Tail estimates for norms of sums of log-concave random vectors
Résumé
We establish new tail estimates for order statistics and for the Euclidean norms of projections of an isotropic log-concave random vector. More generally, we prove tail estimates for the norms of projections of sums of independent log-concave random vectors, and uniform versions of these in the form of tail estimates for operator norms of matrices and their sub-matrices in the setting of a log-concave ensemble. This is used to study a quantity A(k, m) that controls uniformly the operator norm of the sub-matrices with k rows and m columns of a matrix A with independent isotropic log-concave random rows. We apply our tail estimates of A(k, m) to the study of restricted isometry property that plays a major role in the compressive sensing theory.