Rank Tensor Research Articles

Low-rank quaternion tensor completion for color video inpainting via a novel factorization strategy

Recently, a quaternion tensor product named Qt-product was proposed, and then the singular value decomposition and the rank of a third-order quaternion tensor were given. From a more applicable perspective, we extend the Qt-product and propose a novel multiplication principle for third-order quaternion tensor named gQt-product. With the gQt-product, we introduce a brand-new singular value decomposition for third-order quaternion tensors named gQt-SVD and then define gQt-rank and multi-gQt-rank. We prove that the optimal low-rank approximation of a third-order quaternion tensor exists and some numerical experiments demonstrate the low-rankness of color videos. So, we apply the low-rank quaternion tensor completion to color video inpainting problems and present alternating least-square algorithms to solve the proposed low gQt-rank and multi-gQt-rank quaternion tensor completion models. The convergence analyses of the proposed algorithms are established and some numerical experiments on various color video datasets show the high recovery accuracy and computational efficiency of our methods.

A Local Macroscopic Conservative (LoMaC) Low Rank Tensor Method for the Vlasov Dynamics

In this paper, we propose a novel Local Macroscopic Conservative (LoMaC) low rank tensor method for simulating the Vlasov-Poisson (VP) system. The LoMaC property refers to the exact local conservation of macroscopic mass, momentum and energy at the discrete level. This is a follow-up work of our previous development of a conservative low rank tensor approach for Vlasov dynamics (arXiv:2201.10397). In that work, we applied a low rank tensor method with a conservative singular value decomposition to the high dimensional VP system to mitigate the curse of dimensionality, while maintaining the local conservation of mass and momentum. However, energy conservation is not guaranteed, which is a critical property to avoid unphysical plasma self-heating or cooling. The new ingredient in the LoMaC low rank tensor algorithm is that we simultaneously evolve the macroscopic conservation laws of mass, momentum and energy using a flux-difference form with kinetic flux vector splitting; then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables by a conservative orthogonal projection. The algorithm is extended to the high dimensional problems by hierarchical Tuck decomposition of solution tensors and a corresponding conservative projection algorithm. Extensive numerical tests on the VP system are showcased for the algorithm’s efficacy.

A Low Transformed Tubal Rank Tensor Model Using a Spatial-Tubal Constraint for Sample Clustering with Cancer Multi-omics Data

Background: Since each dimension of a tensor can store different types of genomics data, compared to matrix methods, utilizing tensor structure can provide a deeper understanding of multi-dimensional data while also facilitating the discovery of more useful information related to cancer. However, in reality, there are issues such as insufficient utilization of prior knowledge in multi-omics data and limitations in the recovery of low-tubal-rank tensors. Therefore, the method proposed in this article was developed. Objective: In this paper, we proposed a low transformed tubal rank tensor model (LTTRT) using a spatial-tubal constraint to accurately partition different types of cancer samples and provide reliable theoretical support for the identification, diagnosis, and treatment of cancer. Method: In the LTTRT method, the transformed tensor nuclear norm based on the transformed tensor singular value decomposition is characterized by the low-rank tensor, which can explore the global low-rank property of the tensor, resolving the challenge of the tensor nuclear norm-based method not achieving the lowest tubal rank. Additionally, the introduction of weighted total variation regularization is conducive to extracting more information from sequencing data in both spatial and tubal dimensions, exploring cross-correlation features of multiple genomic data, and addressing the problem of overlooking prior knowledge from various perspectives. In addition, the L1-norm is used to improve sparsity. A symmetric Gauss‒Seidel-based alternating direction method of multipliers (sGS-ADMM) is used to update the LTTRT model iteratively. Results: The experiments of sample clustering on multiple integrated cancer multi-omics datasets show that the proposed LTTRT method is better than existing methods. Experimental results validate the effectiveness of LTTRT in accurately partitioning different types of cancer samples. result: The experiments of sample clustering on multiple integrated cancer multi-omics datasets show that the proposed LTTRT method is better than existing methods. Experimental results validate the effectiveness of LTTRT in accurately partitioning different types of cancer samples. Conclusion: The LTTRT method achieves precise segmentation of different types of cancer samples.

