User-friendly Tail Bounds for Sums of Random Matrices

Authors

Joel A. Tropp

Year

2010

Preprint

Source

Foundations of Computational Mathematics

Identifiers

arXiv 1004.4389 DOI 10.1007/s10208-011-9099-z

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.

Summary¶

Presents accessible matrix concentration inequalities for sums of independent random self-adjoint matrices, including matrix analogues of classical scalar bounds. The paper became a standard reference because it packages powerful noncommutative tail tools in user-friendly forms.