, country-of-origin, and vocalization type.", "The age range for speakers is 20.5:39.5 years.", "For the purposes of the competition, the test set is blinded.The Competition Tasks In the A-VB competition, we present four tasks of varying nature utilizing the Hume-VBdata.", "Each explores a different aspect of the affective samples, with our aim to understand more deeply the various strategies for modeling emotion in vocalizations – an ongoing area of research for machine learning.", "A-VB High In the High-Dimensional Emotion Sub-Challenge (A-VB-High), participants are challenged with predicting the intensity of 10 emotions (Awe, Awkwardness, Amusement, Distress, Excitement, Fear, Horror, Sadness, Surprise, and Triumph) associated with each vocal burst as a multi-output regression task.", "Participants will report the mean Concordance Correlation Coefficient (CCC) across all ten emotions.", "A-VB Two In the Two-Dimensional Sub-Challenge (A-VB-Two), participants predict values of arousal and valence (based on 1=unpleasant/subdued, 5=neutral, 9=pleasant/stimulated), derived from the circumplex model for affect [12] as a regression task.", "Participants will report the mean CCC across the two dimensions.", "A-VB Culture The Cross-Cultural High-Dimensional Emotion Sub-Challenge (A-VB-Culture) is a 10-dimensional, 4-country culture-specific emotion intensity regression task.", "In A-VB-Culture, participants are challenged with predicting the intensity of 40 emotions (10 from each culture) as a multi-output regression task.", "The label for each vocal burst consists of a culture-specific gold standard created from the average of annotations from the sample's culture.", "Participants will report the mean CCC across all 40 emotions.", "A-VB Type In the Expressive Burst-Type Sub-Challenge (A-VB-Type), participants are challenged with classifying the type of expressive vocal burst from 8 classes (Gasp, Laugh, Cry, Scream, Grunt, Groan, Pant, Other).", "Participants will report the Unweighted Average Recall (UAR) as a measure of accuracy.", "Table: Baseline scores for A-VB 2022.", "Reporting the mean Concordance Correlation Coefficient (CCC) for the three regression tasks and the Unweighted Average Recall (UAR) across the 8-classes (chance level .125) for A-VB-Type.", "For each task, the best score on the test set is emphasized as the official baseline.", "We report the best scores from 5 seeds.", "General Guidelines To participate in the A-VB  2022 competition, all participants are asked to provide a completed copy of the Hume-VB End-User License Agreement (EULA) (more details can be found on the competition homepagehttp://competitions.hume.ai/avb2022).", "Participants should submit a paper that meets the official ACII guidelines, describing their methods.", "(The A-VB  workshop also accepts contributions on related topics.)", "To obtain test scores, participants should submit their test set predictions to the competition organizers (each team can do this up to 5 times).", "Participants are free to compete in any or all tasks and are encouraged to explore combinations.", "Baseline Experiments For each sub-challenge of the A-VB competition, we provide a baseline system utilizing well-established methods known in audio-based emotion recognition modeling [13], [14], [15].", "We provide reproducible code supporting each baseline system on GitHubhttp://github.com/HumeAI/competitions/tree/main/A-VB2022.", "Feature-based Approach We extract two sets of features, each having precedence in related tasks [16], [17], [18].", "One feature vector is extracted per sample for each feature set.", "Using the openSMILE toolkit [19], we extracted the 6,373-dimensional ComParE set and the 88-dimensional eGeMAPS  set.", "The 2016 COMputational PARalinguistics ChallengE (ComParE) [20] set contains 6,373 static features computed based on functionals from low-level descriptors (LLDs) [21], [16].", "The extended Geneva Minimalistic Acoustic Parameter Set (eGeMAPS) [14], which is smaller in size (88-dimensions), was designed for affective-based computational paralinguistic tasks.", "We apply a standard neural network (NN) for these experiments.", "The NN consists of three fully-connected layers, with layer normalization between each and a leaky rectified linear unit (Leaky ReLU) as the activation function.", "For the regression experiments, sigmoid is applied to the output layer.", "The loss for each task is varied, with multi-label emotion experiments utilizing a combined Mean Square Error (MSE) loss and the classification tasks applying cross-entropy loss, including softmax on the output layer.", "From several experiments for each task, a global learning rate ($lr$ ) and batch size ($bs$ ) is chosen of $lr=10^{-3}$ and $bs=8$ .", "We also apply early stopping (patience of 5, maximum of 25 epochs) to avoid the effects of overfitting the model.", "End-to-End Approach For our end-to-end baseline, we use the multimodal profiling toolkit End2You [15].", "The baseline model comprises a convolutional neural network (CNN) that extracts features from each audio frame and a recurrent neural network (RNN) that extracts temporal features.", "We use the Emo-18 (CNN) network architecture [22], which consists of three cascade blocks of 1-D CNN layers, a Leaky ReLU activation function ($\\alpha =0.1$ ), and max-pooling operations.", "Both convolution and pooling operations are performed in the time domain, using the raw waveform as input.", "Before the final emotion prediction, we exploit temporal patterns in the signals using a 2-layer Long-Short Term Memory (LSTM) network.", "The input audio frame passed to the CNN is $0.1$  sec long, corresponding to a $1\\,600$ dimensional vector, corresponding to the audio sampling rate of 16 kHz.", "Audio signals with a length not divisible by the input length are padded with zeros.", "Our model is trained with the Adam optimization algorithm [23], a batch size of 8, and an initial learning rate of $10^{-4}$ .", "The network weights have been initialized with Kaiming uniform [24] initialization, and the biases are initially set to zero.", "The LSTM network comprises 256 hidden units and is trained with a gradient norm clipping of $5.0$ .", "Finally, we use the MSE loss function and the CCC evaluation metric for the regression tasks.", "For the classification task, we use the cross-entropy loss with UAR as the evaluation metric.", "Figure: Normalized confusion matrix for validation results of A-VB-Type, with eGeMAPS (left) and End2You (right) approaches.", "Discussion of Competition Baselines In tab:results, we provide the baseline results for each of the four sub-challenges of the A-VB competition.", "In all cases, the baseline score is set by the end-to-end approach End2You, with feature-based strategies falling short in all cases.", "For the A-VB-High task, a baseline on the test set of 0.5687 CCC is obtained utilizing the end-to-end, End2You method.", "Of interest here, we see the ComParE features closely following much better than eGeMAPS.", "This suggests that the prosodic- and spectral-based features included with the ComParE set may benefit this task.", "On the other hand, the limited samples available may also restrict the potential performance possible from the End2You method.", "We see similar results for A-VB-Two, with a baseline on the test set of 0.5084 CCC obtained for the mean across the two classes, arousal, and valance.", "Of interest, we find that the score for valance is higher than for arousal, 0.5701 and 0.4468 CCC, respectively.", "Typically, arousal would be easier than valance to model from speech [25].", "However, arousal tends to correlate highly with traits including speech-rate [26], and volume [27].", "With this in mind, this data is non-language based, and we consider that arousal may be more of a challenge in this context, as these samples are mainly single bursts, and volume may be less impacting on the perception of arousal given varied recording environments.", "As with A-VB-High and A-VB-Two, the baseline is set by the End2You approach for A-VB-Culture, with a CCC of 0.4401 CCC on the test set.", "Given the multi-cultural nature of this task, the overall CCC is lower than the others, as some cultures are more difficult to model.", "This deficiency is shown for Venezuela (a mean of 0.3888 CCC) and China (a mean of 0.3870 CCC), possibly due to the smaller sample size and the cultural difference in these samples.", "For the A-VB-Type task, we explore classification for the first time with this data, classifying 8-classes of vocalization types.", "Once again, the End2You approach is set as the baseline (0.4172 UAR on the test set), with a similar margin to the hand-crafted feature-based methods.", "In fig:cm, we can see the confusion matrix for the test results of the baseline system and the eGeMAPS approach.", "The most commonly confused class appears to be `Gasp' in both cases, possibly caused by the class imbalance, given that the `Gasp' class is the most dominant (7,104 samples vs. 4,940 for `Laugh', on the training set).", "Furthermore, we see that the hand-crafted features perform better for some classes, mainly `Screaming' in the case of eGeMAPS; this may indicate that the speech-based features are valuable for this task, supported by their strong performance across tasks.", "Concluding Remarks This contribution introduced the guidelines and baseline scores for the first ACII Affective Vocal Bursts (A-VB ) competition.", "The competition focuses on strategies for computationally modeling emotion in vocal bursts and utilizes a large-scale dataset, the Hume-VB corpus.", "In this year's A-VB , four tasks are presented: (1) A-VB-High, a multi-label regression task utilizing 10 dimensions of emotion, we report a baseline score of 0.5686 CCC for A-VB-High; (2) A-VB-Two, modeling two-dimensions of emotion (arousal and valance), we report a baseline score of CCC of 0.5084 for A-VB-Two; (3) A-VB-Culture in which participants should model 40 emotional dimensions, 10 for each culture, we report a baseline score of 0.4401 CCC for A-VB-Culture; and (4) A-VB-Type, an 8-class classification of vocalization type, we report a baselines score of 0.4172 UAR for A-VB-Type.", "Several aspects can be explored by participants of the A-VB  competition to improve on the provided baselines.", "Namely, exploring the advantages of jointly learning from the various labeling provided and knowledge-based approaches which harness the diversity present in the Hume-VB dataset." ], [ "The Competition Tasks", "In the A-VB competition, we present four tasks of varying nature utilizing the Hume-VBdata.", "Each explores a different aspect of the affective samples, with our aim to understand more deeply the various strategies for modeling emotion in vocalizations – an ongoing area of research for machine learning." ], [ "A-VB High", "In the High-Dimensional Emotion Sub-Challenge (A-VB-High), participants are challenged with predicting the intensity of 10 emotions (Awe, Awkwardness, Amusement, Distress, Excitement, Fear, Horror, Sadness, Surprise, and Triumph) associated with each vocal burst as a multi-output regression task.", "Participants will report the mean Concordance Correlation Coefficient (CCC) across all ten emotions." ], [ "A-VB Two", "In the Two-Dimensional Sub-Challenge (A-VB-Two), participants predict values of arousal and valence (based on 1=unpleasant/subdued, 5=neutral, 9=pleasant/stimulated), derived from the circumplex model for affect [12] as a regression task.", "Participants will report the mean CCC across the two dimensions." ], [ "A-VB Culture", "The Cross-Cultural High-Dimensional Emotion Sub-Challenge (A-VB-Culture) is a 10-dimensional, 4-country culture-specific emotion intensity regression task.", "In A-VB-Culture, participants are challenged with predicting the intensity of 40 emotions (10 from each culture) as a multi-output regression task.", "The label for each vocal burst consists of a culture-specific gold standard created from the average of annotations from the sample's culture.", "Participants will report the mean CCC across all 40 emotions." ], [ "A-VB Type", "In the Expressive Burst-Type Sub-Challenge (A-VB-Type), participants are challenged with classifying the type of expressive vocal burst from 8 classes (Gasp, Laugh, Cry, Scream, Grunt, Groan, Pant, Other).", "Participants will report the Unweighted Average Recall (UAR) as a measure of accuracy.", "Table: Baseline scores for A-VB 2022.", "Reporting the mean Concordance Correlation Coefficient (CCC) for the three regression tasks and the Unweighted Average Recall (UAR) across the 8-classes (chance level .125) for A-VB-Type.", "For each task, the best score on the test set is emphasized as the official baseline.", "We report the best scores from 5 seeds." ], [ "General Guidelines", "To participate in the A-VB  2022 competition, all participants are asked to provide a completed copy of the Hume-VB End-User License Agreement (EULA) (more details can be found on the competition homepagehttp://competitions.hume.ai/avb2022).", "Participants should submit a paper that meets the official ACII guidelines, describing their methods.", "(The A-VB  workshop also accepts contributions on related topics.)", "To obtain test scores, participants should submit their test set predictions to the competition organizers (each team can do this up to 5 times).", "Participants are free to compete in any or all tasks and are encouraged to explore combinations." ], [ "Baseline Experiments", "For each sub-challenge of the A-VB competition, we provide a baseline system utilizing well-established methods known in audio-based emotion recognition modeling [13], [14], [15].", "We provide reproducible code supporting each baseline system on GitHubhttp://github.com/HumeAI/competitions/tree/main/A-VB2022." ], [ "Feature-based Approach", "We extract two sets of features, each having precedence in related tasks [16], [17], [18].", "One feature vector is extracted per sample for each feature set.", "Using the openSMILE toolkit [19], we extracted the 6,373-dimensional ComParE set and the 88-dimensional eGeMAPS  set.", "The 2016 COMputational PARalinguistics ChallengE (ComParE) [20] set contains 6,373 static features computed based on functionals from low-level descriptors (LLDs) [21], [16].", "The extended Geneva Minimalistic Acoustic Parameter Set (eGeMAPS) [14], which is smaller in size (88-dimensions), was designed for affective-based computational paralinguistic tasks.", "We apply a standard neural network (NN) for these experiments.", "The NN consists of three fully-connected layers, with layer normalization between each and a leaky rectified linear unit (Leaky ReLU) as the activation function.", "For the regression experiments, sigmoid is applied to the output layer.", "The loss for each task is varied, with multi-label emotion experiments utilizing a combined Mean Square Error (MSE) loss and the classification tasks applying cross-entropy loss, including softmax on the output layer.", "From several experiments for each task, a global learning rate ($lr$ ) and batch size ($bs$ ) is chosen of $lr=10^{-3}$ and $bs=8$ .", "We also apply early stopping (patience of 5, maximum of 25 epochs) to avoid the effects of overfitting the model." ], [ "End-to-End Approach", "For our end-to-end baseline, we use the multimodal profiling toolkit End2You [15].", "The baseline model comprises a convolutional neural network (CNN) that extracts features from each audio frame and a recurrent neural network (RNN) that extracts temporal features.", "We use the Emo-18 (CNN) network architecture [22], which consists of three cascade blocks of 1-D CNN layers, a Leaky ReLU activation function ($\\alpha =0.1$ ), and max-pooling operations.", "Both convolution and pooling operations are performed in the time domain, using the raw waveform as input.", "Before the final emotion prediction, we exploit temporal patterns in the signals using a 2-layer Long-Short Term Memory (LSTM) network.", "The input audio frame passed to the CNN is $0.1$  sec long, corresponding to a $1\\,600$ dimensional vector, corresponding to the audio sampling rate of 16 kHz.", "Audio signals with a length not divisible by the input length are padded with zeros.", "Our model is trained with the Adam optimization algorithm [23], a batch size of 8, and an initial learning rate of $10^{-4}$ .", "The network weights have been initialized with Kaiming uniform [24] initialization, and the biases are initially set to zero.", "The LSTM network comprises 256 hidden units and is trained with a gradient norm clipping of $5.0$ .", "Finally, we use the MSE loss function and the CCC evaluation metric for the regression tasks.", "For the classification task, we use the cross-entropy loss with UAR as the evaluation metric.", "Figure: Normalized confusion matrix for validation results of A-VB-Type, with eGeMAPS (left) and End2You (right) approaches." ], [ "Discussion of Competition Baselines", "In tab:results, we provide the baseline results for each of the four sub-challenges of the A-VB competition.", "In all cases, the baseline score is set by the end-to-end approach End2You, with feature-based strategies falling short in all cases.", "For the A-VB-High task, a baseline on the test set of 0.5687 CCC is obtained utilizing the end-to-end, End2You method.", "Of interest here, we see the ComParE features closely following much better than eGeMAPS.", "This suggests that the prosodic- and spectral-based features included with the ComParE set may benefit this task.", "On the other hand, the limited samples available may also restrict the potential performance possible from the End2You method.", "We see similar results for A-VB-Two, with a baseline on the test set of 0.5084 CCC obtained for the mean across the two classes, arousal, and valance.", "Of interest, we find that the score for valance is higher than for arousal, 0.5701 and 0.4468 CCC, respectively.", "Typically, arousal would be easier than valance to model from speech [25].", "However, arousal tends to correlate highly with traits including speech-rate [26], and volume [27].", "With this in mind, this data is non-language based, and we consider that arousal may be more of a challenge in this context, as these samples are mainly single bursts, and volume may be less impacting on the perception of arousal given varied recording environments.", "As with A-VB-High and A-VB-Two, the baseline is set by the End2You approach for A-VB-Culture, with a CCC of 0.4401 CCC on the test set.", "Given the multi-cultural nature of this task, the overall CCC is lower than the others, as some cultures are more difficult to model.", "This deficiency is shown for Venezuela (a mean of 0.3888 CCC) and China (a mean of 0.3870 CCC), possibly due to the smaller sample size and the cultural difference in these samples.", "For the A-VB-Type task, we explore classification for the first time with this data, classifying 8-classes of vocalization types.", "Once again, the End2You approach is set as the baseline (0.4172 UAR on the test set), with a similar margin to the hand-crafted feature-based methods.", "In fig:cm, we can see the confusion matrix for the test results of the baseline system and the eGeMAPS approach.", "The most commonly confused class appears to be `Gasp' in both cases, possibly caused by the class imbalance, given that the `Gasp' class is the most dominant (7,104 samples vs. 4,940 for `Laugh', on the training set).", "Furthermore, we see that the hand-crafted features perform better for some classes, mainly `Screaming' in the case of eGeMAPS; this may indicate that the speech-based features are valuable for this task, supported by their strong performance across tasks." ], [ "Concluding Remarks", "This contribution introduced the guidelines and baseline scores for the first ACII Affective Vocal Bursts (A-VB ) competition.", "The competition focuses on strategies for computationally modeling emotion in vocal bursts and utilizes a large-scale dataset, the Hume-VB corpus.", "In this year's A-VB , four tasks are presented: (1) A-VB-High, a multi-label regression task utilizing 10 dimensions of emotion, we report a baseline score of 0.5686 CCC for A-VB-High; (2) A-VB-Two, modeling two-dimensions of emotion (arousal and valance), we report a baseline score of CCC of 0.5084 for A-VB-Two; (3) A-VB-Culture in which participants should model 40 emotional dimensions, 10 for each culture, we report a baseline score of 0.4401 CCC for A-VB-Culture; and (4) A-VB-Type, an 8-class classification of vocalization type, we report a baselines score of 0.4172 UAR for A-VB-Type.", "Several aspects can be explored by participants of the A-VB  competition to improve on the provided baselines.", "Namely, exploring the advantages of jointly learning from the various labeling provided and knowledge-based approaches which harness the diversity present in the Hume-VB dataset." ] ]
2207.03572
[ [ "RELICS: Small-scale Star Formation in Lensed Galaxies at $z = 6-10$" ], [ "Abstract Detailed observations of star forming galaxies at high redshift are critical to understand the formation and evolution of the earliest galaxies.", "Gravitational lensing provides an important boost, allowing observations at physical scales unreachable in unlensed galaxies.", "We present three lensed galaxies from the RELICS survey at $z_{phot} = 6 - 10$, including the most highly magnified galaxy at $z_{phot} \\sim 6$ (WHL0137-zD1, dubbed the Sunrise Arc), the brightest known lensed galaxy at $z_{phot} \\sim 6$ (MACS0308-zD1), and the only spatially resolved galaxy currently known at $z_{phot} \\sim 10$ (SPT0615-JD).", "The Sunrise Arc contains seven star-forming clumps with delensed radii as small as 3 pc, the smallest spatial scales yet observed in a $z>6$ galaxy, while SPT0615-JD contains features measuring a few tens of parsecs.", "MACS0308-zD1 contains a $r\\sim 30$ pc clump with a star formation rate (SFR) of $\\sim 3 M_{\\odot} \\textrm{ yr}^{-1}$, giving it a SFR surface density of $\\Sigma_{SFR} \\sim 10^3 M_{\\odot}\\textrm{ yr}^{-1}\\textrm{ kpc}^{-2}$.", "These galaxies provide a unique window into small scale star formation during the Epoch of Reionization.", "They will be excellent targets for future observations with JWST, including one approved program targeting the Sunrise Arc." ], [ "Introduction", "Deep field observations with the Hubble Space Telescope (HST) have revealed that galaxies at high redshift tend to be smaller [72], [73], [55], [56] and exhibit clumpier structures [74] than local galaxies.", "In field galaxies, these clump structures were found to have typical radii of $\\sim 1$ kpc [24], [26], [25], [33], [31], [30], [34], [35].", "These observations were largely limited by the resolution of HST, which cannot observe smaller scales at high redshift without assistance.", "The magnifying effect of gravitational lensing has opened a new window into small scale star formation in distant galaxies.", "Using HST and strong lensing, many studies have been able to push to scales of $\\sim 100$ pc across a broad range of redshifts [38], [47], [90], [48].", "In certain cases, smaller structures can be observed when galaxies reach particularly high magnifications, leading to detections of clumps measuring tens of pc in radius [79], [80], [92], [36], [91].", "More recently, observations of the highly magnified Sunburst Arc [65], [66] have revealed star forming clumps as small as $\\sim 3$ pc [83].", "Other observations of lensing cluster MACS J0416 have revealed clumps as small as $\\sim 4$ pc [54].", "At redshifts above $z \\sim 1.5$ , these small spatial scale observations are further aided by the decreasing angular diameter distance.", "These observations of smaller structures have put pressure on the dominant explanation of clump formation in the high-redshift universe.", "Recent studies have found evidence of bias towards larger clump radii and masses at low spatial resolution [20], [11].", "Furthermore, while lower-resolution simulations tended to favor larger, more massive ($\\sim 10^9 M_{\\odot }$ ) clumps [52], more recent higher-resolution simulations have found broader mass ranges, and tend to favor smaller ($\\sim 10^7 M_{\\odot }$ ) clumps [77], [51], [58].", "Thus, continued study of strongly magnified galaxies at high redshift is critical to our understanding of early galaxy structures.", "Recently, the study of highly magnified clumps has been pushed to even higher redshift, with [81] reporting a clump measuring 13 pc at $z\\sim 6$ .", "These highly magnified clumps allow exploration of spatial scales otherwise unreachable with current telescopes.", "In particular, they enable studies of clumps on the same scale as local young massive star clusters [62].", "Recent studies of YMCs in local galaxies have found peaks in the distribution of radii around 2-3 pc, with some examples reaching 10 pc in radius [4], [68].", "Local observations of globular clusters have found similar distributions of radii [63].", "Thus YMCs have been proposed as candidate proto-globular clusters, although this remains uncertain [3], [42], [78], [45], [44].", "Highly magnified objects in the distant universe present an opportunity to directly study these possible globular cluster progenitors.", "In this paper, we present HST observations of three highly magnified lensed galaxies at $6 < z \\lesssim 10$ .", "These galaxies all exhibit clumpy morphologies, and the high magnifications allow us to study them in detail.", "In Section we present the HST data used in this study.", "Section presents the lens models used, while Section discusses our measurements of clump radii.", "Spectral energy distribution (SED) fitting is presented in Section .", "We present our results in Section , and we contextualize these results in the subsequent Section .", "Finally, we summarize our conclusions in Section .", "We assume a flat cosmology with $\\Omega _m = 0.3$ , $\\Omega _{\\Lambda } = 0.7$ , and $H_0 = 70 \\textrm { km s}^{-1} \\textrm { Mpc}^{-1}$ throughout." ], [ "Data", "The high-redshift galaxies considered here were discovered in the Reionization Lensing Cluster Survey (RELICS; [14]).", "RELICS observed a total of 41 galaxy clusters with HST, obtaining single-orbit depth in each of the ACS F435W, F606W, and F814W bands along with a total of two orbits split between the WFC3/IR filters F105W, F125W, F140W, and F160W.", "The RELICS clusters were also observed with the Spitzer Space Telescope through the Spitzer-RELICS program (S-RELICS, PI Bradač).", "WHL J013719.8-082841 (henceforth WHL0137$-$ 08 ) was discovered in SDSS imaging as an overdensity of red galaxies by [87], and its redshift was subsequently confirmed at $z = 0.566$ by [86].", "This cluster was ranked as the 31st most massive cluster in the Planck PSZ2 catalog with a Sunyaev-Zeldovich calculated mass of $M_{500} = 9\\times 10^{14} M_{\\odot }$ .", "[70] discovered at 15 long arc at $z_{phot} = 6.2$ lensed by this cluster and dubbed it the Sunrise Arc, prompting follow-up observations with HST (GO 15842, P.I.", "Coe).", "These follow-up images included 5 additional orbits of ACS F814W imaging, and two orbits each of ACS F475W and WFC3/IR F110W imaging, and are presented in detail in [85].", "The Sunrise Arc is shown in Figure REF , and each of its 7 small clump structures are labeled.", "Additionally, this arc contains an extremely magnified lensed star presented in [85].", "WHL0137$-$ 08 received a total of 7 hours of observations in the IRAC 3.6 $\\mu $ m band, and 5 hours in the 4.5$\\mu $ m band.", "The galaxy cluster MACS J0308+2645 (henceforth MACS0308+26 ) was presented as part of the Massive Cluster Survey (MACS; [23]).", "It is at redshift $z = 0.356$ and has an SZ mass of $M_{500} = 10.8\\times 10^{14} M_{\\odot }$ , making it the 12th most massive cluster in the Planck PSZ2 cluster catalog [61].", "This cluster lenses an exceptionally bright galaxy dubbed MACS0308-904, which is the brightest lensed arc in the RELICS sample with an AB magnitude of 23.4 in the F160W filter [70].", "This arc is shown in Figure REF , which and the brightest clump is labeled (1.1).", "Additional substructure may be present within the arc, but this is too faint to measure at present.", "The cluster MACS0308+26 was observed for a total of 5 hours in each Spitzer IRAC band.", "Figure: The lensed arc MACS0308-zD1 at z phot =6.3±0.1z_{phot} = 6.3 \\pm 0.1 is the brightest known lensed galaxy at z>6z > 6.As in Figure , the left panel shows an HST color composite, where the blue channel shows F435W, green is F606W ++ F814W, and red is the WFC3/IR stack of F105W ++ F125W ++ F140W ++ F160W.The right panel shows the hybrid F105W/color image, with a transition threshold of 0.05 e - ^- sec -1 ∼7.6^{-1} \\sim 7.6 nJy, similar to Figure .The clump structures input into our forward model are labeled in blue in the right panel.The brightest clump at the head of the arc (labeled 1.1) has an observed AB magnitude of ∼23\\sim 23.", "Additional substructures may appear with deeper follow-up imaging.The object slightly offset from the tail of the arc is a foreground object, likely either at z∼4z \\sim 4 or a dwarf cluster member galaxy.", "The bright head of the arc lies about 4.3south of the lensing critical curve.", "[69] discovered a $\\sim 2.5$ lensed arc of a galaxy with a photometric redshift of $z_{phot} = 9.9^{+0.8}_{-0.6}$ magnified by the galaxy cluster SPT-CL J0615–5746 (hearafter SPT0615$-$ 57 ).", "The lensed arc is dubbed SPT0615-JD1, and it consists of a total of 5 clumps labeled in Figure REF .", "The cluster was discovered independently by the South Pole Telescope survey (SPT; [88]) and the [60], and it was determined to have a high mass ($M_{500} = 7\\times 10^{14} M_{\\odot }$ ) and high redshift ($z = 0.972$ ).", "SPT0615–57 was followed up with additional HST imaging (GO 15920, P.I.", "Salmon), which obtained an additional one orbit each in F105W and F125W, and two orbits each in F140W and F160W.", "Because of the promising $z\\sim 10$ arc in this field, this cluster received additional Spitzer observations, for a total of 30 hours of exposure time in each of the IRAC bands ($3.6 \\mu m$ and $4.5\\mu m$ ).", "Table: Observed photometry from HST and Spitzer for each arc and the bright clump of MACS0308-zD1 used for SED fitting.All HST images were processed and drizzled to 0.06pixels as described in [14].", "HST photometry for all sources was measured using Source Extractor v2.19.5 [6] following the method detailed in [14].", "Photometry for each full arc, as well as for the brightest individual clumps of both the Sunrise Arc and MACS0308-zD1, are presented in Table REF .", "Disentangling clump fluxes from the background arc can be tricky.", "In this instance, the two brightest clumps are much brighter than the local background arc, making detailed separation of clump from arc less influential.", "We therefore assume that the clump photometry produced from our fiducial Source Extractor parameters is dominated by the clump itself, with negligible contribution from the background arc.", "This assumption would not hold for other clumps, however we only utilize the multiband photometry to fit SEDs to the two brightest clumps.", "A more careful extraction of clump fluxes is therefore not necessary at this time.", "Initial redshifts were measured for all RELICS objects using BPZ [5], [13].", "This technique yields a redshift of $z_{\\text{phot}}= 6.3^{+0.2}_{-0.1}$ for MACS0408-zD1, and we adopt a fiducial redshift of $z 6.3$ for all relevant calculations here.", "Further SED modeling was done for SPT0615-JD1 in [69] and again in [75] to further explore its photometric redshift, along with other physical properties.", "Each method yields a $z\\sim 10$ solution ($z_{phot} = 9.9^{+0.8}_{-0.6}$ for [69], and $z_{phot} = 10.2^{+1.1}_{-0.5}$ for [75]).", "We assume a fiducial redshift of $z = 10.0$ for relevant calculations in this analysis.", "Spitzer flux measurements are detailed in [75], [76].", "While [76] performed many SED fits to high-redshift RELICS galaxies, that paper did not fit the Sunrise Arc or MACS0308-zD1 due to poor constraints on IR fluxes from Spitzer data.", "Additional SED fitting was done for the Sunrise Arc in [85], finding a photometric redshift of $z_{\\text{phot}}= 6.2 \\pm 0.1$ .", "We adopt a fiducial redshift of $z = 6.2$ for relevant calculations for this arc.", "Figure: The ∼2.5\\sim 2.5 long arc SPT0615-JD is the most distant resolved arc observed so far, with multiple photometric redshift estimates putting it solidly at z∼10z\\sim 10.The left panel shows a color composite from HST data, as in previous figures.The right panel shows the F160W band, with each substructure circled and labeled in blue.We identify a total of five clump features within this arc, labeled 1.1 through 1.5.Clumps 1.1 and 1.5, as well as clumps 1.2 and 1.4, appear blended in our HST images, due to their proximity and resolution limits.This arc is located about 9from the lensing critical curve at z∼10z \\sim 10." ], [ "Lens Models", "Proper interpretation of gravitationally lensed galaxies relies on accurate models of the lensing clusters.", "For our present analysis, we utilize published lens models for each lensing cluster, made using the Lenstool [40], [39] and Light-Traces-Mass (LTM; [7], [93], [94]) lens modeling softwares." ], [ "WHL0137", "WHL0137 has a total of five published lens models made using four different lens modeling tools: Lenstool, LTM, Glafic [57] and WSLAP+ [21], [22], the details of which can be found in [85].", "For the present analysis, we focus on the LTM model which produces the most accurate reproduction of the full length of the Sunrise Arc.", "While all models produce a reasonably faithful reproduction of the arc, only LTM simultaneously matches the positions and relative brightnesses of all components.", "Notably, the Lenstool and Glafic models produce much higher magnifications for clump 1.1b than for clump 1.1a, inconsistent with the observed relative fluxes which are consistent within $1\\sigma $ uncertainties in each WFC3/IR band.", "The lens models for WHL0137 are constrained by two photometrically identified multiple image systems: the Sunrise arc at $z \\sim 6.2$ as well as a triply-imaged $z \\sim 3$ galaxy.", "The LTM model predicts magnifications of $\\mu \\sim 60 - 250$ for individual clumps along the arc.", "The rapidly changing magnification in the vicinity of lensing critical curves is the cause of this wide range of magnification predictions, and makes setting a fiducial value of magnification for the full arc difficult.", "Where necessary, we adopt a total magnification of $\\mu = 155 \\pm 13$ for the full arc (inclusive of all three multiple images) and a magnification of $\\mu = 130 \\pm 70$ for the most highly stretched central image.", "The total magnification is calculated using a ratio of observed flux to forward model flux, described in Section ." ], [ "MACS0308", "MACS0308 was modeled by [1] using the LTM software.", "The lens model is constrained by three multiple image systems, each photometrically measured at $z \\sim 2-3$ .", "The bright lensed arc MACS0308-zD1 is not included as a constraint in the lens modeling.", "This lens model predicts a lensing magnification of $\\mu \\sim 20$ for the bright $z\\sim 6$ arc.", "We adopt $\\mu = 20$ as our fiducial magnification for analysis of this arc, however given its extended morphology that magnification will change across the length of the arc.", "Particularly, magnification decreases to the southeast of the brightest clump, meaning the fainter tail of the arc is at lower magnification than the bright clump.", "While the brightest clump is magnified by a factor 22, the furthest tail of the arc is only magnified by a factor 15.", "Its potential counterimage has a model predicted magnification of $\\mu \\sim 2$ ." ], [ "SPT0615", "The lens model for SPT0615 was presented in [59] and utilizes the Lenstool modeling software.", "This model is constrained by three multiple image families, two of which include spectroscopic redshifts.", "This lens model predicts a magnification for SPT0615-JD1 of $\\mu \\sim 4-7$ , varying along the length of the arc.", "We adopt a conservative fiducial magnification for the full arc of $\\mu = 5$ where necessary.", "The lens model of [59] predicts that two additional multiple images of SPT0615-JD1 should be visible.", "These multiple images have yet to be identified in the HST imaging, as they are likely below the detection limits of current observations." ], [ "Clump Modelling", "Gravitational lensing can provide a significant boost in resolution, allowing us to probe much smaller physical scales than would be possible in field galaxies.", "In order to derive the greatest benefit from this increase in resolution, we use a forward modeling technique similar to that described in [37].", "In this section, we describe the forward modeling code, as well as the star formation rate estimate we make using its outputs.", "We then detail the modeling decisions made for each arc individually.", "Figure: Forward model fitting allows detailed study of the substructure of lensed galaxies.", "Here we show HST images of our lensed arcs (left) along with forward model image plane reconstructions of each arc (left middle).", "The weighted residuals, calculated by subtracting the model from the data and dividing by pixel-level uncertainty, are shown in the middle right column.", "The residuals are consistent with noise in each fit, indicating that our forward model is successfully capturing the full arc in each case.", "The source plane models for each galaxy are shown in the right-hand column.", "Note the image stretch causes the wings of the brighter clumps to be more visible, making light from these clumps visible out to 2-3 times the quoted radius (2-3σ\\sigma in the Gaussian profile).", "In the top row depicting the Sunrise Arc, the caustic curves are shown in red in the source plane reconstruction." ], [ "Forward Model", "Our forward modeling technique provides a measurement of the intrinsic size, morphology, and brightness of highly magnified galaxy substructures.", "The forward model begins with the creation of a source plane model.", "This model includes a number of predefined clump structures, along with a larger background structure to represent the diffuse light of the host galaxy.", "Each structure is modeled according to a parametric light profile, and is positioned in the source plane based on the delensed centroid in the image plane.", "Currently, the code can include 2-dimensional Gaussian profiles, and Sersic profiles.", "For this analysis, we primarily model structures with a two-dimensional Gaussian profile.", "Many of the substructures examined here are unresolved or barely resolved, so the simpler Gaussian profile proves sufficient for their characterization.", "For larger resolved structures, Sersic profiles can be used.", "We attempted to fit these structures with Sersic profiles, however we generally found that the additional model complexity did not improve the fitting result.", "Once an initial source plane model has been defined, we create an image plane reconstruction using the lens model deflection maps.", "This image plane model is then convolved with an estimated instrument point spread function (PSF) to match the actual HST data.", "We estimate the PSF using stars observed within our observations.", "We create a PSF convolution kernel by averaging several stars to account for variations in subpixel position variations.", "We intentionally avoid particularly bright stars when generating the PSF kernel, as the diffraction spikes around bright stars would overwhelm the instrument PSF and introduce unnecessary noise in our model.", "To determine the best model parameters, we perform an initial parameter optimization using a downhill simplex algorithm (using Scipy minimize).", "This optimization provides an estimate of the parameter values, but no estimate of the variance of each parameter.", "We therefore next use an MCMC to probe the range of possible solutions and estimate the variance on each model parameter.", "The MCMC is done using the python package emcee [29].", "Before fitting, we set flat priors on the amplitude, width, and ellipticity of each model component.", "The primary function of the priors is to prevent unphysical solutions.", "For example, in the case of unresolved structures, the code can produce arbitrarily small point sources that still reproduce the unresolved image.", "To prevent this behavior, we set a lower limit on the Gaussian width based on the magnification and resolution of the HST images.", "This cutoff improves our estimates of the upper limits on the radii of unresolved objects.", "Currently, the forward model fit analyzes only a single-band image.", "For our analysis, we utilize the bluest filter with sufficient signal to distinguish clump features.", "The appropriate filter choice varies between arcs based on redshift and available image depth.", "Utilizing the bluest available filter ensures the smallest PSF possible, and thus the most precise measurement of clump radii." ], [ "Star Formation Rate Calculation", "The forward model provides a pathway to measure star formation rates for clumps via their rest-frame far-UV luminosity.", "This is particularly useful for faint clumps for which SED fitting does not produce well-constrained results.", "We first use the output of the forward model to calculate the delensed flux density from each clump by integrating over the clump profile.", "In the case of a Gaussian profile, the flux can be calculated as $F = 2\\pi A \\sigma _x \\sigma _y$ where $A$ is the Gaussian amplitude, and $\\sigma _{x,y}$ are the width in either direction.", "For symmetric Gaussian profiles, $\\sigma _x = \\sigma _y$ .", "We then divide this integrated flux by a factor $(1+z)$ to correct for bandwidth compression.", "From here, we calculate SFRs for each clump using the far-UV luminosity – SFR conversion from [49], namely $\\textrm {SFR} = 1.15 \\times 10^{-28} L_\\nu \\text{(FUV)}$ where SFR is in units of $M_{\\odot } \\textrm { yr}^{-1}$ and $L_\\nu $ is in units of ergs s$^{-1} \\textrm { Hz}^{-1}$ .", "This relation assumes a [71] IMF, however [49] calculate a conversion to [12] or [43] IMFs can be made by multiplying by a factor $\\sim 2/3$ .", "Calculating SFR from UV light can be significantly impacted by any amount of dust absorption.", "For these calculations, we assume no dust absorption.", "SED fitting of each galaxy yields fairly low dust attenuation $A_V < 0.1$ mag, indicating that this assumption is reasonable." ], [ "Sunrise Arc", "The Sunrise Arc consists of three multiple images crossing the lensing critical curve in two places.", "The central image has the highest overall magnification as it runs roughly parallel to the critical curve.", "Thus, we use this central image for our forward modeling.", "This arc is best detected in the deep F110W imaging, so we use this band in our forward model fitting.", "We identify a total of seven clump structures within this arc segment, labeled 1.1 through 1.7 (see Figure REF ).", "Clumps are identified as separate segments in the Source Extractor segmentation maps.", "Segments of each arc are then visually inspected in the highest signal-to-noise image (F110W in the case of the Sunrise Arc).", "Each clump appears unresolved in the WFC3/IR imaging, showing no deviation from a PSF-like point source when examined independent of the background arc.", "Thus our radius constraints are upper limits only.", "The arc contains a measurable diffuse component, which we model as a Gaussian with zero ellipticity.", "Given the high lensing shear in this region, only one dimension is probed in great detail, so the round Gaussian profile is sufficient and reduces the total number of parameters needed for the model.", "The length of the arc, coupled with the crowded field in which it is located, lead us to cut out a rectangular region around the arc for fitting purposes.", "This region removes contamination from nearby cluster galaxies, which can overwhelm the fitting code and produce poor results.", "We also choose to cut out the lower-redshift interloper that appears at the southeast end of the central image of the arc (see Fig.", "REF ).", "This interloper would bias the size and brightness measurements for clump 7, causing it to favor a larger and brighter model.", "The resulting best fit for this arc produces a residual that is consistent with background noise (Figure REF )." ], [ "MACS0308-zD1", "We identify one clump feature in MACS0308-zD1, which is the bright clump at the head of the arc.", "We note that the tail exhibits hints of additional clump structures, and indeed this section of the arc is split into multiple components in the Source Extractor segmentation maps.", "However visual inspection determined that the arc tail is too faint to produce reliable clump detections.", "Thus we here model the tail of the arc as a single Gaussian profile with ellipticity less than 0.4.", "Additional substructure may be present beyond the detection limits of these observations.", "We note that the bright clump at the head of the arc appears to have additional structure beyond the Gaussian we use for fitting, particularly a protrusion off the north edge of the clump.", "We did attempt to model this protrusion, however we found it to be highly degenerate with the main clump, leading to increased uncertainties on both clump parameters without a noticeable decrease in residuals.", "There also appears to be a small residual near the center of the clump.", "We attempted to model this using a Sersic profile, which could better model a more peaked light distribution, however this residual remained.", "We thus choose to only include one Gaussian component to describe the bright clump.", "It is likely that additional substructure exists in this clump, indicating that it is made up of multiple smaller unresolved objects." ], [ "SPT0615-JD1", "We identify a total of five clump structures within the SPT0615-JD1 arc using the WFC3/IR F160W image.", "The arc includes three clear features, however the brightest two of these three features appear asymmetrically bright in the HST images.", "We thus determine that these asymmetric clumps are likely blended images of multiple smaller structures.", "Previous studies of highly lensed galaxies have found that many small clumps will appear as such asymmetric larger structures when simulated at lower lensing magnification, supporting our determination to model the arc as multiple smaller clumps [64].", "Including the two additional clumps to describe this arc does introduce some degeneracy between clumps, increasing uncertainties in clump properties.", "However, the inclusion of these additional structures leads to noticeable improvement in the residuals, indicating that the added complexity and degeneracy does improve the final fit.", "There is a diffuse component to this arc, however it is very faint.", "We included this in our model as a Gaussian profile, however it is poorly constrained due to its faintness." ], [ "SED Fitting", "To better understand the properties of the arcs presented here, we perform SED fitting based on available HST and Spitzer photometry.", "We use the SED fitting code BAGPIPES [10] to fit our photometric data, leaving redshift as a free parameter.", "Because each of these objects has a solid photometric redshift estimate in place, we set a redshift range in our fitting of $4 < z < 15$ .", "Previous works [85], [75], [69] have investigated and ruled out lower redshift solutions, justifying our restricted redshift range.", "Additionally, attempted fits with fixed redshifts calculated from previous works did not significantly change the results of our SED fits.", "We use the BAGPIPES default stellar population models from [8], coupled with nebular reprocessing implemented by the photoionization code CLOUDY [27].", "We model each galaxy with an exponential star formation history of the form $SFR \\propto e^{-t/\\tau }$ , allowing $\\tau $ to vary from 100 Myr to 10 Gyr.", "Metallicity is allowed to vary from 0 to $2.5Z_{\\odot }$ , while stellar mass is left with a wide parameter space of $1 < \\log (M_* / M_{\\odot }) < 15$ .", "Age is varied from 1 Myr to the age of the universe, and ionization parameter is varied over the range $-4 < \\log (U) < -2$ .", "Dust extinction is implemented using the [9] law with $A_V < 3$ , and we assume that dust extinction is twice as high around HII regions in the first 10 Myr of their lifetimes.", "Each of these lensed arcs was detected as a series of multiple source segments in Source Extractor, so to perform SED fitting of the full arcs we sum the flux from these segments, as well as summing the uncertainties in quadrature.", "To estimate the intrinsic (delensed) properties of each galaxy, we divide by a fiducial magnification for each arc.", "The fiducial magnifications are discussed in Section .", "The resulting best fit SEDs for each full arc are presented in Figure REF , along with the photometry for each object.", "We attempt to fit each clump's SED individually, however we find that only the brightest clumps of the Sunrise Arc and MACS0308-zD1 produce reliable fits.", "These best fit SEDs are shown along with the source photometry for each clump in Figure REF .", "The remaining clumps of the Sunrise Arc have too low signal to noise to sufficiently constrain the SED, resulting in fractional uncertainties near 100%.", "Meanwhile, the high redshift of SPT0615-JD means that Spitzer IR data is necessary to infer physical parameters from the SED.", "The reduced spatial resolution of the Spitzer images, combined with the faintness of the arc, mean that we cannot reliably extract IR fluxes for individual clumps.", "Meaningful SED fits can therefore not be obtained for clumps within this $z\\sim 10$ arc, only the full arc can be reliably fit." ], [ "Forward Model Radii and SFRs", "Our forward modeling analysis produces direct constraints on clump radii in the source plane, which are presented in Table REF .", "For the Sunrise Arc, each clump appears unresolved in the image plane (PSF-like), so we can only set upper limits on their sizes.", "The upper limits indicate that these are some of the most compact structures thus far observed at $z \\sim 6$ , as most have 68% confidence radius limits less than 5 pc, and 95% confidence limits less than 10 pc.", "Previously, [81] reported a YMC candidate at $z = 6.143$ with upper limit $r < 13$ pc, meanwhile smaller clumps at scales of a few parsecs have been found in local galaxies [62].", "Radii for clumps in the other arcs are not so compact, they are at scales of a few tens of parsecs each.", "These galaxies are not as highly magnified, meaning they may host smaller star clusters that cannot be resolved with HST imaging.", "In particular, the $z\\sim 10$ arc SPT0615-JD1 may include more, smaller clump structures which are blended together to form the larger clumps measured.", "Our forward model provides a useful estimate of star formation rates for each clump based on best-fit intrinsic UV brightness.", "The SFRs for each clump are listed in Table REF , along with their intrinsic UV absolute magnitudes.", "The SFR is shown plotted against clump radius in Figure REF , along with a selection of clumps from previous literature.", "The literature comparison includes clumps at a range of redshifts and magnifications, including results from galaxies lensed by MACS J0416 [82], [54], clumps within the highly magnified Sunbrust Arc [83], and a sample of local ($z = 0$ ) clumps in nearby galaxies [28].", "The clumps presented here stand out as some of the smallest clumps known at high redshift, as well as having the highest star formation rates for their size.", "From the combination of SFR and clump radius, we can calculate the surface density of star formation for each clump as $\\Sigma _{\\textrm {SFR}} = (0.68 * \\textrm {SFR}) / (\\pi r^2)$ , where the factor 0.68 accounts for the fact that the radius is given by the $\\sigma $ of a Gaussian profile, yet the full profile integrated to infinity is used to calculate the SFR.", "From this calculation, we see that the $\\Sigma _{SFR}$ is highest in the bright clump of MACS0308-zD1.1, followed by the brightest clump of the Sunrise Arc (1.1).", "Each of these show SFR densities greater than $1000\\,M_{\\odot } \\textrm { yr}^{-1} \\textrm { kpc}^{-2}$ , notably higher than other clumps analyzed here, indicating that these are particularly dense star forming systems.", "The remainder of the clumps have SFR densities of order a few hundred $M_{\\odot } \\textrm { yr}^{-1} \\textrm { kpc}^{-2}$ .", "It is worth noting that the values of $\\Sigma _{SFR}$ for the Sunrise Arc clumps are best interpreted as lower limits, as the clumps are unresolved in HST imaging.", "If the true radii of these clumps turn out to be smaller than the constraints presented here, they will in turn show higher SFR surface densities." ], [ "BAGPIPES Results", "We can further assess physical parameters of these galaxies using SED fits.", "We fit the combined flux of each arc using BAGPIPES, and the resulting parameters are presented in Table REF .", "We attempted to fit each clump individually, however this only worked for MACS0308-zD 1.1 and WHL0137-zD1.1.", "The rest of the clumps of the Sunrise Arc proved too faint to yield reliable results from SED fitting.", "Meanwhile, the clumps of SPT0615-JD1 rely heavily on Spitzer photometry to constrain the SED.", "The Spitzer resolution is too low to reliably distinguish individual clumps, preventing clump-by-clump SED fitting.", "The SED fitting provides estimates of stellar mass, which the forward modeling cannot.", "The mass measurements for both WHL0137-zD1.1 and MACS0308-zD1.1 suggest these clumps are particularly dense, with a high stellar mass packed into a small region.", "Using the combination of stellar mass and radius, we can calculate a crossing time for each clump following Equation 1 of [32].", "The crossing times for both clumps are less than 1 Myr ($\\sim 0.2$ Myr for WHL0137-zD1.1 and $\\sim 0.5$ Myr for MACS0308-zD1.1).", "This could indicate that the clumps are bound if their ages are much greater than 1 Myr.", "However, there is considerable uncertainty on the ages of these clumps, as stellar population ages are poorly constrained with only the rest-frame UV available to constrain our SED fits.", "We can utilize a few other statistics to assess the possible fates of these clumps.", "For example, theoretical models and simulations have found that clumps reaching a circular velocity of $\\sim 50-100$ km s$^{-1}$ have a strong enough binding energy to be stable to supernova feedback [51], [18], [19].", "We calculate circular velocities ($v_{circ} = \\sqrt{GM/R}$ ) of $130 \\pm 20$ for WHL0137-zD1.1, and $500 \\pm 100$ for MACS0308-zD1.1.", "This suggests that these clumps are stable to disruption by supernova feedback.", "Additionally, previous work has found that clumps with gas surface mass densities $\\Sigma > 300 M_{\\odot } \\textrm {pc}^{-2}$ will be unaffected by radiation pressure feedback [51], [46].", "Here, we find WHL0137-zD1.1 has a surface stellar mass density of $\\Sigma _{M_*} = 4.4 \\pm 0.6 \\times 10^5 M_{\\odot } \\textrm {pc}^{-2}$ , while MACS0308-zD1.1 has $\\Sigma _{M_*} = 6.9 \\pm 1.3 \\times 10^5 M_{\\odot } \\textrm {pc}^{-2}$ .", "Given that the stellar density of a clump must be influenced by the gas density in the parent cloud, the fact that our measured stellar densities are so much greater than the threshold for stability suggests that these clumps may be stable to disruption by radiation pressure feedback.", "Together, these results imply that these clumps are likely to be long-lived systems.", "Table: Forward model results are presented for each clump, as well as the diffuse background arc where applicable.", "For the Sunrise Arc, radius upper limits are quoted at the 68% (95%) confidence interval.", "All other uncertainties are quoted at the 1σ1\\sigma level, and magnification uncertainties are propagated to derived quantities.", "SFR presented in this table is calculated from the delensed UV luminosity, as discussed in the text.", "Note for the Sunrise Arc, magnification uncertainties are calculated using only statistical variations from one lens model, and thus uncertainties are likely underestimated.Table: SED fitting results for each full arc are shown, along with individual clump fits for WHL0137-zD1.1 and MACS0308-zD1.1.", "All other clumps are too faint or too blended to be fit individually.", "Magnification uncertainties are incorporated into quoted uncertainties on all parameters." ], [ "Compact Star Formation at High-$z$", "The star forming clumps presented in this paper are notably compact, particularly within the Sunrise Arc.", "These clumps each have upper limit radii of below 12 pc, which makes them the smallest limits on clump sizes thus far observed at $z > 6$ , smaller than the previous record of $r < 13$ pc presented in [81].", "These small radii put the clumps of the Sunrise Arc squarely in the regime of Young Massive Clusters (YMCs), which have been measured with radii as small as $\\sim 1$ pc [62], [68].", "YMCs in turn are thought to be precursors to globular clusters [78], however there is some debate on this topic as local YMCs have not been found to contain multiple stellar populations indicative of globular clusters [3].", "Either way, the presence of seven highly magnified YMCs in a galaxy at $z > 6$ presents a great opportunity to study high redshift star clusters in detail with future JWST observations.", "The star clusters measured in MACS0308-zD and SPT0615-JD1 are somewhat larger than those in the Sunrise Arc, with radii measuring tens of parsecs.", "This puts these clumps on the larger end of the YMC scale based solely on radius [62], however the stellar mass of MACS0308-zD1.1 ($\\sim 10^9 M_{\\odot }$ indicates that it may be comprised of multiple merging star clusters, or perhaps a dense nucleus of a merging galaxy.", "It is worth noting that the clumps in SPT0615-JD1 may intrinsically be smaller than what is measured here.", "This arc is at a lower magnification ($\\mu \\sim 5$ ) than the other two, and its morphology shows fewer clearly distinguished features.", "This is indicative of multiple smaller clumps blending together to form the structures we observe [64].", "We attempt to model this blending using multiple smaller structures to model the clumps in this arc, but ultimately higher resolution imaging will be needed to better constrain the clump sizes.", "We note also that, as our observations push the limits of what can be observed with HST , our samples run up against completeness limits, as shown by dashed curves in Figure REF .", "These limits are calculated as follows.", "The vertical portion of the curves are calculated from the magnification-corrected diffraction limit, dividing the HST WFC3/IR resolution (0.13) by the magnification.", "For the $z=6$ curve shown, we set the magnified diffraction limit based on the brightest clump in the Sunrise Arc, magnified by a factor $\\mu =215$ .", "We note that one smaller clump exists for this sample, but its magnification is less well constrained.", "The horizontal bottom portion of the curve is calculated from the 5-$\\sigma $ limiting magnitude of our observations, again corrected for the lensing magnification.", "The slanted side of the curve represents the surface brightness limit, calculated at each redshift from cosmological surface brightness dimming: $2.5\\log (1+z)^4$ .", "This surface brightness limit has a slope of 5 mag/dex.", "Additional discussion of these completeness limits can be found in [89].", "Our completeness limits indicate that there may be smaller, fainter structures beyond the reach of our current observations.", "In particular, we cannot rule out that we are seeing only the brightest and most highly star forming clumps, and others that better fit with the trend seen in lower-redshift clumps may exist beyond what we can currently observe.", "For the clumps of SPT0615-JD, we note that all clumps appear on the edge of the completeness limit.", "This again justifies our decision to model several of the detected clumps as multiple, smaller features.", "Again in this arc, additional structure may exist beyond the current limits of our observations.", "Our forward modeling provides a measure of intrinsic luminosity, which can be used to calculate SFR via the scaling relation in [50].", "The resulting SFRs are rather high for each clump given their radii, indicating that these clumps host intense star formation.", "To quantify this, we calculate the SFR surface density (SFRSD) using the measured radius and SFR, finding generally high SFRSDs in each clump.", "In particular, clump 1.1 of MACS0308-zD has $\\Sigma _{SFR} = 900 \\pm 300 M_{\\odot } \\textrm { yr}^{-1}\\textrm { kpc}^{-2}$ , nearing the upper edge of the Kennicutt-Schmidt relation [41], and the regime of maximal Eddington-limited star formation rate calculated by [15].", "The clumps of the Sunrise Arc generally have SFRSDs of a few hundred $M_{\\odot } \\textrm { yr}^{-1}\\textrm { kpc}^{-2}$ , which is still quite high.", "Clump 1.1 in the Sunrise Arc in particular has $\\Sigma _{SFR} = 750 \\pm 140$ , similar to that measured for the clump D1(core) discussed in [81].", "These SFRSDs are also consistent with dense star clusters observed locally [2], [68].", "We note however that the $L_{UV}-$ SFR relation of [50] may break down for the clumps of the Sunrise Arc.", "These clumps show particularly low SFRs, which means that stochastic star formation effects become a greater source of uncertainty, resulting in larger scatter in the $L_{UV}-$ SFR relation [16], [17], [84].", "One particularly interesting question for these YMCs in the early universe is whether they remain bound, going on to form globular clusters, or if they disperse either through tidal disruption or stellar feedback.", "Simulations of larger (100-1000 pc) clumps in galaxies at $z \\sim 2$ have found that these structures tend to be disrupted either by stellar feedback or gravitational interactions within a few hundred Myr at most [58], [53].", "However, other simulations have found that more massive gravitationally bound clumps can survive long-term, forming bound globular clusters [51].", "One useful way to determine if a cluster is bound is to compare its crossing time $t_{cross}$ to its age.", "The ratio of these quantities can provide a useful metric for the boundedness of star clusters, as if a cluster has survived for many crossing times, it is likely gravitationally bound.", "[32], [4].", "Our current data can only provide an estimate of the crossing time for MACS0308-zD1.1 and WHL0137-zD1.1, however both of these are less than one Myr.", "Additional measurements of the stellar mass densities and circular velocities further support the interpretation of these as long-lived objects.", "Recent simulation results have also suggested that external effects within their host galaxies [67], though we do not directly assess these effects here.", "The mass of MACS0308-zD1.1, along with the slight residual in the forward model fit, indicates that it may more likely be comprised of multiple unresolved star clusters, not just one YMC.", "However, given the available evidence, we conclude that WHL0137-zD1.1 is a YMC that appears likely to survive long-term, making it a proto-globular cluster candidate." ], [ "Conclusions", "We present observations of three superlative arcs from the RELICS survey.", "These include the longest $z\\sim 6$ arc (Sunrise), the brightest $z\\sim 6$ arc (MACS0308-zD1), and the most distant resolved arc at $z\\sim 10$ (SPT0615-JD1).", "Each of these arcs show clear substructure on scales ranging from tens of parsecs down to a few parsecs.", "Most of the clumps are likely YMCs, and they all exhibit high star formation rates compared to their small sizes.", "Future observations with JWST will provide greater detail on the inner workings of these superlative lensed galaxies, including determinations of clump ages which in turn determine whether or not a clump is gravitationally bound.", "In particular, an accepted JWST Cycle 1 program (GO-2282) will obtain spectra of many star clusters within the Sunrise Arc, allowing more accurate measures of star formation rates and ages.", "Beyond that, these observations will measure metallicities, and could constrain ionization parameters for each star cluster.", "Together, these observations will better constrain the formation and environment within this $z \\sim 6$ galaxy.", "Similar spectroscopic observations of the other galaxies mentioned herein will be similarly beneficial, providing better constraints on all parameters presented here as well as new measurements of other physical parameters.", "These galaxies will be excellent targets for future observations with JWST as well as other observatories.", "Acknowledgements: The RELICS Hubble Treasury Program (GO 14096) and follow-up programs (GO 15842, GO 15920) consist of observations obtained by the NASA/ESA Hubble Space Telescope (HST).", "Data from these HST programs were obtained from the Mikulski Archive for Space Telescopes (MAST), operated by the Space Telescope Science Institute (STScI).", "Both HST and STScI are operated by the Association of Universities for Research in Astronomy, Inc. (AURA), under NASA contract NAS 5-26555.", "The HST Advanced Camera for Surveys (ACS) was developed under NASA contract NAS 5-32864.", "AZ acknowledges support by Grant No.", "2020750 from the United States-Israel Binational Science Foundation (BSF) and Grant No.", "2109066 from the United States National Science Foundation (NSF), and by the Ministry of Science & Technology, Israel.", "EZ acknowledges funding from the Swedish National Space Agency." ] ]
2207.03532
[ [ "Aerobatic Trajectory Generation for a VTOL Fixed-Wing Aircraft Using\n Differential Flatness" ], [ "Abstract This paper proposes a novel algorithm for aerobatic trajectory generation for a vertical take-off and landing (VTOL) tailsitter flying wing aircraft.", "The algorithm differs from existing approaches for fixed-wing trajectory generation, as it considers a realistic six-degree-of-freedom (6DOF) flight dynamics model, including aerodynamics equations.", "Using a global dynamics model enables the generation of aerobatics trajectories that exploit the entire flight envelope, enabling agile maneuvering through the stall regime, sideways uncoordinated flight, inverted flight etc.", "The method uses the differential flatness property of the global tailsitter flying wing dynamics, which is derived in this work.", "By performing snap minimization in the differentially flat output space, a computationally efficient algorithm, suitable for online motion planning, is obtained.", "The algorithm is demonstrated in extensive flight experiments encompassing six aerobatics maneuvers, a time-optimal drone racing trajectory, and an airshow-like aerobatic sequence for three tailsitter aircraft." ], [ "Supplemental Material", "Video of the experiments can be found at https://aera.mit.edu/projects/TailsitterAerobatics." ], [ "Introduction", "Vertical take-off and landing (VTOL) fixed-wing aircraft combine many of the advantages traditionally associated with either fixed-wing aircraft or rotorcraft.", "They can exceed the range and endurance limitations typical of multicopters, while maintaining the capability to take-off, hover, and land in confined spaces.", "This versatility is relevant to many real-world applications.", "For example, transitioning search and rescue aircraft can cover large areas efficiently and closely inspect (indoor) areas of particular interest.", "Similarly, VTOL delivery drones can safely make time-critical deliveries in remote environments without the need for a dedicated landing area.", "Tailsitter VTOL aircraft transition between hover and forward flight by pitching, so that their rotors transition between lift generation and forward propulsion based on the attitude.", "The tailsitter flying wing omits a tail and vertical surfaces, leading to a relatively simple mechanical design consisting of just a wing, two rotors, and two flaps that function as both elevators and ailerons.", "By placing these flaps in the rotor wash and using differential thrust, the aircraft remains controllable throughout the flight envelope, including static conditions.", "The simple, lightweight design allows a high thrust-to-weight ratio and the absence of a vertical tail surface reduces directional stability, leading to a highly agile and maneuverable aircraft.", "Figure: Loop trajectory reference and flight experiment.In this paper, we show that, under some assumptions, the tailsitter flying wing flight dynamics are differentially flat.", "This entails that the state and input variables can be expressed as a function of a flat output and a finite number of its derivatives [1], [2].", "Based on this flatness transform, we propose an algorithm for generating fast and agile tailsitter trajectories with low computational cost, i.e., suitable for online motion planning applications.", "Our algorithm is capable of generating aerobatics maneuvers that exploit the entire flight envelope of the vehicle, including challenging conditions, such as sideways knife-edge flight and inverted flight, as shown in Fig.", "REF .", "Existing trajectory generation algorithms for fixed-wing aircraft often avoid the relatively complicated flight dynamics and instead use kinematics models.", "For example, an extension of Dubins paths can be used to find the time-optimal trajectory with curvature constraints [3].", "While accurate tracking of the resulting paths is not dynamically feasible due to the instantaneous acceleration changes needed to transition between straight lines and circular arcs, feedback control can be used to maintain a tracking error that is acceptable in calm flight [4].", "When considering fast and agile flight, the aircraft dynamics and control input constraints must be considered in trajectory generation, so that the resulting trajectory is dynamically feasible, i.e., so that it can be accurately tracked in flight.", "Trajectory optimization subject to the six-degree-of-freedom (6DOF) nonlinear flight dynamics model is computationally costly, e.g., optimization of the 4.5 m knife-edge maneuver presented by [5] takes 3–5 minutes of computation time (using direct collocation with twelve states and five control inputs) according to [6].", "Existing works address computational expense in various ways, e.g., by considering only (extended) point-mass equations of motion [7], [8], [9], by using a planner with pre-computed maneuvers [10], [11], by incorporating human-piloted expert demonstrations [12], or by combining multiple simplified local dynamics models [13].", "In practice, these methods may impose limitations on the generated trajectories, especially when planning aerobatic maneuvers that rapidly progress through unconventional flight conditions.", "In the context of trajectory generation, differential flatness enables transformation of trajectories from the flat output space to the state and control input space [1], [14].", "This property is widely leveraged towards computationally efficient trajectory generation and tracking for quadcopters by defining the trajectory in the flat output space consisting of the three-dimensional position and the yaw angle [15], [16], [17].", "Differential flatness of fixed-wing aircraft dynamics has also been considered [14].", "However, the application of differential flatness towards trajectory generation for fixed-wing aircraft has mostly been limited to kinematics or simplified dynamics models.", "Existing works consider path generation and tracking using a differentially flat coordinated flight model [18] and aerobatics maneuvers using an aircraft kinematics model that does not incorporate angle of attack or sideslip angle [19].", "The algorithm presented in [6] is based on the differentially flat coordinated flight model given in [18] and combines Dubins paths with a transverse polynomial offset to obtain smooth trajectories.", "Our proposed method differs from existing flatness-based approaches for fixed-wing trajectory generation, as it considers a global 6DOF flight dynamics model, including aerodynamics equations.", "By using a global dynamics model, we are able to generate aerobatics maneuvers that exploit the entire flight envelope, enabling agile maneuvering through the stall regime, sideways uncoordinated flight, inverted flight etc.", "As we will show, the tailsitter flatness transform has a similar structure as the well-known quadcopter flat transform, in the sense that snap and yaw acceleration roughly correspond to the control inputs.", "Hence, their reduction also increases feasibility of tailsitter trajectories, akin to the premise of minimum-snap trajectory generation algorithms for quadcopters [15], [16].", "This enables the application of similar efficient algorithms for minimum-snap trajectory generation in the flat output space towards generation of tailsitter aerobatics trajectories.", "Our work contains several contributions.", "Firstly, we propose an algorithm for aerobatic trajectory generation for a VTOL fixed-wing aircraft using differential flatness.", "As far as we are aware, this is the first algorithm that uses differential flatness of a realistic fixed-wing flight dynamics model to generate aerobatic flight trajectories.", "Secondly, we show differential flatness of the tailsitter flying wing dynamics model.", "We note that recent work on trajectory-tracking flight control for a tailsitter flying wing also leverages differential flatness of the global dynamics model [20].", "However, this work does not include a method to obtain the control inputs as a function of the higher-order output derivatives, which is necessary for trajectory generation.", "We present computational and experimental results that validate the suitability of the derived flatness transform to determine dynamic feasibility of candidate trajectories.", "Thirdly, we provide extensive experimental results encompassing trajectories and flight tests for (i) six aerobatics maneuvers, (ii) a time-optimal drone racing trajectory at the limit of the vehicle's capability, and (iii) an airshow-like aerobatic sequence for three tailsitter aircraft.", "The outline of this paper is as follows.", "Section presents preliminaries on the tailsitter flying wing flight dynamics and on minimum-snap trajectory generation.", "The tailsitter flying wing flatness transform is derived in Section and its suitability to predict dynamic feasibility of candidate trajectories is assessed in Section .", "Section contains generated trajectories and experimental flight results for aggressive aerobatics maneuvers, a racing trajectory, and a multi-vehicle aerobatic sequence.", "Finally, conclusions are given in Section .", "Our recent work on trajectory-tracking flight control [20] presented a global model of the tailsitter flying wing dynamics based on the $\\varphi $ -theory parameterization introduced by [21].", "In this section, we provide a brief overview of the dynamics model as a preliminary to the derivation of the corresponding flatness transform in Section , which forms the basis of our trajectory generation algorithm.", "The vehicle translational dynamics are given by $\\dot{\\mathbf {x}} &= \\mathbf {v},\\\\\\dot{\\mathbf {v}} &= g\\mathbf {i}_z + m^{-1}\\mathbf {R}^i_\\alpha \\mathbf {f}^\\alpha ,$ where $\\mathbf {x}$ and $\\mathbf {v}$ are respectively the vehicle position and velocity in the world-fixed reference frame, $g$ is the gravitational acceleration, and $m$ is the vehicle mass.", "The vector $\\mathbf {f}^\\alpha $ represents the aerodynamic and thrust force in the vehicle-fixed zero-lift reference frame, and $\\mathbf {R}^i_\\alpha $ is the transformation matrix from this frame to the world-fixed reference frame, which is defined by the columns of the identity matrix $[\\mathbf {i}_x\\;\\mathbf {i}_y\\;\\mathbf {i}_z]$ .", "The zero-lift frame differs from the general body-fixed reference frame, shown in Fig.", "REF , by a $-\\alpha _0$ rotation around $\\mathbf {b}_y$ , where $\\alpha _0$ is the zero-lift angle of attack.", "Figure: Body-fixed reference frame and control inputs.The rotational dynamics are given by $\\dot{\\mathbf { \\xi }} &= \\frac{1}{2}\\mathbf {\\xi }\\circ \\mathbf {\\Omega },\\\\\\dot{\\mathbf {\\Omega }} &= \\mathbf {J}^{-1}(\\mathbf {m} - \\mathbf {\\Omega } \\times \\mathbf {J}\\mathbf {\\Omega }), $ where $\\mathbf {\\Omega }$ is the angular velocity in the body-fixed reference frame, and $\\mathbf {\\xi }$ is the normed quaternion attitude vector.", "The matrix $\\mathbf {J}$ is the vehicle moment of inertia tensor, and $\\mathbf {m}$ represents the aerodynamic and thrust moment in the body-fixed reference frame." ], [ "Force and Moment", "We employ $\\varphi $ -theory parameterization to obtain a global singularity-free model of the aerodynamic force and moment [21].", "The force in the zero-lift axis system is obtained by summing contributions due to thrust, flaps, and wings, as follows: $\\mathbf {f}^\\alpha = \\mathbf {f}^\\alpha _T + \\mathbf {f}^\\alpha _\\delta +\\mathbf {f}^\\alpha _w.$ The thrust force is given by $\\mathbf {f}^\\alpha _T = \\sum _{i=1}^2 \\underbrace{\\left[\\begin{array}{c}\\cos {\\bar{\\alpha }}\\; (1 - c_{D_T})\\\\0\\\\\\sin {\\bar{\\alpha }}\\;(c_{L_T} - 1)\\end{array}\\right]T_i}_{\\mathbf {f}^\\alpha _{T_i}},$ where $\\bar{\\alpha }$ is the sum of $\\alpha _0$ and the thrust angle $\\alpha _{T}$ , $T_i$ is the thrust due to motor $i$ , and the coefficients $c_{D_T}$ and $c_{L_T}$ represent drag and lift due to thrust vector components in the zero-lift axis system, respectively.", "The motor thrust is computed as follows: $T_i = c_T \\omega _i^2\\;\\;\\;\\text{with}\\;\\;i = 1, 2,$ where $c_T$ is the thrust coefficient and $\\omega _i \\ge 0$ is the speed of motor $i$ .", "The force contribution by the flaps is given by $\\mathbf {f}^\\alpha _\\delta = \\sum _{i=1}^{2}\\underbrace{-\\left[\\begin{array}{c}0\\\\0\\\\c^\\delta _{L_T} \\cos {\\bar{\\alpha }}\\; T_i + c^\\delta _{L_V}\\Vert \\mathbf {v}\\Vert \\mathbf {i}_x^\\top \\mathbf {v}^\\alpha \\end{array}\\right]\\delta _i}_{\\mathbf {f}^\\alpha _{\\delta _i}},$ where $\\delta _i$ is the deflection angle of flap $i$ .", "Finally, the wing force contribution is obtained as $\\mathbf {f}^\\alpha _w = -\\left[\\begin{array}{c}c_{D_V}\\mathbf {i}_x^\\top \\mathbf {v}^\\alpha \\\\0\\\\c_{L_V}\\mathbf {i}_z^\\top \\mathbf {v}^\\alpha \\end{array}\\right]\\Vert \\mathbf {v}\\Vert ,$ where $c_{D_V}$ and $c_{L_V}$ are the wing drag and lift coefficients, respectively.", "All aerodynamic coefficients incorporate air density, but could be scaled to account for attitude variations [22].", "We note that (REF ) does not contain any lateral force component, due to the absence of a fuselage and vertical tail surface.", "The main moment contributions are due to the motors and flap deflections $\\mathbf {m} = \\mathbf {m}_T + \\mathbf {m}_\\mu +\\mathbf {m}_\\delta .