Chevet type inequality and norms of submatrices

TL;DR

This paper establishes a Chevet type inequality for isotropic log-concave unconditional matrices, providing sharp bounds on submatrix norms and the Restricted Isometry Constant, with implications for compressed sensing.

Contribution

It introduces a new Chevet type inequality for a specific class of random matrices and applies it to derive optimal bounds on submatrix norms and the Restricted Isometry Constant.

Findings

Sharp upper bounds for submatrix Euclidean norms

Precise estimates for the Restricted Isometry Constant

Limitations of the inequality for general isotropic log-concave matrices

Abstract

We prove a Chevet type inequality which gives an upper bound for the norm of an isotropic log-concave unconditional random matrix in terms of expectation of the supremum of "symmetric exponential" processes compared to the Gaussian ones in the Chevet inequality. This is used to give sharp upper estimate for a quantity $\Gamma_{k,m}$ that controls uniformly the Euclidean operator norm of the sub-matrices with $k$ rows and $m$ columns of an isotropic log-concave unconditional random matrix. We apply these estimates to give a sharp bound for the Restricted Isometry Constant of a random matrix with independent log-concave unconditional rows. We show also that our Chevet type inequality does not extend to general isotropic log-concave random matrices.