arXiv subjects

Dominique Guillot

Publications and source records attributed to Dominique Guillot.

36 records · Page 2Linked to original sources

arXiv2019

A panorama of positivity

This survey contains a selection of topics unified by the concept of positive semi-definiteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or mass distributions). We put emphasis on entrywise operations which preserve positivity, in a variety of guises. Techniques from harmonic analysis, function theory, operator theory, statistics, combinatorics, and group representations are invoked. Some partially forgotten classical roots in metric geometry and distance transforms are presented with comments and full bibliographical references. Modern applications to high-dimensional covariance estimation and regularization are included.

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arXiv2019

Total nonnegativity of GCD matrices and kernels

Let $X = (x_1,\dots,x_n)$ be a vector of distinct positive integers. The $n \times n$ matrix $S = S(X) := (\gcd(x_i,x_j))_{i,j=1}^n$, where $\gcd(x_i,x_j)$ denotes the greatest common divisor of $x_i$ and $x_j$, is called the greatest common divisor (GCD) matrix on $X$. By a surprising result of Beslin and Ligh [Linear Algebra and Appl. 118], all GCD matrices are positive definite. In this paper, we completely characterize the GCD matrices satisfying the stronger property of being totally nonnegative (TN) or totally positive (TP). As we show, a GCD matrix is never TP when $n \geq 3$, and is TN if and only if it is $\textrm{TN}_2$, i.e., all its $2 \times 2$ minors are nonnegative. We next demonstrate that a GCD matrix is $\textrm{TN}_2$ if and only if the exponents of each prime divisor in the prime factorization of the $x_i$s form a monotonic sequence. Reformulated in the language of kernels, our results characterize the subsets of integers over which the kernel $K(x,y) = \gcd(x,y)$ is totally nonnegative. The proofs of our characterizations depend on Gantmacher and Krein's notion of a Green's matrix. We conclude by showing that a GCD matrix is TN if and only if it is a Green's matrix. As a consequence, we obtain explicit formulas for all the minors and for the inverse of totally nonnegative GCD matrices.

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arXiv2018

Simultaneous kernels of matrix Hadamard powers

In previous work [Adv. Math. 298, pp. 325-368, 2016], the structure of the simultaneous kernels of Hadamard powers of any positive semidefinite matrix were described. Key ingredients in the proof included a novel stratification of the cone of positive semidefinite matrices and a well-known theorem of Hershkowitz, Neumann, and Schneider, which classifies the Hermitian positive semidefinite matrices whose entries are $0$ or $1$ in modulus. In this paper, we show that each of these results extends to a larger class of matrices which we term $3$-PMP (principal minor positive).

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arXiv2018

The critical exponent: a novel graph invariant

A surprising result of FitzGerald and Horn (1977) shows that $A^{\circ α} := (a_{ij}^α)$ is positive semidefinite (p.s.d.) for every entrywise nonnegative $n \times n$ p.s.d. matrix $A = (a_{ij})$ if and only if $α$ is a positive integer or $α\geq n-2$. Given a graph $G$, we consider the refined problem of characterizing the set $\mathcal{H}_G$ of entrywise powers preserving positivity for matrices with a zero pattern encoded by $G$. Using algebraic and combinatorial methods, we study how the geometry of $G$ influences the set $\mathcal{H}_G$. Our treatment provides new and exciting connections between combinatorics and analysis, and leads us to introduce and compute a new graph invariant called the critical exponent.

