Rank Tensor Research Articles

Slice-reducible conformal Killing tensors, photon surfaces, and shadows

We generalize our recent method for constructing Killing tensors of the second rank to conformal Killing tensors. The method is intended for foliated spacetimes of arbitrary dimension $m$, which have a set of conformal Killing vectors. It applies to foliations of a more general structure than in previous literature. The primary idea is to start with a Killing tensor reducible in the foliation slices and lift it to the whole manifold using a system of differential equations. The integrability conditions permitting this lift are derived, and a constructive lifting procedure is presented. The resulting conformal Killing tensor may be irreducible. It is shown that subdomains of foliation slices suitable for the method are fundamental photon surfaces if some additional photon region inequality is satisfied. Thus our procedure also opens the way to obtain a simple general analytical expression for the boundary of the gravitational shadow. We apply this technique to electrovacuum, and $\mathcal{N}=2$, 4, 8 supergravity black holes, providing a new easy way to establish the existence of exact and conformal Killing tensors.

Polarizability of Microparticles in Relativistic Field Theory

The article determine a relativistic tensor of the second rank containing the vectors of the electric and magnetic polarization of the medium, based on the Maxwell equations and the determination of the charge density and polarization current of a structural microparticle. Using this tensor and the electromagnetic field tensor, the relativistic lagrangian of the interaction an electromagnetic field with a structural microparticle is obtained, taking into account the polarization of its structural elements. Based on the induced moments, the relativistic lagrangian of the interaction an electromagnetic field with a structural microparticle is constructed with considering the electric and magnetic polarizabilities.

Accelerating Dynamic MRI Reconstruction Using Adaptive Sequentially Truncated Higher-Order Singular Value Decomposition.

Dynamic magnetic resonance imaging (dMRI) plays an important role in cardiac perfusion and functional clinical exams. However, further applications are limited by the speed of data acquisition. A low-rank plus sparse decomposition approach is often introduced for reconstructing dynamic magnetic resonance imaging (dMRI) from highly under-sampling K-space data. In this paper, the reconstruction problem of DMR is transformed into a low-rank tensor plus sparse tensor recovery problem. A sequentially truncated higher-order singular value decomposition method is proposed to quickly approximate the low-rank tensor space structure and learn sparse components by adding a tensor kernel norm to the low-rank tensor and a l1 norm to the sparse tensor to constrain the two parts at the same time. The optimization problem is solved by using the iterative soft-thresholding algorithm; therefore, under the premise of ensuring the accuracy of the data, the amount of computation can be effectively reduced. Compared with the state-of-the-art methods, the experimental results show that the proposed method can achieve better performance in terms of reconstruction speed and reconstruction quality on 3D and 4D dMRI datasets. The multidimensional MRI time series is represented by the tensor tool and decomposed into low rank tensor terms and sparse tensor terms. The low rank spatial structure is captured by the adaptive ST-HOSVD for fast approximation and the sparse component is constrained efficiently with a sparsity transform and l1 norm. The optimization problem is solved by an iterative soft-thresholding algorithm. Through extensive 3D and 4D dMRI experiments, it is demonstrated that our method can achieve superior reconstruction performance and efficiency compared with the other three state-of-theart methods reported in the literature.

BATUDE: Budget-Aware Neural Network Compression Based on Tucker Decomposition

Model compression is very important for the efficient deployment of deep neural network (DNN) models on resource-constrained devices. Among various model compression approaches, high-order tensor decomposition is particularly attractive and useful because the decomposed model is very small and fully structured. For this category of approaches, tensor ranks are the most important hyper-parameters that directly determine the architecture and task performance of the compressed DNN models. However, as an NP-hard problem, selecting optimal tensor ranks under the desired budget is very challenging and the state-of-the-art studies suffer from unsatisfied compression performance and timing-consuming search procedures. To systematically address this fundamental problem, in this paper we propose BATUDE, a Budget-Aware TUcker DEcomposition-based compression approach that can efficiently calculate optimal tensor ranks via one-shot training. By integrating the rank selecting procedure to the DNN training process with a specified compression budget, the tensor ranks of the DNN models are learned from the data and thereby bringing very significant improvement on both compression ratio and classification accuracy for the compressed models. The experimental results on ImageNet dataset show that our method enjoys 0.33% top-5 higher accuracy with 2.52X less computational cost as compared to the uncompressed ResNet-18 model. For ResNet-50, the proposed approach enables 0.37% and 0.55% top-5 accuracy increase with 2.97X and 2.04X computational cost reduction, respectively, over the uncompressed model.

