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In Gurau and Keppler 2022 (), a relation between orthogonal and symplectic tensor models with quartic interactions was proven. In this paper, we provide an alternative proof that extends to polynomial interactions of arbitrary order. We consider tensor models of order D with no symmetry under permutation of the indices that transform in the tensor product of D fundamental representations of O(N) and Sp(N). We explicitly show that the models obey the N to - N duality graph by graph in perturbation theory. [ABSTRACT FROM AUTHOR]

We propose and analyze numerical schemes for the gradient flow of Q-tensor with the quasi-entropy. The quasi-entropy is a strictly convex, rotationally invariant elementary function, giving a singular potential constraining the eigenvalues of Q within the physical range (- 1 / 3 , 2 / 3) . Compared with the potential derived from the Bingham distribution, the quasi-entropy has the same asymptotic behavior and underlying physics. Meanwhile, it is very easy to evaluate because of its simple expression. For the elastic energy, we include all the rotationally invariant terms. The numerical schemes for the gradient flow are built on the nice properties of the quasi-entropy. The first-order time discretization is uniquely solvable, keeping the physical constraints and energy dissipation, which are all independent of the time step. The second-order time discretization keeps the first two properties unconditionally, and the third with an O(1) restriction on the time step. These results also hold when we further incorporate a second-order discretization in space. Error estimates are also established for time discretization and full discretization. Numerical examples about defect patterns are presented to validate the theoretical results. [ABSTRACT FROM AUTHOR]

To enhance the accuracy results of the change detection of buildings utilizing high-resolution remote sensing (HRRS) images, a novel method was proposed by combining both tensor and deep belief network (DBN). To optimize the description of the essential characteristics for changes in buildings, a tensor-based structure covering time-space-spectrum-shadow features integrated into the model (TSSS-Cube) is proposed. The changes occurring as a combination of shadow and spectral features and spatio-temporal autocorrelation at each pixel are represented by a third-order tensor to maintain the structural information and the constraint integrity between them. Then, a restricted Boltzmann machine (TC-RBM) that can be directly used to process TSSS-Cube data is designed, and the support tensor machine (STM) is used to replace the conventional backpropagation neural network at the top of the DBN to construct a multi-tensor deep belief network (MTR-DBN) composing of multi-layer TC-RBMs and an STM classifier. Finally, the multi-layer TC-RBMs in MTR-DBN are trained layer by layer, and the global parameters of the MTR-DBN are optimized by combining a limited number of labeled data and fine-tuning the STM classifier. The implementation of both supervised and unsupervised learning methods comprehensively provides advantages to increase the accuracy result of the MTR-DBN network to detect changes. Three representative different sub-regions are selected from the whole original experimental area respectively for building change detection experiments, and a dataset composed of double-temporal HRRS images in 2012 and 2016 is used as the related experimental dataset. The experimental results show that both a change detection accuracy result with a higher average and better detection efficiency is attained by the proposed method called the MTR-DBN when compared with other similar methods. [ABSTRACT FROM AUTHOR]

Recently, research on the design of relay structure and optimal precoding matrix in massive multiple-input multiple-output (MIMO) relay systems has been proposed in many works assuming that partial or all of the channel state information (CSI) is known. However, in the practical communication systems, the CSI is unknown and needs to be estimated. In particular, channel estimation is more difficult when the relay adopts a hybrid precoding structure. In this paper, we propose an efficient channel estimation scheme based on Tucker-2 tensor model for a multi-way massive MIMO hybrid relay system. The proposed algorithm provides all the CSI involved in the communication links at each user end. Compared with traditional channel estimation algorithms, the proposed algorithm has better estimation performance and requires few iterations to achieve convergence. In addition, we derive the analytical expression of the Cramér-Rao lower bound (CRB) for reference. Simulation results verify the effectiveness of the proposed Tucker-2-based channel estimation approach. [ABSTRACT FROM AUTHOR]

In this paper, a short review of the operator/Feynman diagram/dessin d'enfants correspondence in the rank 3 tensor model is presented, and the cut and join operation is given in the language of dessin d'enfants as a straightforward development. We classify operators of the rank 3 tensor model up to level 5 with dessin d'enfants. (Based on the talk given by R. Y. at the international workshop "Randomness, Integrability and Representation Theory in Quantum Field Theory" at the Osaka City University Media Center on March 25, 2021.) [ABSTRACT FROM AUTHOR]

