Low-Rank Tensor Completion With a New Tensor Nuclear Norm Induced by Invertible Linear Transforms

2019 ◽  
Author(s):  
Canyi Lu ◽  
Xi Peng ◽  
Yunchao Wei
Keyword(s):  
Nuclear Norm ◽  
Low Rank ◽  
Tensor Completion ◽  
Rank Tensor ◽  
Linear Transforms ◽  
Tensor Nuclear Norm

scholarly journals Low-Rank Tensor Completion via Tensor Nuclear Norm With Hybrid Smooth Regularization

IEEE Access ◽  
2019 ◽  
Vol 7 ◽  
pp. 131888-131901
Author(s):  
Xi-Le Zhao ◽  
Xin Nie ◽  
Yu-Bang Zheng ◽  
Teng-Yu Ji ◽  
Ting-Zhu Huang
Keyword(s):  
Nuclear Norm ◽  
Low Rank ◽  
Tensor Completion ◽  
Rank Tensor ◽  
Tensor Nuclear Norm

scholarly journals Tensor p-shrinkage nuclear norm for low-rank tensor completion

2020 ◽  
Vol 387 ◽  
pp. 255-267
Author(s):  
Chunsheng Liu ◽  
Hong Shan ◽  
Chunlei Chen
Keyword(s):  
Nuclear Norm ◽  
Low Rank ◽  
Tensor Completion ◽  
Rank Tensor

scholarly journals Near-optimal sample complexity for convex tensor completion

2018 ◽  
Vol 8 (3) ◽  
pp. 577-619 ◽  
Author(s):  
Navid Ghadermarzy ◽  
Yaniv Plan ◽  
Özgür Yilmaz
Keyword(s):  
Low Rank ◽  
Sample Complexity ◽  
Tensor Completion ◽  
Norm Minimization ◽  
Optimal Sample ◽  
Rank Tensor ◽  
Atomic Norm ◽  
The Matrix ◽  
Tensor Nuclear Norm ◽  
Minimax Rate

Abstract We study the problem of estimating a low-rank tensor when we have noisy observations of a subset of its entries. A rank-$r$, order-$d$, $N \times N \times \cdots \times N$ tensor, where $r=O(1)$, has $O(dN)$ free variables. On the other hand, prior to our work, the best sample complexity that was achieved in the literature is $O\left(N^{\frac{d}{2}}\right)$, obtained by solving a tensor nuclear-norm minimization problem. In this paper, we consider the ‘M-norm’, an atomic norm whose atoms are rank-1 sign tensors. We also consider a generalization of the matrix max-norm to tensors, which results in a quasi-norm that we call ‘max-qnorm’. We prove that solving an M-norm constrained least squares (LS) problem results in nearly optimal sample complexity for low-rank tensor completion (TC). A similar result holds for max-qnorm as well. Furthermore, we show that these bounds are nearly minimax rate-optimal. We also provide promising numerical results for max-qnorm constrained TC, showing improved recovery compared to matricization and alternating LS.

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