On Approximation Algorithm for Orthogonal Low-Rank Tensor Approximation

This work studies solution methods for approximating a given tensor by a sum of R rank-1 tensors with one or more of the latent factors being orthonormal. Such a problem arises from applications such as image processing, joint singular value decomposition, and independent component analysis. Most existing algorithms are of the iterative type, while algorithms of the approximation type are limited. By exploring the multilinearity and orthogonality of the problem, we introduce an approximation algorithm in this work. Depending on the computation of several key subproblems, the proposed approximation algorithm can be either deterministic or randomized. The approximation lower bound is established, both in the deterministic and the expected senses. The approximation ratio depends on the size of the tensor, the number of rank-1 terms, and is independent of the problem data. When reduced to the rank-1 approximation case, the approximation bound coincides with those in the literature. Moreover, the presented results fill a gap left in Yang (SIAM J Matrix Anal Appl 41:1797–1825, 2020), where the approximation bound of that approximation algorithm was established when there is only one orthonormal factor. Numerical studies show the usefulness of the proposed algorithm.

Multi-Mode Tensor Space Clustering Based on Low-Tensor-Rank Representation

Traditional subspace clustering aims to cluster data lying in a union of linear subspaces. The vectorization of high-dimensional data to 1-D vectors to perform clustering ignores much of the structure intrinsic to such data. To preserve said structure, in this work we exploit clustering in a high-order tensor space rather than a vector space. We develop a novel low-tensor-rank representation (LTRR) for unfolded matrices of tensor data lying in a low-rank tensor space. The representation coefficient matrix of an unfolding matrix is tensorized to a 3-order tensor, and the low-tensor-rank constraint is imposed on the transformed coefficient tensor to exploit the self-expressiveness property. Then, inspired by the multi-view clustering framework, we develop a multi-mode tensor space clustering algorithm (MMTSC) that can deal with tensor space clustering with or without missing entries. The tensor is unfolded along each mode, and the coefficient matrices are obtained for each unfolded matrix. The low tensor rank constraint is imposed on a tensor combined from transformed coefficient tensors of each mode, such that the proposed method can simultaneously capture the low rank property for the data within each tensor space and maintain cluster consistency across different modes. Experimental results demonstrate that the proposed MMTSC algorithm can outperform existing clustering algorithms in many cases.

On the Geometry in the Large of Einstein-like Manifolds

Gray has presented the invariant orthogonal irreducible decomposition of the space of all covariant tensors of rank 3, obeying only the identities of the gradient of the Ricci tensor. This decomposition introduced the seven classes of Einstein-like manifolds, the Ricci tensors of which fulfill the defining condition of each subspace. The large-scale geometry of such manifolds has been studied by many geometers using the classical Bochner technique. However, the scope of this method is limited to compact Riemannian manifolds. In the present paper, we prove several Liouville-type theorems for certain classes of Einstein-like complete manifolds. This represents an illustration of the new possibilities of geometric analysis.

Magnetic octupole tensor decomposition and second-order magnetoelectric effect

We discuss the second-order magnetoelectric effect, in which a quadratic or bilinear electric field induces a linear magnetization, in terms of the ferroic ordering of magnetic octupoles. We present the decomposition of a general rank-3 tensor into its irreducible spherical tensors, then reduce the decomposition to the specific case of the magnetic octupole tensor, Mijk=∫μi(r)rjrkd3r. We use first-principles density functional theory to compute the size of the local magnetic multipoles on the chromium ions in the prototypical magnetoelectric Cr2O3, and show that, in addition to the well established local magnetic dipoles and magnetoelectric multipoles, the magnetic octupoles are non-zero. The magnetic octupoles in Cr2O3 have an anti-ferroic arrangement, so the net second-order magnetoelectric response is zero. Therefore they form a kind of hidden order, which could be revealed as a linear magnetic (antiferromagnetic) response to a non-zone-center (uniform) quadratic electric field.

