Random embeddings with an almost Gaussian distortion

Abstract

Let X be a symmetric, isotropic random vector in Rm and let X1...,Xn be independent copies of X. We show that under mild assumptions on ‖X‖2 (a suitable thin-shell bound) and on the tail-decay of the marginals 〈X,u〉, the random matrix A, whose columns are Xi/m exhibits a Gaussian-like behaviour in the following sense: for an arbitrary subset of T⊂Rn, the distortion supt∈T⁡|‖At‖22−‖t‖22| is almost the same as if A were a Gaussian matrix.A simple outcome of our result is that if X is a symmetric, isotropic, log-concave random vector and n≤m≤c1(α)nα for some α>1, then with high probability, the extremal singular values of A satisfy the optimal estimate: 1−c2(α)n/m≤λmin≤λmax≤1+c2(α)n/m.