Higher-order Tensors Research Articles

Global rate optimality of integral curve estimators in high order tensor models: Supplemental material

Global rate optimality of integral curve estimators in high order tensor models: Supplemental material

Multichannel Fault Diagnosis of Planetary Gearbox via Difference-Average Symbol Transition Entropy and Twin Support Higher-Order Tensor Machine

Multichannel Fault Diagnosis of Planetary Gearbox via Difference-Average Symbol Transition Entropy and Twin Support Higher-Order Tensor Machine

Nonconvex Robust High-Order Tensor Completion Using Randomized Low-Rank Approximation.

Within the tensor singular value decomposition (T-SVD) framework, existing robust low-rank tensor completion approaches have made great achievements in various areas of science and engineering. Nevertheless, these methods involve the T-SVD based low-rank approximation, which suffers from high computational costs when dealing with large-scale tensor data. Moreover, most of them are only applicable to third-order tensors. Against these issues, in this article, two efficient low-rank tensor approximation approaches fusing random projection techniques are first devised under the order-d ( d ≥ 3 ) T-SVD framework. Theoretical results on error bounds for the proposed randomized algorithms are provided. On this basis, we then further investigate the robust high-order tensor completion problem, in which a double nonconvex model along with its corresponding fast optimization algorithms with convergence guarantees are developed. Experimental results on large-scale synthetic and real tensor data illustrate that the proposed method outperforms other state-of-the-art approaches in terms of both computational efficiency and estimated precision.

Balanced Unfolding Induced Tensor Nuclear Norms for High-Order Tensor Completion.

The recently proposed tensor tubal rank has been witnessed to obtain extraordinary success in real-world tensor data completion. However, existing works usually fix the transform orientation along the third mode and may fail to turn multidimensional low-tubal-rank structure into account. To alleviate these bottlenecks, we introduce two unfolding induced tensor nuclear norms (TNNs) for the tensor completion (TC) problem, which naturally extends tensor tubal rank to high-order data. Specifically, we show how multidimensional low-tubal-rank structure can be captured by utilizing a novel balanced unfolding strategy, upon which two TNNs, namely, overlapped TNN (OTNN) and latent TNN (LTNN), are developed. We also show the immediate relationship between the tubal rank of unfolding tensor and the existing tensor network (TN) rank, e.g., CANDECOMP/PARAFAC (CP) rank, Tucker rank, and tensor ring (TR) rank, to demonstrate its efficiency and practicality. Two efficient TC models are then proposed with theoretical guarantees by analyzing a unified nonasymptotic upper bound. To solve optimization problems, we develop two alternating direction methods of multipliers (ADMM) based algorithms. The proposed models have been demonstrated to exhibit superior performance based on experimental findings involving synthetic and real-world tensors, including facial images, light field images, and video sequences.

Minimizing Low-Rank Models of High-Order Tensors: Hardness, Span, Tight Relaxation, and Applications

Minimizing Low-Rank Models of High-Order Tensors: Hardness, Span, Tight Relaxation, and Applications

Dynamic Tensor Modeling for Missing Data Completion in Electronic Toll Collection Gantry Systems.

The deployment of Electronic Toll Collection (ETC) gantry systems marks a transformative advancement in the journey toward an interconnected and intelligent highway traffic infrastructure. The integration of these systems signifies a leap forward in streamlining toll collection and minimizing environmental impact through decreased idle times. To solve the problems of missing sensor data in an ETC gantry system with large volumes and insufficient traffic detection among ETC gantries, this study constructs a high-order tensor model based on the analysis of the high-dimensional, sparse, large-volume, and heterogeneous characteristics of ETC gantry data. In addition, a missing data completion method for the ETC gantry data is proposed based on an improved dynamic tensor flow model. This study approximates the decomposition of neighboring tensor blocks in the high-order tensor model of the ETC gantry data based on tensor Tucker decomposition and the Laplacian matrix. This method captures the correlations among space, time, and user information in the ETC gantry data. Case studies demonstrate that our method enhances ETC gantry data quality across various rates of missing data while also reducing computational complexity. For instance, at a less than 5% missing data rate, our approach reduced the RMSE for time vehicle distance by 0.0051, for traffic volume by 0.0056, and for interval speed by 0.0049 compared to the MATRIX method. These improvements not only indicate a potential for more precise traffic data analysis but also add value to the application of ETC systems and contribute to theoretical and practical advancements in the field.

