Structure Tensor Research Articles

Spin Berry curvature of the Haldane model

The feedback of the geometrical Berry phase, accumulated in an electron system, on the slow dynamics of classical degrees of freedom is governed by the Berry curvature. Here, we study local magnetic moments, modelled as classical spins, which are locally exchange coupled to the (spinful) Haldane model for a Chern insulator. In the emergent equations of motion for the slow classical-spin dynamics there is a an additional anomalous geometrical spin torque, which originates from the corresponding spin-Berry curvature. Due to the explicitly broken time-reversal symmetry, this is nonzero but usually small in a condensed-matter system. We develop the general theory and compute the spin-Berry curvature, mainly in the limit of weak exchange coupling, in various parameter regimes of the Haldane model, particularly close to a topological phase transition and for spins coupled to sites at the zigzag edge of the model in a ribbon geometry. The spatial structure of the spin-Berry curvature tensor, its symmetry properties, the distance dependence of its nonlocal elements and further properties are discussed in detail. For the case of two classical spins, the effect of the geometrical spin torque leads to an anomalous non-Hamiltonian spin dynamics. It is demonstrated that the magnitude of the spin-Berry curvature is decisively controlled by the size of the insulating gap, the system size and the strength of local exchange coupling.

Multi‐view side information‐incorporated tensor completion

AbstractTensor completion originates in numerous applications where data utilized are of high dimensions and gathered from multiple sources or views. Existing methods merely incorporate the structure information, ignoring the fact that ubiquitous side information may be beneficial to estimate the missing entries from a partially observed tensor. Inspired by this, we formulate a sparse and low‐rank tensor completion model named SLRMV. The ‐norm instead of its relaxation is used in the objective function to constrain the sparseness of noise. The CP decomposition is used to decompose the high‐quality tensor, based on which the combination of Schatten ‐norm on each latent factor matrix is employed to characterize the low‐rank tensor structure with high computation efficiency. Diverse similarity matrices for the same factor matrix are regarded as multi‐view side information for guiding the tensor completion task. Although SLRMV is a nonconvex and discontinuous problem, the optimality analysis in terms of Karush‐Kuhn‐Tucker (KKT) conditions is accordingly proposed, based on which a hard‐thresholding based alternating direction method of multipliers (HT‐ADMM) is designed. Extensive experiments remarkably demonstrate the efficiency of SLRMV in tensor completion.

Inelastic material formulations based on a co-rotated intermediate configuration—Application to bioengineered tissues

In the field of material modeling, there is a general trend to include more and more complex phenomena in the modeling, making the models’ theoretical derivation and numerical implementation extremely difficult. In particular, modeling inelastic material behavior under finite deformations in a continuum mechanical manner, e.g. to simulate soft biological tissues, remains one of the most challenging tasks. Unfortunately, the multiplicative decomposition of the deformation gradient usually utilized in this context suffers from an inherent rotational non-uniqueness, making it not straightforward to combine this approach with algorithmic differentiation (AD)—a very helpful tool in modern computational mechanics. To address this issue in the growth and remodeling model proposed herein, a novel co-rotated intermediate configuration is introduced. This configuration shares essential characteristics with the intermediate one, but is uniquely defined and applicable to a wide range of inelastic materials. In this regard, the concept of structural tensors, hardening effects, and a thermodynamically consistent derivation are discussed as well. Since the stress-driven growth model presented is based on the approach of homeostatic surfaces by Lamm et al. (2022), a large number of derivatives of potentials and energies are required, which can be elegantly implemented using AD due to the co-rotated formulation. Moreover, fiber remodeling of collagen fibers is taken into account in a stress-driven manner using AD. Finally, qualitative comparisons are made with recently published experiments by Eichinger et al. (2020) in uniaxial and multiaxial settings, revealing the efficient combination of the proposed framework and the material model.

Collective modes of gluons in an anisotropic thermomagnetic medium

We study the collective modes of gluons in an anisotropic thermal medium in the presence of a constant background magnetic field using the hard-thermal-loop perturbation theory. The momentum space anisotropy of the medium has been incorporated through the generalized Romatschke-Strickland form of the distribution function, whereas the magnetic modification arising from the quark loop contribution has been taken into account in the lowest Landau level approximation. We consider two special cases: (i) a spheroidal anisotropy with the anisotropy vector orthogonal to the external magnetic field and (ii) an ellipsoidal anisotropy with two mutually orthogonal vectors describing anisotropies along and orthogonal to the field direction. The general structure of the polarization tensor in both cases is equivalent and consists of six independent basis tensors. We find that the introduction of momentum anisotropy ingrains azimuthal angular dependence in the thermomagnetic collective modes. Our study suggests that the presence of a strong background magnetic field can significantly reduce the growth rate of the unstable modes, which may have important implications in the equilibration of magnetized quark-gluon plasma.

