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+{"text":"\\section{Introduction}\n\nThe inverse obstacle scattering problems are of significant importance in diverse areas of sciences and technology such as non-destructive testing, radar sensing, sonar detection and biomedical imaging (see, e.g. \\cite{Colton}), which are typical exterior inverse scattering problems. However, the interior inverse scattering problems for determining the shape of cavities arise in many practical applications of radar sensing and non-destructive testing (see, e.g. \\cite{Jakubik, Qin1}). In contrast to the typical exterior inverse scattering problems, the interior inverse scattering problems are more complicated to some extent due to the repeated reflections of the scattered waves, and some mathematical\nstudies have been made. In \\cite{Qin1, Zeng, Liu14}, the uniqueness of the inverse cavity scattering with the Dirichlet boundary condition has been established. For the the impedance boundary condition and the mixed boundary condition, the uniqueness results have been  given in \\cite{Qin2, Liu14} and \\cite{Hu}, respectively. In \\cite{Qin}, the authors proposed the method of adding an artificial obstacle to avoid the interior eigenvalues and gave a new proof for the uniqueness of the inverse problems. There have also been some numerical reconstruction algorithms for solving the inverse cavity problems. We refer to \\cite{Qin1, Qin2, Hu, Zeng1, Qin0, Zeng, Liu14, Sun} for the linear sampling method, the regularized Newton iterative method, the decomposition method, the factorization method and the reciprocity gap functional method.\n\nThe above theories and numerical methods are based on the full data (both the intensity and phase). However, in many situations, one can measure only the intensity\/magnitude of the data, which leads to the study of inverse scattering problems with phaseless or intensity-only data.\n\nThe exterior inverse scattering problems with phaseless near-field data have been studied numerically (see, e.g. \\cite{Candes, Candes1, Caorsi, CH17, CFH17, Maleki, Maleki1, Pan, Takenaka}), and few results have been done on the theory of uniqueness for the inverse scattering problems. A recent result on uniqueness in \\cite{Kli14} was related to the reconstruction of a potential with the phaseless near-field data for point sources on a spherical surface and an interval of wave-numbers, which was extended in \\cite{Kli17} for determining the wave speed in generalized 3-D Helmholtz equation. The uniqueness of a coefficient inverse scattering problem with phaseless near-field data has been established in \\cite{KR17}. We also refer to \\cite{KR16, Novikov15, Novikov16} for some recovery algorithms for the inverse medium scattering problems with phaseless near-field data. The stability analysis for linearized near-field phase retrieval in X-ray phase contrast imaging can be found in \\cite{Maretzke}.\n\nFor exterior inverse scattering problems with phaseless far-field data, several uniqueness results have been established. With a priori information, uniqueness on determining the radius of a sound-soft ball was given in \\cite{LZ09}. A method of superposition of incident waves was proposed in \\cite{ZhangBo20171}, which led to the multi-frequency Newton iteration algorithm \\cite{ZhangBo20171, ZhangBo20172} and the fast imaging algorithm \\cite{ZZ18}. Moreover, uniqueness results were established in \\cite{XZZ18a} under some a priori assumptions. Recently, the idea of resorting to the reference ball technique (see, e.g. \\cite{Colton1, Colton2, Li}) in phaseless inverse scattering problems was  proposed by Zhang and Guo in \\cite{ZhangDeyue20181}, and the uniqueness results were established by utilizing the reference ball technique in conjunction with the superposition of incident waves. With the aid of the reference ball technique, the a priori assumptions in \\cite{XZZ18a} can be removed, see \\cite{XZZ18b} for the details. Similar strategies of adding reference objects or sources to the scattering system for different models of phaseless inverse scattering problems can be found in \\cite{DZG19, DLL18, JL18, JLZ18a, JLZ18b, ZhangDeyue20182}. For the numerical algorithms for the shape reconstruction from phaseless data, we refer to \\cite{BLL2013, Bao2016, CH17, Ivanyshyn1, Ivanyshyn2, Ivanyshyn3, KR16, Kress, Lee2016, Li1, LLW17, Shin}.\n\n\nIn this paper, we consider the incident point sources and deal with the uniqueness issue concerning the inverse cavity scattering problems with phaseless total field data.\nWe rigorously prove in this paper that the location and shape of the obstacle as well as its boundary condition can be uniquely determined by the modulus of total fields at an admissible surface. To the best of our knowledge, this is a first uniqueness result in inverse cavity scattering problems with phaseless near-field data.  The main idea here is the utilization of the reference ball technique, superpositions of point sources, the reciprocity relations and the singularity of the total fields. We emphasize that the reference ball technique should be necessary for the phaseless inverse scattering problems for cavities owing to lack of the far-field pattern, and the reference ball can provide some information on the location of the cavity in devising effective numerical inversion schemes in comparison with the exterior inverse scattering problems (see, e.g. \\cite{DZG19}), which will be our future work.\n\nThe rest of this paper is arranged as follows. In the next section, we present an introduction to the model problem. Section \\ref{sec:obstacle} is devoted to the uniqueness results on phaseless inverse cavity scattering problem.\n\n\n\n\\section{Problem setting}\\label{sec:problem_setup}\n\nWe begin this section with the precise formulations of the model cavity scattering problem. Assume $D \\subset\\mathbb{R}^3$ is an open and simply connected domain with\n$C^2$ boundary $\\partial D$.  Denote by $u^i$ the incident field. Then, the interior scattering problem for cavities can be formulated as: to find the scattered field $u^s$ which satisfies the following boundary value problem:\n\\begin{align}\n\\Delta u^s+ k^2 u^s= & 0 \\quad \\mathrm{in}\\ D,\\label{eq:Helmholtz} \\\\\n\\mathscr{B}u= & 0 \\quad \\mathrm{on}\\ \\partial D, \\label{eq:boundary_condition}\n\\end{align}\nwhere $u=u^i+u^s$ denotes the total field and $k>0$ is the wavenumber. Here $\\mathscr{B}$ in \\eqref{eq:boundary_condition} is the boundary operator defined by\n\\begin{equation}\\label{BC}\n\\begin{cases}\n\\mathscr{B}u=u, & \\text{for a sound-soft cavity},  \\\\\n\\mathscr{B}u=\\dfrac{\\partial u}{\\partial \\nu}+ \\lambda u, & \\text{for an impedance cavity},\n\\end{cases}\n\\end{equation}\nwhere $\\nu$ is the unit outward normal to $\\partial D$, and $\\lambda\\in C(\\partial D)$ is the impedance function satisfying $\\Im(\\lambda)\\geq 0$. This boundary condition \\eqref{BC} covers the Dirichlet\/sound-soft boundary condition, the Neumann\/sound-hard boundary condition ($\\lambda=0$), and the impedance boundary condition ($\\lambda\\neq 0$). The existence of a solution to the direct scattering problem \\eqref{eq:Helmholtz}--\\eqref{eq:boundary_condition} is well known (see, e.g. \\cite{Cakoni1, Colton3, Colton}).\n\nNow, we turn to introducing the  interior inverse scattering problem for incident point sources with limited-aperture phaseless near-field data. To this end, we first introduce a reference ball $B$ as an extra artificial object to the scattering system such that $\\overline{B}\\subset\\subset D$ with the impedance boundary condition\n\\begin{eqnarray}\\label{Referenceball}\n\\displaystyle\\frac{\\partial u}{\\partial \\nu}+ i\\lambda_0  u=0\\quad \\mathrm{on}\\ \\partial B,\n\\end{eqnarray}\nwhere $\\lambda_0$ is a positive constant, and the following definition of admissible surfaces.\n\\begin{definition}[Admissible surface]\nAn open surface $\\Gamma$ is called an admissible surface with\nrespect to domain $\\Omega$ if\n\n\\noindent (i) $\\Omega\\subset\\mathbb{R}^3\\backslash\\overline{D}$ is bounded and simply-connected;\n\n\\noindent (ii) $\\partial \\Omega$ is analytic homeomorphic to $\\mathbb{S}^2$;\n\n\\noindent (iii) $k^2$ is not a Dirichlet eigenvalue of $-\\Delta$ in $\\Omega$;\n\n\\noindent (iv) $\\Gamma\\subset\\partial\\Omega$ is a two-dimensional analytic manifold with nonvanishing measure.\n\\end{definition}\n\n\\begin{remark}\nThe artificial obstacle with impedance boundary condition \\eqref{Referenceball} can also be founded in \\cite{Qin} to remove the interior eigenvalues for the direct scattering problems and the reference ball technique has been used in \\cite{Li, Colton1, Colton2} for the exterior inverse scattering problems.\n\\end{remark}\n\\begin{remark}\n    We would like to point out that this requirement for the admissibility of $\\Gamma$ is quite mild and thus can be easily fulfilled. For instance, $\\Omega$ can be chosen as a ball whose radius is less than $\\pi\/k$ and $\\Gamma$ is chosen as an arbitrary corresponding semisphere.\n\\end{remark}\n\nFor a generic point $z\\in D\\backslash\\overline{B}$, the incident field $u^i$ due to the point source located at $z$ is given by\n\\begin{equation*}\nu^i (x, z):=\\frac{\\mathrm{e}^{\\mathrm{i} k|x-z|}}{4\\pi |x-z|}, \\quad x\\in D\\backslash(\\overline{B}\\cup\\{z\\}),\n\\end{equation*}\nwhich is also known as the fundamental solution to the Helmholtz equation. Denote by $u^s(x,z)$  the near-field generated by $D$ and $B$ corresponding to the incident field $u^i(x, z)$. Let $u(x,z)=u^s(x,z)+u^i(x, z)$, $x\\in D\\backslash(\\overline{B}\\cup\\{z\\})$ be the total field.\n\n\nFor two generic and distinct source points $z_1, z_2\\in D\\backslash\\overline{B}$, we denote by\n\\begin{equation}\\label{incident}\n u^i(x; z_1,z_2):=u^i(x, z_1)+u^i(x, z_2),\\quad x\\in D\\backslash(\\overline{B}\\cup\\{z_1\\}\\cup\\{z_2\\}),\n\\end{equation}\nthe superposition of these point sources. Then, by the linearity of direct scattering problem, the near-field co-produced by $D$, $B$ and the incident wave $u^i(x; z_1,z_2)$ is given by\n$$\nu(x;z_1,z_2):=u(x,z_1)+u(x,z_2), \\quad x\\in D\\backslash(\\overline{B}\\cup\\{z_1\\}\\cup\\{z_2\\}).\n$$\n\nWith these preparations,  we formulate the phaseless inverse scattering problems as the following.\n\n\\begin{problem}[Phaseless inverse scattering]\\label{prob:obstacle}\nLet $D$ be the impenetrable cavity with boundary condition $\\mathscr{B}$. Assume that $\\Gamma$ and $\\Sigma$ are admissible surfaces with\nrespect to $\\Omega$ and $G$, respectively, such that\n$\\overline{\\Omega}\\subset\\subset G$ and $\\overline {G}\\subset\\subset D\\backslash\\overline{B}$.  Given the phaseless\nnear-field data\n \\begin{equation*}\n \\begin{array}{ll}\n & \\{|u(x,z_0)|: x\\in \\Sigma\\}, \\\\\n & \\{|u(x,z)|:  x\\in \\Sigma,\\ z\\in \\Gamma\\}, \\\\\n & \\{|u(x,z_0)+u(x,z)|: x\\in \\Sigma,\\ z\\in \\Gamma\\}\n \\end{array}\n \\end{equation*}\n for a fixed wavenumber $k>0$ and a fixed  $z_0\\in D\\backslash(\\overline{B}\\cup\\Gamma\\cup\\Sigma)$, determine the location and shape $\\partial D$ as well as the boundary condition\n $\\mathscr{B}$ for the cavity.\n\\end{problem}\n\nWe refer to Figure \\ref{fig:illustration} for an illustration of the geometry setting of Problem \\ref{prob:obstacle}. The uniqueness of this problem will be analyzed in the next section.\n\n\\begin{figure}\n\t\\centering\n\t\\newdimen\\R\n\t\\R=0.5cm\n\t\\begin{tikzpicture}[thick]\n\t\\pgfmathsetseed{8}\n\t\\draw plot [smooth cycle, samples=6, domain={1:8},  xshift=0cm, yshift=4cm] (\\x*360\/8+5*rnd:0.5cm+7cm*rnd) node at (1.5, 5.5) {$D$};\n\t\n\t\\draw (1.5, 1) circle (0.7cm) [fill=lightgray] node at (1.5,1) {$B$}\n\t\\draw [blue, dashed] (1.5, 3.5) ellipse [x radius=2.2cm, y radius=1.2cm] ;\n\t\n\t\\draw [blue, dashed] (1.5, 3.5) circle (0.6cm);\n\t\\draw node at (1.5,3.4) {$\\Omega$};\n\t\\draw (2.1, 3.5) arc(0:180:0.6cm) [very thick, blue];\n\t\\draw node at (2.2,3.9) {$\\Gamma$};\n\t\\draw node at (1.3,3.8) {$z$};\n\t\\fill [red] (1.,3.85) circle (2pt); \n\t\\draw [->] (0.9, 3.9)--(0.3, 4.3);\n\t\\draw (0.95,4.19) arc(120:180:0.5cm);\n\t\\draw (0.85,4.36) arc(120:185:0.6cm);\n\t\n\t\\fill [red] (-2.6,2.8) circle (2pt); \n\t\\draw (-2.8, 2.6) node   {$z_0$};\n\t\\draw [->](-2.5,2.9)--(-2,3.35);\n\t\\draw (-2.19, 2.9) arc(10:70:0.5cm);\n\t\\draw (-2.05, 2.9) arc(10:70:0.7cm);\n\t\\draw (-1.7, 3.5) node [above] {$u^i(\\cdot; z_0, z)$};\n\t\n\t\\draw [->](4.05, 4.9)-- (3.38,4.3);\n\t\\draw (3.58, 3.1) arc(-35:45:0.7cm) [very thick, blue];\n\t\\draw (3.5, 4) arc(45:80:2.2cm) [very thick, blue];\n\t\\draw node at (3.05,4.) {$x$};\n\t\\fill [red] (3.27,4.19) circle (2pt); \n\t\n\t\\draw node at (3.,3.4) {$G$};\n\t\\draw node at (3.95,3.3) {$\\Sigma$};\n\t\\draw (3.7, 4.9) arc(188:238:0.6cm);\n\t\\draw (3.5, 4.9) arc(188:245:0.8cm);\n\t\n\t\\draw node at (4.9,3.9) {$|u(\\cdot; z_0, z)|$};\n\t\\end{tikzpicture}\n\t\\caption{An illustration of the phaseless inverse scattering problem.} \\label{fig:illustration}\n\\end{figure}\n\n\n\\section{Uniqueness for the phaseless inverse scattering}\\label{sec:obstacle}\n\nNow we present the uniqueness results on phaseless inverse scattering. The following theorem shows that Problem \\ref{prob:obstacle} admits a unique solution, namely, the geometrical and physical information of the scatterer boundary can be simultaneously and uniquely determined from the modulus of near-fields.\n\n\\begin{theorem}\\label{Thm1\n Let $D_1$ and $D_2 $ be two impenetrable cavities with boundary conditions $\\mathscr{B}_1$ and $\\mathscr{B}_2$, respectively.  Assume that $\\Gamma$ and $\\Sigma$ are admissible surfaces with respect to $\\Omega$ and $G$, respectively, such that\n$\\overline{\\Omega}\\subset\\subset G$ and $\\overline {G}\\subset\\subset (D_1\\cap D_2)\\backslash\\overline{B}$. Suppose that the corresponding near-fields satisfy\nthat\n   \\begin{align}\n   |u_1(x,z_0)|= & |u_2(x,z_0)|, \\quad \\forall x  \\in \\Sigma, \\label{obstacle_condition1} \\\\\n   |u_1(x,z)|= & |u_2(x,z)|, \\quad \\forall (x, z) \\in \\Sigma\\times\\Gamma \\label{obstacle_condition2}\n   \\end{align}\n   and\n   \\begin{eqnarray}\\label{obstacle_condition3}\n   |u_1(x,z_0)+u_1(x,z)|=|u_2(x,z_0)+u_2(x,z)|,\\quad \\forall (x, z) \\in \\Sigma\\times\\Gamma\n   \\end{eqnarray}\n   for an arbitrarily fixed $z_0\\in (D_1\\cap D_2)  \\backslash(\\overline{B}\\cup\\Gamma\\cup\\Sigma)$. Then we have\n $D_1=D_2$ and $\\mathscr{B}_1=\\mathscr{B}_2$.\n\\end{theorem\n\\begin{proof}\n    From  \\eqref{obstacle_condition1}, \\eqref{obstacle_condition2} and \\eqref{obstacle_condition3}, we have for all $x\\in\\Sigma, z\\in\\Gamma$\n    \\begin{equation}\\label{Thm1equality1}\n    \\mathrm{Re}\\left\\{u_1(x,z_0) \\overline{u_1(x,z)}\\right\\}\n    =\\mathrm{Re}\\left\\{u_2(x,z_0) \\overline{u_2(x,z)}\\right\\},\n    \\end{equation}\n    where the overline denotes the complex conjugate. According to \\eqref{obstacle_condition1} and \\eqref{obstacle_condition2}, we denote\n    \\begin{equation*}\n        u_j(x,z_0)=r(x,z_0) \\mathrm{e}^{\\mathrm{i} \\alpha_j(x,z_0)},\\quad\n        u_j(x,z)=s(x,z) \\mathrm{e}^{\\mathrm{i} \\beta_j(x,z)},\\quad j=1,2,\n    \\end{equation*}\n    where $r(x,z_0)=|u_j(x,z_0)|$, $s(x,z)=|u_j(x,z)|$, $\\alpha_j(x,z_0)$ and $\\beta_j(x,z)$, are real-valued functions,  $j=1,2$.\n\n    Since $\\Sigma$ is an admissible surface of $G$, by definition 2.1 and the analyticity of $u_j(x,z)$ with respect to $x$, we have $s(x,z)\\not\\equiv 0$\n    for $x \\in \\Sigma, z \\in\\Gamma$. Further, the continuity yields that there exists open sets $\\tilde{\\Sigma}\\subset \\Sigma$ and $\\Gamma_0\\subset\\Gamma$\n    such that $s(x,z)\\neq 0$, $\\forall (x,z)\\in \\tilde{\\Sigma}\\times\\Gamma_0$. Similarly, we have $r(x,z_0)\\not\\equiv 0$ on $\\tilde{\\Sigma}$. Again,\n    the continuity leads to $r(x,z_0)\\neq 0$ on an open set $\\Sigma_0 \\subset \\tilde{\\Sigma}$. Therefore, we have $r(x,z_0)\\neq 0$, $s(x,z)\\neq 0,\\ \\forall (x, z) \\in \\Sigma_0\\times\\Gamma_0$. In addition, taking \\eqref{Thm1equality1} into account, we derive that\n    \\begin{equation*}\n        \\cos[\\alpha_1(x,z_0)-\\beta_{1}(x,z)]=\\cos[\\alpha_2(x,z_0)-\\beta_{2}(x,z)], \\quad \\forall (x, z) \\in \\Sigma_0\\times\\Gamma_0.\n    \\end{equation*}\n    Hence, either\n    \\begin{eqnarray}\\label{Thm1equality2}\n    \\alpha_1(x,z_0)-\\alpha_2(x,z_0)=\\beta_{1}(x,z)-\\beta_{2}(x,z)+ 2m\\pi, \\quad \\forall (x, z) \\in \\Sigma_0\\times\\Gamma_0\n    \\end{eqnarray}\n    or\n    \\begin{eqnarray}\\label{Thm1equality3}\n    \\alpha_1(x,z_0)+\\alpha_2(x,z_0)=\\beta_1(x,z)+\\beta_{2}(x,z)+ 2m\\pi, \\quad \\forall (x, z) \\in \\Sigma_0\\times\\Gamma_0\n    \\end{eqnarray}\n    holds with some $m \\in \\mathbb{Z}$.\n\n    First, we shall consider the case \\eqref{Thm1equality2}. Since $z_0$ is fixed, let us define $\\gamma(x):=\\alpha_1(x,z_0)-\\alpha_2(x,z_0)- 2m\\pi$\n    for $ x \\in \\Sigma_0$, and then, we deduce for all $ (x, z) \\in \\Sigma_0\\times\\Gamma_0$\n    \\begin{equation*}\n        u_1(x,z)=s(x,z)\\mathrm{e}^{\\mathrm{i} \\beta_{1}(x,z)}\n        =s(x,z)\\mathrm{e}^{\\mathrm{i} \\beta_{2}(x,z)+\\mathrm{i} \\gamma(x)}=u_2(x,z)\\mathrm{e}^{\\mathrm{i} \\gamma(x)}.\n    \\end{equation*}\n    From the reciprocity relation \\cite[Theorem 2.1]{Qin} for point sources, we have\n    \\begin{equation*}\n        u_1(z,x)= \\mathrm{e}^{\\mathrm{i} \\gamma(x)}u_2(z,x), \\quad \\forall (x, z) \\in \\Sigma_0\\times\\Gamma_0.\n    \\end{equation*}\nThen, for every $x\\in \\Sigma_0$, by using the analyticity of $u_j(z,x)$($j=1,2$) with respect to $z$, we have $u_1(z,x)= \\mathrm{e}^{\\mathrm{i} \\gamma(x)}u_2(z,x),$   $\\forall z\\in \\partial\\Omega$.\nLet $w(z,x)=u_1(z,x)-\\mathrm{e}^{\\mathrm{i} \\gamma(x)}  u_2(z,x)$ and $D_0=D_1\\cap D_2$, then\n$$\n\\begin{cases}\n\\Delta w+ k^2 w=0 &\\mathrm{in}\\ \\Omega, \\\\\nw=0  &\\mathrm{on}\\ \\partial \\Omega.\n\\end{cases}\n$$\n\nBy the assumption of $\\Omega$ that $k^2$ is not a Dirichlet eigenvalue of $-\\Delta$ in $\\Omega$,  we find $w=0$ in $\\Omega$. Now, the analyticity of\n$u_j(z,x)(j=1,2)$ with respect to $z$ yields\n    \\begin{equation*}\n        u_1(z,x)= \\mathrm{e}^{\\mathrm{i} \\gamma(x)}u_2(z,x),\\quad \\forall z \\in D_0\\backslash (B\\cup \\{x\\}).\n    \\end{equation*}\ni.e., for all $ z \\in D_0\\backslash (B\\cup \\{x\\})$,\n\\begin{equation}\\label{eq:relation}\n        u^s_1(z,x)+\\Phi(z,x)=\\mathrm{e}^{\\mathrm{i} \\gamma(x)} \\left[ u^s_2(z,x)+\\Phi(z,x)\\right].\n\\end{equation}\nBy the Green's formula \\cite[Theorem 2.5]{Colton}, one can readily deduce that $u^s_j(z,x)$ is bounded for $z\\in D_j\\backslash B$ $(j=1,2)$,\nwhich, together with \\eqref{eq:relation}, implies that $(1-\\mathrm{e}^{\\mathrm{i} \\gamma(x)})\\Phi(z,x)$ is bounded for $z \\in D_0\\backslash (B\\cup \\{x\\})$. Hence, by letting $z\\rightarrow x$, we obtain $\\mathrm{e}^{\\mathrm{i} \\gamma(x)} = 1$, and again the reciprocity relation \\cite[Theorem 2.1]{Qin} leads to\n\\begin{equation*}\n    u_{1}^s(x,z) =  u_{2}^s(x,z),\\quad   \\forall (x, z) \\in \\Sigma_0\\times \\partial\\Omega.\n\\end{equation*}\nBy a similar discussion of \\eqref{eq:relation} for $u_1^s(x,z)- u_2^s(x,z)$ on $G$, we have\n\\begin{equation}\\label{coincide}\n    u_{1}^s(x,z) =  u_{2}^s(x,z),\\quad   \\forall (x, z) \\in (D_0\\backslash B)\\times \\partial\\Omega.\n\\end{equation}\n\nNext we are going to show that the case \\eqref{Thm1equality3} does not hold. Suppose that \\eqref{Thm1equality3} is true, then following a similar argument, we see that for every $x\\in \\Sigma_0$, there exists $\\eta(x)$ such that $u_1(z,x)=\\mathrm{e}^{\\mathrm{i} \\eta(x)} \\overline{u_2(z,x)}$ for all $z \\in D_0\\backslash (B\\cup \\{x\\})$, i.e.\n$$\n      u_1^s(z,x)+\\Phi(z,x)=\\mathrm{e}^{\\mathrm{i} \\eta(x)}  \\overline{[u_2^s(z,x) +\\Phi(z,x)]}.\n$$\nThen, from the boundedness of $u_{j}^s(z,x)$, it can be seen that $\\Phi(z,x)-\\mathrm{e}^{\\mathrm{i} \\eta(x)} \\overline{\\Phi(z,x)}$ is bounded for all $z \\in D_0\\backslash (B\\cup \\{x\\})$. Since\n$$\n\t\\Phi(z,x)-\\mathrm{e}^{\\mathrm{i} \\eta(x)} \\overline{\\Phi(z,x)}\n\t=  \\left[1-\\mathrm{e}^{\\mathrm{i} \\eta(x)}\\right]\\frac{\\cos (k|z-x|)}{4\\pi|z-x|}\\\\[2mm]\n\t+\\mathrm{i}\\left[1+\\mathrm{e}^{\\mathrm{i} \\eta(x)}\\right]\\frac{\\sin (k|z-x|) }{4\\pi|z-x|},\n$$\nby letting $z\\to x$, we deduce $\\mathrm{e}^{\\mathrm{i} \\eta(x)}=1$, and thus, $u_1(z,x)= \\overline{u_2(z,x)}$ for $z\n \\in D_0\\backslash (B\\cup \\{x\\})$. Further, by using the  impedance  boundary condition  $\\frac{\\partial u_j(z,x)}{\\partial \\nu}+ \\mathrm{i}\\lambda_0  u_j(z,x)=0, z\\in \\partial B (j=1,2)$, we have\n$$\n\\dfrac{\\partial \\overline{u_2(z,x)}}{\\partial \\nu}+ \\mathrm{i}\\lambda_0  \\overline{u_2(z,x)}\n=\\dfrac{\\partial u_1(z,x)}{\\partial \\nu}+ \\mathrm{i}\\lambda_0  u_1(z,x)=0, \\quad z\\in\\ \\partial B,\n$$\nwhich yields\n$$\n\\dfrac{\\partial u_2(z,x)}{\\partial \\nu}- \\mathrm{i}\\lambda_0  u_2(z,x)=0, \\quad z\\in\\ \\partial B.\n$$\nThis is a contradiction. Hence, the case \\eqref{Thm1equality3} does not hold.\n\nHaving verified \\eqref{coincide}, we complete our proof as the consequences of two existing uniqueness results,  Theorem 2.1 in \\cite{Qin1} and Theorem 3.1 in \\cite{Qin2}.\n\\end{proof}\n\n\n\\begin{remark}\n    We would like to point out that an analogous uniqueness result in two dimensions remains valid after appropriate modifications of the fundamental solution\n    and the admissible surface. So we omit the 2D details.\n\\end{remark}\n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\n\n\n\nD. Zhang and Y. Wang were supported by NSF of China under the grant 11671170. Y. Guo was supported by NSF of China under the grant 11601107, 41474102 and 11671111. The work of J.~Li was partially supported by the NSF of China under the grant No.~11571161 and the Shenzhen Sci-Tech Fund No.~JCYJ20170818153840322.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPeculiar atomic structures such as chains, ladders or the graphene honeycomb\noften exhibit remarkable singular physical properties, as does the\ntriangular cobalt network in Na cobaltates \\cite{Fouassier} which displays\nhigh thermopower \\cite{Terasaki} and superconductivity \\cite{Takada}. There,\nthe control of carrier content of the CoO$_{2}$ planes by varying Na\nconcentration between the planes yields a totally counter-intuitive sequence\nof magnetic properties including anomalous paramagnetism \\cite{Foo}, charge\ndisproportionation \\cite{CoPaper}, metallic antiferromagnetism \\cite%\n{Bayrakci,Helme}.\n\nAn original aspect of these systems is that the Co atoms are stacked between\ntwo triangular oxygen layers with distorted CoO$_{6}$ octahedrons, so that a\nlarge crystal field stabilizes the Co ions in a low spin state by lifting\nthe degeneracy of the cobalt 3$d$ levels. In a Co$^{3+}$ configuration the\nsix lower energy levels ($t_{2g}$) are filled, with a total spin $S=0$,\nwhile Co$^{4+}$ should only retain one hole in the $t_{2g}$ multiplet, with $%\nS=1\/2$, so that original magnetic properties have been anticipated to result\nfrom these local spins associated with charge ordering \\cite{Baskaran}. This\ncharge order intrinsic to the Co planes would depend of $x$ and would yield\nspecific metallic and magnetic properties. For instance, for $x=2\/3$, the 2D\ncharge ordered state would be a honeycomb network of Co$^{3+} $ ions\nintermixed with a triangular lattice of Co$^{4+}$.\n\nWe formerly established experimentally by NMR \\cite{CoPaper} that such a\ncharge order does not occur, but that the charges disproportionates between\nCo$^{3+}$ and cobalt sites with an average formal valence of about 3.5$^{+}$%\n. The differentiation of three Na sites \\cite{NaPaper}, in a phase with $%\nx\\sim 0.7$ (redefined as $x=0.67$ or H67 in ref.~\\cite{ourEPL2008}) implied\nan associated atomic Na ordering with a unit cell larger than that of the\nhoneycomb lattice, in view of the large number of cobalt sites detected by\nNMR \\cite{CoPaper}. This further differentiates the Na cobaltates from the\ncuprates for which copper has a uniform charge while dopants are usually\ndisordered, which  influences the physical properties \\cite{FRA-alloul}.\nIndeed the Na atomic orderings stabilized for specific $x$ values are found\nto play a role in determining the ground state metallic and magnetic\nproperties \\cite{ourEPL2008}. But, while Na ordering has been evidenced by\nvarious experiments \\cite{NaPaper,Zandbergen,Roger}, the actual atomic order\nand its incidence on the local scale electronic properties is still unclear\n\\cite{ourEPL2008}, except for two limiting cases, the $x=1$ filled band\ninsulator and the $x=1\/2$ ordered \"chain-like\" compound \\cite%\n{Bobroff,Yokoi,Mendels}.\n\nLet us point out that the cobalt plane physical properties are influenced by\nthe Na order in the two layers between which the CoO$_{2}$ slab is\nsandwiched, so that any connection between structure and physical properties\n\\textit{requires the knowledge of the 3D structure}. This is in principle\npossible using diffraction techniques and 3D order has indeed been seen by\nneutron scattering \\cite{Roger}, X rays on single crystals \\cite{Shu} and\neven in our powder samples \\cite{ourEPL2008,Lang2}. However, so far,\nmaterials complications linked with phase mixing, difficult accurate\ndetermination of Na content and existence of twins in the Na order have\nprevented the finalization of such studies.\n\nIt is well known that NMR is an ideal technique to link structural and\nelectronic properties, as it allows to measure local magnetic properties and\nis also sensitive to structural properties. Indeed, nuclear spins with $%\nI>1\/2 $ sense the local distribution of charges through the electrostatic\ninteraction which couples the quadrupolar moment $eQ$ of the nuclear charge\nwith the electric field gradient (EFG) at the nuclear site \\cite{Slichter}.\nIn view of the complicated NMR spectra \\cite{CoPaper}, we demonstrate here\nthat the structural and magnetic properties are better sorted out if one\nuses altogether NQR, that is a direct determination of the zero field\nsplitting of the nuclear spin levels \\cite{Slichter} due to the quadrupole\ninteraction. We shall demonstrate below that this approach applied here to\nthe $x=2\/3$ phase allows us to evidence unambiguously that the Na ordering\ndrives an in plane metallic electronic kagom\\'{e} organization \\cite{Mekata}%\n. Efforts to synthesize such geometric structures at the atomic level have\nbeen successful recently on producing a remarkable S=1\/2 \\textit{insulating}\nspin liquid state \\cite{Shores,Mendels3}, but have not allowed so far to\nsynthesize metallic states.\n\n\\section{ Differentiation of sites by NQR and NMR}\n\nWe found that the simplest NMR spectra among the high sodium content phases\n\\cite{ourEPL2008} were observed for $x=0.67$. $^{23}$Na NMR only exhibited\nthree distinct Na sites on powder samples aligned within a polymer matrix by\nan applied field \\cite{NaPaper}. In such samples the crystallite grains were\naligned along their $c$ direction while their $a,b$ directions are at random\nin the perpendicular plane {(the alignment procedure was recalled in\nref.~[18] of our ref.~\\cite{CoPaper})}. The Na lines were not fully resolved,\nwhich could have been attributed to intrinsic sample disorder. In NQR, no\nmagnetic field being applied, the spectra do not depend on the grain\norientations. So the detection, in the same sample, of three well resolved\nnarrow NQR lines (fig.~\\ref{FigNQR}a) with the expected quadrupolar\nfrequencies $\\nu _{Q}$, convinces us that the NMR resolution was limited by\na small distribution of orientations of the crystallites.\n\nFor $^{59}$Co, we detected simple NQR spectra as well, with \\textit{four\nwell resolved sites} (see fig.~\\ref{FigNQR}b), grouped two by two, Co1a and\nCo1b with lower NQR frequencies than Co2a and Co2b. The overlap between the\nNMR spectra of these Co sites results in complicated NMR spectra as shown\nfor instance in fig.~\\ref{FigNMR}. While the NMR differentiation techniques\nused in ref.~\\cite{NaPaper} allowed us to separate there up to five $^{59}$%\nCo NMR signals, one of them was found to result from a small fraction of\npolycrystalline grains of the sample which did not align in the field\n\\footnote{Comparing different batches of samples, we noticed that the part of the\nspectrum corresponding to a large NMR shift for both directions of the field\nwas sample dependent. It was erroneously assigned in the first experiments\nof ref.~\\cite{NaPaper} to a site labelled Co3 with an \\textquotedblleft\nisotropic\\textquotedblright shift. The analysis of the NMR shift data for\nthe four Co sites will be detailed in a forthcoming publication (ref.~\\cite%\n{IrekCo}). The spectra shown in fig.~\\ref{FigNMR} on a better oriented\nsample only retain the four site signals, in agreement with NQR.}.\n\n\\begin{figure}[tbp]\n\\onefigure[width=1\\linewidth]{Fig1EPL.eps}\n\\caption{NQR signals of the various $^{23}$Na and $^{59}$Co sites in Na$%\n_{2\/3}$CoO$_{2}$. (a) For $^{23}$Na, with I=3\/2, a single NQR line is\nexpected per Na site, and indeed we detected three sites with narrow NQR\nlinewidths. (b) For $^{59}$Co, with nuclear spin $I=7\/2$ we display here the\nfour distinct lines corresponding to the higher frequency $(7\/2\\rightarrow\n5\/2)$ transitions.}\n\\label{FigNQR}\n\\end{figure}\n\nThe fractional occupancies of the various sites could be better fixed by\ncomparing first the signal intensities of the sites with neighboring\nquadrupole frequencies. Similar ratios Co1a\/Co1b=1.95(0.1) and\nCo2b\/Co2a=1.9(0.2) were obtained from NQR data of fig.