Simple Relative Deviation Bounds for Covariance and Gram Matrices

TL;DR

This paper introduces non-asymptotic, relative deviation bounds for eigenvalues of empirical covariance and Gram matrices, offering sharper spectrum control than traditional uniform bounds.

Contribution

It presents a general theorem converting uniform bounds into relative bounds, improving spectral analysis of covariance and Gram matrices.

Findings

Provides sharper eigenvalue bounds across the spectrum

Applicable to various settings due to simple analysis

Enhances understanding of eigenvalue deviations in finite samples

Abstract

We provide non-asymptotic, relative deviation bounds for the eigenvalues of empirical covariance and Gram matrices in general settings. Unlike typical uniform bounds, which may fail to capture the behavior of smaller eigenvalues, our results provide sharper control across the spectrum. Our analysis is based on a general-purpose theorem that allows one to convert existing uniform bounds into relative ones. The theorems and techniques emphasize simplicity and should be applicable across various settings.