On the uniqueness and computation of commuting extensions

A tuple (Z1,…,Zp) of matrices of size r×r is said to be a commuting extension of a tuple (A1,…,Ap) of matrices of size n×n if the Zi pairwise commute and each Ai sits in the upper left corner of a block decomposition of Zi (here, r and n are two arbitrary integers with n<r). This notion was discovered and rediscovered in several contexts including algebraic complexity theory (in Strassen's work on tensor rank), in numerical analysis for the construction of cubature formulas and in quantum mechanics for the study of computational methods and the study of the so-called “quantum Zeno dynamics.” Commuting extensions have also attracted the attention of the linear algebra community. In this paper we present 3 types of results:(i)Theorems on the uniqueness of commuting extensions for three matrices or more.(ii)Algorithms for the computation of commuting extensions of minimal size. These algorithms work under the same assumptions as our uniqueness theorems. They are applicable up to r=4n/3, and are apparently the first provably efficient algorithms for this problem applicable beyond r=n+1.(iii)A genericity theorem showing that our algorithms and uniqueness theorems can be applied to a wide range of input matrices.

Self representation based methods for tensor completion problem

Tensor, the higher-order data array, naturally arises in many fields, such as information sciences, seismic data reconstruction, physics, video inpainting and so on. In this paper, we intend to provide a new model to recover a tensor, based on self-representation, for the all-mode unfoldings of the desired tensor, regardless of the tensor rank. The suggested idea generalizes self-representation to tensor and recovers an incomplete tensor by reconstructing one fiber by others in such a way that they all belong to the same subspace. We design least-square and low-rank self-representation algorithms based on the Linearized Alternating Direction Method utilizing this concept. We show that the proposed algorithms converge to the rank-minimization of the incomplete tensor.

FuBay: An Integrated Fusion Framework for Hyperspectral Super-Resolution Based on Bayesian Tensor Ring.

Fusion with corresponding finer-resolution images has been a promising way to enhance hyperspectral images (HSIs) spatially. Recently, low-rank tensor-based methods have shown advantages compared with other kind of ones. However, these current methods either relent to blind manual selection of latent tensor rank, whereas the prior knowledge about tensor rank is surprisingly limited, or resort to regularization to make the role of low rankness without exploration on the underlying low-dimensional factors, both of which are leaving the computational burden of parameter tuning. To address that, a novel Bayesian sparse learning-based tensor ring (TR) fusion model is proposed, named as FuBay. Through specifying hierarchical sprasity-inducing prior distribution, the proposed method becomes the first fully Bayesian probabilistic tensor framework for hyperspectral fusion. With the relationship between component sparseness and the corresponding hyperprior parameter being well studied, a component pruning part is established to asymptotically approaching true latent rank. Furthermore, a variational inference (VI)-based algorithm is derived to learn the posterior of TR factors, circumventing nonconvex optimization that bothers the most tensor decomposition-based fusion methods. As a Bayesian learning methods, our model is characterized to be parameter tuning-free. Finally, extensive experiments demonstrate its superior performance when compared with state-of-the-art methods.

Robust Low-Tubal-Rank Tensor Completion Based on Tensor Factorization and Maximum Correntopy Criterion.

The goal of tensor completion is to recover a tensor from a subset of its entries, often by exploiting its low-rank property. Among several useful definitions of tensor rank, the low tubal rank was shown to give a valuable characterization of the inherent low-rank structure of a tensor. While some low-tubal-rank tensor completion algorithms with favorable performance have been recently proposed, these algorithms utilize second-order statistics to measure the error residual, which may not work well when the observed entries contain large outliers. In this article, we propose a new objective function for low-tubal-rank tensor completion, which uses correntropy as the error measure to mitigate the effect of the outliers. To efficiently optimize the proposed objective, we leverage a half-quadratic minimization technique whereby the optimization is transformed to a weighted low-tubal-rank tensor factorization problem. Subsequently, we propose two simple and efficient algorithms to obtain the solution and provide their convergence and complexity analysis. Numerical results using both synthetic and real data demonstrate the robust and superior performance of the proposed algorithms.

