2012_user-friendly tail bounds for sums of random matrices_FOCM英文版本.pdf

Found Comput Math (2012) 12:389–434 DOI 10.1007/s10208-011-9099-z User-Friendly Tail Bounds for Sums of Random Matrices Joel A. Tropp Received: 16 January 2011 / Accepted: 13 June 2011 / Published online: 2 August 2011 © The Author(s) 2011. This article is published with open access at S Abstract This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hy- potheses 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 classi- cal bounds associated with the names Azuma, Bennett, Bernstein, Chernoff, Hoeffd- ing, and McDiarmid. The matrix inequalities promise the same diversity of applica- tion, ease of use, and strength of conclusion that have made the scalar inequalities so valuable. Keywords Discrete-time martingale · Large deviation · Probability inequality · Random matrix · Sum of independent random variables Mathematics Subject Classification (2000) Primary 60B20 · Secondary 60F10 · 60G50 · 60G42 Communicated by Albert Cohen. J.A. Tropp ( ) Computing Mathematical Sciences, MC 305-16, California Institute of Technology, Pasadena, CA 91125, USA e-mail: jtropp@ 390 Found Comput Math (2012) 12:389–434 1 Introduction Random matrices have come to play a significant role in computational mathemat- ics. This line of research has advanced by using established methods from random matrix theory, but it has also generated difficult questions that cannot be addressed without new tools. Let us summarize some of the challenges that