Covariance Tensor Research Articles

Block structure-based covariance tensor decomposition for group identification in matrix variables

In research fields such as financial market analysis and social network research, understanding variable grouping relationships is fundamental to effective data analysis. This study describes the concept of the covariance tensor and emphasizes its significant role in analyzing matrix variable groupings through block structures. We propose a novel tensor decomposition-based method to exploit these structures for group identification. In addition, we explore the asymptotic properties of our estimators, focusing on the precision of the estimation of the number of groups and the asymptotic convergence of classification error rates to zero. We validate the effectiveness of the method through extensive numerical simulations across diverse data volumes and complexities, affirming its capability in variable grouping.

Fully optimized second-order estimates for the macroscopic behavior and field statistics of particle-reinforced viscoplastic composites

This paper is concerned with the characterization of the macroscopic behavior and statistics for the distribution of the stress and strain-rate fields in composites consisting of random and isotropic suspensions of rigid spherical particles in power-law viscoplastic materials. For this purpose, use is made of the Fully Optimized Second-Order (FOSO) homogenization method (Ponte Castañeda, 2016) in combination with recently developed estimates (Kammer and Ponte Castañeda, 2022) for the macroscopic properties of the associated ‘linear comparison composite’ (LCC). Special attention is devoted to the method’s ability to account for the dependence of the homogenized properties of the nonlinear composite on the Lode angle (third invariant) of the applied loading. It is found that, while, for large particle volume fractions c, the effective flow stress is only weakly dependent on the Lode angle, for dilute volume fractions, the dependence on the Lode angle becomes more pronounced. In the ideally plastic limit, as c tends to zero, the effective yield stress is shown to depend linearly on c for axisymmetric shear, while this dependence becomes weaker with a non-analytic leading-order correction of O(c/lnc) for pure shear loading. This strong dependence on the Lode angle at dilute concentrations is shown to be due to significant differences in the local deformation patterns, which become strongly anisotropic and localize for pure shear conditions, but do not for axisymmetric shear. In turn, the FOSO homogenization method is able to capture the statistical features of these different deformation patterns by providing consistent estimates for the covariance tensor of the strain-rate field fluctuations in the matrix phase, which tend to become more strongly anisotropic for the pure shear case. As c increases, the shear bands are deflected by the randomly dispersed spheres leading to a more isotropic distribution of the stress and strain-rate fields, which is consistent with a weaker Lode angle effect. The estimates can also capture the effect of strong particle interactions, including the existence of a rigidity threshold where the macroscopic flow stress and field fluctuations blow up.

Large-eddy simulation-based reconstruction of turbulence in a neutral boundary layer using spectral-tensor regularization

We propose an efficient method to reconstruct the turbulent flow field in a neutrally stratified atmospheric boundary layer using large-eddy simulation (LES) and a series of lidar measurements. The reconstruction is formulated as a strong four-dimensional variational data assimilation problem, which involves optimizing two competing terms that contribute in the objective functional. The first term is a likelihood term, while the second contains the initial background distribution of turbulent velocity fluctuations and works as a regularization term. However, computing and storing the full background covariance tensor in turbulent flows is time consuming and resource intensive. In the current work, we investigate the possibility of replacing the complex background tensor by simple analytical approximations based on spectral tensors such as the Hunt–Graham–Wilson (HGW) model (Boundary-Layer Meteorol., vol. 85, 1997, pp. 35–52) or the Mann model (J. Fluid Mech., vol. 273, 1994, pp. 141–168). Afterwards, the problem is solved using a quasi-Newton algorithm and preconditioned to enhance the convergence rate. We test the method using virtual lidar measurements collected on a fine reference LES. Results show a super-linear convergence rate of the optimization algorithm to a local minimum and very good agreement between virtual lidar measurements and reconstruction in the scanning region. Furthermore, we demonstrate that incorporating the Saffman energy spectrum ( $E(k) \sim k^2$ where E is the energy spectrum and k is the magnitude of the wavenumber vector) at low wavenumbers into the Mann spectral tensor yields a longer streamwise correlation length, resulting in reduced reconstruction error when compared with the Batchelor spectrum ( $E(k) \sim k^4$ ). Finally, we observe that using the HGW model or Mann model with a Saffman spectrum yields similar results.