$ The moment due to motor thrust is given by $\\mathbf {m}_T = \\left[\\begin{array}{c}l_{T_y} \\mathbf {i}_z^\\top \\mathbf {R}^b_\\alpha (\\mathbf {f}^\\alpha _{T_2} - \\mathbf {f}^\\alpha _{T_1})\\\\c_{\\mu _T} (T_1 + T_2)\\\\l_{T_y} \\mathbf {i}_x^\\top \\mathbf {R}^b_\\alpha (\\mathbf {f}^\\alpha _{T_1} - \\mathbf {f}^\\alpha _{T_2})\\end{array}\\right],$ where $l_{T_y}$ is the moment arm and $c_{\\mu _T}$ is the pitch moment coefficient due to thrust.", "The moment due to motor torque is obtained as follows: $\\mathbf {m}_\\mu = \\left[\\begin{array}{c}\\cos \\alpha _T\\\\0\\\\-\\sin \\alpha _T\\end{array}\\right]\\sum _{i=1}^2 \\mu _i,$ where $\\mu _i = -(-1)^i c_\\mu \\omega _i^2\\;\\;\\;\\text{with}\\;\\;i = 1, 2,$ is the motor torque around the thrust-axis and $c_\\mu $ is the propeller torque coefficient.", "The flap contribution is given by $\\mathbf {m}_\\delta = \\left[\\begin{array}{c}l_{\\delta _y} \\cos {\\alpha _0}\\;\\mathbf {i}_z^\\top (\\mathbf {f}^\\alpha _{\\delta _2} - \\mathbf {f}^\\alpha _{\\delta _1})\\\\l_{\\delta _x} \\mathbf {i}_z^\\top (\\mathbf {f}^\\alpha _{\\delta _1} + \\mathbf {f}^\\alpha _{\\delta _2})\\\\l_{\\delta _y} \\sin {\\alpha _0}\\;\\mathbf {i}_z^\\top (\\mathbf {f}^\\alpha _{\\delta _2} - \\mathbf {f}^\\alpha _{\\delta _1})\\end{array}\\right],$ where $l_{\\delta _y}$ and $l_{\\delta _x}$ are the relevant moment arms.", "Moment contributions due to the freestream velocity and the angular velocity are neglected, as most of these are relatively small for the tailless flying wing and their inclusion may result in a much more complicated expression for the flatness transform.", "An evaluation of the impact of modeling assumptions is provided in Section .", "As we will show in Section , the tailsitter dynamics model—with some simplifications—admits a differentially flat output $\\mathbf {\\sigma }(t)=[\\mathbf {x}(t)^\\top \\;\\psi (t)]^\\top ,$ consisting of four elements: the vehicle position in the world-fixed reference frame $\\mathbf {x}(t)\\in \\mathbb {R}^3$ , and the yaw angle $\\psi (t)\\in \\mathbb {T}$ , where $\\mathbb {T}$ denotes the circle group.", "Consequently, any sufficiently smooth output trajectory satisfies the dynamics (REF ) through (REF ) and, conversely, any state-space trajectory (including aerobatic trajectories with unconventional flight conditions) corresponds to a unique output trajectory (REF ).", "This bijective correspondence can be exploited to generate dynamically feasible aerobatics trajectories without resorting to computationally expensive state-space methods.", "When focusing on aggressive flight trajectories, generation is complicated by the fact that the control input constraints, i.e., the motor speed and flap deflection limits, cannot readily be enforced in the flat output space.", "Widely used algorithms for trajectory generation in the differentially flat output space of the quadcopter dynamics address this difficulty by minimizing snap, i.e., the fourth derivative of position, and yaw acceleration [15].", "In practice, this optimization roughly corresponds to reducing the required control moment and thus to increasing the likelihood that the control input limits are satisfied and the trajectory is feasible.", "In Section , we show that the flatness transform for the tailsitter dynamics has a similar form with control inputs depending on snap and yaw acceleration.", "This makes snap minimization also suitable for generating aggressive tailsitter trajectories.", "Elementary minimum-snap optimization subject to waypoint constraints can be formulated as follows: $\\begin{aligned}&\\underset{\\mathbf {\\sigma }}{\\text{minimize}}& & \\int _{0}^{T} \\left\\Vert \\frac{d^4 \\mathbf {x}}{dt^4}\\right\\Vert ^2 + \\mu _\\psi \\Big (\\frac{d^2 \\psi }{dt^2}\\Big )^2 dt\\\\& \\text{subject to} & & \\mathbf {\\sigma }\\left(\\sum \\nolimits _{j=1}^{i}t_j\\right) = \\tilde{\\mathbf {\\sigma }}_i, \\; i=0,\\;\\dots ,\\;m,\\\\\\end{aligned}$ where $\\mu _\\psi $ is a weighing parameter.", "The nonnegative vector $\\mathbf {t}$ represents the time allocation over the trajectory segments between the $m+1$ waypoints $\\tilde{\\mathbf {\\sigma }}$ that must be attained in order.", "Minimum-snap trajectory generation for quadcopters is widely studied, and various methods to obtain $\\mathbf {t}$ have been proposed [15], [16], [23].", "In principle, our framework for flatness-based trajectory generation is detached from the exact optimization formulation, enabling it to profit from the extensive research on minimum-snap trajectory generation, including extensions such as obstacle avoidance [24].", "In this paper, we use the formulation by [16] to describe the trajectory with piecewise polynomial functions that we define in terms of their derivatives at the waypoints.", "For a given time allocation $\\mathbf {t}$ , the corresponding minimum-snap trajectory is then efficiently obtained in closed form using matrix multiplications, which we conveniently denote as $\\mathbf {\\sigma } = \\mathbf {\\chi }\\left(\\mathbf {t}, {\\tilde{\\sigma }}, \\dot{\\tilde{\\sigma }}, \\ddot{\\tilde{\\sigma }}, \\dots \\right),$ where $\\dot{\\tilde{\\sigma }}$ , $\\ddot{\\tilde{\\sigma }}$ etc.", "denote optional derivative constraints that may be set at some of the waypoints.", "We first minimize snap subject to a rough estimate $\\bar{T}$ of the total trajectory time based on the distance between waypoints, as follows: $\\begin{aligned}&\\underset{\\mathbf {\\sigma }, \\mathbf {t}}{\\text{minimize}}& & \\int _{0}^{\\bar{T}} \\left\\Vert \\frac{d^4 \\mathbf {x}}{dt^4}\\right\\Vert ^2 + \\mu _\\psi \\Big (\\frac{d^2 \\psi }{dt^2}\\Big )^2 dt\\\\& \\text{subject to} & & \\mathbf {\\sigma } = \\mathbf {\\chi }\\left(\\mathbf {t}, {\\tilde{\\sigma }}, \\dot{\\tilde{\\sigma }}, \\ddot{\\tilde{\\sigma }}, \\dots \\right),\\\\&&& \\sum \\nolimits _{j=1}^{m}t_j = \\bar{T}.\\end{aligned}$ In order to obtain aggressive aerobatic trajectories, we then minimize the scale factor $c$ that is applied to the resulting time allocation $\\mathbf {t}$ .", "As such, we obtain the quickest minimum-snap trajectory $\\mathbf {\\sigma } = \\mathbf {\\chi }\\left(c\\mathbf {t}, {\\tilde{\\sigma }}, \\dot{\\tilde{\\sigma }}, \\ddot{\\tilde{\\sigma }}, \\dots \\right)$ that is in the feasible set $\\Sigma _T = \\Big \\lbrace \\mathbf {\\sigma }\\Big |\\mathbf {u}(t) \\in \\mathcal {U}\\;\\;\\;\\;\\forall t \\in \\left[0,T\\right]\\Big \\rbrace ,$ where $\\mathbf {u}$ is the control input trajectory corresponding to $\\mathbf {\\sigma }$ and $\\mathcal {U}$ is the set of permissible control inputs, i.e., the bounded set defined by the minimum and maximum allowed rotor speeds and flap deflections.", "Additionally, we employ the method by [23] to optimize the time allocation $\\mathbf {t}$ using experimental evaluations, as described in Section REF ." ], [ "Differential Flatness Transform", "In recent work on tailsitter flight control, we have shown how the vehicle attitude and angular velocity can be obtained based on the trajectory (REF ) and its derivatives up to yaw rate and jerk (i.e., the third derivative of position) [20].", "In this section, we extend this derivation to obtain the full differential flatness transform, including an expression for the control inputs based on the trajectory derivatives up to yaw acceleration and snap.", "This expression enables us to verify that the motor speeds and flap deflections corresponding to a candidate trajectory are permissible, i.e., that the trajectory is in the feasible set (REF )." ], [ "Attitude", "We first derive expressions for the attitude and collective thrust.", "Rewriting () as $\\mathbf {f}^i = m\\left(\\mathbf {a} - g\\mathbf {i}_z\\right)$ shows that the vehicle attitude and collective thrust are uniquely defined by three major constraints: the yaw angle $\\psi $ , the fact that $\\mathbf {i}_y^\\top \\mathbf {f}^\\alpha = 0$ according to (REF ), and the forces in the vehicle symmetry plane, i.e., $\\mathbf {i}_x^\\top \\mathbf {f}^\\alpha $ and $\\mathbf {i}_z^\\top \\mathbf {f}^\\alpha $ .", "The Euler angles $\\psi $ , $\\phi $ , and $\\theta $ in ZXY rotation sequence are used to describe the attitude.", "These angles form a valid and universal attitude representation with each angle uniquely defined by one of the three constraints given above.", "The angle symbols are also used to refer to rotation matrices between intermediate frames, e.g., the rotation matrix $\\mathbf {R}^\\phi _i$ represents the rotations by $\\psi $ and $\\phi $ .", "The yaw rotation $\\psi \\mathbf {i}_z$ is applied first and defines the direction of the horizontal component of $\\mathbf {b}_y$ .", "Next, constraint (ii) is satisfied by the roll rotation $\\phi = -\\operatorname{atan2}\\left(\\mathbf {i}_y^\\top \\mathbf {R}^\\psi _i \\mathbf {f}^i,\\mathbf {i}_z^\\top \\mathbf {f}^i\\right) + k\\pi $ around the yawed $x$ -axis $\\mathbf {R}^i_\\psi \\mathbf {i}_x$ , where $\\operatorname{atan2}$ is the four-quadrant inverse tangent function.", "Constraint (ii) is satisfied $\\forall k\\in \\lbrace 0,1\\rbrace $ and, in practice, $k$ can be set such that the obtained attitude trajectory is continuous.", "Finally, constraint (iii) is satisfied by equating (REF ) and (REF ) and solving for the collective thrust $T = T_1 + T_2$ and for the pitch rotation angle $\\bar{\\theta }$ from the frame $\\phi $ to the zero-lift reference frame.", "In solving these equations, we neglect the nonminimum phase dynamics due to the direct force contribution by the flaps.", "When combined with feedback control, this approach achieves good trajectory generation and tracking performance for slightly nonminimum phase systems [25].", "The method is simple and avoids the large and quickly changing control actions that exact feedback linearization of the nonminimum phase system may result in [26].", "We note that potentially a flat output of the nonminimum phase dynamics could be used to guarantee stable tracking [27].", "However, this approach requires defining the trajectory in terms of the center of oscillation instead of the vehicle center of mass, leading to difficulty with the relatively complicated 6DOF tailsitter dynamics model.", "We substitute $\\mathbf {f}^\\alpha = \\mathbf {R}^{\\bar{\\theta }}_\\phi \\mathbf {f}^\\phi $ with $\\mathbf {f}^\\phi = \\mathbf {R}^\\phi _i \\mathbf {f}^i$ as well as a similar expression for $\\mathbf {v}^\\alpha $ into (REF ) to obtain $\\operatorname{c}\\!", "{\\bar{\\alpha }}\\; \\left(1 - c_{D_T}\\right) T - c_{D_V} \\Vert \\mathbf {v}\\Vert \\left(\\operatorname{c}\\!", "{\\bar{\\theta }}\\; \\mathbf {i}_x^\\top {\\mathbf {v}^\\phi } - \\operatorname{s}\\!", "{\\bar{\\theta }}\\; \\mathbf {i}_z^\\top \\mathbf {v}^\\phi \\right) =\\\\ \\operatorname{c}\\!", "{{\\bar{\\theta }}}\\;\\mathbf {i}_x^\\top \\mathbf {f}^\\phi -\\operatorname{s}\\!", "{{\\bar{\\theta }}}\\;\\mathbf {i}_z^\\top {\\mathbf {f}^\\phi },$ $\\operatorname{s}\\!", "{\\bar{\\alpha }}\\; (c_{L_T} - 1) T - c_{L_V} \\Vert \\mathbf {v}\\Vert \\left(\\operatorname{s}\\!", "{{\\bar{\\theta }}}\\;\\mathbf {i}_x^\\top \\mathbf {v}^\\phi + \\operatorname{c}\\!", "{{\\bar{\\theta }}}\\;\\mathbf {i}_z^\\top {\\mathbf {v}^\\phi }\\right) =\\\\ \\operatorname{s}\\!", "{{\\bar{\\theta }}}\\;\\mathbf {i}_x^\\top \\mathbf {f}^\\phi + \\operatorname{c}\\!", "{{\\bar{\\theta }}}\\;\\mathbf {i}_z^\\top \\mathbf {f}^\\phi ,$ where $\\operatorname{c}$ and $\\operatorname{s}$ represent cosine and sine, respectively.", "Solving (REF ) and (REF ) for $\\bar{\\theta }$ and $T$ gives ${\\bar{\\theta }} = \\operatorname{atan2}\\\\\\left(\\eta \\left(\\mathbf {i}_x^\\top \\mathbf {f}^\\phi +c_{D_V}\\Vert \\mathbf {v}\\Vert \\mathbf {i}_x^\\top \\mathbf {v}^\\phi \\right) -c_{L_V}\\Vert \\mathbf {v}\\Vert \\mathbf {i}_z^\\top \\mathbf {v}^\\phi -\\mathbf {i}_z^\\top \\mathbf {f}^\\phi ,\\right.\\\\\\left.\\eta \\left(\\mathbf {i}_z^\\top \\mathbf {f}^\\phi +c_{D_V}\\Vert \\mathbf {v}\\Vert \\mathbf {i}_z^\\top \\mathbf {v}^\\phi \\right) +c_{L_V}\\Vert \\mathbf {v}\\Vert \\mathbf {i}_x^\\top \\mathbf {v}^\\phi +\\mathbf {i}_x^\\top \\mathbf {f}^\\phi \\right) + k\\pi ,$ $T = \\frac{1}{\\operatorname{c}\\!", "{\\bar{\\alpha }}\\;\\left(1-c_{D_T}\\right)}\\left(\\operatorname{c}\\!", "{{\\bar{\\theta }}}\\;\\mathbf {i}_x^\\top \\mathbf {f}^\\phi - \\operatorname{s}\\!", "{{\\bar{\\theta }}}\\;\\mathbf {i}_z^\\top \\mathbf {f}^\\phi +\\right.\\\\\\left.", "c_{D_V} \\Vert \\mathbf {v}\\Vert \\left(\\operatorname{c}\\!", "{\\bar{\\theta }}\\; \\mathbf {i}_x^\\top {\\mathbf {v}^\\phi } - \\operatorname{s}\\!", "{\\bar{\\theta }}\\; \\mathbf {i}_z^\\top \\mathbf {v}^\\phi \\right)\\right),$ where $\\eta = \\frac{\\operatorname{s}\\!", "{\\bar{\\alpha }}\\; \\left(c_{L_T} - 1\\right) }{\\operatorname{c}\\!", "{\\bar{\\alpha }}\\;\\left(1-c_{D_T}\\right)}.$ is the ratio of lift and forward force due to thrust.", "Again, the constraint is satisfied $\\forall k\\in \\lbrace 0,1\\rbrace $ and, in practice, $k$ can be set such that the obtained attitude trajectory is continuous.", "Finally, the pitch rotation of the body-fixed reference frame is obtained as $\\theta = \\bar{\\theta }+ \\alpha _0$ ." ], [ "Angular Velocity", "An expression for the angular velocity is obtained by taking the derivative of the Euler angles.", "From (REF ), we obtain $\\dot{\\phi }= -\\frac{\\dot{\\beta }_x\\beta _z-\\beta _x\\dot{\\beta }_z}{\\beta _x^2 + \\beta _z^2},$ where $\\beta _x$ and $\\beta _z$ are respectively the first and second arguments of the $\\operatorname{atan2}$ function, and $\\dot{\\beta }_x &= -\\operatorname{c}\\!", "{\\psi }\\; \\dot{\\psi }\\mathbf {i}_x^\\top \\mathbf {f}^i - \\operatorname{s}\\!", "{\\psi }\\; \\mathbf {i}_x^\\top \\dot{\\mathbf {f}}^i - \\operatorname{s}\\!", "{\\psi }\\;\\dot{\\psi }\\mathbf {i}_y^\\top \\mathbf {f}^i + \\operatorname{c}\\!", "{\\psi }\\;\\mathbf {i}_y^\\top \\dot{\\mathbf {f}}^i,\\\\\\dot{\\beta }_z &= \\mathbf {i}_z^\\top \\dot{\\mathbf {f}}^i,$ with, from the derivative of (REF ), $\\dot{\\mathbf {f}}^i = m\\mathbf {j}.$ Similarly, from (REF ) we obtain $\\dot{\\theta }= \\frac{\\dot{\\sigma }_x\\sigma _z - \\sigma _x\\dot{\\sigma }_z }{\\sigma _x^2 + \\sigma _z^2},$ where $\\sigma _x$ and $\\sigma _z$ are the respective arguments of the $\\operatorname{atan2}$ function, and $\\dot{\\sigma }_x &= \\eta \\left(\\mathbf {i}_x^\\top \\dot{\\mathbf {f}}^\\phi + c_{D_V} \\tau _x \\right) - c_{L_V}\\tau _z - \\mathbf {i}_z^\\top \\dot{\\mathbf {f}}^\\phi ,\\\\\\dot{\\sigma }_z &= \\eta \\left(\\mathbf {i}_z^\\top \\dot{\\mathbf {f}}^\\phi + c_{D_V} \\tau _z \\right) + c_{L_V}\\tau _x + \\mathbf {i}_x^\\top \\dot{\\mathbf {f}}^\\phi $ with $\\tau _x &= \\dot{\\Vert \\mathbf {v}\\Vert } \\mathbf {i}_x^\\top \\mathbf {v}^\\phi + \\Vert \\mathbf {v}\\Vert \\mathbf {i}_x^\\top \\dot{\\mathbf {v}}^\\phi ,\\\\\\tau _z &= \\dot{\\Vert \\mathbf {v}\\Vert } \\mathbf {i}_z^\\top \\mathbf {v}^\\phi + \\Vert \\mathbf {v}\\Vert \\mathbf {i}_z^\\top \\dot{\\mathbf {v}}^\\phi $ and $\\dot{\\Vert \\mathbf {v}\\Vert } &= \\frac{\\mathbf {v}^\\top \\mathbf {a}}{\\Vert \\mathbf {v}\\Vert },\\\\\\dot{\\mathbf {v}}^\\phi &= \\dot{\\mathbf {R}}^\\phi _i\\mathbf {v} + \\mathbf {R}^\\phi _i\\mathbf {a}.$ The expression for the force derivative $\\dot{\\mathbf {f}}^\\phi $ is similar to (REF ).", "As described in Section REF , we neglect the direct force contribution by the flaps.", "Finally, we obtain the angular velocity in the body-fixed reference frame, as follows: $\\mathbf {\\Omega } = \\left[\\begin{array}{c}0\\\\\\dot{\\theta }\\\\0\\end{array}\\right] + \\mathbf {R}^{ \\theta }_\\phi \\left[\\begin{array}{c}\\dot{\\phi }\\\\0\\\\0\\end{array}\\right] + \\mathbf {R}^{ \\theta }_\\psi \\left[\\begin{array}{c}0\\\\0\\\\\\dot{\\psi }\\end{array}\\right].$" ], [ "Motor Speeds and Flap Deflections", "In order to obtain the control inputs, we first derive an expression for the angular acceleration as a function of snap and yaw acceleration.", "By taking the derivative of (REF ), we obtain the following expression for the roll acceleration $\\ddot{\\phi } = \\left(\\beta _x^2+\\beta _z^2\\right)^{-2}\\left(\\left(\\dot{\\beta }_x\\beta _z-\\beta _x\\dot{\\beta }_z\\right)\\left(2\\beta _x\\dot{\\beta }_x+2\\beta _z\\dot{\\beta }_z\\right)-\\right.\\\\\\left.\\left(\\ddot{\\beta }_x\\beta _z-\\beta _x\\ddot{\\beta }_z\\right)\\left(\\beta _x^2+\\beta _z^2\\right)\\right),$ where $\\begin{split}\\ddot{\\beta }_x &= \\left(\\operatorname{s}\\!", "{\\psi }\\;{\\dot{\\psi }}^2 - \\operatorname{c}\\!", "{\\psi }\\;\\ddot{\\psi }\\right)\\mathbf {i}_x^\\top \\mathbf {f}^i - 2 \\operatorname{c}\\!", "{\\psi }\\; \\dot{\\psi }\\mathbf {i}_x^\\top \\dot{\\mathbf {f}}^i - \\operatorname{s}\\!", "{\\psi }\\;\\mathbf {i}_x^\\top \\ddot{\\mathbf {f}}^i\\\\&\\;\\;\\;\\;\\;- \\left(\\operatorname{c}\\!", "{\\psi }\\;{\\dot{\\psi }}^2 + \\operatorname{s}\\!", "{\\psi }\\;\\ddot{\\psi }\\right)\\mathbf {i}_y^\\top \\mathbf {f}^i -2 \\operatorname{s}\\!", "{\\psi }\\;\\dot{\\psi }\\mathbf {i}_y^\\top \\dot{\\mathbf {f}}^i +\\operatorname{c}\\!", "{\\psi }\\;\\mathbf {i}_y^\\top \\ddot{\\mathbf {f}}^i,\\end{split}\\\\\\ddot{\\beta }_z &= \\mathbf {i}_z^\\top \\ddot{\\mathbf {f}}^i$ are obtained as the derivatives of (REF ) and (), and the second force derivative is a function of snap $\\ddot{\\mathbf {f}}^i = m\\mathbf {s}.$ Similarly, by taking the derivative of (REF ) we obtain the pitch acceleration $\\ddot{\\theta }= \\big (\\left(\\ddot{\\sigma }_x\\sigma _z-\\sigma _x\\ddot{\\sigma }_z\\right)\\left(\\sigma _x^2+\\sigma _z^2\\right) -\\\\\\left(\\dot{\\sigma }_x\\sigma _z-\\sigma _x\\dot{\\sigma }_z\\right)\\left(2\\sigma _x\\dot{\\sigma }_x+2\\sigma _z\\dot{\\sigma }_z\\right)\\big ){\\left(\\sigma _x^2+\\sigma _z^2\\right)^{-2}},$ where $\\ddot{\\sigma }_x &= \\eta \\left(\\mathbf {i}_x^\\top \\ddot{\\mathbf {f}}^\\phi + c_{D_V} \\dot{\\tau }_x \\right) - c_{L_V}\\dot{\\tau }_z - \\mathbf {i}_z^\\top \\ddot{\\mathbf {f}}^\\phi ,\\\\\\ddot{\\sigma }_z &= \\eta \\left(\\mathbf {i}_z^\\top \\ddot{\\mathbf {f}}^\\phi + c_{D_V} \\dot{\\tau }_z \\right) + c_{L_V}\\dot{\\tau }_x + \\mathbf {i}_x^\\top \\ddot{\\mathbf {f}}^\\phi $ with $\\dot{\\tau }_x &= \\ddot{\\Vert \\mathbf {v}\\Vert } \\mathbf {i}_x^\\top \\mathbf {v}^\\phi + 2\\dot{\\Vert \\mathbf {v}\\Vert } \\mathbf {i}_x^\\top \\dot{\\mathbf {v}}^\\phi + \\Vert \\mathbf {v}\\Vert \\mathbf {i}_x^\\top \\ddot{\\mathbf {v}}^\\phi ,\\\\\\dot{\\tau }_z &= \\ddot{\\Vert \\mathbf {v}\\Vert } \\mathbf {i}_z^\\top \\mathbf {v}^\\phi + 2\\dot{\\Vert \\mathbf {v}\\Vert } \\mathbf {i}_z^\\top \\dot{\\mathbf {v}}^\\phi + \\Vert \\mathbf {v}\\Vert \\mathbf {i}_z^\\top \\ddot{\\mathbf {v}}^\\phi $ and $\\ddot{\\Vert \\mathbf {v}\\Vert } &= \\frac{\\mathbf {a}^\\top \\mathbf {a} + \\mathbf {v}^\\top \\mathbf {j}}{\\Vert \\mathbf {v}\\Vert } - \\frac{\\mathbf {v}^\\top \\mathbf {a} \\dot{\\Vert \\mathbf {v}\\Vert }}{\\Vert \\mathbf {v}\\Vert ^2},\\\\\\ddot{\\mathbf {v}}^\\phi &= \\ddot{\\mathbf {R}}^\\phi _i\\mathbf {v} + 2\\dot{\\mathbf {R}}^\\phi _i\\mathbf {a} + \\mathbf {R}^\\phi _i\\mathbf {j}.$ The expression for the force second derivative $\\ddot{\\mathbf {f}}^\\phi $ is similar to (REF ).", "We combine the roll acceleration and pitch acceleration obtained from respectively (REF ) and (REF ) with the yaw acceleration $\\ddot{\\psi }$ to obtain the angular acceleration in the body-fixed reference frame.", "We take the derivative of (REF ) to obtain the following expression: $\\dot{\\mathbf {\\Omega }} = \\left[\\begin{array}{c}0\\\\\\ddot{\\theta }\\\\0\\end{array}\\right] + \\dot{\\mathbf {R}}^{ \\theta }_\\phi \\left[\\begin{array}{c}\\dot{\\phi }\\\\0\\\\0\\end{array}\\right] + \\mathbf {R}^{ \\theta }_\\phi \\left[\\begin{array}{c}\\ddot{\\phi }\\\\0\\\\0\\end{array}\\right] +\\\\ \\dot{\\mathbf {R}}^{ \\theta }_\\psi \\left[\\begin{array}{c}0\\\\0\\\\\\dot{\\psi }\\end{array}\\right] + \\mathbf {R}^{ \\theta }_\\psi \\left[\\begin{array}{c}0\\\\0\\\\\\ddot{\\psi }\\end{array}\\right].$ We can now find the moment in the body-fixed reference frame by rewriting (REF ), as follows: $\\mathbf {m} = \\mathbf {J}\\dot{\\mathbf {\\Omega }} + \\mathbf {\\Omega }\\times \\mathbf {J}\\mathbf {\\Omega }.$ Next, we solve (REF ) for the flap deflections and differential thrust $\\Delta T = T_1 - T_2$ .", "We find an expression for $\\Delta T$ by equating $\\mathbf {i}_z^\\top \\left( \\mathbf {m}_T + \\mathbf {m}_\\mu \\right) = \\mathbf {i}_z^\\top \\mathbf {m},$ which assumes that the contribution by $\\mathbf {i}_z^\\top \\mathbf {m}_\\delta $ is negligible.", "Due to the multiplication with $\\sin {\\alpha _0}$ , this assumption typically does not result in significant discrepancies.", "Using $\\mu _1 + \\mu _2 = {c_\\mu }{c_T}\\Delta T$ , we obtain $\\Delta T = {\\mathbf {i}_z^\\top \\mathbf {m}}\\big (- \\operatorname{s}\\!", "{\\alpha _T}\\;\\frac{c_\\mu }{c_T} +\\\\l_{T_y}\\left(\\operatorname{c}\\!", "{\\alpha _0}\\;\\operatorname{c}\\!", "{\\bar{\\alpha }}\\;(1 - c_{D_T}) - \\operatorname{s}\\!", "{\\alpha _0}\\;\\operatorname{s}\\!", "{\\bar{\\alpha }}\\;(c_{L_T}- 1)\\right) \\big )^{-1}.$ The individual thrust values are then given by $T_1 = \\frac{T + \\Delta T}{2},\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;T_2 = \\frac{T - \\Delta T}{2},$ and the motor speeds can be obtained from (REF ).", "For the flap deflections, we deduct $\\mathbf {m}_T$ and $\\mathbf {m}_\\mu $ from $\\mathbf {m}$ to obtain $\\mathbf {m}_\\delta $ , and we rewrite (REF ), as follows: $\\left[\\begin{array}{c}\\delta _1\\\\\\delta _2\\end{array}\\right] = \\left[\\begin{array}{cc}-l_{\\delta _y} \\operatorname{c}\\!", "{\\alpha _0}\\; {\\nu }_1& l_{\\delta _y} \\operatorname{c}\\!", "{\\alpha _0}\\; {\\nu }_2\\\\l_{\\delta _x} {\\nu }_1&l_{\\delta _x} {\\nu }_2\\end{array}\\right]^{-1}\\left[\\begin{array}{c}\\mathbf {i}_x^\\top \\mathbf {m}_\\delta \\\\\\mathbf {i}_y^\\top \\mathbf {m}_\\delta \\end{array}\\right]$ with ${\\nu }_i = -c^\\delta _{L_T} \\cos {\\bar{\\alpha }}\\; T_i - c^\\delta _{L_V}\\Vert \\mathbf {v}\\Vert \\mathbf {i}_x^\\top \\mathbf {v}^\\alpha .$ Note that—since the control inputs cannot instantaneously change—dynamic feasibility of $\\mathbf {\\sigma }$ requires continuity of (REF ), and therefore at least fourth-order continuity of the position $\\mathbf {x}$ and at least second-order continuity of the yaw $\\psi $ ." ], [ "Dynamic Feasibility", "In this section, we evaluate the suitability of the flat transform presented in Section to determine feasibility of a candidate trajectory on the actual vehicle.", "Specifically, the transform is used to obtain the control inputs that are then evaluated according to (REF ).", "For a description of the vehicle parameters and their estimation, the reader is referred to [22]." ], [ "Hover-to-Hover Trajectory", "We consider a single-segment hover-to-hover trajectory with $\\tilde{\\mathbf {\\sigma }}_0 &= \\left[\\begin{array}{cccc}0 \\;\\;\\;\\;\\;\\;\\; & 0& 0& \\psi ^{\\operatorname{start}}\\\\\\end{array}\\right]^\\top ,\\\\\\tilde{\\mathbf {\\sigma }}_1&= \\left[\\begin{array}{cccc}6 \\;\\text{[m]} & 0& 0& \\psi ^{\\operatorname{end}}\\\\\\end{array}\\right]^\\top .$ This trajectory requires large acceleration and simultaneous yawing motion through the transition regime.", "Based on the flatness transform described in Section , we determine the minimal feasible time for the minimum-snap trajectory with various $\\psi ^{\\operatorname{start}}$ and $\\psi ^{\\operatorname{end}}$ .", "An example trajectory is shown in Fig.", "REF .", "Figure REF shows results for the trajectory with yawing motion from $\\psi ^{\\operatorname{start}}$ to $\\psi ^{\\operatorname{end}}$ using the minimal rotation.", "It can be seen that the fastest times are achieved in the center of the figure, around $\\psi ^{\\operatorname{start}}= \\psi ^{\\operatorname{end}}= 0$ rad, which corresponds to forward coordinated flight.", "We observe discontinuity along the yaw direction switching lines, which indicates that, in some cases, it may be beneficial to yaw in the opposite direction.", "However, in practice the difference is typically small, meaning that the minimal rotation that is obtained from optimization in the flat output space is (nearly) optimal.", "We conduct experiments to compare the feasibility boundary from Fig.", "REF to the tracking error of the actual vehicle.", "Figure REF shows the tracking error for the hover-to-hover trajectory in coordinated flight without yaw, i.e., $\\psi ^{\\operatorname{start}}= \\psi ^{\\operatorname{end}}= 0$ rad, and for the same trajectory but with $\\psi ^{\\operatorname{start}}= 0$ rad, $\\psi ^{\\operatorname{end}}= \\pi $ rad.", "Each point on the curves corresponds to a flight experiment.", "As the trajectory time on the horizontal axis increases, the maneuvers become less aggressive, and the tracking error decreases.", "The corresponding feasibility boundaries predicted in Fig.", "REF are indicated by the colored shading, i.e., the shaded areas in the left of the figure correspond to infeasible trajectory times.", "While only a single color is shown at a time, the infeasibility areas continue from their boundary all the way to the vertical axis on the left.", "For the yawing trajectory, the tracking error increases at lower speeds compared to the coordinated flight trajectory, as predicted by the feasibility boundaries.", "We note that these boundaries correspond to the most aggressive trajectories that theoretically can be tracked by the given vehicle dynamics model, neglecting practical factors such as modeling errors and imperfect state estimation and control, so that it is expected that significant tracking error occurs before they are reached.", "The coordinated flight trajectory at the feasibility boundary (2.0 s) attains a maximum speed of 7.6 m/s within 1 s and attains a maximum load of 3.1$g$ .", "It is tracked with less than 1 m position tracking error." ], [ "Circular Trajectory", "In order to evaluate the accuracy of the feasibility prediction at high speed and large sustained acceleration, we use the flatness transform to determine the maximum speed on a circular trajectory with a 3 m radius.", "As shown in Fig.", "REF , we consider two trimmed conditions, coordinated and knife-edge flight, as well as a rolling/yawing motion where $\\psi $ changes at the same rate but in the opposite direction.", "The position and yaw derivatives for evaluation of the feasibility are given in Table REF .", "We perform experiments for all three circular trajectories at various speeds.", "The results are shown in Fig.", "REF , where each point on the curves corresponds to a flight experiment.", "The figure is oriented similarly to Fig.", "REF with the most aggressive, i.e., the highest speed, trajectories towards the left.", "It shows that the flat dynamics model predicts that coordinated flight can be performed up to the highest speed, followed by knife-edge flight, and finally the rolling circle, which has a relatively low maximum speed.", "The position tracking errors obtained from flight experiments agree with this prediction.", "Figure REF shows the expected increase in each position tracking error before the corresponding shaded area is reached.", "Similar behavior can be observed in Fig.", "REF for the yaw tracking error on the coordinated and rolling circle.", "The yaw error in knife-edge flight remains very small, even at high speeds, because—in this condition—the vehicle orientation reduces the sensitivity of yaw to attitude errors and increases the yaw control effectiveness by differential thrust.", "Since the flat transform does not consider lateral forces on the tailless aircraft, the speed in circular knife-edge flight is mostly limited by the maximum thrust.", "In fact, completely neglecting the aerodynamics and solving for the maximum speed $v_{\\max } = \\sqrt{2 c_T \\bar{\\omega }^2\\frac{ r}{m}}$ with $\\bar{\\omega }$ the maximum motor speed, results in only a small overestimation when compared to the maximum speed obtained from the flat transform (9.5 m/s versus 9.2 m/s).", "In flight experiments, the vehicle achieved RMS position and yaw tracking errors of respectively 12.5 cm and 1.1 deg at 8 m/s, approaching the theoretical maximum speed with relatively small tracking error.", "Considering that at least some control input margin must be maintained to enable stabilization of the unstable knife-edge condition (making the theoretical limit unattainable), this affirms that the lateral aerodynamic force must indeed be quite small and can be neglected in the flat dynamics model.", "Considering the comparative results for both trajectories, we can conclude that the differential flatness transform gives a useful prediction of the critical trajectory time or speed where we can expect to observe a stark increase in tracking error on the real vehicle." ], [ "Flight Experiments", "We present extensive experimental results to validate the generated aerobatic trajectories.", "These flight tests demonstrate six types of aerobatic maneuvers, a racing trajectory through a sequence of gates, and an airshow-like aerobatic sequence with three tailsitter aircraft that aggressively maneuver in close proximity to obstacles and to each other.", "Detailed descriptions of the experimental platform, shown in Fig.", "REF , and the trajectory-tracking flight control system are given in [28] and [20], respectively.", "A video of the experiments can be found at https://aera.mit.edu/projects/TailsitterAerobatics.", "Figure: Tailsitter flying wing aircraft used in the experiments." ], [ "Aerobatic Maneuvers", "We first demonstrate how the flatness transform enables generation of aerobatic maneuvers using relatively simple waypoint (derivative) constraints in the trajectory output space.", "The resulting maneuvers exploit the full flight envelope of the tailsitter aircraft, including post-stall and sideways flight, and do not require restrictive assumptions such as coordinated flight and curvature limitations.", "The data shown in Table REF confirms the aerobatic character of the trajectories, which reach speeds up to 8.0 m/s, loads of over 3$g$ , and angular rates that exceed 650 deg/s.", "Conforming to the observations from Section , we found that each maneuver can be slowed down to reduce tracking error but we chose to accept some tracking error in favor of increased aggressiveness.", "The loop trajectory shown in Fig.", "REF consists of five waypoints (of which two coincide) on a vertical circle with 1 m radius, and start and end points constrained to static hover.", "We add tangential velocity constraints to enforce a circular path.", "As shown in Fig.", "REF, the loop trajectory has several feasibility boundaries.", "When flown slowly (i.e., below 2.5 m/s), the trajectory is feasible and flown in hover attitude with $\\theta \\approx {\\pi }{2}$ rad.", "When flown faster (i.e., around 4.5 m/s), the vehicle performs a loop, making a full upward pitch rotation.", "Intermediate speeds (i.e., around 3 m/s) are too slow to perform a loop and require the vehicle to quickly pitch back down at the top of the circular segment, rendering the trajectory infeasible due to flap deflection limits.", "The maximum position tracking error obtained from flight experiments shows a stark increase in this region of infeasibility and also increases as the infeasibility boundary at very high speed (i.e., 5.2 m/s) is approached.", "The trajectory with a maximum speed of 3.8 m/s is shown in Fig.", "REF .", "The loop maneuver is successfully performed in the flight experiment.", "The maximum position error of 74 cm is incurred when exiting the final circular segment." ], [ "Knife-Edge Flight", "Figure REF shows a straight trajectory between static hover start and end points.", "The intermediate waypoints enforce a constant speed of 5 m/s and serve to transition between flight attitudes through the yaw reference $\\psi _{\\operatorname{ref}}$ .", "In the first of the three middle segments, the vehicle transitions from coordinated to knife-edge fight; in the second, it maintains constant knife-edge orientation; and in the third, it transitions back to coordinated flight.", "Performing the transitions while maintaining straight flight at 5 m/s is challenging due to the aerodynamic interactions between vehicle attitude, flap deflections, and rotor speeds.", "As expected, the position tracking error in the flight experiment increases at the transitions.", "Once knife-edge orientation is reached, the position tracking error quickly reduces again.", "The vehicle attitude during knife-edge flight differs somewhat between the reference and experiment trajectories.", "The increased pitch angle in the experiment compensates for the neglected flap force contribution, and the small rotation towards the direction of travel compensates for the nonzero lateral force.", "Finally, we note that the largest position tracking error is incurred close to the end point.", "This error is mainly along the trajectory and is caused by delayed deceleration.", "The maximum path error, i.e., position error with regard to the closest point on the trajectory line, occurs during the second transition and amounts to 47 cm." ], [ "Climbing Turn", "We plan a climbing turn trajectory using four waypoints, as shown in Fig.", "REF .", "The start and end points are constrained to static hover, and the two intermediate waypoints are positioned with only a height difference.", "Using velocity constraints, we enforce straight and coordinated flight at these intermediate waypoints.", "Hence, the entire 270 deg turn and 1 m climb occur between these two waypoints.", "During the turn, the reference trajectory reaches about 90 banking angle, requires nearly the maximum motor speeds of 2500 rad/s, and reaches a peak angular rate of 11.3 rad/s (647 deg/s).", "A peak load of 3.1$g$ is required during the turn, while the loads close to respectively the start and end points reach up to 2.0$g$ .", "Consequently, the vehicle quickly completes the 11.3 m trajectory in 3.1 s, despite starting and ending in static hover.", "In the flight experiment, we observe that, during the turn, a maximum load of 3.4$g$ and a peak angular rate of 10.9 rad/s (625 deg/s) are attained.", "The motors briefly saturate, resulting in some loss of altitude.", "Once the saturation is resolved, the vehicle quickly catches up and reduces the position tracking error to below 20 cm before the turn is exited." ], [ "Immelmann Turn", "The Immelmann turn is a well-known aerobatics and aerial combat maneuver that reverses direction by performing a half loop followed by a half roll.", "We generate the trajectory using static hover start and end points, and four intermediate waypoints.", "The intermediate waypoints enforce constant speed coordinated flight prior to the half loop and constant speed transition from inverted to regular coordinated flight afterward.", "Similar to the loop and knife-edge maneuvers described above, we observe increased error when exiting the loop segment, increased error during transition through uncoordinated flight orientation, and delayed deceleration towards the end point.", "Comparison of the vehicle poses also leads to similar observations of small differences: increased pitch to account for flap force and increased yaw in uncoordinated flight to compensate for the nonzero lateral force.", "The Immelmann turn combines several challenging aspects to exploit the expansive flight envelope of the tailsitter vehicle.", "The maneuver contains large accelerations, inverted flight, and a transition through the entire yaw range (i.e., from $\\psi _{\\operatorname{ref}} = 0$ to $\\psi _{\\operatorname{ref}} = \\pm \\pi $ rad) at a peak angular rate of 9.4 rad/s (538 deg/s) while maintaining a linear speed of 6 m/s.", "Based on snap minimization and differential flatness, the state-space trajectory and corresponding control inputs were generated efficiently and based on only four waypoints.", "The flight experiment shows that the resulting maneuver can be tracked with acceptable position error ($< 60$ cm during the maneuver itself) while approaching the feasibility boundary, as over 90% of the maximum flap deflection is reached during the half roll." ], [ "Split S", "The Split S maneuver, shown in Fig.", "REF , is similar to the Immelmann but performed in opposite order.", "The maneuver starts the top leg in coordinated flight, then transitions to inverted coordinated flight using the yaw reference $\\psi _{\\operatorname{ref}}$ , and ends with a downward half loop that is exited in regular coordinated flight condition.", "The trajectory is generated using similar waypoints as the Immelmann maneuver, albeit with opposite order and velocity direction.", "Compared to the Immelmann turn, a smaller tracking error is achieved, because the flight speed is slightly lower (5 m/s versus 6 m/s) and because the trajectory ends with a relatively long stretch of coordinated flight, leading to a more stable deceleration.", "We note the downward pitch motion during the half loop in both the reference and experiment trajectories.", "By increasing the speed, we can obtain a more traditional Split S maneuver with a positive pitch rate.", "However, this maneuver requires a significantly larger flight volume.", "Figure: Split S maneuver.", "Interval between poses is 1.0 s." ], [ "Differential Thrust Turn", "The differential thrust turn, shown in Fig.", "REF , is an agile flight maneuver in which the vehicle reverses direction without deviating from a straight-line trajectory.", "Unlike more traditional turns, which involve turning on a circular trajectory segment, the turn is performed by reorienting the vehicle using differential thrust and flap deflections, and then applying a large collective thrust to accelerate in the opposite direction.", "The turn itself follows directly from snap minimization based on two coinciding waypoints with opposite velocity and yaw constraints.", "In the flight experiment, a peak angular rate of 8.6 rad/s (493 deg/s) is reached during the turn.", "As shown in the figure, the differential flatness transform is able to accurately predict the vehicle attitude at the midpoint of the turn.", "Figure: Differential thrust turn.", "Interval between poses is 1.5 s." ], [ "Racing Trajectory", "In order to demonstrate agile high-speed flight in close proximity to obstacles, we generate a trajectory through a sequence of four drone racing gates.", "The trajectory, shown in Fig.", "REF , consists of six waypoints: coinciding start and end points constrained to static hover, and four gate waypoints with a directional velocity constraint that enforces flight perpendicular to the gate window.", "Yaw is constrained so that the first three gates are passed in coordinated flight and the final, smaller gate in knife-edge flight, as it is too narrow to accommodate the tailsitter wingspan.", "Instead of using the input constraint (REF ), we scale $\\mathbf {t}$ subject to the experimental feasibility constraint $\\Sigma _T = \\Big \\lbrace \\mathbf {\\sigma }_{\\operatorname{ref}}\\Big | \\Vert \\mathbf {x}_{\\operatorname{ref}}(t) - \\mathbf {x}(t)\\Vert \\le 0.5\\;\\text{m}\\;\\;\\;\\;\\forall t \\in \\left[0,T\\right]\\Big \\rbrace ,$ which guarantees that the vehicle does not collide with any of the gates.", "The resulting trajectory is shown in Fig.", "REF .", "In order to obtain an even faster trajectory, we employ Bayesian optimization (BayesOpt) with experimental evaluations to further optimize $\\mathbf {t}$  [23].", "The BayesOpt algorithm, previously applied to quadrotors, can readily optimize the tailsitter trajectories by virtue of their flatness-based minimum-snap formulation.", "The optimized time allocation $\\mathbf {t}$ is then used to obtain a faster, more aggressive minimum-snap trajectory, shown in Fig.", "REF .", "This trajectory requires 19% less flight time (11.1 s versus 13.7 s for the trajectory shown in Fig.", "REF ) but satisfies the same tracking accuracy constraint (i.e., (REF )).", "It has a maximum speed of 7.1 m/s.", "The aggressive racing trajectory clearly shows how the tailsitter vehicle dynamics are exploited in trajectory generation to enable accurate tracking of fast and agile flight maneuvers.", "In particular, during the knife-edge trajectory segments, differential thrust attitude control enables larger accelerations in the direction of flight.", "The time-optimal trajectory exploits this additional acceleration to significantly increase the flight speed through the final gate.", "Figure: Minimum-snap racing trajectory through gates.", "Start and end points are static hover, and arrows indicate velocity direction constraints." ], [ "Aerobatic Sequence", "As a final demonstration of the consistency and accuracy with which the generated aerobatic trajectories can be flown, we perform an airshow-like aerobatic sequence with three tailsitter aircraft.", "The sequence, shown in the accompanying video and Fig.", "REF , consists of four stages that seamlessly follow each other and incorporate many of the aerobatic maneuvers described in Section REF as well as the racing trajectory described in Section REF .", "During the first stage, the three vehicles synchronously transition from hover to coordinated flight at 5.8 m/s and perform a loop.", "In the second stage, the tailsitters fly the minimum-time racing trajectory through the gates with only 0.7 s separation between successive vehicles.", "The third stage starts with successive transitioning flight from coordinated to knife-edge condition through the center gate, which is followed by synchronous aggressive maneuvers in hover with horizontal accelerations up to 11.5 m/s2 while maintaining 45 cm separation between adjacent vehicles.", "Finally, the three vehicles synchronously perform respectively the Immelmann turn, the differential thrust turn, and a loop through one of the gates.", "Figure: Multi-vehicle aerobatic sequence for three tailsitter aircraft." ], [ "Conclusion", "We proposed the novel application of snap minimization towards aerobatic trajectory generation for a tailsitter flying wing.", "The method plans trajectories in the flat output space, instead of considering computationally expensive optimization on the more complicated state and control input space.", "Through experimental validation, it was shown that the derived flatness transform provides a useful prediction of the critical trajectory time or speed at which a stark increase in tracking error occurs on the real vehicle.", "The proposed algorithm was used to generate trajectories for six aerobatic maneuvers, a race course through several gates, and an airshow-like aerobatic sequence for three tailsitters.", "We found that the real vehicle was indeed capable of accurately tracking these aggressive trajectories and that the vehicle pose and control inputs predicted by the flatness transform closely matched those of the actual vehicle.", "In conclusion, the proposed algorithm accurately plans aerobatic trajectories that exploit the expansive flight envelope of the tailsitter flying wing, without requiring costly optimization on the state and control input space." ], [ "Acknowledgments", "We thank Murat Bronz and John Aleman for the design, fabrication, and assembly of the aircraft used in the experiments.", "This work was supported by the Army Research Office through grant W911NF1910322." ] ]
2207.03524
[ [ "AVDDPG: Federated reinforcement learning applied to autonomous platoon\n control" ], [ "Abstract Since 2016 federated learning (FL) has been an evolving topic of discussion in the artificial intelligence (AI) research community.", "Applications of FL led to the development and study of federated reinforcement learning (FRL).", "Few works exist on the topic of FRL applied to autonomous vehicle (AV) platoons.", "In addition, most FRL works choose a single aggregation method (usually weight or gradient aggregation).", "We explore FRL's effectiveness as a means to improve AV platooning by designing and implementing an FRL framework atop a custom AV platoon environment.", "The application of FRL in AV platooning is studied under two scenarios: (1) Inter-platoon FRL (Inter-FRL) where FRL is applied to AVs across different platoons; (2) Intra-platoon FRL (Intra-FRL) where FRL is applied to AVs within a single platoon.", "Both Inter-FRL and Intra-FRL are applied to a custom AV platooning environment using both gradient and weight aggregation to observe the performance effects FRL can have on AV platoons relative to an AV platooning environment trained without FRL.", "It is concluded that Intra-FRL using weight aggregation (Intra-FRLWA) provides the best performance for controlling an AV platoon.", "In addition, we found that weight aggregation in FRL for AV platooning provides increases in performance relative to gradient aggregation.", "Finally, a performance analysis is conducted for Intra-FRLWA versus a platooning environment without FRL for platoons of length 3, 4 and 5 vehicles.", "It is concluded that Intra-FRLWA largely out-performs the platooning environment that is trained without FRL." ], [ "Introduction", "In recent years, federated learning (FL) and its extension federated reinforcement learning (FRL) have become a popular topic of discussion in the artificial intelligence (AI) community.", "The concept of FL was first proposed by Google with the development of the federated averaging (FedAvg) aggregation method [1].", "FedAvg provided an increase in the performance of distributed systems while also providing privacy advantages when compared to centralized architectures for supervised machine learning (ML) tasks [2], [3], [1].", "FL's core ideology was initially motivated by the need to train ML models from distributed data sets across mobile devices while minimizing data leakage and network usage [1].", "Research on the topics of reinforcement learning (RL) and deep reinforcement learning (DRL) has made great progress over the years; however, there remain important challenges for ensuring the stable performance of DRL algorithms in the real world.", "DRL processes are often sensitive to small changes in the model space or hyper-parameter space, and as such the application of a single trained model across similar systems often leads to control inaccuracies or instability [4], [5].", "In order to overcome the stability challenges that DRL poses, often a model must be manually customized to accommodate the finite differences amongst similar agents in a distributed system.", "FRL aims to overcome the aforementioned issues by allowing agents to share private information in a secure way.", "By utilizing an aggregation method, such as FedAvg [1], systems with many agents can have decreased training times with increased performance.", "Despite the popularity of FL and FRL, to the best of our knowledge at the time of this study, there are no works applying FRL to platoon control.", "In general, there are two types of “models” for AV decision making: vehicle-following modeling and lane-changing modeling [6].", "For the purposes of this study, the vehicle-following approach known as co-operative adaptive cruise control (CACC) is explored.", "Vehicle following models are based on following a vehicle on a single lane road with respect to a leading vehicle's actions [7].", "CACC is a multi-vehicle control strategy where vehicles follow one another in a line known as a platoon, while simultaneously transmitting vehicle data amongst each other [8].", "CACC platoons have been proven to improve traffic flow stability, throughput and safety for occupants [8], [9].", "Traditionally controlled vehicle following models have limited accuracy, poor generalization from a lack of data, and a lack of adaptive updating [7].", "We are motivated by the current state-of-the-art for CACC AV Platoons, along with previous works related to FRL, to apply FRL to the AV platooning problem and observe the performance benefits it may have on the system.", "We propose an FRL framework built atop a custom AV platooning environment in order to analyse FRL's suitability for improving AV platoon performance.", "In addition, two approaches are proposed for applying FRL amongst AV platoons.", "The first proposed method is inter-platoon FRL (Inter-FRL), where FRL is applied to AVs across different platoons.", "The second proposed method is intra-platoon FRL (Intra-FRL), where FRL is applied to AVs within the same platoon.", "We investigate the possibility of Inter-FRL and Intra-FRL as a means to increase performance using two aggregation methods: averaging model weights and averaging gradients.", "Furthermore, the performance of Inter-FRL and Intra-FRL using both aggregation methods is studied relative to platooning environments trained without FRL (no-FRL).", "Finally, we compare the performance of Intra-FRL with weight averaging (Intra-FRLWA) against a platooning environment trained without FRL for platoons of length 3, 4 and 5 vehicles.", "In this subsection, the current state-of-the-art is presented for FRL and DRL applied to AV's.", "In addition the contributions of this paper are presented.", "There are two main areas of research in FRL currently: horizontal federated reinforcement learning (HFRL), and vertical federated reinforcement learning (VFRL).", "HFRL has been selected as the algorithm of choice for the purposes of this study.", "HFRL and VFRL differ with respect to the structure of their environments and aggregation methods.", "All agents in an HFRL architecture use isolated environments.", "It follows that each agent's action in an HFRL system has no effect on the other agents in the system.", "An HFRL architecture proposes the following training cycle for each agent: first, a training step is performed locally, second, environment specific parameters are uploaded to the aggregation server, and lastly, parameters are aggregated according to the aggregation method and returned to each agent in the system for another local training step.", "HFRL may be noted to have similarities to “Parallel RL”.", "Parallel RL is a long studied field of RL, where agent gradients are transferred amongst each other [5], [10], [11].", "Reinforcement learning is often a sequential learning process, and as such data is often non-IID with a small sample space [12].", "HFRL provides the ability to aggregate experience while increasing the sample efficiency, thus providing more accurate and stable learning [13].", "Some of the current works applying HFRL to a variety of applications are summarized below.", "A study by Lim et al.", "aims to increase the performance of RL methods applied to multi-IoT device systems.", "RL models trained on single devices are often unable to control devices in a similar albeit slightly different environment [5].", "Currently, multiple devices need to be trained separately using separate RL agents [5].", "The methods proposed by Lim et al.", "sped up the learning process by 1.5 times for a two agent system.", "In a study by Nadiger et al., the challenges in the personalization of dialogue managers, smart assistants and more are explored.", "RL has proven to be successful in practice for personalized experiences; however, long learning times and no sharing of data limit the ability for RL to be applied at scale.", "Applying HFRL to atari non-playable characters in pong showed a median improvement of  17% for the personalization time [10].", "Lastly, Liu et al.", "discuss RL as a promising algorithm for smart navigation systems, with the following challenges: long training times, poor generalization across environments, and storing data over long periods of time [14].", "In order to address these problems, Liu et al.", "proposed the architecture `Lifelong FRL', which can be categorized as an HFRL problem.", "It is found the Lifelong FRL increased the learning rate for smart navigation system when tested on robots in a cloud robotic system [14].", "The successes of the FedAvg algorithm as a means to improve performance and training times for systems have inspired further research into how aggregation methods should be applied.", "The design of the aggregation method is crucial in providing performance benefits to that of the base case where FRL is not applied.", "The FedAvg [3] algorithm proposed the averaging of gradients in the aggregation method.", "In contrast, Liang et al.", "proposed using model weights in the aggregation method for AV steering control [15].", "Thus, FRL applications can differ based upon the selection of which parameter to use in the aggregation method.", "A study by Zhang et al.", "explores applying FRL to a decentralized DRL system optimizing cellular vehicle-to-everything communication[16].", "Zhang et al.", "utilize model weights in the aggregation method, and describe a weighting factor dividing the sum batch size for all agents by the training batch size for a specific agent[16].", "In addition, the works of Lim et al.", "explore how FRL using gradient aggregation can improve convergence speed and performance on the OpenAI-gym environments CartPole-V0, MountainvehicleContinuous-V0, Pendulum-V0 and Acrobot-V1 [17].", "Lim et al.", "determined that aggregating gradients using FRL creates high performing agents for each of the OpenAI-gym environments relative to models trained without FRL[17].", "In addition, Wang et al.", "apply FRL to heterogeneous edge caching [18].", "Wang et al.", "show the effectiveness of FRL using weight aggregation to improve hit rate, reduce average delays in the network and offload traffic[18].", "Lastly, Huang et al.", "apply FRL using model weight aggregation to Service Function Chains in network function virtualization enabled networks[19].", "Huang et al.", "observe that FRL using model weight aggregation provides benefits to convergence speed, average reward and average resource consumption[19].", "Despite the differences in FRL applications within the aforementioned studies, each study maintains a similar goal: to improve the performance of each agent within the system.", "None of the aforementioned works explore the differences in whether gradient or model weight aggregation is favourable in performance, and many of the works apply FRL to distributed network or communications environments.", "It is the goal of this study to conclude whether model weight or gradient aggregation is favourable for AV platooning, as well as be one of the first (if not the first) to apply FRL to AV platooning." ], [ "Deep reinforcement learning applied to AV platooning", "In recent years, there has been a surge in autonomous vehicle (AV) research, likely due to the technologies potential for increasing road safety, traffic throughput and fuel economy [20], [6].", "Two areas of research are often considered when delving into an AV model: supervised learning or RL [20].", "Driving is considered a multi-agent interaction problem, and due to the large variability of road data, it can be quite challenging (or near impossible) to gather a data set variable enough to train a supervised model [21].", "Driving data is collected from humans, which can also limit an AI's ability to that of human level [6].", "In contrast, RL methods are known to generalize quite well [20].", "RL approaches are model-free and a model may be inferred by the algorithm while training.", "In order to improve the limitations of vehicle following models, DRL has been a steady area of research in the AV community, with many authors contributing works to DRL applied to CACC [22], [8], [9], [23].", "In a study by Lin et al., a DRL framework is designed to control a CACC AV platoon[22].", "The DRL framework uses the deep deterministic policy gradient (DDPG) [24] algorithm and is found to have near-optimal performance [22].", "In addition, Peake et al.", "identify limitations in platooning with regard to the communication in platooning[23].", "Through the application of a multi-agent reinforcement learning process, i.e.", "a policy gradient RL and LSTM network, the performance of a platoon containing 3-5 vehicles is improved upon that of current RL applications to platooning [23].", "Furthermore, Model Predictive Control (MPC) is the current state-of-the-art for real-time optimal control practices [25].", "The study performed by Lin et al.", "applies both MPC and DRL methodologies to the AV platoon problem, observing a DRL model trained using the DDPG algorithm produces merely a 5.8% episodic cost higher than the current state-of-the-art[25].", "The works of Yan et al.", "propose a hybrid approach to the AV platooning problem where the platoon is modeled as a Markov Decision Process (MDP) in order to collect two rewards from the system at each time step simultaneously[26].", "This approach also incorporates jerk, the rate of change of acceleration in the calculation of the reward for each vehicle in order to ensure passenger comfort [26].", "The hybrid strategy led to increased performance to that of the base DDPG algorithm, as the proposed framework switches between using classic CACC modeling and DDPG depending on the performance degradation of the DDPG algorithm [26].", "In another study by Zhu et al., a DRL model is formulated and trained using DDPG to be evaluated against real world driving data.", "Parameters such as time to collision, headway, and jerk were considered in the DRL model's reward function[27].", "The DDPG algorithm provided favourable performance to that of the analysed human driving data, with regard to more efficient driving via reduced vehicle headways, and improved passenger comfort with lower magnitudes of jerk[27].", "As Vehicle-to-Everything (V2X) communications are envisioned to have a beneficial impact on the performance of platoon controllers, the works of Lei et al.", "investigates the value of V2X communications for DRL-based platoon controllers.", "Lei et al.", "emphasizes the trade-off between the gain of including exogenous information in the system state for reducing uncertainty and the performance erosion due to the curse-of-dimensionality.", "When formulating the AV platooning problem as a DRL model DDPG is prominently selected as the algorithm for training.", "DDPG's ability to handle continuous actions space and complex state's is perfect for the CACC platoon problem.", "However, despite the DDPG algorithm's success in literature, there are still instability challenges related to the algorithm along with a time consuming hyper-parameter tuning process to account for the minute differences in vehicle models/dynamics amongst platoons.", "As previously discussed, FRL provides advantages in these areas where information sharing can accelerate performance during training and improve the performance of the system as a whole.", "In addition, the ability to share experience across like models has been proven to allow for fast convergence of models, which further optimizes the performance of DDPG when applied to AV platoons [5].", "To the best of our knowledge, no works at the time of this study existed covering the specific topic of FRL applied to platoon control.", "Many of the works existing on FRL have shown the benefits of FRL with regard to the increased rate of convergence and overall system performance with distributed networks, edge caching and communications [16], [17], [18], [19].", "Furthermore, of the works cited in this study, the works closely related to FRL for platoon control are those of Peake et al.", "and Liang et al.", "[23], [15].", "In contrast to Liang et al., where FedAvg is applied successfully to control the steering angle of a single vehicle, we apply FRL to an AV platooning problem where the control of multiple vehicles' positions and spacing are required [15].", "Peake et al.", "explore multi-agent reinforcement learning and its ability to improve the performance of AV platoons experiencing communication delays[23].", "Although Peake et al.", "are also successful in their approach, there is no specific reference to FRL in the paper[23].", "In addition, a variety of existing works on FRL choose to use either gradients or model weights in the FRL aggregation method.", "This study explores how both aggregation methods can provide benefits to the AV platooning problem and, most importantly, which provides a better result.", "Finally, this study further distinguishes its approach from existing literature by declaring two possible ways to apply FRL to AV platooning: Intra-FRL: where multi-vehicle platoons share data during training to increase the performance of vehicles within the same platoon.", "Inter-FRL: where multi-vehicle platoons share data during training across platoons amongst vehicles in the exact same platoon position to increase performance.", "In contrast to existing literature, where it is common to average the parameters across each model in the system, for Intra-FRL, we propose a directional averaging where follower vehicles incorporate the preceding vehicle parameters in the computation of the gradients or weights.", "Thus, in Intra-FRL, the leading vehicle trains independently of those following.", "The AV platoon provides a unique playground environment suitable for exploring the suitability of FRL as a means to increase the performance of systems with regard to convergence rate and performance.", "In this section, a state space model is formulated and presented for the AV platooning problem.", "Next, the MDP model is presented, outlining the platoon system's state space, action space and reward function.", "Lastly, the FRL DDPG algorithm design and application to AV platooning are described." ], [ "CACC CTHP model formulation", "Consider a platoon $P$ of vehicles $\\mathcal {V}={V_1,V_2,...,V_n}$ where the leader of the platoon is $V_1$ .", "Figure: An example platoon modeled with system parameters.As illustrated in Figure REF , for a general vehicle ($V_i$ ), the position of $V_i$ 's front bumper is defined as $p_i$ .", "The velocity, acceleration and control input of $V_i$ are denoted as $v_i$ , $a_i$ and $u_i$ .", "Furthermore, the acceleration of $V_i$ 's predecessor may be denoted as $a_{i-1}$ .", "The control input for $V_i$ is defined as $u_i$ (whether $V_i$ should accelerate or decelerate).", "$V_i$ 's drive-train dynamics coefficient is defined as $\\tau _i$ , where large values of $\\tau _i$ indicate larger response times for a given input $u_i$ to generate acceleration $a_i$ .", "Lastly, the length of $V_i$ is denoted as $L_i$ .", "The system dynamics for $V_i$ are thus provided below as $\\begin{aligned}\\dot{p_i(t)} &= v_i(t) \\\\\\dot{v_i(t)} &= a_i(t) \\\\\\dot{a_i(t)} &= -\\frac{1}{\\tau _i}a_i(t) + \\frac{1}{\\tau _i}u_i(t) \\\\\\dot{a_{i-1}(t)} &= -\\frac{1}{\\tau _{i-1}}a_{i-1}(t) + \\frac{1}{\\tau _{i-1}}u_{i-1}(t)\\end{aligned}$ The headway $d_i(t)$ in a CACC model is the positional difference of the current vehicle relative to the rear bumper of its leader, which can be derived as [22], [28] $d_i(t) = p_{i-1}(t) - p_i(t) - L_{i-1}.$ In addition, the desired headway $d_{r,i}(t)$ is defined as $d_{r,i}(t) = r_i + h_iv_i(t), $ where $r_i$ is the standstill distance, and $h_i$ is the time-gap for $V_i$ to maintain relative to it's predecessor $V_{i-1}$ .", "The position error $e_{pi}$ and the velocity error $e_{vi}$ are defined as: $\\begin{aligned}e_{pi}(t) &= d_i(t) - d_{r,i}(t) \\\\e_{vi}(t) &= v_{i-1}(t) - v_i(t) \\\\\\end{aligned}$ Therefore, the state of $V_i$ can be defined as $x_i(t) = \\begin{bmatrix}e_{pi}(t) & e_{vi}(t) & a_i(t) & a_{i-1}(t)\\end{bmatrix}^\\top $ , and the derivative of the state is: $\\begin{aligned}\\dot{e_{pi}(t)} &= e_{vi}(t) - h_ia_i(t),\\\\\\dot{e_{vi}(t)} &= a_{i-1}(t) - a_i(t), \\\\\\dot{a_i(t)} &= -\\frac{1}{\\tau _i}a_i(t) + \\frac{1}{\\tau _i}u_i(t), \\\\\\dot{a_{i-1}(t)} &= -\\frac{1}{\\tau _{i-1}}a_{i-1}(t) + \\frac{1}{\\tau _{i-1}}u_{i-1}(t).\\\\\\end{aligned}$ The state space formula for $V_i$ is thus given as $\\dot{x_i(t)} = A_ix_i(t) + B_iu_i(t) + C_iu_{i-1}(t), $ where $A_i$ , $B_i$ , and $C_i$ are defined below as $ \\begin{aligned}A_{i} &= \\begin{bmatrix}0 & 1 & -h_i & 0 \\\\0 & 0 & -1 & 1 \\\\0 & 0 & -\\frac{1}{\\tau _i} & 0 \\\\0 & 0 & 0 & -\\frac{1}{\\tau _{i-1}}\\end{bmatrix} \\qquad B_{i} = \\begin{bmatrix}0 \\\\ 0 \\\\ \\dfrac{1}{\\tau _i} \\\\ 0\\end{bmatrix} \\qquad C_{i} &= \\begin{bmatrix}0 \\\\ 0 \\\\ 0 \\\\ \\frac{1}{\\tau _{i-1}}\\end{bmatrix}.\\end{aligned}$" ], [ "MDP model formulation", "The AV platooning problem can be formulated as an MDP problem, where the optimization objective is to minimize the previously defined $e_{pi}$ , $e_{vi}$ , $u_i$ and lastly jerk." ], [ "2.2.1. State space", "The state space formula (REF ) can be discretized using the forward euler method giving the system equation below $\\begin{split} x_{i,k+1} &= A_{Di}x_{i,k} + B_{Di}u_{i,k} + C_{Di}u_{i-1, k}, \\\\\\end{split}$ where $x_{i,k}=[e_{pi,k}, e_{vi,k}, a_{i,k}, a_{i-1,k}]$ is the observation state for the MDP problem that includes the position error $e_{pi,k}$ , velocity error $e_{vi,k}$ , acceleration $a_{i,k}$ , and the acceleration of the predecessor vehicle $a_{i-1,k}$ at time step $k$ .", "Moreover, $A_{Di}$ , $B_{Di}$ , and $C_{Di}$ are given as $ \\begin{aligned}A_{Di} &= \\begin{bmatrix}1 & T & -Th_i & 0\\\\0 & 1 & -T & T\\\\0 & 0 & -\\dfrac{T}{\\tau _i} + 1 & 0 \\\\0 & 0 & 0 & -\\dfrac{T}{\\tau _{i-1}} + 1\\end{bmatrix} \\qquad B_{Di} = \\begin{bmatrix}0 \\\\ 0 \\\\ \\dfrac{T}{\\tau _i} \\\\ 0\\end{bmatrix} \\qquad C_{Di} &= \\begin{bmatrix}0 \\\\ 0 \\\\ 0 \\\\ \\frac{T}{\\tau _{i-1}}\\end{bmatrix} .\\end{aligned}$" ], [ "Action space", "Each vehicle within a single lane platoon follows the vehicle in front of it, and as such the only action the vehicle may take to maintain a desired headway is to accelerate, or decelerate.", "The action for the system is defined as the control input $u_{i,k}$ to the vehicle." ], [ "Reward function", "The design of a reward in a DDPG system is critical to providing good performance within the system.", "In the considered driving scenario, it is logical to minimize position error, velocity error, the amount of time spent accelerating and the jerkiness of the driving motion.", "The proposed reward thus includes the normalized position error, $e_{pi,k}$ , velocity error $e_{vi,k}$ , control input $u_{i,k}$ and lastly the jerk.", "The vehicle reward $c_{i,k}$ is given below, where $a$ $b$ , $c$ and $d$ are system hyperparameters.", "$ \\small \\begin{aligned}c_{i,k} &= -\\left(a\\frac{|e_{pi,k}|}{\\max (e_{pi,k})} + b\\frac{|e_{vi,k}|}{\\max (e_{vi,k})} + c\\frac{|u_{i,k}|}{\\max (u_{i,k})} + d\\frac{\\dot{|a_{i,k}|}}{2\\max (a_{i,k})}\\right) \\\\\\end{aligned}$ In this section, the design for implementing the FRL DDPG algorithm on the AV platooning problem is presented." ], [ "DDPG model description", "The DDPG algorithm is composed of an actor, $\\mu $ and a critic, $Q$ .", "The actor produces actions $u_t \\in \\mathbf {U}$ given some observation $x_t \\in \\mathbf {X}$ and the critic makes judgements on those actions while training using the Bellman equation [24], [12].", "The actor is updated by the policy gradient [24].", "The critic network uses its weights $\\theta ^q$ to approximate the optimal action-value function $Q(x, u|\\theta ^q)$ [24].", "The actor network uses weights $\\theta ^\\mu $ to represent the agents' current policy $\\mu (x|\\theta ^\\mu )$ for the action-value function [24].", "The actor $\\mu (x): \\mathbf {X} \\xrightarrow{} \\mathbf {U}$ maps the observation to the action.", "Experience replay is used to mitigate the issue of training samples not being independent and identically distributed due to their generation from sequential explorations [24].", "Two additional models, the target actor $\\mu ^{\\prime }$ and critic $Q^{\\prime }$ are used in DDPG to stabilize the training of the actor and critic networks by updating parameters slowly based on the target update coefficient $\\tau $ .", "A sufficient value of $\\tau $ is chosen such that stable training of $\\mu $ and $Q$ is observed.", "Figure REF provides a high level simplified overview of how the DDPG algorithm interacts with a single vehicle in a platoon.", "Figure: High level flow diagram of the DDPG model for a general vehicle V i V_i in a platoon." ], [ "Inter and intra FRL", "Modifications to the base DDPG algorithm are needed in order to implement Inter-FRL and Intra-FRL.", "In order to implement FedAvg the following modifications are required: An FRL server: responsible for averaging the system parameters for use in a global update Model weight aggregation: storing of each model's weights for use in aggregation Model gradient aggregation: storing of each model's gradients for use in aggregation In order to perform FRL, it has been proven that including an update delay between global FRL updates is beneficial for performance [5].", "In addition, turning off FRL partway through training is important to allow each agent to refine their models independently of each other such that they can perform best with respect to their environments [5].", "Lastly, it has also been shown that global updates and local updates should not be performed in the same episode [15].", "Two methods of aggregation are implemented in the system design, Inter-FRL (see Figure REF ), and Intra-FRL (see Figure REF ).", "The proposed system is capable of aggregating both the model weights and gradients for each model so that either type of parameter may be averaged for use in global updates.", "The FRL server has the responsibility of averaging the parameters (model weights or gradients) across each agent in the system.", "Figure: Inter-FRL.Figure: Intra-FRL.The pseudo-code for the Inter/Intra-FRL algorithm is presented in Algorithm 1.", "The system is designed to allow the training of any number of equal length platoons.", "At the lowest level, a DDPG agent exists for each vehicle in each platoon.", "As such, a DRL model must be initialized for each vehicle in the whole system.", "Each DDPG agent trains separately from the others before data is uploaded to the FRL server.", "Federated averaging is applied at a given time delay known as the FRL update delay, while being terminated at a given episode as defined by the cutoff ratio as seen in Table REF .", "Currently, Algorithm 1 is synchronous, and the FRL server is also synchronous.", "1em each platoon $p \\in platoons$ $v\\in vehicles$ initialize replay buffer $R_{i}$ initialize actor $\\mu _{i}$ , critic $Q_{i}$ , target actor $\\mu ^{\\prime }_{i}$ , target critic $Q^{\\prime }_{i}$ $episode \\in training\\_episodes$ $p \\in platoons$ collect all vehicles states $x_{i,k}$ from $p$ $step \\in steps\\_per\\_episode$ $p\\in platoons$ $v\\in vehicles$ collect actions $u_{i,k}$ from actor advance the platoon $p$ , with $u_{i,k}$ collect $(x_{i,k}, x_{i,k+1}, c_{i,k}, terminal)$ from $p$ $p\\in platoons$ $v\\in vehicles$ add $(x_{i,k}, x_{i,k+1}, c_{i,k}, terminal)$ to replay buffer $R_{i}$ FRL update is not required train $\\mu _{i}$ , $Q_{i}$ , $\\mu ^{\\prime }_{i}$ , $Q^{\\prime }_{i}$ locally append gradients of $\\mu _{i}$ and $Q_{i}$ to all_gradients append weights of $\\mu _{i}$ and $Q_{i}$ to all_weights FRL update required gradient averaging enabled avg_gradients $\\xleftarrow{}$ global_update(all_gradients) train $\\mu _{i}$ , $Q_{i}$ using avg gradients weight averaging enabled avg_weights $\\xleftarrow{}$ global_update(all_weights) update weights $\\mu _{i}$ , $Q_{i}$ , $\\mu ^{\\prime }_{i}$ , $Q^{\\prime }_{i}$ using avg weights FnFunction isend global_update(params) upload params to FRL server collect averaged params from FRL server return averaged params FRL applied to an AV platoon.", "1em" ], [ "Experimental Results", "In this section, the experimental setup for applying both Inter and Intra-FRL to the AV platooning environment is presented.", "The AV platooning environment and Inter/Intra FRL algorithms are implemented in Python 3.7 using Tensorflow 2." ], [ "Experimental setup", "The parameters specific to the AV platoon environment are summarized in Table REF .", "The time step interval is $T=0.1 s$ , and each training episode is composed of 600 time steps.", "Furthermore, the coefficients $a$ , $b$ , $c$ and $d$ given in the reward function (REF ) are a means to define how much each component of (REF ) contributes to the calculation of the reward.", "These coefficients may be tuned in order to determine a balance amongst each component, leading to better optimization during training.", "The coefficients were tuned using a grid search strategy and are listed as $a=0.4, b = c = d = 0.2$ .", "Table: Parameters of the AV platoon environment.Each DDPG agent consists of a replay buffer, and networks for the actor, target actor, critic and target critic.", "The actor network contains four layers: an input layer for the state, two hidden layers with 256 and 128 nodes, respectively, and an output layer.", "Both hidden layers use batch normalization and the relu activation function.", "The output layer uses the tanh() activation function.", "The output layer is scaled by the high bound for the control output, in this case 2.5 $m/s^2$ .", "The critic network is structured with two separate input layers for state and action.", "These two layers are concatenated together, and fed into a single hidden layer before the output layer.", "The layer with the state input has 48 nodes, the relu activation function and batch normalization.", "The same is applied for the action layer, but instead with 256 nodes.", "The post concatenation layer uses 304 input nodes, followed by a hidden layer with 128 nodes, again with relu activation and batch normalization applied.", "The output of the critic uses a linear activation function.", "Ornstein-Uhlenbeck noise is applied to the model's predicted action, $u_i$ .", "The structure of the models is presented in Figure REF and REF .", "All except the final layers of the actor and critic networks were initialized within the range $\\left[-\\dfrac{1}{\\sqrt{fan\\:in}},\\:\\dfrac{1}{\\sqrt{fan\\:in}}\\right]$, where-as the final layer is initialized using a random uniform distribution bounded by $[-3\\times 10^{-3},\\:3\\times 10^{-3}]$ .", "Table REF presents the hyperparameters specific to the DDPG algorithm.", "Figure: Actor and critic networks for V i V_i.Table: Hyperparameters for the DDPG algorithm.The hyperparameters specific to Inter and Intra-FRL are presented in Table REF .", "During a training session with FRL, both local updates and FRL updates with aggregated parameters are applied to each DDPG agent in the system.", "FRL updates usually occur at a given frequency known as the FRL update delay, and furthermore, FRL updates may be terminated at a specific training episode as defined by the FRL cutoff ratio.", "The FRL update delay is defined as the time in seconds between FRL updates during a training episode.", "The FRL cutoff ratio is the ratio of the number of episodes where FRL updates are applied divided by the total number of episodes in a training session.", "Note that the aggregation method denotes whether the model gradients or weights are averaged during training using FRL.", "Table: FRL specific initial hyperparameters.For the purposes of this study, an experiment is defined as a training session for a specific configuration of hyper-parameters, using the algorithm defined in Algorithm 1.", "During each experiment training session, model parameters were trained through the base DDPG algorithm or FRL in accordance with Algorithm 1.", "Once training has concluded, a simulation is performed using a custom built evaluator API.", "The evaluator performs simulations for a single 60 second episode using the trained models, calculating the cumulative reward of the model(s) in the experiment.", "The entire project is designed and implemented using Python3, and Tensorflow.", "As previously stated, each vehicle in the platoon is modelled using the CACC CTHP model described in Section 3.", "For the purposes of this study, multiple sets of DRL experiments were conducted, using 4 random seeds (1-4) for training and a single random seed (6) across all evaluations." ], [ "Inter-FRL", "In order to evaluate the effectiveness of Inter-FRL relative to the base case where a DRL model is trained using DDPG without FRL, 4 experiments are conducted without Inter-FRL (no-FRL), and 8 with.", "For each of the 12 conducted experiments, 2 platoons with 2 vehicles each were trained using one of the four random seeds.", "Once training across the four seeds has completed, the cumulative reward for a single evaluation episode is evaluated.", "For the experiments using Inter-FRL, two aggregation methods are examined.", "First, the gradients of each model are averaged during training, and second, the model weights are averaged.", "The multi platoon system trains and shares the aggregated parameters (gradients or weights) amongst vehicles with the same index across platoons.", "The federated server is responsible for performing the averaging, and each vehicle performs a training episode with the averaged parameters in addition to their local training episodes in accordance with the FRL update delay and FRL cutoff ratio (see Table REF ).", "Note that here-after Inter-FRL with gradient aggregation is denoted Inter-FRLGA, and Inter-FRL with weight aggregation is denoted Inter-FRLWA." ], [ "Performance across 4 random seeds", "The performance for each of the systems is calculated by averaging the cumulative reward of each vehicle in the 2 vehicle 2 platoon system, as summarized in Table REF .", "For each of the 3 cases (base case, Inter-FRLGA and Inter-FRLWA), training sessions were run using 4 random seeds.", "In order to determine the highest performing system overall, an average and standard deviation is obtained from the result of training using the 4 random seeds.", "From Table REF , it is observed that both Inter-FRL scenarios using gradient and weight aggregation provide large performance increases to that of the base case.", "Table: Performance after training across 4 random seeds.", "Each simulation result contains 600 time steps." ], [ "Convergence properties", "The cumulative reward is calculated over each training episode, and a moving average is computed over 40 episodes to generate Figure REF -REF .", "It can be seen that the cumulative reward for Inter-FRLWA not only converges more rapidly than both no-FRL and Inter-FRLGA, but Inter-FRLWA also appears to have a more stable training session as indicated by the lower magnitude of the shaded area (the standard deviation across the four random seeds).", "Figure: Average performance across 4 random seeds for a 2 platoon 2 vehicle scenario trained without FRL (Figure , ), with Inter-FRLGA (Figure , ), and with Inter-FRLWA (Figure , ).", "The shaded areas represent the standard deviation across the 4 seeds." ], [ "Test results for one episode", "In Figure REF and REF , a simulation is performed over a single training episode plotting the jerk, along with the control input $u_{i,k}$ , acceleration $a_{i,k}$ , velocity error $e_{vi,k}$ , and position error $e_{pi,k}$ for each platoon.", "There are 2 platoons in the Inter-FRL scenario, and a simulation is provided for each platoon.", "The simulation environment is subject to initial conditions of ($e_{pi} = 1.0\\:m$ , $e_{vi}=1.0\\:m/s$ , $a_i = 0.03\\:m/s^2$ ).", "It can be seen that each DDPG agent for both vehicles within both platoons quickly responds to the platoon leader's control input $u_{i,k}$ to bring the position error, velocity error and acceleration error to 0.", "In addition, each DDPG agent closely approximates the Gaussian random input of the platoon leader, eliminating noise in the response to maintain smooth tracking across the episode.", "Finally, each DDPG agent in the platoon also minimizes the jerk effectively.", "These results are indicative of both a good design of the reward function (REF ), and also a suitable selection of parameters $a, b, c$ and $d$ in (REF ).", "Figure: Results for a specific 60s test episode using the 2 vehicle 2 platoon environment trained using Inter-FRL with weight aggregation.In order to evaluate the effectiveness of Intra-FRL relative to the base AV platooning scenario, 4 experiments are conducted without Intra-FRL (no-FRL), and 8 with.", "For each of the conducted experiments, 1 platoon with 2 vehicles is trained using 4 random seeds.", "A single platoon is required for studying Intra-FRL as parameters are shared amongst vehicles within the platoon (no sharing is performed from vehicle's in one platoon to another).", "Once training across the four seeds is completed, the cumulative reward for a single evaluation episode is evaluated.", "Similar to the experiments using Inter-FRL, two aggregation methods are examined.", "First, the gradients of each model are averaged during training, and second, the model weights are averaged.", "The platoon trains and shares the aggregated parameters (gradients or weights) from vehicle to vehicle such that data is averaged and updated amongst vehicles within the same platoon.", "The federated server is responsible for performing the averaging, and each vehicle performs a training episode with the averaged parameters in addition to their local training episodes in accordance with the FRL update delay and FRL cutoff ratio (see Table REF ).", "Note that here-after Intra-FRL with gradient aggregation is denoted Intra-FRLGA, and Intra-FRL with weight aggregation is denoted Intra-FRLWA." ], [ "Performance across 4 random seeds", "The performance for the platoon is calculated by averaging the cumulative reward generated by the simulation for each of the 4 random seeds and is summarized in Table REF .", "The results in Table REF summarize the performance for no-FRL, Intra-FRLGA, and lastly Intra-FRLWA.", "It is observed that Intra-FRLWA performs most favourably, followed by no-FRL and lastly Intra-FRLGA.", "Table: Performance after training across 4 random seeds.", "Each simulation result contains 600 time steps." ], [ "Convergence properties", "The cumulative reward is calculated over each training episode, and a moving average is computed over 40 episodes to generate Figure REF .", "Similar to the Inter-FRL experiments, Intra-FRLWA shows the most favourable training results.", "In addition, the rate of convergence increases with Intra-FRLWA over no-FRL and Intra-FRLGA.", "Lastly, the stability during training is also shown to be improved as the standard deviation across the four random seeds is much smaller than the other two cases (as evident in the shaded regions of Figure REF ).", "Figure: Average performance across 4 random seeds for 1 platoon 2 vehicle scenario trained without FRL (Figure ), Intra-FRLGA (Figure ), and with Intra-FRLWA (Figure ).", "The shaded areas represent the standard deviation across the 4 seeds." ], [ "Test results for one episode", "A single simulation is performed on an episode plotting the jerk, along with the control input $u_{i,k}$ , acceleration $a_{i,k}$ , velocity error $e_{vi,k}$ , and position error $e_{pi,k}$ .", "Figure REF shows the precise control of Intra-FRLWA on the environment.", "The environment is initialized to the same conditions as that of the Inter-FRLWA scenario ($e_{pi} = 1.0 m$ , $e_{vi}=1.0 m/s$ , $a_i = 0.03 m/s^2$ ), and each DDPG agent in the platoon quickly and precisely tracks the Gaussian random input $u_{i,k}$ from the leader while minimizing position error, velocity error, acceleration , and jerk.", "Much like the Inter-FRLWA scenario, it is observed that a strong optimization of the reward function (Equation 10) has occurred.", "This is an indication of a good design of the reward function in addition to a good balance of parameters $a,b,c$ and $d$ in the reward function.", "Figure: Results for a specific 60s test episode using the 2 vehicle 1 platoon environment trained using Intra-FRLWA." ], [ "Comparison between inter and intra-FRL", "The results for both Inter-FRL and Intra-FRL are summarized in Table REF below.", "Table: Performance after training across 4 random seeds for both Inter and Intra FRL.", "Each simulation result contains 600 time steps.It is clear that using weight aggregation in both Inter-FRL and Intra-FRL is favourable to gradient aggregation.", "In addition, Intra-FRLWA provides the overall best result.", "Intra-FRL likely converges to the best model due to conditions each agent experiences during training.", "For Inter-FRL, the environment is independent and identically distributed.", "For Intra-FRL, each follower's training depends on the policy of the preceding vehicle.", "For the 2 vehicle scenario studied, vehicle 1 will converge prior to vehicle 2 as vehicle 1 learns based on the stochastic random input generated by the platoon leader.", "As vehicle 1 is training, vehicle 2 trains based off the policy of vehicle 1.", "As previously stated, Inter-FRL shares parameters amongst vehicles in the same index across platoons, where-as Intra-FRL provides the advantage of sharing parameters from preceding vehicles to following vehicles.", "Our implementation of Intra-FRL includes a directional parameter averaging.", "For example, vehicle 1 does not train with averaged parameters from the followers, but vehicle 2 has the advantage of including vehicle 1's model in its averaging.", "This directional averaging provides an advantage to vehicle 2, as evidenced by the increased performance in Table REF ." ], [ "Intra-FRL with variant number of vehicles", "An additional factor to consider when evaluating FRL in relation to the no-FRL base scenario is how FRL performs with increasing agents relative to no-FRL.", "In this section, 12 experiments are conducted with no-FRL, and 12 with Intra-FRLWA.", "Each set of 12 experiments for no-FRL and Intra-FRLWA are broken up by number of vehicles and random seed.", "The random seed is selected to be a value between 1 and 4, inclusive.", "In addition, the platoons under study contain either 3, 4, or 5 vehicles.", "Once training has been completed for all experiments, the cumulative reward for each experiment is evaluated using a single simulation episode in which the seed is kept constant.", "Intra-FRLWA is used as the FRL training strategy since Intra-FRLWA was identified to be the highest performing FRL strategy in the previous section." ], [ "Performance with varying number of vehicles", "The performance for each experiment is calculated by taking the average cumulative episodic reward across each vehicle in the platoon at the end of the simulation episode.", "Table REF presents the results for no-FRL and Intra-FRLWA for platoons with 3, 4, and 5 follower vehicles.", "Table REF shows that Intra-FRLWA provides favourable performance in all platoon lengths.", "A notable example of Intra-FRLWA's success is highlighted when considering the poor performance of the 4 vehicle platoon trained with no-FRL using seed 1.", "The Intra-FRLWA training strategy was able to overcome the performance challenges, correcting the poor performance entirely.", "Table: Performance after training across 4 random seeds with varying platoon lengths.", "Each simulation result contains 600 time steps." ], [ "Convergence properties", "The cumulative reward is calculated over each training episode, and a moving average is computed over 40 episodes to generate Figure REF .", "Intra-FRLWA shows favourable training performance to that of the no-FRL scenario for all platoon lengths.", "In addition, the rate of convergence is increased using Intra-FRLWA versus no-FRL.", "Furthermore, the shaded areas corresponding to standard deviation across the seeds are reduced significantly, indicating better stability across the seeds for Intra-FRLWA than no-FRL.", "Last, the overall stability is improved as shown by the large noise reduction during training in Figure REF , REF , REF when compared with no-FRL's Figure REF , REF , REF .", "Figure: Average performance across 4 random seeds for 3 platoons with 3, 4 and 5 followers trained without FRL (Figures , , ), and with Intra-FRLWA (Figure , , ).", "The shaded areas represent the standard deviation across the four seeds." ], [ "Test results for one episode", "As with all previous sections, a single simulation is performed on a 60 second episode plotting the jerk along with the control input $u_{i,k}$ , acceleration $a_{i,k}$ , velocity error $e_{vi,k}$ , and position error $e_{pi,k}$ .", "Figure REF showcases the ability of Intra-FRLWA to control a 5 platoon environment precisely when compared to a platoon trained without Intra-FRLWA.", "The environment for Intra-FRLWA is initialized with the same values as no-FRL, just like all previous experiments: ($e_{pi} = 1.0 m$ , $e_{vi}=1.0 m/s$ , $a_i = 0.03 m/s^2$ ).", "Each DDPG agent trained with Intra-FRLWA quickly and precisely tracks the Gaussian random control input $u_{i,k}$ from the leader minimizing $e_{pi,k}$ , $e_{vi,k}$ , $a_{i,k}$ and jerk.", "In particular, the response for $e_{pi,k}$ and $e_{vi,k}$ in the platoon trained using Intra-FRLWA (Figure REF ) appears to respond to the platoon leader's input quicker and in a much smoother manner than that of the no-FRL scenario (Figure REF ).", "Figure: Results for a specific 60s test episode using the 5 vehicle 1 platoon environment trained using no-FRL (Figure ), and with Intra-FRLWA (Figure ).The large difference in performance for no-FRL versus Intra-FRL can be explained by understanding how Intra-FRLWA works.", "With no-FRL, each agent trains independently, and the inputs to the following vehicles are directly outputted from the predecessors.", "Thus, the followers farther back in the platoon take longer to train as their predecessors' outputs can be highly variable while training.", "As the policies of the predecessors converge, the policy of each follower can then begin to converge.", "This sequential convergence from predecessor to follower can be seen in Figure REF , where the convergence during training is slower for vehicles 4 and 5 than it is for 3, 2 and 1.", "Intra-FRLWA helps to resolve this challenge by allowing vehicles to average their model weights, thus distributing an aggregation of more mature predecessor parameters amongst the platoon." ], [ "Conclusion", "In this paper, we have formulated an AV platooning problem and successfully applied FRL in a variety of methods to AV platooning.", "In addition, we proposed new approaches for applying FRL to AV platoons: Inter-FRL and Intra-FRL.", "By comparing FRL performance with both gradient and weight averaging in the AV platooning scenario, it has been shown that weight averaging was the optimal aggregation method regardless of using Inter-FRL or Intra-FRL.", "Furthermore, it was found that the Intra-FRLWA strategy was most advantageous for applying FRL to AV platooning.", "Finally, it was proven that applying Intra-FRLWA to AV platoons up to 5 vehicles in length provided large performance advantages during and after training when compared to AV platoons that were controlled by DDPG agents trained without FRL.", "These results are backed by simulations performed using models trained across four random seeds, and an additional simulation set with variable platoon sizes.", "The focus of this paper was on decentralized platoon control, where each follower in the platoon trains locally with respect to their individual reward.", "In the future, improvements to the system could be made by implementing weighted averaging in the FRL aggregation method.", "Moreover, in AV platooning, communication delays can be considered in the model to give a more concrete real life example." ] ]