The advanced tensor solution to the problem of inter-domain routing with normalized Quality of Service under hierarchical coordination in a communication network is proposed in the paper. The novelty of the method based on the tensor model lies in the more flexible load balancing over the network due to the presence of requirements to average end-to-end delay of packets. The framework of the method comprises a decomposed flow-based routing model that includes the inter-domain routing interaction conditions and ensures the normalized Quality of Service derived from the tensor model. Considering the mentioned above, the advanced inter-domain Quality of Service routing task was formulated in the optimization form, using the quadratic optimality criterion. The conducted analysis of the numerical research results confirmed the efficiency and adequacy of the proposed method when the desired solutions were obtained during the finite number of iterations under a provision of the normalized Quality of Service. It should be noted that the reduced number of such iterations during the operation of the method will decrease the amount of service traffic transmitted over the network needed for obtaining the final solution in the process of inter-domain routing with normalized Quality of Service. [ABSTRACT FROM AUTHOR]

Background and Aim: The methods to detect brain activation with functional MRI (fMRI), and MRI provide a way to measure the anatomical connections which enable lightning-fast communication among neurons that specialize in different kinds of brain functions. Diffusion tensor imaging (DTI) can measure the direction of bundles of the axonal fibers which are all aligned. Besides mapping white matter fiber tracts, these methods can enable us to detect and characterize white matter disorders in diseases. The objective of this narrative review is to overview current knowledge concerning DTI as one of the prominent popular MRI techniques that provide a planned tool for comprehensive, noninvasive, functional anatomy mapping of the human brain in both research and practical field. This review summarizes the DTI development in recent years concerning the specificity and utility of this technique in brain surgery. Methods and Materials/Patients: The significance of mapping the structure of white matter tracts, constructively the brain’s wiring by visualization and characterization of white matter fasciculi in two and three dimensions enables us to profound how different brain regions are connected and how diseases affect white matter and cause neurological problems. And we noted that while DTI proposes a potent tool to study and visualize white matter, it suffers from inherent artifacts and limitations. Additionally, some materials about the origin of the DTI signal and unique information on white matter and 3D visualization of neuronal tracts have been raised. Results: This article focuses on DTI modality and its computational techniques, and investigates significant considerations in this regard. Moreover, an inspection of the white matter structure and integrity of normal and diseased brains (e.g. multiple sclerosis, stroke, aging, dementia, schizophrenia, etc.) have been raised as a clinical application of tractography. Conclusion: The utilization of advances in diffusion-tensor (DT) imaging techniques considerably enables us to map the white matter tractography (WMT) in the normal brain. These techniques impress the operative decision in a surgical operation, especially concerning cerebral neoplasms. Also, it is possible to judge with the assistance of DTI in each subject. [ABSTRACT FROM AUTHOR]

In this paper, to understand space–time dynamics in the canonical tensor model of quantum gravity for the positive cosmological constant case, we analytically and numerically study the phase profile of its exact wave function in a coordinate representation, instead of the momentum representation analyzed so far. A saddle point analysis shows that Lie group symmetric space–times are strongly favored due to abundance of continuously existing saddle points, giving an emergent fluid picture. The phase profile suggests that spatial sizes grow in "time," where sizes are measured by the tensor-geometry correspondence previously introduced using tensor rank decomposition. Monte Carlo simulations are also performed for a few small N cases by applying a re-weighting procedure to an oscillatory integral which expresses the wave function. The results agree well with the saddle point analysis, but the phase profile is subject to disturbances in a large space–time region, suggesting existence of light modes there and motivating future computations of primordial fluctuations from the perspective of canonical tensor model. [ABSTRACT FROM AUTHOR]

In this work, we present a new versatile 3D multilinear statistical face model, based on a tensor factorisation of 3D face scans, that decomposes the shapes into person and expression subspaces. Investigation of the expression subspace reveals an inherent low-dimensional substructure, and further, a star-shaped structure. This is due to two novel findings. (1) Increasing the strength of one emotion approximately forms a linear trajectory in the subspace. (2) All these trajectories intersect at a single point – not at the neutral expression as assumed by almost all prior works—but at an apathetic expression. We utilise these structural findings by reparameterising the expression subspace by the fourth-order moment tensor centred at the point of apathy. We propose a 3D face reconstruction method from single or multiple 2D projections by assuming an uncalibrated projective camera model. The non-linearity caused by the perspective projection can be neatly included into the model. The proposed algorithm separates person and expression subspaces convincingly, and enables flexible, natural modelling of expressions for a wide variety of human faces. Applying the method on independent faces showed that morphing between different persons and expressions can be performed without strong deformations. [ABSTRACT FROM AUTHOR]

