Rank Tensor Research Articles

Transforms-Based Bayesian Tensor Completion Method for Network Traffic Measurement Data Recovery

Network traffic measurement is regarded as the bedrock of next-generation network systems. Its purpose is to monitor the network traffic and provide data support for traffic engineering. For this reason, monitoring traffic data from a network-wide perspective is particularly important. However, the proliferation of network services has led to the explosive growth of network traffic, which has brought significant challenges to the measure of network-wide traffic. Therefore, how to infer network-wide traffic from partial traffic data is extremely important. In this article, a transforms-based Bayesian tensor completion (TBTC) method is proposed to infer network traffic data. First, the heterogeneous network traffic data with missing entries are organized into observation tensors according to temporal dimensions and other attributes. Second, the sparse hierarchical prior is used to induce lateral slices sparsity of factor tensors, which makes the tubal rank of the observation tensor can be estimated. Further, a variational Bayesian inference method is developed for model learning, and an efficient updating method is presented. Finally, two cases of the linear transforms-based tensor completion model are implemented in the experiments. Experimental results on two real-world network traffic datasets validate that the proposed method can efficiently and accurately recover network traffic data.

Low-Rank Tensor Function Representation for Multi-Dimensional Data Recovery.

Since higher-order tensors are naturally suitable for representing multi-dimensional data in real-world, e.g., color images and videos, low-rank tensor representation has become one of the emerging areas in machine learning and computer vision. However, classical low-rank tensor representations can solely represent multi-dimensional discrete data on meshgrid, which hinders their potential applicability in many scenarios beyond meshgrid. To break this barrier, we propose a low-rank tensor function representation (LRTFR) parameterized by multilayer perceptrons (MLPs), which can continuously represent data beyond meshgrid with powerful representation abilities. Specifically, the suggested tensor function, which maps an arbitrary coordinate to the corresponding value, can continuously represent data in an infinite real space. Parallel to discrete tensors, we develop two fundamental concepts for tensor functions, i.e., the tensor function rank and low-rank tensor function factorization, and utilize MLPs to paramterize factor functions of the tensor function factorization. We theoretically justify that both low-rank and smooth regularizations are harmoniously unified in LRTFR, which leads to high effectiveness and efficiency for data continuous representation. Extensive multi-dimensional data recovery applications arising from image processing (image inpainting and denoising), machine learning (hyperparameter optimization), and computer graphics (point cloud upsampling) substantiate the superiority and versatility of our method as compared with state-of-the-art methods. Especially, the experiments beyond the original meshgrid resolution (hyperparameter optimization) or even beyond meshgrid (point cloud upsampling) validate the favorable performances of our method for continuous representation.

Multidimensional Data Processing With Bayesian Inference via Structural Block Decomposition.

How to handle large multidimensional datasets, such as hyperspectral images and video information, efficiently and effectively plays a critical role in big-data processing. The characteristics of low-rank tensor decomposition in recent years demonstrate the essentials in describing the tensor rank, which often leads to promising approaches. However, most current tensor decomposition models consider the rank-1 component simply to be the vector outer product, which may not fully capture the correlated spatial information effectively for large-scale and high-order multidimensional datasets. In this article, we develop a new novel tensor decomposition model by extending it to the matrix outer product or called Bhattacharya-Mesner product, to form an effective dataset decomposition. The fundamental idea is to decompose tensors structurally in a compact manner as much as possible while retaining data spatial characteristics in a tractable way. By incorporating the framework of the Bayesian inference, a new tensor decomposition model on the subtle matrix unfolding outer product is established for both tensor completion and robust principal component analysis problems, including hyperspectral image completion and denoising, traffic data imputation, and video background subtraction. Numerical experiments on real-world datasets demonstrate the highly desirable effectiveness of the proposed approach.

Denoising multispectral images using non-local rank tensor decomposition and bilateral filtering based on sunflower optimization

ABSTRACT Image denoising is an important pre-processing process in the fields of computer vision and image processing. Traditional denoising techniques blur edges excessively and degrade image quality by removing noise components but failing to maintain clarity. To overcome these problems, this paper proposes a multispectral image denoising strategy combining non-local rank tensor decomposition (NLRTD) and bilateral filtering. To extract patches from noisy images, single-level discrete wavelet transform (DWT) is utilized. Then, similar patches from the extracted images are grouped using spectral clustering. After that, mixed noise is reduced by separating clean images from each clustered group using NLRTD. An optimized bilateral filter using Sunflower optimization (SFO) is used for denoising by preserving edge details and is reconstructed using its constituent parts. The effectiveness of the proposed denoising method is assessed using performance matrices, such as BER, PSNR, MSE, RMSE, SNR and SSIM were 0.8544%, 53.21%, 2.41%, 2.41%, 25.06% and 0.90%, respectively.

