On the interval of fluctuation of the singular values of random matrices
(1509.02322)Abstract
Let $A$ be a matrix whose columns $X1,\dots, XN$ are independent random vectors in $\mathbb{R}n$. Assume that the tails of the 1-dimensional marginals decay as $\mathbb{P}(|\langle Xi, a\rangle|\geq t)\leq t{-p}$ 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 $|Xi|$ 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 $\alpha\in[1, 2]$.
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