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arXiv2016

Model-free consistency of graph partitioning

In this paper, we exploit the theory of dense graph limits to provide a new framework to study the stability of graph partitioning methods, which we call structural consistency. Both stability under perturbation as well as asymptotic consistency (i.e., convergence with probability $1$ as the sample size goes to infinity under a fixed probability model) follow from our notion of structural consistency. By formulating structural consistency as a continuity result on the graphon space, we obtain robust results that are completely independent of the data generating mechanism. In particular, our results apply in settings where observations are not independent, thereby significantly generalizing the common probabilistic approach where data are assumed to be i.i.d. In order to make precise the notion of structural consistency of graph partitioning, we begin by extending the theory of graph limits to include vertex colored graphons. We then define continuous node-level statistics and prove that graph partitioning based on such statistics is consistent. Finally, we derive the structural consistency of commonly used clustering algorithms in a general model-free setting. These include clustering based on local graph statistics such as homomorphism densities, as well as the popular spectral clustering using the normalized Laplacian. We posit that proving the continuity of clustering algorithms in the graph limit topology can stand on its own as a more robust form of model-free consistency. We also believe that the mathematical framework developed in this paper goes beyond the study of clustering algorithms, and will guide the development of similar model-free frameworks to analyze other procedures in the broader mathematical sciences.

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arXiv2016

Schur polynomials and matrix positivity preservers

A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefiniteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg's work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving positivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quotients of Hadamard powers and Schur polynomials, and (c) a description of the joint kernels of Hadamard powers.

math.CO↗

arXiv2016

Matrix positivity preservers in fixed dimension. I

A classical theorem proved in 1942 by I.J. Schoenberg describes all real-valued functions that preserve positivity when applied entrywise to positive semidefinite matrices of arbitrary size; such functions are necessarily analytic with non-negative Taylor coefficients. Despite the great deal of interest generated by this theorem, a characterization of functions preserving positivity for matrices of fixed dimension is not known. In this paper, we provide a complete description of polynomials of degree $N$ that preserve positivity when applied entrywise to matrices of dimension $N$. This is the key step for us then to obtain negative lower bounds on the coefficients of analytic functions so that these functions preserve positivity in a prescribed dimension. The proof of the main technical inequality is representation theoretic, and employs the theory of Schur polynomials. Interpreted in the context of linear pencils of matrices, our main results provide a closed-form expression for the lowest critical value, revealing at the same time an unexpected spectral discontinuity phenomenon. Tight linear matrix inequalities for Hadamard powers of matrices and a sharp asymptotic bound for the matrix-cube problem involving Hadamard powers are obtained as applications. Positivity preservers are also naturally interpreted as solutions of a variational inequality involving generalized Rayleigh quotients. This optimization approach leads to a novel description of the simultaneous kernels of Hadamard powers, and a family of stratifications of the cone of positive semidefinite matrices.

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arXiv2016

Preserving positivity for matrices with sparsity constraints

Functions preserving Loewner positivity when applied entrywise to positive semidefinite matrices have been widely studied in the literature. Following the work of Schoenberg [Duke Math. J. 9], Rudin [Duke Math. J. 26], and others, it is well-known that functions preserving positivity for matrices of all dimensions are absolutely monotonic (i.e., analytic with nonnegative Taylor coefficients). In this paper, we study functions preserving positivity when applied entrywise to sparse matrices, with zeros encoded by a graph $G$ or a family of graphs $G_n$. Our results generalize Schoenberg and Rudin's results to a modern setting, where functions are often applied entrywise to sparse matrices in order to improve their properties (e.g. better conditioning). The only such result known in the literature is for the complete graph $K_2$. We provide the first such characterization result for a large family of non-complete graphs. Specifically, we characterize functions preserving Loewner positivity on matrices with zeros according to a tree. These functions are multiplicatively midpoint-convex and super-additive. Leveraging the underlying sparsity in matrices thus admits the use of functions which are not necessarily analytic nor absolutely monotonic. We further show that analytic functions preserving positivity on matrices with zeros according to trees can contain arbitrarily long sequences of negative coefficients, thus obviating the need for absolute monotonicity in a very strong sense. This result leads to the question of exactly when absolute monotonicity is necessary when preserving positivity for an arbitrary class of graphs. We then provide a stronger condition in terms of the numerical range of all symmetric matrices, such that functions satisfying this condition on matrices with zeros according to any family of graphs with unbounded degrees are necessarily absolutely monotonic.

