Journal articles on the topic 'Tensors methods'
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1
Katoch, Nitish, Bup-Kyung Choi, Ji-Ae Park, In-Ok Ko, and Hyung-Joong Kim. "Comparison of Five Conductivity Tensor Models and Image Reconstruction Methods Using MRI." Molecules 26, no. 18 (September 10, 2021): 5499. http://dx.doi.org/10.3390/molecules26185499.
Abstract:
Imaging of the electrical conductivity distribution inside the human body has been investigated for numerous clinical applications. The conductivity tensors of biological tissue have been obtained from water diffusion tensors by applying several models, which may not cover the entire phenomenon. Recently, a new conductivity tensor imaging (CTI) method was developed through a combination of B1 mapping, and multi-b diffusion weighted imaging. In this study, we compared the most recent CTI method with the four existing models of conductivity tensors reconstruction. Two conductivity phantoms were designed to evaluate the accuracy of the models. Applied to five human brains, the conductivity tensors using the four existing models and CTI were imaged and compared with the values from the literature. The conductivity image of the phantoms by the CTI method showed relative errors between 1.10% and 5.26%. The images by the four models using DTI could not measure the effects of different ion concentrations subsequently due to prior information of the mean conductivity values. The conductivity tensor images obtained from five human brains through the CTI method were comparable to previously reported literature values. The images by the four methods using DTI were highly correlated with the diffusion tensor images, showing a coefficient of determination (R2) value of 0.65 to 1.00. However, the images by the CTI method were less correlated with the diffusion tensor images and exhibited an averaged R2 value of 0.51. The CTI method could handle the effects of different ion concentrations as well as mobilities and extracellular volume fractions by collecting and processing additional B1 map data. It is necessary to select an application-specific model taking into account the pros and cons of each model. Future studies are essential to confirm the usefulness of these conductivity tensor imaging methods in clinical applications, such as tumor characterization, EEG source imaging, and treatment planning for electrical stimulation.
2
Wang, Hai Jun, Fei Yun Xu, and Fei Wang. "Tensor Factorization and Clustering for the Feature Extraction Based on Tucker3 with Updating Core." Advanced Materials Research 308-310 (August 2011): 2517–22. http://dx.doi.org/10.4028/www.scientific.net/amr.308-310.2517.
Abstract:
Aiming at the problems of Tucker3 to large-scale tensor when applied to feature extraction, a new factorization based on Tucker3 is proposed to extract feature from the tensors. First, the large-scale tensor is divided into multiple sub-tensors so as to conveniently compute cores of sub-tensors in parallel mode with Matlab Parallel Computing Toolbox; Then, the cores of each sub-tensor are updated for reducing deviation in calculating and the similar characteristics of sub-tensors are clustered to obtain the features. Experiment results show that this methods is able to extract features rapidly and efficiently.
3
Moes, H., E. G. Sikkes, and R. Bosma. "Mobility and Impedance Tensor Methods for Full and Partial-Arc Journal Bearings." Journal of Tribology 108, no. 4 (October 1, 1986): 612–19. http://dx.doi.org/10.1115/1.3261282.
Abstract:
Mobility and impedance tensors are introduced for full journal bearings. These tensors may replace the well known mobility and impedance vectors. Since tensors apply to arbitrary systems of reference, coordinates rotating with the sleeve, i.e. fixed coordinates, will be introduced and will henceforth replace the unidirectional system in current use. As a consequence, the restriction to full journal bearing applications which was necessary up to now may be withdrawn. The mobility- and impedance methods, as derived for full journal bearings, from now on apply equally well to partial-arc bearings. The mobility- and impedance tensor descriptions, needed in applications of the methods, can be derived in a straightforward manner form the generally applied vector descriptions for full journal bearings. Descriptions for partial-arc bearings will also be presented.
4
Yang, Hye-Kyung, and Hwan-Seung Yong. "Multi-Aspect Incremental Tensor Decomposition Based on Distributed In-Memory Big Data Systems." Journal of Data and Information Science 5, no. 2 (May 20, 2020): 13–32. http://dx.doi.org/10.2478/jdis-2020-0010.
Abstract:
AbstractPurposeWe propose InParTen2, a multi-aspect parallel factor analysis three-dimensional tensor decomposition algorithm based on the Apache Spark framework. The proposed method reduces re-decomposition cost and can handle large tensors.Design/methodology/approachConsidering that tensor addition increases the size of a given tensor along all axes, the proposed method decomposes incoming tensors using existing decomposition results without generating sub-tensors. Additionally, InParTen2 avoids the calculation of Khari–Rao products and minimizes shuffling by using the Apache Spark platform.FindingsThe performance of InParTen2 is evaluated by comparing its execution time and accuracy with those of existing distributed tensor decomposition methods on various datasets. The results confirm that InParTen2 can process large tensors and reduce the re-calculation cost of tensor decomposition. Consequently, the proposed method is faster than existing tensor decomposition algorithms and can significantly reduce re-decomposition cost.Research limitationsThere are several Hadoop-based distributed tensor decomposition algorithms as well as MATLAB-based decomposition methods. However, the former require longer iteration time, and therefore their execution time cannot be compared with that of Spark-based algorithms, whereas the latter run on a single machine, thus limiting their ability to handle large data.Practical implicationsThe proposed algorithm can reduce re-decomposition cost when tensors are added to a given tensor by decomposing them based on existing decomposition results without re-decomposing the entire tensor.Originality/valueThe proposed method can handle large tensors and is fast within the limited-memory framework of Apache Spark. Moreover, InParTen2 can handle static as well as incremental tensor decomposition.
5
Moore, J. G., S. A. Schorn, and J. Moore. "Education Committee Best Paper of 1995 Award: Methods of Classical Mechanics Applied to Turbulence Stresses in a Tip Leakage Vortex." Journal of Turbomachinery 118, no. 4 (October 1, 1996): 622–29. http://dx.doi.org/10.1115/1.2840917.
Abstract:
Moore et al. measured the six Reynolds stresses in a tip leakage vortex in a linear turbine cascade. Stress tensor analysis, as used in classical mechanics, has been applied to the measured turbulence stress tensors. Principal directions and principal normal stresses are found. A solid surface model, or three-dimensional glyph, for the Reynolds stress tensor is proposed and used to view the stresses throughout the tip leakage vortex. Modeled Reynolds stresses using the Boussinesq approximation are obtained from the measured mean velocity strain rate tensor. The comparison of the principal directions and the three-dimensional graphic representations of the strain and Reynolds stress tensors aids in the understanding of the turbulence and what is required to model it.
6
Xue, Zhaohui, Sirui Yang, Hongyan Zhang, and Peijun Du. "Coupled Higher-Order Tensor Factorization for Hyperspectral and LiDAR Data Fusion and Classification." Remote Sensing 11, no. 17 (August 21, 2019): 1959. http://dx.doi.org/10.3390/rs11171959.
