  As low-rank modeling has achieved great success in tensor recovery, many
research efforts devote to defining the tensor rank. Among them, the recent
popular tensor tubal rank, defined based on the tensor singular value
decomposition (t-SVD), obtains promising results. However, the framework of the
t-SVD and the tensor tubal rank are applicable only to three-way tensors and
lack of flexibility to handle different correlations along different modes. To
tackle these two issues, we define a new tensor unfolding operator, named
mode-$k_1k_2$ tensor unfolding, as the process of lexicographically stacking
the mode-$k_1k_2$ slices of an $N$-way tensor into a three-way tensor, which is
a three-way extension of the well-known mode-$k$ tensor matricization. Based on
it, we define a novel tensor rank, the tensor $N$-tubal rank, as a vector whose
elements contain the tubal rank of all mode-$k_1k_2$ unfolding tensors, to
depict the correlations along different modes. To efficiently minimize the
proposed $N$-tubal rank, we establish its convex relaxation: the weighted sum
of tensor nuclear norm (WSTNN). Then, we apply WSTNN to low-rank tensor
completion (LRTC) and tensor robust principal component analysis (TRPCA). The
corresponding WSTNN-based LRTC and TRPCA models are proposed, and two efficient
alternating direction method of multipliers (ADMM)-based algorithms are
developed to solve the proposed models. Numerical experiments demonstrate that
the proposed models significantly outperform the compared ones.