~\\ref{FigNQR}a, after\ncorrecting for the spin-spin $T_{2}$ decay and for the $\\nu ^{2}$ frequency\ndependence of the signal intensity. The $\\nu _{Q}$ values obtained by NQR\nallowed us to reduce as well the input parameters in the analysis of the $%\n^{59}$Co NMR spectra and to confirm this estimate of Co1a\/Co1b =1.85(0.2)\nand that Co2b has definitely a larger intensity than Co2a. To complete these\ncomparisons one needed to determine the relative intensities of the Co1 and\nCo2 sites. The best accuracy has been obtained from a comparison of the\ntotal intensity of the Co1 sites signal to that of the full $^{59}$Co NMR\nspectrum, which gave  Co1\/(Co1+Co2)=0.26(0.04). This allows to obtain the\nfractional occupancies given in table~\\ref{Table1}, which were quite helpful\nin the identification of the structure.\n\n\\begin{table}[tbph]\n\\caption{EFG parameters $\\nu _{Q}$\\ and $\\eta $ for the\ndifferent sites and relative intensities $I_{exp}$ deduced from NQR or NMR\n(see text). $I_{exp}$ can be compared to the intensities $I_{str}$ expected\nfrom the site occupancies $N_{s}$ of the structure depicted in fig.~\\ref{FigStructure}.}\n\\label{Table1}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\raisebox{-1.50ex}[0cm][0cm]{Site} & \\multicolumn{3}{|c|}{Experiment} &\n\\multicolumn{2}{|c|}{Structure} \\\\ \\cline{2-6}\n& \\textit{$\\nu $}$_{Q}$ (MHz) & $\\eta $ & $I_{exp}(\\%)$ & {$N_{s}$} & {$%\nI_{str}(\\%)$} \\\\ \\hline\nNa1 & 1.645(5) & $<0.01$ & 30(5) & 2 & 25 \\\\ \\hline\nNa2a & 1.74(1) & 0.84(2) & 33(5) & 3 & 37.5 \\\\ \\hline\nNa2b & 1.87(1) & 0.89(2) & 37(5) & 3 & 37.5 \\\\ \\hline\nCo1a & 1.193(2) & $<$0.01 & 17(3) & 2 & 16.7 \\\\ \\hline\nCo1b & 1.392(2) & $<$0.01 & 9(2) & 1 & 8.3 \\\\ \\hline\nCo2a & 2.187(3) & 0.36(1) & 25(3) & 3 & 25 \\\\ \\hline\nCo2b & 2.542(3) & 0.36(1) & 49(4) & 6 & 50 \\\\ \\hline\n\\end{tabular}%\n\\end{center}\n\\end{table}\n\n\\section{Lattice structure and site determination}\n\nNa ordering patterns have been suggested both from diffraction experiments\n\\cite{Roger} and calculations restricted to electrostatic interactions\nbetween ions at fixed positions \\cite{Roger,Zhang}. We found that the type\nof Na ordering proposed for $x\\geq 0.75$, which consists in an ordering of\ndroplets of Na2 tri-vacancies in which Na1 trimers are stabilized \\cite%\n{Roger}, cannot explain the present data (see fig.~\\ref{FigStructure} for\nthe usual convention taken for Na1 and Na2 sites). Such calculations are\nsomewhat inconclusive on the possible stable structure for $x=2\/3$. A recent\nquantum chemistry calculation allowing the relaxation of the site positions\n\\cite{Hinuma} suggest for this Na content a relatively simple stable\nstructure for $x=2\/3$, with a 2D unit cell containing 12 cobalt atoms. It\ncorresponds to 2D ordering of Na2 di-vacancies which stabilize a single Na1\nsite at their center in a position with axial symmetry. This unit cell which\nis shown on top of fig.~\\ref{FigStructure}a contains 8 Na for 12 Co atoms in\nthe Co layer.\n\n\\begin{figure}[tbp]\n\\onefigure[width=1\\linewidth]{Fig2EPL.eps}\n\\caption{Asymmetry of the $^{59}$Co NMR spectra. $^{59}$Co NMR spectra\nobtained by sweeping the applied field at fixed frequency and monitoring the\nNMR signal by the magnitude of the spin echo signal. The top spectra are\ntaken with the external field applied parallel to the ab plane, while the\nlower ones are taken with $H\\parallel c$. The black spectra taken with short\ndelay between pulses ($\\tau =7\\mu sec$) give the overall NMR\nsignal including all sites. The red spectra of the non magnetic Co1 sites,\nwhich have longer transverse spin-spin relaxation time, are separated out\nusing a longer delay ($\\tau =200\\mu sec$)\\ Scaling it to\ndeduce it from the full spectrum (as in ref.~\\cite{CoPaper})\nallows to deduce the signal of the Co2 sites displayed as Diff. in the lower\npanel for $H\\parallel c$. We have marked the positions of the\n$(5\/2\\rightarrow 7\/2)$ outer quadrupolar transitions for the Co1 and Co2\nspectra for $H\\parallel c$. For $H\\perp c$ those are still well resolved for\nthe Co1 with an extension reduced by a factor 2 (as marked). This is not the\ncase for the Co2 signals for which $\\eta \\neq 0$.}\n\\label{FigNMR}\n\\end{figure}\n\nWe shall see that we could fit perfectly NMR\/NQR data by choosing the\nstacking pattern of this unit cell shown in fig.~\\ref{FigStructure}a, which\nis in good agreement with the \"rules\" suggested in ref.~\\cite{Hinuma}, $i.e.$\nminimize the number of Co sites with two Na1 above and below. The\ndifferentiation of sites intrinsic to this structure gives 4 Co and 3 Na\nsites, which are distinguished in fig.~\\ref{FigStructure}a and fig.~\\ref%\n{FigStructure}b. We find that six layers of CoO$_{2}$ are required to\ncomplete the 3D unit cell shown in fig.~\\ref{FigStructure}c. This\ncorresponds to a lattice parameter $3c$, where $c$ is the usual unit cell\nlength involving two CoO$_{2}$ layers in the basic hexagonal structure, in\nagreement with the observation done on this phase by X rays \\cite{ourEPL2008}%\n. Although this unit cell contains 72 cobalt, 144 oxygen and 48 sodium\natoms, the local site differentiation of fig.~\\ref{FigStructure}a,b is\nmaintained in all the 3D crystal structure, with the in plane multiplicities\n$N_{s}$ of the 4 Co and 3 Na sites given in table~\\ref{Table1}. The\nagreement found there with the relative intensities of the corresponding\nNMR\/NQR lines allows an unambiguous assignment of the different lines to the\nsites as given in fig.~\\ref{FigStructure}b. The Co1a line which has the\nlower intensity is naturally assigned to the Co site of the 2D unit cell\nwhich has Na1 sites above and below, while Co1b corresponds to the two\ncobalt with one Na1 on one side and three Na2 on the other side (fig.~\\ref%\n{FigStructure}b). The two other sites with multiplicity 3 and 6 are then\nnaturally assigned to Co2a and Co2b, while the relative intensity of\n0.26(0.04) found for the Co1 spectrum agrees perfectly with the expected\n3:12 ratio for the structure of fig.~\\ref{FigStructure}a.\n\nAs for the $^{23}$Na NMR data \\cite{NaPaper}, the lower intensity line\ncorresponds then to the Na1 sites, while the two others with similar\nintensities can be indifferently assigned to Na2a and Na2b. Let us point out\nthat Chou \\textit{et al.} \\cite{Chou2} reported, on a single crystal, a well\nresolved $^{23}$Na spectrum taken at 20K that we could ascribe without\nambiguity to the same H67 phase, from the $^{23}$Na NMR shifts and\nquadrupolar splitting values. They attempted to assign this three line\nspectrum to a phase with $x=0.71$, as obtained by the chemical analysis of\ntheir sample.\\ However the relative intensities reported, 26.5{\\%}, 36.3{\\%}%\n, 37.1{\\%}, correspond extremely well within their 3{\\%} accuracy to the\n2:3:3 ratio for the Na sites, which independently supports the validity of\nthe structure proposed here.\n\n\\begin{figure*}[tp]\n\\begin{center}\n\\onefigure[width=0.75\\linewidth]{Fig3EPL.eps}\n\\end{center}\n\\caption{Three dimensional structure of Na$_{2\/3}$CoO$_{2}$. \\textbf{(a)}\nThe two dimensional Na unit cell proposed in ref.~\\cite{Hinuma} is\nrepresented in the top layer with respect to the 12 initially\nindistinguishable Co atoms of the underlying triangular layer (the oxygen\natoms layers above and below the cobalt layer are omitted). It contains two\nNa1 sites (red) sitting on top of Co and six Na2 sites, topping a triangle\nof Co, with two distinct Na planar environments (light and dark green). The\n3D stacking which fits our data involves a lower Na layer shifted by -2%\n\\textbf{\\textit{a}}. As the Na2 sites of the two layers should project at\nthe centers of distinct Co triangles, \\textbf{\\textit{a}} axis mirroring of\nthe Na pattern is required. This planar Na structure can be considered as an\nordered pattern of di-vacancies in the Na2 plane, which displace a third Na2\non an Na1 site in a three fold symmetric environment (one such di-vacancy is\nillustrated in the lower Na plane, where circles mark the three missing Na2\nsites). \\textbf{(b)} In the upper panel, the Na environments of the four Co\nsites differentiated in this structure are displayed (the smaller size of\nthe orange and yellow balls is meant to distinguish Co sites with Na1\nneighbors). In the lower panel, the 3 Na sites are represented with their Na\nand Co environments deduced from the stacking of cobalt planes shown in\n\\textbf{(a)}. Three fold rotation symmetry is seen for Co1a , Co1b and Na1\nfor which $\\eta $=0. \\textbf{(c)} The four layers of \\textbf{(a)}\nare represented at the top and the 3D stacking of the following Na layers is\npursued downwards by -2\\textbf{\\textit{a}} translations and mirroring, which\nshifts the Co1b site by -2\\textbf{\\textit{a}} between Co layers (red bars).\nThe unit cell is completed after six shifts.}\n\\label{FigStructure}\n\\end{figure*}\n\n\\section{Symmetry of the sites}\n\nAn important cross-test of the validity of this structure is the symmetry of\nthe various sites, which could be probed through that of the EFG\\ tensor.\\\nThe NQR frequencies are indeed determined by the principal values of the EFG\ntensor, which is diagonal in a frame $(X,Y,Z)$ linked with the\ncrystallographic structure $(|V_{ZZ}|\\geq |V_{YY}|\\geq |V_{XX}|)$. Laplace\nequation reduces those into two parameters, the quadrupolar frequency%\n\\begin{equation}\n\\nu _{Q}=3eQV_{ZZ}\/[2I(2I-1)]\n\\end{equation}%\nand the asymmetry parameter%\n\\begin{equation}\n\\eta =(V_{XX}-V_{YY})\/V_{ZZ},\n\\end{equation}%\nwhich vanishes identically for sites with axial symmetry.\n\nFrom fig.~\\ref{FigStructure}b it is easy to see that this is the case for\nthe Na1 site as well as the Co1a and Co1b sites, for which $c$\\textbf{\\ }is\na threefold symmetry axis in the proposed structure. In NMR the weaker\nintensity Na site had been found \\cite{NaPaper} with $\\eta =0.01(1)$,\nconfirming the identification done with the Na1 site, while the Na2a and\nNa2b have large $\\eta $ values in agreement with the non axial structure.\nSimilarly, one can see in fig.~\\ref{FigNMR} that the quadrupole splitting of\nthe $^{59}$Co NMR of the Co1 sites is reduced by a factor two in the $ab$\ndirection with respect to that in the $c$ direction. This is expected for\nsites with a threefold or fourfold symmetry axis for which $%\nV_{XX}=V_{YY}=-V_{ZZ}\/2$, which corresponds to an axial symmetry for the\nEFG. In such a case the quadrupole splitting is identical for all field\ndirections in the $ab$ plane, so that the NMR spectra are still well\nresolved for $H\\perp c$, even for a powder distribution of the $ab$\norientations, as seen in fig.~\\ref{FigNMR}. On the contrary one can see that\nthe $H\\perp c$ spectra of the Co2 sites are less resolved and have a field\nextension comparable with that in the $H\\parallel c$ direction, implying $%\n\\eta \\neq 0$. A full study of the NMR and NQR spectra to be published\nelsewhere allowed us to deduce $\\eta =0.36(1)$ for both Co2 sites.\n\nSo we do find that all the sites with axial symmetry of fig.~\\ref%\n{FigStructure}b have been properly identified, \\textit{which secures then\nthe validity of the proposed structure}. This confirms our $x=0.67(1)$\nestimate for the Na content of this phase and establishes the validity of\nthe relationship between $c$ axis parameter and Na content reported in \\cite%\n{ourEPL2008,Lang2}. {We believe that other calibrations \\cite{Shu,Chou2}\nbased on \\textit{chemical} analyses of samples overestimate the actual\ncontent of the majority phase from which Na tends to be expelled. This is\nparticularly true for single crystal samples synthesized with a large excess\nof Na , while for our single phase ceramic samples reacted in solid state,\nNa excess with respect to nominal content is unlikely.}\n\n\\section{Magnetic properties of the Co sites}\n\nIt has been shown for long that this particular phase has an anomalous Curie\nWeiss behavior extending down to the lowest temperatures \\cite{NaPaper} and\na resistivity which only reaches a $T^{2}$ dependence below 2~K \\cite{LiSY}.\n The local magnetic parameters could be studied through NMR shift data \\cite%\n{CoPaper} or spin lattice relaxation $T_{1}$ measurements which probe\ndirectly the dynamic susceptibilities. Here, to compare in detail the\nmagnetic properties of the four cobalt sites we took advantage of the fact\nthat the NQR lines are fully resolved spectrally in fig.~\\ref{FigNQR},\ncontrary to the NMR spectra of fig.~\\ref{FigNMR} which largely overlap. This\nallowed us to probe independently the $T_{1}$ values of the four sites\nthrough monitoring the recovery of each NQR signal after saturation. The\ntypical data taken at 4.2~K shown in fig.~\\ref{FigT1} immediately allow us\nto evidence that the Co2a and Co2b sites have identical $T_{1}$ values,\n\\textit{which are about 30 times shorter} than that of the non magnetic Co1a\nand Co1b sites. For the latter the $T_{1}$ values are identical as well as\nthe NMR shifts, which have an isotropic orbital component, which allowed us\nto evidence that they correspond to non magnetic Co$^{3+}$ sites \\cite%\n{CoPaper,ourEPL2008}. This illustrates that the magnetic properties are\nsustained by the Co2 sites which sense directly the on site magnetism that\nyields much larger spin lattice relaxation rates (and NMR shifts \\cite%\n{CoPaper}). On the contrary, the Co1 and Na sense the magnetism of the Co2\nsites only through transferred hyperfine couplings. For instance the fact\nthat Na2a and Na2b  display smaller NMR shifts than Na1 \\cite{NaPaper} can\nbe assigned to the larger number of magnetic Co2 near neighbours for the\nlatter (fig.~\\ref{FigStructure}b).\n\n\\begin{figure}[tbp]\n\\onefigure[width=1\\linewidth]{Fig4EPL.eps}\n\\caption{Spin lattice relaxation of the Co sites taken at 4.2K. The\nmagnetization recoveries of the $^{59}$Co NQR signals are seen in (a) to be\nidentical for the lower frequency signals of Co1a and Co1b sites, and\nsimilarly in (b) for the higher frequency signals of Co2a and Co2b. The\nrelaxation rate is about 30 times larger for the latter,\nwhich differentiates markedly the magnetic Co2 sites from the non magnetic\nCo1 sites.}\n\\label{FigT1}\n\\end{figure}\n\nMore importantly the present results prove that Co2a and Co2b sites,\nalthough quite distinguished by their EFG values induced by the local\nstructure, \\textit{are quite identical on their electronic and magnetic\nproperties}. So the Na organization above and below the Co plane allows to\npin Co$^{3+}$ ionic states on the Co1 sites, by lowering their energy levels\nso that holes do only occur on the Co2 sites. For the understanding of the\nelectronic properties one \\textit{should then consider only two types of\nelectronic sites} (3 Co1 and 9 Co2) in a single plane unit cell as shown in\nfig.~\\ref{FigKagome}. There one can notice that the Co2 sites\\textit{\\ are\nquite remarkably organized on a kagom\\'{e}} structure while the Co$^{3+}$\nsites form the complementary triangular lattice. The fractional occupancy\n1:4\\ of the Co1 sites agrees well with the previous estimates of the Co$%\n^{3+} $concentration, which allows us to establish that the partly filled\nCo2 sites bear then a hole concentration of 4\/9 per site, $i.e.$ an\neffective charge Co$^{3.44+}$, similar to that deduced before \\cite%\n{CoPaper,ourEPL2008}, and far less than expected for a Co$^{3+}$\/ Co$^{4+}$\nscenario.\n\n{Let us point out that a similar ordered charge disproportionation (OCD) has\nbeen found in the case of nickelates \\cite{Nickelates}, but with opposite\nmagnetic effects, i.e. local moments on one site out of three and charge\ndelocalisation on the others. The experimental situation here is rather\nanalogous to that found in the cubic CsC$_{60}$ compounds where local\nsinglets are favoured on some C$_{60}$ balls while electrons delocalize on\nthe remaining ones \\cite{CsC60}.}\n\n\\section{Discussion}\n\n{So far, most electrostatic calculations do give indications on possible Na\norderings, but they do not introduce the electronic energies sufficiently\nwell to give any insight on the electronic order associated. The calculation\nof ref.~\\cite{Hinuma} which takes into account the electronic band energy\n(with a variant of the Local Density Approximation called GGA) yields an\natomic structure in agreement with that proposed here. This underlines the\nimportance of the electronic energy in pinning the Na order. Although the\nauthors establish that the same Na structure is stable if a large on site U\nis introduced in the GGA, nothing in the reported results helps so far to\nestablish the stability of the OCD of fig.~\\ref{FigKagome}. The competition\nbetween Hund's energy and band energy which stabilizes local moments in the\ncase of nickelates \\cite{Nickelates} might rather favour the filling of\nselected Co$^{3+}$ sites in the present case. It would certainly be helpful\nto extend then the GGA calculations to check whether this approach is\nsufficient to explain a decrease of the ground sate energy by the OCD, or\nwhether one has to resort to a more refined treatment of the electronic\ncorrelations. The derived band structure would then allow comparisons with\nthe hyperfine and EFG parameters determined from our NMR data. Depending\nwhich bands ($a_{1g}$ and\/or $e_{g}^{^{\\prime }}$) participate to the Fermi\nsurface in the reduced Brillouin zone, might allow one to interpret the\nsmall pockets detected by transport experiments \\cite{Balicas}. It might\nalso be worth to attempt ARPES experiments \\cite{Yang} on ordered Na phases\non the external layer of single crystals to detect their imprint on the\nFermi surface, although this appears an experimental challenge.}\n\nOn theoretical grounds, one would like as well to understand whether the\nKagom\\'{e} structure has any specific role in the physics of cobaltates, as\nhad been underlined by Koshibae and Maekawa \\cite{Koshibae}. These authors\nanticipated that the directionality of the transfer integrals between Co\nsites in the triangular lattice favors electronic \\textit{wave functions}\nrestricted to orbitals organized on a kagom\\'{e} lattice. In a uniformly\ncharged case four such interpenetrating lattices are degenerate, but the Na\nself organization might merely select here one of these kagom\\'{e} lattices?\n\n\\begin{figure}[tbp]\n\\onefigure[width=1\\linewidth]{Fig5EPL.eps}\n\\caption{Two dimensional charge distribution in the Co planes of Na$_{2\/3}$%\nCoO$_{2}$. The resulting 2D structure of the Co planes corresponds to a\nperfect Co2 kagom\\'{e} lattice (fig.~\\ref{FigStructure}c), once the\nminor difference between the electronic properties of Co2a and Co2b sites is\nneglected. The Co1 sites constitute the complementary triangular lattice.}\n\\label{FigKagome}\n\\end{figure}\n\nAs for the magnetic properties of the cobaltates, the experimental evidence\ngiven here that metallicity and magnetism are combined in the correlated\nelectron kagom\\'{e} structure is at odds with most proposals concerning the\nelectronic structure of cobaltates \\cite{Baskaran,Bernhardt,Chou,Marianetti}%\n. It has indeed most often been anticipated that Na2 vacancies pin localized\nspins which would dominate the Curie Weiss magnetic properties. In these\napproaches, close to the ionic Co$^{3+}$\/ Co$^{4+}$ scenario, the metallic\nproperties are associated with quasi non magnetic Co$^{3+}$ sites assumed to\nretain Na configurations similar to that of Na$_{1}$CoO$_{2}$. On the\ncontrary, our results establish that the Na1 sites are linked to non\nmagnetic Co$^{3+}$on the Co1, and that metallicity occurs on the Co2 sites\nwhich, as viewed in fig.~\\ref{FigStructure}b, have a low Na coordinance (3\nor 4) compared to the 6 fold one of Na$_{1}$CoO$_{2}$.\n\nSo the present results give a simple realistic structure on which approaches\ntaking better into account the electronic correlations, such as DMFT \\cite%\n{Marianetti} could be applied.\\ It might even be used as a benchmark to\ndevelop and test the capabilities of electronic structure calculations of\ncorrelated electron systems.This could possibly permit to understand\naltogether why the dominant electronic correlations are ferromagnetic for $%\nx\\geq 2\/3$ \\cite{ourEPL2008} and AF below \\cite{Lang2}. Also with this well\ncharacterized 3D order one should understand whether specific inter-plane\nexchange paths could justify the absence for $x=2\/3$ of the A type AF 3D\norder \\cite{Bayrakci,Helme} found for $x>0.75$. Conversely one might\nwonder whether this difference can be explained by an incidence of the 2D\nfrustration inherent to the kagom\\'{e} structure, establishing for instance\nan underlying spin liquid component as found in the $S=1\/2$ kagom\\'{e}\nlattice \\cite{Mendels3}? The present results not only raise these questions\nbut open a path to solve them, as the experimental approach highlighted\nhere, which combines the structural determination with its local impact on\nmagnetism can be extended to the diverse cobaltate phases with distinct\nground state properties.\n\n\\section{Aknowledgments}\n\nWe acknowledge J.~Bobroff, G.~Collin, G.~Lang, P.~Mendels, M.~Rozenberg and\nF.~Rullier-Albenque for their constant interest, for helpful discussions and\ncomments concerning the manuscript. We acknowledge financial support by the\nANR (NT05-441913) in France. Expenses in Orsay for A.~D. and I.~M. have been supported by the\n\\textquotedblleft Triangle de la Physique\\textquotedblright. T.~Platova\nhas obtained a fellowship from the  E.U. Marie Curie program\n\\textquotedblleft Emergentcondmatphys\\textquotedblright\\ for  part of her\nPhD work performed in Orsay.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nAs the complexity of power distribution systems (PDS) and distributed energy resource (DER) penetration increases, so too does the need for improving diagnostics and reliability. One such problem in this area is high-impedance faults (HIFs). Despite their ubiquity in PDS, HIFs persist without generalized solution for detection due to many characteristics of the HIF itself, including nonlinearity of voltage and current measurements at the monitored line, electrical arc transient spikes, and the inverse relationship between impedance and measurable fault location current \\cite{Incipent}.\n\nVarious strategies have been developed to detect and identify HIFs \\cite{Review}. A popular domain is signal processing, in which methods based on wavelet transform (WT) \\cite{wavelet} are dominant. These techniques use time and frequency information for various power system disturbances, including HIFs. Artificial neural networks (ANNs) have been used, alone and in tandem with WT \\cite{ANN}, to decompose and extract features of current and\/or voltage waveforms of a monitored line under HIF. Essentially, these techniques identify special waveform features indicative of HIFs and aim to discern them from other power system disturbances, such as switching events. An estimation approach using wide-area measurements presented in \\cite{Resapproach} uses normalized residuals, considering only the detectable error component for general fault detection. A PDS model-based approach in \\cite{PDS} identifies HIFs based on impulse signal injections and line impedance analysis. While promising, this method is limited to medium voltage system transformer configurations.\n\nThe proposed model is based on power system state estimation (SE), which has become consolidated as a necessary procedure for real-time power energy system management \\cite{bretas2021cyber}.\nPertinent to this work are SE gross error (GE) analytics, including the detection and identification of errors in measurement data\u2014either nefarious or equipment-based. The proposed HIF detection model in this work is based on the GE analytics of parameter errors \\cite{ParameterProof}, during which network data such as line impedance values do not match those of the PDS due to physical disturbance, system error, or cyber-attack. The proposed work requires minimum modifications to existing SE algorithms. The model detects HIFs regardless of changes in power flow direction due to DERs, as this information is included in the measurement vector.  Further, this work makes no assumptions about system topology or configuration, which are treated as logical inputs to the SE, making it practical for generalized implementation and use in\nindustry.\n\nThe idea for this work is to combine established parameter error detection methods, per-phase weighted least squares SE, and a two-step SE approach to detect HIFs from a wide-area system perspective and, after uncovering information about the error distributions and eliminating the possibility of measurement error, identify the lateral line on which the HIF occurred based on the measurement set fed to the SE process. No training or pattern recognition is required.\n\nThe three specific contributions of this paper towards the state-of-the-art are the following:\n\\begin{itemize}\n    \\item A formal model for HIF detection based on SE.\n    \\item HIF detection that requires no assumptions of a fault or system disturbance having already occurred.\n    \\item A method that is independent of knowledge of the HIF condition with respect to time (e.g., before, during, or after the build-up and shoulder characteristic).\n\\end{itemize}\n\n\n\n\\section{Formal Model}\\label{sec:formalmodel}\n\\subsection{HIF Modeling}\\label{sec:hifmodel}\n\nHIFs exhibit special characteristics which differentiate them from typical short circuit faults \\cite{Review}. These include:\n\\begin{enumerate}\n    \\item \\textbf{Low-magnitude fault currents:}  HIFs are difficult to discern from normal load changing conditions.\n    \\item \\textbf{Intermittent arcing:} the current waveform does not exhibit a steady state waveform, but rather alternates between conduction and non-conduction cycles.\n    \\item \\textbf{Asymmetry and randomness:} the difference between breakdown voltages causes differences in shape and magnitude between the negative and positive half-cycles of the fault current. \n    \\item \\textbf{Nonlinearity:}  arcing causes non-linearity between the voltage and current waveforms at the HIF.\n    \\item \\textbf{Build-up and Shoulder:} an overall gradual increase in fault current until reaching a steady state condition for several cycles.\n\\end{enumerate}\n\n To capture each fault current characteristic, a HIF model was developed, illustrated in Fig. \\ref{HIF Model}. To simulate the build-up and shoulder characteristic, a time-varying resistance $R_f(t)$ was implemented using polynomial regression approximation \\cite{RFT}. To test the detection capabilities of the proposed work, the variation range of the $R_f(t)$ waveforms was restricted such that the fault current remains as low as detectably possible with respect to full load current of the faulted phase.\n\nTwo antiparallel diodes $D_p$ and $D_n$ and two nonequal voltage sources $V_p(t)$ and $V_n(t)$ serve to simulate nonlinearity and the HIF arcing characteristic \\cite{MM}. Intermittency was simulated by selecting a portion of the $R_f(t)$ waveform and inserting a high impedance impulse such that the evolving HIF current waveform is temporarily extinguished before resuming\n\n   \\begin{figure\n\\includegraphics[width=6.5072cm,height = 3cm,scale = 1]{HIF_circuit_horiz.png}\n\\centering\n\\caption{HIF Model \\cite{RFT}}\n\\label{HIF Model}\n\\end{figure}\n\n\\subsection{Innovation-Based State Estimation}\\label{sec:IISE}\nClassical quasi-static state estimation implements a Weighted Least Squares (WLS) method \\cite{bretas2021cyber}. A power system with $n$ buses and $d$ measurements is modeled as a set of non-linear algebraic equations in the measurement model:\n\\begin{equation}\n    \\mathbf{z} = h(\\mathbf{x}) + \\mathbf{e}\n\\label{eq:SE}\\end{equation}\nwhere $\\mathbf{z}\\in\\mathbb{R}^{1 \\times d}$ is the measurement vector, $\\mathbf{x}\\in\\mathbb{R}^{1 \\times N}$ is the vector of state variables, $h:\\mathbb{R}^{1 \\times N}\\rightarrow\\mathbb{R}^{1 \\times d}$ is a continuous nonlinear differentiable function, and $\\mathbf{e}\\in\\mathbb{R}^{1 \\times d}$ is the measurement error vector.  The measurement error, $e_i$ is assumed to have a Gaussian probability distribution with zero mean and standard deviation $\\sigma_i$.  $d$ is the number of measurements and $N=2n-1$ is the number of unknown state variables, namely the complex bus voltages.\n\nIn the classical WLS method, the optimal state vector estimate in \\eqref{eq:SE} is found by minimizing the weighted norm of the residual, represented by the cost function $J(\\mathbf{x})$:\n\\begin{equation} \n\\label{eq:JSE}\nJ(\\mathbf{x})=\\Vert \\mathbf{z}-h(\\mathbf{x})\\Vert _{\\mathbf{R}^{-1}}^{2}=[\\mathbf{z}-h(\\mathbf{x})]^{T}\\mathbf{R}^{-1}[\\mathbf{z}-h(\\mathbf{x})] \n\\end{equation}\nwhere $\\mathbf{R}$ is the covariance matrix of the measurements. For this work, a two-step approach will be adopted for the GE analytic process \\cite{twostep}. In the first step, all measurements are weighted equally proportional to the measurement magnitude, after which GE analytics are performed. After GE processing, the SE is repeated\u2014this time with meter precision values per state-of-the-art methodologies, which consider a specific standard deviation value for each measurement type \\cite{monticelli}.