Tropp, Joel A. (2012) User-Friendly Tail Bounds for Sums of Random Matrices. Foundations of Computational Mathematics, 12 (4). pp. 389-434. ISSN 1615-3375. doi:10.1007/s10208-011-9099-z. https://resolver.caltech.edu/CaltechAUTHORS:20120821-072332716
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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.
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| Additional Information: | © 2011 The Author(s). This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited. Received: 16 January 2011. Accepted: 13 June 2011. Published online: 2 August 2011. Communicated by Albert Cohen. I would like to thank Vern Paulsen and Bernhard Bodmann for some helpful conversations connected with this project. Klas Markström and David Gross provided references to related work. Ben Recht offered some useful comments on the presentation. Yao-Liang Yu proposed the argument in Lemma 6.7. Richard Chen and Alex Gittens have helped me root out (numerous) typographic errors. Finally, let me mention Roberto Oliveira’s elegant work [40] on matrix probability inequalities, which originally spurred me to pursue this project. Research supported by ONR award N00014-08-1-0883, DARPA award N66001-08-1-2065, and AFOSR award FA9550-09-1-0643. | |||||||||||||||
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| Subject Keywords: | Discrete-time martingale, Large deviation, Probability inequality, Random matrix, Sum of independent random variables | |||||||||||||||
| Issue or Number: | 4 | |||||||||||||||
| Classification Code: | MSC (2000): Primary 60B20 · Secondary 60F10 · 60G50 · 60G42 | |||||||||||||||
| DOI: | 10.1007/s10208-011-9099-z | |||||||||||||||
| Record Number: | CaltechAUTHORS:20120821-072332716 | |||||||||||||||
| Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20120821-072332716 | |||||||||||||||
| Official Citation: | User-Friendly Tail Bounds for Sums of Random Matrices Joel A. Tropp pp. 389-434 | |||||||||||||||
| Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||||||||
| ID Code: | 33386 | |||||||||||||||
| Collection: | CaltechAUTHORS | |||||||||||||||
| Deposited By: | Ruth Sustaita | |||||||||||||||
| Deposited On: | 21 Aug 2012 15:19 | |||||||||||||||
| Last Modified: | 09 Nov 2021 21:34 |
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