Anisotropic photon emission from gluon fusion and splitting in a strong magnetic background: The two-gluon one-photon vertex

Having in mind the pre-equilibrium stage in peripheral heavy-ion collisions as a possible scenario for the production of electromagnetic radiation, we compute the two-gluon one-photon vertex in the presence of an intense magnetic field at one-loop order. The quarks in the loop are taken such that two of them occupy the lowest Landau level, with the third one occupying the first excited Landau level. When the field strength is the largest of the energy (squared) scales, the tensor basis describing this vertex corresponds to two of the three-vector particles polarized in the longitudinal direction whereas the third one is polarized in the transverse direction. However, when the photon energy is of the same order as or larger than the field strength, the explicit one-loop computation contains extra tensor structures that spoil the properties of the basis compared with the case when the field strength is the largest of the energy scales, which signals that the calculation is incomplete. Nevertheless, by projecting the result onto the would-be basis, we show that the squared amplitude for processes involving two gluons and one photon exhibits the expected properties such as a preferred in-plane photon emission and a slightly decreasing strength for an increasing magnetic-field strength. We comment on possible venues to improve the one-loop calculation that include accounting for progressive occupation of the three quarks of the lowest and first-excited Landau levels such that, still working in the large field limit, a more complete description can be achieved when the photon energy increases.

General tensor structure for electron scattering in terms of invariant responses

The use of invariant response functions in treatments of electron scattering from hadronic targets is reviewed. Various classes of reaction are treated, building from the simplest (and best known) case of inclusive scattering from unpolarized targets, to more complicated cases involving polarized electrons and possibly polarized spin-1/2 targets. In particular, the general structure of semi-inclusive polarized electron scattering from polarized spin-1/2 targets is emphasized. A summary is presented of how the leptonic and hadronic tensors that enter in the formalism are constructed in a general covariant way in terms of kinematic factors that are frame dependent but model independent and invariant response functions which contain all of the model-dependent dynamics. In the process of reviewing the general problem the relationships to the conventional responses expressed in terms of the frame-dependent helicity components of the exchanged virtual photon are presented.

The structure of symmetric tensor powers of composition algebras

Let [Formula: see text] be a composition algebra which is either the Hamilton quaternion algebra [Formula: see text] or the Cayley octonion algebra [Formula: see text] over [Formula: see text]. In a previous work, the nth symmetric power [Formula: see text] of [Formula: see text] is shown to be a direct sum of central simple algebras, corresponding to the partitions of [Formula: see text] of length [Formula: see text], such that the component corresponding to the partition [Formula: see text] is isomorphic to the component [Formula: see text] of [Formula: see text] corresponding to the partition [Formula: see text] of [Formula: see text]. In this work, we study the building blocks [Formula: see text] of these decompositions. We show that the “local” structure of [Formula: see text], i.e. the complex-like subfields of [Formula: see text], determine both the complement of [Formula: see text] in [Formula: see text] and the trace map of [Formula: see text], induced from the trace map of [Formula: see text]. We also derive a recursive trace formula on the [Formula: see text]’s. We use the “local-global” results to define positive definite symmetric bilinear forms on the vector space [Formula: see text], which has a natural structure of a commutative and associative algebra. Finally, the structure of the central simple algebra [Formula: see text] is described.

Detailed quantification of cardiac ventricular myocardial architecture in the embryonic and fetal mouse heart by application of structure tensor analysis to high resolution episcopic microscopic data.