Structural-Missing Tensor Completion for Robust DOA Estimation with Sensor Failure

Array sensor failure poses a serious challenge to robust direction-of-arrival (DOA) estimation in complicated environments. Although existing matrix completion methods can successfully recover the damaged signals of an impaired sensor array, they cannot preserve the multi-way signal characteristics as the dimension of arrays expands. In this paper, we propose a structural-missing tensor completion algorithm for robust DOA estimation with uniform rectangular array (URA), which exhibits a high robustness to non-ideal sensor failure conditions. Specifically, the signals received at the impaired URA are represented as a three-dimensional incomplete tensor, which contains whole fibers or slices of missing elements. Due to this structural-missing pattern, the conventional low-rank tensor completion becomes ineffective. To resolve this issue, a spatio-temporal dimension augmentation method is developed to transform the structural-missing tensor signal into a six-dimensional Hankel tensor with dispersed missing elements. The augmented Hankel tensor can then be completed with a low-rank regularization by solving a Hankel tensor nuclear norm minimization problem. As such, the inverse Hankelization on the completed Hankel tensor recovers the tensor signal of an unimpaired URA. Accordingly, a completed covariance tensor can be derived and decomposed for robust DOA estimation. Simulation results verify the effectiveness of the proposed algorithm.

Efficiently Differentiating Agonists and Competitive Antagonists for Weak Allosteric Protein-Ligand Interactions with Linear Response Theory.

What makes an agonist and a competitive antagonist? In this work, we aim to answer this question by performing parallel tempering Monte Carlo simulations on the serotonin type 3A (5-HT3A) receptor. We use linear response theory to predict conformational changes in the 5-HT3A receptor active site after weak perturbations are applied to its allosteric binding sites. A covariance tensor is built from conformational sampling of its apo state, and a harmonic approximation allows us to substitute the calculation of ligand-induced forces with the binding site's displacement vector. Remarkably, our study demonstrates the feasibility of effectively discerning between agonists and competitive antagonists for multiple ligands, requiring computationally expensive calculations only once per protein.

Fast real-valued tensor decomposition framework for parameter estimation in FDA-MIMO radar

Frequency diversity array - Multiple input multiple output (FDA-MIMO) radar has a two-dimensional angle-range dependence due to the existence of certain frequency offset between transmitting elements. For obtaining angle-range estimation, a fast real-valued tensor propagator method (PM) for FDA-MIMO radar is developed. The developed approach is based on the real-valued tensor PM, which not only utilizes the original structural information of multidimensional information to improve estimation accuracy, but also eliminates the process of high-order singular value decomposition (HOSVD) on multidimensional data, greatly reducing computational complexity. Firstly, the unitary transformation technique is employed to convert the constructed tensor into a real-valued tensor. Next, construct a covariance tensor to obtain operator matrices in different directions. Then, a signal subspace is constructed using operator matrices. Finally, the selection matrices and the obtained signal subspace are employed to estimate angle and range information. The proposed algorithm can not only achieve parameter estimation at low snapshots, but also has much lower computational complexity than other algorithms at high snapshots. Therefore, the developed approach greatly reduces computational complexity while ensuring estimation accuracy, which enables it to be applied to massive FDA-MIMO radars. Simulation results confirm the accuracy advantage and high-efficiency of our algorithm.

Realizability-preserving time-stepping for the differential Reynolds stress turbulence models

In the context of incompressible turbulence modelling, differential Reynolds stress models allow a better representation of turbulence anisotropy than classical linear eddy viscosity models and are less expensive than large eddy simulations in terms of computational cost. They rely on a modelled transport equation for the Reynolds stress tensor, which is by definition a covariance tensor and must remain symmetric positive semi-definite. However, the solution of the modelled and discretized transport equation may not preserve these mathematical properties. In the present article, a methodology to time-split Reynolds stress R__ terms in the transport equation is proposed using a decomposition theorem which proves the realizability of R__ provided realizable initial conditions. Realizability-preserving time-steppings are designed for several differential Reynolds stress models. The numerical schemes are presented in a finite volume context but may be used with others space discretization schemes. The realizability, precision and robustness of the numerical scheme is verified through the anisotropy time evolution in homogeneous shear-flow simulations with various initial conditions.

