{"text":"\\section{Introduction}\n\nAn $n \\times n$ matrix $M$ over a field $\\Fset$ is said to {\\em represent} a digraph $G=(V,E)$ with vertex set $V = \\{1,2,\\ldots,n\\}$ if $M_{i,i} \\neq 0$ for every $i$, and $M_{i,j}=0$ for every distinct $i,j$ such that $(i,j) \\notin E$. The {\\em minrank} of $G$ over $\\Fset$, denoted ${\\mathop{\\mathrm{minrk}}}_\\Fset(G)$, is the minimum possible rank of a matrix $M \\in \\Fset^{n \\times n}$ representing $G$. The definition is naturally extended to (undirected) graphs by replacing every edge with two oppositely directed edges.\nIt is easy to see that for every graph $G$ the minrank parameter is sandwiched between the independence number and the clique cover number, that is, $\\alpha(G) \\leq {\\mathop{\\mathrm{minrk}}}_\\Fset(G) \\leq \\chi(\\overline{G})$.\nFor example, ${\\mathop{\\mathrm{minrk}}}_\\Fset(K_n)=1$ and ${\\mathop{\\mathrm{minrk}}}_\\Fset(\\overline{K_n})=n$ for every field $\\Fset$.\nThe minrank parameter was introduced by Haemers in 1979~\\cite{Haemers79}, and since then has attracted a significant attention motivated by its various applications in information theory and in theoretical computer science (see, e.g.,~\\cite{Haemers81,BBJK06,Valiant92,Riis07,PudlakRS97,HavivL13,ChlamtacH14}).\n\nIn this work we address the extremal behavior of the minrank parameter of $n$-vertex graphs whose complements are free of a fixed forbidden subgraph.\nFor two graphs $G$ and $H$, we say that $G$ is {\\em $H$-free} if $G$ contains no subgraph, induced or not, isomorphic to $H$.\nFor an integer $n$, a graph $H$, and a field $\\Fset$, let $g(n,H,\\Fset)$ denote the maximum of ${\\mathop{\\mathrm{minrk}}}_\\Fset(G)$ taken over all $n$-vertex graphs $G$ whose complement $\\overline{G}$ is $H$-free.\nOur purpose is to study the quantity $g(n,H,\\Fset)$ where $H$ and $\\Fset$ are fixed and $n$ is growing.\n\n\\subsection{Our Contribution}\n\nWe provide bounds on $g(n,H,\\Fset)$ for various graph families and fields.\nWe start with a simple upper bound for a forest $H$.\n\n\\begin{proposition}\\label{prop:forestIntro}\nFor every integer $n$, a field $\\Fset$, and a nontrivial forest $H$ on $h$ vertices,\n\\[g(n,H,\\Fset) \\leq h-1.\\]\nEquality holds whenever $H$ is a tree and $n \\geq h-1$.\n\\end{proposition}\n\nWe next provide a general lower bound on $g(n,H,\\Fset)$ for a graph $H$ and a finite field $\\Fset$.\nTo state it, we need the following notation.\nFor a graph $H$ with $h \\geq 3$ vertices and $f \\geq 3$ edges define $\\gamma(H) = \\frac{h-2}{f-1}$ and $\\gamma_0(H) = \\min_{H'}{\\gamma(H')}$, where the minimum is taken over all subgraphs $H'$ of $H$ with at least $3$ edges.\n\n\\begin{theorem}\\label{thm:IntroComp}\nFor every graph $H$ with at least $3$ edges there exists $c=c(H)>0$ such that for every integer $n$ and a finite field $\\Fset$,\n\\[g(n,H,\\Fset) \\geq c \\cdot \\frac{n^{1-\\gamma_0(H)}}{\\log (n \\cdot |\\Fset|)} .\\]\n\\end{theorem}\n\nNote that for every finite field $\\Fset$, the quantity $g(n,H,\\Fset)$ grows with $n$ if and only if $H$ is not a forest.\nIndeed, if $H$ is a forest then $g(n,H,\\Fset)$ is bounded by some constant by Proposition~\\ref{prop:forestIntro}, whereas otherwise $H$ satisfies $\\gamma_0(H)<1$ and thus, by Theorem~\\ref{thm:IntroComp}, $g(n,H,\\Fset) \\geq \\Omega(n^\\delta)$ for some $\\delta = \\delta(H)>0$.\nNote further that for the case $H=K_3$, which is motivated by a question in information theory (see Section~\\ref{sec:applications}),\nTheorem~\\ref{thm:IntroComp} implies that\n\\begin{eqnarray}\\label{eq:K_3}\ng(n,K_3,\\Fset) \\geq \\Omega  \\Big ( \\frac{\\sqrt{n}}{\\log n} \\Big )\n\\end{eqnarray}\nfor every fixed finite field $\\Fset$.\nThis is tight up to a $\\sqrt{\\log n}$ multiplicative term (see Proposition~\\ref{prop:K_3}).\n\nTheorem~\\ref{thm:IntroComp} is proved by a probabilistic argument based on the Lov\\'asz Local Lemma~\\cite{LLL75}.\nThe proof involves an approach of Spencer~\\cite{Spencer77} to lower bounds on off-diagonal Ramsey numbers and a technique of Golovnev, Regev, and Weinstein~\\cite{Golovnev0W17} for estimating the minrank of random graphs.\n\nAs our final result, we show that for every non-bipartite graph $H$ there are $H$-free graphs with low minrank over the real field $\\mathbb{R}$.\n\n\\begin{theorem}\\label{thm:IntroNonBi}\nFor every non-bipartite graph $H$ there exists $\\delta=\\delta(H)>0$ such that for every sufficiently large integer $n$, there exists an $n$-vertex $H$-free graph $G$ such that ${\\mathop{\\mathrm{minrk}}}_\\mathbb{R}(G) \\leq n^{1-\\delta}$.\n\\end{theorem}\n\\noindent\nThis theorem is proved by an explicit construction from the family of generalized Kneser graphs, whose minrank was recently studied in~\\cite{Haviv18}.\nIt is known that every $n$-vertex graph $G$ satisfies\n\\begin{eqnarray}\\label{eq:minrk_comp}\n{\\mathop{\\mathrm{minrk}}}_\\Fset(G) \\cdot {\\mathop{\\mathrm{minrk}}}_\\Fset(\\overline{G}) \\geq n\n\\end{eqnarray}\nfor every field $\\Fset$ (see, e.g.,~\\cite[Remark~2.2]{Peeters96}).\nThis combined with the graphs given in Theorem~\\ref{thm:IntroNonBi} implies the following (explicit) lower bound on $g(n,H,\\mathbb{R})$ for non-bipartite graphs $H$.\n\n\\begin{corollary}\\label{cor:IntroNonBi}\nFor every non-bipartite graph $H$ there exists $\\delta=\\delta(H)>0$ such that for every sufficiently large integer $n$,\n$g(n,H,\\mathbb{R}) \\geq n^{\\delta}$.\n\\end{corollary}\n\\noindent\nAs another application of Theorem~\\ref{thm:IntroNonBi}, we disprove a conjecture of Codenotti, Pudl\\'ak, and Resta~\\cite{CodenottiPR00} motivated by Valiant's approach to circuit lower bounds~\\cite{Valiant77} (see Section~\\ref{sec:applications}).\n\n\n\\subsection{Applications}\\label{sec:applications}\n\nThe study of the quantity $g(n,H,\\Fset)$ is motivated by questions in information theory, circuit complexity, and geometry.\nWe gather here several applications of our results.\n\n\\paragraph{Shannon Capacity.}\nFor an integer $k$ and a graph $G$ on the vertex set $V$, let $G^k$ denote the graph on the vertex set $V^k$ in which two distinct vertices $(u_1,\\ldots,u_k)$ and $(v_1,\\ldots,v_k)$ are adjacent if for every $1 \\leq i \\leq k$ it holds that $u_i$ and $v_i$ are either equal or adjacent in $G$.\nThe Shannon capacity of a graph $G$, introduced by Shannon in 1956~\\cite{Shannon56}, is defined as the limit $c(G) = \\lim_{k \\rightarrow \\infty}{(\\alpha(G^k))^{1\/k}}$.\nThis graph parameter is motivated by information theory, as it measures the zero-error capacity of a noisy communication channel represented by $G$.\nAn upper bound on $c(G)$, known as the Lov\\'asz $\\vartheta$-function, was introduced in~\\cite{Lovasz79}, where it was used to show that the Shannon capacity of the cycle on $5$ vertices satisfies $c(C_5)=\\sqrt{5}$, whereas its independence number is $2$.\nHaemers introduced the minrank parameter in~\\cite{Haemers79,Haemers81} and showed that it forms another upper bound on $c(G)$ and that for certain graphs it is tighter than the $\\vartheta$-function.\nIn general, computing the Shannon capacity of a graph seems to be a very difficult task, and its exact value is not known even for small graphs such as the cycle on $7$ vertices.\n\nThe question of determining the largest possible Shannon capacity of a graph with a given independence number is widely open.\nIn fact, it is not even known if the Shannon capacity of a graph with independence number $2$ can be arbitrarily large~\\cite{AlonPowers02}.\nInterestingly, Erd\\\"{o}s, McEliece, and Taylor~\\cite{ErdosMT71} have shown that this question is closely related to determining an appropriate multicolored Ramsey number, whose study in~\\cite{XiaodongZER04} implies that there exists a graph $G$ with $\\alpha(G)= 2$ and $c(G)> 3.199$.\nA related question, originally asked by Lov\\'asz, is that of determining the maximum possible $\\vartheta$-function of an $n$-vertex graph with independence number $2$. This maximum is known to be $\\Theta(n^{1\/3})$, where the upper bound was proved by Kashin and Konyagin~\\cite{KasKon81,Kon81}, and the lower bound was proved by Alon~\\cite{Alon94} via an explicit construction.\nHere we consider the analogue question of determining the maximum possible minrank, over any fixed finite field $\\Fset$, of an $n$-vertex graph with independence number $2$.\nSince the latter is precisely $g(n,K_3,\\Fset)$, our bound in~\\eqref{eq:K_3} implies that the minrank parameter is weaker than the $\\vartheta$-function with respect to the general upper bounds that they provide on the Shannon capacity of $n$-vertex graphs with independence number $2$.\n\n\n\\paragraph{The Odd Alternating Cycle Conjecture.}\nIn 1977, Valiant~\\cite{Valiant77} proposed the matrix rigidity approach for proving superlinear circuit lower bounds, a major challenge in the area of circuit complexity.\nRoughly speaking, the rigidity of a matrix $M \\in \\Fset^{n \\times n}$ for a constant $\\epsilon>0$ is the minimum number of entries that one has to change in $M$ in order to reduce its rank over $\\Fset$ to at most $\\epsilon \\cdot n$. Valiant showed in~\\cite{Valiant77} that matrices with large rigidity can be used to obtain superlinear lower bounds on the size of logarithmic depth arithmetic circuits computing linear transformations.\nWith this motivation, Codenotti, Pudl\\'ak, and Resta~\\cite{CodenottiPR00} raised in the late nineties the Odd Alternating Cycle Conjecture stated below, and proved that it implies, if true, that certain explicit circulant matrices have superlinear rigidity.\nBy an alternating odd cycle we refer to a digraph which forms a cycle when the orientation of the edges is ignored, and such that the orientation of the edges alternates with one exception.\n\\begin{conjecture}[The Odd Alternating Cycle Conjecture~\\cite{CodenottiPR00}]\\label{conj:alternating}\nFor every field $\\Fset$ there exist $\\epsilon >0$ and an odd integer $\\ell$ such that every $n$-vertex digraph $G$ with ${\\mathop{\\mathrm{minrk}}}_\\Fset (G)  \\leq \\epsilon \\cdot n$ contains an alternating cycle of length $\\ell$.\n\\end{conjecture}\n\nCodenotti et al.~\\cite{CodenottiPR00} proved that the statement of Conjecture~\\ref{conj:alternating} does not hold for $\\ell=3$ over any field $\\Fset$. Specifically, they provided an explicit construction of $n$-vertex digraphs $G$, free of alternating triangles, with ${\\mathop{\\mathrm{minrk}}}_\\Fset (G)  \\leq O(n^{2\/3})$ for every field $\\Fset$. For the undirected case, which is of more interest to us, a construction of~\\cite{CodenottiPR00} implies that there are $n$-vertex triangle-free graphs $G$ such that ${\\mathop{\\mathrm{minrk}}}_\\Fset(G) \\leq O(n^{3\/4})$ for every field $\\Fset$ (see~\\cite[Section~4.2]{BlasiakKL13} for a related construction over the binary field as well as for an application of such graphs from the area of index coding). Note that this yields, by~\\eqref{eq:minrk_comp}, that $g(n,K_3,\\Fset) \\geq \\Omega(n^{1\/4})$.\nIn contrast, for the real field and the cycle on $4$ vertices, it was shown in~\\cite{CodenottiPR00} that every $n$-vertex $C_4$-free graph $G$ satisfies ${\\mathop{\\mathrm{minrk}}}_\\mathbb{R} (G) > \\frac{n}{6}$.\nYet, the question whether every $n$-vertex digraph with sublinear minrank contains an alternating cycle of odd length $\\ell \\geq 5$ was left open in~\\cite{CodenottiPR00} for every field.\nOur Theorem~\\ref{thm:IntroNonBi} implies that for every odd $\\ell$ there are (undirected) $C_\\ell$-free graphs $G$ with sublinear ${\\mathop{\\mathrm{minrk}}}_\\mathbb{R}(G)$, and in particular disproves Conjecture~\\ref{conj:alternating} for the real field $\\mathbb{R}$.\n\n\\paragraph{Nearly Orthogonal Systems of Vectors.}\nA system of nonzero vectors in $\\mathbb{R}^m$ is said to be nearly orthogonal if any set of three vectors of the system contains an orthogonal pair.\nIt was proved by Rosenfeld~\\cite{Rosenfeld91} that every such system has size at most $2m$.\nAn equivalent way to state this, is that every $n$-vertex graph represented by a real positive semidefinite matrix of rank smaller than $\\frac{n}{2}$ contains a triangle.\nNote that the positive semidefiniteness assumption is essential in this result, as follows from the aforementioned construction of~\\cite{CodenottiPR00} of $n$-vertex triangle-free graphs $G$ with ${\\mathop{\\mathrm{minrk}}}_\\mathbb{R}(G) \\leq O(n^{3\/4})$.\n\nA related question was posed by Pudl\\'ak in~\\cite{Pudlak02}.\nHe proved there that for some $\\epsilon >0$, every $n$-vertex graph represented by a real positive semidefinite matrix of rank at most $\\epsilon \\cdot n$ contains a cycle of length $5$. Pudl\\'ak asked whether the assumption that the matrix is positive semidefinite can be omitted.\nOur Theorem~\\ref{thm:IntroNonBi} applied to $H = C_5$ implies that there are $C_5$-free graphs $G$ with sublinear ${\\mathop{\\mathrm{minrk}}}_\\mathbb{R}(G)$, and thus answers this question in the negative.\n\n\\subsection{Outline}\nThe rest of the paper is organized as follows.\nIn Section~\\ref{sec:forest} we present the simple proof of Proposition~\\ref{prop:forestIntro}.\nIn Section~\\ref{sec:g_comp} we provide some background on sparse-base matrices from~\\cite{Golovnev0W17} and then prove Theorem~\\ref{thm:IntroComp}.\nIn the final Section~\\ref{sec:non-bip}, we prove Theorem~\\ref{thm:IntroNonBi}.\n\n\\section{Forests}\\label{sec:forest}\n\nIn this section we prove Proposition~\\ref{prop:forestIntro}.\nWe use an argument from one of the proofs in~\\cite{AlonKS05}.\n\n\\begin{proof}[ of Proposition~\\ref{prop:forestIntro}]\nFix a nontrivial $h$-vertex forest $H$ and a field $\\Fset$.\nIt suffices to consider the case where $H$ is a tree, as otherwise $H$ is a subgraph of some $h$-vertex tree $H'$, and since every $H$-free graph is also $H'$-free, we have $g(n,H,\\Fset) \\leq g(n,H',\\Fset)$.\n\nOur goal is to show that every $n$-vertex graph $G$ whose complement $\\overline{G}$ is $H$-free satisfies ${\\mathop{\\mathrm{minrk}}}_\\Fset(G) \\leq h-1$.\nLet $G$ be such a graph.\nWe claim that $\\overline{G}$ is $(h-2)$-degenerate, that is, every subgraph of $\\overline{G}$ contains a vertex of degree at most $h-2$. Indeed, otherwise $\\overline{G}$ has a subgraph $G'$ all of whose degrees are at least $h-1$, and one can find a copy of $H$ in $G'$ as follows: First identify an arbitrary vertex of $G'$ with an arbitrary vertex of $H$, and then iteratively identify a vertex of $G'$ with a leaf added to the being constructed copy of the tree $H$. The process succeeds since $H$ has $h$ vertices and every vertex of $G'$ has degree at least $h-1$.\nAs is well known, the fact that $\\overline{G}$ is $(h-2)$-degenerate implies that $\\overline{G}$ is $(h-1)$-colorable, so we get that ${\\mathop{\\mathrm{minrk}}}_\\Fset(G) \\leq \\chi(\\overline{G}) \\leq h-1$, as required.\n\nWe finally observe that the bound is tight whenever $H$ is a tree and $n \\geq h-1$.\nIndeed, let $G$ be the $n$-vertex complete $\\lceil \\frac{n}{h-1} \\rceil$-partite graph, that has $h-1$ vertices in each of its parts, except possibly one of them.\nIts complement $\\overline{G}$ is a disjoint union of cliques, each of size at most $h-1$, and is thus $H$-free.\nSince $\\alpha(G) = \\chi(\\overline{G})=h-1$, it follows that ${\\mathop{\\mathrm{minrk}}}_\\Fset(G) = h-1$ for every field $\\Fset$, completing the proof.\n\\end{proof}\n\n\\section{A General Lower Bound on $g(n,H,\\Fset)$}\\label{sec:g_comp}\n\nIn this section we prove Theorem~\\ref{thm:IntroComp} and discuss its tightness for $H=K_3$.\nWe start with some needed preparations.\n\n\\subsection{Lov\\'{a}sz Local Lemma}\n\nThe Lov\\'{a}sz Local Lemma~\\cite{LLL75} stated below is a powerful probabilistic tool in Combinatorics (see, e.g.,~\\cite[Chapter~5]{AlonS16}).\nWe denote by $[N]$ the set of integers from $1$ to $N$.\n\n\\begin{lemma}\\label{lemma:lll}[Lov\\'{a}sz Local Lemma~\\cite{LLL75}]\nLet $A_1,\\ldots, A_N$ be events in an arbitrary probability space.\nA digraph $D = (V,E)$ on the vertex set $V = [N]$ is called a dependency digraph for the events $A_1,\\ldots, A_N$ if for every $i \\in [N]$, the event $A_i$ is mutually independent of the events $A_j$ with $j \\neq i$ and $(i,j) \\notin E$.\nSuppose that $D=(V,E)$ is a dependency digraph for the above events and suppose that there are real numbers $x_1,\\ldots,x_N \\in [0,1)$ such that \\[\\Prob{}{A_i} \\leq x_i \\cdot \\prod_{(i,j) \\in E}{(1-x_j)}\\] for all $i \\in [N]$.\nThen, with positive probability no event $A_i$ holds.\n\\end{lemma}\n\n\\subsection{Sparse-base Matrices}\\label{sec:GRW}\n\nHere we review several notions and lemmas due to Golovnev, Regev, and Weinstein~\\cite{Golovnev0W17}.\nFor a matrix $M$ over a field $\\Fset$, let $s(M)$ denote its sparsity, that is, the number of its nonzero entries.\nWe say that a matrix $M$ over $\\Fset$ with rank $k$ contains an $\\ell$-sparse column (row) basis if $M$ contains $k$ linearly independent columns (rows) with a total of at most $\\ell$ nonzero entries.\nWe first state a lemma that provides an upper bound on the number of matrices with sparse column and row bases.\n\n\\begin{lemma}[\\cite{Golovnev0W17}]\\label{lemma:size_M}\nThe number of rank $k$ matrices in $\\Fset^{n \\times n}$ that contain $\\ell$-sparse column and row bases is at most $(n \\cdot |\\Fset|)^{6\\ell}$.\n\\end{lemma}\n\nThe following lemma relates the sparsity of a matrix with nonzero entries on the main diagonal to its rank.\n\n\\begin{lemma}[\\cite{Golovnev0W17}]\\label{lemma:sparsity_M}\nFor every rank $k$ matrix $M \\in \\Fset^{n \\times n}$ with nonzero entries on the main diagonal,\n\\[s(M) \\geq \\frac{n^2}{4k}.\\]\n\\end{lemma}\n\nWe also need the following notion. An {\\em $(n,k,s,\\ell)$-matrix} over a field $\\Fset$ is a matrix in $\\Fset^{n \\times n}$ of rank $k$ and sparsity $s$ that contains $\\ell$-sparse column and row bases and has nonzero entries on the main diagonal. Note that by Lemma~\\ref{lemma:sparsity_M}, an $(n,k,s,\\ell)$-matrix exists only if $s \\geq \\frac{n^2}{4k}$.\nFor integers $n,k,s'$ and a field $\\Fset$ (which will always be clear from the context), let ${\\cal M}_{n,k}^{(s')}$ be the collection that consists of all $(n',k',s',\\frac{2s'k'}{n'})$-matrices over $\\Fset$ for all $n' \\in [n]$ and $k' \\in [k]$ such that $\\frac{k'}{n'} \\leq \\frac{k}{n}$.\nThis collection is motivated by the following lemma.