math.FA↗

arXiv2015

Critical exponents of graphs

The study of entrywise powers of matrices was originated by Loewner in the pursuit of the Bieberbach conjecture. Since the work of FitzGerald and Horn (1977), it is known that $A^{\circ α} := (a_{ij}^α)$ is positive semidefinite for every entrywise nonnegative $n \times n$ positive semidefinite matrix $A = (a_{ij})$ if and only if $α$ is a positive integer or $α\geq n-2$. This surprising result naturally extends the Schur product theorem, and demonstrates the existence of a sharp phase transition in preserving positivity. In this paper, we study when entrywise powers preserve positivity for matrices with structure of zeros encoded by graphs. To each graph is associated an invariant called its "critical exponent", beyond which every power preserves positivity. In our main result, we determine the critical exponents of all chordal/decomposable graphs, and relate them to the geometry of the underlying graphs. We then examine the critical exponent of important families of non-chordal graphs such as cycles and bipartite graphs. Surprisingly, large families of dense graphs have small critical exponents that do not depend on the number of vertices of the graphs.

math.CO↗

arXiv2015

Statistical paleoclimate reconstructions via Markov random fields

Understanding centennial scale climate variability requires data sets that are accurate, long, continuous and of broad spatial coverage. Since instrumental measurements are generally only available after 1850, temperature fields must be reconstructed using paleoclimate archives, known as proxies. Various climate field reconstructions (CFR) methods have been proposed to relate past temperature to such proxy networks. In this work, we propose a new CFR method, called GraphEM, based on Gaussian Markov random fields embedded within an EM algorithm. Gaussian Markov random fields provide a natural and flexible framework for modeling high-dimensional spatial fields. At the same time, they provide the parameter reduction necessary for obtaining precise and well-conditioned estimates of the covariance structure, even in the sample-starved setting common in paleoclimate applications. In this paper, we propose and compare the performance of different methods to estimate the graphical structure of climate fields, and demonstrate how the GraphEM algorithm can be used to reconstruct past climate variations. The performance of GraphEM is compared to the widely used CFR method RegEM with regularization via truncated total least squares, using synthetic data. Our results show that GraphEM can yield significant improvements, with uniform gains over space, and far better risk properties. We demonstrate that the spatial structure of temperature fields can be well estimated by graphs where each neighbor is only connected to a few geographically close neighbors, and that the increase in performance is directly related to recovering the underlying sparsity in the covariance of the spatial field. Our work demonstrates how significant improvements can be made in climate reconstruction methods by better modeling the covariance structure of the climate field.

stat.AP↗

arXiv2015

Differential Calculus on Graphon Space

Recently, the theory of dense graph limits has received attention from multiple disciplines including graph theory, computer science, statistical physics, probability, statistics, and group theory. In this paper we initiate the study of the general structure of differentiable graphon parameters $F$. We derive consistency conditions among the higher Gâteaux derivatives of $F$ when restricted to the subspace of edge weighted graphs $\mathcal{W}_{\bf p}$. Surprisingly, these constraints are rigid enough to imply that the multilinear functionals $Λ: \mathcal{W}_{\bf p}^n \to \mathbb{R}$ satisfying the constraints are determined by a finite set of constants indexed by isomorphism classes of multigraphs with $n$ edges and no isolated vertices. Using this structure theory, we explain the central role that homomorphism densities play in the analysis of graphons, by way of a new combinatorial interpretation of their derivatives. In particular, homomorphism densities serve as the monomials in a polynomial algebra that can be used to approximate differential graphon parameters as Taylor polynomials. These ideas are summarized by our main theorem, which asserts that homomorphism densities $t(H,-)$ where $H$ has at most $N$ edges form a basis for the space of smooth graphon parameters whose $(N+1)$st derivatives vanish. As a consequence of this theory, we also extend and derive new proofs of linear independence of multigraph homomorphism densities, and characterize homomorphism densities. In addition, we develop a theory of series expansions, including Taylor's theorem for graph parameters and a uniqueness principle for series. We use this theory to analyze questions raised by Lovász, including studying infinite quantum algebras and the connection between right- and left-homomorphism densities.

math.CO↗

arXiv2014

Complete characterization of Hadamard powers preserving Loewner positivity, monotonicity, and convexity