Fourth order tensors and covariance tensors

Fourth order tensors and covariance tensors

Joint range, angle and polarization estimation in polarimetric FDA-MIMO radar based on Tucker tensor decomposition

Frequency diverse array multiple-input multiple-output (FDA-MIMO) radar is an emerging technology to offer range-angle-dependent beampattern. Polarimetric FDA-MIMO radar can sense additional polarization information to improve target identification capability. In this article, we investigate the problem of joint range, angle and polarization parameter estimation in a monostatic polarimetric FDA-MIMO radar with an FDA at transmitter and a cross-dipole array at receiver. Unlike the conventional methods in which the multidimensional data structure is rearranged into vectors or matrices by stacking operation, we propose a Tucker tensor decomposition-based scheme, which can reserve the original data structure and avoid spoiling the inherent characteristics of interest, especially when the number of snapshots is small. The third-order tensor model of the observed data is constructed. Two approaches named as Tucker covariance reconstruction and Tucker signal subspace are presented using the fourth-order covariance tensor decomposition. The Cramér–Rao bound for range, angle and polarization estimation is also provided. Numerical experiments demonstrate the superiorities of the proposed approaches. Specifically, two targets with identical range and close angles are effectively distinguished.

A novel framework for in-vivo diffusion tensor distribution MRI of the human brain

Neural tissue microstructure plays an important role in developmental, physiological and pathophysiological processes. Diffusion tensor distribution (DTD) MRI helps probe subvoxel heterogeneity by describing water diffusion within a voxel using an ensemble of non-exchanging compartments characterized by a probability density function of diffusion tensors. In this study, we provide a new framework for acquiring multiple diffusion encoding (MDE) images and estimating DTD from them in the human brain in vivo. We interfused pulsed field gradients (iPFG) in a single spin echo to generate arbitrary b-tensors of rank one, two, or three without introducing concomitant gradient artifacts. Employing well-defined diffusion encoding parameters we show that iPFG retains salient features of a traditional multiple-PFG (mPFG/MDE) sequence while reducing the echo time and coherence pathway artifacts thereby extending its applications beyond DTD MRI. Our DTD is a maximum entropy tensor-variate normal distribution whose tensor random variables are constrained to be positive definite to ensure their physicality. In each voxel, the second-order mean and fourth-order covariance tensors of the DTD are estimated using a Monte Carlo method that synthesizes micro-diffusion tensors with corresponding size, shape, and orientation distributions to best fit the measured MDE images. From these tensors we obtain the spectrum of diffusion tensor ellipsoid sizes and shapes, and the microscopic orientation distribution function (μODF) and microscopic fractional anisotropy (μFA) that disentangle the underlying heterogeneity within a voxel. Using the DTD-derived μODF, we introduce a new method to perform fiber tractography capable of resolving complex fiber configurations. The results revealed microscopic anisotropy in various gray and white matter regions and skewed MD distributions in cerebellar gray matter not observed previously. DTD MRI tractography captured complex white matter fiber organization consistent with known anatomy. DTD MRI also resolved some degeneracies associated with diffusion tensor imaging (DTI) and elucidated the source of diffusion heterogeneity which may help improve the diagnosis of various neurological diseases and disorders.

Divergence-free turbulent inflow data from realistic covariance tensor

Scale-resolving computational fluid dynamics (CFD) methods require carefully constructed boundary conditions to produce accurate results. The inflow data should be unsteady and the successive realizations must follow specific statistics while ideally having a particular correlation in both space and time. A method for generating synthetic correlated stochastic data from uncorrelated sequences is detailed and applied to the problem of inflow turbulence generation for CFD simulations. The technique constructs divergence-free anisotropic random fields with the sensible spectrum and complete complex correlation in space and time. A realistic two-point correlation tensor is inferred from first and second moments and a set of heuristic recommendations based on turbulent flow observations. These statistics are readily available in most practical cases making the technique highly versatile. The approach is computationally efficient with the use of eigendecomposition to reduce the resources required depending on the accuracy needed. Demonstration of the method is provided with the simulation of a turbulent channel flow and a square duct flow, and validation is done against existing numerical data.

