Principal Tensor Research Articles

Multilinear principal component analysis-based tensor decomposition for fabric weave pattern recognition from high-dimensional streaming data

Multilinear principal component analysis-based tensor decomposition for fabric weave pattern recognition from high-dimensional streaming data

Superposed Atomic Representation for Robust High-Dimensional Data Recovery of Multiple Low-Dimensional Structures

This paper proposes a unified Superposed Atomic Representation (SAR) framework for high-dimensional data recovery with multiple low-dimensional structures. The data can be in various forms ranging from vectors to tensors. The goal of SAR is to recover different components from their sum, where each component has a low-dimensional structure, such as sparsity, low-rankness or be lying a low-dimensional subspace. Examples of SAR include, but not limited to, Robust Sparse Representation (RSR), Robust Principal Component Analysis (RPCA), Tensor RPCA (TRPCA), and Outlier Pursuit (OP). We establish the theoretical guarantee for SAR. To further improve SAR, we also develop a Weighted SAR (WSAR) framework by paying more attention and penalizing less on significant atoms of each component. An effective optimization algorithm is devised for WSAR and the convergence of the algorithm is rigorously proved. By leveraging WSAR as a general platform, several new methods are proposed for high-dimensional data recovery. The experiments on real data demonstrate the superiority of WSAR for various data recovery problems.

Research on the Evaluation and Influencing Factors of China’s Provincial Employment Quality Based on Principal Tensor Analysis

The research on the quality of employment in China holds immense significance for attaining high-quality employment development. Firstly, enhancing the quality of employment facilitates the optimization of labor resource allocation and enhances economic efficiency. Secondly, high-quality employment serves as a fundamental pillar for social equity and stability. Lastly, continual enhancement of employment quality caters to the requirements of social development and plays a crucial role in promoting economic transformation and achieving sustainable development. However, what is the current situation of employment quality in China? How can we scientifically measure employment quality? What are the key factors for the development of employment quality? This study aimed to use spatiotemporal tensor data to measure the level of employment quality in China’s provinces and analyzed the magnitude and direction of its influencing factors in the spatiotemporal dimension. Taking thirty provinces, autonomous regions, and municipalities directly under the central government in China from 2011 to 2020 as the research objects, the employment quality evaluation system was constructed from six dimensions: employment environment, employment status, employability, labor remuneration, social security, and labor relations. The employment quality index data were expressed in the form of three-order, high-dimensional tensor spatiotemporal data, and the employment quality of China’s provinces was measured from the spatiotemporal perspective by using principal tensor analysis. Then, the visual analysis of the development and change process of employment quality was carried out. The spatial autocorrelation analysis of employment quality was carried out, and the time–space dual-fixed-effect model of the spatial Durbin model was selected to analyze the direction and magnitude of the influence factors of employment quality on the selected and neighboring provinces. The research showed that: (1) The overall level of employment quality in China was not high, the employment quality varied greatly among provinces, and the employment quality development gap among provinces showed a trend of widening. (2) The development of employment quality in western China was relatively fast, while the development of employment quality in central China showed insufficient stamina. (3) Sichuan Province had a strong radiation effect on the development of employment quality in neighboring provinces, and Beijing and Tianjin had a strong siphon effect on the development of employment quality in neighboring provinces. (4) The level of industrialization and informatization promoted the development of employment quality in China’s provinces, while the industrial structure had a significant negative effect on the development of employment quality. According to the research findings, the following policy recommendations are proposed: (1) strengthen inter-provincial cooperation and exchange, (2) emphasize support for the central and western regions, (3) fully leverage the radiation effect of Sichuan while optimizing the siphon effect of Beijing and Tianjin, and (4) enhance industrialization and information technology levels, as well as adjust the industrial structure.

DEM investigation of strength and critical state behaviours of granular materials under true triaxial loadings

