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

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 $\mathbb{P}(|⟨X_i,a⟩|\ge t)\le C{t}^{-p}$ uniformly in $a\in S^{n-1}$ and $i\le N$. 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 empirical covariance matrices 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 $\alpha \in (1,2]$.