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An Extension of Yuan's Lemma to fourth-order tensor System

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2019年第3期180卷 803-810页

作者： Yang, Qingzhi Zhou, Yang Yang, Yuning Kashgar Univ Sch Math & Stat Kashgar 844008 Peoples R China Nankai Univ Sch Math Sci Tianjin 300071 Peoples R China Nankai Univ LPMC Tianjin 300071 Peoples R China Guangxi Univ Sch Math Sci Nanning 530004 Peoples R China

Yuan's lemma is a basic proposition on homogeneous quadratic function systems. In this note, we extend Yuan's lemma to the fourth-order tensor system. We first give generalized definition of positive semidefinite of fourth-order tensor, and based on it, an extension of Yuan's lemma is proposed. As an application, we establish the strong duality result of a class of quadratic semidefinite programming problems using the extended Yuan's lemma.

关键词： Yuan's lemma S-lemma fourth-order tensor Symmetry Positive semidefinite Quadratic semidefinite programming

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A Fully Automatic Ocular Artifact Removal from EEG Based on fourth-order tensor Method

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BIOMEDICAL ENGINEERING LETTERS 2014年第1期4卷 55-63页

作者： Ge, Sunan Han, Min Hong, Xiaojun Dalian Univ Technol Fac Elect Informat & Elect Engn Dalian 116023 Peoples R China Dalian Med Univ Affiliated Hosp 1 Dalian 116011 Peoples R China

Purpose The aim of this paper is to propose a fully automatic system using the underdetermined blind source separation (UBSS) method and kurtosis to remove ocular artifacts (OAs) from scalp electroencephalogram (EEG). Methods The fully automatic system about OAs rejection is devised with the fourth-order tensor method (FOOBI). Firstly, the FOOBI method decomposes multiple EEG channels into a relative large number of source components. The kurtosis value is used to identify the ocular components in these source components. Then, the free-ocular sources components are reconstructed to EEG without OAs. Results The simulations show that the FOOBI method can completely separate the ocular signals from the observed signals. The data that have got rid of the OAs are used to classify with the epileptic EEG The classification accuracy acquired by FOOBI method is better than the independent component analysis (ICA). Conclusions The results inferred that the FOOBI method can not only completely remove the OAs from the observed signals but also preserve an amount of useful information of EEG Compared with the ICA method, this fully automatic system is more suitable to remove OAs.

关键词： EEG Ocular artifacts Independent component analysis fourth-order tensor

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Smoothed Sparse Blind DOA Estimation for MC-CDMA Systems Based on tensor Decomposition

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IEEE WIRELESS COMMUNICATIONS LETTERS 2023年第9期12卷 1548-1552页

作者： Wang, Yuexian Liu, Chao Luo, Shuai Liu, Xiangyang Tellambura, Chintha Northwestern Polytech Univ Sch Elect & Informat Xian 710072 Peoples R China Univ Alberta Dept Elect & Comp Engn Edmonton AB T6G 1H9 Canada

We propose a blind direction of arrival (DOA) estimation method for multicarrier code-division multiple access (MC-CDMA) systems, which does not require knowledge of the spreading codes of transmitters. Our method involves reconstructing a fourth-order tensor with observations from the antenna array at the receiver. We then apply higher-order singular value decomposition (HOSVD) to separate the signal subspace from the noise subspace. To enhance the sparsity of the solution, we develop a smoothed l(0)-norm sparse reconstruction method by designing continuous functions with reweighted vectors. Finally, DOA estimates are obtained by searching the spectrum of the sparse solution using an iterative method that ensures fast convergence. Numerical simulations show its superiority over previous work.

关键词： Direction-of-arrival estimation Multicarrier code division multiple access Codes tensors Wireless communication Receiving antennas Sparse matrices Multicarrier code-division multiple access (MC-CDMA) direction of arrival (DOA) fourth-order tensor higher-order singular value decomposition (HOSVD) smoothed l ₀-norm

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fourth order tensors and covariance tensors

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ANNALS OF FUNCTIONAL ANALYSIS 2023年第3期14卷 1-15页

作者： Bai, Jinxuan Wang, Jun Xu, Changqing Suzhou Univ Sci & Technol Sch Math Sci Suzhou Peoples R China

In this paper, we investigate the invertibility of the fourth-order cubic tensors and present several necessary and sufficient conditions for such tensors to be invertible. We also introduce tensors in statistics and use the fourth-order tensors to simplify the expressions of the higher order derivatives of a multivariate function. Finally, we define the covariance tensor of a random matrix X as a fourth-order tensor D[X] and show that D[X] is positive definite if X is square symmetric.