Abstract:
Hyperspectral and light detection and ranging (LiDAR) data fusion and classification has been an active research topic, and intensive studies have been made based on mathematical morphology. However, matrix-based concatenation of morphological features may not be so distinctive, compact, and optimal for classification. In this work, we propose a novel Coupled Higher-Order Tensor Factorization (CHOTF) model for hyperspectral and LiDAR data classification. The innovative contributions of our work are that we model different features as multiple third-order tensors, and we formulate a CHOTF model to jointly factorize those tensors. Firstly, third-order tensors are built based on spectral-spatial features extracted via attribute profiles (APs). Secondly, the CHOTF model is defined to jointly factorize the multiple higher-order tensors. Then, the latent features are generated by mode-n tensor-matrix product based on the shared and unshared factors. Lastly, classification is conducted by using sparse multinomial logistic regression (SMLR). Experimental results, conducted with two popular hyperspectral and LiDAR data sets collected over the University of Houston and the city of Trento, respectively, indicate that the proposed framework outperforms the other methods, i.e., different dimensionality-reduction-based methods, independent third-order tensor factorization based methods, and some recently proposed hyperspectral and LiDAR data fusion and classification methods.
7
Hajarian, Masoud. "Solving coupled tensor equations via higher order LSQR methods." Filomat 34, no. 13 (2020): 4419–27. http://dx.doi.org/10.2298/fil2013419h.
Abstract:
Tensors have a wide application in control theory, data mining, chemistry, information sciences, documents analysis and medical engineering. The material here is motivated by the development of the efficient numerical methods for solving the coupled tensor equations (A1*M X *N B1 + C1 *M Y *N D1 = E1, A2 *M X *N B2 + C2 *M Y *N D2 = E2, with Einstein product. We propose the tensor form of the LSQR methods to find the solutions X and Y of the coupled tensor equations. Finally we give some numerical examples to illustrate that our proposed methods are able to accurately and efficiently find the solutions of tensor equations with Einstein product.
8
Zhong, Guoqiang, and Mohamed Cheriet. "Large Margin Low Rank Tensor Analysis." Neural Computation 26, no. 4 (April 2014): 761–80. http://dx.doi.org/10.1162/neco_a_00570.
Abstract:
We present a supervised model for tensor dimensionality reduction, which is called large margin low rank tensor analysis (LMLRTA). In contrast to traditional vector representation-based dimensionality reduction methods, LMLRTA can take any order of tensors as input. And unlike previous tensor dimensionality reduction methods, which can learn only the low-dimensional embeddings with a priori specified dimensionality, LMLRTA can automatically and jointly learn the dimensionality and the low-dimensional representations from data. Moreover, LMLRTA delivers low rank projection matrices, while it encourages data of the same class to be close and of different classes to be separated by a large margin of distance in the low-dimensional tensor space. LMLRTA can be optimized using an iterative fixed-point continuation algorithm, which is guaranteed to converge to a local optimal solution of the optimization problem. We evaluate LMLRTA on an object recognition application, where the data are represented as 2D tensors, and a face recognition application, where the data are represented as 3D tensors. Experimental results show the superiority of LMLRTA over state-of-the-art approaches.
9
Shi, Qiquan, Jiaming Yin, Jiajun Cai, Andrzej Cichocki, Tatsuya Yokota, Lei Chen, Mingxuan Yuan, and Jia Zeng. "Block Hankel Tensor ARIMA for Multiple Short Time Series Forecasting." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 04 (April 3, 2020): 5758–66. http://dx.doi.org/10.1609/aaai.v34i04.6032.
Abstract:
This work proposes a novel approach for multiple time series forecasting. At first, multi-way delay embedding transform (MDT) is employed to represent time series as low-rank block Hankel tensors (BHT). Then, the higher-order tensors are projected to compressed core tensors by applying Tucker decomposition. At the same time, the generalized tensor Autoregressive Integrated Moving Average (ARIMA) is explicitly used on consecutive core tensors to predict future samples. In this manner, the proposed approach tactically incorporates the unique advantages of MDT tensorization (to exploit mutual correlations) and tensor ARIMA coupled with low-rank Tucker decomposition into a unified framework. This framework exploits the low-rank structure of block Hankel tensors in the embedded space and captures the intrinsic correlations among multiple TS, which thus can improve the forecasting results, especially for multiple short time series. Experiments conducted on three public datasets and two industrial datasets verify that the proposed BHT-ARIMA effectively improves forecasting accuracy and reduces computational cost compared with the state-of-the-art methods.
10
DE AZCÁRRAGA, J. A., and A. J. MACFARLANE. "COMPILATION OF RELATIONS FOR THE ANTISYMMETRIC TENSORS DEFINED BY THE LIE ALGEBRA COCYCLES OF su(n)." International Journal of Modern Physics A 16, no. 08 (March 30, 2001): 1377–405. http://dx.doi.org/10.1142/s0217751x01003111.
Abstract:
This paper attempts to provide a comprehensive compilation of results, many new here, involving the invariant totally antisymmetric tensors (Omega tensors) which define the Lie algebra cohomology cocycles of su (n), and that play an essential role in the optimal definition of Racah–Casimir operators of su (n). Since the Omega tensors occur naturally within the algebra of totally antisymmetrized products of λ-matrices of su (n), relations within this algebra are studied in detail, and then employed to provide a powerful means of deriving important Omega tensor/cocycle identities. The results include formulas for the squares of all the Omega tensors of su (n). Various key derivations are given to illustrate the methods employed.
11
Foulger, Gillian R., and Bruce R. Julian. "Earthquakes and errors: Methods for industrial applications." GEOPHYSICS 76, no. 6 (November 2011): WC5—WC15. http://dx.doi.org/10.1190/geo2011-0096.1.
Abstract:
The high accuracies and realistic confidence assessments demanded for seismic monitoring of hydraulic fracturing work require specialist experimental approaches. These include seismic network design based on quantitative modeling, high-quality instrument deployments, and accurate and detailed crustal models. Confidence estimates must take into account uncertainties about crustal structure, which may dominate error budgets. Earthquake size should be expressed in terms of scalar seismic moment or the associated moment magnitude [Formula: see text], which is related to fundamental physical source processes, and not as traditional earthquake magnitudes. Representing earthquake mechanisms in terms of seismic moment tensors allows for processes such as volume changes and complex types of shearing that are important in hydrocarbon and geothermal reservoirs. Traditional fault-plane solutions are based on simplifying assumptions such as shear slip on a planar faults, and isotropic crustal structures, which may introduce large uncertainties. Quantitative assessment of confidence regions for moment-tensor source mechanisms, a newly emerging field, is important for distinguishing computational artifacts from real physical phenomena. We review methods currently available for realistic error estimation for earthquake locations and moment tensors, with particular emphasis on surface sensor arrays in geothermal areas.
12
Breuer, Kevin, Markus Stommel, and Wolfgang Korte. "Analysis and Evaluation of Fiber Orientation Reconstruction Methods." Journal of Composites Science 3, no. 3 (July 4, 2019): 67. http://dx.doi.org/10.3390/jcs3030067.
