Statistics Papers

Document Type

Journal Article

Date of this Version

5-2012

Publication Source

Journal of Multivariate Analysis

Volume

107

Start Page

24

Last Page

39

DOI

10.1016/j.jmva.2011.11.008

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: 1⁄n log p → 0, 1⁄n log p → β ∈ (0, ∞), and 1⁄n log p → ∞. 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.

Keywords

coherence, correlation coefficient, limiting distribution, maximum, phase transition, random matrix, sample correlation matrix, Chen-Stein mathoud

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Date Posted: 27 November 2017

This document has been peer reviewed.