This paper investigates the strength and critical state (CS) behaviours of granular materials via DEM simulations of true triaxial drained tests under three different loading modes including constant b (intermediate stress ratio) tests with constant σ1′ (major principal stress), constant p′ (mean pressure) and constant σ3′ (minor principal stress) respectively. To this end, a series of samples are generated with the same particle size distribution, and with the confining stresses ranging from 100 kPa to 900 kPa. The CS is achieved for all samples. Both the macroscopic behaviours and the microscopic behaviours are examined and compared considering different loading modes, confining stresses and intermediate stress ratios (b). The critical state lines (CSLs) are found to be unique and independent of the loading modes, but dependent on the b values. The CSLs with b=0 and b=1 form the two boundaries of CSLs respectively beyond which CSLs under all other b tests cannot go beyond. Six different strength criteria are examined and compared in terms of both peak and CS failures. The Mohr–Coulomb strength criterion is found to be only suitable for axisymmetric loading conditions. The Lade-Duncan criterion is only suitable for describing peak strengths, which is dependent on the loading modes and confining stresses. The Satake criterion and Matsuoka-Nakai criterion are the more appropriate strength criteria for describing CS failures, indicating that the CS values of both the Satake parameter and the Matsuoka-Nakai parameter describe an inherent property that characterizes the CS failure for a given type of soil. The CS mechanical coordination number is fitted by a curved line for a given b value, which is unique regardless of the loading modes. The peak and CS values of both major and minor principal fabric tensors decrease with increasing b values, while they increase with increasing b values for the intermediate principal fabric tensor.

Mechanistic driving force for martensite decay in rolling contact fatigue

Martensite decay due to large scale microstructural alterations is the main fatigue process which occurs during extensive rolling contact fatigue in bearing applications. Despite its practical and scientific importance, the full description of the controlling phenomena is still incomplete. Here, we demonstrate that the main mechanistic driving force for martensite decay comes from the positive components of the principal subsurface stress tensor. Such a conclusion is derived using the concept of Continuum Damage Mechanics where the damage rate is controlled by the elastic energy release and plastic strain range. Modeling results are compared with experiments and a good quantitative agreement is obtained. The analysis presented here illuminates the mechanistic reasons on why the fastest material decay occurs at the depth corresponding to the so-called maximum orthogonal shear stress.

Aligned fields double copy to Kerr-NUT-(A)dS

We find Abelian gauge fields that double copy to a large class of black hole spacetimes with spherical horizon topology known as the Kerr-NUT-(A)dS family. Using a multi-Kerr-Schild prescription, we extend the previously-known double copy structure for arbitrarily rotating general dimension black holes, to include NUT charges and an arbitrary cosmological constant. In all cases, these single copy gauge fields are ‘aligned fields’, because their nonzero components align with the principal tensor which generates the Killing structure of the spacetime. In five dimensions, we additionally derive the same single-copy field strengths via the Weyl double copy procedure.

Calculation of molecular g-tensors by sampling spin orientations of generalised Hartree-Fock states

The variational inclusion of spin–orbit coupling in self-consistent field (SCF) calculations affords single-determinant wave functions that completely break spin symmetry. The individual components of the molecular g-tensor are commonly obtained from separate SCF solutions that align the magnetic moment along one of the three principal tensor axes. However, this strategy raises the question if energy differences between solutions are relevant, or how convergence is achieved if the principal axis system is not determined by molecular symmetry. The present work resolves these issues by a simple two-step procedure akin to the generator coordinate method (GCM). First, the orientation of the magnetic moment is constrained in generalised Hartree-Fock (GHF) calculations. Then, non-orthogonal configuration interaction between GHF determinants yields a Kramers doublet for the calculation of the complete g-tensor. Alternatively, diagonalisation in a basis spanned by spin rotations of a single GHF determinant results in qualitatively correct g-tensors by eliminating errors related to spin contamination. For small first-row molecules, these approaches are evaluated against experimental data and full configuration interaction results. It is further demonstrated for two fictitious tetrahedral and species that a GCM strategy can describe the spin–orbit splitting of orbitally-degenerate ground states, which may lead to negative g-values.

Chiral anomalies in black hole spacetimes

We study the properties of chiral anomalies in a wide class of spacetimes which possess a principal Killing-Yano tensor. This class includes metrics of charged rotating black holes as a special physically important case. The spacetimes which admit a principal Killing-Yano tensor possess a number of remarkable properties. In particular, such spacetimes have two commuting Killing vectors and a Killing tensor responsible for their hidden symmetries. We calculate the gravitational and electromagnetic contributions to the axial anomaly currents in the spacetime of a charged rotating black hole, and demonstrate that the equation for the chiral anomaly current has special solutions which respect both explicit and hidden symmetries. Two of these solutions have the form of currents propagating along two principal null directions, which are null eigenvectors of the Riemann tensor. These solutions describe chiral currents for the incoming and outgoing polarization fluxes. It is demonstrated that these principal chiral currents can be written explicitly in the form which contains the off-shell metric coefficients and their derivatives. We discuss conditions where the principle chiral anomaly current is regular at the horizon and the axes of symmetry. We demonstrate that for states where the current vanishes at the past horizon and at the past null infinity, there exist chirality fluxes at both the future horizon and future infinity. The latter is directly related to the polarization asymmetry of Hawking radiation for massless spinning particles. We also calculate the Chern-Simons currents for both gravitational and electromagnetic chiral anomalies in the black hole spacetime, and discuss the properties of the chirality fluxes associated with these currents.

