Matrix concentration inequalities via the method of exchangeable pairs

TL;DR

This paper develops new exponential and polynomial concentration inequalities for the spectral norm of random matrices using an extension of Stein's method of exchangeable pairs, generalizing classical scalar inequalities to the matrix setting.

Contribution

It introduces a matrix extension of Chatterjee's scalar concentration theory, providing novel bounds for sums of independent and dependent random matrices.

Findings

Derived exponential concentration inequalities for matrix spectral norms.

Established polynomial moment inequalities for matrix spectral norms.

Extended classical scalar inequalities to the matrix setting for dependent and independent matrices.

Abstract

This paper derives exponential concentration inequalities and polynomial moment inequalities for the spectral norm of a random matrix. The analysis requires a matrix extension of the scalar concentration theory developed by Sourav Chatterjee using Stein's method of exchangeable pairs. When applied to a sum of independent random matrices, this approach yields matrix generalizations of the classical inequalities due to Hoeffding, Bernstein, Khintchine and Rosenthal. The same technique delivers bounds for sums of dependent random matrices and more general matrix-valued functions of dependent random variables.