Higher-order Tensors Research Articles

Midcourse correction of Earth-Moon distant retrograde orbit transfer trajectories based on high-order state transition tensors

Midcourse correction of Earth-Moon distant retrograde orbit transfer trajectories based on high-order state transition tensors

The Temperature-Induced Deflection Data Missing Recovery of a Cable-Stayed Bridge Based on Bayesian Robust Tensor Learning

Changes in the deflection of cable-stayed bridges due to thermal effects may adversely affect the bridge structure and reflect the degradation of bridge performance. Therefore, complete deflection field data are important for bridge health monitoring. A strong linear correlation has been found between temperature-induced deflections in different positions of the same span of a cable-stayed bridge in many studies, which make the deflection data matrix/tensor have a low-rank structure. Therefore, it is appropriate to use a low-rank matrix/tensor learning to model the temperature–deflection field of a cable-stayed bridge. Moreover, to avoid disturbing the recovery results via abnormal data (e.g., baseline shift and outliers), a Bayesian robust tensor learning method is proposed to extract the spatio-temporal characteristics of the bridge temperature–deflection field. The missing data recovery and abnormal data cleaning are achieved simultaneously in the process of reconstructing the temperature-induced field via tensor learning. The performance of the method is verified with actual continuous monitoring data from a cable-stayed bridge. The experimental results show that low-order tensor (i.e., matrix) learning has a good recovery and cleaning performance. The extension to higher-order tensor learning is proposed to extract the spatial symmetry of the sensor locations, which is experimentally proven to have better missing recovery and abnormal data cleaning performance.

Automatic tractography and segmentation using finsler geometry based on higher-order tensor fields

Background and objective: We focus on three-dimensional higher-order tensorial (HOT) images using Finsler geometry. In biomedical image analysis, these images are widely used, and they are based on the diffusion profiles inside the voxels. The diffusion information is stored in the so-called diffusion tensor D. Our objective is to present new methods revealing the architecture of neural fibers in presence of crossings and high curvatures. After tracking the fibers, we achieve direct 3D image segmentation to analyse the brain’s white matter structures. Methods: To deal with the construction of the underlying fibers, the inverse of the second-order diffusion tensor D, understood as the metric tensor D−1, is commonly used in DTI modality. For crossing and highly curved fibers, higher order tensors are more relevant, but it is challenging to find an analogue of such an inverse in the HOT case. We employ an innovative approach to metrics based on higher order tensors to track the fibers properly. We propose to feed the tracked fibers as the internal initial contours in an efficient version of 3D segmentation. Results: We propose a brand-new approach to the inversion of a diffusion HOT, and an effective way of fiber tracking in the Finsler setting, based on innovative classification of the individual voxels. Thus, we can handle complex structures with high curvatures and crossings, even in the presence of noise. Based on our novel tractography approach, we also introduce a new segmentation method. We feed the detected fibers as the initial position of the contour surfaces to segment the image using a relevant active contour method (i.e., initiating the segmentation from inside the structures). Conclusions: This is a pilot work, enhancing methods for fiber tracking and segmentation. The implemented algorithms were successfully tested on both synthetic and real data. The new features make our algorithms robust and fast, and they allow distinguishing individual objects in complex structures, even under noise.

Tracking online low-rank approximations of higher-order incomplete streaming tensors

In this paper, we propose two new provable algorithms for tracking online low-rank approximations of high-order streaming tensors with missing data. The first algorithm, dubbed adaptive Tucker decomposition (ATD), minimizes a weighted recursive least-squares cost function to obtain the tensor factors and the core tensor in an efficient way, thanks to an alternating minimization framework and a randomized sketching technique. Under the canonical polyadic (CP) model, the second algorithm, called ACP, is developed as a variant of ATD when the core tensor is imposed to be identity. Both algorithms are low-complexity tensor trackers that have fast convergence and low memory storage requirements. A unified convergence analysis is presented for ATD and ACP to justify their performance. Experiments indicate that the two proposed algorithms are capable of streaming tensor decomposition with competitive performance with respect to estimation accuracy and runtime on both synthetic and real data.

