Restricted isometry property of random subdictionaries

TL;DR

This paper investigates the statistical restricted isometry property (StRIP) of deterministic sensing matrices, providing new conditions under which these matrices can reliably recover sparse signals with fewer measurements.

Contribution

The paper introduces new sufficient conditions for the StRIP of deterministic matrices based on mutual coherence, reducing the required number of measurements to O(k) for many matrix families.

Findings

Many deterministic matrices satisfy StRIP with m=O(k)

Improves previous bounds of m=Θ(k log N) or m=Θ(k log k)

Provides examples of matrices with proven StRIP property

Abstract

We study statistical restricted isometry, a property closely related to sparse signal recovery, of deterministic sensing matrices of size $m \times N$. A matrix is said to have a statistical restricted isometry property (StRIP) of order $k$ if most submatrices with $k$ columns define a near-isometric map of ${\mathbb R}^k$ into ${\mathbb R}^m$. As our main result, we establish sufficient conditions for the StRIP property of a matrix in terms of the mutual coherence and mean square coherence. We show that for many existing deterministic families of sampling matrices, $m=O(k)$ rows suffice for $k$-StRIP, which is an improvement over the known estimates of either $m = \Theta(k \log N)$ or $m = \Theta(k\log k)$. We also give examples of matrix families that are shown to have the StRIP property using our sufficient conditions.