Quaternion tensor train rank minimization with sparse regularization in a transformed domain for quaternion tensor completion

The tensor train rank (TT-rank) has achieved promising results in tensor completion due to its ability to capture the global low-rankness of higher-order (>3) tensors. On the other hand, in recent times, quaternions have demonstrated their remarkable suitability as a framework for encoding color pixels, leading to exceptional performance across a range of color image processing tasks. In this paper, we encode the three channels of color pixels using the three imaginary parts of quaternions, leveraging the structural advantages of quaternions to fully preserve the potential relationships between color pixel channels. Subsequently, we extend the TT-rank to higher-order quaternion tensors to capture the global low-rank structure of higher-dimensional data. Specifically, the quaternion tensor train (QTT) decomposition is presented, and based on that the quaternion TT-rank (QTT-rank) is naturally defined. In addition, to utilize the local sparse prior of the quaternion tensor, a general and flexible transform framework is defined. Combining both the global low-rank and local sparse priors of the quaternion tensor, we propose a novel quaternion tensor completion model, i.e., QTT-rank minimization with sparse regularization in a transformed domain. Furthermore, to facilitate QTT-rank minimization for processing color images and enhancing its performance with color videos, we extend KA, a tensor augmentation method, to quaternion tensors, introducing quaternion KA (QKA). Numerical experiments conducted on color image and color video inpainting tasks showcase the superiority of the proposed method over state-of-the-art alternatives.

Fast parameterless prototype-based co-clustering

Tensor co-clustering algorithms have been proven useful in many application scenarios, such as recommender systems, biological data analysis and the analysis of complex and evolving networks. However, they are significantly affected by wrong parameter configurations, since, at the very least, they require the cluster number to be set for each mode of the matrix/tensor, although they typically have other algorithm-specific hyper-parameters that need to be fine-tuned. Among the few known objective functions that can be optimized without setting these parameters, the Goodman–Kruskal τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}—a statistical association measure that estimates the strength of the link between two or more discrete random variables—has proven its effectiveness in complex matrix and tensor co-clustering applications. However, its optimization in a co-clustering setting is tricky and, so far, has leaded to very slow and, at least in some specific but not unfrequent cases, inaccurate algorithms, due to its normalization term. In this paper, we investigate some interesting mathematical properties of τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}, and propose a new simplified objective function with the ability of discovering an arbitrary and a priori unspecified number of good-quality co-clusters. Additionally, the new objective function definition allows for a novel prototype-based optimization strategy that enables the fast execution of matrix and higher-order tensor co-clustering. We show experimentally that the new algorithm preserves or even improves the quality of the discovered co-clusters by outperforming state-of-the-art competing approaches, while reducing the execution time by at least two orders of magnitude.

Multilinear subspace learning for Person Re-Identification based fusion of high order tensor features

Video surveillance image analysis and processing is an important field in computer vision and among the challenging tasks for Person Re-Identification (PRe-ID). The latter aims at finding a target person who has already been identified and appeared on a camera network using a powerful description of their pedestrian images. The success of recent research on person PRe-ID is largely backed into the effective features extraction and representation with a powerful learning of these features to correctly discriminate pedestrian images. To this end, two powerful features, Convolutional Neural Network (CNN) and Local Maximal Occurrence (LOMO) are modeled on a multidimensional data in the proposed method, High-Dimensional Feature Fusion (HDFF). Specifically, a new tensor fusion scheme is introduced to take advantage and combine two types of features in the same tensor data even if its dimensions are not the same. To improve the accuracy, we use Tensor Cross-View Quadratic Analysis (TXQDA) to perform multilinear subspace learning followed by the Cosine similarity for matching. TXQDA efficiently ensures the learning ability and reduces the high dimensionality resulting from high-order tensor data. The effectiveness of the proposed method is verified through experiments on three challenging widely-used PRe-ID datasets namely, VIPeR, GRID, and PRID450S. Extensive experiments show that the proposed method performs very well when compared with recent state-of-the-art methods.

Higher-order tensor renormalization group study of the J_{1}-J_{2} Ising model on a square lattice.

Phase transitions of the J_{1}-J_{2} Ising model on a square lattice are studied using the higher-order tensor renormalization group (HOTRG) method. This system involves a competition between the ferromagnetic interaction J_{1} and antiferromagnetic interaction J_{2}, and in previous studies, weak first-order and second-order transitions were observed near the ratio g=J_{2}/|J_{1}|=1/2. It has also been suggested that the universality class of the second-order phase transition connected to the first-order transition line for g>1/2 belongs to the Ashkin-Teller class, which is characterized by a continuously varying critical exponent with g, as predicted by field-theoretical and other studies. Our results, based on the HOTRG calculations for significantly larger sizes, indicate that the region of the first-order transition is marginally narrower than that in previous studies. Furthermore, it is suggested that the region where the critical exponent changes does not necessarily coincide with the Ashkin-Teller region.

