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 $\mathbb P(|\langle X_i, a\rangle|\geq t)\leq C t^{-p}$ 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 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 $\alpha \in (1,2]$.