Encoding a Many-Body Potential Energy Surface into a Grid-Based Matrix Product Operator.

An efficient algorithm for compressing a given many-body potential energy surface (PES) of molecular systems into a grid-based matrix product operator (MPO) is proposed. The PES is once represented by a full-dimensional or truncated many-body expansion form, which is obtained by ab initio calculations at each grid mesh point, and then all terms in the expansion are compressed and merged into a single MPO while maintaining the bond dimension of the MPO as small as possible. It was shown that the ab initio PES of the H2CO was compressed by more than 2 orders of magnitude in the size of the site operators without loss of accuracy. By the use of grid basis, the tensor rank of the site operators of the MPO is reduced from four to three due to the diagonal nature of the position-dependent operators on grid basis, which significantly reduces the computational cost of the tensor contractions required in the real and imaginary time evolution of the matrix product state (MPS) wave functions with the grid-based MPO (Grid-MPO) Hamiltonian. Similar to other grid-based methods, Grid-MPO is easily applicable to any kinds of potentials of molecular systems, such as analytical empirical model potentials expressed by position operators and ab initio potentials, if the values at the grid points are available. Using the Grid-MPO combined with the MPS, we calculated the time correlation function of the Eigen cation to predict the infrared spectrum and compared with the experimental and the previous theoretical studies. The actual scaling with the size of systems was examined for the multidimensional Henon-Heiles Hamiltonian. It was shown that the method is considerably accelerated by the graphic processing unit (GPU) because the sizes of site operators were kept small and all tensors were able to be stored on the VRAM of a GPU.

Semi-inclusive production of spin- 3/2 hadrons in deep inelastic scattering

We investigate the production of spin-3/2 hadrons in semi-inclusive deep inelastic lepton-nucleon scatterings. The complete differential cross section is derived through the kinematic analysis and expressed in terms of 288 structure functions, corresponding to all polarization configurations and azimuthal modulations. For an unpolarized lepton beam, half of the 192 structure functions have nonzero leading order contributions in the parton model, among which 42 are from rank-3 tensor polarized fragmentation functions of the hadron. For a polarized lepton beam, one third of the 96 structure functions contribute at the leading order and 14 of them are from rank-3 tensor polarized fragmentation functions. In addition to the formalism, we perform a model estimation of the spin transfer to a ShLLL polarized hadron and sizable asymmetry is expected. Therefore, these newly defined observables for the production of a spin-3/2 hadron in a deep inelastic scattering process can be explored in future experiments to understand nucleon spin structures and spin-dependent fragmentation functions. Published by the American Physical Society 2024

Robust low tubal rank tensor recovery using discrete empirical interpolation method with optimized slice/feature selection

Robust low tubal rank tensor recovery using discrete empirical interpolation method with optimized slice/feature selection

TENSOR AND VECTOR APPROACHES TO OBJECTS RECOGNITION BY INVERSE FEATURE FILTERS

The investigation of the extraction of image objects features by filters based on tensor and vector data presentation is considered. The tensor data is obtained as a sum of rank-one tensors, given by the tensor product of the vector of lexicographic representation of image fragments pixels with itself. The accumulated tensor is approximated by one rank tensor obtained using singular values decomposition. It has been shown that the main vector of the decomposition can be considered as the object feature vector. The vector data is obtained by accumulating analogous vectors of image fragments pixels. The accumulated vector is also considered as an object feature. The filter banks of a set of objects are obtained by regularized inversion of the matrices compiled by object features vectors. Optimized regularization of the inversion is used to expand the regions of object features capture with minimal error. The object fragments and corresponding feature vectors are selected through a training iterative process. The tensor and vector approaches create two channels for recognition. High efficiency of object recognition can be achieved by choosing the filter capture band and creating filter branches according to the given bands. The filters create a convolutional network to recognize a set of objects. It has been shown that the obtained filters have an advantage over known correlation filters when recognizing objects with small fragments.