\n\n\\begin{lemma}[\\cite{Golovnev0W17}]\\label{lemma:M->M'}\nEvery matrix in $\\Fset^{n \\times n}$ with rank at most $k$ and nonzero entries on the main diagonal has a principal sub-matrix that lies in ${\\cal M}_{n,k}^{(s')}$ for some $s'$.\n\\end{lemma}\n\nNow, for integers $n,k,s'$, let ${\\cal P}_{n,k}^{(s')}$ be the collection that consists of all pairs $(M,R)$ such that, for some $n' \\in [n]$, $M$ is an $n' \\times n'$ matrix in ${\\cal M}_{n,k}^{(s')}$ and $R$ is an $n'$-subset of $[n]$.\nObserve that Lemma~\\ref{lemma:M->M'} implies that for every digraph $G$ on the vertex set $[n]$ with ${\\mathop{\\mathrm{minrk}}}_\\Fset(G) \\leq k$ there exist $s'$ and a pair $(M,R)$ in ${\\cal P}_{n,k}^{(s')}$ such that $M$ represents the induced subgraph $G[R]$ of $G$ on $R$, with respect to the natural order of the vertices in $R$ (from smallest to largest).\n\nThe following lemma provides an upper bound on the size of ${\\cal P}_{n,k}^{(s')}$.\n\n\\begin{lemma}\\label{lemma:size_P}\nFor every integers $n,k,s'$, $|{\\cal P}_{n,k}^{(s')}| \\leq (n \\cdot |\\Fset|)^{24s'k\/n}$.\n\\end{lemma}\n\n\\begin{proof}\nTo bound the size of ${\\cal P}_{n,k}^{(s')}$, we consider for every $n' \\in [n]$ and $k' \\in [k]$ such that $\\frac{k'}{n'} \\leq \\frac{k}{n}$ the pairs $(M,R)$ where $M$ is an $(n',k',s',\\frac{2s'k'}{n'})$-matrix and $R$ is an $n'$-subset of $[n]$.\nBy Lemma~\\ref{lemma:size_M} there are at most $(n' \\cdot |\\Fset|)^{12s'k'\/n'}$ such matrices $M$, each of which occurs in $n \\choose {n'}$ pairs of ${\\cal P}_{n,k}^{(s')}$. It follows that\n\\begin{eqnarray*}\n|{\\cal P}_{n,k}^{(s')}| & \\leq & \\sum_{n',k'}{ {n \\choose n'} \\cdot (n' \\cdot |\\Fset|)^{12s'k'\/n'}}\n \\leq  n^2 \\cdot \\max_{n',k'} \\big ( n^{n'} \\cdot (n' \\cdot |\\Fset|)^{12s'k'\/n'} \\big ) \\\\\n& \\leq & \\max_{n',k'} \\big ( n^{3n'} \\cdot (n' \\cdot |\\Fset|)^{12s'k'\/n'} \\big )\n \\leq \\max_{n',k'} \\big ( (n \\cdot |\\Fset|)^{3n'+12s'k'\/n'} \\big )\\\\\n& \\leq & \\max_{n',k'} \\big ( (n \\cdot |\\Fset|)^{12s'k'\/n' +12s'k'\/n'} \\big )\n\\leq  (n \\cdot |\\Fset|)^{24s'k\/n},\n\\end{eqnarray*}\nwhere in the fifth inequality we have used the relation $s' \\geq \\frac{n'^2}{4k'}$ from Lemma~\\ref{lemma:sparsity_M}, and in the sixth we have used $\\frac{k'}{n'} \\leq \\frac{k}{n}$.\n\\end{proof}\n\n\\subsection{Proof of Theorem~\\ref{thm:IntroComp}}\n\nWe prove the following theorem and then derive Theorem~\\ref{thm:IntroComp}.\nRecall that for a graph $H$ with $h \\geq 3$ vertices and $f \\geq 3$ edges, we denote $\\gamma(H) = \\frac{h-2}{f-1}$.\nWe also let $\\exp(x)$ stand for $e^x$.\n\n\\begin{theorem}\\label{thm:Comp}\nFor every graph $H$ with at least $3$ edges there exists $c=c(H)>0$ such that for every integer $n$ and a finite field $\\Fset$,\n\\[g(n,H,\\Fset) \\geq c \\cdot \\frac{n^{1-\\gamma(H)}}{\\log (n \\cdot |\\Fset|)} .\\]\n\\end{theorem}\n\n\\begin{proof}\nFix a graph $H$ with $h \\geq 3$ vertices and $f \\geq 3$ edges and denote $\\gamma = \\gamma(H) = \\frac{h-2}{f-1} > 0$.\nThe proof is via the probabilistic method. Let $\\vec{G} \\sim \\vec{G}(n,p)$ be a random digraph on the vertex set $[n]$ where each directed edge is taken randomly and independently with probability $p$. Set $q=1-p$.\nLet $G$ be the (undirected) graph on $[n]$ in which two distinct vertices $i,j$ are adjacent if both the directed edges $(i,j)$ and $(j,i)$ are included in $\\vec{G}$. Notice that every two distinct vertices are adjacent in $G$ with probability $p^2$ independently of the adjacencies between other vertex pairs.\n\nTo prove the theorem, we will show that for a certain choice of $p$ the random graph $G$ satisfies with positive probability that its complement $\\overline{G}$ is $H$-free and that ${\\mathop{\\mathrm{minrk}}}_{\\Fset}(G) > k$, where\n\\begin{eqnarray}\\label{eq:k}\nk = c_1 \\cdot \\frac{n^{1-\\gamma}}{\\ln{(n \\cdot |\\Fset|)}}\n\\end{eqnarray}\nfor a constant $c_1>0$ that depends only on $H$.\nTo do so, we define two families of events as follows.\n\nFirst, for every set $I \\subseteq [n]$ of size $|I|=h$, let $A_I$ be the event that the induced subgraph of $\\overline{G}$ on $I$ contains a copy of $H$. Observe that\n\\[\\Prob{}{A_I} \\leq h! \\cdot (1-p^2)^f = h! \\cdot (1-(1-q)^2)^f \\leq h! \\cdot (2q)^f.\\]\n\nSecond, consider the collection ${\\cal P} = \\cup_{s' \\in [n^2]}{{\\cal P}_{n,k}^{(s')}}$ (see Section~\\ref{sec:GRW}).\nRecall that every element of ${\\cal P}$ is a pair $(M,R)$ such that, for some $n' \\in [n]$, $M$ is an $n' \\times n'$ matrix over $\\Fset$ and $R$ is an $n'$-subset of $[n]$.\nDenote $N_{s'} = |{\\cal P}_{n,k}^{(s')}|$.\nBy Lemma~\\ref{lemma:size_P}, combined with~\\eqref{eq:k}, we have\n\\begin{eqnarray}\\label{eq:N_s'}\nN_{s'} \\leq (n \\cdot |\\Fset|)^{24s'k\/n} = \\exp(24c_1 \\cdot s' \\cdot n^{-\\gamma}).\n\\end{eqnarray}\nLet ${\\cal S} = \\{s' \\in [n^2] \\mid N_{s'} \\geq 1\\}$.\nBy Lemma~\\ref{lemma:sparsity_M}, for every $s' \\in {\\cal S}$ and an $n' \\times n'$ matrix of rank $k'$ in ${\\cal M}_{n,k}^{(s')}$ where $n' \\in [n]$, $k' \\in [k]$, and $\\frac{k'}{n'} \\leq \\frac{k}{n}$, we have that\n\\begin{eqnarray}\\label{eq:s'}\ns' \\geq \\frac{n'}{4} \\cdot \\frac{n'}{k'} \\geq \\frac{n'}{4} \\cdot \\frac{n}{k} = n' \\cdot \\frac{n^{\\gamma} \\cdot \\ln (n \\cdot |\\Fset|)}{4c_1}. \\end{eqnarray}\nNow, for every pair $(M,R) \\in {\\cal P}$, let $B_{M,R}$ be the event that the matrix $M$ represents over $\\Fset$ the induced subgraph $\\vec{G}[R]$ of $\\vec{G}$ on $R$ with respect to the natural order of the vertices in $R$.\nFor $M$ to represent $\\vec{G}[R]$ we require that for every distinct $i, j$ such that $M_{i,j} \\neq 0$, there is an edge in $\\vec{G}$ from the $i$th to the $j$th vertex of $R$. Hence, for $M \\in \\Fset^{n' \\times n'}$ of sparsity $s'$ and an $n'$-subset $R$ of $[n]$,\n\\[\\Prob{}{B_{M,R}} = p^{s'-n'} \\leq p^{s'\/2} = (1-q)^{s'\/2} \\leq \\exp(-qs'\/2),\\] where for the first inequality we have used the inequality $s' \\geq 2n'$ which follows from~\\eqref{eq:s'} for every sufficiently large $n$.\n\nWe claim that it suffices to prove that with positive probability none of the events $A_I$ and $B_{M,R}$ holds.\nIndeed, this implies that there exists an $n$-vertex digraph $\\vec{G}$ that does not satisfy any of these events.\nSince the $A_I$'s are not satisfied it immediately follows that the complement $\\overline{G}$ of the (undirected) graph $G$ associated with $\\vec{G}$ is $H$-free.\nWe further claim that ${\\mathop{\\mathrm{minrk}}}_\\Fset (G) > k$.\nTo see this, assume by contradiction that there exists a matrix $M \\in \\Fset^{n \\times n}$ of rank at most $k$ that represents $G$, and thus, in particular, represents $\\vec{G}$.\nBy Lemma~\\ref{lemma:M->M'}, such an $M$ has a principal $n' \\times n'$ sub-matrix $M' \\in {\\cal M}_{n,k}^{(s')}$ for some $n'$ and $s'$.\nHence, for some $n'$-subset $R$ of $[n]$, the matrix $M'$ represents $\\vec{G}[R]$ with respect to the natural order of the vertices in $R$, in contradiction to the fact that the event $B_{M',R}$ with $(M',R) \\in {\\cal P}$ does not hold.\n\nTo prove that with positive probability none of the events $A_I$ and $B_{M,R}$ holds, we apply the Lov\\'{a}sz Local Lemma (Lemma~\\ref{lemma:lll}).\nTo this end, construct a (symmetric) dependency digraph $D=(V,E)$ whose vertices represent all the events $A_I$ and $B_{M,R}$, and whose edges are defined as follows.\n\\begin{itemize}\n  \\item An $A_I$-vertex and an $A_{I'}$-vertex are joined by edges (in both directions) if $|I \\cap I'| \\geq 2$. Notice that the events $A_I$ and $A_{I'}$ are independent when $|I \\cap I'| < 2$.\n  \\item An $A_I$-vertex and a $B_{M,R}$-vertex are joined by edges if there are distinct $i,j \\in I \\cap R$ for which the entry of $M$ that corresponds to the edge $(i,j)$ is nonzero. Notice that the events $A_I$ and $B_{M,R}$ are independent when such $i$ and $j$ do not exist.\n  \\item Every two distinct $B_{M,R}$-vertices are joined by edges.\n\\end{itemize}\nClearly, each event is mutually independent of all other events besides those adjacent to it in $D$, and thus $D$ is a dependency digraph for our events.\nObserve that every $A_I$-vertex is adjacent to at most ${h \\choose 2} \\cdot {n \\choose {h-2}} \\leq {h \\choose 2} \\cdot n^{h-2}$ $A_{I'}$-vertices.\nAdditionally, every $B_{M,R}$-vertex, where $M$ is an $n' \\times n'$ matrix of sparsity $s'$, is adjacent to at most $(s'-n') \\cdot {n \\choose {h-2}} < s' \\cdot n^{h-2}$ $A_{I}$-vertices.\nFinally, every vertex of $D$ is adjacent to at most $N_{s'}$ $B_{M,R}$-vertices with $M \\in {\\cal M}_{n,k}^{(s')}$ (that is, $s(M) = s'$).\n\nTo apply Lemma~\\ref{lemma:lll} we assign a number in $[0,1)$ to each vertex of $D$.\nDefine\n\\[ q = c_2 \\cdot n^{-\\gamma},~~~~x = c_3 \\cdot n^{-\\gamma \\cdot f},~~~~\\mbox{and}~~~~x_{s'} = \\exp(-c_4 \\cdot s' \\cdot n^{-\\gamma})~~~~\\mbox{for every $s' \\in {\\cal S}$},\\]\nwhere $c_2,c_3,c_4>0$ are constants, depending only on $H$, to be determined.\nWe assign the number $x$ to every $A_I$-vertex, and the number $x_{s'}$ to every $B_{M,R}$-vertex with $s(M)=s'$.\nWe present now the conditions of Lemma~\\ref{lemma:lll}.\nFor every $A_I$-vertex, recalling that $\\Prob{}{A_I} \\leq h! \\cdot (2q)^f$, we require\n\\begin{eqnarray}\\label{eq:lll_A}\nh! \\cdot (2q)^f \\leq x \\cdot (1-x)^{{h \\choose 2} \\cdot n^{h-2}} \\cdot \\prod_{s' \\in {\\cal S}}{(1-x_{s'})^{N_{s'}}}.\n\\end{eqnarray}\nSimilarly, for every $B_{M,R}$-vertex with $s(M)=s'$, recalling that $\\Prob{}{B_{M,R}} \\leq \\exp(-qs'\/2)$, we require\n\\begin{eqnarray}\\label{eq:lll_B}\n\\exp(-qs'\/2) \\leq x_{s'} \\cdot (1-x)^{s' \\cdot n^{h-2}} \\cdot \\prod_{s' \\in {\\cal S}}{(1-x_{s'})^{N_{s'}}}.\n\\end{eqnarray}\n\nTo complete the proof, it suffices to show that the constants $c_1,c_2,c_3,c_4>0$ can be chosen in a way that satisfies the inequalities~\\eqref{eq:lll_A} and~\\eqref{eq:lll_B}. Consider the following three constraints:\n\\begin{enumerate}\n  \\item\\label{itm:1} $c_2 > 2 \\cdot (2c_3+c_4)$,\n  \\item\\label{itm:2} $c_3 \\geq h! \\cdot (2c_2)^f \\cdot \\exp(3)$, and\n  \\item\\label{itm:3} $c_4 \\geq 32 \\cdot c_1$.\n\\end{enumerate}\nIt is easy to see that it is possible to choose the constants under the above constraints. Indeed, by $f \\geq 3$, for a sufficiently small choice of $c_2>0$ one can take $c_3$ with, say, an equality in Item~\\ref{itm:2} so that some $c_4>0$ satisfies Item~\\ref{itm:1}. Then, $c_1$ can be chosen as a positive constant satisfying Item~\\ref{itm:3}.\nWe show now that such a choice satisfies~\\eqref{eq:lll_A} and~\\eqref{eq:lll_B} for every sufficiently large $n$. Note that we use below several times the inequality $1-\\alpha \\geq \\exp(-2\\alpha)$, which holds for any $\\alpha \\in [0,1\/2]$.\n\nFirst, use~\\eqref{eq:N_s'} and the condition $c_4 \\geq 32 \\cdot c_1$ to obtain that\n\\[ \\sum_{s' \\in {\\cal S}}{x_{s'} \\cdot N_{s'}} \\leq \\sum_{s' \\in {\\cal S}}{\\exp((24c_1-c_4) \\cdot s' \\cdot n^{-\\gamma})} \\leq \\sum_{s' \\in {\\cal S}}{\\exp(-8c_1\\cdot s' \\cdot n^{-\\gamma})} \\leq\n\\sum_{s' \\in {\\cal S}}{\\exp(-2\\ln n)} \\leq 1, \\]\nwhere the third inequality follows by $s' \\geq \\frac{ n^{\\gamma} \\cdot \\ln (n \\cdot |\\Fset|)}{4c_1}$ which we get from~\\eqref{eq:s'}, and the fourth by $| {\\cal S}| \\leq n^2$.\nConsidering the term $\\prod_{s' \\in {\\cal S}}{(1-x_{s'})^{N_{s'}}}$, which appears in both~\\eqref{eq:lll_A} and~\\eqref{eq:lll_B},\nwe derive that\n\\[ \\prod_{s' \\in {\\cal S}}{(1-x_{s'})^{N_{s'}}} \\geq \\prod_{s' \\in {\\cal S}}{\\exp(-2x_{s'} \\cdot N_{s'})} = \\exp \\Big (-2 \\cdot \\sum_{s' \\in {\\cal S}}{x_{s'} \\cdot N_{s'}} \\Big ) \\geq \\exp(-2).\\]\nFor inequality~\\eqref{eq:lll_A}, observe that\n\\begin{eqnarray*}\nx \\cdot (1-x)^{{h \\choose 2} \\cdot n^{h-2}} \\cdot \\prod_{s' \\in {\\cal S}}{(1-x_{s'})^{N_{s'}}}\n& \\geq & x \\cdot \\exp \\Big ( -2x \\cdot {h \\choose 2} \\cdot n^{h-2} \\Big ) \\cdot \\exp(-2) \\\\\n& = & c_3 \\cdot n^{-\\gamma \\cdot f} \\cdot \\exp \\Big (-2c_3 \\cdot n^{-\\gamma \\cdot f} \\cdot {h \\choose 2} \\cdot n^{h-2} -2 \\Big) \\\\\n& \\geq & h! \\cdot (2c_2)^f \\cdot n^{-\\gamma \\cdot f} \\cdot \\exp \\Big(1-2c_3 \\cdot {h \\choose 2} \\cdot n^{-\\gamma} \\Big) \\\\\n& \\geq & h! \\cdot (2q)^f,\n\\end{eqnarray*}\nwhere for the second inequality we use $c_3 \\geq h! \\cdot (2c_2)^f \\cdot \\exp(3)$ and $\\gamma = \\frac{h-2}{f-1}$,\nand for the third we use the assumption that $n$ is sufficiently large.\nFor inequality~\\eqref{eq:lll_B}, observe that\n\\begin{eqnarray*}\nx_{s'} \\cdot (1-x)^{s' \\cdot n^{h-2}} \\cdot \\prod_{s' \\in {\\cal S}}{(1-x_{s'})^{N_{s'}}}\n& \\geq & x_{s'} \\cdot \\exp (-2x \\cdot s' \\cdot n^{h-2} ) \\cdot \\exp (-2) \\\\\n& = & \\exp(-c_4 \\cdot s' \\cdot n^{-\\gamma}) \\cdot \\exp (-2 c_3 \\cdot n^{-\\gamma \\cdot f} \\cdot s' \\cdot n^{h-2} ) \\cdot \\exp (-2) \\\\\n& = & \\exp ( -(2c_3+c_4) \\cdot s' \\cdot n^{-\\gamma} -2) \\\\\n& \\geq & \\exp (-(c_2\/2) \\cdot s' \\cdot n^{-\\gamma}) \\\\\n& = & \\exp (-qs'\/2),\n\\end{eqnarray*}\nwhere for the second equality we again use the definition of $\\gamma$, and for the second inequality we use the condition $c_2 > 2 \\cdot (2c_3+c_4)$, the fact that $s' \\cdot n^{-\\gamma} = \\omega(1)$ by~\\eqref{eq:s'}, and the assumption that $n$ is sufficiently large. This completes the proof.\n\\end{proof}\n\nWe can derive now Theorem~\\ref{thm:IntroComp}. Recall that $\\gamma_0(H) = \\min_{H'}{\\gamma(H')}$, where the minimum is over all subgraphs $H'$ of $H$ with at least $3$ edges.\n\n\\begin{proof}[ of Theorem~\\ref{thm:IntroComp}]\nFor a graph $H$ with $h \\geq 3$ vertices and $f \\geq 3$ edges, let $H'$ be a subgraph of $H$ with at least $3$ edges such that $\\gamma_0(H) = \\gamma(H')$.\nBy Theorem~\\ref{thm:Comp} there exists $c>0$ such that\n\\[g(n,H',\\Fset) \\geq c \\cdot \\frac{n^{1-\\gamma_0(H)}}{\\log (n \\cdot |\\Fset|)}\\]\nfor every integer $n$ and a finite field $\\Fset$. Since every $H'$-free graph is also $H$-free, it follows that $g(n,H,\\Fset) \\geq g(n,H',\\Fset)$ and we are done.\n\\end{proof}\n\n\\subsection{The Minrank of Graphs with Small Independence Number}\n\nFor an integer $t \\geq 3$, $g(n,K_t,\\Fset)$ is the maximum possible minrank over $\\Fset$ of an $n$-vertex graph with independence number smaller than $t$. For this case we derive the following corollary.\n\\begin{corollary}\\label{cor:K_t}\nFor every $t \\geq 3$ there exists $c=c(t)>0$ such that for every integer $n$ and a finite field $\\Fset$,\n\\[g(n,K_t,\\Fset) \\geq c \\cdot \\frac{n^{1-\\frac{2}{t+1}}}{\\log (n \\cdot |\\Fset|)} .\\]\n\\end{corollary}\n\n\\begin{proof}\nApply Theorem~\\ref{thm:IntroComp} to the graph $H = K_t$, and notice that $\\gamma_0(K_t) = \\gamma(K_t) = \\frac{t-2}{{t \\choose 2}-1} = \\frac{2}{t+1}$.\n\\end{proof}\n\nFor $H=K_3$, we observe that our lower bound on $g(n,K_3,\\Fset)$ is nearly tight.\n\\begin{proposition}\\label{prop:K_3}\nThere exist constants $c_1,c_2>0$ such that for every integer $n$ and a finite field $\\Fset$,\n\\[ c_1 \\cdot \\frac{\\sqrt{n}}{\\log (n \\cdot |\\Fset|)} \\leq g(n,K_3,\\Fset) \\leq  c_2 \\cdot \\sqrt{\\frac{n}{\\log n}}.\\]\n\\end{proposition}\n\n\\begin{proof}\nFor the lower bound apply Corollary~\\ref{cor:K_t} with $t=3$.\nTo prove the upper bound we need a result of Ajtai et al.~\\cite{AjtaiKS80} which says that every triangle-free $n$-vertex graph has an independent set of size $\\Omega(\\sqrt{n \\cdot \\log n})$. By repeatedly omitting such independent sets it follows that the chromatic number of such a graph is $O(\\sqrt{n \/ \\log n})$.\nNow, let $G$ be an $n$-vertex graph whose complement $\\overline{G}$ is triangle-free.\nWe get that ${\\mathop{\\mathrm{minrk}}}_{\\Fset}(G) \\leq \\chi(\\overline{G}) \\leq O(\\sqrt{n\/\\log n})$, as required.\n\\end{proof}\n\n\\section{Non-bipartite Graphs}\\label{sec:non-bip}\n\nIn this section we show that for every non-bipartite graph $H$ there are $H$-free graphs with low minrank over $\\mathbb{R}$, confirming Theorem~\\ref{thm:IntroNonBi}.\nWe start with the case where $H$ is an odd cycle, and since every non-bipartite graph contains an odd cycle the general result follows easily.\nThe proof is by an explicit construction from the following family of graphs.\n\n\\begin{definition}\\label{def:Kneser}\nFor integers $m \\leq s \\leq d$, the graph $\\Kneser{d}{s}{m}$ is defined as follows: the vertices are all the $s$-subsets of $[d]$, and two distinct sets $A,B$ are adjacent if $|A \\cap B| < m$.\n\\end{definition}\n\nThe minrank of such graphs over finite fields was recently studied in~\\cite{Haviv18} using tools from~\\cite{AlonBS91}.\nThe proof technique of~\\cite{Haviv18} can be used for the real field as well, as shown below.\n\n\\begin{proposition}\\label{prop:minrk_Kneser}\nFor every integers $m \\leq s \\leq d$,\n\\[{\\mathop{\\mathrm{minrk}}}_{\\mathbb{R}}(\\Kneser{d}{s}{m}) \\leq \\sum_{i=0}^{s-m}{d \\choose i}.\\]\n\\end{proposition}\n\n\\begin{proof}\nLet $f: \\{0,1\\}^d \\times \\{0,1\\}^d \\rightarrow \\mathbb{R}$ be the function defined by\n\\[ f(x,y) = \\prod_{j=m}^{s-1}{ \\Big ( \\sum_{i=1}^{d}{x_i y_i -j}\\Big )}\\]\nfor every $x,y \\in \\{0,1\\}^d$.\nExpanding $f$ as a linear combination of monomials, the relation $z^2 = z$ for $z \\in \\{0,1\\}$ implies that one can reduce to $1$ the exponent of each variable occuring in a monomial. It follows that $f$ can be represented as a multilinear polynomial in the $2d$ variables of $x$ and $y$. By combining terms involving the same monomial in the variables of $x$, one can write $f$ as\n\\[ f(x,y) = \\sum_{i=1}^{R}{g_i(x) h_i(y)} \\]\nfor an integer $R$ and functions $g_i, h_i : \\{0,1\\}^d \\rightarrow \\mathbb{R}$, $i \\in [R]$, such that the $g_i$'s are distinct multilinear monomials of total degree at most $s-m$ in $d$ variables. It follows that $R \\leq \\sum_{i=0}^{s-m}{d \\choose i}$.\n\nNow, let $M_1$ and $M_2$ be the $2^d \\times R$ matrices whose rows are indexed by $\\{0,1\\}^d$ and whose columns are indexed by $[R]$, defined by $(M_1)_{x,i} = g_i(x)$ and $(M_2)_{x,i} = h_i(x)$. Then, the matrix $M = M_1 \\cdot M_2^T$ has rank at most $R$ and for every $x,y \\in\\{0,1\\}^d$ it holds that $M_{x,y} = f(x,y)$.