Co-positivity of tensors and boundedness-from-below conditions of CP conserving two-Higgs-doublet potential

In this paper, the analytic sufficient and necessary conditions are obtained for the CP conserving two-Higgs-doublet potential to be bounded from below by using the co-positivity of tensors. This is achieved by treating the potential as a fourth-order homogeneous polynomial about the moduli of the two Higgs doublet fields, where the “angles” is described as the misalignment of the two doublets, then solving three minimum problems with respect to the misalignment. Finally, the analytic conditions are established with the help of the corresponding theory and methods of higher-order tensors.

Hilbert’s energy–momentum tensor extended

In classical field models, Hilbert’s energy–momentum tensor for a matter field is defined by a variational derivative of a Lagrangian with respect to the metric tensor. Constitutive relationships between fundamental field variables are a subject that is dealt with by a wide range of models in solid-state physics. In this case, a higher-order constitutive tensor replaces the second-order metric tensor. A similar premetric description with a linear constitutive relation has recently been provided for classical field models of gravity and electrodynamics. In this paper, we analyze the extension of the Hilbert definition of the energy–momentum tensor to models with general linear constitutive law. In order to treat integrals on a differential manifold covariantly, we need the differential forms description. It follows that the Lagrangian, electromagnetic current, and energy–momentum current must all be represented as twisted 4-forms, 3-forms, and vector-valued 3-forms, respectively. The constitutive relation then appears as a linear map on forms. We derive a commutative variation identity for an arbitrary linear map, enabling direct variation processes without having to deal with the individual components. We present evidence for the close relationship between the explicit form of the energy–momentum current of Hilbert’s type and the commutative-variation identity. A variety of field models with a general linear constitutive law are subject to this construction.

A novel recursive least-squares adaptive method for streaming tensor-train decomposition with incomplete observations

Tensor tracking which is referred to as online (adaptive) decomposition of streaming tensors has recently gained much attention in the signal processing community due to the fact that many modern applications generate a huge number of multidimensional data streams over time. In this paper, we propose an effective tensor tracking method via the tensor-train format for decomposing high-order incomplete streaming tensors. On the arrival of new data, the proposed algorithm minimizes a weighted least-squares objective function accounting for both missing values and time-variation constraints on the underlying tensor-train cores, thanks to the recursive least-squares filtering technique and the block coordinate descent framework. Our algorithm is fully capable of tensor tracking from noisy, incomplete, and high-dimensional observations in both static and time-varying environments. Its tracking ability is validated with several experiments on both synthetic and real data.

Bounding nonlinear stretching about spacecraft trajectories using tensor eigenpairs

Local linearization is often used to enable analytical methods in spacecraft dynamics, navigation, and control. However, the linear approximation becomes inaccurate at large distances from the linearization point and/or when the underlying dynamical system is strongly nonlinear. As a result, linearization-based methods like the finite time Lyapunov exponent (FTLE) can underestimate stretching in phase space. Alternatively, this paper presents a semianalytical method to quantify nonlinear stretching: the eigenpairs of higher-order tensors are used to bound the maximum stretching of a state deviation over time. The tenors in question are derived from the Taylor series expansion of a nonlinear solution flow about a reference trajectory, i.e., the state transition tensor (STT) expansion. It is well known that the eigenvalues of the Cauchy–Green tensor (CGT) bound the stretching of state deviations over time for a linear system, and the largest eigenvalue can be related to the FTLE. This paper will present higher-order, nonlinear analogs to the CGT and the FTLE to more accurately bound nonlinear stretching for applications in guidance, navigation, and control.

Multiphase elasto-plastic mean-field homogenisation and its consistent linearisation

The aim of this work is to develop a model based on incremental mean-field homogenisation methods that allows the simulation of composites with an arbitrary number of elasto-plastic material phases. The model is implemented in an implicit finite-element program, whereby, both matrix and inhomogeneities exhibit elasto-plastic material behaviour. Moreover, the implementation of the resulting constitutive equations is based on a consistent linearisation using higher order tensors. The proposed homogenisation method is evaluated by comparing it with existing methods from the literature in the context of two metal matrix composites. The target results used for comparison are obtained through computational homogenisation using FEM, taken from the literature. The overall findings indicate that the developed Mori-Tanaka method yields favourable results when compared to the existing approaches from the literature. Furthermore, a macroscopic tensile specimen with different microscopic morphologies is simulated using FEM and various mean-field homogenisation methods, including Mori-Tanaka, Voigt, Reuss, and Voigt-Reuss-Hill average. This example illustrates how the different microscopic morphologies and homogenisation methods influence the macroscopic response. Additionally, it demonstrates the successful implementation and correct linearisation of the proposed homogenisation method.