Rigorous biaxial limit of a molecular-theory-based two-tensor hydrodynamics

We consider a two-tensor hydrodynamics derived from the molecular model, where high-order tensors are determined by closure approximation through the maximum entropy state or the quasi-entropy. We prove the existence and uniqueness of local in time smooth solutions to the two-tensor system. Then, we rigorously justify the connection between the molecular-theory-based two-tensor hydrodynamics and the biaxial frame hydrodynamics. More specifically, in the framework of Hilbert expansion, we show the convergence of the solution to the two-tensor hydrodynamics to the solution to the frame hydrodynamics.

Optimizing Tensor Programs on Flexible Storage

Tensor programs often need to process large tensors (vectors, matrices, or higher order tensors) that require a specialized storage format for their memory layout. Several such layouts have been proposed in the literature, such as the Coordinate Format, the Compressed Sparse Row format, and many others, that were especially designed to optimally store tensors with specific sparsity properties. However, existing tensor processing systems require specialized extensions in order to take advantage of every new storage format. In this paper we describe a system that allows users to define flexible storage formats in a declarative tensor query language, similar to the language used by the tensor program. The programmer only needs to write storage mappings, which describe, in a declarative way, how the tensors are laid out in main memory. Then, we describe a cost-based optimizer that optimizes the tensor program for the specific memory layout. We demonstrate empirically significant performance improvements compared to state-of-the-art tensor processing systems.

Research on multi-context aware recommendation methods based on tensor factorization

Compared to the traditional recommender systems, context-aware recommender systems are more in line with actual application contexts. However, the existing researches are mostly focused on single context-aware recommendation, such as time-aware recommendation or location-aware recommendation, and lack of in-depth research on multi-context-aware recommendation. Therefore, we proposed a recommendation method of high-order tensor factorization based on multi-context-aware. First, on the basis of analyzing the influence of context on users’ interest preferences, the sensitivity of users to multiple contexts was detected using statistical methods. For context-sensitive users, four-dimensional tensors and feature matrices used to solve data sparsity were constructed based on rating matrix and situational information. And then the stochastic gradient descent algorithm was used for iterative calculation to fill in missing data values and carry out parameter optimization. For context-insensitive users, we used matrix factorization to predict users’ interest preferences. Finally, we tested and validated our method on a multi-context-aware movie dataset, and the experimental results show that the proposed method could effectively reduce the prediction error and improve the recommendation quality.

DHOST gravity in ultra-diffuse galaxies—Part II: NGC 1052-DF4 and Dragonfly 44

Ultra-Diffuse galaxies are a family of gravitational systems with quite varied properties, counting both objects highly deficient in dark matter and others which are dark matter-dominated. Such a variety of behaviors might be a problem for both the standard dark matter paradigm and for alternative theories of gravity. Here we consider two systems, NGC1052-DF4 and Dragonfly 44, which are respectively dark matter deficient and dark matter dominated, in the context of Degenerate Higher-Order Scalar Tensor theories exhibiting a partial breaking of the Vaishtein screening mechanism for which they might have an influence not only on cosmological scales but also on astrophysical ones, thus mimicking dark matter. For NGC1052-DF4 the galaxy dynamics might be successfully described only by a stellar component and, at least at the scales which are probed, the content of dark matter is quite low. We also show that our alternative gravity model is consistent with data and is statistically equivalent to a standard General Relativity dark matter scenario, and it might even replace dark matter. On the contrary, Dragonfly 44 requires dark matter both in General Relativity and in our extended theory. When the latter is considered only as a cosmological dark energy fluid, it is statistically fully reliable and equivalent to General Relativity. But when we try to use it to substitute dark matter, although we get good fits to the data, the constraints on the theoretical parameters are in sharp contrast with those derived from more stringent probes from the stellar scales.

