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June 2011 Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices
T. Tony Cai, Tiefeng Jiang
Ann. Statist. 39(3): 1496-1525 (June 2011). DOI: 10.1214/11-AOS879

Abstract

Testing covariance structure is of significant interest in many areas of statistical analysis and construction of compressed sensing matrices is an important problem in signal processing. Motivated by these applications, we study in this paper the limiting laws of the coherence of an n × p random matrix in the high-dimensional setting where p can be much larger than n. Both the law of large numbers and the limiting distribution are derived. We then consider testing the bandedness of the covariance matrix of a high-dimensional Gaussian distribution which includes testing for independence as a special case. The limiting laws of the coherence of the data matrix play a critical role in the construction of the test. We also apply the asymptotic results to the construction of compressed sensing matrices.

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T. Tony Cai. Tiefeng Jiang. "Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices." Ann. Statist. 39 (3) 1496 - 1525, June 2011. https://doi.org/10.1214/11-AOS879

Information

Published: June 2011
First available in Project Euclid: 13 May 2011

zbMATH: 1220.62066
MathSciNet: MR2850210
Digital Object Identifier: 10.1214/11-AOS879

Subjects:
Primary: 60F05 , 62H12
Secondary: 60F15 , 62H10

Keywords: Chen–Stein method , Coherence , compressed sensing matrix , covariance structure , Law of Large Numbers , Limiting distribution , Maxima , Moderate deviations , mutual incoherence property , Random matrix , sample correlation matrix

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 3 • June 2011
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