A remark on weaken restricted isometry property in compressed sensing

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 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.