\n\nFinally, let $V$ be the vertex set of $\\Kneser{d}{s}{m}$, that is, the collection of all $s$-subsets of $[d]$, and identify every vertex $A \\in V$ with an indicator vector $c_A \\in \\{0,1\\}^d$ in the natural way. We claim that the matrix $M$ restricted to $V \\times V$ represents the graph $\\Kneser{d}{s}{m}$. Indeed, for every $A,B \\in V$ we have\n\\[M_{c_A, c_B} = f(c_A,c_B) = \\prod_{j=m}^{s-1}{ \\Big ({|A \\cap B| -j}\\Big )}.\\]\nHence, for every $A \\in V$ we have $|A|=s$ and thus $M_{c_A,c_A} \\neq 0$, whereas for every distinct non-adjacent $A,B \\in V$ we have $m \\leq |A \\cap B|\\leq s-1$ and thus $M_{c_A,c_B} = 0$. Since the restriction of $M$ to $V \\times V$ has rank at most $R$ it follows that ${\\mathop{\\mathrm{minrk}}}_\\mathbb{R}(\\Kneser{d}{s}{m}) \\leq R$, and we are done.\n\\end{proof}\n\nWe turn to identify graphs $\\Kneser{d}{s}{m}$ with no short odd cycles.\nFor this purpose, take an even integer $d$, $s = \\frac{d}{2}$, and $m = \\epsilon \\cdot d$ for a small constant $\\epsilon>0$.\nEvery path in these graphs is a sequence of $\\frac{d}{2}$-subsets of $[d]$ such that the intersection size of every two consecutive sets is small. This implies, for a sufficiently small $\\epsilon$, that the sets in the even positions of the path are almost disjoint from the first set, whereas the sets in the odd positions of the path share with it many elements, hence such a graph contains no short odd cycle.\nThis is shown formally in the following lemma.\n\n\\begin{lemma}\\label{lemma:cycle_K}\nLet $\\ell \\geq 3$ be an odd integer.\nFor every even integer $d$ and an integer $m \\leq \\frac{d}{2\\ell}$, the graph $\\Kneser{d}{\\frac{d}{2}}{m}$ contains no odd cycle of length at most $\\ell$.\n\\end{lemma}\n\n\\begin{proof}\nFix an odd integer $\\ell \\geq 3$, an even integer $d$, and an integer $m \\leq \\frac{d}{2\\ell}$.\nWe prove that for every odd integer $\\ell'$, such that $3 \\leq \\ell' \\leq \\ell$, the graph $\\Kneser{d}{\\frac{d}{2}}{m}$ contains no cycle of length $\\ell'$.\nFor such an $\\ell'$, let $A_1,A_2,\\ldots,A_{\\ell'}$ be a sequence of $\\ell'$ vertices in the graph, i.e., $\\frac{d}{2}$-subsets of $[d]$. Assuming that for every $i \\leq \\ell'-1$ the vertices $A_i$ and $A_{i+1}$ are adjacent in the graph, that is, $|A_i \\cap A_{i+1}| < m$, our goal is to show that $A_1$ and $A_{\\ell'}$ are not.\n\nTo this end, we argue that for every $i$, such that $0 \\leq i \\leq \\frac{\\ell'-1}{2}$, we have\n\\begin{eqnarray}\\label{eq:A_i}\n|A_1 \\cap A_{2i+1}| \\geq \\frac{d}{2}-2i \\cdot m.\n\\end{eqnarray}\nWe prove this claim by induction on $i$. The case $i=0$ follows immediately from $|A_1|=\\frac{d}{2}$.\nAssume that~\\eqref{eq:A_i} holds for $i-1$, that is, $|A_1 \\cap A_{2i-1}| \\geq \\frac{d}{2}-(2i-2) \\cdot m$.\nObserve that this implies that\n\\begin{eqnarray*}\n|A_1 \\cap A_{2i}| &=& | A_1 \\cap A_{2i} \\cap A_{2i-1} | + | A_1 \\cap A_{2i} \\cap \\overline{A_{2i-1}} | \\\\\n& \\leq & |A_{2i-1} \\cap A_{2i}| + | A_1 \\cap \\overline{A_{2i-1}} | \\\\\n& \\leq & m + |A_1|- | A_1 \\cap A_{2i-1} | \\\\\n& \\leq & m + \\frac{d}{2} - \\Big ( \\frac{d}{2}-(2i-2) \\cdot m \\Big ) = (2i-1) \\cdot m,\n\\end{eqnarray*}\nwhere in the second inequality we have used $|A_{2i-1} \\cap A_{2i}| < m$.\nWe proceed by proving~\\eqref{eq:A_i} for $i$. Observe that\n\\begin{eqnarray*}\n|A_1 \\cap A_{2i+1}| &=& |A_{2i+1}| - | \\overline{A_1} \\cap A_{2i+1} | \\\\\n&=& |A_{2i+1}| - | \\overline{A_1} \\cap A_{2i+1} \\cap A_{2i} | - | \\overline{A_1} \\cap A_{2i+1} \\cap \\overline{A_{2i}} | \\\\\n& \\geq & \\frac{d}{2} - m - | \\overline{A_1} \\cap \\overline{A_{2i}} |,\n\\end{eqnarray*}\nwhere we have used $|A_{2i} \\cap A_{2i+1}| < m$.\nNotice that\n\\[ |\\overline{A_1} \\cap \\overline{A_{2i}}| = d - |A_1 \\cup A_{2i}| = d-(|A_1|+|A_{2i}|-|A_1 \\cap A_{2i}|) = |A_1 \\cap A_{2i}|. \\]\nIt follows that\n\\[|A_1 \\cap A_{2i+1}| \\geq \\frac{d}{2}-m-|A_1 \\cap A_{2i}| \\geq \\frac{d}{2}-m-(2i-1)\\cdot m = \\frac{d}{2}-2i\\cdot m,\\]\ncompleting the proof of~\\eqref{eq:A_i}.\n\nFinally, applying~\\eqref{eq:A_i} to $i = \\frac{\\ell'-1}{2}$, using the assumption $m \\leq \\frac{d}{2\\ell}$, we get that\n\\[|A_1 \\cap A_{\\ell'}|  \\geq \\frac{d}{2}-(\\ell'-1) \\cdot m = \\frac{d}{2}-\\ell' \\cdot m+m \\geq  \\frac{d}{2}-\\ell \\cdot m +m \\geq m,\\]\nhence $A_1$ and $A_{\\ell'}$ are not adjacent in the graph $\\Kneser{d}{\\frac{d}{2}}{m}$. It thus follows that the graph contains no cycle of length $\\ell'$, as desired.\n\\end{proof}\n\nEquipped with Proposition~\\ref{prop:minrk_Kneser} and Lemma~\\ref{lemma:cycle_K}, we obtain the following.\n\n\\begin{theorem}\\label{thm:Cycles}\nFor every odd integer $\\ell \\geq 3$ there exists $\\delta = \\delta(\\ell) >0$ such that for every sufficiently large integer $n$, there exists an $n$-vertex graph $G$ with no odd cycle of length at most $\\ell$ such that\n\\[{\\mathop{\\mathrm{minrk}}}_{\\mathbb{R}}(G) \\leq n^{1-\\delta}.\\]\n\\end{theorem}\n\n\\begin{proof}\nFix an odd integer $\\ell \\geq 3$.\nFor an integer $d$ divisible by $2 \\ell$, consider the graph $G = \\Kneser{d}{\\frac{d}{2}}{m}$ where $m = \\frac{d}{2 \\ell}$.\nBy Lemma~\\ref{lemma:cycle_K}, $G$ contains no odd cycle of length at most $\\ell$.\nAs for the minrank, Proposition~\\ref{prop:minrk_Kneser} implies that\n\\[{\\mathop{\\mathrm{minrk}}}_{\\mathbb{R}}(G) \\leq \\sum_{i=0}^{d\/2-m}{d \\choose i} \\leq 2^{H(\\frac{1}{2}-\\frac{m}{d}) \\cdot d} = 2^{H(\\frac{1}{2}-\\frac{1}{2\\ell}) \\cdot d},\\]\nwhere $H$ stands for the binary entropy function.\nSince $G$ has $|V| = {d \\choose {d\/2}} = 2^{(1-o(1)) \\cdot d}$ vertices, for any $\\delta>0$ such that $H(\\frac{1}{2}-\\frac{1}{2\\ell}) < 1-\\delta$ we have ${\\mathop{\\mathrm{minrk}}}_{\\mathbb{R}}(G) \\leq |V|^{1-\\delta}$ for every sufficiently large integer $d$.\nThe proof is completed by considering, for every sufficiently large integer $n$, some $n$-vertex subgraph of the graph defined above, where $d$ is the smallest integer divisible by $2\\ell$ such that $n \\leq {d \\choose {d\/2}}$.\n\\end{proof}\n\n\nNow, Theorem~\\ref{thm:IntroNonBi} follows easily from Theorem~\\ref{thm:Cycles}.\n\n\\begin{proof}[ of Theorem~\\ref{thm:IntroNonBi}]\nLet $H$ be a non-bipartite graph. Then, for some odd integer $\\ell \\geq 3$, the cycle $C_\\ell$ is a subgraph of $H$.\nBy Theorem~\\ref{thm:Cycles}, there exists $\\delta > 0$ such that for every sufficiently large integer $n$, there exists an $n$-vertex $C_\\ell$-free graph $G$ satisfying ${\\mathop{\\mathrm{minrk}}}_{\\mathbb{R}}(G) \\leq n^{1-\\delta}$.\nSince every $C_\\ell$-free graph is also $H$-free, the result follows.\n\\end{proof}\n\n\\begin{remark}\nAs mentioned in the introduction, Theorem~\\ref{thm:IntroNonBi} implies a lower bound on $g(n,H,\\mathbb{R})$ for every non-bipartite graph $H$ (see Corollary~\\ref{cor:IntroNonBi}).\nWe note that upper bounds on certain Ramsey numbers can be used to derive upper bounds on $g(n,H,\\Fset)$ for a general field $\\Fset$.\nFor example, it was shown in~\\cite{ErdosFRS78}\nthat for every $\\ell \\geq 3$, every $n$-vertex $C_\\ell$-free graph has an independent set of size $\\Omega( n^{1-1\/k} )$ for $k = \\lceil \\frac{\\ell}{2} \\rceil$ (see~\\cite{CaroLRZ00,Sudakov02} for slight improvements).\nBy repeatedly omitting such independent sets it follows that the chromatic number of such a graph is $O(n^{1\/k})$.\nThis implies that every $n$-vertex graph $G$ whose complement is $C_\\ell$-free satisfies ${\\mathop{\\mathrm{minrk}}}_{\\Fset}(G) \\leq \\chi(\\overline{G}) \\leq O(n^{1\/k})$, hence $g(n,C_\\ell,\\Fset) \\leq O(n^{1\/k})$.\n\\end{remark}\n\n\n\\section*{Acknowledgements}\nWe are grateful to Alexander Golovnev and Pavel Pudl\\'ak for useful discussions and to the anonymous referees for their valuable suggestions.\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\nTheoretical calculations have predicted that dark matter density distribution would be altered by a massive black hole \\citep{Gondolo,Merritt2,Gnedin,Merritt,Sadeghian,Nampalliwar}. The conservation of angular momentum and energy would naturally force dark matter to form a dense spike (i.e. a cusp-like density profile) \\citep{Gondolo,Merritt2,Gnedin,Sadeghian}. Generally speaking, dark matter density around a black hole would eventually follow a simple power-law form: $\\rho_{\\rm DM} \\propto r^{-\\gamma}$, where $r$ is the radial distance from the black hole and $\\gamma$ is the spike index. The value of $\\gamma$ is model-dependent, which can range from $\\gamma=1.5$ to $\\gamma=2.5$ \\citep{Merritt2,Gnedin,Sadeghian,Fields,Lacroix}. Since the dark matter density profile is a singular form in $r$, the dark matter density near black hole would be very high (i.e. a dense spike).\n\nBased on this theoretical prediction, if dark matter can self-annihilate to give gamma-ray photons, one can expect that the annihilation gamma-ray signals would be greatly enhanced because the annihilation rate is proportional to $\\rho_{\\rm DM}^2$. A lot of attention has been paid specifically for the dark matter density spike surrounding galactic supermassive black holes \\citep{Gondolo,Gnedin,Fields,Bertone,Shapiro} and intermediate-mass black holes \\citep{Lacroix,Chan}. Various studies have been done to examine the possible enhanced gamma-ray signals, especially near the supermassive black hole in the Milky Way galaxy \\citep{Fields,Shapiro}. However, no promising signals have been observed to verify the theoretical prediction \\citep{Fields}. However, it does not mean that the dark matter density spike model is wrong. The negative result in gamma-ray observations could be due to the following reasons: 1. the rest mass of dark matter particles is very large, 2. the annihilation cross section is very small, or 3. dark matter particles do not self-annihilate.\n\nOn the other hand, observations and studies of the two closest black hole low-mass X-ray binaries (BH-LMXBs), A0620-00 and XTE J1118+480, have provided very precise measurements for many important physical parameters, including the orbital period $P$, observed radial velocity of the companion star $K$, orbital inclination $i$, black hole mass $M_{\\rm BH}$, and the mass of the companion star $m$ (or the mass ratio $q=m\/M_{\\rm BH}$) \\citep{McClintock,Neilsen,Cantrell,Grunsven,Khargharia,Zurita,Cherepashchuk} (see Table 1 for the measured values). The companion star is orbiting the black hole in a nearly circular orbit for each binary. In particular, observations have revealed abnormally fast orbital decays in the two BH-LMXBs: $\\dot{P}=-0.60 \\pm 0.08$ ms yr$^{-1}$ for A0620-00 and $\\dot{P}=-1.90 \\pm 0.57$ ms yr$^{-1}$ for XTE J1118+480 \\citep{Gonzalez}. These decays are two orders of magnitude larger than the one expected with gravitational wave radiation \\citep{Chen,Chen2}. Standard theories only predict $\\dot{P} \\sim -0.02$ ms yr$^{-1}$ \\citep{Gonzalez3}. Two major proposals have been suggested recently to account for the fast orbital decay. The first one is related to the magnetic braking of the companion star. If the surface magnetic field of the companion star is very strong (e.g. $\\ge 10^4$ G), the coupling between the magnetic field and the winds from the companion star driven by X-ray irradiation from the black hole would decrease the orbital period through tidal torques \\citep{Chen,Justham}. However, this model requires a significant mass loss from the binary system, which has not been observed \\citep{Gonzalez}. The second proposal suggests that the tidal torque between the circumbinary disk and the binary can efficiently extract the orbital angular momentum from the binary to cause the orbital decay \\citep{Chen}. Nevertheless, simulations show that the predicted mass transfer rate and the circumbinary disk mass are much greater than the inferred values from observations \\citep{Chen}. Although a few recent studies suggesting the resonant interaction between the binary and a surrounding circumbinary disk could produce the observed orbital period decays \\citep{Chen2,Xu}, the calculated initial mass and effective temperature of the companion stars somewhat do not match the observations \\citep{Chen2}. Therefore, it is still a mystery for the abnormally fast orbital decays in the two BH-LMXBs.\n\nBeside the annihilation rate, the dynamical friction due to dark matter density spike would also be very large. If a star is moving inside a collisionless dark matter background, the star would exert a gravitational force to pull the dark matter particles towards it. Then a concentration of the dark matter particles would locate behind the star and exert a collective gravitational force on the star. This collective gravitational force would slow down the star and the resulting effect is called dynamical friction. The idea of dynamical friction was proposed by Chandrasekhar more than 70 years ago \\citep{Chandrasekhar}. However, very surprisingly, the dynamical frictional effect in BH-LMXBs has not been seriously examined in previous studies. Most of the related studies are focusing on the compact binary systems \\citep{Antonini,Eda,Pani,Yue,Dai,Li,Becker,Speeney,Kavanagh}. Also, no previous study has realized the possible observable consequence of dark matter density spike surrounding a stellar-mass black hole. In this letter, we will discuss the observed fast orbital decays in the two closest BH-LMXBs with the idea of dynamical friction of dark matter density spike.\n\n\\section{The dynamical friction model}\nConsider a typical BH-LMXB system. The low-mass companion star with mass $m<1M_{\\odot}$ is orbiting a central black hole with mass $M_{\\rm BH}$ much greater than the stellar mass. The central black hole is almost stationary at the center of the system. If a dark matter density spike is surrounding the central black hole, the companion star would experience the dynamical friction exerted by dark matter. The energy loss due to dynamical friction would decrease the orbital period $P$ of the companion star.\n\n The energy loss due to dynamical friction is given by \\citep{Chandrasekhar,Yue}:\n\\begin{equation}\n\\dot{E}=- \\frac{4\\pi G^2\\mu^2 \\rho_{\\rm DM} \\xi(\\sigma) \\ln \\Lambda}{v},\n\\end{equation}\nwhere $\\mu$ is the reduced mass of the BH-LMXB, $\\ln \\Lambda \\approx \\ln (\\sqrt{M_{\\rm BH}\/m})$ is the Coulomb Logarithm \\citep{Kavanagh}, $v$ is the orbital velocity, and $\\xi(\\sigma)$ is a numerical factor which depends on the distribution function and the velocity dispersion $\\sigma$ of dark matter. If we assume a Maxwell's distribution for dark matter and take $\\sigma=200$ km\/s, we will have $\\xi(\\sigma) \\sim 0.9$. However, as the information about dark matter is uncertain, we simply assume $\\xi(\\sigma)=1$. The orbital velocity can be determined by the observed radial velocity $K$ and the orbital inclination $i$: $v=K\/\\sin i$.\n\nUsing the Keplerian relation $P^2=4\\pi^2a^3\/G(M_{\\rm BH}+m)$ with $a$ being the radius of the orbital motion, we can write\n\\begin{equation}\n\\frac{\\dot{P}}{P}=\\frac{3 \\dot{a}}{2a}=-\\frac{3 \\dot{E}}{2E},\n\\end{equation}\nwhere $E=-GM_{\\rm BH}m\/2a$ is the total mechanical energy. Therefore, the orbital decay rate can be expressed in terms of the observed parameter set \\{ $q$, $K$, $i$, $P$, $M_{\\rm BH}$ \\} by:\n\\begin{equation}\n\\dot{P}=- \\frac{12\\pi qGP \\ln \\Lambda}{(1+q)^2(K\/\\sin i)} \\left[\\frac{GM_{\\rm BH}(1+q)P^2}{4\\pi^2} \\right]^{1\/3} \\rho_{\\rm DM},\n\\end{equation}\nwhere $q=m\/M_{\\rm BH}$ is the mass ratio.\n\nFollowing the dark matter density spike theory, dark matter would re-distribute to form a density spike around the black hole in the BH-LMXB within the spike radius $r_{\\rm sp}$. We follow the standard assumption $r_{\\rm sp}=0.2r_{\\rm in}$ used in many other studies \\citep{Fields,Eda}, where $r_{\\rm in}$ is the radius of black hole's sphere of influence. Outside $r_{\\rm sp}$, the dark matter density would follow the local dark matter density of their respective positions in the Milky Way. The dark matter density around the black hole with mass $M_{\\rm BH}$ can be modeled by the following profile \\citep{Lacroix}:\n\\begin{equation}\n\\rho_{\\rm DM}=\\left\\{\n\\begin{array}{ll}\n0 & {\\rm for }\\,\\,\\, r\\le 2R_s \\\\\n\\rho_0 \\left(\\frac{r}{r_{\\rm sp}} \\right)^{-\\gamma} & {\\rm for }\\,\\,\\, 2R_s <r \\le r_{\\rm sp}, \\\\\n\\rho_0 & {\\rm for}\\,\\,\\, r>r_{\\rm sp} \\\\\n\\end{array}\n\\right.\n\\end{equation}\nwhere $R_s=2GM_{\\rm BH}\/c^2$, and $\\rho_0$ is the local dark matter density. When the distance from the black hole is larger than the spike radius $r_{\\rm sp}$, we assume that the dark matter density would follow back to the local dark matter density. By taking the reference value at the solar position $\\rho_{\\odot}=0.33 \\pm 0.03$ GeV cm$^{-3}$ \\citep{Ablimit} and following the Navarro-Frenk-White dark matter density profile \\citep{Navarro}, the local dark matter densities of A0620-00 and XTE J1118+480 can be determined by their respective positions \\citep{Gonzalez2}: $\\rho_0=0.29 \\pm 0.03$ GeV cm$^{-3}$ for A0620-00 and $\\rho_0=0.34 \\pm 0.03$ GeV cm$^{-3}$ for XTE J1118+480.\n\nThe radius of influence can be determined by \\citep{Merritt,Merritt2}:\n\\begin{equation}\nM_{\\rm DM}(r\\le r_{\\rm in})=\\int_0^{r_{\\rm in}}4 \\pi r^2\\rho_{\\rm DM}dr=2M_{\\rm BH}.\n\\end{equation}\nTherefore, the spike radius $r_{\\rm sp}$ is also a function of $M_{\\rm BH}$.  Note that the spike density profile assumed here is not an ad hoc parametrization, but follows from theoretical calculations \\citep{Gondolo,Sadeghian}. It is mainly determined by the black hole mass.\n\nThe spike index $\\gamma$ is the only free parameter in this analysis. For a spike of collisionless dark matter that forms about an adiabatically growing black hole, we have $\\gamma=2.25-2.5$ \\citep{Gondolo,Fields}. However, if gravitational scattering of stars is important, the stellar heating effect would drive the value of $\\gamma$ down to a minimum value $\\gamma=1.5$ \\citep{Merritt2,Gnedin}. Such a change in the spike index depends on the heating time scale, which is given by \\citep{Merritt2}:\n\\begin{eqnarray}\nt_{\\rm heat}&=&\\frac{\\sqrt{3\\pi} \\Gamma(0.5)M_{\\rm BH}}{18m \\ln \\Lambda} \\left(\\frac{GM_{\\rm BH}}{r_{\\rm in}^3} \\right)^{-1\/2}=1.2 \\times 10^{15}~{\\rm s} \\nonumber\\\\\n&&\\times \\left(\\frac{M_{\\rm BH}}{5M_{\\odot}} \\right)^{1\/2}\\left(\\frac{r_{\\rm in}}{5~\\rm pc} \\right)^{3\/2} \\left(\\frac{m}{M_{\\odot}} \\right)^{-1} \\left(\\frac{\\ln \\Lambda}{3} \\right)^{-1},\n\\end{eqnarray}\nHere, a constant stellar density and an initial dark matter spike index $\\gamma=2.5$ are assumed \\citep{Merritt2}. Generally speaking, for the age of the black hole $t_{\\rm BH} \\ge t_{\\rm heat}$, the spike index would approach the minimum value $\\gamma=1.5$ more likely.