The mammalian heart, which is one of the first organs to form and function during embryogenesis, develops from a simple tube into a complex organ able to efficiently pump blood towards the rest of the body. The progressive growth of the compact myocardium during embryonic development is accompanied by changes in its structural complexity and organisation. However, how myocardial myoarchitecture develops during embryogenesis remain poorly understood. To date, analysis of heart development has focused mainly on qualitative descriptions using selected 2D histological sections. High resolution episcopic microscopy (HREM) is a novel microscopic imaging technique that enables to obtain high-resolution three-dimensional images of the heart and perform detailed quantitative analyses of heart development. In this work, we performed a detailed characterization of the development of myocardial architecture in wildtype mice, from E14.5 to E18.5, by means of structure tensor analysis applied to HREM images of the heart. Our results shows that even at E14.5, myocytes are already aligned, showing a gradual change in their helical angle from positive angulation in the endocardium towards negative angulation in the epicardium. Moreover, there is gradual increase in the degree of myocardial organisation concomitant with myocardial growth. However, the development of the myoarchitecture is heterogeneous showing regional differences between ventricles, ventricular walls as well as between myocardial layers, with different growth patterning between the endocardium and epicardium. We also found that the percentage of circumferentially arranged myocytes within the LV significantly increases with gestational age. Finally, we found that fractional anisotropy (FA) within the LV gradually increases with gestational age, while the FA within RV remains unchanged.

Using tensor contractions to derive the structure of slice-wise multiplications of tensors with applications to space–time Khatri–Rao coding for MIMO-OFDM systems

The slice-wise multiplication of two tensors is required in a variety of tensor decompositions (including PARAFAC2 and PARATUCK2) and is encountered in many applications, including the analysis of multidimensional biomedical data (EEG, MEG, etc.) or multi-carrier multiple-input multiple-output (MIMO) systems. In this paper, we propose a new tensor representation that is not based on a slice-wise (matrix) description, but can be represented by a double contraction of two tensors. Such a double contraction of two tensors can be efficiently calculated via generalized unfoldings. It leads to new tensor models of the investigated system that do not depend on the chosen unfolding (in contrast to matrix models) and reveal the tensor structure of the data model, such that all possible unfoldings can be seen at the same time. As an example, we apply this new concept to the design of new receivers for multi-carrier MIMO systems in wireless communications. In particular, we consider MIMO-orthogonal frequency division multiplexing (OFDM) systems with and without Khatri–Rao coding. The proposed receivers exploit the channel correlation between adjacent subcarriers, require the same amount of training symbols as traditional OFDM techniques, but have an improved performance in terms of the symbol error rate. Furthermore, we show that the spectral efficiency of the Khatri–Rao-coded MIMO-OFDM can be increased by introducing cross-coding such that the “coding matrix” also contains useful information symbols. Considering this transmission technique, we derive a tensor model and two types of receivers for cross-coded MIMO-OFDM systems using the double contraction of two tensors.

Search for Sparse Solutions of Super-Large Systems with a Tensor Structure

Search for Sparse Solutions of Super-Large Systems with a Tensor Structure

Time-Varying Channel Estimation Scheme for Uplink MU-MIMO in 6G Systems

Channel estimation is a challenging issue for millimeter wave (mmWave) and massive multiple-input multiple-output (MIMO) in the future sixth generation (6G) wireless systems, where the conventional estimation schemes may fail to track the fast varying channels, especially in high-speed mobile scenarios. In this paper, a novel tensor-based uplink channel estimation scheme is proposed for multi-user MIMO (MU-MIMO) systems over time-varying channels. In the proposed scheme, a low-overhead pilot transmission scheme is designed to track the varying channel. The received uplink signal at the base station (BS) is formulated as a third-order tensor which admits a CANDECOMP/PARAFAC (CP) model. The CP decomposition issue is then solved using blind matrix decomposition, in which the special structures of signals in the time dimension and the matrix subspace are utilized. By exploiting low-rank structure of the signal tensor, the channel parameters (angles of arrival/departure, path gains, and Doppler shifts) are estimated from the factor matrices. Moreover, the proposed scheme is theoretically analyzed and it is guaranteed with low pilot overhead. Simulation results verify that the proposed scheme can outperform the compressed sensing (CS) based scheme and the iteration-based scheme in terms of accuracy and stability. The uniqueness of the proposed scheme is also verified in our simulation.

Infrared Maritime Small Target Detection Based on Multidirectional Uniformity and Sparse-Weight Similarity

Infrared maritime target detection is a key technology in the field of maritime search and rescue, which usually requires high detection accuracy. Despite the promising progress of principal component analysis methods, it is still challenging to detect small targets of unknown polarity (bright or dark) with strong edge interference. Using the partial sum of tubal nuclear norm to estimate low-rank background components and weighted l1 norm to estimate sparse components is an effective method for target extraction. In order to suppress the strong edge interference, considering that the uniformity of the target scattering field is significantly higher than that of the background scattering field in the eigenvalue of the structure tensor, a prior weight based on the multidirectional uniformity of structure tensor eigenvalue was proposed and applied to the optimization model. In order to detect targets with unknown polarity, the images with opposite polarity were substituted into the optimization model, respectively, and the sparse-weight similarity is used to judge the polarity of the target. In order to make the method more efficient, the polarity judgment is made in the second iteration, and then, the false iteration will stop. The proposed method is compared with nine advanced baseline methods on 14 datasets and shows significant strong robustness, which is beneficial to engineering applications.

