Compressed Sensing Recovery via Nonconvex Shrinkage Penalties

TL;DR

This paper investigates nonconvex shrinkage penalties like p-shrinkage and firm thresholding in compressed sensing, proving conditions for exact sparse recovery and convergence of related algorithms, improving upon traditional methods.

Contribution

It introduces theoretical guarantees for exact recovery using nonconvex shrinkages and proves convergence of the iterative p-shrinkage algorithm.

Findings

Exact recovery guaranteed under broad conditions

Convergence of iterative p-shrinkage established

Nonconvex penalties outperform traditional methods in some cases

Abstract

The $\ell^0$ minimization of compressed sensing is often relaxed to $\ell^1$, 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 $\ell^0$ 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.