A remark on weaken restricted isometry property in compressed sensing

TL;DR

This paper demonstrates that a weakened version of the restricted isometry property (RIP) is sufficient for stable and robust sparse recovery in compressed sensing, broadening its applicability under weak moment assumptions.

Contribution

It proves the weakened RIP ensures stable and robust recovery by connecting it with the robust width property, extending its relevance in compressed sensing.

Findings

Weakened RIP suffices for stable sparse recovery.

Linking weakened RIP with robust width property enhances understanding.

Applicable to various compressed sensing scenarios.

Abstract

The restricted isometry property (RIP) has become well-known in the compressed sensing community. Recently, a weaken version of RIP was proposed for exact sparse recovery under weak moment assumptions. In this note, we prove that the weaken RIP is also sufficient for \textsl{stable and robust} sparse recovery by linking it with a recently introduced robust width property in compressed sensing. Moreover, we show that it can be widely apply to other compressed sensing instances as well.