\nThe solution of \\eqref{eq:JSE} is obtained through the iterative Newton-Raphson method. The linearized form of \\eqref{eq:SE} becomes:\n\n\\begin{equation} \\Delta \\mathbf{z}=\\mathbf{H}\\Delta \\mathbf{x}+\\mathbf{e} \\label{SElin} \\end{equation}\nwhere $\\mathbf{H}=\\frac{\\partial h}{\\partial \\mathbf{x}}$ is the Jacobian matrix of $h$ at the current state estimate $\\mathbf{x}^*$, $\\Delta \\mathbf{z}=\\mathbf{z}-h(\\mathbf{x}^*)=\\mathbf{z}-\\mathbf{z}^*$ is the measurement vector correction and $\\Delta \\mathbf{x}=\\mathbf{x}-\\mathbf{x}^*$ is the state vector correction.  The WLS solution can be understood geometrically \\cite{geom} as the projection of $\\Delta \\mathbf{z}$ onto the Jacobian space $\\mathfrak{R}$($H$) by a linear projection matrix $\\mathbf{K}$, i.e. $\\Delta\\hat{\\mathbf{z}}=\\mathbf{K}\\Delta\\mathbf{z}$. \n\n\n\\begin{equation}\\label{eq:P}\n\\mathbf{K} = \\mathbf{H}(\\mathbf{H}^{T}\\mathbf{R}^{-1}\\mathbf{H})^{-1}\\mathbf{H}^{T}\\mathbf{R}^{-1}.\n\\end{equation}\n\n\n\n\n\n\n\n\nBy decomposing the measurement vector space into a direct sum of $\\mathfrak{R}$($H$) and $\\mathfrak{R}($H$)^{\\perp}$, it is then possible to decompose the measurement error vector $\\mathbf{e}$ into two components:\n\\begin{equation} \\label{eq:e}\n\\mathbf{e} = \\underbrace{\\mathbf{K}\\mathbf{e}}_{\\mathbf{e_U}} + \\underbrace{(I-\\mathbf{K})\\mathbf{e}}_{\\mathbf{e_D}}.\n\\end{equation}\n\nThe component $\\mathbf{e_D}$ is the detectable error, which is equivalent to the residual in the classical WLS model. The component $\\mathbf{e_U}$ is the undetectable or masked component of the error.\nTo quantify the impact of the undetectable error, the measurement Innovation Index ($II$) is introduced \\cite{II}:\n\\begin{equation}\\label{eq:10}\n{II}_{i} = \\frac{\\Vert{e^i_D}\\Vert_{\\mathbf{R}^{-1}}}{\\Vert{e^i_U}\\Vert_{\\mathbf{R}^{-1}}} = \\frac{\\sqrt{1-K_{ii}}}{\\sqrt{K_{ii}}}.\n\\end{equation}\n\nLow Innovation Index means that a large component of the error is not reflected in the residual alone. The residual will therefore be very small even if there is a GE. From (\\ref{eq:e})  and (\\ref{eq:10}), the composed measurement error ($CME$) can be expressed in terms of the residual and the innovation index, after which the normalized $CME^N$ is obtained:\n\\begin{equation}\\label{eq:11}\nCME_i = r_i\\left(\\sqrt{1+\\frac{1}{{II_i}^2}}\\right) \\Rightarrow CME_i^N = \\frac{CME_i}{\\sigma_i}.\n\\end{equation}\nwhere $\\sigma_i$ is the $i$-th measurement standard deviation.\n\n\n\n$CME$ values are estimated from real and synthetic measurements (SM), after which bad data analysis \\cite{bretas2021cyber} is performed. Real measurements are defined as those obtained from sensors. SM are created artificially per the methodology in \\cite{synth}. SM are created in low redundancy areas considering measurement $II$ and $n$-tuple of critical measurements, serving to maintain a high global redundancy level $GRL$ (number of measurements divided by the number of state variables). This in turn increases the degrees of freedom such that the $\\chi^2$ distribution used for hypothesis testing tends to a normal distribution. SM are calculated based on the previous set of state estimates. The location and type of SM generated are strategically chosen based on the $VI$ of a given bus $i$:\n\n\\begin{equation}\\label{eq:vi}\nVI_i = \\sqrt{\\frac{1}{L}\\sum_{l=1}^{L}\\sum_{j=1}^{J}(S_{CME}(l,j)-S_r(l,j)))^2 }\n\\end{equation}\nwhere $L$ is the set of measurements affiliated with bus $i$, $J$ is the set of all measurements, and $S_{CME}$ and $S_r$ are the sensitivity matrices ($S = 1-K$) of the $CME$ and residual respectively. The $VI$ metric is used to identify buses susceptible to undetectable errors and create SM for those buses.\n\n$\\chi^2$ hypothesis testing is used as the GE analytic for bad data detection in the measurement set. The $CME$-based objective function value (\\ref{eq:chi-squared}) is compared to a $\\chi^2$ threshold, which is based on a chosen probability $p$ (typically $p=0.95$) and the degrees of freedom $d$ of the measurement model:\n\\begin{equation} \\label{eq:chi-squared}\nJ_{CME}(\\hat{\\mathbf{x}})=\\sum_{i=1}^d \\left[\\frac{CME_i}{\\sigma_i}\\right]^2 > \\chi^2_{d,p}.\n\\end{equation}\n\nIf the value of $J_{CME}$ is greater than the $\\chi^2$ threshold, then a GE is detected. If this occurs, the second SE step is invoked, during which the SE is re-run with meter precision values for each measurement type. Next, following \\cite{ParameterProof}, bad data is identified through $CME^N$ analysis. \n\nIt is also important to know which phase\nwas affected by the line-to-ground HIF. Thus, a per-phase SE approach is employed \\cite{perphase}. In this method, the presented SE is performed, after which the vector of residuals is decomposed into three vectors\u2014one for each phase $A$, $B$, and $C$, i.e., $\\Delta \\mathbf{z}_A$, $\\Delta \\mathbf{z}_B$, $\\Delta \\mathbf{z}_C$. One can then obtain Jacobian matrices for each phase $i$; $\\mathbf{H}_i = -\\partial\\Delta \\mathbf{z}_i\/\\Delta \\mathbf{x}_i$. The covariance matrix of the measurements $\\mathbf{R}$ is similarly decomposed to obtain per-phase weight matrices $\\mathbf{R}^{-1}_i$. The projection matrices for each phase $i$ can then be calculated along with phase analysis quantities $II_i$, $CME_i$, and $CME_i^N$. Accordingly, a respective $J_{CME}(\\hat{\\mathbf{x}})$ is obtained for each phase, abbreviated as $J_{A}(\\hat{\\mathbf{x}})$, $J_{B}(\\hat{\\mathbf{x}})$, and $J_{C}(\\hat{\\mathbf{x}})$. The $\\chi^2$ distribution then uses the number of measurements of each phase $i$ $(mi)$ as degrees of freedom. Thus, the HIF-affected phase $i$ can be identified if $J_{i}(\\hat{\\mathbf{x}}) > \\chi^2_{mi,p}$. \nThe SE then communicates with protection systems to locate and isolate the HIF of the affected feeder at relay times of 0.3-0.5 s, however SE could be run locally on relay\/recloser phasor measurement units (PMUs), further expediting the process \\cite{relay}. \n\n\\subsection{Parameter Error Analysis}\\label{sec:paramerror}\nTo obtain accurate state vector estimates during the SE process, PDS parameters must be pre-established, such as line impedance values. Parameter errors occur when the parameter values used for SE do not match those of the physical system. For this work, the properties of parameter errors are of particular interest. The proposed model, after all, must differentiate between measurement errors and parameter errors caused by the HIF. Measurement errors are identified individually when comparing their $CME^N$ to a threshold $\\beta$ (typically equal to three standard deviations of a given measurement, i.e., $\\beta$ = 3). If a GE is detected, the $CME^N$s are analyzed. If an isolated measurement is found to have a $CME^N$ above the threshold, it is determined to have a measurement error.\n\n\\begin{figure\n\\includegraphics[width=7.5072cm,height = 4cm,scale = 1]{33diagram.png}\n\\centering\n\\caption{33-Bus Distribution System and HIF Locations}\n\\label{33bus}\n\\end{figure}\n\nIn contrast to measurement errors, parameter errors spread out to the measurement functions containing the parameter in error, a formal proof of which is presented in \\cite{ParameterProof}. Indeed, the measurement model in \\eqref{eq:SE} does not consider the possibility of parameter data errors. If one instead considers a correct measurement $z_i$ with a parameter in error $p_i$, one obtains a new measurement model: \n\\begin{equation}\n    z_i = h(x,p_i) + e_i \n\\label{PEModel}\\end{equation}\n\nThe function model (\\ref{PEModel}) can further be developed into a Taylor series form: $z_i = h_{i,0} + \\frac{\\partial h_i}{\\partial p}(x_0,p_0)\\Delta p_i$, after which the parameter error $\\Delta p_i$ is obtained: $\\Delta p_i = \\frac{z_i-h_{i,0}}{\\mathbf{H}_{p,0}}$, where $\\mathbf{H}_{p,0}$ is the Jacobian of parameters. During GE analytics, if it is found that there are measurements $i$-$j$ and $j$-$i$\n(real or reactive power flows) and measurements $i$ and $j$ (real or\nreactive power injections) with $CME^N$ above the threshold value $\\beta$, then the branch $i$-$j$ is suspicious of having a parameter error. \n\n\n\n\n\n\n\n\n\\section{Case Study}\\label{casestudy}\nValidation was performed in Simulink on the 12.66 kV 33-bus distribution system from Baran and Wu \\cite{33bus}. Three HIF scenarios are presented. At Bus 20, 10, and 28, phases $A$, $B$, and $C$ respectively are subject to a line-to-ground HIF lasting 1.0 second using the Fig. \\ref{HIF Model} model at the lateral line from bus to load. The measurement plan consisted of 195 measurements, leading to $GRL = 3$. A figure of the 33-bus system and proposed HIF locations is included in Fig. \\ref{33bus}. To ensure that HIF current is low enough to be categorized as such and that the immediate substation current measurement is low enough so as not to exceed typical overcurrent relay thresholds \\cite{overcurrentrelays}, current measurements were recorded, illustrated in Fig. \\ref{Current Comparisons} , where substation measurement in this context refers to the recorded current flow from Bus 9 to 10. Tables I, II, and III quantify the low current displacement of the affected phase.\n\n\\begin{table}[h]\n\\caption{Bus 20 Lateral HIF Current Measurements}\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabular}{ | m{22em} | m{1.2cm}| } \n  \\hline\n  Phase $A$ I$_{RMS}$ pre-fault & 20.35 A \\\\ \n  \\hline\n  Phase $A$ I$_{RMS}$ post-fault when $J_{A}(\\hat{\\mathbf{x}}) > \\chi^2_{195,0.95}$ & 23.73 A \\\\ \n  \\hline\n Phase $A$ I$_{RMS}$ displacement & 3.38 A \\\\ \n  \\hline\n  Smallest detectable HIF I$_{RMS}$ & 4.08 A \\\\ \n  \\hline\n\\end{tabular}\n\\end{center}\n\\medskip\n\\caption{Bus 10 Lateral HIF Current Measurements}\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabular}{ | m{22em} | m{1.2cm}| } \n  \\hline\n  Phase $B$ I$_{RMS}$ pre-fault & 48.65 A \\\\ \n  \\hline\n  Phase $B$ I$_{RMS}$ post-fault when $J_{B}(\\hat{\\mathbf{x}}) > \\chi^2_{195,0.95}$ & 52.37 A \\\\ \n  \\hline\n Phase $B$ I$_{RMS}$ displacement & 3.72 A \\\\ \n  \\hline\n  Smallest detectable HIF I$_{RMS}$ & 4.78 A \\\\ \n  \\hline\n\\end{tabular}\n\\end{center}\n\\medskip\n\\caption{Bus 28 Lateral HIF Current Measurements}\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabular}{ | m{22em} | m{1.2cm}| } \n  \\hline\n  Phase $C$ I$_{RMS}$ pre-fault & 86.07 A \\\\ \n  \\hline\n  Phase $C$ I$_{RMS}$ post-fault when $J_{C}(\\hat{\\mathbf{x}}) > \\chi^2_{195,0.95}$ & 89.26 A \\\\ \n  \\hline\n Phase $C$ I$_{RMS}$ displacement & 3.19 A \\\\ \n\n\n  \\hline\n  Smallest detectable HIF I$_{RMS}$ & 5.92 A \\\\ \n  \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\nSensor measurements are obtained from the Simulink model, after which the first step of the two-step quasi-static SE is conducted and the first set of SM are generated. Because all measurements are considered as possibly having GEs, each measurement standard deviation is a percentage of the measurement magnitude $(\\sigma_i = |z_i|\/100)$ \\cite{convprop}. Random errors with distribution $X \\sim \\mathcal{N}(0,1)$ are applied to all measurements.\n\n  \\begin{figure\n\\includegraphics[width=4.3705cm,height = 3.66445733cm,scale = 1]{Bus10FaultRMS.png}\n\\includegraphics[width=4.3705cm,height = 3.66445733cm,scale = 1]{Bus10FaultSin.png}\n\\includegraphics[width=4.3705cm,height = 3.66445733cm,scale = 1]{Bus10Substation.png}\n\\centering\n\\caption{Comparison between HIF and Substation current measurements for a Phase $B$ Fault at the Bus 10 lateral line}\n\\label{Current Comparisons}\n\\end{figure}\n\nFor a given phase $i$, if $J_{i}(\\hat{\\mathbf{x}}) > \\chi^2_{mi,p}$, a GE is detected. $J_{CME}(\\hat{\\mathbf{x}})$ evolution for the first SE step during a HIF between Bus 28 and its load is included in Fig. \\ref{Jc}; only if $J_{i}(\\hat{\\mathbf{x}}) > \\chi^2_{mi,p}$ does the procedure advance to the second SE step, during which the SE is repeated but with meter precision values for each measurement type. After obtaining a descending order list of $CME^N$, \\cite{ParameterProof}, the errors can be classified as either measurement errors or a suspected HIF with parameter error spread. Tables IV, V, and VI catelogue the $CME^N$ distributions, including only those above the detection threshold $\\beta = 3$. \n\n\\begin{figure\n\\centering\n\\includegraphics[width=7.2cm,height = 5.4cm,scale = 1]{Jc_Figure_28.png}\n\\caption{$J_{CME}$ evolution for a Phase $C$ lateral HIF at Bus 28}\n\\label{Jc}\n\\end{figure}\n\nAccording to the list in Table IV, for example, the phase $A$ injections measurements and associated power flow measurements at Bus 20 presented six $CME^N$ values above the threshold $\\beta = 3$, indicative of a HIF at the lateral line from Bus 20 to its load. The highest $CME^N$ corresponds to the real power injection P:20 at the bus of the lateral HIF, followed by the  $i$-$j$ and $j$-$i$ real power flow measurements, P:20-19 and P:19-20, immediately upstream to Bus 20. The next highest error is also associated with Bus 20\u2014the reactive power injection Q:20. Finally, the last set of $CME^N$s above the threshold $\\beta = 3$ are the two $i$-$j$ and $j$-$i$ real power flow measurements immediately downstream to Bus 20, P:21-20 and P:20-21. This $CME^N$ distribution is consistent with parameter error spread, as opposed to measurement error \\cite{ParameterProof}. Similar measurement function error spread is observed for the Bus 10 and Bus 28 cases (Tables V and VI, respectively). For all three cases, the highest $CME^N$ is the real power injection at the bus of the lateral fault. Monte Carlo simulations were run for the same measurement sets used in Tables IV-VI but with varying sets of noise; the highest $CME^N$ was the respective case's real power injection at the HIF bus location 100\\% of the time.\n\n\\begin{table}[]\n\\caption{Bus 20 Lateral HIF $CME^N$ Distribution}\n\\centering\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabular}{ccc}\n\\hline\n\\multicolumn{3}{c}{\\begin{tabular}[c]{@{}c@{}}$J_{A}(\\hat{\\mathbf{x}}) = 250.7226 \\ \\textgreater \\ C = \\chi^2_{195,0.95} = 228.5799 \\Rightarrow\\ $Bad data detected!\\end{tabular}} \\\\ \\hline\n\\multicolumn{3}{c}{$CME^N$ Descending List}                       \\\\\\hline                                                                                      \nMeasurement                                           & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ $II$                                            & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ $CME^N$                                          \\\\\\hline\nP:20                                                  & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1.6388                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 12.5684                                          \\\\\nP:20-19                                               & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1.4147                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -4.5078                                           \\\\\nP:19-20                                               & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1.4058                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 4.2587                                          \\\\\nQ:20                                                  & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1.7335                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -3.9474                                           \\\\\nP:21-20                                               & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1.1959                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 3.1522                                          \\\\\nP:20-21                                                  & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1.1954                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -3.0565                                \\\\\\hline         \n\\end{tabular}\n\\end{table}\n\n\\begin{table}[]\n\\caption{Bus 10 Lateral HIF $CME^N$ Distribution}\n\\centering\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabular}{ccc}\n\\hline\n\\multicolumn{3}{c}{\\begin{tabular}[c]{@{}c@{}}$J_{B}(\\hat{\\mathbf{x}}) = 241.7247 \\ \\textgreater \\ C = \\chi^2_{195,0.95} = 228.5799 \\Rightarrow\\ $Bad data detected!\\end{tabular}} \\\\ \\hline\n\\multicolumn{3}{c}{$CME^N$ Descending List}                       \\\\\\hline                                                                                      \nMeasurement                                           & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ $II$                                            & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ $CME^N$                                          \\\\\\hline\nP:10                                                  & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1.2862                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 11.7488                                          \\\\\nP:09-10                                               & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1.5342                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 4.2983                                           \\\\\nP:10-09                                               & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1.5433                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -3.8723                                          \\\\\nP:11-10                                               & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1.5546                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 3.3604                                           \\\\\nP:10-11                                               & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1.5534                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -3.2135                                          \\\\\nP:09                                                  & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1.0557                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 3.0204                                             \\\\\\hline         \n\\end{tabular}\n\\end{table}\n\n\\begin{table}[]\n\\caption{Bus 28 Lateral HIF $CME^N$ Distribution}\n\\centering\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabular}{ccc}\n\\hline\n\\multicolumn{3}{c}{\\begin{tabular}[c]{@{}c@{}}$J_{C}(\\hat{\\mathbf{x}}) = 242.1354$ \\ \\textgreater \\ $C = \\chi^2_{195,0.95} = 228.5799 \\Rightarrow\\ $Bad data detected!\\end{tabular}} \\\\ \\hline\n\\multicolumn{3}{c}{$CME^N$ Descending List}                       \\\\\\hline                                                                                      \nMeasurement                                           & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ $II$                                            & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ $CME^N$                                          \\\\\\hline\nP:28                                                  & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 0.9316                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 13.8367                                          \\\\\nP:29-28                                               & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1.4273                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 6.2406                                           \\\\\nP:27-28                                               & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1.7423                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 5.7956                                          \\\\\nP:28-29                                               & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1.4639                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -4.8387                                           \\\\\nP:28-27                                               & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 1.7292                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -4.5188                                          \\\\\nQ:28                                                  & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 0.3208                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -3.7285                                           \\\\\nP:27                                                  & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 0.9033                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 3.7171                                           \\\\\nQ:28                                                  & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 0.4279                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ -3.4318                                          \\\\\nP:29                                               & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 2.1124                                          & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 3.1529                                      \\\\\\hline         \n\\end{tabular}\n\\end{table}\n\n\n\\section{Summary and Conclusions}\\label{sec:conclusion}\nThis paper presents a parameter error modeling approach for HIF detection under a two-step, per-phase SE framework. Overall, the presented model was successful in detecting low current magnitude HIFs and generating identifiable $CME^N$ error distributions following parameter error methodology. It should be noted that this work does assume synchronized PMU measurements. Sufficient granularity is required to capture the wide-area measurement set needed for the SE process and $\\chi^2$ hypothesis test. The first SE step allowed for detection at realistically low HIF currents. The second step revealed HIF parameter error distributions, as opposed to measurement error(s). The error spread of power flow and injection measurements near the HIF, as well as the prominence of errors pertaining to the bus at which the lateral HIF occurred, demonstrate detection and, for future work, location capabilities for HIFs under a parameter error modeling framework.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the context of network analysis the clique model, dating back at least to the work of Luce and Perry \\cite{luce1949method} about social networks, refers to subsets with every two elements in a direct relationship. The problem of finding maximal cliques has numerous applications in domains including telecommunication networks, biochemistry, financial networks, and scheduling (\\cite{bomze1999maximum}, \\cite{wu2015review}). From an optimization perspective, this problem has been the subject of extensive studies stimulating new research directions in both continuous and discrete optimization (see, e.g., \\cite{bomze1998standard}, \\cite{bomze1999maximum}, \\cite{bomze2000copositive}, \\cite{stozhkovcontinuous}). The Motzkin-Straus quadratic formulation \\cite{motzkin1965maxima} in particular has motivated several algorithmic approaches (see \\cite{bomze1997evolution}, \\cite{hungerford2019general} and references therein) to the maximum clique problem, beside being of independent interest for its connection with Tur\\'{a}n's theorem \\cite{aigner2010proofs}. \\\\\nSince the strict requirement that every two elements have a direct connection is often not satisfied in practice, many relaxations of the clique model have been proposed (see, e.g., \\cite{pattillo2013clique} for a survey). In this work we are interested in $s$-defective cliques (\\cite{chen2021computing}, \\cite{trukhanov2013algorithms}, \\cite{yu2006predicting}), allowing up to $s$ links to be missing, and introduced in \\cite{yu2006predicting} for the analysis of protein interaction networks obtained with large scale techniques subject to experimental errors. \\\\ \nIn this paper, we first define a regularized version of a cubic continuous formulation for the maximum \\mbox{$s$-defective} clique problem  proposed in \\cite{stozhkovcontinuous}. We then apply variants of the classic Frank-Wolfe (FW) method  \\cite{frank1956algorithm} to this formulation. \\\\ \nFW variants are a class of first order optimization methods widely used in the optimization and machine learning communities (see, e.g., \\cite{clarkson2010coresets}, \\cite{jaggi2013revisiting}, \\cite{lacoste2015global}, \\cite{kerdreux2020affine} and references therein) thanks to their sparse approximation properties, their weaker requirement of a linear minimization oracle instead of a projection oracle with respect to proximal gradient like methods, and their ability to quickly identify the support of a solution. These identification properties, first proved qualitatively for the Frank Wolfe method with in face directions (FDFW, \\cite{bashiri2017decomposition}, \\cite{freund2017extended}, \\cite{guelat1986some}) in the strongly convex setting \\cite{guelat1986some}, were recently revisited for the away-step Frank-Wolfe (AFW)  with quantitative bounds (\\cite{bomze2020active}, \\cite{garber2020revisiting}) and extended to non convex objectives (\\cite{bomze2019first}, \\cite{bomze2020active}). \\\\\nAs we will see in the paper, the support identification properties of FW variants are especially suited for our maximal $s$-defective clique formulation, since in this case the optimization process can stop as soon as the support of a solution is identified. \\\\\nOur contributions can be summarized as follows:\n\\begin{itemize}\n\t\\item We solve the spurious solution problem for the maximum $s$-defective clique formulation proposed in \\cite{stozhkovcontinuous} by introducing a regularized version, for which we prove equivalence between local maximizers and maximal $s$-defective cliques. In particular, no postprocessing algorithms are needed to derive the desired structure from a local solution. Our work develops along the lines of analogous results proved for regularized versions of the Motzkin - Straus quadratic formulation (\\cite{bomze1999maximum}, \\cite{hungerford2019general}).\n\t\\item We prove that the FDFW applied to our formulation identifies the support of a maximal $s$-defective clique in a finite number of iterations.\n\t\\item We propose a tailored Frank-Wolfe variant for the $s$-defective clique  formulation at hand exploiting its product domain structure. This method retains the identification properties of the FDFW while significantly outperforming it in numerical tests.  \n\\end{itemize}\nThe paper is organized as follows: after \ngiving some basic notation and preliminaries\nin Section \\ref{Sec:Not}, we study the regularized maximum $s$-defective clique formulation in Section \\ref{Sec:Reg}. We then analyze, in Section \\ref{Sec:FDFW}, the FDFW algorithm and prove that it identifies a maximal $s$-defective clique in finite time. In Section \\ref{Sec:FWvar}, we describe our tailored FW variant and prove that it shares similar identification properties as FDFW. Finally, in Section~\\ref{Sec:NumRes}, we report some numerical results showing the practical effectiveness of the proposed approach.  In order to improve readability,\nsome technical details and numerical result tables are deferred to a small appendix.\n\\section{Notation and preliminaries}\\label{Sec:Not}\nFor any integers $a,b$ we denote by $[a\\! :  \\! b]$ the set of all integers $k$ satisfying $a\\le k \\le b$. \n\nFor a vector $r \\in \\R^d$, the $d$-dimensional Euclidean space, and a set $A \\subset [1\\!:\\! d]$,  we denote with $r_A$ the components of $r$ with indexes in $A$; $e$ is always a vector with all components equal to 1, and dimension clear from the context. Similarly, we denote by $e_i$ the $i$-th column of an appropriately sized identity matrix. \n\nLet $\\G = (V, E)$ be a graph with  vertices $V$ and  and edges $E$, $n = |V|$,\n$\\ad$ the adjacency matrix of $\\G$, and let  $\\bG = (V, \\bE)$ the complementary graph. \nThe notation we use largely overlaps with the one introduced in \\cite{stozhkovcontinuous}. For $s \\in \\mathbb{N}$ with $s\\le |\\bE|$ we define \n$$D_s(\\G) = \\{  y \\in  \\{0, 1\\}^{\\bE} \\ | \\  e^{\\intercal}y \n\\leq s \\}\\, ,$$\nrepresenting the set of \"fake edges\" to be added to the graph in order to complete an $s$-defective clique, and its continuous relaxation as\n$$D'_s(\\G) =  \\{  y \\in  [0, 1]^{\\bE} \\ | \\  e^{\\intercal}y \n\\leq s \\} \\, .$$ For $y \\in D'_s(\\G)$ we define the induced adjacency matrix $A(y)\\in \\R^{n \\times n}$ as\t\n$$\nA(y)_{ij} = \n\\begin{cases}\n\ty_{ij} \\ &\\text{if} \\ \\{i, j\\} \\in \\bE \\, ,\t\\\\\n\t0 \\ &\\text{if} \\\t\\{i, j\\} \\notin E \\, .\n\\end{cases}\n$$\nFor $y \\in D_s(\\G)$ in particular we define $\\G(y)$ as the graph with adjacency matrix $A_\\G + A(y)$, that is the  graph where we add to $\\G$ the edge $\\{i, j\\}$ whenever $y_{ij} = 1$. We also define $E(i)$ and $E^y(i)$ as the neighbors of $i$ in $\\G$ and $\\G(y)$ respectively. \\\\  \nLet $\\p=\\Delta_{n-1} \\times D'_s(\\G) $, with $\\Delta_{n-1}$ the \n$(n-1)$-dimensional simplex.