Limiting Laws of Coherence of Random Matrices with Applications to Testing Covariance Structure and Construction of Compressed Sensing Matrices
Tony Cai and Tiefeng Jiang
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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 papers:
Cai, T., Wang, L. & Xu, G. (2010).
Stable recovery of sparse signals and an oracle inequality.
IEEE Transactions on Information Theory 56, 3516-3522.Cai, T. & Jiang, T. (2012).
Phase transition in limiting distributions of coherence of high-dimensional random matrices.
J. Multivariate Analysis 107, 24-39.