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

Yu, Quan; Zhang, Xinzhen; Huang, Zheng-Hai

doi:10.48550/arxiv.2012.05065

Cited by 3 publications

(4 citation statements)

References 42 publications

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“…Both categories of algorithms necessitate the underlying time series data to adhere to a low-rank structure as per their respective decomposition methodologies. A proof presented in [25] establishes that:…”

Section: Comparison With Prior Workmentioning

confidence: 95%

High-Dimensional Vector Autoregressive Time Series Modeling via Tensor Decomposition

Wang¹,

Lian²,

Zheng

et al. 2019

SSRN Journal

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The classical vector autoregressive model is a fundamental tool for multivariate time series analysis. However, it involves too many parameters when the number of time series and lag order are even moderately large. This paper proposes to rearrange the coefficient matrices of the model into a tensor form such that the parameter space can be restricted in three directions simultaneously via tensor decomposition. The proposed method substantially expands the capacity of vector autoregressive modeling for a large number of time series. In contrast, the widely used reduced-rank regression method can restrict the parameter space in only one direction. Moreover, to handle high-dimensional time series, this paper considers imposing sparsity on factor matrices to improve the interpretability and estimation efficiency, which leads to a sparsity-inducing estimator. For the low-dimensional case, we derive asymptotic properties of the proposed least squares estimator and introduce an alternating least squares algorithm. For the high-dimensional case, we establish non-asymptotic properties of the sparsity-inducing estimator and propose an ADMM-based algorithm for regularized estimation. Simulation experiments and a real data example demonstrate the advantages of the proposed approach over various existing methods.

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Section: Comparison With Prior Workmentioning

confidence: 95%

High-Dimensional Vector Autoregressive Time Series Modeling via Tensor Decomposition

Wang¹,

Lian²,

Zheng

et al. 2019

SSRN Journal

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show abstract

“…In this section, Algorithm 1 is used to restore color images and multispectral images to evaluate its performance, and compared with the following four data completion methods, namely TCTF [13], Tmac [36], TCTFTVT [34] and MTRTC [37]. In order to quantitatively evaluate the image quality restored by each method, PSNR [38], SSIM [39] and CPU time are used as numerical indicators, and PSNR and SSIM are defined as follows:…”

Section: Numerical Experimentsmentioning

confidence: 99%

Variable T-Product and Zero-Padding Tensor Completion with Applications

Yan

Luo

et al. 2023

Preprint

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The T-product method based upon Discrete Fourier Transformation (DFT) has found wide applications in engineering, in particular, in image processing. In this paper, we propose variable T-product, and apply the Zero-Padding Discrete Fourier Transformation (ZDFT), to convert third order tensor problems to the variable Fourier domain. An additional positive integer parameter is introduced, which is greater than the tubal dimension of the third order tensor. When the additional parameter is equal to the tubal dimension, the ZDFT reduces to DFT. Then, we propose a new tensor completion method based on ZDFT and TV regularization, called VTCTF-TV. Extensive numerical experiment results on visual data demonstrate that the superior performance of the proposed method. In addition, the zero-padding is near the size of the original tubal dimension, the resulting tensor completion can be performed much better.

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“…We evaluate our method on the widely used YUV Video Sequences 4 . Each sequence contains at least 150 frames and we use the first 30 frames of each.…”

Section: Video Inpaintingmentioning

confidence: 99%

“…L OW rank tensor recovery problem has gotten a lot of attention during the last decade. Furthermore, low rank tensor can be recovered efficiently using tensor (matrix) factorization [1], [2], [3], [4], [5], [6] and tensor rank minimization methods [7], [8], [9], [10], [11], [12], [13], [14], respectively. In this paper, we consider the tensor rank minimization problem.…”

Section: Introductionmentioning

confidence: 99%

Efficient Recovery of Low Rank Tensor via Triple Nonconvex Nonsmooth Rank Minimization

Yu¹

2022

Preprint

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A tensor nuclear norm (TNN) based method for solving the tensor recovery problem was recently proposed, and it has achieved state-of-the-art performance. However, it may fail to produce a highly accurate solution since it tends to treats each frontal slice and each rank component of each frontal slice equally. In order to get a recovery with high accuracy, we propose a general and flexible rank relaxation function named double weighted nonconvex nonsmooth rank (DWNNR) relaxation function for efficiently solving the third order tensor recovery problem. The DWNNR relaxation function can be derived from the triple nonconvex nonsmooth rank (TNNR) relaxation function by setting the weight vector to be the hypergradient value of some concave function, thereby adaptively selecting the weight vector.To accelerate the proposed model, we develop the general inertial smoothing proximal gradient method. Furthermore, we prove that any limit point of the generated subsequence is a critical point. Combining the Kurdyka-Lojasiewicz (KL) property with some milder assumptions, we further give its global convergence guarantee. Experimental results on a practical tensor completion problem with both synthetic and real data, the results of which demonstrate the efficiency and superior performance of the proposed algorithm.

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

Cited by 3 publications

References 42 publications

High-Dimensional Vector Autoregressive Time Series Modeling via Tensor Decomposition

High-Dimensional Vector Autoregressive Time Series Modeling via Tensor Decomposition

Variable T-Product and Zero-Padding Tensor Completion with Applications

Efficient Recovery of Low Rank Tensor via Triple Nonconvex Nonsmooth Rank Minimization

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