Compressed sensing recovery via nonconvex shrinkage penalties

The minimization of compressed sensing is often relaxed to , which yields easy computation using the shrinkage mapping known as soft thresholding, and can be shown to recover the original solution under certain hypotheses. Recent work has derived a general class of shrinkages and associated nonconvex penalties that better approximate the original penalty and empirically can recover the original solution from fewer measurements. We specifically examine p-shrinkage and firm thresholding . In this work, we prove that given data and a measurement matrix from a broad class of matrices, one can choose parameters for these classes of shrinkages to guarantee exact recovery of the sparsest solution. We further prove convergence of the algorithm iterative p-shrinkage (IPS) for solving one such relaxed problem.