Tensor Space Research Articles

Nonlocality of Star-Shaped Correlation Tensors Based on the Architecture of a General Multi-Star-Network

In this work, we study the nonlocality of star-shaped correlation tensors (SSCTs) based on a general multi-star-network MSN(m,n1,…,nm). Such a network consists of 1+m+n1+⋯+nm nodes and one center-node A that connects to m star-nodes B1,B2,…,Bm while each star-node Bj has nj+1 star-nodes A,C1j,C2j,…,Cnjj. By introducing star-locality and star-nonlocality into the network, some related properties are obtained. Based on the architecture of such a network, SSCTs including star-shaped probability tensors (SSPTs) are proposed and two types of localities in SSCTs and SSPTs are mathematically formulated, called D-star-locality and C-star-locality. By establishing a series of characterizations, the equivalence of these two localities is verified. Some necessary conditions for a star-shaped CT to be D-star-local are also obtained. It is proven that the set of all star-local SSCTs is a compact and path-connected subset in the Hilbert space of tensors over the index set ΔS and has least two types of star-convex subsets. Lastly, a star-Bell inequality is proved to be valid for all star-local SSCTs. Based on our inequality, two examples of star-nonlocal MSN(m,n1,…,nm) are presented.

Geometry of tree-based tensor formats in tensor Banach spaces

In the paper ‘On the Dirac–Frenkel Variational Principle on Tensor Banach Spaces’, we provided a geometrical description of manifolds of tensors in Tucker format with fixed multilinear (or Tucker) rank in tensor Banach spaces, that allowed to extend the Dirac–Frenkel variational principle in the framework of topological tensor spaces. The purpose of this note is to extend these results to more general tensor formats. More precisely, we provide a new geometrical description of manifolds of tensors in tree-based (or hierarchical) format, also known as tree tensor networks, which are intersections of manifolds of tensors in Tucker format associated with different partitions of the set of dimensions. The proposed geometrical description of tensors in tree-based format is compatible with the one of manifolds of tensors in Tucker format.

A Tensor Method based on Enhanced Tensor Nuclear Norm and Hypergraph Laplacian Regularization for Pan-Cancer Omics Data Analysis.

As a powerful data representation technique, tensor robust principal component analysis (TRPCA) has been widely used for clustering and feature selection tasks. However, it ignores the significant difference in singular values of tensor data and the manifold information contained in different views, thereby causing serious degradation of conventional TRPCA performance. In this paper, a novel tensor method based on enhanced tensor nuclear norm and hypergraph Laplacian regularization (ETHLR) is developed to address the above problem. ETHLR can jointly learn the prior knowledge of singular values and high-order manifold structures in the unified tensor space and the view-specific feature spaces, respectively. Specifically, the enhanced tensor nuclear norm, namely, the weighted tensor Schatten p-norm, is used to shrink the singular values by fully considering the salient difference information of singular values and the complementary information embedded in the tensor space; the hypergraph Laplacian constraint helps encode high-order geometric structures among multiple samples in the nonlinear view-specific feature space. Furthermore, we employ inexact augmented Lagrange multipliers (ALM) to optimize the ETHLR method. Numerous experiments on pan-cancer omics data show that the superiority of ETHLR over several state-of-the-art competitors.

Lie Bialgebras on the Rank Two Heisenberg–Virasoro Algebra

The rank two Heisenberg–Virasoro algebra can be viewed as a generalization of the twisted Heisenberg–Virasoro algebra. Lie bialgebras play an important role in searching for solutions of quantum Yang–Baxter equations. It is interesting to study the Lie bialgebra structures on the rank two Heisenberg–Virasoro algebra. Since the Lie brackets of rank two Heisenberg–Virasoro algebra are different from that of the twisted Heisenberg–Virasoro algebra and Virasoro-like algebras, and there are inner derivations (from itself to its tensor space) which are hidden more deeply in its interior algebraic structure, some new techniques and strategies are employed in this paper. It is proved that every Lie bialgebra structure on the rank two Heisenberg–Virasoro algebra is triangular coboundary.

