The 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
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.
Ann. Statist., Volume 39, Number 3 (2011), 1496-1525.
First available in Project Euclid: 13 May 2011
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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
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
- Supplementary material: Additional technical proofs. We give complete proofs for some technical lemmas used in the proofs of the main results.