Phase transition in limiting distributions of coherence of high-dimensional random matrices

https://doi.org/10.1016/j.jmva.2011.11.008Get rights and content
Under an Elsevier user license
open archive

Abstract

The coherence of a random matrix, which is defined to be the largest magnitude of the Pearson correlation coefficients between the columns of the random matrix, is an important quantity for a wide range of applications including high-dimensional statistics and signal processing. Inspired by these applications, this paper studies the limiting laws of the coherence of n×p random matrices for a full range of the dimension p with a special focus on the ultra high-dimensional setting. Assuming the columns of the random matrix are independent random vectors with a common spherical distribution, we give a complete characterization of the behavior of the limiting distributions of the coherence. More specifically, the limiting distributions of the coherence are derived separately for three regimes: 1nlogp0, 1nlogpβ(0,), and 1nlogp. The results show that the limiting behavior of the coherence differs significantly in different regimes and exhibits interesting phase transition phenomena as the dimension p grows as a function of n. Applications to statistics and compressed sensing in the ultra high-dimensional setting are also discussed.

AMS 2000 subject classifications

primary
62H12
60F05
secondary
60F15
62H10

Keywords

Coherence
Correlation coefficient
Limiting distribution
Maximum
Phase transition
Random matrix
Sample correlation matrix
Chen–Stein method

Cited by (0)