Approximate Message Passing Algorithm for Nonconvex Regularization

In this paper, we study the sparse signal reconstruction with nonconvex regularization, mainly focusing on two popular nonconvex regularization methods, minimax concave penalty (MCP) and smoothly clipped absolute deviation (SCAD). An approximate message passing (AMP) algorithm is an effective method for signal reconstruction. Based on the AMP algorithm, we propose an improved MCP iterative thresholding algorithm and an improved SCAD iterative thresholding algorithm. Furthermore, we analyze the convergence of the new algorithms and provide a series of experiments to assess the performance of the new algorithms. The experiments show that the new algorithms based on AMP have stronger reconstruction capabilities, higher phase transition for sparse signal reconstruction, and better variable selection ability than the original MCP iterative thresholding algorithm and the original SCAD iterative thresholding algorithm.