Tensor Completion via Fully-Connected Tensor Network Decomposition with Regularized Factors

The recently proposed fully-connected tensor network (FCTN) decomposition has a powerful ability to capture the low-rankness of tensors and has achieved great success in tensor completion. However, the FCTN decomposition-based method is highly sensitive to the choice of the FCTN-rank and can not provide satisfactory results in the recovery of local details. In this paper, we propose a novel tensor completion model by introducing a factor-based regularization to the framework of the FCTN decomposition. The regularization provides a robust performance to the choice of the FCTN-rank and simultaneously enforces the global low-rankness and the local continuity of the target tensor. More specifically, by illustrating that the unfolding matrices of the FCTN factors can be reasonably assumed to be of low-rank in the gradient domain and further imposing a low-rank matrix factorization (LRMF) on them, the proposed model enhances the robustness to the choice of the FCTN-rank. By employing a Tikhonov regularization to the LRMF factors, the proposed model promotes the local continuity and preserves local details of the target tensor. To solve the optimization problem associated with the proposed model, we develop an efficient proximal alternating minimization (PAM)-based algorithm and theoretically demonstrate its convergence. To reduce the running time of the developed algorithm, we design an automatic rank-increasing strategy. Numerical experimental results demonstrate that the proposed method outperforms its competitors.