Geometry of log-concave Ensembles of random matrices and approximate reconstruction

TL;DR

This paper investigates the Restricted Isometry Property of random matrices with log-concave rows, introducing a new parameter and deriving tail estimates and deviation inequalities to analyze their geometric properties for approximate reconstruction.

Contribution

It introduces a novel parameter $\Gamma_{k,m}$ and provides new tail estimates and deviation inequalities for isotropic log-concave vectors, advancing understanding of matrix geometry in compressed sensing.

Findings

Established bounds for the operator norm of sub-matrices with log-concave rows.

Derived new tail estimates for order statistics of isotropic log-concave vectors.

Provided deviation inequalities for norms of projections of isotropic log-concave vectors.

Abstract

We study the Restricted Isometry Property of a random matrix $\Gamma$ with independent isotropic log-concave rows. To this end, we introduce a parameter $\Gamma_{k,m}$ that controls uniformly the operator norm of sub-matrices with $k$ rows and $m$ columns. This parameter is estimated by means of new tail estimates of order statistics and deviation inequalities for norms of projections of an isotropic log-concave vector.