A Factorized Column Sparse Approach to the CP Rank Regularized Tensor Optimization Problem

ABSTRACT Tensor optimization problems with rank regularization have attracted significant attentions in recent years due to their extensive applications in various fields. In this paper, we focus on the CANDECOMP/PARAFAC (CP) rank regularized tensor optimization problem and equivalently transform it into a column sparse regularized problem, that is, the problem with a regularization to describe the column sparsity of CP factor matrices. Moreover, we study the relationships between the original problem and the transformed problem in the sense of global and local minimizers. To solve the nonconvex and nonsmooth transformed problem, we design an inertial block coordinate descent (iBCD) algorithm and establish its global convergence. Finally, we apply the proposed iBCD algorithm to solve tensor CP decomposition and low‐CP‐rank tensor completion problems. Numerical experiments on both synthetic data and real‐world images validate the promising performance of our proposed method compared with several excellent methods for solving the CP rank regularized tensor optimization problem.