Tensor Coefficients Research Articles

Light propagation in 2PN approximation in the monopole and quadrupole field of a body at rest: Initial value problem

The light trajectory in the gravitational field of one body at rest with monopole and quadrupole structure is determined in the second post-Newtonian (2PN) approximation. The terms in the geodesic equation for light rays are separated into time-independent tensorial coefficients and four kind of time-dependent scalar functions. Accordingly, the first and second integration of geodesic equation can be reduced in each case to only four kind of scalar master integrals. These integrals can be solved in closed form by recurrence relations. The 2PN terms of monopole and quadrupole contribute less than $1$ nano-arcsecond to the total light deflection. There are, however, enhanced terms in the 2PN light deflection, both in case of monopole and quadrupole. These enhanced 2PN terms are caused by the use of an impact vector which is indispensable for modeling of real astrometric measurements. In case of grazing light rays at Jupiter and Saturn, the enhanced 2PN terms, caused by the quadrupole structure of the body, amount up to 0.95 micro-arcseconds and 0.29 micro-arcseconds, respectively. Thus, the 2PN quadrupole terms are relevant for high-precision astrometry on the sub-micro-arcsecond scale of accuracy.

Regularized high dimension low tubal-rank tensor regression

Tensor regression models are of emerging interest in diverse fields of social and behavioral sciences, including neuroimaging analysis, neural networks, image processing and so on. Recent theoretical advancements of tensor decomposition have facilitated significant development of various tensor regression models. The focus of most of the available literature has been on the Canonical Polyadic (CP) decomposition and its variants for the regression coefficient tensor. A CP decomposed coefficient tensor enables estimation with relatively small sample size, but it may not always capture the underlying complex structure in the data. In this work, we leverage the recently developed concept of tubal rank and develop a tensor regression model, wherein the coefficient tensor is decomposed into two components: a low tubal rank tensor and a structured sparse one. We first address the issue of identifiability of the two components comprising the coefficient tensor and subsequently develop a fast and scalable Alternating Minimization algorithm to solve the convex regularized program. Further, we provide finite sample error bounds under high dimensional scaling for the model parameters. The performance of the model is assessed on synthetic data and is also used in an application involving data from an intelligent tutoring platform.

Improved estimation in tensor regression with multiple change-points

In this paper, we consider an estimation problem about the tensor coefficient in a tensor regression model with multiple and unknown change-points. We generalize some recent findings in five ways. First, the problem studied is more general than the one in context of a matrix parameter with multiple change-points. Second, we develop asymptotic results of the tensor estimators in the context of a tensor regression with unknown change-points. Third, we construct a class of shrinkage tensor estimators that encompasses the unrestricted estimator (UE) and the restricted estimator (RE). Fourth, we generalize some identities which are crucial in deriving the asymptotic distributional risk (ADR) of the tensor estimators. Fifth, we show that the proposed shrinkage estimators perform better than the UE. The additional novelty of the established results consists in the fact that the dependence structure of the errors is as weak as that of an L2-mixingale. Finally, the theoretical results are corroborated by the simulation findings and our methods are applied to analyse MRI and fMRI datasets.

Hyperspectral Image Mixed Denoising Using Difference Continuity-Regularized Nonlocal Tensor Subspace Low-Rank Learning

With the rapid advancement of spectrometers, the imaging range of the electromagnetic spectrum starts growing narrower. The reduction of electromagnetic wave energy received in a single wavelength range leads more complex noise into the generated hyperspectral image (HSI), thus causing a severe cripple in the accuracy of subsequent applications. The requirement for the HSI mixed denoising algorithm’s accuracy is further lifted. To address this challenge, in this letter, we propose a novel difference continuity-regularized nonlocal tensor subspace low-rank learning (named DNTSLR) method for HSI mixed denoising. Technically, the original high-dimensional HSI data was first projected into a low-dimensional subspace spanned by a spectral difference continuous basis instead of an orthogonal basis, so the data continuity of the restored HSI spectrum and tensor low-rankness was guaranteed. Then, a cube matching strategy was employed to stack the nonlocal tensor patches from the projected coefficient tensor, and a shrinkage algorithm was used to approximate the low-rank coefficient tensor. Eventually, the subspace low-rank learning algorithm was designed to alternately separate the noise tensor and restore the latent clean low-rank HSI tensor. Extensive experiments on multiple open datasets validate that the proposed method realizes the state-of-the-art denoising accuracy for HSI.

