User-Friendly Tail Bounds for Sums of Random Matrices

Concepts

Natural SciencesMathematicsDecision TheoryProbability TheoryStochastic ProcessesLarge DeviationsMarkov ProcessesProbability InequalitiesChernoff BoundEigenvalue BoundsLarge Deviation InequalitiesMartingale InequalitiesMartingalesProbability Inequalities For Sums Of Random VariablesRandom MatricesTail Bounds

TL;DR

Matrix concentration inequalities aim to offer diverse applications, ease of use, and strong conclusions comparable to their scalar counterparts. The paper develops noncommutative generalizations of classical scalar concentration bounds for sums of independent random self‑adjoint matrices. By extending Azuma, Bennett, Bernstein, Chernoff, Hoeffding, and McDiarmid inequalities to the matrix setting, the authors derive new probability inequalities for eigenvalues and norms. These inequalities provide simple, verifiable hypotheses and strong large‑deviation bounds for the maximum eigenvalue, extend to norms of rectangular sums, and yield insights into 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.