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

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.

References in corpus (2)

• Order statistics and concentration of l_r norms for log-concave vectors
• Geometry of log-concave Ensembles of random matrices and approximate reconstruction

Cited by in corpus (7)

• A short proof of Paouris' inequality
• Moment estimates for convex measures
• Geometry of log-concave Ensembles of random matrices and approximate reconstruction
• Weak and strong moments of l_r-norms of log-concave vectors
• Restricted Khinchine inequality
• Modified Paouris inequality
• Estimating the covariance of random matrices