A sparse and spectral smooth regularized low-rank tensor decomposition method for hyperspectral target detection

ABSTRACT In recent years, hyperspectral target detection methods have been widely used in military or civilian fields. Many high-performance hyperspectral target detection methods have been proposed, and tensor decomposition-based methods have attracted the attention of researchers. However, the non-uniqueness of tensor rank and the interference of noise will reduce the effect of target detection. In order to solve these problems, a sparse and spectral smooth regularized low-rank tensor decomposition method for hyperspectral target detection is proposed in this paper. The proposed method adds the low-rank regularization to the factor matrix of the tensor decomposition framework, which can reduce the adverse effect of information redundancy on the detection effect of the algorithm. Furthermore, the spectral smooth regularization is added to the spectral factor matrix to suppress noise and the sparse regularization is added to the core tensor to solve the problems of non-uniqueness of the Tucker tensor decomposition, which can improve the effect of target detection. On this basis, it simplifies the calculation process and reduces the complexity of the algorithm by using the conjugate gradient algorithm and the tensor Kronecker product. In this paper, the algorithm is tested on four real hyperspectral data sets and compared with six state-of-the-art algorithms. The experimental results show that targets detected by the proposed method are more obvious and the background is more pure.

Non-minimum tensor rank Gabidulin codes

The tensor rank of some Gabidulin codes of small dimension is investigated. In particular, we determine the tensor rank of any rank metric code equivalent to an 8-dimensional Fq-linear generalized Gabidulin code in Fq4×4. This shows that such a code is never minimum tensor rank. In this way, we detect the first infinite family of Gabidulin codes which are not minimum tensor rank.

Analysis method for 3D power spectrum of projected tensor fields with fast estimator and window convolution modeling: An application to intrinsic alignments

Rank-2 tensor fields of large-scale structure, e.g. a tensor field inferred from shapes of galaxies, open up a window to directly access 2-scalar, 2-vector and 2-tensor modes, where the scalar fields can be measured independently from the standard density field that is traced by distribution of galaxies. Here we develop an estimator of the multipole moments of power spectra for the three-dimensional tensor field, taking into account the projection onto plane perpendicular to the line-of-sight direction. To do this, we find that a convenient representation of the power spectrum multipoles can be obtained by the use of the associated Legendre polynomials in the form which allows for the fast Fourier transform estimations under the local plane-parallel (LPP) approximation. The formulation also allows us to obtain the Hankel transforms to connect the two-point statistics in Fourier and configuration space, which are needed to derive theoretical templates of the power spectrum including convolution of a survey window. To validate our estimators, we use the simulation data of the projected tidal field assuming a survey window that mimics the BOSS-like survey footprint. We show that the LPP estimators fairly well recover the multipole moments that are inferred from the global plane-parallel approximation. We find that the survey window causes a more significant change in the multipole moments of projected tensor power spectrum at $k\lesssim 0.1\,h{\rm Mpc}^{-1}$ from the input power spectrum, than in the density power spectrum. Nevertheless, our method to compute the theory template including the survey window effects successfully reproduces the window-convolved multipole moments measured from the simulations. The analysis method presented here paves the way for a cosmological analysis using three-dimensional tensor-type tracers of large-scale structure for current and future surveys.

Compressed Channel Estimation for IRS-Assisted Millimeter Wave OFDM Systems: A Low-Rank Tensor Decomposition-Based Approach

We consider the problem of downlink channel estimation for intelligent reflecting surface (IRS)-assisted millimeter Wave (mmWave) orthogonal frequency division multiplexing (OFDM) systems. By exploring the inherent sparse scattering characteristics of mmWave channels, we show that the received signals can be expressed as a low-rank third-order tensor that admits a tensor rank decomposition, also known as canonical polyadic decomposition (CPD). A structured CPD-based method is then developed to estimate the channel parameters. Our analysis reveals that the training overhead required by our proposed method is as low as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {O}(U^{2})$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${U}$ </tex-math></inline-formula> denotes the sparsity of the cascade channel. Simulation results are provided to illustrate the efficiency of the proposed method.

