Adaptive total variation and second-order total variation-based model for low-rank tensor completion

Recently, low-rank regularization has achieved great success in tensor completion. However, only considering the global low-rankness is not sufficient, especially for a low sampling rate (SR). Total variation (TV) is introduced into low-rank tensor completion (LRTC) problem to promote the local smoothness by incorporating the first-order derivatives information. However, TV usually leads to undesirable staircase effects. To alleviate these staircase effects, we suggest a first- and second-order TV-based parallel matrix factorization model for LRTC problem, which integrates the local smoothness and global low-rankness by simultaneously exploiting the first- and second-order derivatives information. To solve the proposed model, an efficient proximal alternating optimization (PAO)-based algorithm is developed with theoretical guarantee. Moreover, we suggest a regularization parameter selection strategy to automatically update two regularization parameters, which is able to take advantage of the best properties of each of the two regularization terms. Extensive experiments on different tensor data show the superiority of the proposed method over other methods, particularly for extremely low SRs.