Effective and robust infrared small target detection based on directional improved structure tensor

Small target detection under complicated background is a research hotspot and challenging problem. To increase the ability of small target detection under complicated background, a simple but effective target detection approach based on directional improved structure tensor is designed. Firstly, considering that small target region will be omni-directional while background edge clutters will tend to expand in some specific directions, the directional structure tensor (DST) is constructed by incorporating eight-direction filter template. Then, the directional positive small eigenvalue of the structure tensor (DPSTSE) is computed to effectively perceive and locate the junctions of small target and surrounding background in all directions. Finally, the DPSTSE maps in all directions are integrated together to distinguish small targets from strong background clutter. Experiments illustrate that the presented method is better than some advanced algorithms in clutter suppression ability and target detection effect. Moreover, the presented method has high computational efficiency and potential for real-time processing.

Tensor-Based Multi-Modality Feature Selection and Regression for Alzheimer's Disease Diagnosis.

The assessment of Alzheimer's Disease (AD) and Mild Cognitive Impairment (MCI) associated with brain changes remains a challenging task. Recent studies have demonstrated that combination of multi-modality imaging techniques can better reflect pathological characteristics and contribute to more accurate diagnosis of AD and MCI. In this paper, we propose a novel tensor-based multi-modality feature selection and regression method for diagnosis and biomarker identification of AD and MCI from normal controls. Specifically, we leverage the tensor structure to exploit high-level correlation information inherent in the multi-modality data, and investigate tensor-level sparsity in the multilinear regression model. We present the practical advantages of our method for the analysis of ADNI data using three imaging modalities (VBM-MRI, FDG-PET and AV45-PET) with clinical parameters of disease severity and cognitive scores. The experimental results demonstrate the superior performance of our proposed method against the state-of-the-art for the disease diagnosis and the identification of disease-specific regions and modality-related differences. The code for this work is publicly available at https://github.com/junfish/BIOS22.

On the direct and reverse multiplicative decompositions of deformation gradient in nonlinear anisotropic anelasticity

In this paper we discuss nonlinear anisotropic anelasticity formulated based on the two multiplicative decompositions F=FeFa and F=FaFe. Using the Bilby–Kröner–Lee decomposition F=FeFa one can define a Riemannian material manifold (the natural configuration of an anelastic body) whose metric explicitly depends on the anelastic deformation Fa. We call this the global material intermediate configuration. Deformation is a map from this Riemannian manifold to the flat ambient space. Using the reverse decomposition F=FaFe, the reference configuration is a (flat) submanifold of the Euclidean ambient space, while the global intermediate configuration is a Riemannian manifold whose metric explicitly depends on the elastic deformation Fe. We call this the global spatial intermediate configuration. We show that the direct F=FeFa and reverse F=FaFe decompositions correspond to the same anelastic motion if and only if Fe and Fe are equal up to local isometries of the reference configuration. We discuss the constitutive equations of anisotropic anelastic solids in terms of both intermediate configurations. It is shown that the two descriptions of anelasticity are equivalent in the sense that the Cauchy stresses calculated using them are identical. We note that, unlike isotropic solids, for an anisotropic solid the material metric is not sufficient for describing the constitutive behavior of the solid; the energy function explicitly depends on Fa (or Fa) through the structural tensors.

Multiscale contrast enhancement method for small infrared target detection

Infrared search and track (IRST) systems have been applied extensively in the civil and military fields. However, fast, effective, and robust dim and small infrared target detection remains a challenging research topic. Dim and small targets are easily obscured in different backgrounds or noises, which results in a high false alarm rate. A novel strategy is proposed for small infrared target detection to achieve a high true positive rate and robust detection. First, the prominent edge structures of the original image are obtained by constructing a structure tensor. Subsequently, a Gaussian bandpass filter with multiscale windows is used to obtain an infrared image that contains complex edges and small targets. An infrared image with an enhanced signal-to-clutter ratio is achieved by combining with the two methods, whereby the noise can be effectively reduced. Thereafter, candidate regions that contain the target and a smaller number of edges with higher brightness than that of the target are obtained using relaxed threshold segmentation. Finally, a Gaussian gradient contrast method is proposed in which the highlighted edges are removed so that only small infrared targets remain. Extensive experimental results demonstrate that the proposed method can achieve a significant optimization of the true positive and false alarm rates, as well as improved efficiency and robustness.