Principal component analysis- and tensor decomposition-based unsupervised feature extraction to select more suitable differentially methylated cytosines: Optimization of standard deviation versus state-of-the-art methods

In contrast to RNA-seq analysis, which has various standard methods, no standard methods for identifying differentially methylated cytosines (DMCs) exist. To identify DMCs, we tested principal component analysis and tensor decomposition-based unsupervised feature extraction with optimized standard deviation, which has been shown to be effective for differentially expressed gene (DEG) identification. The proposed method outperformed certain conventional methods, including those that assume beta-binomial distribution for methylation as the proposed method does not require this, especially when applied to methylation profiles measured using high throughput sequencing. DMCs identified by the proposed method also significantly overlapped with various functional sites, including known differentially methylated regions, enhancers, and DNase I hypersensitive sites. The proposed method was applied to data sets retrieved from The Cancer Genome Atlas to identify DMCs using American Joint Committee on Cancer staging system edition labels. This suggests that the proposed method is a promising standard method for identifying DMCs.

Cl@Si20X20 cages: evaluation of encapsulation nature, structural rigidity, and 29Si-NMR patterns using relativistic DFT calculations.

The experimental characterization of Cl@Si20 endohedral clusters, featuring different ligands such as [Cl@Si20H20]- (1) [Cl@Si20H12Cl8]- (2), and [Cl@Si20Cl20]- (3), provides insight into the variable encapsulation environment for chloride anions. The favorable formation of such species enables the evaluation of the encapsulation nature and the role of the inner anion in the rigidity of the overall cluster. Our results show a sizable interaction which increases as -66.7, -100.8, and -130.3 kcal mol-1 from 1 to 3, respectively, featuring electrostatic character. The orbital interaction involves 3p-Cl → Si20X20 and 3s-Cl → Si20X20 charge transfer channels and a slight contribution from London dispersion-type interactions. These results show that the inner bonding environment can be modified by the choice of exobonded ligands. Moreover, 29Si-NMR parameters are depicted in terms of the chemical shift anisotropy (CSA), leading to a strong variation of the three principal tensor components (δ11, δ22, δ33), unraveling the origin of the experimental 29Si-NMR chemical shift (δiso) differences along the given series. Thus, the Si20 cage is a useful template to further evaluate different environments for encapsulating atomic species.

Projection in genomic analysis: A theoretical basis to rationalize tensor decomposition and principal component analysis as feature selection tools.

Identifying differentially expressed genes is difficult because of the small number of available samples compared with the large number of genes. Conventional gene selection methods employing statistical tests have the critical problem of heavy dependence of P-values on sample size. Although the recently proposed principal component analysis (PCA) and tensor decomposition (TD)-based unsupervised feature extraction (FE) has often outperformed these statistical test-based methods, the reason why they worked so well is unclear. In this study, we aim to understand this reason in the context of projection pursuit (PP) that was proposed a long time ago to solve the problem of dimensions; we can relate the space spanned by singular value vectors with that spanned by the optimal cluster centroids obtained from K-means. Thus, the success of PCA- and TD-based unsupervised FE can be understood by this equivalence. In addition to this, empirical threshold adjusted P-values of 0.01 assuming the null hypothesis that singular value vectors attributed to genes obey the Gaussian distribution empirically corresponds to threshold-adjusted P-values of 0.1 when the null distribution is generated by gene order shuffling. For this purpose, we newly applied PP to the three data sets to which PCA and TD based unsupervised FE were previously applied; these data sets treated two topics, biomarker identification for kidney cancers (the first two) and the drug discovery for COVID-19 (the thrid one). Then we found the coincidence between PP and PCA or TD based unsupervised FE is pretty well. Shuffling procedures described above are also successfully applied to these three data sets. These findings thus rationalize the success of PCA- and TD-based unsupervised FE for the first time.

Quadraxial metamaterial.

We study the dispersion of electromagnetic waves in a spatially dispersive metamaterial with Lorentz-like dependence of principal permittivity tensor components on the respective components of the wave vector performing the analysis of isofrequency contours. The considered permittivity tensor describes a triple non-connected wire medium. It is demonstrated that the metamaterial has four optic axes in the frequency range below the artificial plasma frequency. The directions of the optical axes do not depend on frequency and coincide with the diagonals of quadrants. The conical refraction effect is observed for all four optic axes.

