Tensor Ring Decomposition

Zhao, Qibin; Zhou, Guoxu; Xie, Shengli; Zhang, Liqing; Cichocki, Andrzej

doi:10.48550/arxiv.1606.05535

Cited by 110 publications

(187 citation statements)

References 48 publications

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“…When λ = 0, (1.1) transforms into tensor completion problem. Unlike the matrix case, there exist different kinds of tensor rank, such as Tucker rank [12], multi-rank and tubal rank [13], tensor train (TT) rank [14], and tensor ring (b) TR decomposition (c) FCTN decomposition (a) TT decomposition (TR) rank [15], which are derived from the corresponding tensor decompositions. In general, the minimization of tensor rank is NP-hard [16], the convex/nonconvex relaxation of tensor rank or the low-rank tensor decomposition is usually used instead of the minimization of tensor rank.…”

Section: Introductionmentioning

confidence: 99%

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Fully-Connected Tensor Network Decomposition for Robust Tensor Completion Problem

Liu,

Zhao,

Song

et al. 2021

Preprint

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The robust tensor completion (RTC) problem, which aims to reconstruct a low-rank tensor from partially observed tensor contaminated by a sparse tensor, has received increasing attention. In this paper, by leveraging the superior expression of the fully-connected tensor network (FCTN) decomposition, we propose a FCTN-based robust convex optimization model (RC-FCTN) for the RTC problem. Then, we rigorously establish the exact recovery guarantee for the RC-FCTN. For solving the constrained optimization model RC-FCTN, we develop an alternating direction method of multipliers (ADMM)-based algorithm, which enjoys the global convergence guarantee. Moreover, we suggest a FCTN-based robust nonconvex optimization model (RNC-FCTN) for the RTC problem. A proximal alternating minimization (PAM)-based algorithm is developed to solve the proposed RNC-FCTN. Meanwhile, we theoretically derive the convergence of the PAM-based algorithm. Comprehensive numerical experiments in several applications, such as video completion and video background subtraction, demonstrate that proposed methods are superior to several state-of-the-art methods.

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Section: Introductionmentioning

confidence: 99%

“…TR decomposition [15] (as shown in Fig. 1.1 (b)) decomposes an N th-order tensor X ∈ R I1×I2×•••×I N into a circular multilinear product of a list of third-order core tensors, and the element-wise form is expressed as…”

Section: Introductionmentioning

confidence: 99%

Fully-Connected Tensor Network Decomposition for Robust Tensor Completion Problem

Liu,

Zhao,

Song

et al. 2021

Preprint

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show abstract

“…The Tucker-rank has been applied in LRTC problem by minimizing its convex surrogate [14] or non-convex surrogates [15,16]. Moreover, a series of tensor network decomposition-based ranks are proposed, such as tensor train (TT)rank [17], tensor ring (TR)-rank [18], and fully-connected tensor network (FCTN)-rank [19]. All of them have been achieved great success in higher-order LRTC [19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Transform Induced Tensor Nuclear Norm for Tensor Completion

Li¹,

Zhao²,

Ji³

et al. 2021

Preprint

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The linear transform-based tensor nuclear norm (TNN) methods have recently obtained promising results for tensor completion. The main idea of this type of methods is exploiting the low-rank structure of frontal slices of the targeted tensor under the linear transform along the third mode. However, the lowrankness of frontal slices is not significant under linear transforms family. To better pursue the low-rank approximation, we propose a nonlinear transform-based TNN (NTTNN). More concretely, the proposed nonlinear transform is a composite transform consisting of the linear semi-orthogonal transform along the third mode and the element-wise nonlinear transform on frontal slices of the tensor under the linear semi-orthogonal transform, which are indispensable and complementary in the composite transform to fully exploit the underlying low-rankness. Based on the suggested low-rankness metric, i.e., NTTNN, we propose a low-rank tensor completion (LRTC) model. To tackle the resulting nonlinear and nonconvex optimization model, we elaborately design the proximal alternating minimization (PAM) algorithm and establish the theoretical convergence guarantee of the PAM algorithm. Extensive experimental results on hyperspectral images, multispectral images, and videos show that the our method outperforms linear transform-based state-of-the-art LRTC methods qualitatively and quantitatively.

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“…Tensor network (TN) [40], [41], one of the powerful tools in physics, is broadly applied to solving large-scale optimization problems and provides graphical computation interpretation. TD models based on TN representation has recently attracted much attention, including tensor train (TT) [42] representation and tensor ring (TR) [43] representation, which can overcome the Curse of Dimensionality [40] with linear storage cost O(N IR 2 ). Lately, TN is exploited to explore more complex topology structures [42], [43], [44] for TD, e.g., Projected Entangled Pair States (PEPS) [45] and Fully-Connected TN (FCTN) [46].…”

mentioning

confidence: 99%

“…TD models based on TN representation has recently attracted much attention, including tensor train (TT) [42] representation and tensor ring (TR) [43] representation, which can overcome the Curse of Dimensionality [40] with linear storage cost O(N IR 2 ). Lately, TN is exploited to explore more complex topology structures [42], [43], [44] for TD, e.g., Projected Entangled Pair States (PEPS) [45] and Fully-Connected TN (FCTN) [46]. However, most of the available high-dimensional data suffer the difficulties of mode-dimension unbalance [47] (e.g., filters in convolution neural networks), and mode-correlation discrepancy (e.g., spatial modes in color image are strongly intercorrelated but their weakly correlated with channel mode).…”

mentioning

confidence: 99%

Multi-Tensor Network Representation for High-Order Tensor Completion

Nie,

Wang,

Lai

2021

Preprint

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This work studies the problem of high-dimensional data (referred to as tensors) completion from partially observed samplings. We consider that a tensor is a superposition of multiple low-rank components. In particular, each component can be represented as multilinear connections over several latent factors and naturally mapped to a specific tensor network (TN) topology. In this paper, we propose a fundamental tensor decomposition (TD) framework: Multi-Tensor Network Representation (MTNR), which can be regarded as a linear combination of a range of TD models, e.g., CANDECOMP/PARAFAC (CP) decomposition, Tensor Train (TT), and Tensor Ring (TR). Specifically, MTNR represents a high-order tensor as the addition of multiple TN models, and the topology of each TN is automatically generated instead of manually pre-designed. For the optimization phase, an adaptive topology learning (ATL) algorithm is presented to obtain latent factors of each TN based on a rank incremental strategy and a projection error measurement strategy. In addition, we theoretically establish the fundamental multilinear operations for the tensors with TN representation, and reveal the structural transformation of MTNR to a single TN. Finally, MTNR is applied to a typical task, tensor completion, and two effective algorithms are proposed for the exact recovery of incomplete data based on the Alternating Least Squares (ALS) scheme and Alternating Direction Method of Multiplier (ADMM) framework. Extensive numerical experiments on synthetic data and real-world datasets demonstrate the effectiveness of MTNR compared with the start-of-the-art methods.

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Tensor Ring Decomposition

Cited by 110 publications

References 48 publications

Fully-Connected Tensor Network Decomposition for Robust Tensor Completion Problem

Fully-Connected Tensor Network Decomposition for Robust Tensor Completion Problem

Nonlinear Transform Induced Tensor Nuclear Norm for Tensor Completion

Multi-Tensor Network Representation for High-Order Tensor Completion

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