Reconstruction of Sparse Signals by Minimization of Nonconvex Penalties

Abstract

In this paper, we introduce a signal recovery method using nonconvex sparse penalties in compressed sensing. We deal with nonconvex sparse penalties called Smoothly Clipped Ab solute Deviation (SCAD) and Minimax Concave Penalty (MCP). The form of these penalties change depending on the regularization parameter, which we call the nonconvexity parameter,and SCAD and MCP coincide with the ℓ₁ penalty at a certain limit of the nonconvexity param eter. We introduce an approximate message passing (AMP) algorithm to solve the minimiza tion problem of the nonconvex sparse penalties. The corresponding theoretical analysis shows that the nonconvex penalty minimization method gives better recovery performance than the ℓ₁-minimization method as the nonconvexity parameter decreases. However, the small nonconvexity parameter induces the difficulty in the convergence of AMP. We show that this difficulty is caused by the vanishing basin of attraction to the fixed point of AMP, and mitigate the difficulty by introducing a method called non-convexity control (Sakata and Obuchi (2021)).