Mathematics of Information and Applications Seminar: Ben Adock

Compressed sensing concerns the recovery of signals and images from seemingly incomplete data sets. Introduced nearly a decade ago, it has since become an intensive area of research in applied mathematics, engineering and computer science. However, many practical problems in which compressed sensing is applied, e.g. imaging, are not fully explained by existing theory. In this talk I will present a new framework for compressed sensing that seeks to bridge this gap. This framework is based on replacing some standard principles of compressed sensing with new local notions; specifically, sparsity in levels, local coherence in levels and multilevel random subsampling. I will demonstrate a series of near-optimal recovery guarantees based on these local concepts that explains the effectiveness of compressed sensing in such applications. Moreover, this framework is not just useful in understanding existing compressed sensing approaches. In the final part of the talk I will demonstrate how leveraging local sparsity through appropriately-designed locally incoherent sensing matrices leads to substantially improved compressed sensing techniques in a range of other applications.