\n\n\\section{Results}\n\\subsection{Constraints on the spike index}\nThe analytic formula gives $\\dot{P}$ in terms of the precisely measured parameters \\{ $q$, $K$, $i$, $P$, $M_{\\rm BH}$ \\}. We find that the typical values of these parameters \\{ $0.05$, $500$ km\/s, $45^{\\circ}$, $1$ day, $5M_{\\odot}$ \\} in BH-LMXBs can give $\\dot{P} \\sim -1$ ms yr$^{-1}$ for a typical dark matter spike density $\\rho_{\\rm DM} \\sim 10^{-13}$ g cm$^{-3}$. For our two target BH-LMXBs, we put the corresponding measured parameters and the observed orbital decay rates to constrain the dark matter densities at the respective companion stellar orbits (with radius $a$): $\\rho_{\\rm DM}(a) \\approx 7.65^{+1.62}_{-1.43} \\times 10^{-13}$ g cm$^{-3}$ (A0620-00) and $\\rho_{\\rm DM}(a) \\approx 1.60^{+1.51}_{-0.73} \\times 10^{-11}$ g cm$^{-3}$ (XTE J1118+480). Following our dark matter density spike model and involving the uncertainties of the measured parameters, we get $\\gamma=1.71^{+0.01}_{-0.02}$ for A0620-00 and $\\gamma=1.85^{+0.04}_{-0.04}$ for XTE J1118+480 (see Fig.~1 for the general relation between $\\dot{P}$ and $\\gamma$).\n\nAs mentioned above, theoretical predictions give $1.5 \\le \\gamma \\le 2.5$ \\citep{Merritt2,Gnedin,Fields,Lacroix}. If the effects of baryons or stellar heating are important, the spike index might be close to the smallest extreme value $\\gamma=1.5$ \\citep{Merritt2,Gnedin,Fields}. In fact, this stellar heating effect is due to the dynamical friction between stars and dark matter. Therefore, in a BH-LMXB, the continuous gravitational scattering between the companion star and dark matter might provide the similar stellar heating effect to reduce the spike index to a smaller value. Nevertheless, the case for BH-LMXB is somewhat different from the stellar heating scenario discussed in \\citet{Merritt2,Gnedin}. There is only one companion object in a BH-LMXB while many stars are involved in the stellar heating scenario. However, recent simulations show that the dynamical friction of the companion object with a large mass ratio $q$ would increase the kinetic energy of the dark matter particles in the halo and somewhat decrease the dark matter density \\citep{Kavanagh}, which apparently reduces the spike index. Although this is not identical to the stellar heating scenario, both processes involve the dynamical friction to re-distribute the dark matter density.\n\n Using the stellar heating scenario as an analogy, we expect that the spike index might be smaller if $t_{\\rm BH} \\ge t_{\\rm heat}$. Using Eq.~(6), the heating time scales for A0620-00 and XTE J1118+480 are $t_{\\rm heat}=3.5\\times 10^{15}$ s and $t_{\\rm heat}=6.1\\times 10^{15}$ s respectively. Although we do not know the ages of the black holes in the BH-LMXBs, we can assume $t_{\\rm BH} \\le P\/\\dot{P}$. It is because if $t_{\\rm BH}>P\/\\dot{P}$, it should be highly improbable for us to observe A0620-00 and XTE J1118+480 now as both systems would have collapsed very likely within the cosmological age of 13.7 Gyr ($4.3\\times 10^{17}$ s), unless a significant change of mass transfer rate has occurred. If we follow this assumption, we can find that the upper limit of $t_{\\rm BH}$ for the A0620-00 black hole is the same order of magnitude as the heating time scale while the upper limit of $t_{\\rm BH}$ for the XTE J1118+480 black hole is about 20 times smaller than the heating time scale (see Table 2). This may explain why the spike index for A0620-00 is smaller. Therefore, our results reveal a consistent picture for the dark matter spike model and provide a very good explanation for the abnormally fast orbital decay in the two closest BH-LMXBs. Note that our major conclusion still holds even if the stellar heating scenario is not a good analogy.\n\n\\subsection{The effect of dark matter annihilation}\nWe did not assume any dark matter annihilation in the above discussion. If dark matter annihilation rate is large enough, the central dark matter density would approach the constant saturation density $\\rho_{\\rm sat}=m_{\\rm DM}\/\\langle \\sigma v \\rangle t_{\\rm BH}$ \\citep{Lacroix} when $\\rho_0(r\/r_{\\rm in})^{-\\gamma}> \\rho_{\\rm sat}$, where $m_{\\rm DM}$ is the mass of a dark matter particle and $\\langle \\sigma v \\rangle$ is the annihilation cross section. If the orbital decays originate from the dynamical friction of dark matter with the saturation density (i.e. the orbital radius is smaller than the saturation radius), we can determine the upper limits of dark matter mass for this particular scenario. Taking the thermal annihilation cross section $\\langle \\sigma v \\rangle=2.2\\times 10^{-26}$ cm$^3$\/s predicted by standard cosmology \\citep{Steigman} and the upper limits of $t_{\\rm BH}$, we can get $m_{\\rm DM} \\le 14$ GeV for A0620-00 and $m_{\\rm DM} \\le 48$ GeV for XTE J1118+480 if the companion stars are moving in the dark matter saturation density region. In other words, if $m_{\\rm DM}>48$ GeV, the companion stars in both systems would be orbiting the corresponding black hole in the dark matter density spike region. Since many recent stringent constraints of thermal annihilating dark matter indicate $m_{\\rm DM} \\ge 100$ GeV \\citep{Ackermann,Chan2,Abazajian,Regis}, the dark matter density would not be saturated at the orbital positions in A0620-00 and XTE J1118+480.\n\n\\section{Discussion}\nThe existence of dark matter density spike surrounding a black hole has been suggested for more than two decades. However, no smoking-gun evidence has been obtained from observations. Here, we show that the effect of dynamical friction due to dark matter density spike can satisfactorily explain the fast orbital decay in the two closest BH-LMXBs.  The resultant spike index is $\\gamma=1.7-1.8$, which is close to the value predicted by the stellar heating model ($\\gamma=1.5$) \\citep{Merritt2,Gnedin}. Although the BH-LMXBs considered here are not identical to the stellar heating scenario discussed in \\citet{Gnedin}, recent simulations of the compact-object inspirals show that the motion of the companion object would affect the distribution of the dark matter density spike surrounding an intermediate-mass black hole, especially for the mass ratio $q>10^{-3}$ \\citep{Kavanagh}. Therefore, we may also see similar results of the stellar heating effect in the BH-LMXBs. Note that although the dark matter density is changing in time during re-distribution, the dynamical friction expression used in Eq.~(1) is still applicable because the change is very slow in time \\citep{Kavanagh}. An overall consistent picture can be described as follows. When the black hole in a BH-LMXB is formed, the surrounding dark matter would be re-distributed to form a density spike (probably with an initial spike index $\\gamma \\approx 2-2.5$) \\citep{Gondolo}. However, the dynamical friction between dark matter and the companion star eventually help re-distribute the dark matter density spike again to reduce the spike index to approach $\\gamma=1.7-1.8$. The orbital period is also decreasing with a fast rate $\\sim 1$ ms yr$^{-1}$ due to dynamical friction. If the age of the black hole is larger than the heating time scale, the final spike index may change to a smaller value.\n\nWe can get very small uncertainties in $\\gamma$ because the uncertainties of the measured parameters are very small, especially for A0620-00.  The uncertain factor $\\xi(\\sigma)$ would only change the resulting spike index slightly. Generally speaking, our results may suggest a possible evidence of the existence of dark matter density spike surrounding a black hole. It also suggests that a dark matter density spike might exist around a stellar-mass black hole ($M_{\\rm BH} \\sim 1-10M_{\\odot}$), but not only around a supermassive black hole \\citep{Gondolo,Merritt2,Gnedin,Lacroix2} or an intermediate-mass black hole \\citep{Lacroix,Dai,Li} as suggested in the past literature. Since no previous study has focused on the case of dark matter density spike around a stellar-mass black hole, the effect of dark matter dynamical friction has also been neglected. In fact, one recent study has proposed that the electron excess detected by the DAMPE experiment might originate from the annihilating dark matter density spike in A0620-00 \\citep{Chan3}. Therefore, analyzing the effect of dark matter dynamical friction in BH-LMXBs would open a new independent way for investigating the dark matter distribution near stellar-mass black holes.\n\nMoreover, if dark matter annihilation effect is important so that the central dark matter density becomes saturated, we can calculate the upper limits of dark matter mass for this particular scenario. Since the calculated upper limits of thermal annihilating dark matter mass $m_{\\rm DM}$ are generally smaller than the lower limits constrained from recent multi-wavelength studies, the companion stars should be orbiting inside the dark matter density spike rather than the saturation density. In other words, the effect of annihilation is not important in constraining the spike index.\n\nIn fact, analyzing the effect of dynamical friction of dark matter density spike in a binary system is not a new idea. Nevertheless, most of the related studies have focused on the binaries of the compact objects (e.g. black hole binaries) rather than the BH-LMXB systems \\citep{Eda,Pani,Yue,Dai,Li,Becker,Speeney,Kavanagh}. In compact binaries, both gravitational radiation and dynamical friction of dark matter are significant. Therefore, gravitational wave detection might be required to reveal the nature of dark matter, which might contribute extra uncertainties in the constrained parameters. Since optical and X-ray observations can give very precise measurements for most of the important physical parameters in BH-LMXBs, we anticipate that analyzing BH-LMXBs can better reveal the nature of the dark matter density spike surrounding a black hole. There are at least 18 black hole X-ray binaries in our Galaxy \\citep{Chen}, which can give rich information to constrain the nature of dark matter. For example, one nearby black hole X-ray binary Nova Muscae 1991 also shows an abnormally fast orbital decay $\\dot{P}=-20.7 \\pm 12.7$ ms yr$^{-1}$, although the uncertainty is quite large \\citep{Gonzalez3}. Future high quality measurements may be helpful to further confirm the existence of dark matter density spike in these black hole X-ray binaries. This kind of analyses would open an entirely new window for observations and theoretical studies to investigate dark matter astrophysics \\citep{Bertone2}.\n\n\\begin{table}\n\\caption{The measured parameters of A0620-00 and XTE J1118+480.}\n\\begin{tabular}{ |l|l|l|}\n \\hline\\hline\n        & A0620-00 & XTE J1118+480 \\\\\n \\hline\n  $M_{\\rm BH}$ & $5.86 \\pm 0.24 M_{\\odot}$ \\citep{Grunsven} & $7.46^{+0.34}_{-0.69}M_{\\odot}$ \\citep{Gonzalez} \\\\\n  $q$ & $0.060 \\pm 0.004$ \\citep{Grunsven} & $0.024 \\pm 0.009$ \\citep{Khargharia} \\\\\n  $K$ (km\/s) & $435.4 \\pm 0.5$ \\citep{Neilsen} & $708.8 \\pm 1.4$ \\citep{Khargharia} \\\\\n  $i$ & $54^{\\circ}.1 \\pm 1^{\\circ}.1$ \\citep{Grunsven} & $73^{\\circ}.5 \\pm 5^{\\circ}.5$ \\citep{Khargharia} \\\\\n  $P$ (day) & $0.32301415(7)$ \\citep{Gonzalez} & $0.16993404(5)$ \\citep{Gonzalez} \\\\\n  $\\dot{P}$ (ms yr$^{-1}$) & $-0.60 \\pm 0.08$ \\citep{Gonzalez} & $-1.90 \\pm 0.57$ \\citep{Gonzalez} \\\\\n  $d$ (kpc) & $1.06 \\pm 0.12$ \\citep{Gonzalez2} & $1.70 \\pm 0.10$ \\citep{Gonzalez2} \\\\\n \\hline\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{ The orbital radius $a$, the calculated dark matter density at $a$, the spike index $\\gamma$, the radius of influence $r_{\\rm in}$, the heating time scale $t_{\\rm heat}$, and the upper limit of the black hole age $t_{\\rm BH}$ for each BH-LMXB based on the dark matter density spike model.}\n\\begin{tabular}{ |l|l|l|}\n \\hline\\hline\n         & A0620-00 & XTE J1118+480 \\\\\n\\hline\n$a$ (AU) & $0.0169^{+0.0003}_{-0.0002}$ & $0.0118^{+0.0002}_{-0.0004}$ \\\\\n$\\rho_{\\rm DM}(a)$ (g cm$^{-3}$) & $7.65^{+1.62}_{-1.43} \\times 10^{-13}$  & $1.60^{+1.51}_{-0.73} \\times 10^{-11}$  \\\\\n$\\gamma$ &  $1.71^{+0.01}_{-0.02}$ & $1.85^{+0.04}_{-0.04}$ \\\\\n$r_{\\rm in}$ (pc) & $5.41^{+0.10}_{-0.09}$ & $5.34^{+0.02}_{-0.06}$ \\\\\n$t_{\\rm heat}$ (s) & $3.5 \\times 10^{15}$ & $6.1 \\times 10^{15}$ \\\\\n$t_{\\rm BH}$ (s) & $\\le 1.7\\times 10^{15}$ & $\\le 3.5 \\times 10^{14}$ \\\\\n  \\hline\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}\n\\vskip 10mm\n\\includegraphics[width=140mm]{power2.eps}\n\\caption{The black and red solid lines indicate the relation between $\\gamma$ and $\\dot{P}$ for A0620-00 and XTE J1118+480 respectively. The horizontal dashed lines and dotted lines represent the mean values and the $1\\sigma$ limits of the observed orbital decay rates (black: A0620-00; red: XTE J1118+480).}\n\\label{Fig1}\n\\vskip 5mm\n\\end{figure}\n\n\\section{Acknowledgements}\nWe thank the anonymous referees for useful comments. The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. EdUHK 18300922).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\nMost of the energy released in the nuclear fission process appears in the kinetic energy of the fission fragments.\n  A first order estimate of the magnitude of the total kinetic energy release is that of the Coulomb energy of the fragments at scission, i.e., \n\\begin{equation}\nV_{Coul}=\\frac{Z_{1}Z_{2}e^{2}}{r_{1}+r_{2}}\n\\end{equation}\nwhere Z$_{n}$, r$_{n}$ are the atomic numbers and radii of fragments 1 and 2.  Recognizing that the fragments are deformed at scission, one can re-write equation 1 as \n\\begin{equation}\nTKE=\\frac{Z_{1}Z_{2}e^{2}}{1.9(A_{1}^{1\/3}+A_{2}^{1\/3})}\n\\end{equation}\nwhere the coefficient 1.9 (instead of the usual 1.2 - 1.3) represents the fragment deformation.  For symmetric fission, Z$_{1}$=Z$_{2}$=Z\/2 and A$_{1}$ =A$_{2}$=A\/2, then we have\n\\begin{equation}\nTKE = (0.119)\\frac{Z^{2}}{A^{1\/3}}MeV\n\\end{equation}\nTrajectory calculations \\cite{raja} for alpha particle emission in fission have shown that the fission\n fragments are in motion at scission with a pre-scission kinetic energy of 7.3 MeV and an additive term representing this motion is needed. \n Thus we have the ``Viola systematics\" \\cite{vic} that say \n\\begin{equation}\nTKE = (0.1189\\pm 0.0011)\\frac{Z^{2}}{A^{1\/3}}+7.3(\\pm 1.5)MeV\n\\end{equation}\n\n\nThe deformed scission point fragments will contract to their equilibrium deformations and the energy\n stored in deformation will be converted into internal excitation energy.  Thus we can define a related quantity,  the total excitation energy , TXE, in fission as\n\\begin{equation}\nTXE=Q-TKE\n\\end{equation}where Q is the mass-energy release.  One quickly realizes that these quantities depend\n on the mass split in fission which in turn, at low excitation energies, may reflect the fragment nuclear structure.  The TXE is the starting point for calculations of the prompt neutron and gamma emission in fission, the yields of beta emitting fission fragments, reactor anti-neutrino spectra, etc.  As such, it is a fundamental property of all fissioning systems and sadly not very well known.\n\nAs a practical matter, one needs to know the dependence of the TKE and TXE on neutron \nenergy for the neutron induced fission of technologically important actinide fissioning systems\n like $^{233}$U(n,f),$^{235}$U(n,f), and $^{239}$Pu(n,f).  The first question we might pose is\n whether the TKE should depend on the excitation energy of the fissioning system. \n Does the energy brought in by an incident neutron in neutron induced fission appear\n in the fragment excitation energy or does it appear in the total kinetic energy? \n In a variety of experiments, one finds that increasing the excitation energy of the \nfissioning system does not lead to significant increases in the TKE of the fission \nfragments or changes in the fragment separation at scission. \\cite{VH}.  However,\n there may be more subtle effects that render this statement false in some circumstances. \n For example, we expect, on the basis of the Coulomb energy systematics given above, \n that the TKE will be proportional to changes in the fission mass splits which in turn can depend on the excitation energy.\n    \nFor the technologically important reaction $^{235}$U(n,f), Madland \\cite{dave} summarizes the known data \\cite{straede, meadows, muller}with the following equations\n\\begin{equation}\n\\left\\langle T_{f}^{tot}\\right\\rangle =\\left( 170.93\\pm 0.07\\right) -\\left(\n0.1544\\pm 0.02\\right) E_{n}(MeV)\n\\end{equation}\n\\begin{equation}\n\\left\\langle T_{p}^{tot}\\right\\rangle =\\left( 169.13\\pm 0.07\\right) -\\left(\n0.2660\\pm 0.02\\right) E_{n}(MeV) \n\\end{equation}\nwhere E$_{n}$ is the energy of the incident neutron and T$_{f}^{tot}$and T$_{p}^{tot}$ are the\n average total fission fragment kinetic energy (before neutron emission) and the average fission\n product kinetic energy after neutron emission, respectively.  These quantities are related by the relation\n\\begin{equation}\n\\left\\langle T_{p}^{tot}(E_{n}\\right\\rangle =\\left\\langle\nT_{f}^{tot}(E_{n}\\right\\rangle \\left[ 1-\\frac{\\overline{\\nu _{p}}(E_{n})}{2A}%\n\\left( \\frac{\\left\\langle A_{H}\\right\\rangle }{\\left\\langle\nA_{L}\\right\\rangle }+\\frac{\\left\\langle A_{L}\\right\\rangle }{\\left\\langle\nA_{H}\\right\\rangle }\\right) \\right] \n\\end{equation}\nThese data show a modest decrease in TKE with increasing excitation energy for the neutron\n energy interval E$_{n}$ =1-9 MeV.  There is no clearly identified changes in the TKE values\n near the second chance fission threshold, a feature that is important in semi-empirical models\n of fission such as represented by the GEF code.\\cite{khs}  \n\nIn this paper, we report the results of measuring the total kinetic energy release in the neutron \ninduced fission of $^{235}$U for neutron energies E$_{n}$ = 3.2 -50  MeV.  