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Low-rank tensor singular value decomposition model for hyperspectral image super-resolution

JOURNAL OF ELECTRONIC IMAGING 2020年第4期29卷

作者： Zou, Changzhong Huang, Xusheng Fuzhou Univ Coll Math & Comp Sci Fuzhou Peoples R China

We propose a method for hyperspectral image (HSI) super-resolution by designing a tensor singular value decomposition (t-SVD) and three-dimensional total variation (3D-TV) regularization terms. The super-resolution method is designed as an optimization problem whose cost function consists of a data-fidelity term, the low-rank representation term by t-SVD, and the 3D-TV regularization term. The sparse representation term is used to enhance the low-rank quality to unify the spectrum and space of HSI. Furthermore, the 3D-TV regularization term exploits the spectral and spatial similarity between adjacent pixels of HSI. Then we develop an effective algorithm for solving the resulting optimization by the alternative direction method of multipliers. The results on the simulated and the real data demonstrate that the proposed method is competitive with other state-of-the-art methods. (C) 2020 SPIE and IS&T

关键词： hyperspectral image tensor singular value decomposition three-dimensional-TV alternating direction method

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A Randomized Algorithm for tensor singular value decomposition Using an Arbitrary Number of Passes

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JOURNAL OF SCIENTIFIC COMPUTING 2024年第1期98卷 1-25页

作者： Ahmadi-Asl, Salman Phan, Anh-Huy Cichocki, Andrzej Skolkovo Inst Sci & Technol Ctr Artificial Intelligence Technol Moscow Russia Polish Acad Sci Syst Res Inst Warsaw Poland

Efficient and fast computation of a tensor singular value decomposition (t-SVD) with a few passes over the underlying data tensor is crucial because of its many potential applications. The current/existing subspace randomized algorithms need (2q + 2) passes over the data tensor to compute a t-SVD, where q is a non-negative integer number (power iteration parameter). In this paper, we propose an efficient and flexible randomized algorithm that can handle any number of passes q, which not necessary need be even. The flexibility of the proposed algorithm in using fewer passes naturally leads to lower computational and communication costs. This advantage makes it particularly appropriate when our task calls for several tensor decompositions or when the data tensors are huge. The proposed algorithm is a generalization of the methods developed for matrices to tensors. The expected/average error bound of the proposed algorithm is derived. Extensive numerical experiments on random and real-world data sets are conducted, and the proposed algorithm is compared with some baseline algorithms. The extensive computer simulation experiments demonstrate that the proposed algorithm is practical, efficient, and in general outperforms the state of the arts algorithms. We also demonstrate how to use the proposed method to develop a fast algorithm for the tensor completion problem.

关键词： tensor singular value decomposition Randomization Pass-efficient algorithms

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Low-rank plus sparse joint smoothing model based on tensor singular value decomposition for dynamic MRI reconstruction

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MAGNETIC RESONANCE IMAGING 2023年 104卷 1-9页

作者： Liu, Xiaotong He, Jingfei Mi, Chenghu Zhang, Xiaoyue Hebei Univ Technol Sch Elect & Informat Engn Tianjin Key Lab Elect Mat & Devices 340 Xiping Rd Tianjin 300401 Peoples R China Hebei Univ Technol Sch Elect & Informat Engn 340 Xiping Rd Tianjin 300401 Peoples R China

Dynamic magnetic resonance imaging (DMRI) is an important medical imaging modality, but the long imaging time limits its practical applications. This paper proposes a low-rank plus sparse joint smoothing model based on tensor singular value decomposition (T-SVD) to reconstruct DMR images from highly under-sampled k-t space data. The low-rank plus sparse tensor (L + S) model decomposes the DMR data into a low-rank and sparse tensor, which naturally fits the dynamic MR images characteristics and exploits the spatiotemporal correlation of DMRI data to improve reconstruction effect. T-SVD is utilized in the L + S model to maintain the intrinsic structure of the low-rank tensor and further enhance the low-rank property. In addition, considering the global multidimensional smoothness of the DMR images, the proposed method joint tensor total variation (TTV) constraints to utilize the smoothness of DMR images to obtain more reconstruction details while protecting the global structure. We conducted experiments on the dynamic cardiac datasets, and the experiment results show that the proposed method has superior performance to several state-of-the-art imaging methods.

关键词： Dynamic magnetic resonance imaging tensor singular value decomposition Low rank plus sparsity model tensor total variation constraint Smoothness

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Fast tensor singular value decomposition Using the Low-Resolution Features of tensors 20

Fast Tensor Singular Value Decomposition Using the Low-Resol...

