Annals of Statistics
- Ann. Statist.
- Volume 39, Number 3 (2011), 1496-1525.
Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices
Full-text: Open access
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.
Article information
Source
Ann. Statist., Volume 39, Number 3 (2011), 1496-1525.
Dates
First available in Project Euclid: 13 May 2011
Permanent link to this document
https://projecteuclid.org/euclid.aos/1305292044
Digital Object Identifier
doi:10.1214/11-AOS879
Mathematical Reviews number (MathSciNet)
MR2850210
Zentralblatt MATH identifier
1220.62066
Subjects
Primary: 62H12: Estimation 60F05: Central limit and other weak theorems
Secondary: 60F15: Strong theorems 62H10: Distribution of statistics
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
Citation
Cai, T. Tony; Jiang, Tiefeng. Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices. Ann. Statist. 39 (2011), no. 3, 1496--1525. doi:10.1214/11-AOS879. https://projecteuclid.org/euclid.aos/1305292044
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Supplemental materials
- Supplementary material: Additional technical proofs. We give complete proofs for some technical lemmas used in the proofs of the main results.Digital Object Identifier: doi:10.1214/11-AOS879SUPP
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