User-friendly tail bounds for sums of random matrices

TL;DR

This paper introduces new, easy-to-verify probability inequalities for sums of independent random matrices, providing strong bounds on their eigenvalues and norms, with broad applications in matrix analysis.

Contribution

It offers novel, noncommutative tail bounds for sums of random matrices that are simple to apply and extend classical scalar inequalities to the matrix setting.

Findings

Provides bounds on the maximum eigenvalue of matrix sums

Derives tail bounds for the norm of rectangular matrices

Includes results on matrix-valued martingales

Abstract

This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands, and they deliver strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of random rectangular matrices follow as an immediate corollary. The proof techniques also yield some information about matrix-valued martingales. In other words, this paper provides noncommutative generalizations of the classical bounds associated with the names Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid. The matrix inequalities promise the same diversity of application, ease of use, and strength of conclusion that have made the scalar inequalities so valuable.