Tensor \(N\)-tubal rank and its convex relaxation for low-rank tensor recovery. (English) Zbl 1459.68181
Summary: The recent popular tensor tubal rank, defined based on tensor singular value decomposition (t-SVD), yields promising results. However, its framework is applicable only to three-way tensors and lacks the flexibility necessary to handle different correlations along different modes. To tackle these two issues, we define a new tensor unfolding operator, named mode-\( k_1 k_2\) tensor unfolding, as the process of lexicographically stacking all mode-\( k_1 k_2\) 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. On this basis, we define a novel tensor rank, named the tensor \(N\)-tubal rank, as a vector consisting of the tubal ranks of all mode-\( k_1 k_2\) 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 the tensor nuclear norm (WSTNN). Then, we apply the 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.
MSC:
| 68T05 | Learning and adaptive systems in artificial intelligence |
| 15A69 | Multilinear algebra, tensor calculus |
| 62H25 | Factor analysis and principal components; correspondence analysis |
Keywords:
low-rank tensor recovery (LRTR); mode-\( k_1 k_2\) tensor unfolding; tensor \(N\)-tubal rank; weighted sum of tensor nuclear norm (WSTNN); alternating direction method of multipliers (ADMM)References:
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