Abstract:
The calculation of the fiber orientation of short fiber-reinforced plastics with the Fokker–Planck equation requires a considerable numerical effort, which is practically not feasible for injection molding simulations. Therefore, only the fiber orientation tensors are determined, i.e., by the Folgar–Tucker equation, which requires much less computational effort. However, spatial fiber orientation must be reconstructed from the fiber orientation tensors in advance for structural simulations. In this contribution, two reconstruction methods were investigated and evaluated using generated test scenarios and experimentally measured fiber orientation. The reconstruction methods include spherical harmonics up to the 8th order and the method of maximum entropy, with which a Bingham distribution is reconstructed. It is shown that the quality of the reconstruction depends massively on the original fiber orientation to be reconstructed. If the original distribution can be regarded as a Bingham distribution in good approximation, the method of maximum entropy is superior to spherical harmonics. If there is no Bingham distribution, spherical harmonics is more suitable due to its greater flexibility, but only if sufficiently high orders of the fiber orientation tensor can be determined exactly.
13
El Guide, Mohamed, Alaa El Ichi, Khalide Jbilou, and Rachid Sadaka. "On tensor GMRES and Golub-Kahan methods via the T-product for color image processing." Electronic Journal of Linear Algebra 37 (July 23, 2021): 524–43. http://dx.doi.org/10.13001/ela.2021.5471.
Abstract:
The present paper is concerned with developing tensor iterative Krylov subspace methods to solve large multi-linear tensor equations. We use the T-product for two tensors to define tensor tubal global Arnoldi and tensor tubal global Golub-Kahan bidiagonalization algorithms. Furthermore, we illustrate how tensor-based global approaches can be exploited to solve ill-posed problems arising from recovering blurry multichannel (color) images and videos, using the so-called Tikhonov regularization technique, to provide computable approximate regularized solutions. We also review a generalized cross-validation and discrepancy principle type of criterion for the selection of the regularization parameter in the Tikhonov regularization. Applications to image sequence processing are given to demonstrate the efficiency of the algorithms.
14
Selvan, Raghavendra, Silas Ørting, and Erik B. Dam. "Locally orderless tensor networks for classifying two- and three-dimensional medical images." Machine Learning for Biomedical Imaging 1, MIDL 2020 (March 23, 2021): 1–21. http://dx.doi.org/10.59275/j.melba.2021-g65b.
Abstract:
Tensor networks are factorisations of high rank tensors into networks of lower rank tensors and have primarily been used to analyse quantum many-body problems. Tensor networks have seen a recent surge of interest in relation to supervised learning tasks with a focus on image classification. In this work, we improve upon the matrix product state (MPS) tensor networks that can operate on one-dimensional vectors to be useful for working with 2D and 3D medical images. We treat small image regions as orderless, squeeze their spatial information into feature dimensions and then perform MPS operations on these locally orderless regions. These local representations are then aggregated in a hierarchical manner to retain global structure. The proposed locally orderless tensor network (LoTeNet) is compared with relevant methods on three datasets. The architecture of LoTeNet is fixed in all experiments and we show it requires lesser computational resources to attain performance on par or superior to the compared methods.
15
Selvan, Raghavendra, Erik B. Dam, Søren Alexander Flensborg, and Jens Petersen. "Patch-based Medical Image Segmentation using Matrix Product State Tensor Networks." Machine Learning for Biomedical Imaging 1, IPMI 2021 (February 24, 2022): 1–24. http://dx.doi.org/10.59275/j.melba.2022-d1f5.
Abstract:
Tensor networks are efficient factorisations of high dimensional tensors into network of lower order tensors. They have been most commonly used to model entanglement in quantum many-body systems and more recently are witnessing increased applications in supervised machine learning. In this work, we formulate image segmentation in a supervised setting with tensor networks. The key idea is to first lift the pixels in image patches to exponentially high dimensional feature spaces and using a linear decision hyper-plane to classify the input pixels into foreground and background classes. The high dimensional linear model itself is approximated using the matrix product state (MPS) tensor network. The MPS is weight-shared between the non-overlapping image patches resulting in our <em>strided tensor network</em> model. The performance of the proposed model is evaluated on three three 2D- and one 3D- biomedical imaging datasets. The performance of the proposed tensor network segmentation model is compared with relevant baseline methods. In the 2D experiments, the tensor network model yeilds competitive performance compared to the baseline methods while being more resource efficient.
16
Neukirch, Maik, Savitri Galiana, and Xavier Garcia. "Appraisal of magnetotelluric galvanic electric distortion by optimizing amplitude and phase tensor relations." GEOPHYSICS 85, no. 3 (April 29, 2020): E79—E98. http://dx.doi.org/10.1190/geo2019-0359.1.
Abstract:
The introduction of the phase tensor marked a major breakthrough in the analysis and treatment of electric field galvanic distortion in the magnetotellurics method. Recently, the phase tensor formulation has been extended to a complete impedance tensor decomposition by introducing the complementary amplitude tensor, and both tensors can be further parameterized to represent geometric properties such as dimensionality, strike angle, and macroscopic anisotropy. Both tensors are characteristic for the electromagnetic induction phenomenon in the conductive subsurface with its specific geometric structure. The central hypothesis is that this coupling should result in similarities in both tensor’s geometric parameters, skew, strike, and anisotropy. A synthetic example illustrates that the undistorted amplitude tensor parameters are more similar to the phase tensor than increasingly distorted ones and provides empiric evidence for the predictability of the proposed hypothesis. Conclusions drawn are reverse engineered to produce an objective function that minimizes when amplitude and phase tensor parameter dissimilarity is, along with any present distortion, minimal. A genetic algorithm with such an objective function is used to systematically seek the distortion parameters necessary to correct any affected amplitude tensor and, thus, impedance data. The successful correction of a large synthetic impedance data set with random distortion further supports the central hypothesis and serves as comparison to the state-of-the-art. The classic BC87 data set sites lit007/ lit008 and lit901/ lit902 have been noted by various authors to contain significant distortion and a 3D regional response, thus invalidating current distortion analysis methods and eluding geologic interpretation. Correction of the BC87 responses based on the present hypothesis conforms to the regional geology.
17
Zhang, Lipeng, Peng Zhang, Xindian Ma, Shuqin Gu, Zhan Su, and Dawei Song. "A Generalized Language Model in Tensor Space." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 7450–58. http://dx.doi.org/10.1609/aaai.v33i01.33017450.
Abstract:
In the literature, tensors have been effectively used for capturing the context information in language models. However, the existing methods usually adopt relatively-low order tensors, which have limited expressive power in modeling language. Developing a higher-order tensor representation is challenging, in terms of deriving an effective solution and showing its generality. In this paper, we propose a language model named Tensor Space Language Model (TSLM), by utilizing tensor networks and tensor decomposition. In TSLM, we build a high-dimensional semantic space constructed by the tensor product of word vectors. Theoretically, we prove that such tensor representation is a generalization of the n-gram language model. We further show that this high-order tensor representation can be decomposed to a recursive calculation of conditional probability for language modeling. The experimental results on Penn Tree Bank (PTB) dataset and WikiText benchmark demonstrate the effectiveness of TSLM.