Tensor Principal Component Analysis in High Dimensional CP Models

The CP decomposition for high dimensional non-orthogonal spiked tensors is an important problem with broad applications across many disciplines. However, previous works with theoretical guarantee typically assume restrictive incoherence conditions on the basis vectors for the CP components. In this paper, we propose new computationally efficient composite PCA and concurrent orthogonalization algorithms for tensor CP decomposition with theoretical guarantees under mild incoherence conditions. The composite PCA applies the principal component or singular value decompositions twice, first to a matrix unfolding of the tensor data to obtain singular vectors and then to the matrix folding of the singular vectors obtained in the first step. It can be used as an initialization for any iterative optimization schemes for the tensor CP decomposition. The concurrent orthogonalization algorithm iteratively estimates the basis vector in each mode of the tensor by simultaneously applying projections to the orthogonal complements of the spaces generated by other CP components in other modes. It is designed to improve the alternating least squares estimator and other forms of the high order orthogonal iteration for tensors with low or moderately high CP ranks, and it is guaranteed to have second or higher order convergence when the error of any given initial estimator is bounded by a small constant. Our theoretical investigation provides estimation accuracy and convergence rates for the two proposed algorithms. Both proposed algorithms are applicable to deterministic tensor, its noisy version, and the order- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2K$ </tex-math></inline-formula> covariance tensor of order- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> tensor data in a factor model with uncorrelated factors. Simulation experiments demonstrate significant practical superiority of our approach over existing methods.

Capacity Performance of Tensor Multi-Domain Communication Systems With Discrete Signalling Constellations

Modern communication systems employ multi-domain modulation and coding techniques for effectively exploiting all available resources. Hence in such systems the transmit and receive signals have an inherent multi-domain structure which can be represented using tensors. This work considers the capacity of higher order tensor channels associated with such multi-domain communication systems when the elements of the input tensor are constrained to be drawn from discrete signalling constellations. We establish a relationship between the tensor gradient of the mutual information and the error covariance tensor associated with the minimum mean squared error estimator at the receiver. This relation is used to iteratively find a multi-linear precoder at the input which achieves capacity of the tensor channel under the signalling constellation constraints. Through numerical examples, we show the convergence behavior of the proposed precoder, and compare the capacity achieved under different constellations with the capacity when the input is Gaussian. Further, we exploit the tensor formulation of the problem to find the channel capacity under a variety of different power constraints spanning across several domains. At high SNR, the constellation constraints saturates the capacity while at low SNRs, the constellation constraints are not too relevant, and the power constraints dominate and limit the performance. The capacity saturation level depends on the input order and distribution.

Generation of turbulent inflow data from realistic approximations of the covariance tensor

This study presents a novel synthetic inflow generator capable of producing a random field matching a realistic set of two-point statistics with minimal input. The method is based on two main elements. The first element is a procedure to infer realistic two-point covariance tensors from readily available data (e.g., freestream velocity, boundary layer thickness, and turbulence intensity) by a preliminary Reynolds-averaged Navier–Stokes simulation with an explicit algebraic Reynolds stress model closure. The second element is an efficient eigen-decomposition step of the two-point correlation tensor, which determines a set of modes. The modal decomposition guarantees the spatial correlation in transversal directions, while the temporal correlation/streamwise spatial correlation is obtained by digital filters based on longitudinal and transversal spectra of a realistic shape and Taylor's hypothesis. The instantaneous inlet flow field is obtained by a linear combination of the modes via uncorrelated random weights with unit variance. The modes are generated in a computationally inexpensive pre-processing step. Compared to existing inflow generation methods that try to match given two-point statistics, the proposed method relieves the burden of obtaining data from direct numerical simulation (DNS) or experiments, while the complexity of the eigenvalue problem that needs to be solved is reduced. The proposed method is shown to produce a realistic turbulent channel flow and a realistic turbulent boundary layer by the large-eddy simulation, which contains statistics that are in good agreement with results from DNS. The proposed inflow generator features, cost-effectiveness, robustness, and potential for generalization to complex geometries.