Reinterpreting deformed Heisenberg algebras

Minimal and maximal uncertainties of position measurements are widely considered possible hallmarks of low-energy quantum as well as classical gravity. While General Relativity describes interactions in terms of spatial curvature, its quantum analogue may also extend to the realm of curved momentum space as suggested, e.g. in the context of Relative Locality in Deformed Special Relativity. Drawing on earlier work, we show in an entirely Born reciprocal, i.e. position and momentum space covariant, way that the quadratic Generalized Extended Uncertainty principle can alternatively be described in terms of quantum dynamics on a general curved cotangent manifold. In the case of the Extended Uncertainty Principle the curvature tensor in position space is proportional to the noncommutativity of the momenta, while an analogous relation applies to the curvature tensor in momentum space and the noncommutativity of the coordinates for the Generalized Uncertainty Principle. In the process of deriving this map, the covariance of the approach constrains the admissible models to an interesting subclass of noncommutative geometries which has not been studied before. Furthermore, we reverse the approach to derive general anisotropically deformed uncertainty relations from general background geometries. As an example, this formalism is applied to (anti)-de Sitter spacetime.

Infrared object detection via patch‐tensor model and image denoising based on weighted truncated Schatten‐p norm minimization

Infrared object detection via patch‐tensor model and image denoising based on weighted truncated Schatten‐<i>p</i> norm minimization

K- constant type Kahler and Nearly Kahler manifolds for conharmonic curvature tensor

The constant of permanence conharmonic type kahler and nearly kahler manifold conditions are obtained when the Nearly Kahler manifold is a manifold conharmonic constant type (K). and also proved that M nearly kahler manifold of pointwise constant holomorphic sectional conharmonic (PHKm(X)) – curvature) curvature tensor if the components of holomorphic sectional (HS- curvature) curvature tensor in the adjoint G-structure space that satisfies condition .

Enhanced tensor low-rank representation learning for multi-view clustering

Multi-view subspace clustering (MSC), assuming the multi-view data are generated from a latent subspace, has attracted considerable attention in multi-view clustering. To recover the underlying subspace structure, a successful approach adopted recently is subspace clustering based on tensor nuclear norm (TNN). But there are some limitations to this approach that the existing TNN-based methods usually fail to exploit the intrinsic cluster structure and high-order correlations well, which leads to limited clustering performance. To address this problem, the main purpose of this paper is to propose a novel tensor low-rank representation (TLRR) learning method to perform multi-view clustering. First, we construct a 3rd-order tensor by organizing the features from all views, and then use the t-product in the tensor space to obtain the self-representation tensor of the tensorial data. Second, we use the ℓ1,2 norm to constrain the self-representation tensor to make it capture the class-specificity distribution, that is important for depicting the intrinsic cluster structure. And simultaneously, we rotate the self-representation tensor, and use the tensor singular value decomposition-based weighted TNN as a tighter tensor rank approximation to constrain the rotated tensor. For the challenged mathematical optimization problem, we present an effective optimization algorithm with a theoretical convergence guarantee and relatively low computation complexity. The constructed convergent sequence to the Karush–Kuhn–Tucker (KKT) critical point solution is mathematically validated in detail. We perform extensive experiments on four datasets and demonstrate that TLRR outperforms state-of-the-art multi-view subspace clustering methods.

Lower bounds on the rank and symmetric rank of real tensors

We lower bound the rank of a tensor by a linear combination of the ranks of three of its unfoldings, using Sylvester's rank inequality. In a similar way, we lower bound the symmetric rank by a linear combination of the symmetric ranks of three unfoldings. Lower bounds on the rank and symmetric rank of tensors are important for finding counterexamples to Comon's conjecture. A real counterexample to Comon's conjecture is a tensor whose real rank and real symmetric rank differ. Previously, only one real counterexample was known, constructed in a paper of Shitov. We divide the construction into three steps. The first step involves linear spaces of binary tensors. The second step considers a linear space of larger decomposable tensors. The third step is to verify a conjecture that lower bounds the symmetric rank, on a tensor of interest. We use the construction to build an order six real tensor whose real rank and real symmetric rank differ.

Proper Generalized Decomposition using Taylor expansion for non-linear diffusion equations

For a physical problem described by a parameterized mathematical model, different configurations of the problem require computing the solution over a range of parameters in order to study the phenomenon when parameters change. In other words, it is a process of looking for a continuum of solutions of the equation, relative to these parameters, in order to find the ones that fit the experimental data. However, solving a direct problem for each parametric configuration will generate a cascade of direct problems, which will cost a huge amount of time, especially when we deal with non-linear equations. Therefore, the parametric solution is a suitable alternative strategy to compute the solution of the equation. In this paper, we will use the Proper Generalized Decomposition (PGD) method to solve non-linear diffusion equations and produce parametric solutions. To treat the non-linear functions, we will not use the Discrete Empirical Interpolation Methods (DEIM), which has proven their utility, but the non-linear terms will be replaced by their Taylor series expansion up to an order m. This will produce a new model, which we call here the ”developed equation” and therefore the PGD is applied on. Polynomial equations appear for each tensor element computation. While space and time tensor elements’ equations are to be solved using Finite Elements Methods (FEM) and Borel–Padé–Laplace (BPL) integrator respectively, Newton solver is used for tensors relative to the parameters’ equations. Here, rational polynomial functions arise for parametric tensor elements, which are known to extrapolate solutions. Numerical simulations are done for a non-linear diffusion equation with exponential diffusion coefficient as first trial, and with a magnetic diffusion coefficient as a second one.