Tensor network based finite-size scaling for two-dimensional Ising model

We propose a scheme to perform tensor network based finite-size scaling analysis for two-dimensional classical models. In the tensor network representation of the partition function, we use higher-order tensor renormalization group (HOTRG) method to coarse grain the weight tensor. The renormalized tensor is then used to construct the approximated transfer matrix of an infinite strip of finite width. By diagonalizing the transfer matrix we obtain the correlation length, the magnetization, and the energy density which are used in finite-size scaling analysis to determine the critical temperature and the critical exponents. As a benchmark we study the two-dimensional classical Ising model. We show that the critical temperature and the critical exponents can be accurately determined. With HOTRG bond dimension $D=70$, the absolute errors of the critical temperature $T_c$ and the critical exponent $\nu$, $\beta$ are at the order of $10^{-7}, 10^{-5}$, $10^{-4}$ respectively. Furthermore, the results can be systematically improved by increasing the bond dimension of the HOTRG method. Finally, we study the length scale induced by the finite cut-off in bond dimension and elucidate its physical meaning in this context.

Determination of Klinkenberg and higher-order correction tensors for slip flow in porous media

The flow fields in a porous medium under no slip conditions contain all the physical information to predict the Klinkenberg slip correction tensor, without requiring the solution of additional problems. For situations in which higher-order terms are needed to predict the apparent permeability tensor, the flow problem with no-slip and considering a first-order slip are enough to build an excellent estimate using a Pad\'e approximant.

Detecting moving targets in active sonar echograph of harbor environment using robust high-order flux tensor.

An important goal of an active sonar system is to detect and track underwater intruders such as frogmen, unmanned underwater vehicles, etc. Unfortunately, the intruders appear visually as a small fluctuating "blob" against the high-level fluctuating background caused by multipath propagation and reverberation in the harbor environment, making it difficult to be distinguished. Classical motion features well developed in computer vision cannot cope with an underwater environment. Thus, this paper presents a robust high-order flux tensor (RHO-FT) to characterize the small underwater moving targets against high-level fluctuating background. According to the dynamic behavior of active clutter from real-world harbor environment, we first classify it into two main types: (1) dynamic clutter but with relatively consistent spatial-temporal variation in a certain neighborhood; (2) sparkle clutter presenting completely random flashing. Then starting from the classical flux tensor, we develop a statistical high-order computation to handle the former followed by a spatial-temporal connected component to suppress the latter to achieve higher robustness. Experiments on a set of real-world harbor datasets demonstrate the effectiveness of our RHO-FT.

Robust to Rank Selection: Low-Rank Sparse Tensor-Ring Completion.

Tensor-ring (TR) decomposition was recently studied and applied for low-rank tensor completion due to its powerful representation ability of high-order tensors. However, most of the existing TR-based methods tend to suffer from deterioration when the selected rank is larger than the true one. To address this issue, this article proposes a new low-rank sparse TR completion method by imposing the Frobenius norm regularization on its latent space. Specifically, we theoretically establish that the proposed method is capable of exploiting the low rankness and Kronecker-basis-representation (KBR)-based sparsity of the target tensor using the Frobenius norm of latent TR-cores. We optimize the proposed TR completion by block coordinate descent (BCD) algorithm and design a modified TR decomposition for the initialization of this algorithm. Extensive experimental results on synthetic data and visual data have demonstrated that the proposed method is able to achieve better results compared to the conventional TR-based completion methods and other state-of-the-art methods and, meanwhile, is quite robust even if the selected TR-rank increases.

Stress-Charge Nonlinear Physical Description and Tensor Symmetries for Piezoelectric Materials.