Two-Dimensional DOA Estimation for Coprime Planar Arrays Based on Self-Correlation Tensor

In the coprime planar array (CPA), the existing tensor DOA estimation has the problem that the statistics are not fully utilized. We propose a two-dimensional DOA estimation method based on tensor self-correlation, which realizes the high-resolution and high-precision joint estimation of elevation angle and azimuth angle. Firstly, we represent the received signals of two subarrays with tensors and then obtain the self-correlation covariance tensor of the subarrays themselves and the cross-correlation covariance tensor of the two subarrays. Then, we extract the covariance tensor corresponding to the maximum continuous virtual array and prove the expression of the maximum continuous virtual array aperture of the proposed method. Compared with the existing methods, the proposed method effectively improves the maximum aperture of the continuous virtual array. Finally, the signal subspace is solved by tensor expansion and tensor decomposition. Simulation results show that under the same conditions, the proposed method has higher estimation accuracy and degree of freedom than the cross-correlation tensor method, and the resolution is also improved significantly.

High Resolution Ex Vivo Diffusion Tensor Distribution MRI of Neural Tissue

Neural tissue microstructure plays a key role in developmental, physiological and pathophysiological processes. In the continuing quest to characterize it at ever finer length scales, we use a novel diffusion tensor distribution (DTD) paradigm to probe microstructural features much smaller than the nominal MRI voxel size. We first assume the DTD is a normal tensor variate distribution constrained to lie on the manifold of positive definite matrices, characterized by a mean and covariance tensor. We then estimate the DTD using Monte Carlo signal inversion combined with parsimonious model selection framework that exploits a hierarchy of symmetries of mean and covariance tensors. High resolution multiple pulsed field gradient (mPFG) MRI measurements were performed on a homogeneous isotropic diffusion phantom (PDMS) for control, and excised visual cortex and spinal cord of macaque monkey to investigate the capabilities of DTD MRI in revealing neural tissue microstructural features using strong gradients not typically available in clinical MRI scanners. DTD-derived stains and glyphs, which disentangle size, shape, and orientation heterogeneities of microscopic diffusion tensors, are presented for all samples along with the distribution of the mean diffusivity (MD) within each voxel. We also present a new glyph to visualize the symmetric (kurtosis) and asymmetric parts of the fourth-order covariance tensor. An isotropic mean diffusion tensor and zero covariance tensor was found for the isotropic PDMS phantom, as expected, while the covariance tensor (both symmetric and asymmetric parts) for neural tissue was non-zero indicating that the kurtosis tensor may not be sufficient to fully describe the microstructure. Cortical layers were clearly delineated in the higher moments of the MD spectrum consistent with histology, and microscopic anisotropy was detected in both gray and white matter of neural tissue. DTD MRI captures crossing and splaying white matter fibers penetrating into the cortex, and skewed fiber diameter distributions in the white matter tracts within the cortex and spinal cord. DTD MRI was also shown to subsume diffusion tensor imaging (DTI) while providing additional microstructural information about tissue heterogeneity and microscopic anisotropy within each voxel.

Structured Tensor Reconstruction for Coherent DOA Estimation

Existing tensor-based coherent direction-of-arrival (DOA) estimation methods adopting spatial smoothing to decorrelate the coherent tensor statistics usually lead to a poor decorrelation performance. In this letter, we propose a structured tensor reconstruction method for two-dimensional coherent DOA estimation, which then avoids the inefficient spatial smoothing. In particular, after investigating the structural property of the four-dimensional incoherent covariance tensor, we propose a tensorial Hermitian Toeplitz mapping rule to reconstruct a structured covariance tensor from the rank-deficient coherent covariance tensor statistics. It is theoretically proved that, the reconstructed covariance tensor admits a decorrelated canonical polyadic model with a tensorial Hermitian Toeplitz structure, whose decomposition ensures a closed-form coherent DOA estimation. The effectiveness of the proposed method is verified by simulations.