2207.03484
[ [ "The Complexity of Proportionality Degree in Committee Elections" ], [ "Abstract Over the last few years, researchers have put significant effort into understanding of the notion of proportional representation in committee election.", "In particular, recently they have proposed the notion of proportionality degree.", "We study the complexity of computing committees with a given proportionality degree and of testing if a given committee provides a particular one.", "This way, we complement recent studies that mostly focused on the notion of (extended) justified representation.", "We also study the problems of testing if a cohesive group of a given size exists and of counting such groups." ], [ "Introduction", "If we consider a parliamentary election where about $45\\%$ voters support party $A$ , $30\\%$ support party $B$ , and the remaining $25\\%$ support party $C$ , then there are well-understood ways of assigning seats to the parties in a proportional manner.", "However, if instead of naming the supported parties the voters can approve each candidate individually, e.g., depending on some nonpartisan agendas, then the situation becomes less clear.", "While such free-form elections are not popular in political settings, they do appear in the context of artificial intelligence.", "For example, they can be used to organize search results [28], assure fairness in social media [7], help design online Q&A systems [15], or suggest movies to watch [12].", "As a consequence, seeking formal understanding of proportionality in multiwinner elections is among the most active branches of computational social choice [18].", "In this paper we extend this line of work by analyzing the computational complexity of one of the recent measures of proportionality, the proportionality degree, introduced by [2].", "We consider the model of elections where, given a set of candidates, each of the $n$ voters specifies which candidates he or she approves, and the goal is to choose a size-$k$ subset of the candidates, called the winning committee.", "Since the committee is supposed to represent the voters proportionally, [1] proposed the following requirement: For each positive integer $\\ell $ and each group of $\\ell \\cdot {n}{k}$ voters who agree on at least $\\ell $ common candidates, the committee should contain at least $\\ell $ candidates approved by at least one member of the group.", "In other words, such a group, known as an $\\ell $ -cohesive group, deserves at least $\\ell $ candidates, but it suffices that a single member of the group approves them.", "If this condition holds, then we say that the committee provides extended justified representation (provides EJR; if we restrict our attention to $\\ell = 1$ , then we speak of providing justified representation, JR).", "The key to the success of this notion is that committees providing EJR always exist and are selected by voting rules designed to provide proportional representation, such as the proportional approval voting rule (PAV) of [29].", "Many researchers followed [1] either in defining new variants of the justified representation axioms [25], [23] or in analyzing and designing rules that would provide committees satsifying these properties [2], [26], [5].", "Others study these notions experimentally [4] or analyze the restrictions that they impose [19].", "Yet, JR and EJR are somewhat unsatisfying.", "After all, to provide them it suffices that a single member of each cohesive group approves enough committee members, irrespective of all the other voters.", "To address this issue, [2] introduced the notion of proportionality degree (PD).", "They said that the satisfaction of a voter is equal to the number of committee members that he or she approves and they considered average satisfactions of the voters in cohesive groups.", "More precisely, they said that a committee has PD $f$ , where $f$ is a function from positive integers to nonnegative reals, if for each $\\ell $ -cohesive group, the average satisfaction of the voters in this group is at least $f(\\ell )$ .", "So, establishing the PD of a rule gives quantitative understanding of its proportionality, whereas JR and EJR only give qualitative information.", "That said, [2] did show that if a committee provides EJR then it has PD at least $f(\\ell ) = {\\ell -1}{2}$ , so the two notions are related.", "Further, they showed that PAV committees have PD $f(\\ell ) = \\ell -1$ and [27] established good bounds on the PDs of numerous other proportionality-oriented rules.", "In particular, these results allow one to order the rules according to their theoretical guarantees, from those providing the strongest ones to those providing the weakest.", "We extend this line of work by studying the complexity of problems pertaining to the proportionality degree: We show that, in general, deciding if a committee with a given PD exists is both ${\\mathrm {NP}}$ -hard and ${\\mathrm {coNP}}$ -hard and we suspect it to be ${\\mathrm {NP}}^{\\mathrm {NP}}$ -complete (but we only show membership).", "Nonetheless, we find the problem to be ${\\mathrm {NP}}$ -complete for (certain) constant PD functions.", "These results contrast those for JR and EJR, for which analogous problems are in ${\\mathrm {P}}$ .", "We show that verifying if a given committee provides a given PD is ${\\mathrm {coNP}}$ complete, which is analogous to the case of EJR.", "We also provide ILP formulations that may allow one to compute PDs in practice (thus, one could use them to establish an empirical hierarchy of proportionality among multiwinner voting rules).", "We show that many of our problems are polynomial-time solvable for the candidate interval and voter interval domains of preferences [9] and are fixed-parameter tractable for the parameterizations by the number of candidates or the number of voters.", "We also study the complexity of finding and counting cohesive groups." ], [ "Preliminaries", "For an integer $t$ , we write $[t]$ to denote the set $\\lbrace 1, \\ldots , t\\rbrace $ .", "An election $E = (C,V)$ consists of a finite set $C$ of candidates and a finite collection $V$ of voters.", "Each voter $v \\in V$ is endowed with a set $A(v) \\subseteq C$ of candidates that he or she approves.", "Analogously, for each candidate $c$ we write $A(c)$ to denote the set of voters that approve $c$ ; value $|A(c)|$ is known as the approval score of $c$ .", "The elections considered in the $A(\\cdot )$ notation will always be clear from the context." ], [ "Multiwinner Voting Rules.", "A multiwinner voting rule is a function that given an election $E = (C,V)$ and a committee size $k \\le |C|$ outputs a family of winning committees, i.e., a family of size-$k$ subsets of $C$ .", "(While in practice some form of tie-breaking is necessary, theoretical studies usually disregard this issue.)", "Generally, we do not focus on specific rules, but the following three provide appropriate context for our discussions (we assume that $E = (C,V)$ is some election and we seek a committee of size $k$ ): Multiwinner Approval Voting (AV) selects size-$k$ committees whose members have highest total approval score.", "Intuitively, AV selects committees of individually excellent candidates.", "The Approval-Based Chamberlin–Courant rule (CC) selects those size-$k$ committees that maximize the number of voters who approve at least one member of the committee.", "Originally, the CC rule was introduced by [8] and its approval variant was discussed, e.g., by [24] and [3].", "CC selects committees of diverse candidates, that cover as many voters as possible.", "Proportional Approval Voting (PAV) selects those size-$k$ committees $S$ that maximize the value $\\sum _{v \\in V} {w(|S \\cap A(v)|)}$ , where for each natural number $t$ we have $w(t) = \\sum _{j=1}^t {1}{j}$ .", "PAV selects committees that, in a certain sense, represent the voters proportionally; see, e.g., the works of [6] and [20].", "The rule is due to [29].", "AV, CC, and PAV are examples of the so-called Thiele rules [29], [20], but there are also many other rules, belonging to other families.", "For more details, we point the reader to the survey of [18].", "Classifying multiwinner rules as focused on individual excellence, diversity, or proportionality is due to [11]." ], [ "(Extended) Justified Representation.", "Let $E$ be an election with $n$ voters and let $k$ be the committee size.", "For an integer $\\ell \\in [k]$ , called the cohesiveness level, we say that a group of voters forms an $\\ell $ -cohesive group if (a) the group consists of at least $\\ell \\cdot {n}{k}$ voters, and (b) there are at least $\\ell $ candidates approved by each member of the group.", "Intuitively, $\\ell $ -cohesive groups are large enough to demand representation by at least $\\ell $ candidates (as they include a large-enough proportion of the voters) and they can name these $\\ell $ candidates (as there are at least $\\ell $ common candidates that they approve).", "Thus many proportionality axioms focus on satisfying such demands.", "In particular, we are interested in the notions of (extended) justified representation, due to [1].", "Definition 1 Let $E = (C,V)$ be an election, let $k$ be a committee size, and let $S$ be some committee: We say that $S$ provides justified representation (JR) if each 1-cohesive group contains at least one voter who approves at least one member of $S$ .", "We say that $S$ provides extended justified representation (EJR) if for each $\\ell \\in [k]$ , each $\\ell $ -coheseive group contains at least one voter that approves $\\ell $ members of $S$ .", "Researchers also consider other proportionality axioms, such as the notion of proportional justified reperesentation (PJR), due to [25], and the recently introduced axiom of fully justified representation (FJR), due to [23].", "JR is the weakest of these (in the sense that if a committee satisfies any of the other ones then it also provides JR), followed by PJR, EJR, and FJR.", "We focus on JR and EJR as they will suffice for our purposes.", "For every election and every committee size there always exists at least one committee providing EJR (thus, also JR).", "Indeed, all CC committees provide JR and all PAV committees also provide EJR [1], but AV committees may fail to provide (E)JR." ], [ "Proportionality Degree.", "Our main focus is on the notion of a proportionality degree of a committee, introduced by [2].", "Let us consider some voter $v$ and a committee $S$ .", "We define $v$ 's satisfaction with $S$ as $|A(v) \\cap S|$ , i.e., the number of committee members that $v$ approves.", "Definition 2 Let $E$ be an election, let $S$ be a committee of size $k$ , and let $f \\colon [k] \\rightarrow \\mathbb {R}$ be a function.", "We say that $S$ has proportionality degree $f$ if for each $\\ell $ -cohesive group of voters $X$ (where $\\ell \\in [k]$ ) the average satisfaction of the voters in $X$ is at least $f(\\ell )$ .", "In other words, if a committee has a certain proportionality degree $f$ for a given election, then members of the cohesive groups in this election are guaranteed at least a certain average level of satisfaction.", "We are interested in several special types of proportionality degree (PD) functions: We say that $f$ is a nonzero PD if $f(\\ell ) > 0$ for all $\\ell $ .", "We say that $f$ is a unit PD if $f(\\ell ) = 1$ for all $\\ell $ .", "We say that $f$ is nearly perfect PD if $f(\\ell ) = \\ell -1$ for all $\\ell $ .", "We say that $f$ is a perfect PD if $f(\\ell ) = \\ell $ for all $\\ell $ .", "One can verify that every CC committee (or, in fact, every JR committee) has nonzero PD, and [2] have shown that every PAV committee has nearly perfect PD.", "It is also known that if a committee provides EJR then, at least, it has proportionality degree $f$ such that $f(\\ell ) = {\\ell -1}{2}$ [25].", "Yet there exist elections for which no committee has unit PD or perfect PD [2].", "For a detailed analysis of proportionality degrees of various multiwinner rules, we point to the work of [27].", "Finally, we note that by saying that a certain rule has PD $f$ we only indicate a lower bound on its performance.", "Consequently, for many rules we can say that they provide several different proportionality degrees, as in the case of PAV, which provides both nearly perfect PD and some nonzero PD (nonzero PD offers a stronger guarantee than the nearly perfect one for $\\ell = 1$ )." ], [ "Computational Complexity.", "We assume knowledge of classic and parameterized computational complexity theory, including classes ${\\mathrm {P}}$ and ${\\mathrm {NP}}$ , the notions of hardness and completeness for a given complexity class, and FPT algorithms.", "Occasionally, we also refer to the ${\\mathrm {coNP}}$ class and to higher levels of the Polynomial Hierarchy.", "Given a problem $X$ from ${\\mathrm {NP}}$ , where we ask if a certain mathematical object exists, we write $\\#X$ to denote its counting variant, where we ask for the number of such objects.", "Counting problems belong to the class ${\\mathrm {\\#P}}$ and it is commonly believed that if a counting problem is ${\\mathrm {\\#P}}$ -complete then it cannot be solved in polynomial time.", "We mention that ${\\mathrm {\\#P}}$ -completeness is defined via Turing reductions [30] and not many-one reductions, as in the case of ${\\mathrm {NP}}$ -completeness.", "A counting problem $\\#X$ Turing-reduces to a counting problem $\\#Y$ , denoted $\\#X \\le _{fp}^{T} \\#Y$ , if there is an algorithm that solves $\\#X$ in polynomial time, provided it has access to $\\#Y$ as an oracle (i.e., provided that it has a subroutine for solving $\\#Y$ in constant time)." ], [ "Computational Aspects of JR, EJR, and PD.", "There are polynomial-time algorithms that given an election and a committee size compute committees which provide JR [1] or EJR [2].", "There is also a polynomial-time algorithm that given a committee verifies if it provides JR.", "The same task for EJR is ${\\mathrm {coNP}}$ -complete [1].", "In this paper we answer analogous questions for the case of the proportionality degree." ], [ "Finding and Counting Cohesive Groups", "As cohesive groups lay at the heart of JR, EJR, and PD, we start our discussion by analyzing the hardness of finding them.", "More precisely, we consider the following problem.", "Definition 3 An instance of the $\\textsc {Cohesive-Group}$ problem consists of an election $E$ , a committee size $k$ , and a positive integer $\\ell $ .", "We ask if $E$ contains an $\\ell $ -cohesive group.", "Somewhat disappointingly, this problem is ${\\mathrm {NP}}$ -complete.", "This follows via a reduction inspired by that provided by [1] to show that testing if a given committee provides EJR is ${\\mathrm {coNP}}$ -complete (we include the proof for the sake of completeness, as some of our further hardness proofs follow by reductions from Cohesive-Group).", "Theorem 3.1 Cohesive-Group is ${\\mathrm {NP}}$ -complete We observe that Cohesive-Group is in ${\\mathrm {NP}}$ : Given an election $E$ with $n$ voters, committee size $k$ , and cohesiveness level $\\ell $ , it suffices to nondeterministically guess a group of at least $\\ell \\cdot {n}{k}$ voters and check that the intersection of their approval sets contains at least $\\ell $ candidates.", "To show ${\\mathrm {NP}}$ -hardness, we give a reduction from the ${\\mathrm {NP}}$ -complete Balanced-Biclique problem [17].", "The input for Balanced-Biclique consists of a bipartite graph $G$ and a nonnegative integer $k$ .", "The vertices of $G$ are partitioned into two sets, $L(G)$ and $R(G)$ , and we write $E(G)$ to denote the set of $G$ 's edges; each edge connects a vertex from $L(G)$ with a vertex from $R(G)$ .", "We ask if there is a size-$k$ subset of $L(G)$ and a size-$k$ subset of $R(G)$ such that each vertex from the former is connected with each vertex from the latter.", "Such two sets are jointly referred to as a $k$ -biclique of $G$ .", "Given an instance of Balanced-Biclique, we form an instance of Cohesive-Group as follows.", "We construct an election $E^{\\prime }$ , where $R(G)$ is the set of candidates and $L(G)$ is a collection of voters.", "A voter $\\ell _i \\in L(G)$ approves a candidate $r_j \\in R(G)$ if $\\ell _i$ and $r_j$ are connected in $G$ .", "We extend $E^{\\prime }$ by adding $\\max (||L(G)| - |R(G)|,0)$ candidates not approved by any voter, we set the committee size to be $k^{\\prime } = |L|$ , and we let the desired cohesiveness level be $\\ell ^{\\prime } = k$ .", "This completes the construction.", "Note that each $\\ell ^{\\prime }$ -cohesive group in our election consists of at least $\\ell ^{\\prime } \\frac{|L|}{k^{\\prime }} = k \\frac{|L|}{|L|} = k$ voters who approve at least $k$ common candidates.", "Focusing on exactly $k$ voters and $k$ candidates, we see that such a group exists if and only if $G$ has a $k$ -biclique.", "This completes the proof.", "On the positive side, [1] gave a polynomial-time algorithm for deciding if an election contains a 1-cohesive group (we refer to this variant of the problem as One-Cohesive-Group): It suffices to check if there is a candidate $c$ for whom $|A(c)| \\ge {n}{k}$ , where $n$ is the total number of voters and $k$ is the committee size.", "If such a candidate $c$ exists, then the voters from $A(c)$ form a 1-cohesive group; otherwise, there are no 1-cohesive groups.", "Corollary 3.2 ([1]) One-Cohesive-Group is in ${\\mathrm {P}}$ .", "We complement the above results by considering the complexity of #Cohesive-Group, i.e., the problem of counting cohesive groups.", "If we had an efficient algorithm for this problem, then we could also derive an efficient procedure for sampling cohesive groups uniformly at random [16], which would be quite useful.", "Indeed, we could use it, e.g., to experimentally study the distribution of cohesive groups in elections.Formally, an approximate counting algorithm would suffice to obtain a nearly uniform sampling procedure.", "Our results do not preclude existence of such an algorithm, but we leave studies in this direction for future work.", "Naturally, #Cohesive-Group is intractable—namely, ${\\mathrm {\\#P}}$ -complete—since even deciding if a single cohesive group exists is hard.", "More surprisingly, the same holds for 1-cohesive groups.", "Theorem 3.3 #One-Cohesive-Group is ${\\mathrm {\\#P}}$ -complete An intuition as to why finding a single 1-cohesive group is easy but counting them is hard is as follows.", "Using the argument from Corollary REF , for each candidate we can count (in polynomial time) the number of 1-cohesive groups whose members approve this candidate.", "Yet, if we simply added these values, then some groups could be counted multiple times.", "If we used the inclusion-exclusion principle, then we would get the correct result, but doing so would take exponentially many arithmetic operations.", "To give a formal proof of Theorem REF , we use the following intermediate problem, which captures counting cohesive groups that consist of a given number of voters.", "Definition 4 In the #Fixed-Size-Cohesive-Group problem (the #FSCG problem) we are given an election $E = (C, V)$ , a committee size $k$ , and two positive integers, $\\ell $ and $x$ .", "We ask how many $\\ell $ -cohesive groups that consist of exactly $x$ voters are there in $E$ .", "Given an instance $(E,k,\\ell ,x)$ of #FSCG, by $\\#\\text{LCG}(E, k, \\ell , x)$ we mean the number of $\\ell $ -cohesive groups of size $x$ from election $E$ for committee size $k$ .", "If we omit parameter $x$ , then we mean to total number of $\\ell $ -cohesive groups, irrespective how many voters they include.", "In the next proposition we show that #FSCG is computationally equivalent to #CohesiveGroup (the proof is in Appendix ).", "Proposition 3.4 #Cohesive-Group $\\le _{\\mathrm {T}}^{\\mathrm {fp}}$ #FSCG and #FSCG $\\le _{\\mathrm {T}}^{\\mathrm {fp}}$ #Cohesive-Group.", "We define problem #One-FSCG by fixing $\\ell = 1$ in the definition of #FSCG.", "Proposition REF also holds for the case of #One-FSCG and #One-Cohesive-Group (indeed, we never modify $\\ell $ in the proposition's proof).", "We use the computational equivalence of #One-FSCG and #One-Cohesive-Group to prove Theorem REF .", "[Proof of Theorem REF ] A problem belongs to ${\\mathrm {\\#P}}$ if its value can be expressed as the number of accepting paths of a polynomial-time nondeterministic Turing machine.", "For #One-Cohesive-Group it suffices that such a machine guesses a group of voters, verifies if they form an $\\ell $ -cohesive group (which can be done deterministically in polynomial time), and accepts if so.", "Hence, #One-Cohesive-Group is in ${\\mathrm {\\#P}}$ .", "To show ${\\mathrm {\\#P}}$ -hardness of #One-Cohesive-Group, we give a reduction from $\\textsc {\\#Set-Cover}$ to $\\textsc {\\#One-FSCG}$ .", "The latter problem is well-known to be ${\\mathrm {\\#P}}$ -complete, and the former is computationally equivalent to #One-Cohesive-Group.", "Let $(U, \\mathcal {S}, k)$ be an instance of the $\\textsc {\\#Set-Cover}$ problem, where $U = \\lbrace u_1, \\ldots , u_n\\rbrace $ is a universe, $\\mathcal {S}= \\lbrace S_1, \\ldots , S_m\\rbrace $ is a family of subsets of $U$ , and $k$ is a positive integer.", "The question is how many combinations of at most $k$ subsets from $\\mathcal {S}$ sum up to the universe $U$ .", "We create an instance of $\\textsc {\\#One-FSCG}$ with an election $E^{\\prime } = (C^{\\prime }, V^{\\prime })$ , such that the candidates correspond to the elements of $U$ and the voters correspond to the elements of $\\mathcal {S}$ (hence, we can speak both of a universe element $u_i$ and a candidate $u_i$ , or of a set $S_j$ and voter $S_j$ ).", "For each candidate $u_i$ and voter $S_j$ , $u_i$ is approved by voter $S_j$ if element $u_i$ does not belong to the set $S_j$ .", "Further, we extend $E^{\\prime }$ by adding $max(m-n, 0)$ new candidates not approved by anyone.", "Altogether, the number of candidates in $E^{\\prime }$ is at least $m$ .", "We set the size of the final committee to be $k^{\\prime } = m$ .", "We also write $n^{\\prime }$ to denote the number of voters in $E^{\\prime }$ ; naturally, we have $n^{\\prime } = m$ .", "Due to the definition of a 1-cohesive group, its size must be at least ${n^{\\prime }}{k^{\\prime }} = {m}{m} =1$ .", "Thus, every group of voters that approve at least one common candidate is a 1-cohesive group in our election.", "Let us consider a 1-cohesive group of size $x^{\\prime } \\le k$ and let us call it $T$ .", "By definition, there is at least one candidate approved by all members of $T$ .", "Let us call her $c^{\\prime }$ .", "This means that $c^{\\prime }$ is not included in any set $S_j$ corresponding to the voters from $T$ .", "Hence, the union of these sets is different from $U$ .", "On ther other hand, if a group $R$ of $x^{\\prime } \\le k$ voters does not form an 1-cohesive group, then the sets corresponding to the voters from this group do sum up to the universe $U$ .", "Indeed, for each candidate $c$ there is a voter in $R$ who does not approve $c$ , which means that the corresponding set includes her.", "As a consequence, each member of $U$ belongs to at least one set corresponding to a voter from $R$ .", "Above observations mean that families of subsets from $\\mathcal {S}$ sum up to the universe $U$ if and only if the voter groups that correspond to these families are not 1-cohesive.", "Since for a positive integer $x$ there are ${|\\mathcal {S}| \\atopwithdelims ()x} = {m \\atopwithdelims ()x}$ size-$x$ families of sets from $\\mathcal {S}$ , we conclude that the answer for our instance of #Set-Cover is $\\sum _{x=1}^{k} { ({|\\mathcal {S}| \\atopwithdelims ()x} - \\#\\text{LCG}(E^{\\prime }, 1, x)) }$ .", "This completes the proof." ], [ "Computing a Committee with a Given PD", "In this section we focus on the complexity of deciding if a committee with a given proportionality degree exists.", "At first, this problem may seem trivial as for each election there is a committee with nearly perfect PD [2].", "Yet, we find that the answer is quite nuanced.", "This stands in sharp contrast to analogous decision questions for JR and EJR, which are trivial (a committee with the desired property always exists so the algorithm always accepts).", "Formally, we consider the following problem.", "Definition 5 In the PD-Committee problem we are given an election $E$ , a committee size $k$ , and a function $f \\colon [k] \\rightarrow \\mathbb {Q}$ , specified by listing its values.", "We ask if $E$ has a size-$k$ committee with proportionality degree at least $f$ .", "We find that PD-Committee is both ${\\mathrm {coNP}}$ -hard and ${\\mathrm {NP}}$ -hard.", "For the former result, we use the fact that for a given $\\ell $ , the $f(\\ell )$  value of a PD function is binding only if the given election contains $\\ell $ -cohesive groups.", "Theorem 4.1 PD-Committee is both ${\\mathrm {NP}}$ -hard and ${\\mathrm {coNP}}$ -hard We will show ${\\mathrm {NP}}$ -hardness in Theorem REF and here we focus on ${\\mathrm {coNP}}$ -hardness.", "To this end, we give a reduction from Cohesive-Group to the complement of PD-Committee.", "Let $(E, k, \\ell )$ be our input instance, where $E = (C,V)$ is an election, $k$ is the committee size, and $\\ell $ is the cohesiveness level.", "The question is if there exists an $\\ell $ -cohesive group for election $E$ with committee size $k$ .", "For convenience, we set $n = |V|$ , and $m = |C|$ .", "We create an instance of the complement of PD-Committee as follows.", "Let $s$ be the smallest integer such that $s \\cdot k > m$ .", "We form an election $E^{\\prime }$ by first copying $E$ and then adding (a) $s \\cdot k$ new candidates who are not approved by any voters and (b) $(s-1) \\cdot n$ new voters who do not approve any candidates.", "Altogether, in $E^{\\prime }$ we have $n^{\\prime } = s \\cdot n$ voters, and $m^{\\prime } = m + s \\cdot k$ candidates.", "Further, we set the committee size to be $k^{\\prime } = s \\cdot k$ and we let the PD function $f$ be such that for $i < \\ell $ we have $f(i) = 0$ and for $i \\ge \\ell $ we have $f(i) = k^{\\prime }$ .", "This completes the construction.", "Note that the minimum size of an $\\ell $ -cohesive group in $E^{\\prime }$ is equal to the minimum size of an $\\ell $ -cohesive group in $E$ , because $\\frac{\\ell \\cdot n^{\\prime }}{k^{\\prime }} = \\frac{\\ell \\cdot s \\cdot n}{s \\cdot k} =\\frac{\\ell \\cdot n}{k}$ .", "Thus every $\\ell $ -cohesive group from $E$ is also an $\\ell $ -cohesive group for $E^{\\prime }$ and vice versa.", "Further, each size-$k^{\\prime }$ committee must contain at least one new candidate, because $k^{\\prime } = s \\cdot k > m$ .", "Yet, the new candidates are not approved by any voter and, so, if $E^{\\prime }$ has some $\\ell $ -cohesive group, then its average satisfaction must be strictly below $f(\\ell ) = k^{\\prime }$ .", "This means that if $E^{\\prime }$ has a committee with PD $f$ then there are no $\\ell $ -cohesive groups in $E^{\\prime }$ (and, thus, there are no cohesive groups in $E$ ).", "In other words, the answer for the PD-Committee instance is “yes” if and only if the answer for the Cohesive-Group instance is “no.” Since, by Theorem REF , the latter is ${\\mathrm {NP}}$ -complete, the former is ${\\mathrm {coNP}}$ -hard.", "Since PD-Committee is both ${\\mathrm {NP}}$ -hard and ${\\mathrm {coNP}}$ -hard, it is unlikely that it is complete for either of these classes (we would have ${\\mathrm {NP}}= {\\mathrm {coNP}}$ if it were).", "Indeed, we suspect that it is complete for ${\\mathrm {NP}}^{\\mathrm {NP}}$ and we show that it belongs to this class.", "An ${\\mathrm {NP}}^{\\mathrm {NP}}$ -hardness result remains elusive, unfortunately.", "Theorem 4.2 PD-Committee is in ${\\mathrm {NP}}^{\\mathrm {NP}}$ .", "Consider an instance $(E,k,f)$ of PD-Committee.", "It is a “yes”-instance exactly if there exists a size-$k$ committee such that for every $\\ell \\in [k]$ , every $\\ell $ -cohesive group has average satisfaction at least $f(\\ell )$ .", "We can verify that this holds by first nondeterministically guessing the committee and then asking the oracle if there is a cohesive group for which the constrained implied by the PD function is failed (since computing an average satisfaction of a given cohesive group can be done in polynomial time, this task belongs to ${\\mathrm {NP}}$ ).", "We accept if the oracle answers “yes” and we reject otherwise.Note that this way our nondeterministic machine makes only a single query to the oracle.", "While one could worry that this might mean that our problem is somehow “easy” for ${\\mathrm {NP}}^{\\mathrm {NP}}$ , this is not the case.", "Indeed, it is well-known that every problem in ${\\mathrm {NP}}^{\\mathrm {NP}}$ can be solved by a nondeterministic machine with a single oracle call.", "While PD-committee seems very hard in general, for some classes of PD functions it is significantly easier.", "As an extreme example, for nearly perfect ones it is trivially in ${\\mathrm {P}}$ because PAV winning committees always have nearly perfect PD.", "We consider the following restricted variants of PD-Committee: In Constant-PD-Committee we require the desired PD functions to be constant, in Unit-PD-Committee we require them to take value 1 for each argument, and in Perfect-PD-Committee we require them to be perfect.", "We find that both Constant-PD-Committee and Unit-PD-Committee are ${\\mathrm {NP}}$ -complete and, thus, likely much easier than the general variant.", "To establish these results, it suffices to show membership in ${\\mathrm {NP}}$ for the former and ${\\mathrm {NP}}$ -hardness for the latter.", "Theorem 4.3 Constant-PD-Committee is in ${\\mathrm {NP}}$ .", "Consider an instance $(E,k,f)$ of Constant-PD-Committee, where $E = (C,V)$ is an election, $k$ is the committee size, and $f$ is a constant PD function.", "Since $f$ is a constant function, there is a value $x$ such that for each $\\ell \\in [k]$ we have $f(\\ell ) = x$ .", "To show that Constant-PD-Committee is in ${\\mathrm {NP}}$ , we give a polynomial-time algorithm that given such an instance and size-$k$ committee $W$ verifies if $W$ has PD $f$ .", "Let $n = |V|$ be the number of voters.", "For each candidate $c \\in C$ , we define ${{\\mathrm {sat}}}(c)$ to be the average satisfaction of $\\lceil \\frac{n}{k}\\rceil $ members of $A(c)$ that are least satisfied with $W$ ; if $A(c)$ contains fewer than $\\frac{n}{k}$ voters then we set ${{\\mathrm {sat}}}(c) = +\\infty $ .", "We set $y = \\min _{c \\in C}{{\\mathrm {sat}}}(c)$ .", "If $y = +\\infty $ then election $E$ has no cohesive groups and $W$ has PD $f$ trivially.", "Otherwise, $y$ is the smallest average satisfaction that a 1-cohesive group from $E$ has for $W$ (indeed, every 1-cohesive group must have at least $\\lceil \\frac{n}{k} \\rceil $ members and for each $c \\in C$ , each 1-cohesive group whose members approve $c$ has satisfaction at least ${{\\mathrm {sat}}}(c)$ ).", "For each $\\ell \\in [k]$ , each $\\ell $ -cohesive group also has satisfaction at least $y$ (each such group also is a 1-cohesive group and, so, also has average satisfaction at least $y$ ).", "Thus, if $y \\ge x$ then we accept and otherwise we reject.", "This algorithm runs in polynomial time.", "Theorem 4.4 $\\textsc {Unit-PD-Committee}$ is ${\\mathrm {NP}}$ -hard We give a reduction from a variant of the classic X3C problem, which we call RX3C and which is well-known to be ${\\mathrm {NP}}$ -complete [14]: An instance of RX3C consists of a universe set $U = \\lbrace u_1, u_2, ... , u_{3k}\\rbrace $ and a family $\\mathcal {S}= \\lbrace S_1, S_2, ... , S_{3k}\\rbrace $ of size-3 subsets of $U$ , each element from $U$ belongs to exactly three sets from $\\mathcal {S}$ , and we ask if there exist $k$ subsets from $\\mathcal {S}$ which sum up to the universe $U$ .", "We form an instance of Unit-PD-Committee with an election $E$ , committee size $k$ , and unit PD function.", "We let the sets from $\\mathcal {S}$ be the candidates in $E$ , and we let the universe elements be the voters.", "A voter $u_i$ approves a candidate $S_j$ if $u_i \\in S_j$ .", "This completes the construction.", "We note that all cohesive groups in $E$ contain exactly three voters and have cohesiveness level one.", "This holds because each candidate is approved by exactly three voters and this is also the lower bound on the size of 1-cohesive groups in $E$ (indeed, ${3k}{k}=3$ ).", "It is clear that if there exist $k$ subsets from $\\mathcal {S}$ which sum up to $U$ , then the corresponding candidates form a committee which has average satisfaction at least 1.", "Indeed, for each voter there is at least one candidate in the committee that he or she approves (in fact, exactly one).", "Otherwise the selected sets would not sum up to $U$ .", "As a consequence, the average satisfaction of each (1-)cohesive group with the committee is at least 1.", "Next, let us show that if there exists a committee $W$ of size $k$ such that each cohesive group has average satisfaction at least 1, then there is a collection $T$ of $k$ sets from $\\mathcal {S}$ that sum up to $U$ (i.e., there is an exact cover of $U$ ).", "Let $B$ be the sum of the total satisfactions of all the $3k$ 1-cohesive groups in $E$ .", "Since each 1-cohesive group has average satisfaction at least one, its total satisfaction is at least 3.", "There are $3k$ such groups, so we have that $B$ is at least $9k$ .", "Moreover, $B$ is equal to $9k$ exactly if each 1-cohesive group has average satisfaction equal to 1.", "However, each committee member is approved by exactly three voters, and each of these voters belongs to exactly three 1-cohesive groups.", "Hence $B = 9k$ and each 1-cohesive group has average satisfaction equal to 1.", "Consider some set $S_j = \\lbrace u_{j_1}, u_{j_2}, u_{j_3}\\rbrace $ such that candidate $S_j$ is a member of committee $W$ .", "Naturally, $\\lbrace u_{j_1}, u_{j_2}, u_{j_3}\\rbrace $ is a 1-cohesive group, all its member approve $S_j$ , and, so, its average satisfaction is at least 1.", "Indeed, by previous discussion we know that it is exactly 1.", "Hence, for each voter in $\\lbrace u_{j_1}, u_{j_2}, u_{j_3}\\rbrace $ , candidate $S_j$ is the only member of $W$ that he or she approves.", "If we repeat this reasoning for every member of $W$ , we find that each of them is approved by exactly three voters and no two of them are approved by the same voters.", "This means that $W$ corresponds to an exact cover of $U$ .", "The proof is complete.", "Corollary 4.5 Both Constant-PD-Committee and Unit-PD-Committee are ${\\mathrm {NP}}$ -complete.", "As all the cohesive groups in the election constructed in the proof of Theorem REF have cohesiveness level 1, we have a stronger result: Given a PD function $f$ such that $f(1) = 1$ , it is ${\\mathrm {NP}}$ -hard to decide if there is a committee with proportionality degree $f$ .", "In particular, we have the next corollary.", "Corollary 4.6 Perfect-PD-Committee is ${\\mathrm {NP}}$ -hard.", "We can extend Theorem REF to work for any positive integer constant $x$ and functions $f$ such that $f(1) = x$ .", "For example, for $x = 2$ it suffices to extend the constructed election with three voters that do not approve anyone and with a single candidate who is approved by all the other voters.", "It would also be interesting to consider functions $f$ such that $f(1)$ is a constant between 0 and 1, but we leave it for future work.", "The above results are nicely aligned with existing polynomial-time algorithms for computing committees with guarantees on their PD.", "For example, there are polynomial-time algorithms for computing EJR committees, and EJR committees are guaranteed to have PD $f$ such that $f(\\ell ) =\\frac{\\ell -1}{2}$  [25].", "As we see, $f(1) = 0$ (though this could be improved very slightlySince the committee provides EJR, and thus JR, this zero could be replaced by $\\frac{1}{{n}{k}} = \\frac{k}{n}$ , where $n$ is the number of voters and $k$ is the committee size.", "This follows from the fact that in each 1-cohesive group of size $\\frac{n}{k}$ there is at least one candidate who approves at least one voter.).", "As we have shown, extending the algorihtm to find committees with PD functions $f$ such that $f(1) = 1$ (whenever such committees exist) would not be possible in polynomial time (assuming ${\\mathrm {P}}\\ne {\\mathrm {NP}}$ )." ], [ "Computing the PD of a Given Committee", "Sometimes, instead of computing a committee with a specified PD, we would like to establish the PD of an already existing one.", "For example, this would be the case if we wanted to experimentally compare how well the committees provided by various voting rules represent the voters.", "One way to proceed would be as follows: For a given election $E$ and committee $W$ , consider each cohesiveness level $\\ell $ and, using binary search, find value $f(\\ell )$ , $0 \\le f(\\ell ) \\le |W|$ , such that each $\\ell $ -cohesive group has average satisfaction at least $f(\\ell )$ , but for every $\\varepsilon > 0$ there exists an $\\ell $ -cohesive group with average satisfaction below $f(\\ell )+\\varepsilon $ (or there are no $\\ell $ -cohesive groups in this election).", "Using binary search to compute this value is possible because in an election with $n$ voters and committee size $k$ , there are at most $O(kn^2)$ different average satisfaction values of cohesive groups (each cohesive group can have total satisfaction between 0 and $nk$ , and each cohesive group can have at most $n$ voters).", "Running such binary search requires the ability to solve the following problem.", "Definition 6 In the PD-Failure problem we are given an election $E$ , a committee $W$ , a cohesiveness level $\\ell $ , and a nonnegative rational threshold $y \\le k$ .", "We ask if $E$ contains an $\\ell $ -cohesive group whose average satisfaction for $W$ is lower than $y$ .", "As one may expect, this problem is ${\\mathrm {NP}}$ -complete.", "Membership in ${\\mathrm {NP}}$ follows by nondeterministically guessing an $\\ell $ -cohesive group and checking if its average satisfaction is below $y$ .", "To show ${\\mathrm {NP}}$ -hardness, we note that setting $y$ to an impossible-to-achieve value makes the problem equivalent to testing if an $\\ell $ -cohesive group exists.", "Theorem 5.1 PD-Failure is ${\\mathrm {NP}}$ -complete.", "Membership in ${\\mathrm {NP}}$ was already argued in the paragraph above the theorem statement.", "To prove ${\\mathrm {NP}}$ -hardness, we show a reduction from $\\textsc {Cohesive-Group}$ to $\\textsc {PD-Failure}$ .", "Let $(E, k, \\ell )$ be an instance of the $\\textsc {Cohesive-Group}$ problem, where $E$ is an election consisting of candidates $C$ and voters $V$ , $k$ is the size of the final committee, and $\\ell $ is the cohesiveness level.", "We ask if there exists an $\\ell $ -cohesive group for election $E$ with committee size $k$ .", "To create a $\\textsc {PD-Failure}$ instance, we use the same election $E$ , the same committee size $k$ , and the same cohesiveness level $\\ell $ , but we add $k$ fresh candidates who are not approved by any voter and select them to the committee $W^{\\prime }$ .", "Further, we set the PD threshold $y=k$ .", "From the above observation we conclude that if there exists any valid $\\ell $ -cohesive group, then its average satisfaction with $W^{\\prime }$ is 0, which is strictly less than $y$ .", "Therefore if there exists an $\\ell $ -cohesive group with average satisfaction lower than $y$ , then this group is also an $\\ell $ -cohesive group for the $\\textsc {Cohesive-Group}$ instance.", "Further, if there are no $\\ell $ -cohesive groups for the $\\textsc {PD-Failure}$ instance, then the $\\textsc {Cohesive-Group}$ instance doesn't have any $\\ell $ -cohesive groups either.", "Since the $\\textsc {Cohesive-Group}$ problem is ${\\mathrm {NP}}$ -complete, the $\\textsc {PD-Failure}$ problem is in ${\\mathrm {NP}}$ and we reduced the $\\textsc {Cohesive-Group}$ problem to the $\\textsc {PD-Failure}$ problem, the $\\textsc {PD-Failure}$ problem is also ${\\mathrm {NP}}$ -complete." ], [ "ILP Formulation", "Fortunately, in practice we may be able to solve instances of our problem by expressing them as integer linear programs (ILPs) and solving them using off-the-shelf software.", "Specifically, let us consider an instance of PD-Failure with election $E = (C,V)$ , committee $W$ , cohesiveness level $\\ell $ , and threshold $y$ .", "We set $m = |C|$ , $n = |V|$ , and $k = |W|$ .", "For convenience, let $A$ be the binary matrix of approvals for $E$ , that is, we have $a_{ij} = 1$ if the $i$ -th voter approves the $j$ -th candidate, and we have $a_{ij} = 0$ otherwise.", "We note that if there is an $\\ell $ -cohesive group $X$ whose satisfaction for $W$ is below $y$ , then there is such a group of size exactly $s = \\lceil \\ell \\cdot {n}{k} \\rceil $ (e.g., consider $X$ and remove sufficiently many voters who approve the most members of $W$ ).", "To form our ILP instance, we first specify the variables: For each $i \\in [n]$ , we have a binary variable $x_i$ , with the intention that $x_i = 1$ if the $i$ -th voter is included in the sought cohesive group, and $x_i = 0$ otherwise.", "For each $j \\in [m]$ , we have a binary variable $y_i$ , with the intention that $y_j = 1$ if all the voters in the group specified by variables $x_1, \\ldots , x_n$ approve the $j$ -th candidate, and $y_j = 0$ otherwise.", "We refer to the voters (to the candidates) whose $x_i$ ($y_j$ ) variables are set to 1 as selected.", "Next, we specify the constraints.", "Foremost, we ensure that we select exactly $s$ voters and at least $\\ell $ candidates: $& \\textstyle \\sum _{i=1}^{n} {x_i} = s, & \\text{and}& &\\textstyle \\sum _{j=1}^{m} {y_j} \\ge \\ell .$ Then, we ensure that each selected voter approves all the selected candidates.", "For each $j \\in [m]$ , we form constraint: $\\textstyle \\sum _{i=1}^{n} {a_{ij} \\cdot x_i} \\ge s \\cdot y_j.$ If the $j$ -th candidate is not selected, then this inequality is satisfied trivially.", "However, if the $j$ -th candidate is selected, then the sum on the left-hand side must be at least $s$ , i.e., there must be at least $s$ selected voters who approve the $j$ -th candidate.", "Since there are exactly $s$ selected voters, all of them must approve the $j$ -th candidate.", "Finally, we ensure that the average satisfaction of the selected voters is below $y$ , by adding constraint $\\textstyle \\frac{1}{s}\\sum _{i=1}^{n} \\sum _{j \\in W} {a_{ij} \\cdot x_i} < y.$ If there is an assignment that satisfies these constraints, then the selected voters form an $\\ell $ -cohesive group with average satisfaction below $y$ .", "Otherwise, no such group exists." ], [ "Verification", "For a comparison with previous studies regarding JR and EJR, we also consider the following verification problem.", "Definition 7 In the PD-Verification problem we are given an election $E$ , a committee $W$ , a PD function $f$ , and we ask if $W$ has proportionality degree $f$ .", "As PD-Verification is very closely related to the complement of PD-Failure, we find that it is ${\\mathrm {coNP}}$ -complete (we give the formal proof in Appendix ).", "Theorem 5.2 PD-Verification is ${\\mathrm {coNP}}$ -complete.", "Testing if a committee provides EJR is ${\\mathrm {coNP}}$ -complete as well [1], so in this respect PD and EJR are analogous.", "There is also a polynomial-time algorithm for testing if a committee provides JR, and in the PD world this corresponds to a polynomial-time algorithm for checking if a committee admits a given constant PD function.", "Such an algorithm was included as part of the proof of Theorem REF .", "Corollary 5.3 PD-Verification for a constant PD functions (provided as input) is in ${\\mathrm {P}}$ ." ], [ "Dealing With Computational Hardness", "In this section we consider circumventing the computational hardness of our problems by studying their parameterized complexity and by considering structured elections." ], [ "Fixed-Parameter Tractability", "Our two main problems, PD-Committee and PD-Failure, are fixed-parameter tractable with respect to the number of candidates and the number of voters.", "For PD-Failure and the parameterization by the number of candidates, we proceed similarly as in the proof of Theorem REF .", "Namely, for each set $R$ of candidates we consider the set $V(R)$ of all the voters that approve members of $R$ , one-by-one remove from this set the voters with the highest satisfation, and watch if at any point we obtain an $\\ell $ -cohesive group with average satisfaction below the required value.", "Using a similar approach, and trying every possible committee, we also obtain an algorithm for PD-Committee.", "For the parameterization by the number of voters, we solve our problems by forming ILP instances and solving them using the classic algorithm of [21].", "This is possible because with $n$ voters there are at most $2^n$ cohesive groups and each candidate has one of $2^n$ types (where the type of a candidate is the set of voters that approve him; candidates with the same type are interchangeable).", "Theorem 6.1 There are FPT algorithms for PD-Committee and PD-Failure both for the parameterization by the number of candidates and for the parameterization by the number of voters.", "Let us first consider the parameterization by the number of candidates and the PD-Failure problem.", "Our input consists of an election $E = (C,V)$ , committee $W$ of size $k$ , cohesiveness level $\\ell $ , and rational threshold $y$ .", "Let $m$ be the number of candidates and let $n$ be the number of voters.", "For each subset of $\\ell $ candidates, we find a group of $\\ell \\cdot \\frac{n}{k}$ voters who are least satisfied with $W$ .", "If the lowest satisfaction among such groups is below $y$ then we accept and otherwise we reject.", "The correctness and fixed-parameter tractability follow immediately.", "For parameterization by the number of candidates and the PD-Committee problem, it suffices to try all committees and for each of them (and each cohesiveness level) use the algorithm for PD-Failure to check if it indeed achieves required PD.", "Next let us move on to the parameterization by the number of voters and the PD-Failure problem.", "We use the same notation as in the argument above for parameterization by the number of candidates.", "It suffices to consider every subset of voters, check if it is an $\\ell $ -cohesive group, and verify if its average satisfaction is below $y$ .", "For the case of PD-Committee and parameterization by the number of voters, we employ integer linear programming.", "Let $E = (C,V)$ be the input election with $m$ candidates and $n$ voters.", "We seek a committee of size $k$ , with PD $f$ .", "There are $2^n$ subsets of the voters, and we order them in some way, so for each $i \\in [2^n]$ we can speak of the $i$ -th subset.", "For each such subset, we say that a candidate has type $i$ if she is approved exactly by the voters from the $i$ -th subset (and only by them).", "For each $i \\in [2^n]$ we let $c_i$ be the number of type-$i$ candidates and we form a variable $x_i$ , with the intended meaning that $x_i$ is the number of type-$i$ candidates in the committee.", "We introduce the following constraints: For each $i$ , we require that $x_i \\le c_i$ , i.e., that we do not select more type-$i$ candidates than available.", "We require that $\\sum _{i \\in [2^n]} x_i = k$ , i.e., we ensure that we select a committee of size exactly $k$ .", "For each $\\ell \\in [k]$ and each $\\ell $ -cohesive group $S$ of voters (due to our parameterization, we can enumerate them all), we form the following constraint: $\\sum _{v \\in S} \\sum _{i \\in [2^n]} x_i \\cdot [\\text{$v$ approves type-$i$ candidates}] \\ge |S| \\cdot f(\\ell ),$ where we use the Iverson bracket notation (i.e., for a true/false statement $F$ , by $[F]$ we mean 1 is $F$ is true and we mean 0 otherwise).", "This constraint ensures that each cohesive group has required level of average satisfaction.", "We solve this ILP instance using the classic algorithm of [21].", "Since the number of variables is $2^n$ and $n$ is the parameter, doing so is possible in FPT time.", "This completes the proof.", "Testing if a committee provides EJR is also fixed-parameter tractable for the parameterizations considered in Theorem REF .", "So, from this point of view, dealing with PD is not harder than dealing with EJR.", "Finally, the problem of counting cohesive groups (and, thus, also the problem of deciding if groups with particular cohesiveness level exist) also is fixed-parameter tractable for our parameters.", "For parameterization by the number of candidates, we can use the inclusion-exclusion principle, and for the parameterization by the number of voters we can explicitly look at each subset of voters.", "Theorem 6.2 There are FPT algorithms for (#)Cohesive-Group, for the parameterizations by the number of candidates and by the number of voters.", "Let us first consider the parameterization by the number of voters and the #Cohesive-Group problem.", "Our input consists of an election $E = (C,V)$ , committee $W$ of size $k$ , and cohesiveness level $\\ell $ .", "Let $m$ be the number of candidates and let $n$ be the number of voters.", "Initially, we have a counter set to zero.", "For each subset of at least $\\ell \\cdot \\frac{n}{k}$ voters, we compute the set of candidates that are approved by all these voters.", "If this set has size at least $\\ell $ then we increase the counter and otherwise we do not.", "At the end, the counter contains the desired answer.", "For the Cohesive-Group problem it is enough to check if the counter is above 0.", "For the parameterization by the number of candidates and the Cohesive-Group problem, for each subset of $\\ell $ candidates we calculate the number of voters that approve all these candidates and accept if it is at least $\\ell \\cdot \\frac{n}{k}$ , we reject if we do not accept for any subset.", "For the #Cohesive-Group problem, we can use the inclusion-exclusion principle." ], [ "Structured Preferences", "Next we consider two domains of structured preferences, introduced by [9].", "Such domains are interesting because, on the one hand, they capture some realistic scenarios, and, on the other hand, by assuming them it is often possible to provide polynomial time algorithms for problems that in general are intractable.", "Definition 8 ([9]) An election $E = (C,V)$ has candidate interval (CI) preferences (voter interval preferences, VI) if it is possible to order the candidates (the voters) so that for each voter $v$ (for each candidate $c$ ) the set $A(v)$ (the set $A(c)$ ) is an interval w.r.t.", "this order.", "For an example of CI preferences, consider a political election where candidates are ordered according to the left-to-right spectrum of opinions and the voters approve ranges of candidates whose opinions are close enough to their own.", "[9] gave algorithms for deciding if a given election has CI or VI preferences, and for computing appropriate orders of candidates or voters.", "Thus, for simplicity, we assume that these orders are provided together with our input elections.", "We mention that a number of other preference domains are considered in the literature—see, e.g., the works of [31] and [13]—but the CI and VI ones are by far the most popular.", "For a very detailed discussion of structured domains, albeit in the world of ordinal preferences, we point to the survey of [10].", "Unfortunately, even for CI and VI elections we do not know how to solve the PD-Committee problem in polynomial-time.In particular, the approach of [22] based on solving totally unimodular ILP instances does not seem to work here.", "Nonetheless, we do have polynomial-time algorithms for the PD-Failure problem.", "Theorem 6.3 PD-Failure restricted to either CI or VI elections is in ${\\mathrm {P}}$ .", "First, we give an algorithm for the CI case.", "Our input consists of an election $E = (C,V)$ , where $C = \\lbrace c_1, \\ldots , c_m\\rbrace $ and $V = (v_1, \\ldots , v_n)$ , a size-$k$ committee $W$ , cohesiveness level $\\ell $ , and threshold value $y$ .", "Without loss of generality, we assume that $E$ is CI with respect to the order $c_1 \\lhd c_2 \\lhd \\cdots \\lhd c_m$ .", "Since $E$ is a CI election, we observe that if $X$ is some cohesive group whose all members approve some two candidates $c_i$ and $c_j$ , $i \\le j$ , then all members of $X$ also approve candidates $c_{i+1}, \\ldots , c_{j-1}$ .", "For each $i \\le m-\\ell +1$ , let $X(i)$ be the set of all voters who approve each of the candidates $c_{i}, \\ldots , c_{i+\\ell -1}$ .", "By the preceding argument, we see that every $\\ell $ -cohesive group can be obtained by taking some set $X(i)$ and (possibly) removing some of its members.", "Our algorithm proceeds as follows.", "For each set $X(i)$ , we form a set $Y(i)$ by taking $X(i)$ and removing all but $\\lceil \\ell \\cdot {n}{k} \\rceil $ voters that are least satisfied with $W$ .", "(If a given $X(i)$ contains fewer than $\\lceil \\ell \\cdot {n}{k} \\rceil $ voters then we set $Y(i) = \\emptyset $ and we assume that the average satisfaction of its voters is $+\\infty $ .)", "If there is some $i$ such that the average satisfaction of the voters in $Y(i)$ is below $y$ , then we accept (indeed, we have just found an $\\ell $ -cohesive group with average satisfaction below $y$ ).", "If there is no such $Y(i)$ , then we reject (we do so because each nonempty $Y(i)$ has the lowest average satisfaction among all the $\\ell $ -cohesive groups that can be obtained by removing voters from $X(i)$ ).", "Correctness and polynomial running time follow immediately.", "Now let us consider the VI case.", "We use the same notation as before, except that we assume that $E$ is VI with respect to the voter order $v_1 \\lhd v_2 \\lhd \\cdots \\lhd v_n$ .", "We use the same algorithm as in the CI case, but for the $X(i)$ sets defined as follows (let $s = \\lceil \\ell \\cdot {n}{k}\\rceil $ ): For each $i \\in [n-s+1]$ , we let $X(i) = \\lbrace v_{i}, v_{i+1}, \\ldots , v_{j}\\rbrace $ , where $j$ is the largest value such that $|A(v_{i}) \\cap A(v_{j})| \\ge \\ell $ (if $v_i$ approves fewer than $\\ell $ candidates then $X(i)$ is empty).", "The algorithm remains correct because, as in the CI case, every $\\ell $ -cohesive group is a subset of some $X(i)$ .", "Similar reasoning and observations as in the above proof also give the algorithms for counting cohesive groups (and, thus, for deciding their existence).", "Theorem 6.4 (#)Cohesive-Group restricted to either CI or VI elections is in ${\\mathrm {P}}$ .", "We prove Theorem REF via the following two theorems (they suffice due to Proposition REF ).", "Theorem 6.5 There is a polynomial-time algorithm for the $\\#\\textsc {FSCG}$ problem under the VI restriction Let $(E, k, \\ell , x)$ be a $\\#\\textsc {FSCG}$ VI instance, where $E$ is an election with candidates $C$ and voters $V$ , $k$ is the committee size, $\\ell $ is the cohesiveness level, and $x$ is the size of cohesive groups.", "We ask how many $\\ell $ -cohesive groups of size $x$ are there in election $E$ .", "We assume that $V = (v_1, \\ldots , v_n)$ and the election is VI for this order of the candidates.", "We observe that if voters $v_i$ and $v_j$ approve candidate $c_p$ , then each voter $v_k$ between $v_i$ and $v_j$ also approves $c_p$ , because under VI each candidate is approved by a consecutive segment of voters.", "As a result, if $v_i$ and $v_j$ have at least $\\ell $ common candidates, then each voter $v_k$ between $v_i$ and $v_j$ also approves these candidates.", "By $\\mathit {smallestCG}(v_i, \\ell , x)$ we mean the number of $\\ell $ -cohesive groups of size $x$ in which voter $v_i$ has the smallest index.", "Then, the sum of the $\\mathit {smallestCG}$ values over all the voters is the final answer.", "Now let us show how to calculate $\\mathit {smallestCG}(v_i, \\ell , x)$ .", "Given a voter $v_i$ , a group cohesiveness level $\\ell $ , and an integer $x$ , we find the greatest index $j$ such that voter $v_j$ still has at least $\\ell $ common candidates with $v_i$ .", "If $v_j$ does not exist or the number of voters in range $[v_i,v_j]$ is lower than $x$ , then return 0.", "Otherwise, we select the voter $v_i$ and $x-1$ other voters from $[v_{i+1},v_j]$ ; we can do it in ${j-i} \\atopwithdelims (){x-1}$ ways and this is the value we output.", "This completes the proof.", "Theorem 6.6 There is a polynomial-time algorithm for the $\\#\\textsc {FSCG}$ problem under the CI restriction Let $(E, k, \\ell , x)$ be a $\\#\\textsc {FSCG}$ CI instance, where $E$ is an election with candidates $C$ and voters $V$ , $k$ is the committee size, $\\ell $ is the cohesiveness level, and $x$ is the size of cohesive groups.", "We ask how many $\\ell $ -cohesive groups of size $x$ are there in election $E$ .", "We assume that $C = \\lbrace c_1, \\ldots , c_m\\rbrace $ and the election is CI for candidate order $c_1,c_2, \\ldots , c_m$ .", "We observe that if candidates $c_i$ and $c_j$ are approved by voter $v_p$ , then each candidate $c_k$ between $c_i$ and $c_j$ is also approved by $v_p$ , because under CI each voter approves a consecutive segment of candidates.", "As a result, for each $\\ell $ -cohesive group the candidates approved by all its members form a consecutive segment.", "By $\\mathit {smallestCG}(c_j, \\ell , x)$ we mean the number of $\\ell $ -cohesive groups of size $x$ in which candidate $c_j$ is the commonly approved candidate with the smallest index.", "Then, the sum of the $\\mathit {smallestCG}$ values through all the candidates is the final answer.", "Now let us show how to calculate the function $\\mathit {smallestCG}(c_j, \\ell , x)$ .", "Assume we are given candidate $c_j$ , group cohesiveness level $\\ell $ , and an integer $x$ .", "Let $L_1$ be the set of voters that approve all the candidates from $\\lbrace c_j, c_{j+1}, ... , c_{j+\\ell -1}\\rbrace $ and at least one candidate which has index strictly smaller than $j$ .", "Similarly, let $L_2$ be the set of voters that approve all the candidates from $\\lbrace c_j, c_{j+1}, ... , c_{j+\\ell -1}\\rbrace $ and do not approve any candidate whose index is strictly smaller than $j$ .", "Both values can be calculated in polynomial-time by a single iteration through election $E$ .", "Now let us point out that each $\\ell $ -cohesive group accounted for in $\\mathit {smallestCG}(c_j, \\ell , x)$ must consist of at least one voter included in $L_2$ and some voters included in $L_1$ .", "As we do not know how many voters we should take from the first part, we iterate through all possible partition sizes.", "Thus, the number of $\\ell $ -cohesive groups of size $x$ whose members' smallest index of a commonly approved candidate is $j$ , is equal to: $\\textstyle \\sum _{k=1}^{\\min (|L_2|,x)} { { |L_2| \\atopwithdelims ()k } \\cdot { |L_1| \\atopwithdelims (){x-k} } }$ This completes the proof.", "Similar approach shows that testing if a committee provides EJR can be done in polynomial time for CI or VI elections (to the best of our knowledge, this is a folk result)." ], [ "Perfect PD in CI/VI Elections?", "[2] have shown that for each election and each committee size there is a committee with a nearly perfect PD, but there are scenarios where committees with perfect PDs do not exist.", "Unfortunately, this remains true even if the elections are CI and VI at the same time.", "Example 1 Consider an election $E = (C,V)$ , where $C =\\lbrace c_1, \\ldots , c_7\\rbrace $ , and $V = (v_1, \\ldots , v_{15})$ .", "The committee size is $k = 5$ and the approval sets are as follows: Table: NO_CAPTION Clearly, the election is both CI and VI.", "We see that ${n}{k} = 3$ and, thus, for each $i \\in [7]$ , voters $v_{2i-1}, v_{2i}, v_{2i+1}$ form a cohesive group (for candidate $c_i$ ).", "Now consider some size-$k$ committee.", "If it does not contain some candidate $c_i$ , then the 1-cohesive group $\\lbrace v_{2i-1}, v_{2i}, v_{2i+1}\\rbrace $ must have average satisfaction below 1.", "Indeed, altogether members of this group give at most five approvals, of which three go to $c_i$ .", "Thus, without $c_i$ , the average satisfaction is at most $\\frac{2}{3} < 1$ .", "However, since the committee size is five and there are seven candidates, for each committee there is some 1-cohesive group with satisfaction below 1.", "Thus there is no committee with a perfect PD for this election and committee size five." ], [ "Conclusions and Future Work", "We have shown that computing committees with a given proportionality degree is, apparently, more difficult than computing EJR committees, but verification problems for these two notions have the same complexity.", "Two most natural directions of future work would be to establish the exact complexity of the PD-Committee problem and experimentally analyze PDs of committees provided by various voting rules." ], [ "Acknowledgments", "This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 101002854).", "Figure: NO_CAPTION" ], [ "Proof of Proposition ", "We prove Proposition REF via Lemmas REF and REF below.", "Lemma A.1 #Cohesive-Group $\\le _{\\mathrm {T}}^{\\mathrm {fp}}$ #FSCG Let $(E, k, \\ell )$ be an instance of #Cohesive-Group, where $E$ is an election instance with $m$ candidates and $n$ voters, $k$ is the committee size, and $\\ell $ is the cohesiveness level.", "To count the number of $\\ell $ -cohesive groups, it suffices to sum up the number of $\\ell $ -cohesive groups of each possible size $x$ .", "From the definition of an $\\ell $ -cohesive group, we know that its size must be at least $\\ell \\cdot \\frac{n}{k}$ .", "It is also clear that its size cannot exceed the number of voters.", "Thus, $\\#\\text{LCG}(E, \\ell , k) = \\sum _{x = \\lceil \\ell \\cdot \\frac{n}{k} \\rceil }^{n}{\\#\\text{LCG}(E, \\ell , k, x)}$ .", "This concludes the argument.", "We also have a reduction in the reverse direction, thus obtaining computational equivalence between #Cohesive-Group and #FSCG.", "Lemma A.2 #FSCG $\\le _{\\mathrm {T}}^{\\mathrm {fp}}$ #Cohesive-Group Let $(E, k, \\ell , x)$ be an instance of #FSCG, where $E$ is an election with $m$ candidates and $n$ voters, $k$ is the size of the final committee, $\\ell $ is the cohesiveness level, and $x$ is the size of the considered groups.", "We assume that $\\ell \\cdot \\frac{n}{k} \\le x \\le n$ and $1 \\le \\ell \\le m$ as otherwise we would immediately output zero as the answer.", "We create an election $E^{\\prime }$ which, initially, is a copy of $E$ .", "Next, we extemed $E^{\\prime }$ by adding $n \\cdot x \\cdot (x+1) - n$ new voters that do not approve any candidates and $n \\cdot x \\cdot (x+1) \\cdot m - m$ new candidates that are not approved by any voter.", "Thus, in $E^{\\prime }$ we have $n^{\\prime } = n \\cdot x \\cdot (x+1)$ voters and $m^{\\prime } = n \\cdot x \\cdot (x+1) \\cdot m$ candidates.", "The aim of adding the new voters and candidates is to establish the lower bound on the size of the cohesive groups.", "If we were able to count the number of $\\ell $ -cohesive groups with the size greater or equal to $x$ and those with the size strictly greater than $x$ , then the difference between these two values would be the number of $\\ell $ -cohesive groups with size $x$ .", "To use the idea from the preceding paragraph, we define $k_1^{^{\\prime }} = \\ell \\cdot n \\cdot x$ and $k_2^{^{\\prime }} = \\ell \\cdot n \\cdot (x+1)$ .", "It should be clear that $1 \\le k_1^{^{\\prime }} < k_2^{^{\\prime }} \\le m^{\\prime }$ .", "Further, we note that $\\ell \\cdot \\frac{n^{\\prime }}{k_1^{^{\\prime }}} = \\ell \\cdot \\frac{n \\cdot x \\cdot (x+1)}{\\ell \\cdot n \\cdot x} = x+1$ and, so, $\\#\\text{LCG}(E^{\\prime }, k_1^{^{\\prime }})$ is equal to the number of $\\ell $ -cohesive groups in $E^{\\prime }$ with size at least $x+1$ .", "Similarly, as $\\ell \\cdot \\frac{n^{\\prime }}{k_2^{^{\\prime }}} = \\ell \\cdot \\frac{n \\cdot x \\cdot (x+1)}{\\ell \\cdot n \\cdot (x+1)} = x$ , we have that $\\#\\text{LCG}(E^{\\prime }, k_2^{^{\\prime }})$ is equal to the number of $\\ell $ -cohesive groups in $E^{\\prime }$ with size at least $x$ .", "Furthermore, as newly created voters in $E^{\\prime }$ do not approve any candidates and newly created candidates in $E^{\\prime }$ are not approved by any voter, each $\\ell $ -cohesive group in $E^{\\prime }$ consists of the voters from $E$ approving only the candidates from $E$ , so it is also a valid $\\ell $ -cohesive group in $E$ .", "Thus, $\\#\\text{LCG}(E, k, \\ell ) = \\#\\text{LCG}(E^{\\prime }, k_1^{^{\\prime }},\\ell ) - \\#\\text{LCG}(E, k_2^{^{\\prime }},\\ell )$ .", "This completes the proof as we have shown a polynomial-time algorithm that solves #FSCG using oracle access to #Cohesive-Group." ], [ "Proof of Theorem ", "Theorem  REF PD-Verification is ${\\mathrm {coNP}}$ -complete.", "Let us show that $\\overline{\\textsc {PD-Verification}}$ is in ${\\mathrm {NP}}$ .", "Given a $\\overline{\\textsc {PD-Verification}}$ instance, we guess a group of voters and a cohesiveness level $\\ell $ .", "We can verify in polynomial time whether these voters form an $\\ell $ -cohesive group.", "Then we calculate the average satisfaction of the group and compare it with the given threshold.", "If the voters form an $\\ell $ -cohesive group and their average satisfaction is lower than the threshold, then the selected voters witness $\\overline{\\textsc {PD-Verification}}$ .", "Therefore $\\overline{\\textsc {PD-Verification}}$ is in ${\\mathrm {NP}}$ and $\\textsc {PD-Verification}$ is in ${\\mathrm {coNP}}$ .", "We show a reduction from the $\\textsc {PD-Failure}$ problem to the complement of the $\\textsc {PD-Verification}$ problem.", "Let $(E, W, k, \\ell , y)$ be a $\\textsc {PD-Failure}$ instance, where $E$ is an election, $W$ is a final committee of size $k$ , $\\ell $ is the cohesiveness level, and $y$ is a nonnegative real threshold $y \\le k$ .", "We ask if there exists an $\\ell $ -cohesive group whose average satisfaction is lower than $y$ .", "We create a $\\textsc {PD-Verification}$ instance as follows.", "We have the same election $E$ and the same committee $W$ .", "We set the PD function to be $f(\\ell ) = y$ , and 0 otherwise.", "Suppose that the answer for the $\\textsc {PD-Verification}$ instance is “no”.", "Then, for some $\\ell ^{\\prime }$ there exists an $\\ell ^{\\prime }$ -cohesive group $S$ whose average satisfaction is lower than $f(\\ell ^{\\prime })$ .", "It is quite clear that each $\\ell ^{\\prime }$ -cohesive group has average satisfaction at least 0, regardless of a selected committee.", "It means that $\\ell ^{\\prime }$ must be equal to $\\ell $ .", "Therefore $S$ is an $\\ell $ -cohesive group and has average satisfaction lower than $f(\\ell ^{\\prime }) = f(\\ell ) = y$ .", "Thus the answer for the $\\textsc {PD-Failure}$ instance is “yes”.", "Suppose that the answer for the $\\textsc {PD-Verification}$ instance is “yes”.", "Then, for each $\\ell ^{\\prime }$ and each $\\ell ^{\\prime }$ -cohesive group its average satisfaction is at least $f(\\ell ^{\\prime })$ .", "In particular, it means that each $\\ell $ -cohesive group has average satisfaction at least $f(\\ell ) = y$ .", "Therefore there does not exist an $\\ell $ -cohesive group that has average satisfaction lower than $y$ .", "Thus the answer for the $\\textsc {PD-Failure}$ instance is “no”.", "Since the $\\textsc {PD-Verification}$ problem is in ${\\mathrm {coNP}}$ , the $\\textsc {PD-Failure}$ problem is ${\\mathrm {NP}}$ -complete, and we reduced the $\\textsc {PD-Failure}$ problem to the complement of the $\\textsc {PD-Verification}$ problem, the $\\textsc {PD-Verification}$ problem is ${\\mathrm {coNP}}$ -complete." ] ]
2207.03518
