A Novel Two-Parameter Penalty: Relaxation Degree Analysis and Sparse Signal Recovery

Abstract

In this article, we introduce a nonconvex two-parameter penalty function $P_{a,p}$, parameterized by $a\in (0,\infty )$ and $p\in (0,1]$, and the relaxation degree RD${}_P$ for a separable nonconvex penalty function $P$. Based on $P_{a,p}$, we further propose the $P_{a,p}$ minimization framework for sparse signal recovery. This framework generalizes the TL1 minimization model established by S. Zhang and J. Xin (corresponding to the special case $p=1$) and provides a unified and flexible family of nonconvex penalty functions for sparse signal recovery. Using the sparse convex-combination technique, we establish both exact and stable sparse signal recovery under the restricted isometry property (RIP). To efficiently solve the resulting nonconvex optimization problem, we apply a modified iteratively re-weighted least squares method and the difference of convex functions algorithm (DCA) to develop the IRLSTLp algorithm for unconstrained $P_{a,p}$ minimization and prove some convergence results. Finally, some numerical experiments are conducted to show the flexibility of the $P_{a,p}$ minimization framework, the robustness of the IRLSTLp, and also the utility of the relaxation degree.