Sparse signal recovery via infimal convolution based penalty

Abstract

In this paper, we propose a non-convex penalty function for sparse signal recovery by using infimal convolution approximation. First, we show that this penalty function is between the ℓ1 norm and the difference of ℓ1 and ℓ2 norm (ℓ1−2 norm), thus it can retain the advantages of these two norms at the same time, which means that it can induce the sparsity effectively for the low-amplitude components as the ℓ1 norm and relieve underestimating the high-amplitude components as the ℓ1−2 norm. Second, we employ two iterative methods to solve the non-convex penalty minimization. One is based on the difference of convex algorithm (DCA), which uses the alternating direction method of multipliers (ADMM) to solve the subproblem. And the other one employs the forward–backward splitting (FBS) algorithm, which can be solved by using the derived closed-form solutions. We also show that these two algorithms converge to a stationary point satisfying the first-order optimality condition. The experimental results demonstrate the effectiveness of the proposed method by comparing with some other penalty functions.