LIMITING LAWS OF COHERENCE OF RANDOM MATRICES WITH APPLICATIONS TO TESTING COVARIANCE STRUCTURE AND CONSTRUCTION OF COMPRESSED SENSING MATRICES

• 成果类型: Article
• 署名作者: Cai, T. Tony; Jiang, Tiefeng
• 署名单位: University of Pennsylvania; University of Minnesota System; University of Minnesota Twin Cities
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/11-AOS879
• 发表日期: 2011
• 页码: 1496-1525
• 关键词: sample correlation-matrices; dimensional feature space; DANTZIG SELECTOR; asymptotic-distribution; statistical estimation; MULTIVARIATE-ANALYSIS; LARGEST EIGENVALUE; stable recovery; largest entries; sparse signals

摘要

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 x 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.