18
Wang, Wenzhe, Jingjing Zheng, Li Zhao, Huiling Chen, and Xiaoqin Zhang. "A Non-Local Tensor Completion Algorithm Based on Weighted Tensor Nuclear Norm." Electronics 11, no. 19 (October 9, 2022): 3250. http://dx.doi.org/10.3390/electronics11193250.
Abstract:
In this paper, we proposed an image inpainting algorithm, including an interpolation step and a non-local tensor completion step based on a weighted tensor nuclear norm. Specifically, the proposed algorithm adopts the triangular based linear interpolation algorithm firstly to preform the observation image. Second, we extract the non-local similar patches in the image using the patch match algorithm and rearrange them to a similar tensor. Then, we use the tensor completion algorithm based on the weighted tensor nuclear norm to recover the similar tensors. Finally, we regroup all these recovered tensors to obtain the final recovered image. From the image inpainting experiments on color RGB images, we can see that the performance of the proposed algorithm on the peak signal-to-noise ratio (PSNR) and the relative standard error (RSE) are significantly better than other image inpainting methods.
19
Shcherbakova, Elena M., Sergey A. Matveev, Alexander P. Smirnov, and Eugene E. Tyrtyshnikov. "Study of performance of low-rank nonnegative tensor factorization methods." Russian Journal of Numerical Analysis and Mathematical Modelling 38, no. 4 (August 1, 2023): 231–39. http://dx.doi.org/10.1515/rnam-2023-0018.
Abstract:
Abstract In the present paper we compare two different iterative approaches to constructing nonnegative tensor train and Tucker decompositions. The first approach is based on idea of alternating projections and randomized sketching for factorization of tensors with nonnegative elements. This approach can be useful for both TT and Tucker formats. The second approach consists of two stages. At the first stage we find the unconstrained tensor train decomposition for the target array. At the second stage we use this initial approximation in order to fix it within moderate number of operations and obtain the factorization with nonnegative factors either in tensor train or Tucker model. We study the performance of these methods for both synthetic data and hyper-spectral image and demonstrate the clear advantage of the latter technique in terms of computational time and wider range of possible applications.
20
Gao, Tong, Hao Chen, and Wen Chen. "MCMS-STM: An Extension of Support Tensor Machine for Multiclass Multiscale Object Recognition in Remote Sensing Images." Remote Sensing 14, no. 1 (January 2, 2022): 196. http://dx.doi.org/10.3390/rs14010196.
Abstract:
The support tensor machine (STM) extended from support vector machine (SVM) can maintain the inherent information of remote sensing image (RSI) represented as tensor and obtain effective recognition results using a few training samples. However, the conventional STM is binary and fails to handle multiclass classification directly. In addition, the existing STMs cannot process objects with different sizes represented as multiscale tensors and have to resize object slices to a fixed size, causing excessive background interferences or loss of object’s scale information. Therefore, the multiclass multiscale support tensor machine (MCMS-STM) is proposed to recognize effectively multiclass objects with different sizes in RSIs. To achieve multiclass classification, by embedding one-versus-rest and one-versus-one mechanisms, multiple hyperplanes described by rank-R tensors are built simultaneously instead of single hyperplane described by rank-1 tensor in STM to separate input with different classes. To handle multiscale objects, multiple slices of different sizes are extracted to cover the object with an unknown class and expressed as multiscale tensors. Then, M-dimensional hyperplanes are established to project the input of multiscale tensors into class space. To ensure an efficient training of MCMS-STM, a decomposition algorithm is presented to break the complex dual problem of MCMS-STM into a series of analytic sub-optimizations. Using publicly available RSIs, the experimental results demonstrate that the MCMS-STM achieves 89.5% and 91.4% accuracy for classifying airplanes and ships with different classes and sizes, which outperforms typical SVM and STM methods.
21
Wimalawarne, Kishan, Makoto Yamada, and Hiroshi Mamitsuka. "Scaled Coupled Norms and Coupled Higher-Order Tensor Completion." Neural Computation 32, no. 2 (February 2020): 447–84. http://dx.doi.org/10.1162/neco_a_01254.
Abstract:
Recently, a set of tensor norms known as coupled norms has been proposed as a convex solution to coupled tensor completion. Coupled norms have been designed by combining low-rank inducing tensor norms with the matrix trace norm. Though coupled norms have shown good performances, they have two major limitations: they do not have a method to control the regularization of coupled modes and uncoupled modes, and they are not optimal for couplings among higher-order tensors. In this letter, we propose a method that scales the regularization of coupled components against uncoupled components to properly induce the low-rankness on the coupled mode. We also propose coupled norms for higher-order tensors by combining the square norm to coupled norms. Using the excess risk-bound analysis, we demonstrate that our proposed methods lead to lower risk bounds compared to existing coupled norms. We demonstrate the robustness of our methods through simulation and real-data experiments.
22
Betten, J. "Mathematical Modelling of Materials Behavior Under Creep Conditions." Applied Mechanics Reviews 54, no. 2 (March 1, 2001): 107–32. http://dx.doi.org/10.1115/1.3097292.
Abstract:
This article will provide a short survey of some recent advances in the mathematical modelling of materials behavior under creep conditions. The mechanical behavior of anisotropic solids requires a suitable mathematical modelling. The properties of tensor functions with several argument tensors constitute a rational basis for a consistent mathematical modelling of complex material behavior. This article presents certain principles, methods, and recent successful applications of tensor functions in creep mechanics. The rules for specifying irreducible sets of tensor invariants and tensor generators for material tensors of rank two and four are also discussed. Furthermore, it is very important that the scalar coefficients in constitutive and evolutional equations are determined as functions of the integrity basis and experimental data. It is explained in detail that these coefficients can be determined by using tensorial interpolation methods. Some examples for practical use are discussed. Finally, we have carried out our own experiments to examine the validity of the mathematical modelling. Furthermore, an overview of some important experimental investigations in creep mechanics of other scientists has been provided. There are 243 references cited in this review article.
23
Chen, Xi’ai, Zhen Wang, Kaidong Wang, Huidi Jia, Zhi Han, and Yandong Tang. "Multi-Dimensional Low-Rank with Weighted Schatten p-Norm Minimization for Hyperspectral Anomaly Detection." Remote Sensing 16, no. 1 (December 24, 2023): 74. http://dx.doi.org/10.3390/rs16010074.
Abstract:
Hyperspectral anomaly detection is an important unsupervised binary classification problem that aims to effectively distinguish between background and anomalies in hyperspectral images (HSIs). In recent years, methods based on low-rank tensor representations have been proposed to decompose HSIs into low-rank background and sparse anomaly tensors. However, current methods neglect the low-rank information in the spatial dimension and rely heavily on the background information contained in the dictionary. Furthermore, these algorithms show limited robustness when the dictionary information is missing or corrupted by high level noise. To address these problems, we propose a novel method called multi-dimensional low-rank (MDLR) for HSI anomaly detection. It first reconstructs three background tensors separately from three directional slices of the background tensor. Then, weighted schatten p-norm minimization is employed to enforce the low-rank constraint on the background tensor, and LF,1-norm regularization is used to describe the sparsity in the anomaly tensor. Finally, a well-designed alternating direction method of multipliers (ADMM) is employed to effectively solve the optimization problem. Extensive experiments on four real-world datasets show that our approach outperforms existing anomaly detection methods in terms of accuracy.