\nThe objective of the $s$-defective clique relaxation defined in~\\cite{stozhkovcontinuous} is \n\\begin{equation}\n\tf_{\\G}(z)=\tf_{\\G}(x, y) := x^{\\intercal}[\\ad + A(y)]x \\,  ,\\quad z=(x,y)\\in \\p\n\\end{equation}\nso that when $A(y) = 0$ one retrieves Motzkin-Straus quadratic formulation. \\\\\n\\section{A regularized maximum $s$-defective clique formulation}\\label{Sec:Reg}\nHere we consider the problem     \n\\begin{equation} \\label{deq:P}\n\t\\max\\{ h_{\\G}(z)\\ | \\ \tz\\in \\p \\}\\, , \\tag{P} \t\n\\end{equation}\nwhere $h_{\\G}: \\p \\rightarrow \\R_{> 0}$ is a regularized version of $f_{\\G}$:\n$$h_{\\G}(z)= h_{\\G}(x, y) := f_{\\G}(x, y) + \\frac{\\alpha}{2} \\n{x}^2 + \\frac{\\beta}{2} \\n{y}^2 $$\nfor some $\\alpha \\in (0, 2)$ and $\\beta > 0$. In particular, when $y = 0$ the objective $h_{\\G}$ corresponds to  the quadratic regularized maximal clique formulation introduced in \\cite{bomze1997evolution}. \\\\\nFor non-empty $C \\subseteq V$ let $x^{(C)} = \\frac 1{|C|}\\, \\sum_{i\\in C}e_i$\nbe the characteristic vector in $\\Delta_{n  - 1}$ of the clique $C$, \tand $$\\Delta^{(C)}= \\{ x\\in \\Delta_{n-1} \\ | \\ x_i=0\\mbox{ for all }i\\in V\\setminus C\\}$$ \nbe the minimal face of $\\Delta_{n  - 1}$ containing  $x^{(C)}$ in its relative interior. \\\\\nFor $p \\in \\p$ we define as $T_{\\p}(p)= \\{ v-p:v\\in \\p\\}$ as the cone of feasible directions at $p$ in $\\p$, while for $r \\in \\mathbb{R}^{|V| + |\\bE|}$ we define  $T_{\\p}^0(p, r)$ as the intersection between $T_{\\p}(p)$ and the plane orthogonal to $r$:\n\\begin{equation*}\n\tT_{\\p}^0(p, r) = \\{d \\in T_{\\p}(p) \\ | \\ \\Sc{d}{r} = 0\\} \\, .\n\\end{equation*}\nWe now prove that there is a one to one correspondence between (strict) local maxima of $h_{\\G}$ and $s$-defective cliques coupled together with $s$ fake edges including the one missing on the clique.\\\\ Recall that in our \npolytope-constrained setting, (second order) sufficient conditions for the local maximality of $p \\in \\p$ are (see, e.g., \\cite{bertsekas1997nonlinear}) \n\\begin{equation}\n\t\\Sc{\\nabla \\hg(p)}{d} \\leq 0 \\tx{ for all } d \\in T_{\\p}(p)\n\\end{equation}\nand \n\\begin{equation}\n\td^{\\intercal} D^2\\hg(p) d  < 0 \\tx{ for all } d \\in T_{\\p}^0(p, \\nabla \\hg(p)) \\, .\n\\end{equation}\nIn the rest of the article we use $\\M_s(\\G)$ to denote the set of strict local maximizers of $h_{\\G}$.\n\\begin{prop}[characterization of local maxima for $h_{\\G}$] \\label{p:lm}\n\tThe following are equivalent: \n\t\\begin{itemize}\n\t\t\\item[(i)] $p \\in \\p$ is a local maximizer for $h_{\\G}(x,y)$;\n\t\t\\item[(ii)] $p \\in \\M_s(\\G)$;\n\t\t\\item[(iii)] $p = (x^{(C)}, y^{(p)})$ where   \n\t\t$s=e^{\\intercal}y^{(p)} \\in \\N$, with $C$ an $s$-defective clique in $\\G$ which is also a maximal clique in $\\G(y^{(p)})$, and $y^{(p)} \\in D_s(\\G)$ such that  $y^{(p)}_{ij} = 1$ for every $\\{i, j\\} \\in {C \\choose 2} \\cap\\bE$.\t\t\n\t\\end{itemize}\n\tIn either of these equivalent cases, we have\n\t\\begin{equation}\\label{loval}h_{\\G}(p) = 1 - \\frac{2 - \\alpha}{2|C|} + s\\,  \\frac{\\beta}{2} \\, .\n\t\\end{equation}\n\\end{prop}\n\\begin{proof}\n\tLet $p = (x^{(p)}, y^{(p)}) \\in \\p$, $g = \\nabla \\hg(p), H = D^2\\hg(p)$. \\\\\n\t(ii) $\\Rightarrow$ (i) is trivial. \\\\\t\n\t(i) $\\Rightarrow$ (iii). If $s:= e^{\\intercal}\n\t{y^{(p)}}$ were fractional, then for some $\\{i, j\\} \\in \\bar{E}$ we would have $y^{(p)}_{ij} < 1$. Furthermore \n\t\\begin{equation} \\label{eq:dph}\n\t\t\\frac{\\partial\\hg (p)}{\\partial y_{ij}}  = 2x^{(p)}_i x^{(p)}_j + \\beta y^{(p)}_{ij} \\geq 0, \\quad \t\\frac{\\partial\\hg (p)}{\\partial^2y_{ij}}   = \\beta > 0 \\, .\n\t\\end{equation}\n\tThus for $\\varepsilon >0$ small enough we have $\\hg (p + \\varepsilon e_{ij}) > \\hg(p)$ with $p + \\varepsilon e_{ij} \\in \\p$, which means that $p$ is not a local maximizer. Hence $s\\in \\N$ and obviously $s\\le |\\bE|$ as well as $y^{(p)}\\in D'_s(\\G) $.\\\\\n\tAssume now  by contradiction that $p$ is a local maximizer but $y^{(p)} \\notin D_s(\\G)$. Then for two distinct edges $\\{i, j\\}$, $\\{l, m\\} \\in \\bar{E}$ we must have $y^{(p)}_{ij}, y^{(p)}_{lm} \\in (0, 1)$. Let $d = (0, e_{ij} - e_{lm})$. Since $\\pm d$ are both feasible directions and $p$ is a local maximizer, necessarily $\\Sc{g}{d} = 0$. But we also have\n\t\\begin{equation}\n\t\td^{\\intercal} H d  = \\frac{\\partial \\hg(p)}{\\partial^2 y_{ij}} + \\frac{\\partial \\hg(p)}{\\partial^2 y_{lm}} - 2\\, \\frac{\\partial \\hg(p)}{\\partial y_{ij} \\partial y_{lm}}  = 2\\beta > 0 \\, .\n\t\\end{equation}  \n\tso that again for $\\varepsilon>0$ small enough $\\hg (p + \\varepsilon d) > \\hg(p)$ with $p + \\varepsilon d \\in \\p$, a contradiction. \\\\\n\tWe proved that if $p$ is a local maximizer, then $s=e^{\\intercal} {y^{(p)}} \\in \\N$ and  $y^{(p)} \\in D_s(\\G)$. But $x^{(p)}$ must be a local maximizer for the function $x \\mapsto h_{\\G}(x, y^{(p)})$, which is (up to a constant) a regularized maximal clique relaxation for the augmented graph $\\G (y^{(p)})$. Thus by well known results (see, e.g., \\cite{hungerford2019general}, \\cite{bomze1997evolution}) we must have $x = x^{(C)}$ with $C$ a maximal clique in $\\G(y^{(p)})$. In particular, since $\\G(y^{(p)})$ is defined by adding $s$ edges to $\\G$, $C$ must be an $s$-defective clique in $\\G$. \\\\\n\t(iii) $\\Rightarrow$ (ii). For a fixed $p = (x^{(C)}, y^{(p)})$ with $C, y^{(p)}$ satisfying the conditions of point (iii) let $\\bar{C} = V\\sm C$, $S = \\tx{supp}(y^{(p)})$ and $\\bar{S} = \\bar{E} \\sm S$. We abbreviate $E^{(p)}(i)=E^y(i)$ with $y=y^{(p)}$. For every $i \\in V$ we have \n\t\\begin{equation}\n\t\tg_i = \\alpha x^{(C)}_i + \\sum_{j \\in E^{(p)}(i)} 2x^{(C)}_j\n\t\\end{equation}\n\tIn particular for $i \\in C$ \n\t\\begin{equation}\\label{peq:gic}\n\t\tg_i = \\frac{\\alpha}{|C|} + \\sum_{j \\in C \\sm \\{ i \\}} 2x^{(C)}_j = \\frac{1}{|C|}(\\alpha + 2|C| - 2)\n\t\\end{equation}\n\tand for every $i \\in \\bc$\n\t\\begin{equation} \\label{peq:gbic}\n\t\tg_i =  \\sum_{j \\in E^{(p)}(i)\\cap C} 2x^{(C)}_j \\le \\frac{2|C|-2}{|C|}\n\t\\end{equation}\n\twhere we used $x^{(C)}_j = 1\/|C|$ for every $j \\in C$, $x^{(C)}_j = 0$ otherwise. \\\\\n\tFor $\\{i, j\\} \\in \\bE$ we have \n\t\\begin{equation}\n\t\tg_{ij} = \\beta y^{(p)}_{ij} + 2x^{(C)}_i x^{(C)}_j\n\t\\end{equation}\n\tand in particular $g_{ij} = 0$ for $\\{i, j\\} \\in \\bar{S}$, while for $\\{i, j\\} \\in S$\n\t\\begin{equation}\n\t\tg_{ij} = \\beta + 2 x^{(C)}_i x^{(C)}_j \\geq \\beta > 0 \\, ,\n\t\\end{equation}\n\twhere we used $y^{(p)}_{ij} = 1$ for $\\{i, j\\} \\in S$, $0$ otherwise, and \n\t$x^{(C)}_i x^{(C)}_j = 0$ for $\\{i, j\\} \\in \\bar{S}\\subseteq \\bE$. \\\\\n\tLet $d$ be a feasible direction from $p$, so that $d = v - p$ with $v \\in \\p$. Let $\\sigma_S = \\sum_{\\{i, j\\} \\in S} g_{ij}$, $\\sigma_C = \\sum_{i \\in C} v_i $ \\\\ $= 1 - \\sum_{i \\in \\bc} v_i \\in [0, 1]$, $m_{\\bar{C}} = \\max_{i \\in \\bar{C}}g_i$, so that by \\cref{peq:gbic} we have $m_{\\bar{C}} \\le \\frac{2|C|-2}{|C|}$.\n\tThen\n\t\\begin{equation} \\label{peq:gp}\n\t\t\\Sc{g}{p} = \\sum_{i \\in \\bar{C}} x^{(C)}_i g_i + \\sum_{i \\in C} x^{(C)}_i g_i + \\sum_{(i, j ) \\in S} y^{(p)}_{ij} g_{ij} = \\frac{1}{|C|} \\sum_{i \\in C} g_i +  \\sum_{\\{i, j\\} \\in S} g_{ij} = \\frac{1}{|C|}(\\alpha + 2|C| - 2) + \\sigma_S\n\t\\end{equation}\n\twhere we used \\cref{peq:gic} in the last equality. We also have\n\t\\begin{equation} \\label{peq:gv}\n\t\t\\Sc{g_V}{v_V} =\t\\Sc{g_C}{v_C} + \\Sc{g_{\\bc}}{v_{\\bc}} \\leq \\frac{\\alpha + 2|C| - 2}{|C|} \\sigma_C + (1 - \\sigma_C) m_{\\bc} \\leq \\frac{\\alpha + 2|C| - 2}{|C|} \n\t\\end{equation}\n\twhere we used \\cref{peq:gic} together with the H\\\"older inequality in the first inequality, $m_{\\bc} \\le \\frac{2|C|-2}{|C|}$ in the second inequality and $\\sigma_C \\leq 1$. Finally, \n\t\\begin{equation} \\label{peq:ge}\n\t\t\\Sc{g_{\\bar{E}}}{v_{\\bar{E}}} = \\Sc{g_S}{v_S} + \\Sc{g_{\\bs}}{v_{\\bs}} = \\Sc{g_S}{v_S} \\leq \\sigma_S\n\t\\end{equation}\n\twhere we used $g_{\\bar{S}} = 0$ in the second equality, and $v_i \\leq 1$ for every $i \\in \\bE$ in the inequality. We can conclude \n\t\\begin{equation} \\label{peq:1c}\n\t\t\\Sc{g}{d} = \\Sc{g_V}{v_V} + \\Sc{g_{\\bar{E}}}{v_{\\bar{E}}} - \\Sc{g}{p} \\leq 0  \n\t\\end{equation}\n\twhere we used \\cref{peq:ge}, \\cref{peq:gp} and \\cref{peq:gv} in the inequality. We have equality iff there is equality both in \\cref{peq:gv} and \\cref{peq:ge}, and thus iff $v = (x^{(v)}, y^{(v)})$ with $\\tx{supp}(x^{(v)}) = C$ and $y^{(v)} = y^{(p)}$. In particular $p$ is a first order stationary point with \n\t\\begin{equation}\n\t\tT_{\\p}^0(p, g) = \\{d \\in T_{\\p}(p) \\ | \\ d = v - p, v_{\\bc} = 0, v_{\\bar{E}} = p_{\\bar{E}} \\} = \\{d \\in T_{\\p}(p) \\ | \\ d_{\\bc} = d_{\\bar{E}} = 0\\} \\, .\n\t\\end{equation}\n\tLet $H_C$ be the submatrix of $H$ with indices in $C$. We have, for $(i, j) \\in C^2$ with $i \\neq j$, $H_{ij} = 1$ since $C$ is a clique in the augmented graph $\\G(y_p)$, while $H_{ii} = \\alpha$ for every $i \\in V$ and in particular for every $i \\in C$. This proves \n\t\\begin{equation}\n\t\tH_C = 2e e^{\\intercal} + (\\alpha - 2) \\mathbb{I} \\, .\n\t\\end{equation}\n\tNow if $T_{\\p}^0(p, g) \\ni d \\neq 0$ we have \n\t\\begin{equation}\\label{peq:2c}\n\t\td^{\\intercal} H d = d_C^{\\intercal} H_C d_C = d_C^{\\intercal}(2e e^{\\intercal} + (\\alpha - 2) \\mathbb{I}) d_C = (\\alpha - 2) \\n{d_C}^2 < 0\n\t\\end{equation}\n\twhere we used $d_{\\bc} = d_{\\bE} = 0$ in the first equality, $e^{\\intercal} d_C = e^{\\intercal} (v_V-p_V)=1-1=0$ in the third one. This proves the claim, since we have sufficient conditions for local optimality thanks to \\cref{peq:1c} and \\cref{peq:2c}. \n\\end{proof}\nAs a corollary, the global optimum of $h_{\\G}$ is achieved on maximum $s$-defective cliques.\n\\begin{cor} \\label{c:globalm}\n\tThe global maximizers of $h_{\\G}(z)$ are all the points $p$ of the form $p = (x^{C^*}, y^{(p)})$ where $C^*$ is an $s$-defective clique of maximum cardinality, and $y^{(p)} \\in D_s(\\G)$ such that $e^{\\intercal} y^{(p)} = s$.\n\\end{cor}\n\\begin{proof}\n\tLet $p = (x^{(C)}, y^{(p)})$ a local maximizer for $h_{\\G}(z)$. Then its objective value is, by~\\cref{loval},\n\t$h_{\\G}(p) = 1 - \\frac{2 - \\alpha}{2|C|} + s\\, \\frac{\\beta}{2} $, which is\n\t(globally) maximized when $|C|$ is as large as possible, because $2 - \\alpha > 0$ by assumption.\n\\end{proof}\nThanks to \\cref{p:lm}, for every $p \\in \\M_s(\\G)$ we can define $y^{(p)} \\in D_s(\\G) $ and a maximal clique $C$ of $\\G(y^{(p)})$ such that $p = (x^{(C)}, y^{(p)})$. We now recall that the face of a polytope $\\Q$ exposed by a gradient $g \\in \\R^n$ is defined as\n\\begin{equation}\n\t\\F_e(g) = \\argmax_{w \\in Q} \\Sc{g}{w}.\n\\end{equation}\nWith this notation, we prove that the face of $\\p$ exposed by the gradient in $p \\in \\M_s(\\G)$ is simply the product between $\\Delta^{(C)}$ and the singleton $\\{y^{(p)}\\}$. This property, sometimes referred to as strict complementarity, is of key importance to prove identification results for Frank-Wolfe variants (see \\cite{bomze2019first}, \\cite{bomze2020active}, \\cite{garber2020revisiting}, and the discussion of external regularity in~\\cite[Section~5.3]{bomze2002sirev}).\n\\begin{lem}\\label{l:sc}\n\tLet $p = (x^{(C)}, y^{(p)}) \\in \\M_s(\\G)$. Then the face exposed by $\\nabla h_{\\G}(p)$ coincides with the minimal face $\\F(p)$ of $\\p$ containing $p$:\n\t\\begin{equation} \\label{eq:fep}\n\t\t\\F_{e}(\\nabla h_{\\G}(p)) = \\F(p) = \\Delta_{n - 1}^{(C)} \\times \\{y^{(p)}\\} \\, .\n\t\\end{equation}\n\\end{lem}\n\\begin{proof}\n\tTo start with, the second equality follows from the fact that $y^{(p)}$ is a vertex of $D_s'(\\G)$ and that $\\Delta_{n - 1}^{(C)}$ is the minimal face of $\\Delta_{n  - 1}$ containing $x^{(C)}$. The first equality is then equivalent to proving that for every vertex $a = (a_x, a_y)$ of $\\p$ with $a\\in \\p \\sm \\F(p)$ we have $\\lambda_a(p) < 0$. Given that stationarity conditions must hold in $\\Delta_{n - 1}$ and $D_s'(\\G)$ separately,\n\t$\\lambda_a(p) < 0$ if and only if\n\t\\begin{subequations}\n\t\t\\begin{align}\n\t\t\t\\lambda^x_{a}(p):=& \\Sc{\\nabla_x h_{\\G}(p)}{(a_x - x^{(C)})} \\leq 0\\, , \\label{eq:xc}\\\\\n\t\t\n\t\t\n\t\t\n\t\t\t\\lambda^{y}_a(p):= &\\Sc{\\nabla_y h_{\\G}(p)}{(a_y - y^{(p)})} \\leq 0 \\, , \\label{eq:yc}\n\t\t\\end{align}\t\n\t\\end{subequations}\n\tand at least one of these relations must be strict. Since $a$ is a vertex of $\\p$,  $a_x = e_l$ with $l \\in [1:n]$ and $a_y \\in D_s(\\G)$, while  $a \\notin \\F(p)$ implies $l \\notin C$ or $a_y \\neq y^{(p)}$. If $l \\in C$ then $\\lambda^x_{a}(p) = 0$ by stationarity conditions, otherwise\n\t\\begin{equation} \\label{eq:x1piece}\n\t\t\\Sc{\\nabla_x h_{\\G}(p)}{x^{(C)}} = 2 (x^{(C)})^{\\intercal}[A + A(y^{(p)})] x^{(C)} + \\alpha \\n{x^{(C)}}^2 = 2 - \\frac{2 - \\alpha}{|C|}\n\t\\end{equation}  \n\tand\n\t\\begin{equation} \\label{eq:x2piece}\n\t\t\\Sc{\\nabla_x h_{\\G}(p)}{a_x} = \\frac{\\partial}{\\partial x_l} h_{\\G}(p) = \\alpha x_l  + \\sum_{j \\in C \\cap E^{(p)}(l)}2 x_j = 2\\frac{|C \\cap E^{(p)}(l)|}{|C|} \\leq 2 - \\frac{2}{|C|} \\, ,\n\t\\end{equation} \n\twhere we used $a_x = e_l$ in the first equality, $l \\notin C$ together with $x_j = 1\/|C|$ for every $j \\in C$ in the third equality, and the maximality of the clique $C$ in the augmented graph $\\G(y^{(p)})$ in the inequality. Combining \\cref{eq:x1piece} and \\cref{eq:x2piece}, we obtain\n\t\\begin{equation}\n\t\t\\Sc{\\nabla_x h_{\\G}(p)}{(a_x - x^{(C)})} \\leq - \\frac{\\alpha}{|C|} < 0 \\, ,\n\t\\end{equation}\n\twhich proves that \\cref{eq:xc} holds with strict inequality if $l\\notin C$, or else with equality if $l\\in C$. \\\\\n\tIn a similar vain we proceed with~\\cref{eq:yc}. If $a_y=y^{(p)}$ then~\\cref{eq:yc} holds with equality but then $l\\in V\\sm C$ and we are done. So suppose $a_y\\neq y^{(p)}$, and consider the supports $S_y = \\{\\{i, j\\} \\in \\bar{E} \\ | \\ (a_y)_{ij} = 1\\}$ and $S_p = \\{\\{i, j\\} \\in \\bar{E} \\ | \\ y^{(p)}_{ij} = 1\\}$. Since $a_y\\in D_s(\\G)$, we have $|S_y|\\leq s$ and on the other hand, by Proposition~\\ref{p:lm}(iii), $|S_p| =s$. As $S_y$ and $S_p$ must be distinct, we conclude $S_y\\sm S_p\\neq \\emptyset$.\n\tFurthermore, by~\\cref{eq:dph} for every $\\{i, j\\}$ in $A_p$ we have\n\t\\begin{equation} \\label{eq:apy1}\n\t\t\\frac{\\partial}{\\partial y_{ij}} h_{\\G}(p) \\geq \\beta y^{(p)}_{ij} = \\beta > 0 \\, ,\n\t\\end{equation}\n\twhile for every $\\{i, j\\}$ in $A_y \\sm A_p$ we have\n\t\\begin{equation} \\label{eq:apy2}\n\t\t\\frac{\\partial}{\\partial y_{ij}} h_{\\G}(p)  = 0\n\t\\end{equation}\n\tbecause $y^{(p)}_{ij} = 0$ by definition of $A_p$ and $x^{(C)}_{i} x^{(C)}_j = 0$ since, again invoking~\\cref{p:lm}(iii), $\\{i, j\\}\\in \\bar E\\setminus {C \\choose 2}$. So we can finally prove \\cref{eq:yc} by observing\n\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t\t& \\Sc{\\nabla_y h_{\\G}(p)}{(a_y - y^{(p)})}  = \\sum_{\\{i, j\\} \\in A_y} \\frac{\\partial}{\\partial y_{ij}} h_{\\G}(p) - \\sum_{\\{i, j\\} \\in A_p} \\frac{\\partial}{\\partial y_{ij}} h_{\\G}(p) \\\\ = & \\sum_{\\{i, j\\} \\in A_y \\sm A_p} \\frac{\\partial}{\\partial y_{ij}} h_{\\G}(p) - \\sum_{\\{i, j\\} \\in A_p \\sm A_y} \\frac{\\partial}{\\partial y_{ij}} h_{\\G}(p) = \n\t\t\t-  \\sum_{\\{i, j\\} \\in A_p \\sm A_y} \\frac{\\partial}{\\partial y_{ij}} h_{\\G}(p) < 0\n\t\t\\end{aligned}\n\t\\end{equation}\n\twhere we used \\cref{eq:apy2} in the third equality and \\cref{eq:apy1} together with $A_p\\sm A_y \\neq \\emptyset$ in the inequality. \n\\end{proof}\n\n\\section{Frank-Wolfe method with in face directions} \\label{Sec:FDFW}\n\nLet $\\Q = \\tx{conv}(A) \\subset \\R^n $ with $|A| < +\\infty$.  In this section, we consider the FDFW for the solution of the smooth constrained optimization problem\n\\begin{equation*}\n\t\\max \\{ f(w) \\ | \\ w \\in \\Q \\} \\, .\n\\end{equation*}\nIn particular, $\\{w_k\\}$ is always a sequence generated by the FDFW applied to the polytope $\\Q$ with objective $f$.\nFor $w \\in \\Q$ we denote with $\\F(w)$ the minimal face of $\\Q$ containing $w$. \t\n\\begin{algorithm}\n\t\\caption{Frank-Wolfe method with in face directions (FDFW) on a polytope}\n\t\\label{alg:FW}\n\t\\begin{algorithmic}[1]\n\t\t\\STATE{\\textbf{Initialize} $w_0 \\in \\Q$, $k := 0$}\n\t\t\n\t\t\\STATE{Let $s_k \\in \\argmax_{y \\in \\Q}\\Sc{\\nabla f(w_k)}{y}$ and $d_k^{\\mathcal{FW}} := s_k - w_k$.} \\label{st:FW}\n\t\t\\IF{$w_k$ is stationary}\n\t\t\\STATE{STOP}\n\t\t\\ENDIF\n\t\t\\STATE{Let $v_k \\in \\argmin_{y \\in \\F(w_k)} \\Sc{\\nabla f(w_k)}{y}$ and $d_k^{\\mathcal{FD}} := w_k - v_k$.}\n\t\t\\IF{$\\Sc{\\nabla f(w_k)}{d_k^{\\FW}} \\geq \\Sc{\\nabla f(w_k)}{d_k^{\\FD}}$}\n\t\t\\STATE{$d_k := d_k^{\\FW}$}\n\t\t\\ELSE\n\t\t\\STATE{$d_k := d_k^{\\FD}$}\\label{st:inface}\n\t\t\\ENDIF\n\t\t\\STATE{Choose the stepsize $\\alpha_k \\in (0, \\alpha_{k}^{\\max}]$ with a suitable criterion}\\label{st:stepsize}\n\t\t\\STATE{Update: $w_{k+1} := w_k + \\alpha_k d_k$}\n\t\t\\STATE{Set $k : = k+1$. Go to step 2.}\n\t\\end{algorithmic}\n\\end{algorithm}\nThe FDFW at every iteration chooses between the classic FW direction $d_k^{\\FW}$ calculated at \\cref{st:FW} and the in face direction $d_k^{\\mathcal{FD}}$ calculated at \\cref{st:inface} with the criterion in \\cref{st:stepsize}. The classic FW direction points toward the vertex maximizing the scalar product with the current gradient, or equivalently the vertex maximizing the first order approximation $w \\mapsto f(w_k) + \\Sc{\\nabla f(w_k)}{w}$ of the objective $f$. The in face direction $d_k^{\\mathcal{FD}}$ is always a feasible direction in $\\F(w_k)$ from $w_k$, and it points away from the vertex of the face minimizing the first order approximation of the objective. When the algorithm performs an in face step, we have that the minimal face containing the current iterate either stays the same or its dimension drops by one. The latter case occurs when the method performs a maximal feasible in face step (i.e., a step with $\\alpha_k = \\alpha_k^{\\max}$ and $d_k = d_k^{\\FD}$), generating a point on the boundary of the current minimal face. As we will see formally in  \\cref{p:FDFWli}, this drop in dimension is what allows the method to quickly identify low dimensional faces containing solutions.\\\\\nWe often require the following lower bound on the stepsizes:\n\\begin{equation} \\label{deq:alphalb}\n\t\\alpha_k \\geq  \\bar{\\alpha}_k := \\min(\\alpha_k^{\\max}, \\cC\\frac{\\Sc{\\nabla f(w_k)}{d_k}}{\\n{d_k}^2}) \\tag{S1}\n\\end{equation}\nfor some $\\cC > 0$. Furthermore, for some convergence results we need the following sufficient increase condition for some constant $\\rho >0$:\n\\begin{equation} \\label{eq:deltaflb}\n\t\\fun(w_k + \\alpha_k d_k) - \\fun(w_k) \\geq \\rho \\bar{\\alpha}_k \\, \\Sc{\\nabla \\fun(w_k)}{d_k} \\tag{S2} \\, .\n\\end{equation}\nAs explained in the Appendix (see \\cref{alphacond}), these conditions generalize properties of exact and Armijo line search. \\\\\n\nWe also define the multiplier functions $\\lambda_a$ for $a \\in A$, $w\\in \\R^n$ as\n\\begin{equation}\n\t\\lambda_a(w) = \\Sc{\\nabla f(w)}{(a - w)} \\, .\n\\end{equation}\nWe adapt the well known FW gap  (\\cite{bomze2020active}, \\cite{lacoste2016convergence}) to the maximization case, thus obtaining the following measure of stationarity\n\\begin{equation} \\label{def:gz}\n\tG(w) :=  \\max_{y \\in \\Q} \\Sc{\\nabla f(w)}{(w-y)} = \\max_{a \\in A} \\Sc{\\nabla f(w)}{(w - a)} = \\max_{a \\in A}- \\lambda_a(w) \\, ,\n\\end{equation}\nas well as an \"in face\" gap \n\\begin{equation} \\label{def:gfz}\n\tG_{\\F}(w) = \\max (G(w), \\max_{b \\in \\F(w) \\cap A} \\lambda_b(w)) \\, .\n\\end{equation}\nUsing these definitions, we have\n\\begin{equation} \\label{eq:gfz}\n\t\\begin{aligned}\n\t\t&\\Sc{\\nabla f(w_k)}{d_k} = \\max (\\Sc{\\nabla f(w_k)}{d_k^{\\FW}}, \\Sc{\\nabla f(w_k)}{d_k^{\\FD}}) \\\\\n\t\t= &\\max (G(w_k), \\max_{y \\in \\F(w_k)} \\Sc{\\nabla f(w_k)}{(w_k - y)}) =  G_{\\F}(w_k) \\, ,\t\n\t\\end{aligned}\n\\end{equation}\nwhere in the second equality we used\n\\begin{equation} \\label{eq:dkfw}\n\t\\begin{aligned}\n\t\t\\Sc{\\nabla f(w_k)}{d_k^{\\FW}} = \\max_{y \\in Q} \\Sc{\\nabla f(w_k)}{(y - w_k)}\t&\t\n\t\\end{aligned}\n\\end{equation} \nand in the third equality  \n\\begin{equation} \\label{eq:fdl}\n\t\\Sc{\\nabla f(w_k)}{d_k^{\\FD}} = \\max_{b \\in \\F(w_k)} \\Sc{\\nabla f(w_k)}{(w_k - b)} = \\max_{b \\in \\F(w_k) \\cap A} - \\lambda_b(w_k) \\, . \n\\end{equation}\nFrom the definitions it also immediately follows\n\\begin{equation}\n\tG_{\\F}(w) \\geq G(w) \\geq 0\n\\end{equation}\nwith equality iff $w$ is a stationary point. \\\\\nIn order to obtain a local identification result, we need to prove that under certain conditions the method does consecutive maximal in face steps, thus identifying a low dimensional face containing a minimizer. First, in the following lemma we give an upper bound for the maximal feasible stepsize. \n\\begin{lem} \\label{l:alphakineq}\n\tIf $w_k$ is not stationary, then $\\alpha_k \\leq G(w_k) \/ G_{\\F}(w_k)$. \n\\end{lem}\n\\begin{proof}\n\tNotice that since $w_k$ is not stationary we have $G(w_k) > 0$ and therefore also $G_{\\F}(w_k) > 0$. Now \n\t\\begin{equation*}\n\t\t\\Sc{\\nabla f(w_k)}{(w_k + \\alpha_kd_k)} \\leq \\max_{y \\in \\Q} \\Sc{\\nabla f(w_k)}{y} = \t\\Sc{\\nabla f(w_k)}{(w_k +  d_k^{\\FW})} = \\Sc{\\nabla f(w_k)}{w_k} + G(w_k) \\, ,\n\t\\end{equation*} \n\twhere in the inequality we used $w_k + \\alpha_k d_k \\in \\Q$. Subtracting $\\Sc{\\nabla f(w_k)}{w_k}$ on both sides we obtain\n\t\\begin{equation}\n\t\t\\alpha_k \\Sc{\\nabla f(w_k)}{d_k} \\leq G(w_k) \\, .\n\t\\end{equation}\n\tand the thesis follows by applying \\cref{eq:gfz} to the LHS.\n\\end{proof}\nWe can now prove a local identification result.\n\\begin{prop}[FDFW local identification] \\label{p:FDFWli}\n\tLet $p$ be a stationary point for $f$ defined on $\\Q$ and assume that \\cref{deq:alphalb} holds. We have the following properties:\n\t\\begin{itemize}\n\t\t\\item[(a)]there exists $r^*(p) > 0$ such that if $w_{k} \\in B_{r^*(p)}(p) \\cap \\F_e(\\nabla f(p))$ then $w_{k + 1} \\in \\F_e(\\nabla f(p))$;\n\t\t\\item[(b)] for any $\\delta > 0$ there exists $r(\\delta, p) \\leq \\delta$ such that if $w_{k} \\in B_{r(\\delta, p)}(p)$ then  $w_{k + j} \\in \\F_e(\\nabla f(p)) \\cap B_{\\delta}(p)$ for some $j \\leq \\dim (\\F(w_{k}))$.\n\t\\end{itemize} \n\\end{prop}\n\\begin{proof}\n\t(a) Notice that by definition of exposed face and stationarity conditions\n\t\\begin{equation} \\label{eq:lam}\n\t\t\\lambda_a(p) \\leq 0\n\t\\end{equation}\n\tfor every $a \\in A$, with equality iff $a \\in \\F_e(\\nabla f(p))$. Then by continuity we can take $r^*(p)$ small enough so that $\\lambda_a(w) < 0$ for every $a \\in A \\sm (A \\cap \\F_e(\\nabla f(p)))$. Under this condition, if $w_k \\in B_{r^*(p)}(p)$ then the method cannot select a FW direction pointing toward an atom outside the exposed face $\\F_e(\\nabla f(p))$, because all the atoms maximizing the RHS of \\cref{def:gz} must necessarily be in $\\F_e(\\nabla f(p))$. In particular if $w_k \\in  B_{r^*(p)}(p)\\cap \\F_e(\\nabla f(p))$ then the method selects either an in face direction or a FW direction pointing toward a vertex in $\\F_e(\\nabla f(p))$. In both cases, $w_{ k + 1} \\in \\F_e(\\nabla f(p))$. \\\\\n\t\\newcommand{\\rp}[1]{r^{(#1)}(\\delta, p) }\n\t(b) Let $D$ be the diameter of $\\Q$. We now consider $r^{(0)}(\\delta, p) \\leq \\min(\\delta, r^*(p))$ such that, for every $w \\in B_{r^{(0)}(\\delta, p)}(p)$\n\t\\begin{equation} \\label{eq:c1}\n\t\t\\max_{a \\in A} \\lambda_a(w) < \\min_{b \\in A \\sm  \\F_e(\\nabla f(p))}\\min(-\\lambda_b(w), \\frac{\\cC}{D^2} \\lambda_b(w)^2) \t\\, .\t\n\t\\end{equation}\n\tAs we will see in the rest of the proof this upper bound together with \\cref{l:alphakineq} ensures in particular that the FDFW performs maximal in face steps in $B_{r^{(0)}(\\delta, p)}(p) \\sm \\F_e(\\nabla f(p))$.\tFurthermore, \\cref{eq:c1} can always be satisfied thanks to \\cref{eq:lam} and by the continuity of multipliers. We then define recursively for $1\\leq l \\leq n$ a sequence $r^{(l)}(\\delta, p) \\leq \\rp{l - 1}$ of radii small enough so that, for \n\t\\begin{equation} \\label{eq:Ml}\n\t\tM_l = \\sup_{w \\in B_{(l)}(p) \\sm \\F_e(\\nabla f(p))} G(w)\/G_{\\F}(w) \\, ,\t\n\t\\end{equation}\n\twith $B_{(l)}(p) := B_{{r^{(l)}}(\\delta, p)}(p)$ we have\n\t\\begin{equation} \\label{eq:c2}\n\t\tr^{(l)}(\\delta, p) + DM_l < r^{(l - 1)}(\\delta, p) \\, .\n\t\\end{equation}\n\tAgain this sequence can always be defined thanks to the continuity of multipliers. Finally, we define $r(\\delta, p) = \\rp{n}$. \\\\\n\tGiven these definitions, when $w_k \\in B_{(l)}(p) \\subset B_{(0)}(p) $ and $w_k $ is not in $ \\F_e(\\nabla f(p))$ an in face direction is selected, because\n\t\\begin{equation}\n\t\t\\Sc{\\nabla f(w_k)}{d_k^{\\FW}} = \\max_{a \\in A} \\lambda_a(w) < \\min_{b \\in A \\sm  \\F_e(\\nabla f(p))} - \\lambda_b(w) \\leq  \\max_{b \\in \\F(w_k) \\cap A} - \\lambda_b(w_k) = \\Sc{\\nabla f(x_k)}{d_k^{\\FD}} \\, ,\n\t\\end{equation} \n\twhere we used \\cref{eq:c1} in the first inequality, $w_k \\notin \\F_{e}(p)$ in the second, and \\cref{eq:fdl} in the second equality. \n\tWe now want to prove that in this case $\\alpha_k$ is maximal reasoning by contradiction. On the one hand, we have\n\t\\begin{equation} \\label{eq:algl}\n\t\t\\alpha_k \\geq \\cC \\frac{\\Sc{\\nabla f(x_k)}{d_k}}{\\n{d_k}^2} \\geq \\frac{\\cC}{D^2} \\Sc{\\nabla f(x_k)}{d_k} = \\frac{\\cC}{D^2} G_{\\F}(w_k)\n\t\\end{equation}\n\twhere we used the assumption \\cref{deq:alphalb} in the first inequality, $\\n{d_k} \\leq D$ in the second and $G_{\\F}(w_k) =\\Sc{\\nabla f(x_k)}{d_k^{\\FD}} $ together with $d_k = d_k^{\\FD}$ in the last one. \\\\\n\tOn the other hand,  \n\t\\begin{equation}\\label{eq:inegzk}\n\t\t\\begin{aligned}\n\t\t\tG(w_k) = \\max_{a \\in A} \\lambda_a(w_k) < &  \\frac{\\cC}{D^2} \\min_{b \\in A \\sm  \\F_e(\\nabla f(p))} \\lambda_b(w)^2  \\leq  \\frac{\\cC}{D^2} \\max_{b \\in \\F(w_k)} \\lambda_b(w)^2 \\\\ \n\t\t\t= & \\frac{\\cC}{D^2}(\\Sc{\\nabla f(w_k)}{d_k})^2 = \\frac{\\cC}{D^2} G_{\\F}(w_k)^2\n\t\t\\end{aligned}\n\t\\end{equation}\n\twhere we used \\cref{eq:c1} in the first inequality, $w_k \\notin  \\F_e(\\nabla f(p))$ in the second, \\cref{eq:fdl} together with $d_k = d_k^{\\FD}$ in the second equality, and \\cref{eq:gfz} in the third equality. \\\\\n\tThe inequality \\cref{eq:inegzk} leads us to a contradiction with the lower bound on $\\alpha_k$ given by \\cref{eq:algl}, since it implies\n\t\\begin{equation} \\label{eq:alphakup}\n\t\t\\alpha_k \\leq \\frac{G(w_k)}{G_{\\F}(w_k)} < \\frac{\\cC}{D^2} G_{\\F}(w_k) \\, ,\n\t\\end{equation}\n\twhere we applied \\cref{l:alphakineq} in the first inequality and \\cref{eq:inegzk} in the second.\n\t\\\\ \n\tAssume now $w_k \\in B_{(n)}(p)$. We prove by induction that, for every $j \\in [-1 : \\tx{dim}(\\F(w_k)) - 1]$, if $\\{w_{k + i}\\}_{0 \\leq i \\leq  j} \\cap \\F_e(\\nabla f(p)) = \\emptyset$  then $w_{k + j + 1} \\in B_{(n - j - 1)}(p)$. \n\tFor $j = - 1$ we have $w_k \\in B_{(n)}(p)$ by assumption. Now if $\\{w_{k + i}\\}_{0 \\leq i \\leq j} \\cap \\F_e(\\nabla f(p)) = \\emptyset$ we have\n\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t\t&\t\\n{w_{k + j + 1} - p} \\leq \\n{w_{k + j} - p} + \\n{w_{k + j + 1} - w_{k + j}} < \\rp{n - j}  +  \\n{w_{k + j + 1} - w_{k + j}}  \\\\\n\t\t\t= & \\rp{n - j} + \\alpha_k \\n{d_k}\t\\leq   \\rp{n - j} +  D\\frac{G(w_k)}{G_{\\F}(w_k)}\t\\leq   \\rp{n - j} + D M_{n - j} < \t\\rp{n - j - 1} \\, ,\n\t\t\\end{aligned}\n\t\\end{equation}\n\twhere we used the inductive hypothesis $w_{k + j} \\in B_{(n - j)}(p)$ in the second inequality, \\cref{l:alphakineq} in the third inequality,  \\cref{eq:Ml} in the fourth inequality and the assumption \\cref{eq:c2} in the last one.  In particular $w_{k + j + 1} \\in B_{(n - j - 1)}(p)$ and the induction is completed. \\\\\n\tSince $B_{(n - j)}(p) \\subset B_{(0)}(p)$, if  $w_{k + j} \\in (B_{(n - j)}(p) \\sm \\F_e(\\nabla f(p))$ then $\\alpha_{k + j}$ must be maximal and therefore $\\tx{dim}(\\F(w_{k + j + 1})) < \\tx{dim}(\\F(w_{k + j}))$. But starting from the index $k$ the dimension of the current face can decrease at most $\\tx{dim}(\\F(w_{k})) < n$ times in consecutive steps, so there must exists $j \\in [0, \\tx{dim}(\\F(w_k))]$ such that $w_{k + j} \\in \\F_{e}(\\nabla f(p))$. Taking the minimum $j$ satisfying this condition we also obtain $w_{k + j} \\in B_{(0)}(p) \\subset B_{\\delta}(p)$.\n\\end{proof}\n\\newcommand{\\rs}{r^{(s)}}\nA straightforward adaptation of results from \\cite{bomze2020active} implies convergence to the set of stationary points for the FDFW.\n\\begin{prop} \\label{p:fdfwconv}\t\t\n\tIf \\cref{deq:alphalb} and \\cref{eq:deltaflb} hold, then all the limit points of the FDFW are contained in the set of stationary points of $f$.\n\\end{prop}\n\\begin{proof}\n\tThe proof presented in the special case of the simplex in \\cite{bomze2020active}, where the FDFW coincides with the away-step Frank-Wolfe, extends to generic polytopes in a straightforward way. \n\\end{proof}\n\\newcommand{\\tr}{\\tilde{r}(p)}\nIn the next lemma we improve the FDFW local identification result given in \\cref{p:FDFWli} under an additional strong concavity assumption for the face containing the solution, satisfied in particular by $h_{\\G}$.\n\\begin{lem} \\label{eq:scv} \n\tLet $p$ be a stationary point for $f$ restricted to $\\Q$. Assume that \\cref{deq:alphalb} holds and that $f$ is strongly concave\\footnote{in fact, we only need strict concavity of $f$ here.