Multi-mode tensor train factorization with spatial-spectral regularization for third-order tensor completion

The tensor train (TT) factorization and its associated TT rank have been gaining attention in recent years due to their ability to express the low-rankness and mode correlations of higher-order tensors. However, these methods are not sufficient to characterize the low-rankness along each mode of third-order tensors. To address this, we generalized the tensor train factorization to the mode-k tensor train factorization and introduced a multi-mode tensor train (MTT) rank. We then proposed a novel low-MTT-rank tensor completion model that combines multi-mode TT factorization with spatial-spectral smoothness regularization. To solve this model, we developed an efficient proximal alternating minimization (PAM) algorithm. Numerical experiments on visual data show that the proposed MTT3R method outperforms other methods in terms of visual and quantitative measures.