Generalized notions of sparsity and restricted isometry property. Part I: A unified framework
This work provides a theoretical foundation for analyzing inverse problems in machine learning and signal processing, though it appears incremental as it builds on existing RIP approaches.
The authors tackled the problem of extending the restricted isometry property (RIP) to broader sparsity models, such as those in compressed sensing and low-rank tensors, by proposing generalized notions of sparsity and a unified framework, with results extending prior work to commutative and noncommutative function spaces and affine group actions.
The restricted isometry property (RIP) is an integral tool in the analysis of various inverse problems with sparsity models. Motivated by the applications of compressed sensing and dimensionality reduction of low-rank tensors, we propose generalized notions of sparsity and provide a unified framework for the corresponding RIP, in particular when combined with isotropic group actions. Our results extend an approach by Rudelson and Vershynin to a much broader context including commutative and noncommutative function spaces. Moreover, our Banach space notion of sparsity applies to affine group actions. The generalized approach in particular applies to high order tensor products.