Limiting Laws of Coherence of Random Matrices with Applications to Testing Covariance Structure and Construction of Compressed Sensing Matrices
Tony Cai and Tiefeng Jiang
- 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 law of large numbers and 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. The asymptotic results is also applied to the construction of compressed sensing matrices.
- Paper: pdf file.
- Supplement: Additional technical arguments are contained in the supplement.
- Other related paper:
Cai, T., Wang, L. & Xu, G. (2010).
Stable recovery of sparse signals and an oracle inequality.
IEEE Transactions on Information Theory , to appear.
- Cai, T. & Jiang, T. (2011).
Phase transition in limiting distributions of coherence of high-dimensional random matrices.
J. Multivariate Analysis, to appear.