English

Phase Transition in Limiting Distributions of Coherence of High-Dimensional Random Matrices

Probability 2011-02-16 v1

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×pn\times p random matrices for a full range of the dimension pp 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\frac{1}{n}\log p \to 0, 1nlogpβ(0,)\frac{1}{n}\log p \to \beta\in (0, \infty), and 1nlogp\frac{1}{n}\log p \to\infty. The results show that the limiting behavior of the coherence differs significantly in different regimes and exhibits interesting phase transition phenomena as the dimension pp grows as a function of nn. 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}
}