User-Friendly Tail Bounds for Sums of Random Matrices

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 boun...

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Bibliographic Details
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 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
Online Access:Get full text
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Summary: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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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-011-9099-z