The method used for the \nmeasurement is the 2E method, i.e., measurement of the kinetic energies of the two coincident fission \nproducts using semiconductor detectors.  The time of flight of the neutrons inducing fission was measured, \nallowing deduction of their energy.  The details of the experiment are discussed in Section II while the \nexperimental results and a comparison of the results \nwith various models and theories is made in Section III with conclusions being summarized in Section IV.\n\n\\section{Experimental}\n\nThis experiment was carried out at the Weapons Neutron Research Facility (WNR) at the Los Alamos Neutron Science Center (LANSCE) at the Los Alamos  National Laboratory \\cite{Lis, Liso}. ``White spectrum\" neutron beams were generated from an unmoderated tungsten spallation source using the 800 MeV proton beam from the LANSCE linac.  The  experiment was located on the 15R  beam line (15$^{\\circ}$-right with respect to the proton beam).  The calculated (MCNPX) ``white spectrum \" at the target position is shown in figure 1.  \\cite{snow} The proton beam is pulsed allowing one to measure the time of flight (energy) of the neutrons arriving at the experimental area.\n\nA schematic diagram of the experimental apparatus is shown in figure 2. The neutron beam was collimated to a 1 cm diameter at the entrance to the experimental area. At the entrance to the scattering chamber, the beam diameter was measured to be 1.3 cm. A fission ionization chamber \\cite{steve} was used to continuously monitor the absolute neutron beam intensities. The $^{235}$U target and the Si PIN diode fission detectors were housed in an  evacuated, thin-walled aluminum scattering chamber. The scattering chamber was located $\\sim$ 3.1 m from the collimator, and $\\sim$ 11 m from the neutron beam dump.  The center of the scattering chamber was located 16.46 m from the production target.\n\nThe $^{235}$U target consisted of a deposit of $^{235}$UF$_{4}$ on a thin C backing.  The thickness of the $^{235}$U was 175.5 $\\mu$g $^{235}$U\/cm$^{2}$ while the backing thickness was 100 $\\mu$g\/cm$^{2}$.  The isotopic purity of the $^{235}$U was 99.91 $\\%$.  The target was tilted at 50 $^{\\circ}$ with respect to the incident beam.\n\nFission fragments were detected by two arrays of Si PIN photodiodes (Hamamatsu S3590-09) arranged on opposite sides of the beam.  The area of the individual PIN diodes was 1 cm$^{2}$. The distance of the detectors from the target varied with angle from 2.60 cm to 4.12 cm.   The coincident detector pairs were at approximately 45, 60, 90, 115, and 135 $^{\\circ}$.  The alpha particle energy resolution of the diodes was 18 keV for the 5475 keV line of $^{241}$Am. \n\nThe time of flight of each interacting neutron was measured using a timing pulse from a Si PIN diode and the accelerator RF signal.  Absolute calibrations of this time scale were obtained from the photofission peak in the fission spectra and the known flight path geometry.  \n\nThe energy calibration of the fission detectors was done with a $^{252}$Cf source.  We have used the traditional Schmitt method  \\cite{hal}.  Some have criticized this method especially for PIN diodes.  However with our limited selection of detectors, we were unable to apply the methods of \\cite{moz} to achieve a robust substitute for the Schmitt method.\n\nThe measured fragment energies have be to be corrected for energy loss in the $^{235}$UF$_4$ deposit and the C backing foil. This correction was done by scaling the energy loss correction given by the Northcliffe-Schilling energy loss tables \\cite{NS} to a measured mean energy loss of collimated beams of light and heavy $^{252}$Cf fission fragments in 100 $\\mu$ g\/cm$^{2}$ C foils. The scaling factor that was used was a linear function of mass using the average loss of the heavy and light fission fragments as anchor points. The correction factors at the anchor points were 1.24 and 1.45 for the heavy and light fragments, respectively. Similar factors were obtained if the SRIM code \\cite{srim} was used to calculate dE\/dx. These large deviation factors from measured to calculated fission fragment stopping powers have been observed in the past \\cite{Knyazheva}, and represent the largest systematical uncertainty in the determination of the kinetic energies.  \n\n\n\\section{Results and Discussion}\n\nThe measured average post-neutron emission fission product total kinetic energy release for the $^{235}$U(n,f) reaction(Table 1) is shown in Figure 3 along with other data and predictions\n\\cite{gunn, kapoor,  stevenson}.  The evaluated post-neutron emission data from Madlund \\cite{dave} are shown as a dashed line while the individual pre-neutron emission measurements of \\cite{muller} are shown as points.   The point at E$_{n}$ =14 MeV is the average of \\cite{gunn} and \\cite{stevenson}. The slope of the measured TKE release (this work) is in rough agreement with the previous measurements \\cite{dave} at lower energies. Also shown are the predictions of the GEF model \\cite{khs}.  GEF is a semi-empirical model of fission that provides a good description of fission observables using a modest number of adjustable parameters. The dashed line in Figure 1 is a semi-empirical equation (TKE = 171.5 -0.1E* for E* $>$ 9 MeV) suggested by Tudora et al. \\cite{tudy} Qualitatively the decrease in TKE with increasing neutron energy reflects the increase in symmetric fission (with its lower associated TKE release) with increasing excitation energy.  This general dependence is reflected in the GEF code predictions with the slope of our data set being similar to  the predictions of the GEF model but with  the absolute values of the TKE release being  substantially less. \n\nIn Figure 4, we show some typical TKE distributions along with Gaussian representations of the data.  In general, the TKE distributions appear to be Gaussian in shape.  This is in contrast to previous studies \\cite{PR,D} which showed a sizable skewness in the distributions.\n\nIn Figure 5, we show the dependence of the measured values of the variance of the TKE distributions as a function of neutron energy along with the predictions of the GEF model of the same quantity. The measured variances are larger than expected.  \nAt low energies (near the second chance fission threshold) the observed variances show a dependence on neutron energy similar to that predicted by the GEF model, presumably reflecting the changes in variance with decreasing mass asymmetry.  At higher energies (11-50 MeV) the variances are roughly constant with changes in neutron energy.  Models \\cite{poop} would suggest that most of the variance of the TKE distribution is due to fluctuations in the nascent fragment separation at scission.  The constancy of the variances  is puzzling.\n\nUsing the Q values predicted by the GEF code, one can make a related plot (Fig. 6) of the TXE values in the $^{235}$U(n,f) reaction.  The ``bump\" in the TXE at lower neutron energies is pronounced and the dependence of the TXE upon neutron energy agrees with the GEF predictions although the absolute values are larger.\n\n\\section{Conclusions}\n\nWe conclude that : (a) For the first time, we have measured the TKE release and its variance for the technologically important $^{235}$U(n,f) reaction over a large range of neutron energies (3.2 - 50 MeV).  (b) The dependence of the TKE upon E$_{n}$ seems to agree with semi-empirical models although the absolute value does not.  (c) Understanding the variance and its energy dependence for the TKE distribution remains a challenge.\n\n\\begin{acknowledgments}\n\nThis work was supported in part by the\nDirector, Office of Energy Research, Division of Nuclear \nPhysics of the Office of High Energy and Nuclear Physics \nof the U.S. Department of Energy\nunder Grant DE-FG06-97ER41026.  One of us (WL) wishes to  thank the [Department of Energy's]\n Institute for Nuclear Theory at the University of Washington for its hospitality\n and the Department of Energy for partial support during the completion of this work.\n This work has benefited from the use of the Los Alamos Neutron Science Center at the Los Alamos National Laboratory.  This facility is funded by the U. S. Department of Energy under DOE Contract No. DE-AC52-06NA25396.\n \n \\end{acknowledgments}\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nThe Milky Way (MW) galaxy is one of the most important laboratories for studying galaxy formation and cosmology,\ngiven the abundant information available from its well-resolved constituents \\citep{Bland-Hawthorn2016a}.\nIn the current hierarchical structure formation framework,\nthe properties of a galaxy are tightly connected to the properties of its dark matter halo.\nTo place the MW in the context of cosmological galaxy formation,\none usually relies on the estimated size of the MW halo according to a certain definition of the halo boundary and the corresponding enclosed mass.\n\nDespite many efforts dedicated to measuring the mass distribution \nin the virialized region of the MW halo \\citep{Wang2019b} in observations, \nmuch less attention has been paid to the very outskirts beyond the formal virial radius.\nIn addition to the normally higher incompleteness and larger measurement errors for tracers at large distances,\nthe lack of equilibrium in this region also blocks dynamical modeling attempts based on the steady-state assumption~\\citep{oPDFI,oPDFII} and thus requires better theoretical understanding.\n\nConventionally, most studies use the classical\nvirial definition (or its variants) derived from the spherical collapse model \\citep{Gunn1972},\nwhich marks out a radius by a fixed enclosed overdensity under some idealized assumptions.\nHowever, a halo in the real universe is not abruptly separated from the neighboring environment at this specific radius.\nIn fact, the mass distribution within and around a halo is a continuous mixture of\nthe virialized content, the infalling materials, and background materials receding with the rest of the universe.\nThis fact has inspired people to further investigate other boundaries better separating these components\n(see \\citealt{Fong2020} for a detailed summary),\nsuch as the splashback radius \\citep{Adhikari2014,Diemer2014,Diemer2017,Aung2021}, \nthe depletion radius \\citep{Fong2020},\nand the turnaround radius \\citep[e.g.,][]{Gunn1972,Cuesta2008,Pavlidou2014,Faraoni2015},\nfrom the inside out.\nUnlike the spherical overdensity-based definition,\nthe latter boundaries are more directly associated with dynamical processes,\nand hence detectable from the kinematics of tracers \\citep[e.g.,][]{Deason2020,Bose2020,Tomooka2020}. \nThis is a particular advantage because \nwe can measure the velocity of tracers, e.g., nearby galaxies, even at a large distance,\nbut cannot observe the density directly.\n\nThe different halo radii definitions also serve to provide different insights on the structure and evolution of halos. In a recent work, \\citet{Fong2020} introduced the \\textit{inner depletion radius}, $r_{\\mathrm{id}}$, defined at the location of the maximum mass inflow rate, as the outer edge of the \\emph{growing} part of a halo. \nPractically, this radius is identifiable at the location of the maximum infall velocity (see Fig.~11 of \\citealt{Fong2020}) which is the approach we follow in this work.\nWith $r_{\\mathrm{id}}$ defined at the maximum inflow location, matter within $r_{\\mathrm{id}}$ gets deposited onto the halo as the infall rate slows down towards the inner halo.\nOutside this radius, however, matter is being pumped into the halo and gradually depleted due to the increasing infall rate towards the inner region. \nThis process leads to the formation of a relatively flat shoulder in the density profile and a trough in the bias profile around the $r_{\\mathrm{id}}$ scale (\\citealt{Fong2020}).\nThus, this location marks the transition between the halo being built up and the environment being depleted by halo accretion.\nMoreover, the enclosed density within this radius is found to have an approximately universal value, which enables us to easily estimate the enclosed mass.\n\nFrom the perspective of particle orbits, $r_{\\mathrm{id}}$ can be interpreted as a boundary enclosing a more complete population of splashback orbits than the customary \\textit{splashback radius} defined at the steepest slope radius, $r_{\\rm sp}$. The latter is based on the steepening in the slope resulted from the buildup of particles at their first orbital apogees, but it is found to enclose only about 75\\% of the splashback orbits \\citep{Diemer2017}.\nHence, $r_{\\mathrm{id}}$ is normally outside the splashback radius, $r_{\\rm sp}$, with $r_{\\mathrm{id}}\\approx 1.7 \\sim 2.6 r_{\\rm sp}$.\\footnote{This relation is converted combining the relations of $r_{\\mathrm{id}}\\approx 0.85 r_{\\rm cd}$ and $r_{\\rm cd}=2-3 r_{\\rm sp}$ in \\citet{Fong2020}, where $r_{\\rm cd}$ is the characteristic depletion radius defined at the minimum bias.}\nInterestingly, this scale is shown to be very close to (or $\\sim\\!\\! 15$ percent smaller than) the location of the minimum in the halo bias profile \\citep{Han2018} around the trans-linear scale and  almost identical to the optimal halo exclusion radius measured by \\citet{Garcia2020} that defines the geometrical boundary of non-overlapping halos in the halo model description of the large-scale structure.\n\nCompared with the virial radius, $r_{\\mathrm{id}}$ is roughly located at the $1.6 \\R{200m}$, where the $\\R{200m}$ is the radius within which the average density is 200 times the mean background density.%\n\\footnote{\nSimilarly, $\\R{200c}$ and $\\R{vir}$ are defined as the radius within which the average density is 200 and $\\Delta_\\mathrm{vir}$ times the critical density of the universe, respectively, where $\\Delta_\\mathrm{vir}$ is the virial overdensity predicted from the spherical collapse model \\citep{Bryan1998}.\n}\nBy definition, the inner depletion radius at maximum infall is enclosed within the turnaround radius where the radial velocity reaches zero. The turnaround radius is of important dynamical significance as it separates infalling material from the expansion of the universe, and can serve as a probe of both halo evolution and the background cosmology~\\citep[e.g.,][]{Gunn1972,Cuesta2008,Pavlidou2014,Faraoni2015}.\n\nIn this work, we present the first measurement of the inner depletion radius of the MW\nusing the motion of nearby dwarf galaxies, along with the turnaround radius measured from the same data set.\nAlthough these radii were first introduced based on dark matter, galaxies are found to closely trace the underlying phase space structures of dark matter \\citep[e.g.,][]{Han2020,Deason2020} especially in the outskirts of haloes. As a result, we will use galaxies as tracers to probe these radii. The measurements are then compared directly with those using galaxies in hydrodynamical simulations, as well as with previous results using dark matter particles.\nUsing the scaling relation learned from halos in simulations,\nthe enclosed masses within these boundaries are also estimated. As these boundaries directly quantify the ongoing evolution of the MW halo, the measurements can provide crucial information for better placing the MW into a cosmological context of halo evolution and galaxy formation.\n\nThe structure of this letter is as follows.\nWe present the measurements of the MW's outer edges in \\refsec{sec:mw}, \ninterpret the results with simulations in \\refsec{sec:validate},\ncompare them with previous measurements in \\refsec{sec:compare},\nand summarize in \\refsec{sec:conclusion}.\nIn addition, we provide the details of measuring the velocity profile in Appendix \\ref{sec:gp}\nand selecting simulation sample in Appendix \\ref{sec:simu}.\n\n\n\\section{The outer edges of the MW}\\label{sec:mw}\n\n\nWe use nearby galaxies within $3\\mathrm{Mpc}$ of the MW, compiled from the catalog of the Local Volume galaxies \n\\citep{Karachentsev2013,Karachentsev2019}%\n\\footnote{\\url{http:\/\/www.sao.ru\/lv\/lvgdb\/tables.php}, updated on 2020-08-12}\nand the catalog of Nearby Dwarf Galaxies \\citep{McConnachie2012}%\n\\footnote{\\url{http:\/\/www.astro.uvic.ca\/\\~alan\/Nearby\\_Dwarf\\_Database.html}, updated on 2021-01-19}.\nThe observed Heliocentric line-of-sight velocities are converted into radial velocities in the Galactocentric rest frame.\nThe proper motions from the catalog of Nearby Dwarf Galaxies \n(mostly measured by \\citealt{McConnachie2020}) are used for the conversion when available.\nFor the remaining galaxies, we ignore their proper motions in the conversion considering their large distance.\nThe observational error of the line-of-sight velocity is typically smaller than several $\\mathrm{km \\, s}^{-1}$,\nwhich is negligible in this task compared with the bulk motion at several tens or hundreds of $\\mathrm{km \\, s}^{-1}$ level.\n\n\n\n\\begin{figure}[bt]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{LG_gals_vtot.pdf}\n\\includegraphics[width=0.47\\textwidth]{mw_profile_new.pdf}\n\\caption{%\n    Top panel:\n    Radial velocities of galaxies within 3 Mpc of the MW.\n    Galaxies within $600\\mathrm{kpc}$ from M31 are marked as\n    open circles and discarded in the analysis.\n    The mean velocity profile (green solid curve) and its $1\\sigma$ uncertainty (green band) are computed from the remaining galaxies (filled circles).\n    The measured MW edges including\n    the \\textit{inner depletion radius} (i.e., location of maximum infall), $r_\\mathrm{id}$, \n    and the \\textit{turnaround radius}, $r_\\mathrm{ta}$, are indicated by star symbols.\n    The Hubble flow, $\\vr=H_0 r$, is shown by the dotted line for reference.\n    Bottom panel:\n    The MW mass profile.\n    The star symbols indicate the estimated MW masses within the corresponding edges based on their typical enclosed densities in simulation. The estimates calibrated using a fiducial sample (black) or LG-like sample (gray, slightly shifted horizontally for clarity) of halos in simulation are shown separately (see the text for detail). Previous measurements of the inner MW mass profile using stars, globular clusters, and satellite galaxies \n    are shown for comparison. As an extrapolation of the inner profile \\citep{Li2020},  the long-dashed (dash-dotted) curve shows the mean mass profile of the fiducial (LG-like) halos in the TNG100 simulation.