  Tensor train (TT) factorization and corresponding TT rank, which can well
express the low-rankness and mode correlations of higher-order tensors, have
attracted much attention in recent years. However, TT factorization based
methods are generally not sufficient to characterize low-rankness along each
mode of third-order tensor. Inspired by this, we generalize the tensor train
factorization to the mode-k tensor train factorization and introduce a
corresponding multi-mode tensor train (MTT) rank. Then, we proposed a novel
low-MTT-rank tensor completion model via multi-mode TT factorization and
spatial-spectral smoothness regularization. To tackle the proposed model, we
develop an efficient proximal alternating minimization (PAM) algorithm.
Extensive numerical experiment results on visual data demonstrate that the
proposed MTTD3R method outperforms compared methods in terms of visual and
quantitative measures.
