Sharp nonasymptotic bounds on the norm of random matrices with independent entries

TL;DR

This paper derives sharp, nonasymptotic bounds on the spectral norm of random matrices with independent entries, improving previous results and applicable to various distributions, with implications for spectral edge behavior.

Contribution

The paper introduces improved nonasymptotic bounds on the spectral norm of random matrices with independent entries, including optimal bounds and extensions to different distributions.

Findings

Bounds are optimal and match lower bounds under mild conditions.

Results apply to rectangular, sub-Gaussian, and heavy-tailed matrices.

Bounds accurately capture the spectral edge and phase transition phenomena.

Abstract

We obtain nonasymptotic bounds on the spectral norm of random matrices with independent entries that improve significantly on earlier results. If $X$ is the $n\times n$ symmetric matrix with $X_{ij}\sim N(0,b_{ij}^2)$, we show that \[\mathbf{E}\Vert X\Vert \lesssim\max_i\sqrt{\sum_jb_{ij}^2}+\max _{ij}\vert b_{ij}\vert \sqrt{\log n}.\] This bound is optimal in the sense that a matching lower bound holds under mild assumptions, and the constants are sufficiently sharp that we can often capture the precise edge of the spectrum. Analogous results are obtained for rectangular matrices and for more general sub-Gaussian or heavy-tailed distributions of the entries, and we derive tail bounds in addition to bounds on the expected norm. The proofs are based on a combination of the moment method and geometric functional analysis techniques. As an application, we show that our bounds immediately yield the correct phase transition behavior of the spectral edge of random band matrices and of sparse Wigner matrices. We also recover a result of Seginer on the norm of Rademacher matrices.