Tensor robust PCA with nonconvex and nonlocal regularization

Tensor robust principal component analysis (TRPCA) is a classical way for low-rank tensor recovery, which minimizes the convex surrogate of tensor rank by shrinking each tensor singular value equally. However, for real-world visual data, large singular values represent more significant information than small singular values. In this paper, we propose a nonconvex TRPCA (N-TRPCA) model based on the tensor adjustable logarithmic norm. Unlike TRPCA, our N-TRPCA can adaptively shrink small singular values more and shrink large singular values less. In addition, TRPCA assumes that the whole data tensor is of low rank. This assumption is hardly satisfied in practice for natural visual data, restricting the capability of TRPCA to recover the edges and texture details from noisy images and videos. To this end, we integrate nonlocal self-similarity into N-TRPCA, and further develop a nonconvex and nonlocal TRPCA (NN-TRPCA) model. Specifically, similar nonlocal patches are grouped as a tensor and then each group tensor is recovered by our N-TRPCA. Since the patches in one group are highly correlated, all group tensors have strong low-rank property, leading to an improvement of recovery performance. Experimental results demonstrate that the proposed NN-TRPCA outperforms existing TRPCA methods in visual data recovery. The demo code is available at https://github.com/qguo2010/NN-TRPCA.

Embedded Feature Selection on Graph-Based Multi-View Clustering

Recently, anchor graph-based multi-view clustering has been proven to be highly efficient for large-scale data processing. However, most existing anchor graph-based clustering methods necessitate post-processing to obtain clustering labels and are unable to effectively utilize the information within anchor graphs. To solve these problems, we propose an Embedded Feature Selection on Graph-Based Multi-View Clustering (EFSGMC) approach to improve the clustering performance. Our method decomposes anchor graphs, taking advantage of memory efficiency, to obtain clustering labels in a single step without the need for post-processing. Furthermore, we introduce the l2,p-norm for graph-based feature selection, which selects the most relevant data for efficient graph factorization. Lastly, we employ the tensor Schatten p-norm as a tensor rank approximation function to capture the complementary information between different views, ensuring similarity between cluster assignment matrices. Experimental results on five real-world datasets demonstrate that our proposed method outperforms state-of-the-art approaches.

Randomized two-sided subspace iteration for low-rank matrix and tensor decomposition

The low-rank approximation of big data matrices and tensors plays a pivotal role in many modern applications. Although, a truncated version of the singular value decomposition (SVD) furnishes the best approximation, its computation is challenging on modern, multicore architectures. Recently, the randomized subspace iteration has shown to be a powerful tool in approximating large-scale matrices. In this paper we present a two-sided variant of the randomized subspace iteration. Novel in our work is the exploitation of the unpivoted QR factorization, rather than the SVD, for factorizing the compressed matrix. Hence our algorithm is a randomized rank-revealing URV decomposition. We prove the rank-revealingness of our algorithm by establishing bounds for the singular values as well as the other blocks of the compressed matrix. We further provide bounds on the error of the low-rank approximations of the proposed algorithm, in both 2- and Frobenius norm. In addition, we employ the proposed algorithm to efficiently compute low rank tensor decompositions: we present two randomized algorithms, one for the truncated higher-order SVD, and the other for the tensor SVD. We conduct numerical tests on (i) various classes of matrices, and (ii) synthetic tensors and real datasets to demonstrate the efficacy of the proposed algorithms.

Error-robust multi-view subspace clustering with nonconvex low-rank tensor approximation and hyper-Laplacian graph embedding

Tensor based subspace clustering, a method aimed at partitioning multi-view data into distinct clusters through the tensor low-rank representation, has gained widespread popularity. Despite the effectiveness of tensor singular value decomposition (t-SVD) based nuclear norm in extracting high-dimensional information from multiple views, certain limitations persist. Firstly, while the commonly used l2,1 norm enhances clustering robustness, it remains susceptible to the impact of high-intensity noises. Secondly, the t-SVD based nuclear norm provides a biased estimation of tensor rank. Thirdly, existing methods either fail to preserve manifold structures from the original space or only extract binary relationships between data points. To address these challenges and enhance clustering performance in both high-intensity noisy and real-world conditions, an error-robust multi-view subspace clustering with nonconvex low-rank tensor approximation and hyper-Laplacian graph embedding (EMSC-NLTHG) method is proposed. Specifically, Cauchy loss function which reduces sensitivity to larger noises and outliers is introduced. Based on the Cauchy loss function, we develop the Cauchy pseudo norm to represent the reconstruction error in clustering to enhance the robustness. Moreover, rather than uniformly weighting all singular values in the t-SVD nuclear norm, a nonconvex tensor nuclear norm is adopted to add weights adaptively and approximate the true tensor rank. Additionally, hypergraph is embedded in the representation matrices to preserve local manifold structures and extract the complex multivariate relationships between data points. Extensive experiments demonstrate that the proposed EMSC-NLTHG method consistently outperforms state-of-the-art techniques by approximately 18% to 40% on datasets corrupted by high-intensity noises and around 1% to 27% on eight real-world datasets across six popular evaluation metrics.

