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

TL;DR

This paper analyzes the limiting distributions of the coherence in high-dimensional random matrices, revealing phase transition phenomena across different growth regimes of the dimension p relative to n, with implications for statistics and signal processing.

Contribution

It provides a complete characterization of the asymptotic behavior of coherence for random matrices with spherical columns across various high-dimensional regimes, highlighting phase transitions.

Findings

Limiting distributions differ across three regimes of p growth.

Phase transition phenomena occur as p increases relative to n.

Results have applications in high-dimensional statistics and compressed sensing.

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 $n\times p$ 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: $\frac{1}{n}\log p \to 0$, $\frac{1}{n}\log p \to \beta\in (0, \infty)$, and $\frac{1}{n}\log p \to\infty$. The results show that the limiting behavior of the coherence differs significantly in different regimes and exhibits interesting phase transition phenomena as the dimension $p$ grows as a function of $n$. Applications to statistics and compressed sensing in the ultra high-dimensional setting are also discussed.