Truncated tensor Schatten [formula omitted]-norm based approach for spatiotemporal traffic data imputation with complicated missing patterns

Rapid advances in sensor, wireless communication, cloud computing and data science have brought unprecedented amount of data to assist transportation engineers and researchers in making better decisions. However, traffic data in reality often has corrupted or incomplete values due to detector and communication malfunctions. Data imputation is thus required to ensure the effectiveness of downstream data-driven applications. To this end, numerous tensor-based methods treating the imputation problem as the low-rank tensor completion (LRTC) have been attempted in previous works. To tackle rank minimization, which is at the core of the LRTC, most of aforementioned methods utilize the tensor nuclear norm (NN) as a convex surrogate for the minimization. However, the over-relaxation issue in NN refrains it from desirable performance in practice. In this paper, we define an innovative nonconvex truncated Schatten p-norm for tensors (TSpN) to approximate tensor rank and impute missing spatiotemporal traffic data under the LRTC framework. We model traffic data into a third-order tensor structure of time intervals×locations (sensors)×days and introduce four complicated missing patterns, including random missing and three fiber-like missing cases according to the tensor mode-n fibers. Despite nonconvexity of the objective function in our model, we derive the global optimal solutions by integrating the alternating direction method of multipliers (ADMM) with generalized soft-thresholding (GST). In addition, we design a truncation rate decay strategy to deal with varying missing rate scenarios. Comprehensive experiments are finally conducted using four different types of real-world spatiotemporal traffic datasets, which demonstrate that the proposed LRTC-TSpN method performs well under various missing cases even with high rates of data loss, meanwhile outperforming other state-of-the-art tensor-based imputation models in almost all scenarios. In general, the performance of the proposed method achieves up to a 52% improvement in mean absolute error (MAE) compared to baseline methods even when the missing ratio is as high as 90%. Moreover, both theoretical and empirical evidences indicate that our method has robust and efficient convergence performances.

The Low-Rank Approximation of Fourth-Order Partial-Symmetric and Conjugate Partial-Symmetric Tensor

We present an orthogonal matrix outer product decomposition for the fourth-order conjugate partial-symmetric (CPS) tensor and show that the greedy successive rank-one approximation (SROA) algorithm can recover this decomposition exactly. Based on this matrix decomposition, the CP rank of CPS tensor can be bounded by the matrix rank, which can be applied to low-rank tensor completion. Additionally, we give the rank-one equivalence property for the CPS tensor based on the SVD of matrix, which can be applied to the rank-one approximation for CPS tensors.

Existence and Emergent Dynamics of Quadratically Separable States to the Lohe Tensor Model

A tensor is a multidimensional array of complex numbers, and the Lohe tensor model is an aggregation model on the space of tensors with the same rank and size. It incorporates previously well-studied aggregation models on the space of low-rank tensors such as the Kuramoto model, Lohe sphere, and matrix models as special cases. Due to its structural complexities in cubic interactions for the Lohe tensor model, explicit construction of solutions with specific structures looks daunting. Recently, we obtained completely separable states by associating rank-1 tensors. In this paper, we further investigate another type of solutions, namely “quadratically separable states" consisting of tensor products of matrices and their component rank-2 tensors are solutions to the double matrix model whose emergent dynamics can be studied using the same methodology of the Lohe matrix model.

TOP-ALCM: A novel video analysis method for violence detection in crowded scenes

Despite the Video Violence Detection (VVD) plays a critical role in video surveillance, it is not trivial in crowded scenes due to the complexity and diversity of violence. Generally, the most typical features of violence are its drastic, disordered, and chaotic motion in contrast to non-violence. To capture these features for violence analysis in a video clip, we propose a novel Angle-level Co-occurrence Matrix (ALCM). Given a video volume, we treat it as a tensor of rank 3, which consists of a bound of fibers in one plane. ALCM records the co-occurrence of two specific quantized angle levels between fibers with their neighbors, which is the distribution of the co-occurrence of fiber pairs with specific similarities in one plane of the tensor of rank 3. To completely characterize the violence in the volume, we compute three ALCMs for three orthogonal planes to form a TOP-ALCM, respectively. We also propose both conventional and deep-learning-based VVD frameworks, in which the former one leverages the features such as entropy, homogeneity, and energy computed from TOP-ALCM for classification, while the latter one directly uses CNN to classify the TOP-ALCMs. Experimental results analysis demonstrates that the proposed TOP-ALCM outperforms the state-of-the-art methods for VVD.