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20th IEEE International Conference on Machine Learning and Applications (ICMLA)

作者： Ozdemir, Cagri Hoover, Randy C. Caudle, Kyle South Dakota Mines Dept Comp Sci & Engn Rapid City SD 57701 USA South Dakota Mines Dept Math Rapid City SD USA

ISBN: (纸本)9781665443371

The tensor singular value decomposition (t-SVD) based on an algebra of circulants is an effective multilinear subspace learning technique for dimensionality reduction and data classification. Unfortunately, the computational cost associated with computing the t-SVD can become prohibitively expensive, particularly when dealing with very large data sets. In this paper, we present a computationally efficient approach for estimating the t-SVD by capitalizing on the correlations of the data in the temporal dimension. The approach proceeds by extending our prior work on fast eigenspace decompositions by transforming the tensor data from the spatial domain to the spectral domain in order to obtain reduced order harmonic tensor. The t-SVD can then be applied in the transform domain thereby significantly reducing the computational burden. Experimental results which are presented on the extended Yale-B, COIL-100, and MNIST data sets show the proposed method provides considerable computational savings with the approximated subspaces that are nearly the same as the true subspaces as computed via the t-SVD.

关键词： fast tensor singular value decomposition tensor singular value decomposition multilinear subspace learning

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Quantum tensor singular value decomposition

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JOURNAL OF PHYSICS COMMUNICATIONS 2021年第7期5卷

作者： Wang, Xiaoqiang Gu, Lejia Lee, Heung-wing Zhang, Guofeng Hong Kong Polytech Univ Dept Appl Math Hong Kong Peoples R China

tensors are increasingly ubiquitous in various areas of applied mathematics and computing, and tensor decompositions are of practical significance and benefit many applications in data completion, image processing, computer vision, collaborative filtering, etc. Recently, Kilmer and Martin propose a new tensor factorization strategy, tensor singular value decomposition (t-svd), which extends the matrix singular value decomposition to tensors. However, computing t-svd for high dimensional tensors costs much computation and thus cannot efficiently handle large scale datasets. Motivated by advantage of quantum computation, in this paper, we present a quantum algorithm of t-svd for third-order tensors and then extend it to order-p tensors. We prove that our quantum t-svd algorithm for a third-order N dimensional tensor runs in time.(N polylog(N)) if we do not recover classical information from the quantum output state. Moreover, we apply our quantum t-svd algorithm to context-aware multidimensional recommendation systems, where we just need to extract partial classical information from the quantum output state, thus achieving low time complexity.

关键词： tensor singular value decomposition quantum singular value estimation t-svd t-product quantum Fourier transform

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Low-rank tensor recovery based on nonconvex logarithmic regularization factor

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INTERNATIONAL JOURNAL OF WAVELETS MULTIRESOLUTION AND INFORMATION PROCESSING 2025年

作者： Li, Shuo Gao, Yi Zhu, Xiaolong North Minzu Univ Sch Math & Informat Sci Yinchuan 750021 Ningxia Peoples R China

This paper studies the low-rank tensor recovery (LRTR) problem under the framework of tensor singular value decomposition (t-SVD). The t-SVD avoids the inherent information loss associated with tensor matricization. Conventional LRTR based on t-SVD approaches generally rely on nuclear norm minimization. However, in real-world applications, higher singular values are related to the most informative regions of an image. In light of this, we introduce an innovative nonconvex regularization term that treats different singular values of tensors distinctly. Aiming at the problem of pixel missing in the image, this paper introduces a new low-rank tensor completion model based on the nonconvex logarithmic regularization term. Besides, for the problem of the image denoising issue, this paper establishes a new tensor robust principal component analysis model. Two algorithms based on the alternating direction method of multipliers (ADMMs) are proposed to solve the proposed tensor recovery model. Experimental results on real images and videos demonstrate the superiority of the proposed model, especially in the aspect of denoising, where it exhibits significant advantages.

关键词： tensor singular value decomposition low-rank tensor completion tensor robust principal component analysis

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Fast tensor robust principal component analysis with estimated multi-rank and Riemannian optimization

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APPLIED INTELLIGENCE 2025年第1期55卷 1-16页

作者： Zhu, Qile Wu, Shiqian Fang, Shun Wu, Qi Xie, Shoulie Agaian, Sos Wuhan Univ Sci & Technol Sch Informat Sci & Engn Wuhan 430000 Peoples R China Henan Acad Sci Inst Adv Displays & Imaging Zhengzhou 450000 Peoples R China Jiangxi Univ Finance & Econ Sch Informat Management Nanchang 330000 Peoples R China Agcy Sci Technol & Res Inst Infocomm Res Singapore 138632 Singapore CUNY Grad Ctr New York NY 10314 USA CUNY Coll Staten Isl New York NY 10314 USA

Motivated by the fact that tensor robust principal component analysis (TRPCA) and its variants do not utilize the actual rank value, which limits the recovery performance, and their computational costs are always monumental for large-scale tensor recovery, a fast TRPCA is proposed to recover the low-rank tensor and sparse tensor by estimating the multi-rank vector and adopting Riemannian optimization strategy in this paper. Specifically, a fast multi-rank estimation of low-rank tensor is proposed by modifying the Gershgorin disk theorem-based matrix rank estimation. An innovative TRPCA with Estimated Multi-Rank (TRPCA-EMR) is proposed to eliminate hyperparameter tuning by imposing strict multi-rank equality constraints. Additionally, Riemannian optimization is employed to project each frontal slice of the tensor in Fourier domain onto a low multi-rank manifold in efficiently coping with tensor singular value decomposition (t-SVD) and reduce computational complexity. Experimental results on synthetic and real-world tensor datasets demonstrate TRPCA-EMR's superior efficiency and effectiveness compared to existing methods, confirming its potential for practical applications and reassuring its reliability.