\n    The error bars or shades correspond to the 68\\% confidence intervals.\n}\n\\label{fig:local_group}\n\\end{figure}\n\n\nThe Galactocentric distances and radial velocities, $\\{r, \\vr\\}$, of these galaxies are shown in\n\\reffig{fig:local_group}. \nIn this work, we exclude galaxies within 600 $\\mathrm{kpc}$ from M31 (about $1.5 \\R{200m, M31}$) to reduce the potential influence of our massive neighbor. \nWe have also checked that our results are not very sensitive to the particular choice of this radius of exclusion.\nChanging the exclusion radius from 550 to 850 kpc only leads to a variation $\\lesssim 2\\%$ in the measured edges, while using a smaller value (e.g., 400 kpc) leads to slightly larger estimates (by factors of 5\\% in $r_{\\mathrm{id}}$ and 10\\% in $r_{\\mathrm{ta}}$). Note the six dwarf galaxies (Eridanus 2, Leo T, Pheonix, NGC 6822, Leo A, and Cetus) that lie in our inferred infall zone between 300 and 840 kpc are clearly not affiliated with M31, considering their large angular separation and distance from M31.\n\nIn order to extract the mean radial velocity profile, we model the distribution of radial velocities as a Gaussian distribution with a mean velocity, $\\bar v_r(r)$, and a velocity dispersion, $\\sigma_r (r)$. To obtain smooth estimates of the two, we adopt an iterative Gaussian process regression~\\citep{Rasmussen2005} method, which we briefly outline here but leave further details to Appendix \\ref{sec:gp}. Specifically, we first extract a rough estimate of the mean velocity profile $\\bar v_r(r)$ assuming a constant $\\sigma_r$ using Gaussian process regression. The estimated $\\bar v_r(r)$ profile is then combined with the observed velocities to obtain a radial-dependent velocity dispersion profile, $\\sigma_r(r)$. Finally, the $\\bar v_r(r)$ profile and its uncertainty is refined by fitting a Gaussian process with the estimated $\\sigma_r(r)$ profile as the noise term, in addition to a kernel that determines the uncertainty on the mean profile $\\bar v_r(r)$. By this process, we self-consistently obtain smooth estimates of $\\bar v_r(r)$, $\\sigma_r(r)$ as well as the uncertainty on $\\bar v_r(r)$.\n\nThe fitted $\\bar v_r(r)$ profile and its uncertainty are shown in the top panel of \\reffig{fig:local_group}.%\n\\footnote{See also Fig.~11 of \\citet{Deason2020} for a similar figure, where the $\\bar v_r(r)$ profile was obtained via the Savitzky--Golay smoothing algorithm and a slightly different galaxy sample. However, \\citet{Deason2020} focused on the slope of the $\\bar v_r$ profile rather than $\\bar v_r$ itself.}\nThe inner part of the profile is flat and consistent with zero net radial flow, as expected for the virialized part of the halo where the density remains largely static. On the largest scale, the positive radial velocity is dominated by the Hubble expansion of the universe. The profile crosses zero at the turnaround radius $r_{\\mathrm{ta}} \\simeq 840\\, \\mathrm{kpc}$, within which matter starts to fall towards the halo. Within this infall zone but outside the virialized region, the mean $\\vr$ profile exhibits a clear minimum that defines the depletion radius $r_{\\mathrm{id}} \\simeq 560\\, \\mathrm{kpc}$. The matter in between $r_{\\mathrm{id}}$ and $r_{\\mathrm{ta}}$ is being pumped into the region inside $r_{\\mathrm{id}}$, so $r_{\\mathrm{id}}$ unveils precisely the border where the MW is feeding on the environment. The amplitude of the maximum infall velocity is relatively small compared to the scatter of the velocities, revealing the MW halo is only growing at a very low rate.\n\nThe Gaussian process also enables a probabilistic way to asses the uncertainty in measuring the two characteristics, as it provides a posterior distribution of the entire profile.\nWe sample $10^4$ random realizations from the posterior of the velocity profile and measure the halo edges respectively.\nIn most ($>95\\%$) realizations, an infall region is detectable with $300\\mathrm{kpc} < r_{\\mathrm{id}} < 1000 \\mathrm{kpc}$.\nTaking their average and dispersion, we locate the inner depletion radius at $r_{\\mathrm{id}}=559\\pm 107\\, \\mathrm{kpc}$\nand turnaround radius at $r_{\\mathrm{ta}}=839\\pm 121\\, \\mathrm{kpc}$. \nThe maximum infall velocity is estimated to be $v_\\mathrm{inf, max}=-46_{-39}^{+24}\\mathrm{km s^{-1}}$, suggesting that our tentative detection of the infall zone is only at a marginal significance at about 2 $\\sigma$ level. \nThis is due to both the at most weak infall zone around the MW and the size of the uncertainty given the limited tracer sample size, the latter of which can be reduced by enlarging the nearby galaxy sample in future observation. Despite this, the infall region is also clearly detectable using other smoothing techniques such as the moving average or the Savitzky-\u2013Golay smoothing algorithm \\citep{Deason2020}.\n\nIt is worth pointing out that the above turnaround radius encloses the M31 (at $r=780 \\mathrm{kpc}$),  the MW's massive companion.\nThough the M31 and its satellites are excluded from the analysis, \nthe M31 can perturb the velocity flow pattern in the vicinity and make the isovelocity surface anisotropic (e.g., \\citealt{Deason2020}).\nTherefore, our estimate of the turnaround radius should be viewed as a rough estimate in an spherically averaged sense. \n\n\n\\section{Interpreting the measurements with simulations}\\label{sec:validate} \n\nThe above measurements are compared with those of simulated halos in the state-of-the-art cosmological hydrodynamical simulation Illustris TNG100 as detailed in Appendix~\\ref{sec:simu}.\nFollowing similar procedures to those in \\refsec{sec:mw},\nfor each MW-sized halo in TNG100, \nwe identify the turnaround radius, $r_{\\mathrm{ta}}$, \nas the furthest zero velocity radius along the mean radial velocity profile\nand the inner depletion radius, $r_{\\mathrm{id}}$, as the furthest local minimum point within $r_{\\mathrm{ta}}$.\n\nUnlike the MW, for some halos (especially low-mass ones) we fail to locate a $r_{\\mathrm{id}}$\nbeyond the halo virial radius\n$\\R{vir}$ due to the lack of an infall region in the velocity profile \n(see also e.g., \\citealt{Cuesta2008,Fong2020}).\nWe exclude those halos without a detectable infall zone ($n=1517$) from the parent sample ($n=4681$) of MW-sized halos. We emphasize that the differing strength of the infall zone around halos of a given mass is itself an important diagnostic of the dynamical state and environment of the halo. By definition, halos without an infall region have halted their mass growth while those with one are still accreting.\n\nOur MW is observed to be embedded in a relatively cold environment dynamically, which we find to have a significant influence on the outer halo profile. To make a fair comparison, we select a \\emph{fiducial} sample of halos ($n=2153$) with similar masses and dynamical environments to the MW. Out of the fiducial sample, we further select an \\emph{LG-like} sample ($n=35$) with the additional requirement of having a close massive companion as detailed in Appendix~\\ref{sec:enviro}. \n\n\n\\begin{figure*}[hbtp]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{radius_density.pdf}\n\\caption{%\n  Halo edges and corresponding mean enclosed densities of simulated halos.\n  The fiducial halo sample and the LG-like (paired) halos\n  are shown as solid circles and squares, respectively.\n  The fiducial sample is further divided into three halo mass bins, which are shown as open circles.\n  The symbols and error bars correspond to the median and the $50\\pm34$th percentiles, respectively, with those of the fiducial sample also indicated by horizontal lines and bands for ease of comparison.\n  In the top panels, the measurements of MW edges are also shown as star symbols for reference.\n}\n\\label{fig:estimate}\n\\end{figure*}\n\n\\begin{figure}[hbtp]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{mah.pdf}\n\\caption{%\n  Median mass growth history of the LG-like halos.\n  The sample is divided into two equal subsets by $r_\\mathrm{id}\/\\R{200m}$.\n  The shaded bands show the interval between the 20th to 80th percentiles.\n}\n\\label{fig:mah}\n\\end{figure}\n\nFor the fiducial sample, as shown in \\reffig{fig:estimate},\n$r_{\\mathrm{id}} \\sim 1.6 \\R{200m}$ with a mean enclosed density $\\bar\\rho(<r_{\\mathrm{id}}) \\sim 60 \\rho_{\\mathrm{m}}$\nfor the whole mass range explored.\nIt is consistent with \\citet{Fong2020}, where $r_{\\mathrm{id}}$ is identified using dark matter particles.\nAs to the turnaround radius, we have $r_{\\mathrm{ta}} \\sim 2.5 \\R{200m}$\nand $\\bar\\rho(<r_{\\mathrm{ta}}) \\sim 20 \\rho_{\\mathrm{m}}$ with little mass dependence.\nThis seems consistent with the idealized spherical collapse model,\nwhich predicts a constant overdensity within $r_{\\mathrm{ta}}$.\nHowever, \\citet{Fong2020} found a much stronger mass dependence\nthat $\\bar\\rho(<r_{\\mathrm{ta}})$ is significantly higher for smaller halos in their sample.\nThis is probably because some smaller halos reside in the vicinity of massive halos,\nwhose tidal force can reduce $r_{\\mathrm{ta}}$ and thus increase the density enclosed.%\n\\footnote{%\n  Note that $r_{\\mathrm{id}}$ is located at a smaller radius, hence less influenced by such effect.\n}\nIn contrast, here we only consider the halos in a dynamically cold environment with a detectable infall zone, resulting in the absence of a mass dependence.\n\nA massive companion halo like the M31 in the LG can perturb the velocity flow pattern in the vicinity and make the isovelocity surface anisotropic (e.g., \\citealt{Deason2020}).\nNevertheless, our estimate still provides a meaningful quantification of the halo edges in a spherically averaged sense.\nAs shown in \\reffig{fig:estimate}, the halo edges of LG-like halos are typically slightly larger \nthan those in the fiducial sample: $r_{\\mathrm{id}}\\sim 2\\R{200m}$ and $r_{\\mathrm{ta}} \\sim 3\\R{200m}$.\nIt leads to a smaller density within $r_{\\mathrm{id}}$, $\\bar\\rho(<r_{\\mathrm{id}})\\sim 40 \\rho_{\\mathrm{m}}$, \nbecause the density is a decreasing function of radius.\n$\\bar\\rho(<r_{\\mathrm{ta}})$ is less affected due to the mass compensated at this scale\nby the companion halo.\n\nBased on the characteristic densities enclosed within halo edges,\nit is straightforward to estimate the corresponding MW masses,\n$\\hat M(<r_{\\mathrm{id}}) = 60 \\rho_{\\mathrm{m}} \\times \\frac{4}{3}\\pi r_{\\mathrm{id}}^3$ and\n$\\hat M(<r_{\\mathrm{ta}}) = 20 \\rho_{\\mathrm{m}} \\times \\frac{4}{3}\\pi r_{\\mathrm{ta}}^3$.\nConsidering the dispersion among the enclosed densities for the fiducial sample,\nthe uncertainties in the above mass estimates are about $0.25$ dex and $0.2$ dex, respectively.\nNote that this uncertainty actually represents the total uncertainty,\nincluding both the halo-to-halo scatter and the uncertainty due to imprecise determination of the halo edges. The resulting mass estimates are shown in \\reffig{fig:local_group}. Calibrating the enclosed densities using the fiducial sample, we obtain $M(<r_{\\mathrm{id}})=1.6_{-0.6}^{+1.9}\\times 10^{12} M_\\odot$ and $M(<r_{\\mathrm{ta}})=1.9_{-0.5}^{+1.5}\\times 10^{12} M_\\odot$, while calibrating on LG-like halos leads to a lower estimate of $M(<r_{\\mathrm{id}})=1.2_{-0.5}^{+1.2}\\times 10^{12} M_\\odot$. We emphasize that our new measurements of the outermost edges are independent from the inner halo measurements, and the enclosed masses within these radii can be potentially improved with a better theoretical understanding of the outer halo profile.\n\nThe observed $r_{\\mathrm{id}}$ and $r_{\\mathrm{ta}}$ for our MW are about 1.5 and 2.3 times of the $\\R{200m,MW}=364\\pm19\\,\\mathrm{kpc}$ measured by \\citet{Li2020}.\nSurprisingly, this measured $r_{\\mathrm{id}}\/\\R{200m}\\approx 1.5$ for our MW is closer to that of the fiducial sample rather than the LG-like halos, even though they are still consistent within the uncertainties. This slight discrepancy may be interpreted as indicating the unique formation history of our MW compared with typical simulated ones in the LG-like sample. To demonstrate this, we divide the LG-like halos into two equal subsamples by $r_{\\mathrm{id}}\/\\R{200m}$ and compare their mass growth histories in \\reffig{fig:mah}.\nHalos with a smaller $r_{\\mathrm{id}}$ seem to have accreted more mass at an earlier time on average, consistent with \\cite{Fong2020}.\nWe have to point out that the limited sample size and the large uncertainties in our current analysis prevent us from making a strong conclusion. Nevertheless, these results illustrate a promising way to constrain the MW formation history in greater details with a larger LG galaxy sample in the future.\n\n\\section{Comparisons with previous measurements of the MW mass profile} \\label{sec:compare}\nIn \\reffig{fig:local_group}, we also show \nprevious measurements of the inner halo profile (see \\citealt{Wang2019b} for a more comprehensive review) using halo stars \\citep{Xue2008,Huang2016, Ablimit2017, Zhai2018,Deason2020a}, globular clusters \n\\citep{Sohn2018,Watkins2018,Vasiliev2018,Eadie2018},\nand satellite galaxies \\citep{Watkins2010a,Callingham2020,Cautun2019,Li2020,Fritz2020}.\nIn particular, \\citet{Li2020} recently measured the mass profile of the MW outer halo\nusing satellite galaxies within 300 kpc\nand obtained $M_\\mathrm{200c}=1.23_{-0.18}^{+0.21}\\times 10^{12} M_\\odot$ with a concentration of $c=9.4_{ -2.1}^{ +2.8}$.\nAssuming an NFW profile, these are equivalent to $M_\\mathrm{200m}=1.58\\pm 0.25\\times 10^{12} M_\\odot$ and $\\R{200m}=364\\pm 19\\,\\mathrm{kpc}$ or $M_\\mathrm{vir}=1.43\\pm 0.23\\times 10^{12} M_\\odot$ and $\\R{vir}=297\\pm 16\\,\\mathrm{kpc}$.\n\nOur estimated outer halo masses are further compared with the extrapolated mass profile of the MW based on above measurements of the inner halo mass in Fig.~\\ref{fig:local_group}.\nIt is known that the outer density profiles of halos at $r > \\R{200m}$ are remarkably\nuniversal when the radius is normalized by $\\R{200m}$ (\\citealt{Diemer2014}, see also \\reffig{fig:vsig_profile} in Appendix),\nwhich allows us to make profile extrapolation with a reasonable precision.\nWe rescale the density profiles of the aforementioned simulated halos\nto the $\\R{200m,MW}$ measured in \\citet{Li2020}, \nas $\\rho_\\mathrm{scaled}(r')=\\rho_\\mathrm{original}(\\frac{r'}{\\R{200m,MW}}\\R{200m})$. \nTo take the uncertainty in $\\R{200m,MW}$ into account, the $\\R{200m,MW}$ value used to rescale each halo is drawn randomly from the posterior distribution of $\\R{200m,MW}$ each time. \nThe extrapolated profiles for the fiducial and LG-like halos are quite close within $r_{\\mathrm{id}}$,\nwhile the mass at larger scale for the LG-like halos is significantly higher due to the presence of the companion halo.\nBoth profiles are consistent with the mass estimates at our measured outer edges, although slightly closer to those adopting the fiducial enclosed densities. Note that the enclosed density within $r_{\\mathrm{id}}$ depends mostly on the location of $r_{\\mathrm{id}}\/\\R{200m}$ and is not sensitive to how the halo is selected as the profiles are largely universal around this scale. The slightly lower $M(<r_{\\mathrm{id}})$ estimated adopting LG-like depletion density compared with the extrapolated mass profile can thus be viewed as an alternative representation of the smaller $r_{\\mathrm{id}}\/\\R{200m}$ in the observation compared with that of the simulated LG-like halos. \n\n\nOur measurements of the halo edges are also consistent with previous results at scales beyond the virial radius.\nThe measured $r_{\\mathrm{id}}$ is about 1.9 times of the MW caustic radius $r_\\mathrm{sp} = 292 \\pm 61 \\mathrm{kpc}$ measured by \\citet{Deason2020}.\nIt is consistent with \\citet{Fong2020},\nwho found that $r_{\\mathrm{id}}$ is located at $1.7 \\sim 2.6$ typically.\nThe turnaround radius of the MW is slightly smaller than the estimated LG turnaround radius at about $1\\mathrm{Mpc}$ \n(e.g., \\citealt{Karachentsev2002}) as expected.\nOur mass estimates within the depletion and turnaround radii are also consistent \nwith previous measurements of the ``total'' mass of the Local Group using various methods,\nincluding the timing argument (e.g., \\citealt{Li2008},  $5.3_{-1.7}^{+2.5} \\times 10^{12} M_{\\odot}$), simulation-based statistics of the MW-M31 motions (e.g., \\citealt{Gonzalez2014}, $ 4.2^{+3.4}_{-2.0}\\times 10^{12} M_{\\odot}$) and dynamical modeling of the Local Group galaxies (e.g., \\citealt{Penarrubia2016}, $ 2.64^{+0.42}_{-0.38}\\times 10^{12} M_{\\odot}$;\n\\citealt{Shaya2017}, $5.15^{\\pm0.35} \\times 10^{12} M_{\\odot}$),\nthough it might be worth further investigating \nwhat radii such estimates correspond to \\citep[e.g.,][]{Penarrubia2017}.\n\n\n\\section{Summary} \\label{sec:conclusion}\n\nFollowing recent developments in characterizing the boundary of halos \\citep{Fong2020},\nwe measure for the first time the inner depletion radius (aka.\\ the maximum infall radius)\nand the turnaround radius of the MW halo\nand the corresponding enclosed masses. \nThis inner depletion radius is expected to be the natural boundary demarcating \nthe transition between a growing halo and the environment being depleted,\nwhile the turnaround separates the infall region from the receding background.\n\nUsing the radial motion of nearby dwarf galaxies, \nwe estimate the inner depletion radius of the MW halo to be $r_\\mathrm{id}=559\\pm 107 \\mathrm{kpc}$ \nand the turnaround radius to be $r_\\mathrm{ta}=839\\pm 121 \\mathrm{kpc}$.\nOur detection of the infall zone is only at a marginal significance at about 2 $\\sigma$ level due to both the at most weak infall zone around the Milky Way and the size of the uncertainty given the limited tracer sample size. The latter can be improved by enlarging the nearby galaxy sample in future observation. With more observational data, it might also be possible to measure the anisotropic depletion and turnaround surfaces that better characterize the influence from the neighboring M31 halo.