A Modified Proximal Gradient Algorithm for Nonconvex Regularized Least Squares in Compressed Sensing

In this paper, we propose a proximal gradient with truncation and pursuit (PGTP) to solve the least-squares problem with nonconvex sparsity-inducing penalties in compressed sensing. When the proximal mapping of a penalty is a limited shrinkage thresholding operator, under some condition regarding the $ 3s$th order restricted isometry property and the stepsize, the PGTP algorithm can approach the best $s$-sparse approximation for a general signal. Furthermore, we obtain a variant of the hard thresholding pursuit (HTP) algorithm (called HTP_modify) that has a better RIP-based sufficient condition for achieving the best $s$-sparse approximation recovery effect $(\delta _{3s}< 0.5931)$ than those yielded by HTP and its variants when the stepsize is 1. We show that HTP_modify is merely the PGTP algorithm under some conditions. We conclude with some numerical experiments to demonstrate the high recovery performance of PGTP.