Low‐rank approximation of tensors via sparse optimization

The goal of this paper is to find a low rank approximation for a given tensor. Specifically, we also give a computable strategy on calculating the rank of a given tensor, based on approximating the solution of a NP-hard problem. In this paper, we formulate a sparse optimization via $l_1$-regularization to find low rank approximation of tensors. To solve this sparse optimization problem, we propose a rescaling algorithm of proximal alternating minimization and study the theoretical convergence of this algorithm. Furthermore, we discuss the probabilistic consistency of the sparsity result and suggest a way to choose the regularization parameter for practical computation. In the simulation experiment, the performance of our algorithm supports that our method provides an efficient approximation of the rank-one component number in tensors. Moreover, this algorithm can be also applied to surveillance video for extracting background information.