Entrywise powers of symmetric matrices preserving positivity, monotonicity or convexity with respect to the Loewner ordering arise in various applications, and have received much attention recently in the literature. Following FitzGerald and Horn [J. Math. Anal. Appl., 1977], it is well-known that there exists a critical exponent beyond which all entrywise powers preserve positive definiteness. Similar phase transition phenomena have also recently been shown by Hiai (2009) to occur for monotonicity and convexity. In this paper, we complete the characterization of all the entrywise powers below and above the critical exponents that are positive, monotone, or convex on the cone of positive semidefinite matrices. We then extend the original problem by fully classifying the positive, monotone, or convex powers in a more general setting where additional rank constraints are imposed on the matrices. We also classify the entrywise powers that are super/sub-additive with respect to the Loewner ordering. Finally, we extend all the previous characterizations to matrices with negative entries. Our analysis consequently allows us to answer a question raised by Bhatia and Elsner (2007) regarding the smallest dimension for which even extensions of the power functions do not preserve Loewner positivity.

math.FA↗

arXiv2014

Loewner positive entrywise functions, and classification of measurable solutions of Cauchy's functional equations

Entrywise functions preserving Loewner positivity have been studied by many authors, most notably Schoenberg and Rudin. Following their work, it is known that functions preserving positivity when applied entrywise to positive semidefinite matrices of all dimensions are necessarily analytic with nonnegative Taylor coefficients. When the dimension is fixed, it has been shown by Vasudeva and Horn that such functions are automatically continuous and sufficiently differentiable. A natural refinement of the aforementioned problem consists of characterizing functions preserving positivity under rank constraints. In this paper, we begin this study by characterizing entrywise functions which preserve the cone of positive semidefinite real matrices of rank $1$ with entries in a general interval. Classifying such functions is intimately connected to the classical problem of solving Cauchy's functional equations, which have non-measurable solutions. We demonstrate that under mild local measurability assumptions, such functions are automatically smooth and can be completely characterized. We then extend our results by classifying functions preserving positivity on rank $1$ Hermitian complex matrices.

math.FA↗

arXiv2014

On fractional Hadamard powers of positive block matrices

Entrywise powers of matrices have been well-studied in the literature, and have recently received renewed attention in the regularization of high-dimensional correlation matrices. In this paper, we study powers of positive semidefinite block matrices $(H_{st})_{s,t=1}^n$ with complex entries. We first characterize the powers $α\in\mathbb{R}$ such that the blockwise power map $(H_{st}) \mapsto (H_{st}^α)$ preserves Loewner positivity. The characterization is obtained by exploiting connections with the theory of matrix monotone functions developed by Loewner. Second, we revisit previous work by Choudhury [Proc. AMS 108] who had provided a lower bound on $α$ for preserving positivity when the blocks $H_{st}$ pairwise commute. We completely settle this problem by characterizing the full set of powers preserving positivity in this setting. Our characterizations generalize previous work by FitzGerald-Horn, Bhatia-Elsner, and Hiai from scalars to arbitrary block size, and in particular, generalize the Schur Product Theorem. Finally, a natural and unifying framework for studying the case of diagonalizable blocks consists of replacing real powers by general characters of the complex plane. We thus classify such characters, and generalize our results to this more general setting. In the course of our work, given $β\in\mathbb{Z}$, we provide lower and upper bounds for the threshold power $α>0$ above which the complex characters $re^{iθ}\mapsto r^αe^{iβθ}$ preserve positivity when applied entrywise to positive semidefinite matrices. In particular, we completely resolve the $n=3$ case of a question raised in 2001 by Xingzhi Zhan. As an application, we extend previous work by de Pillis [Duke Math. J. 36] by classifying the characters $K$ of the complex plane for which the map $(H_{st})_{s,t=1}^n \mapsto (K({\rm tr}(H_{st})))_{s,t=1}^n$ preserves positivity.