On the Phase Space of Fourth-Order Fiber-Orientation Tensors

Fiber-orientation tensors describe the relevant features of the fiber-orientation distribution compactly and are thus ubiquitous in injection-molding simulations and subsequent mechanical analyses. In engineering applications to date, the second-order fiber-orientation tensor is the basic quantity of interest, and the fourth-order fiber-orientation tensor is obtained via a closure approximation. Unfortunately, such a description limits the predictive capabilities of the modeling process significantly, because the wealth of possible fourth-order fiber-orientation tensors is not exploited by such closures, and the restriction to second-order fiber-orientation tensors implies artifacts. Closures based on the second-order fiber-orientation tensor face a fundamental problem – which fourth-order fiber-orientation tensors can be realized? In the literature, only necessary conditions for a fiber-orientation tensor to be connected to a fiber-orientation distribution are found. In this article, we show that the typically considered necessary conditions, positive semidefiniteness and a trace condition, are also sufficient for being a fourth-order fiber-orientation tensor in the physically relevant case of two and three spatial dimensions. Moreover, we show that these conditions are not sufficient in higher dimensions. The argument is based on convex duality and a celebrated theorem of D. Hilbert (1888) on the decomposability of positive and homogeneous polynomials of degree four. The result has numerous implications for modeling the flow and the resulting microstructures of fiber-reinforced composites, in particular for the effective elastic constants of such materials. Based on our findings, we show how to connect optimization problems on fourth-order fiber-orientation tensors to semi-definite programming. The proposed formulation permits to encode symmetries of the fiber-orientation tensor naturally. As an application, we look at the differences between orthotropic and general, i.e., triclinic, fiber-orientation tensors of fourth order in two and three spatial dimensions, revealing the severe limitations inherent to orthotropic closure approximations.

A novel Siamese network object tracking algorithm based on tensor space mapping and memory-learning mechanism

The tracker is a core component of the tracking algorithm, but it is difficult to identify the object, which is a challenge to improve the tracking accuracy. This paper proposes a Siamese network-based tracking algorithm based on tensor space mapping and memory-learning mechanisms. Firstly, the source image is mapped to the tensor space to serialize the feature distributions. Then the gating mechanism is used to extract the association information about the adjacent state, which guides the update of the subsequent state, and the interactive information on the objects is used to locate the object. On this basis, a memory-learning module is built to traverse and extract the fine-grained features, which can filter the semantic information of the object learned by the tracker. As a result, the tracking accuracy is enhanced. The experiments show that the proposed algorithm has better performance than that of the comparison methods in the OTB100 data set and the VOT data set.

Spatial–Temporal Tensor Graph Convolutional Network for Traffic Speed Prediction

Accurate traffic speed prediction is crucial for the guidance and management of urban traffic, which at the same time requires a model with a satisfactory computational burden and memory space in applications. In this paper, we propose a factorized Spatial-Temporal Tensor Graph Convolutional Network for traffic speed prediction. Traffic networks are modeled and unified into a graph tensor that integrates spatial and temporal information simultaneously. We extend graph convolution into tensor space and propose a tensor graph convolution network to extract more discriminating features from spatial-temporal graph data. We further introduce Tucker decomposition and derive a factorized tensor convolution to reduce the computational burden, which performs separate filtering in small-scale space, time, and feature modes. Besides, we can benefit from noise suppression of traffic data when discarding those trivial components in the process of tensor decomposition. Extensive experiments on the three real-world datasets demonstrate that our method is more effective than traditional prediction methods, and achieves state-of-the-art performance.

