Estimating the covariance of random matrices

TL;DR

This paper extends covariance estimation techniques from vectors to matrices, providing bounds on eigenvalues of sums of log-concave matrices and generalizing the law of large numbers for positive semi-definite matrices.

Contribution

It introduces a matrix version of covariance estimation results, including eigenvalue bounds for sums of log-concave matrices, advancing understanding of matrix concentration phenomena.

Findings

Eigenvalue bounds for sums of log-concave matrices

Extension of covariance estimation to matrix setting

Quantified law of large numbers for positive semi-definite matrices

Abstract

We extend to the matrix setting a recent result of Srivastava-Vershynin about estimating the covariance matrix of a random vector. The result can be in- terpreted as a quantified version of the law of large numbers for positive semi-definite matrices which verify some regularity assumption. Beside giving examples, we dis- cuss the notion of log-concave matrices and give estimates on the smallest and largest eigenvalues of a sum of such matrices.