\n\n\nApplying the same analysis to groups with similar masses and dynamical environments in the hydrodynamical simulation Illustris TNG100,  \nwe find that $r_{\\mathrm{id}} \\sim 1.6 \\R{200m}$ for the whole mass range explored,\nwhile the mean enclosed  density $\\bar\\rho(<r_{\\mathrm{id}}) \\sim 60 \\rho_{\\mathrm{m}}$,\nwhich is consistent with \\citet{Fong2020}.\nAs to the turnaround radius, we have $r_{\\mathrm{ta}} \\sim 2.5 \\R{200m}$\nand $\\bar\\rho(<r_{\\mathrm{ta}}) \\sim 20 \\rho_{\\mathrm{m}}$ with little mass dependence.\nThe constant enclose density allows us to estimate the MW mass as\n$M(<r_{\\mathrm{id}})=1.6_{-0.6}^{+1.9}\\times 10^{12} M_\\odot$ and $M(<r_{\\mathrm{ta}})=1.9_{-0.5}^{+1.5}\\times 10^{12} M_\\odot$. The estimates are consistent with the mass profile constrained \nat smaller radii \\citep[e.g.,][]{Li2020} and the LG mass constrained at larger scale (see \\refsec{sec:compare}).\n\nWhen the selection criterion in the simulation is tightened to select paired halos similar to our LG, we find a slightly lower enclosed density associated with a larger maximum infall radius. Taking this enclosed density as a reference,\nthe MW mass within the boundary is expected to be \n$M(<r_{\\mathrm{id}})=1.2_{-0.5}^{+1.2}\\times 10^{12} M_\\odot$.\n\nThese measurements directly quantify the ongoing evolution of the MW outer halo and provide constraints on the current dynamical state and mass content of the MW independent of internal dynamics. The tentative detection of the infall zone indicates the MW halo is likely still accreting mass from the surrounding environment rather than having halted its growth. The slightly smaller $r_{\\mathrm{id}}\/\\R{200m}$ of the observed MW halo compared with those of LG-analogies in the simulation can be potentially interpreted as suggesting the distinct formation history of our MW. For example, a smaller $r_{\\mathrm{id}}\/\\R{200m}$ indicates a smaller infall region and might imply inefficient recent mass growth. However, we emphasize that the difference is still very weak given the large measurement uncertainties, which could be improved with a larger nearby galaxy sample in the future. Improved theoretical understandings of the connections between $r_{\\mathrm{id}}$ and other halo properties can also help to better interpret the measurements. Nevertheless, we expect these measured outermost characteristic radii can serve as alternative selection variables when placing our MW halo into the context of cosmological galaxy formation.\n\n\n\\acknowledgments\n\nWe are very grateful to Hongyu Gao, Matthew Fong and Wenting Wang for their helpful discussions.\n\nThis work is supported by NSFC (11973032, 11890691, 12022307, 11621303), National Key Basic Research and Development Program of \nChina (No.\\ 2018YFA0404504), 111 project (No.\\ B20019), and the science research grants from the China Manned Space Project (No.\\ CMS-CSST-2021-A03, CMS-CSST-2021-B03). We gratefully acknowledge the support of the MOE Key Lab for Particle Physics,\nAstrophysics and Cosmology, Ministry of Education. The computation of this work is partly done on the \\textsc{Gravity} supercomputer at the Department of Astronomy, Shanghai Jiao Tong University. \n\nThe IllustrisTNG simulations were undertaken with compute time awarded by the Gauss Centre for Supercomputing (GCS) under GCS Large-Scale Projects GCS-ILLU and GCS-DWAR on the GCS share of the supercomputer Hazel Hen at the High Performance Computing Center Stuttgart (HLRS), as well as on the machines of the Max Planck Computing and Data Facility (MPCDF) in Garching, Germany.\n\nThis research has made use of NASA's Astrophysics Data System\nand adstex (\\url{https:\/\/github.com\/yymao\/adstex}).\n\n\\textit{Software:} \n  Astropy \\citep{AstropyCollaboration2013},\n  scikit-learn \\citep{Pedregosa2011},\n  Numpy \\citep{Walt2011}, \n  Scipy \\citep{Oliphant2007},\n  Matplotlib \\citep{Hunter2007}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\nState-of-the-art object detection performance on the COCO benchmark~\\cite{coco} has been pushed from 30\\% AP to 58\\% since the development of popular object detectors~\\cite{ssd, yolo, yolov3, rcnn, fast_rcnn, fasterrcnn, mrcnn, cai2018cascade, htc, retinanet, centernet, fpn, hrnet, panet, nasfpn, spinenet, Du2020EfficientSB, efficientdet, detr, bochkovskiy2020yolov4, zhang2020resnest, copypaste,wang2021scaledyolov4, liu2021swin}. In addition to pursuing high accuracies, the research community also cares about the speed-accuracy Pareto curve of object detection systems~\\cite{huang2017speed}. The performance improvements not only come from the novel model architectures proposed in the literature but also from improved scaling, and modern training and inference methods\\footnote{\\url{https:\/\/ai.facebook.com\/blog\/advancing-computer-vision-research-with-new-detectron2-mask-r-cnn-baselines\/}}.\nMany techniques that contribute to a large portion of performance improvements are not emphasized enough or are overlooked in the literature. Some of these details can conflate the performance of current state-of-the-art detection systems.\n\nOur work aims to carefully study these techniques and tease apart where most of the improvements are coming from in modern detection systems. We carefully ablate the impact of the commonly used techniques in current state-of-the-art object detection and instance segmentation systems from two angles: 1) minor architectural improvements, including Squeeze-and-Excitation modules~\\cite{senet}, activation functions\\cite{silu} and model stem\\cite{he2019bag};\n2) training and inference methods, including stronger data augmentation, better model regularization\\cite{dropconnect}, longer training schedules and \\texttt{float16} benchmarking.\n\nLastly, we propose a simple yet effective model scaling method for object detection and instance segmentation. Based on our empirical results, we find that only scaling the input image resolution and backbone depth is already quite effective. We apply this strategy to RetinaNet~\\cite{retinanet} and RCNN~\\cite{fasterrcnn,mrcnn} models and name them RetinaNet-RS and RCNN-RS.\n\n\nWe evaluate our RetinaNet-RS and Cascade RCNN-RS models on the COCO dataset~\\cite{coco} and the Waymo Open dataset~\\cite{waymo} (WOD). Our results reveal that the above mentioned architectural changes and training\/inference methods improve detection baselines by \\textcolor{blue}{7.7\\%} AP while reducing inference time by \\textcolor{blue}{30\\%}.\nBy further adopting the proposed scaling strategy, we present two families of detectors. In particular, our Cascade RCNN-RS model adopting a ResNet152-FPN backbone achieves 52.9\\% AP on COCO at 119ms per image on a V100 GPU. Our Cascade RCNN-RS adopting a SpineNet143L backbone achieves 53.6\\% AP on COCO and 71.2 AP\/L1 on WOD. \n\n\nOur contributions are:\n\n\\begin{itemize}\n    \\item We identify the key architectural changes, training methods and inference methods that significantly improve object detection and instance segmentation systems in speed and accuracy.\n    \\item We highlight the key implementation details and establish new baselines for RetinaNet and Cascade RCNN models.\n    \\item We provide two object detection model families as strong new baselines for future research, \\textbf{RetineNet-RS} and \\textbf{Cascade RCNN-RS}.\n    \\item We explore speed-accuracy trade-off between one-stage RetinaNet and two-stage RCNN models.\n   \n\\end{itemize}\n\n\n\\setlength{\\tabcolsep}{4pt}\n\\begin{table}[t]\n\\vspace{0mm}\n\\begin{center}{\n\\begin{tabular}{l | l |l}\n\\toprule\n Model & AP \\textcolor{gray}{(\\%)} &  Latency \\textcolor{gray}{(ms)} \\\\\n \\midrule\nRetinaNet baseline &  37.0 & 41\\\\\n+ \\texttt{float16} &  37.0 & 23 (\\textcolor{blue}{-18})\\\\\n+ SJ \\& 350-epoch &  40.5 (\\textcolor{blue}{+3.5}) & 23 \\\\\n+ SD \\& 600-epoch & 41.5 (\\textcolor{blue}{+1.0}) & 23 \\\\\n+ SiLU activation & 42.5 (\\textcolor{blue}{+1.0}) & 27 (\\textcolor{red}{+4}) \\\\\n+ SE & 44.0 (\\textcolor{blue}{+1.5}) & 29 (\\textcolor{red}{+2})\\\\\n+ ResNet-D & 44.7 (\\textcolor{blue}{+0.7}) & 29\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\caption{\\textbf{Ablation study of the modern techniques discussed in this paper.} Results are reported using a RetinaNet detector with a ResNet-50 backbone at $640\\times 640$ input resolution on COCO \\texttt{val2017}. SJ: scale jittering. SD: stochastic depth. SE: Squeeze-and-Excitation. ResNet-D: ResNet-D style stem. We show that there is a 7.7\\% AP improvement by adopting all the techniques while reducing inference latency by 30\\%.}\n\\label{tab:resnet_modifications}\n\\vspace{-0mm}\n\\end{table}\n\n\\section{Methodology}\\label{sec:methodology}\n\\subsection{Modified ResNet Backbone}\\label{sec:resnet_modifications}\nWe modify the standard ResNet~\\cite{resnet} architecture in three ways to improve its performance with small computational costs. Bello~\\etal~\\cite{bello2021revisiting} has demonstrated Squeeze-and-Excitation module~\\cite{senet} and ResNet-D stem~\\cite{inceptionv2,bagoftricks} are both effective for classification model. Recent detection systems, \\eg,~\\cite{efficientdet, bochkovskiy2020yolov4}, show the above two changes and a non-linear activation function like Sigmoid Linear Unit activation are effective on improving detection performance.\n\n\\paragraph{\\bf Squeeze-and-Excitation:} We apply the Squeeze-and-Excitation module to all  all residual blocks in the ResNet architecture. Following~\\cite{senet}, one attention module is placed after the final 1$\\times$1 convolutional layer, but before merging the residual with the shortcut connection. A squeeze ratio of $0.25$ is used for all experiments. \n\n\\paragraph{\\bf ResNet-D stem:} We follow~\\cite{inceptionv2,bagoftricks} and modify the original ResNet stem to the ResNet-D stem. In summary, we replace the first $7\\times7$ convolutional layer at feature dimension $64$ with three $3\\times3$ convolutional layers at feature dimension $32, 32, 64$, respectively. The first $3\\times3$ convolution has a stride of 2. Batch normalization and activation layers are applied after each convolutional layer.\n\n\\paragraph{\\bf Sigmoid Linear Unit activation:} The Sigmoid Linear Unit (SiLU)~\\cite{GELU}, computed as $f(x)=x\\cdot \\sigma(x)$, has shown promising results as a replacement of the ReLU activation. In this work, we replace all ReLU activations in the model architecture (backbone, FPN and detection heads) with the SiLU activation.\n\n\n\\setlength{\\tabcolsep}{4pt}\n\\begin{table}[b]\n\\begin{center}{\n\\begin{tabular}{c | c c}\n\\toprule\n Model Scale & Resolution & Backbone\\\\\n \\midrule\nScale 1 & $512\\times512$ & ResNet-50 \\\\\nScale 3 & $640\\times640$ & ResNet-50 \\\\\nScale 4 & $640\\times640$ & ResNet-101 \\\\\nScale 5 & $768\\times768$ & ResNet-101 \\\\\nScale 6 & $768\\times768$ & ResNet-152 \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\caption{\\textbf{RetinaNet-RS scaling method}. A simple scaling method to scale up RetinaNet detection systems by changing only the input resolution and ResNet backbone depth.}\n\\label{tab:scaling_ret}\n\\vspace{-0mm}\n\\end{table}\n\n\\setlength{\\tabcolsep}{4pt}\n\\begin{table}[!ht]\n\\begin{center}{\n\\begin{tabular}{c | c c}\n\\toprule\n Model Scale & Resolution & Backbone\\\\\n\\midrule\nScale 1 & $512\\times512$ & ResNet-50 \\\\\nScale 2 & $640\\times640$ & ResNet-50 \\\\\nScale 3 & $768\\times768$ & ResNet-50 \\\\\nScale 4 & $768\\times768$ & ResNet-101 \\\\\nScale 5 & $896\\times896$ & ResNet-101 \\\\\nScale 6 & $896\\times896$ & ResNet-152 \\\\\nScale 7 & $1024\\times1024$ & ResNet-152 \\\\\nScale 8 & $1280\\times1280$ & ResNet-152 \\\\\nScale 9 & $1280\\times1280$ & ResNet-200 \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\caption{\\textbf{RCNN-RS scaling method}. A simple scaling method to scale up RCNN models by changing only the input resolution and ResNet backbone depth.}\n\\label{tab:scaling_cas}\n\\vspace{-0mm}\n\\end{table}\n\n\\subsection{Training and Inference Methods}\\label{sec:training_benchmark_settings}\n\n\\paragraph{Strong data augmentation:} We apply horizontal flipping and image scale jittering with a random scale between [0.1, 2.0] is used as our main data augmentation strategy. The same scale jittering strategy is used and studied in~\\cite{spinenet,efficientdet,copypaste}. For example, if the output image size is $640\\times640$, we first resize the image to randomly between $64\\times64$ and $1280\\times1280$ and then pad or crop the resized image to $640\\times640$.\n\n\\paragraph{Strong regularization:} We apply 4e-5 weight decay and stochastic depth~\\cite{dropconnect} with 0.2 initial drop rate for model regularization. We set the drop rate for a network block based on its depth in the network. The final drop rate of one block is calculated by multiplying the initial drop rate with the block order divided by the total number of blocks.\n\n\\paragraph{Longer training schedule:} The strong data augmentation and regularization methods are combined with a longer training schedule to fully train a model to convergence. On different datasets, we keep increasing the training epochs until we find the best schedule.\n\n\\paragraph{Inference methods:} For inference we use the same square image size as training. We resize the longer side of an image to the target size and pad zeros to keep aspect ratio. We measure inference speed on a Tesla V100 GPU with batch size 1 under settings that only includes the model forward pass time and forward pass plus post-processing (\\eg, non-maximum suppression) time. We report both latency measured with \\texttt{float16} precision and \\texttt{float32} precision. Further inference time speedup can be achieved by TensorRT optimization, which is not used in this work.\n\n\n\\subsection{Model Scaling Method}\\label{sec:scaling_method}\nWe present a simple yet effective scaling method for the one-stage RetinaNet detectors and the two-satge RCNN detectors. The compounding scaling rule in EfficientDet~\\cite{tan2019efficientnet,efficientdet} scales up input resolution together with model depth and feature dimensions for all model components including the backbone, FPN and detection heads. We find that only scaling up the model in input resolution and backbone depth is quite effective for most stages of the speed-accuracy Pareto curve, while being significantly simpler. We empirically control the image resolution and backbone model, then perform a grid search as shown in Table~\\ref{tab:resnet_modifications} to determine the Pareto curve.\n\nFor RetinaNet, we scale up input resolution from $512$ to $768$ and the ResNet backbone depth from $50$ to $152$. As RetinaNet performs dense one-stage object detection, we find scaling up input resolution leads to large resolution feature maps hence more anchors to process. This results in a higher capacity dense prediction heads and expensive NMS. We stop at input resolution $768\\times768$ for RetinaNet.\nThe scaling method is presented in Table~\\ref{tab:scaling_ret}. We name the rescaled RetinaNet models RetinaNet-RS.\n\nScaling up input resolution for RCNN models is more effective than one-stage detectors. RCNN uses a two-stage object detection mechanism.\nThe first region proposal stage is typically lightweight and class-agnostic, thus the input resolution does not place too much overhead at the first stage.\nThe second stage always processes a fixed number of proposals generated from the first stage. We design the scaling method to scale up input resolution from $512$ to $1280$ and the ResNet backbone depth from $50$ to $200$. The scaling method for RCNN models is presented in Table~\\ref{tab:scaling_cas} and the re-scaled model family is named as RCNN-RS.\n\n\\newcommand{\\light}[1]{\\textcolor{gray}{#1}}\n\\begin{table*}[!ht]\\centering\n\\small\n\\begin{tabular}{c c  c | ccc| c | c c c}\n\\toprule\n \\multicolumn{1}{c}{\\multirow{2}{*}{Backbone}} & \\multicolumn{1}{c}{\\multirow{2}{*}{Detector}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{Resolution}}& \\multicolumn{3}{c|}{Latency \\light{(ms)}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\hspace{5mm}AP\\hspace{5mm}}} & \\multicolumn{1}{c}{\\multirow{2}{*}{AP$_{S}$}}& \\multicolumn{1}{c}{\\multirow{2}{*}{AP$_{M}$}} & \\multicolumn{1}{c}{\\multirow{2}{*}{AP$_{L}$}} \\\\\n  &  & &\n \\multicolumn{1}{c}{\\text{FP16}}  & \\multicolumn{1}{c}{\\text{FP16$^{\\dagger}$}} & \\multicolumn{1}{c|}{\\text{FP32}} & \\multicolumn{1}{c|}{} \\\\\n\n \\midrule\n  ResNet50-FPN & RetinaNet & 512$\\times$512&  22 & 14 & \\light{28}  & 42.0 & \\light{22.7} & \\light{47.1}   & \\light{58.4}  \\\\\n  ResNet50-FPN & RetinaNet & 640$\\times$640&  29 & 17 & \\light{44}  &  44.7 &  \\light{27.0}  & \\light{48.5}  &  \\light{60.0} \\\\\n  ResNet101-FPN & RetinaNet & 640$\\times$640&  38 & 24 & \\light{57} &46.1 &  \\light{28.0}  &  \\light{50.3}  &  \\light{62.3} \\\\\n  ResNet101-FPN & RetinaNet & 768$\\times$768&  49 & 30 & \\light{76} & 46.8 &  \\light{29.6} &  \\light{50.6}  &  \\light{62.8} \\\\\n  ResNet152-FPN & RetinaNet & 768$\\times$768&  60 & 38 & \\light{87}  & 47.7 & \\light{30.4}  &  \\light{52.2} & \\light{63.6}   \\\\\n \\midrule\n  ResNet50-FPN & Cascade FRCNN & 512$\\times$512&  35 & 27 & \\light{59} & 46.2 &  \\light{26.2}  & \\light{50.8} & \\light{63.9}  \\\\\n  ResNet50-FPN & Cascade FRCNN & 640$\\times$640&  39 & 31 & \\light{67} & 47.9 &  \\light{30.0}  &  \\light{52.3}  &  \\light{64.3} \\\\\n  ResNet50-FPN & Cascade FRCNN & 768$\\times$768&  43 & 33 & \\light{75}&  49.1 & \\light{31.5}  &  \\light{53.0}  &  \\light{64.5} \\\\\n  ResNet101-FPN & Cascade FRCNN & 768$\\times$768&  53 & 41 & \\light{87} & 50.1 & \\light{33.4}   & \\light{53.9}   & \\light{65.2}  \\\\\n  ResNet101-FPN & Cascade FRCNN & 896$\\times$896&  62 & 48 & \\light{100} & 50.8 & \\light{34.0} & \\light{54.5}   & \\light{65.6}  \\\\\n  ResNet152-FPN & Cascade FRCNN & 896$\\times$896&  73 & 59 & \\light{116} & 51.3 & \\light{33.9}  &  \\light{55.0}  & \\light{65.9}  \\\\\n  ResNet152-FPN & Cascade FRCNN & 1024$\\times$1024&  84 & 70 & \\light{136} & 52.1 & \\light{35.3}  & \\light{55.8}  & \\light{66.5}  \\\\\n  ResNet152-FPN & Cascade FRCNN & 1280$\\times$1280&  114 & 96 & \\light{181} & 52.6 & \\light{36.7}  & \\light{55.8}   & \\light{66.6}  \\\\\n \n \n\\bottomrule\n\\end{tabular}\n\\caption{Result comparisons on COCO \\texttt{val2017} of RetinaNet-RS (first group) and Cascade FRCNN-RS (second group). We report end-to-end latency including post-processing (\\eg NMS) on a Tesla V100 GPU with \\texttt{float16} precision (FP16) and \\texttt{float32} precision (FP32). FP16$^\\dagger$ represents model latency in \\texttt{float16} without measuring post-processinig ops (NMS).