Holistic Feature Reconstruction-Based 3-D Attention Mechanism for Cross-Spectral Periocular Recognition

Recognition of individuals in cross-spectral environments involves situations where probe and gallery images are captured in distinct wavelength ranges and hence differ significantly in terms of appearance and illumination. In case of heterogeneous periocular images, alleviating this wide appearance gap and learning to extract illumination-invariant features from such local regions become cumbersome, giving rise to a challenging research problem. In this work, we design a novel holistic feature reconstruction-based attention module (H-FRAM) to refine and generate discriminative convolutional features. In contrast to existing spatial and channel attention mechanisms that compute 2-D or 1-D attention weights respectively, H-FRAM calculates the importance of each feature location by performing multi-linear principal component analysis on full 3-D tensor space. In H-FRAM, we claim that the feature reconstruction error, computed in a holistic manner, plays a crucial role in determining the relevance of feature locations. This reconstruction error is found by projecting the input feature map onto a multi-dimensional eigenspace that captures most of the important variations. To our knowledge, this is the first work which explores subspace learning approach in the context of 3-D attention mechanism to capture discriminative feature information. We have created an in-house cross-spectral periocular dataset containing visible and near-infrared images from 200 classes. The images are captured in unconstrained acquisition setup involving unsupervised eye and head movements as well as accessory variations (face masks and eyeglasses). Extensive experiments and ablation studies show that the proposed network achieves state-of-the-art recognition performances on the existing and in-house datasets for heterogeneous and homogeneous periocular recognition as well as heterogeneous face recognition.

AN ALGEBRAIC DECOMPOSITION OF HEMITROPIC PSEUDOTENSORS IN N DIMENSIONS AND APPLICATIONS TO MICROPOLAR CONTINUUM THEORIES

The paper is devoted to problems related to algebraic decompositions and coordinate representations of tensors with constant components, hemitropic tensors and pseudotensors. Such objects of tensor algebra are interesting from the micropolar continuum mechanics viewpoint. Algebraic properties and coordinate representations of tensors and pseudotensors with constant components are discussed. The base examples of tensors with constant components usable in continuum mechanics are given. An algebraic algorithm for coordinate representations of tensors and pseudotensors with constant components proposed by prof. G.B. Gurevich is highlighted and employed.The notions of fully isotropic, conventionally isotropic, unconventionally isotropic, semi-isotropic (demitropic, hemitropic, chiral) fourth rank tensors and pseudotensors are proposed. The coordinate representations of a Cartesian hemitropic fourth rank tensor in three-dimensional space are obtained in terms of the Kronecker delta and metric tensor products. Based on an unconventional definition of a hemitropic fourth rank tensor, general coordinate representations in dimensions in terms of the Kronecker deltas and metric tensors are given. A comparison of an arbitrary hemitropic fourth rank tensor and a tensor with constant components is carried out. A general form of the elastic potential of a linear anisotropic micropolar elastic continuum is obtained by the pseudotensor technique. Obtained anisotropic micropolar potential is reduced to a hemitropic one by proposed coordinate representations of fourth rank tensors. Coordinate representations for constitutive tensors and pseudotensors usable in mathematical modeling of linear hemitropic micropolarelastic continua are obtained as a modification of pseudotensors with constant components in three-dimensional space.

Annihilating Entanglement Between Cones

Every multipartite entangled quantum state becomes fully separable after an entanglement breaking quantum channel acted locally on each of its subsystems. Whether there are other quantum channels with this property has been an open problem with important implications for entanglement theory (e.g., for the distillation problem and the PPT squared conjecture). We cast this problem in the general setting of proper convex cones in finite-dimensional vector spaces. The max-entanglement annihilating maps transform the k-fold maximal tensor product of a cone {textsf{C}}_1 into the k-fold minimal tensor product of a cone {textsf{C}}_2, and the pair ({textsf{C}}_1,{textsf{C}}_2) is called resilient if all max-entanglement annihilating maps are entanglement breaking. Our main result is that ({textsf{C}}_1,{textsf{C}}_2) is resilient if either {textsf{C}}_1 or {textsf{C}}_2 is a Lorentz cone. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation: As a warm-up, we use the multiplication tensors of real composition algebras to construct a finite family of generalized distillation protocols for Lorentz cones, containing the distillation protocol for entangled qubit states by Bennett et al. (Phys Rev Lett 76(5):722, 1996) as a special case. Then, we construct an infinite family of protocols using solutions to the Hurwitz matrix equations. After proving these results, we focus on maps between cones of positive semidefinite matrices, where we derive necessary conditions for max-entanglement annihilation similar to the reduction criterion in entanglement distillation. Finally, we apply results from the theory of Banach space tensor norms to show that the Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied.