Facial age estimation using tensor based subspace learning and deep random forests

Recently, the estimation of facial age has attracted much attention. This letter extends and improves a recently developed method (Guehairia et al., 2020) for fusing multiple deep facial features for age estimation. This method was based on deep random forests. We propose a new pipeline that integrates tensor-based subspace learning before applying DRFs. Deep face features of a training set are represented as a 3D tensor. Multi-linear Whitened Principal Component (MWPCA) and Tensor Exponential Discriminant (TEDA) are used to extract the most discriminative information. The tensor subspace features are then fed into DRFs to predict age. Experiments conducted on five public face databases show that our method can compete with many state-of-the-art methods.

Hyperspectral denoising based on the principal component low-rank tensor decomposition

Abstract Due to the characteristics of hyperspectral images (HSIs), such as their high spectral resolution and multiple continuous narrow bands, HSI technology has become widely used in fields such as target recognition, environmental detection, and agroforestry detection. HSIs are subject, for various reasons, to noise in the processes of data acquisition and transmission. Therefore, the denoising of HSIs is very necessary and important. In this article, according to the characteristics of HSIs, an HSI denoising model combining principal component analysis (PCA) and CANDECOMP/PARAFAC decomposition (CP decomposition) is proposed, which is called PCA-TensorDecomp. First, we use PCA to reduce the dimension of HSI signals by obtaining the first K principal components and get the principal composite components. The low-rank part corresponding to the first K principal components is considered the characteristic signal. Then, low-rank CP decomposition is carried out, to denoise the first principal components and the remaining minor components, the secondary composite components, which contain a large amount of noise. Finally, the inverse PCA is then used to restore the HSIs denoised, such that the effect of comprehensive denoising is achieved. To test the effectiveness of the improved algorithm introduced in this article, we compare it with several methods on simulated and real hyperspectral data. The results of the analysis herein indicate that the proposed algorithm possesses a good denoising effect.

Astrophysically viable Kerr-like spacetime

Based on carefully selected physical requirements, we develop and analyze a rotating regular black hole with asymptotically Minkowski core. This Kerr-like geometry possesses the full ``Killing tower'' of nontrivial Killing tensor, Killing-Yano tensor, and principal tensor. The Hamilton-Jacobi equation, Klein-Gordon equation, and Maxwell's equations are separable. Energy-condition violating physics is pushed into an arbitrarily small region in the deep core. Extraction of astrophysical observables falsifiable/verifiable by the observational community is straightforward.

Slowly rotating black holes with exact Killing tensor symmetries

We present a novel family of slowly rotating black hole solutions in four, and higher dimensions, that extend the well known Lense-Thirring spacetime and solve the field equations to linear order in rotation parameter. As "exact metrics" in their own right, the new (non-vacuum) spacetimes feature the following two remarkable properties: i) near the black hole horizon they can be cast in the, manifestly regular, Painlev\'e-Gullstrand form and ii) they admit exact Killing tensor symmetries. We show these symmetries are inherited from the principal Killing-Yano tensor of the exact rotating black hole geometry in the slow rotation limit. This provides a missing link as to how the exact hidden symmetries emerge as rotation is switched on. Remarkably, in higher dimensions the novel generalized Lense-Thirring spacetimes feature a rapidly growing number of exact irreducible rank-2, as well as higher-rank, Killing tensors -- giving a first example of a physical spacetime with more hidden than explicit symmetries.

The eye of the storm: a regular Kerr black hole

We analyse in some detail a highly tractable non-singular modification of the Kerr geometry, dubbed the “eye of the storm” — a rotating regular black hole with an asymptotically Minkowski core. This is achieved by “exponentially suppressing” the mass parameter in the Kerr spacetime: m → m e-ℓ/r . The single suppression parameter ℓ quantifies the deviation from the usual Kerr spacetime. Some of the classical energy conditions are globally satisfied, whilst certain choices for ℓ force any energy-condition-violating physics into the deep core. The geometry possesses the full “Killing tower” of principal tensor, Killing-Yano tensor, and nontrivial Killing tensor, with associated Carter constant; hence the Hamilton-Jacobi equations are separable, and the geodesics integrable. Both the Klein-Gordon equation and Maxwell's equations are also separable on this candidate spacetime. The tightly controlled deviation from Kerr renders the physics extraordinarily tractable when compared with analogous candidates in the literature. This spacetime will be amenable to straightforward extraction of astrophysical observables falsifiable/ verifiable by the experimental community.