关键词： fourth-order tensor High order moment Random matrix Covariance tensor

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Two irreducible functional bases of isotropic invariants of a fourth-order three-dimensional symmetric and traceless tensor

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MATHEMATICS AND MECHANICS OF SOLIDS 2019年第10期24卷 3092-3102页

作者： Chen, Zhongming Chen, Yannan Qi, Liqun Zou, Wennan Hangzhou Dianzi Univ Sch Sci Dept Math Hangzhou Zhejiang Peoples R China South China Normal Univ Sch Math Sci Guangzhou Guangdong Peoples R China Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China Nanchang Univ Inst Adv Study Nanchang Jiangxi Peoples R China

The elasticity tensor is one of the most important fourth-order tensors in mechanics. fourth-order three-dimensional symmetric and traceless tensors play a crucial role in the study of the elasticity tensor. In this paper, we present two isotropic irreducible functional bases for a fourth-order three-dimensional symmetric and traceless tensor. One of them is exactly the minimal integrity basis introduced by Smith and Bao in 1997. It has nine homogeneous polynomial invariants of degrees two, three, four, five, six, seven, eight, nine and ten, respectively. We prove that it is also an irreducible functional basis. The second irreducible functional basis also has nine homogeneous polynomial invariants. It has no quartic invariant but has two sextic invariants. The other seven invariants are the same as those of the Smith-Bao basis. Hence, the second irreducible functional basis is not contained in any minimal integrity basis.

关键词： Invariant irreducible functional basis symmetric and traceless tensor fourth-order tensor

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A systematic approach to reduce the independent tensor components by symmetry transformations: a commented translation of "tensors and Crystal Symmetry" by Carl Hermann

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CONTINUUM MECHANICS AND THERMODYNAMICS 2021年第4期33卷 1603-1620页

作者： Gluge, Rainer Assmus, Marcus Otto von Guericke Univ Inst Mech Fac Mech Engn Univ Pl 2 D-39106 Magdeburg Germany Univ Bremen Fac Prod Engn Dept Appl & Struct Mech Biol Garten 2 D-28359 Bremen Germany

We here present a faithful translation of Carl Hermann's important 1934 paper "tensoren und Kristallsymmetrie". This work, originally published in German language, is transferred into English while the p... 详细信息

关键词： Crystal symmetry Symmetry groups Elasticity fourth-order tensor Hermann’ s theorem

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Z-Eigenvalues-Based Sufficient Conditions for the Positive Definiteness of fourth-order tensors

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BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY 2020年第2期43卷 1069-1093页

作者： He, Jun Liu, Yanmin Xu, Guangjun Zunyi Normal Coll Sch Math Zunyi 563006 Guizhou Peoples R China

New Z-eigenvalue localization sets for Z-eigenvalues of fourth-order tensors are given. As applications, sharper upper bounds for the Z-spectral radius of weakly symmetric nonnegative fourth-order tensors are obtained. Some Gershgorin-type results for Z-eigenvalues of structured fourth-order tensors are obtained. Then, some Z-eigenvalues-based sufficient conditions for the positive definiteness of fourth-order tensors are also presented. Finally, numerical examples are given to verify the efficiency of our results.

关键词： fourth-order tensor Z-eigenvalue Positive definiteness

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HARDI Segmentation via fourth-order tensors and Anisotropy Preserving Similarity Measures

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JOURNAL OF MATHEMATICAL IMAGING AND VISION 2019年第8期61卷 1221-1234页

作者： Kaushik, Sumit Slovak, Jan Masaryk Univ Fac Sci Dept Math & Stat Kotlarska 2 CS-61137 Brno Czech Republic

