Matrix Poincar\'e inequalities and concentration

TL;DR

This paper extends classical scalar Poincaré inequalities to the matrix setting, establishing matrix concentration inequalities for various measures and introducing new matrix trace inequalities with broad applications.

Contribution

It introduces a matrix Poincaré inequality framework that leads to matrix concentration results and new trace inequalities, expanding the scope of concentration phenomena to matrices.

Findings

Matrix Poincaré inequalities imply exponential matrix concentration.

Application to Gaussian, product, and Strong Rayleigh measures.

First matrix concentration results for negatively dependent variables.

Abstract

We show that any probability measure satisfying a Matrix Poincar\'e inequality with respect to some reversible Markov generator satisfies an exponential matrix concentration inequality depending on the associated matrix carr\'e du champ operator. This extends to the matrix setting a classical phenomenon in the scalar case. Moreover, the proof gives rise to new matrix trace inequalities which could be of independent interest. We then apply this general fact by establishing matrix Poincar\'{e} inequalities to derive matrix concentration inequalities for Gaussian measures, product measures and for Strong Rayleigh measures. The latter represents the first instance of matrix concentration for general matrix functions of negatively dependent random variables.