A DEIM Tucker tensor cross algorithm and its application to dynamical low-rank approximation

We introduce a Tucker tensor cross approximation method that constructs a low-rank representation of a d-dimensional tensor by sparsely sampling its fibers. These fibers are selected using the discrete empirical interpolation method (DEIM). Our proposed algorithm is referred to as DEIM fiber sampling (DEIM-FS). For a rank-r approximation of an O(Nd) tensor, DEIM-FS requires access to only dNrd−1 tensor entries, a requirement that scales linearly with the tensor size along each mode. We demonstrate that DEIM-FS achieves an approximation accuracy close to the Tucker-tensor approximation obtained via higher-order singular value decomposition at a significantly reduced cost. We also present DEIM-FS (iterative) that does not require access to singular vectors of the target tensor unfolding and can be viewed as a black-box Tucker tensor algorithm. We employ DEIM-FS to reduce the computational cost associated with solving nonlinear tensor differential equations (TDEs) using dynamical low-rank approximation (DLRA). The computational cost of solving DLRA equations can become prohibitive when the exact rank of the right-hand side tensor is large. This issue arises in many TDEs, especially in cases involving non-polynomial nonlinearities, where the right-hand side tensor has full rank. This necessitates the storage and computation of tensors of size O(Nd). We show that DEIM-FS results in significant computational savings for DLRA by constructing a low-rank Tucker approximation of the right-hand side tensor on the fly. Another advantage of using DEIM-FS is to significantly simplify the implementation of DLRA equations, irrespective of the type of TDEs. We demonstrate the efficiency of the algorithm through several examples including solving high-dimensional partial differential equations.

Countably many asymptotic tensor ranks

In connection with recent work on gaps in the asymptotic subranks of complex tensors the question arose whether the number of nonnegative real numbers that arise as the asymptotic subrank of some complex tensor is countable. In this short note we settle this question in the affirmative, for all tensor invariants that are algebraic in the sense that they are invariant under field automorphisms of the complex numbers.

RG flows and fixed points of O(N)r models

By means of ϵ and large N expansions, we study generalizations of the O(N) model where the fundamental fields are tensors of rank r rather than vectors, and where the global symmetry (up to additional discrete symmetries and quotients) is O(N)r, focusing on the cases r ≤ 5. Owing to the distinct ways of performing index contractions, these theories contain multiple quartic operators, which mix under the RG flow. At all large N fixed points, melonic operators are absent and the leading Feynman diagrams are bubble diagrams, so that all perturbative fixed points can be readily matched to full large N solutions obtained from Hubbard-Stratonovich transformations. The family of fixed points we uncover extend to arbitrary higher values of r, and as their number grows superexponentially with r, these theories offer a vast generalization of the critical O(N) model.We also study sextic O(N)r theories, whose large N limits are obscured by the fact that the dominant Feynman diagrams are not restricted to melonic or bubble diagrams. For these theories the large N dynamics differ qualitatively across different values of r, and we demonstrate that the RG flows possess a numerous and diverse set of perturbative fixed points beginning at rank four.

Imputed quantile vector autoregressive model for multivariate spatial–temporal data

AbstractImputing missing values in multivariate spatial–temporal data is important in many fields. Existing low rank tensor learning methods are popular for handling this task but are sensitive to high level of skewness. The aim of this paper is to develop an alternative method with robustness and high imputation accuracy for multivariate spatial–temporal data. In view of the fact that quantile regression is robust to noises and outliers, we propose an imputed quantile vector autoregressive (IQVAR) model. IQVAR can simultaneously impute missing values and estimate parameters of quantile vector autoregressive model. The objective function includes check loss and nuclear norm penalization. We develop an ADMM (Alternating Direction Method of Multipliers) algorithm to solve the resulting optimization problem. Simulation studies and real data analysis are conducted to verify the efficiency of IQVAR. Compared with other approaches, IQVAR is more robust and accurate.