} in $\\F_e(\\nabla f(p))$. Then, for a neighborhood $U(p)$ of $p$, if $w_0 \\in U(p)$:\n\t\\begin{itemize}\n\t\t\\item[(a)] if $\\{f(w_k)\\}$ is increasing, there exists $k \\in [0\\! :\\! \\tx{dim}(\\F(w_0))]$ such that $w_{k + i} \\in \\F_e(\\nabla f(p))$ for every $i \\geq 0$;\n\t\t\\item[(b)] if in addition also \\cref{eq:deltaflb} holds, then  $\\{w_{k + i}\\}_{i \\geq 0}$ converges to $p$. \t\t\n\t\\end{itemize} \n\\end{lem}\n\\begin{proof}\n\t(a)\tLet $\\mu$ be the strong concavity constant of $f$ restricted to $\\F_e(\\nabla f(p))$, so that \n\t\\begin{equation}\\label{eq:Ff}\n\t\tf(w) \\leq f(p) - \\frac{\\mu}{2}\\n{w - p}^2\n\t\\end{equation}\n\tfor every $w \\in \\F_e(\\nabla f(p))$. For $\\varepsilon = \\frac{\\mu r^{*}(p)^2}{2}$, let $\\mathcal{L}_{\\varepsilon}$ be the superlevel of $f$ for $f(p) - \\varepsilon$:\n\t\\begin{equation}\n\t\t\\mathcal{L}_{\\varepsilon} = \\{y \\in \\Q \\ | \\ f(y) > f(p) - \\varepsilon\\} \\, .\n\t\\end{equation}\n\tLet now $\\bar{r} = r(\\delta, p)$ defined as in \\cref{p:FDFWli}, with $\\delta = r^*(p)$. By \\cref{eq:Ff} it follows $\\mathcal{L}_{\\varepsilon} \\cap \\F_e(\\nabla f(p)) \\subset B_{r^*(p)}(p)$. Assume now $w_0 \\in U(p)$ with $U(p) =B_{\\bar{r}}(p) \\cap \\mathcal{L}_{\\varepsilon}$. By applying \\cref{p:FDFWli} we obtain that there exists $k \\in [0\\! : \\!\\tx{dim}(\\F(w_0))]$ such that $w_{k}$ is in $ \\F_e(\\nabla f(p))\\cap B_{r^*(p)}(p)$. But since $f(w_k) \\geq f(w_0) > f(p) - \\varepsilon$ we have the stronger condition $w_k \\in \\mathcal{L}_{\\varepsilon} \\cap  \\F_e(\\nabla f(p))$.To conclude, notice that the sequence cannot escape from this set, because for $i \\geq 0$ $w_{k + i} \\in \\mathcal{L}_{\\varepsilon}$ implies that also $w_{k + i + 1}$ is in $\\mathcal{L}_{\\varepsilon}$, and $w_{k + i} \\in \\mathcal{L}_{\\varepsilon} \\cap  \\F_e(\\nabla f(p)) \\subset B_{r^*(p)}(p)\\cap  \\F_e(\\nabla f(p)) $ implies that also $w_{k + i + 1}$ is in $ \\F_e(\\nabla f(p))$.    \\\\\n\t(b) By point (a) $\\{w_{k + i}\\}_{i \\geq 0}$ is contained in $ \\F_e(\\nabla f(p))$.  But by assumption $f$ is strongly concave in $ \\F_e(\\nabla f(p))$ with $p$ global maximum and the only stationary point. To conclude it suffices to apply \\cref{p:fdfwconv}.\n\\end{proof}\n\\begin{cor} \\label{cor:fwglob}\n\tLet $\\{w_k\\}$ be a sequence generated by the FDFW and assume that at least one limit point $p$ is stationary and such that $f$ is strongly concave in $\\F_e(\\nabla f(p))$. Then under the conditions \\cref{deq:alphalb} and \\cref{eq:deltaflb} on the stepsize we have $w_k \\rightarrow p $ with $w_k \\in \\F_e(\\nabla f(p))$ for $k$ large enough.\n\\end{cor}\n\\begin{proof}\n\tFollows from \\cref{eq:scv} by observing that the sequence must be ultimately contained in $U(p)$. \n\\end{proof}\nWe can now prove local convergence and identification for the FDFW applied to the $s - $defective maximal clique formulation \\cref{deq:P}.\n\\begin{prop}[FDFW local convergence] \\label{p:fdfwlocal}\n\tLet $p = (x^{(C)}, y^{(p)}) \\in \\M_s(\\G)$, let $\\{z_k\\}$ be a sequence generated by the FDFW. Then under \\cref{deq:alphalb} there exists a neighborhood $U(p)$ of $p$ such that if $\\bar{k} := \\min \\{k \\in \\N_{0} \\ | \\ z_k \\in U(p) \\}$ we have the following properties:\n\t\\begin{itemize}\n\t\t\\item[(a)] if $h_{\\G}(z_k)$ is monotonically increasing, then $\\supp(z_{k}) = C$ and $y_k = y^{(p)}$ for every $k \\geq \\bar{k} + \\dim \\F(w_{\\bar{k}})$;\n\t\t\\item[(b)]  if \\cref{eq:deltaflb} also holds then $z_{k} \\rightarrow p$.\n\t\\end{itemize}\n\\end{prop}\n\\begin{proof}\n\tLet $A(p) = A_{\\G} + A(y^{(p)})$. Then for $x \\in \\Delta^{(C)}$\n\t\\begin{equation} \\label{eq:cdelta}\n\t\t\\begin{aligned}\n\t\t\t& x^{\\intercal}A(p)x = \\sum_{(i, j) \\in V^2} x_i A(p)_{ij} x_j = \\sum_{i \\in C} x_i (\\sum_{j \\in C} A(p)_{ij} x_j)  =  \\sum_{i \\in C} x_i (\\sum_{j \\in C \\sm \\{i\\} } x_j) \\\\\n\t\t\t= & \\sum_{i \\in C} (x_i \\sum_{j \\in C} x_j - x_i^2) =  (\\sum_{i \\in C} x_i)^2 - \\sum_{i \\in C} x_i^2 \\, ,\n\t\t\\end{aligned}\n\t\\end{equation}\n\twhere in the first equality we used $\\tx{supp}(x) = C$, in the second that $C$ is a clique n $G(y^{(p)})$, and $\\sum_{i \\in C}x_i = \\sum_{i \\in V} x_i = 1$. \\\\\n\tObserve now that the function $x \\mapsto h_{\\G}(x, y^{(p)})$ is strongly concave in $\\Delta^{(C)}$. Indeed for $x \\in \\Delta^{(C)}$ \n\t\\begin{equation} \\label{eq:sc}\n\t\t\\begin{aligned}\n\t\t\th_{\\G}(x, y^{(p)}) = x^{\\intercal}A(p)x + \\frac{\\alpha}{2}\\n{x}^2 + \\frac{\\beta}{2}\\n{y^{(p)}}^2  = & (\\sum_{i \\in C} x_i)^2 - \\sum_{i \\in C} x_i^2 +  \\frac{\\alpha}{2}\\n{x}^2 + \\frac{\\beta}{2}\\n{y^{(p)}}^2 \\\\\n\t\t\t= & 1 - (1 - \\frac{\\alpha}{2})\\sum_{i \\in C} x_i^2 +  \\frac{\\beta}{2}\\n{y^{(p)}}^2 \\, ,\t\n\t\t\\end{aligned}\n\t\\end{equation}\n\twhere in the second equality we used \\cref{eq:cdelta}. The RHS of \\cref{eq:sc} is strongly concave in $x$ since $\\alpha \\in (0, 2)$ so that $- (1 - \\alpha\/2) \\in (- 1, 0)$. This together with \\cref{l:sc} gives us the necessary assumptions to apply \\cref{eq:scv}. \n\\end{proof}\nAs a corollary, we have the following global convergence result under the mild assumption that the set of limit points contains one local minimizer.\n\\begin{cor}[FDFW global convergence]\n\tLet $\\{z_k\\}$ be a sequence generated by the FDFW and assume that at least one limit point $p = (x^{(C)}, y^{(p)})$ of $\\{z_k\\}$ is in $\\M_s(\\G)$. Then under the conditions \\cref{deq:alphalb} and \\cref{eq:deltaflb} on the stepsize we have $z_k \\rightarrow p $ with $\\supp(x_k) \\subset C$ and $y_k = y_p$ for $k$ large enough.\n\\end{cor}\n\\begin{proof}\n\tFollows from \\cref{cor:fwglob} where all the necessary assumptions are satisfied as for \\cref{p:fdfwlocal}.\n\\end{proof}\n\n\\begin{rem}\n\tWhen the sequence converges to a first order stationary point $z^* = (x^*, y^*)$ which is not a local maximizer, one can use the procedure described in \\cite{stozhkovcontinuous} to obtain an $s$-defective clique $C$ and $y \\in D_s(\\G)$ with $h_{\\G}((x^{(C)}, y)) > h(z^*)$. The cost of the procedure is $O(|\\tx{supp}(x^*)|^2)$.\n\\end{rem}\n\\section{A Frank-Wolfe variant for $s$-defective clique}\\label{Sec:FWvar}\nAs can be seen from numerical results, one drawback of the standard FDFW applied to the $s$-defective clique  formulation \\cref{deq:P} is the slow convergence of the high dimensional $y$ component. Since this component is \"tied\" to the $x$ component, it is not possible to speed up the convergence by changing the regularization term without compromising the quality of the solution.  Motivated by this challenge, we introduce a tailored Frank-Wolfe variant, namely FWdc,  for the maximum $s$-defective clique formulation \\cref{deq:P}, which exploits the product domain structure of the problem at hand by employing separate updating rules for the two blocks.   \\\\\n\\setcounter{ALC@unique}{0}\t\n\\begin{algorithm}\n\t\\caption{Frank-Wolfe variant for $s$-defective clique}\n\t\\label{alg:BCFW}\n\t\\begin{algorithmic}[1]\n\t\t\\STATE{\\textbf{Initialize} $z_0:=(x_0, y_0) \\in \\p$, $k := 0$}\n\t\t\\IF{$z_k$ is stationary}\n\t\t\\STATE{STOP}\n\t\t\\ENDIF\n\t\t\\STATE{Compute $x_{k + 1}$ applying one iterate of \\cref{alg:FW} with $w_0 = x_k$ and $f(w)=  h_{\\G}(w, y_k)$.} \\label{st:xk1}\n\t\t\\STATE{Let $y_{k+1} \\in \\argmax_{y \\in D'_s(\\G)} \\Sc{\\nabla_y h_{\\G}(x_k, y_k)}{y} $.} \\label{st:yk1}\n\t\t\\STATE{Set $k : = k+1$. Go to step 2.}\n\t\\end{algorithmic}\n\\end{algorithm}\nIn particular, at every iteration the method alternates a FDFW step on the $x$ variables (\\cref{st:xk1}) with a full FW step on the $y$ variable (\\cref{st:yk1}), so that $y_k$ is always chosen in the set of vertices $D_s(\\G)$ of $D'_s(\\G)$. Furthermore, as we prove in the next proposition, $\\{y_k\\}$ is ultimately constant. This allows us to obtain convergence results by applying the general properties of the FDFW proved in the previous section to the $x$ component. \n\\begin{prop} \\label{p:algo2}\n\tIn the FWdc if $h_{\\G}(z_k)$ is increasing then $\\{y_k\\}$ can change at most $\\frac{2}{\\beta} - \\frac{2 - \\alpha}{\\beta |C^*|} + s$ times, with $C^*$ $s$-defective clique of maximal cardinality.  \n\\end{prop}\n\\begin{proof}\n\tAssume that $y_k$ and $y_{k + 1}$ are distinct vertices of $D_s'(\\G)$. Then \n\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t\t&\th_{\\G}(z_{k + 1}) - h_{\\G}(z_k) \\geq \\Sc{\\nabla h_{\\G}(z_k)}{(z_{k + 1} - z_k)} + \\frac{\\beta}{2}\\n{z_{k + 1} - z_k}^2 \\\\  = &\\Sc{\\nabla_y h_{\\G}(z_k)}{(y_{k + 1} - y_k)} + \\frac{\\beta}{2} \\n{y_{k + 1} - y_k}^2 \\geq \\frac{\\beta}{2} > 0 \\,  \t\n\t\t\\end{aligned}\n\t\\end{equation}\n\twhere we used the $\\beta - $strong convexity of $y \\mapsto h_{\\G}(x , y)$ in the first inequality, $x_k = x_{k + 1}$ in the equality, $y_{k + 1} \\in \\argmax_{y \\in \\p} \\Sc{\\nabla_y h_{\\G}(z_k)}{y}$ and the fact that the distance between vertices of $D_s'(\\G)$ is at least 1 in the second inequality. \\\\\n\tTherefore $y_k$ can change at most\n\t\\begin{equation*}\n\t\t\\max_{z \\in \\p} \\frac{2(h_{\\G}(z) - h_{\\G}(z_0))}{\\beta} \\leq \\max_{z \\in \\p} \\frac{2h_{\\G}(z)}{\\beta} = \\frac{1 - 1\/|C^*| + \\alpha\/2|C^*| + s\\beta\/2 }{\\beta\/2} = \\frac{2}{\\beta} + \\frac{\\alpha - 2}{\\beta|C^*|} + s\n\t\\end{equation*}\n\ttimes, where we used $h_{\\G} \\geq 0$ in the first inequality, and \\cref{c:globalm} in the second inequality. \n\\end{proof}\n\\begin{cor} \\label{cor:algo2conv}\n\tLet $\\{z_k\\}$ be a sequence generated by Algorithm 2. \n\t\\begin{itemize}\n\t\t\\item[1.]  If conditions \\cref{deq:alphalb} and \\cref{eq:deltaflb} hold on the stepsizes, then $\\{z_k\\}$ converges to the set of stationary points.\n\t\t\\item[2. ]If the stepsize is given by exact line search or Armijo line search and the set of limit points of $\\{z_k\\}$ is finite, then $z_k \\rightarrow p$ with $p$ stationary. \n\t\\end{itemize} \n\\end{cor}\n\\begin{proof}\n\tAs a corollary of \\cref{p:algo2}, an application of Algorithm 2 reduces, after a finite number of changes for the variable $y$, to an application of the FDFW on the simplex for the optimization of a quadratic objective. After noticing that on the simplex the FDFW coincides with the AFW, point 1 follows directly from \\cref{p:fdfwconv}, and point 2 follows from \\cite[Theorem 4.5]{bomze2020active}. \n\\end{proof}\nFor a clique $C$ of $\\G(y)$ different from $\\G$ we define $m(C, \\G(y))$ as \n\\begin{equation}\n\t\\min_{v \\in V \\sm C} |C| - |E^y(v) \\cap C| \\, ,\n\\end{equation}\nthat is the minimum number of edges needed to increase by 1 the size of the clique. \\\\\nWe now give an explicit bound on how close the sequence $\\{x_k\\}$ generated by \\cref{alg:BCFW} must be to $x^{(C)}$ for the identification to happen.\n\\newcommand{\\lmax}{\\delta_{\\max}}\n\\begin{prop}\n\tLet $\\{z_k\\}$ be a sequence generated by  \\cref{alg:BCFW}, $\\bar{y} \\in D^s(\\G) $, $C$ be a clique in $\\G(\\bar{y})$,  let $\\lmax$ the maximum eigenvalue of the adjacency matrix $\\bar{A} : = A_{\\G} + A(\\bar{y})$.\n\tLet $\\bar{k}$ be a fixed index in $\\N_0$, $I^c$ the components of $\\tx{supp}(x_{\\bar{k}})$ with index not in $C$ and let $L := 2 \\delta_{\\max} + \\alpha$. Assume that $y_{\\bar{k} + j} = \\bar{y} $ is constant for $0 \\leq j \\leq |I^c|$, that \\cref{deq:alphalb} holds for $\\cC = 1\/L$, and that \n\t\\begin{equation} \\label{eq:sdc}\n\t\t\\n{x_{\\bar{k}} - x^{(C)}}_1 \\leq  \\frac{m_{\\alpha}(C, \\G(y_{\\bar{k}}))}{m_{\\alpha}(C,  \\G(y_{\\bar{k}})) + 2|C|\\lmax + |C|\\alpha }\n\t\\end{equation}\n\tfor $m_{\\alpha}(C,  \\G(y_{\\bar{k}})) = m(C,  \\G(y_{\\bar{k}})) - 1 + \\alpha\/2$.\n\tThen $\\tx{supp}(x_{\\bar{k} + |I^c|})  = C$.\n\\end{prop}\n\\begin{proof}\n\tSince $y_k$ does not change for $k \\in [\\bar{k}\\! :\\! \\bar{k} + |I^c|]$, Algorithm 2 corresponds to an application of the AFW to the simplex $\\Delta_{n - 1}$ on the variable $x$. For $1 \\leq i \\leq n$ let $\\lambda_{i}(x) = \\frac{\\partial}{\\partial x_i} h_{\\G}(x, y_{\\bar{k}})$ be the multiplier functions associated to the vertices of the simplex, and let \n\t\\begin{equation}\n\t\t\\lambda_{\\min} = \\min_{i \\in V \\sm C} - \\lambda_i(x^{(C)}) \\, ,\n\t\\end{equation}\n\tbe the smallest negative multiplier with corresponding index not in $C$. Let $L'$ be a Lipschitz constant for $\\nabla_x h_{\\G}(x, y)$ with respect to the variable $x$. By \\cite[Theorem 3.3]{bomze2020active} if \n\t\\begin{equation} \\label{deq:bomze}\n\t\t\\n{x_{\\bar{k}} - x^{(C)}}_1 < \\frac{\\lambda_{\\min}}{\\lambda_{\\min} + 2L'} \n\t\\end{equation}\n\twe have the desired identification result. \\\\\t\t\n\tWe now prove that we can take $L'$ equal to $L$ in the following way:\n\t\\begin{equation}\n\t\t\\n{\\nabla_x h_{\\G}(x', y_{\\bar{k}}) - \\nabla_x h_{\\G}(x, y_{\\bar{k}})} = \\n{2\\bar{A}(x' - x) + \\alpha(x' - x)} \\leq (2 \\lmax + \\alpha) \\n{x' - x} \\, ,\n\t\\end{equation}\n\twhere we used $\\nabla_x h_{\\G}(x, y) = 2\\bar{A}x + \\alpha x$ in the equality. As for the multipliers, for $i \\in V \\sm C$ we have the lower bound\n\t\\begin{equation} \\label{lambdaineq}\n\t\t\\begin{aligned}\n\t\t\t-\t\\lambda_i(x^{(C)}) = \\Sc{\\nabla_x h_{\\G}(x^{(C)}, y_{\\bar{k}})}{(x^{(C)} - e_i)}  = & \\frac{- 2|C\\cap E^{y_{\\bar{k}}}(i)| + 2|C| -  2 + \\alpha}{|C|} \\\\\n\t\t\t\\geq & \\frac{2m_{\\alpha}(C, \\G(y_{\\bar{k}}))}{|C|} \n\t\t\\end{aligned}\n\t\\end{equation}\n\tby combining  \\cref{eq:x1piece} and \\cref{eq:x2piece} in the second equation. We can now bound $\\lambda_{\\min}$ from below:\n\t\\begin{equation} \\label{deq:lm}\n\t\t\\lambda_{\\min} = \\min_{i \\in V \\sm C} - \\lambda_i(x^{(C)}) \\geq \\min_{i \\in V \\sm C}\\frac{2|C| - 2|C\\cap E^y_{\\bar{k}}(i)| - 2 + \\alpha}{|C|} \\geq  \\frac{2m_{\\alpha}(C, \\G(y_{\\bar{k}}))}{|C|} \\, ,\n\t\\end{equation}\n\twhere we applied \\cref{lambdaineq} in the inequality. \n\tFinally, we have\n\t\\begin{equation} \\label{deq:ri}\n\t\t\\frac{\\lambda_{\\min}}{\\lambda_{\\min} + 2L} \\leq  \\frac{m_{\\alpha}(C, \\G(y_{\\bar{k}}))}{m_{\\alpha}(C, \\G(y_{\\bar{k}})) + 2|C|\\lmax + |C|\\alpha} \\, \n\t\\end{equation}\n\twhere we applied \\cref{lambdaineq} together with \\cref{deq:lm} in the inequality. The thesis follows applying \\cref{deq:ri} to the RHS of \\cref{deq:bomze}.\n\\end{proof}\n\\begin{rem} \\label{r:dmax}\n\tIt is a well known result that for any graph the maximal eigenvalue $\\delta_{\\max}$  of the adjacency matrix is less than or equal to $d_{\\max}$, the maximum degree of a node (see, e.g., \\cite{cvetkovic1990largest}). Then condition \\cref{eq:sdc} can be replaced by\n\t\\begin{equation}\n\t\t\\n{x_{\\bar{k}} - x^{(C)}}_1 \\leq  \\frac{m_{\\alpha}(C, \\G(y_{\\bar{k}}))}{m_{\\alpha}(C, \\G(y_{\\bar{k}})) + 2|C|d_{\\max} + |C|\\alpha} \\, .\n\t\\end{equation}\n\\end{rem}\n\\section{Numerical results}\\label{Sec:NumRes}\nIn this section we report on a numerical comparison of the methods. We remark that, even though these methods only find maximal $s$-defective cliques, they can still be applied as a heuristic to derive lower bounds on the maximum $s$-defective clique within a global optimization scheme.\t\nWith our tests, we aim to achieve the followings:\n\\begin{itemize}\n\t\\item empirically verify the active set identification property of the proposed methods;\n\t\\item prove that the proposed FW variant  is faster than the FDFW on these problems, while mantaining the same solution quality;\n\t\\item make a preliminary comparison between the FW methods and the CONOPT solver used in \\cite{stozhkovcontinuous}. \n\\end{itemize}\n\nIn the tests, the regularization parameters were set to $\\alpha = 1$ and $\\beta = 2\/n^2$. An intuitive motivation for this choice of $\\beta$ can be given by imposing that the missing edges for an identified $s$-defective clique are always included in the support of the  FW vertex. Formally, if $x_k = x^{(C)}$ with $C$ an $s$-defective clique and $(y_k)_{ij} = 0$ with $\\{i, j\\} \\in {C \\choose 2}$ we want to ensure that the FW vertex $s_k = (x^{(s_k)}, y^{(s_k)})$ is such that $y^{(s_k)}_{ij} = 1$. Now for $\\{l, m\\} \\notin {C \\choose 2}$ and assuming $|C| < n$ (otherwise $C = V$ and the problem is trivial) we have\n\\begin{equation}\\label{ppyh}\n\t\\frac{\\partial}{\\partial y_{ij}} h_{\\G}(x_k, y_k) =\t\\frac{2}{|C|^2} > \\frac{2}{n^2} = \\beta \\geq  \\frac{\\partial}{\\partial y_{lm}} h_{\\G}(x_k, y_k)\n\\end{equation}\nwhere the first equality and the last inequality easily follow from \\cref{eq:dph}. From \\cref{ppyh} it is then immediate to conclude that $\\{i, j\\}$ must be in the support of $y^{(s_k)}$. \\\\\nWe used the stepsize $\\alpha_{k} = \\bar{\\alpha}_k$  with $\\bar{\\alpha}_k$ given by \\cref{deq:alphalb} for $\\cC= 2$, corresponding to an estimate of $0.5$ for the Lipschitz constant $L$ of $\\nabla h_{\\G}$. A gradient recycling scheme was adopted to use first order information more efficiently (see \\cite{rinaldi2020avoiding} for details).  The code was written in MATLAB and the tests were performed on an Intel Core i7-10750H CPU 2.60GHz, 16GB RAM. \\\\\nThe 50 graph instances we used in the tests are taken from the Second DIMACS Implementation Challenge \\cite{johnson1993cliques}. These graphs are a common benchmark to assess the performance of algorithms for maximum (defective) clique problems (see references in \\cite{stozhkovcontinuous}), and the particular instances we selected coincide with the ones employed in \\cite{stozhkovcontinuous} in order to ensure a fair comparison at least for the quality of the solutions. Following the rule adopted in \\cite{stozhkovcontinuous}, for each triple $(\\G, s, \\mathcal{A})$ with $\\G$ a graph from the 50 instances considered, $s \\in [1\\! : \\! 4]$, $\\mathcal{A}$ the FDFW or the FWdc, we set a global time limit of 600 seconds and employed a simple restarting scheme with up to 100 random starting points. The algorithms always completed 100 runs within the time limit, with the exception of 2 instances (see \\cref{tab:2}). For both algorithms the $x$ component of the starting point was generated with MATLAB's function rand and then normalized dividing it by its sum. An analogous rule was applied to generate the $y$ component for the starting point of the FDFW, while for the FWdc the $y$ component was simply initialized to 0. For the stopping criterion, two conditions are required: the current support of the $x$ components coincides with an $s$-defective clique, and the FW gap is less than or equal to $\\varepsilon:= 10^{-3}$.\nIn the experiments, both algorithms always terminated having identified an $s$-defective clique, thus providing an empirical verification of the results we proved in this paper.  \\\\\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.6\\textwidth]{FDFWvsFWdcmc}\n\t\\caption{$\\mathcal{A} i$ is the box plot of the maximum clique found within the 600 seconds\/ 100 runs limit for each instance by the method $\\mathcal{A}$ for $s = i$ divided by the maximum cardinality clique of the instance. }\n\t\\label{fig:1}\n\\end{figure}\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.6\\textwidth]{FDFWvsFWdcT}\n\t\\caption{$\\mathcal{A} i$ is the box plot of the average running time for each instance for the method $\\mathcal{A}$ and $s = i$.}\n\t\\label{fig:2}\n\\end{figure}\n\nIn the boxplots, each series consists of 50 values corresponding to aggregate data for the runs performed on the 50 instances. The data for the CONOPT solver are taken from \\cite{stozhkovcontinuous}. The red lines represent the median of the values in each series, and the boxes extend from the 25th percentile $q_1$  of the observed data to the 75th percentile $q_3$. The whiskers cover all the other values in a range of $[q_1 - w (q_3 - q_1), q_3 + w(q_3 - q_1)]$, with the coefficient $w$ equal to $2.7$ times the standard deviation of the values. \\\\\nIn \\cref{fig:1}, the bar $\\mathcal{A}i$ represents the distribution of the maximum cardinality of the $s$-defective clique found by method $\\mathcal{A}$ with $s = i$, divided by the maximum clique cardinality of the instance. Notice that some data points are greater than 1, as expected since for $s > 0$ the cardinality of an $s$-defective clique can exceed the maximum clique cardinality. While the variance is higher for the max cliques found by the CONOPT solver, no significant differences can be seen for the median size of max cliques. \\\\\nIn \\cref{fig:2}, $\\mathcal{A}i$ represents the distribution of average running times in seconds (on a logarithmic scale, explainng the asymmetry of the box plots) of method $\\mathcal{A}$ for $s = i$. Here we can see that FWdc outperforms FDFW by about 1.5 orders of magnitude, which in turns outperforms the CONOPT solver by about 1 order of magnitude. This is even more impressive if we take into account the fact that the CONOPT solver is written in C++, while FW variants are written in MATLAB. \\\\\n\n\n\n\\section{Appendix}\n\\subsection{Line searches} \\label{sa:ls}\nHere we briefly report some relevant results about line searches proved in \\cite{bomze2020active} showing a connection between well known line searches and conditions \\cref{deq:alphalb}, \\cref{eq:deltaflb}. \\\\\nRecall that the exact line search stepsize is given by\n\\begin{equation} \\label{eq:ls}\n\t\\alpha_k \\in \\argmax_{\\alpha \\in [0, \\alpha_{k}^{\\max}]} f(w_k + \\alpha d_k) \\, .\n\\end{equation}\nThe stronger condition $\\alpha_k = \\max \\{ \\argmax_{\\alpha \\in [0, \\alpha_{k}^{\\max}]} f(w_k + \\alpha d_k) \\}$\tis required for \\cite[Theorem 4.3]{bomze2019first}, used to prove \\cref{cor:algo2conv},   \\\\\nThe stepsize $\\alpha_k$ given by the Armijo line search always satisfies the condition\n\\begin{equation} \\label{Armijo}\n\tf(w_k + \\alpha_k d_k) - f(w_k) \\geq c_1 \\alpha_k \\Sc{\\nabla f(w_k)}{d_k} \\, ,\n\\end{equation}\nfor some constant $c_1 \\in (0, 1)$. This stepsize is produced by considering a sequence $\\{\\beta_k^{(j)}\\}_{j \\in \\mathbb{N}_0}$ of tentative stepsizes given by $\\beta_k^{(0)} = \\alpha_{k}^{\\max}$, $\\beta_{k}^{(j + 1)} = \\gamma \\beta_{k}^{(j)}$, with $\\gamma \\in (0, 1)$, and taking the largest tentative stepsize satisfying \\cref{Armijo}. \\\\ \nWe report here for completeness results from \\cite{bomze2020active} proving that Armijo and exact line search satisfy conditions \\cref{deq:alphalb} and \\cref{eq:deltaflb}.\n\\begin{lem}\\label{alphacond}\n\tConsider a sequence $\\{w_k\\}$ in $\\Q$ such that $w_{k  + 1} = w_{k} + \\alpha_k d_k$ with $\\alpha_k \\in \\R_{\\geq 0}$, $d_k\\in \\R^n$. Assume that $d_k$ is a proper ascent direction in $w_k$, i.e. $\\Sc{\\nabla f(w_k)}{d_k} > 0$. \n\t\\begin{itemize}\n\t\t\\item[1.] If $\\alpha_k$ is given by exact line search, then \\cref{deq:alphalb} and \\cref{eq:deltaflb} are satisfied with $\\cC = \\frac{1}{L}$ and $\\rho = \\frac{1}{2}$. \n\t\t\\item[2.] If $\\alpha_k$ is given by the Armijo line search described above, then \\cref{deq:alphalb}  and \\cref{eq:deltaflb} are satisfied with $\\cC= \\frac{2\\gamma(1 - c_1)}{L}$ and $\\rho = c_1\\min\\{1,2\\gamma (1- c_1)\\}<1.$ \n\t\\end{itemize}\n\\end{lem}\n\\begin{proof}\n\tPoint 1 follows from \\cite[Lemma B.1]{bomze2020active} and point 2 follows from \\cite[Lemma B.3]{bomze2020active}.\n\\end{proof}\n\\clearpage\n\\subsection{Detailed numerical results} \\label{s:dnr}\nWe report in this section detailed numerical results for each of the 50 graphs we used in our tests.\n\\begin{table}[tbph]\n\t{\\footnotesize\n\t\t\\caption{Clique sizes for the FDFW}\n\t\t\\resizebox{\\textwidth}{!}{\\begin{tabular}{|l|SSS|SSS|SSS|SSS|}\n\t\t\t\t\\hline\n\t\t\t\tGraph          &     & {$s=1$}  &       &     & {$s=2$}  &       &     & {$s =3$}  &       &     & {$s =4$}  &       \\\\ \\cline{2-13} \n\t\t\t\t& \\multicolumn{1}{c}{Max} & \\multicolumn{1}{c}{Mean} & \\multicolumn{1}{c|}{Std} & \\multicolumn{1}{c}{Max} & \\multicolumn{1}{c}{Mean} & \\multicolumn{1}{c|}{Std} & \\multicolumn{1}{c}{Max} & \\multicolumn{1}{c}{Mean} & \\multicolumn{1}{c|}{Std} & \\multicolumn{1}{c}{Max} & \\multicolumn{1}{c}{Mean} & \\multicolumn{1}{c|}{Std} \\\\ \\hline \n\t\t\t\tbrock200\\_1    & 19   & 15.6  & 1.3  & 21   & 17.0  & 1.4  & 22   & 17.9  & 1.2  & 21   & 18.6  & 1.3  \\\\\n\t\t\t\tbrock200\\_2    & 10   & 8.0   & 0.9  & 11   & 9.0   & 0.9  & 11   & 9.5   & 0.8  & 12   & 9.5   & 1.0  \\\\\n\t\t\t\tbrock200\\_3    & 13   & 10.2  & 1.1  & 14   & 11.5  & 1.0  & 14   & 12.2  & 0.9  & 14   & 12.5  & 0.9  \\\\\n\t\t\t\tbrock200\\_4    & 15   & 11.9  & 1.2  & 16   & 13.2  & 1.1  & 17   & 14.0  & 1.1  & 17   & 14.3  & 1.3  \\\\\n\t\t\t\tbrock400\\_1    & 21   & 17.5  & 1.4  & 22   & 18.8  & 1.3  & 23   & 20.0  & 1.3  & 24   & 20.9  & 1.2  \\\\\n\t\t\t\tbrock400\\_2    & 21   & 17.8  & 1.4  & 22   & 18.9  & 1.2  & 23   & 20.0  & 1.4  & 24   & 20.9  & 1.1  \\\\\n\t\t\t\tbrock400\\_3    & 21   & 17.8  & 1.5  & 22   & 19.0  & 1.3  & 23   & 20.2  & 1.2  & 24   & 20.9  & 1.2  \\\\\n\t\t\t\tbrock400\\_4    & 21   & 17.7  & 1.5  & 22   & 18.7  & 1.3  & 23   & 20.0  & 1.3  & 24   & 21.1  & 1.3  \\\\\n\t\t\t\tc-fat200-1     & 12   & 11.5  & 1.1  & 12   & 11.6  & 0.9  & 12   & 11.6  & 0.9  & 12   & 11.7  & 0.8  \\\\\n\t\t\t\tc-fat200-2     & 24   & 20.8  & 3.1  & 24   & 21.9  & 2.0  & 24   & 22.2  & 1.2  & 24   & 22.3  & 0.9  \\\\\n\t\t\t\tc-fat200-5     & 58   & 45.7  & 11.7 & 58   & 54.4  & 5.0  & 58   & 56.2  & 3.0  & 58   & 56.7  & 2.3  \\\\\n\t\t\t\tc-fat500-1     & 14   & 13.6  & 1.1  & 14   & 13.7  & 0.7  & 14   & 13.7  & 0.7  & 14   & 13.7  & 0.7  \\\\\n\t\t\t\tc-fat500-2     & 26   & 25.1  & 2.0  & 26   & 25.4  & 1.4  & 26   & 25.6  & 1.1  & 26   & 25.6  & 0.9  \\\\\n\t\t\t\tc-fat500-5     & 64   & 55.2  & 10.1 & 64   & 59.7  & 5.2  & 64   & 62.2  & 2.6  & 64   & 62.4  & 3.1  \\\\\n\t\t\t\tc-fat500-10    & 126  & 99.8  & 20.8 & 126  & 114.8 & 11.8 & 126  & 123.7 & 4.1  & 126  & 125.2 & 1.0  \\\\\n\t\t\t\thamming6-2     & 32   & 22.6  & 4.6  & 32   & 23.3  & 4.8  & 32   & 16.2  & 5.5  & 32   & 13.7  & 5.6  \\\\\n\t\t\t\thamming6-4     & 4    & 3.7   & 0.5  & 5    & 3.7   & 0.5  & 4    & 3.2   & 0.7  & 4    & 2.7   & 0.9  \\\\\n\t\t\t\thamming8-2     & 121  & 83.3  & 14.5 & 122  & 86.1  & 15.1 & 123  & 87.1  & 15.3 & 123  & 88.1  & 15.2 \\\\\n\t\t\t\thamming8-4     & 14   & 10.5  & 1.2  & 15   & 11.9  & 1.1  & 15   & 12.4  & 1.0  & 15   & 12.7  & 1.0  \\\\\n\t\t\t\thamming10-2 \\tablefootnote{The time limit was reached in 29 runs}      & 454  & 309.6 & 42.1 & 456  & 313.9 & 43.2 & 468  & 315.5 & 44.4 & 468  & 317.5 & 45.3 \\\\\n\t\t\t\thamming10-4    & 31   & 27.8  & 1.3  & 32   & 29.1  & 1.3  & 33   & 30.3  & 1.2  & 34   & 31.4  & 1.3  \\\\\n\t\t\t\tjohnson8-2-4   & 4    & 2.4   & 1.1  & 4    & 2.2   & 1.2  & 4    & 2.1   & 1.2  & 4    & 2.0   & 1.2  \\\\\n\t\t\t\tjohnson8-4-4   & 14   & 9.5   & 1.6  & 14   & 9.9   & 1.6  & 14   & 9.9   & 1.8  & 14   & 7.6   & 2.3  \\\\\n\t\t\t\tjohnson16-2-4  & 8    & 7.8   & 0.4  & 9    & 8.5   & 0.6  & 9    & 8.4   & 0.6  & 9    & 8.3   & 0.6  \\\\\n\t\t\t\tjohnson32-2-4  & 16   & 14.9  & 0.7  & 17   & 15.9  & 0.8  & 17   & 16.4  & 0.7  & 18   & 17.3  & 0.8  \\\\\n\t\t\t\tkeller4        & 11   & 8.3   & 0.9  & 12   & 9.5   & 0.7  & 12   & 10.0  & 1.0  & 13   & 9.9   & 0.9  \\\\\n\t\t\t\tkeller5        & 20   & 17.2  & 1.1  & 20   & 18.3  & 1.0  & 21   & 19.4  & 0.9  & 23   & 20.4  & 1.0  \\\\\n\t\t\t\tMANN\\_a9       & 16   & 14.6  & 1.0  & 17   & 11.1  & 2.9  & 17   & 9.4   & 3.1  & 17   & 8.0   & 2.9  \\\\\n\t\t\t\tMANN\\_a27      & 118  & 117.6 & 0.5  & 119  & 118.6 & 0.6  & 120  & 119.5 & 0.7  & 121  & 120.4 & 0.7  \\\\\n\t\t\t\tMANN\\_a45 \\tablefootnote{\\label{nrun2} The time limit was reached in 13 runs}     & 331  & 330.5 & 0.5  & 332  & 331.5 & 0.7  & 333  & 332.4 & 0.8  & 334  & 333.3 & 0.6  \\\\\n\t\t\t\tp\\_hat300-1    & 8    & 6.4   & 0.8  & 9    & 7.3   & 0.7  & 9    & 7.7   & 0.7  & 9    & 7.9   & 0.8  \\\\\n\t\t\t\tp\\_hat300-2    & 25   & 20.8  & 1.6  & 24   & 21.6  & 1.4  & 26   & 22.3  & 1.6  & 27   & 22.6  & 1.8  \\\\\n\t\t\t\tp\\_hat300-3    & 33   & 28.9  & 1.7  & 34   & 30.1  & 1.9  & 36   & 30.9  & 1.9  & 36   & 32.0  & 1.9  \\\\\n\t\t\t\tp\\_hat500-1    & 9    & 7.2   & 0.9  & 10   & 8.1   & 0.8  & 10   & 8.7   & 0.8  & 11   & 9.2   & 0.8  \\\\\n\t\t\t\tp\\_hat500-2    & 35   & 29.7  & 2.1  & 35   & 30.6  & 2.1  & 37   & 31.5  & 2.2  & 37   & 32.1  & 2.2  \\\\\n\t\t\t\tp\\_hat500-3    & 46   & 41.9  & 2.1  & 48   & 42.7  & 2.5  & 49   & 43.6  & 2.4  & 51   & 44.8  & 2.5  \\\\\n\t\t\t\tp\\_hat700-1    & 9    & 7.4   & 0.8  & 11   & 8.4   & 0.8  & 12   & 8.9   & 0.8  & 11   & 9.6   & 0.7  \\\\\n\t\t\t\tp\\_hat700-2    & 41   & 36.5  & 2.1  & 43   & 37.2  & 2.3  & 44   & 38.2  & 2.4  & 44   & 39.1  & 2.1  \\\\\n\t\t\t\tp\\_hat700-3    & 57   & 52.1  & 2.5  & 60   & 53.0  & 2.5  & 61   & 54.0  & 2.3  & 61   & 55.3  & 2.7  \\\\\n\t\t\t\tsan200\\_0.7\\_1 & 17   & 15.6  & 0.5  & 18   & 16.8  & 0.4  & 18   & 17.2  & 0.7  & 19   & 17.3  & 0.8  \\\\\n\t\t\t\tsan200\\_0.7\\_2 & 13   & 12.9  & 0.4  & 14   & 14.0  & 0.2  & 15   & 14.5  & 0.6  & 16   & 15.0  & 0.6  \\\\\n\t\t\t\tsan200\\_0.9\\_1 & 46   & 45.4  & 0.5  & 47   & 46.6  & 0.6  & 48   & 47.6  & 0.7  & 49   & 48.5  & 0.7  \\\\\n\t\t\t\tsan200\\_0.9\\_2 & 39   & 35.3  & 2.2  & 43   & 36.4  & 2.4  & 41   & 37.2  & 2.4  & 42   & 38.0  & 2.6  \\\\\n\t\t\t\tsan200\\_0.9\\_3 & 32   & 28.2  & 2.0  & 33   & 28.8  & 2.2  & 35   & 29.7  & 2.6  & 35   & 30.0  & 2.3  \\\\\n\t\t\t\tsan400\\_0.5\\_1 & 8    & 6.5   & 0.9  & 9    & 9.0   & 0.0  & 10   & 9.9   & 0.3  & 11   & 10.5  & 0.6  \\\\\n\t\t\t\tsan400\\_0.7\\_1 & 22   & 20.6  & 0.7  & 22   & 21.9  & 0.3  & 23   & 23.0  & 0.2  & 24   & 23.9  & 0.2  \\\\\n\t\t\t\tsan400\\_0.7\\_2 & 16   & 15.3  & 0.7  & 17   & 17.0  & 0.2  & 18   & 18.0  & 0.0  & 19   & 18.8  & 0.4  \\\\\n\t\t\t\tsan400\\_0.7\\_3 & 13   & 12.1  & 1.0  & 14   & 13.9  & 0.4  & 15   & 14.9  & 0.2  & 16   & 15.6  & 0.5  \\\\\n\t\t\t\tsanr200\\_0.7   & 16   & 13.1  & 1.2  & 17   & 14.4  & 1.1  & 18   & 15.5  & 1.1  & 18   & 15.9  & 1.2  \\\\\n\t\t\t\tsanr200\\_0.9   & 37   & 32.4  & 2.0  & 39   & 33.7  & 2.1  & 40   & 34.7  & 2.2  & 40   & 35.7  & 2.0 \\\\ \\hline\n\t\t\\end{tabular}}\n\t}\n\\end{table}\n\n\\begin{table}[h]\n\t\n\t\\centering\n\t\\caption{Running times for the FDFW}\n\t\\resizebox{0.8\\textwidth}{!