Certifying the restricted isometry property is hard

TL;DR

This paper proves that verifying whether a matrix satisfies the restricted isometry property in compressed sensing is computationally infeasible (NP-hard), indicating no efficient testing method exists unless P equals NP.

Contribution

It establishes the NP-hardness of testing the restricted isometry property, a fundamental condition in compressed sensing, highlighting computational limitations.

Findings

Testing RIP is NP-hard

Efficient RIP testing is unlikely unless P=NP

Implications for compressed sensing algorithms

Abstract

This paper is concerned with an important matrix condition in compressed sensing known as the restricted isometry property (RIP). We demonstrate that testing whether a matrix satisfies RIP is NP-hard. As a consequence of our result, it is impossible to efficiently test for RIP provided P \neq NP.