Limiting Laws of Coherence of Random Matrices with Applications to Testing Covariance Structure and Construction of Compressed Sensing Matrices

TL;DR

This paper investigates the limiting behavior of the coherence of high-dimensional random matrices and applies these results to covariance testing and the design of compressed sensing matrices.

Contribution

It derives the limiting laws of matrix coherence in high dimensions and uses these laws for covariance structure testing and compressed sensing matrix construction.

Findings

Established the law of large numbers for matrix coherence

Derived the limiting distribution of coherence in high dimensions

Applied results to covariance testing and compressed sensing design

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 $n\times p$ 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.