Matrix regularizing effects of Gaussian perturbations

TL;DR

This paper investigates how Gaussian noise influences the spectral properties of Hermitian matrices, providing bounds on eigenvalue distributions and inverse norms, with implications for understanding matrix regularization effects.

Contribution

It offers new bounds on eigenvalue counts and inverse norms for matrices perturbed by Gaussian ensembles, extending understanding of regularization effects.

Findings

Bound on the mean number of eigenvalues in an interval

Tail bounds for Frobenius and operator norms of the inverse

Estimates on the probability of multiple eigenvalues in an interval

Abstract

The addition of noise has a regularizing effect on Hermitian matrices. This effect is studied here for $H=A+V$, where $A$ is the base matrix and $V$ is sampled from the GOE or the GUE random matrix ensembles. We bound the mean number of eigenvalues of $H$ in an interval, and present tail bounds for the distribution of the Frobenius and operator norms of $H^{-1}$ and for the distribution of the norm of $H^{-1}$ applied to a fixed vector. The bounds are uniform in $A$ and exceed the actual suprema by no more than multiplicative constants. The probability of multiple eigenvalues in an interval is also estimated.