The gap between the null space property and the restricted isometry property

TL;DR

This paper explores the relationship between the null space property (NSP) and the restricted isometry property (RIP) in compressed sensing, demonstrating that RIP is fundamentally stronger than NSP through theoretical analysis and robust recovery guarantees.

Contribution

It introduces the concept of RIP-NSP matrices and proves that RIP is strictly stronger than NSP, providing new insights into their differences.

Findings

RIP implies NSP, but not vice versa.

RIP-NSP matrices can achieve robust recovery similar to RIP matrices.

The paper establishes that RIP is fundamentally stronger than NSP.

Abstract

The null space property (NSP) and the restricted isometry property (RIP) are two properties which have received considerable attention in the compressed sensing literature. As the name suggests, NSP is a property that depends solely on the null space of the measurement procedure and as such, any two matrices which have the same null space will have NSP if either one of them does. On the other hand, RIP is a property of the measurement procedure itself, and given an RIP matrix it is straightforward to construct another matrix with the same null space that is not RIP. %Furthermore, RIP is known to imply NSP and therefore RIP is a strictly stronger assumption than NSP. We say a matrix is RIP-NSP if it has the same null space as an RIP matrix. We show that such matrices can provide robust recovery of compressible signals under Basis pursuit which in many applicable settings is comparable to the guarantee that RIP provides. More importantly, we constructively show that the RIP-NSP is stronger than NSP with the aid of this robust recovery result, which shows that RIP is fundamentally stronger than NSP.