[ [ "Anomalous transport in itinerant van der Waals ferromagnets\n Fe$_n$GeTe$_2$ (\\emph{n}=3, 4, 5)" ], [ "Abstract Ferromagnetic (FM) semimetals Fe$_n$GeTe$_2$(n=3, 4, 5), exhibit several symmetry-protected band-crossing points or lines near the Fermi energy (E$_F$) and these topological properties of energy bands lead to interesting transport properties.", "We study these materials employing the first-principle calculations and the tight-binding Hamiltonian constructed by fitting the parameters of the first principles calculation.", "In the presence of spin-orbit coupling (SOC) for n=3,5 a large Berry curvature (BC) concentrated on the nodal lines is observed.", "The consequence of the correlation of the topological nodal line and magnetic moments on anomalous Hall conductivity (AHC) $\\sigma_{xy}$ and anomalous Nernst conductivity (ANC) $\\alpha_{xy}$ have been investigated.", "We find $\\sigma_{xy}=150$ S/cm for n=3, 295 S/cm for n=4, and 90 S/cm for n=5 at 0 K, while the ANC is observed as $\\alpha_{xy}=0.55$ A/Km for n=3, 0.10 A/Km for n=5, and 0.80 A/Km for n=4, at the E$_F$ at room temperature.", "Our calculated AHC values at 0 K, i.e., 150 S/cm for Fe$_3$GeTe$_2$ and 90 S/cm Fe$_5$GeTe$_2$, are consistent with the experimentally reported values.", "Also the experimentally reported value of ANC for Fe$_5$GeTe$_2$ is close to our calculated value at room temperature, i.e., 0.10 A/Km." ], [ "INTRODUCTION", "Recent experimental discovery of two and three dimensional non-spatial symmetry protected topological insulating materials and their prospective applications inspired further prediction of a group of novel states protected by different symmetries, i.e., crystalline symmetry over the last few years[1], [2], [3].", "Furthermore, the concept of topological band insulator has been extended to topological semimetals (TSMs)[4], [5], [6], [7], [8], [9], [10].", "TSMs are described by crossing valance and conduction bands in the Brillouin zone (BZ); for example, the nodes formed by touching the conduction and valance bands at a discrete point are the Weyl/Dirac semimetals since the Dirac/Weyl equations govern their low excitation behavior[9], [10], [11].", "The recent addition is topological nodal line semimetals (TNLSMs), where bands cross each other along a line or closed-loop in BZ and, in principle, it can exist in quasi-2D[12] and 3D systems[13].", "In TNSLMs, the degenerate crossing point of the conduction and the valence bands near E$_F$ are protected by the crystal and time-reversal symmetries (TRS).", "Perturbations on the Hamiltonian cannot lift the degeneracy without breaking any of its symmetries.", "By breaking time-reversal or spin parity symmetry, these systems may have fully gapped nodal lines or gapped into nodal points.", "For example, the first principles calculations demonstrate that the electronic band structure of TaAs [14], [15] in the absence of SOC exhibits two nodal lines, which are protected by mirror reflection and spin-rotation symmetries.", "Each nodal line gaps into three pairs of Weyl nodes in the presence of SOC.", "A double nodal line in SrIrO$_3$ is another example of nodal lines gapping into a pair of Dirac nodes when a mirror reflection symmetry is broken [16], [17].", "These nontrivial topological energy bands lead to many exotic phenomena, such as the AHE and ANE.", "ANE is a thermoelectric counterpart of the AHE, both of which are associated with the BC[18].", "Moreover, compared to AHE, which investigates the Berry curvature of the whole Fermi sea, ANE is exposed to the BC near the Fermi Energy(E$_F$ ).", "Consequently, with an enhanced Berry curvature at the nodal line near the E$_F$ , ANE evolves as the general term of the total Nernst signal in TNLSMs.", "Recently, the van der Waals (vdW) ferromagnet Fe$_3$ GeTe$_2$ has been demonstrated to be a magnetic variant of TNLSM, wherein the large BC induced AHC has been attributed to the presence of a gapped nodal line near E$_F$[19].", "Later related iron rich compounds in the series Fe$_4$ GeTe$_2$ and Fe$_5$ GeTe$_2$ were prepared[19].", "Interestingly, both are vdW compounds and exhibit ferromagnetic behavior.", "This motivated us to investigate the interplay between magnetism and topology in the magnetic vdW materials series, Fe$_n$ GeTe$_2$ (n=3, 4, 5) in order to understand their potential anomalous transport properties.", "Figure: Three stable vdW structures in the series of Fe 3 _3GeTe 2 _2 for n=3, 4, 5.", "The upper panels (a), (b), and (c) are side views of the structures Fe 3 _3GeTe 2 _2, Fe 4 _4GeTe 2 _2, and Fe 5 _5GeTe 2 _2 respectively.", "While the lower panels(d), (e), and (f) are corresponding top views and d represents the inter-layer distance.Owing to the dearth of vdW ferromagnets, Fe$_n$ GeTe$_2$ have recently attracted a lot of attention.", "This series of compounds are also technologically important due to the large Curie temperature (T$_C$ ); T$_C$ = 230 K,270 K and 293 K forFe$_3$ GeTe$_2$ [20], [19], Fe$_4$ GeTe$_2$[21] and Fe$_5$ GeTe$_2$[22], respectively.", "Chen et al.", "showed that T$_C$ of Fe$_5$ GeTe$_2$ could be further enhanced to 478 K upon Ni-doping[23].", "The monolayer of these systems shows interesting magnetic behavior, and the thin atomic layer can be tuned using the femtosecond laser[24].", "Fe$_3$ GeTe$_2$ also indicated towards high electronic correlation in terms of the emergence of Kondo behavior[25].", "The Density functional theory study of Fe$_3$ GeTe$_2$ and Fe$_4$ GeTe$_2$ monolayers show that these systems have large magnetic anisotropy under an electric field[26].", "Fe$_5$ GeTe$_2$ shows butterfly type of magneto-resistance[27].", "The structural characteristics of Fe$_n$ GeTe$_2$ could be understood in terms of the triangular Fe-layers and Fe-Fe dumbbells.", "In Fe$_3$ GeTe$_2$ , Fe1 makes a triangular lattice when considered together with Ge,which can be viewed as honeycomb lattice.", "Fe2-Fe2 dumbbells align perpendicular to the layer, centered at each hexagon of the honeycomb, with one Fe2 above the hexagon and another Fe2 below the hexagon.", "Fe$_3$ GeTe$_2$ belongs to space group P63/mmc (No.194).", "Fe$_4$ GeTe$_2$ consists of two types of dumbbells each displaced with respect to the other along the direction perpendicular to the vdW layer.", "Fe$_4$ GeTe$_2$ has a rhombohedral structure with space group R$\\bar{3}$ m (No.166).", "The structure of Fe$_5$ GeTe$_2$ also consists of two types of dumbbells as in the case of Fe$_4$ GeTe$_2$ , in addition to a triangular layer of Fe.", "The Space group of Fe$_5$ GeTe$_2$ is P3m1 (No.156)[19].", "These materials also have C$_{3y}$ symmetry in common in addition to a mirror or C$_2$ symmetry.", "The top and side views of Fe$_n$ GeTe$_2$ structures have been depicted in Fig.REF .", "where different Fe atoms are distinguished with separate colors.", "In this manuscript, we perform a comparative study of the electronic properties of Fe$_n$ GeTe$_2$ (n=3, 4, 5) to explore the origin of the different topologies in Fe$_n$ GeTe$_2$ .", "We also study AHE and ANE using a tight-binding Hamiltonian constructed by fitting parameters with density functional theory (DFT).", "The magnetic moments of all these materials are close to each other; therefore, our main focus is to highlight the correlation between topology and magnetism.", "The remainder of this paper is organized as follows.", "In Sec.", "we briefly discuss the computational details, followed by a discussion on the magnetism of these three materials in Sec.", "and Sec.", "and Sec.", "summarise our results on the nontrivial band topology, nodal lines, and corresponding Berry curvature distribution.We present our paper's primary focus, anomalous transport, in Sec.", "and conclude with a summary and conclusion in Sec.", "Figure: (a) The total DOS for Fe 3 _3GeTe 2 _2 as a function of (E-E F _F) is plotted where the blue and red curves represent the total DOS for the majority spin carrier (solid) and minority spin carrier(dotted)of Fe1 and Fe2, respectively.", "The magnetic moments, M Fe1 _{Fe1} M Fe2 _{Fe2}, are shown with the black (green) curve, the main contribution from the majority spins.", "(b) Similarly the total DOS for Fe 4 _4GeTe 2 _2 as a function of (E-E F _F) is plotted where the blue and red curves represent the total DOS for the majority spin carrier (solid) and minority spin carrier(dotted)of Fe1 and Fe2, respectively.", "The magnetic moments, M Fe1 _{Fe1} M Fe2 _{Fe2}, are shown with the black (green) curve.", "(c)the total DOS for Fe 5 _5GeTe 2 _2 as a function of (E-E F _F) is plotted where the red, green, blue, deep green, orange curves represent the total DOS for the majority spin carrier (solid) and minority spin carrier (dotted) of Fe1 and Fe2, Fe3,Fe4, Fe5 respectively." ], [ "Theory and Computational Details", "We examine the electronic structures and transport properties of Fe$_n$ GeTe$_2$ in the scheme of DFT.", "First principles calculations are performed using the VASP package [28] based on the generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof (PBE) for the exchange-correlation functional.", "A plane-wave basis set with a kinetic energy cutoff of 600 eV is considered while performing first-principles calculations.", "Furthermore, we have used the BZ sampling, a 6$\\times $ 6$\\times $ 1 k-point mesh forFe$_3$ GeTe$_2$ and Fe$_5$ GeTe$_2$ , and a 6$\\times $ 6$\\times $ 6 k-point mesh for Fe$_4$ GeTe$_2$ .", "The Gaussian smearing method is acquired for broadening the Fermi surface with a width of 0.05 eV.", "Both cell parameters and internal atomic positions were fully relaxed until the forces on all atoms were smaller than 0.01 eV/A.", "To explore the nontrivial band topology and the intrinsic AHE, the tight-binding Hamiltonian was constructed with the maximally localized Wannier functions [29], [30].", "The intrinsic AHC is computed using the linear-response Kubo formula approach in the clean limit, and a 500$\\times $ 500$\\times $ 500 k-grid in the BZ was used for the integral of the AHC." ], [ "Magnetic Properties", "In this section, the magnetic properties of all three vdW compounds Fe$_3$ GeTe$_2$ , Fe$_4$ GeTe$_2$ and Fe$_5$ GeTe$_2$ are studied and we calculate spin resolved total density of states (DOS) to understand the magnetic moments.", "We notice that the d-orbitals of Fe atoms dictate the magnetism; therefore, we show only the DOS of the d-orbitals where down DOS is shown as negative of its value.", "We calculate the total magnetic moment of Fe atoms as a difference of total up and down spin electrons filled up to E$_F$ and density of up or down electron at any energy is proportional to the DOS of up and down spins at that energy.", "Fe$_3$ GeTe$_2$ : In this compound, there are two in-equivalent Fe atomic sites denoted as Fe1 and Fe2, as shown in Fig.REF (a), Fe1 lies in a-b plane and arranges on hexagonal structure, whereas Fe2 lies perpendicular to this plane.", "Energy dependence spin up (solid line) and down ( dashed line) DOS are shown in Fig.REF (a): red lines represent the DOS of Fe1 atoms, whereas the blue line represents DOS of Fe2 atoms.", "The partial DOS calculations suggest that the DOS of all d-sub-orbitals are spread near the E$_F$ , therefore, electrons of all the sub-orbitals are delocalised.", "The DOS of Fe1 compared to Fe2 has a higher magnitude at -2.5 eV below the E$_F$ , whereas the DOS of Fe2 has a marginally larger value compared to Fe2 between the energy range 2 eV to the E$_F$ .", "We note that the difference between up and down DOS of Fe2 atoms have higher a spread than that of Fe1 atoms; therefore, we expect a higher magnetic moment for Fe1 than Fe2.", "The magnetic moments M$_{Fe1}$ and M$_{Fe2}$ for Fe1 and Fe2 atoms are shown as solid black and green lines for Fe1 and Fe2 atoms in Fig.REF (a).", "The magnetic moment contribution of Fe1 is higher than that of Fe2 and these are M$_{Fe1}$ =2.5 $\\mu _B$ /f.u.", "and M$_{Fe2}$ =1.5 $\\mu _B$ /f.u.", "respectively at zero temperature.", "In Fe$_3$ GeTe$_2$ , the average contribution magnetic moment of Fe is M$_{Fe}$ =2.12 $\\mu _B$ /f.u and it matches very well with experimentally reported value[19].", "Figure: The band structure of Fe n _nGeTe 2 _2 without and with SOC.", "After opening a gap, there is a Berry curvature along symmetry lines.", "(a) For Fe 3 _3GeTe 2 _2, the crossing point at K, 0.03 eV above the Fermi energy.", "The gap is opened below the Fermi energy due to SOC.", "(b) The crossing point is between K and M points at the Fermi energy for Fe 4 _4GeTe 2 _2.", "(c) The crossing point at K point below the Fermi energy in Fe 5 _5GeTe 2 _2.", "With SOC, the gap opens : (d) at the K point below the Fermi energy for Fe 3 _3GeTe 2 _2, (e) between K and M point at the Fermi energy for Fe 4 _4GeTe 2 _2, and (f) at the K point below the Fermi energy.", "A large negative Berry curvature: (g) at the K point for Fe 3 _3GeTe 2 _2, (h) between K and M point for Fe 4 _4GeTe 2 _2, and (i) at the K point for Fe 5 _5GeTe 2 _2.Fe$_4$ GeTe$_2$ : This system also has two in-equivalent Fe sites, labeled as Fe1 and Fe2 in Fig.REF (b).", "Fe2 atoms lying below the Te atom of hexagonal plaquettes and Fe1 atom is present in hexagonal plaquettes.", "The up and down spin DOS are plotted as a function of energy with solid and dotted line.", "Red and blue colour represent DOS of Fe1 and Fe2 d-orbitals.", "The Fe2 $d$ -bands are more dispersed compared to Fe1.", "The DOS of Fe1 has a larger amplitude at -2.8 eV below E$_F$ for spin up electron than Fe2 DOS, as shown in Fig.REF (b).", "The magnetic moments M$_{Fe1}$ and M$_{Fe2}$ for atoms Fe1 and atoms Fe2 are shown as solid black and green lines for Fe1 and Fe2 atoms in Fig.REF (b).", "Magnetic moments of Fe1 and Fe2 are 1.7 and 2.47$\\mu _B$ /f.u and the average magnetic moment of Fe is M$_{Fe}$ =2.084 $\\mu _B$ /f.u, which agrees well with the experimental value[19] .", "Figure: For Fe 3 _3GeTe 2 _2 with SOC: (a) 2D Fermi surface using spectral function A(k x _x,k y _y, k z _z=0, E f _f).", "(b) Energy gap, ΔE(k x ,k y ,k z =0)=(E a (k x ,k y ,k z =0)-E b (k x ,k y ,k z =0))\\Delta E(k_x,k_y,k_{z}=0) = (E_{a}(k_x,k_y,k_{z}=0)-E_{b}(k_x,k_y,k_{z}=0)), between two crossing bands (a(red) and b(blue)) and the high symmetry points are also shown.", "(c) The difference in Berry curvature distribution, ΔΩ z (k x ,k y ,k z =0)=(Ω z a (k x ,k y ,k z =0)-Ω z b (k x ,k y ,k z =0))\\Delta \\Omega _z(k_x,k_y,k_z=0)=(\\Omega _{z}^{a}(k_x,k_y,k_z=0)-\\Omega _{z}^{b}(k_x,k_y,k_z=0)) of two crossing bands, has a large negative value at the K point (red spot).", "(d) Energy gap, ΔE(k y ,k z ,k x =0)\\Delta E(k_y,k_z,k_{x}=0), and gapped nodal at k y {k}_y=1.2.", "(e) the difference of the Berry curvature distribution between two bands, ΔΩ z (k y ,k z ,k x =0)\\Delta \\Omega _z(k_y,k_z, k_{x}=0).Fe$_5$ GeTe$_2$ : It contains five in-equivalent Fe sites which are shown in Fig.REF (c) and two atoms Fe1 and Fe4 are in the same a-b plane.", "Fe2 and Fe3 are also in same plane and have a similar chemical environments, but have different bonding along the c-axis.", "d-orbitals of these atoms are also highly hybridized.", "The DOS of all five Fe d-orbitals are shown in Fig.REF (c) and DOS of Fe1 and Fe4 looks similar, whereas it is distinct for other three Fe atoms.", "The Green curve represents the large peak in up DOS of Fe2 at -3.0 eV below $E_F$ shows localization magnetic moment and it is contributed from d$_{z^2}$ sub-orbitals.", "Fe2 atoms have the highest magnetic moments and Fe5 and Fe3 have the lowest contribution.", "Fe5 and Fe3 d-orbitals have higher DOS for both up and down spins near the $E_F$ .", "The colored thick lines represent the magnetic moments for all five Fe atoms and the largest magnetic moment is M$_{Fe2}$ =2.55$\\mu _B$ /f.u.", "for Fe2.", "Magnetic moment of Fe1 and Fe4 are nearly the same M$_{Fe1}$ =2.4$\\mu _B$ /f.u.", "and M$_{Fe4}$ =2.3$\\mu _B$ /f.u.", "respectively, whereas the magnetic moment of Fe$_3$ and Fe5 are M$_{Fe3}$ =1.29$\\mu _B$ /f.u.", "and M$_{Fe5}$ =1.84$\\mu _B$ /f.u.", "The average magnetic moment per Fe atom is M$_{Fe}$ =2.064$\\mu _B$ /f.u., which agrees with the experimental reported value.", "[31]." ], [ "Nontrivial band topology", "We study the topology of energy bands of these three materials and analyze the energy band crossovers.", "The contributions of various sub-orbitals of Fe d-orbitals to bands near E$_F$ are also investigated.", "Fe$_3$ GeTe$_2$ : The energy dispersion curve near E$_F$ is shown in Fig.REF (a) without SOC and in Fig.REF (b) with SOC for this material.", "As shown in Fig.3(a), in the absence of SOC, two bands, represented by 'a' (thick red line) and 'b' ( thick black line), cross at point K, 0.03 eV above E$_F$ .", "These two bands are solely contributed from d orbitals of Fe1 and Fe2 atoms.", "These two bands are solely contributed from d orbitals of Fe1 and Fe2 atoms.", "We observe that the bands crossing resemble Mexican-hat shapes induced by the Rashba effect with quenched spins, as reported in the literature[19].", "The degeneracy at the K point arises due to C$_{2y}$ symmetry in the crystal and on the application of SOC along 001 direction, lifts this degeneracy due to breaking of the time-reversal symmetry.", "We also find some other band crossings which contribute large BC contribution which lie along the high symmetry direction, M-$\\Gamma $ and $K$ -$\\Gamma $ , as shown in Fig.REF (a) .", "Non-trivial crossover points open a gap in the presence of the SOC as shown in Fig.REF (b).", "Therefore, we investigate the BC (It is defined in Eqn (1) in the Sec. .)", "of these bands in the above mentioned high symmetry directions.", "The total BC has a large magnitude corresponding to these at nontrivial band crossing close to the E$_F$ , as shown in Fig.REF (c).", "It shows the largest negative BC at the K point and relatively smaller values along the M-$\\Gamma $ , and $K$ -$\\Gamma $ .", "Fe$_4$ GeTe$_2$ :It has a rhombohedral structure, and the space group is R$\\bar{3}$ m. This material also has three-fold rotational symmetry about the z-axis (c-axis) and two-fold rotation about the y-axis.", "All the energy bands close to the E$_F$ are contributed by Fe atoms, Many energy bands cross close to E$_F$ , as shown in Fig.REF (d), and these bands are formed from the d$_{xz}$ ,d$_{yz}$ , and d$_{x^2-y^2}$ sub-orbitals of Fe atoms.", "The nontrivial degeneracies near the E$_F$ are along the K-M and K-$\\Gamma $ symmetry points in the absence of the SOC.", "However, the presence of the SOC along 001 breaks the TR symmetry and lifts the degeneracy, as shown in Fig.REF (e).", "The BC in this system shows a large negative value along the K-$\\Gamma $ symmetry point, as shown in Fig.REF (f).", "Fe$_5$ GeTe$_2$ : This material belongs to P3m1 symmetry group and has three-fold rotational symmetry along the z-axis (c-axis) and mirror symmetry along the y-axis.", "Most energy bands are contributed from the d$_{xz}$ ,d$_{yz}$ and d$_{x_{2}-y_{2}}$ sub-orbitals of Fe atoms.", "A prominent band crossing -0.11 eV below E$_F$ is detected at the K point, and the participating bands are occupied by minority spins, as shown in Fig.REF (g).", "As was mentioned earlier, the crossing point is encircled with a red dotted circle in the absence of the SOC.", "There are two other crossings close to K points along the K-M and K-$\\Gamma $ , 0.1 eV below the E$_F$ .", "But in the presence of the SOC, gap is opened at these points as shown in Fig.REF (h).", "The Berry curvature along the K-M and K-$\\Gamma $ symmetry points are shown in Fig.REF (i).", "We notice a large negative berry curvature at the K point and two other crossings around, the K point as mentioned earlier.", "Figure: For Fe 5 GeTe2Fe_5GeTe2 with SOC: 2D Fermi surface, the energy gap, ΔE(k y ,k z ,k x =0)\\Delta E(k_y,k_z,k_x=0), and the difference of Berry curvature distribution are plotted in (a),(b), and (c), respectively." ], [ "Nodal line and Berry curvature distribution", "This section highlights the band crossing feature by the 2D Fermi surface using the spectral function A(${\\bf k}$ , $E_f$ ) computed from wanniertools[32], the energy gap, $\\Delta E$ (${\\bf k}$ ), between the band crossing, and the difference of the z component of the Berry curvature, $\\Delta \\Omega _z({\\bf k})$ , at each point of the BZ.", "We usually observe large BC at the crossing points or the nodal lines.", "We show our results in the $k_z$ =0, $k_x$ =0 and $k_y$ =0 planes.", "Fe$_3$ GeTe$_2$ : Fig.REF (a) shows the FS plot in $k_z=0$ plane.", "The calculated FS consists of a circular-shaped pocket centered at the six K-point and a hexagonal-shaped pocket centered at $\\Gamma $ .", "The red region shows a high spectral function value, and the blue has values close to zero, i.e., a small spectral function.", "The spectral function's intense value reveals the band's degeneracy and crossing between different bands.", "Most of the region is blue, i.e., low, low spectral function regime, except the red line on $\\Gamma $ points and six k points exhibit high spectral function.", "The yellow color represents finite gaps, whereas the white and red regions show tiny gaps opened at the high symmetry points due to the SOC.", "In Fig.REF (b), gap distribution , $\\Delta E(k_x,k_y,k_z=0)$ , is shown, and a gap smaller than 10$^{-5}$ eV is considered zero.", "Mainly the white region is at K points, and corresponding to this gap, there is a finite BC, as shown in Fig.REF (c) with the red spot.", "We also studied the energy gap in $k_x=0$ plane and observed a gapped line, as shown in Fig.REF (d) by a white line (at K$_y$ =1.2 Å$^{-1}$ ) called a gapped nodal line.", "The BC is large at this nodal line, and we show it in Fig.REF (e) by a red line(at K$_y$ =1.2 Å$^{-1}$ ).", "Figure: For Fe5 with SOC : (a) 2D Fermi surface, (b) ΔE(k x ,k y ,k z =0)\\Delta E(k_x,k_y,k_z=0), (c)ΔΩ z (k x ,k y ,k z =0)\\Delta \\Omega ^z(k_x,k_y,k_z=0), (d) ΔE(k y ,k z ,k x =0)\\Delta E(k_y,k_z,k_x=0) and (e) ΔΩ z (k y ,k z ,k x =0)\\Delta \\Omega ^z(k_y,k_z,k_x=0)Fe$_4$ GeTe$_2$ : In Fig.REF (a), 2D FS using the spectral function is plotted in the K$_x$ =0 plane, and there is a $\\Gamma $ -centered hexagonal-shaped and six K and M point-centred circular-shaped FS.", "Like the previous system, the red region shows a high spectral function value indicating band degeneracy and the crossing point.", "The blue has values close to zero, i.e., a small spectral function.", "The intense value of the spectral function at the red dots around the $\\Gamma $ points and red lines around K and M points indicate band crossing and band degeneracy.", "The higher symmetry around M points has consequences on the transport properties.", "The SOC opens tiny gaps around the degenerate points at $\\Gamma $ , along the K-M direction, as shown in Fig.REF (b).", "White spots between the K-M line and $\\Gamma $ points correspond to gapped regions, and BC at these points are high, as shown in Fig.REF (c) by red spots.", "The gaps and BC are also calculated in the K$_y=0$ plane, but there is not much feature associated with it.", "Gaps and finite BC are scattered in the first BZ; therefore, we safely conclude the absence of a nodal line.", "Fe$_5$ GeTe$_2$ : $Fe_{3}GeTe_{2}$ and $Fe_{5}GeTe_{2}$ have similar symmetries; therefore, the spectral function behavior and its symmetry are expected to be similar, as can be seen in Fig.REF (a) of the contour plot of FS.", "Most of the region is blue with a low spectral function value, except the red line with a high spectral function value in the neighborhood of $\\Gamma $ points and six K points, reflecting the band degeneracy and the band crossing.", "In Fig.REF (b), gap distribution, $\\Delta E(k_x,k_y)$ , is shown in the first BZ, and blue and white regions show tiny gaps opened at the high symmetry points due to the SOC.", "Mainly the white region is around the K points, and corresponding to this gap, there is a finite BC, as shown in Fig.REF (c) with the red spot.", "In K$_x$ =0 plane, white stripes represent the gaped nodal line as shown in Fig.REF (d), and there is a large BC associated with it as shown in Fig.REF (e) by a red and blue line(at K$_y$ =1.2 Å$^{-1}$ )." ], [ "Anomalous transport properties", "We also study anomalous transport properties described by a tight-binding Hamiltonian, $H$ , constructed by fitting its parameters with DFT results.", "As discussed in the previous section, these materials exhibit a large BC on the nodal lines in the presence of SOC.", "The BC (${\\bf \\Omega }$ ) acts as a magnetic monopole in the momentum space, giving rise to anomalous Hall and heat conductivities.", "The computation of the BC requires the energy eigenvalues $\\epsilon _{n{\\bf k}}$ and the energy eigenstates $\\; \\vert n({\\bf k)} \\rangle $ where ${\\bf k}$ and n are the wave vector and band index respectively.", "${{\\bf \\Omega }({\\bf k})}$ is given as: $\\begin{aligned}& {\\bf \\Omega ^{n} ({\\bf k})} = \\\\ & i\\sum _{n \\ne m} \\frac{\\langle n({\\bf k}) \\vert \\nabla _{\\bf k}H({\\bf k}) \\vert m({\\bf k})\\rangle \\times \\langle m({\\bf k}) \\vert \\nabla _{\\bf k}H({\\bf k}) \\vert n({\\bf k}) \\rangle }{(\\epsilon _{n {\\bf k}}-\\epsilon _{m\\bf k})^2}.\\end{aligned}$ The sum of the BC defines the anomalous Hall conductivity ($\\sigma _{xy}$ ) for all the occupied bands up to the Fermi energy E$_F$ .", "$\\sigma _{xy}$ is defined as: $\\sigma _{xy} = -{\\frac{e^2}{\\hbar }} \\sum _{n}\\int \\frac{d^3 k}{(2\\pi )^3}\\Omega ^n_z({\\bf k})f_n({\\bf k})$ where f$_n(\\vec{k})$ is the Fermi distribution function corresponding to the nth eigenstate and $\\Omega _z^n({\\bf k})$ is a z-component of $\\bf {\\Omega }^n ({\\bf k}).$ The BC-effect also manifests in thermoelectric transport is driven by a statistical force, for example, temperature gradient.", "In the presence of a temperature gradient, the local current of carriers acquires an additional term from the carrier's magnetic moment in the presence of a non-uniform distribution.", "The extra term, an extrinsic Hall current ${\\bf j}_{in}$ , can be written in terms of the BC as, ${\\bf j}_{in}=-\\frac{{\\bf \\nabla } T}{T} \\times \\frac{e}{h}\\sum _{n}\\int {} {d{\\bf k}} \\Omega ^n({\\bf k})[ (\\epsilon _{n\\bf k}-\\mu )f_n({\\bf k})&& \\nonumber \\\\+k_B T log(1+e^{\\beta (\\epsilon _{n\\bf k}-\\mu )}]$ the Nernst conductivity $\\alpha _{xy}$ can be extracted using $j_x=\\alpha _{xy}(-\\Delta T_y)$ and the $\\alpha _{xy}$ can be written as $\\alpha _{xy} = -\\frac{1}{e} \\int d\\epsilon \\frac{-df(\\epsilon )}{d\\mu } \\sigma _{xy}(\\epsilon ) \\frac{(\\epsilon -\\mu )}{T} $ where intrinsic anomalous Hall conductivity $\\sigma _{xy}(E_f)$ at zero temperature with Fermi-energy $E_f$ .", "Figure: (a) Red, green, blue curve represent Energy (E-E F E-E_F) dependence of the AHC For Fe 3 _3GeTe 2 _2, Fe 4 _4GeTe 2 _2 and Fe 5 _5GeTe 2 _2 respectively.", "(b) Red, green, blue curve represent Energy (E-E F E-E_F) dependence of the ANE For Fe 3 _3GeTe 2 _2, Fe 4 _4GeTe 2 _2 and Fe 5 _5GeTe 2 _2 respectively.All three materials show the variation in AHC as (E-E$_F$ ) varies, which is prominent in n = 3, 5 compared to n=4 and for n = 3 as the energy (E-E$_F$ ) varies, AHC first decreases from 300 S/cm to -400 S/cm at (E-E$_F$ ) = 0.1 eV and then increases as (E-E$_F$ ) approaches zero and acquires a value of 150 S/cm.", "This material reveals the large BC at the nodal line, as seen in Fig.REF (c), which causes a high value of AHC.", "The SOC opens the gap in the energy bands below the Fermi energy, $E_F$ , which contributes to a large variation in BC.", "Further, we can tune its values by doping, for example, up to 600 S/cm by electron doping and 450 S/cm by hole doping.", "Our result is consistent with the experimental value of AHC, 150 S/cm, at a low temperature.", "In n=4 the six-fold degeneracy around K and M points lying close to E$_F$ contributes to the large value of AHC, 295 S/cm at (E-E$_F$ ) = 0.4 eV decrease to -100S/cm at (E-E$_F$ ) = 0.3 eV and at E$_F$ its value reaches to 300 S/cm.", "In the case of $Fe_{5}GeTe_{2}$ , the various bands contribute to BC, which is responsible for fluctuating AHC with (E-E$_F$ ).", "The calculated AHC is 90 S/cm and most of the contribution comes from the nodal line below the E$_F$ .", "We now discuss ANE, which is intimately related to AHE via anomalous thermoelectric response tensor and emerges when charge carriers acquire an anomalous transverse velocity in a longitudinal temperature gradient and a finite BC.", "AHE probes the BC of the whole Fermi sea, while ANE is sensitive to the BC of the near Fermi energy.", "As a result, ANE may become the dominant term of the total Nernst signal in this series of materials with an enhanced BC near the Fermi energy.", "Fig.REF (b) shows ANE at room temperature and as a function of (E-E$_F$ ) for all three materials.", "The behavior of Fe$_3$ GeTe$_2$ and Fe$_5$ GeTe$_2$ are similar but opposite to Fe$_4$ GeTe$_2$ .", "Near Fermi energy E-E$_F$ = 0, Fe$_4$ GeTe$_2$ has a large value while Fe$_3$ GeTe$_2$ and Fe$_5$ GeTe$_2$ have the small value and are close to each other.", "This suggests that band crossing near the Fermi energy in all these materials is different, as evident in the band structure calculation discussed in the previous section.", "At room temperature, the anomalous Nernst conductivity is observed 0.55 A/Km for Fe$_3$ GeTe$_2$ and 0.10 A/Km for Fe$_5$ GeTe$_2$ .", "The observed Nernst conductivities at room temperature are 0.55 A/Km for Fe3GeTe2, 0.10 A/Km for Fe$_5$ GeTe$_2$ , and 0.8 A/Km for Fe$_5$ GeTe$_2$ .", "The experimentally observed value of ANE for Fe$_5$ GeTe$_2$ , 0.15 A/Km, is close to our calculated value." ], [ "Summary and Conclusions", "In this manuscript, we compare the electronic properties of Fe$_n$ GeTe$_2$ (n=3, 4, 5) using the ab-initio theory and showed these materials are semimetal and have interesting topological electronic energy bands and finite magnetization.", "Fe$_3$ GeTe$_2$ and Fe$_5$ GeTe$_2$ have a nodal line in the electronic band, whereas the Fe$_4$ GeTe$_2$ no-trivial energy band crossing in the neighborhood of the K and M high symmetry points.", "All these three systems have three-fold rotational symmetry about the c-axis and either two-fold mirror symmetry or rotational symmetry perpendicular to this axis and these symmetries get reflected in spectral densities at the Fermi-surface as shown in Fig.REF (a), Fig.REF (b) and Fig.REF (c).", "We also notice that Fe$_3$ GeTe$_2$ system shows extra non-trivial degenerate points along $K-\\Gamma $ and $M-\\Gamma $ high symmetry line other than the bands crossing at the $K$ points found in ref.", "[19].", "We also show that in the presence of the SOC the time-reversal symmetry breaks and the energy degeneracies at high symmetry points get lifted.", "We observe the finite magnetic moments of Fe atoms which are consistent with the experimental results in the literature [19].", "The finite magnetic moments and non-zero BC in momentum space at these non-degenerate points lead to the anomalous Hall and Nernst conductivity.", "The experimentally reported AHC [19] at the low temperature is consistent with our predicted value, 150 S/cm and 90 S/cm, for Fe$_3$ GeTe$_2$ and Fe$_5$ GeTe$_2$ , respectively.", "Among the three Fe$_4$ GeTe$_2$ has the largest AHC, 295 S/cm, due to the higher contribution of the BC stemming from the near degenerate energy bands around K and M symmetry points.", "We obtain the ANC at the room temperature, 0.55 A/Km, 0.80 A/Km, and 0.10 A/Km, for Fe$_3$ GeTe$_2$ , Fe$_4$ GeTe$_2$ , and Fe$_5$ GeTe$_2$ , respectively.", "The theoretical value of ANC, 0.10 A/Km, for Fe$_5$ GeTe$_2$ is close to the experimentally reported value.", "The large ANC in these materials offers promising applications as the new generation of thermoelectric energy conversion devices.", "In summary, we have studied the electronic band structure of Fe$_n$ GeTe$_2$ (n=3, 4, 5) materials and explored the topology in energy bands.", "The magnetic moments of all three materials are close but our results showed that anomalous transport properties are not the same since the nontrivial topological points in the energy bands having large BC appearing at the different symmetry points in the BZ in each system." ], [ "Acknowledgement", "M.K.", "thanks Professor Atindra Nath Pal, Professor Prabhat Mandal and Professor Thirupathaiah Setti for fruitful discussion.", "J.S.", "thanks U.G.C for financial support.", "M.K thanks SERB for financial support through Grant Sanction No.CRG/2020/000754.", "N.K.", "thanks SERB for financial support through Grant Sanction No.CRG/2021/002747." ] ]
2207.03547
[ [ "X-haul Outage Compensation in 5G/6G Using Reconfigurable Intelligent\n Surfaces" ], [ "Abstract 5G network operators consider the dense deployment of small base-stations (SBSs) to increase network coverage and capacity.", "Hence, operators face the challenge of X-hauling, i.e., backhauling or fronthauling, their traffic to the core network.", "Also, SBSs densification will increase the possibility of failure of these X-haul links.", "To cope with this problem, an X-haul outage compensation scheme with the assistance of Reconfigurable Intelligent Surfaces (RIS) is proposed to mitigate or at least alleviate the effect of X-haul failure.", "The RIS is a newly adopted technology that is able to improve the performance of wireless networks.", "In this paper, we present and evaluate an X-haul outage compensation scheme based on placing a number of RIS panels in pre-planned locations to mitigate the effect of X-haul failure.", "This evaluation is done using frequencies below and beyond 6 GHz.", "Based on our analytical results, the proposed RIS scheme shows that placing a sufficient number of RIS elements in proximity to the failed SBS under certain conditions can help acquire the same X-haul rate before the occurrence of the failure.", "Also, we show that for high X-haul spectral density, the RIS-assisted transmission with a certain number of elements can be more energy-efficient than line-of-sight and non-line-of-sight transmissions.", "Finally, the system's energy efficiency is addressed with and without RIS, and the optimal number of RIS reflecting elements is derived." ], [ "Introduction", "The 5G Radio Access Network is commercially deployed in many countries.", "Service providers are faced with new technologies, access methods, applications, and traffic compared to past generations.", "They will troubleshoot both known and never-before-seen issues in previous cellular generations or the rigorous R&D validation environment.", "5G introduces millimeter-waves (mmWaves), wider bandwidths, massive multi-antenna schemes, Small Base-stations (SBSs), and network function virtualization and orchestration [1].", "According to the small cell forum [2], deployments of 5G SBSs will rise steeply to reach 5.2 million in 2025.", "Besides, the international organization named GSMA, which represents the interests of more than 750 cellular operators and manufacturers (www.gsma.com), states that 5G total connections will pass the 1 billion mark by the end of 2024.", "As adoption grows, 5G revenues will swell, reaching $1.15 trillion by 2025.", "Therefore, SBSs densification in a given area is a promising technique that provides a huge capacity gain and brings SBSs closer to Users Equipment (UEs).", "This means that 5G/6G networks will have to handle more SBSs and X-haul, i.e., backhaul or fronthaul, links than before [3].", "A survey [4] showed that 56% of operators consider backhaul as one of the most significant challenges in 5G/6G networks.", "This is mainly because most of the SBSs do not always have access to wired backhaul.", "Self-healing is the execution of actions that keep the network operational and/or prevent disruptive problems from arising.", "Self-healing, or specifically, cell outage compensation, executes actions to mitigate or, at least, alleviate the effect of most failures.", "In the literature, a significant amount of work [5]-[7] addressed the mitigation of the failure of the BS itself.", "However, few works addressed the X-haul, i.e., backhaul/fronthaul, outage compensation.", "The 5G-xHaul project [8] proposed the use of a redundant mmWave X-haul link.", "This solution is excellent for low latency applications; however, it will consume double the resources.", "The authors in [9] proposed to use additional radio named self-healing radio to be used only in case of backhaul failure.", "To the best of our knowledge, our work is the first to address RIS usage to mitigate backhaul failure.", "Recently, RIS has arisen as a promising technology that reconfigures the propagation environment to improve wireless networks' performance.", "RIS is a planar surface that contains a large number of passive, low-cost reflecting elements.", "Each element is able to induce a change in the amplitude and/or the phase of the incident signal, which proactively changes the wireless channel via its controllable reflecting surface.", "This provides a new degree of freedom for the enhancement of the 5G performance and paves the way to a programmable wireless environment.", "Since RIS eliminates the use of transmitting RF chains and operates in short-range, it can be densely deployed with scalable cost and low energy consumption, yet without the need for interference management among RIS elements [10], [11].", "The three surveys, [12], [13], [14], and the references therein, cover different aspects of RIS technology, including channel modeling and estimation, performance evaluation, optimization, resource allocation, and early trials of the technology.", "It is worth noticing that DOCOMO NTT [15] conducted the world's first prototype trial of dynamic metasurfaces using 28 GHz radio signals.", "This means that we expect to see an RIS panel as a commercial product over the upcoming few months.", "Also, the authors in [16] evaluate the RIS placement near the user and near the BS.", "They also proposed a new hybrid RIS deployment strategy by combining the previous two strategies to deliver the best signal-noise-ratio to the end-user.", "In [17], the authors considered deploying RIS panels for wireless backhauling of multiple BSs connected in a mesh topology.", "They showed that the RIS-mesh backhauling has several desired features that can be exploited to overcome some of the backhauling challenges.", "Based on the authors' conclusion in [18] where they showed that with very high rates and many RIS elements, RIS-assisted communications could beat decode-and-forward relay communications, we propose to use RIS to help mitigate the failure of the X-haul links in dense urban areas.", "Our proposed scheme aims to provide the failed BS with an alternate X-haul connection from any neighboring BS with the RIS aid.", "Our scheme will be analytically verified to highlight our proposed X-haul scheme's practicality, which is compared to Line-of-sight (LoS) and Non-LoS (NLoS) alternate X-hauling.", "The main contributions of our proposed X-haul outage scheme can be summarized as follows: The total received power, power consumption, and spectral density in the presence of RIS are derived.", "A closed-form expression is derived for the optimal number of RIS reflecting elements with respect to the total consumed power, considering the dissipated power per RIS element caused by the circuitry required for the adaptive phase shift.", "We present the benefit of using RIS for X-haul outage compensation and its optimal location with respect to the source and the destination.", "Also, we show that at a given distance and with a large number of RIS reflecting elements, the RIS can fully recover the failed X-haul rate.", "The rest of the paper is organized as follows.", "Section II presents the X-haul outage compensation scheme.", "Section III introduces the proposed system model with a detailed analytical model.", "In Section IV, we provide an analysis of the X-haul rate and power in addition to finding the optimal number of RIS elements.", "In Section V, we present the numerical results and analysis of the proposed scheme.", "Finally, we conclude the paper in Section VI." ], [ "X-haul Outage Compensation Scheme", "We consider an ultra-dense network configuration for our X-haul outage compensation scheme where SBSs are distributed in the street level in the presence of different shaped buildings.", "For clarification, we define three types of SBSs: 1) the Donor SBS (DBS), 2) the Failed SBS (FBS), and the neighboring SBS (NBS).", "The DBS is considered as the BS that provides the X-haul connection to the FBS before the occurrence of the failure.", "This connection can be wired or wireless.", "Although our scheme is activated after the occurrence of the failure, we consider the connection from the DBS to the FBS and compare it to the compensation connection(s) to understand the degree of recovery from the failure using this scheme.", "The FBS is the BS that lost its X-haul connection from the DBS.", "There are many reasons for this connection to fail, including equipment failure from the DBS side.", "Finally, the NBS is the BS that will heal the failed X-haul connection of the FBS.", "We assume that there is no LoS between NBS and FBS.", "Fig.", "REF shows the system model where we consider the previously mentioned three SBSs in addition to several buildings, one of them is equipped with an RIS panel.", "We have three different connection scenarios here in this figure: 1) DBS-FBS, 2) NBS-FBS, and 3) NBS-RIS-FBS.", "For the first scenario, i.e., DBS-FBS, this is considered as the no-failure scenario where the FBS is receiving its X-haul LoS connection from DBS.", "The remaining two scenarios are going to occur after the X-haul failure between DBS and FBS.", "For the second scenario, i.e., NBS-FBS (NLoS), the NBS will provide the alternate X-haul connection to the FBS.", "Since there is no LoS between the two SBSs (based on our assumption), this connection is considered to be NLoS.", "For the third scenario, i.e., NBS-RIS-FBS, the NBS will provide the alternate X-haul connection to the FBS with the RIS panel's assistance.", "Note that for this scenario, the connection from NBS to FBS is NLoS; however, the connections from the NBS to RIS and from the RIS to FBS are LoS.", "It is obvious that scenario one will give the highest X-haul rate, given the LoS connectivity, followed by scenario three, and the lowest rate is achievable in scenario two.", "Scenarios one and two are considered upper and lower X-haul rate bounds to the third scenario.", "The third scenario is the same as the second scenario with the addition of the X-haul rate acquired by the FBS from the directed reflection coming from the RIS panel.", "Moreover, we would like to investigate where we have to place the RIS panel and how many RIS elements need to be used to achieve an LoS like X-haul rate or, in other words, how we can recover the failed X-haul or so that the X-haul failure will not affect the achievable rate of the users associated with the FBS.", "Figure:  : The system model." ], [ "System Model", "We consider the communication before the X-haul failure to be between the DBS and the FBS.", "The LoS communication between DBS and FBS (denoted as subscript DF) using single antenna, can be expressed as: $y_{DF}^{LoS}=h_{DF}^{LoS}\\sqrt{p}s + \\tilde{n},$ where $h_{DF}^{LoS}$ is the deterministic flat-fading channel gain between DBS and FBS, $p$ is the transmitted power, s is the unit power information signal, and $\\tilde{n}$ is the receiver noise, which is assumed to be additive white Gaussian noise.", "The X-haul achievable rate of this channel can be expressed as: $R_{DF}^{LoS}= \\log _2(1+\\frac{p|h_{DF}^{LoS}|^2}{\\sigma ^2})$ where $\\sigma ^2$ is the noise variance.", "This X-haul rate is considered as the maximum achievable capacity by the FBS (before failure) acquired from the DBS.", "Note that once this X-haul connection fails, the FBS will not be able to serve its users until it acquires another X-haul connection.", "Note that this paper is concerned mainly with the proof of the concept, this is why we didn't consider the interference from the neighboring BSs.", "However, we are considering the interference in a future extension of this paper.", "This paper proposes that the healing X-haul connection will be acquired from an NBS where there is no LoS between the FBS and the NBS.", "In this case, the healing X-haul capacity between the FBS and the NBS is expressed as: $R_{NF}^{NLoS}= \\log _2(1+\\frac{p|h_{NF}^{NLoS}|^2}{\\sigma ^2})$ where $h_{NF}^{NLoS}$ is the deterministic flat-fading channel gain between NBS and FBS.", "It is worth pointing out that if there is a LoS between the FBS and the NBS, our proposed approach will give a better X-haul rate.", "This is why we build our model based on the worst case, i.e., no LoS.", "Consider an RIS presence that will aid the communication between the NBS and the FBS.", "The RIS is equipped with $N_x$ discrete reflecting elements.", "In this case, the channel from NBS to RIS is indicated by $\\textbf {h}_{NI}^{NLoS} \\in \\mathbb {C}$ , where $[\\textbf {h}_{NI}^{NLoS}]_n$ denotes the nth component.", "The channel between the RIS and the NBS is denoted by $\\textbf {h}_{IF}^{NLoS} \\in \\mathbb {C}$ , where $[\\textbf {h}_{IF}^{NLoS}]_n$ denotes the nth component.", "Each element in the RIS has a size that is smaller than the wavelength $\\lambda $ ; consequently, the incoming signal is scattered with a constant gain in all directions of interest [19].", "The following diagonal matrix represents the properties of the RIS: $\\mathbf {\\Phi } = diag(\\epsilon e^{j\\phi _1}, \\dots , \\epsilon e^{j\\phi ^{N_x}})$ where $\\epsilon \\in (0,1]$ is the fixed amplitude reflection coefficient and $\\phi _1, \\dots , \\phi _N$ are the phase shift variables which are controlled/optimized by the RIS.", "The received signal between the NBS and the FBS now consists of two components; 1) the NLoS component based on the direct communication between NBS and FBS and 2) the reflected signal from the RIS and received by the FBS.", "Hence, the received signal is given as: $y_{NIF}^{NLoS} = (h_{NF}^{NLoS} + \\mathbf {h}_{NI}^T \\mathbf {\\Phi } \\mathbf {h}_{IF})\\sqrt{p}s + n,$ We consider the channel deterministic since the RIS elements reflect passively using a certain phase shift, and these elements do not have any signal processing capabilities.", "The destination, i.e., FBS, is supposed to know the channel, and the phase shift can be optimized.", "This assumption is considered since the channel estimation for RIS is non-trivial; however, the recent work presented in [20] introduced a general framework for estimating the channel.", "Also, the authors in [21] proposed to include non-uniformly distributed active elements in the RIS panel in order to train/estimate the channel by leveraging deep learning tools.", "The X-haul capacity of the communication between NBS and FBS with the support of RIS is given by: $R_{NIF} = &\\underset{\\phi _1 , \\dots , \\phi _N}{\\text{max}} \\log _2 \\bigg (1+ \\frac{p|h_{NF}^{NLoS} + \\mathbf {h}_{NI}^T \\mathbf {\\Phi } \\mathbf {h}_{IF}|^2}{\\sigma ^2}\\bigg )\\\\= & \\log _2 \\bigg (1+\\frac{p(|h_{NF}^{NLoS}| + \\epsilon \\sum _{n=1}^{N_x} |[\\mathbf {h}_{NI}]_n[\\mathbf {h}_{IF}]_n|)^2}{\\sigma ^2}\\bigg )$ where $N_x$ is the number of reflecting elements in the RIS panel.", "The steps of deriving Eq.", "() from Eq.", "(REF ) can be found in [11].", "It is worth pointing out that $\\mathbf {h}_{NI}^T \\mathbf {\\Phi } \\mathbf {h}_{IF} = \\epsilon \\sum _{n=1}^{N_x} e^{j\\phi _n} [\\mathbf {h}_{NI}]_n[\\mathbf {h}_{IF}]_n$ .", "The maximum X-haul capacity is achieved when the phase shifts are selected as follows: $\\phi _n = arg(h_{NF}) - arg([\\mathbf {h}_{NI}]_n[\\mathbf {h}_{IF}]_n) $ This phase shift will give the same phase as $h_{NF}$ to every term in the sum.", "This proof follows the steps in [11].", "Note that the term $[\\mathbf {h}_{NI}]_n[\\mathbf {h}_{IF}]_n$ contains the transmit beamforming and the NBS-RIS channel.", "which can be considered as the RIS effective channel perceived by nth element.", "Hence, Eq.", "(REF ) recommends that the nth phase shift must be tuned so that the phase of the signal that passes through the RIS is aligned with that of the signal over the NBS-FBS to achieve the best coherent combining at FBS.", "To present the X-haul achievable rate equations in a more compact form, we introduce the notation $h_{NF}$ = $\\sqrt{\\beta _{NF}}$ , $h_{NI}$ = $\\sqrt{\\beta _{NI}}$ , $h_{IF}$ = $\\sqrt{\\beta _{IF}}$ , and $\\frac{1}{N}\\sum _{n=1}^{N_x}|[\\mathbf {h}_{NI}]_n[\\mathbf {h}_{IF}]_n|$ = $\\sqrt{\\beta _{NIF}}$ .", "Hence, Eqs.", "(REF ), (REF ), and () can be rewritten as: $R_{DF}^{LoS}= \\log _2(1+\\frac{p\\beta _{DF}^{LoS}}{\\sigma ^2})$ $R_{NF}^{NLoS}= \\log _2(1+\\frac{p\\beta _{NF}^{NLoS}}{\\sigma ^2})$ $R_{NIF} = \\log _2 (1+\\frac{p\\bigg (\\sqrt{\\beta _{NF}^{NLoS}} + \\epsilon ~N_x \\sqrt{\\beta _{NIF}^{LoS}}\\bigg )^2}{\\sigma ^2})$ The 3GPP Urban Micro (UMi) is used to model the channel gains [22] given that the used carrier frequency is the 3 GHz C-band.", "We also used 28 GHz mmW band for comparison purpose.", "As modeled in the previous equations, we will use both the LoS and NLoS versions of the model, which are defined for distances greater than or equal to 10 m. We denote the antenna gain of the transmitter BS and receiver BS as $G_t$ and $G_r$ , respectively.", "Hence, the channel gain as a function of the distance for LoS and NLoS can be expressed as: $\\beta ^{LoS}(d)[dB] = G_t + G_r -37.5 -22\\log _{10}(\\frac{d}{d_o})$ $\\beta ^{NLoS}(d)[dB] = G_t + G_r -35.1 -36.7\\log _{10}(\\frac{d}{d_o})$ where $d_o$ is the reference distance, which is considered in the UMi model as 1m.", "Note that in case of LoS between NBS-RIS-FBS, the channel gain $\\beta _{NIF}^{LoS}$ can be expressed as: $\\beta _{NIF}^{LoS} = \\beta _{NI}^{LoS}\\beta _{IF}^{LoS}$" ], [ "Analysis of X-haul Rate and Power", "We consider that the FBS requires a minimum X-haul rate, i.e., $\\bar{R}$ , to satisfy its users' requirements.", "Based on that, the X-haul capacity for equations (REF )-(REF ) can identify the required power for each of the three scenarios, i.e., DBS-FBS LoS, NBS-FBS NLoS, and NBS-RIS-FBS NLoS scenarios.", "For each scenario, the required power is given as follows: $p_{DF}^{LoS} = (2^{\\bar{R}} - 1) \\frac{\\sigma ^2}{\\beta _{DF}^{LoS}}$ $p_{NF}^{NLoS} = (2^{\\bar{R}} - 1) \\frac{\\sigma ^2}{\\beta _{NF}^{NLoS}}$ $p_{NIF}(N) = (2^{\\bar{R}} - 1) \\frac{\\sigma ^2}{(\\sqrt{\\beta _{NF}^{NLoS}} + \\epsilon N \\sqrt{\\beta _{NIF}^{LoS}})^2}$ The total power consumption, $P^{tot}$ , of the BS consists of two components; 1) the transmit power and 2) power dissipated in hardware components.", "The total power for the first scenario, i.e., DBS-FBS, is given as follows: $P_{DF}^{tot} = \\frac{p_{DF}^{LoS}}{\\nu } + P_D + P_F$ where $\\nu \\in (0,1]$ is the power amplifier efficiency while $P_D$ and $P_F$ are the power dissipated in the hardware at the DBS and FBS, respectively.", "Similarly, the total power for the second scenario, i.e., NBS-FBS, can be expressed as: $P_{NF}^{tot} = \\frac{p_{NF}^{NLoS}}{\\nu } + P_N + P_F$ where $P_N$ is the hardware dissipated power at the NBS.", "Finally, for the third scenario, i.e., NBS-RIS-FBS, the total power is given as: $P_{NIF}^{tot}(N) = \\frac{p_{NIF}(N)}{\\nu } + P_N + P_F + NP_e$ where $P_e$ is the dissipated power per RIS element caused by the circuitry required for the adaptive phase shift.", "To find the optimal number of RIS elements given a certain X-haul rate $\\bar{R}$ , an optimization problem can be solved aiming to minimize the total power of the RIS scenario.", "The total power is a convex function in N since $\\frac{\\partial ^2 }{\\partial N^2} P_{NIF}^{tot}(N) > 0$ .", "Then the optimal number of RIS elements, $N_x^*$ can be obtained from $\\frac{\\partial }{\\partial N} P_{NIF}^{tot}(N) = 0$ which can be evaluated as: $N^* = \\@root 3 \\of {\\frac{(2^{\\bar{R}}-1)\\sigma ^2}{\\epsilon ^2 \\beta _{NIF}^{LoS}P_e}} - \\frac{1}{\\epsilon }\\sqrt{\\frac{\\beta _{NF}^{NLoS}}{\\beta _{NIF}^{LoS}}}$ It is worth pointing out that the above solution is optimal under the assumption that $\\beta _{NIF}^{LoS}$ is independent of $N_x$ .", "Also, the optimal number of RIS elements in (REF ) in general is not an integer, hence, the true optimal solution is either ${N^*}$ or ${N^*}$ .", "Note that in the presence of LoS between NBS-RIS and RIS-FBS, $\\beta _{NIF}^{LoS}$ is independent of $N_x$ and hence $\\beta _{NIF}^{LoS} = \\beta _{NI}^{LoS}\\beta _{IF}^{LoS}$ ." ], [ "Simulation Results", "This section presents the numerical results generated using MATLAB to illustrate our proposed scheme's effectiveness and investigate the benefits of utilizing RIS to mitigate X-haul failures in 5G/6G networks.", "The simulation model consists of one DBS, one NBS, and one FBS each of them is having a single antenna.", "Also, we assume that only one building's facade is equipped with RIS, as shown earlier in Fig.", "REF .", "To clarify the comparison, we assumed that the distance between the DBS-FBS is the same as that of NBS-FBS.", "Fig.", "REF shows the simulation setup where the distance between the NBS and the FBS is fixed and equals 100 m. Also, the vertical distance, d, between the RIS and both NBS and FBS is fixed to two values; 15 m and 60 m. We vary the horizontal distance between the FBS/NBS and the RIS, i.e., $d_{IF}$ , (the RIS is moving on the x-axis from $d_{IF}$ = 0 m and $d_{IF}$ = 200 m) to evaluate our model.", "The simulation parameters are summarized in Table I.", "We are considering three different scenarios where; the first one is before failure between DBS and FBS (there is an LoS link between them).", "The second scenario considers the failure of the X-haul link transmitted from the DBS.", "In this case, NBS will heal FBS via an NLoS link only.", "The third scenario is based on the same setup as the second scenario.", "Besides, the RIS is added to aid the healing X-haul transmission from NBS to FBS.", "For all generated figures, we used the carrier 3 GHz carrier frequency except for Fig.", "REF we used both 3GHz and 28 GHz.", "Fig.", "REF shows the X-haul achievable rate when changing the RIS panel position on the x-axis from x=0 m to x=200 m and h = 15 m. The NBS-FBS LoS and the NBS-FBS NLoS curves are not varying with the x-axis because the distance between NBS and FBS is fixed and equals 100 m. The threshold rate is an arbitrary value chosen to represent the acceptable X-haul rate during failure (can be tuned by the network operator).", "However, we choose it to be around 50% of the rate before failure.", "The X-haul rate before the occurrence of failure, i.e., DBS-FBS LoS curve (also known as NBS-FBS LoS), is around 3.25 bits/s/Hz.", "Once the failure occurs and without RIS, the NLoS X-haul link between the NBS and the FBS, i.e., NBS-FBS NLoS curve, is very close to 0 bits/s/Hz, which shows that without using the RIS panel, the FBS will not achieve the minimum required threshold rate and hence will not be able to serve its users.", "Figure:  : The Simulation Setup.The three double hump-shaped curves represent the X-haul achievable rate when using RIS with 200, 500, and 1000 reflecting elements.", "It is observed that there are two peaks at x = 0 m and x = 100 m where the NBS and FBS are located, respectively.", "This shows that the optimal location for the RIS is to be placed near either of them.", "Therefore, the optimal location for the RIS panel is either near the transmitter or the receiver.", "Also, for N = 200, we can achieve the X-haul threshold rate at these two locations only.", "When N = 500, the X-haul achievable rate exceeds that of the LoS rate at x = 0 m and x = 100 m. This occurs under the condition that all reflecting elements are dedicated to reflecting the received signal from the NBS to the FBS.", "When doubling the number of reflecting elements, the X-haul achievable rate outperforms that of the LoS even if the RIS panel is not located at the two identified optimal locations.", "Although 1000 reflecting elements may sound uncanny, covering a building's facade by an RIS panel, similar to Fig.", "REF , will accommodate 1000 reflecting elements or more.", "The only drawback of including a large number of elements is interfacing them to the smart controller; however, this issue can be alleviated by controlling each cluster of ten elements or more by one interface or limiting the available phase shifts.", "Table: : Simulation ParametersFig.", "REF has the same simulation setup as Fig.", "REF except that the vertical distance between the FBS/NBS and the RIS panel is increased from 15 m to 60 m. This increase changed the double hump-shaped curves into concave curves, which means that the RIS panel should be placed in the mid-distance between the source and destination.", "This infers that the optimal location of the RIS panel is heavily dependent on the vertical distance h. Also, the only RIS panel that satisfies the threshold rate is the one equipped with 1000 reflecting elements.", "Fig.", "REF shows the transmit power needed by the DBS/NBS to deliver an X-haul rate of $\\bar{R}=4$ bit/s/Hz to the FBS for two different carrier frequencies, i.e., 3 GHz and 28 GHz.", "The NBS-FBS NLoS link is consuming most of the power of the NBS to achieve the target X-haul rate.", "Meanwhile, the RIS with 1000 reflecting elements requires the least amount of transmit power and outperforms the amount of power needed for LoS when the RIS panel is located anywhere between the NBS and the FBS.", "When the number of reflecting elements is reduced, more power is needed to achieve the same target X-haul rate.", "It is worth noting that when the RIS panel and NBS or FBS are aligned on the same vertical line, the transmit power is minimal.", "For 28 GHz related results, the transmit power increases dramatically due to the high attenuation of this range carrier frequencies exceeding the threshold power, i.e., 30 dBm, of the SBS.", "Using Multiple-Input-Multiple-Output (MIMO) or massive MIMO antennas with active beamforming can reduce the needed transmit power to acceptable limits.", "Meanwhile, without using MIMO, the RIS panel with 1000 reflecting elements can achieve the target X-haul rate without exceeding the maximum power of the SBS, given that the panel is located anywhere between x = 0 m or x = 100 m. Fig.", "REF investigates the Energy Efficiency (EE) performance for the considered RIS-based scenario.", "The EE is defined as the ratio between the X-haul achievable rate and total power consumption, i.e., $EE=BW*R/P_{tot}$ where BW is the transmission bandwidth.", "Figure:  : Simulation results for X-haul achievable rate and power using RISFigure:  : The energy efficiency vs. spectral efficiencyFig.", "REF shows the EE versus the X-haul achievable rate when optimizing the number of RIS elements with respect to the transmit power based on Eqs.", "(20) and (21).", "It is shown that the optimal number of RIS elements is zero when the X-haul rate is 5 bit/s/Hz or less.", "This means that it is not recommended to use RIS if the FBS's required X-haul spectral efficiency is less than or equal to 5 bit/s/Hz.", "As the X-haul spectral efficiency increases beyond 5 bit/s/Hz, the EE of the NBS-RIS-FBS scenario outperforms that of the NBS-FBS scenario.", "Although the DBS-FBS LoS scenario is the most EE, it degrades dramatically by increasing the spectral density.", "When the spectral density is greater than 20 bit/s/Hz, the NBS-RIS-FBS scenario outperforms the LoS scenario.", "Also, on the second y-axis, the number of RIS elements increases exponentially, inferring the impracticality of using RIS when the X-haul rate is greater than 25 bit/s/Hz.", "In addition, when increasing the required X-haul rate from 20 bit/s/Hz to 25 bit/s/Hz, the optimal number of reflecting elements will increase from 1000 to 3000 reflecting elements." ], [ "Conclusions", "5G cellular networks are being deployed to promise that they will deliver significantly faster and more responsive mobile broadband experiences.", "However, based on the 5G network's densifying concept, the number of failures of X-haul links will increase.", "We proposed Reconfigurable Intelligent Surfaces (RIS) usage to help mitigate X-haul failure/outages in dense and ultra-dense networks.", "Our proposed scheme, along with the presented analytical results, showed that using RIS can mitigate or at least alleviate the effect of X-haul failures.", "Moreover, using the RIS with a certain number of reflecting elements, our proposed solution can deliver approximately the same X-haul rate achieved before failure.", "We also showed that under certain conditions, the proposed RIS X-haul solution could deliver more rate and be more energy-efficient than the line-of-sight and the non-line-of-sight solutions.", "Finally, if the associated issues with increasing the number of reflecting elements are resolved, the RIS panel with a large number of reflecting elements, i.e., 1000 or more, can compensates any failed X-haul link regardless the frequency band used, i.e., below or above 6 GHz, given that the panel is placed in its optimal location." ] ]
2207.03582
[ [ "Semi-analytical near-Earth objects propagation: the orbit history of\n (35107) 1991 VH and (175706) 1996 FG3" ], [ "Abstract The propagation of small bodies in the Solar system is driven by the combination of planetary encounters that cause abrupt changes in their orbits and secular long-term perturbations.", "We propose a propagation strategy that combines both of these effects into a single framework for long-term, rapid propagation of small bodies in the inner Solar System.", "The analytical secular perturbation of Jupiter is interrupted to numerically solve planetary encounters, which last a small fraction of the simulation time.", "The proposed propagation method is compared to numerical integrations in the Solar system, effectively capturing properties of the numerical solutions in a fraction of the computational time.", "We study the orbital history of the Janus mission targets: (35107) 1991 VH and (175706) 1996 FG3, obtaining a stochastic representation of their long-term dynamics and frequencies of very close encounters.", "Over the last million years the probability of a strongly perturbing flyby is found to be small." ], [ "Introduction", "As remnants of primordial planetary formation, Near Earth Objects (NEOs) are relevant targets for scientific exploration.", "The number of discovered NEOs is expected to continue increasing with future surveys, offering new opportunities to the scientific community [19].", "The interest and availability of specific objects is assessed with long term predictions of their orbits.", "Trajectories of Near Earth Objects are dominated by close encounters with the inner Solar System planets and secular perturbations [25].", "Planetary encounters cause a high dependence on the initial conditions as the flybys cause neighboring trajectories to diverge [41].", "This divergence of the dynamics also causes the uncertainty to grow very rapidly.", "Thus, an analytical model that captures the main dynamical effects can avoid the computational cost of using high-fidelity models while capturing the overall statistical evolution of the orbits accurately.", "The semi-analytical propagation of asteroids allows the rapid propagation of NEO orbits, in the interest of the analysis of large databases of NEOs.", "The long-term study of asteroid orbits has been achieved in the past using a wide variety of analytical, semi-analytical and numerical methods.", "Analytical methods are based on the study of the gravity potential to obtain secular and resonant perturbations [30].", "Semi-analytical methods are used to map orbital elements to the locations of linear secular resonances, which are resonances involving one planetary and one asteroid frequency [26], [24].", "Both types of solutions represent the dynamics of asteroids in the absence of planetary encounters by averaging the perturbing potential.", "On the other hand, previous studies focus on the accumulation of planetary encounters in contrast to numerical integration [6].", "The effect of close encounters on the orbit of asteroids can be computed using analytical [34], semi-analytical or numerical methods.", "Semi-analytical solutions [1] allow the computation of flybys treating the planet as a perturbing force in the Lagrange Planetary Equations.", "Specific numerical integrators are convenient to propagate orbits of asteroids in the long-term, in which symplecticity is desired along with the capacity to accurately solve close encounters [46], [5].", "Under multiple resonances asteroids start to encounter planets while their eccentricity increases.", "This increase often causes the asteroids to eventually collide with the Sun, planets or to be ejected from the Solar System on a hyperbolic orbit [9], [13], [28], [6], [27].", "In this paper we aim to provide a simulation framework for the propagation of particles in the Solar System.", "Our approach consists in the analytical propagation of the particle until a close encounter is found.", "The propagation is stopped when the trajectory is close to a planet, then the close encounter is evaluated numerically.", "The evaluation of the encounters is based on a quadrature of the Lagrange Planetary Equations (LPE) around the closest approach date.", "After the encounter the analytical propagation of the orbit is resumed.", "The propagation under secular perturbations provides a realistic prediction of when the next encounter can occur as the orbit of the asteroid drifts between different regions of the inner Solar System.", "This approach reduces substantially the computational time of solutions obtained entirely by numerical integration while providing deeper insight into the dynamics.", "The use of the analytical secular model allows the prediction of long-term properties of the asteroid dynamics.", "Eccentricities, inclinations and angles of asteroid and planets drift secularly.", "Thus, we can propagate the minimum orbit intersection distance (MOID).", "The MOID constrains the minimum closest approach distance between the asteroid and the planets and defines if asteroids are potentially hazardous (PHAs).", "The long-term dynamics of the orbits of NEOs and the MOID are studied by sampling a large number of virtual asteroids from their uncertainty distributions.", "We use the semi-analytical propagation of these asteroids to show the stochastic nature of the orbital evolution of NEOs.", "The semi-analytical propagation allows us to track the encounters experienced by asteroids in the inner Solar System, which can perturb the physical properties of asteroids.", "The orbits of binary asteroids can be disrupted by a very close encounter [22].", "In this paper we study the orbital history of the targets of the exploration mission Janus [39]: the two binaries (35107) 1991 VH and (175706) 1996 FG3.", "The stochastic long-term dynamics in the last million years are modelled by sampling a large number of particles from their current orbit uncertainties.", "We model the evolution of these statistical distributions by a random walk in semi-major axis, eccentricity and inclination and a uniform distribution in longitude of perihelion.", "Then, we compute the probability that (35107) 1991 VH and (175706) 1996 FG3 could have been potentially disrupted by a close encounter in this period of a million years.", "This section presents the scope of the paper and is followed by the background of this work in section .", "Next, section describes the methodology including a detailed study of flybys evaluation and the derivation of an analytical N-body secular problem solution.", "Section shows examples of the long-term propagation of asteroids and how the long-term dynamics can be characterized stochastically.", "Section discusses the limitations of the semi-analytical propagation tool.", "Section studies the orbital history of the Janus targets and the frequency of close encounters.", "Last, section concludes by evaluating the aspects in which this methodology proves beneficial, questions that remained unanswered, and future work." ], [ "Background", "The long-term dynamics of NEOs are governed by their gravitational interactions with the other bodies of the Solar System.", "The effects of the most massive and external planets have timescales of millennia.", "However, planetary close encounters can abruptly change an orbit over a timescale of days.", "The accumulation of such planetary encounters cause the orbits of NEOs to be chaotic [41].", "This section describes this phenomenon in more detail.", "The evaluation of close encounters is necessary for the propagation of NEOs, hence the variety of possible flybys is demonstrated later for the validation of the method.", "Many asteroids experience long periods of time without flybys.", "The dominant dynamics in those periods of time are the secular perturbations from massive planets in the Solar System.", "Likewise, the orbits of the planets evolve secularly over similar timescales.", "The Laplace-Lagrange secular theory qualitatively describes the evolution of the elements of the planets at any distant time in the future or past.", "As for the asteroid, the secular solution from external perturbers represents the orbital dynamics of asteroids between encounters.", "The presence of repeated encounters is one of the main characteristics of the long-term propagation of asteroids in the inner Solar System.", "Repeated close encounters cause a random walk in the elements of the asteroids.", "Very close encounters occur less frequently but change substantially the orbits of NEOs, modifying predictions on the long-term evolution of their orbits.", "Thus, we propose an informed analytical propagation of the orbits while characterizing planetary close encounters.", "The proposed methodology is born from the combination of these two dynamical regimes: the long-term effects of secular dynamics and the frequent changes in elements experienced in planetary encounters.", "Considering the secular drift of the asteroid we model the seasonal variation of the possible encounters with planets." ], [ "Chaotic dynamics in the inner Solar System", "An accurate description of the evolution of orbits of near-Earth asteroids beyond a few centuries is challenging.", "This is because the succession of planetary encounters disperse neighboring trajectories to become chaotic [41].", "Small deviations in the orbital period change the timing of the flybys, spreading the uncertainty along the Line of Variation [29].", "After successive flybys the resulting imaginary stream of particles is spread in highly non-linear distributions.", "For this reason the study of long-term dynamics is often left to a statistical analysis requiring a large number of particles and computational efforts.", "In this context we propose the use of this semi-analytical tool to obtain long-term simulations in short computational times.", "Figure: Chaotic dynamics of (35107) 1991 VH as obtained from numerical integration.", "Each axis represents the variation from the initial value of elements pairs: (left) semi-major axis-eccentricity, (center) semi-major axis-inclination, (right) argument of the node-argument of perihelion in degrees.", "The orbital evolution is shown at four instants of time: initial (first row), after 500 years (second row), after 5000 years (third row) and after 1 million years (bottom).We exemplify the sensitivity to initial conditions in a numerical integration of asteroid (35107) 1991 VH, which is one of the two targets of mission Janus [39], a NASA SIMPLEx mission.", "Figure REF shows 1440 particles generated from the uncertainty in the orbit solution of (35107) 1991 VH, which is included in appendix .", "These particles are propagated in the N-body integrator IAS15 [37] including the Solar System planets from Venus to Neptune.", "The particles are propagated for a million years, although in this section we study in more detail the distributions after shorter periods of time.", "After 500 years the initial normal distribution already becomes a stream of particles.", "While the variation in the elements from the nominal is similar for all the particles, there is a dispersion orders of magnitude smaller that represents the stream of particles.", "After 5000 years, the distribution becomes completely different: the presence of planetary encounters disperses the particles around the initial orbit.", "The variation in eccentricities and inclinations has a secular component.", "However, the variation on the argument of the node and argument of perihelion is dominantly secular after a few millennia.", "After a million years, the particles are spread along a large region of near-Earth space.", "In argument of perihelion and ascending node we observe that the distribution becomes almost uniform in the whole 2D angular space.", "The secular drift in the arguments defines the possibility of encounters over time.", "For this reason, it is important to characterize this drift and the secular cycles under the perturbation of the large bodies of the Solar System.", "When encounters are possible with the inner Solar System planets, these need to be accounted as perturbers of the orbit evolution.", "The stochastic nature of the long-term dynamics of NEOs under close encounters implies that the precise determination of their position after hundreds of thousands of years is unachievable.", "However, we can still collect statistics that give us insight on their orbital history.", "Another implication is that the inclusion of higher order dynamics is shadowed under the stochastic dispersion caused by the main gravitational perturbations.", "For example, the magnitude of the Yarkovsky effect is typically $~10^{-4}$ au/Myr [44], [33], which is still two orders of magnitude smaller than a typical dispersion after 10,000yr under repeated close encounters, as observed in the example of Figure REF .", "In section REF we show that (35107) 1991 VH is not under a particularly high frequency of close encounters compared to other NEOs.", "Similarly, relativistic effects can have a non-negligible effect in the secular rates of the argument of perihelion.", "These are usually measured in arcseconds per year or century, and typical values are 1-2 orders of magnitude smaller than the typical secular periods of the order of  100,000 years [3].", "Even if the secular rate has an error, the presence of encounters already causes the distributions to become uniform in argument of the node and perihelion after a few secular periods." ], [ "NEO close encounters in the inner Solar System", "Flybys can occur with multiple planets over short periods of time.", "Even if the encounters are with the same planet, the closest approach distance and relative velocities can change depending on the timing of the flyby.", "The geometry of the flyby is constrained by the heliocentric elements of the asteroid.", "If shallow encounters are considered, the position in the asteroid orbit in which the planet is encountered can significantly change the relative velocity.", "These variations are not well captured by analytical theories, but the proposed propagation tool aims to accurately model these variations.", "These are different regimes of flybys in which the evaluation tool needs to be accurate.", "Figure: Relative velocity at closest approach of flybys generated from the propagation of NEOs with semi-major axis smaller than 2 au for 50 years.", "The symbol indicates the planet that the asteroid is encountering and the horizontal axis shows respectively closest approach distance (au), inclination (deg) and eccentricity.In order to broadly show the diversity in flybys that different NEOs experience, we generate a list of flybys that will be used to validate the evaluation of close encounters.", "From the database of NEOs we select the ones with semi-major axis smaller than 2 au [20].", "Then, we propagate their positions using the secular model for 50 years.", "For such a brief period of time the change in the elements is insignificant for our purposes.", "Figure REF shows more than 30,000 flybys generated with the described method.", "Shallow encounters are much more frequent than the very close encounters that cause large variations in orbit elements.", "Thus, we want to consider them even if their individual contribution is not as significant.", "The range of possible relative velocities in figure REF depends on the planet in question, with increasing maximum relative velocity for the planet closest to the Sun.", "The relative velocity is defined by the heliocentric orbit of the asteroid, with an increasing range of possible values depending on the inclination and eccentricity of the orbit.", "Overall, after millions of years asteroids experience a variety of encounters that can be computed with different methods.", "With this purpose the list of generated flybys is used to decide the method to compute the post-encounter elements of flybys.", "In section we compute the error of different close encounter evaluation methods referenced to numerical integration of the trajectories." ], [ "Methodology", "This semi-analytical propagation tool consists in the following process.", "First, the orbit of the asteroid is propagated by an analytical secular solution.", "This perturbed motion is interrupted when an encounter is found with a nearby planet.", "Then, the trajectory during the planetary encounter is modeled using a numerical method.", "Next, the secular propagation is continued until the subsequent encounter.", "The simplest way to find encounters is to track the distance between the asteroid and planets at all times.", "While the regions in which encounters are possible are determined by the geometry of the asteroid around the central body, searching at all times is the most generic approach.", "The state of the planets is obtained from the secular solution of the 8 main planets interacting with each other.", "The state of the small body is corrected given the secular dynamics model.", "Once we determine the initial conditions of the encounter, the change in orbit elements is computed through the proposed numerical procedure.", "There are many methods to compute planetary encounters available in the literature.", "Analytical solutions for Keplerian elements before and after close encounters in Öpik's Theory [34] were extended for multiple applications by [43], [42].", "However, these analytical expressions are constrained to encounters that are very close and small bodies that are not co-moving with the planet.", "Asteroid and planet are co-moving when they have a small inclination and at least one of the node crossings close to the planet orbit.", "We name shallow encounters those with large close approach distance but non-negligible effects.", "Shallow encounters are very frequent and influence the long-term evolution of small bodies in the Solar System.", "In order to account for shallow encounters, semi-analytical methodologies can be used to map before and after encounter conditions [1].", "These methods are based on the quadrature of Lagrange Planetary Equations around the encounter.", "In this work we derive a quadrature of Lagrange Planetary Equations in Delaunay elements which is solved using a numerical integration scheme.", "In the case of extremely close or slow encounters we solve the encounters numerically.", "Once the solution of most encounters is obtained satisfactorily we focus efforts in the computation of the perturbed motion of the asteroid in absence of encounters.", "We evaluate the solution of N-bodies interacting secularly to generate the orbits