}{\\begin{tabular}{|l|SS|SS|SS|SS|}\n\t\t\t\\hline\n\t\t\tGraph          &     \\multicolumn{2}{c|}{$s =1$}  &        \\multicolumn{2}{c|}{$s =2$}       &     \\multicolumn{2}{c|}{$s =3$}   &     \\multicolumn{2}{c|}{$s =4$}         \\\\ \\cline{2-9} \n\t\t\t& \\multicolumn{1}{c}{Time} & \\multicolumn{1}{c|}{Std} & \\multicolumn{1}{c}{Time} & \\multicolumn{1}{c|}{Std} & \\multicolumn{1}{c}{Time} & \\multicolumn{1}{c|}{Std} & \\multicolumn{1}{c}{Time} & \\multicolumn{1}{c|}{Std} \\\\ \\hline \n\t\t\tbrock200\\_1    & 0.053  & 0.0013 & 0.179  & 0.0271  & 0.227  & 0.0468 & 0.261  & 0.0341  \\\\\n\t\t\tbrock200\\_2    & 0.167  & 0.0297 & 0.218  & 0.0336  & 0.263  & 0.0727 & 0.327  & 0.1844  \\\\\n\t\t\tbrock200\\_3    & 0.156  & 0.0216 & 0.196  & 0.0350  & 0.254  & 0.0697 & 0.318  & 0.1616  \\\\\n\t\t\tbrock200\\_4    & 0.151  & 0.0178 & 0.189  & 0.0321  & 0.231  & 0.0483 & 0.278  & 0.0777  \\\\\n\t\t\tbrock400\\_1    & 0.677  & 0.0989 & 0.968  & 0.0246  & 1.250  & 0.1431 & 1.495  & 0.0679  \\\\\n\t\t\tbrock400\\_2    & 0.591  & 0.0197 & 0.983  & 0.0939  & 1.258  & 0.1185 & 1.493  & 0.0812  \\\\\n\t\t\tbrock400\\_3    & 0.595  & 0.0201 & 0.972  & 0.0316  & 1.242  & 0.0518 & 1.493  & 0.1350  \\\\\n\t\t\tbrock400\\_4    & 0.597  & 0.0190 & 0.961  & 0.0352  & 1.258  & 0.1030 & 1.519  & 0.1454  \\\\\n\t\t\tc-fat200-1     & 0.181  & 0.0057 & 0.318  & 0.0080  & 0.348  & 0.0082 & 0.376  & 0.0119  \\\\\n\t\t\tc-fat200-2     & 0.180  & 0.0079 & 0.314  & 0.0123  & 0.345  & 0.0169 & 0.379  & 0.0208  \\\\\n\t\t\tc-fat200-5     & 0.148  & 0.0071 & 0.264  & 0.0174  & 0.317  & 0.0247 & 0.356  & 0.0251  \\\\\n\t\t\tc-fat500-1     & 1.488  & 0.0249 & 2.441  & 0.0446  & 2.728  & 0.0458 & 3.016  & 0.0515  \\\\\n\t\t\tc-fat500-2     & 1.519  & 0.0257 & 2.579  & 0.1139  & 2.891  & 0.1459 & 3.256  & 0.1961  \\\\\n\t\t\tc-fat500-5     & 1.600  & 0.0641 & 2.408  & 0.1294  & 2.868  & 0.2600 & 3.311  & 0.4613  \\\\\n\t\t\tc-fat500-10    & 1.664  & 0.1425 & 2.436  & 0.3427  & 2.719  & 0.5139 & 3.073  & 0.5811  \\\\\n\t\t\thamming6-2     & 0.008  & 0.0011 & 0.015  & 0.0036  & 0.020  & 0.0037 & 0.026  & 0.0107  \\\\\n\t\t\thamming6-4     & 0.013  & 0.0005 & 0.024  & 0.0009  & 0.025  & 0.0018 & 0.027  & 0.0013  \\\\\n\t\t\thamming8-2     & 0.172  & 0.0350 & 0.267  & 0.0515  & 0.346  & 0.0604 & 0.428  & 0.0645  \\\\\n\t\t\thamming8-4     & 0.172  & 0.0296 & 0.312  & 0.0283  & 0.368  & 0.0487 & 0.409  & 0.0330  \\\\\n\t\t\thamming10-2  & 19.686 & 8.3243 & 21.857 & 10.8597 & 22.468 & 9.8968 & 25.991 & 11.2051 \\\\\n\t\t\thamming10-4    & 5.295  & 0.4853 & 8.365  & 0.5668  & 10.680 & 0.7274 & 13.257 & 0.7791  \\\\\n\t\t\tjohnson8-2-4   & 0.003  & 0.0005 & 0.005  & 0.0005  & 0.007  & 0.0030 & 0.009  & 0.0049  \\\\\n\t\t\tjohnson8-4-4   & 0.011  & 0.0009 & 0.020  & 0.0044  & 0.024  & 0.0033 & 0.030  & 0.0067  \\\\\n\t\t\tjohnson16-2-4  & 0.037  & 0.0048 & 0.065  & 0.0048  & 0.080  & 0.0030 & 0.097  & 0.0041  \\\\\n\t\t\tjohnson32-2-4  & 1.235  & 0.3833 & 1.547  & 0.2246  & 1.900  & 0.1825 & 2.325  & 0.2487  \\\\\n\t\t\tkeller4        & 0.081  & 0.0163 & 0.164  & 0.0733  & 0.209  & 0.1159 & 0.253  & 0.1667  \\\\\n\t\t\tkeller5        & 2.354  & 0.3020 & 3.819  & 0.6645  & 5.140  & 1.3110 & 6.438  & 2.0281  \\\\\n\t\t\tMANN\\_a9       & 0.005  & 0.0013 & 0.012  & 0.0081  & 0.018  & 0.0132 & 0.027  & 0.0260  \\\\\n\t\t\tMANN\\_a27      & 1.757  & 0.3220 & 2.504  & 0.5624  & 3.362  & 0.7169 & 4.371  & 0.8807  \\\\\n\t\t\tMANN\\_a45  & 44.945 & 5.8122 & 54.466 & 8.6046  & 67.499 & 9.7882 & 85.228 & 11.6795 \\\\\n\t\t\tp\\_hat300-1    & 0.348  & 0.0133 & 0.600  & 0.0408  & 0.634  & 0.0592 & 0.683  & 0.0746  \\\\\n\t\t\tp\\_hat300-2    & 0.323  & 0.0105 & 0.548  & 0.0177  & 0.631  & 0.0246 & 0.717  & 0.0754  \\\\\n\t\t\tp\\_hat300-3    & 0.262  & 0.0182 & 0.401  & 0.0197  & 0.466  & 0.0237 & 0.534  & 0.0332  \\\\\n\t\t\tp\\_hat500-1    & 1.209  & 0.0944 & 1.996  & 0.2460  & 2.293  & 0.8532 & 2.449  & 0.5990  \\\\\n\t\t\tp\\_hat500-2    & 1.353  & 0.0713 & 2.295  & 0.0999  & 2.836  & 0.1464 & 3.261  & 0.1831  \\\\\n\t\t\tp\\_hat500-3    & 1.369  & 0.1112 & 2.161  & 0.1798  & 2.535  & 0.1734 & 2.803  & 0.1708  \\\\\n\t\t\tp\\_hat700-1    & 2.549  & 0.1642 & 4.080  & 0.3388  & 4.468  & 0.8064 & 4.943  & 1.0512  \\\\\n\t\t\tp\\_hat700-2    & 2.765  & 0.1417 & 5.040  & 0.2496  & 6.340  & 0.2623 & 7.396  & 0.3281  \\\\\n\t\t\tp\\_hat700-3    & 3.084  & 0.2889 & 4.929  & 0.5101  & 5.871  & 0.5302 & 6.607  & 0.5871  \\\\\n\t\t\tsan200\\_0.7\\_1 & 0.159  & 0.0454 & 0.277  & 0.0429  & 0.358  & 0.0659 & 0.481  & 0.0947  \\\\\n\t\t\tsan200\\_0.7\\_2 & 0.271  & 0.0732 & 0.402  & 0.1320  & 0.404  & 0.1214 & 0.452  & 0.0791  \\\\\n\t\t\tsan200\\_0.9\\_1 & 0.128  & 0.0218 & 0.220  & 0.0302  & 0.296  & 0.0282 & 0.373  & 0.0332  \\\\\n\t\t\tsan200\\_0.9\\_2 & 0.129  & 0.0171 & 0.207  & 0.0225  & 0.260  & 0.0278 & 0.319  & 0.0295  \\\\\n\t\t\tsan200\\_0.9\\_3 & 0.105  & 0.0134 & 0.183  & 0.0186  & 0.237  & 0.0392 & 0.319  & 0.0786  \\\\\n\t\t\tsan400\\_0.5\\_1 & 1.605  & 0.1003 & 2.178  & 0.5510  & 2.011  & 0.6413 & 2.860  & 0.8604  \\\\\n\t\t\tsan400\\_0.7\\_1 & 1.706  & 0.6290 & 2.266  & 0.6993  & 2.469  & 0.1244 & 2.956  & 0.1424  \\\\\n\t\t\tsan400\\_0.7\\_2 & 1.684  & 0.5364 & 2.069  & 0.7020  & 2.133  & 0.3488 & 2.497  & 0.1083  \\\\\n\t\t\tsan400\\_0.7\\_3 & 1.713  & 0.3159 & 2.131  & 0.7214  & 2.020  & 0.6281 & 2.244  & 0.4235  \\\\\n\t\t\tsanr200\\_0.7   & 0.103  & 0.0045 & 0.186  & 0.0248  & 0.227  & 0.0476 & 0.297  & 0.0961  \\\\\n\t\t\tsanr200\\_0.9   & 0.108  & 0.0110 & 0.173  & 0.0175  & 0.220  & 0.0178 & 0.283  & 0.0223 \\\\ \\hline\n\t\\end{tabular}}\n\t\n\t\\label{tab:2}\n\\end{table}\n\n\n\\clearpage\n\n\\begin{table}[h]\n\t\\caption{Clique sizes for the FWdc}\n\t\\resizebox{\\textwidth}{!}{\\begin{tabular}{|l|SSS|SSS|SSS|SSS|}\n\t\t\t\\hline\n\t\t\tGraph          &     & {$s =1$}  &       &     & {$s =2$}  &       &     & {$s =3$}  &       &     & {$s =4$}  &       \\\\ \\cline{2-13} \n\t\t\t& \\multicolumn{1}{c}{Max} & \\multicolumn{1}{c}{Mean} & \\multicolumn{1}{c|}{Std} & \\multicolumn{1}{c}{Max} & \\multicolumn{1}{c}{Mean} & \\multicolumn{1}{c|}{Std} & \\multicolumn{1}{c}{Max} & \\multicolumn{1}{c}{Mean} & \\multicolumn{1}{c|}{Std} & \\multicolumn{1}{c}{Max} & \\multicolumn{1}{c}{Mean} & \\multicolumn{1}{c|}{Std} \\\\ \\hline \n\t\t\tbrock200\\_1    & 21   & 18.2  & 1.01  & 21   & 18.2  & 0.90  & 21   & 18.5  & 1.02  & 22  & 18.7  & 1.05  \\\\\n\t\t\tbrock200\\_2    & 10   & 8.6   & 0.91  & 11   & 8.9   & 0.81  & 11   & 9.3   & 0.91  & 12  & 9.5   & 0.86  \\\\\n\t\t\tbrock200\\_3    & 13   & 11.4  & 0.87  & 14   & 11.5  & 0.88  & 15   & 11.9  & 1.02  & 14  & 12.0  & 0.97  \\\\\n\t\t\tbrock200\\_4    & 16   & 13.3  & 1.01  & 16   & 13.7  & 0.93  & 16   & 14.0  & 1.01  & 17  & 14.1  & 1.14  \\\\\n\t\t\tbrock400\\_1    & 24   & 20.7  & 1.10  & 25   & 21.4  & 1.30  & 24   & 21.3  & 1.25  & 25  & 21.7  & 1.33  \\\\\n\t\t\tbrock400\\_2    & 24   & 20.9  & 1.12  & 25   & 21.3  & 1.26  & 26   & 21.4  & 1.32  & 25  & 21.8  & 1.16  \\\\\n\t\t\tbrock400\\_3    & 24   & 20.7  & 0.97  & 24   & 21.0  & 1.18  & 25   & 21.2  & 1.09  & 24  & 21.4  & 1.06  \\\\\n\t\t\tbrock400\\_4    & 23   & 20.6  & 1.16  & 23   & 21.1  & 1.07  & 24   & 21.5  & 1.07  & 25  & 21.6  & 1.38  \\\\\n\t\t\tc-fat200-1     & 12   & 11.4  & 1.69  & 12   & 10.8  & 2.64  & 12   & 8.7   & 4.31  & 12  & 7.9   & 4.49  \\\\\n\t\t\tc-fat200-2     & 24   & 21.2  & 4.76  & 24   & 20.3  & 6.25  & 24   & 18.7  & 7.63  & 24  & 17.2  & 8.45  \\\\\n\t\t\tc-fat200-5     & 58   & 55.1  & 7.08  & 58   & 53.3  & 9.88  & 58   & 52.0  & 12.88 & 58  & 53.3  & 10.66 \\\\\n\t\t\tc-fat500-1     & 14   & 13.5  & 1.13  & 14   & 12.7  & 2.69  & 14   & 10.8  & 4.57  & 14  & 9.5   & 5.24  \\\\\n\t\t\tc-fat500-2     & 26   & 25.5  & 2.47  & 26   & 24.5  & 5.23  & 26   & 22.8  & 7.65  & 26  & 21.6  & 8.73  \\\\\n\t\t\tc-fat500-5     & 64   & 60.8  & 10.85 & 64   & 61.7  & 8.62  & 64   & 58.8  & 14.73 & 64  & 56.2  & 18.38 \\\\\n\t\t\tc-fat500-10    & 126  & 122.6 & 12.02 & 126  & 119.8 & 17.20 & 126  & 118.0 & 24.57 & 126 & 115.7 & 29.14 \\\\\n\t\t\thamming6-2     & 32   & 28.6  & 4.58  & 32   & 27.9  & 4.63  & 32   & 27.5  & 4.40  & 32  & 27.4  & 4.09  \\\\\n\t\t\thamming6-4     & 4    & 3.7   & 0.46  & 5    & 4.2   & 0.61  & 6    & 4.4   & 0.66  & 6   & 4.8   & 0.63  \\\\\n\t\t\thamming8-2     & 128  & 121.1 & 9.18  & 128  & 120.2 & 9.07  & 128  & 118.8 & 10.29 & 128 & 116.9 & 11.86 \\\\\n\t\t\thamming8-4     & 16   & 12.7  & 2.68  & 16   & 12.6  & 2.31  & 16   & 12.5  & 2.24  & 17  & 12.8  & 2.28  \\\\\n\t\t\thamming10-2    & 512  & 498.9 & 14.52 & 512  & 497.0 & 15.59 & 512  & 495.5 & 17.16 & 512 & 493.9 & 18.06 \\\\\n\t\t\thamming10-4    & 36   & 31.6  & 2.93  & 36   & 32.2  & 2.67  & 37   & 32.1  & 2.80  & 37  & 32.9  & 2.32  \\\\\n\t\t\tjohnson8-2-4   & 4    & 4.0   & 0.00  & 5    & 4.9   & 0.27  & 5    & 5.0   & 0.10  & 6   & 5.3   & 0.47  \\\\\n\t\t\tjohnson8-4-4   & 14   & 11.9  & 1.73  & 14   & 11.7  & 1.46  & 14   & 11.8  & 1.35  & 15  & 11.9  & 1.18  \\\\\n\t\t\tjohnson16-2-4  & 8    & 8.0   & 0.00  & 9    & 9.0   & 0.00  & 9    & 9.0   & 0.00  & 10  & 9.8   & 0.39  \\\\\n\t\t\tjohnson32-2-4  & 16   & 16.0  & 0.00  & 17   & 17.0  & 0.00  & 17   & 17.0  & 0.00  & 18  & 17.8  & 0.41  \\\\\n\t\t\tkeller4        & 12   & 9.3   & 1.15  & 12   & 9.7   & 0.81  & 13   & 10.1  & 0.80  & 13  & 10.6  & 0.82  \\\\\n\t\t\tkeller5        & 27   & 20.7  & 1.77  & 26   & 21.1  & 1.72  & 26   & 21.5  & 1.50  & 27  & 21.5  & 1.55  \\\\\n\t\t\tMANN\\_a9       & 17   & 16.3  & 0.67  & 18   & 16.8  & 0.71  & 19   & 17.4  & 0.72  & 19  & 17.6  & 0.82  \\\\\n\t\t\tMANN\\_a27      & 120  & 118.2 & 0.43  & 120  & 119.2 & 0.37  & 121  & 120.1 & 0.43  & 122 & 121.1 & 0.38  \\\\\n\t\t\tMANN\\_a45      & 332  & 331.0 & 0.17  & 333  & 332.0 & 0.17  & 334  & 333.0 & 0.20  & 335 & 334.0 & 0.22  \\\\\n\t\t\tp\\_hat300-1    & 8    & 6.9   & 0.68  & 9    & 7.1   & 0.79  & 9    & 7.4   & 0.75  & 9   & 7.5   & 0.85  \\\\\n\t\t\tp\\_hat300-2    & 26   & 21.9  & 1.25  & 25   & 22.0  & 1.20  & 25   & 22.2  & 1.18  & 26  & 22.2  & 1.27  \\\\\n\t\t\tp\\_hat300-3    & 35   & 31.4  & 1.35  & 34   & 31.8  & 1.32  & 35   & 31.8  & 1.30  & 36  & 32.2  & 1.19  \\\\\n\t\t\tp\\_hat500-1    & 10   & 7.9   & 0.81  & 10   & 8.1   & 0.83  & 10   & 8.1   & 0.90  & 11  & 8.4   & 0.96  \\\\\n\t\t\tp\\_hat500-2    & 35   & 31.6  & 1.84  & 35   & 31.9  & 1.73  & 36   & 31.8  & 1.79  & 35  & 31.8  & 1.57  \\\\\n\t\t\tp\\_hat500-3    & 48   & 44.8  & 1.58  & 49   & 44.8  & 1.76  & 49   & 45.1  & 1.73  & 49  & 45.4  & 1.67  \\\\\n\t\t\tp\\_hat700-1    & 9    & 8.0   & 0.70  & 10   & 8.2   & 0.78  & 10   & 8.4   & 0.78  & 10  & 8.6   & 0.77  \\\\\n\t\t\tp\\_hat700-2    & 44   & 39.9  & 1.77  & 43   & 40.0  & 1.77  & 44   & 40.2  & 1.81  & 44  & 40.0  & 2.00  \\\\\n\t\t\tp\\_hat700-3    & 62   & 57.0  & 1.94  & 60   & 57.6  & 1.73  & 61   & 57.5  & 1.85  & 62  & 57.9  & 1.94  \\\\\n\t\t\tsan200\\_0.7\\_1 & 18   & 16.6  & 0.84  & 19   & 17.1  & 1.23  & 20   & 18.0  & 1.17  & 21  & 18.7  & 1.44  \\\\\n\t\t\tsan200\\_0.7\\_2 & 15   & 13.1  & 0.26  & 15   & 14.0  & 0.14  & 16   & 14.7  & 0.49  & 16  & 15.2  & 0.42  \\\\\n\t\t\tsan200\\_0.9\\_1 & 65   & 48.9  & 4.23  & 65   & 49.4  & 3.94  & 68   & 50.1  & 4.11  & 70  & 50.6  & 3.85  \\\\\n\t\t\tsan200\\_0.9\\_2 & 52   & 39.5  & 2.74  & 52   & 39.9  & 2.53  & 55   & 40.8  & 3.32  & 55  & 41.5  & 3.35  \\\\\n\t\t\tsan200\\_0.9\\_3 & 36   & 33.4  & 1.24  & 36   & 33.7  & 1.30  & 38   & 34.1  & 1.43  & 39  & 34.4  & 1.51  \\\\\n\t\t\tsan400\\_0.5\\_1 & 9    & 8.1   & 0.30  & 10   & 9.0   & 0.32  & 10   & 9.6   & 0.50  & 12  & 10.1  & 0.57  \\\\\n\t\t\tsan400\\_0.7\\_1 & 23   & 21.8  & 0.77  & 24   & 22.5  & 0.86  & 25   & 22.9  & 1.30  & 25  & 23.5  & 1.73  \\\\\n\t\t\tsan400\\_0.7\\_2 & 23   & 17.4  & 1.16  & 20   & 17.9  & 0.78  & 21   & 18.5  & 0.93  & 21  & 19.0  & 0.75  \\\\\n\t\t\tsan400\\_0.7\\_3 & 17   & 15.1  & 0.93  & 18   & 15.6  & 0.85  & 18   & 16.2  & 0.92  & 19  & 16.6  & 0.88  \\\\\n\t\t\tsanr200\\_0.7   & 17   & 14.9  & 0.86  & 18   & 15.2  & 1.09  & 17   & 15.6  & 0.89  & 19  & 15.8  & 0.99  \\\\\n\t\t\tsanr200\\_0.9   & 41   & 37.5  & 1.79  & 41   & 37.5  & 1.67  & 42   & 38.1  & 1.75  & 43  & 38.3  & 1.73 \\\\ \\hline\n\t\\end{tabular}}\n\t\n\\end{table}\t\n\\pagebreak\n\\begin{table}[h]\n\t\\caption{Running times for the FWdc}\t\n\t\\resizebox{\\textwidth}{!}{\\begin{tabular}{|l|SS|SS|SS|SS|}\n\t\t\t\\hline\n\t\t\tGraph          &     \\multicolumn{2}{c|}{$s =1$}  &        \\multicolumn{2}{c|}{$s =2$}       &     \\multicolumn{2}{c|}{$s =3$}   &     \\multicolumn{2}{c|}{$s =4$}         \\\\ \\cline{2-9} \n\t\t\t& \\multicolumn{1}{c}{Time} & \\multicolumn{1}{c|}{Std} & \\multicolumn{1}{c}{Time} & \\multicolumn{1}{c|}{Std} & \\multicolumn{1}{c}{Time} & \\multicolumn{1}{c|}{Std} & \\multicolumn{1}{c}{Time} & \\multicolumn{1}{c|}{Std} \\\\ \\hline \n\t\t\tbrock200\\_1    & 0.0060 & 0.00560 & 0.0060 & 0.00064 & 0.0064 & 0.00070 & 0.0069 & 0.00083 \\\\\n\t\t\tbrock200\\_2    & 0.0045 & 0.00053 & 0.0051 & 0.00041 & 0.0051 & 0.00050 & 0.0053 & 0.00042 \\\\\n\t\t\tbrock200\\_3    & 0.0044 & 0.00045 & 0.0050 & 0.00056 & 0.0052 & 0.00047 & 0.0052 & 0.00038 \\\\\n\t\t\tbrock200\\_4    & 0.0045 & 0.00039 & 0.0053 & 0.00055 & 0.0055 & 0.00054 & 0.0057 & 0.00052 \\\\\n\t\t\tbrock400\\_1    & 0.0122 & 0.00088 & 0.0142 & 0.00157 & 0.0141 & 0.00082 & 0.0144 & 0.00080 \\\\\n\t\t\tbrock400\\_2    & 0.0129 & 0.00066 & 0.0133 & 0.00118 & 0.0144 & 0.00095 & 0.0147 & 0.00072 \\\\\n\t\t\tbrock400\\_3    & 0.0129 & 0.00060 & 0.0138 & 0.00098 & 0.0139 & 0.00057 & 0.0146 & 0.00096 \\\\\n\t\t\tbrock400\\_4    & 0.0130 & 0.00065 & 0.0136 & 0.00100 & 0.0142 & 0.00069 & 0.0146 & 0.00067 \\\\\n\t\t\tc-fat200-1     & 0.0065 & 0.00061 & 0.0070 & 0.00080 & 0.0068 & 0.00052 & 0.0071 & 0.00074 \\\\\n\t\t\tc-fat200-2     & 0.0060 & 0.00100 & 0.0067 & 0.00109 & 0.0067 & 0.00098 & 0.0067 & 0.00091 \\\\\n\t\t\tc-fat200-5     & 0.0071 & 0.00216 & 0.0081 & 0.00237 & 0.0084 & 0.00261 & 0.0089 & 0.00259 \\\\\n\t\t\tc-fat500-1     & 0.0351 & 0.00254 & 0.0370 & 0.00188 & 0.0366 & 0.00210 & 0.0364 & 0.00243 \\\\\n\t\t\tc-fat500-2     & 0.0311 & 0.00360 & 0.0331 & 0.00374 & 0.0334 & 0.00355 & 0.0334 & 0.00314 \\\\\n\t\t\tc-fat500-5     & 0.0296 & 0.00621 & 0.0330 & 0.00730 & 0.0343 & 0.00599 & 0.0347 & 0.00592 \\\\\n\t\t\tc-fat500-10    & 0.0390 & 0.01353 & 0.0400 & 0.01617 & 0.0422 & 0.01685 & 0.0433 & 0.01748 \\\\\n\t\t\thamming6-2     & 0.0014 & 0.00012 & 0.0021 & 0.00017 & 0.0023 & 0.00027 & 0.0026 & 0.00029 \\\\\n\t\t\thamming6-4     & 0.0009 & 0.00015 & 0.0011 & 0.00014 & 0.0012 & 0.00014 & 0.0013 & 0.00015 \\\\\n\t\t\thamming8-2     & 0.0204 & 0.00162 & 0.0245 & 0.00195 & 0.0287 & 0.00272 & 0.0302 & 0.00308 \\\\\n\t\t\thamming8-4     & 0.0060 & 0.00055 & 0.0065 & 0.00059 & 0.0068 & 0.00075 & 0.0069 & 0.00075 \\\\\n\t\t\thamming10-2    & 1.0562 & 0.06427 & 1.0825 & 0.06712 & 1.0816 & 0.07577 & 1.0717 & 0.07835 \\\\\n\t\t\thamming10-4    & 0.0762 & 0.00268 & 0.0805 & 0.00400 & 0.0805 & 0.00379 & 0.0823 & 0.00342 \\\\\n\t\t\tjohnson8-2-4   & 0.0004 & 0.00006 & 0.0006 & 0.00009 & 0.0007 & 0.00013 & 0.0008 & 0.00013 \\\\\n\t\t\tjohnson8-4-4   & 0.0011 & 0.00009 & 0.0015 & 0.00017 & 0.0018 & 0.00021 & 0.0020 & 0.00027 \\\\\n\t\t\tjohnson16-2-4  & 0.0023 & 0.00028 & 0.0029 & 0.00036 & 0.0034 & 0.00041 & 0.0037 & 0.00055 \\\\\n\t\t\tjohnson32-2-4  & 0.0218 & 0.00218 & 0.0240 & 0.00229 & 0.0248 & 0.00232 & 0.0259 & 0.00282 \\\\\n\t\t\tkeller4        & 0.0034 & 0.00034 & 0.0044 & 0.00053 & 0.0044 & 0.00065 & 0.0051 & 0.00062 \\\\\n\t\t\tkeller5        & 0.0428 & 0.00207 & 0.0446 & 0.00206 & 0.0455 & 0.00239 & 0.0460 & 0.00236 \\\\\n\t\t\tMANN\\_a9       & 0.0010 & 0.00013 & 0.0014 & 0.00017 & 0.0016 & 0.00024 & 0.0018 & 0.00031 \\\\\n\t\t\tMANN\\_a27      & 0.0910 & 0.00610 & 0.1029 & 0.00723 & 0.1096 & 0.00723 & 0.1150 & 0.00699 \\\\\n\t\t\tMANN\\_a45      & 2.3774 & 0.07529 & 2.4419 & 0.10145 & 2.4544 & 0.08325 & 2.4712 & 0.09195 \\\\\n\t\t\tp\\_hat300-1    & 0.0071 & 0.00029 & 0.0078 & 0.00068 & 0.0080 & 0.00046 & 0.0083 & 0.00040 \\\\\n\t\t\tp\\_hat300-2    & 0.0085 & 0.00031 & 0.0096 & 0.00061 & 0.0103 & 0.00080 & 0.0104 & 0.00062 \\\\\n\t\t\tp\\_hat300-3    & 0.0095 & 0.00046 & 0.0110 & 0.00070 & 0.0119 & 0.00072 & 0.0126 & 0.00092 \\\\\n\t\t\tp\\_hat500-1    & 0.0163 & 0.00041 & 0.0169 & 0.00039 & 0.0170 & 0.00043 & 0.0171 & 0.00057 \\\\\n\t\t\tp\\_hat500-2    & 0.0208 & 0.00129 & 0.0224 & 0.00133 & 0.0228 & 0.00133 & 0.0235 & 0.00131 \\\\\n\t\t\tp\\_hat500-3    & 0.0243 & 0.00085 & 0.0266 & 0.00100 & 0.0278 & 0.00125 & 0.0281 & 0.00100 \\\\\n\t\t\tp\\_hat700-1    & 0.0298 & 0.00070 & 0.0307 & 0.00091 & 0.0309 & 0.00109 & 0.0311 & 0.00079 \\\\\n\t\t\tp\\_hat700-2    & 0.0394 & 0.00108 & 0.0421 & 0.00211 & 0.0427 & 0.00143 & 0.0431 & 0.00161 \\\\\n\t\t\tp\\_hat700-3    & 0.0466 & 0.00151 & 0.0498 & 0.00181 & 0.0508 & 0.00177 & 0.0520 & 0.00144 \\\\\n\t\t\tsan200\\_0.7\\_1 & 0.0049 & 0.00059 & 0.0056 & 0.00071 & 0.0061 & 0.00066 & 0.0064 & 0.00068 \\\\\n\t\t\tsan200\\_0.7\\_2 & 0.0048 & 0.00068 & 0.0063 & 0.00093 & 0.0073 & 0.00134 & 0.0081 & 0.00148 \\\\\n\t\t\tsan200\\_0.9\\_1 & 0.0086 & 0.00076 & 0.0108 & 0.00104 & 0.0118 & 0.00107 & 0.0129 & 0.00149 \\\\\n\t\t\tsan200\\_0.9\\_2 & 0.0075 & 0.00053 & 0.0092 & 0.00075 & 0.0101 & 0.00079 & 0.0113 & 0.00073 \\\\\n\t\t\tsan200\\_0.9\\_3 & 0.0070 & 0.00058 & 0.0084 & 0.00067 & 0.0092 & 0.00069 & 0.0098 & 0.00084 \\\\\n\t\t\tsan400\\_0.5\\_1 & 0.0116 & 0.00056 & 0.0128 & 0.00094 & 0.0141 & 0.00172 & 0.0142 & 0.00136 \\\\\n\t\t\tsan400\\_0.7\\_1 & 0.0130 & 0.00056 & 0.0143 & 0.00077 & 0.0152 & 0.00101 & 0.0158 & 0.00110 \\\\\n\t\t\tsan400\\_0.7\\_2 & 0.0128 & 0.00064 & 0.0139 & 0.00068 & 0.0147 & 0.00082 & 0.0157 & 0.00138 \\\\\n\t\t\tsan400\\_0.7\\_3 & 0.0125 & 0.00043 & 0.0136 & 0.00059 & 0.0143 & 0.00072 & 0.0148 & 0.00084 \\\\\n\t\t\tsanr200\\_0.7   & 0.0049 & 0.00061 & 0.0057 & 0.00071 & 0.0058 & 0.00064 & 0.0062 & 0.00078 \\\\\n\t\t\tsanr200\\_0.9   & 0.0070 & 0.00053 & 0.0087 & 0.00072 & 0.0095 & 0.00078 & 0.0119 & 0.00090 \\\\ \\hline\n\t\\end{tabular}}\n\\end{table}\t\n\n\\clearpage\n\n\\bibliographystyle{plain}\t\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMatrix concentration inequalities describe the probability\nthat a random matrix is close to its expected value, with\ndeviations measured by the $\\ell_2$ operator norm.  These results have\nhad a profound impact on a wide range of areas in computational mathematics\nand statistics.  See the monograph~\\cite{Tro15:Introduction-Matrix}\nfor an introduction to the subject and its applications.\n\nAs did the field of scalar concentration, matrix concentration theory\nbegan with simple models, such as sums of independent\nrandom matrices~\\cite{LP86:Inegalites-Khintchine,Rud99:Random-Vectors,AW02:Strong-Converse,Tro12:User-Friendly}\nand matrix-valued martingale sequences~\\cite{PX97:Noncommutative-Martingale,Oli09:Concentration-Adjacency,Tro11:Freedmans-Inequality}.\nIn recent years, researchers have sought to develop results that hold for\na wider class of random matrix models.  Some initial successful efforts were based on\nexchangeable pairs techniques~\\cite{MJCFT14:Matrix-Concentration,PMT16:Efron-Stein-Inequalities},\nbut these methods do not address all cases of interest.\n\nResearchers have also tried to extend scalar\nconcentration techniques based on functional inequalities. An early attempt, by Chen \\& Tropp~\\cite{CT14:Subadditivity-Matrix}, demonstrates\nthat (traces of) matrix variance and entropy quantities are subadditive, which \nleads to some Poincar{\\'e} and modified log-Sobolev inequalities.  A number of authors,\nincluding~\\cite{CH16:Characterizations-Matrix,CHT17:Exponential-Decay,CH19:Matrix-Poincare},\nhave pursued this line of work.\nUnfortunately, these approaches have not\nbeen sufficient to reproduce all the results that have been established in the simpler models.\n\nVery recently, Aoun et al.~\\cite{ABY19:Matrix-Poincare} have\nshown that a matrix form of the Poincar{\\'e} inequality implies\nsubexponential concentration of a random matrix with respect to the $\\ell_2$\noperator norm.  We believe that this is the first instance\nwhere a matrix functional inequality leads unconditionally\nto a matrix concentration result (with respect to the operator norm).\nNevertheless, it remains an open question to deduce a full spectrum\nof matrix concentration results from matrix functional inequalities.\n\nIn this paper, we improve on~\\cite{ABY19:Matrix-Poincare}\nby demonstrating that the ordinary scalar Poincar{\\'e}\ninequality also leads to subexponential concentration with respect\nto the operator norm.\nOur argument has some elements in common with the work in\n\\cite{ABY19:Matrix-Poincare}, but we have found a route to avoid\nmost of the technical difficulty of their approach.\n\nThe basic idea is to bound the trace of an odd function\n(for example, the hyperbolic sine)\nof the random matrix using a symmetrization argument.\nThe variance appears, and \nthe Poincar{\\'e} inequality\nyields a bound on the variance in terms of the\nDirichlet form.  Last,\nwe apply a new matrix chain rule inequality\nfor the Dirichlet form to\nobtain a moment comparison.\nAfter this paper was written, we learned\nthat Bobkov \\& Ledoux proposed\na similar argument in the scalar\ncase~\\cite[Sec.~4]{BL97:Poincares-Inequalities}.\n\n\n\nAs in the scalar setting, Poincar{\\'e} inequalities may not capture\nthe strongest concentration properties that are possible.  In a companion\npaper~\\cite{HT20:Matrix-Concentration}, we demonstrate that\n\\emph{local} Poincar{\\'e} inequalities lead to the optimal\nsubgaussian concentration results.  The analysis in~\\cite{HT20:Matrix-Concentration}\ncaptures most of the previous results on matrix concentration,\nbut it involves more technical machinery.  We also refer the\nreader to work of Junge \\& Zeng~\\cite{JZ15:Noncommutative-Martingale}\nthat contains similar results in the fully noncommutative setting.\n\nAn expert reader may still wonder about the role of (modified)\nlog-Sobolev inequalities in establishing matrix concentration\ninequalities.  At the time of writing, it is not clear how\nto obtain a matrix analog of the log-Sobolev inequality\nthat would imply matrix concentration results in the\nsame spirit as the ones in this paper or the related\nworks~\\cite{JZ15:Noncommutative-Martingale,ABY19:Matrix-Poincare,HT20:Matrix-Concentration}.\n\n\n\n\n\\section{Main result}\n\nThis section summarizes our notation and the setting for our problem.\nIt highlights our primary result on matrix concentration, and it\ngives a number of examples.  In the next section,\nwe comment on the relationship with previous work.\n\n\n\\subsection{Notation}\n\nLet $\\mathbbm{H}_d$ be the real linear space of $d \\times d$ self-adjoint complex matrices,\nequipped with the $\\ell_2$ operator norm $\\norm{\\cdot}$.\nWe work with both the ordinary trace, $\\operatorname{tr}$, and the normalized trace, $\\operatorname{\\bar{\\trace}} := d^{-1} \\operatorname{tr}$\non the space $\\mathbbm{H}_d$.\nMatrices, and occasionally vectors, are written in boldface italic.\nIn particular, $\\mtx{f}$ and $\\mtx{g}$ refer to functions taking values in $\\mathbbm{H}_d$.\nThe cone $\\mathbbm{H}_d^+$ contains the positive semidefinite matrices,\nand the symbol $\\preccurlyeq$ refers to the semidefinite order.\n\nGiven a function $\\phi : \\mathbbm{R} \\to \\mathbbm{R}$ taking real values,\nwe extend it to a function $\\phi : \\mathbbm{H}_d \\to \\mathbbm{H}_d$\non self-adjoint matrices by means of the spectral resolution:\n$$\n\\mtx{A} = \\sum_{\\lambda \\in \\mathrm{spec}(\\mtx{A})} \\lambda \\, \\mtx{P}_{\\lambda} \\in \\mathbbm{H}_d\n\\quad\\text{implies}\\quad\n\\phi(\\mtx{A}) = \\sum_{\\lambda \\in \\mathrm{spec}(\\mtx{A})} \\phi(\\lambda) \\, \\mtx{P}_{\\lambda} \\in \\mathbbm{H}_d.\n$$\nWhenever we apply a scalar function, such as a power\nor a hyperbolic function, to a matrix, we are referring\nto the standard matrix function.  Nonlinear functions\nbind before the trace.\n\nWe use familiar notation from probability. The operator $\\Expect$\nreturns the expectation, and $\\Prob{\\cdot}$ is the probability\nof an event.  The symbol $\\sim$ means ``has the distribution.''\nNonlinear functions bind before the expectation.\n\n\n\n\n\n\n\\subsection{Random matrices}\n\nLet $\\Omega$ be a Polish space, equipped with a probability measure $\\mu$,\nand write $\\Expect_{\\mu}$ for the integral with respect to the measure $\\mu$.\nConsider a $\\mu$-integrable matrix-valued function $\\mtx{f} : \\Omega \\to \\mathbbm{H}_d$ on the state space $\\Omega$.\nBy drawing a random variable $Z \\sim \\mu$, we can construct a random matrix\n$\\mtx{f}(Z)$.  Our goal is to understand the concentration of $\\mtx{f}(Z)$\naround its mean $\\Expect_{\\mu} \\mtx{f}$.\n\n\\begin{example}[Gaussians]\nConsider the Gaussian space $(\\mathbbm{R}^n, \\gamma_n)$ of $n$-dimensional\nreal vectors equipped with the standard normal measure $\\gamma_n$.\nSuppose we are interested in a matrix-valued function  $\\mtx{f}(\\vct{X})$\nof a standard normal random vector $\\vct{X} \\sim \\gamma_n$.\nA familiar example~\\cite[Chap.~5]{Tro15:Introduction-Matrix} is the matrix Gaussian series\n\\begin{equation} \\label{eqn:gauss-series}\n\\mtx{f}(\\vct{X}) = \\sum_{i=1}^n X_i \\mtx{A}_i\n\\quad\\text{where $\\mtx{X} \\sim \\gamma_n$ and $\\mtx{A}_1, \\dots, \\mtx{A}_n \\in \\mathbbm{H}_d$ are fixed.}\n\\end{equation}\nWe will use the Gaussian case as a running example to illustrate\nthe concepts that we introduce.\n\\end{example}\n\n\n\\subsection{Markov processes}\n\nSuppose that we can identify an ergodic, reversible, time-homogeneous Markov process\n$(Z_t : t \\geq 0) \\subset \\Omega$ with initial\nvalue $Z_0$ and stationary measure $\\mu$.\nThis induces a matrix-valued Markov process\n$( \\mtx{f}(Z_t) : t \\geq 0 ) \\subset \\mathbbm{H}_d$.\n\nBy ergodicity, for any point $z \\in \\Omega$,\nwe have the limit $\\Expect[ \\mtx{f}(Z_t) \\, | \\, Z_0 = z ] \\to \\Expect_{\\mu} \\mtx{f}$\nas $t \\to \\infty$.\nThe results in this paper build on\nthe intuition that a random matrix $\\mtx{f}(Z)$\nwith $Z \\sim \\mu$ concentrates sharply about\nits mean when the matrix-valued Markov process\ntends quickly to equilibrium.\n\n\n\\begin{example}[Gaussians]\nWe can construct a reversible Markov process $(\\vct{X}_t : t \\geq 0) \\subset \\mathbbm{R}^n$,\ncalled the \\emph{Ornstein--Uhlenbeck (OU) process},\nby means of the stochastic differential equation\n$$\n\\diff{\\vct{X}}_t = - {\\vct{X}}_t \\idiff{t} + \\sqrt{2} \\idiff{\\vct{B}}_t\n\\quad\\text{with initial value $\\vct{X}_0 \\in \\mathbbm{R}^n$,}\n$$\nwhere $( \\vct{B}_t : t \\geq 0 ) \\subset \\mathbbm{R}^n$ is random vector whose coordinates are\nindependent Brownian motions.  The stationary measure of\nthe OU process is the standard normal distribution $\\gamma_n$.\n\\end{example}\n\n\n\n\n\n\\subsection{Derivatives and energy}\n\n\n\nTo understand how quickly a matrix Markov process $\\mtx{f}(Z_t)$\nconverges to stationarity, we introduce notions\nof the ``squared derivative'' and the ``energy'' of the function $\\mtx{f}$.\n\nInspired by~\\cite{ABY19:Matrix-Poincare}, \nwe define the matrix \\emph{carr{\\'e} du champ operator} by the formula\n\\begin{equation} \\label{eqn:carre-limit}\n\\mtx{\\Gamma}(\\mtx{f})(z) := \\lim_{t \\downarrow 0} \\frac{1}{2t} \\Expect \\big[ \\big(\\mtx{f}(Z_t) - \\mtx{f}(Z_0) \\big)^2 \\,\\big|\\, Z_0 = z \\big] \\in \\mathbbm{H}_d^+\n\\quad\\text{for $z \\in \\Omega$.}\n\\end{equation}\nIn many instances, the carr{\\'e} du champ $\\mtx{\\Gamma}(\\mtx{f})$ has a natural interpretation\nas a squared derivative of $\\mtx{f}$.\nThe expectation of the carr{\\'e} du champ is called the \\emph{matrix Dirichlet form}:\n\\begin{equation} \\label{eqn:dirichlet-limit}\n\\bm{\\mathcal{E}}(\\mtx{f}) := \\lim_{t \\downarrow 0} \\frac{1}{2t} \\Expect_{Z \\sim \\mu} \\big[ \\big(\\mtx{f}(Z_t) - \\mtx{f}(Z_0)\\big)^2 \\,\\big|\\, Z_0 = Z \\big] \\in \\mathbbm{H}_d^+.\n\\end{equation}\nThe Dirichlet form $\\bm{\\mathcal{E}}(\\mtx{f})$ reflects the total energy of the function $\\mtx{f}$.\n\nIn a general setting, it requires some care to make sense of the\ndefinitions~\\eqref{eqn:carre-limit} and~\\eqref{eqn:dirichlet-limit}.\nWithout further comment, we restrict our attention to a ``nice'' class of functions\nwhere the limit in~\\eqref{eqn:carre-limit} exists pointwise and in $L_1(\\mu)$\nand where calculus operations are justified.\nBy approximation, our main results on concentration hold for a wider\nclass of functions.\n\n\n\\begin{example}[Gaussians]\nAccording to~\\cite[Prop.~5.5]{ABY19:Matrix-Poincare},\nthe matrix carr{\\'e} du champ operator and matrix Dirichlet form of the OU process\nare determined by\n$$\n\\mtx{\\Gamma}(\\mtx{f})(\\vct{x}) = \\sum_{i=1}^n (\\partial_i \\mtx{f}(\\vct{x}))^2\n\t\\quad\\text{for $\\vct{x} \\in \\mathbbm{R}^n$}\n\\quad\\text{and}\\quad\n\\bm{\\mathcal{E}}(\\mtx{f}) = \\sum_{i=1}^n \\Expect_{\\gamma_n} (\\partial_i \\mtx{f})^2.\n$$\nThe interpretations as a squared derivative and an energy are evident,\nand it is easy to check when the carr{\\'e} du champ is defined.\n\n\n\nThe matrix Gaussian series~\\eqref{eqn:gauss-series} provides\nan illustration:\n$$\n\\mtx{\\Gamma}(\\mtx{f}) = \\bm{\\mathcal{E}}(\\mtx{f}) = \\sum_{i=1}^n \\mtx{A}_i^2.\n$$\nThis quantity is familiar from work on matrix concentration for Gaussian series~\\cite[Chap.~4]{Tro15:Introduction-Matrix}.