math.FA↗

arXiv2013

The critical exponent conjecture for powers of doubly nonnegative matrices

Doubly non-negative matrices arise naturally in many setting including Markov random fields (positively banded graphical models) and in the convergence analysis of Markov chains. In this short note, we settle a recent conjecture by C.R. Johnson et al. [Linear Algebra Appl. 435 (2011)] by proving that the critical exponent beyond which all continuous conventional powers of $n$-by-$n$ doubly nonnegative matrices are doubly nonnegative is exactly $n-2$. We show that the conjecture follows immediately by applying a general characterization from the literature. We prove a stronger form of the conjecture by classifying all powers preserving doubly nonnegative matrices, and proceed to generalize the conjecture for broad classes of functions. We also provide different approaches for settling the original conjecture.

math.CA↗

arXiv2013

Functions preserving positive definiteness for sparse matrices

We consider the problem of characterizing entrywise functions that preserve the cone of positive definite matrices when applied to every off-diagonal element. Our results extend theorems of Schoenberg [Duke Math. J. 9], Rudin [Duke Math. J. 26], Christensen and Ressel [Trans. Amer. Math. Soc., 243], and others, where similar problems were studied when the function is applied to all elements, including the diagonal ones. It is shown that functions that are guaranteed to preserve positive definiteness cannot at the same time induce sparsity, i.e., set elements to zero. These results have important implications for the regularization of positive definite matrices, where functions are often applied to only the off-diagonal elements to obtain sparse matrices with better properties (e.g., Markov random field/graphical model structure, better condition number). As a particular case, it is shown that \emph{soft-thresholding}, a commonly used operation in modern high-dimensional probability and statistics, is not guaranteed to maintain positive definiteness, even if the original matrix is sparse. This result has a deep connection to graphs, and in particular, to the class of trees. We then proceed to fully characterize functions which do preserve positive definiteness. This characterization is in terms of absolutely monotonic functions and turns out to be quite different from the case when the function is also applied to diagonal elements. We conclude by giving bounds on the condition number of a matrix which guarantee that the regularized matrix is positive definite.

math.CA↗

arXiv2012

Iterative Thresholding Algorithm for Sparse Inverse Covariance Estimation

The L1-regularized maximum likelihood estimation problem has recently become a topic of great interest within the machine learning, statistics, and optimization communities as a method for producing sparse inverse covariance estimators. In this paper, a proximal gradient method (G-ISTA) for performing L1-regularized covariance matrix estimation is presented. Although numerous algorithms have been proposed for solving this problem, this simple proximal gradient method is found to have attractive theoretical and numerical properties. G-ISTA has a linear rate of convergence, resulting in an O(log e) iteration complexity to reach a tolerance of e. This paper gives eigenvalue bounds for the G-ISTA iterates, providing a closed-form linear convergence rate. The rate is shown to be closely related to the condition number of the optimal point. Numerical convergence results and timing comparisons for the proposed method are presented. G-ISTA is shown to perform very well, especially when the optimal point is well-conditioned.

stat.CO↗

arXiv2011

Retaining positive definiteness in thresholded matrices

Positive definite (p.d.) matrices arise naturally in many areas within mathematics and also feature extensively in scientific applications. In modern high-dimensional applications, a common approach to finding sparse positive definite matrices is to threshold their small off-diagonal elements. This thresholding, sometimes referred to as hard-thresholding, sets small elements to zero. Thresholding has the attractive property that the resulting matrices are sparse, and are thus easier to interpret and work with. In many applications, it is often required, and thus implicitly assumed, that thresholded matrices retain positive definiteness. In this paper we formally investigate the algebraic properties of p.d. matrices which are thresholded. We demonstrate that for positive definiteness to be preserved, the pattern of elements to be set to zero has to necessarily correspond to a graph which is a union of disconnected complete components. This result rigorously demonstrates that, except in special cases, positive definiteness can be easily lost. We then proceed to demonstrate that the class of diagonally dominant matrices is not maximal in terms of retaining positive definiteness when thresholded. Consequently, we derive characterizations of matrices which retain positive definiteness when thresholded with respect to important classes of graphs. In particular, we demonstrate that retaining positive definiteness upon thresholding is governed by complex algebraic conditions.

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