of the planets.", "Then, we obtain the perturbed motion of the asteroid including only the planets relevant to its secular influence.", "Taking into account the influence of only Jupiter is a valid generic approach to estimate the secular dynamics of NEOs [45], [35], [11].", "In this work we use the Laplace-Lagrange secular model.", "The secular rates as obtained by the analytical theory are compared to numerical integration to validate the range of validity of the solution.", "This defines a range of applicability of the tool, as we discuss later.", "In this section we compare the individual pieces of the semi-analytical propagation tool to numerical methods.", "Last, we compare the combined semi-analytical propagation tool with trajectories obtained through numerical integration and evaluate the computational efficiency of the method." ], [ "Analytical secular dynamics of multibody systems", "The dynamical landscape of the Solar System is complex with gravitational interactions between all planets.", "This landscape leads to resonances and secular motion in asteroids in the system.", "Well inside the inner Solar System, the dynamics are dominantly secular.", "The secular solution of a planetary system formed by N-planets can be obtained analytically to first order in inclinations and eccentricities and in the absence of resonances.", "This section derives an implementation of the solution following the procedure in Chapter 7 of [32].", "The perturbing potential is written for the $N$ bodies considered.", "Then Lagrange Planetary Equations are used to compute the equations of motion of the elements of each particle, leading to a system of differential equations solved together.", "The secular model is obtained as follows: (1) The perturbing potential is split in a direct part and an indirect part based on the dependency on fast angles, (2) then the perturbing potential is expanded in Keplerian Elements.", "(3) The important terms of the expansion are selected based on the averaging principle.", "(4) The terms are rewritten in semi-equinoctial elements to ease the solution of the global system of equations.", "(5) Take the necessary partials to solve the set of Lagrange Planetary Equations.", "The perturbing potential experienced by a mass $j$ by a second mass $k$ is: $R_{jk} = \\frac{G m_k}{a_{k}} \\left( {R_{jk}}_D + {R_{jk}}_{I} \\right)$ Where $a_{k}$ is the semi-major axis of body the external body.", "The perturbing potential is separated in the direct ${R_{jk}}_D$ and indirect ${R_{jk}}_{I}$ parts: $\\begin{aligned}[c]{R_{jk}}_D = \\frac{a_k}{\\left| {r}_j-{r}_k \\right|}\\end{aligned}\\qquad \\qquad \\begin{aligned}[c]{R_{jk}}_I = - \\frac{a_k^2}{a_j} \\frac{{r}_j\\cdot {r}_k}{\\left| {r}_k \\right|^3}\\end{aligned}$ The separation is convenient to expand in the ratio of semi-major axes $\\alpha _{jk}$ as well as sines and cosines of $\\left\\lbrace \\varpi _j,\\Omega _j,\\lambda _j,\\varpi _k, \\Omega _k,\\lambda _k,\\right\\rbrace $ .", "The ratio of semi-major axes is $\\alpha _{jk} = a_j/a_k$ if the perturber is external, or $\\alpha _{jk} = a_j/a_k$ if the perturber is internal.", "All the terms that depend on the longitudes $\\left\\lbrace \\lambda ,\\lambda _j\\right\\rbrace $ are of short-period, so it can be argued that they do not contribute to the averaged potential $R_j$ .", "The secular potential lowest order in eccentricities and inclinations is: $R_j = R_{0,j} + R_{1,j} = \\sum ^{N}_{k=1,k\\ne j} G m_k \\frac{1}{2a_k} b^{(0)}_{1/2}\\left( \\alpha _{jk} \\right) + R_{1,j}$ $ R_{1,j} = n_ja^2_j \\left[ \\frac{1}{2}A_{jj}e^2_j + \\frac{1}{2}B_{jj}I^2_j + \\sum ^{N}_{\\begin{array}{c}k=1 \\\\ k\\ne j\\end{array}} A_{jk} e_je_k\\cos { \\left( \\varpi _j-\\varpi _k\\right)} + B_{jk} I_jI_k\\cos { \\left( \\Omega _j-\\Omega _k\\right)} \\right]$ Where $\\bar{\\alpha }_{jk} = a_j/a_k$ if the perturber is external, or $\\bar{\\alpha }_{jk} = 1$ if the perturber is internal.", "The coefficients $A_{jj}, A_{jk}, B_{jj}, B_{jk}$ are: $A_{jk} = - n_j \\frac{1}{4} \\frac{m_k}{m_c+m_j} \\alpha _{jk} \\bar{\\alpha }_{jk} b^{(2)}_{3/2}\\left( \\alpha _{jk} \\right)$ $B_{jk} = + n_j \\frac{1}{4} \\frac{m_k}{m_c+m_j} \\alpha _{jk} \\bar{\\alpha }_{jk} b^{(1)}_{3/2}\\left( \\alpha _{jk} \\right)$ $A_{jj} = + n_j \\frac{1}{4} \\sum ^{N}_{k=1,k\\ne j} \\frac{m_k}{m_c+m_j} \\alpha _{jk} \\bar{\\alpha }_{jk} b^{(1)}_{3/2}\\left( \\alpha _{jk} \\right) = \\sum ^{N}_{k=1,k\\ne j} B_{jk}$ $B_{jj} = - n_j \\frac{1}{4} \\sum ^{N}_{k=1,k\\ne j} \\frac{m_k}{m_c+m_j} \\alpha _{jk} \\bar{\\alpha }_{jk} b^{(1)}_{3/2}\\left( \\alpha _{jk} \\right) = - \\sum ^{N}_{k=1,k\\ne j} B_{jk}$ where the coefficients $b^{(k)}_{s}$ are Laplace Coefficients.", "More details on their computation can be found in Appendix .", "The coefficients $A_{jj}, A_{jk}, B_{jj}, B_{jk}$ form the matrices ${A}$ and ${B}$ .", "We can rewrite the potential in semi-equinoctial elements, $\\begin{aligned}[c]h_j&=e_j \\sin \\varpi _j\\\\k_j&=e_j \\cos \\varpi _j\\\\\\end{aligned}\\qquad \\qquad \\begin{aligned}[c]p_j&=I_j \\sin \\Omega _j\\\\q_j&=I_j \\sin \\Omega _j\\\\\\end{aligned}$ the potential becomes: $R_{1,j} = n_ja^2_j \\left[ \\frac{1}{2} A_{jj}(h_j^2+k_j^2) + \\frac{1}{2} B_{jj}(p_j^2+q_j^2) + \\sum ^{N}_{k=1,k\\ne j} A_{jk} (h_jh_k + k_jk_k) + B_{jk} (p_jp_k + q_jq_k) \\right]$ Our complete set of states includes the mean anomaly at epoch $\\sigma _j$ and $L_j=\\sqrt{GMa_j}$ .", "The equations of motion become: $\\begin{aligned}[c]\\dot{p}_j&=\\frac{1}{n_ja^2_j} \\frac{\\partial R_j}{\\partial q_j}\\\\\\dot{q}_j&=-\\frac{1}{n_ja^2_j} \\frac{\\partial R_j}{\\partial p_j}\\\\\\end{aligned}\\qquad \\qquad \\begin{aligned}[c]\\dot{h}_j&=\\frac{1}{n_ja^2_j} \\frac{\\partial R_j}{\\partial k_j}\\\\\\dot{k}_j&=-\\frac{1}{n_ja^2_j} \\frac{\\partial R_j}{\\partial h_j}\\\\\\end{aligned}\\qquad \\qquad \\begin{aligned}[c]\\dot{L}_j&= \\frac{\\partial R_j}{\\partial \\sigma _j}\\\\\\dot{\\sigma }_j&=- \\frac{\\partial R_j}{\\partial L_j}\\\\\\end{aligned}$ The solution of $h_j,k_j,p_j,q_j$ only depends on $R_{1,j}$ .", "For this reason the perturbing potential is often only expressed with those components.", "However, if we want the solution of the mean anomaly at epoch $\\sigma _j$ it is necessary to take into account $R_{0,j}$ .", "In the process of averaging the terms that would effect the semi-major axis are removed, meaning that under this assumption that element remains constant.", "The solution of $h_j,k_j,p_j,q_j$ is: $\\begin{aligned}[c]h_j(t)&=\\sum ^{N}_{i=1} e_{ji} \\sin \\left( g_it+\\beta _i \\right)\\\\k_j(t)&=\\sum ^{N}_{i=1} e_{ji} \\cos \\left( g_it+\\beta _i \\right)\\\\\\end{aligned}\\qquad \\qquad \\begin{aligned}[c]p_j(t)&=\\sum ^{N}_{i=1} I_{ji} \\sin \\left( f_it+\\gamma _i \\right)\\\\q_j(t)&=\\sum ^{N}_{i=1} I_{ji} \\cos \\left( f_it+\\gamma _i \\right)\\\\\\end{aligned}$ where two sets of eigenvalue problems are solved for $e_{ji}, I_{ji}, f_i, g_i$ .", "The frequencies $g_i$ are the eigenvalues of ${A}$ , and the frequencies $f_i$ are the eigenvalues of ${B}$ .", "$e_{ji}$ and $I_{ji}$ are related to the eigenvectors of ${A}$ and ${B}$ , but need to be solved with $\\beta _i,\\gamma _i$ given a set of initial conditions.", "In order to solve for $e_{ji}, I_{ji}, \\beta _i,\\gamma _i$ we proceed as follows.", "From the matrices of normalized eigenvectors $\\bar{e}_{ji}, \\bar{I}_{ji}$ and the initial conditions ${h},{k},{p},{q}$ we form: $\\begin{aligned}[c]{h} &= \\bar{e}_{ji} \\left[ S_i \\sin \\beta _i \\right] \\\\{k} &= \\bar{e}_{ji} \\left[ S_i \\cos \\beta _i \\right] \\\\\\end{aligned}\\qquad \\qquad \\begin{aligned}[c]{p} &= \\bar{I}_{ji} \\left[ T_i \\sin \\gamma _i \\right] \\\\{q} &= \\bar{I}_{ji} \\left[ T_i \\cos \\gamma _i \\right] \\\\\\end{aligned}$ These are four linear systems of equations, where $S_i,T_i$ are the scaling factors of each eigenvector.", "Solving for the combined factors $\\left[ S_i \\sin \\beta _i \\right],\\left[ S_i \\cos \\beta _i \\right],\\left[ T_i \\sin \\gamma _i \\right]$ and $\\left[ T_i \\cos \\gamma _i \\right]$ we can reconstruct the vectors $e_{ji}, I_{ji}$ and the phase angles $\\beta _i,\\gamma _i$ .", "lcccccc[b!]", "1 Initial conditions of the Solar System propagation in Figure REF .", "0pt Planet $a$ (au) $e$ $i$ (deg) $\\Omega $ (deg) $\\omega $ (deg) $M_0$ (deg) Mercury 0.39703 0.21337 6.936 48.264 31.991 52.745 Venus 0.73096 0.012687 3.378 76.799 45.020 16.566 Earth 1.0030 0.018402 0.001 154.979 296.322 8.654 Mars 1.5177 0.093083 1.852 49.461 288.507 322.879 Jupiter 5.1904 0.047388 1.305 100.514 273.897 353.761 Saturn 9.5499 0.05412 2.487 113.612 339.598 91.261 Uranus 19.207 0.04628 0.772 73.997 96.864 189.506 Neptune 30.109 0.0091006 1.770 131.780 265.440 291.693 From ephemeris DE431 at Epoch: JD$_0 = 2455562.5$ (2011 January 1) TDB Figure: Semi-equinoctial elements of the inner Solar System planets obtained using three models.", "The analytical secular model derived is shown in blue, the integration of the 9BP of the main planets and the Sun in red and the ephemeris file DE431 is shown in grey.", "The initial conditions are obtained from the time average of the ephemeris file over the first orbital periods of the planets.Figure REF shows the solution of eq.", "REF for Mercury, Venus, Earth and Mars as perturbed mutually and from the rest of planets of the Solar System.", "This model is compared to two other models for 15,000 years into the past.", "The first one is a numerical integration of the N-body problem taking into account the main 8 planets of the Solar System and the Sun.", "Then, we also compare to the planetary ephemerides DE431 [10].", "While the complete ephemerides models show the short period effects, the secular component is modeled by the two simplified models.", "The initial conditions of the 9BP integration and the secular theory are obtained by averaging the full ephemeris model for two orbit periods.", "As a result, the initial conditions visually appear to be off from the mean of the full ephemeris solution, but they are the actual time average.", "The short-term components have a significant effect in the evolution of $h_j,k_j$ .", "In the case of $p_j,q_j$ , the secular component is the dominant effect of the evolution.", "Because of the assumptions of small eccentricities and inclinations, the predicted frequencies are not perfectly accurate, as observed in the drift between the 9BP solution and the analytical theory of figure REF , especially in $p_j,q_j$ .", "A similar agreement in these elements is found for the gas giants.", "However, only the inner Solar System planets are shown as they are the bodies that are encountered by near-Earth objects.", "Thus, these are the planets for which we want to guarantee an accurate model of their secular dynamics.", "The analytical propagation of Mercury drifts the most from the full ephemeris solution, although it is the least relevant inner planet.", "Close encounters with Mercury are unfrequent and have a small effect, as Mercury is the least massive planet and it is encountered with very high relative velocity.", "As a result of the averaging of the perturbing potential, the semi-major axis of the bodies remains constant.", "The complete set of secular solutions includes the mean anomaly at epoch $\\sigma _j$ .", "Short term applications benefit from the improved characterization of the position of the bodies in their orbits.", "Solving for $\\sigma _j$ is straightforward if we ignore the contribution of $R_{1,j}$ , which has a small effect compared to $R_{0,j}$ .", "The equation of motion for $\\sigma _j$ becomes: $\\dot{\\sigma }_j=- \\frac{\\partial R_j}{\\partial L_j} =- \\frac{2}{n_{j} a_{j}} \\frac{\\partial R_{0,j}}{\\partial a_j}$ and the solution depends on whether the perturber is external or internal: $\\dot{\\sigma }_j = - \\sum ^{N}_{k=1,k\\ne j} \\frac{Gm_k}{n_{j} a^2_{j}} \\bar{c}_{jk} D b^{(0)}_{1/2}$ where $\\bar{c}_{jk}=a_j/a^2_k$ in the case of an external perturber and $\\bar{c}_{jk}=-1/a_j$ if the perturber is internal.", "The solution of the equation is simply a constant drift given by the rate $\\dot{\\sigma }$ .", "This element completes the set of elements of the secular model." ], [ "Analytical secular dynamics of near-Earth asteroids", "The secular dynamics of asteroids can be modelled as a particular case of the secular dynamics of multibody systems described above.", "In the present work we apply this solution to the evolution of the asteroid under the external perturbation of Jupiter.", "The solutions of equation REF simplify in the case of a system of 2 bodies with a massless internal body.", "We follow the same process to obtain the solution.", "Matrices $A_{jk}$ and $B_{jk}$ simplify to: $\\begin{aligned}[c]A_{jk} = \\left(\\begin{matrix}B_{12} & A_{12}\\\\0 & 0\\end{matrix}\\right)\\end{aligned}\\qquad \\qquad \\begin{aligned}[c]B_{jk} = \\left(\\begin{matrix}-B_{12} & B_{12}\\\\0 & 0\\end{matrix}\\right)\\end{aligned}$ where the subindexes $1,2$ correspond respectively to the massless particle and the external perturber.", "The coefficients of the matrices are found as in equations REF -REF above.", "The solution to the eigenvalue problem yields the secular frequencies of the secular propagation $g_1=B_{12}$ , $g_2=0$ , $f_1=-B_{12}$ , $f_2=0$ .", "As expected, the elements of the perturber $h_2,k_2,p_2,k_2$ remain constant.", "The eigenvectors are the columns of the matrices: $\\begin{aligned}[c]\\bar{e}_{jk} = \\left(\\begin{matrix}1 & \\kappa \\\\0 & 1\\end{matrix}\\right)\\end{aligned}\\qquad \\qquad \\begin{aligned}[c]\\bar{i}_{jk} \\left(\\begin{matrix}1 & \\frac{\\sqrt{2}}{2}\\\\0 & \\frac{\\sqrt{2}}{2}\\end{matrix}\\right)\\end{aligned}$ where the constant $\\kappa $ is found as the ratio between Laplace coefficients: $\\kappa = \\frac{A_{12}}{-B_{12}} = \\frac{b^{(2)}_{3/2}}{b^{(1)}_{3/2}}$ Note that the vector ($e_{12},e_{22}$ ) is not normalized.", "This is not necessary because in the process of obtaining the integration constants from the initial conditions the scaling of the eigenvectors is found.", "The solution of the elements of the massless particle becomes: $ \\begin{aligned}[c]h_1{(t)} &= S_1 \\sin {(g_1 t + \\beta _1)} + \\kappa h_2 \\\\k_1{(t)} &= S_1 \\cos {(g_1 t + \\beta _1)} + \\kappa k_2 \\\\\\end{aligned}\\qquad \\qquad \\begin{aligned}[c]p_1{(t)} &= T_1 \\sin {(f_1 t + \\gamma _1)} + p_2 \\\\q_1{(t)} &= T_1 \\cos {(f_1 t + \\gamma _1)} + q_2 \\\\\\end{aligned}$ with constants of integration: $\\begin{aligned}[c]S^2_1 &= e^2_{1,0} + \\kappa ^2 e^2_2 - 2 \\kappa e_{1,0} e_2 \\cos {(\\varpi _{1,0} - \\varpi _2)} \\\\T^2_1 &= i^2_{1,0} + i^2_2 - 2 i_{1,0} i_2 \\cos {(\\Omega _{1,0} - \\Omega _2)} \\\\\\end{aligned}\\qquad \\qquad \\begin{aligned}[c]\\tan \\beta _1 &= \\frac{h_{1,0} - \\kappa h_2}{k_{1,0} - \\kappa k_2} \\\\\\tan \\gamma _1 &= \\frac{p_{1,0} - p_2}{q_{1,0} - q_2} \\\\\\end{aligned}$ The time evolution of the Keplerian elements set can be obtained from the relationships with the semi-equinoctial set in equation REF .", "The solutions of $\\varpi (t),\\Omega (t)$ are the secular drift with frequencies $g_1,f_1$ that are equal with opposite signs.", "The solutions of $e(t),i(t)$ are oscillations with frequencies $g_1,f_1$ as obtained from the development of eccentricity $e_1 (t)=\\sqrt{h^2_1 (t) + k^2_1 (t)}$ and inclination $i_1 (t) =\\sqrt{p^2_1 (t) + q^2_1 (t)}$ .", "The maximum and minimum values of eccentricity and inclination are: $\\begin{aligned}[c]e^2_{1,min} &= S^2_1 + \\kappa ^2 e^2_2 - 2 S_1 \\kappa e_2 \\\\e^2_{1,max} &= S^2_1 + \\kappa ^2 e^2_2 + 2 S_1 \\kappa e_2 \\\\\\end{aligned}\\qquad \\qquad \\begin{aligned}[c]i^2_{1,min} &= T^2_1 + i^2_2 - 2 T_1 i_2 \\\\i^2_{1,max} &= T^2_1 + i^2_2 - 2 T_1 i_2 \\\\\\end{aligned}$ The secular model is computed for the fictitious asteroid of Case 1 of table REF with the perturbation of Jupiter given by the elements of table REF .", "These cases are used later to demonstrate the propagation tool.", "For a nominal eccentricity of 0.15 the minimum eccentricity is 0.14946 and maximum is 0.17466.", "For a nominal inclination of 10 degrees, the minimum inclination is 7.41823 degrees and maximum is 10.02508 degrees.", "The characteristic period of the secular motion $T_{sec}$ is 154,116 years.", "This model assumes small eccentricities and inclinations.", "While these conditions are usually not fulfilled, it is important to remark that eccentricity and inclination are under frequent disturbance due to close encounters.", "Most importantly, the secular drift in $\\Omega ,\\omega $ controls the evolution of the possible planetary encounters.", "The assumptions on the heliocentric orbit of the asteroid for the analytical secular perturbation solution are not always fulfilled among the NEO population.", "In this section we show that the analytical theory represents the dynamics of the perturbation by Jupiter.", "For this reason, we integrated the orbits of 4462 NEOs with $e<0.7$ and $i<0.5$ rad for 50,000 years.", "Note that in the solution of equation REF if the terms of the external perturber are small the solution tends to a linear drift of the angles $\\Omega , \\varpi $ .", "In addition, given the relationship between the frequencies $g$ and $f$ , the relationship between the arguments rates is $\\dot{\\omega }=-2\\dot{\\Omega }$ .", "Figure: Error in the secular rates of near-Earth objects as obtained by linear regression of the propagation using Laplace-Lagrange secular theory and compared to numerical integration of the three-body problem in percent.", "Secular rates in ascending node (left) and argument of perihelion (center) as function of the initial conditions semi-major axis and inclination.", "Secular rates in the angles as obtained with the two methods and function of the initial semi-major axis (right).", "The dashed lines indicate the region in which we compute the average errors.", "This region includes the initial conditions used throughout the paper, indicated as cross marks.Figure REF shows the secular rates computed from linear regression of the time histories of $\\Omega $ ,$\\omega $ .", "Note that from an initially larger list of NEOs, a significant fraction (12%) was discarded because either eccentricity or inclination were larger than 0.5.", "An additional 11% of the solutions were discarded because the error in the regression was too large or during the propagation close encounters with Jupiter moved the orbit of the NEO to a completely different location than the initial conditions.", "While the linear regression secular rates are not equivalent to the frequencies of the analytical solution, they serve as a comparison between the two dynamical models.", "The error is computed in percent relative to the rate measured from the regression of the numerically integrated trajectories, given by: $E(\\dot{\\omega }) = 100 \\left| \\frac{\\dot{\\omega }_{sec} - \\dot{\\omega }_{3BP}}{\\dot{\\omega }_{3BP}} \\right|$ As observed in Figure REF the rates obtained with the two methods agree towards the smaller end of semi-major axis.", "Since these are near-Earth objects, the condition of being in the vicinity of Earth means that eccentricity increases with semi-major axis.", "We can see that past 1.5-2 au the difference between the two models is increased, as well as the secular rates values themselves also increase.", "This difference is also appreciated in the rates as function of semi-major axis, in which we show the agreement in the NEO region.", "Using the current model we find secular rates for 60% of the population with an error less than 30% in both $\\dot{\\omega }$ and $\\dot{\\varpi }$ .", "If we limit the application of the secular theory to semi-major axes between 0.8-1.4 au (dashed region in Figure REF ) we find that this agreement improves to 88%.", "It is important to note that the examples chosen to demonstrate the semi-analytical propagation tool fall within this region.", "Outside of this region we can verify if the secular rates found are reliable by using numerical integration.", "This test integration must be long enough to observe the secular rates, but still orders of magnitude shorter than the time-scales that we can more efficiently study using the semi-analytical propagation.", "At larger semi-major axis the effect of mean-motion resonances becomes important, and that Lidov-Kozai dynamics may better represent the dynamics for large eccentricities and inclinations [25], [31].", "The implementation of additional analytical long-term dynamics models to model any generic asteroid is left as future work." ], [ "Finding the subsequent encounter", "The analytical propagation of particles is interrupted when an encounter with a planet is detected.", "In principle it is not necessary to track the distance to planets at all times, since the regions in which encounters are possible are determined by the geometry of the asteroid around the central body.", "If the inclination relative to the planet is high, then the encounters are only possible in the vicinity of the ascending and descending node.", "However, the most generic approach is to track the distance between the asteroid and the crossing planets at all times.", "Thus, the results in this work follow the latter approach to find encounters within a closest approach distance of 0.1 au.", "When the two bodies are close, the unperturbed closest approach distance is found using a bisection method where the function is the derivative of the distance as obtained by finite differences.", "This process results in less evaluations of the relative distance function based on the heliocentric elements of the bodies.", "The elements of the planets and the asteroid are propagated using the secular solution at the date of start of the encounter, which is defined below.", "The transition between models consists in the conversion between the sets of elements, obtaining the necessary Keplerian elements in the process.", "These are semi-equinoctial elements for the analytical perturbed propagation as in equation REF and Delaunay elements for the quadrature of the Lagrange planetary equations." ], [ "Evaluation of planetary encounters", "Close encounters are commonly solved using the analytical Öpik theory [34].", "While this theory requires the least computational resources, its accuracy is limited to specific circumstances.", "The quadrature of Lagrange Planetary Equations can be used to solve close encounters [1].", "In this work we derive a solution using this method for generic close encounters using Delaunay elements.", "The two methods are compared to the integration of the three body problem from the same date and during the same period of time." ], [ "Öpik theory of close encounters", "An analytical solution to the planetary close encounter problem was derived by Öpik [34].", "This solution was extended and studied in detail by [43], [42].", "The encounter solution uses a linearized mapping from orbital elements to a planetocentric Cartesian frame, that is later expressed in B-plane coordinates.", "Then, the encounter is assumed instantaneous and the incoming asymptote and B-plane both rotate.", "The new B-plane coordinates are mapped back to the orbit elements space.", "The analytical solution is derived for a hyperbolic flyby around a point secondary mass.", "This mapping between B-plane coordinates and orbit elements is linearized in the impact parameter.", "Thus, the encounter must be close for the method to be reliable.", "Additionally, if the inclination is small the relative velocity coordinates become undefined.", "A possible way to avoid this is by using a method sometimes referred as pseudo-Öpik [14].", "In this case the relative velocity vector is computed directly and defines the turn angle $\\gamma $ at the time of closest approach: $\\tan {\\frac{\\gamma }{2}} = \\frac{m}{bU^2}$ where $m$ is the mass of the planet in units of the mass of the Sun, $b$ is the impact parameter and $U^2$ is the relative velocity in units of the circular velocity of the planet.", "Here we use the unperturbed trajectory of the planet and asteroid to find these quantities.", "That is, the impact parameter and relative velocity are found as the planetocentric distance and velocity at closest approach.", "Figure: Logarithm of the errors in the computation of the final Keplerian elements given by pseudo-Öpik theory (PÖpik - first and second rows) and the quadrature of LPE (QLPE - third and fourth rows).", "The list of flybys used is shown in figure and generated as described in the text.", "The flybys are represented in the plane of relative velocity at closest approach V inf V_{inf} (km s -1 ^{-1}) and distance of closest approach (au).", "The dashed area in the QLPE error plots separates the region in which the encounters are computed using an integration of the 3BP during the semi-analytical propagation." ], [ "Lagrange Planetary Equations", "The proposed computation of the encounter effect is computed as follows.", "The variation in elements over the encounter event is obtained from a quadrature of the Lagrange Planetary Equations assuming the geometry of the unperturbed flyby.", "The elements used are obtained from the secular propagation of the asteroid.", "The Lagrange Planetary Equations describe the evolution of orbit elements due to a perturbing potential.", "The derivation of Lagrange Planetary Equations can be found in some references for different sets of orbit elements [4], [38].", "In general, they have the form of: $\\mathbf {\\dot{D}} = \\left[L (\\mathbf {D})\\right] \\frac{\\partial R}{\\partial \\mathbf {D}}(\\mathbf {D},t)$ where $\\mathbf {D}$ is the set of elements of choice and $L(\\mathbf {D})$ is a function of the elements that depends on the chosen elements.", "In this case the perturbing potential $R$ is the gravitational potential of the encountered planet.", "Note that without further simplifying assumptions the partials of the perturbing potential are function of the elements and function of time.", "For the elements of choice we take the partial derivatives that relate the set of elements to Keplerian elements $\\mathbf {K} = [a,e,i,\\Omega ,\\omega ,\\sigma ]$ and Cartesian coordinates.", "From the orbital elements representations available to choose, the current implementation uses the Delaunay elements: $\\begin{aligned}[c]L&=\\sqrt{\\mu a}\\\\G&=L\\sqrt{1-e^2}\\\\H&=G \\cos {i}\\end{aligned}\\qquad \\qquad \\begin{aligned}[c]l&=\\sigma \\\\g&=\\omega \\\\h&=\\Omega \\end{aligned}$ The Lagrange Planetary Equations with the perturbing potential of Equation REF with $j$ being the asteroid and $k$ the encountered planet can be written as: $\\begin{aligned}[c]\\frac{\\mathrm {d} L}{\\mathrm {d} t}&=\\frac{\\partial R}{\\partial {r}}\\frac{\\partial {r}}{\\partial \\mathbf {K}}\\frac{\\partial \\mathbf {K}}{\\partial l}\\\\\\frac{\\mathrm {d} G}{\\mathrm {d} t}&=\\frac{\\partial R}{\\partial {r}}\\frac{\\partial {r}}{\\partial \\mathbf {K}}\\frac{\\partial \\mathbf {K}}{\\partial g}\\\\\\frac{\\mathrm {d} H}{\\mathrm {d} t}&=\\frac{\\partial R}{\\partial {r}}\\frac{\\partial {r}}{\\partial \\mathbf {K}}\\frac{\\partial \\mathbf {K}}{\\partial h}\\end{aligned}\\qquad \\qquad \\begin{aligned}[c]\\frac{\\mathrm {d} l}{\\mathrm {d} t}&=-\\frac{\\partial R}{\\partial {r}}\\frac{\\partial {r}}{\\partial \\mathbf {K}}\\frac{\\partial \\mathbf {K}}{\\partial L}\\\\\\frac{\\mathrm {d} g}{\\mathrm {d} t}&=-\\frac{\\partial R}{\\partial {r}}\\frac{\\partial {r}}{\\partial \\mathbf {K}}\\frac{\\partial \\mathbf {K}}{\\partial G}\\\\\\frac{\\mathrm {d} h}{\\mathrm {d} t}&=-\\frac{\\partial R}{\\partial {r}}\\frac{\\partial {r}}{\\partial \\mathbf {K}}\\frac{\\partial \\mathbf {K}}{\\partial H}\\end{aligned}$ The proposed solution is the integration of these differential equations around the encounter date $t_{e}$ and assuming the unperturbed geometry of the flyby.", "Hence, the asteroid coordinates are obtained from the heliocentric elements secularly propagated until the start of integration date $D_0$ and the quadrature is only a function of time: $\\Delta {\\mathbf {D}}=\\int _{t_{e}-\\delta t}^{t_{e}+\\delta t}\\mathbf {\\dot{D}}(\\mathbf {D}_0,t)dt$ The integration is conducted for a fraction of the orbit period around the closest approach distance.", "This fraction is a constant set large enough such that the effect of the encounter is captured completely.", "In this work we use a self-coded fast quadrature function based on the midpoint rule and a total integration time of 20% of the orbital period.", "This method avoids the frame transformation to the center of the planet, since it considers the planet as an external perturber of the asteroid motion around the Sun.", "For this reason, it is not possible to obtain a closed form solution of the integral.", "Nonetheless, this approach does not imply further assumptions that limit its range of applicability.", "Future work will be done in finding the optimal set for this application.", "This approach is accurate for the vast majority of encounters, but it is less accurate for the closest ones, as we explore in the following section.", "Using the list of flybys generated in figure REF we computed the errors of Öpik theory and the quadrature of LPE compared to the solution of the encounter using the three-body problem.", "The error $E(K)$ is relative to the variation and in percent, given by: $E(K) = 100 \\frac{\\Delta K_{QLPE} - \\Delta K_{3BP}}{\\Delta K_{3BP}}$ The results of this evaluation are described in figure REF .", "Using pseudo-Öpik theory there is a region in the space of relative velocity and closest approach distance in which flybys can be computed accurately.", "However, this region is not constant for all Keplerian elements.", "In addition, most flybys are not computed correctly using this method.", "Slow flybys break the assumption in Öpik theory that the behavior during the flyby is modeled by the two-body hyperbolic interaction.", "Many of the faster flybys occur with Venus and Mercury.", "The two-body hyperbolic flyby model fails to characterize the effect of flybys with these less massive planets even if they are faster.", "Using the quadrature of the Lagrange Planetary Equations 99% of the flybys list are computed with less than 3% of error, and more than 88% with less than 0.1% error.", "The flybys that are not computed accurately with this method are very close and with a slow relative velocity.", "These flybys can break the assumption of the unperturbed geometry of the flyby.", "Given that these encounters also cause significant variations in the elements, these infrequent encounters are solved using a three-body problem integration in Cartesian coordinates.", "The criteria to solve these encounters using the alternative method is by defining three threshold regions in the encounter parameters: with very small $V_\\infty $ , very small closest approach distance, and a combination of both close to zero.", "This process simplifies the detection of collisions with the planets during the numerical integration in Cartesian space in the heliocentric frame." ], [ "Semi-analytical propagation vs. numerical integration", "In the previous sections we validate the individual pieces of the semi-analytical propagation tool.", "Once combined, we want to compare the resulting trajectories to trajectories obtained using numerical integration.", "With this purpose we generate a fictitious NEO population and propagate their orbits using both methods.", "The fictitious NEO population we define consists in normal distributions for the perihelion distance, eccentricity, inclination.", "The distributions are centered respectively around $0.8$ au, $0.2, 10\\deg $ and with standard deviations of $0.05$ au, $0.05, 3\\deg $ .", "Arguments of the node, perihelion and initial mean anomalies are defined as uniform distributions in the $[0,360]$ degrees range.", "We sample 1000 particles from these distributions as our test set for the comparison between the two methodologies.", "We setup the numerical integration of the asteroid orbits considering the planets as third body perturbers.", "The model we use for the orbits of the planets is the secular theory developed in section REF .", "Figure REF shows the results of the propagation using both methods as well as the distributions that defined the initial conditions.", "In addition, we find that the cumulative distribution of the mean number of encounters versus closest approach distance is matched very closely.", "After 200,000 years, the presence of planetary encounters causes a dispersion of the initial distributions.", "The distribution obtained through semi-analytical propagation is able to track this drift.", "The main difference between the results using the two methods is that the numerically integrated distribution shows a small drift in the center of the distribution, but very similar dispersions.", "In terms of the longitude of the perihelion, the resulting distributions after 200,000 years remain uniform using either of the two propagation methods.", "The other significant difference between the two methods is in the required the computational time, which is discussed next.", "Figure: Comparison between the 1000 trajectories obtained using numerical integration (blue) and semi-analytical propagation (orange) of a fictitious population of NEOs after 200,000 years.", "The probability density function is shown for the perihelion distance, inclination and argument of perihelion.", "The analytical probability density function of the initial conditions is shown as a continuous red line.", "On the right, the mean number of encounters found as function of the closest approach distance for both methods." ], [ "Computational time of the semi-analytical propagation tool", "The semi-analytical propagation of near-Earth asteroids reduces the computational time required to obtain long-term trajectories.", "The use of numerical techniques is limited almost exclusively to the computation of planetary encounters, that represent a fraction of the simulation time.", "In order to estimate the speedup of the propagation we generate a fictitious population of NEOs and propagate them for 100,000 years with the semi-analytical propagation tool and with numerical integration.", "The semi-analytical propagation tool uses the secular solution of the Solar System to compute the orbits of the planets over long periods of time.", "Then, planetary encounters can occur with any planet.", "The secular propagation of the asteroid is derived as described in the text accounting only for Jupiter, which accurately represents the asteroid motion in the absence of resonances.", "The semi-analytical propagation of 100,000 years with this setup is computed in around 5 seconds.", "In order to account for these effects in numerical integrations we use the N-Body problem integrator IAS15 [37] with all the planets and the asteroid.", "The simulation is setup using high-level programming code that runs libraries in more efficient low-level code.", "This is the case for both numerical integrations and semi-analytical propagation, running in the same 2.5GHz Intel Core i7 processor.", "The result is a speed-up of x500-x1000 of the semi-analytical propagation tool as compared to the numerical integration.", "The current implementation allows room for significant speed-up that is left for future work.", "The perturbed long-term propagation could be extended to use other suitable models of interest.", "The computational cost is not expected to increase while we use analytical solutions of these long-term perturbations." ], [ "Semi-analytical propagation results", "In this section we demonstrate the semi-analytical propagation tool in a variety of scenarios.", "First, we want to compare the semi-analytical model with trajectories obtained through numerical integration.", "Matching very accurately trajectories obtained with more complex models is outside the scope of the comparison.", "Even if the models were identical, trajectories under encounters are very sensitive to the initial conditions and under small perturbations they diverge into different paths.", "This effect was visualized in figure REF using only numerical integration.", "For this reason, long term simulations may focus on the statistical analysis of the dynamical evolution rather than individual trajectories.", "Throughout the section, the simulations include Jupiter as the only planet secularly perturbing the asteroids.", "All the inner Solar System bodies are considered to evaluate planetary encounters.", "These are secularly evolving due to mutual perturbations and the perturbations of the outer Solar System planets, as described in section 3.a.." ], [ "Short-term propagation of near-Earth objects using different models", "The asteroid chosen for the tool demonstrations is the binary (35107) 1991 VH.", "The first reference trajectory is obtained from the HORIZONS system of JPL [12].", "The second model is the numerical integration of the asteroid motion under the influence of the Sun, Earth and Jupiter.", "Jupiter is the main driver of the secular motion, which is observed as a linear drift in the argument of perihelion and argument of the ascending node.", "Given the current orbit of (35107) 1991 VH, it only experiences planetary encounters with Earth in the next few centuries.", "In Figure REF , we compare these two models from numerical integration with the present semi-analytical propagation tool.", "In the three trajectories we observe similar behavior, although it manifests differently in every element.", "First, there is a close agreement in the encounter dates of the Sun-Earth-Jupiter integration and the encounters found by the propagation tool that shows in all elements.", "The encounter dates can be distinguished as the discontinuities in the trajectories, especially in the semi-analytical propagation trajectory.", "The variation of the semi-major axis is characterized by a random walk from the planetary encounters.", "In both eccentricity and inclination there is a relevant role of the encounters with an additional secular component that the secular model is able to model.", "The dynamics of the argument of the ascending node are dominated by the secular drift.", "There is not a significant effect that can be perceived by the planetary encounters and this is expected when the encounters occur with a unique planet and close to the node.", "What we observe is that the secular dynamics including the complete effects of Earth and Jupiter are very similar, and the analytical secular drift is off by about a degree after the 500 years of the propagation.", "The secular drift rate in the argument of perihelion is not as trivial to compare since it must be observed between encounters, although good agreement is found too.", "Last, the mean anomaly at epoch evolves over time with an increasing amplitude present in all three models.", "Figure: Close encounters in the semi-analytical propagation of the fictitious NEOs Cases 1-3 with initial conditions given in table .", "Closest approach distance d CA d_{CA} (au) of the encounters and MOID with the planets.", "Dots indicate close encounters, solid lines indicate the MOID.", "Color code indicates the planets: (Green) Venus, (Blue) Earth, (Red) Mars.lccccccc[h!]", "2 Near-Earth objects used as example cases for the demonstration of the semi-analytical propagation tool.", "0pt Asteroid $a (au)$ $e$ $i$ (deg) $\\Omega $ (deg) $\\omega $ (deg) $M_0$ (deg) JD (TBD) Case 1 1.1 0.15 10 90 90 90 2451545.0 Case 2 1.2 0.35 40 90 90 90 2451545.0 Case 3 1.3 0.5 10 90 270 90 2451545.0 Case 4 0.95 0.07 20 90 90 90 2451545.0 Case 5 0.9 0.25 15 90 90 90 2451545.0 1991 VH 1.1373 0.14426 13.912 139.37 206.88 302.39 2456902.5 1996 FG3 1.0543 0.34987 1.9903 299.88 23.930 147.277 2454796.5 The elements of asteroids 1991 VH and 1996 FG3 were retrieved from HORIZONS [12].", "Using DE431 and SB431-N16.", "1991 VH: Orbit solution date 2021 April 15, 1996 FG3: Orbit solution date of 2021 April 26." ], [ "Long-term propagation and the MOID", "The minimum orbit intersection distance (MOID) indicates what the minimum distance between any two points of the two heliocentric Keplerian orbits is.", "In this case we focus on the orbit of Earth and the orbit of the asteroid.", "The MOID is also used as one of the criteria to define an asteroid as a potentially hazardous asteroid.", "There are many algorithms available in the literature to compute the MOID [15], [47], [2].", "In this paper we use the tool derived in [17], [16] based on an asymptotic approach.", "The MOID constrains the minimum distance of a possible close encounter.", "In other words, the periods with a large MOID are absent of close encounters.", "Three examples are used to visualize time histories of close encounters and the evolution of the MOID for 100,000 years.", "These are obtained for high and low eccentricities and inclinations, and the initial conditions are found in table REF .", "Figure: Close encounters in the semi-analytical propagation of the fictitious NEO Cases 1-3 with initial conditions given in table .", "V ∞ V_\\infty at closest approach of the encounter (km s -1 ^{-1}).", "Color code indicates the planets: (Green) Venus, (Blue) Earth, (Red) Mars.The distributions of closest approach distances are shown in figure REF and the unperturbed relative velocity $V_\\infty $ at those encounters is found in figure REF .", "Case 1 is an example of a NEO with relatively low eccentricity and inclination.", "In these conditions, close encounters are only possible with Earth and at a low relative velocity.", "The MOID oscillates secularly with long periods of low MOID.", "Case 2 is an example of an opposite scenario in which both eccentricity and inclination are large.", "In the secular evolution of the MOID this translates in short periods of low MOID and long periods absent from encounters.", "This is scenario in which the semi-analytical propagation of the asteroid allows a rapid propagation until the next period of frequent encounters.", "Case 2 faces high velocity close encounters with Venus, Earth and Mars.", "Case 3 is an example of a NEO with high eccentricity and low inclination.", "This combination of factors leads to a large frequency of close encounters with the inner planets.", "In this case, encounters are very frequent with Venus, Earth and Mars.", "The close encounters experienced by Case 3 are with a relative velocity smaller than in Case 2 given the reduced inclination.", "Even under the elevated frequency of close encounters, the secular signature of the MOID is maintained.", "The structure shows until the event of an energetic close encounter.", "The occurrence of such encounters is just a matter of probability of having the right timing during the low-MOID intervals of the secular propagation." ], [ "Statistics of long-term propagation", "The chaotic nature of the dynamics implies that the study of the orbital evolution over long timescales should be done statistically.", "Given the uncertainty in the orbit solution of an asteroid, we can sample a large number of particles and study the dynamical paths that the different particles take.", "Because of the sensitivity to initial conditions in planetary encounters, very well determined distributions diverge in a few centuries to widely different paths.", "We demonstrate these effects by propagation of 500 particles that sample uncertainty distributions around a nominal asteroid orbit for 500,000 years.", "We inspected 7 examples: first, in detail, orbit solution of (35107) 1991 VH; then, the orbit solution of (175706) 1996 FG3; last, the previous Cases 1-5 of table REF with artificial orbit uncertainties as described in appendix .", "Figure: 500,000 year Monte Carlo semi-analytical propagation of asteroid (35107) 1991 VH.", "Initial conditions are given in table as obtained from HORIZONS and uncertainties in the distribution are obtained from JPL’s SSD/CNEOS Small-Body DataBase as described in Appendix .", "Elements shown are perihelion, eccentricity, inclination, argument of the ascending node, argument of perihelion and minimum orbit intersection distance (MOID).", "Grey lines show individual simulations, black lines are the median of the 500 simulations of each parameter shown.Figure: Semi-analytical propagation of asteroid Cases 1-5, (35107) 1991 VH, (175706) 1996 FG3 for 500,000 years.", "Histograms at the initial time, after 100,000 years and at the final time.", "Initial conditions are given in table as obtained from HORIZONS and uncertainties in the distribution are obtained from JPL’s SSD/CNEOS Small-Body DataBase as described in Appendix .Figure REF shows the time histories of the individual runs of the cloud of points originally neighboring (35107) 1991 VH, showing that the cloud of particles distributes over a wide region of near-Earth space.", "On the order of hundreds of thousand years, the dispersion is accomplished by the less frequent very close encounters.", "Eccentricity and inclination show the secular component but are dispersed by the presence of encounters.", "The dynamics of argument of ascending node and argument of perihelion remain mainly secular with a degree of dispersion because of the presence of encounters.", "Note that the initial uncertainty on the orbit of (35107) 1991 VH is very small as shown while demonstrating the chaotic dynamics nature in Figure REF of the background section.", "As it was observed in the detailed analysis of a shorter simulation in figure REF , the mean anomaly at epoch changes completely with small changes in the semi-major axis.", "This fact reflects in the long-term simulations as a complete uniformization after just a few centuries.", "The binary (35107) 1991 VH currently presents a MOID that is decreasing.", "This means that after a few millenia the probability of experiencing very close encounters increases.", "In the statistical analysis, this probability shows in that a fraction of the fictitious asteroids experience such encounters.", "We observe that towards the end of the simulation there is a large dispersion in inclination and perihelion distance.", "By the end of the simulation, the angles $\\Omega -\\omega $ are dispersed along a linear drift as caused by numerous close encounters that not only change these angles, but modify the secular dynamic frequencies.", "The long-term dynamics of 6 more examples are integrated for 500,000 years.", "These are the Cases that we used to illustrate the long-term dynamics with initial conditions in Table REF .", "The case of the binary (175706) 1996 FG3 is also added to the discussion, as it is the other target of exploration of the Janus mission [39].", "The uncertainties of (175706) 1996 FG3 and (35107) 1991 VH are sampled based on their publicly available orbit solutions.", "In the case of the fictitious asteroids we used an arbitrary distribution.", "Both approaches are explained in detail in the appendix .", "The statistical distributions after 100,000 and 500,000 years are shown in Figure REF .", "The combined final and initial distributions are shown in Figure REF .", "The recorded number of encounters are shown in Figure REF and more details on the statistical distributions over time are shown in Figure REF .", "Next, we describe the dynamical evolution of these test cases.", "In Figure REF , the 500 virtual asteroids that are generated at the initial time per case are in the same bar of the histogram.", "The effect of repeated encounters causes the distributions to spread along near-Earth space.", "This dispersion is clearly shown in the perihelion distance in all cases, with a general trend of a decrease in the distance.", "Figure REF additionally shows this spread of all the cases together.", "The presence of mean motion resonances in the inner Solar System can protect asteroids from close encounters [28].", "In the semi-analytical propagation of near-Earth asteroids the orbits may drift to these regions, as observed in Figure REF .", "The resonance regions are found for semi-major axes larger than $a=1$ au.", "In these cases encounters with the Earth stop occurring for a period of time.", "However, this clustering of particles is not found in numerically integrated populations or in the discovered population of near-Earth asteroids.", "Thus, we investigate the trajectories obtained through NBP dynamics.", "The process of obtaining the secular long-term perturbation eliminates the short-period perturbations.", "The latter perturbations cause an oscillation in the orbit elements of non-negligible amplitude.", "In order to measure the influence of this effect, we included an analytical oscillation in the semi-major axis with the frequency of the orbital period and an amplitude of the order of 0.01 au.", "This extension completely eliminates the artificial mean motion resonance regions.", "The analytical characterization of the short-period perturbation is left as future work, as its contribution must be considered for the complete set of orbital elements and is not expected to significantly modify the obtained distributions.", "Figure: Dispersion in the distributions of the 5 asteroid in table , (175706) 1996 FG3 and (35107) 1991 VH after 100,000 years (top) and 500,000 years (bottom).", "Semi-major axis vs. eccentricity (left) and semi-major axis vs. inclination (right).", "Initial uncertainty distributions detailed in appendix and shown in red.", "See text for the explanation of the resonances found in the semi-analytical propagation.Figure: Number of encounters experienced by the 5 test cases given in table , (175706) 1996 FG3 and (35107) 1991 VH during 500,000 years.", "Mean of the distributions of 500 particles (solid lines) and 1-σ\\sigma bounds (shadowed area).In general, we observe that asteroids encountering the planets more frequently disperse their distributions faster.", "This is the case for the Janus targets and Case 1, that present the largest standard deviations increase in the simulation time (Fig.", "REF ).", "This dispersion is not only shown in the elements, but also in the number of encounters (Fig.", "REF ).", "The evolution of the distribution as caused by encounters could be modelled as a random walk.", "If this hypothesis is true, then the standard deviation in the population increases linearly with the square root of time.", "Figure REF tests graphically this hypothesis for semi-major axis, eccentricity and inclination.", "Initially in all cases there is a fast increase in the standard deviations.", "After the few first millennia, some of the distributions follow the hypothesis of the linear relationship $\\sigma \\propto \\sqrt{t}$ , especially in the semi-major axis.", "In eccentricity and inclination, we observe that there is a secular component in the evolution of the distribution.", "The secular component is observed in both the evolution of the standard deviation and the mean of the variations (Fig.", "REF ).", "These are shown with respect to the initial values to have a common reference in the comparison of cases.", "The dispersion in semi-major axis modifies the secular rates of the drift in argument if perihelion and argument of the node.", "This effect combined with the direct variation of the angles during planetary encounters leads to the uniformization of the distribution in $\\omega , \\Omega $ .", "In the duration of our simulations of 500,000 most cases approach this uniformization as we showed in figure REF .", "Figure: Statistical evolution of the distributions of the 5 test cases given in table , (175706) 1996 FG3 and (35107) 1991 VH during 500,000 years.", "Standard deviation of semi-major axis, eccentricity, inclination as function of the number of encounters (top row), square root of time (center row).", "Variation of the mean value over time with respect of the initial value of semi-major axis, eccentricity and inclination (bottom row).We can compute more rigorously whether the distributions can be considered uniform by conducting the chi-squared test on the longitude of perihelion $\\varpi = \\omega + \\Omega $ .", "Figure REF shows the result of the chi-squared test of a uniform distribution over the simulation time for all cases.", "If the p-value is larger than our threshold of $p=0.05$ , we can consider that our null hypothesis of the uniform distribution of $\\varpi $ is true.", "Cases 1, 3, 4, (175706) 1996 FG3 and (35107) 1991 VH reach this threshold, while Cases 2 and 5 do not approach the significance by the end of the simulation time.", "It is interesting to show the evolution of the p-value compared to the mean number of encounters.", "In figure REF we show how the distribution of Case 3 tends to the uniformization in $\\varpi $ with less encounters than other cases that achieve this distribution earlier in the simulation time.", "This is expected since this case experiences more frequent encounters with the most massive planets and with a slower relative velocity, which means that the impact of these encounters in the dispersion of their distributions is larger.", "From the general trends that we observed, the only case that is very different is Case 2, that experiences much fewer encounters.", "As we illustrate in Figure REF , Case 2 experiences close encounters much less frequently than the other cases.", "The relative velocity is also larger in this case, which means that the effects of the encounters are not as strong.", "The case of (175706) 1996 FG3 is more difficult to fit in the general description of the dynamics, as the close encounters do not cause such a fast dispersion of the distribution.", "This binary asteroid is studied in more detail in section .", "Figure: P-value of the chi-squared test of the uniform distribution of the longitude of perihelion ϖ=ω+Ω\\varpi = \\omega + \\Omega for the 5 test cases given in table , (175706) 1996 FG3 and (35107) 1991 VH during 500,000 years.", "P-value is shown as function of the mean number of encounters (top) and function of time (bottom)." ], [ "Discussion", "The semi-analytical propagation tool shows the main dynamical effects observed in long-term numerical integration of the inner Solar System.", "The secular drifts caused by Jupiter move the asteroids between the vicinities of the different planets of the inner Solar System.", "This effect causes a seasonal presence of strong close encounters that can disturb both the orbit of the asteroid and its physical properties.", "While the time-scales of these events is of millions of years [8], if we sample a large enough number of particles we can measure the probabilities of collisions or the potential disruption of other physical properties of asteroids.", "The measurement of the collision probabilities was outside the scope of this paper, but in this work a few collisions were detected in the uncertainty sampling of the asteroids.", "The analytical modelling of the dynamics far from the planets was done using the Laplace-Lagrange theory, which works well in a large fraction of the NEO population.", "For this reason we defined a region in near-Earth space in which the secular model works best, as shown in Figure REF .", "However, we could extend the modelling in the regions of large eccentricity and inclination.", "In the previous section we describe the low frequency of encounters that is characteristic of asteroids with high inclination, specifically with Case 2.", "An asteroid of these characteristics would be likely to be dominated by the Lidov-Kozai effect, in which there is an exchange between high inclination-low eccentricity periods and low inclination-high excentricity periods.", "This would mean that Case 2 evolves to become a case closer to Case 3, in which encounters are more frequent.", "The use of analytical models of the Lidov-Kozai model [21] for the perturbed propagation is left for future work.", "Using the semi-analytical propagation tool we observe the stochastic nature of the dynamics.", "However, the effect is different on each of the elements.", "While in semi-major axis we observe what could be described as a random-walk process, the angles $\\Omega $ and $\\omega $ become uniformly distributed.", "Eccentricity and inclination show a mixed effect between a random-walk that adds dispersion to the distribution and the oscillations driven by the secular theory." ], [ "Orbit histories of the Janus mission targets", "The binary asteroids (175706) 1996 FG3 and (35107) 1991 VH are the targets of the exploration mission Janus [39].", "The long-term orbital dynamics of the asteroids are studied in this section with two goals.", "First, characterizing the stochastic nature of their long-term dynamics.", "Then, estimating the probability that they have been potentially disrupted by a very close encounter.", "With these purposes, we study their orbital origins by sampling 1000 particles and propagating them for 1Myr into the past using the semi-analytical propagation tool.", "Figure: Orbit history of (35107) 1991 VH in the last million years.", "Initial conditions are given in table as obtained from HORIZONS and uncertainties in the distribution are obtained from JPL’s SSD/CNEOS Small-Body DataBase as described in Appendix .", "Elements shown are perihelion distance, eccentricity, inclination, argument of the ascending node, argument of perihelion and minimum orbit intersection distance (MOID) with Venus, Earth and Mars.", "Grey lines show individual simulations, black lines are the median of the 1000 simulations of each parameter shown.", "Only 50 example runs and the median of the full distribution are shown in orbit elements (Top 2 rows).The orbit time histories of (35107) 1991 VH and (175706) 1996 FG3 are shown respectively in figure REF and figure REF .", "For clarity, we show only a subset of the runs and the median of the full distribution of 1000 runs.", "The minimum orbit intersection distance (MOID) is shown for the inner Solar System planets Venus, Earth and Mars.", "These metrics show when close encounters with these planets are possible.", "The presence of frequent close encounters causes the dispersion of the orbit histories.", "This feature manifests in the orbit history of (175706) 1996 FG3, in which the period of very low Venus MOID corresponds with a dispersion in the overall statistical representation of the orbit.", "Figure: Orbit history of (175706) 1996 FG3 in the last million years.", "Initial conditions are given in table as obtained from HORIZONS and uncertainties in the distribution are obtained from JPL’s SSD/CNEOS Small-Body DataBase as described in Appendix .", "Elements shown are perihelion distance, eccentricity, inclination, argument of the ascending node, argument of perihelion and minimum orbit intersection distance (MOID) with Venus, Earth and Mars.", "Grey lines show individual simulations, black lines are the median of the 1000 simulations of each parameter shown.", "Only 100 example runs and the median of the full distribution are shown in orbit elements (Top 2 rows).Figure: Histograms of the orbit history of (35107) 1991 VH and (175706) 1996 FG3 at the initial time, 100,000 years ago and 1Myr ago.", "Initial conditions are given in table as obtained from HORIZONS and uncertainties in the distribution are obtained from JPL’s SSD/CNEOS Small-Body DataBase as described in Appendix .Similarly to the long-term dynamics into the future studied in section , the initially very close distribution becomes a wide statistical distribution when propagated far into the past.", "In Figure REF we show histograms of the orbit elements and the number of encounters recorded below a closest approach distance threshold of 0.1 au.", "The orbit evolution of (35107) 1991 VH is mostly a spread around the initial conditions.", "However, (175706) 1996 FG3 is in a particular initial orbit with low inclination.", "On average, the very low inclination and high eccentricities drift toward a smaller eccentricity and higher inclination that are more frequent in the secular cycle.", "In both cases, the longitude of the perihelion becomes uniformly distributed.", "In the next section we characterize this uniformization process." ], [ "Stochastic modelling of the long-term dynamics", "In section we show that we can model the long-term dynamics with a random walk in semi-major axis, eccentricity and inclination.", "In addition, the latter two present also the influence of the oscillations of the secular theory.", "We also want to study the uniformization in the longitude of the perihelion, as this process occurs with time but also with a repeated number of close encounters.", "The random-walk model can be characterized by a linear increase in standard deviation with the square root of time.", "Figure REF shows the standard deviation of the 1000 Monte Carlo experiment that we conducted into the past of the two Janus targets (35107) 1991 VH and (175706) 1996 FG3.", "We fit a linear model to the standard deviation evolution, and measure the slopes to compare the evolution of the two targets.", "In the case of (35107) 1991 VH we avoid the use of the full simulation time, as the standard deviation bends from the initial linear increase.", "The slower increase after this period occurs when the distribution migrates from a configuration with slower encounters.", "The opposite case occurs with inclinations, in which the rapid increase of (175706) 1996 FG3 from the low-inclination initial regime slows down after the initial growth.", "The measured slopes are reported in table REF , showing that the random walk of (175706) 1996 FG3 is faster in semi-major axis and inclination.", "However, because of the bends in the progression after a few hundred thousand years, the final standard deviations are not substantially larger than the ones of (35107) 1991 VH after the million years into the past in eccentricity and inclination.", "Figure: Random walk statistical modelling of the evolution of semi-major axis, eccentricity, inclination of (35107) 1991 VH (top) and (175706) 1996 FG3 (bottom).", "Standard deviation of the 1000 Monte Carlo runs as function of the square root of time and random walk model fit.cccccc[htb!]", "3 Stochastic modelling of the long-term dynamics of the Janus targets 0pt Target 3cRandom walk $k_{x}$ constant ($s_{x}=k_{x}\\sqrt{t}$ ) 2c$\\varpi $ Uniformization Name $k_{a}$ $k_{e}$ $k_{i}$ Time Encounters $< 0.1$ au (au/$\\sqrt{yr}$ ) $\\cdot 10^{-3}$ (1/$\\sqrt{yr}$ ) $\\cdot 10^{-3}$ (deg/$\\sqrt{yr}$ ) $\\cdot 10^{-3}$ $yr\\cdot 10^{3}$ Number (35107) 1991 VH 0.2661 0.1799 6.0055 -434 11800 (175706) 1996 FG3 0.3688 0.0944 11.598 -797 29800 The process of uniformization of the longitude of perihelion is shown in figure REF .", "We conduct the chi-squared test of the uniform distribution over the 1Myr simulation, to find when the hypothesis of the uniform distribution is significant.", "In table REF we show the first time in which this criterion is satisfied, both in time and mean number of encounters: -434,000 years and a mean of 11800 encounters for (35107) 1991 VH, and -797,000 years and a mean of 29800 encounters for (175706) 1996 FG3.", "The uniformization of (35107) 1991 VH is faster than the uniformization of (35107) 1991 VH in both time and mean number of encounters.", "It is remarkable that (35107) 1991 VH takes a much lower mean number of encounters.", "This is explained by the faster relative velocities of the encounters of (175706) 1996 FG3 and a larger fraction occuring with Mars, a less massive planet.", "The relative velocities of a few of the recorded flybys are shown in figures REF and REF in the context of studying the probability that a close encounter could potentially disrupt the binaries.", "The comparison between the two binary systems highlights how the effect of the encounters depends on the relative velocities and the mass of the planets.", "In general, slow encounters and with larger planets are more efficient at causing the distributions to become uniform.", "However, depending on the heliocentric orbit these encounters may be more or less frequent.", "Thus, leveraging both effects is required to obtain a stochastic representation of the long-term dynamics of NEOs under frequent encounters.", "Figure: P-value of the chi-squared test of the uniform distribution of the longitude of perihelion ϖ=ω+Ω\\varpi = \\omega + \\Omega of (35107) 1991 VH (top) and (175706) 1996 FG3 (bottom).", "P-value is shown over the past 1Myr (left) and as function of the mean number of encounters (center).", "The mean number of encounters of the 1000 Monte Carlo runs is shown over time the corresponding with 1-σ\\sigma bounds (right)." ], [ "Potentially disruptive planetary encounters", "The two binary targets of the Janus mission present different relative orbits as observed by radar and photometry [36], [23], showing that (175706) 1996 FG3 is in a stable orbital state and (35107) 1991 VH is in a chaotic state.", "The perturbed state of (35107) 1991 VH could be explained by a recent very close encounter with the inner Solar System planets [18].", "Thus, it is of interest to characterize the frequency of such encounters in the orbital history of asteroid binaries.", "Using the semi-analytical propagation tool we obtain the history of flybys over a long time period of time.", "The perturbation in the orbit of a binary system during a planetary close encounter is studied in detail as described in [22].", "In this section we combine both results to predict the frequency of a disrupting flyby.", "The effect of the close encounter on the binary can be modelled as an impulsive variation in the binary Keplerian elements.", "In [22] the effect of close encounters to singly synchronous binary asteroids is studied.", "The variation in semi-major axis, eccentricity, and inclination obtained with numerical methods was compared to analytical expressions for the impulsive variation in binary Keplerian elements [7].", "We used these analytical expressions as they provide an estimate of the variation as function of the relative velocity and distance of closest approach.", "Figure: Potentially disruptive encounters recorded by the 1000 virtual (35107) 1991 VH binaries during a million years into the past.", "The background contour represents the logarithm of the mean variation in binary eccentricity during a close encounter with a planet: Earth (blue, top) or Mars (red, bottom).", "The encounters found during the propagation in this threshold are shown by their closest approach distance (au) and V ∞ V_\\infty (km s -1 ^{-1}).", "The radius of the planet is shown as a dashed line.", "The initial uncertainty distribution of (35107) 1991 VH is detailed in appendix .For every binary and encountered planet we can generate contours of the variation of the eccentricity.", "In Figures REF and REF we show these contours respectively for (35107) 1991 VH and (175706) 1996 FG3.", "In every figure we highlight two levels, a variation of 0.1 in eccentricity, and a variation of 1, which would mean that the binary is completely separated.", "The probability of disruption in the binary orbit increases as the relative velocity and closest approach distance are reduced.", "Using the semi-analytical propagation tool we track the close encounters below the threshold of 0.003 au, above which the variation in the binary Keplerian elements becomes negligible.", "For each binary we generate 1000 trajectories for a million years into the past.", "All the encounters that are found in this threshold are plotted in figures REF and REF and separated by encountered planet.", "The encounters potentially disruptive recorded for (35107) 1991 VH with Earth and Mars are shown in figure REF .", "Less than 1% of the recorded encounters are with Venus so the map with this planet is not included.", "The relative velocities of the flybys are mostly found between 5 and 15 km s$^{-1}$ .", "In the case of (175706) 1996 FG3 this range of possible relative velocities is larger in all the planets.", "In addition, many encounters are found with quite slower $V_\\infty $ , which means that even if they are not as close, they can still potentially cause a disruption.", "As we observed in figure REF , (175706) 1996 FG3 experiences more frequent encounters.", "However, the regions in which the encounters are potentially disruptive depend on the current orbital configurations of the binaries.", "In this case, (175706) 1996 FG3 requires closer and slower encounters to obtain the same mean variation in binary elements.", "Figure: Potentially disruptive encounters recorded by the 1000 virtual (175706) 1996 FG3 binaries during a million years into the past.", "The background contour represents the logarithm of the mean variation in binary eccentricity during a close encounter with a planet: Venus (green, top), Earth (blue, center) or Mars (red, bottom).", "The encounters found during the propagation in this threshold are shown by their closest approach distance (au) and V ∞ V_\\infty (km s -1 ^{-1}).", "The radius of the planet is shown as a dashed line.", "The initial uncertainty distribution of (175706) 1996 FG3 is detailed in appendix .Considering the orbit history in a million years, a non-negligible probability exists that both binaries have been potentially disrupted at some point.", "However, it is possible that the signature of these potential disruptions vanish if there is energy dissipation in the system.", "Thus it is relevant to study the probability that the binary orbits have potentially been disrupted recently in the orbit histories.", "We can study the first fraction of the long-term secular periods, in which the particles still have not mixed.", "Figure REF shows the history of potentially disruptive encounters in the recent periods of frequent encounters.", "The last period of possible very close encounters that we find for (35107) 1991 VH starts beyond the last 10,000 years.", "As determined by a mean variation in binary eccentricity of 0.1, we find that 61 of the 1000 test runs experience a potential disruption in the period of time up until the last 30,000 years.", "When we incorporate the next period of close encounters, the probability of a potential disruption increases to 131 out of 1000 runs in the last 60,000 years.", "Given the current orbit state of (175706) 1996 FG3, we find very close encounters with Venus in the last millennia.", "Because of the more restrictive closest approach distance for a potential disruption to happen, only 10 out of the 1000 test runs experienced this potential disruption in this period of frequent encounters.", "If we consider the last 10,000 years, then the probability increases to 54 cases in which at least one potentially disruptive encounter was found.", "If we keep increasing the time in which we consider all the potentially disruptive encounters, we can estimate the probability that a potentially disruptive encounter occurs.", "These probabilities are shown in figure REF with the corresponding 95% confidence intervals.", "The probability of suffering a disruptive encounter of (175706) 1996 FG3 increases faster than the probability of (35107) 1991 VH.", "This is explained by the significantly higher number of recorded close encounters.", "Thus, it is not possible to explain the chaotic state of (35107) 1991 VH only from the long-term probability of experiencing such encounters.", "However, a low probability in recent times combined with the incapability to dissipate the perturbation in a long time could explain the chaotic state of (35107) 1991 VH.", "Thus, future work will be done in the lines of characterizing the timescales of the dissipation of perturbations due to close encounters.", "Figure: (Top) Potentially disruptive encounters time history in the most recent periods of close encounters of (35107) 1991 VH and (175706) 1996 FG3.", "In the most recent 50,000 years, (35107) 1991 VH experiences Earth and Mars encounters.", "In the latest 10,000 years, (175706) 1996 FG3 experiences Venus and Earth encounters.", "The close encounters below the line of mean variation in binary eccentricity of 0.1 are highlighted with a black circle and the dashed line marks the radius of Earth.", "(Bottom) Probability of disruption based on the number of encounters found below thresholds of mean Δe=0.1\\Delta e=0.1 for low disruption and Δe=1\\Delta e=1 for high disruption.", "Confidence intervals on the probability of disruption prediction are shown in dashed lines." ], [ "Conclusions", "In this paper we present a rapid semi-analytical propagation tool for asteroids in the inner Solar System.", "The tool combines an analytical solution for the secular dynamics and the evaluation of planetary encounters.", "The derived solution of planetary encounters proves to accurately model the effect of the majority of flybys that asteroids experience in the inner Solar System.", "The long-term effect of the perturbation by Jupiter is captured by the analytical secular solutions in a large fraction of the NEO population.", "The combination with detection and evaluation of close encounters recreates the full dynamics as we demonstrate for the case of (35107) 1991 VH.", "The description of the orbits of NEOs in long-term timescales must be done statistically.", "We showed how the different elements can be represented by different distributions, and how the time it takes for the elements to become statistical depends on the frequency of close encounters.", "Through the sampling of different NEO cases we inspect the stochastic models that represent the long-term evolution of the orbital elements.", "The use of a fast semi-analytical propagation tool allows an efficient study of the dynamics of asteroids.", "We studied in detail the orbital histories of the Janus mission targets: (35107) 1991 VH and (175706) 1996 FG3.", "We characterized the encounters that can cause a potentially disruption of the binary orbits and computed the frequency of such encounters.", "Additional modeling of the effects of close encounters to other physical properties of asteroids will allow the study of the frequency of disruptive events.", "These are just a few potential examples of the benefits of a fast propagation tool for Solar System studies in the fashion of the presented tool." ], [ "Initial Conditions and uncertainties", "The statistical representation of the uncertainty in the orbit solutions can be done using the covariance matrix.", "This information is available for multiple asteroids in JPL’s SSD/CNEOS Small-Body DataBase [40].", "The covariance matrices for (175706) 1996 FG3 and (35107) 1991 VH that are used in this work are: ccccccc[h] 4 Initial covariance of the orbit of NEO binary (35107) 1991 VH 0pt 1991 VH e q (au) $t_p$ (TDB) $\\Omega $ (deg) $\\omega $ (deg) $i$ (deg) e 3.0691e-16 -3.5095e-16 -8.8918e-14 -7.2651e-15 -5.2217e-14 3.6255e-15 q (au) -3.5095e-16 4.0175e-16 1.0484e-13 8.2268e-15 6.1722e-14 -4.1006e-15 $t_p$ (TDB) -8.8918e-14 1.0484e-13 7.5479e-11 -7.8662e-12 6.4146e-11 -3.8102e-12 $\\Omega $ (deg) -7.2651e-15 8.2268e-15 -7.8662e-12 3.104e-11 -3.4282e-11 -4.876e-12 $\\omega $ (deg) -5.2217e-14 6.1722e-14 6.4146e-11 -3.4282e-11 8.1951e-11 1.9766e-12 $i$ (deg) 3.6255e-15 -4.1006e-15 -3.8102e-12 -4.876e-12 1.9766e-12 8.4244e-12 As obtained from JPL’s SSD/CNEOS Small-Body DataBase [40].", "Using DE431 and SB431-N16, orbit solution date 2021 April 15 for epoch JD = 2456902.5. ccccccc[h] 5 Initial covariance of the orbit of NEO binary (175706) 1996 FG3 0pt 1996 FG3 e q (au) $t_p$ (TDB) $\\Omega $ (deg) $\\omega $ (deg) $i$ (deg) e 1.2362e-16 -1.3026e-16 7.9966e-15 -4.1649e-14 3.9787e-14 -2.3726e-14 q (au) -1.3026e-16 1.3752e-16 -9.0726e-15 4.7436e-14 -4.5397e-14 2.5036e-14 $t_p$ (TDB) 7.9966e-15 -9.0726e-15 2.9877e-12 6.6638e-11 -6.6225e-11 -2.4698e-12 $\\Omega $ (deg) -4.1649e-14 4.7436e-14 6.6638e-11 7.0256e-09 -6.9647e-09 -6.8055e-11 $\\omega $ (deg) 3.9787e-14 -4.5397e-14 -6.6225e-11 -6.9647e-09 6.9046e-09 6.7284e-11 $i$ (deg) -2.3726e-14 2.5036e-14 -2.4698e-12 -6.8055e-11 6.7284e-11 6.7136e-12 As obtained from JPL’s SSD/CNEOS Small-Body DataBase [40].", "Using DE431 and SB431-N16, orbit solution date 2021 April 26 for epoch JD = 2454796.5.", "In the case of the artificial cases used to illustrate the long-term dynamics, we set the covariance matrix to be a diagonal matrix of 1e-6 in the Keplerian set $\\lbrace a,e,i,\\Omega ,\\omega ,\\sigma \\rbrace $ .", "While this is orders of magnitude larger than the uncertainties of (175706) 1996 FG3 and (35107) 1991 VH, the uncertainty without further observations increases exponentially after only tens of encounters.", "Thus, it is adequate for the studies in long-term simulations of this work.", "The individual particles are sampled considering a multidimensional normal distribution centered around the nominal values shown in Table REF .", "Then, we use the Cholesky factorization of the covariance matrices to add the corresponding perturbation from the nominal." ], [ "Computation of Laplace coefficients", "The expansion of the potential requires the computation of Laplace coefficients, as introduced by Laplace (1798).", "[4], [32] detail both the expansion and computation of coefficients.", "In the case of the expansion in equation REF : $\\left\\langle R_{j} \\right\\rangle = n_ja^2_j \\left[ \\frac{1}{2}A_{jj}e^2_j + \\frac{1}{2}B_{jj}I^2_j + \\sum ^{N}_{\\begin{array}{c}k=1 \\\\ k\\ne j\\end{array}} A_{jk} e_je_k\\cos { \\left( \\varpi _j-\\varpi _k\\right)} + B_{jk} I_jI_k\\cos { \\left( \\Omega _j-\\Omega _k\\right)} \\right]$ The coefficients $A_{jk}$ ,$B_{jk}$ ,$B_{jj}$ and $A_{jj}$ are: $A_{jk} = - n_j \\frac{1}{4} \\frac{m_k}{m_c+m_j} \\alpha _{jk} \\bar{\\alpha }_{jk} b^{(2)}_{3/2}\\left( \\alpha _{jk} \\right)$ $B_{jk} = + n_j \\frac{1}{4} \\frac{m_k}{m_c+m_j} \\alpha _{jk} \\bar{\\alpha }_{jk} b^{(1)}_{3/2}\\left( \\alpha _{jk} \\right)$ $A_{jj} = \\sum ^{N}_{k=1,k\\ne j} B_{jk}$ $B_{jj} = - \\sum ^{N}_{k=1,k\\ne j} B_{jk}$ The definition of Laplace coefficient is: $\\frac{1}{2} b_{s}^{(k)} (\\alpha ) = \\frac{1}{2 \\pi } \\int _{0}^{2 \\pi } \\frac{\\cos { (k\\psi ) }d\\psi }{(1-2\\alpha \\cos \\psi + \\alpha ^2)^s}$ This integral can be rewritten in a series expansion that simplifies the computation of the Laplace coefficients numerically as function of the rising factorial or Pochhammer symbol.", "However, it is found to be computationally more efficient to compute the quadrature integral above.", "There are many recursion and derivative rules that avoid computing the coefficients based on the definition.", "These expressions use the nomenclature of $D$ being the derivative operator $\\frac{\\mathrm {d} }{\\mathrm {d} \\alpha }$ .", "$b_{s+1}^{(j)} = \\frac{s+j}{s} \\frac{(1+\\alpha ^2)}{(1-\\alpha ^2)^2}b_{s}^{(j)} - \\frac{2(j-s+1)}{s} \\frac{\\alpha }{(1-\\alpha ^2)^2} b_{s}^{(j+1)}$ $b_{s+1}^{(j+1)} = \\frac{j}{j-s} \\left( \\alpha + \\frac{1}{\\alpha } \\right)b_{s+1}^{(j)} - \\frac{j+s}{j-s} b_{s+1}^{(j-1)}$ $D b_s^{(j)} = s \\left( b_{s+1}^{(j-1)} - 2\\alpha b_{s+1}^{(j)} + b_{s+1}^{(j+1)} \\right)$ $D^n b_s^{(j)} = s \\left( D^{n-1} b_{s+1}^{(j-1)} - 2\\alpha D^{n-1} b_{s+1}^{(j)} + D^{n-1} b_{s+1}^{(j+1)} - 2(n-1) D^{n-2} b_{s+1}^{(j)}\\right)$" ] ]
2207.03527
