scholarly journals On the Synergy between Nonconvex Extensions of the Tensor Nuclear Norm for Tensor Recovery

Signals ◽  
2021 ◽  
Vol 2 (1) ◽  
pp. 108-121
Author(s):  
Kaito Hosono ◽  
Shunsuke Ono ◽  
Takamichi Miyata
Keyword(s):  
Nuclear Norm ◽  
Low Rank ◽  
Tensor Rank ◽  
Simple Method ◽  
Rank Tensor ◽  
Special Cases ◽  
Tensor Recovery ◽  
Two Parameters ◽  
Tensor Nuclear Norm ◽  
Complex Methods

Low-rank tensor recovery has attracted much attention among various tensor recovery approaches. A tensor rank has several definitions, unlike the matrix rank—e.g., the CP rank and the Tucker rank. Many low-rank tensor recovery methods are focused on the Tucker rank. Since the Tucker rank is nonconvex and discontinuous, many relaxations of the Tucker rank have been proposed, e.g., the sum of nuclear norm, weighted tensor nuclear norm, and weighted tensor schatten-p norm. In particular, the weighted tensor schatten-p norm has two parameters, the weight and p, and the sum of nuclear norm and weighted tensor nuclear norm are special cases of these parameters. However, there has been no detailed discussion of whether the effects of the weighting and p are synergistic. In this paper, we propose a novel low-rank tensor completion model using the weighted tensor schatten-p norm to reveal the relationships between the weight and p. To clarify whether complex methods such as the weighted tensor schatten-p norm are necessary, we compare them with a simple method using rank-constrained minimization. It was found that the simple methods did not outperform the complex methods unless the rank of the original tensor could be accurately known. If we can obtain the ideal weight, p=1 is sufficient, although it is necessary to set p<1 when using the weights obtained from observations. These results are consistent with existing reports.

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 Hyperspectral Image Recovery Using Non-Convex Low-Rank Tensor Approximation

2020 ◽  
Vol 12 (14) ◽  
pp. 2264
Author(s):  
Hongyi Liu ◽  
Hanyang Li ◽  
Zebin Wu ◽  
Zhihui Wei
Keyword(s):  
Hyperspectral Image ◽  
Convex Relaxation ◽  
Principal Component ◽  
Nuclear Norm ◽  
Low Rank ◽  
Low Rank Approximation ◽  
Rank Tensor ◽  
Alternating Direction ◽  
Tensor Approximation ◽  
Tensor Nuclear Norm

Low-rank tensors have received more attention in hyperspectral image (HSI) recovery. Minimizing the tensor nuclear norm, as a low-rank approximation method, often leads to modeling bias. To achieve an unbiased approximation and improve the robustness, this paper develops a non-convex relaxation approach for low-rank tensor approximation. Firstly, a non-convex approximation of tensor nuclear norm (NCTNN) is introduced to the low-rank tensor completion. Secondly, a non-convex tensor robust principal component analysis (NCTRPCA) method is proposed, which aims at exactly recovering a low-rank tensor corrupted by mixed-noise. The two proposed models are solved efficiently by the alternating direction method of multipliers (ADMM). Three HSI datasets are employed to exhibit the superiority of the proposed model over the low rank penalization method in terms of accuracy and robustness.

scholarly journals Exact Low Tubal Rank Tensor Recovery from Gaussian Measurements

2018 ◽  
Author(s):  
Canyi Lu ◽  
Jiashi Feng ◽  
Zhouchen Lin ◽  
Shuicheng Yan
Keyword(s):  
Degrees Of Freedom ◽  
Low Complexity ◽  
Nuclear Norm ◽  
Low Rank ◽  
Robust Pca ◽  
L1 Norm ◽  
Rank Tensor ◽  
Atomic Norm ◽  
Tensor Recovery ◽  
The Matrix

The recent proposed Tensor Nuclear Norm (TNN) [Lu et al., 2016; 2018a] is an interesting convex penalty induced by the tensor SVD [Kilmer and Martin, 2011]. It plays a similar role as the matrix nuclear norm which is the convex surrogate of the matrix rank. Considering that the TNN based Tensor Robust PCA [Lu et al., 2018a] is an elegant extension of Robust PCA with a similar tight recovery bound, it is natural to solve other low rank tensor recovery problems extended from the matrix cases. However, the extensions and proofs are generally tedious. The general atomic norm provides a unified view of low-complexity structures induced norms, e.g., the l1-norm and nuclear norm. The sharp estimates of the required number of generic measurements for exact recovery based on the atomic norm are known in the literature. In this work, with a careful choice of the atomic set, we prove that TNN is a special atomic norm. Then by computing the Gaussian width of certain cone which is necessary for the sharp estimate, we achieve a simple bound for guaranteed low tubal rank tensor recovery from Gaussian measurements. Specifically, we show that by solving a TNN minimization problem, the underlying tensor of size n1×n2×n3 with tubal rank r can be exactly recovered when the given number of Gaussian measurements is O(r(n1+n2−r)n3). It is order optimal when comparing with the degrees of freedom r(n1+n2−r)n3. Beyond the Gaussian mapping, we also give the recovery guarantee of tensor completion based on the uniform random mapping by TNN minimization. Numerical experiments verify our theoretical results.

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