24
Bocci, Cristiano, Luca Chiantini, and Giorgio Ottaviani. "Refined methods for the identifiability of tensors." Annali di Matematica Pura ed Applicata (1923 -) 193, no. 6 (May 24, 2013): 1691–702. http://dx.doi.org/10.1007/s10231-013-0352-8.
25
Guo, Pei-Chang. "New regularization methods for convolutional kernel tensors." AIMS Mathematics 8, no. 11 (2023): 26188–98. http://dx.doi.org/10.3934/math.20231335.
Abstract:
<abstract><p>Convolution is a very basic and important operation for convolutional neural networks. For neural network training, how to bound the convolutional layers is a currently popular research topic. Each convolutional layer is represented by a tensor, which corresponds to a structured transformation matrix. The objective is to ensure that the singular values of each transformation matrix are bounded around 1 by changing the entries of the tensor. We propose three new regularization terms for a convolutional kernel tensor and derive the gradient descent algorithm for each penalty function. Numerical examples are presented to demonstrate the effectiveness of the algorithms.</p></abstract>
26
Roy, George. "Complementarity of experimental and numerical methods for determining residual stress states." Powder Diffraction 24, S1 (June 2009): S3—S10. http://dx.doi.org/10.1154/1.3139050.
Abstract:
Residual stress states in engineering structures are usually determined by measuring components of stress tensors with depth below the material surface. There are destructive and nondestructive methods to measure strain tensor components and convert them into stress tensor components by a variety of techniques derived from constitutive (material) equations. In this study, four methods for determining the strain tensor components are presented: X-ray diffraction method (XRDM), magnetic Barkhausen noise method (MBNM), hole drilling method (HDM), and cut-and-section method (CSM); the first two are nondestructive, and the third and fourth are semidestructive and destructive, respectively. A complementarity of the experimental and two numerical methods such as boundary element method and finite element method is explained. An application of the experimental and numerical methods to measure residual stress states in an industrial component, an L-shaped part of a supporting column in a high voltage structure, is presented.
27
Belova, O. O. "Generalized bilinear connection on the space of centered planes." Differential Geometry of Manifolds of Figures, no. 53 (2022): 20–32. http://dx.doi.org/10.5922/0321-4796-2022-53-3.
Abstract:
We continue to study the space of centered planes in projective space . In this paper, we use E. Cartan's method of external forms and the group-theoretical method of G. F. Laptev to study the space of centered planes of the same dimension. These methods are successfully applied in physics. In a generalized bundle, a bilinear connection associated with a space is given. The connection object contains two simplest subtensors and subquasi-tensors (four simplest and three simple subquasi-tensors). The object field of this connection defines the objects of torsion, curvature-torsion, and curvature. The curvature tensor contains six simplest and four simple subtensors, and curvature-torsion tensor contains three simplest and two simple subtensors. The canonical case of a generalized bilinear connection is considered.
28
Bai, Dongjian, and Feng Wang. "New methods based $ \mathcal{H} $-tensors for identifying the positive definiteness of multivariate homogeneous forms." AIMS Mathematics 6, no. 9 (2021): 10281–95. http://dx.doi.org/10.3934/math.2021595.
Abstract:
<abstract><p>Positive definite polynomials are important in the field of optimization. $ \mathcal{H} $-tensors play an important role in identifying the positive definiteness of an even-order homogeneous multivariate form. In this paper, we propose some new criterion for identifying $ \mathcal{H} $-tensor. As applications, we give new conditions for identifying positive definiteness of the even-order homogeneous multivariate form. At last, some numerical examples are provided to illustrate the efficiency and validity of new methods.</p></abstract>
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Song, Hong-Mei, Shi-Wei Wang, and Guang-Xin Huang. "Tensor Conjugate-Gradient methods for tensor linear discrete ill-posed problems." AIMS Mathematics 8, no. 11 (2023): 26782–800. http://dx.doi.org/10.3934/math.20231371.
Abstract:
<abstract><p>This paper presents three types of tensor Conjugate-Gradient (tCG) methods for solving large-scale linear discrete ill-posed problems based on the t-product between third-order tensors. An automatic determination strategy of a suitable regularization parameter is proposed for the tCG method in the Fourier domain (A-tCG-FFT). An improved version and a preconditioned version of the tCG method are also presented. The discrepancy principle is employed to determine a suitable regularization parameter. Several numerical examples in image and video restoration are given to show the effectiveness of the proposed tCG methods.</p></abstract>
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Zhu, Yada, Jingrui He, and Rick Lawrence. "Hierarchical Modeling with Tensor Inputs." Proceedings of the AAAI Conference on Artificial Intelligence 26, no. 1 (September 20, 2021): 1233–39. http://dx.doi.org/10.1609/aaai.v26i1.8283.
Abstract:
In many real applications, the input data are naturally expressed as tensors, such as virtual metrology in semiconductor manufacturing, face recognition and gait recognition in computer vision, etc. In this paper, we propose a general optimization framework for dealing with tensor inputs. Most existing methods for supervised tensor learning use only rank-one weight tensors in the linear model and cannot readily incorporate domain knowledge. In our framework, we obtain the weight tensor in a hierarchical way — we first approximate it by a low-rank tensor, and then estimate the low-rank approximation using the prior knowledge from various sources, e.g., different domain experts. This is motivated by wafer quality prediction in semiconductor manufacturing. Furthermore, we propose an effective algorithm named H-MOTE for solving this framework, which is guaranteed to converge. The time complexity of H-MOTE is linear with respect to the number of examples as well as the size of the weight tensor. Experimental results show the superiority of H-MOTE over state-of-the-art techniques on both synthetic and real data sets.
31
Suh, Young Jin, Carlo Alberto Mantica, Uday Chand De, and Prajjwal Pal. "Pseudo B-symmetric manifolds." International Journal of Geometric Methods in Modern Physics 14, no. 09 (August 2, 2017): 1750119. http://dx.doi.org/10.1142/s0219887817501195.
Abstract:
In this paper, we introduce a new tensor named [Formula: see text]-tensor which generalizes the [Formula: see text]-tensor introduced by Mantica and Suh [Pseudo [Formula: see text] symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. Then, we study pseudo-[Formula: see text]-symmetric manifolds [Formula: see text] which generalize some known structures on pseudo-Riemannian manifolds. We provide several interesting results which generalize the results of Mantica and Suh [Pseudo [Formula: see text] symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. At first, we prove the existence of a [Formula: see text]. Next, we prove that a pseudo-Riemannian manifold is [Formula: see text]-semisymmetric if and only if it is Ricci-semisymmetric. After this, we obtain a sufficient condition for a [Formula: see text] to be pseudo-Ricci symmetric in the sense of Deszcz. Also, we obtain the explicit form of the Ricci tensor in a [Formula: see text] if the [Formula: see text]-tensor is of Codazzi type. Finally, we consider conformally flat pseudo-[Formula: see text]-symmetric manifolds and prove that a [Formula: see text] spacetime is a [Formula: see text]-wave under certain conditions.