关键词： tensor robust principal component analysis tensor multi-rank tensor singular value decomposition Riemannian optimization Background subtraction

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Adaptive sampling with tensor leverage scores for exact low-rank third-order tensor completion

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APPLIED MATHEMATICAL MODELLING 2025年 138卷

作者： Chen, Xuan Jiang, Tai-Xiang Hu, Yexun Yu, Jinjin Ng, Michael K. Southwestern Univ Finance & Econ Sch Comp & Artificial Intelligence Chengdu Peoples R China Kash Inst Elect & Informat Ind Kashi Peoples R China Minist Educ Engn Res Ctr Intelligent Finance Chengdu Peoples R China Hong Kong Baptist Univ Dept Math Hong Kong Peoples R China

tensor completion aims at estimating the missing entries from the incomplete observation. Under the tensor singular value decomposition framework, the exact recovery of a low-tubal-rank third- order tensor could be achieved via convex optimization with high probability if the tensor satisfies the tensor incoherence condition. In this work, we show that, when the random selection of entries is made adaptive to a distribution which is dependent on the coherence structure of the tensor, any low-tubal-rank tensor of the size n x n x n with tubal-rank r can be exactly recovered with high probability from as few as O(rn(2) log(2)(n)) randomly chosen entries. In practice, tensor leverage scores are not known a priori, and we design a two-phase adaptive sampling strategy to obtain the leverage scores. Numerical experiments on synthetic and real-world third-order tensor data sets are used to validate our theoretical results and illustrate that the tensor recovery performance of the proposed two-phase adaptive sampling scheme is better than that of the other state-of-the-art methods.

关键词： tensor completion Adaptive sampling tensor singular value decomposition Low-rank

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Learning amore compact representation for low-rank tensor completion

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NEUROCOMPUTING 2025年 617卷

作者： Li, Xi-Zhuo Jiang, Tai-Xiang Yang, Liqiao Liu, Guisong Southwestern Univ Finance & Econ Sch Comp & Artificial Intelligence Chengdu Sichuan Peoples R China Kash Inst Elect & Informat Ind Kashi Peoples R China Southwestern Univ Finance & Econ Engn Res Ctr Intelligent Finance Minist Educ Chengdu Sichuan Peoples R China

Transform-based tensor nuclear norm (TNN) methods have gained considerable attention for their effectiveness in addressing tensor recovery challenges. The integration of deep neural networks as nonlinear transforms has been shown to significantly enhance their performance. Minimizing transform-based TNN is equivalent to minimizing the l(1) norm of singular values in the transformed domain, which can be interpreted as finding a sparse representation with respect to the bases supported by singular vectors. This work aims to advance deep transform-based TNN methods by identifying amore compact representation through learnable bases, ultimately improving recovery accuracy. We specifically employ convolutional kernels as these learnable bases, demonstrating their capability to generate more compact representation, i.e., sparser coefficients of real-world tensor data compared to singular vectors. Our proposed model consists of two key components: a transform component, implemented through fully connected neural networks (FCNs), and a convolutional component that replaces traditional singular matrices. Then, this model is optimized using the ADAM algorithm directly on the incomplete tensor in a zero-shot manner, meaning all learnable parameters within the FCNs and convolution kernels are inferred solely from the observed data. Experimental results indicate that our method, with this straightforward yet effective modification, outperforms state-of-the-art approaches on video and multispectral image recovery tasks.

关键词： tensor completion tensor singular value decomposition Nonlinear transform Convolution Multi-dimensional image

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tensor neural network models for tensor singular value decompositions

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2020年第3期75卷 753-777页

作者： Wang, Xuezhong Che, Maolin Wei, Yimin Hexi Univ Sch Math & Stat Zhangye 734000 Peoples R China Southwest Univ Finance & Econ Sch Econ Math Chengdu 611130 Peoples R China Fudan Univ Sch Math Sci Shanghai 200433 Peoples R China Fudan Univ Shanghai Key Lab Contemporary Appl Math Shanghai 200433 Peoples R China

tensor decompositions have become increasingly prevalent in recent years. Traditionally, tensors are represented or decomposed as a sum of rank-one outer products using either the CANDECOMP/PARAFAC, the Tucker model, or some variations thereof. The motivation of these decompositions is to find an approximate representation for a given tensor. The main propose of this paper is to develop two neural network models for finding an approximation based on t-product for a given third-order tensor. Theoretical analysis shows that each of the neural network models ensures the convergence performance. The computer simulation results further substantiate that the models can find effectively the left and right singular tensor subspace.

关键词： tensor decomposition singular value decomposition tensor singular value decomposition tensor neural networks Asymptotic stability

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