THE FEATURES OF LAMINAR FLUID MOTION IN THE IMPELLER OF THE CENTRIFUGAL POROUS PUMP

A method for solving the problem of the motion of an incompressible fluid in the impeller of a porous pump is considered. Within the framework of the proposed method, theoretical studies were further developed, aimed at improving the analytical dependencies, which make it possible to evaluate the changes in pressure and kinematic parameters during the laminar motion of fluid in the cavity of a porous body. It is assumed that the porous body has anisotropic properties. The accepted design scheme of fluid movement in the impeller does not take into account the energy loss when the fluid turns from the axial direction to the radial direction, as well as possible losses due to vortex-like phenomena in the inlet section. The system of equations describing the motion of a liquid in a porous wheel is written in a polar coordinate system under the assumption that there is no change in the parameters of the liquid with respect to the angle of rotation. In the laminar mode of motion, the filtration characteristics are expressed as symmetric tensors of the second rank. Equations are obtained for the static pressure of the liquid and the filtration rate in the porous impeller of a centrifugal pump. The results obtained by calculation, as well as previously conducted experimental studies, indicate that they are accurate enough for practice.

Reduction of general one-loop integrals using auxiliary vector

As a key method to deal with loop integrals, Integration-By-Parts (IBP) method can be used to do reduction as well as establish the differential equations for master integrals. However, when talking about tensor reduction, the Passarino-Veltman (PV) reduction method is also widely used for one-loop integrals. Recently, we have proposed an improved PV reduction method, i.e., the PV reduction method with auxiliary vector R, which can easily give analytical reduction results for any tensor rank. However, our results are only for integrals with propagators with power one. In this paper, we generalize our method to one-loop integrals with general tensor structures and propagators with general powers. Our ideas are simple. We solve the generalised reduction problem by combining differentiation over masses and proper limit of reduction with power-one propagators. Finally, we demonstrate our method with several examples. With the result in this paper, we have shown that our improved PV-reduction method with auxiliary vector is a self-completed reduction method for one-loop integrals.

Learning with tree tensor networks: Complexity estimates and model selection

Tree tensor networks, or tree-based tensor formats, are prominent model classes for the approximation of high-dimensional functions in computational and data science. They correspond to sum-product neural networks with a sparse connectivity associated with a dimension tree and widths given by a tuple of tensor ranks. The approximation power of these models has been proved to be (near to) optimal for classical smoothness classes. However, in an empirical risk minimization framework with a limited number of observations, the dimension tree and ranks should be selected carefully to balance estimation and approximation errors. We propose and analyze a complexity-based model selection method for tree tensor networks in an empirical risk minimization framework and we analyze its performance over a wide range of smoothness classes. Given a family of model classes associated with different trees, ranks, tensor product feature spaces and sparsity patterns for sparse tensor networks, a model is selected (\`a la Barron, Birg\'e, Massart) by minimizing a penalized empirical risk, with a penalty depending on the complexity of the model class and derived from estimates of the metric entropy of tree tensor networks. This choice of penalty yields a risk bound for the selected predictor. In a least-squares setting, after deriving fast rates of convergence of the risk, we show that our strategy is (near to) minimax adaptive to a wide range of smoothness classes including Sobolev or Besov spaces (with isotropic, anisotropic or mixed dominating smoothness) and analytic functions. We discuss the role of sparsity of the tensor network for obtaining optimal performance in several regimes. In practice, the amplitude of the penalty is calibrated with a slope heuristics method. Numerical experiments in a least-squares regression setting illustrate the performance of the strategy.

Generalizing eleven-dimensional supergravity

We develop a procedure to reproduce the ten-dimensional generalized supergravity equations from T-duality covariant equations, that facilitates generalization to U-duality covariant formulations of eleven-dimensional supergravity. The latter leads to a modification of the eleven-dimensional supergravity equations with terms that contain a rank-2 tensor field $J^{mn}$ which is the eleven-dimensional analog of the non-unimodularity Killing vector $I^m$ in ten dimensions.