Quasinormal modes of Proca fields in a Schwarzschild-AdS spacetime

We present new results concerning the Proca massive vector field in a Schwarzschild-AdS black hole geometry. We provide a first principles analysis of Proca vector fields in this geometry using both the vector spherical harmonic (VSH) separation method and the Frolov-Krtou\v{s}-Kubiz\v{n}\'{a}k-Santos (FKKS) method that separates the relevant equations in spinning geometries. The analysis in the VSH method shows, on one hand, that it is arduous to separate the scalar-type from the vector-type polarizations of the electric sector of the Proca field, and on the other hand, it displays clearly the electric and the magnetic mode sectors. The analysis in the FKKS method is performed by taking the nonrotating limit of the Kerr-AdS spacetime, and shows that the ansatz decouples the polarizations in the electric mode sector even in the nonrotating limit. On the other hand, it captures only two of the three possible polarizations, the magnetic mode sector is missing. The reason for the absence of this polarization is related to the degeneracy of the principal tensor in static spherical symmetric spacetimes. The degrees of freedom and quasinormal modes in both separation methods of the Proca field are found. The frequencies of the quasinormal modes are also computed. For the electric mode sector in the VSH method the frequencies are found through an extension, which substitutes number coefficients by matrix coefficients, of the Horowitz-Hubeny numerical procedure, whereas for the magnetic mode sector in the VSH method and the electric sector of the FKKS method it is shown that a direct use of the procedure can be made. The values of the quasinormal mode frequencies obtained for each method are compared and showed to be in good agreement with each other. This further supports the analytical approaches presented here for the behavior of the Proca field in a Schwarzschild-AdS black hole background.

Fault diagnosis method of vehicle engine via HOSVD–HOALS hybrid algorithm-based multi-dimensional feature extraction

Engine faults, which are difficult to be accurately diagnosed, seriously affect the normal running of vehicles. To solve this problem, a novel fault diagnosis method via HOSVD–HOALS​ hybrid algorithm-based multi-dimensional feature extraction was proposed for a vehicle engine. First, multiple source signals-based engine status samples with the form of third-order tensors were constructed to retain the correlation among sample data. Then, by analyzing the high-dimensional characteristics of the constructed tensor samples, the Tucker decomposition was employed to realize the dimension reduction of the samples. Simultaneously, combining the high order singular value decomposition (HOSVD) and high order alternation least square (HOALS), a hybrid algorithm was proposed to solve the optimal low-dimensional core tensors and features of the constructed tensor samples. Finally, based on the extracted features, the fault pattern of the vehicle engine was recognized by using the derived tensor-based K-nearest neighbor (K-NN) and fuzzy C-mean (FCM) algorithms, respectively. Research results show that the average accuracy of the fault diagnosis via HOSVD–HOALS hybrid algorithm-based multi-dimensional feature extraction can reach 96.25% which is the highest compared with that via the principal component analysis (PCA), HOSVD, HOALS, and direct tensor sample method, respectively, and the computation time required for the fault diagnosis is greatly shortened, which can provide theoretical and technical support for the fault diagnosis of vehicle engines.

Equatorial Electronic Structure in the Uranyl Ion: Cs2UO2Cl4 and Cs2UO2Br4.

Electric field gradient (EFG) tensors in the equatorial plane of the linear UO22+ ion have been measured by nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR) experiments and computed by relativistic Kohn-Sham methods with and without environment embedding for Cs2UO2Cl4 and Cs2UO2Br4. This approach expands the possibilities for probing the electronic structure in uranyl complexes beyond the strongly covalent U-O bonds. The combined analyses find that one of the two largest principal EFG tensor components at the halogen sites points along the U-X bond (X = Cl, Br), and the second is parallel to the UO22+ ion; in Cs2UO2Cl4, the components are nearly equal in magnitude, whereas in Cs2UO2Br4, due to short-range bromide-cesium interactions, the equatorial component is dominant for one pair of Br sites and the axial component is larger for the second pair. The directions and relative magnitudes of the field gradient principal axes are found to be sensitive to the σ and π electron donation by the ligands and the model of the environment. Chlorine-35 NQR spectra of 235U-depleted and 235U-enriched Cs2UO2Cl4 exhibited no uranium-isotope-dependent shift, but the resonance of the depleted sample displayed a 58% broader line width.