\n\\end{example}\n\n\n\n\n\n\n\\subsection{Trace Poincar{\\'e} inequalities}\n\nThe \\emph{matrix variance} of a function $\\vct{f} : \\Omega \\to \\mathbbm{H}_d$\nwith respect to the distribution $\\mu$ is defined as\n\\begin{equation}\\label{eqn:matrix-var}\n\\Var_{\\mu}[ \\mtx{f} ] :=\n\t\\Expect_{\\mu}\\big[ (\\mtx{f} - \\Expect_{\\mu}\\mtx{f} )^2 \\big]\n\t= \\Expect_{\\mu}[ \\vct{f}^2 ] - (\\Expect_{\\mu}  \\vct{f} )^2\n\t\\in \\mathbbm{H}_d^+.\n\\end{equation}\nAs in the scalar case, the variance reflects fluctuations\nof the random matrix $\\mtx{f}(Z)$ about its mean, where\nthe random variable $Z \\sim \\mu$.\n\nWe say that the Markov process satisfies a \\emph{trace Poincar{\\'e} inequality}\nwith constant $\\alpha > 0$ if\n\\begin{equation} \\label{eqn:trace-poincare}\n\\operatorname{tr} \\Var_{\\mu}[ \\mtx{f} ]\n\t\\leq \\alpha \\cdot \\operatorname{tr} \\bm{\\mathcal{E}}(\\mtx{f})\n\t\\quad\\text{for all $\\mtx{f} : \\Omega \\to \\mathbbm{H}_d$.} \\end{equation}\nIn other words, the trace variance of $\\mtx{f}(Z)$ is controlled\nby the energy in the function $\\mtx{f}$.  The inequality~\\eqref{eqn:trace-poincare}\nprovides a way to quantify the ergodicity of the Markov process.\n\nAs it happens, the trace Poincar{\\'e} inequality is equivalent to\nan ordinary Poincar{\\'e} inequality.  We are grateful to Ramon\nVan Handel for this observation.  The same result has recently\nappeared in the independent work of Garg et al.~\\cite{GKS20:Scalar-Poincare}.\n\n\\begin{proposition}[Equivalence of Poincar{\\'e} inequalities] \\label{prop:equiv}\nConsider a Markov process $(Z_t : t \\geq 0) \\subset \\Omega$\nwith stationary measure $\\mu$.  The following are equivalent:\n\n\\begin{enumerate}\n\\item\t\\textbf{Scalar Poincar{\\'e}.}  For all $f : \\Omega \\to \\mathbbm{R}$, it holds that\n$\\Var_{\\mu}[ f ] \\leq \\alpha \\cdot \\bm{\\mathcal{E}}( f )$.\n\n\\item\t\\textbf{Trace Poincar{\\'e}.}  For all $d \\in \\mathbbm{N}$ and all $\\mtx{f} : \\Omega \\to \\mathbbm{H}_d$,\nit holds that $\\operatorname{tr} \\Var_{\\mu}[ \\mtx{f} ] \\leq \\alpha \\cdot \\operatorname{tr} \\bm{\\mathcal{E}}( \\mtx{f})$.\n\\end{enumerate}\n\n\\noindent\nThe Poincar{\\'e} constant $\\alpha \\geq 0$ is the same for both cases.\n\\end{proposition}\n\n\\begin{proof}\nIt is evident that the validity of the trace Poincar{\\'e} inequality for all $d \\in \\mathbbm{N}$\nimplies the scalar Poincar{\\'e} inequality.  For the reverse implication, it suffices to\nconsider a real matrix-valued function $\\mtx{f} : \\Omega \\to \\mathbbm{H}_d(\\mathbbm{R})$ with zero mean.\nFor vectors $\\vct{u}, \\vct{v} \\in \\mathbbm{R}^d$, define the scalar function\n$g(z) = \\ip{ \\vct{u} }{ \\mtx{f}(z) \\, \\vct{v} } \\in \\mathbbm{R}$.  Apply the scalar\nPoincar{\\'e} inequality to $g$ and invoke the definition~\\eqref{eqn:dirichlet-limit}\nof the Dirichlet form.  Thus,\n\\begin{align*}\n\\Expect_{\\mu} \\ip{ \\vct{u} }{ \\mtx{f}(z) \\, \\vct{v} }^2\n\t\\leq \\alpha \\cdot \\lim_{t \\downarrow 0} \\frac{1}{2t} \\Expect_{Z \\sim \\mu} \\big[\n\t\\ip{ \\vct{u} }{ (\\mtx{f}(Z_t) - \\mtx{f}(Z_0)) \\, \\vct{v} }^2 \\, \\big\\vert\\, Z_0 = Z \\big].\n\\end{align*}\nInstate this inequality with $\\vct{v} = \\mathbf{e}_i$ for each $i = 1, \\dots, d$\nand sum over $i$ to arrive at\n$$\n\\ip{\\vct{u}}{ \\smash{\\Var_{\\mu}[ \\mtx{f} ]} \\, \\vct{u}}\n\t\\leq \\ip{ \\vct{u} }{ \\bm{\\mathcal{E}}(\\mtx{f}) \\, \\vct{u} }.\n$$\nAverage over $\\vct{u} \\sim \\textsc{uniform}\\{\\pm 1\\}^d$ to reach the trace Poincar{\\'e}\ninequality.  To extend this argument to complex matrices, apply the same approach\nto the real and imaginary parts of the inner product.\n\\end{proof}\n\n\n\nThe main result of this paper is that concentration properties\nof the random matrix $\\mtx{f}(Z)$ follow from the\ntrace Poincar{\\'e} inequality~\\eqref{eqn:trace-poincare}\nor, equivalently, the scalar Poincar{\\'e} inequality.\n\n\n\\begin{example}[Gaussians]\nIt is well known that the OU process satisfies the Poincar{\\'e}\ninequality with constant $\\alpha = 1$.  Thus, it satisfies\nthe trace Poincar{\\'e} inequality~\\eqref{eqn:trace-poincare}\nwith $\\alpha = 1$.  For an alternative proof, see~\\cite[Thm.~1.2]{ABY19:Matrix-Poincare}.\n\\end{example}\n\n\n\n\n\n\n\n\\subsection{Subexponential concentration and expectation bounds}\n\nWe are now prepared to present our main result.\nIt demands several hypotheses,\nwhich will be enforced throughout the paper.\n\n\\begin{assumption}[Conditions] \\label{ass:main}\nWe assume that\n\\begin{enumerate}\n\\item\tThe Markov process $(Z_t : t \\geq 0) \\subset \\Omega$ is reversible\nand homogeneous, with initial value $Z_0$ and stationary measure $\\mu$.\n\\item\tThe process admits a trace Poincar{\\'e} inequality~\\eqref{eqn:trace-poincare}\nwith constant $\\alpha$.  Equivalently, the process admits a scalar Poincar{\\'e}\ninequality with the same constant $\\alpha$.\n\n\\item\tThe class of valid functions is suitably restricted so that manipulations of\nexpectations, limits, and derivatives are justified.\n\\end{enumerate}\n\\end{assumption}\n\nUnder Assumption~\\ref{ass:main},\nwe will deduce \nsubexponential concentration of the random matrix $\\mtx{f}(Z)$\naround its mean $\\Expect_{\\mu} \\mtx{f}$, where we measure the size of deviations\nwith the $\\ell_2$ operator norm $\\norm{\\cdot}$.  \n\n\\begin{theorem}[Subexponential Concentration] \\label{thm:main}\nEnforce Assumption~\\ref{ass:main}.\nLet $\\mtx{f} : \\Omega \\to \\mathbbm{H}_d$ be a matrix-valued function, and define the variance proxy\n\\begin{equation*} \\label{eqn:vf}\nv_{\\mtx{f}} := \\norm{ \\norm{ \\mtx{\\Gamma}(\\mtx{f})(z) } }_{L_{\\infty}(\\mu)}.\n\\end{equation*}\nFor all $\\lambda > 0$,\n\\begin{equation} \\label{eqn:main-tail}\n\\mathbbm{P}_{\\mu}\\big\\{ \\norm{ \\smash{\\mtx{f} - \\operatorname{\\mathbb{E}}_{\\mu} \\mtx{f}} } \\geq \\sqrt{\\alpha v_{\\mtx{f}}} \\cdot \\lambda  \\big\\}\n\t\\leq 6 d \\cdot \\mathrm{e}^{- \\lambda}.\n\t\\end{equation}\nIn particular,\n\\begin{equation} \\label{eqn:main-expect}\n\\Expect_{\\mu} \\norm{ \\smash{ \\mtx{f} - \\operatorname{\\mathbb{E}}_{\\mu} \\mtx{f} } }\n\t\\leq \\log( 6 \\mathrm{e} d ) \\cdot \\sqrt{\\alpha v_{\\mtx{f}}}.\n\t\\end{equation}\n\\end{theorem}\n\n\\noindent\nThe proof of Theorem~\\ref{thm:main} appears in Section~\\ref{sec:exp-moments}\nafter we present some more background on matrix-valued Markov processes.\nNote that we have made no effort to refine constants.\n\nThe main point of the tail bound~\\eqref{eqn:main-tail} is that\nthe random matrix $\\mtx{f}(Z)$ exhibits exponential concentration on the scale $\\sqrt{\\alpha v_{\\mtx{f}}}$.\nThe variance proxy $v_{\\mtx{f}}$ is analogous to a global bound on\nthe Lipschitz constant of $\\mtx{f}$.\nBe aware that we cannot achieve tail decay faster than exponential\nunder the sole assumption of a Poincar{\\'e} inequality, so this\napproach may not capture the strongest possible concentration.\nThe leading constant in~\\eqref{eqn:main-tail} reflects\nthe ambient dimension $d$ of the matrix;\nthis feature is typical of matrix concentration bounds.\n\n\n\nThe expectation bound~\\eqref{eqn:main-expect} shows\nthat the average value of $\\norm{ \\smash{\\mtx{f} - \\Expect_{\\mu} \\mtx{f}} }$ is proportional to the\nsquare root $\\sqrt{v_{\\vct{f}}}$ of the variance proxy and to the logarithm of the ambient dimension.\nFor many examples, the optimal bound contains the \\emph{square-root} of the logarithm,\nbut the result~\\eqref{eqn:main-expect} is nontrivial and informative.\n\n\n\n \n\n\n\\begin{example}[Gaussian Series]\nTheorem~\\ref{thm:main} applies to a matrix-valued function of a standard normal vector.\nFor instance, according to~\\eqref{eqn:main-expect}, a matrix Gaussian series satisfies the expectation bound\n$$\n\\Expect \\norm{ \\sum_{i=1}^n X_i \\mtx{A}_i } \\leq \\log(6 \\mathrm{e} d) \\cdot \\norm{ \\sum_{i=1}^n \\mtx{A}_i^2 }^{1\/2}.\n$$\nModulo the constant and the power on the logarithm, this bound is qualitatively correct for worst-case examples. On the other hand, the subexponential tail bound~\\eqref{eqn:main-tail} does not reproduce the actual\nsubgaussian behavior.  See~\\cite[Chap.~5]{Tro15:Introduction-Matrix} for discussion.\nExample~\\ref{ex:chaos} describes an application to Gaussian chaos that requires\ntools that are more delicate than Theorem~\\ref{thm:main}.\n\\end{example}\n\n\\begin{remark}[Extensions]\nThe bounds in Theorem~\\ref{thm:main} can be refined in several ways.\nWe can replace the variance proxy $v_{\\mtx{f}}$ with less stringent measures of the size of the carr{\\'e}\ndu champ $\\mtx{\\Gamma}(\\mtx{f})$, such as the expected Schatten $q$-norm with $q \\approx \\log d$. \nIt is also possible to replace the ambient dimension $d$ with a measure\nof the intrinsic dimension of the random matrix $\\mtx{f}(Z)$. \nSee Section~\\ref{sec:poly-moments}.\n\\end{remark}\n\n\\begin{remark}[Rectangular case]\nBy a standard formal argument, we can extend all the results here to\na function $\\mtx{h}: \\Omega \\to \\mathbb{M}_{d_1 \\times d_2}$ that takes\nvalues in the $d_1 \\times d_2$ complex matrices.\nTo do so, we simply apply our results to the self-adjoint function\n$$\n\\mtx{f}(z) = \\begin{bmatrix} \\mtx{0} & \\mtx{h}(z) \\\\ \\mtx{h}(z)^* & \\mtx{0} \\end{bmatrix}\n\t\\in \\mathbbm{H}_{d_1 + d_2}\n\\quad\\text{for $z \\in \\Omega$.}\n$$\nSee~\\cite[Sec.~2.1.17]{Tro15:Introduction-Matrix} for details.\n\\end{remark}\n\n\n\n\n\\subsection{Examples}\n\nTo indicate the scope of Theorem~\\ref{thm:main},\nlet us present some more examples.  Most of these\nexamples are actually known to exhibit subgaussian\nmatrix concentration, but there is\nat least one case (Section~\\ref{sec:scp})\nwhere the results here are currently the best available.\n\n\n\\subsubsection{Log-concave measures}\n\nThe Gaussian case is a particular example of a more general result for log-concave measures.\nSuppose that $J : \\mathbbm{R}^n \\to \\mathbbm{R}$ is a strongly convex function that satisfies\n$\\operatorname{Hess} J \\succcurlyeq \\eta \\mathbf{I}$ uniformly.  Construct the probability\nmeasure $\\mu$ on $\\mathbbm{R}^n$ whose density is proportional to $\\mathrm{e}^{-J}$.\nIn this example, we briefly discuss concentration of matrix-valued functions\n$\\mtx{f}(\\vct{X})$ where $\\vct{X} \\sim \\mu$.  This model is interesting\nbecause it captures a type of negative dependence.\n\nThe appropriate Markov process $(\\vct{X}_t : t \\geq 0)$ evolves with\nthe stochastic differential equation\n$$\n\\diff{\\vct{X}}_t = - \\nabla} \\newcommand{\\subdiff}{\\partial J(\\vct{X}_t) \\idiff{t} + \\sqrt{2} \\idiff{\\vct{B}}_t\n\\quad\\text{with initial value $\\vct{X}_0 \\in \\mathbbm{R}^n$,}\n$$\nwhere $( \\vct{B}_t : t \\geq 0 ) \\subset \\mathbbm{R}^n$ is Brownian motion.\nThe stationary distribution is $\\mu$, and the matrix carr{\\'e} du champ is $$\n\\mtx{\\Gamma}(\\mtx{f})(\\vct{x}) = \\sum_{i=1}^n (\\partial_i \\mtx{f}(\\vct{x}))^2\n\t\\quad\\text{for $\\vct{x} \\in \\mathbbm{R}^n$.}\n$$\nIt is well known that these diffusions satisfy a Poincar{\\'e} inequality\nwith constant $\\alpha = 1\/\\eta$; see~\\cite[Cor.~4.8.2]{BGL14:Analysis-Geometry}.\nTherefore, Theorem~\\ref{thm:main} applies.\n\n\\textit{A fortiori}, these log-concave measures also satisfy a local Poincar{\\'e} inequality,\nwhich leads to subgaussian matrix concentration~\\cite[Sec.~2.12.2]{HT20:Matrix-Concentration}.\n\n\n\\subsubsection{Riemannian manifolds with positive curvature}\n\nMore generally, let $(M, \\mathfrak{g})$ be a compact Riemannian manifold with co-metric $\\mathfrak{g}$.\nThe manifold carries a canonical Riemannian probability measure $\\mu_\\mathfrak{g}$.\nThe diffusion whose infinitesimal generator is the Laplace--Beltrami operator\n$\\Delta_{\\mathfrak{g}}$ is called the Brownian motion on the manifold.  This\nis a reversible, ergodic Markov process.  Its matrix carr{\\'e} du champ takes\nthe form\n\\begin{equation} \\label{eqn:manifold-gamma}\n\\mtx{\\Gamma}(\\mtx{f})(z) = \\sum_{i,j} g^{ij}(z) \\, (\\partial_i \\mtx{f}(z)) \\, (\\partial_j \\mtx{f}(z))\n\\quad\\text{for $\\mtx{f} : \\Omega \\to \\mathbbm{H}_d$.}\n\\end{equation}\nThe co-metric $g$ and the partial derivatives $\\partial_i \\mtx{f}$ are computed\nwith respect to local coordinates.  See~\\cite{BGL14:Analysis-Geometry} for\nan introduction to diffusions on manifolds; the companion paper~\\cite{HT20:Matrix-Concentration}\ntreats matrix-valued functions on manifolds.\n\nSuppose that the manifold has uniformly positive Ricci curvature, where the Ricci tensor\nhas eigenvalues bounded below by $\\rho$.  By now, it is a classic fact that the\nBrownian motion on this manifold satisfies the scalar Poincar{\\'e} inequality\nwith constant $\\alpha = \\rho^{-1}$.  See~\\cite[Sec.~4.8]{BGL14:Analysis-Geometry}.\n\n\\begin{example}[Sphere]\nFor $n \\geq 2$, the unit sphere $\\mathbb{S}^{n} \\subset \\mathbbm{R}^{n+1}$ is a Riemannian submanifold\nof $\\mathbbm{R}^{n+1}$.  Its canonical measure is the uniform probability distribution,\nand the carr{\\'e} du champ of the Brownian motion on the sphere\nis computed using~\\eqref{eqn:manifold-gamma}.\nThe sphere has positive Ricci curvature with $\\rho = n - 1$,\nso it admits a Poincar{\\'e} inequality with $\\alpha = (n-1)^{-1}$.\nThus, matrix-valued functions on the sphere satisfy exponential matrix\nconcentration.\n\\end{example}\n\nA similar story can be told about every positively curved manifold.\nIn fact, in this setting, we even have subgaussian matrix concentration\nbecause of the stronger arguments in~\\cite{HT20:Matrix-Concentration}.\n\n\\subsubsection{Products}\n\nConsider a probability space $(\\Omega, \\mu)$.  It is common to work with multivariate\nfunctions defined on the product space $(\\Omega^n, \\mu^{\\otimes n})$.  There is\na standard construction~\\cite[Sec.~2.3.2]{VH16:Probability-High} of a\nMarkov process on the product space.  Aoun et al.~\\cite{ABY19:Matrix-Poincare}\nverify that the (matrix) carr{\\'e} du champ of this process is\n$$\n\\mtx{\\Gamma}(\\mtx{f})(\\vct{z})\n\t= \\frac{1}{2} \\sum_{i=1}^n \\Expect_{Z \\sim \\mu} \\big[ \\big( \\mtx{f}(z_1, \\dots, z_n) - \\mtx{f}(z_1, \\dots, z_{i-1}, Z,  z_{i+1}, \\dots, z_n) \\big)^2 \\big].\n$$\nThis is the sum of squared discrete derivatives, each averaging over perturbations in a single coordinate.\nThe variance proxy $v_{\\mtx{f}}$ takes the form\n$$\nv_{\\mtx{f}} = \\operatorname{ess\\,sup}\\nolimits_{\\mtx{z} \\in \\Omega^n} \\frac{1}{2} \\norm{ \\sum_{i=1}^n \\Expect_{Z \\sim \\mu} \\big[ \\big( \\mtx{f}(z_1, \\dots, z_n) - \\mtx{f}(z_1, \\dots, z_{i-1}, Z,  z_{i+1}, \\dots, z_n) \\big)^2 \\big] }\n$$\nThe variance proxy coincides with the matrix bounded difference that arises in Paulin et al.~\\cite{PMT16:Efron-Stein-Inequalities}.\nAoun et al.~prove that the Markov process satisfies a trace Poincar{\\'e} inequality~\\eqref{eqn:trace-poincare}\nwith constant $\\alpha = 1$.  Therefore, Theorem~\\ref{thm:main} yields a suboptimal version\nof the exponential matrix Efron--Stein inequality~\\cite[Thm.~4.3]{PMT16:Efron-Stein-Inequalities}.\nSee~\\cite{ABY19:Matrix-Poincare,HT20:Matrix-Concentration} for more details.\n\n\\begin{remark}[Bernstein concentration?]\nIn the scalar case, Bobkov \\& Ledoux~\\cite[Cor.~3.2]{BL97:Poincares-Inequalities} have\nshown that a function $f : \\Omega^n \\to \\mathbbm{R}$ on a product space $(\\Omega^n, \\mu^{\\otimes n})$\nexhibits Bernstein-type concentration when $\\mu$ admits a Poincar{\\'e} inequality  with constant $\\alpha$.\nIn detail,\n$$\n\\Prob{ \\abs{f  - \\Expect f} > t } \\leq \\exp\\left( \\frac{-\\mathrm{c} t^2}{v + Bt} \\right)\n\\quad\\text{with}\\quad\nv = \\norm{ \\sum_{i=1}^n (\\partial_i f)^2 }_{L_{\\infty}} \\quad\\text{and}\\quad\nB = \\max\\nolimits_i \\norm{ \\partial_i f }_{L_{\\infty}}.\n$$\nThe constant $\\cnst{c}$ depends on the constant $\\alpha$ in the Poincar{\\'e} inequality.\nOur approach does not seem powerful enough to transfer this insight to the matrix setting.\nNevertheless, our paper~\\cite{HT20:Matrix-Concentration} demonstrates\nthat a \\emph{local} Poincar{\\'e} inequality is sufficient to achieve\nBernstein concentration.\n\\end{remark}\n\n\n\n\\subsubsection{Stochastic covering property}\n\\label{sec:scp}\n\nAoun et al.~\\cite{ABY19:Matrix-Poincare} have considered a model for negatively\ndependent functions on the hypercube $\\{0, 1\\}^n$, namely the\nclass of measures with the \\emph{stochastic covering property} (SCP).\nFor a $k$-homogeneous measure $\\mu$ with the SCP, it is possible to construct a\nMarkov process that satisfies the trace Poincar{\\'e} inequality~\\eqref{eqn:trace-poincare}\nwith constant $2k$.  Thus, Theorem~\\ref{thm:main} applies.\nSee~\\cite{PP14:Concentration-Lipschitz,HS19:Modified-Log-Sobolev,ABY19:Matrix-Poincare}\nfor a more complete discussion of this example.\n\n\\begin{remark}[Subgaussian concentration?]\nAlthough the Markov process associated with an SCP measure\nsatisfies a log-Sobolev inequality, we do not know\nif it satisfies the local Poincar{\\'e} inequality that we would need to activate\nthe subgaussian concentration inequalities in~\\cite{HT20:Matrix-Concentration}.\n\\end{remark}\n\n\n\\subsubsection{Expander graphs}\n\nLet $G = (\\Omega, E)$ be a $k$-regular, connected, undirected graph.\nWe can construct a Markov process $(Z_t : t \\geq 0) \\subset \\Omega$\nby taking a continuous-time random walk on the vertex set $\\Omega$.\nThe stationary measure $\\mu$ of the random walk is the uniform\nmeasure on $\\Omega$.  The carr{\\'e} du champ operator takes the form\n$$\n\\mtx{\\Gamma}(\\mtx{f})(z) = \\frac{1}{2k} \\sum_{(z',z) \\in E} \\big(\\mtx{f}(z') - \\mtx{f}(z)\\big)^2.\n$$\nIn other words, the ``squared gradient'' is just the half the average squared\ndifference between the matrix at the current vertex and its $k$ neighbors.\n$$\nv_{\\mtx{f}} := \\max_{z \\in \\Omega} \\norm{ \\frac{1}{2k} \\sum_{(z',z)\\in E} \\big(\\mtx{f}(z') - \\mtx{f}(z) \\big)^2 }.\n$$\nThe variance proxy is just the maximum ``squared gradient''\nat any vertex.\n\nAssume that the Markov process satisfies the (scalar) Poincar{\\'e}\ninequality, Proposition~\\ref{prop:equiv}(1), with constant $\\alpha$.\nIn this case, we say that $G$ is an $\\alpha$-expander graph.\nAccording to Theorem~\\ref{thm:main}, a matrix-valued function\n$\\mtx{f} : \\Omega \\to \\mathbbm{H}_d$ on an $\\alpha$-expander graph satisfies a\nsubexponential matrix concentration inequality:\n$$\n\\Prob{ \\norm{ \\smash{\\mtx{f} - \\Expect_\\mu \\mtx{f}} }\\geq \\sqrt{\\alpha v_{\\mtx{f}}} \\cdot \\lambda }\n\t\\leq 6d \\cdot \\mathrm{e}^{-\\lambda}.\n$$\nTherefore, local control over the fluctuations yields global concentration around the mean.\nThe number of vertices where the function departs from its mean is controlled by\nthe quality $\\alpha$ of the expander.\n\n\n\nThis example is potentially interesting because the Markov process\ndoes not satisfy a dimension-independent (modified)\nlog-Sobolev inequality~\\cite[Sec.~5]{BT06:Modified-Logarithmic}.\nIndeed, to achieve the functional inequality\n$$\n\\mathrm{Ent}_{\\mu}(f) \\leq\n\t\\beta \\cdot \\mathcal{E}(f, \\log f)\n\t\\quad\\text{for all $f : \\Omega \\to \\mathbbm{R}$},\n$$\nit is necessary that $\\beta \\geq \\mathrm{const} \\cdot \\log( \\# \\Omega )$.\nSee~\\cite[Sec.~4.3]{Mun19:Li-Yau-Inequality} for related\nresults on curvature-dimension conditions of Bakry--{\\'E}mery type.\n\n\n\\begin{remark}[Matrix Expander Chernoff]\nAlthough the modified log-Sobolev inequality fails,\nit is still possible to establish subgaussian concentration\nfor the ergodic averages of a matrix-valued random walk\non an expander graph~\\cite{GLSS18:Matrix-Expander}.\n\\end{remark}\n\n\n\n\n\\section{Related work}\n\n\n\\subsection{Markov processes}\n\nMuch of the classical research on Markov processes concerns the relationship between\nthe geometry of the state space and the behavior of\ncanonical diffusion processes (e.g., Brownian motion\non a Riemannian manifold).\nFor an introduction, we recommend\nthe lecture notes~\\cite{VH16:Probability-High}.  A more comprehensive\nsource is the treatise~\\cite{BGL14:Analysis-Geometry}.  \n\nMatrix-valued Markov processes first arose in the mathematical\nphysics literature as a model for the evolution of a quantum system.\nSome of the foundational works include Davies~\\cite{Dav69:Quantum-Stochastic}\nand Lindblad~\\cite{Lin76:Generators-Quantum}.\nQuantum information theory has provided a new impetus\nfor studying matrix-valued Markov processes;\nsee~\\cite{KT13:Quantum-Logarithmic}\nfor a discussion and some background references.\n\nHere, we are interested in a mixed classical-quantum\nsetting, where a classical Markov process drives\na matrix-valued function.  Surprisingly, this model\ndoes not seem to have received much attention\nuntil the last few years.\nSee Cheng et al.~\\cite{CHT17:Exponential-Decay}\nfor a more expansive framework that includes this case.\nOther foundational results appear\nin~\\cite{ABY19:Matrix-Poincare,HT20:Matrix-Concentration}.\n\n\\subsection{Functional inequalities}\n\nIn the scalar setting, the connection between functional inequalities,\nconvergence of Markov processes, and concentration\nis a long-standing topic of research.  References\ninclude~\\cite{Led01:Concentration-Measure,BLM13:Concentration-Inequalities,\nBGL14:Analysis-Geometry,VH16:Probability-High}.\n\nFunctional inequalities for matrices were\noriginally formulated in the mathematical physics literature;\nfor example, see the work of Gross~\\cite{Gro75:Hypercontractivity-Logarithmic}.\nThe application of functional inequalities to the\nergodicity of matrix-valued Markov processes dates\nback at least as far as the\npapers~\\cite{MOZ98:Dissipative-Dynamics,OZ99:Hypercontractivity-Noncommutative}.\n\nFunctional inequalities in the mixed classical-quantum setting\nseem to have a more recent vintage.  Chen \\& Tropp~\\cite{CT14:Subadditivity-Matrix}\nformulated subadditivity properties for tracial entropy-like\nquantities, including the trace variance~\\eqref{eqn:matrix-var}.\nThey showed that these properties imply some\nSobolev and modified log-Sobolev-type inequalities for random matrices,\nand they obtained some restricted matrix concentration inequalities.\nSome of the partial results from~\\cite{CT14:Subadditivity-Matrix}\nwere completed in~\\cite{HZ15:Characterization-Matrix,PV15:Joint-Convexity,CH16:Characterizations-Matrix}.\n\nCheng et al.~\\cite{CHT17:Exponential-Decay} developed a framework for studying Markov\nprocesses in the mixed classical-quantum setting (and beyond),\nand they showed an equivalence between tracial log-Sobolev\ninequalities and exponential ergodicity of the trace entropy.\nFurther results and implications for concentration appear in~\\cite{CH19:Matrix-Poincare}.\nAt present, we do not have a full picture of the relationships\nbetween matrix functional inequalities and matrix concentration.\n\nVan Handel (personal communication) has pointed out that we can derive\nnonlinear matrix concentration for functions of a log-concave measure\nwith a strongly convex potential by combining Pisier's method~\\cite{Pis86:Probabilistic-Methods}, the (sharp)\nnoncommutative Khintchine inequality~\\cite{Buc01:Operator-Khintchine,Tro16:Expected-Norm},\nand Caffarelli's contraction theorem~\\cite{Caf00:Monotonicity-Properties}.\nThis approach gives subgaussian concentration, which is better than we can obtain via\nthe trace Poincar{\\'e} inequality for log-concave measures, but\nit apparently does not extend beyond this setting.\n\n\n\n\\subsection{From Poincar{\\'e} to concentration}\n\nIt has been recognized for about 40 years that Poincar{\\'e} inequalities\nimply exponential concentration. Gromov \\& Milman~\\cite[Thm.~4.1]{GM83:Topological-Application}\nprove such a theorem in the context of Riemannian manifolds.\nThe standard argument, a recursive estimate of the moment generating function, is attributed\nto Aida \\& Stroock~\\cite{AS94:Moment-Estimates}.  For a textbook presentation,\nsee~\\cite[Sec.~3.6]{BLM13:Concentration-Inequalities}.\n\nConsider a matrix Poincar{\\'e} inequality of the form\n\\begin{equation} \\label{eqn:matrix-poincare}\n\\Var_{\\mu}[\\mtx{f}]\t\n\\preccurlyeq \\alpha \\cdot \\bm{\\mathcal{E}}(\\mtx{f}).\n\\end{equation}\nThe argument in Proposition~\\ref{prop:equiv} shows that\nthis matrix Poincar{\\'e} inequality~\\eqref{eqn:matrix-poincare}\nis also equivalent to a scalar Poincar{\\'e} inequality\nwith the same constant $\\alpha$.\nThe papers~\\cite{CH16:Characterizations-Matrix,CHT17:Exponential-Decay,CH19:Matrix-Poincare}\ndemonstrate that~\\eqref{eqn:matrix-poincare} leads to some inequalities for\nthe matrix variance~\\eqref{eqn:matrix-var} and its trace,\nbut these approaches do not lead to matrix concentration\ninequalities like Theorem~\\ref{thm:main}.\n\n\nAoun, Banna, and Youssef~\\cite{ABY19:Matrix-Poincare} have recently\nshown that the matrix Poincar{\\'e} inequality~\\eqref{eqn:matrix-poincare}\ndoes imply exponential concentration of a random matrix about\nits mean with respect to the $\\ell_2$ operator norm.\nModulo constants, their result is equivalent with the tail bound~\\eqref{eqn:main-tail},\nbut it is weaker than the bounds in Theorem~\\ref{thm:poly-moment}.\nThe proof in~\\cite{ABY19:Matrix-Poincare} is a direct analog of the argument of Aida \\& Stroock~\\cite{AS94:Moment-Estimates}.\nBut, in the matrix setting, the recursive estimate requires some heavy lifting.\nAnother contribution of the paper~\\cite{ABY19:Matrix-Poincare} is to establish that some\nparticular matrix-valued Markov processes satisfy~\\eqref{eqn:matrix-poincare}.\nNevertheless, Proposition~\\ref{prop:equiv} indicates that no additional\neffort is required for this end.\n\nOur approach is similar in spirit to the work of Aoun et al.~\\cite{ABY19:Matrix-Poincare},\nbut we use a symmetrization argument to avoid the difficult recursion.\nFor related work in the scalar setting, see~\\cite[Sec.~4]{BL97:Poincares-Inequalities}.\nIn a companion paper~\\cite{HT20:Matrix-Concentration}, we\nshow that \\emph{local} Poincar{\\'e} inequalities lead to much\nstronger ergodicity and concentration properties.  The theory\nin the companion paper is significantly more involved than\nthe development here, so we have chosen to separate them.\nFor results on (local) Poincar{\\'e} inequalities in\nnoncommutative probability spaces,\nsee Junge \\& Zeng~\\cite{JZ15:Noncommutative-Martingale}.\n\n\n\n\n\n\n\n\n\\section{Subadditivity}\n\nTo control the variance of a function of several independent variables,\nit is helpful to understand the influence of each individual variable.\nAs in the scalar setting, the matrix variance can be bounded by\na sum of conditional variances.  We can control each conditional\nvariance by a conditional application of a trace Poincar{\\'e} inequality.\nFor simplicity, we focus on the case that is relevant to our proof,\nbut these results hold in greater generality\n(products of more than two spaces that may carry different measures).\nSome of the material in this section is drawn\nfrom~\\cite{CT14:Subadditivity-Matrix,CH16:Characterizations-Matrix}.\nSee~\\cite{Led96:Talagrands-Deviation} for some of the classic\nwork on subadditivity and concentration. \n\n\\subsection{Influence of a coordinate}\n\nConsider the product space $(\\Omega^2, \\mu \\otimes \\mu)$.\nWe want to study how individual coordinates affect the\nbehavior of a matrix-valued function $\\mtx{g} : \\Omega^2 \\to \\mathbbm{H}_d$.\n\nFirst, introduce notation for the expectation of the function\nwith respect to each coordinate:\n\\begin{align*}\n\\Expect_{1}[ \\mtx{g} ] (z_2) := \\Expect_{Z \\sim \\mu}[ \\mtx{g}(Z, z_2) ] \\in \\mathbbm{H}_d \\quad\\text{for all $z_1 \\in \\Omega$;} \\\\\n\\Expect_{2}[ \\mtx{g} ] (z_1) := \\Expect_{Z \\sim \\mu}[ \\mtx{g}(z_1, Z) ] \\in \\mathbbm{H}_d \\quad\\text{for all $z_2 \\in \\Omega$.}\n\\end{align*}\nThe coordinate-wise variance is the positive-semidefinite random matrix\n$$\n\\Var_i[ \\mtx{g} ] := \\Expect_{i} \\big[ (\\mtx{g} - \\Expect_i \\mtx{g} )^2 \\big] \\in \\mathbbm{H}_d^+\n\t\\quad\\text{for $i = 1, 2$.}\n$$\nThis matrix reflects the fluctuation in the $i$th coordinate,\nwith the other coordinate held fixed.\n\nSimilarly, we can introduce the coordinate-wise carr{\\'e} du champ operator\nand Dirichlet form:\n\\begin{align}\n\\mtx{\\Gamma}_1(\\mtx{g})(z_1, z_2) &:= \\lim_{t \\downarrow 0} \\frac{1}{2t}\n\t\\Expect \\big[ \\big( \\mtx{g}(Z_t, z_2) - \\mtx{g}(Z_0, z_2) \\big)^2 \\, \\big| \\, Z_0 = z_1 \\big] \\in \\mathbbm{H}_d^+; \\label{eqn:cond-carre} \\\\\n\\bm{\\mathcal{E}}_1(\\mtx{g})(z_2) &:= \\lim_{t \\downarrow 0} \\frac{1}{2t} \\Expect_{Z \\sim \\mu} \n\t\\big[ \\big( \\mtx{g}(Z_t, z_2) - \\mtx{g}(Z_0, z_2) \\big)^2 \\, \\big| \\, Z_0 = Z \\big] \\in \\mathbbm{H}_d^+. \\label{eqn:cond-dirichlet}\n\\end{align}\nAs usual, $(Z_t : t \\geq 0)$ is a realization of the Markov process with initial\nvalue $Z_0$. We make analogous definitions for the second coordinate $i = 2$.\n\nLast, we extend the carr{\\'e} du champ operator and the Dirichlet form\nto bivariate functions:\n\\begin{align}\n\\mtx{\\Gamma}(\\mtx{g})\n\t&:= \\mtx{\\Gamma}_1(\\mtx{g})\n\t+ \\mtx{\\Gamma}_2(\\mtx{g}) \\in \\mathbbm{H}_d^+;  \\label{eqn:bivar-carre} \\\\\n\\bm{\\mathcal{E}}(\\mtx{g})\n\t&:= \\operatorname{\\mathbb{E}}_{\\mu \\otimes \\mu}\\big[ \\bm{\\mathcal{E}}_1(\\mtx{g})\n\t+ \\bm{\\mathcal{E}}_2(\\mtx{g}) \\big]  \\in \\mathbbm{H}_d^+.  \\label{eqn:bivar-dirichlet}\t\n\\end{align}\nThese formulas have a heuristic interpretation: the squared derivative\nof a bivariate function is the sum of the squared partial derivatives.\n\n\n\n\\subsection{Trace variance is subadditive}\n\nObserve that the trace variance is controlled\nby the sum of the coordinate-wise variances.