On the singular values of random matrices

Abstract

We present an approach that allows one to bound the largest and smallest singular values of an $N\times n$ random matrix with iid rows, distributed according to a measure on ${\mathbb{R}}^n$ that is supported in a relatively small ball and linear functionals are uniformly bounded in $L_p$ for some $p>8$, in a quantitative (non-asymptotic) fashion. Among the outcomes of this approach are optimal estimates of $1\pm c\sqrt{n/N}$ not only in the case of the above mentioned measure, but also when the measure is log-concave or when it a product measure of iid random variables with "heavy tails".