The restricted isometry property for time-frequency structured random matrices

TL;DR

This paper proves the restricted isometry property for finite-dimensional Gabor systems with random windows, showing improved bounds on the sparsity level for stable signal recovery in time-frequency structured matrices.

Contribution

It establishes the RIP for Gabor systems with random windows, improving previous bounds and developing new chaos process bounds.

Findings

RIP holds for s ≤ c n^{2/3} / log^2 n in Gabor systems

Improved bounds over previous quadratic scaling estimates

Develops bounds for a related chaos process

Abstract

We establish the restricted isometry property for finite dimensional Gabor systems, that is, for families of time--frequency shifts of a randomly chosen window function. We show that the $s$-th order restricted isometry constant of the associated $n \times n^2$ Gabor synthesis matrix is small provided $s \leq c \, n^{2/3} / \log^2 n$. This improves on previous estimates that exhibit quadratic scaling of $n$ in $s$. Our proof develops bounds for a corresponding chaos process.