32
Felício Fuck, Rodrigo, and Ilya Tsvankin. "Analysis of the symmetry of a stressed medium using nonlinear elasticity." GEOPHYSICS 74, no. 5 (September 2009): WB79—WB87. http://dx.doi.org/10.1190/1.3157251.
Abstract:
Velocity variations caused by subsurface stress changes play an important role in monitoring compacting reservoirs and in several other applications of seismic methods. A general way to describe stress- or strain-induced velocity fields is by employing the theory of nonlinear elasticity, which operates with third-order elastic (TOE) tensors. These sixth-rank strain-sensitivity tensors, however, are difficult to manipulate because of the large number of terms involved in the algebraic operations. Thus, even evaluation of the anisotropic symmetry of a medium under stress/strain proves to be a challenging task. We employ a matrix representation of TOE tensors that allows computation of strain-related stiffness perturbations from a linear combination of [Formula: see text] matrices scaled by the components of the strain tensor. In addition to streamlining the numerical algorithm, this approach helps to predict strain-induced symmetry using relatively straightforward algebraic considerations. For example, our analysis shows that a transversely isotropic (TI) medium acquires orthorhombic symmetry if one of the principal directions of the strain tensor is aligned with the symmetry axis. Otherwise, the strained TI medium can become monoclinic or even triclinic.
33
SCHÄFER, MARCO, IDRISH HUET, and HOLGER GIES. "ENERGY-MOMENTUM TENSORS WITH WORLDLINE NUMERICS." International Journal of Modern Physics: Conference Series 14 (January 2012): 511–20. http://dx.doi.org/10.1142/s2010194512007647.
Abstract:
We apply the worldline formalism and its numerical Monte-Carlo approach to computations of fluctuation induced energy-momentum tensors. For the case of a fluctuating Dirichlet scalar, we derive explicit worldline expressions for the components of the canonical energy-momentum tensor that are straightforwardly accessible to partly analytical and generally numerical evaluation. We present several simple proof-of-principle examples, demonstrating that efficient numerical evaluation is possible at low cost. Our methods can be applied to an investigation of positive-energy conditions.
34
Hameduddin, Ismail, Dennice F. Gayme, and Tamer A. Zaki. "Perturbative expansions of the conformation tensor in viscoelastic flows." Journal of Fluid Mechanics 858 (November 6, 2018): 377–406. http://dx.doi.org/10.1017/jfm.2018.777.
Abstract:
We consider the problem of formulating perturbative expansions of the conformation tensor, which is a positive definite tensor representing polymer deformation in viscoelastic flows. The classical approach does not explicitly take into account that the perturbed tensor must remain positive definite – a fact that has important physical implications, e.g. extensions and compressions are represented similarly to within a negative sign, when physically the former are unbounded and the latter are bounded from below. Mathematically, the classical approach assumes that the underlying geometry is Euclidean, and this assumption is not satisfied by the manifold of positive definite tensors. We provide an alternative formulation that retains the conveniences of classical perturbation methods used for generating linear and weakly nonlinear expansions, but also provides a clear physical interpretation and a mathematical basis for analysis. The approach is based on treating a perturbation as a sequence of successively smaller deformations of the polymer. Each deformation is modelled explicitly using geodesics on the manifold of positive definite tensors. Using geodesics, and associated geodesic distances, is the natural way to model perturbations to positive definite tensors because it is consistent with the manifold geometry. Approximations of the geodesics can then be used to reduce the total deformation to a series expansion in the small perturbation limit. We illustrate our approach using direct numerical simulations of the nonlinear evolution of Tollmien–Schlichting waves.
35
McDowell, S. A. C., and A. D. Buckingham. "Electric nuclear shielding tensors by finite-field methods." Journal of the Chemical Society, Faraday Transactions 88, no. 22 (1992): 3281. http://dx.doi.org/10.1039/ft9928803281.
36
Oseledets, I. V., and D. V. Savost’yanov. "Minimization methods for approximating tensors and their comparison." Computational Mathematics and Mathematical Physics 46, no. 10 (October 2006): 1641–50. http://dx.doi.org/10.1134/s0965542506100022.
37
ANIELLO, P., J. CLEMENTE-GALLARDO, G. MARMO, and G. F. VOLKERT. "CLASSICAL TENSORS AND QUANTUM ENTANGLEMENT I: PURE STATES." International Journal of Geometric Methods in Modern Physics 07, no. 03 (May 2010): 485–503. http://dx.doi.org/10.1142/s0219887810004300.
Abstract:
The geometrical description of a Hilbert space associated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here, we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.
38
Gao, Tong, Hao Chen, and Junhong Lu. "Coupled Heterogeneous Tucker Decomposition: A Feature Extraction Method for Multisource Fusion and Domain Adaptation Using Multisource Heterogeneous Remote Sensing Data." Remote Sensing 14, no. 11 (May 26, 2022): 2553. http://dx.doi.org/10.3390/rs14112553.
Abstract:
To excavate adequately the rich information contained in multisource remote sensing data, feature extraction as basic yet important research has two typical applications: one of which is to extract complementary information of multisource data to improve classification; and the other is to extract shared information across sources for domain adaptation. However, typical feature extraction methods require the input represented as vectors or homogeneous tensors and fail to process multisource data represented as heterogeneous tensors. Therefore, the coupled heterogeneous Tucker decomposition (C-HTD) containing two sub-methods, namely coupled factor matrix-based HTD (CFM-HTD) and coupled core tensor-based HTD (CCT-HTD), is proposed to establish a unified feature extraction framework for multisource fusion and domain adaptation. To handle multisource heterogeneous tensors, multiple Tucker models were constructed to extract features of different sources separately. To cope with the supervised and semi-supervised cases, the class-indicator factor matrix was built to enhance the separability of features using known labels and learned labels. To mine the complementarity of paired multisource samples, coupling constraint was imposed on multiple factor matrices to form CFM-HTD to extract multisource information jointly. To extract domain-adapted features, coupling constraint was imposed on multiple core tensors to form CCT-HTD to encourage data from different sources to have the same class centroid. In addition, to reduce the impact of interference samples on domain adaptation, an adaptive sample-weighting matrix was designed to autonomously remove outliers. Using multiresolution multiangle optical and MSTAR datasets, experimental results show that the C-HTD outperforms typical multisource fusion and domain adaptation methods.
39
Sucharitha, B., and Dr K. Anitha Sheela. "Compression of Hyper Spectral Images using Tensor Decomposition Methods." International Journal of Circuits, Systems and Signal Processing 16 (October 7, 2022): 1148–55. http://dx.doi.org/10.46300/9106.2022.16.138.