This contribution is not intended as a review but, by suggestion of the editors, as a glimpse ahead into the realm of dually weighted tensor models for quantum gravity. This class of models allows one to consider a wider class of quantum gravity models, in particular one can formulate state sum models of spacetime with an intrinsic notion of foliation. The simplest one of these models is the one proposed by Benedetti and Henson [1], which is a matrix model formulation of two-dimensional Causal Dynamical Triangulations (CDT). In this paper we apply the Functional Renormalization Group Equation (FRGE) to the Benedetti-Henson model with the purpose of investigating the possible continuum limits of this class of models. Possible continuum limits appear in this FRGE approach as fixed points of the renormalization group flow where the size of the matrix acts as the renormalization scale. Considering very small truncations, we find fixed points that are compatible with analytically known results for CDT in two dimensions. By studying the scheme dependence of our results we find that precision results require larger truncations than the ones considered in the present work. We conclude that our work suggests that the FRGE is a useful exploratory tool for dually weighted matrix models. We thus expect that the FRGE will be a useful exploratory tool for the investigation of dually weighted tensor models for CDT in higher dimensions.

Background and Aim: The methods to detect brain activation with functional MRI (fMRI), and MRI provide a way to measure the anatomical connections which enable lightning-fast communication among neurons that specialize in different kinds of brain functions. Diffusion tensor imaging (DTI) can measure the direction of bundles of the axonal fibers which are all aligned. Besides mapping white matter fiber tracts, these methods can enable us to detect and characterize white matter disorders in diseases. The objective of this narrative review is to overview current knowledge concerning DTI as one of the prominent popular MRI techniques that provide a planned tool for comprehensive, noninvasive, functional anatomy mapping of the human brain in both research and practical field. This review summarizes the DTI development in recent years concerning the specificity and utility of this technique in brain surgery. Methods and Materials/Patients: The significance of mapping the structure of white matter tracts, constructively the brain’s wiring by visualization and characterization of white matter fasciculi in two and three dimensions enables us to profound how different brain regions are connected and how diseases affect white matter and cause neurological problems. And we noted that while DTI proposes a potent tool to study and visualize white matter, it suffers from inherent artifacts and limitations. Additionally, some materials about the origin of the DTI signal and unique information on white matter and 3D visualization of neuronal tracts have been raised. Results: This article focuses on DTI modality and its computational techniques, and investigates significant considerations in this regard. Moreover, an inspection of the white matter structure and integrity of normal and diseased brains (e.g. multiple sclerosis, stroke, aging, dementia, schizophrenia, etc.) have been raised as a clinical application of tractography. Conclusion: The utilization of advances in diffusion-tensor (DT) imaging techniques considerably enables us to map the white matter tractography (WMT) in the normal brain. These techniques impress the operative decision in a surgical operation, especially concerning cerebral neoplasms. Also, it is possible to judge with the assistance of DTI in each subject.

Abstract It has turned out that the tensor expansion model has better approximation to the objective function than models of the normal second Taylor expansion. This paper conducts a study of the tensor model for nonlinear equations and it includes the following: (i) a three dimensional symmetric tensor trust-region subproblem model of the nonlinear equations is presented; (ii) the three dimensional symmetric tensor is replaced by interpolating function and gradient values from the most recent past iterate, which avoids the storage of the three dimensional symmetric tensor and decreases the workload of the computer; (iii) the limited BFGS quasi-Newton update is used instead of the second Jacobian matrix, which generates an inexpensive computation of a complex system; (iv) the global convergence is proved under suitable conditions. Numerical experiments are done to show that this proposed algorithm is competitive with the normal algorithm.

The tag-based recommendation systems that are built based on tensor models commonly suffer from the data sparsity problem. In recent years, various weighted-learning approaches have been proposed to tackle such a problem. The approaches can be categorized by how a weighting scheme is used for exploiting the data sparsity – like employing it to construct a weighted tensor used for weighing the tensor model during the learning process. In this paper, we propose a new weighted-learning approach for exploiting data sparsity in tag-based item recommendation system. We introduce a technique to represent the users’ tag preferences for leveraging the weighted-learning approach. The key idea of the proposed technique comes from the fact that users use different choices of tags to annotate the same item while the same tag may be used to annotate various items in tag-based systems. This points out that users’ tag usage likeliness is different and therefore their tag preferences are also different. We then present three novel weighting schemes that are varied in manners by how the ordinal weighting values are used for labelling the users’ tag preferences. As a result, three weighted tensors are generated based on each scheme. To implement the proposed schemes for generating item recommendations, we develop a novel weighted-learning method called as WRank (Weighted Rank). Our experiments show that considering the users' tag preferences in the tensor-based weighting-learning approach can solve the data sparsity problem as well as improve the quality of recommendation. [ABSTRACT FROM AUTHOR]