}\n\\label{tab:mainresults}\n\\vspace{-0mm}\n\\end{table*}\n\n\\subsection{Detection Framework}~\\label{sec:det_frameworks}\n\\subsubsection{RetinaNet-RS}\n\\label{sec:retinanet}\n\\paragraph{\\bf Detection head:} We follow the standard RetinaNet~\\cite{retinanet} head design. In brief, we use 4 $3\\times3$ convolutional layers at feature dimension $256$ in the box and classification subnets before the final prediction layers. Each convolutional layer is followed by a batch norm layer and a SiLU activation. The convolutional layers are shared across all feature levels in the detection head while the batch norm layers are not shared. We place 3 anchors at aspect ratios $[1.0, 2.0, 0.5]$ at each pixel location and set the base anchor size to $3.0$. The focal loss parameters $\\alpha$ and $\\gamma$ are set to 0.25 and 1.5, respectively. \n\n\\paragraph{\\bf Feature extractor:} RetinaNet-RS uses the backbone, \\eg, modified ResNet-50\/101\/152 described in Section~\\ref{sec:resnet_modifications}. A standard FPN~\\cite{fpn} at feature dimension $256$ is applied after the backbone to extract multi-scale features from level $P_3$ to $P_7$.\n\n\n\\subsubsection{Cascade RCNN-RS}\\label{sec:cascade_rcnn}\nWe use Cascade RCNN, one of the strongest RCNN detectors, as our two-stage detection framework in this work. \n\\paragraph{\\bf RPN head:} For our Cascade RCNN-RS experiments we generally follow the implementation from~\\cite{cai2018cascade}. For the RPN head, we use two 3$\\times$3 convolutional layers at feature dimension $256$ and the same anchor settings as RetinaNet mentioned in Section~\\ref{sec:retinanet}. We use $500$ proposals for training and $1000$ proposals for inference.\n\n\\paragraph{\\bf Box regression head:} We use two settings for the box regression heads, one for regular-size models and one for large-size models. For regular-size models, we implement two cascaded heads with increasing IoU thresholds $0.6$ and $0.7$. Each head has 4 3$\\times$3 convolutional layers at feature dimensions $256$ and one fully connected layer at feature dimension $1024$ before the final prediction layer.\nWe importantly note for getting good performance improvements class agnostic bounding box regression must be used. This is where for the box regression heads only 4 bounding box coordinates are predicted instead of $4 \\times \\text{(number of classes)}$. \n\n\n\\paragraph{\\bf Instance segmentation head:} We use four 3$\\times$3 convolutional layers and one 3$\\times$3 stride-2 deconvolutional layer at feature dimension $256$ before the final prediction layer in the instance segmentation head. \n\n\\paragraph{\\bf Feature extractor:} We first study the performance of the ResNet-50\/101\/152\/200 model family and the EfficientNet B1 to B7 model family with the regular-size Cascade RCNN framework. To scale up ResNet based models, we use the scaling method described in Table~\\ref{tab:scaling_cas}. To scale up EfficientNet based models, we follow the compound scaling rule introduced in~\\cite{efficientdet}. A standard FPN is attached to ResNet and EfficientNet backbones to extract $P_3$ to $P_7$ multi-scale features. To obtain the best performance, we adopt the SpineNet-143\/143L backbones. The SpineNet-143L backbone uniformly scales up feature dimension of all convolutional layers in SpineNet-143 by $1.5\\times$.\n\n\\begin{figure}\n    \\includegraphics[width=1.0\\columnwidth]{imgs\/flopsvsap.pdf}\n    \\caption{\\textbf{Performance comparison of Cascade MRCNN-RS models adopting ResNet-FPN, EfficientNet-FPN and SpineNet backbones.} Results for all models are generated under the same settings described in Section~\\ref{sec:exp_coco_inst}. We show ResNet-FPN and SpineNet backbones outperform EfficientNet-FPN backbones at all model scales.}\n    \\label{fig:inst_seg_curves}\n\\vspace*{-0mm}\n\\end{figure}\n\\begin{figure}\n    \\includegraphics[width=1.0\\columnwidth]{imgs\/precision_latency.pdf}\n    \\caption{\\textbf{Speed improvements with \\texttt{float16} precision.} Inference with \\texttt{float16} leads to a 1.5$\\times$ to 1.7$\\times$ speed boost for RetinaNet-RS and Cascade FRCNN-RS models. Latency numbers are reported on a V100 GPU.}\n\\label{fig:precision_comp}\n\\vspace{-0mm}\n\\end{figure}\n\n\\section{Experimental Results}\\label{sec:experiments}\n\\subsection{Experimental settings}\n\\paragraph{COCO dataset:}\nWe first evaluate our models on the popular COCO benchmark~\\cite{coco}. All models are trained on the \\texttt{train2017} split and evaluated on the \\texttt{val2017} split. Besides the settings described in Section~\\ref{sec:training_benchmark_settings}, we generally follow~\\cite{spinenet,Du2020EfficientSB,efficientdet} to train models from scratch on COCO \\texttt{train2017} with synchronized batch normalization and SGD with a 0.9 momentum rate. Unless noted, all models are trained with a batch size of 256 for 600 epochs on TPUv3 devices~\\cite{jouppi2017tpu}. We apply a step-wise decay learning rate schedule with an initial learning rate 0.28 that decays to $0.1\\times$ and $0.01\\times$ at the last 25 epoch and the last 10 epoch. A linear learning rate warm-up is applied over the first 5 epochs. Our main results for bounding box detection and instance segmentation are reported on COCO \\texttt{val2017}.\n\n\\subsection{COCO Bounding Box Detection}\\label{sec:exp_coco_box}\nOur results of RetinaNet-RS and Cascade Faster RCNN-RS (Cascade FRCNN-RS) on the COCO bounding box detection task are presented in Table~\\ref{tab:mainresults} and Figure~\\ref{fig:page1_fig}.\n\nFrom Figure~\\ref{fig:page1_fig} we can see that the new training methods improve COCO AP by 4.5\\% with no inference cost and the architectural changes improve COCO AP by another 3.2\\% with insignificant computational cost. Figure~\\ref{fig:page1_fig} further shows that at high-computation regime (\\eg when using large model scale and large input size), the two-stage Cascade RCNN-RS models outperform the one-stage RetinaNet-RS models on the speed-accuracy pareto curve. At low-computation regime, RetinaNet-RS is more efficient than Cascade FRCNN-RS.\n\n\\begin{table*}\\centering\n\\small\n\\begin{tabular}{c c  | c c| c | c c c}\n\\toprule\n \n  \\multicolumn{1}{c}{\\multirow{2}{*}{Backbone}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{Resolution}} & \\multicolumn{2}{c|}{Latency \\light{(ms)}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\hspace{5mm}AP\\hspace{5mm}}} & \\multicolumn{1}{c}{\\multirow{2}{*}{AP$_{S}$}}& \\multicolumn{1}{c}{\\multirow{2}{*}{AP$_{M}$}} & \\multicolumn{1}{c}{\\multirow{2}{*}{AP$_{L}$}} \\\\\n  &  &\n \\multicolumn{1}{c}{\\text{FP16}}  & \\multicolumn{1}{c|}{\\text{FP32}} & \\multicolumn{1}{c|}{} \\\\\n\n \\midrule\n  EfficientNetB1-FPN  & 640$\\times$640& 42 &\\light{71}&  45.5 &  \\light{24.3}  &  \\light{48.8}  &  \\light{65.0}  \\\\\n  EfficientNetB2-FPN  & 768$\\times$768& 48 &\\light{78}&  46.9 & \\light{26.4}   & \\light{49.8}   &  \\light{66.5}  \\\\\n  EfficientNetB3-FPN  & 896$\\times$896& 54&\\light{90}& 48.9 & \\light{29.2}  & \\light{52.0}  & \\light{67.2}   \\\\\n  EfficientNetB4-FPN  & 1024$\\times$1024& 67&\\light{109}&  50.5 &  \\light{30.5}  & \\light{53.1}   & \\light{69.2}  \\\\\n  EfficientNetB5-FPN  & 1280$\\times$1280&  90&\\light{158} &  52.0 &  \\light{32.3}  & \\light{55.3}   & \\light{69.7}   \\\\\n  EfficientNetB6-FPN  & 1280$\\times$1280&  101&\\light{179}&  52.5 &  \\light{33.0}  &  \\light{55.4}  & \\light{69.9}   \\\\\n  EfficientNetB7-FPN  & 1536$\\times$1536& 149&\\light{263} & 52.6 & \\light{32.2}  & \\light{55.8}   &  \\light{70.2} \\\\\n \\midrule\n  ResNet50-FPN  & 512$\\times$512&  41&\\light{69}&  46.1 &  \\light{23.7}  &  \\light{50.1}  & \\light{67.1}   \\\\\n  ResNet50-FPN  & 640$\\times$640&  45&\\light{77}&  48.4 &  \\light{27.4}  & \\light{51.5}   & \\light{67.9}    \\\\\n  ResNet50-FPN  & 768$\\times$768&  50&\\light{85}&  49.4 &  \\light{29.4}  & \\light{52.6}   & \\light{68.3}  \\\\\n  ResNet101-FPN & 768$\\times$768&  60&\\light{97}&  50.3 & \\light{29.7} & \\light{53.9} &  \\light{69.7}   \\\\\n  ResNet101-FPN & 896$\\times$896&  68&\\light{109}&  51.1 &  \\light{31.4}  & \\light{54.3}  & \\light{69.8}    \\\\\n  ResNet152-FPN & 896$\\times$896&  79&\\light{125}&  51.8 &  \\light{32.0}  & \\light{55.1} & \\light{70.0} \\\\\n  ResNet152-FPN & 1024$\\times$1024&  90&\\light{148}&  52.4 &  \\light{32.9}  &  \\light{55.3}  & \\light{70.0}  \\\\\n  ResNet152-FPN & 1280$\\times$1280&  119&\\light{191}&  52.9 &  \\light{33.5}  &  \\light{56.7}  &  \\light{70.3}  \\\\\n  ResNet200-FPN &  1280$\\times$1280&  149&\\light{232}&  53.1 & \\light{33.9}  & \\light{56.2}   & \\light{70.3}  \\\\\n  \\midrule\n  SpineNet143 & 1280$\\times$1280&  109&\\light{175}&  53.0 &  \\light{33.8}  & \\light{55.8} & \\light{70.5}  \\\\\n  SpineNet143L & 1280$\\times$1280&  144&\\light{234}&  53.6 & \\light{34.5} & \\light{56.7} & \\light{70.6}  \\\\\n  \n\\bottomrule\n\\end{tabular}\n\\vspace*{-0mm}\n\\caption{\\textbf{Result comparisons on COCO \\texttt{val2017} among Cascade MRCNN-RS models adopting ResNet-FPN, SpineNet and EfficientNet-FPN backbones.} All results are generated in the same codebase. We report end-to-end latency including NMS on a Tesla V100 GPU with \\texttt{float16} precision (FP16) and \\texttt{float32} precision (FP32).}\n\\label{tab:instace_seg_reuslts}\n\\vspace{-0mm}\n\\end{table*}\n\n\\subsection{COCO Instance Segmentation}\\label{sec:exp_coco_inst}\nCascade Mask RCNN-RS (Cascade MRCNN-RS) is used as our two-stage instance segmentation framework. We compare three backbones: ResNet-FPN, EfficeintNet-FPN and SpineNet. All models are trained and evaluated in the same codebase and benchmarked on a Tesla V100 GPU under the same settings. Results are presented in Table~\\ref{tab:instace_seg_reuslts} and Figure~\\ref{fig:inst_seg_curves}.\n\nAdopting the same Cascade MRCNN-RS framework and trained under the same settings, ResNet-FPN backbones are able to achieve a better speed-accuracy Pareto curve than the EfficientNet-FPN backbones at all model scales. The Cascade MRCNN-RS model adopts a ResNet200-FPN backbone at input resolution 1280 achieves 53.1\\% AP. By replacing the backbone with SpineNet-143L, we further improve the AP to 53.6\\% with a slightly faster speed.\n\n\\section{Understanding Performance Improvements}\n\\subsection{Effectiveness of Modern Techniques}\nWe conduct ablation studies for the modern training methods and architectural modifications with a RetinaNet-RS model that adopts a ResNet50-FPN backbone at 640 input resolution. As shown in Table~\\ref{tab:resnet_modifications}, training with large scale jittering for 350 epochs improves AP by 3.5\\%. Stronger model regularization and a longer 600-epoch training schedule improves AP by 1.0\\%. The modern training methods improve AP by 4.5\\% without introducing any computational costs. For architectural modifications, replacing ReLU with SiLU activation improves AP by 1.0\\%, Plugging in Squeeze-and-Excitation modules improve AP by 1.5\\% and adopting the ResNet-D stem improves AP by another 0.7\\%. The three modifications improve AP by 3.2\\% while introducing insignificant computational costs. The results are also presented in Figure~\\ref{fig:page1_fig}.\n\n\\setlength{\\tabcolsep}{4pt}\n\\begin{table*}[!h]\n\\begin{center}{\n\\begin{tabular}{c | c c  c c c c}\n\\toprule\nBackbone \\textbackslash Resolution & 512 & 640 & 768 & 896 & 1024 & 1280 \\\\\n\\midrule\nR50-FPN & 46.1 \\light{(69)} & 48.4 \\light{(77)} & 49.4 \\light{(85)} & 50.0 \\light{(94)} & 50.2 \\light{(106)} & 50.8 \\light{(132)} \\\\\nR101-FPN & 47.5 \\light{(75)} & 49.3 \\light{(86)} & 50.3 \\light{(97)} & 51.1 \\light{(109)} & 51.6 \\light{(123)} & 51.9 \\light{(160)} \\\\\nR152-FPN & 47.8 \\light{(83)} & 49.7 \\light{(96)} & 50.6 \\light{(109)} & 51.8 \\light{(125)} & 52.3 \\light{(148)} & 52.9 \\light{(193)} \\\\\nR200-FPN & 48.3 \\light{(92)} & 50.4 \\light{(109)} & 51.5 \\light{(127)} & 52.2 \\light{(148)} & 52.6 \\light{(174)} & 53.1 \\light{(232)} \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\caption{\\textbf{ResNet depth vs input resolution.} We report COCO \\texttt{val2017} AP of different ResNet backbones and the corresponding inference latency on V100 GPU (numbers in parentheses) with \\texttt{float32} precision. All models adopt the Cascade MRCNN-RS detector.}\n\\label{tab:resnet_resolution}\n\\vspace{-0mm}\n\\end{table*}\n\n\n\\begin{figure}[!h]\n    \\includegraphics[width=1.0\\columnwidth]{imgs\/nms_latency.pdf}\n    \\caption{\\textbf{Speed comparisons with or without measuring post-processing ops for RetinaNet-RS and Cascade FRCNN-RS.} Latency numbers are reported on a V100 GPU.}\n\\label{fig:nms_comp}\n\\vspace{-0mm}\n\\end{figure}\n\n\\subsection{Latency benchmarking settings}\n\\paragraph{Precision float16:} Inference in 16-bit floating-point types make model run faster and use less memory. This subsection compares RetinaNet-RS and Cascade FRCNN-RS models using \\texttt{float32} precision and \\texttt{float16} precision for model inference. By casting model weights and input images from \\texttt{float32} to \\texttt{float16} precision, the speed of model forward pass can be boosted to 1.5$\\times$ to 1.7$\\times$. Inference latency and speed improvements of all models are presented in Figure~\\ref{fig:precision_comp} and Table~\\ref{tab:mainresults},~\\ref{tab:instace_seg_reuslts}.\n\\paragraph{Post-processing:} Post-processing ops such as NMS takes a non-negligible portion of latency. We show the speed of our RetinaNet-RS and Cascade FRCNN-RS models with and without measuring post-processing ops in Table~\\ref{tab:mainresults} and the pareto curve comparisons in Fig.~\\ref{fig:nms_comp}.\n\n\\subsection{ResNet Depth vs Input Resolution}\nThis section analyzes the efficiency of the proposed scaling method that scales model depth and input resolution at the same time. We present the performance of Cascade MRCNN-RS models that adopt ResNet-50\/101\/152\/200 backbones and trained at input resolution 512, 640, 768, 896, 1024 and 1280. We show that a better scaling efficiency can be achieved when scaling model depth and input resolution together. At low input resolutions (\\ie 512 or 640), the ResNet-50 backbone is more effective than deeper ResNet variants. At slight larger input resolutions (\\ie, 768 or 896), the ResNet-101 backbone is more effective. At high input resolutions (\\ie, 1024 or 1280) the ResNet-152 backbone is the most effective choice. Further scaling model depth to ResNet-200 does not improve the speed-accuracy Pareto curve. The results are shown in Table~\\ref{tab:resnet_resolution}.\n\n\\setlength{\\tabcolsep}{4pt}\n\\begin{table}[!h]\n\\begin{center}{\n\\begin{tabular}{c | c c c c}\n\\toprule\n Jitter Scale & 90-ep & 200-ep & 400-ep & 600-ep\\\\\n \\midrule\n\\text{[1.0, 1.0]} & 41.9 & 39.3 & 38.4 & 35.7 \\\\\n\\text{[0.8, 1.2]} & 43.8 & 45.0 & 44.6 & 44.3\\\\\n\\text{[0.1, 2.0]} & 43.9 & 47.0 & 47.6 & 47.9\\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\caption{\\textbf{Jitter scales vs training epochs.} We show that model performance significantly benefits from using aggressive scale jittering with longer training schedules.}\n\\label{tab:jitter_epoch}\n\\vspace{-0mm}\n\\end{table}\n\n\\setlength{\\tabcolsep}{4pt}\n\\begin{table*}[h]\n\\begin{center}{\n\\begin{tabular}{c  | c c | c c}\n\\toprule\n Model & Train Resolution & Eval Resolution & AP\\slash L1 & AP\\slash L2\\\\\n \\midrule\n RetinaNet-RS & 896$\\times$1664 & 896$\\times$1664 & 64.5 & 56.1 \\\\\n\\midrule\n FRCNN-RS & 896$\\times$1664 & 896$\\times$1664 & 68.1 & 60.0\\\\\n FRCNN-RS & 896$\\times$1664 & 1024$\\times$1920 & 69.5 & 61.2\\\\\n FRCNN-RS \n & 896$\\times$1664 & 1280$\\times$2400 & 70.0 & 63.4\\\\\n\\midrule\n Cascade FRCNN-RS & 896$\\times$1664 & 896$\\times$1664 & 68.6 & 60.7\\\\\n Cascade FRCNN-RS & 896$\\times$1664 & 1024$\\times$1920 & 70.5 & 61.8\\\\\n Cascade FRCNN-RS & 896$\\times$1664 & 1280$\\times$2400 & 71.2 & 62.9\\\\\n\n\\bottomrule\n\\end{tabular}\n}\n\\end{center}\n\\caption{\\textbf{Results on Waymo Open Dataset.} We evaluate different detectors adopting the SpineNet143L backbone.}\n\\label{tab:waymo}\n\\end{table*}\n\n\n   \n\n\n\n\\subsection{Scale Jittering vs Training Schedule}\nApplying large scale jittering augmentation potentially results in more unique training samples during the training procedure, which allows the model to benefit from a longer training schedule. We empirically show that the best model performance can be achieved by enabling large scale jittering to train models for an up to 600 epoch schedule. To study the effectiveness, we compare three scale jittering setups, [0.1, 2.0], [0.8, 1.2], and [1.0. 1.0], on 90-epoch, 200-epoch, 400-epoch, and 600-epoch model training schedules. The results are shown in Table~\\ref{tab:jitter_epoch}.\n\n\\subsection{Impact of Non-maximum Suppression}\nIn this paper, the detection frameworks (RetinaNet and R-CNN) employ non-maximum suppression (NMS) to remove duplicate detection. It becomes a bottleneck as we keep improving train the training technologies and model architecture. Figure~\\ref{fig:page1_fig} and~\\ref{fig:nms_comp} compares the inference time with and without the NMS operation. The non-maximum suppression takes about ~40\\% of inference time in RetinaNet and 15-30\\% of inference time in Cascade Faster R-CNN. A well optimized software library, \\eg, TensorRT, or the detectors without NMS, \\eg,~\\cite{carion2020endtoend,centernet}, can save the computation overhead.\n\n\n\\section{Conclusion}\\label{sec:conclusion}\nIn this work, we dissects the performance improvements coming from minor architecture modifications and training\/inference methods. We design a family of models using a simple scaling strategy that only changes the backbone capacity and image resolution. We find the speed-accuracy Pareto curve is already a strong baseline to recent object detection systems. We hope this study will help the research community understand the bottleneck and performance improvements of modern object detection systems.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}