Hyperspectral Image Restoration via Subspace-Based Nonlocal Low-Rank Tensor Approximation

In this letter, we present a subspace-based nonlocal low-rank tensor approximation framework (SNLRTA) for hyperspectral image (HSI) restoration. The proposed method consists of a subspace learning method to achieve an accurate subspace characterization of HSI and a nonlocal low-rank tensor approximation to take spatial nonlocal self-similarity into consideration. Specifically, the HSI first exploits residual statistics on median filtered image to estimate a robust subspace. Laplacian scale mixture (LSM) modeling is then investigated to model tensor coefficients from overlapping cubes in low-rank subspace. Both the hidden scale parameters and the sparse coefficients therein are adaptively shrink, characterizing the sparsity of similar patches. Meanwhile, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula> data fidelity facilitates the implicit detection of outliers after median filtering. Substantiated by extensive experimental results, the proposed method outperforms several state-of-the-art approaches on mixed noise removal, qualitatively and quantitatively.

An Effective Tensor Regression with Latent Sparse Regularization

As data acquisition technologies advance, longitudinal analysis is facing challenges of exploring complex feature patterns from high-dimensional data and modeling potential temporally lagged effects of features on a response. We propose a tensor-based model to analyze multidimensional data. It simultaneously discovers patterns in features and reveals whether features observed at past time points have impact on current outcomes. The model coefficient, a k-mode tensor, is decomposed into a summation of k tensors of the same dimension. We introduce a so-called latent F-1 norm that can be applied to the coefficient tensor to performed structured selection of features. Specifically, features will be selected along each mode of the tensor. The proposed model takes into account within-subject correlations by employing a tensor-based quadratic inference function. An asymptotic analysis shows that our model can identify true support when the sample size approaches to infinity. To solve the corresponding optimization problem, we develop a linearized block coordinate descent algorithm and prove its convergence for a fixed sample size. Computational results on synthetic datasets and real-life fMRI and EEG datasets demonstrate the superior performance of the proposed approach over existing techniques.

マルチスケール構造体の熱応力問題に対する形状最適設計

This study proposes a shape optimization method to minimize the maximum thermal stress of the microstructures in a multiscale structure caused by thermal loading. Using the homogenization method, the homogenized elastic tensors and the homogenized thermal expansion coefficient tensors for the microstructures are calculated to link the microstructures to the macrostructure. The KS function is used to avoid discontinuities and singularities in the maximum thermal stress. Based on the problem formulation, the shape gradient functions are derived for shape optimization.

Acoustic radiation force and radiation torque beyond particles: Effects of nonspherical shape and Willis coupling.

Acoustophoresis mainly deals with the manipulation of subwavelength scatterers in an incident acoustic field. The geometric details of manipulated particles are often neglected by replacing them with equivalent symmetric geometries such as spheres, spheroids, cylinders, or disks. It has been demonstrated that geometric asymmetry, represented by Willis coupling terms, can strongly affect the scattering of a small object; hence neglecting these terms may miss important force contributions. In this work, we present a generalized formalism of acoustic radiation force and radiation torque based on the polarizability tensor, where Willis coupling terms are included to account for geometric asymmetry. Following Gorkov's approach, the effects of geometric asymmetry are explicitly formulated as additional terms in the radiation force and torque expressions. By breaking the symmetry of a sphere along one axis using intrusion and protrusion, we characterize the changes in the force and torque in terms of partial components, associated with the direct and Willis coupling coefficients of the polarizability tensor. We investigate the cases of standing and traveling plane waves and show how the equilibrium positions and angles are shifted by these additional terms. We show that while the contributions of asymmetry to the force are often negligible for small particles, these terms greatly affect the radiation torque. Our presented theory, providing a way of calculating radiation force and torque directly from polarizability coefficients, shows that it is essential to account for shape of objects undergoing acoustophoretic manipulation, with important implications for applications such as the manipulation of biological cells.

Расчет нелинейно-оптических свойств кристаллов ниобата лития

The coefficients of the second-order nonlinear optical tensor for lithium niobate crystal with a composition close to stoichiometric were calculated. The calculated results show that the contribution of the Li-O group to the optical effects of the second harmonic was greater than the contribution of the Nb-O group. The results also showed that the most efficient frequency conversion occurred along the polar axis of the crystal

The diffusion and anomalous diffusion of charged particles in the plateau regime of toroidal plasma

The diffusions and anomalous diffusions of charged particles in the plateau regime of toroidal plasma with a Maxwellian velocity distribution are studied. The transport theory of plasma is discussed under the toroidal coordinate system and the magnetic field is assumed to be axisymmetric. Based on the diffusion flux obtained in the toroidal plasma, we derive the diffusion coefficient tensor and thermal diffusion coefficient tensor, where two angular diffusion coefficients are the anomalous diffusions, depending on the radial position and having nothing to do with the magnetic field. All the radial transport coefficients and the normal angular coefficients are inversely proportional to square of the gyrofrequency and so to square of the magnetic induction. The numerical analyses show clearly dependences of these complex transport coefficients on the temperature, the magnetic fluid stability safe factor and the effective collision frequency, respectively.