Tensor Kalman Filter and Its Applications

Kalman filter is one of the most important estimation algorithms, which estimates certain unknown variables given the measurements observed over time subject to a dynamic system, for many applications in science and engineering including environmental science, ecometrics, robotics, financial analysis, data mining, etc. It is often necessary to characterize multiple relationships among various kinds of signals/data in tensor form. The conventional Kalman filter paradigm is based on the low-dimensional state-space representation, which is restricted by the state-transition, observation-model, process-noise covariance, and observation-noise covariance matrices. However, we often need to express some or all of them in terms of tensors in practice. Very lately, the aforementioned Kalman filter in tensor form was tackled using tensor decomposition but the exact estimator has never been established so far. In this work, we propose a new generalized Kalman filter framework consisting of state, state-transition model, observation-model, process-noise covariance, and observation-noise covariance tensors of arbitrary orders by applying the ShermanMorrisonWoodbury identity and block tensor inverse, which we call "Tensor Kalman Filter" (TKF). Our proposed new approach can produce the exact Kalman filter estimator without any need of tensor decomposition (approximation). The pertinent computational- and memory-complexity studies are also provided in this paper.

SubTTD: DOA Estimation via Sub-Nyquist Tensor Train Decomposition

Conventional tensor direction-of-arrival (DOA) estimation methods for sparse arrays apply canonical polyadic decomposition (CPD) to the high-order coarray covariance tensor for retrieving angle information. However, due to the low convergence rate of CPD-based algorithms for high-order tensors, these methods suffer from a high computation cost. To address this issue, a sub-Nyquist tensor train decomposition (SubTTD)-based DOA estimation method is proposed for a three-dimensional (3-D) sparse array, where an augmented virtual array is derived from the sub-Nyquist tensor statistics. To reduce computational complexity of processing the 6-D coarray covariance tensor, the proposed SubTTD model efficiently decomposes it into a train of head matrix, 3-D core tensors, and tail matrix. Based on that, a core tensor decomposition and a change-of-basis transformation for the head matrix are designed to retrieve canonical polyadic factors of the coarray covariance tensor for DOA estimation. The computational efficiency of the proposed method is theoretically analyzed, and its effectiveness is verified via simulations.

Matrix moments of the diffusion tensor distribution and matrix-variate Gamma approximation

Diffusion MRI allows for the non-invasive investigation of the microscopic architecture of biological tissues in vivo. While being sensitive to microstructural tissue changes, diffusion MRI yields poor specificity regarding the cause of these changes. This specificity can be enhanced by using (i) signal representations that retrieve global diffusion properties of the voxel content from the features of the measured diffusion signal or (ii) models that relate these features to predefined sub-voxel tissue-specific compartments. Either on the voxel scale or within diffusion compartments, tissue microstructure can be described using a diffusion tensor distribution (DTD) P(D), which captures a collection of microscopic diffusion environments. A convenient way to estimate DTDs relies on choosing a plausible parametric functional form approximating them. In particular, the mean diffusion tensor 〈D〉 and covariance tensor C associated with this choice enable derivation of measures quantifying specific microstructural diffusion properties. However, high-dimensional DTDs can be quite intractable in practice. In this tutorial paper, we present the matrix moments of P(D), which allow computation of the mean diffusion tensor, covariance tensor and higher-order statistical tensors for any arbitrary functional choice approximating P(D). In turn, these statistical tensors are related to common diffusion measures, such as the mean diffusivity, microscopic diffusion anisotropy and variance of isotropic diffusivities. Using the well-known case of the matrix-variate Gaussian distribution to validate the matrix moments, we then use these tools on the more complex non-central matrix-variate Gamma distribution, computing its covariance tensor for the first time. Finally, we utilize this covariance tensor to design a new voxel-scale signal representation, the matrix-variate Gamma approximation, which we evaluate in vivo and compare to q-space trajectory imaging (QTI) in silico. While the matrix-variate Gamma approximation possesses important limitations explained by the structure of its newly computed covariance tensor, it presents a greater accuracy than QTI in systems exhibiting high orientational order, which suggests that non-central matrix-variate Gamma distributions appropriately describe anisotropic diffusion compartments, as in the DIAMOND model. The framework of matrix moments promotes a more widespread use of matrix-variate distributions as plausible approximations of the DTD by alleviating their intractability, thereby facilitating their design and validation.