Phase Transition in Limiting Distributions of Coherence of High-Dimensional Random Matrices
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 random matrices for a full range of the dimension 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: , , and . The results show that the limiting behavior of the coherence differs significantly in different regimes and exhibits interesting phase transition phenomena as the dimension grows as a function of . Applications to statistics and compressed sensing in the ultra high-dimensional setting are also discussed.
Keywords
Cite
@article{arxiv.1102.2926,
title = {Phase Transition in Limiting Distributions of Coherence of High-Dimensional Random Matrices},
author = {Tony Cai and Tiefeng Jiang},
journal= {arXiv preprint arXiv:1102.2926},
year = {2011}
}