\n\n\\begin{fact}[Trace variance: Subadditivity] \\label{fact:var-decomp}\nLet $\\mtx{g} : \\Omega^2 \\to \\mathbbm{H}_d$ be a matrix-valued function on the product\nspace $(\\Omega^2, \\mu \\otimes \\mu)$.  Then\n$$\n\\operatorname{tr} \\Var_{\\mu \\otimes \\mu}[ \\mtx{g} ] \\leq \\Expect_2 \\operatorname{tr} \\Var_1[ \\mtx{g} ] + \\Expect_1 \\operatorname{tr} \\Var_2[ \\mtx{g} ]. \n$$\n\\end{fact}\n\nThis result is due to Chen \\& Tropp~\\cite{CT14:Subadditivity-Matrix},\nwho showed that other matrix functions are also subadditive.  Later,\nCheng \\& Hsieh~\\cite{CH16:Characterizations-Matrix} noticed that\nan analogous result holds without the trace.  Similar decompositions\nare also valid for functions on the $n$-fold product $(\\Omega^n, \\mu^{\\otimes n})$.\n\n\\begin{proof} The proof is the same as in the scalar setting.\nWriting $\\Expect = \\Expect_{\\mu \\otimes \\mu}$ for the total expectation,\n\\begin{align*}\n\\Expect \\Var_{\\mu \\otimes \\mu}[ \\mtx{g} ]\n\t:= \\Expect  \\big[ (\\mtx{g} - \\Expect \\mtx{g})^2 \\big]\n\t&= \\Expect  \\big[ (\\mtx{g} - \\Expect_1 \\mtx{g})^2 + (\\Expect_1 \\mtx{g} - \\Expect_{1} \\Expect_{2} \\mtx{g})^2 \\big] \\\\\n\t&\\preccurlyeq \\Expect [ (\\mtx{g} - \\Expect_1 \\mtx{g})^2 ]\n\t+ \\Expect [ (\\mtx{g} - \\Expect_2 \\mtx{g})^2 ] \\\\\n\t&= \\Expect_2  \\Var_1[ \\mtx{g} ] + \\Expect_1  \\Var_2[ \\mtx{g} ].\n\\end{align*}\nThe first line relies on orthogonality of the conditionally centered\nrandom matrices.  The second line requires the operator convexity\nof the square, applied conditionally.  Last, take the trace.\n\\end{proof}\n\n\n\\subsection{Trace Poincar{\\'e} inequalities are subadditive}\n\nIf the Markov process satisfies a trace Poincar{\\'e} inequality,\nthen the variance of a bivariate function\nalso satisfies a trace Poincar{\\'e} inequality.\n\n\\begin{proposition}[Trace Poincar{\\'e}: Subadditivity] \\label{prop:poincare-subadd}\nSuppose that the Markov process satisfies a trace Poincar{\\'e} inequality~\\eqref{eqn:trace-poincare}\nwith constant $\\alpha$.  Let $\\mtx{g} : \\Omega^2 \\to \\mathbbm{H}_d$ be a suitable bivariate matrix-valued function.\nThen\n$$\n\\operatorname{tr} \\Var_{\\mu \\otimes \\mu}[ \\mtx{g} ] \t\\leq \\alpha \\cdot \\operatorname{tr} \\bm{\\mathcal{E}}(\\mtx{g}).\n$$\n\\end{proposition}\n\n\\begin{proof}\nStart with Fact~\\ref{fact:var-decomp}.  Apply the trace Poincar{\\'e}\ninequality~\\eqref{eqn:trace-poincare} coordinate-wise to control\neach of the two coordinate-wise variances.  Last, introduce the definition~\\eqref{eqn:bivar-dirichlet}\nof the Dirichlet form for a bivariate function.\n\\end{proof}\n\n\n\n\\section{Chain rule inequality for the Dirichlet form}\n\nThe key new tool in our approach is a simple trace inequality for the matrix\nDirichlet form that shows how it interacts with composition.\n\n\\begin{proposition}[Chain rule inequality] \\label{prop:chain-rule}\nEnforce Assumption~\\ref{ass:main}.\nLet $\\phi : \\mathbbm{R} \\to \\mathbbm{R}$ be a scalar function whose squared derivative $\\psi = (\\phi')^2$ is convex.\nThen\n$$\n\\operatorname{tr} \\bm{\\mathcal{E}}( \\phi(\\mtx{f})) \n\t= \\Expect_{\\mu} \\operatorname{tr} \\mtx{\\Gamma}(\\phi(\\mtx{f}))\n\t\\leq \\Expect_{\\mu} \\operatorname{tr} \\big[ \\mtx{\\Gamma}(\\mtx{f}) \\, \\psi(\\mtx{f}) \\big]\n\t\\quad\\text{for all suitable $\\mtx{f} : \\Omega \\to \\mathbbm{H}_d$.}\n$$\nIn particular,\n$$\n\\operatorname{tr} \\bm{\\mathcal{E}}( \\phi(\\mtx{g})) \n\t= \\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\mtx{\\Gamma}(\\phi(\\mtx{g}))\n\t\\leq \\Expect_{\\mu \\otimes \\mu} \\operatorname{tr}  \\big[\\mtx{\\Gamma}(\\mtx{g}) \\, \\psi(\\mtx{g}) \\big]\n\t\\quad\\text{for all suitable $\\mtx{g} : \\Omega^2 \\to \\mathbbm{H}_d$.}\n$$\n\\end{proposition}\n\n\\noindent\nThe proof of Proposition~\\ref{prop:chain-rule} consumes the rest\nof this section.\n\nFor context, recall that the carr{\\'e} du champ operator $\\Gamma$ of a (scalar-valued, reversible) diffusion process\nsatisfies a chain rule~\\cite[Sec.~1.11]{BGL14:Analysis-Geometry}:\n$$\n\\Gamma( \\phi(f) ) = \\Gamma( f ) \\, \\phi'(f)^2\n\\quad\\text{for smooth $f$ and $\\phi$.}\n$$\nProposition~\\ref{prop:chain-rule} provides a substitute for this relation for an arbitrary reversible Markov process\nthat takes matrix values.  In exchange for the wider applicability,\nwe need some additional averaging (provided by the Dirichlet form);\nwe must restrict our attention to functions $\\phi$ with a convexity property;\nand the equality is relaxed to an inequality.\nIn the scalar case, Proposition~\\ref{prop:chain-rule} is related\nto the Stroock--Varopoulos inequality~\\cite{Str84:Introduction-Theory,Var85:Hardy-Littlewood-Theory}.\n\n\n\n\n\\subsection{Mean-value inequality for trace functions}\n\nThe argument hinges on an elementary trace inequality for deterministic matrices. This result is obtained by lifting a numerical inequality\nto self-adjoint matrices.  A very similar statement~\\cite[Lem.~3.4]{MJCFT14:Matrix-Concentration}\nanimates the exchangeable pairs approach to matrix concentration,\nwhich is motivated by work in the scalar setting~\\cite{Cha07:Steins-Method}.\n\n\n\n\\begin{lemma}[Mean-value trace inequality] \\label{lem:mvti}\nLet $\\mtx{A}, \\mtx{B} \\in \\mathbbm{H}_d$.  Let $\\phi : \\mathbbm{R} \\to \\mathbbm{R}$ be a scalar function\nwhose squared derivative $\\psi = (\\phi')^2$ is convex.  Then\n$$\n\\operatorname{tr} \\big[ \\big(\\phi(\\mtx{A}) - \\phi(\\mtx{B})\\big)^2 \\big]\n\t\\leq \\frac{1}{2} \\operatorname{tr} \\big[ (\\mtx{A} - \\mtx{B})^2 \\big( \\psi(\\mtx{A}) + \\psi(\\mtx{B}) \\big) \\big].\n$$\n\\end{lemma}\n\n\\begin{proof}\nLet $a, b \\in \\mathbbm{R}$.  The fundamental theorem of calculus and Jensen's inequality together deliver the relations\n\\begin{align*}\n\\big(\\phi(a) - \\phi(b)\\big)^2 &= (a-b)^2 \\left[ \\int_0^1 \\diff{\\tau} \\, \\phi'\\big(\\tau a + (1-\\tau) b \\big) \\right]^2 \\\\\n\t&\\leq (a-b)^2 \\int_0^1 \\diff{\\tau} \\,\\psi\\big( \\tau a + (1-\\tau) b\\big) \\\\\n\t&\\leq (a-b)^2 \\int_0^1 \\diff{\\tau} \\, \\big[ \\tau \\psi(a) + (1-\\tau) \\psi(b) \\big]\n\t= \\frac{1}{2} (a-b)^2 \\big(\\psi(a) + \\psi(b)\\big).\n\\end{align*}\nThe generalized Klein inequality~\\cite[Prop.~3]{Pet94:Survey-Certain} allows us to lift\nthis numerical fact to a trace inequality for matrices $\\mtx{A}, \\mtx{B} \\in \\mathbbm{H}_d$.\n\\end{proof}\n\n\\subsection{Proof of Proposition~\\ref{prop:chain-rule}}\n\nThe result follows from a short calculation.\nFirst, we use the definition~\\eqref{eqn:dirichlet-limit}\nof the Dirichlet form as a limit:\n\\begin{align*}\n\\operatorname{tr} \\bm{\\mathcal{E}}( \\phi(\\mtx{f}) )\n\t&= \\lim_{t \\downarrow 0} \\frac{1}{2t} \\Expect_{Z \\sim \\mu} \\big[\\operatorname{tr}\\big[ \\big(\\phi(\\mtx{f}(Z_t)) - \\phi(\\mtx{f}(Z_0)) \\big)^2 \\big] \\, \\big| \\, Z_0 = Z \\big] \\\\\n\t&\\leq \\lim_{t \\downarrow 0} \\frac{1}{4t} \\Expect_{Z \\sim \\mu} \\operatorname{tr} \\big[ \\big(\\mtx{f}(Z_t) - \\mtx{f}(Z_0) \\big)^2 \\big(\\psi(\\mtx{f}(Z_t)) + \\psi(\\mtx{f}(Z_0)) \\big) \\, \\big| \\, Z_0 =Z \\big] \\\\\n\t&= \\lim_{t \\downarrow 0} \\frac{1}{2t} \\Expect_{Z \\sim \\mu} \\operatorname{tr} \\big[ (\\mtx{f}(Z_t) - \\mtx{f}(Z_0))^2 \\, \\psi(\\mtx{f}(Z_0)) \\, \\big| \\, Z_0 = Z \\big] \\\\\n\t&= \\Expect_{Z \\sim \\mu} \\operatorname{tr} \\Big[ \\lim_{t \\downarrow 0} \\frac{1}{2t} \\Expect \\Big[ (\\mtx{f}(Z_t) - \\mtx{f}(Z_0))^2 \\, \\big| \\, Z_0 = Z \\Big] \\, \\psi(\\mtx{f}(Z)) \\Big] \\\\\n\t&= \\Expect_{\\mu} \\operatorname{tr} \\big[ \\mtx{\\Gamma}(\\mtx{f}) \\, \\psi(\\mtx{f}) \\big].\n\\end{align*}\nThe inequality is Lemma~\\ref{lem:mvti}.  To reach the third line, we use the\nfact that $(Z_0, Z_t)$ is an exchangeable pair for each $t \\geq 0$, a consequence of the reversibility\nof the Markov process $(Z_t : t \\geq 0)$ and the fact that $Z_0 \\sim \\mu$.\nLast, we condition on the value of $Z_0$, invoke dominated convergence to pass the expectation through\nthe limit, and we apply the definition~\\eqref{eqn:carre-limit} of the carr{\\'e} du champ operator.\n\n\n\n\n\\section{Exponential moments}\n\\label{sec:exp-moments}\n\n\nOur main technical result is a bound for the exponential moments of a general matrix-valued\nfunction on the state space.  In contrast to the usual approach of bounding the moment generating\nfunction, we will compute the expectation of a hyperbolic function.  \n\\begin{theorem}[Exponential moments] \\label{thm:exp-moment}\nEnforce Assumption~\\ref{ass:main}.\nLet $\\mtx{f} : \\Omega \\to \\mathbbm{H}_d$ be a function with $\\Expect_{\\mu} \\mtx{f} = \\mtx{0}$.\nFor $\\theta > 0$,\n\\begin{equation} \\label{eqn:exp-moment}\n\\Expect_{\\mu} \\operatorname{tr} \\cosh(\\theta \\mtx{f})\n\t\\leq d \\cdot \\left[ 1 + \\frac{\\alpha \\theta^2 \\operatorname{\\bar{\\trace}} \\bm{\\mathcal{E}}(\\mtx{f})}{(1 - \\alpha v_{\\mtx{f}} \\theta^2\/2)_+} \\right].\n\\end{equation}\nThe variance proxy $v_{\\mtx{f}}$ is defined in~\\eqref{eqn:vf},\nand $\\operatorname{\\bar{\\trace}}$ is the normalized trace.\n\\end{theorem}\n\nThe proof of Theorem~\\ref{thm:exp-moment} occupies the rest of the section.\nBut first, we use this moment bound to derive our main result, Theorem~\\ref{thm:main}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:main} from Theorem~\\ref{thm:exp-moment}]\nWithout loss, assume that $\\Expect_{\\mu} \\mtx{f} = \\mtx{0}$.\nWe use the matrix moment method~\\cite{AW02:Strong-Converse,Oli10:Sums-Random}:\n\\begin{align*}\n\\mathbbm{P}_{\\mu} \\left\\{ \\norm{ \\mtx{f} } \\geq \\lambda \\right\\}\n\t&\\leq \\inf_{\\theta > 0} \\frac{1}{\\cosh(\\theta \\lambda)} \\cdot \\Expect_\\mu \\cosh(\\theta \\norm{\\mtx{f}})\n\t= \\inf_{\\theta > 0} \\frac{2}{\\mathrm{e}^{\\theta \\lambda} + \\mathrm{e}^{-\\theta \\lambda}} \\cdot \\Expect_{\\mu} \\norm{ \\smash{\\cosh(\\theta \\mtx{f})} } \\\\\n\t&\\leq \\inf_{\\theta > 0} \\frac{2}{\\mathrm{e}^{\\theta \\lambda}} \\cdot \\Expect_{\\mu} \\operatorname{tr} \\cosh(\\theta \\mtx{f})\n\t\\leq 2d \\cdot \\inf_{\\theta > 0} \\mathrm{e}^{-\\theta \\lambda} \\cdot \\left[ 1 + \\frac{\\alpha \\theta^2 \\operatorname{\\bar{\\trace}} \\bm{\\mathcal{E}}(\\mtx{f})}{(1 - \\alpha v_{\\mtx{f}} \\theta^2\/2)_+} \\right] \\\\\n\t&\\leq 2d \\cdot \\mathrm{e}^{- \\lambda \/ \\sqrt{\\alpha v_{\\mtx{f}}}} \\cdot \\left[ 1 + \\frac{2 \\operatorname{\\bar{\\trace}} \\bm{\\mathcal{E}}(\\mtx{f})}{ v_{\\mtx{f}} } \\right]\n\t\\leq 6 d \\cdot \\mathrm{e}^{- \\lambda \/ \\sqrt{\\alpha v_{\\mtx{f}}}}.\n\\end{align*}\nThe first inequality is Markov's.  The second relation is the spectral mapping theorem.\nThe $\\ell_2$ operator norm of a positive-definite matrix is obviously bounded by its trace.\nThen invoke Theorem~\\ref{thm:exp-moment} to control the moment.\nWe have chosen $\\theta = (\\alpha v_{\\mtx{f}})^{-1\/2}$.  Last, we have noted that\n$\\operatorname{\\bar{\\trace}} \\bm{\\mathcal{E}}(\\mtx{f}) \\leq \\norm { \\bm{\\mathcal{E}}(\\mtx{f})} \\leq v_{\\mtx{f}}$.\nTo finish the proof of the tail bound~\\eqref{eqn:main-tail},\nmake the change of variables $\\lambda \\mapsto \\lambda \\sqrt{\\alpha v_{\\mtx{f}}}$.\n\n\nThe expectation bound~\\eqref{eqn:main-expect} follows when we integrate the tail bound~\\eqref{eqn:main-tail},\ntaking into account that the probability is also bounded by one; we omit the details.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:exp-moment}}\n\nOur goal is to develop an exponential moment bound for a\nfunction $\\mtx{f} : \\Omega \\to \\mathbbm{H}_d$ that satisfies\n$\\Expect_{\\mu} \\mtx{f} = \\mtx{0}$.  We will need to work\nwith both the hyperbolic sine and cosine,\npassing between them using simple identities.\nAfter writing this paper, we learned that this proof is\na matrix analog of an argument proposed by\nBobkov \\& Ledoux~\\cite[Sec.~4]{BL97:Poincares-Inequalities}.\n\n\n\n\\subsubsection{Symmetrization}\n\nThe first step is to symmetrize the function.  Let $Z, Z' \\in \\Omega$\nbe independent random variables, each with distribution $\\mu$.\nSince $\\mtx{f}(Z')$ has mean zero, a conditional application\nof Jensen's inequality yields\n\\begin{equation} \\label{eqn:coshf-bd}\n\\Expect \\operatorname{tr} \\sinh^2(\\theta \\mtx{f}(Z))\n\t\\leq \\Expect \\operatorname{tr} \\sinh^2( \\theta (\\mtx{f}(Z) - \\mtx{f}(Z')))\n\t=: \\Expect \\operatorname{tr} \\sinh^2( \\theta \\mtx{g}(Z,Z')).\n\\end{equation}\nIndeed, since $\\sinh^2$ is convex, the function $\\operatorname{tr} \\sinh^2$ is also convex~\\cite[Prop.~2]{Pet94:Survey-Certain}.\nWe have defined the antisymmetric function $\\mtx{g}(z, z') = \\mtx{f}(z) - \\mtx{f}(z')$ for $z, z' \\in \\Omega$.\n\n\\subsubsection{From moments to variance}\n\\label{sec:exp-mom-var}\n\nNext, we will write the expectation as a variance.\nConsider the odd function $\\phi(s) = \\sinh(\\theta s)$.\nFirst, we claim that\n$$\n\\Expect \\sinh( \\theta \\mtx{g}(Z, Z') ) = \\mtx{0}.\n$$\nIndeed, using the antisymmetry of $\\mtx{g}$\nand the oddness of $\\phi$,\n$$\n\\Expect \\sinh( \\theta \\mtx{g}(Z, Z') ) = \\Expect \\sinh( - \\theta \\mtx{g}(Z', Z) )\n\t= - \\Expect \\sinh( \\theta \\mtx{g}(Z', Z) )\n\t= - \\Expect \\sinh( \\theta \\mtx{g}(Z, Z') ).\n$$\nThe last identity holds because $(Z, Z')$ is exchangeable.\n\nAs an immediate consequence,\n$$\n\\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\sinh^2( \\theta \\mtx{g} )\n\t= \\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\sinh^2( \\theta \\mtx{g} ) - \\operatorname{tr}\\big[ \\big(\\Expect_{\\mu \\otimes \\mu} \\sinh(\\theta \\mtx{g}) \\big)^2 \\big]\n\t=  \\operatorname{tr} \\Var_{\\mu \\otimes \\mu}[ \\sinh(\\theta \\mtx{g}) ].\n$$\nThe appearance of the variance gives us access to Poincar{\\'e} inequalities.\n\n\\subsubsection{Poincar{\\'e} inequality}\n\nTo continue, we apply Proposition~\\ref{prop:poincare-subadd}, the trace Poincar{\\'e}\ninequality for bivariate functions:\n\\begin{equation} \\label{eqn:exp-mom-pf-2}\n\\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\sinh^2( \\theta\\mtx{g} )\n\t\\leq \\alpha \\cdot \\operatorname{tr} \\bm{\\mathcal{E}}( \\sinh(\\theta \\mtx{g}) )\n\t\\leq \\alpha \\theta^2 \\cdot \\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\big[ \\mtx{\\Gamma}(\\mtx{g}) \\cosh^2(\\theta \\mtx{g}) \\big].\n\\end{equation}\nThe second inequality is the chain rule, Proposition~\\ref{prop:chain-rule},\nfor the Dirichlet form.  To activate it, we note\nthat the squared derivative of $\\phi(s) = \\sinh(\\theta s)$\nis the convex function $\\psi(s) = \\theta^2 \\cosh^2(\\theta s)$.\n\n\n\n\\subsubsection{Moment comparison}\n\nA moment comparison argument allows us to isolate the exponential moment.\nDefine the variance proxy of the bivariate function:\n\\begin{equation} \\label{eqn:vg}\nv_{\\mtx{g}} := \\norm{ \\norm{ \\mtx{\\Gamma}(\\mtx{g}) } }_{L_{\\infty}(\\mu \\otimes \\mu)}.\n\\end{equation}\nContinuing from~\\eqref{eqn:exp-mom-pf-2}, the identity $\\cosh^2 = 1 + \\sinh^2$ implies that\n\\begin{align*}\n\\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\sinh^2( \\theta\\mtx{g} )\n\t&\\leq \\alpha \\theta^2 \\cdot \\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\mtx{\\Gamma}(\\mtx{g}) + \\alpha \\theta^2 \\cdot \\Expect_{\\mu \\otimes \\mu} \\operatorname{tr}\\big[\\mtx{\\Gamma}(\\mtx{g}) \\sinh^2(\\theta \\mtx{g}) \\big] \\\\\n\t&\\leq \\alpha \\theta^2 \\cdot \\operatorname{tr} \\bm{\\mathcal{E}}(\\mtx{g}) + \\alpha \\theta^2 v_{\\mtx{g}} \\cdot \\Expect_{\\mu\\otimes \\mu} \\operatorname{tr} \\sinh^2(\\theta \\mtx{g}).\n\\end{align*}\nThe second inequality is just the usual operator-norm bound for the trace.\nRearrange this identity to arrive at\n\\begin{equation} \\label{eqn:coshg-bd}\n\\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\sinh^2( \\theta \\mtx{g} )\n\t\\leq \\frac{\\alpha \\theta^2 \\operatorname{tr} \\bm{\\mathcal{E}}(\\mtx{g})}{(1 - \\alpha v_{\\mtx{g}} \\theta^2)_+}.\n\\end{equation}\nIt remains to revert to the original function $\\mtx{f}$.\n\n\\subsubsection{Comparison of carr{\\'e} du champs}\n\nLet us compute the Dirichlet form $\\bm{\\mathcal{E}}(\\mtx{g})$ and the variance proxy $v_{\\mtx{f}}$\nin terms of the original function $\\mtx{f}$.\nTo that end, observe that the coordinate-wise carr{\\'e} du champ~\\eqref{eqn:cond-carre}\nsatisfies\n\\begin{align*}\n\\mtx{\\Gamma}_1(\\mtx{g})(z,z')\n\t&= \\Expect \\lim_{t \\downarrow 0} \\frac{1}{2t} \\big[ \\big(\\mtx{g}(Z_t,z') - \\mtx{g}(Z_0,z')\\big)^2 \\, \\big| \\, Z_0 = z \\big] \\\\\n\t&= \\Expect \\lim_{t \\downarrow 0} \\frac{1}{2t} \\big[ \\big(\\mtx{f}(Z_t) - \\mtx{f}(Z_0) \\big)^2 \\, \\big| \\, Z_0 = z \\big]\n\t= \\mtx{\\Gamma}(\\mtx{f})(z).\n\\end{align*}\nA similar calculation reveals that $\\mtx{\\Gamma}_2(\\mtx{g})(z,z') = \\mtx{\\Gamma}(\\mtx{f})(z')$.\nThus, the bivariate carr{\\'e} du champ~\\eqref{eqn:bivar-carre} takes the form\n\\begin{equation} \\label{eqn:bivar-carre-sym}\n\\mtx{\\Gamma}(\\mtx{g})(z,z') = \\mtx{\\Gamma}_1(\\mtx{g})(z,z') + \\mtx{\\Gamma}_2(\\mtx{g})(z,z')\n\t= \\mtx{\\Gamma}(\\mtx{f})(z) + \\mtx{\\Gamma}(\\mtx{f})(z').\n\\end{equation}\nAs a consequence, the Dirichlet form can be calculated as\n\\begin{equation} \\label{eqn:dirform-gf}\n\\bm{\\mathcal{E}}(\\mtx{g}) = \\Expect_{\\mu \\otimes \\mu} \\mtx{\\Gamma}(\\mtx{g})\n\t= \\Expect_{\\mu \\otimes \\mu} \\big[\\mtx{\\Gamma}(\\mtx{f})(z) + \\mtx{\\Gamma}(\\mtx{f})(z') \\big]\n\t= 2\\bm{\\mathcal{E}}(\\mtx{f}).\n\\end{equation}\nThe variance proxy~\\eqref{eqn:vg} of the bivariate function satisfies\n\\begin{equation} \\label{eqn:vg-vf}\nv_{\\mtx{g}} = \t\\norm{ \\norm{ \\mtx{\\Gamma}(\\mtx{g})(z,z') } }_{L_{\\infty}(\\mu \\otimes \\mu)}\n\t\t\\leq \t\t\\norm{ \\norm{ \\mtx{\\Gamma}(\\mtx{f})(z) } }_{L_{\\infty}(\\mu)}\n\t+ \t\\norm{ \\norm{ \\mtx{\\Gamma}(\\mtx{f})(z') } }_{L_{\\infty}(\\mu)}\n\t= 2 v_{\\mtx{f}}.\n\\end{equation}\nThe last relation is the definition~\\eqref{eqn:vf} of the variance proxy $v_{\\mtx{f}}$.\n\n\n\\subsubsection{Endgame}\n\nCombining~\\eqref{eqn:coshf-bd} and~\\eqref{eqn:coshg-bd}, we see that\n\\begin{equation*}\n\\Expect_{\\mu} \\operatorname{tr} \\sinh^2(\\theta \\mtx{f})\n\t\\leq  \\frac{\\alpha \\theta^2 \\operatorname{tr} \\bm{\\mathcal{E}}(\\mtx{g})}{(1 - \\alpha v_{\\mtx{g}} \\theta^2)_+}\n\t\\leq \\frac{2 \\alpha \\theta^2 \\operatorname{tr} \\bm{\\mathcal{E}}(\\mtx{f})}{(1 - 2 \\alpha v_{\\mtx{f}} \\theta^2)_+}.\n\\end{equation*}\nWe have also used the relations~\\eqref{eqn:dirform-gf} and~\\eqref{eqn:vg-vf}\nfrom the last section.\n\nTo compete the proof of~\\eqref{eqn:exp-moment},\ninvoke the identity\n$\\sinh^2(s) = (\\cosh(2 s) - 1)\/2$ to see that\n\\begin{align*}\n\\Expect_{\\mu} \\operatorname{tr} \\cosh(2 \\theta \\mtx{f})\n\t\t\\leq d + \\frac{4 \\alpha \\theta^2 \\operatorname{tr} \\bm{\\mathcal{E}}(\\mtx{f})}{(1 - 2 \\alpha v_{\\mtx{f}} \\theta^2)_+}.\n\\end{align*}\nFinally, introduce the normalized trace, $\\operatorname{\\bar{\\trace}}$,\nand make the change of variables $\\theta \\mapsto \\theta\/2$ to arrive at\n$$\n\\Expect_{\\mu} \\operatorname{tr} \\cosh(\\theta \\mtx{f})\n\t\\leq d \\cdot \\left[ 1 + \\frac{\\alpha \\theta^2 \\operatorname{\\bar{\\trace}} \\bm{\\mathcal{E}}(\\mtx{f})}{(1 - \\alpha v_{\\mtx{f}} \\theta^2 \/ 2)_+} \\right].\n$$\nThis is the stated result.\n\n\n\n\n\n\n\n\n \t\n\n\\section{Polynomial Moments}\n\\label{sec:poly-moments}\n\nBy a simple variation on the proof of Theorem~\\ref{thm:exp-moment},\nwe can also obtain bounds for the polynomial moments of a random matrix.  \n\n\\begin{theorem}[Polynomial moments] \\label{thm:poly-moment}\nEnforce Assumption~\\ref{ass:main}.\nLet $\\mtx{f} : \\Omega \\to \\mathbbm{H}_d$ be a function with $\\Expect_{\\mu} \\mtx{f} = \\mtx{0}$.\nFor $q = 1$ and $q \\geq 1.5$,\n\\begin{equation} \\label{eqn:poly-moment}\n\\big( \\Expect_{\\mu} \\operatorname{tr} \\abs{\\mtx{f}}^{2q} \\big)^{1\/(2q)}\n\t\\leq \\sqrt{2 \\alpha q^2} \\cdot \\big( \\Expect_{\\mu} \\operatorname{tr} \\mtx{\\Gamma}(\\mtx{f})^q \\big)^{1\/(2q)}.\n\\end{equation}\n\\end{theorem}\n\n\n\nBy combining Theorem~\\ref{thm:poly-moment} with the moment method, we can obtain\nprobability bounds for $\\norm{\\mtx{f}}$.  Let us summarize how these results compare\nwith the main result, Theorem~\\ref{thm:main}.  Observe that Theorem~\\ref{thm:poly-moment}\ngives a bound on the Schatten $2q$-norm of the random matrix $\\mtx{f}(Z)$\nin terms of the Schatten $2q$-norm of $\\mtx{\\Gamma}(\\mtx{f})^{1\/2}$.\nWe have the relation\n\\begin{equation} \\label{eqn:poly-unif-1}\n\\big( \\Expect_{\\mu} \\operatorname{tr} \\mtx{\\Gamma}(\\mtx{f})^{q} \\big)^{1\/(2q)}\n\t\\leq d^{1\/(2q)} \\cdot \\norm{ \\norm{ \\mtx{\\Gamma}(\\mtx{f})(z) } }_{L_{\\infty}(\\mu)}^{1\/2}\n\t= d^{1\/(2q)} \\cdot \\sqrt{ v_{\\mtx{f}} }.\n\\end{equation}\nTherefore, Theorem~\\ref{thm:poly-moment} potentially yields stronger bounds\nthan Theorem~\\ref{thm:exp-moment}.\n\nIn particular, Theorem~\\ref{thm:poly-moment} applies even when\n$\\mtx{\\Gamma}(\\mtx{f})$ is not uniformly bounded.\nExample~\\ref{ex:chaos} illustrates why this flexibility is valuable. \nIn Section~\\ref{sec:poly-var},\nwe show that slightly better polynomial moment bounds are possible\nwhen $\\mtx{\\Gamma}(\\mtx{f})$ is uniformly bounded.\n\n\n\\begin{remark}[Missing Parameters]\nTheorem~\\ref{thm:poly-moment} also holds for $q \\in (1, 1.5)$, with an\nextra factor of $\\sqrt{2}$ on the right-hand side.  The proof uses\na variant of Proposition~\\ref{prop:chain-rule} that only requires\nthe function $\\psi$ to be monotone.\n\\end{remark}\n\n\n\\begin{example}[Gaussian Chaos]\n\\label{ex:chaos}\nConsider the matrix Gaussian chaos\n$$\n\\vct{f}(\\vct{X}) = \\sum_{i, j = 1}^n X_i X_j \\, \\mtx{A}_{ij}\n\\quad\\text{where $\\vct{X} \\sim \\gamma_n$ and $\\mtx{A}_{ij} = \\mtx{A}_{ji} \\in \\mathbbm{H}_d$.}\n$$\nTo bound the trace moments of $f(\\vct{X})$,\nobserve that the carr{\\'e} du champ takes the form\n$$\n\\mtx{\\Gamma}(\\mtx{f})(\\vct{x})\n\t= \\sum_{i=1}^n ( \\partial_i \\mtx{f}(\\vct{x}) )^2\n\t= 4 \\sum_{i=1}^n \\left( \\sum_{j=1}^n x_j \\mtx{A}_{ij} \\right)^2.\n$$\nEvidently, $\\mtx{\\Gamma}(\\mtx{f})$ is unbounded,\nso Theorem~\\ref{thm:main} does not apply.\nBut Theorem~\\ref{thm:poly-moment} yields\n\\begin{equation} \\label{eqn:gauss-chaos-bd}\n\\big( \\Expect_{\\gamma_n} \\operatorname{tr} \\abs{\\mtx{f}}^{2q} \\big)^{1\/(2q)}\n\t\\leq \\sqrt{8 q^2} \\cdot \\left(\n\t\\Expect_{\\gamma_n} \\operatorname{tr} \\left[ \\sum_{i=1}^n \\left( \\sum_{j=1}^n X_j \\mtx{A}_{ij} \\right)^2 \\right]^q \\right)^{1\/(2q)}.\n\\end{equation}\nWe have used the fact that the Poincar{\\'e} constant of the OU process is $\\alpha = 1$.\nFurther bounds can be obtained by applying Theorem~\\ref{thm:poly-moment} repeatedly.\nIn the matrix setting, there are obstacles that prevent us from simplifying~\\eqref{eqn:gauss-chaos-bd}\ncompletely (related to the fact that the partial transpose operator is not completely bounded).\n\nIn the scalar case $d = 1$, we can obtain more transparent conclusions.  Consider the\nscalar-valued Gaussian chaos\n$$\nf(\\mtx{X}) = \\sum_{i,j=1}^n X_i X_j a_{ij}.\n$$\nThe most satisfying outcome takes place when $\\mtx{A} = [ a_{ij} ]$ is positive semidefinite.  \nIn this case, the result~\\eqref{eqn:gauss-chaos-bd} implies that\n$$\n\\begin{aligned}\n\\big( \\Expect_{\\gamma_n} \\abs{f}^{2q} \\big)^{1\/(2q)}\n\t&\\leq \\sqrt{8q^2} \\cdot \\left( \\Expect_{\\gamma_n} \\left[ \\sum_{i=1}^n \\left( \\sum_{j=1}^n X_j a_{ij} \\right)^2 \\right]^q \\right)^{1\/(2q)} \\\\\n\t&= \\sqrt{8q^2} \\cdot \\left( \\Expect_{\\gamma_n} \\left[ \\sum_{j,k=1}^n X_j X_k (\\mtx{A}^2)_{jk} \\right]^q \\right)^{1\/(2q)} \\\\\n\t&\\leq \\sqrt{8 q^2 \\norm{\\mtx{A}}} \\cdot \\big(\\Expect_{\\gamma_n} \\abs{f}^q\\big)^{1\/2q}\n\t\\leq \\sqrt{8 q^2 \\norm{\\mtx{A}}} \\cdot \\big(\\Expect_{\\gamma_n} \\abs{f}^{2q} \\big)^{1\/4q}. \n\\end{aligned}\n$$\nSolving, we obtain the bound\n$$\n\\big( \\Expect_{\\gamma_n} \\abs{f}^{2q} \\big)^{1\/(2q)}\n\t\\leq 8 q^2 \\norm{\\mtx{A}}.\n$$\nThis result gives a suboptimal estimate for large moments of the Gaussian chaos;\nas $q \\to \\infty$, the moments should grow in proportion to $q \\norm{\\mtx{A}}$ rather than $q^2 \\norm{\\mtx{A}}$.\nFor example, see~\\cite[Cor.~3.9]{LT91:Probability-Banach}.\n\\end{example}\n\n\n\n\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:poly-moment}}\n\nFor a parameter $q = 1$ or $q \\geq 1.5$, we wish to estimate the Schatten $2q$-norm\nof a function $\\mtx{f} : \\Omega \\to \\mathbbm{H}_d$ that satisfies $\\Expect_{\\mu} \\mtx{f} = \\mtx{0}$.\nThe argument has the same structure as Theorem~\\ref{thm:exp-moment}.\n\nFirst, we symmetrize.  Let $Z,Z' \\in \\Omega$ be independent random variables,\neach with distribution $\\mu$.  Jensen's inequality implies that\n\\begin{equation} \\label{eqn:powf-bd}\n\\Expect \\operatorname{tr} \\abs{ \\mtx{f} }^{2q}\n\t\\leq \\Expect \\operatorname{tr} \\abs{ \\mtx{f}(Z) - \\mtx{f}(Z') }^{2q}\n\t=: \\Expect \\operatorname{tr} \\abs{\\mtx{g}(Z,Z')}^{2q}.\n\\end{equation}\nSince $\\abs{\\cdot}^{2q}$ is convex, the function $\\operatorname{tr} \\abs{\\cdot}^{2q}$\nis also convex~\\cite[Prop.~2]{Pet94:Survey-Certain}.\n\nDefine the signed moment function $\\phi(s) := \\sgn(s) \\cdot \\abs{s}^{q}$,\nwhich is odd.  Note that its squared derivative $\\psi(s) := (\\phi'(s))^2 = q^2 \\abs{s}^{2(q-1)}$ is convex. \nSince $\\mtx{g}(z,z') = \\mtx{f}(z) - \\mtx{f}(z')$ is antisymmetric,\n\\begin{align*}\n\\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\abs{\\mtx{g}}^{2q}\n\t&= \\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\phi(\\mtx{g})^2 \\\\\n\t&= \\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\phi(\\mtx{g})^2 - \\operatorname{tr} \\big[ \\big( \\Expect_{\\mu \\otimes \\mu} \\phi(\\mtx{g}) \\big)^2 \\big]\n\t= \\operatorname{tr} \\Var_{\\mu \\otimes \\mu}[ \\phi(\\mtx{g}) ].\n\\end{align*}\nApply the bivariate trace Poincar{\\'e} inequality, Proposition~\\ref{prop:poincare-subadd}:\n$$\n\\operatorname{tr} \\Var_{\\mu \\otimes \\mu}[ \\phi(\\mtx{g}) ]\n\t\\leq \\alpha \\cdot \\operatorname{tr} \\bm{\\mathcal{E}}( \\phi(\\mtx{g}) )\n\t\\leq \\alpha q^2 \\cdot \\Expect_{\\mu \\otimes \\mu} \\operatorname{tr}\\big[ \\mtx{\\Gamma}(\\mtx{g}) \\, \\psi(\\mtx{g}) \\big].\n$$\nThe second bound is the chain rule inequality, Proposition~\\ref{prop:chain-rule}.\nIn summary,\n\\begin{equation} \\label{eqn:q-to-q-1}\n\\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\abs{\\mtx{g}}^{2q}\n\t\\leq \\alpha q^2 \\cdot \\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\big[\\mtx{\\Gamma}(\\mtx{g}) \\abs{\\mtx{g}}^{2(q-1)} \\big].\n\\end{equation}\nThis formula allow us to perform a moment comparison.\n\nTo isolate the carr{\\'e} du champ $\\mtx{\\Gamma}(\\vct{g})$, invoke\nH{\\\"o}lder's inequality for the Schatten norms: $$\n\\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\abs{\\mtx{g}}^{2q}\n\t\\leq \\alpha q^2 \\big( \\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\mtx{\\Gamma}(\\mtx{g})^q \\big)^{1\/q}\n\t \\big( \\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\abs{\\mtx{g}}^{2q} \\big)^{(q-1)\/q}.\n$$\nRearrange the last display, and use the initial bound~\\eqref{eqn:powf-bd} to arrive at\n$$\n\\big( \\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\abs{\\mtx{f}}^{2q} \\big)^{1\/(2q)}\n\t\\leq \\big( \\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\abs{\\mtx{g}}^{2q} \\big)^{1\/(2q)}\n\t\\leq \\sqrt{\\alpha q^2} \\cdot \\big( \\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\mtx{\\Gamma}(\\mtx{g})^q \\big)^{1\/(2q)}.\n$$\nTo finish the proof of~\\eqref{eqn:poly-moment}, recall the expression~\\eqref{eqn:bivar-carre-sym}\nfor the carr{\\'e} du champ $\\mtx{\\Gamma}(\\mtx{g})$.  Thus,\n$$\n\\big( \\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\mtx{\\Gamma}(\\mtx{g})^q \\big)^{1\/q}\n\t= \\big( \\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\abs{ \\mtx{\\Gamma}(\\mtx{f})(z) + \\mtx{\\Gamma}(\\mtx{f})(z') }^q \\big)^{1\/q}\n\t\\leq 2 \\big( \\Expect_{\\mu} \\operatorname{tr} \\mtx{\\Gamma}(\\mtx{f})^q  \\big)^{1\/q}.\n$$\nThis point follows from the triangle inequality. The argument is complete.\n\n\n\n\\subsection{A variant of the argument}\n\\label{sec:poly-var}\n\nThe \\emph{intrinsic dimension} of a positive-semidefinite matrix is $$\n\\operatorname{intdim}(\\mtx{A}) := \\frac{\\operatorname{tr}(\\mtx{A})}{\\norm{\\mtx{A}}}\n\\quad\\text{for $\\mtx{A} \\in \\mathbbm{H}_d^+$.}\n$$\nWe also set $\\operatorname{intdim}(\\mtx{0}) = 0$.\nFor a nonzero matrix $\\mtx{A}$, the intrinsic dimension\nsatisfies $1 \\leq \\operatorname{intdim}(\\mtx{A}) \\leq \\operatorname{rank}(\\mtx{A})$.\nIt can be interpreted as a continuous measure of the rank.\n\nSuppose that $q$ is a natural number.\nIf we use the uniform bound~\\eqref{eqn:vg} for the carr{\\'e} du champ\ninstead of H{\\\"o}lder's inequality, we can apply the bound~\\eqref{eqn:q-to-q-1}\niteratively to obtain\n$$\n\\Expect_{\\mu \\otimes \\mu} \\operatorname{tr} \\abs{\\mtx{g}}^{2q}\n\t\\leq \\operatorname{intdim}(\\bm{\\mathcal{E}}(\\mtx{g})) \\cdot \\alpha^q q! \\cdot v_{\\mtx{g}}^q.\n$$\nThe latter estimate improves over the uniform inequality that follows from\nTheorem~\\ref{thm:poly-moment} and~\\eqref{eqn:poly-unif-1}.\n\n\n\n\\section*{Acknowledgments}\n\nRamon Van Handel offered valuable feedback on a preliminary version of this work,\nand we are grateful to him for the proof of Proposition~\\ref{prop:equiv}.\nDH was funded by NSF grants DMS-1907977 and DMS-1912654.\nJAT gratefully acknowledges funding from ONR awards N00014-17-12146 and N00014-18-12363,\nand he would like to thank his family for their support in these difficult times.\n\n\n\\bibliographystyle{myalpha}\n\\newcommand{\\etalchar}[1]{$^{#1}$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}