Smoothing fast proximal gradient algorithm for the relaxation of matrix rank regularization problem

Abstract

This paper proposes a general inertial smoothing proximal gradient algorithm for solving the Capped-ℓ1 exact continuous relaxation regularization model proposed by Yu and Zhang (2022) [29]. The proposed algorithm incorporates different extrapolations into the gradient and proximal steps. It is proved that, under some general parameter constraints, the singular values of any accumulation point of the sequence generated by the proposed algorithm have the common support set, and the zero singular values can be achieved in a finite number of iterations. Furthermore, any accumulation point is a lifted stationary point of the relaxation model. Numerical experiments illustrate the efficiency of the proposed algorithm on synthetic and real data, respectively.