User-Friendly Tail Bounds for Sums of Random Matrices

• Published in: Foundations of computational mathematics Vol. 12; no. 4; pp. 389 - 434
• Main Author: Tropp, Joel A.
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.08.2012; Springer; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Computer Science; Economics; Eigenvalues; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Probability; Random matrix; 60F10; 60G42; Discrete-time martingale; Sum of independent random variables; 60G50; Probability inequality; 60B20; Large deviation
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-011-9099-z

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.