Abstract:
Tensor decomposition methods have beenrecently identified as an effective approach for compressing high-dimensional data. Tensors have a wide range of applications in numerical linear algebra, chemo metrics, data mining, signal processing, statics, and data mining and machine learning. Due to the huge amount of information that the hyper spectral images carry, they require more memory to store, process and send. We need to compress the hyper spectral images in order to reduce storage and processing costs. Tensor decomposition techniques can be used to compress the hyper spectral data. The primary objective of this work is to utilize tensor decomposition methods to compress the hyper spectral images. This paper explores three types of tensor decompositions: Tucker Decomposition (TD_ALS), CANDECOMP/PARAFAC (CP) and Tucker_HOSVD (Higher order singular value Decomposition) and comparison of these methods experimented on two real hyper spectral images: the Salinas image (512 x 217 x 224) and Indian Pines corrected (145 x 145 x 200). The PSNR and SSIM are used to evaluate how well these techniques work. When compared to the iterative approximation methods employed in the CP and Tucker_ALS methods, the Tucker_HOSVD method decomposes the hyper spectral image into core and component matrices more quickly. According to experimental analysis, Tucker HOSVD's reconstruction of the image preserves image quality while having a higher compression ratio than the other two techniques.
40
De Paris, Alessandro. "Seeking for the Maximum Symmetric Rank." Mathematics 6, no. 11 (November 12, 2018): 247. http://dx.doi.org/10.3390/math6110247.
Abstract:
We present the state-of-the-art on maximum symmetric tensor rank, for each given dimension and order. After a general discussion on the interplay between symmetric tensors, polynomials and divided powers, we introduce the technical environment and the methods that have been set up in recent times to find new lower and upper bounds.
41
Wang, Shuangyue, and Ziyan Luo. "Sparse Support Tensor Machine with Scaled Kernel Functions." Mathematics 11, no. 13 (June 24, 2023): 2829. http://dx.doi.org/10.3390/math11132829.
Abstract:
As one of the supervised tensor learning methods, the support tensor machine (STM) for tensorial data classification is receiving increasing attention in machine learning and related applications, including remote sensing imaging, video processing, fault diagnosis, etc. Existing STM approaches lack consideration for support tensors in terms of data reduction. To address this deficiency, we built a novel sparse STM model to control the number of support tensors in the binary classification of tensorial data. The sparsity is imposed on the dual variables in the context of the feature space, which facilitates the nonlinear classification with kernel tricks, such as the widely used Gaussian RBF kernel. To alleviate the local risk associated with the constant width in the tensor Gaussian RBF kernel, we propose a two-stage classification approach; in the second stage, we advocate for a scaling strategy on the kernel function in a data-dependent way, using the information of the support tensors obtained from the first stage. The essential optimization models in both stages share the same type, which is non-convex and discontinuous, due to the sparsity constraint. To resolve the computational challenge, a subspace Newton method is tailored for the sparsity-constrained optimization for effective computation with local convergence. Numerical experiments were conducted on real datasets, and the numerical results demonstrate the effectiveness of our proposed two-stage sparse STM approach in terms of classification accuracy, compared with the state-of-the-art binary classification approaches.
42
Мурашкин, Евгений Валерьевич, and Юрий Николаевич Радаев. "Covariantly constant tensors in Euclid spaces. Elements of the theory." Вестник Чувашского государственного педагогического университета им. И.Я. Яковлева. Серия: Механика предельного состояния, no. 2(52) (December 12, 2022): 106–17. http://dx.doi.org/10.37972/chgpu.2022.52.2.012.
Abstract:
В настоящей работе обсуждаются вопросы ковариантного постоянства тензоров и псевдотензоров (в том числе, двухточечных) произвольной валентности и веса в Евклидовом пространстве. Приводятся минимально необходимые сведения из алгебры и анализа псевдотензоров в пространствах Евклида. Выясняются общие условия ковариантного постоянства псевдотензоров. Рассматриваются примеры ковариантно постоянных тензоров и псевдотензоров из многомерной геометрии. Речь, в частности, идет о фундаментальном ориентирующем псевдоскаляре, целые степени которого удовлетворяет условию ковариантного постоянства. Обсуждаются свойства и способы координатного представления тензоров ковариантно постоянных тензоров ипсевдотензоров четвертого ранга. На основе неконвенционального определения полуизотропного тензора четвертого ранга приводится координатное представление в терминах дельт Кронекера и метрических тензоров. Устанавливаются условия ковариантного постоянства полуизотропных тензоров четвертого ранга. In this paper, we discuss the covariant constancy of tensors and pseudotensors (including two-point ones) of arbitrary valency and weight in Euclidean space. The requisite notions and equations from algebra and analysis of pseudotensors in Euclidean spaces are given. The general conditions for the covariant constancy of pseudotensors are highlighted. Examples of covariantly constant tensors and pseudotensors from multidimensional geometry are considered. In particular, a fundamental orienting pseudoscalar whose integer powers satisfy the condition of covariant constancy is introduced. The properties and methods of coordinate representation of covariantly constant tensors and pseudotensors of the fourth rank are discussed. Based on an unconventional definition of a semi-isotropic tensor of the fourth rank, a coordinate representation in terms of Kronecker deltas and metric tensors is given. Conditions for the covariant constancy of semi-isotropic tensors of the fourth rank are derived.
43
Dogariu, Laura-Maria, Cristian-Lucian Stanciu, Camelia Elisei-Iliescu, Constantin Paleologu, Jacob Benesty, and Silviu Ciochină. "Tensor-Based Adaptive Filtering Algorithms." Symmetry 13, no. 3 (March 15, 2021): 481. http://dx.doi.org/10.3390/sym13030481.
Abstract:
Tensor-based signal processing methods are usually employed when dealing with multidimensional data and/or systems with a large parameter space. In this paper, we present a family of tensor-based adaptive filtering algorithms, which are suitable for high-dimension system identification problems. The basic idea is to exploit a decomposition-based approach, such that the global impulse response of the system can be estimated using a combination of shorter adaptive filters. The algorithms are mainly tailored for multiple-input/single-output system identification problems, where the input data and the channels can be grouped in the form of rank-1 tensors. Nevertheless, the approach could be further extended for single-input/single-output system identification scenarios, where the impulse responses (of more general forms) can be modeled as higher-rank tensors. As compared to the conventional adaptive filters, which involve a single (usually long) filter for the estimation of the global impulse response, the tensor-based algorithms achieve faster convergence rate and tracking, while also providing better accuracy of the solution. Simulation results support the theoretical findings and indicate the advantages of the tensor-based algorithms over the conventional ones, in terms of the main performance criteria.
44
Belova, Olga. "The space of centered planes and generalized bilinear connection." Filomat 37, no. 25 (2023): 8455–64. http://dx.doi.org/10.2298/fil2325455b.