[ [ "RWT-SLAM: Robust Visual SLAM for Highly Weak-textured Environments" ], [ "Abstract As a fundamental task for intelligent robots, visual SLAM has made great progress over the past decades.", "However, robust SLAM under highly weak-textured environments still remains very challenging.", "In this paper, we propose a novel visual SLAM system named RWT-SLAM to tackle this problem.", "We modify LoFTR network which is able to produce dense point matching under low-textured scenes to generate feature descriptors.", "To integrate the new features into the popular ORB-SLAM framework, we develop feature masks to filter out the unreliable features and employ KNN strategy to strengthen the matching robustness.", "We also retrained visual vocabulary upon new descriptors for efficient loop closing.", "The resulting RWT-SLAM is tested in various public datasets such as TUM and OpenLORIS, as well as our own data.", "The results shows very promising performance under highly weak-textured environments." ], [ "INTRODUCTION", "Visual simultaneous localization and mapping (SLAM) is an essential task for mobile robots navigation as well as in AR/VR.", "Nowadays feature based SLAM algorithms are popular for their high performances and computational efficiency.", "However, robust SLAM under highly weak-textured environments still remains a challenging problem.", "Hand-crafted features such as SIFT [1], ORB [2] and Shi-Tomas [3] are not able to extract reliable keypoints for matching in regions with highly low texture or motion blur.", "In fact, these extreme environments are difficult for whatever the state-of-art feature-based methods [4] or direct methods [5][6].", "For some textureless but well structured environment, some solutions are proposed to strengthen the feature matching by integrating line or plane features, e.g.", "Stereo-PLSLAM [7], SuperLine [8] and Structure-SLAM [9].", "However, these methods rely heavily on visible structures like object edges and planes.", "In environments with little structure and texture, tracking may still fail.", "On the other hand, there is a trend to replace hand-crafted features with deep features to improve the robustness of the SLAM systems.", "Trained with large mount of diversified data, the deep learning-based methods can operate on full-size image and jointly compute pixel-wise interest points and the associated descriptors.", "Experiments in [10] indicate that SuperPoint can produce more distinctive descriptors than classical methods and the interest point detector is on par with hand-crafted features.", "Similarly, GCN [11] uses a recurrent neural network to predict the location of keypoints as well as their descriptions for camera motion estimation.", "GCNv2 [12] simplifies the network for efficiency and incorporates the learned features into ORB-SLAM2 [4] to form GCN-SLAM.", "Although these deep learning-based methods perform better than the traditional ones in complex environments, when dealing with highly weak-textured scenes, there is still much room for improvement.", "Figure: A matching example in areas of weak-texture using keypoints and descriptors of our system.", "The point correspondence in top and bottom images are produced by KNN matcher before and after applying the feature mask, respectively.Recently, several works [13][14] propose to directly predict pixel-wise matches for a pair of images by using CNN.", "LoFTR [14] utilizes transformer [15] with self-attention and cross-attention mechanisms to produce dense matches.", "They are able to produce large quantities of matches under low textured environments without using any feature descriptor.", "This motivates us to induce this state-of-the-art method into visual SLAM.", "However, when purpose of accurate localization of SLAM is concerned, the detector-free matching becomes a drawback.", "The reason lies in three aspects.", "Firstly, although LoFTR can produce dense matches for a pair of images, the repetitiveness of the matching pairs across the neighboring images is not guaranteed.", "This makes it difficult for further optimization of the features’ position across multiple frames, which modern SLAM systems high rely on.", "Secondly, popular SLAM systems like ORB-SLAM2 [4] rely on feature descriptors to match the feature points to the local map, and use PnP algorithm to produce good initial estimation for poses.", "Lack of feature descriptors makes this stage infeasible.", "Lastly, despite the dense matching, the pixel positions produced by the detector-free methods are not as accurate as the features extracted from those detector based methods such as ORB.", "In particular, lots of matching outliers can be observed in highly low textured regions.", "In this paper, we propose a novel visual SLAM system named RWT-SLAM, which corresponds to very Robust SLAM in Weak-Textured environments.", "We modify LoFTR network in order to produce distinctive feature descriptors.", "To integrate the new features into the popular ORB-SLAM framework, we develop feature masks to filter out the unreliable matches and employ KNN to strengthen the robustness of the matching.", "We also trained visual vocabulary upon new descriptors for loop closing.", "The resulting RWT-SLAM is tested in various highly weak-textured environments and shows very promising performance.", "The contributions of this paper are summarized as follows: A novel full visual SLAM system capable of robustly working under highly weak-textured environments is proposed.", "To the best of our knowledge, it is the first SLAM algorithm which can achieve successful tracking on all sequences with either no structure or no texture labels in TUM dataset [35].", "We demonstrate that a detector-free network aimimg at producing dense feature matches can be reformed for SLAM applications.", "We extract coarse and fine level deep features from LoFTR [14] to construct the final descriptors.", "Comprehensive experiments are carried out on public TUM RGB-D [35], OpenLORIS [36] and our own dataset, demonstrating the superior localization and robustness performance under extremely weak-textured environments.", "Traditional visual SLAM algorithms can be roughly divided into two classes: direct (photometric-based) methods and indirect (feature-based) methods.", "The direct approaches estimate motion from phtotometric changes of the image while indirect methods rely on a subset of features of the image.", "LSD-SLAM [5], a direct monocular method, can build semi-dense consistent maps for large-scale scenes.", "The sparse direct visual odometry DSO [6] omits the smoothness prior used in other direct methods and instead samples pixels evenly throughout the images.", "Most indirect methods rely on PTAM [16], which is the first algorithm splitting the tracking and mapping thread separately to achieve better performance.", "The popular ORB-SLAM [17] employs ORB [2] features and incorporates three threads, i.e., tracking, local mapping and loop closing to fullfill the task.", "In particular, SVO [18] is a semi-direct odometry extracting sparse features and operating directly on pixel intensities around the features.", "Comparing with the feature-based methods, indirect approaches perform better on images with low or repetitive textures.", "However, they are much more sensitive to illumination changes of the environment and more difficult to initialization." ], [ "Deep learning based SLAM", "Many deep learning based SLAM methods are proposed in recent years.", "They are usually formed by replacing one or more modules of the traditional SLAM framework with deep networks [19][20][21][12][22].", "To improve depth predictions that are used to initialize the SLAM system, CNN-SLAM [19] incorporates a depth estimation network within the popular LSD-SLAM framework to produce dense scene reconstructions with metric scale.", "DS-SLAM [21] utilizes a semantic segmentation network to filter out features on moving objects, achieving better localization accuracy in highly dynamic environments.", "The most similar methods to ours are GCN-SLAM [12] and DXSLAM [22].", "In GCN-SLAM [12], keypoints and binary descriptors produced by the neuro-network named GCNv2 is employed to replace ORB [2] features used in ORB-SLAM2 [4].", "DXSLAM [22] uses keypoints and descriptors generated by a pretrained HF-Net to improve system's performance.", "Compared with GCN-SLAM and DXSLAM, we obtain the features from a modified transformer based feature matching network LoFTR and integrate them into the ORB-SLAM framework.", "The resulting algorithm can achieve much higher performance than its counterparts under highly weak-textured environments.", "Other works have attempted to train end-to-end SLAM systems [23][24].", "Most of them are not full SLAM systems, but focus on small scale reconstruction on several frames.", "They do not have key modules of the modern SLAM system such as loop closure and global bundle adjustment, which limits the accuracy of the system.", "Recently, DeepSLAM [25] integrate three sub-network to imitate full modules of SLAM.", "However, they need to be trained for each benchmark from scratch, limiting the generalization ability for practical use." ], [ "Deep Feature Matching", "The deep feature matching methods can be roughly divided into two categories.", "The first one focuses on learning to produce keypoints and their descriptors simultaneously [10][26][27][28][29].", "SuperPoint [10] proposes a self-supervised strategy to train a fully convolutional network for joint keypoints detection and description.", "In D2-Net [26], a local maxima within and across feature maps are selected as interest points and the descriptors are generated from the same feature maps.", "R2D2 [27] utilizes a predictor of discriminativeness to avoid ambiguous areas.", "The second category is detector-free methods which directly learn dense matches or descriptors without explicit keypoint detection phase [30][31][32][13][14].", "NCNet [30] uses 4D cost volumes to enumerate all possible matches between a pair of images.", "More recently, DRC-Net[31] follows this line and trains a CNN in a coarse-to-fine manner with synthetic transformations.", "In [32], a novel weakly-supervised framework is proposed by using solely relative poses between images to learn descriptors.", "Inspired by SuperGlue [33] which is a detector-based feature matching method, LoFTR [14] proposes a transformer based detector-free design to produce dense matches for various complex environments.", "It is able to produce large quantities of pixel level matches even under highly weak structured or textured environments.", "Figure: The keypoint and descriptor generation pipeline.", "Coarse-level and fine-level feature maps are generated within the LoFTR, where we further extract the coarse and fine level descriptors and concatenate them together to construct the final descriptors for the feature points.In this section, we introduce our RWT-SLAM algorithm in detail.", "The generation of keypoint descriptors from the detector-free network LoFTR [14] is introduced in part A.", "Then the reformation of ORB-SLAM2 [4] to make it adaptive to the new feature points is described in part B." ], [ "Feature Descriptors", "The descriptor generation pipeline is illustrated in Fig.2.", "Given a pair of images $I_{A}$ and $I_{B}$ , LoFTR [14] establishes coarse-level matches with coarse features and then refines the matches to the fine scale with a coarse-to-fine scheme.", "Coarse-level descriptors: As suggested in Fig.", "2, the coarse features $\\hat{F_{A}}$ and $\\hat{F_{B}}$ are from the fourth blocks of the CNN with 256 channels with $1/8$ scale of the original image.", "They are fed into a transformer with attentional aggregation and receptive filed expansion for further feature extraction.", "Then a matching module is utilized to construct coarse-level matches with an outlier rejection.", "Based on the transformed features $\\hat{F_{A}^{^{\\prime }}}$ and $\\hat{F_{B}^{^{\\prime }}}$ , the score matrix S whose elements $S_{\\hat{i},\\hat{j}}$ can be computed by the inner product of corresponding feature vectors $f_{\\hat{i}}^{A}$ and $f_{\\hat{j}}^{B}$ with $S_{\\hat{i},\\hat{j}}=f_{\\hat{i}}^{A}\\cdot f_{\\hat{j}}^{B}, \\quad where \\quad (f_{\\hat{i}}^{A},f_{\\hat{j}}^{B})\\in (\\hat{F_{A}^{^{\\prime }}},\\hat{F_{B}^{^{\\prime }}})$ Inspired by the matching strategy, we notice that the transformed coarse features can be utilized to produce descriptors for a match $(\\hat{i},\\hat{j})$ .", "Therefore, the corresponding items on the coarse feature maps $\\hat{F_{A}^{^{\\prime }}}$ and $\\hat{F_{B}^{^{\\prime }}}$ are separately saved as the coarse-level descriptors, namely $f_{\\hat{i}}^{A},f_{\\hat{j}}^{B} \\in \\mathbb {R}^{256}$ .", "Thanks to the self-attention and cross-attention mechanisms of the transformer module, the coarse-level descriptors encode the unique description of a local 8×8 image patch while maintaining global receptive filed of the whole image.", "Fine-level descriptors: LoFTR [14] operates on fine-level features to obtain sub-pixel matches in a coarse-to-fine manner.", "The fine-level features are cropped within a local window of size $5\\times 5$ centered at $\\hat{i}$ and $\\hat{j}$ respectively.", "Then a fine-level processing and matching pipeline is applied to get a sub-pixel coordinate $\\tilde{j}$ on the second image $I_{B}$ .", "The final matched positions $(i,j)$ can be formulated as: ${\\begin{array}{c}i=8\\times \\hat{i}\\\\j=8\\times \\hat{j}+2\\times \\tilde{j},\\end{array}}$ We select the center feature vector of $\\tilde{F^{s}_{A}}$ as the fine-level descriptor on image $I_{A}$ for position $i$ .", "The fine-level descriptor on image $I_{B}$ is obtained by interpolating $\\tilde{F^{s}_{B}}$ bilinearly at position $\\tilde{j}$ , as shown in the right part of Fig.", "2.", "At this point, we have obtained the fine-level descriptors $f_{\\tilde{i}}^{A},f_{\\tilde{j}}^{B} \\in \\mathbb {R}^{128}$ .", "Finally, by concatenating the coarse and fine level descriptors, the complete hierarchical descriptors that capture both high and low level textures of the matched feature $(i,j)$ can be obtained and recorded as: $\\begin{aligned}f_{i}^{A}=f_{\\hat{i}}^{A} \\oplus f_{\\tilde{i}}^{A} \\qquad \\in \\mathbb {R}^{384}\\\\f_{j}^{B}=f_{\\hat{j}}^{B} \\oplus f_{\\tilde{j}}^{B} \\qquad \\in \\mathbb {R}^{384}\\end{aligned}$" ], [ "Framework of RWT-SLAM", "The system overview of our RWT-SLAM is illustrated in Fig.", "3.", "Given the current frame, the keypoints and descriptors from the reformed LoFTR are fed into the classical ORB-SLAM framework.", "To make our RWT-SLAM more adaptive to the extreme environment, we make more improvements on the ORB-SLAM framework.", "Feature extraction and filtering.", "LoFTR [14] directly produces dense feature matches for a pair of images.", "However, the distribution of the matches can change drastically across images, especially for the scene with large illumination changes or weak texture.", "When applying them on successive input frames with weak texture, the repetitiveness of the feature points across the frames can be low, which is fatal for SLAM.", "To produce enough feature points with wide distribution, we copy the current frame and concatenate them at the channel dimension before feeding them into LoFTR.", "Generally for each $8\\times 8$ image patch, one feature point and corresponding descriptor can be obtained.", "The feature points we obtained have sufficient quantity while at the cost of less distinctiveness.", "If there is some structure in the environment, which is true for common corridors or offices, the structure information can be used to select more distinctive feature points.", "We rely on canny edge detection followed by Hough transformation to form feature masks with the shape of thick line.", "Only the feature points within the mask are maintained for matching because they are more distinctive and are able to produce more accurate matching.", "An example showing the matching results before and after the use of feature mask is presented in Fig.", "1.", "Some examples of the feature masks are also shown in Fig.", "4, where the width of the line is set to 20 pixels.", "Tracking.", "In ORB-SLAM2 [4], the motion estimation is achieved by frame to frame keypoints tracking and pose-only bundle adjustment.", "Once the keypoints and the descriptors are obtained, the system relies on a progressive methods for frame to frame tracking.", "First, by assuming a constant velocity the keypoints of the previous frame are projected to the current frame and matches are searched in a local window.", "If that fails, the system will try to find matches using bag of words (BoW) among current frame and the referenced keyframes.", "In our system, we replace these two matching strategies with a standard K nearest-neighbor search followed by a ratio test.", "The KNN matcher establishes stable matches by computing the Euclidean distance between the descriptors.", "With the learned descriptors, the brute-force matcher is more suitable for images with low-texture.", "Vocabulary traning.", "A visual vocabulary is trained offline in ORB-SLAM2 [4] to accelerate matching process in tracking, relocalization and loop closing.", "We adopt DBoW3 framework to build the new visual vocabulary based on the descriptors produced in part A, and Bovisa 2008-09-01 [34] is used for vocabulary training.", "The vocabulary is produced in binary form, which is more efficient for use during system initialization and image matching." ], [ "EXPERIMENTS", "We carried out extensive experiments on various datasets to verify the effectiveness of our SLAM system.", "Similar to GCN-SLAM and DXSLAM, all experiments are conducted on RGB-D data.", "The TUM RGB-D dataset [35] contains sequences with highly low texture, which are very challenging to most existing visual SLAM algorithms.", "We further perform experiments on OpenLORIS-Scene datasets [36] and our own dataset, evaluating the lifelong capabilities and localization performance under low-textured environments.", "Figure: The system overview of our RWT-SLAM.We use the pretrained LoFTR model provided by the author to generate the keypoints and descriptors.", "All experiments are performed on a computer with an intel i7-10700k CPU and NVIDIA 1650 GPU.", "We sample the images at an interval of 5 frames for the RWT-SLAM system and interpolate the trajectory for the other frames in real time.", "Our system works at frame rates around 8Hz and can be accelerated with more powerful GPUs.", "Figure: From left to right: The estimated trajectory with error, the tracked keypoints and the feature masks for fr3strnotexnear (top) and fr3nostrnotexnear (below) sequences." ], [ "Evaluation on TUM RGB-D dataset", "The TUM RGB-D dataset is an excellent dataset with accurate ground truth from motion capture system, which is very suitable for evaluating the localization accuracy.", "We select several sequences in fr3 which have weak or even no-texture for testing.", "RWT-SLAM is built upon ORB-SLAM2 [4].", "For comparison, we also execute GCN-SLAM [12] and DXSLAM [22], who are both based on ORB-SLAM2 and use learned features at the front-end.", "Absolute Trajectory Error(ATE) are used for quantitative evaluation.", "The experiments are conducted with $640\\times 480$ resolution except for GCN-SLAM, where $320\\times 160$ resolutions is adopted since this is the best configuration according to their original paper.", "Table: RMSE RESULTS ON TUM RGB-D DATASETFigure: Per-sequence results on OpenLORIS-Scence dataset.", "Each black dot on the top line indicates the start point of one sequence in a scene.", "For each algorithm, blue dots represent successful initialization and blue lines represent the duration of successful tracking.", "The percentage value on the top left of each scene is the average correct rate CR ∞ \\mathrm {CR}^{\\infty }, with larger meaning more robust.", "The average ATE RMSE is on the bottom right, with smaller meaning more accurate.The comparison results are listed in Table I, where we can see our method succeeds tracking in all sequences.", "To the best of our knowledge, this is the first visual SLAM algorithm who can survive all of these harsh sequences.", "For fr3cabinet, there is only a large cabinet in an empty room.", "Hand crafted keypoints like ORB can be only extracted around the corners or the edges of the cabinet so it is easy to lose tracking.", "fr3strnotexfar and fr3strnotextnear have some structures but with little texture.", "The last two sequences are of both little structure and little texture, causing failure for most of the algorithms.", "Comparing with other learning-based or hand-crafted keypoints, our system are highly reliable in these extreme environments.", "The trajectory errors, the tracked keypoints as well as the feature masks for two of these sequences are also illustrated in Fig.", "4.", "Meanwhile, we notice that for the sequences that ORB-SLAM2 or GCN-SLAM can survive, RWT-SLAM does not achieve the best accuracy.", "This may due to the more accurate feature position the ORB or GCN produces once they can survive." ], [ "Experiment on OpenLORIS Dataset", "There are five scenes in OpenLORIS-Scene dataset, including office, corridor, home, café and market.", "Some scenes are very challenging because of featureless walls and low illumination, making it impossible to finish the entire tracking for most visual SLAM algorithms.", "We compare our algorithm with ORB-SLAM and other deep feature based SLAM methods.", "GCN-SLAM dose not conduct experiment on this dataset and we run the source code of it with the default configurations provided by their paper.", "Since OpenLORIS provides IMU data, we also test ORB-SLAM3 [37] which is a state-of-the-art visual-inertial SLAM system for comparison.", "The success rate $\\mathrm {CR}^{\\infty }$ and the RMSE error of the successful part of the trajectory are used as evaluating metric.", "The results are shown in Fig.", "5, where we can see that RWT-SLAM achieves the longest tracking life on all of these scenes.", "When the most challenging corridor and home scenes which are highly weak-textured are concerned, our RWT-SLAM outperforms the GCN-SLAM who ranks the second by $5.90\\%$ and $20.37\\%$ in $\\mathrm {CR}^{\\infty }$ respectively.", "It is interesting to find that with the help of IMU, ORB-SLAM3 does not performs better than ORB-SLAM2 in most of the sequences.", "This may due to the difficulty of the initialization of the IMU in these environments.", "As to the localization accuracy, our system is comparable to the GCN-SLAM, while inferior than the feature tailored system like ORB-SLAM and DXSLAM.", "The phenomenon is similar to section IV-A.", "Nevertheless, our goal is to survive longer and achieve higher robustness under challenging weak-textured scenes.", "Figure: The estimated trajectory for RWT-SLAM, GCN-SLAM and ORB-SLAM2 in our own dataset.", "We mark the starting and ending points of the trajectory with the star and thick dot respectively.", "Our method maintains tracking and gives reasonable localization trajectory for all of the sequences while the other two either fail tracking or output wrong trajectory.Figure: Dense point cloud reconstructions for the Outdoor road and Stairs sequences.", "The point cloud is obtained by projecting the keyframe’s depth maps with the estimated poses from RWT-SLAM.Table: ABLATION STUDIES OF RWT-SLAM." ], [ "Qualitative Results on our own dataset", "To further demonstrate the performance of RWT-SLAM in real weak-texture areas, we use an Intel RealSense D455 RGB-D camera and collect four sequences under different conditions: a) walking through a corridor, turning 180 degrees at the end of the corridor and going back; b) going into a room looking at the floor and white walls; c) walking on a road outdoor forming a closed loop in the end; d) going upstairs and down back to the start point.", "Since there is no metric ground truth available, the experiments in this section serves as a complementary testing under real challenging scenes.", "We compare our system with GCN-SLAM and ORB-SLAM2 and the estimated trajectory results are shown in Fig.", "6.", "As shown in Fig.", "6a, ORB-SLAM2 cannot cope with 180 degrees turn and lost tracking at the top right of the trajectory.", "GCN-SLAM gives wrong rotation prediction and trajectory deviates from the actual waking route while our system keeps reasonable tracking through the sequence.", "In room sequence, GCN-v2 and ORB features decrease dramatically when encountering walls and floors with weak-texture, leading to incorrect localization stating from the marked red circle shown in Fig.", "6b.", "As shown in Fig.", "6c, in outdoor sequence tracking fails immediately for ORB-SLAM2, and GCN-SLAM loses tracking at the third turn.", "Fig.", "6d shows great robustness of RWT-SLAM when dealing with the images containing large featureless areas and motion blurs.", "Our RWT-SLAM is the only one surviving all of the four challenging sequences with reasonable localization accuracy.", "The reconstructed map for the outdoor and stair sequences are shown in Fig.", "7." ], [ "Ablation Studies", "In order to demonstrate the effectiveness of each component proposed, we conduct ablation studies on TUM RGB-D dataset.", "The results of different configurations of RWT-SLAM are shown in Table II.", "In term of the feature descriptor, RWT-SLAM with coarse-level descriptors only can survive all of the five sequences but with lower localization accuracy, while with fine-level descriptors alone fails in most of the sequences.", "This is because coarse-level descriptors have large receptive field but with rough resolution, which leads to worse tracking.", "Fine-level descriptors only captures small receptive filed which is not distinctive enough for matching.", "Comparing to the full configuration, removing the feature mask module will decrease the localization accuracy.", "KNN matcher is also essential for the method in the sense that without it only two sequences can be successful treated." ], [ "CONCLUSIONS", "In this paper, a robust visual SLAM system based on deep feature is proposed to deal with highly weak-textured environments.", "Different from the existing detector-based deep features, the state-of-art detector-free network LoFTR is modified to generate dense interest points and the corresponding descriptors.", "Thanks to the large receptive field provided by the attentional aggregation of the network, we are able to obtain high quality descriptors for scenes with little texture and structure.", "To make the feature more adaptive to challenging scenes, we present feature masks and retrain the vocabulary for the classical ORB-SLAM framework.", "KNN matching is also employed during tracking to strengthen the feature matching under extreme environments.", "Experimental results on various public datasets and our own data prove the success of our system." ] ]
2207.03539
[ [ "Hyper-Universal Policy Approximation: Learning to Generate Actions from\n a Single Image using Hypernets" ], [ "Abstract Inspired by Gibson's notion of object affordances in human vision, we ask the question: how can an agent learn to predict an entire action policy for a novel object or environment given only a single glimpse?", "To tackle this problem, we introduce the concept of Universal Policy Functions (UPFs) which are state-to-action mappings that generalize not only to new goals but most importantly to novel, unseen environments.", "Specifically, we consider the problem of efficiently learning such policies for agents with limited computational and communication capacity, constraints that are frequently encountered in edge devices.", "We propose the Hyper-Universal Policy Approximator (HUPA), a hypernetwork-based model to generate small task- and environment-conditional policy networks from a single image, with good generalization properties.", "Our results show that HUPAs significantly outperform an embedding-based alternative for generated policies that are size-constrained.", "Although this work is restricted to a simple map-based navigation task, future work includes applying the principles behind HUPAs to learning more general affordances for objects and environments." ], [ "Introduction", "The American psychologist James Gibson first proposed the idea of object affordances [2], namely, that an object is perceived not only by the object's visual features, but also by the potential motor actions it affords.", "The idea that a single image of an object can suggest action plans to achieve particular goals has important implications for computer vision-based robotic agents: it can allow zero-shot generalization of learned skills to new objects and environments.", "Can such a capability be realized by computer vision systems?", "In reinforcement learning, action plans are formalized in terms of a policy function $\\pi (s)\\rightarrow a$ that maps the current state $s$ of the agent to the desired action $a$ .", "Policy-based techniques, widely used in reinforcement learning [5], aim to find strategies that maximize the expected reward received from the environment.", "In most realistic scenarios, agents are required to be adept in many different tasks and good policies often exploit shared structure defined by task similarity and the dynamics of the environment.", "For example, in the case of navigation, policies that guide an agent to two different goals in close proximity should agree for most states $s$ .", "Similarly, policies for different but closely-related environments should also be related.", "To extrapolate policies in novel environments, humans often rely on visual cues.", "In navigation tasks, humans can utilize a visual map to navigate in a previously unknown space.", "To formalize this correspondence between visual (and potentially other) auxiliary information and action policies, we propose the concept of a universal policy function (UPF) $\\pi _{E}(s,g)$ , where $g$ is the goal and $E$ a description of the environment, obtained, for example, from a visual map.", "UPFs are related to general value functions [6], [4], but besides generalizing value functions to novel goals, we additionally tackle the challenge of transferring policies to novel environments, a new dimension which introduces significant complexity.", "Although UPFs can be approximated by any generic neural network, training such models based on a limited set of goal and environment samples can be challenging.", "An agent with limited computational capacity and communication bandwidth faces another challenge; storing or transmitting a monolithic model with lots of parameters can be unwieldy or even impossible.", "Such constraints are frequently encountered in designing edge devices.", "In this paper, we propose Hyper-Universal Policy Approximators (HUPAs) which, to our knowledge, is the first framework for mapping visual information, e.g., a single map image, to universal policy functions for agents with computational and communication constraints.", "HUPAs leverage the modularity properties of hypernetworks [1] to generate small environment-conditional, policy functions from an image (or other auxiliary information).", "These policy functions can then be easily transmitted to the agent and fine-tuned.", "We compare HUPAs to the traditional embedding alternative and demonstrate a significant performance gap between the two approaches when the generated policy network is size-constrained.", "While we focus on a simple map-based navigation task, our results are the first to highlight the advantage of hypernetworks in representing UPFs." ], [ "Universal Policy Approximation", "In the context of a Markov Decision Process (MDP), we denote a UPF by $\\pi _{E}(s,g)\\rightarrow a$ , where $g$ is the goal state, $s$ the current state, $a$ the chosen action, and $E$ a context vector that serves as a description of the environment.", "This description contains any auxiliary information that can be useful in conditioning the resulting policy.", "It can be given explicitly (e.g., image of a map for map-based navigation, an embedding-based natural language description, etc.)", "or implicitly (inferred from local observations such as images from a robot's on-board camera).", "The goal is to efficiently generate policies $\\pi _E$ conditioned on a specific novel environment description $E$ .", "We make the following assumptions for our setting: (a) the agent/edge device is limited in computational capacity, (b) it operates within a specific context/environment $E$ and (c) has access to a device with large computational capacity on the cloud.", "A straightforward way to approximate $\\pi _{E}$ is to use a large network $\\phi $ hosted on the cloud to obtain a context embedding $\\phi (E)$ .", "This embedding is then transmitted to the edge device.", "The edge device hosts a size-limited network $P$ that takes as input $x=[s,g,\\phi (E)]$ and predicts the appropriate action for each $x$ .", "We refer to this technique as the “embedding” approach.", "Note that instead of transmitting the whole embedding vector it suffices to transmit its projection to the first layer of $P$ which is usually smaller in size.", "A diagram for the embedding approach is shown in Figure REF .", "As an alternative, we propose Hyper-Universal Policy Approximators (HUPAs), based on hypernetworks [3].", "A hypernetwork is a network that generates parameters for another network called the primary network.", "The hypernetwork $H$ , which is hosted on the cloud, takes as input the context vector and outputs a set of parameters $\\theta _E = H(E)$ .", "These parameters are then transmitted to the edge device and are used to parameterize a small primary network $P$ .", "The HUPA model is shown in Figure REF .", "Recent results on the modularity of hypernetworks show that they are provably more efficient approximators of functions of this form compared to the embedding approach [1].", "Figure: Embedding versus Hypernet-based Approaches for Predicting Action Policies from a Single Environmental Input.Figure: Zero-Shot Learning of Policies from Single Images: After training on known EE and gg samples, a HUPA (a) predicts policies for novel goals (b) and novel environments (c) from a single input map image.", "The goal is marked by a red square on the right-side policy maps; action vectors are color-coded by angle." ], [ "Dataset and Metrics", "We evaluate HUPAs on a novel, artificially generated dataset based on a two-dimensional navigation task.", "The agent is given a map $E$ , a starting coordinate $s$ and must navigate to a goal $g$ .", "The maps consist of discretized tiles which are either open or occupied by a wall.", "The action space of the agent consists of 8 actions/angles $a \\in \\lbrace \\frac{k\\pi }{4}: k \\in \\lbrace 1,...,8\\rbrace \\rbrace $ that transfer the agent to one of the neighboring tiles (up, down, left, right and the diagonals) as long as they are open.", "Our goal is to learn a HUPA that, given previously unseen maps and goals, can generate policies that transfer the agent successfully to the target location.", "Since the action space is discrete, the generated primary network $P$ takes the form of a classifier." ], [ "Map Generation", "For our experiments we use a nine room generalization of the four-room environment commonly used in RL navigation scenarios.", "In this regime, each map $E$ is derived from a base map that contains rooms arranged in a $3\\times 3$ grid separated by walls.", "Each room consists of a $9\\times 9$ grid of open tiles, and neighbouring rooms are connected by doors in the center of the separating wall.", "There are twelve doors in total.", "To generate a new map from the base map, three doors are chosen to be blocked off under the constraint that the space remains connected.", "This results in 164 potential maps.", "From each map all possible start and goal coordinates are sampled to create the dataset.", "By representing states $s,g$ by their two-dimensional coordinates $(x,y)$ the maps can be conveniently represented as images, as we do in the following experiments.", "HUPAs function as zero-shot policy generators for previously unseen environments based on these single images (see Figure REF )." ], [ "Reachability Ratio", "While canonical accuracy serves as a good measure of how close our generated policies are to the ground truth policy, it does not encapsulate the usability of a policy.", "The successful transition of the agent to the goal state relies on a sequence of several correctly predicted actions, a requirement that is not captured by plain accuracy.", "As an example a generated policy can leave a whole room disconnected from the goal, whereas most of the state-action pairs inside the room are predicted correctly (for examples of such generated policies, see Figure REF ).", "A natural way to properly assess the quality of a generated policy is to evaluate the fraction of states from which it successfully guides the agent to the goal.", "We call this the Reachability Ratio (RR) metric.", "To that end we construct an incidence graph where each state $s$ is represented by a node $n_s$ .", "For each such state we add an edge $n_s\\rightarrow n_{\\pi _E(s,g)}$ that corresponds to the immediate transition that results from following the policy at $s$ .", "Let $C_g$ be the connected component of this graph that contains $n_g$ .", "Then $C_g$ contains all the nodes from which the policy successfully reaches the goal.", "Hence the Reachability Ratio of a set $T$ of test goals is the average fraction of nodes in the target set “accessible” to each state in the map: $\\textrm {RR}(T) = \\frac{1}{|S|}\\sum _{g \\in S}\\frac{|T \\cap C_g|}{|T|}$ where $C_g$ is computed via the reachability graph of $\\pi _E$ ." ], [ "Experiments", "For the embedding function $\\phi $ , we used a convolutional neural network with four residual blocks, followed by a fully connected bottleneck layer.", "An additional layer outputs the embedding vector $\\phi (E)$ .", "The hypernetwork had almost identical structure, except for an additional layer that generates the parameters $\\theta _E$ .", "For a fair comparison we adjust the size of the embedding vector accordingly, while keeping the size of the primary network fixed, to approximately equate the total number of parameters.", "The primary network consists of three hidden layers with the same number of neurons, followed by an output layer that predicts the movement direction.", "We compare the two approaches on primary networks with $16,32,64,128$ and 256 neurons.", "The resulting models have $315K ,544K ,1.4M ,4.7M$ and $17.6M$ parameters respectively.", "We used 50 maps and $40\\%$ of the states at random as our training set.", "We used early stopping on a validation set of 5 novel maps and $10\\%$ novel goals, with patience equal to 10.", "For evaluation we used 20 test maps and the remaining goals.", "Figure REF shows examples of generated policies represented by their reachability graphs.", "Figures REF and REF show high-quality policies for known maps and known/unknown goals.", "Figure REF shows successful zero-shot policy generation for unknown maps and goals.", "Finally Figure REF shows examples of policies that fail to correctly understand the structure of the map, leaving whole rooms disconnected from the goal.", "However, these rooms are disconnected due to only a few erroneous decisions.", "Fine-tuning the generated policies using reinforcement learning based on a few episodes could potentially correct these errors.", "We quantitatively compared HUPAs to the embedding alternative both in terms of accuracy and reachability.", "As seen in Figure REF , HUPAs significantly outperform the baseline in both metrics.", "The effect is more pronounced for more constrained primary networks, with the embedding model approaching the performance of HUPAs as the parameter count becomes significantly higher.", "Figure: Reachability Graph Examples: Examples of reachability graphs in policies predicted by the HUPA model for unseen maps and goals.", "Walls are denoted by yellow, open tiles by purple, while the red node marks the goal.", "Green nodes are states from which the goal is reachable under the predicted policy.", "Blue nodes do not reach the goal.", "The failure example in (d) (right) illustrates the importance of using reachability compared to accuracy, and also the potential for fixing the policy via few-shot RL.Figure: Generalization to Novel Maps and Goals: Performance of HUPAs on novel maps and goal compared to the embedding approach: policy accuracy (left); goal reachability (right).To evaluate the robustness of our approach, we varied the number of training maps and goal states.", "We chose the 128 neuron model for our primary architecture.", "As seen in Figure REF , the hypernetwork generalizes well even when the map/goal training set is sparse.", "HUPAs consistently outperform the embedding model in all sparsity settings.", "Figure: Robustness to Map and Goal Sparsity in Training: Goal reachability as a function of percentage of total possible goals (left) and maps (right) used for training." ], [ "Conclusion", "We proposed HUPAs, a hypernetwork-based model for zero-shot generation of whole action policies from a single image, thereby suggesting a neural network implementation of Gibson's affordances.", "We demonstrated the significant advantage of HUPAs over the traditional embedding approach for size-constrained primary networks.", "Future work includes fine-tuning and meta-learning the generated policies, applying HUPAs to more complex settings and inferring environment descriptions directly from observations." ] ]
2207.03593
[ [ "Winning the lottery with neurobiology: faster learning on many cognitive\n tasks with fixed sparse RNNs" ], [ "Abstract RNNs are often used as models of biological brain circuits and can solve a variety of difficult problems requiring memory, error-correction, or selection \\cite{hopfield1982neural, maass2002real,maass2011liquid}.", "However, fully-connected RNNs contrast structurally with their biological counterparts, which are extremely sparse ($\\sim 0.1$\\%).", "Practical deployment of large RNNs is often limited due to requirements of long training times and large memory requirements.", "Motivated by brains, where neural connectivity is constrained by distance along cortical sheets and other synaptic wiring costs, we introduce locality masked RNNs (LM-RNNs) that utilize task-agnostic predetermined graphs with sparsity as low as 4\\%.", "We make three contributions: First, we show that LM-RNNs can perform as well as their fully-connected counterparts, without \\emph{a posteriori} construction of the best sparse subnetwork.", "Second, we find that LM-RNNs train faster with more data-efficiency in a multitask setting relevant to cognitive systems neuroscience, often achieving better asymptotic performance.", "Third, we contribute a new cognitive multi-task battery, Mod-Cog, consisting of 132 tasks that expands by $\\sim 7$-fold the number of tasks and task-complexity of an existing commonly used set of tasks \\cite{yang2019task}, showing that while LM-RNNs can solve the simple \\cite{yang2019task} tasks with a small pool of unconnected autapses, the expanded task-set produces richer solutions." ], [ "Introduction", "As models get more complex and larger, various techniquess have been developed to tackle the problem of prohibitively large resources required to train and deploy these models.", "Several methods have tried to get around this issue using techniques like pruning, first introduced in [18], [31], [13], low-rank approximation[11] and distillation[9].", "These results demonstrate that it is in principle possible to have equally good solutions with far fewer parameters.", "However there do not exist any computationally cheap methods to find these solutions.", "Inspired by neuroscience, we modify the architecture of a vanilla fully-connected RNN in one particular fashion — noting that neurons in the brain are arranged on two-dimensional cortical sheets and wiring length costs constrain connectivity, we construct a fixed sparse graph chosen by allowing local connections among neurons laid on a two-dimensional sheet.", "We then use this graph for the RNN, by only training weights between nodes that correspond to edges on the graph, and setting all other weights to zero.", "We refer to an RNN in this set up as a `Locality Masked RNN' (LM-RNN).", "Another motivation for our work comes from modeling grid cell networks in systems neuroscience [15], where it was shown analytically that fixed and local topographic connectivity encouraged the formation of modules.", "Exploiting such simple locality constraints in LM-RNNs, when applied to multitask regimes relevant to cognitive systems neuroscience, results in distinct advantages: these networks require far fewer parameters to train, there is no cost to performance relative to an unconstained network, and in fact learning is more rapid, sample-efficient and can acheive higher asymptotic performance.", "In contrast to other works in this domain, we do not need any sophisticated pruning methods, any algorithms to construct and modify sparse skeletons nor any training data.", "We focus our results on multitasking regimes for RNNs, by training them to simultaneously learn many cognitive tasks.", "We expect our results on the improvements conferred by LM-RNNs to primarily apply in this multitask learning setting, where the recognition of modular structure across tasks is important for generalization and effective learning.", "We make 3 main contributions: We show that LM-RNNs can perform as well or better than dense networks, when accounting for the total number of nodes or the total number of synapses.", "Locality masking is thus an effecient prescription for choosing sparse subnetworks in a task agnostic and data independent fashion while still acheiving high performance.", "We show that LM-RNNs reach this high performance faster and with lesser training than dense networks, indicating that sparse networks may be preferable to dense networks in memory and data-limited regimes for learning multiple tasks We show that the tasks defined in [39] (a commonly used set of 20 tasks[33], [2], [4], [27], [14] to study representations in networks performing many cognitive tasks) can be solved by a small pool of unconnected autapses, which is essentially a feedforward structure.", "We then introduce Mod-Cog, a large battery consisting of 132 tasks inspired by cognitive science problems such as interval estimation, mental navigation and sequence generation which provides a useful setting to examine multitask learning and representation across tasks relevant to cognitive systems neuroscience." ], [ "Related work", "Several works have tried to operationalize the idea of sparsity.", "These methods can be roughly categorized into 2 main classes: Dense-to-sparse Ref.", "[8] experimentally showed that training followed by pruning and retraining can give sparse networks with no loss of accuracy and ignited an interest in pruning methods.", "Following this work, the lottery ticket hypothesis states that dense, randomly-initialized, feed-forward networks contain subnetworks (“winning tickets”) that — when trained in isolation — reach test accuracy comparable to the original network in a similar number of iterations [6].", "The best method to identify such winning tickets is Iterative Magnitude-based Pruning (IMP) [6], [7], which is computationally expensive and has to be run thoroughly for every different network.", "It has also been shown that parameters of the sparse initialization distribution and sign of weights at initialization are important factors [40] which determine winning tickets.", "Overall, iterative pruning and retraining methods involve 3 steps: (1) pre-training a dense all-to-all model, (2) pruning synapses based on some criteria, and (3) re-training the pruned model to improve performance.", "This cycle needs to be done atleast once and in many cases, multiple times, to get good performance.", "So this procedure requires at least the same training cost as training a dense model and often even more than that.", "Other methods involving ways to encourage sparsity during the training process include $L_1$ (Lasso) regularization[38], $L_0$ regularization[23], [35] and pruning weights below a certain threshold which increases during training[32].", "Another recent method, STR, smoothly induces sparsity while learning pruning thresholds thereby obtaining a non-uniform sparsity budget[17].", "Unfortunately, all of the aforementioned methods require training the original dense network, in varying amounts, thus precluding the benefits that can be obtained by having exact sparsity on the computation during training.", "A class of methods which do not involve training data like SynFlow[36], GraSP[37], SNIP[19], [20] and FORCE[3] have been studied for only feedforward networks.", "In the context of neuroscience, recent work has shown that effective pruning in recurrent networks can be done by biologically plausible algorithms based on noise correlations between the presynaptic and postsynaptic neuron[30].", "Sparse-to-sparse Another line of work concerning sparse-to-sparse training is most relevant to our study.", "This involves using a sparse interaction graph which is used to mask gradient updates.", "Older works maintained a static graph [28] and dealt only with feedforward networks but newer methods such as dynamic sparse training (DST)[5], [22] have been proposed for both feedforward networks and RNNs which dynamically improve the sparse graph and provide better performance.", "All of these methods involve changing the topology of the sparse graph during training.", "Ref.", "[22] considers Erdos-Renyi (ER) type static sparse RNNs but for relatively denser values of sparsity (0.53 and 0.67) and more complex architectures like stacked LSTMs and RHNs [41].", "Here we explore more extreme values of sparsity ($\\sim 5\\%$ and below) and show that they are optimal.", "Static sparse networks have also found common usage in reservoir computing architectures, where large sparse networks are preferred to fully-connected networks to increase heterogeneity across nodes and allow for “richer” dynamics[12], [24]." ], [ "Locality masked RNNs (LM-RNNs)", "We restrict ourselves to simple RNNs for interpretability in the context of systems neuroscience.", "Our RNNs follow dynamics defined by: $h_{t+1} &= (1-\\alpha )h_{t} + \\alpha f(W h_{t} + O i_{t} + b_{i}) \\\\o_{t+1} &= R h_{t+1} + b_{o}$ Corresponding to the biological arrangement on neurons on a two-dimensional cortical sheet, we arrange the nodes of an RNN on the lattice points of a two-dimensional plane, as shown in Fig.", "REF a.", "Then, we constrain the weights for recurrent connections within the nodes of the RNN to be always zero for pairs nodes that lie at a euclidean distance of larger than $d$ .", "The training of the RNN then proceeds in the usual fashion by using back propogation of the loss to update the unconstrained weights.", "We refer to such an RNN as a Locality masked RNN (LM-RNN).", "These networks are trained on many cognitive tasks using supervised learning with a cross entropy loss [39] (see Appendix for more details).", "In effect, the LM-RNN consists of a sparse subgraph of the fully connected network, with each node connected to the $\\sim \\pi d^2$ nearest nodes to it.", "We posit that this sparse subgraph is a “winning lottery ticket”, such that when trained in isolation the LM-RNN acheives comparable performance to a fully recurrently trained network.", "Moreover, we will demonstrate that these winning lottery tickets perform better in a more data-efficient manner than fully connected counterparts with a similar number of nodes or a similar number of synapses.", "This approach can be implemented easily in all training frameworks and is agnostic to the specific optimization algorithm being used." ], [ "LM-RNNs learn the `", "We apply LM-RNNs to a dataset used commonly in systems neuroscience, a set of 20 cognitive tasks introduced in [39], which we henceforth refer to as the Yang tasks.", "Each of these 20 tasks are constructed on the same input stimulus modality — two rings of input units are used that support a single activity bump, encoding a one-dimensional circular variable (which could represent direction of motion, for example).", "Along with the two rings, two additional inputs are fed into the RNN: first, a one-hot encoded rule input vector, indicating which task is to be performed; and second, a fixation input, a decrease of which is treated a `go' signal for the RNN to provide the appropriate output.", "The expected output for each task is a response direction, which is again encoded in a ring of output units (which could represent, for example, a reach or saccade direction).", "A schematic of the setup of the RNN is shown in Fig.", "REF c. For each trial of each task, the inputs are drawn probabilistically from the same distribution across trials that uniformly samples all input directions across both input modalities.", "Figure: (a) An autapse network from Fig.", "trained on the original 20 Yang tasks showing the formation of 12 specialized clusters; (b) The weights of model including the diagonal hidden to hidden matrix which clearly show the existence of no modular structure.We compare LM-RNNs with different values of $d$ against fully-connected RNNs with the same number of neurons (and hence many more parameters; cf.", "Fig.", "REF a) and fully-connected RNNs of a smaller size but with the same number of parameters (cf.", "Fig.", "REF b).", "In all cases, LM-RNNs for any value of $d$ are far more sample-efficient and learn the tasks with same asymptotic performance as compared to the fully-connected counterparts.", "We also compare the performance of these models at the same fixed number of gradient steps early in training to show sample efficiency differences (Fig.", "REF a, inset)." ], [ "LM-RNNs show that `", "While we demonstrated that LM-RNNs at all $d$ perform better than fully-connected networks, we particularly note that $d=0$ , (i.e., a `network' where each node is only connected to itself; in this case the network is simply a pool of disconnected autapses) also performs better than a fully-connected network in terms of learning speed while reaching the same asymptotic performance as a larger fully connected network.", "Remarkably, as seen in Fig.", "REF , this pool of autapses continues to show `modularity' in the network through the nodal-task-variance based metrics used in [39] — however this apparent modularity clearly cannot be a result of any modular structures in the network due to the absence of any inter-node network connections.", "While these results are in themselves indicative of the advantages conferred by LM-RNNs, we note that the Yang tasks are evidently too simplistic to make any strong claims, since they do not even require a network of neurons to accomplish the task." ], [ "To perform a more robust demonstration of the utility of LM-RNNs, we construct a battery of new tasks, inspired by cognitive science problems such as interval estimation, mental navigation and sequence generation.", "We build this series of tasks on the neurogym framework[29] as a set of modular and compositional extensions to the 20 Yang tasks, such that the same input and output space can be utilized across all tasks.", "In particular, we build extensions, that incorporate additional complexity in two main forms: interval estimation, and sequence generation.", "In the case of interval estimation, we consider the set of `delay' based tasks in the Yang tasks.", "In the Yang tasks, 12 out of 20 tasks involve a delay period in the presented input, wherein the network is expected to persistently hold the input presented in an internal memory before performing a task-relevant computation.", "To incorporate interval estimation in these tasks, we require the output to be discplaced with respect to the orignally expected output by a magnitude dependent on the length of the delay period.", "To this end, we choose the delay length randomly from a uniform distribution (as opposed to the fixed delay length considered in Yang tasks).", "As representative examples, in Fig.", "REF a,b we show the inputs and expected outputs for two different delay period lengths in the DlyGo_IntL task, constructed as an interval estimation extension to the DlyGo task from Yang tasks.", "For each of the 12 tasks, the interval-dependent-discplacement may be either of clockwise or anti-clockwise, resulting in the introduction of 24 new tasks.", "In the case of sequence generation, the output of each task is modified to not be a single static direction, but instead a time-varying output corresponding to a drifting direction starting at a particular point (dependent on the particular task).", "The direction of drift can be changed dependent on the particular task.", "As representative examples, in Fig.", "REF c,d we show the inputs and expected outputs for the Go_SeqL and Go_SeqR tasks, constructed as an sequence generation extensions to the Go task from Yang tasks.", "This introduces 40 new tasks based on the earlier set of 20 tasks, with the output of each task drifting either clockwise or anti-clockwise.", "This completes the construction of the 64 new tasks that we use in conjunction with the original 20 tasks as the tasks used for our main set of results hereafter in this paper.", "We refer to this set of 84 tasks as Mod-Cog.", "We note however that our modifications to the tasks are modular in nature (which is similar in spirit to the already existing modular subtask structure in the Yang tasks).", "This allows for an additional extension of 48 more tasks that may be generated by a composition of the sequence generation and interval estimation extensions (such as the DlyGo_IntL_SeqR task shown in Fig.", "REF e).", "For simplicity, we do not use these additional 48 tasks in our main results; however they are included in the repository of tasks that we provide at github link (will be inserted upon acceptance; provided as .zip file in supplementary material).", "The rule input used for Mod-Cog is encoded as a one-hot vector, similar to the setup used in [39].", "This ensures that the rule input in not used as a signal to help decompose tasks into having common subtasks.", "To demonstrate that Mod-Cog is significantly harder than Yang tasks, we demonstrate in Fig.", "REF f that a pool of autapses is incapable of acheiving significant performance levels, in sharp contrast with the Yang tasks.", "Moreover, this is independent of the number of autapses — even pools that are $\\sim 6$ times larger than those necessary for solving Yang tasks are unable to produce larger than 50% performance accuracy." ], [ "RNN and LM-RNN performance on ", "As earlier, we examine the performance of LM-RNNs with varying $d$ on Mod-Cog, and compare them with fully-connected RNNs as a function of the number of gradient steps in training (cf.", "Fig.", "REF a).", "Here again we see that LM-RNNs, for appropriately chosen values of $d$ train significantly more rapidly as compared to fully connected networks.", "Due to the increased task complexity (as evidenced by Fig.", "REF f), locality masks corresponding to very small values of $d$ result in suboptimal performance (which is nonetheless similar to the $d\\rightarrow \\infty $ corresponding to the fully connected network).", "Instead, intermediate small values $d$ outperform all other values of $d$ , as shown in Fig.", "REF b, indicating an optimal nontrivial locality mask $d$ .", "For smaller networks, the optimal value of $d$ corresponds to increasingly larger fractions of all edges in the network (cf.", "Fig.", "REF d), indicating that the optimal may depend more directly on the complexity of the tasks to be solved, rather than scaling with the number of nodes in the network.", "Remarkably, we note that for most LM-RNNs, a random sparse graph with the same number of incoming edges from each node performs almost as well as the graph chosen through locality masking, as shown in Fig.", "REF c. In each case the locality mask performs slightly better or as well as the random sparse graph, while remaining significantly better than dense fully-connected networks.", "For the case of small values of $d$ , we hypothesize that the slight improvement of locality masks may arise from the presence of disconnected components in random sparse graphs in networks with low degrees." ], [ "Controls – comparisons with various baselines for sparsity", "We perform comparisons across 3 baselines.", "First, fully-connected networks with the same number of nodes; second, fully-connected networks with the same number of synapses; and third, fully-connected networks with the same number of nodes, but with an additional $L_1$ regularization term in the loss function to promote the discovery of sparse solutions through training.", "As we demonstrate in Table REF , LM-RNNs reach a higher performance earlier than all other comparable models.", "While sparse networks acheived through $L_1$ regularized models perform better than fully-connected models at some choices of regularization strengths, they acheive this better performance slower than LM-RNNs with static sparsity.", "Thus while sparsity is clearly benefecial to improved performance in the multitask setting of Mod-Cog, choosing this sparsity in a fixed, task-independent fashion at the start of training results in faster training.", "Table: Comparison of LM-RNN performance as compared with other networks with similar number of trainable parameters and nonzero weights.", "L 1 L_1 RNNs are fully-connected RNNs with an L 1 L_1 regularization to promote sparse solutions.", "In this case, the number of nonzero weights is counted as the number of weights above a certain threshold in the trained RNN." ], [ "Discussion", "Through our results, we have demonstrated that a simple fixed sparse graph can provide a large increase in the sample-efficiency of the training process in single-layer recurrent networks.", "These results could in principle be extended to other architectures and tasks but we restricted our experiments to simple RNNs and cognitive tasks for their relevance to systems and computational neuroscience.", "We have found that given a certain amount of task complexity and network size, there is an optimal amount of locality masking that provides the most benefit to learning, both in terms of sample efficiency and asymptotic performance.", "While we have shown this result for recurrent networks, similar results on an optimal value of sparsity have been derived analytically for cerebellum-like feedforward architectures [1], [21].", "It remains an open question to theoretically investigate the relationship between task complexity and the amount of sparsity that is needed: if the network is too sparse, it will not have enough expressivity to perform well on the dataset, while if the network is too dense, the benefits provided by sparsity will not be exploited.", "Although our matrices are extremely sparse and would be well suited to sparse matrix representations, we still maintain dense matrix dataypes for all of the training and evaluation processes since standard libraries like PyTorch do not have native support for these data structures.", "As sparse matrices get more common and their usefulness more apparent, as has been pointed out before [22], it will be very useful for native deep learning software and hardware implementations on GPUs to exploit the potential efficiencies of very sparse matrix structures." ], [ "Acknowledgements", "We thank Guangyu Robert Yang for open-sourcing the 20 cognitive tasks set and the NeuroGym framework [29] released under the MIT License.", "This work was supported by the MathWorks Science Fellowship to MK, the Simons Foundation through the Simons Collaboration on the Global Brain, the ONR, and the Howard Hughes Medical Institute through the Faculty Scholars Program to IRF." ], [ "Methods and Hyperparameters", "PyTorch was used for all simulations.", "All networks were trained with supervised learning using a cross entropy loss.", "The optimizer used was Adam [16] with a learning rate of $10^{-3}$ .", "Each trial was drawn randomly and independently and the tasks were randomly interleaved.", "The Neurogym [29] environment was used to create tasks and the training data for the model.", "For every performance curve, we trained 15 RNNs with different random seeds and used the averaged curve for plotting and computing the optimal locality mask sizes.", "An expanded description of the Mod-Cog tasks and how to create them will be made available at the Github repository after acceptance." ], [ "Formation of clusters in LM-RNNs", "To measure the number of clusters that the hidden nodes of the RNN partition into to solve a task, we use first use agglomerative hierarchical clustering https://docs.scipy.org/doc/scipy/reference/generated/scipy.cluster.hierarchy.linkage.html to obtain a linkage tree for the variance of each node across different inputs for a given task.", "Then, the silhouette score [34] is evaluated for each possible linkage-based cluster partition and the partition with the highest score is selected to represent the clustering of the RNN.", "Through figures REF ,REF ,REF ,REF , REF , REF we show that LM-RNNs at the optimal locality mask size $d$ form a more optimal number of clusters that correspond better to the modular subtask structure present.", "For example, for LM-RNN($N=50^2, d=10$ ) Fig.REF , a single module (module #15) is involved soleley with right-ward sequence generation.", "In contrast, for the fully connected RNN($N=2500$ ) Fig.REF , multiple modules (module #18, #23, #24, #25) are required for the same subtask.", "Figure: Performance curves of models trained with varying values of sparsity (L1) regularization showing that a fixed local mask (of similar sparsity, cf.", "Table ) is better.Figure: (a) Performance at 10k gradient steps for networks of sizes (400,1225,2500) on 84 tasks showing optimal locality mask sizes; (b) for LM-RNNs with 20×2020\\times 20 nodes, the number of clusters obtained by agglomerative hierarchical clustering over training shows that the more optimal locality masked networks discover a lower, more appropriate number of clusters which seems to qualitatively correspond to the modular subtask structures in Mod-Cog.Figure: An LM-RNN(N=20x20,d=2) trained on 84 Mod-Cog tasks showing the formation of clustersFigure: An LM-RNN(N=20x20,d=6) trained on 84 Mod-Cog tasks showing the formation of clustersFigure: An RNN(N=400) trained on 84 Mod-Cog tasks showing the formation of clustersFigure: 24 Clusters formed in an LM-RNN(N = 50 ×\\times 50, d = 10) trained on 84 Mod-Cog tasks.Figure: 33 Clusters formed in an RNN(N=2500) trained on 84 Mod-Cog tasks, contrast with previous figure where the number of clusters is smaller." ] ]
2207.03523
[ [ "Deriving instrumental point spread functions from partially occulted\n images" ], [ "Abstract The point-spread function (PSF) of an imaging system describes the response of the system to a point source.", "Accurately determining the PSF enables one to correct for the combined effects of focussing and scattering within the imaging system, and thereby enhance the spatial resolution and dynamic contrast of the resulting images.", "We present a semi-empirical semi-blind methodology to derive a PSF from partially occulted images.", "We partition the two-dimensional PSF into multiple segments, set up a multi-linear system of equations, and directly fit the system of equations to determine the PSF weight in each segment.", "The algorithm is guaranteed to converge towards the correct instrumental PSF for a large class of occultations, does not require a predefined functional form of the PSF, and can be applied to a large variety of partially occulted images, such as within laboratory settings, regular calibrations within a production line or in the field, astronomical images of distant clusters of stars, or partial solar eclipse images.", "We show that the central weight of the PSF, which gives the percentage of photons that are not scattered by the instrument, is accurate to bettern than 1.2%.", "The mean absolute percentage error between the reconstructed and true PSF is usually between 0.5% and 5% for the entire PSF, between 0.5% and 5% for the PSF core, and between 0.5% and 3% for the PSF tail." ], [ "Introduction", "The resolution and dynamic response of an imaging system, be it for light or for matter waves, is ideally given by the diffraction limit the system.", "In practice, it is further degraded by any distortions due to the imaging components and by the internal structure of the imaging system, which together we dub imperfections.", "These imperfections can arise from the frequency dependence of the refractive index of the imaging components, the microroughness of mirrors or lenses, design flaws such as internal reflections or a misalignment of the imaging components, or from additional imaging analysis components such as optical filters.", "The point-spread function (PSF) of the imaging system, equal to the impulse response of the instrument to a point source, describes the combined effects of the focusing properties of the imaging system along with all the imperfections of the system.", "Accurately determining the PSF enables one to correct for these imperfections and can significantly improve the effective resolution and dynamic contrast of the imaging system.", "Reliably determining the PSF of an imaging system is still a challenging task.", "There are three main approaches for determining a PSF.", "First, one can create a computational model of the entire imaging system with all its components.", "This requires accurate and detailed knowledge of the physical instrument and enables one to build a quantitative model for the PSF.", "As the PSF depends on the focus selected, this task is usually only performed for static imaging systems such as remote observing instruments on satellites.", "However, it can account neither for unknown imperfections of the imaging system nor for its temporal evolution due, for instance, to exposure to the radiation field in interplanetary space.", "Instruments for which this has been performed include the Hubble space telescope [1], [2], [3], Chandra [4], [5], [6], the Atmospheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory [7], and the Nuclear Spectroscopic Telescope Array [8], [9].", "Second, one can empirically determine the PSF by observing the impulse response of a sup-pixel point source.", "Such point sources can be quantum dots or sub-resolution fluorescent beads in laboratory settings, or distant stars in astronomy.", "This empirical PSF represents the PSF at a specific time and can be either directly used to correct images for scientific analysis or to calibrate a theoretical model of the imaging system [10], [11], [12], [13], [14], [9], [4].", "Determining the PSF empirically provides a good estimate of the PSF core, which describes the short-distance scattered light and is related to image blurring.", "But the intensity of a point source is usually too low to enable one to measure the tail of the PSF accurately, which correspond to the long-distance scattered fraction of the light.", "Although long-distance scattering is several orders of magnitudes weaker than short-distance scattering, the accumulated effect over the field of view can result in significant offset intensities in the darkest image regions.", "Furthermore, scattering over the entire image can also significantly reduce peak intensities of very bright point-like sources in the image.", "Both effects reduce the dynamic contrast of the image and result in incorrect image intensities.", "Lastly, one can reconstruct the PSF by comparing an observed image with the true image, i.e., an image unaffected by instrumental effects of the imaging system.", "The true image, however, is in general not known.", "Therefore, blind reconstruction techniques have been developed to simultaneously reconstruct the true image together with the true PSF from a single observed image.", "This process is ill-posed and usually does not have a unique solution.", "The result depends on the first guess of the PSF and the blind reconstruction algorithm chosen.", "Typically the algorithms constrain the inversion problem by adding a priori information in the form of a regularization parameter, such as Tikhonov's regularization [15] or the total variance regularization [16], [17].", "These algorithms are mostly fast and can significantly improve the image quality.", "However, since their solution is not unique and the a priori information employed is an assumption, the reconstructed PSF is not necessarily the correct instrumental PSF.", "One needs to take great care when using such a PSF to correct other images for instrumental effects.", "Here, we present a semi-empirical semi-blind deconvolution methodology to accurately determine the instrumental PSF using partially occulted images.", "Using a partial occultation dramatically increases the overall number of photons involved compared to point-source-based empirical methodologies.", "This enables our algorithm to fit both the core and the tail of the PSF with a high level of precision.", "As the true intensities in the occulted region are known a priori to be zero, our algorithm is not entirely blind.", "This enables our algorithm to converge towards the correct instrumental PSF.", "Additionally, a priori knowledge on the shape of the PSF, such as from any observed diffraction patterns, can easily be incorporated.", "Furthermore, our approach enables one to revise existing high-quality PSFs derived from other approaches, to test their fidelity, and to add missing components.", "Thus, the algorithm is an advancement based on empirical point-source-based methods and blind deconvolution methods.", "The rest of the paper is structured as follows.", "In Section , we first describe the mathematical concept of the algorithm.", "In Section  , we give details regarding the implementation.", "In Section , we showcase the convergence for several test cases, and in Section , we summarize our results." ], [ "Concept", "In the following sections, we describe the mathematical concept of our algorithm, how it can be used to derive or revise a PSF from partially occulted images, and under which conditions our algorithm is guaranteed to converge towards the true PSF." ], [ "General", "The PSF is equivalent to the instrumental impulse response to a point source.", "With $I_\\text{t}$ being the true intensity of a point source at a location $\\textbf {r$ t $}$ in the image plane, the observed intensity $I_\\text{o}$ within an infinitesimally small image segment $dA$ at a location ${r}$ in the image plane is given by Io,${r}$ = It,${r_\\text{t}}$ psf(r-rt) dA + = It,${r}$ -${\\Delta r}$ psf(r) dA+ , where $\\text{psf}({r} - {r_\\text{t}})$ is the instrumental scattering function giving the fractional number of photons that are scattered from their expected location ${r_\\text{t}}$ in the image plane into an area of size $dA$ located at ${r} - {r_\\text{t}}$ in the image plane, ${\\Delta r} = {r} - {r_\\text{t}}$ , and $\\epsilon $ is a noise component in the observed signal.", "The integral of the PSF over the entire image plane $P$ is one, $\\iint _{P} dA\\ \\text{psf}({\\Delta r}) =1, $ as long as we assume perfect reflectivity of the optical components, i.e., that no photons are lost.", "We interpret an image as the superposition of numerous point sources and approximate the PSF to be shift-invariant, i.e., that the PSF does not depend on the location in the image plane.", "The observed intensity in an image pixel is given by $I_\\text{t,{r}} \\ \\text{psf}({\\Delta r}=0)$ plus the scattered light from all other point sources located at distances $\\left| {\\Delta r} \\right|>0 $ from the observed pixel, i.e., Io,${r}$ = P dA It,${r}$ -${\\Delta r}$ psf(r) + = r It,${r}$ -${\\Delta r}$ psfr + = S psfS r in S It,${r}$ -${\\Delta r}$ + .", "In the second line of Equation REF , we have discretized the PSF in the summation into segments having the size of one image pixel.", "The summation goes over the entire image plane described by the vector ${\\Delta r}$ .", "The coefficients $\\text{psf}_{{\\Delta r}}$ give the number of photons which are scattered into the direction ${\\Delta r}$ into one pixel.", "In the third line, we have discretized the PSF into segments $S$ by aggregating PSF coefficients $\\text{psf}_{{\\Delta r}}$ over regions where the PSF varies slowly; $\\text{psf}_S$ denotes the mean PSF weight of the segment.", "This segmentation is an approximation and converges to the exact solution when the size of the PSF segments $S$ become sufficiently small.", "The error involved in this approximation is negligible compared to other errors in our methodology that we describe below.", "As the PSF decreases rapidly within the core of the PSF and slowly in its tail, one would typically choose small PSF segments in the PSF core region and larger segments in the PSF tail region.", "Equation REF defines the main concept of our algorithm.", "Under the presumption that both the true image $I_\\text{t}$ and the observed image $I_\\text{o}$ are known very accurately, that the noise level $\\epsilon $ is small, and that there are more observed pixels $I_{\\text{o},{r}}$ than PSF coefficients $\\text{psf}_S$ , the PSF coefficients $\\text{psf}_S$ can be determined by a multi-linear fit between the true and observed intensities.", "For the reminder of this section, we presume the noise level to be negligible, which is further discussed in Section .", "Next, we investigate the requirements on suitable images for determining the PSF in laboratory settings.", "There are two trivial kinds of images where we know the true intensities in advance: (1) occulted images, and (2) images with a uniform intensity.", "In the first case, the true intensities in the occulted area are zero.", "In the second case, the true intensity of each pixel is equal, and Equation REF simplifies to $I_\\text{o,${r}$} = I_\\text{t,${r}$} \\sum _{{\\Delta r}} \\text{psf}_{{\\Delta r}} = I_\\text{t,${r}$} .", "$ Thus, the observed intensity is the true intensity for uniformly illuminated images.", "This is because each image location in a uniformly illuminated image has scattered away exactly the same number of photons away as it receives from all other locations.", "Equation REF is also approximately true for partial image illuminations as long as the uniformly illuminated area is much larger than the area of influence of the PSF.", "Only close to the edge of the illuminated area do the true and observed intensities start significantly to differ.", "Combining these two cases, if follows that the true intensities are also well known for large parts of an image that is partially occulted and that is uniformly illuminated in the unocculted area.", "We construct a first estimate for the true image of such a partially occulted image by setting the intensities in the occulted pixels to zero.", "The observed intensities in the occulted pixels can now be used to fit for an estimate of the PSF coefficients $\\text{psf}_{{\\Delta r}>0}$ using Equation REF , and $\\text{psf}_{{\\Delta r}=0}$ can be derived from Equation REF .", "The core of the PSF will not be fitted accurately by this first fit, as the true intensities near the illuminated edge are not well known in advance.", "This issue can be resolved by an iterative approach.", "Using the estimate of the PSF, we deconvolve the image and subsequently set the intensities in the occulted pixels in the deconvolved image again to zero, which yields a better approximation of the true image.", "Afterwards, we use this newly derived approximation to the true image together with the observed intensities of the occulted pixels in the original image to derive the next approximations of the PSF, and iterate until the approximation of the true image and the PSF both converge.", "In this approach, the accuracy of the final PSF depends only on the closeness of the final deconvolved image to the true image in the illuminated portion of the image plane.", "Finally, we explore the accuracy and speed of convergence of the PSF core and tail in more detail.", "The coefficients describing the tail of the PSF are related to long-distance scattered photons, i.e., they mostly originate from deep within the illuminated region where $I_\\text{o}$ is almost exactly $I_\\text{t}$ (see Equation REF ).", "Therefore, the coefficients related to the derived PSF tail are expected to be highly reliable.", "For the core of the PSF, which depends on the quality of the estimates of the true intensities in the edge region, the situation is more complex.", "In the first iteration, the assumed true intensities in the edge region are underestimated, because the photons from the occulted region have not yet been redistributed to the illuminated part.", "As the true intensities are underestimated but the observed intensities are fixed, it follows from Equation REF that the coefficients of the PSF core are initially overestimated.", "In the second iteration, this overestimation results in those photons scattered into the occulted region being redistributed into the illuminated region, particularly for those photons close to the edge region.", "This greatly improves the quality of the estimate of the true intensities, and consequently results in good estimates of the PSF coefficients by the second iteration.", "Subsequent iterations further adjust the PSF coefficients until the estimate of the true intensities in the edge region and the associated PSF coefficients both converge." ], [ "Improving existing PSFs", "Existing or published PSFs can also potentially be improved using this methodology.", "Usually, the coefficients related to the core region are well known either from theoretical models or from observations of point sources, whereas the coefficients describing the PSF tail are difficult to fit.", "In our methodology, the coefficients describing the PSF tail depend mostly on the well-known true intensities deep within the illuminated region, and only weakly on the less constrained true intensities in the edge region.", "Thus, an adaption of our methodology enables one to revise the tail coefficients of an otherwise well-known PSF.", "Let us denote the known PSF as $\\overline{\\text{psf}}$ , and the missing portion of the PSF describing the tail as $\\widetilde{\\text{psf}}$ .", "Since photons scattered far away, which are related to the PSF tail, have not been accounted for in the known PSF, the derived true intensities $I_\\text{t,derived}$ of point sources were underestimated by the number of long-distance scattered photons, $I_\\text{t,derived} = I_\\text{t} \\left(1 - \\sum _{{\\Delta r}>0} \\widetilde{\\text{psf}}_{{\\Delta r}}\\right) .