Decomposing tensor spaces via path signatures

The signature of a path is a sequence of tensors whose entries are iterated integrals, playing a key role in stochastic analysis and applications. The set of all signature tensors at a particular level gives rise to the universal signature variety. We show that the parametrization of this variety induces a natural decomposition of the tensor space via representation theory, and connect this to the study of path invariants. We also reveal certain constraints that apply to the rank and symmetry of a signature tensor.

A new formula for the determinant and bounds on its tensor and Waring ranks

Abstract We present a new explicit formula for the determinant that contains superexponentially fewer terms than the usual Leibniz formula. As an immediate corollary of our formula, we show that the tensor rank of the $n \times n$ determinant tensor is no larger than the $n$ -th Bell number, which is much smaller than the previously best-known upper bounds when $n \geq 4$ . Over fields of non-zero characteristic we obtain even tighter upper bounds, and we also slightly improve the known lower bounds. In particular, we show that the $4 \times 4$ determinant over ${\mathbb{F}}_2$ has tensor rank exactly equal to $12$ . Our results also improve upon the best-known upper bound for the Waring rank of the determinant when $n \geq 17$ , and lead to a new family of axis-aligned polytopes that tile ${\mathbb{R}}^n$ .

Pion gravitational form factors in the QCD instanton vacuum. I

The pion form factors of the QCD energy-momentum tensor (EMT) are studied in the instanton liquid model (ILM) of the QCD vacuum. In this approach, the breaking of conformal symmetry is encoded in the form of stronger-than-Poisson fluctuations in the number of instantons. For the trace of the EMT, it is shown that the gluonic trace anomaly term contributes half the pion mass, with the other half coming from the quark-mass-dependent σ term. The Q2 dependence of the form factors is governed by glueball and scalar meson exchanges. For the EMT, the spin-0 (trace) and spin-2 (traceless rank-2 tensor) form factors are computed at next-to-leading order in the instanton density using effective quark operators. Relations between the gluon and quark contributions to the EMT form factors are derived. The form factors are also expressed in terms of the pion light-front wave functions in the ILM. The results at the low resolution scale of the inverse instanton size are evolved to higher scales using the renormalization group equation. The ILM results compare well with those of recent lattice QCD calculations. Published by the American Physical Society 2024

Nonlinear Optics Through the Field Tensor Formalism

AbstractThe “field tensor” is the tensor product of the electric fields of the interacting waves during a sum‐ or difference‐frequency generation nonlinear optical interaction. It is therefore a tensor describing light interacting with matter, the latter being characterized by the “electric susceptibility tensor.” The contracted product of these two tensors of equal rank gives the light‐matter interaction energy, whether or not propagation occurs. This notion having been explicitly or implicitly present from the early pioneering studies in nonlinear optics, its practical use has led to original developments in many highly topical theoretical or experimental situations, at the microscopic as well macroscopic level throughout a variety of coherent or non‐coherent processes. The aim of this review article is to rigorously explain the field tensor formalism in the context of tensor algebra and nonlinear optics in terms of a general time‐space multi‐convolutional development, using spherical tensors, with components expressed in the frame of a common basis set of irreducible tensors, or Cartesian tensors. A wide variety of media are considered, including biological tissues and their imaging, artificially engineered by various combinations of optical and static electric fields, with the two extremes of all‐optical and purely electric poling, and also bulk single crystals.

Tensor rank and other multipartite entanglement measures of graph states

Tensor rank and other multipartite entanglement measures of graph states

Nonsymmetrical thermal conductivity along the director field in ferroelectric nematic liquid crystals.

