Tail estimates for norms of sums of log-concave random vectors

TL;DR

This paper develops new tail estimates for norms of sums and projections of log-concave random vectors, with applications to matrix operator norms and the Restricted Isometry Property in Compressive Sensing.

Contribution

It introduces novel tail bounds for projections and sums of log-concave vectors, extending to matrix norms and advancing understanding of the Restricted Isometry Property.

Findings

Established tail estimates for Euclidean norms of projections of log-concave vectors

Derived bounds for operator norms of sub-matrices in log-concave ensembles

Applied results to analyze the Restricted Isometry Property in Compressive Sensing

Abstract

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.