Hirschman and Widder introduced a class of Polya frequency functions given by linear combinations of one-sided exponential functions. The members of this class are probability densities, and the class is closed under convolution but not under pointwi
se multiplication. We show that, generically, a polynomial function of such a density is a Polya frequency function only if the polynomial is a homothety, and also identify a subclass for which each positive-integer power is a Polya frequency function. We further demonstrate connections between the Maclaurin coefficients, the moments of these densities, and the recovery of the density from finitely many moments, via Schur polynomials.
A matrix-compression algorithm is derived from a novel isogenic block decomposition for square matrices. The resulting compression and inflation operations possess strong functorial and spectral-permanence properties. The basic observation that Hadam
ard entrywise functional calculus preserves isogenic blocks has already proved to be of paramount importance for thresholding large correlation matrices. An array of applications to current investigations in computational matrix analysis arises, touching concepts such as symmetric statistical models, hierarchical matrices and coherent matrix organization induced by partition trees.
In the past decade, significant progress has been made to generalize classical tools from Fourier analysis to analyze and process signals defined on networks. In this paper, we propose a new framework for constructing Gabor-type frames for signals on
graphs. Our approach uses general and flexible families of linear operators acting as translations. Compared to previous work in the literature, our methods yield the sharp bounds for the associated frames, in a broad setting that generalizes several existing constructions. We also examine how Gabor-type frames behave for signals defined on Cayley graphs by exploiting the representation theory of the underlying group. We explore how natural classes of translations can be constructed for Cayley graphs, and how the choice of an eigenbasis can significantly impact the properties of the resulting translation operators and frames on the graph.
The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on kernels with arbitrary domains, it is shown that such
a composition operator maps the set of totally non-negative kernels to itself if and only if the function is constant or linear, or just linear if it preserves total positivity. Symmetric kernels are also discussed, with a similar outcome. These classification results are a byproduct of two matrix-completion results and the second theme: an extension of A.M. Whitneys density theorem from finite domains to subsets of the real line. This extension is derived via a discrete convolution with modulated Gaussian kernels. The third theme consists of analyzing, with tools from harmonic analysis, the preservers of several families of totally non-negative and totally positive kernels with additional structure: continuous Hankel kernels on an interval, Polya frequency functions, and Polya frequency sequences. The rigid structure of post-composition transforms of totally positive kernels acting on infinite sets is obtained by combining several specialized situations settled in our present and earlier works.
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 distribu
tions). 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.
A surprising result of FitzGerald and Horn (1977) shows that $A^{circ alpha} := (a_{ij}^alpha)$ is positive semidefinite (p.s.d.) for every entrywise nonnegative $n times n$ p.s.d. matrix $A = (a_{ij})$ if and only if $alpha$ is a positive integer or
$alpha geq n-2$. 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.
We prove that the only entrywise transforms of rectangular matrices which preserve total positivity or total non-negativity are either constant or linear. This follows from an extended classification of preservers of these two properties for matrices
of fixed dimension. We also prove that the same assertions hold upon working only with symmetric matrices; for total-positivity preservers our proofs proceed through solving two totally positive completion problems.
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 positiv
e 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).
We classify all functions which, when applied term by term, leave invariant the sequences of moments of positive measures on the real line. Rather unexpectedly, these functions are built of absolutely monotonic components, or reflections of them, wit
h possible discontinuities at the endpoints. Even more surprising is the fact that functions preserving moments of three point masses must preserve moments of all measures. Our proofs exploit the semidefiniteness of the associated Hankel matrices and the complete monotonicity of the Laplace transforms of the underlying measures. As a byproduct, we characterize the entrywise transforms which preserve totally non-negative Hankel matrices, and those which preserve all totally non-negative matrices. The latter class is surprisingly rigid: such maps must be constant or linear. We also examine transforms in the multivariable setting, which reveals a new class of piecewise absolutely monotonic functions.
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.