The synthesis of ferroelectric nematic liquid crystals (FNLCs) concludes the long wait for their existence and potential usage in multiple liquid crystal based applications. In FNLCs, electric polarization in the nematic phase significantly decreases the switching time of in-on display pixels. In this article, we report the occurrence of translation symmetry breaking for heat propagation along the director field n[over ̂] in the ferroelectric nematic phase. Due to the C_{∞V} symmetry of such a phase and close similarity to the bent-core polar liquid crystal phase, a rank-3 tensor describes its scalar order parameter and algebraic deductions. The finite element simulations show the occurrence of the nonsymmetrical thermal conductivity along n[over ̂]. The preferential heat transport in FNLCs can allow them to work as an all-thermal monophase non-nanostructured single-material thermal rectifier. We expect that this study will contribute towards the FNLCs application as functional layers and inks.

A low-rank support tensor machine for multi-classification

In recent decades, there has been an increasing demand for effectively handling high-dimensional multi-channel tensor data. Due to the inability to utilize internal structural information, Support Vector Machine (SVM) and its variations struggle to classify flattened tensor data, consequently resulting in the ‘curse of dimensionality’ issue. Furthermore, most of these methods can not directly apply to multiclass datasets. To overcome these challenges, we have developed a novel classification method called Multiclass Low-Rank Support Tensor Machine (MLRSTM). Our method is inspired by the well-established low-rank tensor hypothesis, which suggests a correlation between each channel of the feature tensor. Specifically, MLRSTM adopts the hinge loss function and introduces a convex approximation of tensor rank, the order-d Tensor Nuclear Norm (order-d TNN), in the regularization term. By leveraging the order-d TNN, MLRSTM effectively exploits the inherent structural information in tensor data to enhance generalization performance and avoid the curse of dimensionality. Moreover, we develop the Alternating Direction Method of Multipliers (ADMM) algorithm to optimize the convex problem inherent in training MLRSTM. Finally, comprehensive experiments validate the excellent performance of MLRSTM in tensor multi-classification tasks, showcasing its potential and efficacy in handling high-dimensional multi-channel tensor data.

The interplay between bounded ranks of tensors arising from partitions

AbstractLet be integers. Using a fragmentation technique, we characterise ‐tuples of non‐empty families of partitions of such that it suffices that an order‐ tensor has bounded ‐rank for each for it to have bounded ‐rank. On the way, we prove power lower bounds on suitable products of diagonal tensors, providing a qualitative answer to a question of Naslund.

Large N universality of 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 superconformal index and AdS black holes

We study the large N limit of the matrix models associated with the superconformal indices of four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 superconformal field theories. We find that for a large class of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 superconformal gauge theories, the superconformal indices in the large N limit of such theories are dominated by the ‘parallelogram’ saddle, providing O(N2) free energy for the generic value of chemical potentials. This saddle corresponds to BPS black holes in AdS5 whenever a holographic dual description is available. Our saddle applies to a large class of gauge theories, including ADE quiver gauge theories, and the theories with rank-2 tensor matters. Our analysis works for most N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 superconformal gauge theories that admit a suitable large N limit while keeping the flavor symmetry fixed. We also find ‘multi-cut’ saddle points, which correspond to the orbifolded Euclidean black holes in AdS5.

Ferumoxytol-Enhanced 5D Multiphase Steady-State Imaging Using Rotating Cartesian K-Space With Low-Rank Reconstruction for Pediatric Congenital Heart Disease.

