On the interval of fluctuation of the singular values of random matrices

TL;DR

This paper investigates the spectral properties of random matrices with independent columns having heavy-tailed distributions, establishing conditions under which they satisfy the Restricted Isometry Property and approximate covariance matrices.

Contribution

It extends existing results by providing new bounds for matrices with heavy-tailed entries, including exponential decay cases, and analyzes their spectral behavior.

Findings

Matrices with heavy-tailed columns satisfy RIP under certain conditions.

Empirical covariance matrices closely approximate true covariance matrices.

Sharp bounds are established for decay rates of tail distributions.

Abstract

Let $A$ be a matrix whose columns $X_1,\dots, X_N$ are independent random vectors in $\mathbb{R}^n$. Assume that the tails of the 1-dimensional marginals decay as $\mathbb{P}(|\langle X_i, a\rangle|\geq t)\leq t^{-p}$ uniformly in $a\in S^{n-1}$ and $i\leq N$. Then for $p>4$ we prove that with high probability $A/{\sqrt{n}}$ has the Restricted Isometry Property (RIP) provided that Euclidean norms $|X_i|$ are concentrated around $\sqrt{n}$. We also show that the covariance matrix is well approximated by the empirical covariance matrix and establish corresponding quantitative estimates on the rate of convergence in terms of the ratio $n/N$. Moreover, we obtain sharp bounds for both problems when the decay is of the type $ \exp({-t^{\alpha}})$ with $\alpha \in (0,2]$, extending the known case $\alpha\in[1, 2]$.