Nonlinear piezoelectric materials are raised as a great replacement for devices that require low power consumption, high sensitivity, and accurate transduction, fitting with the demanding requirements of new technologies such as the Fifth-Generation of telecommunications (5G), the Internet of Things (IoT), and modern radio frequency (RF) applications. In this work, the state equations that correctly predict the nonlinear piezoelectric phenomena observed experimentally are presented. Furthermore, we developed a fast methodology to implement the state equations in the main FEM simulation software, allowing an easy design and characterization of this type of device, as the symmetry structures for high-order tensors are shown and explained. The operation regime of each high-order tensor is discussed and connected with the main nonlinear phenomena reported in the literature. Finally, to demonstrate our theoretical deductions, we used the experimental measurements, which presented the nonlinear effects, which were reproduced through simulations, obtaining maximum percent errors for the effective elasticity constants, relative effective permittivity, and resonance frequencies of 0.79%, 2.9%, and 0.3%, respectively, giving a proof of the potential of the nonlinear state equations presented for the unifying of all nonlinear phenomena observed in the piezoelectric devices.

Local and global magnetization on the Sierpiński carpet.

The phase transition of the classical Ising model on the Sierpiński carpet, which has the fractal dimension log_{3}^{}8≈1.8927, is studied by an adapted variant of the higher-order tensor renormalization group method. The second-order phase transition is observed at the critical temperature T_{c}^{}≈1.478. Position dependence of local functions is studied through impurity tensors inserted at different locations on the fractal lattice. The critical exponent β associated with the local magnetization varies by two orders of magnitude, depending on lattice locations, whereas T_{c}^{} is not affected. Furthermore, we employ automatic differentiation to accurately and efficiently compute the average spontaneous magnetization per site as a first derivative of free energy with respect to the external field, yielding the global critical exponent of β≈0.135.

High-efficiency 3D black-blood thoracic aorta imaging with patch-based low-rank tensor reconstruction.

Three-dimensional (3D) black-blood (BB) vessel wall imaging is a promising noninvasive imaging technique for assessing thoracic aortic diseases. We aimed to develop and evaluate a fast thoracic aorta vessel wall imaging method with patch-based low-rank tensor (Pt-LRT) reconstruction using the 3D-modulated variable flip angle fast-spin echo (vFA-FSE) sequence. The Pt-LRT technique adopts a low-rank tensor image model with regularization to explore the local low-rankness and nonlocal redundancies of the images to assess the thoracic aorta vessel wall. It uses high-order tensors to capture correlations between data in multiple dimensions and reconstructs images from highly undersampled data. For this study, 12 healthy participants and 2 patients with thoracic aortic diseases were evaluated at 3T magnetic resonance (MR). The reconstruction results were compared to the traditional generalized autocalibrating partially parallel acquisitions (GRAPPA) and ℓ1-SPIRiT reconstruction to assess the feasibility of the proposed framework. Quantitative analyses of the vessel wall thickness (VWT), internal diameter (ID), lumen area (LA), and contrast-to-noise ratio (CNR) between the lumen and vessel wall were performed on all healthy participants. Results demonstrated no significant differences between the GRAPPA and the proposed Pt-LRT in VWT, ID, or LA of the aorta (P<0.05). A higher mean CNR was attained with 3D patch-based low-rank tensor reconstruction than with ℓ1-SPIRiT reconstruction (49.4±10.8 vs. 38.9±8.2). The proposed 3D BB thoracic aorta vessel wall imaging method can reduce the scan time and produce an image quality that is in good agreement with the conventional GRAPPA acquisition, which takes approximately more than 8 min. This study also shows that the proposed Pt-LRT method substantially improves the visualization and sharpness of the vessel wall and the definition of the tissue boundary compared to the imaging obtained with ℓ1-SPIRiT.

Calculation of critical exponents on fractal lattice Ising model by higher-order tensor renormalization group method.