Predefined-time convergent neural networks for solving the time-varying nonsingular multi-linear tensor equations

Equation solving is one of the main subjects in various research fields. In this study, we consider tensor equations with time-varying nonsingular coefficient tensors and right-hand side vectors, which generalize the time-invariant tensor equations and also the time-varying or time-invariant linear equations studied by many researchers. Four complex-valued neural networks termed as ZNN-I, ZNN-II, WsbpPTZNN-I and WsbpPTZNN-II are proposed for solving this problem. We show that, under certain conditions, the four models converge to a solution of the original tensor equations. Moreover, we prove that WsbpPTZNN-I and WsbpPTZNN-II models have predefined-time convergence property, which means that they converge to a true solution in a prescribed time. The upper bounds of the convergence time are also given. For comparison, the gradient-based model, termed as GNN, is also introduced. Through many numerical tests, we demonstrate the validity of our theoretical analysis of these models. The simulation results also illustrate the effectiveness and reliability of our proposed neural network models and show that their convergence performance can be improved by using the predefined-time convergent WsbpPTZNN models.

Random tensors, propagation of randomness, and nonlinear dispersive equations

We introduce the theory of random tensors, which naturally extends the method of random averaging operators in our earlier work (Deng et al. in: Invariant Gibbs measures and global strong solutions for the nonlinear Schrödinger equations in dimension two, arXiv:1910.08492), to study the propagation of randomness under nonlinear dispersive equations. By applying this theory we establish almost-sure local well-posedness for semilinear Schrödinger equations in the full subcritical range relative to the probabilistic scaling (Theorem 1.1). The solution we construct has an explicit expansion in terms of multilinear Gaussians with adapted random tensor coefficients. As a byproduct we also obtain new results concerning regular data and long-time solutions, in particular Theorem 1.6, which provides long-time control for random homogeneous data, demonstrating the highly nontrivial fact that the first energy cascade happens at a much later time than in the deterministic setting. In the random setting, the probabilistic scaling is the natural scaling for dispersive equations, and is different from the natural scaling for parabolic equations. Our theory of random tensors can be viewed as the dispersive counterpart of the existing parabolic theories (regularity structures, para-controlled calculus and renormalization group techniques).

A new FV scheme and fast cell-centered multigrid solver for 3D anisotropic diffusion equations with discontinuous coefficients

In this paper, an efficient cell-centered extrapolation cascadic multigrid (CEXCMG) method is proposed for solving large linear system of equations resulting from finite volume (FV) discretizations of three dimensional (3D) anisotropic diffusion equations with discontinuous coefficients. For cell-centered FV schemes, the values at vertex need to be approximated often by a linear combination of neighboring cell-centered values. In the literature, the weighted coefficients are obtained by solving local linear system of equations which is costly in 3D. One of the novelties of this paper is a new approach for obtaining vertex values by interpolating the cell-centered ones, which avoids solving local linear system of equations even with arbitrary diffusion tensors. Another main novelty of this paper is a new cascadic multigrid solver based on a prolongation operator, the newly developed explicit gradient transfer method, and a splitting extrapolation operator for solving 3D anisotropic diffusion equations with discontinuous coefficients. Numerical experiments are presented to demonstrate the efficiency and robustness of the CEXCMG method in terms of the mesh size and the contrast in the coefficients of the anisotropic diffusion tensor.

Evaluation approach for whole dose distribution in clinical cases using spherical projection and spherical harmonics expansion: spherical coefficient tensor and score method.

Whole dose distribution results from well-conceived treatment plans including patient-specific (location, size and shape of tumor, etc.) and facility-specific (clinical policy and goal, equipment, etc.) information. To evaluate the whole dose distribution efficiently and effectively, we propose a method to apply spherical projection and real spherical harmonics (SH) expansion, thus leading to the expanded coefficients as a rank-2 tensor, SH coefficient tensor, for every patient-specific dose distribution. To verify the feature of this tensor, we introduce Isomap from the manifold learning method and multi-dimensional scaling (MDS). Subsequently, we obtained the MDS distance representing similarity, η, and the SH score, ζ, which is a Frobenius norm of the SH coefficient tensor. These were then validated in the intensity-modulated radiation therapy (IMRT) data sets of: (i) 375 mixing treated regions, (ii) 135 head and neck (HN), and (iii) 132 prostate cases, respectively. The MDS map indicated that the SH coefficient tensor enabled a quantitative feature extraction of whole dose distributions. In particular, the SH score systematically detected irregular cases as the deviation higher than +1.5 standard deviations (SD) from the average case, which matched up with clinically irregular case that required very complicated dose distributions. In summary, the proposed SH coefficient tensor is a useful representation of the whole dose distribution. The SH score from the SH coefficient tensor is a convenient and simple criterion used to characterize the entire dose distributions, which is not dependent on the data set.

