Generalized notions of sparsity and restricted isometry property. Part II: Applications
This work provides theoretical extensions of RIP for compressed sensing and signal processing, though it appears incremental as it builds directly on Part I.
The paper explores how the restricted isometry property (RIP) applies to generalized sparsity and group measurements, showing that optimizing over Banach spaces can reduce required measurements, extending RIP to group-structured measurements with optimal scaling, and obtaining infinite-dimensional RIP for Fourier measurements with smoothness assumptions.
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.