Abstract:
We continue to study the space of centered planes in n-dimension projective space. We use E. Cartan?s method of external forms and the group-theoretical method of G. F. Laptev to study the space of centered planes of the same dimension. These methods are successfully applied in physics. In a generalized bundle, a bilinear connection associated with a space is given. The connection object contains two simplest subtensors and subquasi-tensors (four simplest and three simple subquasi-tensors). The object field of this connection defines the objects of torsion S, curvature-torsion T, and curvature R. The curvature tensor contains six simplest and four simple subtensors, and curvature-torsion tensor contains three simplest and two simple subtensors. The canonical case of a generalized bilinear connection is considered. We realize the strong Lumiste?s affine clothing (it is an analog of the strong Norde?s normalization of the space of centered planes). Covariant differentials and covariant derivatives of the clothing quasi-tensor are described. The covariant derivatives do not form a tensor. We present a geometrical characterization of the generalized bilinear connection using mappings.
45
YANUSHKEVICH, V. "ELECTROMAGNETIC METHODS OF SEARCHING AND DEFINITION OF HYDROCARBON DEPOSITS." HERALD OF POLOTSK STATE UNIVERSITY. Series С FUNDAMENTAL SCIENCES 39, no. 11 (November 15, 2022): 80–88. http://dx.doi.org/10.52928/2070-1624-2022-39-11-80-88.
Abstract:
The article studies the characteristics of an anisotropic medium over hydrocarbons with the complex use
of electromagnetic geo-exploration methods. The simulation of the components of dielectric permittivity tensors
in the mode of amplitude-modulated, frequency-modulated, amplitude-frequency-modulated and radio pulse signals
is carried out. The frequencies of the electron plasma and electron cyclotron resonances for the specified
regimes are established. A study was made of the influence of modes of probing signals on the characteristics
of an anisotropic medium above deposits and the components of the dielectric tensor. The effect of particle concentration
variation on the real components of the components of the dielectric permittivity of an anisotropic
medium above the UVZ is analyzed. Recommendations are given for improving the methods of electrical exploration
and their application for prospecting geophysics.
46
Gavrilyuk, Ivan, Boris Khoromskij, and Eugene Tyrtyshnikov. "Preface to the special issue, CMAM 2011, no. 3." Computational Methods in Applied Mathematics 11, no. 3 (2011): 272. http://dx.doi.org/10.2478/cmam-2011-0014.
Abstract:
Abstract In the recent years, multidimensional numerical simulations with tensor-structured data formats have been recognized as the basic concept for breaking the "curse of dimensionality". Modern applications of tensor methods include the challenging high-dimensional problems of material sciences, bio-science, stochastic modeling, signal processing, machine learning, and data mining, financial mathematics, etc. The guiding principle of the tensor methods is an approximation of multivariate functions and operators with some separation of variables to keep the computational process in a low parametric tensor-structured manifold. Tensors structures had been wildly used as models of data and discussed in the contexts of differential geometry, mechanics, algebraic geometry, data analysis etc. before tensor methods recently have penetrated into numerical computations. On the one hand, the existing tensor representation formats remained to be of a limited use in many high-dimensional problems because of lack of sufficiently reliable and fast software. On the other hand, for moderate dimensional problems (e.g. in "ab-initio" quantum chemistry) as well as for selected model problems of very high dimensions, the application of traditional canonical and Tucker formats in combination with the ideas of multilevel methods has led to the new efficient algorithms. The recent progress in tensor numerical methods is achieved with new representation formats now known as "tensor-train representations" and "hierarchical Tucker representations". Note that the formats themselves could have been picked up earlier in the literature on the modeling of quantum systems. Until 2009 they lived in a closed world of those quantum theory publications and never trespassed the territory of numerical analysis. The tremendous progress during the very recent years shows the new tensor tools in various applications and in the development of these tools and study of their approximation and algebraic properties. This special issue treats tensors as a base for efficient numerical algorithms in various modern applications and with special emphases on the new representation formats.
47
Hached, Mustapha, Khalide Jbilou, Christos Koukouvinos, and Marilena Mitrouli. "A Multidimensional Principal Component Analysis via the C-Product Golub–Kahan–SVD for Classification and Face Recognition." Mathematics 9, no. 11 (May 29, 2021): 1249. http://dx.doi.org/10.3390/math9111249.
Abstract:
Face recognition and identification are very important applications in machine learning. Due to the increasing amount of available data, traditional approaches based on matricization and matrix PCA methods can be difficult to implement. Moreover, the tensorial approaches are a natural choice, due to the mere structure of the databases, for example in the case of color images. Nevertheless, even though various authors proposed factorization strategies for tensors, the size of the considered tensors can pose some serious issues. Indeed, the most demanding part of the computational effort in recognition or identification problems resides in the training process. When only a few features are needed to construct the projection space, there is no need to compute a SVD on the whole data. Two versions of the tensor Golub–Kahan algorithm are considered in this manuscript, as an alternative to the classical use of the tensor SVD which is based on truncated strategies. In this paper, we consider the Tensor Tubal Golub–Kahan Principal Component Analysis method which purpose it to extract the main features of images using the tensor singular value decomposition (SVD) based on the tensor cosine product that uses the discrete cosine transform. This approach is applied for classification and face recognition and numerical tests show its effectiveness.
48
Bachmayr, Markus, and Reinhold Schneider. "Iterative Methods Based on Soft Thresholding of Hierarchical Tensors." Foundations of Computational Mathematics 17, no. 4 (April 29, 2016): 1037–83. http://dx.doi.org/10.1007/s10208-016-9314-z.
49
Ramzan, Muhammad, Ali Othman, and Neville R. Watson. "Comparison of assessment methods for tensors of nonlinear devices." Electric Power Systems Research 219 (June 2023): 109237. http://dx.doi.org/10.1016/j.epsr.2023.109237.
50
Lohmann, Christoph. "Algebraic flux correction schemes preserving the eigenvalue range of symmetric tensor fields." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 3 (May 2019): 833–67. http://dx.doi.org/10.1051/m2an/2019006.
Abstract:
This work extends the algebraic flux correction (AFC) paradigm to finite element discretizations of conservation laws for symmetric tensor fields. The proposed algorithms are designed to enforce discrete maximum principles and preserve the eigenvalue range of evolving tensors. To that end, a continuous Galerkin approximation is modified by adding a linear artificial diffusion operator and a nonlinear antidiffusive correction. The latter is decomposed into edge-based fluxes and constrained to prevent violations of local bounds for the minimal and maximal eigenvalues. In contrast to the flux-corrected transport (FCT) algorithm developed previously by the author and existing slope limiting techniques for stress tensors, the admissible eigenvalue range is defined implicitly and the limited antidiffusive terms are incorporated into the residual of the nonlinear system. In addition to scalar limiters that use a common correction factor for all components of a tensor-valued antidiffusive flux, tensor limiters are designed using spectral decompositions. The new limiter functions are analyzed using tensorial extensions of the existing AFC theory for scalar convection-diffusion equations. The proposed methodology is backed by rigorous proofs of eigenvalue range preservation and Lipschitz continuity. Convergence of pseudo time-stepping methods to stationary solutions is demonstrated in numerical studies.