Tensor Network Rewriting Strategies for Satisfiability and Counting

We provide a graphical treatment of SAT and #SAT on equal footing. Instances of #SAT can be represented as tensor networks in a standard way. These tensor networks are interpreted by diagrams of the ZH-calculus: a system to reason about tensors over C in terms of diagrams built from simple generators, in which computation may be carried out by transformations of diagrams alone. In general, nodes of ZH diagrams take parameters over C which determine the tensor coefficients; for the standard representation of #SAT instances, the coefficients take the value 0 or 1. Then, by choosing the coefficients of a diagram to range over B, we represent the corresponding instance of SAT. Thus, by interpreting a diagram either over the boolean semiring or the complex numbers, we instantiate either the decision or counting version of the problem. We find that for classes known to be in P, such as 2SAT and #XORSAT, the existence of appropriate rewrite rules allows for efficient simplification of the diagram, producing the solution in polynomial time. In contrast, for classes known to be NP-complete, such as 3SAT, or #P-complete, such as #2SAT, the corresponding rewrite rules introduce hyperedges to the diagrams, in numbers which are not easily bounded above by a polynomial. This diagrammatic approach unifies the diagnosis of the complexity of CSPs and #CSPs and shows promise in aiding tensor network contraction-based algorithms.

A homotopy method for solving multilinear systems with strong completely positive tensors

A basic problem in both pure and applied mathematics is solving various kinds of equations. In recent years, multilinear systems of equations have received a great deal of attention and active research. However, most studies are focused on those multilinear systems with the tensors involved being strong M-tensors. It is well known that many tensors from real problems are non-negative, including strong completely positive tensors. In this paper, we consider the multilinear system of equations whose coefficient tensors are strong completely positive tensors of even order. First we show that this multilinear system has a unique real solution. Secondly, we propose a homotopy approximation method for solving the multilinear system, which has global convergence, and it possesses locally superlinear convergence under certain conditions. Preliminary numerical experiment results show that the proposed algorithm is effective.

Analytic tadpole coefficients of one-loop integrals

One remaining problem of unitarity cut method for one-loop integral reduction is that tadpole coefficients can not be straightforward obtained through this way. In this paper, we reconsider the problem by applying differential operators over an auxiliary vector R. Using differential operators, we establish the corresponding differential equations for tadpole coefficients at the first step. Then using the tensor structure of tadpole coefficients, we transform the differential equations to the recurrence relations for undetermined tensor coefficients. These recurrence relations can be solved easily by iteration and we can obtain analytic expressions of tadpole coefficients for arbitrary one-loop integrals.

Multivariate Temporal Point Process Regression

Point process modeling is gaining increasing attention, as point process type data are emerging in a large variety of scientific applications. In this article, motivated by a neuronal spike trains study, we propose a novel point process regression model, where both the response and the predictor can be a high-dimensional point process. We model the predictor effects through the conditional intensities using a set of basis transferring functions in a convolutional fashion. We organize the corresponding transferring coefficients in the form of a three-way tensor, then impose the low-rank, sparsity, and subgroup structures on this coefficient tensor. These structures help reduce the dimensionality, integrate information across different individual processes, and facilitate the interpretation. We develop a highly scalable optimization algorithm for parameter estimation. We derive the large sample error bound for the recovered coefficient tensor, and establish the subgroup identification consistency, while allowing the dimension of the multivariate point process to diverge. We demonstrate the efficacy of our method through both simulations and a cross-area neuronal spike trains analysis in a sensory cortex study.

Structural tensor-on-tensor regression with interaction effects and its application to a hot rolling process

This paper proposes a method of Structural Tensor-On-Tensosr regression considering the Interaction effects (STOTI). To alleviate the curse of dimensionality and resolve computational challenge, the STOTI method describes the specific structure of the main and interaction effect tensors indicated by the prior knowledge of the data using corresponding regularization terms on their appropriate modes. We designed an ADMM consensus algorithm to estimate these coefficient tensors. Extensive simulations and a real case study of the hot rolling process verified the superiority of the proposed method in terms of estimation and prediction accuracy.

Updates to the one-loop provider NLOX

We describe the 1.2 update to NLOX, a computer program for calculations in high-energy particle physics. New features since the 1.0 release and other changes are described, along with usage documentation.