Harnessing Tensor Structures—Multi-Mode Reservoir Computing and Its Application in Massive MIMO

In this paper, we introduce a new neural network (NN) structure, multi-mode reservoir computing (Multi-Mode RC). It inherits the dynamic mechanism of RC and processes the forward path and loss optimization of the NN using tensor as the underlying data format. Multi-Mode RC exhibits less complexity compared to conventional RC structures (e.g. single-mode RC), and offers comparable generalization performance to its single-mode counterpart. Furthermore, we introduce an alternating least square-based learning algorithm as well as the associated theoretical analysis for Multi-Mode RC. The result can be utilized to guide the configuration of NN parameters to sufficiently circumvent over-fitting issues. As a key application, we consider the symbol detection task in multiple-input-multiple-output (MIMO) orthogonal-frequency-division-multiplexing (OFDM) systems with massive MIMO employed at the base stations (BSs). Thanks to the tensor structure of massive MIMO-OFDM signals, our online learning-based symbol detection method generalizes well in terms of bit error rate even using a limited online training set. Evaluation results suggest that the Multi-Mode RC-based learning framework can efficiently and effectively combat practical constraints of wireless systems (i.e. channel state information (CSI) errors and hardware non-linearity) to enable robust and adaptive communications over the air.

Concentration estimates for random subspaces of a tensor product and application to quantum information theory

Given a random subspace Hn chosen uniformly in a tensor product of Hilbert spaces Vn ⊗ W, we consider the collection Kn of all singular values of all norm one elements of Hn with respect to the tensor structure. A law of large numbers has been obtained for this random set in the context of W fixed and the dimension of Hn, Vn tending to infinity at the same speed by Belinschi, Collins, and Nechita [Commun. Math. Phys. 341(3), 885–909 (2016)]. In this paper, we provide measure concentration estimates in this context. The probabilistic study of Kn was motivated by important questions in quantum information theory and allowed us to provide the smallest known dimension for the dimension of an ancilla space, allowing for Minimum Output Entropy (MOE) violation. With our estimates, we are able, as an application, to provide actual bounds for the dimension of spaces where the violation of MOE occurs.

Universal treatment of the reduction for one-loop integrals in a projective space

Recently a nice work about the understanding of one-loop integrals has been done in [1] using the tricks of the projective space language associated to their Feynman parametrization. We find this language is also very suitable to deal with the reduction problem of one-loop integrals with general tensor structures as well as propagators with arbitrary higher powers. In this paper, we show that how to combine Feynman parametrization and embedding formalism to give a universal treatment of reductions for general one-loop integrals, even including the degenerated cases, such as the vanishing Gram determinant. Results from this method can be written in a compact and symmetric form.

A variational multiscale method for natural convection of nanofluids

The notion of enhanced thermal convection via particle laden fluids has been around for a long time. Technological challenges associated with the development of micro to nano particles with desired properties and their uniform dispersion in the base fluid have been a bottleneck. Relatively recently, the advent of modern manufacturing techniques from micro to nanoscales have rekindled interest in this class of fluids for innovative applications in advanced engineering systems.Buoyancy-induced convection and heat transfer involves conservation laws of mass, momentum, and energy. The mathematical model is comprised of two-way coupled system of mixed-field and convection-dominated partial differential equations. A stabilized method for nonlinearly coupled system is presented, and a systematic approach to develop the sub-grid scale (SGS) physics-based models is described. Explicit structure of the stabilization tensor is derived and it is shown to preserve nonlinear coupling in the SGS models that plays a critical role when nonlinear coupling of mechanical and thermal fields leads to anisotropy across the scales. The formulation is variationally consistent and results in optimal spatial convergence rates on structured meshes for linear triangles and bilinear quadrilaterals. Consistent linearization of the nonlinear system of equations yields quadratic rate of convergence of nonlinear iterations in the Newton-Raphson method. The method is tested on problems with increasing level of complexity to highlight the mathematical attributes of the method and its range of applicability.