Recently, more and more multi-view knowledge graphs with various important attributes are being constructed and applied, such as temporal information, geolocation, commonsense knowledge, multi-language and multi-modal. Knowledge graphs (KGs) have been widely applied in question answering systems, drug discovery, information retrieval, and so on. Knowledge graph representation learning (KGRL) is emerging as the most effective approach to tasks such as knowledge graph completion, entity alignment, knowledge reasoning and knowledge query. However, most previous work on KGs and KGRL are mainly designed for traditional KGs and only consider single-view, and the KGRL methods for special multi-view KGs faces the challenges of lacking a unified framework, insufficient interactive information and underutilizing important attributes. To bridge this gap, in this paper, we proposes a multi-view knowledge graphs (e.g., temporal KGs) representation learning method based on tensor tucker model (MvTuckER), which models the multi-view KGs from tuples to a n th-order binary tensor, and adopts the tensor n -mode product to capture complete interaction between and within different views. We also introduce low-rank and sparse approximations of core tensors to balance expressivity and complexity of MvTuckER. Moreover, we theoretically illustrate that our model is fully expressive for modeling multi-view KGs, discuss the working mechanism of MvTuckER from the perspective of logical operation, and explain how to migrate and extend our method to other types of views. We conducted extensive experiments on three multi-view knowledge graphs, and obtain 4.7%, 2.7%, and 5.3% effective improvement in Hit@1 respectively, which proves that the proposed MvTuckER can achieve state-of-the-art performance. • The various views are treated equally and encoded into a unified latent space. • Easily migrate to other views and introduce more views. • Low-rank approximation of core tensor guarantees support for largerscale KGs. • Fully expressive for the multi-view KGs, and interpretability. [ABSTRACT FROM AUTHOR]

The advanced tensor solution to the problem of inter-domain routing with normalized Quality of Service under hierarchical coordination in a communication network is proposed in the paper. The novelty of the method based on the tensor model lies in the more flexible load balancing over the network due to the presence of requirements to average end-to-end delay of packets. The framework of the method comprises a decomposed flow-based routing model that includes the inter-domain routing interaction conditions and ensures the normalized Quality of Service derived from the tensor model. Considering the mentioned above, the advanced inter-domain Quality of Service routing task was formulated in the optimization form, using the quadratic optimality criterion. The conducted analysis of the numerical research results confirmed the efficiency and adequacy of the proposed method when the desired solutions were obtained during the finite number of iterations under a provision of the normalized Quality of Service. It should be noted that the reduced number of such iterations during the operation of the method will decrease the amount of service traffic transmitted over the network needed for obtaining the final solution in the process of inter-domain routing with normalized Quality of Service.

Deformation prediction is the basis of deformation control in manufacturing process planning. This paper presents an on-line part deformation prediction method using a deep learning model during numerical control machining process, which is different from traditional methods based on finite element simulation of stress release prior to the actual machining process. A fourth-order tensor model is proposed to represent the continuous part geometric information, process information, and monitoring information, which is used as the input to the deep learning model. A deep learning framework with a conventional neural network and a recurrent neural network has been constructed and trained by monitored deformation data and process information associated with interim part geometric information. The proposed method can be generalised for different parts with certain similarities and has the potential to provide a reference for an adaptive machining control strategy for reducing part deformation. The proposed method was validated by actual machining experiments, and the results show that the prediction accuracy has been improved compared with existing methods. Furthermore, this paper shifts the difficult problem of residual stress measurement and off-line deformation prediction to the solution of on-line deformation prediction based on deformation monitoring data. [ABSTRACT FROM AUTHOR]

A new subspace minimization conjugate gradient method based on tensor model is proposed and analysed. If the objective function is close to a quadratic, we construct a quadratic approximation model in a two-dimensional subspace to generate the search direction; otherwise, we construct a tensor model. It is remarkable that the search direction satisfies the sufficient descent property. We prove the global convergence of the proposed method under mild assumptions. Numerical comparisons are given with well-known CGOPT and CG_DESCENT and show that the proposed algorithm is very promising. [ABSTRACT FROM AUTHOR]

A new type of stepsize, which was recently introduced by Liu et al. (Optimization 67(3):427–440, 2018), is called approximately optimal stepsize and is very efficient for gradient method. Interestingly, all gradient methods can be regarded as gradient methods with approximately optimal stepsizes. In this paper, we present an efficient gradient method with approximately optimal stepsize based on tensor model for unconstrained optimization. In the proposed method, if the objective function is not close to a minimizer and a quadratic function on a line segment between the current and latest iterates, then a tensor model is exploited to generate approximately optimal stepsize for gradient method. Otherwise, quadratic approximation models are constructed to generate approximately optimal stepsizes for gradient method. The global convergence of the proposed method is established under weak conditions. Numerical results indicate that the proposed method is very promising. [ABSTRACT FROM AUTHOR]