Measure What Should be Measured: Progress and Challenges in Compressive Sensing

Abstract

Is compressive sensing overrated? Or can it live up to our expectations? What will come after compressive sensing and sparsity? And what has Galileo Galilei got to do with it? Compressive sensing has taken the signal processing community by storm. A large corpus of research devoted to the theory and numerics of compressive sensing has been published in the last few years. Moreover, compressive sensing has inspired and initiated intriguing new research directions, such as matrix completion. Potential new applications emerge at a dazzling rate. Yet some important theoretical questions remain open, and seemingly obvious applications keep escaping the grip of compressive sensing. In this paper I discuss some of the recent progress in compressive sensing and point out key challenges and opportunities as the area of compressive sensing and sparse representations keeps evolving. I also attempt to assess the long-term impact of compressive sensing.

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• Structured Compressed Sensing: From Theory to Applications
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• A Perturbation Inequality for the Schatten-$p$ Quasi-Norm and Its Applications to Low-Rank Matrix Recovery
• Nonlinear compressed sensing based on composite mappings and its pointwise linearization
• $\alpha {\ell }_1-\beta {\ell }_2$ sparsity regularization for nonlinear ill-posed problems
• Information Theory and Moduli of Riemann Surfaces
• A sparse-sampling approach for the fast computation of matrices: application to molecular vibrations