The rotating Cartesian k-space multiphase steady-state imaging with contrast (ROCK-MUSIC) pulse sequence enables acquisition of whole-heart, cardiac phase-resolved images in pediatric congenital heart disease (CHD) without reliance on the ventilator gating signal. Multidimensional reconstruction with low rank tensor (LRT) has shown promise for resolving complex cardiorespiratory motion. To enhance ROCK-MUSIC by resolving cardiorespiratory phases using LRT reconstruction and to enable semi-automatic hyperparameter tuning by developing an image quality scoring model. Retrospective. Thirty patients (45% female, age 2 days to 6.7 years) with CHD. 3-T, four-dimensional (4D) spoiled gradient recalled echo sequence. Eigenvector-based iTerative Self-consistent Parallel Imaging Reconstruction (ESPIRiT) served as the reference comparison for LRT reconstruction. A 4-point Likert scale was used for cardiac and vascular image quality scoring based on cardiac chamber definition, lumen signal uniformity, vascular margin clarity, and motion artifact. Ejection fraction and ventricular volumes were assessed in 16 patients. Signal-to-noise ratio (SNR), contrast-to-noise ratio (CNR), and edge sharpness were computed. Intraclass correlation coefficients, Wilcoxon signed-rank test, Bland-Altman. A P-value <0.05 was considered statistically significant. Relative to ESPIRiT, LRT images received significantly higher cardiac (2.81 ± 0.57 vs. 3.19 ± 0.54) and vascular (2.81 ± 0.60 vs. 3.36 ± 0.53) image quality scores. Image quality scoring with semi-automated hyperparameter tuning showed strong correlations (R2 = 0.748) among image quality, SNR, and septal sharpness. Comparison of ejection fraction and volumetry derived from ESPIRiT, and LRT showed no significant systematic difference (P = 0.32). Integration of low-rank reconstruction with ROCK-MUSIC acquisition may be feasible, and semi-automatic hyperparameter tuning could be effective for generating cardiorespiratory resolved images. 2 TECHNICAL EFFICACY: Stage 1.

Tensor double arc-tangent norm minimization for multi-view clustering

Tensor structures are widely utilized in clustering tasks due to their ability to represent high-order relationships among multi-view data. The rank and decomposition of tensors are essential characteristics that have recently received considerable attention. Most approaches approximate tensor rank using the tensor nuclear norm (TNN) and non-convex rank functions. We introduce a multi-view clustering method based on an enhanced self-expression with a tensor double arc-tangent norm. Initially leveraging tensor singular value decomposition, this method impartially approximates the true rank of the tensor, avoiding rank overestimation. The enhanced self-expression further utilizes the latent connections among the original data. Then, we propose an improved alternating direction method of multipliers (ADMM) for rapid optimization of the model. Experiments conducted on six datasets confirm the effectiveness of the proposed model.

Deflation Conjecture and Local Dimensions of Brent Equations

In this paper, a classical deflation process raised by Dayton, Li and Zeng is realized for the Brent equations, which provides new bounds for local dimensions of the solution set. Originally, this deflation process focuses on isolated solutions. We generalize it to the case of irreducible components and a related conjecture is given. We analyze its realization and apply it to the Brent equations. The decrease of the nullities is easily observed. So the deflation process can be served as a useful tool for determining the local dimensions. In addition, our result implies that along with the decrease of the tensor rank, the singular solutions will become more and more.

Robust tensor recovery via a nonconvex approach with ket augmentation and auto‐weighted strategy

AbstractIn this article, we introduce a nonconvex tensor recovery approach, which employs the powerful ket augmentation technique to expand a low order tensor into a high‐order one so that we can exploit the advantage of tensor train (TT) decomposition tailored for high‐order tensors. Moreover, we define a new nonconvex surrogate function to approximate the tensor rank, and develop an auto‐weighted mechanism to adjust the weights of the resulting high‐order tensor's TT ranks. To make our approach robust, we add two mode‐unfolding regularization terms to enhance the model for the purpose of exploring spatio‐temporal continuity and self‐similarity of the underlying tensors. Also, we propose an implementable algorithm to solve the proposed optimization model in the sense that each subproblem enjoys a closed‐form solution. A series of numerical results demonstrate that our approach works well on recovering color images and videos.