A three-operator splitting algorithm for nonconvex sparsity regularization

Sparsity regularization has been widely adopted in many fields, such as signal and image processing and machine learning. In this paper, we mainly consider nonconvex minimization problems involving three terms, for applications such as sparse signal recovery and low rank matrix recovery. We employ a three-operator splitting proposed by Davis and Yin [Set-Valued Var. Anal., 25 (2017), pp. 829-858] (namely, DYS) to solve the resulting possibly nonconvex problems and develop the convergence theory for this three-operator splitting algorithm in the nonconvex case. We show that when the step size is chosen less than a computable threshold, the whole sequence converges to a stationary point. By defining a new decreasing energy function associated with the DYS method, we establish the global convergence of the whole sequence and a local convergence rate under an additional assumption that F, G, and H are semialgebraic. We also provide sufficient conditions for the boundedness of the generated sequence. Finally, some numerical experiments are conducted to compare the DYS algorithm with some classical efficient algorithms for sparse signal recovery and low rank matrix completion. The numerical results indicate that DYS outperforms the existing methods for these applications.