The spectral norm of Gaussian matrices with correlated entries

TL;DR

This paper provides a non-asymptotic bound on the spectral norm of Gaussian matrices with correlated entries, improving existing inequalities by removing extraneous logarithmic factors in certain cases.

Contribution

It introduces a sharp, covariance-dependent bound on the spectral norm of Gaussian matrices, refining the noncommutative Khintchine inequality.

Findings

The bound is sharp in some cases.

It removes the log d factor in the inequality.

Provides a covariance-based estimate for spectral norms.

Abstract

We give a non-asymptotic bound on the spectral norm of a $d\times d$ matrix $X$ with centered jointly Gaussian entries in terms of the covariance matrix of the entries. In some cases, this estimate is sharp and removes the $\sqrt{\log d}$ factor in the noncommutative Khintchine inequality.