A three-operator splitting algorithm for nonconvex sparsity regularization

TL;DR

This paper introduces a three-operator splitting algorithm for nonconvex sparsity regularization problems, providing convergence guarantees and demonstrating superior performance in sparse signal and low-rank matrix recovery tasks.

Contribution

It develops a convergence theory for the nonconvex three-operator splitting method and shows its effectiveness over existing algorithms in practical applications.

Findings

Convergence to stationary points under step size conditions

Global convergence with a new energy function

Outperforms classical algorithms in numerical experiments

Abstract

Sparsity regularization has been largely applied 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 the applications such as: sparse signal recovery and low rank matrix recovery. We employ a three-operator splitting proposed by Davis and Yin (called 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 if the step size is chosen less than a computable threshold, then 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 this energy function is a Kurdyka-$\L$ojasiewicz function. 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 method outperforms the exsiting methods for these specific applications.