$ Following Equation REF , this underestimation of the derived true intensities combined with the fixed observed intensities resulted in an overestimation of the known PSF coefficients by the factor $\\left(1 - \\sum _{{\\Delta r}} \\widetilde{\\text{psf}}_{{\\Delta r}}\\right)^{-1} $ .", "By discretizing the unknown PSF describing the tail into segments, this factor becomes $\\left( 1 - \\sum _S n_S\\ \\widetilde{\\text{psf}}_S\\right)^{-1}$ , where $n_S$ is the number of pixels in a PSF segment.", "Combining the unknown PSF coefficients of the tail $\\widetilde{\\text{psf}}_S$ with the known PSF coefficients $\\overline{\\text{psf}}_{{\\Delta r}}$ corrected for their overestimation yields $I_\\text{o,${r}$} = \\sum _{S} \\widetilde{\\text{psf}}_S \\sum _{{\\Delta r} \\text{ in } S} I_\\text{t,${r}$-${\\Delta r}$} + \\left( 1 - \\sum _S n_S\\ \\widetilde{\\text{psf}}_S\\right) \\sum _{{\\Delta r}} I_\\text{t,${r}$-${\\Delta r}$} \\ \\overline{\\text{psf}}_{{\\Delta r}}.", "$ This equation can be fitted to the observed and true intensities analogous to Equation REF to obtain the unknown tail coefficients, $\\widetilde{\\text{psf}}_S$ .", "The final revised PSF coefficients are then given by $\\text{psf}_{{\\Delta r}} = \\widetilde{\\text{psf}}_{S |_{{\\Delta r} \\text{ in } S}} + \\left( 1 - \\sum _S n_S\\ \\widetilde{\\text{psf}}_S\\right) \\ \\overline{\\text{psf}}_{{\\Delta r}} .$ We note that the solution to this fit, i.e., the newly derived PSF coefficients $\\widetilde{\\text{psf}}_S$ , is degenerate, i.e., that there are several solutions for the $\\widetilde{\\text{psf}}_S$ that result in the same composed PSF coefficients $\\text{psf}_{{\\Delta r}}$ .", "This becomes clear when looking at an example where the known PSF is equivalent to the true PSF.", "In this case, there are two solutions for the newly derived PSF coefficients $\\widetilde{\\text{psf}}_S$ : (1) the newly fitted PSF coefficients $\\widetilde{\\text{psf}}_S$ are all zero, and (2) the newly fitted PSF coefficients $\\widetilde{\\text{psf}}_S$ are equivalent to the true PSF coefficients.", "Both fit solutions result in the same final assembled PSF coefficients $\\text{psf}_{{\\Delta r}}$ .", "Furthermore, we note that our formulation preserves diffraction patterns and the structure of the PSF core explained by the known PSF.", "The newly derived true intensities increase by a factor of $ \\left( 1 - \\sum _S n_S\\ \\widetilde{\\text{psf}}_S\\right)^{-1}$ while the known PSF coefficients decrease by $\\left( 1 - \\sum _S n_S\\ \\widetilde{\\text{psf}}_S\\right)$ .", "These terms cancel, and the predicted image intensities from the known PSF, which describes the PSF core and diffraction patterns around a point source, remain constant." ], [ "Uniqueness of the solution", "We now discuss if the methodology for deriving the instrumental PSF described in Section REF always converges to the correct solution, or, in other words, if the result of this procedure is unique.", "We are able to show that the solution is unique for a large class of occultation masks that are homogeneously illuminated by assuming that the solution is not unique and showing that this assumption results in a contradiction.", "If the solution is not unique, then two different sets of assumed true images and PSFs, $\\lbrace I_{1,\\text{t}}, \\text{psf}_1 \\rbrace $ and $\\lbrace I_{2,\\text{t}},\\text{psf}_2\\rbrace $ , exist that result in this same observed image $I_{\\text{o}}$ .", "First, we focus on the uniquness of the true images.", "Without loss of generality, we assume that the true intensity in the illuminated portion of the image $I_{1,t} > I_{2,t}$ .", "We define the $\\Theta $ function to be zero if a pixel ${x}$ is in the occulted region and one if the pixel is in the illuminated region, $\\Theta ({x}) = {\\left\\lbrace \\begin{array}{ll}0 \\quad \\text{for}\\ I_\\text{\\lbrace 1,2\\rbrace ,t,{x}} = 0, \\\\1 \\quad \\text{for}\\ I_\\text{\\lbrace 1,2\\rbrace ,t,{x}} > 0.\\end{array}\\right.", "}$ We also define the location ${r}$ to be in the occulted region of the image.", "Subtracting Equation REF for the second true image, $I_{2,t}$ from Equation REF for the first true image, $I_{1,t}$ , yields $0 \\ =\\ \\left( I_\\text{1,t} \\sum _{S} \\sum _{({\\Delta r}>0) \\text{ in } S} \\Theta ({r}-{\\Delta r}) \\ \\text{psf}_{1,S} \\right) \\ -\\ \\left( I_\\text{2,t} \\sum _{S} \\sum _{({\\Delta r}>0) \\text{ in } S} \\Theta ({r}-{\\Delta r}) \\ \\text{psf}_{2,S} \\right).", "$ We have omitted the terms for ${\\Delta r} = 0$ , as in this case $\\Theta ({r}) = 0$ .", "Since $I_\\text{1,t} > I_\\text{2,t}$ , it follows that either S (r>0) in S (r-r) psf1,S    =    S (r>0) in S (r-r) psf2,S    = 0 or S (r>0) in S ar (r-r) psf1,S    <    S (r>0) in S ar (r-r) psf2,S.", "We have multiplied Condition REF with an arbitrary positive factor $a_{{r}}$ , so that the inequality keeps its sign, as a preparation for our subsequent discussion.", "Condition REF  or REF have to be fulfilled for each occulted pixel ${r}$ .", "Therefore, Condition REF defines a system of equations and Condition REF a system of inequalities, where the number of occulted pixels ${r}$ determines the number of lines in the system of equations or system of inequalities, and the number of PSF coefficients $\\text{psf}_S$ determines the number of variables.", "We first lead Condition REF to a contradiction.", "As long as the system of equations of Condition REF is well-defined, i.e., as long as each PSF coefficient $\\text{psf}_S$ with ${\\Delta r}>0$ appears at least once in the system of equations and as long as we have at least as many occulted pixels as PSF coefficients, the solution of this system of equations is unique.", "It follows that the trivial solution $\\text{psf}_{1,S_{|{\\Delta r} >0}} = \\text{psf}_{2,S_{|{\\Delta r} >0}} = 0$ is the only solution.", "This equation infers that the PSF does not scatter any photons.", "However, there are scattered photons in the occulted region; therefore, Condition REF results in a contradiction.", "The argument for Condition REF is more complex.", "To lead Condition REF to a contradiction, we aim at transforming Condition REF into $\\sum _{S} \\sum _{({\\Delta r}>0) \\text{ in } S} \\text{psf}_{1,S} < \\sum _{S} \\sum _{({\\Delta r}>0) \\text{ in } S} \\text{psf}_{2,S}.", "$ In Inequality REF , the summation runs over the entire PSF except for the center pixel of the PSF.", "Therefore, Inequality REF infers that $\\text{psf}_1$ scatters a smaller percentage of its intensity away than $\\text{psf}_2$ .", "A smaller scattering rate implies that the observed intensities are closer to the true intensities, i.e., that $I_{1,t} - I_{\\text{o}} < I_{2,t} - I_{\\text{o}}$ , and it follows that $I_{1,t} < I_{2,t}$ .", "However, we have assumed $I_{1,t} > I_{2,t}$ .", "Therefore, if the system of inequalities in Condition REF can be transformed into Inequality REF , this will result in a contradiction.", "Then, neither Condition REF nor Condition REF can be fulfilled, and it would follow that the assumption $I_{1,t} > I_{2,t}$ was wrong and that consequentially $I_{1,t} = I_{2,t}$ , i.e., that the solution for the true image is unique.", "Next, we investigate the likelihood for the existence of this transformation.", "Condition REF can be transformed to Inequality REF by a linear combination of the inequalities in Condition REF .", "The $a_{{r}}$ are the coefficients for this linear combination, and all the $a_{{r}}$ have to be non-negative so that Condition REF , as a sum of the Inequalities of Condition REF , does not change its sign.", "Therefore, we aim at finding a valid set $a_{{r}}$ that transforms Condition REF to Inequality REF .", "A sufficient condition is that a set $a_{{r}}$ exists that transforms each individual coefficient $\\text{psf}_{S}$ from Condition REF to the corresponding one of Inequality REF .", "Summing all terms containing a given PSF coefficient $\\text{psf}_S$ over ${r}$ , i.e., over the lines of the system of inequalities in Condition REF , and comparing it with Inequality REF defines the wanted transformation, $\\sum _{{r}} \\sum _{({\\Delta r}>0) \\text{ in } S} a_{{r}} \\Theta ({r}-{\\Delta r}) \\text{psf}_S = \\sum _{({\\Delta r}>0) \\text{ in } S} \\text{psf}_S.", "$ Equation REF is a system of equations, where the PSF coefficients $\\text{psf}_S$ define the lines and the $a_{{r}}$ are the variables.", "We usually have significantly fewer PSF coefficient to fit than occulted pixels in the image.", "Therefore, we have significantly fewer lines in the System of Equations REF than variables $a_{{r}}$ , and consequentially System of Equation REF is strongly underdetermined and the solution for the $a_{{r}}$ is degenerate.", "In the following, we explore the probability that for at least one of these degenerate solutions all the $a_{{r}}$ are non-negative.", "Let us denote the probability that a single coefficient $a_{{r}}$ is non-negative as $p$ , the number of occulted pixels as $n_\\text{oc}$ , the number of PSF segments to fit as $n_{\\text{seg}}$ , and the number of solutions as $n_\\text{sol}$ .", "A lower boundary for the number of solutions $n_\\text{sol}$ can be estimated using combinatorics.", "Having $n_{\\text{seg}}$ lines in the System of Equations REF , we need exactly $n_{\\text{seg}}$ variables, i.e., occulted pixels, to derive a single solution.", "Having $n_\\text{oc}$ occulted pixels in total, the number of possibilities to draw $n_{\\text{seg}}$ occulted pixels from $n_\\text{oc}$ occulted pixels is $n_\\text{sol} = \\frac{n_\\text{oc}!", "}{\\left(n_\\text{oc} - n_{\\text{seg}}\\right)!\\ n_{\\text{seg}}!}.", "$ For a single solution derived from $n_{\\text{seg}}$ occulted pixels, the probability that all $a_{{r}}$ are non-negative is $p^{n_{\\text{seg}}}$ .", "The probability that not a single solution exists where all the $a_{{r}}$ are non-negative is $\\left(1 - p^{n_{\\text{seg}}}\\right)^{n_\\text{sol}}$ .", "It follows that the probability $P$ that at least one solution exists where all $a_{{r}}$ are non-negative is $P(p, n_\\text{oc}, n_\\text{sol}) = 1 - \\left(1 - p^{n_{\\text{seg}}}\\right)^{n_\\text{sol}}.", "$ Combining Equation REF and Equation REF enables one to derive a rough estimate of the probability that at least one solution exists where all the $a_{{r}}$ are non-negative, or, alternatively, to estimate how many occulted pixel in an image are required so that the probability is higher than a certain threshold.", "In Figure REF , we plot the required number of occulted pixels $n_\\text{oc}$ so that the probability for the existence of a set of non-negative $a_{{r}}$ is higher than 99 versus the number of PSF segments $n_{\\text{seg}}$ to fit.", "The results are plotted for an assumed probability $p$ of 50, 10, and 1.", "For 100 PSF segments and an assumed probability of $p={50}{}$ , we find that at least 133 occulted pixels are required.", "When we assume $p={1}{}$ , i.e., that the probability to obtain a non-negative $a_{{r}}$  is very small, 3908 occulted pixels are required.", "As we usually have hundreds of thousands of occulted pixels in an image, it follows that it is highly probable that a set of non-negative $a_{{r}}$ exists that solves Equation REF , and, consequentially, that the solution for the true image will be unique.", "Finally, we check if the solution for the PSF is unique.", "With $I_{1,t} = I_{2,t}$ , Equation REF can be rewritten as $\\sum _{S} \\sum _{({\\Delta r}>0) \\text{ in } S} \\Theta ({r}-{\\Delta r}) \\ \\text{psf}_{1,S} = \\sum _{S} \\sum _{({\\Delta r}>0) \\text{ in } S} \\Theta ({r}-{\\Delta r}) \\ \\text{psf}_{2,S}.", "$ Condition REF is a system of equations with one line for each occulted pixel ${r}$ .", "As long as this system of equations is well-defined, the solution for the variables $\\text{psf}_S$ is unique.", "It follows that the trivial solution $\\text{psf}_{1,S} \\ = \\ \\text{psf}_{2,S}$ is the only solution.", "Thus, $\\text{psf}_1 =\\text{psf}_2$ , and the solution for the PSF is unique.", "We note that the argument in this section is valid as long as Condition REF can be transformed to Condition REF .", "Our requirement that the coefficients $a_{{r}}$ in Equation REF are non-negative is a stronger constraint than necessary.", "However, it guarantees that Condition REF has the same sign as Condition REF and thus that Equation REF is a valid transformation.", "As Equation REF is independent of the PSF coefficients, this makes the entire argument independent of the a priori unknown PSF coefficients.", "This enables one to verify, independently of the unknown PSF coefficients, if a given occultation mask together with a given discretization of the PSF guarantees a unique solution.", "The solution is unique if a set of non-negative $a_{{r}}$ exists which solves System of Equations REF .", "There is one simple cases where the uniqueness of the PSF is guaranteed: Exactly one pixel in the center of the image is illuminated.", "Then, for each occulted pixel,in the summation over the illuminated pixels in Inequality REF exactly one $\\Theta ({r} - {\\Delta r})$ is one.", "Having only one PSF coefficient per line remaining in the system of inequalities enables one to rewrite the System of Inequalities REF as Inequality REF , guaranteeing the uniqueness.", "In this section, we delineate the main stages for an implementation of our algorithm.", "The general procedure is as follows: (a) discretize the PSF into segments, (b) create an occultation mask, (c) illuminate the occultation mask, (d) identify the occulted pixels, (e) derive an approximation for the true image of the illuminated occultation mask, (f) set up the system of equations defined by Equation REF , (g) fit the PSF from this system of equations, (h) postprocess the PSF, (i) deconvolve the occulted image with the PSF to derive a revised approximation of the true image of the illuminated occultation mask, and (j) repeat from steps (d)-(i) until the approximations of the true image and the PSF both converges.", "The final PSF can then be used to correct for instrumental effects of other images taken by the same imaging instrument.", "In the following paragraphs, we will explain each of these steps in more detail.", "In Figure REF , we visualize these steps by showing a schematic of the procedure; each panel has the label of the corresponding step or steps in our procedure." ], [ "(a) Discretizing the PSF", "First, we discretize the two-dimensional PSF spatially into segments (see Fig.", "REF (a)).", "Each segment corresponds to one PSF coefficient to be fitted.", "As the values of the PSF coefficients in the core of the PSF usually decreases rapidly within a few pixels from the peak, the core of the PSF should be determined at the full resolution of the detector.", "The PSF coefficients in the tail usually vary much more slowly and larger regions in the PSF tail can be aggregated to a single PSF coefficient.", "This aggregation improves the signal-to-noise (S/N) level when fitting the PSF coefficients later on and reduces the required computation time.", "A priori knowledge, such as diffraction patterns, can also be included, in which case each segment in the diffraction pattern should be assigned a PSF coefficient to be fitted." ], [ "(b) Creating occultation masks", "Our algorithm works with all kinds of partially occulted images.", "In laboratory settings, one can create various occultation masks that are specialized for different tasks.", "To resolve the core of the PSF, we recommend using an occultation mask containing several pinholes.", "Setting the pinhole diameters to a size of several pixels enhances the approximation of the true intensities in the pinholes and thereby the overall accuracy of our algorithm.", "If a diffraction pattern is apparent, the distance between the pinholes should be large enough so that the different diffraction patterns are distinguishable.", "Larger pinhole sizes are preferred as it results in more intensity in the diffraction pattern.", "To fit the tail of the PSF accurately, we suggest an occultation mask containing a single large hole, which drastically increases the total number of photons available.", "The concepts of these occultation masks can also be combined into a single occultation mask for convenience (see Fig.", "REF (b))." ], [ "(c) Illuminating the occultation mask", "Next, we uniformly illuminate a screen, set the occultation mask between the screen and the imaging device, and record the resulting partially occulted image (see Fig.", "REF (c))." ], [ "(d) Identifying occulted pixels", "Pixels at the edge of the illuminated area will have photons scattered into the occulted area.", "However, there are no photons scattered back from the occulted area to the illuminated edge to counterbalance the photon loss.", "Consequently, the intensities decrease across the edge between the illuminated area and occulted area.", "As most photons are scattered over short distances and as the total numbers of photons is conserved, the edge can be estimated to be roughly at the 50 level of the illuminated edge intensities.", "We recommend defining occulted pixels to be at least one pixel away from the 50 intensity level of the edge to avoid partially illuminated pixels." ], [ "(e) Deriving an approximation for the true image of the illuminated occulation mask", "The true image is defined as the observed image with the intensities of the occulted pixels set to zero (see Fig.", "REF (e))." ], [ "(f) Setting up the system of equations", "Each occulted pixel intensity $I_{{r}}$ gives one line in the system of equations defined by Equation REF (see Fig.", "REF (f)).", "For each occulted pixel, the integrated intensities $I_\\text{t,${r}$-${\\Delta r}$}$ in the illuminated regions of the presumed true image associated with each PSF segment $\\text{psf}_S$ have to be derived (see Equation REF ).", "Therefore, the computational cost scales as $n_\\text{oc} n_\\text{il}$ , where $n_\\text{il}$ is the number of illuminated pixels in the true image.", "With $n_\\text{oc}$ and $n_\\text{il}$ both being typically at the order of e6, the computational costs can become expansive.", "To reduce the computation time, for occulted pixels that are far enough from the illuminated edge, we recommend grouping several contiguous occulted pixels into a superpixel that has their average intensity.", "This merging also increases the S/N in the observed intensity of the occulted superpixel.", "We note that the system of equations to be solved is not limited to using a single image.", "Multiple images involving possibly different occultation masks can also be used to set up a combined system of equations.", "This enables one to determine the PSF by using several specialized occultation masks, i.e., one mask for deriving the PSF core, one mask for fitting the diffraction patterns, and one mask to determine the PSF tail." ], [ "(g) Fit the PSF", "Having set up the system of equations, we derive a multi-linear fit to the system of equations to determine the PSF coefficients (see Fig.", "REF (g)).", "To optimize the fitting process, the PSF coefficients, which define the columns of the system of equations, should be constrained to allow for only non-negative values.", "Furthermore, as the observed intensities in the occulted pixels can be expected to vary over several orders of magnitudes from the illuminated edge to far within the occulted region, each line of the system of equations should be normalized by its observed intensity.", "There are several procedures by which this system of equations can be fit: When the number of PSF coefficients is small, one can perform a simple multi-linear fit to determine the weights in the PSF segments.", "When the number of PSF cofficients is large, one has to prevent the fitting algorithm from terminating in a local minimum.", "The easiest way to do this is to start the fitting process at a low resolution, i.e., a small number of coefficients to be fitted, and to iteratively increase the resolution.", "For this purpose, we recommend placing a low-resolution adaptive grid on top of the discretized PSF.", "Each node in the adaptive grid becomes a support coefficient.", "The PSF coefficients are linked to the support coefficients by a spline interpolation.", "At each iteration, we fit the support coefficients to the system of equations.", "Subsequently, we increase the resolution of the adaptive grid and iterate until the full resolution given by the PSF discretization is reached.", "This guarantees a reasonable initialization of the fit coefficients at each iteration, speeds up the fitting process, and reduces the risk of terminating in a local minimum.", "When noise is present, one has to avoid overfitting, i.e., fitting the specific solution of the noise-dependent image instead of a general solution.", "There are two ways to circumvent overfitting: First, instead of using all lines of the system of equations at once, one can use a random subset, fit the PSF coefficients, and repeat this procedure many times.", "The final PSF coefficients are then given as the mean of the individual fits.", "Second, one can add a regularization parameter to the fitting process, such as the Ridge regularization [18] or Lasso regularization [19].", "Then, each column of the system of equations has to divided (i.e., normalized) by the estimated size of the associated PSF coefficient beforehand, so that the regularization works with the same strength on all coefficients.", "After having obtained the final fit coefficients, the normalization of the fit coefficients has to be removed.", "When revising the coefficients of a known PSF by Equation REF , the solution for the fitted missing PSF coefficients is degenerate.", "In general, one is interested in the solution that least modifies the known PSF, i.e., the solution where the fit coefficients describing the missing portion of the PSF are minimal.", "This can be achieved by adding the sum of the fit coefficients describing the missing portion of the PSF as a regularization parameter to the fit.", "When the analytical form of the PSF can be guessed, e.g., from a previous fit of the PSF or from theoretical work, one can re-parameterize the PSF coefficients by this analytical expression as an initial step.", "This enables one to fit for the free parameters in the analytical expression of the theoretical PSF.", "Finally, we note that the system of equations is usually strongly overdetermined, as there are typically more occulted pixels than PSF coefficients to determine.", "Using only a subset of the system of equations is typically enough to reliably constrain the fit and to greatly speed up the calculation.", "For this, the occulted pixels in this subset need to be evenly distributed over the entire occulted region, as pixels close to the illuminated edge mostly affect the quality of the core of the fitted PSF, while those far away constrain the tail of the PSF." ], [ "(h) Postprocessing the PSF", "Postprocessing the PSF is not strictly required as its effect on the reconstruction of true images is usually negligible.", "But it enables one to derive a continuous approximation of the discretized PSF.", "For this task, we recommend performing a large number of iterations of Laplacian smoothing on the PSF, where the weights within each segment have to be renormalized after each iteration to match the fitted weight.", "The larger the sizes of the PSF segments are, the more iterations are needed to derive the a good continuous approximation.", "This guarantees a smooth transition between the PSF segments while keeping the weights of the PSF segments correct (see Fig.", "REF (h))." ], [ "(i) Updating the approximation of the true image", "Next, the original image has to be deconvolved with the derived PSF to acquire an improved approximation for the true image (see Fig.", "REF (i)).", "The deconvolution algorithm used has to retain the total image intensity as well as sharp edges, making the Richardson-Lucy algorithm a good choice [20].", "We note that the image deconvolution involved in this step is an ill-posed problem and limits one to the accuracy of the deconvolution algorithm chosen.", "Therefore, even for the fully converged PSF solution, deviations from the true PSF can be expected.", "These deviations, however, are usually small.", "The overall accuracy will be further investigated in Section ." ], [ "(j) Iterating", "Finally, we repeat Steps (d)-(i) of this entire procedure until the true image and the PSF both converge.", "Having derived the instrumental PSF, we are able to reconstruct additional true images by deconvolving the associated observed images with the fitted instrumental PSF." ], [ "Numerical experiments", "In this section, we study the stability and accuracy of our algorithm.", "For five test cases, we convolve a numerical true image of an illuminated occultation mask with a true PSF to derive an observed illuminated occultation mask.", "Afterwards, we apply our algorithm on this observed illuminated occultation mask to reconstruct the PSF, and analyze the accuracy by comparing the reconstructed PSF with the true PSF.", "In Section REF , we investigate in detail the accuracy of our algorithm for a simple cylindrically symmetric PSF as a proof of concept, and in Section REF , we analyze the noise stability of our algorithm on this PSF.", "In Section REF , we show reconstruction results for a cylindrically symmetric PSF where the functional form is known and only free parameters have to be determined, for an elliptical PSF, for a PSF that contains a diffraction pattern, and for the revised PSF of AIA [21], [22].", "These test cases all have an image sizes of $1024\\times 1024$ pixels.", "The tests were run using an NVIDIA GeForce RTX 2080 Ti graphics processing unit, i.e., a consumer graphics card, to set up the system of equations, for deconvolving the images, and for postprocessing the images, and on an AMD Ryzen 9 3950X processor for fitting the system of equations.", "Table REF lists the runtimes and the details of the configurations.", "Typical runtimes for our algorithm are 15seconds to 3minutes per iteration, with $\\le 5$ iterations required for convergence." ], [ "Proof of concept", "As a first test case, which is presented in Figure REF , we define a cylindrically symmetric PSF composed of a Gaussian core with a $\\sigma = 10~\\text{pixels}$ and a power law tail, $\\text{psf}(r) = {2.4e-4}\\ \\exp {\\left(-\\frac{r^2}{200}\\right)} + \\frac{0.5}{1 + 61 r^2}, $ where $r$  is the distance to the PSF center (Fig.", "REF (a)).", "This PSF scatters about 50 of the photons over the image plane.", "We use an occultation mask that contains one large hole with a radius of 50 pixels in its center and 48 pinholes each with a size of 1 pixel distributed over the occultation mask.", "The true image of the illuminated occultation mask is the occultation mask with the intensities in each pixel within the holes set to 10000 digitial numbers (DNs).", "The observed image of the illuminated occultation mask is the true image convolved with the PSF.", "The observed image is shown in Figure REF (b); each of the holes exhibits a halo due to the scattered light.", "To reconstruct the PSF from the observed illuminated occultation mask, we discretize the PSF into shells.", "We define the core region of the PSF as the area within 10 pixels around the PSF center and the tail region as the remaining PSF area.", "We describe the core region of the PSF with 10 concentric shells, each having a width of $0.5$  pixels, and then 5 additional concentric shells each with a width of 1 pixel.", "The PSF tail region is segmented into 25 concentric shells in logarithmic steps.", "This results in a total of 40 PSF coefficients to determine (Fig.", "REF (c)).", "Next, we approximate the true illuminated occultation mask by setting the intensities in the occulted pixels to 0DNs.", "We set up System of Equations REF , randomly draw $10^4$  lines from the approximately $10^6$  lines of the System of Equations, and fit the PSF coefficients by a simple multi-linear fit.", "We assemble the PSF using the fitted PSF coefficients and derive a continuous approximation of the PSF by apply $10^4$  iterations of Laplacian smoothing.", "This large number of iterations of Laplacian smoothing accounts for the large size of the PSF segments in our PSF tail.", "Finally, we deconvolve the observed illuminated occultation mask with this smoothed PSF to derive the next approximation of the true illuminated occultation mask.", "To be able to study the rate and quality of convergence of our algorithm, we iterate this procedure 100 times, and arrive at the final PSF.", "We now analyze the accuracy of the reconstructed PSF.", "We show the fitted PSF at the first, second, fifth, and 100th iteration together with the true PSF in Figure REF (a).", "In the first iteration, the PSF is overestimated, but from the second iteration on, the fitted PSF shows excellent agreement with the overall shape of the true PSF over all eight order of magnitudes.", "The mean of the absolute percentage error (MAPE) of the fitted to the true PSF coefficients is shown in Figure REF (d).", "There, we plot the MAPE of the PSF center coefficient, of the PSF core region, of the PSF tail region and of the entire PSF versus the iteration number.", "By the second iteration, the MAPE of the entire PSF decreases to 1.5, and by the third iteration, the MAPE can be considered to have converged to 1.3.", "The MAPE of the PSF center coefficient converges to 0.2, where we remind the reader that the PSF center coefficient determines the percentage of photons that are not scattered.", "The MAPE of the PSF core region converges to 0.7, and the MAPE of the PSF tail region to 1.7.", "These values show that all the PSF segments are fitted accurately over all eight orders of magnitude.", "In Figure REF (e), we show the percentage error for the PSF weights along the PSF cross section.", "Starting from the second iteration, the errors in the PSF core region are below 2, while the errors in the tail region oscillate around the true solution with an amplitude of 3.", "The larger amplitude in the last oscillation at the very end of the tail is an effect of the discretization of the PSF.", "As there is no further outer shell, the boundary condition for the smoothing is not well defined, resulting in larger maximum deviations for the outermost shell.", "These oscillations, however, have almost no effect on real-world applications as their effect averages out.", "A more relevant parameter is the accumulated error, i.e., the total signed error of the PSF weights from the PSF center up to a given distance to the PSF center.", "This corresponds to the absolute error in the number of photons that are scattered up to that given distance, and is shown in Figure REF (f).", "For each distance, the absolute value of the accumulated error is always smaller than 0.1, which shows the high fidelity in the spatial distribution of the scattered photons.", "Next, we evaluate the quality of the reconstructed image.", "In Figure REF (g), we show the reconstructed image, i.e, the observed illuminated occultation mask deconvolved with the final PSF.", "The intensity in the large hole is homogeneous without showing a halo, and the 48 pinholes appear as point sources with a size of one pixel.", "The reconstructed intensities along a horizontal slice through the center of the image are plotted in Figure REF (h) for the first, second, fifth, and final iteration of our algorithm.", "The intensities in the occulted areas along the slice were originally between 1.3 and 1700DNs in the observed image (cyan line), but they become negligible from the second iteration onward (orange, green, and black line; the orange and green lines are mostly covered by the black line).", "The peak intensities in the pinholes were 5000DNs in the observed image, compared to 10000DNs in the true image.", "Deconvolving the observed image with the fitted PSF increases the peak intensities to between 9650 and 9950DNs.", "The same effect is apparent in the large hole.", "The observed intensities ranged from 70008750DNs for the pixels within the large hole, compared to 10000DNs in the true image.", "Deconvolving the observed image increases the intensities to between 9975 and 10006DNs.", "Finally, we focus on the intensity drop at the illuminated edges.", "At the edge of the pinholes in the deconvolved image, the intensities drop from 9950DNs to <0.05DNs within one pixel.", "At the edge of the large hole in the deconvolved image, the intensities drop from 9975DNs to 2DN within one pixel.", "Thus, we find excellent reconstructions of the pinholes, the large hole, and the edge of the illuminated areas." ], [ "Noise stability", "In our methodology, the PSF is fitted from the intensities in the approximated true image and the observed intensities in the occulted regions.", "In the occulted regions, the signal is in general low, and consequentially the S/N can be low.", "Here, we analyse the effect of image noise on the quality of the fitted PSF, and test two methodologies for mitigating image noise: clustering pixels into a superpixel and using multiple fit repetitions.", "Image noise can affect the quality of the fitted PSF core, the fitted PSF tail, or both.", "In the PSF core region, the PSF weights typically fall off rapidly, and the most important occulted pixels for fitting these weights are occulted pixels close to the illuminated edge.", "As the intensities in the occulted region close to the illuminated edge decreases rapidly, these occulted pixels cannot be clustered without affecting the quality of the reconstructed PSF.", "Therefore, to improve the quality of the PSF core, averaging multiple fit repetitions is the preferred technique.", "For the PSF tail region, the most important occulted pixels for fitting these weights are those at locations far from the illuminated edge.", "Far from the illuminated pixels, the observed intensity decreases slowly, and therefore many of these occulted pixels can be clustered to improve the S/N in these pixels.", "Hence, to improve the quality of the PSF tail, clustering is the preferred technique.", "In the following, we simulate the effect of image noise and investigate its effect on the quality of the reconstructed PSF for the PSF and occultation mask of Section REF .", "The average observed intensity in the occulted region is 14DNs, with a maximum intensity of 1700DNs close to the edge to the illuminated region and a minimum intensity of 1.3DNs at the edges of the detector.", "Therefore, when a constant noise level is present, the S/N varies greatly from the edge of the illuminated region to the edge of the detector.", "Here, we define the S/N level as the average intensity in the occulted image region over the noise level.", "Assuming S/N levels of 1, 10, and 100, the corresponding Poisson noise levels are 14DNs, 1.4DNs, and 0.14DNs.", "For each S/N level, we select the intensities in the observed image from the associated Poisson distributions to create the noisy images shown in (Fig REF (a)-(c)) and subsequently re-derive the PSF.", "Using these images, we test two noise mitigating techniques: (1) we cluster $3\\times 3$  occulted pixels into a superpixel when the minimum distance to the illuminated pixels is at least 15 pixels, $7\\times 7$  occulted pixels when the minimum distance is at least 35 pixels, and $15\\times 15$  occulted pixels when the minimum distance is at least 75 pixels.", "The minimum distance from the illuminated edge chosen corresponds to ten times half the edge length of the superpixel.", "This choice satisfies the condition that the intensities in the clustered pixels have to vary slowly across the superpixels.", "And (2), we repeat the fitting procedure 1000 times and derive the final fit as the average of the individual fits.", "As a baseline for the noise mitigation techniques, we use a simple multi-linear fit without noise mitigation.", "For maximum noise mitigation, we combine the cluster technique with the fit repetition technique.", "In Figure REF (d), we show the fitted PSFs for S/N levels of 1, 10, and 100 for the baseline configuration.", "At a S/N level of 1, the fitted PSF shows strong oscillations around the true PSF.", "The amplitude of the oscillations decreases with increasing S/N level.", "The evolution of the MAPE between the fitted and true PSF coefficients with increasing S/N level is shown in Figure REF (e).", "At a S/N level of 1 the MAPE is 53; it decreases to 21 at a S/N level of 10 and to 9 at a S/N level of 100.", "When applying the clustering technique alone, the MAPE is 21 for a S/N of 1, 10 for a S/N of 10, and 3 for a S/N of 100.", "When applying the fit repetition technique alone, the MAPE is 10 for a S/N of 1,  5 for a S/N of 10, and 2 for a S/N of 100.", "The MAPEs for the combined clustering and fit repetition configuration performs about as well as the fit repetition technique alone.", "For comparison, the MAPE without noise is 1.3 (see Section REF ).", "In Figure REF (f), we plot the fitted PSFs for S/N levels of 1, 10, and 100 for the combined clustering and fit repetition configuration.", "For a S/N level of 1, only moderate oscillations are apparent, and for S/N levels of 10 and 100, the oscillations around the true solution almost vanish.", "In Figure REF (g), we show the associated deviations of the fitted PSF weights from the true PSF weights along a cross-section through the PSF center.", "For a S/N level of 1, the fitted PSF oscillates with an average amplitude of 20 around the true PSF, for a S/N of 10, the average amplitude of the oscillations is about 5, and for a S/N of 100, the average amplitude of the oscillations is about 4.", "For comparison, the amplitude of the oscillations without noise is 3 (see Section REF ).", "These cases show that noise mitigation techniques can successfully be applied to generate good fits even for mediocre S/N levels.", "For cases where the S/N level is too low for generating good results even after applying noise mitigation techniques, we recommend improving the S/N level by either increasing the exposure time of the image or by enlarging the size of the holes to increase the total numbers of photons available.", "In general, we recommend good S/N levels in the occulted region; the higher the S/N, the more accurate the reconstructed PSF will be." ], [ "Further examples", "Here, we present 4 additional examples of how our algorithm can be used to determine the instrumental PSF.", "In the first example, we determine free parameters of a PSF whose functional form can be guessed.", "In the second example, we fit the weights of an elliptical PSF.", "In the third example, we derive the weights in a PSF containing a diffraction pattern.", "And in the fourth example, we revise the PSF of a satellite imager.", "These examples are presented in Figure REF , whereby each row shows one example.", "The first column shows the observed illuminated occultation masks of these examples, the second column shows the discretized PSFs, the third column shows the fitted PSFs, and the fourth column compares the weights of the fitted PSFs and the weights of the true PSFs along a PSF cross-section.", "In the first example, shown in Figure REF (a)-(d), we take as the true PSF the PSF of Sections REF and REF , defined by a Gaussian core and a powerlaw tail.", "We select an occultation mask which contains in its center a single large hole with a radius of 50 pixels, and discretize the PSF into 200 shells.", "We assume that the functional relationship of this PSF can already be guessed, e.g., from a previous fit of the PSF analog to Section REF , and aim at determining the coefficients of the Gaussian core and powerlaw tail.", "To do so, we parametrize the PSF coefficients $\\text{psf}_S$ as a Gaussian function superposed with a powerlaw, $\\text{psf}_S(r) = A\\ \\exp { \\left( -\\frac{r^2}{2\\ B^2} \\right)} + \\frac{C}{D + r^E}.", "$ When fitting the System of Equations REF , we fit for the coefficients $A$ , $B$ , $C$ , $D$ , and $E$ .", "The fit results in A = 2.4282(5)e-4, B = 10.01(2), C = 4.700(2)e-2, D = 1.0802(4), E = 1.99726085(5), where the one-sigma fitting uncertainties in the last digit are given in the parenthesis.", "These fitted values are very close to the true parameters $A={2.3942e-4}$ , $B= 10$ , $C={4.732e-2} $ , $D=1$ , and $E=-2$ .", "The MAPE of the fitted to the true PSF weights is 0.4.", "In the second example, shown in Figure REF (e)-(h), we define an elliptical true PSF by $\\text{psf}(x, y) = {2.4e-4}\\ \\exp {\\left(-\\frac{0.25 x^2 +y^2}{200}\\right)} + \\frac{0.5}{1 + 61 (0.25 x^2+y^2)}, $ where $x$ and $y$ are the horizontal and vertical distances from the PSF center.", "This PSF scatters 54 of the photons away from its center.", "The occultation mask contains a single large hole with a radius of 50 pixels.", "We discretize the PSF into shell segments, where we divide the 40 cylindrical shells from Section REF into segments having an angular width of 30, resulting in 480 PSF segments.", "We fit for the PSF weights by randomly drawing 10000 lines from the system of equations, performing a multi-linear fit using the adaptive grid method described in Section .", "The result is presented in Figure REF (g), showing a smooth, elliptical PSF.", "In Figure REF (h), we plot the fitted and the true PSF weights along the horizontal and vertical axis through the PSF center.", "The weights of the fitted and true PSF agree very well.", "The MAPE between fitted and true weights derived from all PSF segments is 4.3.", "In the third example, shown in Figure REF (i)-(l), we define the true PSF to have a Gaussian core, a powerlaw tail, and a diffraction pattern, defined by psf(x, y) = 1.76e-4 (-x2 +y2200) + 0.3671 + 61 (x2+y2) + + {ll 0.367 ( 0.1 x2 + y2)0.1 x2+y2    for -1yx = 4 - 8     and | x2+y2 mod 18 pixel | 4 pixel 0 else ..", "This PSF scatters 73 of the photons away from its center.", "The occultation mask contains a large hole with a radius of 50 pixels and 48 pinholes each with a size of 1 pixel.", "We discretize the PSF into shells and additionally assign each location in the diffraction pattern its own segment.", "To fit for the PSF weights, we use a simple multi-linear fit.", "In the postprocessing step, only the shells are smoothed, and the fitted weights of the diffraction pattern are re-inserted afterwards.", "The resulting PSF is presented in Figure REF (k).", "In Figure REF (l), we show the fitted and true PSF weights in the direction through the diffraction pattern, which agree well.", "The MAPE is of the fitted PSF from the true PSF 1.4.", "In the fourth example, shown in Figure REF (m)-(p), we revise the instrumental PSF for AIA, which observes the solar atmosphere in 10 filters at extreme ultraviolet, ultraviolet, and visible wavelength at a resolution of $4096\\times 4096$ pixels and a cadence of 1224 seconds.", "The instrumental PSF is known to contain a diffraction pattern of two crosses originating from spectral filters, and the theoretical PSF weights in the diffraction pattern have been confirmed by studies of flares, i.e., strong localized energy outbreaks in the solar atmosphere which act as strong point sources.", "Nevertheless, in solar eclipse images, a small number of counts is still measured within the occulted area even after deconvolving the image with the instrumental PSF.", "Therefore, we assume that the weights in the PSF tail, responsible for long-distance scattered light, are underestimated.", "To revise the instrumental PSF, we use an eclipse image in the 193 filter taken on 15 May 2012, where the eclipse serves as external occulter, and rebin the image to a resolution of $1024\\times 1024$ pixels.", "We discretize the PSF into 40 shells, fit the missing PSF weights using Equation REF , and average the results of 10 fit repetitions.", "The result is presented in Figure REF (o), showing the original diffraction pattern superposed with the newly fitted smoothed shells.", "In Figure REF (p), we plot for the instrumental PSF and our revised PSF the shell-averaged weights of the PSF versus their distance from the PSF center.", "In both PSFs, the diffraction pattern is clearly visible as peaks in the PSF weights.", "However, in the revised PSF, the PSF weights in the PSF tail outside of the diffraction pattern are significantly larger.", "In the original image, the average intensity of the solar image was 71DNs, and the average counts in the eclipse region was 1.3DNs.", "Deconvolving the image with the PSF provided by the instrument team increases the average intensity of the solar image to 72DNs and reduces the counts in the eclipse region to 1.1DNs.", "Deconvolving the image with our revised PSF increases the counts of the solar image to 73DNs and diminishes the counts in the eclipse region to 0.18DNs." ], [ "Summary", "We have presented a semi-empirical semi-blind algorithm that enables one to determine accurately the instrumental PSF of an imaging system from partially occulted images.", "Our algorithm converges towards the true PSF solution both in the core and the tail of the PSF, is easy to implement, noise resistant, and does not require a point source.", "Furthermore, the method enables one to fit an arbitrary PSF independent of the functional relationship of the PSF, and is only dependent on the initial segmentation chosen for the PSF.", "Our algorithm combines both advantages of empirical algorithms and blind-deconvolution algorithms.", "Similar to blind-deconvolution algorithms, the utilization of entire images that are partially occulted enables one to derive the PSF in an automated manner.", "Using entire images further boosts the total number of photons available, which enables one to fit both the PSF core and the PSF tail simultaneously and accurately.", "Similar to empirical algorithms, we utilize the information on where the true image is zero.", "In our algorithm, this enables the algorithm to converge to the true solution.", "We have tested our algorithm on five numerical use cases: a cylindrically symmetric PSF, a cylindrically symmetric PSF where the functional form of the PSF was provided in advance and only free parameters in the functional form had to be fit, an elliptical PSF, a PSF containing a diffraction pattern, and the PSF of a real imager onboard a satellite.", "For these studies, we have used occultation masks that contain a large hole and multiple pinholes to provide the partial occultation for the first four cases and a solar eclipse as external occultation for the latter case.", "Typical runtimes of our algorithm for images containing $1024\\times 1024$  pixels are 15seconds to 3minutes per iteration, depending on the numbers of parameters to fit, with typically less than five iterations required for convergence.", "The MAPE of the fitted to the true PSF coefficients was less than 5 for all test cases.", "Our algorithm works with any type of partially occulted or partially illuminated images.", "In addition to calibrating instruments in laboratory settings, another strength of our algorithm is in calibrating instruments for which calibration exposures are taken on a regular basis (e.g., for medical X-ray imagers), or for imagers which only need a one-time calibration and which take partially occulted images by chance.", "These latter include imagers on satellites, where the field of view might be partially blocked by debris or other satellites when looking down to Earth, or by solar eclipses when looking towards the Sun.", "One can also use images of stars and distant galaxies.", "Although their field of view is not externally occulted, the intensity is a priori known to be negligible for every pixel not containing a star.", "This makes our algorithm versatile for accurately determining the PSF of imagers in many diverse situations." ] ]
2207.03487
[ [ "Engines for predictive work extraction from memoryful quantum stochastic\n processes" ], [ "Abstract Quantum information-processing techniques enable work extraction from a system's inherently quantum features, in addition to the classical free energy it contains.", "Meanwhile, the science of computational mechanics affords tools for the predictive modeling of non-Markovian classical and quantum stochastic processes.", "We combine tools from these two sciences to develop a technique for predictive work extraction from non-Markovian stochastic processes with quantum outputs.", "We demonstrate that this technique can extract more work than non-predictive quantum work extraction protocols, on one hand, and predictive work extraction without quantum information processing, on the other.", "We discover a phase transition in the efficacy of memory for work extraction from quantum processes, which is without classical precedent.", "Our work opens up the prospect of machines that harness environmental free energy in an essentially quantum, essentially time-varying form." ], [ "Decomposition of free energy in quantum patterns", "Consider a finite portion of the quantum pattern: $\\rho ^{(1:L)} = \\mathrm {Tr}_{\\mathbb {Z}\\setminus \\lbrace \\ell \\rbrace _1^L }(\\rho _{\\dots A_{-1} A_0 A_1 \\dots }) ~.$ If each subsystem is non-interacting and the $\\ell ^\\text{th}$ subsystem has a reference equilibrium Gibbs state $\\gamma ^{(\\ell )}$ , then the nonequilibrium free energy for this portion of the pattern is given by $\\mathcal {F}^{(1:L)}&=F_\\text{eq}^{(1:L)}+ k_\\text{B}T \\text{D} \\Bigl [ \\rho ^{(1:L)} \\Vert \\bigotimes _{\\ell = 1}^L \\gamma ^{(\\ell )} \\Bigr ] \\\\&=F_\\text{eq}^{(1:L)}+ k_\\text{B}T \\bigl [ \\mathrm {Tr}(\\rho ^{(1:L)} \\ln \\rho ^{(1:L)}) - \\sum _{\\ell =1}^L \\mathrm {Tr}(\\rho ^{(\\ell )} \\ln \\gamma ^{(\\ell )}) \\bigr ]\\\\&=F_\\text{eq}^{(1:L)} +k_\\text{B}T \\underbrace{ \\text{D} [ \\rho ^{(1:L)} \\Vert \\bigotimes _{\\ell =1}^L \\rho ^{(\\ell )} ]}_\\text{total correlation}+ k_\\text{B}T \\sum _{\\ell =1}^L \\text{D} [ \\rho ^{(\\ell )} \\Vert \\gamma ^{(\\ell )} ]~,$ where $F_\\text{eq}^{(1:L)}$ is the equilibrium free energy.", "We recognize that $\\text{D} [ \\rho ^{(1:L)} \\Vert \\bigotimes _{\\ell =1}^L \\rho ^{(\\ell )} ]$ is the total correlation within the quantum pattern, while $\\text{D} [ \\rho ^{(\\ell )} \\Vert \\gamma ^{(\\ell )}] $ is the local nonequilibrium addition to free energy.", "Each of these factors contributes uniquely to the free energy.", "When operating sequentially on each subsystem, quantum pattern engines must leverage past information to harvest the free energy in the correlations.", "In the main text, we suppose each non-interacting subsystem has the same local Hamiltonian, which implies that the reference Gibbs states are also all the same: $\\gamma ^{(\\ell )} = \\gamma $ .", "The general decomposition here shows that both quantum and classical correlations contribute to the extractable nonequilium addition to free energy.", "However, in the main text, the quantum pattern is assumed to be classically-generated, despite having non-orthogonal states.", "The inter-time quantum correlations are thus restricted to quantum discord, with no inter-time entanglement [22]." ], [ "Synchronizing to a memoryfull quantum source", "Inferring the latent state of a known memoryfull quantum source allows maximal work extraction when operating serially on the quantum states of the process.", "The optimal state of knowledge, given a sequence of observations $o_1 o_2 \\dots o_t$ obtained via interventions on the sequence of quantum systems $\\sigma ^{(x_1)}, \\sigma ^{(x_2)}, \\dots \\sigma ^{(x_t)}$ is the conditional probability distribution induced by these interventions, ${\\mathbf {}}_t := \\Pr (S_t | O_1 \\dots O_t = o_1 \\dots o_t, S_0 \\sim {\\mathbf {}})$ .", "The last condition $S_0 \\sim {\\mathbf {}}$ means that the initial latent state of the generator is distributed as ${\\mathbf {}}$ .", "Recall that ${\\mathbf {}}= {\\mathbf {}}\\sum _{x \\in \\mathcal {X}} T^{(x)}$ is the stationary distribution over the states of the generator.", "Thus, ${\\mathbf {}}_0 = {\\mathbf {}}$ , and the optimal state of knowledge is seen to be recursive: ${\\mathbf {}}_t& := \\Pr (S_t | O_1 \\dots O_t = o_1 \\dots o_t, S_0 \\sim {\\mathbf {}}) \\\\& =\\Pr (S_t | O_t = o_t, S_{t-1} \\sim {\\mathbf {}}_{t-1} ) ~.$ Figure: Bayesian network showing the structure of conditional independencies among latent states S t S_t of the quantum source, the type X t X_t of quantum state produced, the observable O t O_t attained from interaction, and the state of knowledge K t K_t that influences the work extraction protocol.Further manipulation, using the rules of probability and the conditional independencies indicated in the Bayesian network depicted in Fig.", "REF , allow us to express the optimal state of knowledge in terms of both conditional work distributions and simple linear algebraic manipulations of the generative HMM representing the memoryfull source.", "We find ${\\mathbf {}}_t& =\\Pr (S_t | O_t = o_t, S_{t-1} \\sim {\\mathbf {}}_{t-1} ) \\\\& =\\sum _{x \\in \\mathcal {X} }\\Pr (S_t , X_t = x | O_t = o_t, S_{t-1} \\sim {\\mathbf {}}_{t-1} ) \\\\& =\\sum _{x \\in \\mathcal {X} }\\Pr ( X_t = x | O_t = o_t, S_{t-1} \\sim {\\mathbf {}}_{t-1} )\\Pr (S_t | X_t = x , S_{t-1} \\sim {\\mathbf {}}_{t-1} ) \\\\& =\\sum _{x \\in \\mathcal {X} }\\Pr ( X_t = x | O_t = o_t, S_{t-1} \\sim {\\mathbf {}}_{t-1} )\\frac{{\\mathbf {}}_{t-1} T^{(x)}}{{\\mathbf {}}_{t-1} T^{(x)}\\mathbf {1}} \\\\& =\\frac{\\sum _{x \\in \\mathcal {X} }\\Pr ( X_t = x , O_t = o_t | S_{t-1} \\sim {\\mathbf {}}_{t-1} ){\\mathbf {}}_{t-1} T^{(x)} / {\\mathbf {}}_{t-1} T^{(x)}\\mathbf {1} }{\\sum _{x^{\\prime } \\in \\mathcal {X} }\\Pr ( X_t = x^{\\prime } , O_t = o_t | S_{t-1} \\sim {\\mathbf {}}_{t-1} )} \\\\& =\\frac{\\sum _{x \\in \\mathcal {X} }\\Pr ( O_t = o_t | X_t = x, S_{t-1} \\sim {\\mathbf {}}_{t-1} ) \\, {\\mathbf {}}_{t-1} T^{(x)} }{\\sum _{x^{\\prime } \\in \\mathcal {X} }\\Pr ( O_t = o_t | X_t = x^{\\prime }, S_{t-1} \\sim {\\mathbf {}}_{t-1} ) \\, {\\mathbf {}}_{t-1} T^{(x^{\\prime })} \\mathbf {1}}~.$ If we introduce a new random variable $K_t$ to denote the optimally updated state of knowledge about the latent state of the pattern generator, then we can replace the condition $S_{t-1} \\sim {\\mathbf {}}_{t-1}$ with $K_{t-1} = {\\mathbf {}}_{t-1}$ .", "The condition on the state of knowledge is relevant to the extent that the choice of POVM is influenced by the state of knowledge." ], [ "Using this to build a predictive work-extraction engine", "Rather than repeatedly calculating these ideal belief states on the fly for a specific realization of the process, we can alternatively systematically build up the set of all such belief states, together with the observation-induced transitions among them, to inform the design of an autonomous engine.", "There will be both a set of transient belief states and a set of recurrent belief states.", "Both of these sets may be either finite or infinite.", "In the case that only finitely many belief states are induced by observations, we can explicitly build out the transition structure among them.", "If there are infinitely many such states, then we would need to truncate unlikely states in the design of our finite physical engine [26].", "The physical memory system of our proposed engine should have at least one distinguishable state corresponding to every observation-induced belief state.", "In fact, the memory must encode both the belief state and the most recent energy of the battery, so that conditioning on the new state of the battery is sufficient to supply the change in battery energy.", "These will likely be encoded with some finite precision, to avoid storing real numbers.", "Conditioned on the state of the memory encoding ${\\mathbf {}}$ , the work extraction protocol will operate jointly on the quantum system, thermal reservoirs, and battery, to optimally extract work from the expected state $\\xi = \\sum _{x \\in \\mathcal {X}} {\\mathbf {}}T^{(x)} \\mathbf {1} \\sigma ^{(x)}$ .", "The subsequently observed work value $w$ uniquely updates the memory from the state encoding ${\\mathbf {}}$ to the state encoding ${\\mathbf {}}^{\\prime } = \\frac{\\sum _{x \\in \\mathcal {X} }\\Pr ( W_t = w | X_t = x, S_{t-1} \\sim {\\mathbf {}}) \\, {\\mathbf {}}T^{(x)} }{\\sum _{x^{\\prime } \\in \\mathcal {X} }\\Pr ( W_t = w | X_t = x^{\\prime }, S_{t-1} \\sim {\\mathbf {}}) \\, {\\mathbf {}}T^{(x^{\\prime })} \\mathbf {1}}$ .", "Once the next quantum system arrives, the predictive quantum work extraction cycle begins again." ], [ "Proof of Thm. 1: Work extraction, in the limit of zero entropy production", "Work extraction in the limit of zero entropy production is important since it extracts all extractable work from a quantum state.", "It thus indicates the best possible scenario, against which other efforts can be compared.", "In the limit of zero-entropy-production work extraction from $\\rho ^*$ , the net unitary time evolution of the system–battery–baths supersystem must take a special form.", "In particular, the state of the battery will change deterministically when the initial state of the system is an eigenstate $\\mathinner {|{\\lambda }\\rangle }$ of $\\rho ^* = \\sum _\\lambda \\lambda \\mathinner {|{\\lambda }\\rangle } \\mathinner {\\langle {\\lambda }|}$ , almost-surely independent of the initial realization of the reservoirs.", "This implies that the net unitary time evolution will be of the form $U = \\sum _{\\varepsilon , \\lambda , r}\\Bigl (\\mathinner {|{\\varepsilon + w^{(\\lambda )}}\\rangle } \\otimes \\mathinner {|{f_\\varepsilon (\\lambda , r)}\\rangle }\\Bigr ) \\Bigl (\\mathinner {\\langle {\\varepsilon }|} \\otimes \\mathinner {\\langle {\\lambda }|} \\otimes \\mathinner {\\langle {r}|}\\Bigr )$ for some $w^{(\\lambda )} \\in \\mathbb {R}$ .", "Above, $\\mathinner {|{\\varepsilon }\\rangle }$ and $\\mathinner {|{\\varepsilon + w^{(\\lambda )} }\\rangle }$ are energy eigenstates of the work reservoir, while $\\mathinner {|{r}\\rangle }$ is an energy eigenstate of the thermal baths.", "It will be useful in the following to note that $\\mathinner {\\langle {f_\\varepsilon (\\lambda , r) | f_\\varepsilon (\\lambda ^{\\prime }, r^{\\prime })}\\rangle } = \\delta _{\\lambda , \\lambda ^{\\prime }} \\delta _{r,r^{\\prime }}$ since unitary operations map orthogonal states to orthogonal states.", "The form of the unitary Eq.", "(REF ) effectively assumes that the energy of the battery is well above its ground state.", "Some interesting nuances have recently been explored for batteries close to their ground state (see, e.g., Ref.", "[32]), which would affect the statistics of the work-extraction values, but we avoid that regime here to instead focus on the best-possible scenario.", "One way to determine $w^{(\\lambda )}$ is via the initial-state dependence of entropy production.", "Let $\\mathinner {\\langle {\\Sigma }\\rangle }_\\rho $ denote the expectation value for entropy production, given initial system-state $\\rho $ , under the fixed work-extraction protocol optimized for $\\rho ^*$ .", "Since all initial states map to $\\gamma $ by the end of the work-extraction protocol, we know from Ref.", "[30] that $\\mathinner {\\langle {\\Sigma }\\rangle }_{\\sigma } - \\mathinner {\\langle {\\Sigma }\\rangle }_{\\rho ^*} = k_\\text{B}\\text{D}[ \\sigma \\Vert \\rho ^*] ~.$ In this case, $\\mathinner {\\langle {\\Sigma }\\rangle }_{\\rho ^*} = 0$ and $T \\mathinner {\\langle {\\Sigma }\\rangle }_{\\sigma } = \\mathinner {\\langle {\\tilde{W}}\\rangle }_\\sigma - \\Delta \\mathcal {F}_t = \\mathinner {\\langle {\\tilde{W}}\\rangle }_\\sigma + k_\\text{B}T \\text{D}[\\sigma \\Vert \\gamma ]$ , where $\\mathcal {F}_t$ is the nonequilibrium free energy at time $t$ , and $\\tilde{W}$ is the work exerted, which is just the negative of the extractable work.", "Hence, with $\\beta =(k_\\text{B}T)^{-1}$ , $\\beta \\mathinner {\\langle {\\tilde{W}}\\rangle }_\\sigma &= \\text{D}[ \\sigma \\Vert \\rho ^*] - \\text{D}[ \\sigma \\Vert \\gamma ] \\\\&= \\mathrm {Tr}(\\sigma \\ln \\gamma ) - \\mathrm {Tr}(\\sigma \\ln \\rho ^*) ~.$ In particular, let $\\sigma = \\mathinner {|{\\lambda }\\rangle } \\mathinner {\\langle {\\lambda }|}$ , and note that $\\ln \\gamma = \\ln (e^{-\\beta H} / Z) = \\beta (F-H)$ .", "This yields $\\mathinner {\\langle {\\tilde{W}}\\rangle }_{\\mathinner {|{\\lambda }\\rangle } \\mathinner {\\langle {\\lambda }|}} = F - \\mathinner {\\langle {\\lambda | H | \\lambda }\\rangle } - k_\\text{B}T \\ln \\lambda $ .", "The deterministic work-extraction value, given initial pure state $\\mathinner {|{\\lambda }\\rangle }$ , must be the same as its expected value $w^{(\\lambda )}= -\\mathinner {\\langle {\\tilde{W}}\\rangle }_{\\mathinner {|{\\lambda }\\rangle } \\mathinner {\\langle {\\lambda }|}} $ , and is thus given by $w^{(\\lambda )}= \\mathinner {\\langle {\\lambda | H | \\lambda }\\rangle } + k_\\text{B}T \\ln \\lambda - F ~.$ The probability of obtaining the work-extraction value $w$ , given any input state $\\sigma = \\sum _{\\lambda , \\lambda ^{\\prime }} \\mathinner {|{\\lambda }\\rangle } \\mathinner {\\langle {\\lambda ^{\\prime }}|} \\mathinner {\\langle {\\lambda | \\sigma | \\lambda ^{\\prime } }\\rangle }$ , can be calculated as $\\Pr (W = w | \\sigma )&=\\mathrm {Tr}\\Bigl [ \\bigl ( \\mathinner {|{\\varepsilon _0 + w}\\rangle } \\mathinner {\\langle {\\varepsilon _0 + w}|} \\otimes I \\bigr ) U \\bigl ( \\mathinner {|{\\varepsilon _0}\\rangle } \\mathinner {\\langle {\\varepsilon _0}|} \\otimes \\sigma \\otimes \\mathinner {|{r}\\rangle } \\mathinner {\\langle {r}|} \\bigr ) U^\\dagger \\Bigr ] \\\\&= \\sum _{\\lambda , \\lambda ^{\\prime }}\\mathinner {\\langle {\\lambda | \\sigma | \\lambda ^{\\prime } }\\rangle }\\mathrm {Tr}\\Bigl [ \\bigl ( \\mathinner {|{\\varepsilon _0 + w}\\rangle } \\mathinner {\\langle {\\varepsilon _0 + w}|} \\otimes I \\bigr ) U \\bigl ( \\mathinner {|{\\varepsilon _0}\\rangle } \\mathinner {\\langle {\\varepsilon _0}|} \\otimes \\mathinner {|{\\lambda }\\rangle } \\mathinner {\\langle { \\lambda ^{\\prime } }|} \\otimes \\mathinner {|{r}\\rangle } \\mathinner {\\langle {r}|} \\bigr ) U^\\dagger \\Bigr ] \\\\&= \\sum _{\\lambda , \\lambda ^{\\prime }}\\mathinner {\\langle {\\lambda | \\sigma | \\lambda ^{\\prime } }\\rangle }\\mathrm {Tr}\\Bigl [ \\bigl ( \\mathinner {|{\\varepsilon _0 + w}\\rangle } \\mathinner {\\langle {\\varepsilon _0 + w}|} \\otimes I \\bigr ) \\bigl ( \\mathinner {|{\\varepsilon _0 + w^{(\\lambda )}}\\rangle } \\otimes \\mathinner {|{f_{\\varepsilon _0}(\\lambda , r)}\\rangle } \\bigr ) \\bigl ( \\mathinner {\\langle {\\varepsilon _0 + w^{(\\lambda ^{\\prime })}}|} \\otimes \\mathinner {\\langle {f_{\\varepsilon _0}(\\lambda ^{\\prime }, r)}|} \\bigr ) \\Bigr ] \\\\&= \\sum _{\\lambda , \\lambda ^{\\prime }}\\mathinner {\\langle {\\lambda | \\sigma | \\lambda ^{\\prime } }\\rangle }\\delta _{w, w^{(\\lambda )}}\\delta _{w, w^{(\\lambda ^{\\prime })}}\\mathinner {\\langle {f_{\\varepsilon _0}(\\lambda ^{\\prime }, r) | f_{\\varepsilon _0}(\\lambda , r) }\\rangle } \\\\&= \\sum _{\\lambda , \\lambda ^{\\prime }}\\mathinner {\\langle {\\lambda | \\sigma | \\lambda ^{\\prime } }\\rangle }\\delta _{w, w^{(\\lambda )}}\\delta _{w, w^{(\\lambda ^{\\prime })}}\\delta _{\\lambda , \\lambda ^{\\prime }} \\\\&= \\sum _{\\lambda }\\mathinner {\\langle {\\lambda | \\sigma | \\lambda }\\rangle }\\delta _{w, w^{(\\lambda )}}$ independent of the initial energy state of the battery $\\mathinner {|{\\varepsilon _0}\\rangle }$ , and almost-surely independent of the initial realization $\\mathinner {|{r}\\rangle }$ of the thermal reservoirs.", "We see that $\\Pr (W = w | \\sigma ) = 0$ unless $w \\in \\lbrace w^{(\\lambda )}\\rbrace _\\lambda $ .", "The probability distribution over these allowed work-extraction values is $\\Pr (W = w^{(\\lambda )}| \\sigma )&= \\sum _{\\lambda ^{\\prime }}\\mathinner {\\langle {\\lambda ^{\\prime } | \\sigma | \\lambda ^{\\prime } }\\rangle }\\delta _{w^{(\\lambda )}, w^{(\\lambda ^{\\prime })}} ~.$ When there is some entropy production, the probability density of work extraction will have more diffuse peaks.", "However, for sufficiently low entropy production, the peaks will still be well separated and, so, effectively discrete for the purpose of Bayesian updating.", "In case $\\rho ^*$ has degenerate eigenvalues, the above sums should be taken over the eigenstates, rather than the eigenvalues.", "In particular, we can rewrite the spectral decomposition as $\\rho ^* = \\sum _{n} \\lambda _n \\mathinner {|{n}\\rangle } \\mathinner {\\langle {n}|}$ .", "With a slight change in notation, the allowed work-extraction values can then be written as $w^{(n)} = \\mathinner {\\langle {n| H | n}\\rangle } + k_\\text{B}T \\ln \\lambda _n - F ~,$ which occur with probability $\\Pr (W = w^{(n)} | \\sigma )&= \\sum _{m}\\mathinner {\\langle {m | \\sigma | m }\\rangle }\\delta _{w^{(n)}, w^{(m)}} ~.$ In the main text, we've avoided this notation so that eigenstates $\\mathinner {|{n}\\rangle }$ and $\\mathinner {|{m}\\rangle }$ are not confused with computational-basis states." ], [ "The work extraction protocol of Skrzypczyk ", "For self containment, here we give a brief summary of the Skrzypczyk et al.", "work-extraction protocol [11], utilized in our numerical simulations.", "Ref.", "[11] should be consulted for further details.", "The work extraction protocol of Ref.", "[11], proceeds in two stages.", "The first stage (isentropically) puts the system into a mixture of energy eigenstates, using the battery to offset internal energy changes.", "In the second stage, the system is coupled with the ambient heat bath to slowly (in a sequence of $N$ steps) relax it into a thermal state, again using the battery to absorb all energy offsets.", "Consider a system and a weight initially in an uncoupled state $\\rho ^* \\otimes \\rho _w$ .", "Let $\\rho ^*=\\sum _n \\lambda _n\\mathinner {|{\\lambda _n}\\rangle }\\mathinner {\\langle {\\lambda _n}|}$ be a spectral decomposition of the system state $\\rho ^*$ , with $\\lambda _n \\ge \\lambda _{n+1}$ .", "Step 1 of the work extraction protocol maps the eigenvectors $\\mathinner {|{\\lambda _n}\\rangle }$ into the energy eigenstates of the system, via the unitary operator $V=\\sum _n \\mathinner {|{E_n}\\rangle }\\mathinner {\\langle {\\lambda _n}|}\\otimes \\Gamma _{h_n},$ where $h_n=\\mathinner {\\langle {\\lambda _n}|}H_s\\mathinner {|{\\lambda _n}\\rangle }-E_n$ accounts for the difference in energy in the two states and $\\Gamma $ is an operator to raise the potential energy of the weight.", "$V$ therefore conserves energy on average.", "The resultant state will be $\\rho _{\\text{SW}}=\\sum _n \\lambda _n \\mathinner {|{E_n}\\rangle }\\mathinner {\\langle {E_n}|}\\otimes \\Gamma _{h_n}\\rho _w\\Gamma _{h_n}^\\dag $ In Step 2, the resultant state $\\rho _{SW}$ will go through a sequence of transformation to reach the Gibbs state, $\\gamma =\\sum _n e^{-\\beta E_n}/Z \\mathinner {|{E_n}\\rangle }\\mathinner {\\langle {E_n}|}$ .", "At each sub-step of the transformation, the relative probabilities of two levels will be adjusted by $\\delta p$ towards the Gibbs distribution over energy eigenstates.", "Suppose we now focus on the occupation probability of the ground state and first excited state, $\\mathinner {|{E_0}\\rangle }$ and $\\mathinner {|{E_1}\\rangle }$ .", "In the first sub-step of thermalization, the system will interact with a bath qubit, $\\rho _B=\\frac{q_0}{q_0+q_1}\\mathinner {|{0}\\rangle }\\mathinner {\\langle {0}|}+\\frac{q_1}{q_0+q_1}\\mathinner {|{1}\\rangle }\\mathinner {\\langle {1}|}$ , where $q_0=\\lambda _0-\\delta p$ and $q_1=\\lambda _1+\\delta p$ .", "A unitary transformation is then used to swap the occupation statistics of the bath qubit and the system qubit; in doing so, the battery's energy level rises or drops to conserve the total energy.", "The transformation can be described as $\\mathinner {|{E_0}\\rangle }_S\\mathinner {|{1}\\rangle }_B\\mathinner {|{x}\\rangle }_W \\longleftrightarrow \\mathinner {|{E_1}\\rangle }_S\\mathinner {|{0}\\rangle }_B\\mathinner {|{x+h}\\rangle }_W$ where $h=k_BT\\log {\\frac{q_0}{q_1}}-(E_1-E_0)$ .", "The process then repeats itself by varying the value of $q_0$ and $q_1$ until $\\rho _{SW}$ reaches a Gibbs state.", "It is provable that this protocol is able to extract all the free energy of the system up to $\\mathcal {O}(\\delta p^2)$ ." ], [ "Simulation", "Here we elaborate on the the method of simulations.", "We considered a string of output with length $n=5000$ produced by the the perturbed-coin process.", "The possible emissions are $\\sigma ^{(0)}=\\mathinner {|{0}\\rangle }\\mathinner {\\langle {0}|}$ and $\\sigma ^{(1)}=\\mathinner {|{\\psi }\\rangle }\\mathinner {\\langle {\\psi }|}$ where $\\mathinner {|{\\psi }\\rangle }=\\sqrt{r}\\mathinner {|{0}\\rangle }+\\sqrt{1-r}\\mathinner {|{1}\\rangle }$ .", "Here we considered two pure states—this however is not necessary: mixed states can be used too.", "For the thermalization, the number of SWAP operations (between system and tailored baths) was chosen to be $N=200$ due to limitation in computational power.", "One can refer to Fig.", "REF to see that as $N\\rightarrow \\infty $ the protocol indeed extracts all the free energy from the system.", "After the extraction, the change in the battery system is measured and recorded.", "As mentioned in a previous section, if $N\\rightarrow \\infty $ , the work distribution will converge to a set of $\\delta $ -functions.", "If $N$ is finite, the probability distribution of the work measured becomes more diffuse, as shown in Fig.", "REF .", "The state of knowledge is updated via Bayesian inference, conditioned on the observed work value.", "This inference step is notably absent from the memoryless approach, where the state of knowledge effectively remains at the initial state of ignorance ${\\mathbf {}}_t={\\mathbf {}}$ .", "For the classical protocol, we assume that energetic coherences are inaccessible.", "In this case, our simulations utilize the Skrzypczyk work-extraction protocol tailored to the decohered state $\\rho _t^* = \\xi _t^\\text{dec}$ .", "However, initial decoherence in any other basis, besides $\\xi _t$ 's eigenbasis, would likewise underperform compared to the memory-assisted quantum protocol.", "The overcommitment protocol differs from the rest as it tailors the extraction protocol for the state with the highest probability of emission.", "The probability of emission can be calculated from the state of knowledge ${\\mathbf {}}_t T^{(x)}{\\mathbf {1}}$ .", "The graphs in Panels (c) and (d) of Fig REF display only positive work-extraction values on the vertical axis; hence most of the data points for overcommitment are not shown owing to its bad performance." ], [ "Expected work extraction from the four approaches", "Here, we derive analytical expressions for the expectation value of work extraction from the various approaches compared in the main text.", "We derive these expressions for the ($p$ , $r$ )-parametrized family of perturbed-coin processes of classically correlated quantum states discussed in the main text.", "Recall that the quantum states $(\\sigma ^{(x)} )_{x\\in \\mathcal {X} }$ are the outputs of a Mealy HMM with labeled transition matrices $( T^{(x)} )_{x \\in \\mathcal {X}}$ .", "An element of the labeled transition matrix $T_{s \\rightarrow s^{\\prime }}^{(x)} = \\Pr (X_t = x, S_{t} = s^{\\prime } | S_{t-1} = s)$ gives the joint probability of producing quantum state $\\sigma ^{(x)}$ and arriving at latent state $s^{\\prime }$ , given that the HMM begins in state $s$ .", "For the perturbed-coin example, the HMM's labeled transition matrices are $T^{(0)} =\\begin{bmatrix}1-p & 0 \\\\p & 0\\end{bmatrix}\\qquad \\text{and }\\quad T^{(1)} =\\begin{bmatrix}0 & p \\\\0 & 1-p\\end{bmatrix} ~.$ The stationary distribution over the latent states is ${\\mathbf {}}=\\left[\\frac{1}{2},\\frac{1}{2}\\right]$ and the two different quantum states created are $\\sigma ^{(0)} = \\mathinner {|{0}\\rangle } \\mathinner {\\langle {0}|} =\\begin{bmatrix}1 & 0 \\\\0 & 0\\end{bmatrix}\\quad \\text{and }\\quad \\sigma ^{(1)} = \\mathinner {|{\\psi }\\rangle } \\mathinner {\\langle {\\psi }|} =\\begin{bmatrix}r & \\sqrt{r (1-r) } \\\\\\sqrt{r (1-r)} & 1 - r\\end{bmatrix} ~.$ For any state of knowledge, ${\\mathbf {}}_t=[\\frac{1}{2}+\\epsilon _t, \\frac{1}{2}-\\epsilon _t]$ parameterized by $\\epsilon _t \\in [-\\frac{1}{2},\\frac{1}{2}]$ , the induced expected state is $\\xi _t&= \\rho ^{(\\epsilon _t)} :=\\sum _x\\begin{bmatrix}\\tfrac{1}{2} + \\epsilon _t & \\tfrac{1}{2} - \\epsilon _t\\end{bmatrix} T^{(x)} \\mathbf {1} \\sigma ^{(x)}=\\tfrac{1}{2}\\begin{bmatrix}1 + r + \\epsilon _t^{\\prime } \\sqrt{1-r} \\,\\, & \\sqrt{r (1-r) } - \\epsilon _t^{\\prime } \\sqrt{r} \\\\\\sqrt{r (1-r) } - \\epsilon _t^{\\prime } \\sqrt{r} \\,\\, & 1 - r - \\epsilon _t^{\\prime } \\sqrt{1-r}\\end{bmatrix} ~,$ where $\\epsilon ^{\\prime } :=2 \\epsilon (1-2p) \\sqrt{1-r}\\; \\propto \\epsilon $ ." ], [ "Memoryassisted quantum processing", "In the memory-assisted quantum approach, we utilize work-extraction protocols that are thermodynamically optimized for the expected quantum state $\\rho _t^* = \\xi _t = \\rho ^{(\\epsilon _t)} ~.$ Recall that the eigenvalues and eigenstates of $\\rho _t^*$ play a prominent role in the work-extraction statistics.", "We find that the eigenvalues of $\\rho ^{(\\epsilon )}$ are $\\lambda _{\\pm }^{(\\epsilon )}= \\tfrac{1}{2} \\pm \\tfrac{1}{2} \\sqrt{r + \\epsilon ^{\\prime 2}}~,$ with corresponding eigenstates $\\mathinner {|{\\lambda _{\\pm }^{(\\epsilon )}}\\rangle } =2^{-1/2}\\Bigl [ r + \\epsilon ^{\\prime 2} \\mp \\bigl (r + \\epsilon ^{\\prime } \\sqrt{1-r} \\, \\bigr ) \\sqrt{r + \\epsilon ^{\\prime 2} } \\, \\Bigr ]^{-1/2}\\begin{bmatrix}\\sqrt{r(1-r)} - \\epsilon ^{\\prime } \\sqrt{r} \\\\- r -\\epsilon ^{\\prime } \\sqrt{1-r} \\pm \\sqrt{r+\\epsilon ^{\\prime 2}}\\end{bmatrix}~.$ For all times after $t=0$ , the update rule for belief states simplifies to the following ${\\mathbf {}}_{t+1} \\Bigr |_{{\\xi _t = \\rho ^{(\\epsilon _t)}} \\atop W_{t+1} = w^{(\\lambda _{\\pm }^{(\\epsilon _t)})} }& =\\frac{\\sum _{x \\in \\mathcal {X} }\\mathinner {\\langle { \\lambda _{\\pm }^{(\\epsilon _t)} }|} \\sigma ^{(x)} \\mathinner {|{ \\lambda _{\\pm }^{(\\epsilon _t)}}\\rangle } \\,{\\mathbf {}}_{t} T^{(x)}}{\\lambda _{\\pm }^{(\\epsilon _t)}} ~,$ which can be expressed explicitly in terms of $p$ , $r$ , and $\\epsilon _t$ .", "When $\\epsilon _t=0$ , we find that ${\\mathbf {}}_{t+1}=[\\frac{1}{2},\\frac{1}{2}] = {\\mathbf {}}$ .", "I.e., the stationary distribution is a fixed point for this dynamic over belief states.", "Because of this, we break the initial symmetry by setting $\\epsilon $ to a small non-zero value to obtain useful knowledge.", "In other words, for the very first work-extraction protocol, we choose some $\\rho _0^* \\ne \\xi _0$ to avoid an unstable fixed point of the update rule.", "However, for all subsequent time steps, we choose $\\rho _t^* = \\xi _t$ .", "For the perturbed coin, the metadynamic of the belief state in the long run will yield two different results, depending on which regime the system is in.", "We will dub them as “memory-apathetic regime\" and “memory-advantageous regime\", and their respective metadynamic is shown in Fig.", "REF .", "Figure: Steady-state metadynamic among recurrent states of knowledge for the perturbed-coin process, in the memory-assisted quantum approach.", "Panel (a) represents the memory-apatheticregion of parameter space, where the sole recurrent state is the stationary distribution over latent states.Panel (b) represents the steady-state metadynamic throughout the memory-advantageous region of parameter space.", "The transition probabilities are governed by the eigenvalues of the induced expected state.The reason for this separation comes from the shape of their update function.", "For the memory-apathetic region, the update function has gradient less than unity, making $\\epsilon =0$ an attractor(Fig.", "REF ).", "For the memory-advantageous region, the gradient of the update function exceeds unity, therefore making $\\epsilon =0$ a repellor, at the same time two other points become part of a new attractor (Fig.", "REF ).", "Figure: Panel (a) shows the shape of the update functions in the memory-apathetic regime.", "Panel (b) shows the evolution of ϵ\\epsilon over time.", "Panel (c) shows the work-extraction values over 100 sequential time steps.Figure: Panel (a) shows the shape of the update functions in the memory-advantageous regime.", "Panel (b) shows the evolution of ϵ\\epsilon over time.", "Panel (c) shows the work-extraction values over 100 sequential time steps.In the long run, transient belief states die out, leaving only the steady-state dynamics among the recurrent states of knowledge; any initial distribution over belief states generically converges to the stationary measure $\\pi _{\\mathcal {K}}$ .", "Hence the steady-state rate of work extraction is given by $\\lim _{t \\rightarrow \\infty }\\langle {W_t}\\rangle = k_\\text{B}T \\langle \\text{D} [ \\xi _t \\Vert \\gamma ]\\rangle _{\\Pr (K_t) = \\pi _{\\mathcal {K}}} ~,$ The expected extracted work for the memory-agnostic region coincide with that of memoryless extraction and is given by $\\left\\langle W^{\\text{apathetic}}\\right\\rangle = k_\\text{B}T\\text{D}[\\xi _0 \\Vert \\gamma ]= \\left\\langle W^{\\text{memoryless}} \\right\\rangle ~.$ On the other hand, in the regime where memory enhances the performance of the protocol, the stationary distribution over the two recurrent belief states ${\\mathbf {}}$ and ${\\mathbf {}}^{\\prime }$ is $\\pi _{\\mathcal {K}} =\\frac{1}{\\lambda ^{({\\mathbf {}})}_++\\lambda ^{({\\mathbf {}}^{\\prime })}_+} [\\lambda ^{({\\mathbf {}}^{\\prime })}_+,\\lambda ^{({\\mathbf {}})}_+] ~,$ where $\\lambda ^{({\\mathbf {}})}_+$ is shorthand for $\\lambda ^{(g({\\mathbf {}}))}_+$ with $g({\\mathbf {}}_t) := {\\mathbf {}}_t\\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}- 1/2 = \\epsilon _t$ .", "Hence, the work extraction rate is given by $\\left\\langle W^{\\text{advantage}} \\right\\rangle =\\frac{k_\\text{B}T }{\\lambda ^{({\\mathbf {}})}_++\\lambda ^{({\\mathbf {}}^{\\prime })}_+} \\Bigl ( \\lambda ^{({\\mathbf {}})}_+\\text{D}[\\rho ^{({\\mathbf {}})} \\Vert \\gamma ]+\\lambda ^{({\\mathbf {}}^{\\prime })}_+\\text{D}[\\rho ^{({\\mathbf {}}^{\\prime })} \\Vert \\gamma ] \\Bigr ) ~.$" ], [ "Classical approach", "The derivation for the memory-assisted classical approach is similar to that of the memory-assisted quantum approach illustrated above.", "However rather than operating on the induced expected state $\\xi _t$ , the classical approach uses work-extraction protocols that are thermodynamically optimized for the decohered state $\\rho _t^* = \\xi _t^{\\text{dec}}=\\frac{1}{2}\\begin{bmatrix}1 + r + \\epsilon _t^{\\prime } \\sqrt{1-r} \\,\\, & 0 \\\\0 \\,\\, & 1 - r - \\epsilon _t^{\\prime } \\sqrt{1-r}\\end{bmatrix} ~.$ The eigenstates of $\\rho _t^*$ are thus $\\mathinner {|{0}\\rangle }$ and $\\mathinner {|{1}\\rangle }$ , independent of time in this case.", "In the classical approach, ${\\mathbf {}}$ is no longer a fixed point of the belief-state update maps.", "The metadynamic of belief in the classical case behaves as a reset processes illustrated in Fig.", "REF .", "Unlike the quantum case with only two recurrent belief states, the classical protocol induces an infinite set of refurrent belief states.", "To construct a finite-state autonomous engine, we could choose to truncate those states within some small $\\delta $ distance from another recurrent state, or truncate belief states with negligible probability, with vanishing work-extraction penalty.", "We find that the work-extraction rate can again be computed by averaging the relative entropy—now between the decohered expected state and thermal state—over all recurrent states of knowledge: $\\langle W_t^\\text{classical} \\rangle =k_\\text{B}T \\langle \\text{D}[\\xi _t^\\text{dec} \\Vert \\gamma ] \\rangle _{\\Pr (K_t^\\text{classical})} ~.$ Figure: Metadynamic of the belief state for the classical protocol.", "Panel (a) shows the exact transitions which resembles that of a reset process.", "Panel (b) shows the approximation used for computer simulation which allows us to get arbitrarily close to the full process in (a)." ], [ "Overcommitment to the most likely outcome", "The “overcommitment” approach used for comparison in the main text bets exclusively on the most likely outcome in $\\lbrace \\sigma ^{(x)} \\rbrace _x$ .", "The expected thermodynamic cost of misaligned expectations during work extraction can be quantified exactly via the relative entropy $\\text{D} [ \\rho _0 \\Vert \\alpha _0 ]$ between the actual input $\\rho _0$ and the anticipated input $\\alpha _0$ that the protocol is optimal for, if we assume that the final state is independent of the initial state [30], [31].", "Hence, if we design the protocol for a pure state, but operate on a mixed state, we will encounter divergent thermodynamic penalties.", "Accordingly, we can observe divergent thermodynamic costs when we design the Skrzypczyk work extraction protocol to be optimal for operation on a pure state.", "Using the Skrzypczyk protocol (with $N$ relaxation steps) to extract work from the pure state bet upon, we see that the first bath state swapped with the system for energy extraction is not exactly pure, but rather satisfies $\\gamma _\\text{B} = (1 - \\frac{e^{-\\beta E_0}}{N(e^{-\\beta E_0} + e^{-\\beta E_1})}) \\mathinner {|{0}\\rangle } \\mathinner {\\langle {0}|}+ \\frac{e^{-\\beta E_1}}{N(e^{-\\beta E_0} + e^{-\\beta E_1})} \\mathinner {|{1}\\rangle } \\mathinner {\\langle {1}|}$ .", "(Recall that $H$ is the Hamiltonian for the system, not of the bath.)", "Any purity of the actual input beyond this initial bath purity is wasted.", "The input state leading to minimal entropy production under this protocol is thus a unitary rotation of $\\gamma _\\text{B}$ .", "Thus, for this use case of the Skrzypczyk protocol, the minimally dissipative state $\\alpha _0$ becomes pure as $N \\rightarrow \\infty $ .", "As $N \\rightarrow \\infty $ , we observe the battery's final expected energy diverging (but only logarithmically in $N$ ) to negative infinity, when this protocol acts on any other state.", "I.e., $\\langle W \\rangle \\sim - k_\\text{B}T \\ln N$ .", "More specifically, we can leverage Eqs.", "(REF ) and (REF ) to calculate the expected value of work for the overcommitment approach.", "We find that $\\Pr (W=w^{(\\lambda _\\text{min} )} | \\sigma ^{(\\text{argmin}_{x} {\\mathbf {}}_t T^{(x)} \\mathbf {1} )}) = |\\mathinner {\\langle {1 | \\psi }\\rangle }|^2 = 1-r ~.$ With $\\lambda _\\text{min} = \\frac{e^{-\\beta E_1}}{N(e^{-\\beta E_0} + e^{-\\beta E_1})}$ , $w^{(\\lambda _\\text{min})} \\sim - k_\\text{B}T \\ln N$ , and $\\text{min}_x {\\mathbf {}}_t T^{(x)} \\mathbf {1} \\sim \\text{min}(p,1-p)$ when ${\\mathbf {}}_t$ is close to either latent state, we anticipate that the overcommited work penalty diverges as $- k_\\text{B}T (1-r) \\min (p, 1-p) \\ln N$ , as observed.", "Interestingly, for a finite number of bath interactions, some work can be extracted on average within certain regimes.", "But other regions of parameter space would yield very negative work-extraction averages.", "Unlike the other approaches, the expectation value of work in the overcommitment approach cannot be written as a relative entropy.", "Hence, whereas the other approaches were guaranteed to have non-negative work extraction on average, the overcommitment approach enjoys no such guarantee of non-negativity.", "Indeed in the limit of many bath interactions, the overcommitment approach leads to infinitely negative work extraction." ], [ "Non-Markovian generators", "Unlike the classical case, the classical control symbols $X_t$ are hidden from direct observation when the process emits non-orthogonal quantum states.", "The latent-state generators may thus be referred to as `doubly-hidden Markov models'.", "Accordingly, even if the intermediary $X_t$ process is Markovian, this would not directly imply a meaningful sense of quantum Markovianity of the outputs.", "Nevertheless, there is some sense in which processes with non-Markovian control outputs $X_t$ have more deeply hidden structure.", "To benchmark the performance of the memory-assisted protocol on a process with higher Markov order of the control symbols $X_t$ , the 2-1 golden-mean process was chosen for comparison.", "The time-averaged density matrix of the memoryless approach for both models is kept the same, $\\xi _0=\\frac{1}{2}(\\sigma ^{(0)}+\\sigma ^{(1)})$ .", "The comparison is shown in Fig.", "REF , where we see that more work is extracted from the non-Markovian generator.", "This example suggests that memory can become even more important for enabling work extraction from non-Markovian generators of quantum processes.", "The reasoning is most transparent in the case of energy-degenerate Hamiltonians, where the maximal extractable work depends directly on the von Neumann entropy rate of each input process: $\\beta ( \\mathcal {F}^{(1:L)} - F_\\text{eq}^{(1:L)} ) / L&= \\ln d- S_\\text{vN} \\bigl ( \\rho ^{(1:L)} \\bigr ) / L\\qquad \\text{if } H \\propto I ~,$ as can be derived from Eq.", "(REF ).", "Here $S_\\text{vN}(\\rho ) = -\\mathrm {Tr}(\\rho \\ln \\rho )$ is the von Neumann entropy while $S_\\text{vN} \\bigl ( \\rho ^{(1:L)} \\bigr ) / L$ is the von Neumann entropy rate.", "Unfortunately, there is no known closed-form expression for the von Neumann entropy rate from classical memoryfull generators of non-orthogonal quantum states.", "Nevertheless, non-Markovianity of the control symbol $X_t$ loosely suggests lower apparent von Neumann entropy rate, and thus more extractable work, once longer lengths $L$ are taken into account.", "Figure: Comparison of average work extracted between 2-1 golden-mean and perturbed-coin processes, varying the nonorthogonality parameter rr.", "Blue and red dots represent the memory-assisted quantum approach on golden mean and perturbed coin respectively; Green line represents the memoryless approach." ] ]
