Concentration of Measure and Large Random Matrices with an application to Sample Covariance Matrices

TL;DR

This paper introduces a new probabilistic framework based on concentration of measure theory for analyzing large random matrices, especially sample covariance matrices, with broad applications in statistical learning.

Contribution

It develops a versatile set of tools for extending classical random matrix results using concentration of measure concepts, moving beyond independent entry assumptions.

Findings

Extended classical results to concentrated vectors

Applicable to sample covariance matrices in high dimensions

Broad potential applications in statistical learning

Abstract

The present work provides an original framework for random matrix analysis based on revisiting the concentration of measure theory from a probabilistic point of view. By providing various notions of vector concentration ($q$-exponential, linear, Lipschitz, convex), a set of elementary tools is laid out that allows for the immediate extension of classical results from random matrix theory involving random concentrated vectors in place of vectors with independent entries. These findings are exemplified here in the context of sample covariance matrices but find a large range of applications in statistical learning and beyond, thanks to the broad adaptability of our hypotheses.