Decomposition of Riemannian Recurrent Curvature Tensor Manifolds of First Order

Takano [6] premeditated decomposition of curvature tensor in a recurrent Riemannian space. After that, Negi and Bisht [3] defined and deliberated decomposition of recurrent curvature tensor fields in a Kaehlerian manifolds of first order. We have calculated the decomposition of Riemannian recurrent curvature tensor manifolds of first order and some theorems established using the decomposition tensor field.

Memory surfaces separating the processes of monotonous and cyclic loads

The processes of elastoplastic deformation of structural materials can consist of a sequence of monotonous and cyclic loading regimes, in which peculiar effects and features arise. Mathematical modeling of such processes, as well as resource assessment and forecasting, is a very difficult task. In addition, the analysis of transient processes from cyclic to monotonous and from monotonous to cyclic shows the need to separate these processes. Based on the analysis of the results of experimental studies of samples of stainless steel 12Х18Н10T under a rigid (controlled deformation) deformation process, which is a sequence of monotonic and cyclic loading modes, under conditions of uniaxial tension-compression at normal temperature, the features and differences in the processes of monotonic and cyclic loading are revealed. To describe these features and separate the processes of monotonic and cyclic loading modes in the theories of plastic flow with combined hardening, various variants of memory surfaces are introduced. An analysis of the results of experimental studies of stainless steel showed that in the space of the plastic strain tensor, the size of the memory surface is determined by the range of plastic strains, and the position of the center is determined by the values of average plastic strains under cyclic loading. Various variants of the memory surface are considered, their capabilities and disadvantages are identified, and the most adequate variant of the memory surface is determined. To confirm the operability of this version of the memory surface, together with the equations of the Bondar plasticity model, the calculated and experimental results were compared and a reliable agreement was obtained between these results both in terms of the kinetics of the stress-strain state and in terms of the number of cycles to failure.

TESPDA-SEI: Tensor embedding substructure preserving domain adaptation for specific emitter identification

For the specific emitter identification (SEI) with few or no labels, domain adaptation make the model respond quickly with the help of empirical information. However, the more extreme case is that there are so few labeled samples in the source domain that it is difficult to train an excellent recognition model. In fact, it is more valuable to make full use of these limited label information. This work aims at proposing an unsupervised domain adaptation (UDA)-based method to accommodate the typical case of no labels in the target domain and small samples in the source domain when new devices are first introduced. The basic principle is to learn tensor embedding shared feature space and preserving inter-class substructure, which perform feature space mapping under the joint source and target domain led by mapping error minimize in the source domain. Specifically, this tensor embedding substructure preserving domain adaptation (TESPDA) consist of three parts, tensor invariant subspace learning, substructure preserving feature space mapping and pseudo-label prediction, which are used to learn inter-class substructure after tensor space mapping and identify the predict labels for the target domain. Finally, experiments are conducted on the real-word ADS-B dataset to demonstrate the effectiveness of the TESPDA method.

Planar fiber orientation: Jeffery, non-orthotropic closures, and reconstructing distribution functions

Many models for fiber suspensions use the second-order orientation tensor to represent the orientation state. This requires that the fourth-order orientation tensor be approximated using a closure formula. The prediction of non-linear properties also requires reconstructing an approximate orientation distribution function. All existing fourth-order closures and most reconstructions have orthotropic symmetry, which limits their accuracy. In this paper, closures and reconstruction are explored for planar fiber orientation, using a recent parameterization of fourth-order orientation tensors by Bauer and Böhlke (2021, 2022a, 2022b). We show that the orientation distribution for fibers that follow Jeffery’s equation, together with the associated natural closure, form the central axis of the space of fourth-order orientation tensors. A new family of non-orthotropic planar closure approximations is created by including the rate-of-deformation tensor as an additional independent variable. These closures are shown to be substantially more accurate than any planar orthotropic closure. We also explore the Jeffery, Bingham, ellipse radius, and fourth-order maximum entropy (ME4) functions for reconstructing the orientation distribution. When compared to distributions from the Folgar-Tucker model, ME4 using the exact fourth-order orientation tensor gives the most accurate reconstruction, while ME4 with the fourth-order tensor from the new non-orthotropic closure is nearly as good. • The Jeffery distribution is central to planar fourth-order orientation tensors. • The first-ever non-orthotropic closure approximations are developed. • Non-orthotropic closures are more accurate than any previous planar closure. • Fourth-order maximum entropy distributions give the most accurate reconstruction.