2207.03480
[ [ "Realistic HI scale heights of Milky Way-mass galaxies in the FIREbox\n cosmological volume" ], [ "Abstract Accurately reproducing the thin cold gas discs observed in nearby spiral galaxies has been a long standing issue in cosmological simulations.", "Here, we present measurements of the radially resolved HI scale height in 22 non-interacting Milky Way-mass galaxies from the FIREbox cosmological volume.", "We measure the HI scale heights using five different approaches commonly used in the literature: fitting the vertical volume density distribution with a Gaussian, the distance between maximum and half-maximum of the vertical volume density distribution, a semi-empirical description using the velocity dispersion and the galactic gravitational potential, the analytic assumption of hydrostatic equilibrium, and the distance from the midplane which encloses $\\gtrsim$60 per cent of the HI mass.", "We find median HI scale heights, measured using the vertical volume distribution, that range from ~100 pc in the galactic centres to ~800 pc in the outskirts and are in excellent agreement with recent observational results.", "We speculate that the presence of a realistic multiphase interstellar medium, including cold gas, and realistic stellar feedback are the drivers behind the realistic HI scale heights." ], [ "Introduction", " Producing realistic disc galaxies in cosmological scale simulations has been a long-standing problem for numerical astrophysics .", "Only in the past decade have cosmological zoom-in simulations and volumes been able to reproduce the rotation curves and thin stellar discs seen in late-type spiral galaxies , , [2], , [1], , , , , .", "This success if often attributed to the (stellar) feedback models used, as the thickness of the stellar disc is sensitive to the feedback strength .", "Resolving the gas scale height has been identified as another crucial part of reproducing realistic Milky Way-like, $$ galaxies .", "Observations indicate that the discs of galaxies are very thin, with scale heights of $\\sim $ 100 pc in the centre, flaring to $\\lesssim $ 1 kpc at large galactic radii , , [3], , .", "found average scale heights within the optical radius of the spiral galaxies studied of $\\sim $ 0.35 $\\pm $ 0.16 kpc for the galaxies and $\\sim $ 0.68 $\\pm $ 0.58 kpc for The Nearby Galaxy Survey .", "By contrast, recent cosmological (zoom) simulations which investigated the content of $$ galaxies found the discs to be much thicker.", "scale heights of the Auriga galaxies within the optical radius range from $\\sim $ 1${-} 10~$ .", "Similarly, the EAGLE galaxies have discs with scale heights > 1.5 kpc, which [4] interpreted as consequence of too strong feedback.", "The scale height or thickness of gas discs is relevant for determining their susceptibility to gravitational instability .", "Whether a (super)bubble driven by supernovae (SNe) can break out of the galactic disc and pollute the circumgalactic medium (CGM) with metals, or whether it stalls and deposits all ejecta in the gas disc of the galaxy also depends on the scale height of the cold gas , , .", "Thus the scale height has direct implications for the star formation and feedback cycle of galaxies, as well as the metal pollution of the CGM.", "In turn, feedback will affect the structure [4].", "Therefore, the scale height can be a sensitive test for sub-grid physics, as it is unclear a priori whether well validated sub-grid models like -2 will be able to reproduce observed scale heights.", "Accurately modelling (spiral) galaxies and their gas content is also highly relevant with respect to the upcoming Square Kilometer Array observatory and its precursors.", "Both to make higher fidelity predictions, and to be able to better interpret observational results.", "Encouragingly, thin cold gas discs have been found in several $$ galaxies simulated with the -2 model , , , but no systematic exploration with a focus on the HI has been conducted yet.", "The cosmological volume , with its high spatial resolution ($\\sim $ 20 pc), and multiphase ISM, that yields 20–30 $$ galaxies,is ideally suited to systematically analyse the discs of $$ galaxies.", "In this letter, we focus on their scale heights in particular, measuring them with the variety of different methods used in the literature." ], [ "FIREbox", " In this letter we study galaxies drawn from the (22.1 Mpc)$^3$ box cosmological volume simulation that is part of the Feedback In Realistic Environments () projecthttps://FIRE.northwestern.edu/.The simulation was run with the meshless-finite-mass code Gizmohttp://www.tapir.caltech.edu/~phopkins/Site/GIZMO.html and the -2 sub-grid physics.", "Specifically, these simulations use the cooling and heating rates valid for temperatures ranging from $10 {-} 10^9~$ , including heating and photoionisation from a UV Background, naturally resulting in a multiphase interstellar medium (ISM) that includes a cold gas phase.", "Stars form with a local efficiency of 100 per cent from gas that is molecular, self-gravitating, Jeans unstable and above a density threshold of $n\\ge 300~$ .", "A variety of stellar feedback channels are included, namely SN Type II and Ia, stellar winds from massive OB and evolved AGB stars, as well as photoionization, photoelectric heating and radiation pressure.", "has a mass resolution of $m_{\\rm b}=6.3\\times 10^4~$ ($m_{\\rm DM}=3.3\\times 10^5~$ ), fixed gravitational softening for the stars (12 pc) and dark matter (80 pc), and adaptive gravitational softening for the gas down to a softening length of 1.5 pc (all in proper units).", "Galaxies were identified at $z=0$ using the AMIGA halo finder , .", "We restrict our analysis to centrals in the virial mass range $11.85 < \\log _{10}(/) < 12.48$ , such that the mean of the resultant 27 galaxies matches the Milky Way $= 1.3\\pm 0.1\\times 10^{12}$ from [6].", "To better match the observational samples, we visually inspect the galaxies and exclude those that are interacting or undergoing a merger.", "The stellar masses of our final sample of 22 galaxies range from $3.5\\times 10^{10}$ to $1.6\\times 10^{11}~$ , in good agreement with the mass range of the galaxies and the higher mass THINGS galaxies studied by , while [3] also include some lower mass THINGS galaxies.", "To obtain the mass fraction of each gas particle, we first calculate the $$ fraction of the neutral gas following , then subtract the molecular fraction, as well as contributions from metals and Helium, from the neutral gas fraction of each particle." ], [ "Measuring the HI scale height", " The scale height of real galaxies can only be measured directly for edge-on galaxies, where the vertical gas distribution can be fit with a Gaussian .", "Below, we briefly discuss the different ways the scale height can be measured for galaxies at lower inclinations.", "Assuming that the gas disc is in hydrostatic equilibrium, with turbulent pressure[3] explicitly assume the gas to be vertically isothermal.", "However, it has also been shown in numerical simulations with effective feedback that the turbulent pressure dominates over the thermal pressure [5].", "balancing the effect of gravity, it is possible to derive an analytic estimate of the scale height [3].", "$\\centering (R) \\approx \\sigma _{\\rm HI}(R)/\\sqrt{4\\pi G\\times \\left(\\rho (R,0) - \\frac{1}{2\\pi G}\\frac{v_{\\rm c}(R)}{R}\\frac{\\partial v_{\\rm c}(R)}{\\partial R}\\right)},$ where $$ is the scale height of the disc, $\\sigma _{\\rm HI}$ is the mass-weighted line-of-sight velocity dispersion in the direction perpendicular to the disc plane, $v_{\\rm c}$ is the circular velocity and $\\rho (R,0)$ is the mean mass-weighted density at the galaxy mid-plane, all as a function of galactocentric radius $R$ .", "We calculate all quantities as running mean in overlapping radial bins of 1 kpc width.", "The analytic approximation neglects the contribution of the gas self-gravity to the gravitational potential.", "To include gas self-gravity, and thus determine $$ more accurately, one needs to solve for the volume density numerically.", "It is done by solving the equation of the hydrostatic equilibrium condition under the constraints of the rotation curve and velocity dispersion, iterating until the so derived estimates converge.", "This method has been used in two subtly-different ways in the literature.", "The first approach assumes that the vertical volume density profile has a Gaussian shape that depends on the scale height, i.e.", "$\\rho (z) \\propto \\exp (-z^2/(2^2))$ .", "It was used by e.g.", "[3] to measure the scale heights of 12 THINGS galaxies.", "To mimic this method, we fit the mass-weighted vertical volume density profile of gas with $z\\le 2~$ in radial annuli of 1 kpc width, with a Gaussian function to measure $$ .", "With the half-width half maximum (HWHM) approach, no functional form is assumed for the volume density.", "The scale height is defined as the distance between the midplane (maximum volume density) and one of the points where the volume density reaches its half-maximum.", "It was used by e.g.", "to measure $$ in 9 THINGS galaxies.", "We determine this distance for the mass-weighted vertical volume density profile of gas with $z\\le 2~$ in radial annuli of 1 kpc width.", "Another way to estimate $$ using a semi-empirical formula was originally derived by , assuming hydrostatic equilibrium but taking into account observational results regarding the gravitational potentials of galaxies.", "We follow the parametrisation of , who used this equation to measure $$ for the galaxies.", "The scale height can be expressed as: $\\centering h(R) = \\frac{\\sigma _{\\rm HI}^{2}(R)}{\\pi G \\Sigma _{\\rm total}} \\times \\left({1 + \\sigma ^{2}_{\\rm HI, 0}}/{\\left[2v_{\\rm max}^{2} \\times \\left(\\frac{M_{\\rm dyn}}{M_{\\rm HI} + }\\right)^2\\right]}\\right)^{-1},$ where $\\sigma _{\\rm HI, 0}$ is the central velocity dispersion, $\\Sigma _{\\rm total}$ is the total surface density within the gas layer, $v_{\\rm max}$ is the peak of the galaxy's rotation curve, $M_{\\rm dyn}$ the total (dynamical) mass, $M_{\\rm HI}$ the total mass of and $$ the total stellar mass.", "We calculate all radially dependent quantities in radial annuli of 1 kpc width and the masses within $0.1$ of the galaxy, in analogy to the optical radius used by .", "Finally, measure the scale height as the distance from the midplane where the mass enclosed within a cylinder spanning $\\pm 30~$ from the midplane of the galaxy equals 75 per cent of the mass.", "We calculate the distance from the midplane that encloses 63 per cent (one e-folding) of the mass within $30~$ above and below the midplane in radial annuli of 1 kpc width, to determine how this method of measuring $$ compares to the other approaches used in the (observational) literature." ], [ "Properties of FIREbox galaxies", " Figure: Properties of the 22 box galaxies considered in this letter: velocity dispersion (top left), circular velocity (top right), mass surface density (bottom left) and volume density (bottom right) as a function of radius.", "Thin grey lines show the radial profiles of the individual galaxies.", "The blue line indicates the sample median, and the blue shaded region the 16th-to-84th percentile.", "The dashed navy line in the top left plot indicates a velocity dispersion of 1010~.We show the radial profiles of several galaxy properties (velocity dispersion, surface density, mass-weighted volume density at the midplane and rotation velocity curves), relevant for calculating the scale height either analytically (Equation REF ) or using the semi-empirical equation (REF ), in Figure REF .", "The median $\\sigma _{\\rm HI}$ radial profile peaks at the centre of the galaxy with 40 $$ , declining with increasing galactocentric radius to a value of $\\sim 17~$ for $\\rm R<15$ kpc, where it increases again to $\\sim 27~$ .", "The median consistently lies above the often assumed canonical value of $\\sigma _{\\rm HI}\\sim 10~$ , also seen in the galaxies .", "However the velocity dispersion for the box galaxies is calculated for all gas, including the warm component, for which found an average velocity dispersion of $16.8~$ in good agreement with the median $\\sigma _{\\rm HI}$ at intermediate radii.", "The rise of the median velocity dispersion in the central 5 kpc is indicative of shear-driven turbulence induced by a bulge (see e.g.", "Figure 9 in ).", "The median rotation curve in the top right panel shows the characteristic bump expected from a spheroidal component around $R = 2~$ , supporting this interpretation.", "The velocity dispersion profiles of the THINGS galaxies with bulges studied in [3] and qualitatively match the radial profiles in Figure REF , although the central velocity dispersions of the THINGS galaxies reach $\\sim 30~$ at most.", "The surface density is calculated as the mass enclosed in overlapping radial annuli of width 1 kpc, divided by the area of the annulus.", "The median $\\Sigma _{\\rm HI}$ stays approximately constant at $\\sim $ 7.5 $^{-2}$ out to a galactocentric radius of 8 kpc, before declining to a density $<1^{-2}$ at a median disc radius $R_{\\rm HI}\\approx $ $20^{+9}_{-7}$ kpc, in good agreement with observed $\\Sigma _{\\rm HI}$ profiles , , .", "The median mass-weighted volume density at the midplane peaks at 18 $$ in the galactic centre and declines steeply with increasing galactocentric radius, falling below 1 $$ at a radius of 13 kpc." ], [ "scale height of FIREbox galaxies", " Figure: Radial profiles of the scale heights of the box galaxies estimated from Equation , , a Gaussian fit to the vertical volume density profile, distance from the galaxy mid-plane that encloses 63 per cent of the mass and the vertical distance between the maximum and half of the maximum volume density, from top left to bottom left respectively, with observations from , , and over-plotted, where the same method of estimating the scale height is used.", "Coloured lines indicate the median, the dark shaded regions the error on the median determined via bootstrapping and the light shaded regions the 16th-to-84th percentile variation of the data.", "The bottom right panel combines the median measurements of observations and simulations presented in the other five panels.", "The scale heights measured from the vertical volume density distribution are in excellent agreement with observational measurements.Figure REF shows the scale height of the 22 $$ box galaxies measured using the different approaches outlined in Section .", "The top three and bottom two leftmost panels show the measurements for individual galaxies, with median, errror on the median determined via bootstrapping, and 16th-to-84th percentile variation for the sample indicated by the coloured line and dark and light shaded regions respectively, compared with the observational result using the same method where applicable.", "The bottom right panel shows a compilation of the median measurements from the different methods, simulations and observations.", "All median scale height profile obtained from the simulated galaxies show disc flaring, i.e.", "follow the trend of increasing $$ with increasing galactocentric radius.", "However, the different methods vary by more than an order of magnitude in their estimate of the central $$ , the amount of disc flaring predicted, as well as the radial gradient of the profile.", "Using the vertical profile of the mass-weighted volume density to determine the $$ , either by fitting the gas distribution with a Gaussian, or calculating the HWHM, yields results in excellent agreement with measurements from the THINGS galaxies that use the same methodology [3], .", "The HWHM method finds a lower central scale height compared to the Gaussian fit (60 and 85 pc respectively), and predicts a steeper increase to a scale height of 500 pc at $\\rm R=8~$ , before oscillating and gradually increasing to 900 pc at $\\rm R=25~$ .", "The scale height obtained from the Gaussian fit increases more shallowly with R, reaching 800 pc at $\\rm R=14~$ , staying nearly constant at larger radii (reaching 1 kpc at $\\rm R=25~$ ).", "The analytic determination of $$ agrees well with the volume density estimates and the observations for $\\rm R \\le 15~$ , but predicts too much flaring at larger galactocentric radii, likely driven by the increase of $\\sigma _{\\rm HI}$ beyond this radius.", "Determining the scale height from Equation REF yields a median central scale height of 13 pc, increasing to 100 pc at a galactocentric radius of 12 kpc, and flaring further until reaching a scale height of $\\approx 0.9~$ at $\\rm R>17.5~$ .", "The median of the simulated galaxies lies significantly below the median measurement, that only shows a very modest increase from 200 to 350 pc, for $R<12~$ .", "Due to the large flaring predicted, the median of the simulated galaxies lies within the 16th-to-84th percentile of at larger radii.", "The large discrepancies at small galactocentric radii are a consequence of the strong dependence of $$ on $\\sigma _{\\rm HI, 0}$ in Equation REF , as the central velocity dispersions of the box $$ galaxies far exceed the $\\sim 10~$ of the galaxiesHowever, the observations have coarse resolution and lack $\\sigma $ measurements in their centres, making it difficult to determine how much of the discrepancy between the observations and simulations is due to (potential) morphological differences or strong -2 feedback potentially further enhancing the velocity dispersion.. At large galactocentric radii the strong flaring is driven is driven by the increase in the median $\\sigma _{\\rm HI}$ from 17 to 27 $$ (see the top left panel of Figure REF ).", "Defining the scale height as the distance from the midplane that encloses 63 per cent of the total mass in a hollow cylinder of height 60 kpc systematically overpredicts $$ compared to all other methods of obtaining a scale height, with a median scale height of 350 pc at the galactic centre, increasing to 10 kpc by a galactocentric radius of 25 kpc.", "The total mass enclosed depends on the height of the cylinder, thus using a smaller distance from the midplane would yield lower scale height estimates." ], [ " Why does FIREbox produce realistic HI scale heights?", " We can speculate what makes box so successful at producing the thin discs shown in Figure REF based on possible explanations put forward in the literature.", "In some cases , high resolution has been crucial to reproducing thin (stellar) discs.", "However, as we show in Appendix , using the vertical volume density profile to determine the scale height yields good agreement for simulations differing by nearly two orders of magnitude in spatial resolution (minimum gas softening ranging from 1.5 to 100 pc).", "This suggests that the other modelling choices are more relevant for producing realistic galaxy discs.", "However, the minimum gravitational softening likely needs to be lower than the scale height, to reproduce thin scale heights.", "This is consistent with the results, as the lowest resolution re-run has a minimum gravitational force softening below the smallest measured $$ of $\\sim $ 100 pc.", "suggest that using an effective equation of state instead of directly modelling the multiphase ISM leads to too thick stellar discs in the EAGLE galaxies.", "also report a 0.5 dex variation in scale height when adjusting the parameters of their equation of state model.", "The -2 model includes cooling down to temperatures of 10 K, thus we can self-consistently model a multiphase ISM including cold gas, plausibly a major factor in producing realistic gas scale heights.", "Many authors have argued that the inclusion of more sophisticated feedback models such as superbubble feedback or early stellar feedback are key to produce realistic $$ galaxies (e.g.", ", [2], ).", "This suggests that the early stellar feedback implemented in -2 could be crucial to producing thin gas discs, by pre-processing the ISM gas, thus enabling the effective driving of localised SNe outflows even at coarse mass resolution (see Figure REF ), as opposed to SNe depositing more of their energy and momentum within the ISM, increasing the local pressure and thereby increasing the overall scale height.", "Apart from dedicated new simulations, it might be informative to see whether other simulations with a cold ISM but no early stellar feedback (e.g.", "EMP-Pathfinder; ) and those with early stellar feedback also show thin gas discs." ], [ "Concluding remarks", " We measure the scale height of 22 Milky Way-mass galaxies from the box pathfinder volume, using five different methods.", "We find that the two methods using the vertical volume density distribution (Gaussian fit and HWHM) are in excellent agreement with each other and the observations.", "We speculate that the combination of detailed modelling of the cold ISM and early stellar feedback are responsible for these realistically thin gas discs, a trend that is independent of resolution.", "The analytic hydrostatic equilibrium assumption works well for the central $\\sim 15~$ , but overpredicts the scale height compared to observations and measurements from the vertical volume density distribution at large radii, due to the rising velocity dispersion.", "The equation underpredicts the scale height in the central region of the galaxies, due to the high central velocity dispersion of the galaxies, but is in good agreement with the volume density estimates at large radii.", "Using the mass enclosed as an estimate consistently overpredicts the gas scale height at all radii, although the general trend of disc flaring at large radii is present as well.", "Turbulent theory predicts that the scale height is a transition scale, where turbulence changes from three-dimensional turbulence on small scales to two-dimensional turbulence on larger scales [1].", "This could cause a break in the power spectra of galaxies .", "Since box produces galaxies with realistic scale heights, it is well suited to study turbulence and its drivers and other processes that depend on the scale height.", "Futhermore, this makes box a great resource for more in-depth studies of the content of galaxies in general, and for making predictions for upcoming observatories such as the SKA." ], [ "Acknowledgements", "JG, RF and LM gratefully acknowledge financial support from the Swiss National Science Foundation (grant no CRSII5_193826).", "RF acknowledges financial support from the Swiss National Science Foundation (grant no PP00P2_194814 and 200021_188552).", "AW received support from: NSF via CAREER award AST-2045928 and grant AST-2107772; NASA ATP grant 80NSSC20K0513; HST grants AR-15809, GO-15902, GO-16273 from STScI.", "CAFG was supported by NSF through grants AST-1715216, AST-2108230, and CAREER award AST-1652522; by NASA through grants 17-ATP17-006 7 and 21-ATP21-0036; by STScI through grants HST-AR-16124.001-A and HST-GO-16730.016-A; by CXO through grant TM2-23005X; and by the Research Corporation for Science Advancement through a Cottrell Scholar Award.", "We acknowledge PRACE for awarding us access to MareNostrum at the Barcelona Supercomputing Center (BSC), Spain.", "This work was supported in part by a grant from the Swiss National Supercomputing Centre (CSCS) under project IDs s697 and s698.", "We acknowledge access to Piz Daint at the Swiss National Supercomputing Centre, Switzerland under the University of Zurich's share with the project ID uzh18.", "This work made use of infrastructure services provided by S3IT (www.s3it.uzh.ch), the Service and Support for Science IT team at the University of Zurich.", "The data underlying this article will be shared on reasonable request to the corresponding author." ] ]
2207.03493
[ [ "Inferring Structural Parameters of Low-Surface-Brightness-Galaxies with\n Uncertainty Quantification using Bayesian Neural Networks" ], [ "Abstract Measuring the structural parameters (size, total brightness, light concentration, etc.)", "of galaxies is a significant first step towards a quantitative description of different galaxy populations.", "In this work, we demonstrate that a Bayesian Neural Network (BNN) can be used for the inference, with uncertainty quantification, of such morphological parameters from simulated low-surface-brightness galaxy images.", "Compared to traditional profile-fitting methods, we show that the uncertainties obtained using BNNs are comparable in magnitude, well-calibrated, and the point estimates of the parameters are closer to the true values.", "Our method is also significantly faster, which is very important with the advent of the era of large galaxy surveys and big data in astrophysics." ], [ "Introduction", "Despite their morphological diversity and complexity, the approximate light distribution of galaxies can be well-described by analytic fitting functions with a limited number of free parameters, such as their orientation, size (radius), light concentration, total brightness, etc.", "Measurements of these parameters allow for a quantitative comparison of different galaxy populations and the derivation of empirical scaling relations [4], which in turns facilitates the testing of galaxy formation models.", "Traditionally, these parameters are measured using galaxy profile fitting software (two widely used options being GALFIT; [11] and Imfit; [5]) that performs a $\\chi ^2$ minimization between the chosen analytic model and a given galaxy image, to derive the best-fit parameters.", "Despite their success, these codes have their limitations, too: they are not optimized to fit a large number of galaxies quickly, and they usually require some manual intervention (for example, the selection of good initial model parameters).", "The low speed is an even more significant problem if one wants to obtain accurate estimates of the uncertainties associated with those measurements, for example via bootstrap resampling, or by using a Markov-Chain Monte Carlo (MCMC) approach to sample the parameter posterior distribution.", "Large galaxy surveys, such as the Dark Energy Survey (DES)https://www.darkenergysurvey.org/ and the upcoming Legacy Survey of Space and Time (LSST)https://https://www.lsst.org/ on the Vera C. Rubin Observatory, observe hundreds of millions (the former) to tens of billions (the latter) of galaxies.", "With the advent of these surveys, fast, automated, and reliable methods for measuring the structural parameters of galaxies are needed.", "Deep learning methods are well-suited to tackle this problem since, once trained, they are able to make predictions on new, unseen, examples very quickly.", "Indeed, several works [14], [1], [8] have demonstrated that Convolutional Neural Networks (CNNs), trained on simulated galaxy images are significantly faster, and almost as accurate as the standard profile fitting methods in predicting galaxy parameters.", "However, those works used standard, deterministic, neural networks, that output single-point estimates and thus are unable to quantify the uncertainty associated with their predictions.", "A rigorous uncertainty quantification is imperative for studies of challenging, low signal-to-noise objects, such as low-surface-brightness galaxies (LSBGs).", "LSBGs, defined as galaxies with a central brightness at least a magnitude fainter than that of the ambient dark sky, are challenging to observe and galaxy surveys have only recently started to produce large LSBG catalogs [6], [13], [17], althought they are expected to dominate the galaxy population.", "LSBGs are a target of future surveys, in the quest of understanding the galaxy formation process in a relatively unexplored regime.", "In this work, we explore the use of Bayesien Neural Networks [15] for the problem of LSBG structural parameter estimation with simultaneous uncertainty quantification.", "BNNs output posterior probability distributions instead of point estimates for their predictions, and thus they naturally offer a way to quantify the uncertainties associated with neural network predictions.", "Specifically, we use a simulated dataset of DES-like LSBGs, to train, validate, and test a convolutional BNN model and compare the speed and accuracy of its predictions with those obtained using pyImfitpyImfit is a Python wrapper around Imfit., for a single-component Sérsic light-profile model.", "Our code and simulated data related to this work are available at: https://github.com/dtanoglidis/BNN_LSBGs_ICML." ], [ "Simulated Data", "We use PyImfit to create a simulated dataset of 170,000 LSBG images.", "Each image has dimensions $64\\times 64$ pixels, and it has two components: a uniform background $I(x,y)=I_{\\mbox{\\scriptsize {sky}}}$ , and a Sérsic function [12] that describes the surface-brightness profile of the galaxies: $I_{\\mbox{\\scriptsize {gal}}}(r) = I_e \\exp \\left\\lbrace -b_n\\left[\\left(\\frac{r}{r_e}\\right)^{1/n}-1\\right] \\right\\rbrace ,$ where $r=(x^2+y^2/q^2)^{1/2}$ , $(x,y)$ are the coordinates with origin at the center of the image, and $q$ is the axis ratio.", "The axis ratio is connected to the ellipticity as $q=1-\\epsilon $ .", "The other free parameters of the model are: the effective half-light radius, $r_e$ , the surface brightness at the effective radius, $I_e$ , the Sérsic index, $n$ , that controls the shape of the light distribution, and the position angle that defines the orientation of the galaxy profile.", "As for the value of $b_n$ (not a free parameter of the model), pyImfit uses the approximation by [3].", "We want our simulated images to resemble real LSBG images.", "For that reason we sample parameters uniformly, from a range that roughly corresponds to the bulk of the LSBGs discovered by DES, as described in [13]: Position angle, PA $\\in [0,180]$ degrees, Ellipticity, $\\epsilon \\in [0.05,0.7]$ , Sérsic index, $n \\in [0.5,1.5]$ , Surface brightness, $I_e \\in [24.3,25.5]$ mag/arcsec$^2$ , Effective radius, $r_e \\in [2.5,6.0]$ arcsec.", "Furthermore, we assume a pixel to angular scale conversion 1 pix = 0.263 arcsec (as in DES), and background sky surface brightness $I_{\\mbox{\\scriptsize {sky}}} = 22.23$ mag/arcsec$^2$ [9].", "At each pixel we randomly assign a number of photons (counts) drawn from a Poisson distribution with mean the one predicted from the total surface brightness model $I_{\\mbox{\\scriptsize {tot}}}=I_{\\mbox{\\scriptsize {gal}}}+I_{\\mbox{\\scriptsize {sky}}}$ .", "In Fig.", "REF we present a small subset of the simulated galaxy images." ], [ "Bayesian Neural Networks (BNNs)", "Standard neural networks, once trained, output a single point estimate prediction every time the same example is presented to the network.", "Thus, deterministic neural networks are unable to capture uncertainties in their predictions.", "In BNNs, the single weights are being replaced by appropriate probability distributions, that can be subsequently used to provide a measure of how (un)certain a model is in its predictions.", "Given a training dataset, ${\\cal {D}}=(\\mathbf {x},\\mathbf {y})$ , training a BNN consists of finding the posterior distribution of the weights, $p(w|{\\cal {D}})$ .", "The exact evaluation of this posterior distribution is computationally intractable, since it requires multi-dimensional integration over the weight values.", "One approach (variational inference; VI) is to approximate the true posterior with a distribution $q(w|\\theta )$ of a known form (e.g., a Gaussian), with free parameters $\\theta $ to be learned.", "The goal is to select the parameters to minimize the difference between the true and approximate posteriors.", "It can be shown that this is equivalent to minimizing the (negative) evidence lower bound (ELBO) loss function [2]: ${\\cal {L}}({\\cal {D}},\\theta ) = \\mathbb {E}_{q(w|\\theta )}\\left[\\log q(w|\\theta ) - \\log p(w)p({\\cal {D}}|w)\\right].$ Once trained, random weights can be drawn from the approximate posterior $q(w|\\theta )$ , and the network can give predictions on new examples presented to it." ], [ "Implementation and Training", "Our BNN architecture consists of five (probabilistic) 2D convolutional layers, each one followed by a Max Pooling layer, and a (probabilistic) dense layer following the convolutional part.", "The model output is a multivariate ($n=5$ dimensions, equal to the five free parameters of the Serśic model described in Sec. )", "Gaussian, with full covariance.", "For the interested reader, we present a schematic overview and more details about the architecture in Appendix .", "We implement our network using the Keras and Tensorflow Probability (TFP) Python libraries.", "Before training, we randomly split the full simulated dataset to a training (150k), a validation (10k), and a test (10k) set.", "We perform training with a learning rate $\\eta =0.2$ (Adadelta optimizer), for 150 epochs, using a batch size of 64.", "During training we observed no signs of overfitting.", "We utilized the 25 GB high-RAM Nvidia P100 GPUs available through the Google Colab Pro.", "The training took approximately three hours to complete." ], [ "Parameter posteriors", "In Fig.", "REF we present the predicted posterior distributions for the five parameters of the Sérsic model, described in Sec.", ", for a simulated galaxy at the bright end of the surface brightness distribution ($I_e = 24.4$ mag/arcsec$^2$ , panel (a)), and one at the faint end ($I_e = 25.3$ mag/arcsec$^2$ , panel (b)).", "We present the predictions of the BNN model (red contours) and those from the pyImfit model, using two different estimation methods: bootstrap resampling (green contours), and Markov chain Monte Carlo (MCMC; blue contours).", "To get the BNN posteriors, we stack together the output distributions from 400 forward passes (predictions) of the model, for each one of the LSBG images.", "We see that the constraints on the parameters are tighter (as in the case of the brighter LSBG) or comparable (as in the case of the fainter LSBG) to those obtained using pyImfit.", "Although we present only two examples here, we have confirmed that this is true for a larger number of randomly selected LSBG images.", "In terms of speed, obtaining the posterior distribution for each one of the examples using BNNs was significantly faster than running MCMC ($\\sim 1$ minute vs $\\sim $ 6 minutes), and comparable in time to the bootstrap method.", "The real gain in time comes when one wants to process a large number of galaxy images simultaneously; for example, obtaining full posterior estimates for 1000 LSBGs using BNNs takes $\\sim 7$ minutes on our machine, while processing the same number of images using pyImfit and bootstrap resampling would had taken $\\sim 16$ hours." ], [ "Calibration", "We have showed that BNNs fit Sérsic model parameters with tighter or similar uncertainties to those produced by pyImfit.", "In order to be interpreted as confidence intervals, we have to demonstrate that a $x\\%$ interval contains the true value $x\\%$ of the time – in other words, that the posterior is well-calibrated.", "To investigate that, we consider the parameter posterior predictions on 1000 simulated LSBGs drawn from the test set.", "Following [16], [10], we calculate, at different posterior percentile levels, the fraction of LSBGs with true values within the limits of that percentile interval.", "For well-calibrated posteriors, those two quantities (percentile and fraction) should be equal.", "Here, we show only the (marginalized) posterior for the effective radius, $r_e$ .", "However, similar results are found for the other parameters; the interested reader can see those plots in Appendix .", "As we can see in Fig.", "REF , the posterior produced by the BNN is statistically consistent with being perfectly calibrated (the error band was calculated by performing a bootstrap resampling of the test set used for the calculations)." ], [ "Comparison of point estimates", "We have demonstrated that the BNN model outputs well-calibrated errors, and we have seen examples where we compared the output parameter posteriors from the BNN and the pyImfit algorithm.", "We now compare the point estimate (mean) prediction from the BNN with the best fit parameter output from pyImfit.", "In Fig.", "REF we plot the true value of the effective radius parameter (for the same 1000 simulated LSBGs as in the previous section) vs the predicted one, using both methods.", "The point estimates produced by the BNN method tend to be closer to the true value, as indicated by the higher coefficient of determination ($R^2 = 0.83$ for BNN vs $R^2 = 0.54$ for pyImfit), and it performs significantly better for higher effective radius values.", "In Appendix we present similar plots for the other Sérsic model parameters." ], [ "Discussion and Future Work", "We have used a Bayesian Neural network model to predict structural parameters of LSBGs in simulated galaxy images.", "We compare the posterior parameter predictions from the BNN method with those from a profile-fitting algorithm (pyImfit) for simulated LSBGs and we show that the BNN gives comparable or even tighter parameter constraints.", "We furthermore show that the uncertainties estimated using the BNN method are well calibrated, and that, for a sample of simulated LSBGs, the BNN gives better point-estimate parameter predictions (higher coefficient of determinations) compared to those from pyImfit.", "A significant strength of our BNN method is its speed.", "For example, it can predict the full posterior distribution of the five-parameter Sérsic model for 1000 LSBGs images within $\\sim 7$ minutes (on the machine used here); using pyImfit and the bootstrap resampling methods to get parameter constraints for the same number of images, would require $\\sim 16$ hours.", "An important next step, which we plan to address in future work, is to test the performance of our method on real LSBG images and investigate ways to improve it if necessary (for example by re-training on real data, as in [14]).", "Indeed, the case presented here, where both training and testing was done on very simple simulated data, can be significantly different from real data.", "However, some preliminary investigation on applying the model trained here on real LSBGs has shown promising results.", "Other areas of future investigation include testing different BNN architectures, testing the performance of the model on data outside of the training range, and a more rigorous comparison of the performance (parameter constraints and speed) between BNNs and pyImfit." ], [ "Acknowledgements", "We would like to thank Alex Ji, Brian Nord, Ji Won Park, and the anonymous referees for helpful suggestions.", "We acknowledge the Deep Skies Lab as a community of multi-domain experts and collaborators who’ve facilitated an environment of open discussion, idea-generation, and collaboration.", "This community was important for the development of this project.", "This material is based upon work supported by the National Science Foundation under Grant No.", "AST-2006340.", "This work was partially funded by Fermilab LDRD 2018-052.", "A. Ćiprijanović is partially supported by the High Velocity Artificial Intelligence grant as part of the Department of Energy High Energy Physics Computational HEP sessions program.", "This work was supported by the University of Chicago and the Department of Energy under section H.44 of Department of Energy Contract No.", "DE-AC02-07CH11359 awarded to Fermi Research Alliance, LLC." ], [ "BNN Architecture", "In Fig.", "REF we present a schematic overview of the BNN architecture used in this work.", "As we mentioned in the main text, we used the Kerashttps://keras.io/ library on a TensorFlowhttps://www.tensorflow.org/ backend, and the Tensorflow Probabilityhttps://www.tensorflow.org/probability extension of it, for the probabilistic layers.", "The architecture consists of five probabilistic convolutional layers (convolution2DFlipout, provided by TensorFlow Probability).", "The number of filters and the kernel size used in each layer can be seen in the figure.", "Each convolutional layer is followed by a Max Pooling layer.", "After flattening we have a dense layer (DelseFlipout).", "The output is a multidimensional Gaussian (the five parameters our model tries to learn), with full covariance that allows to capture the correlations between the parameters." ], [ "Calibration and Point Estimate Comparison Plots", "In Sec.", "REF , we presented the calibration plot for the BNN posterior for the effective radius, $r_e$ , and the comparison of the BNN point-estimate predictions with those coming from pyImfit, for the same parameter.", "We showed that the BNN errors are well-calibrated, and that the BNN point-estimates are closer to the true value compared to those of the pyImfit for the effective radius parameter.", "The figures presented in this Appendix demonstrate that these conclusions hold for other model parameters, too, with the notable exception of the position angle (PA).", "For the PA, the network seems to be confused by the artificial discontinuity at the 0 to 180 degrees boundary (see Fig.", "7d).", "In future iterations of this work, we plan to remove this discontinuity by reparameterizing the position angle.", "Figure: Calibration curves for the BNN posteriors of the four parameters of the Sérsic model, not presented in the main text.", "Expect for the position angle (PA) parameter, which the BNN seems to give slightly underconfident predictions, the calibration curves of the other parameters indicate perfect calibration, within the statistical uncertainty.Figure: Comparison of the parameter predictions (point estimates) from the BNN model (blue dots) and pyImfit (red dots) versus the ground truth values, for the Sérsic model parameters not presented in the main text." ] ]
2207.03471
[ [ "Finite-rate sparse quantum codes aplenty" ], [ "Abstract We introduce a methodology for generating random multi-qubit stabilizer codes based on solving a constraint satisfaction problem (CSP) on random bipartite graphs.", "This framework allows us to enforce stabilizer commutation, X/Z balancing, finite rate, sparsity, and maximum-degree constraints simultaneously in a CSP that we can then solve numerically.", "Using a state-of-the-art CSP solver, we obtain convincing evidence for the existence of a satisfiability threshold.", "Furthermore, the extent of the satisfiable phase increases with the number of qubits.", "In that phase, finding sparse codes becomes an easy problem.", "Moreover, we observe that the sparse codes found in the satisfiable phase practically achieve the channel capacity for erasure noise.", "Our results show that intermediate-size finite-rate sparse quantum codes are easy to find, while also demonstrating a flexible methodology for generating good codes with custom properties.", "We therefore establish a complete and customizable pipeline for random quantum code discovery that can be geared towards near to mid-term quantum processor layouts." ], [ "Introduction", "Quantum error-correcting codes are an essential prerequisite of reliable quantum computing.", "While codes like surface codes [10], [18], [30] and color codes [26], [25] are sufficient to encode and protect a small constant number of qubits, their performance degrades rapidly when performing computations with more qubits [15], [12].", "While quantum error-correcting codes with asymptotically good performance have been known to exist for more than 20 years [14], [7], these codes require measurement of operators whose weight increases with block size.", "This is a major limitation to efficient implementation of measurement circuits for these codes.", "More recently, Gottesman showed that finite-rate quantum low-density parity-check (LDPC) codes can be used to build constant-overhead fault-tolerant quantum computers [21].", "Quantum LDPC codes are stabilizer codes with bounded-weight stabilizers.", "That is, they involve only bounded-weight measurements.", "This seminal result together with the plethora of asymptotically optimal classical LDPC code constructions [19], [28] led to a surge of research toward quantum LDPC codes inspired by their classical counterparts.", "This resulted in many quantum LDPC code constructions such as hypergraph product codes [31], homological product codes [11] and fiber bundle codes [24].", "Finally, more than 20 years after the introduction of the first quantum error-correcting codes, Panteleev and Kalachev [27] introduced the first family of quantum LDPC codes with optimal asymptotic performance.", "These constructions are based on homological products of classical or quantum error-correcting codes.", "While there is some flexibility in the choice of the input codes, these methods are hard to adapt to precise connectivity limitations of near to mid-term quantum devices.", "Furthermore, they are generally more relevant for a larger number of qubits.", "For example, recent numerical studies of hypergraph product codes [22], [32] involve a few thousands to hundreds of thousands of qubits.", "Other constructions such as building codes from low-depth random circuits [23], [13], have been proposed but, to our knowledge, no procedure generates the stabilizer operators directly while considering arbitrary physical limitations.", "In this work, we introduce a methodology for the generation of an arbitrary number of stabilizer operators directly for any given number of qubits.", "We reformulate code generation as a constraint satisfaction problem (CSP) on random bipartite graphs of varying edge inclusion probability, whose vertices correspond to qubits and stabilizers.", "The constraints imposed correspond to desired code properties, including, but not limited to, stabilizer commutation, $X/Z$ balancing, finite rate, sparsity, and maximum-degree bounds.", "The resulting CSP is akin to paradigmatic NP-complete problems like random $k$ -SAT and random 2-coloring of $k$ -uniform hypergraphs.", "Although these CSPs are hard to solve in the worst case, they often display a transition between a hard phase, where problem instances are hard to solve on average, and an easy phase, within which typical instances can be solved in polynomial time, as a function of some parameter [6], [5].", "This motivates us to ask whether our code generation CSP on random bipartite graphs also exhibits a transition to an easy phase as a function of some parameter, which in the present case we choose to be the edge inclusion probability of the sampled graphs.", "To define our problem instances, we sample bipartite graphs at random with varying edge inclusion probability, define appropriate constraints on qubit and stabilizer vertices that ensure our desired code properties, and then solve the resulting instances using a state-of-the-art CSP solver.", "We discover a satisfiability threshold at a critical edge inclusion probability that decreases with increasing number of qubits.", "We therefore show that the CSP has an easy phase, whose extent grows with increasing number of qubits and within which we readily find finite-rate stabilizer codes.", "Furthermore, we find that the resulting codes are much sparser than the initial graphs.", "Going a step further, we show that we can guarantee code sparsity via maximum-degree constraints, at the expense of a computational overhead.", "Finally, we demonstrate that the codes we obtain using our methodology achieve the erasure channel capacity.", "We therefore establish a complete and customizable pipeline for random quantum code discovery that can be geared towards near- to mid-term quantum processor layouts.", "The rest of this paper is organized as follows.", "In Section , we recall notions of quantum coding theory and stabilizer code construction.", "In Section , we introduce our methodology for the generation of stabilizer codes through solving a CSP on random bipartite graphs, detailing the implementation of constraints.", "Finally, we present numerical results on code discovery and near-optimal error correction performance for the erasure channel in Section ." ], [ "Stabilizer codes", "Given the $n$ -qubit Pauli group $\\mathcal {P}_n$ , a stabilizer group is a commuting subgroup of $\\mathcal {P}_n$ that does not include the $-I$ operator.", "A stabilizer group $\\mathcal {S}$ defines the stabilizer code [20] $\\mathcal {C}(\\mathcal {S}) = {{\\psi }S {\\psi } = {\\psi },\\forall S \\in \\mathcal {S}}.$ That is, a stabilizer code is the common $+1$ eigenspace of the operators of a stabilizer group.", "We often define a stabilizer group using a set of generators $g(\\mathcal {S})= {S_1, S_2, ..., S_m}$ with $S_i \\in \\mathcal {S}$ .", "A family of codes is a finite-rate family if there exists a constant $c > 0$ such that $\\frac{n - m}{n} > c$ for $n\\rightarrow \\infty $ .", "Finite-rate code families are crucial for achieving large-scale fault-tolerant quantum computation since they allow encoding a constant ratio of logical qubits to physical qubits with vanishing error rate.", "This is in contrast with zero-rate code families such as surface codes, which can only encode a small constant number of logical qubits without significantly degrading error-correcting performance [12].", "A common class of stabilizer codes are Calderbank-Shor-Steane (CSS) codes [14], [29].", "These are codes for which there exists a set of generators such that each of them is a product of only $I$ and $X$ or only $I$ and $Z$ .", "While the techniques introduced in this work are applicable to both CSS and non-CSS stabilizer codes alike, in what follows we focus mainly on CSS codes as this simplifies the presentation of our methodology.", "We discuss the case of non-CSS stabilizer codes in Sec. .", "Below we make use of the graphical representation of stabilizer codes based on Tanner graphs, illustrated in Figure REF .", "A Tanner graph $T = (Q \\cup g(\\mathcal {S}), E)$ is a bipartite graph that contains edge ${q, S} \\in E$ if and only if the stabilizer generator $S$ acts as $X$ , $Y$ or $Z$ on qubit $q$ .", "The degree of a vertex is its number of neighbors.", "In error-correction terms, the degree of a stabilizer corresponds to its weight and the degree of a qubit corresponds to the number of stabilizers acting on it." ], [ "Stabilizing edge coloring", "To introduce our algorithmic approach to principled search for finite-rate stabilizer codes, we reformulate the task as a graph coloring problem.", "Our strategy is to start with a bipartite graph $G = (Q \\cup g(\\mathcal {S}), E_0)$ , called the support graph, then search for an edge coloring $l : E_0 \\rightarrow {I, X, Y, Z}$ for which all stabilizers commute.", "We call such a coloring a stabilizing edge coloring.", "The result of this procedure is a Tanner graph $T = (Q \\cup g(\\mathcal {S}), E)$ where the edge $e \\in E$ iff $l(e) \\ne I$ and hence $E \\subseteq E_0$ .", "Deciding whether there exists a stabilizing edge coloring for a given support graph $G$ is equivalent to determining whether there exists a valid stabilizer code whose Tanner graph is a subgraph of $G$ as defined above.", "As formulated above, stabilizing edge coloring admits trivial solutions.", "For example, assigning the same color to all edges is always a valid solution, but is of little interest for quantum error correction.", "To avoid such trivial solutions, or other undesirable solutions, we must introduce additional constraints.", "We represent a constraint as a pair $(F, L)$ where $F \\subseteq E_0$ and $L \\subseteq {I, X, Y, Z}^{|F |}$ .", "A coloring $l$ satisfies a constraint $(F, L)$ for $F = {e_1 , \\dots , e_{|F|}}$ if $(l(e_1 ), .", ".", ".", ", l(e_{|F|})) \\in L$ .", "We can now define the constrained stabilizing edge coloring problem we are interested in.", "Definition 1 ((Constrained) stabilizing edge coloring) Given a bipartite graph $G = (Q \\cup g(\\mathcal {S}), E_0 )$ and a set of constraints $\\mathcal {F} = {(F_1 , L_1), \\ldots , (F_{|\\mathcal {F}|}, L_{|\\mathcal {F}|})}$ with $F_i \\subseteq E_0$ and $L_i \\subseteq {I, X, Y, Z}^{|F_i|}$ , find a stabilizing edge coloring $l : E_0 \\rightarrow {I, X, Y, Z}$ satisfying $\\mathcal {F}$ .", "The problem is constrained whenever $\\mathcal {F}$ is not empty.", "Table REF presents estimates of $|\\mathcal {F}|$ for the instances we study numerically.", "This defines a rich class of constraint satisfaction problems.", "In the following section, we provide concrete realisations of unconstrained and constrained stabilizing edge coloring which we solve to generate non-trivial CSS codes.", "Figure: Graphical representation of the commutation constraints for a pair of stabilizerssharing three qubits.", "(a)The support graph GG.The circles and squares respectively represent qubit and stabilizers.", "(b)The boolean variables and constraints assuring the commutation.Filled vertices correspond to variables.", "In particular circles are either activatoror Pauli variables while the squares are auxiliary variables.The circle constraints correspond to Equation (),the square constraints to Equations () and ()and the half-circle constraints to Equation ().The numbers within variable vertices illustrate a valid assignment.", "(c)The resulting Tanner graph according to the variable assignment." ], [ "CSS codes from constraints on boolean variables", "For CSS codes, we simplify stabilizing edge coloring by assigning Pauli values $l_p : g(\\mathcal {S})\\rightarrow {X, Z}$ to stabilizer vertices and boolean values $l_a : E_0 \\rightarrow {0, 1}$ to edges.", "Then, an edge $e \\in E_0$ has color $l(e) = I$ when $l_a(e) = 0$ and color $l_p(e)$ otherwise.", "We say that an edge $e$ is active if $l_a(e) = 1$ .", "In this representation, the active neighborhood of a stabilizer $S$ is the set of qubits for which $\\lbrace q, S\\rbrace \\in E_0$ and $l_a(\\lbrace q,S \\rbrace ) = 1$ .", "Then, two stabilizers with different Pauli values commute if the intersection of their active neighborhoods has even cardinality.", "To solve the stabilizing edge coloring problem numerically, we represent the coloring functions $l_p$ and $l_a$ with boolean variables.", "To each edge $e \\in E_0$ , we assign a boolean variable $v_{a}(e) \\in {0, 1}$ such that $l_a(e) = v_{a}(e)$ .", "To each stabilizer $S \\in g(\\mathcal {S})$ , we assign a boolean variable $v_{p}(S)$ such that $l_p(S) = X$ if $v_{p}(S) = 1$ and $l(S) = Z$ otherwise.", "We call $v_{a}$ an activator variable and $v_{p}$ a Pauli variable.", "We also introduce some auxiliary boolean variables.", "The auxiliary variable $v_{s}(S, S^{\\prime }) = 1$ when the action of $S$ and $S^{\\prime }$ is the same.", "The variable $v_{e}(S, S^{\\prime }) = 1$ when $S$ and $S^{\\prime }$ have an even overlap.", "Finally, the variable $v_{b}(q, S, S^{\\prime }) = 1$ when both edges ${q, S}$ and ${q, S^{\\prime }}$ are active.", "We are now ready to introduce the boolean constraints that define the stabilizing edge coloring problem for CSS codes.", "In the rest of this section, we represent a constraint as a boolean function $c_{}: {0, 1}^* \\rightarrow {0, 1}$ acting on a subset of the boolean variables $v_{a}$ , $v_{p}$ , $v_{s}$ , $v_{e}$ , and $v_{b}$ .", "An assignment $x \\in {0, 1}^*$ satisfies a constraint if $c_{}(x) = 1$ .", "Boolean constraints allow us to represent the stabilizing edge coloring constraints of Definition REF .", "Stabilizers $S$ and $S^{\\prime }$ commute if they satisfy at least one of the following conditions: they have the same non-trivial action ($X$ or $Z$ ) or, as discussed previously, they act non-trivially on an even number of common-neighbor qubits.", "We can thus write a commutation constraint $c_{c}(S, S^{\\prime })$ as $c_{c}(S, S^{\\prime })=v_{s}(S, S^{\\prime }) \\vee v_{e}(S, S^{\\prime }).$ That the variable $v_{s}(S, S^{\\prime })=1$ when the action of $S$ and $S^{\\prime }$ is the same is enforced by the constraint $c_{s}(S, S^{\\prime })=v_{s}(S, S^{\\prime }) \\oplus v_{p}(S) \\oplus v_{p}(S^{\\prime }).$ Similarly, to ensure that $v_{e}(S, S^{\\prime }) = 1$ when $S$ and $S^{\\prime }$ have an even overlap, we add the constraint $c_{e}(S, S^{\\prime })=v_{e}(S, S^{\\prime })\\oplus [\\bigoplus _{q \\in \\eta (S) \\cap \\eta (S^{\\prime })} v_{b}(q, S, S^{\\prime })],$ where $\\eta (v)$ is the set of neighbors of vertex $v$ in $G$ .", "Finally, to ensure that $v_{b}(q, S, S^{\\prime }) = 1$ when both edges ${q, S}$ and ${q, S^{\\prime }}$ are active, we add the constraint $c_{b}(q,S, S^{\\prime })=&\\lnot v_{b}(q, S, S^{\\prime }) \\oplus \\\\&[v_{a}({q, S}) \\wedge v_{a}({q, S^{\\prime }})].$ Equations (REF ) to (REF ) define a set of constraints on activator, Pauli and auxiliary variables.", "We draw the variables and constraints for a small example in Figure REF .", "When simultaneously satisfied, they ensure that the coloring function $l$ defines a stabilizing edge coloring.", "Thus, any variable assignment for which all constraints evaluate to 1 yields a valid CSS code.", "In Section REF , we present how we find such assignments." ], [ "Extra constraints for good codes", "In this section, we introduce extra constraints to restrict the search to better codes.", "We represent these contraints using integer linear functions $c_{}: \\mathbb {Z}^* \\rightarrow \\mathbb {Z}$ restricted to $D \\subseteq \\mathbb {Z}$ .", "That is, an assignment $x \\in \\mathbb {Z}^*$ satisfies constraint $c$ if $c(x) \\in D$ .", "We define these constraints by extending the boolean variables of the previous section to be integer variables.", "We first add lower bounds on the number of stabilizers of each kind connected to a qubit.", "For each edge ${q, S} \\in E_0$ , we add a variable $v_{X}({q, S})$ with value 1 if the edge is active and the corresponding stabilizer is of the $X$ type.", "This is enforced by the constraint $c_{X}(q,S)=\\lnot v_{X}(q, S) \\oplus [v_{a}({q, S}) \\wedge v_{p}(S)].$ Then, we impose that the number of variables $v_{X}({q, S})$ with value 1 for $S \\in \\eta (q)$ is at least $\\delta _q$ .", "That is, we add the constraint $c_{X}(q) =\\sum _{S \\in \\eta (q)} v_{X}(q, S)\\ge \\delta _q$ for each qubit $q$ .", "We use similar constraints to lower bound the number of $Z$ stabilizer generators per qubit.", "These constraints together with the commutation constraints are the first set of constraints we numerically study in the following section.", "Subsequently, we add more constraints to search for codes with improved decoding performances.", "We impose to each stabilizer $S \\in g(\\mathcal {S})$ that the number of variables $v_{a}({q, S})$ with value 1 for $q \\in \\eta (S)$ is at least $\\delta _s$ .", "Then, to keep the codes sparse, we impose that at most $\\Delta _S$ edges per stabilizer are active.", "These are both enforced by constraints akin to Equation (REF ).", "Finally, we impose that the number of stabilizers of each kind is balanced by enforcing that $\\sum _{S \\in g(\\mathcal {S})} v_{p}(S) = \\left\\lfloor \\frac{|g(\\mathcal {S})|}{2}\\right\\rfloor .$ Figure: Satisfiability phase diagram for the commutation and minimum qubitdegree constraints.For each pixel we generate 100 support graphs, then solve the corresponding stabilizing edge coloring instance for each graph.For each support graph, we run the CSP solver on four CPU cores running at 2.4 GHz for up to four hours.A green pixel indicates a combination of edge inclusion probability and number of qubits for which the solver is able to solve less than 10% of the instances within the allocated timeout.In the rest of the phase diagram, where more than 90% of the instances are solved for each combination of parameters,a blue pixel indicates more satisfiable than unsatisfiable instances, whereasan orange pixel indicates the opposite: more unsatisfiable than satisfiable instances." ], [ "Constraint satisfaction problem solver", "To obtain CSS codes from the aforementioned constraints, we use the OR-Tools library [3].", "We decompose the constraints of Equations (REF ), (REF ), and (REF ) into a constant number of OR constraints and we leave those of Equations (REF ) and (REF ) as XOR constraints.", "Both OR and XOR constraints, as well as the linear constraints used to enforce the minimum and maximum degree and balancing, are native to the library.", "We provide our implementation in an online repository [4]." ], [ "Phase transition", "To search for codes with $n$ qubits and $m$ stabilizer generators, we start by building random support graphs with the corresponding numbers of vertices using the Erdős–Rényi model [17].", "That is, we sample the graphs with an edge inclusion probability of $\\gamma $ .", "We use $G_{n,m,\\gamma }$ to denote the corresponding random bipartite graph generator.", "Our goal is to find sparse codes for given $n,m$ , and $\\gamma $ , or obtain a proof that such codes are statistically unlikely.", "One can see that if $E \\subseteq E^{\\prime }$ and $G = (V, E) \\in \\mathcal {P}$ , where $\\mathcal {P}$ denotes the existence of at least one stabilizing edge coloring, then for any graph $G^{\\prime } = (V, E^{\\prime })$ , we have $G^{\\prime } \\in \\mathcal {P}$ .", "Thus, $\\Pr [G_{n,m,\\gamma } \\in \\mathcal {P}]\\le \\Pr [G_{n,m,\\gamma ^{\\prime }} \\in \\mathcal {P}]$ when $\\gamma \\le \\gamma ^{\\prime }$ and there exists a threshold function $\\gamma ^*(n)$ such that [9] $\\lim _{n\\rightarrow \\infty }\\Pr [G_{n,m,\\gamma } \\in \\mathcal {P}]={\\left\\lbrace \\begin{array}{ll}0 & \\gamma (n)/\\gamma ^*(n) \\rightarrow 0, \\\\1 & \\gamma (n)/\\gamma ^*(n) \\rightarrow \\infty .\\end{array}\\right.", "}$ Our numerical results, discussed below, suggest this threshold is a non-increasing function of the number of qubits.", "We start by imposing only the commutation constraints and a minimum qubit degree.", "That is, we impose $\\delta _q = 3$ , $\\delta _s = 0$ and $\\Delta _s = \\infty $ for graphs with fixed ratio $m/n = 9/10$ , yielding codes with rate $1/10$ .", "If no solution was found and no proof of unsatisfiability was produced within a timeout, we label the instance as unknown.", "We note that most instances are solved in a fraction of the timeout, except close to the threshold.", "In Table REF we give estimates for the number and weight (in number of variables) of each type of constraint in the CSS stabilizing edge coloring for the Erdős–Rényi model.", "Table: Expected constraint occurrences and weights.The data is obtained considering there are 𝒪(m 2 )\\mathcal {O}(m^2) pairs of stabilizerseach expected to share 𝒪(nγ 2 )\\mathcal {O}(n\\gamma ^2) common qubits.Figure REF illustrates that the threshold $\\gamma ^*$ is decreasing with the number of qubits.", "While we cannot definitively claim that the threshold does not start increasing for larger block sizes, we expect that finite-size effects are insignificant for larger numbers of qubits.", "This implies that $\\gamma ^*$ either decreases monotonically or it plateaus to a value of at most $15\\%$ .", "The unknown region demarcates the parameter regime where typical instances of the CSP become hard to solve and our calculations time out.", "Crucially, Figure REF shows that the densities $\\frac{E}{mn}$ of the Tanner graphs of the codes go to zero as the number of qubits increase and are much lower than the edge inclusion probabilites of the support graphs.", "We observe similar results for different values of $m/n$ .", "Figure: Densities of the resulting codes for commutation and minimum qubit degreeconstraints.For the support graphs,we plot the minimum edge inclusion probability in the satisfiable region.For the codes,we plot the average density over all solutions with the same number of qubits found in the satisfiable phase of Figure .We now restrict the problem further to the regime of quantum LDPC codes.", "That is, we impose $\\delta _q = 3$ , $\\delta _s = 6$ and $\\Delta _S = 20$ together with the stabilizer balancing constraint.", "Figure REF indicates that the threshold function is non-increasing when increasing the number of qubits.", "We thus once again find a satisfiable phase that expands with increasing number of qubits and within which it is statistically likely that quantum LDPC codes exist and are easy to find." ], [ "Decoding experiment", "We showed numerical evidence that even if it is generally hard to find commuting sets of random stabilizer operators, it is relatively easy to generate such a set from a random bipartite graph as long as the edge inclusion probability of the graph is above some threshold function.", "However, this result implies nothing about the decoding performance of the code generated this way.", "Figure: Satisfiability phase diagram with qubit degree, stabilizer degree, and balancing constraints.All parameters and details are the same as in Figure .In this section, we probe the performance of the codes we find, using the erasure channel as suggested in [16], [23].", "The erasure channel is particularly useful since there exists a maximum-likelihood decoder that runs in polynomial time.", "In contrast, we do not yet have a good universal decoder for the depolarizing channel.", "The single-qubit erasure channel, $\\mathcal {E}_p(\\rho ) =(1 - p) \\rho \\otimes {0}+ p \\frac{I}{2} \\otimes {1},$ erases a qubit with probability $p$ by replacing it with a maximally mixed single-qubit state.", "The second register indicates whether a qubit has been erased.", "Since $\\frac{I}{2} = \\frac{1}{4}(I\\rho I + X \\rho X + Y \\rho Y + Z \\rho Z),$ the multi-qubit version of the channel can be written as $\\mathcal {E}^{n}_p(\\rho ) =\\sum _{e \\in {0, 1}^n}\\Pr [e](\\frac{1}{4^{|e|}}\\sum _{E \\in P_n(e)} E\\rho E)\\otimes {e},$ where $P_n(e)$ is the set of $n$ -qubit Pauli operators with a trivial action on every qubit $q_i$ for which $e_i = 0$ .", "Then, after a measurement of the second register identifying erasure positions, the channel reduces to a Pauli channel on the erased qubits where each error is equally likely.", "Therefore, from a measurement of the syndrome, a maximum likelihood decoder searches for any Pauli operator restricted to the erased region with the appropriate syndrome.", "The success probability of this decoder is the inverse of the number of logical operators that cannot be moved out of the region of erased qubits by multiplying with a stabilizer.", "This probability can be computed by Gaussian reduction as described in [23].", "The capacity, i.e.", "the maximum rate of information, of the erasure channel [8] is computed from the probability of erasure.", "That is, the capacity of the channel with erasure probability $p$ is $R_{\\max } = 1 - 2p$ .", "Inverting this relation, we observe that a code family with rate $R$ can be used to protect information as long as the erasure probability is at most $\\frac{1 - R}{2}$ .", "Figure REF illustrates that the rate $1/10$ codes obtained with our methodology achieve this limit.", "In other words, we have a procedure to construct random sparse stabilizer codes that are essentially capacity-achieving for the erasure channel.", "Figure: Failure rate for maximum-likelihood decoding of the erasure channel.For each system size, we show the lowest failure rates amongst all codes found." ], [ "Discussion and outlook", "In this work, we reformulate the search for finite-rate sparse stabilizer codes as a constraint satisfaction problem, which we call stabilizing edge coloring, that is amenable to solution with state-of-the-art CSP solvers.", "We believe future work can exploit the connection between quantum error correction and constraint programming that we establish here in order to guide the search for new families of random stabilizer codes.", "We note that even though here we limit our analysis to CSS codes for the sake of simplicity, the method we introduce is easily adaptable to more general stabilizer codes.", "For example, one can assign an extra variable to each edge to represent its Pauli value instead of labelling the stabilizer vertices directly.", "Furthermore, the method is flexible in the sense that it is simple incorporate constraints that take into account multiple factors on the same footing, including, for example, hardware limitations like qubit layout and connectivity.", "In this work we had to strike a balance between computational resources and a reasonable timeframe for completion of numerical calculations.", "Targeted searches for larger codes within the satisfiable phase using more resources are a straightforward direction for future work.", "Impressive progress in the performance of solvers in the last decades [2], [1] means that larger codes could soon be discoverable by our methodology with moderate resources.", "Customized CSP solvers that resolve the types of constraints involved in stabilizing edge coloring could also boost the search for codes with desirable properties.", "It is also interesting to study different random graph constructions as input to CSP solvers instead of the uniformly sampled graphs we used.", "This could lead to sparser initial graphs with structure favorable to the discovery of good quantum codes.", "Finally, we were able to use our approach to construct a family of finite-rate codes achieving optimal threshold for the erasure channel.", "This result, together with many recent stabilizer code constructions, motivate the search for more generally applicable quantum decoders for more complex noise channels such as the depolarizing channel and correlated Pauli noise.", "This would allow us to probe the performance of random code constructions more thoroughly." ], [ "Acknowledgments", "This work was supported by the Ministère de l'Économie et de l'Innovation du Québec via its Research Chair in Quantum Computing and a Natural Sciences and Engineering Research Council of Canada Discovery grant.", "MT is supported by a Canada Graduate Scholarship from the Natural Sciences and Engineering Research Council of Canada.", "We acknowledge Calcul Québec and Compute Canada for computing resources.", "We thank the members of the QuICoPhy Theory Lab for valuable discussions." ] ]
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