A Nonconvex Nonsmooth Regularization Method for Compressed Sensing and Low-Rank Matrix Completion

TL;DR

This paper introduces a nonconvex, nonsmooth regularization approach for compressed sensing and low-rank matrix completion, providing convergence analysis and recovery guarantees, with experimental validation of its effectiveness.

Contribution

It develops an alternating minimization algorithm for nonconvex regularized models in CS and MC, with convergence proofs and stability analysis under NSP and RIP conditions.

Findings

Algorithm converges to a critical point.

Provides stable recovery guarantees under NSP and RIP.

Experimental results demonstrate improved performance.

Abstract

In this paper, nonconvex and nonsmooth models for compressed sensing (CS) and low rank matrix completion (MC) is studied. The problem is formulated as a nonconvex regularized leat square optimization problems, in which the l0-norm and the rank function are replaced by l1-norm and nuclear norm, and adding a nonconvex penalty function respectively. An alternating minimization scheme is developed, and the existence of a subsequence, which generate by the alternating algorithm that converges to a critical point, is proved. The NSP, RIP, and RIP condition for stable recovery guarantees also be analysed for the nonconvex regularized CS and MC problems respectively. Finally, the performance of the proposed method is demonstrated through experimental results.