An Introduction to Compressed Sensing

Abstract

Compressed sensing and many research activities associated with it can be seen as a framework for signal processing of low-complexity structures. A cornerstone of the underlying theory is the study of inverse problems with linear or nonlinear measurements. Whether it is sparsity, low-rankness, or other familiar notions of low complexity, the theory addresses necessary and sufficient conditions behind the measurement process to guarantee signal reconstruction with efficient algorithms. This includes consideration of robustness to measurement noise and stability with respect to signal model inaccuracies. This introduction aims to provide an overall view of some of the most important results in this direction. After discussing various examples of low-complexity signal models, two approaches to linear inverse problems are introduced which, respectively, focus on the recovery of individual signals and recovery of all low-complexity signals simultaneously. In particular, we focus on the former setting, giving rise to so-called nonuniform signal recovery problems. We discuss different necessary and sufficient conditions for stable and robust signal reconstruction using convex optimization methods. Appealing to concepts from non-asymptotic random matrix theory, we outline how certain classes of random sensing matrices, which fully govern the measurement process, satisfy certain sufficient conditions for signal recovery. Finally, we review some of the most prominent algorithms for signal recovery proposed in the literature.