Tensor generalized canonical correlation analysis

Regularized Generalized Canonical Correlation Analysis (RGCCA) is a general statistical framework for multiblock data analysis. RGCCA enables deciphering relationships between several sets of variables and subsumes many well-known multivariate analysis methods as special cases. However, RGCCA only deals with vector-valued blocks, disregarding their possible higher-order structures. This paper presents Tensor GCCA (TGCCA), a new method for analyzing higher-order tensors with canonical vectors admitting an orthogonal rank-R CP decomposition. Moreover, two algorithms for TGCCA, based on whether a separable covariance structure is imposed or not, are presented along with convergence guarantees. The efficiency and usefulness of TGCCA are evaluated on simulated and real data and compared favorably to state-of-the-art approaches.

A hierarchical multivariate denoising diffusion model

Multivariable time-series prediction based on the denoising diffusion probabilistic model (DDPM) highlights a major challenge in improving prediction robustness and ensuring relatively low computing costs. This paper proposes a novel hierarchical multivariate denoising diffusion model called HMD-Diffusion to address this challenge. First, the model uses a robust Gauss–Newton (RGN) algorithm-based sampling strategy to reduce the sampling computation cost of the diffusion model and a Gated Attention Unit (GAU)-based mechanism to improve the multivariate prediction ability with minimal quality loss. Second, for data pre-processing, we introduce a dynamic variable coefficient embedding layer to achieve good adaptability to higher-order tensors and higher prediction accuracy in the initial stages. Moreover, the interval prediction architecture is embedded in the DDPM for the first time, which reduces the influence of interference factors in multivariate time-series prediction on the prediction accuracy of hierarchical models. The experimental results in two real time-series prediction scenarios show an improvement in the predictive performance of the proposed model compared to other benchmark models.

On a simplified multi-layered plate model of growth: Asymptotic analyses and numerical implementation

In this paper, a simplified multi-layered plate model is proposed, which can be used to study the growth-induced deformations of multi-layered hyperelastic plates. First, we introduce a full-form multi-layered plate theory of growth, which has a relatively complicated plate equation system. To simplify the plate equations, systematic asymptotic analyses are conducted with some specified magnitudes of growth functions and surface tractions. It is found that the terms containing a high-order stress tensor always have higher asymptotic orders. By dropping these higher-order terms from the original plate equations, a simplified multi-layered plate model can be derived. Despite its simplicity, this new plate model is valid in a wide range of growth and loading conditions. To be prepared for practical applications, the associated weak forms of the simplified plate equations are derived and implemented in a finite element software. Some typical examples are further studied to show the efficiency of the plate model. By virtue of the plate model, the growth deformations of multi-layered plate samples are simulated, which show very good consistency with the results of 3D volume model. Especially, the distributions of residual stresses on the interfaces of different layers can be predicted accurately, which has an important meaning for the design of soft devices with multi-layered plate structures.

Global Rate Optimality of Integral Curve Estimators in High Order Tensor Models

Global Rate Optimality of Integral Curve Estimators in High Order Tensor Models

Doubly coupled nested tensor decompositions with application to multirelay multicarrier MIMO communication networks

Coupled tensor decompositions have emerged as a promising approach to analyze large dimensional datasets in the context of signal processing applications. In this paper, the general concept of doubly coupled decomposition (DCD) for high-order tensors is first proposed, extending the idea of coupled decompositions to doubly coupled nested structures which result from the contraction of two sets of tensors, each set depending on a specific mode. Two new decompositions are defined, the so-called doubly coupled nested Tucker decomposition (DCNTD) and doubly coupled nested PARAFAC decomposition (DCNPD). Uniqueness of these DCDs is analyzed. In a second part, we show how these DCDs can be used to model multirelay multicarrier MIMO cooperative communication networks with two different tensor codings at the source and relay nodes. Exploiting the multilinear structure of the received signals and assuming the coding tensors are known at the destination, semi-blind closed-form receivers are developed for jointly estimating the channels and transmitted symbols. The proposed receivers use Khatri-Rao and Kronecker product factorization algorithms. Identifiability conditions for system parameter estimation and design of the tensor codes are also addressed. Monte Carlo simulation results illustrate the performance improvement of the proposed DCD-based systems over existing state-of-the-art ones.