A Corrected Tensor Nuclear Norm Minimization Method for Noisy Low-Rank Tensor Completion

Zhang, Xiongjun; Ng, Michael K.

doi:10.1137/18m1202311

Cited by 43 publications

(22 citation statements)

References 58 publications

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“…, and N 0 initialized to 0, k = 0, k max = 150. while not converged and k < k max do Update Y (k+1) via (6); Update Z (k+1) via (7); Update Q (k+1) via (8); Update V (k+1) via (9); Update X (k+1) via (11); Update M (k+1) , (k+1) , (k+1) , and N (k+1) via (12); end while Output: The completed tensor X .…”

Section: Algorithmmentioning

confidence: 99%

“…Low-rank tensor completion (LRTC) is a next generation of low-rank matrix completion (LRMC), which has been a hot problem of research in many fields, such as color image inpainting [1], magnetic resonance imaging (MRI) data recovery [2], video processing [3], and hyperspectral/multispectral image (HSI/MSI) processing [4]- [8]. With the purpose of recovering a low-rank tensor from its partial observation, the core problem of LRTC is to accurately characterize the inherent low-rank structure of a tensor [9]- [11].…”

Section: Introductionmentioning

confidence: 99%

“…This implies that the Tucker-rank, SNN-based methods and the TT rank all could not preserve the intrinsic structure of the target tensors well. Recently, based on tensor-tensor product (t-product) [20], a novel tool named tensor singular value decomposition (t-SVD) and its induced tensor tubal rank and tensor multi rank have been proposed [11], [20], [29]. Moreover, being a convex surrogate for multi rank, the tensor nuclear norm (TNN) has shown its effectiveness to keep the intrinsic structure of tensors [30]- [33].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Zhao

Nie

et al. 2019

IEEE Access

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As a convex surrogate of tensor multi rank, recently the tensor nuclear norm (TNN) obtains promising results in the tensor completion. However, only considering the low-tubal-rank prior is not enough for recovering the target tensor, especially when the ratio of available elements is extremely low. To address this problem, we suggest a novel low-rank tensor completion model by exploiting both low-tubal-rankness and smoothness. Especially, motivated by the capability of framelet preserving details, we characterize the spatial smoothness by framelet regularization and the smoothness of the third mode by total variation (TV) regularization. The resulting convex optimization problem is efficiently tackled by a carefully designed alternating direction method of multipliers (ADMM) algorithm. Extensive numerical results including color images, videos, and fluorescence microscope images validate the superiority of our method over the competing methods. INDEX TERMS Low-rank tensor completion, tensor nuclear norm, framelet, total variation, alternating direction method of multipliers.

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Section: Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Zhao

Nie

et al. 2019

IEEE Access

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“…For tubal rank minimization with third‐order tensors, Jiang et al 37 proposed a convex model combined the tubal nuclear norm with tensor

ℓ_{1}

norm for tensor RPCA and analyzed the exact recovery conditions, provided that its tube rank is not too large and the corruptions are reasonably sparse. Similar work based on tubal rank and its transformed formulation for robust tensor recovery can be referred to References 38‐43. However, these methods are only efficient for third‐order tensors and many real‐world data are higher‐order, for example, color videos are fourth‐order tensors.…”

Section: Introductionmentioning

confidence: 99%

Robust tensor train component analysis

Zhang

2021

Numerical Linear Algebra App

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Summary Robust Principal Component Analysis plays a key role in various fields such as image and video processing, data mining, and hyperspectral data analysis. In this paper, we study the problem of robust tensor train (TT) principal component analysis from partial observations, which aims to decompose a given tensor into the low TT rank and sparse components. The decomposition of the proposed model is used to find the hidden factors and help alleviate the curse of dimensionality via a set of connected low‐rank tensors. A relaxation model is to minimize a weighted combination of the sum of nuclear norms of unfolding matrices of core tensors and the tensor ℓ1 norm. A proximal alternating direction method of multipliers is developed to solve the resulting model. Furthermore, we show that any cluster point of the convergent subsequence is a Karush‐Kuhn‐Tucker point of the proposed model under some conditions. Extensive numerical examples on both synthetic data and real‐world datasets are presented to demonstrate the effectiveness of the proposed approach.

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“…By analysis in [15,16,28,42], the t-product can be computed by some block diagonal matrices of smaller sizes, which makes a significant reduction of computational cost. Later, a corrected tensor nuclear norm minimization method was proposed in [39] for noisy observations.…”

Section: Introductionmentioning

confidence: 99%

Multi-Tubal Rank of Third Order Tensor and Related Low Rank Tensor Completion Problem

Yu¹,

Zhang²,

Huang³

2020

Preprint

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Recently, a tensor factorization based method for a low tubal rank tensor completion problem of a third order tensor was proposed, which performed better than some existing methods. Tubal rank is only defined on one mode of third order tensor without low rank structure in the other two modes. That is, low rank structures on the other two modes are missing. Motivated by this, we first introduce multi-tubal rank, and then establish a relationship between multi-tubal rank and Tucker rank. Based on the multi-tubal rank, we propose a novel low rank tensor completion model. For this model, a tensor factorization based method is applied and the corresponding convergence analysis is established. In addition, spatio-temporal characteristics are intrinsic features in video and internet traffic tensor data. To get better performance, we make full use of such features and improve the established tensor completion model. Then we apply tensor factorization based method for the improved model. Finally, numerical results are reported on the completion of image, video and internet traffic data to show the efficiency of our proposed methods. From the reported numerical results, we can assert that our methods outperform the existing methods.

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A Corrected Tensor Nuclear Norm Minimization Method for Noisy Low-Rank Tensor Completion

Cited by 43 publications

References 58 publications

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

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

Robust tensor train component analysis

Multi-Tubal Rank of Third Order Tensor and Related Low Rank Tensor Completion Problem

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