Matrix Infinitely Divisible Series: Tail Inequalities and Their Applications

In this paper, we study tail inequalities of the largest eigenvalue of a matrix infinitely divisible (i.d.) series, which is a finite sum of fixed matrices weighted by i.d. random variables. We obtain several types of tail inequalities, including Bennett-type and Bernstein-type inequalities. This al...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 66; no. 2; pp. 1099 - 1117
Main Authors: Zhang, Chao, Gao, Xianjie, Hsieh, Min-Hsiu, Hang, Hanyuan, Tao, Dacheng
Format: Journal Article
Language:English
Published: New York IEEE 01.02.2020
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Compressed sensing
Constraints
Covariance matrices
Economic models
Eigenvalues
Eigenvalues and eigenfunctions
Gaussian distribution
Inequalities
Inequality
infinitely divisible distribution
largest eigenvalue
Linear matrix inequalities
Lower bounds
Optimization
Orthogonality
Poisson distribution
Random matrix
Random variables
restricted isometry property
Series (mathematics)
Statistical analysis
tail inequality
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary:In this paper, we study tail inequalities of the largest eigenvalue of a matrix infinitely divisible (i.d.) series, which is a finite sum of fixed matrices weighted by i.d. random variables. We obtain several types of tail inequalities, including Bennett-type and Bernstein-type inequalities. This allows us to further bound the expectation of the spectral norm of a matrix i.d. series. Moreover, by developing a new lower-bound function for Q(s) = (s + 1) log(s + 1) - s that appears in the Bennett-type inequality, we derive a tighter tail inequality of the largest eigenvalue of the matrix i.d. series than the Bernstein-type inequality when the matrix dimension is high. The resulting lower-bound function is of independent interest and can improve any Bennett-type concentration inequality that involves the function Q(s). The class of i.d. probability distributions is large and includes Gaussian and Poisson distributions, among many others. Therefore, our results encompass the existing work on matrix Gaussian series as a special case. Lastly, we show that the tail inequalities of a matrix i.d. series have applications in several optimization problems including the chance constrained optimization problem and the quadratic optimization problem with orthogonality constraints. In addition, we also use the resulting tail bounds to show that random matrices constructed from i.d. random variables satisfy the restricted isometry property (RIP) when it acts as a measurement matrix in compressed sensing.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2019.2951759