Hidden cliques and the certification of the restricted isometry property

TL;DR

This paper explores the computational difficulty of verifying the restricted isometry property in sensing matrices used in compressed sensing, linking it to the hidden clique problem, and proposes an improved checking algorithm.

Contribution

It establishes the hardness of approximating restricted isometry parameters and introduces an enhanced algorithm for verifying this property.

Findings

Verifying restricted isometry is computationally hard under certain assumptions.

Approximation of isometry parameters cannot be achieved efficiently.

An improved algorithm for checking the restricted isometry property is proposed.

Abstract

Compressed sensing is a technique for finding sparse solutions to underdetermined linear systems. This technique relies on properties of the sensing matrix such as the restricted isometry property. Sensing matrices that satisfy this property with optimal parameters are mainly obtained via probabilistic arguments. Deciding whether a given matrix satisfies the restricted isometry property is a non-trivial computational problem. Indeed, we show in this paper that restricted isometry parameters cannot be approximated in polynomial time within any constant factor under the assumption that the hidden clique problem is hard. Moreover, on the positive side we propose an improvement on the brute-force enumeration algorithm for checking the restricted isometry property.