Generalized notions of sparsity and restricted isometry property. Part II: Applications

Abstract

The restricted isometry property (RIP) is a universal tool for data recovery. We explore the implication of the RIP in the framework of generalized sparsity and group measurements introduced in the Part I paper. It turns out that for a given measurement instrument the number of measurements for RIP can be improved by optimizing over families of Banach spaces. Second, we investigate the preservation of difference of two sparse vectors, which is not trivial in generalized models. Third, we extend the RIP of partial Fourier measurements at optimal scaling of number of measurements with random sign to far more general group structured measurements. Lastly, we also obtain RIP in infinite dimension in the context of Fourier measurement concepts with sparsity naturally replaced by smoothness assumptions.