The critical behavior of the Ising model on a fractal lattice, which has the Hausdorff dimension log_{4}12≈1.792, is investigated using a modified higher-order tensor renormalization group algorithm supplemented with automatic differentiation to compute relevant derivatives efficiently and accurately. The complete set of critical exponents characteristic of a second-order phase transition was obtained. Correlations near the critical temperature were analyzed through two impurity tensors inserted into the system, which allowed us to obtain the correlation lengths and calculate the critical exponent ν. The critical exponent α was found to be negative, consistent with the observation that the specific heat does not diverge at the critical temperature. The extracted exponents satisfy the known relations given by various scaling assumptions within reasonable accuracy. Perhaps most interestingly, the hyperscaling relation, which contains the spatial dimension, is satisfied very well, assuming the Hausdorff dimension takes the place of the spatial dimension. Moreover, using automatic differentiation, we have extracted four critical exponents (α, β, γ,andδ) globally by differentiating the free energy. Surprisingly, the global exponents differ from those obtained locally by the technique of the impurity tensors; however, the scaling relations remain satisfied even in the case of the global exponents.

Wilson’s disease classification using higher-order Gabor tensors and various classifiers on a small and imbalanced brain MRI dataset

Wilson’s disease classification using higher-order Gabor tensors and various classifiers on a small and imbalanced brain MRI dataset

A Novel Method of Efficient Max-min Metric for Classification

Distance metric learning is an important method to study distance metrics that reflect the interaction between features and labels. Because of the high computational complexity and the fact that existing studies on algorithms that measure the similarities with Euclidean distances cannot reflect the real correlations between pairs of samples, learning a suitable distance metric is quite demanding for many data mining tasks. This paper innovatively proposes an extended efficient max-min metric (EMM) that maximizes the total distance between different pairs and minimizes the total distance between similar pairs as much as possible. Simultaneously, the adoption of the local preserving projection framework changes the solution process of the algorithm and improves the speed of the algorithm without losing accuracy. Because traditional EMM only considers pairwise constraints and ignores sample distribution, this study extends EMM based on sample distribution and successfully solves the multi-manifold problem. In the process of data realization, compared with the vector representation method, the use of high-order tensors will make the image representation more accurate and natural. To maintain the structure of higher-order tensors, a tensor-efficient max-min metric (TEMM) is proposed. In order to prove the accuracy and superiority of the research method in this paper, a large number of experiments have been carried out on image processing. The experimental results show that the method proposed in this paper has a good effect.

Tensor slow feature analysis and its applications for batch process monitoring

How to establish efficient monitoring models for complex batch processes is a challenging problem because of the three-dimensional (3-D) data array and serious dynamics. The unfolding operation in most methods would destroy the batch data structure and increase the dimensions of modeling data obviously. Meanwhile, it is more practical to perform within-batch detection rather than end-of-batch detection. In this work, a novel high-order tensor slow feature analysis model is proposed to handle the 3-D and dynamical issues simultaneously. The proposed model can be solved by iteratively tackling two sub-optimal problems. Within-batch detection is performed based on the properly defined monitoring statistics to timely detect the abnormal situation. The multi-phase property is further considered in this work to improve detection accuracy. A simulated case and an industrial case are taken to show the merits of the proposed method.

Rank-Adaptive Time Integration of Tree Tensor Networks

A rank-adaptive integrator for the approximate solution of high-order tensor differential equations by tree tensor networks is proposed and analyzed. In a recursion from the leaves to the root, the integrator updates bases and then evolves connection tensors by a Galerkin method in the augmented subspace spanned by the new and old bases. This is followed by rank truncation within a specified error tolerance. The memory requirements are linear in the order of the tensor and linear in the maximal mode dimension. The integrator is robust to small singular values of matricizations of the connection tensors. Up to the rank truncation error, which is controlled by the given error tolerance, the integrator preserves norm and energy for Schrödinger equations, and it dissipates the energy in gradient systems. Numerical experiments with a basic quantum spin system illustrate the behavior of the proposed algorithm.