Matrix Chaos Inequalities and Chaos of Combinatorial Type

TL;DR

This paper develops general matrix concentration inequalities for matrix chaoses, extending existing tools to handle polynomial-based random matrices systematically and improving bounds in various applications.

Contribution

It introduces systematic matrix concentration inequalities for matrix chaoses and identifies a special combinatorial family with easily computable parameters, unifying and enhancing prior bounds.

Findings

Provides general bounds for matrix chaoses.

Identifies a family of combinatorial matrix chaoses with simple parameters.

Improves bounds for applications like graph matrices and sum-of-squares analysis.

Abstract

Matrix concentration inequalities and their recently discovered sharp counterparts provide powerful tools to bound the spectrum of random matrices whose entries are linear functions of independent random variables. However, in many applications in theoretical computer science and in other areas one encounters more general random matrix models, called matrix chaoses, whose entries are polynomials of independent random variables. Such models have often been studied on a case-by-case basis using ad-hoc methods that can yield suboptimal dimensional factors. In this paper we provide general matrix concentration inequalities for matrix chaoses, which enable the treatment of such models in a systematic manner. These inequalities are expressed in terms of flattenings of the coefficients of the matrix chaos. We further identify a special family of matrix chaoses of combinatorial type for which the flattening parameters can be computed mechanically by a simple rule. This allows us to provide a unified treatment of and improved bounds for matrix chaoses that arise in a variety of applications, including graph matrices, Khatri-Rao matrices, and matrices that arise in average case analysis of the sum-of-squares hierarchy.