In this work, we discuss the higher-order tensors appearing in high angular resolution diffusion tensor imaging, and we have tested two segmentation methods, the Riemannian spectral clustering and the deformable models, using several projections of the fourth-order tensors to the second-order ones, and diverse similarity measures on them. High angular resolution diffusion imaging has proved its effectiveness in modeling white matter brain structures along with the fiber intersection regions, which is of high importance in brain research. Along with other known projections, we observe that the diagonal components of the flattened fourth-order tensors also live in the well-known Riemannian symmetric space of symmetric positive-definite matrices. We discuss and compare several natural approximations of the distance on the latter space to be used in clustering and segmentation algorithms. The results show that some of the projections unfold the geometry of the higher-order tensors very well, and we also propose the exploitation of the spherical linear interpolation spectral quaternion metric, which proves to be very effective. The latter claims are supported by experimental comparison of the effectiveness of our algorithms with the more usual logarithmic Euclidean and spectral quaternionic metrics, in particular in the presence of noise. Our methods allow to distinguish individual objects in complex structures with high curvatures and crossings.

关键词： Diffusion tensor imaging Riemannian symmetric space Quaternions Similarity measures Dimension reduction fourth-order tensor

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Rectangular array of electromagnetic vector sensors: tensor modelling/decomposition and DOA-polarisation estimation

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IET SIGNAL PROCESSING 2019年第7期13卷 689-699页

作者： Ahmed, Tanveer Zhang Xiaofei Zheng Wang Pan Gong Nanjing Univ Aeronaut & Astronaut Coll Elect & Informat Engn Nanjing 210016 Jiangsu Peoples R China Southeast Univ State Key Lab Millimeter Waves Nanjing 210096 Jiangsu Peoples R China Southeast Univ Natl Mobile Commun Res Lab Nanjing 210096 Jiangsu Peoples R China

In this study, the authors propose a fast quadrilinear decomposition algorithm for estimation of the directions-of-arrival and polarisations of the incident sources via a uniform rectangular array of electromagnetic vector sensors (EMVSs). Conventional quadrilinear alternating least squares (QALS), involves computationally intensive Khatri-Rao products in each iteration, to update the parameter matrices (factors). Moreover, QALS is more likely to fall in a local minimum and tends to take more steps before an acceptable solution, which further slows down the convergence and often mis-converges, thereby yielding meaningless results. To preserve the quadrilinearity, they arrange the measurements as a four-dimensional (4D) data (fourth-order tensor), from which a third-order sub-tensor (3D slice) can be obtained by fixing one index along any dimension. These slices are used to create new cost functions that are alternately minimised while updating the factors until convergence. They show that the rows of parameter matrices form the diagonal elements of a tensor, which capture the internal quadrilinearity of data and significantly reduce the cost function in few iterations only. Simulation results verify that the authors' algorithm holds faster convergence, does not mis-converge, provides parameter estimation accuracy similarly to the QALS and superior of the Estimation of Signal Parameters via Rotational Invariance Technique and propagator method.

关键词： parameter estimation tensors iterative methods least squares approximations direction-of-arrival estimation array signal processing matrix algebra internal quadrilinearity cost function parameter estimation accuracy QALS Signal Parameters electromagnetic vector sensors DOA-polarisation estimation fast quadrilinear decomposition algorithm 3D slice third-order sub-tensor fourth-order tensor four-dimensional data acceptable solution local minimum parameter matrices intensive Khatri-Rao products uniform rectangular array incident sources polarisations directions-of-arrival

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tensor representation of magnetostriction for all crystal classes

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MATHEMATICS AND MECHANICS OF SOLIDS 2019年第9期24卷 2814-2843页

作者： Federico, Salvatore Consolo, Giancarlo Valenti, Giovanna Univ Calgary Dept Mech & Mfg Engn 2500 Univ Dr NW Calgary AB T2N 1N4 Canada Univ Messina Dept Math Comp Phys & Earth Sci Messina Italy Univ Messina Dept Engn Messina Italy

A systematic representation of the fourth-order magnetostriction tensor for all crystal classes is presented, based on the formalism proposed by Walpole (Proc R Soc Lond Ser A 1984;391: 149-179). This representation allows the general, unconstrained case, as well as the case in which the magnetostrictive strain is assumed to be isochoric, to be studied. The knowledge of the fourth-order magnetostriction tensor enables the stress-free magnetostrictive strain tensor as well as the scalar strain in a given direction to be calculated.

关键词： Magnetostriction fourth-order tensor tensor algebra strain crystal symmetry material symmetry

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