Gittens, Alex A. and Tropp, Joel A. (2011) Tail Bounds for All Eigenvalues of a Sum of Random Matrices. ACM Technical Reports, 201402. California Institute of Technology , Pasadena, CA. (Unpublished) http://resolver.caltech.edu/CaltechAUTHORS:20140828084239607

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Abstract
This work introduces the minimax Laplace transform method, a modification of the cumulantbased matrix Laplace transform method developed in [Tro11c] that yields both upper and lower bounds on each eigenvalue of a sum of random selfadjoint matrices. This machinery is used to derive eigenvalue analogs of the classical Chernoff, Bennett, and Bernstein bounds. Two examples demonstrate the efficacy of the minimax Laplace transform. The first concerns the effects of column sparsification on the spectrum of a matrix with orthonormal rows. Here, the behavior of the singular values can be described in terms of coherencelike quantities. The second example addresses the question of relative accuracy in the estimation of eigenvalues of the covariance matrix of a random process. Standard results on the convergence of sample covariance matrices provide bounds on the number of samples needed to obtain relative accuracy in the spectral norm, but these results only guarantee relative accuracy in the estimate of the maximum eigenvalue. The minimax Laplace transform argument establishes that if the lowest eigenvalues decay sufficiently fast, Ω(ε^(2)κ^2_ℓ ℓ log p) samples, where κ_ℓ = λ_1(C)/λ_ℓ(C), are sufficient to ensure that the dominant ℓ eigenvalues of the covariance matrix of a N(0,C) random vector are estimated to within a factor of 1 ± ε with high probability.
Item Type:  Report or Paper (Technical Report)  

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Additional Information:  Research supported by ONR awards N000140810883 and N000141110025, AFOSR award FA95500910643, and a Sloan Fellowship.  
Group:  Applied & Computational Mathematics  
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Record Number:  CaltechAUTHORS:20140828084239607  
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:20140828084239607  
Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  49011  
Collection:  CaltechACMTR  
Deposited By:  Sydney Garstang  
Deposited On:  29 Aug 2014 20:47  
Last Modified:  29 Apr 2019 23:03 
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