Statistical guaranteed noisy tensor recovery by fusing low-rankness on all orientations in frequency–original domains

Low-rank tensor recovery faces challenges in accurately defining the low-rankness of a tensor. Most existing definitions typically focus on one domain alone — either the original or frequency domain. Additionally, certain definitions often exhibit limitations in their sensitivity to orientation variation. To overcome these challenges, we define a novel tensor rank, the Orientation Invariant Hybrid Rank (OIHR). This rank fuses rank information across all orientations in both frequency and original domains. Employing its convex approximation, the Orientation Invariant Hybrid Nuclear Norm (OIHNN), we propose a general tensor recovery model. We further explore the statistical performance of the estimator based on this model, establishing a deterministic upper bound on the estimation error under generic noise. Furthermore, non-asymptotic upper bounds under Gaussian noise are separately derived for two specific cases: tensor compressive sensing and tensor completion. Finally, we propose the algorithm to solve the model. Extensive experiments on both synthetic and real data are conducted to validate the statistical guarantees and verify the effectiveness of our algorithm.

Tensor recovery using the tensor nuclear norm based on nonconvex and nonlinear transformations

Recently, the tensor completion problem has witnessed significant advancements with the introduction of the tensor nuclear norm based on self-supervised nonlinear transformations (S2NTNN). However, the tensor nuclear norm serves as a convex substitute for the tubal rank of tensors, often resulting in suboptimal solutions to the original problem. In this work, we propose a novel method, called Nonconvex Nonlinear transformation-based Tensor Nuclear Norm (N2TNN), which builds upon nonlinear transformations and introduces a generalized nonconvex function to better exploit the prior information of underlying tensors. Additionally, we utilize regularization techniques, namely spatial total variation (TV) and spatial spectral total variation (SSTV), to enhance both the details and edge information. Extensive numerical experiments unequivocally validate the superior performance of our proposed method when compared to other state-of-the-art techniques in terms of both numerical accuracy and visual quality.

A fast correction approach to tensor robust principal component analysis

Tensor robust principal component analysis (TRPCA) is a useful approach for obtaining low-rank data corrupted by noise or outliers. However, existing TRPCA methods face certain challenges when it comes to estimating the tensor rank and the sparsity accurately. The commonly used tensor nuclear norm (TNN) may lead to sub-optimal solutions due to the gap between TNN and the tensor rank. Additionally, the ℓ1-norm is not an ideal estimation of the ℓ0-norm, and solving TNN minimization can be computationally intensive because of the tensor singular value thresholding (t-SVT) scheme. To address these issues, a method called fast correction TNN (FC-TNN) is proposed for TRPCA. In contrast to existing methods, FC-TNN introduces a correction term to bridge the gap between TNN and the tensor rank. Furthermore, a new correction term is employed for the ℓ1-norm to achieve the desired solution. To improve computational efficiency, the Chebyshev polynomial approximation (CPA) technique is presented for computing t-SVT without requiring tensor singular value decomposition (t-SVD). The CPA technique is incorporated into the alternating direction method of multipliers (ADMM) algorithm to solve the proposed model effectively. Theoretical analysis demonstrates that FC-TNN offers a lower error bound compared to TNN under certain conditions. Extensive experiments conducted on various tensor-based datasets illustrate that the proposed method outperforms several state-of-the-art methods.

Derivations of the stress-strain relations for viscoanelastic media and the heat equation in irreversible thermodynamic with internal variables

Abstract By using a procedure of classical irreversible thermodynamics with internal variable (CIT-IV), some possible interactions among heat conduction and viscous-anelastic flows for rheological media are studied. In particular, we introduce as internal variables a second rank tensor ε α β ( 1 ) \varepsilon _{\alpha \beta }^{(1)} that is contribution to inelastic strain and a vector ξα that influences the thermal transport phenomena and we derive the phenomenological equations for these variables in the anisotropic and isotropic cases. The stress-strain equations, the general flows and the temperature equation in visco-anelastic processes are obtained and when the medium is isotropic, we obtain that the total heat flux J (q) can be split in two parts: a first contribution J (0), governed by Fourier law, and a second contribution J (1), obeying Maxwell-Cattaneo-Vernotte (M-C-V) equation.