The dimension-free structure of nonhomogeneous random matrices

TL;DR

This paper establishes a universal, dimension-free characterization of the expected Schatten norms of nonhomogeneous Gaussian matrices, resolving a conjecture and extending to non-Gaussian cases with applications in random graphs.

Contribution

It provides a dimension-free formula for the expected Schatten norms of nonhomogeneous Gaussian matrices, settling a conjecture and extending to non-Gaussian matrices with practical applications.

Findings

Dimension-free bounds for expected Schatten norms.

Complete characterization of Gaussian matrices as bounded operators on ℓ₂.

Extensions to non-symmetric and non-Gaussian matrices with applications.

Abstract

Let $X$ be a symmetric random matrix with independent but non-identically distributed centered Gaussian entries. We show that $$ \mathbf{E}\|X\|_{S_p} \asymp \mathbf{E}\Bigg[ \Bigg(\sum_i\Bigg(\sum_j X_{ij}^2\Bigg)^{p/2}\Bigg)^{1/p} \Bigg] $$ for any $2\le p\le\infty$, where $S_p$ denotes the $p$-Schatten class and the constants are universal. The right-hand side admits an explicit expression in terms of the variances of the matrix entries. This settles, in the case $p=\infty$, a conjecture of the first author, and provides a complete characterization of the class of infinite matrices with independent Gaussian entries that define bounded operators on $\ell_2$. Along the way, we obtain optimal dimension-free bounds on the moments $(\mathbf{E}\|X\|_{S_p}^p)^{1/p}$ that are of independent interest. We develop further extensions to non-symmetric matrices and to nonasymptotic moment and norm estimates for matrices with non-Gaussian entries that arise, for example, in the study of random graphs and in applied mathematics.