The Experts below are selected from a list of 2718 Experts worldwide ranked by ideXlab platform
Lei Wang - One of the best experts on this subject based on the ideXlab platform.
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the real Eigenpairs of symmetric tensors and its application to independent component analysis
IEEE Transactions on Systems Man and Cybernetics, 2021Co-Authors: Lei Wang, Xiurui GengAbstract:It has been proved that the determination of independent components (ICs) in the independent component analysis (ICA) can be attributed to calculating the Eigenpairs of high-order statistical tensors of the data. However, previous works can only obtain approximate solutions, which may affect the accuracy of the ICs. In addition, the number of ICs would need to be set manually. Recently, an algorithm based on semidefinite programming (SDP) has been proposed, which utilizes the first-order gradient information of the Lagrangian function and can obtain all the accurate real Eigenpairs. In this article, for the first time, we introduce this into the ICA field, which tends to further improve the accuracy of the ICs. Note that the number of Eigenpairs of symmetric tensors is usually larger than the number of ICs, indicating that the results directly obtained by SDP are redundant. Thus, in practice, it is necessary to introduce second-order derivative information to identify local extremum solutions. Therefore, originating from the SDP method, we present a new modified version, called modified SDP (MSDP), which incorporates the concept of the projected Hessian matrix into SDP and, thus, can intellectually exclude redundant ICs and select true ICs. Some cases that have been tested in the experiments demonstrate its effectiveness. Experiments on the image/sound blind separation and real multi/hyperspectral image also show its superiority in improving the accuracy of ICs and automatically determining the number of ICs. In addition, the results on hyperspectral simulation and real data also demonstrate that MSDP is also capable of dealing with cases, where the number of features is less than the number of ICs.
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npsa nonorthogonal principal skewness analysis
IEEE Transactions on Image Processing, 2020Co-Authors: Xiurui Geng, Lei WangAbstract:Principal skewness analysis (PSA) has been introduced for feature extraction in hyperspectral imagery. As a third-order generalization of principal component analysis (PCA), its solution of searching for the local maximum skewness direction is transformed into the problem of calculating the Eigenpairs (the eigenvalues and the corresponding eigenvectors) of a coskewness tensor. By combining a fixed-point method with an orthogonal constraint, the new Eigenpairs are prevented from converging to the same previously determined maxima. However, in general, the eigenvectors of the supersymmetric tensor are not inherently orthogonal, which implies that the results obtained by the search strategy used in PSA may unavoidably deviate from the actual Eigenpairs. In this paper, we propose a new nonorthogonal search strategy to solve this problem and the new algorithm is named nonorthogonal principal skewness analysis (NPSA). The contribution of NPSA lies in the finding that the search space of the eigenvector to be determined can be enlarged by using the orthogonal complement of the Kronecker product of the previous eigenvector with itself, instead of its orthogonal complement space. We also give a detailed theoretical proof on why we can obtain the more accurate Eigenpairs through the new search strategy by comparison with PSA. In addition, after some algebraic derivations, the complexity of the presented algorithm is also greatly reduced. Experiments with both simulated data and real multi/hyperspectral imagery demonstrate its validity in feature extraction.
Huiqing Xie - One of the best experts on this subject based on the ideXlab platform.
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a receptance method for robust and minimum norm partial quadratic eigenvalue assignment
Mechanical Systems and Signal Processing, 2021Co-Authors: Huiqing XieAbstract:Abstract This paper considers the partial quadratic eigenvalue assignment problem (PQEAP) in active vibration control using state feedback. A receptance method is proposed for PQEAP such that the norms of feedback matrices and the condition number of close-loop system are simultaneously minimized. The real form of the proposed method is developed. The proposed method only uses receptance matrices and the unwanted Eigenpairs of open-loop system. The system matrices and the unchanged Eigenpairs of open-loop system are not required. The efficiency of the proposed method is illustrated by some numerical examples.
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an iterative method for simultaneously computing the derivatives of simple and multiple Eigenpairs of a quadratic eigenvalue problem
Journal of Computational and Applied Mathematics, 2019Co-Authors: Huiqing XieAbstract:Abstract A new iterative method is proposed to compute the first-order and high-order derivatives of several Eigenpairs of a quadratic eigenvalue problem (QEP). The derivatives of simple and semi-simple Eigenpairs are simultaneously computed. The proposed method avoids to solve many different linear systems with the size of the QEP. Only the Eigenpairs to be differentiated are required in our method. Convergence properties of the proposed iterative method are established. In order to accelerate the proposed iterative method, vector epsilon algorithm is applied and then an acceleration algorithm for our iterative method is presented and analyzed. Numerical experiments show that the proposed method is efficient for the computation of the first-order and high-order derivatives of Eigenpairs of a QEP, especially for several eigenpair derivatives in moderate and large QEP.
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simultaneous iterative method for the derivatives of several Eigenpairs of unsymmetric damped systems
Mechanical Systems and Signal Processing, 2015Co-Authors: Huiqing XieAbstract:Abstract In this paper, a new iterative method is proposed to calculate a few eigenpair derivatives of damped systems. The proposed method simultaneously computes the derivatives of several simple eigenvalues and their corresponding eigenvectors. Eigenpair derivatives are directly determined in n-space. Moreover, our method does not require the left eigenvectors. Convergence theory of the proposed method is given. On these grounds, using vector ϵ -algorithm, acceleration techniques for our method are provided and analyzed. Finally some numerical experiments are reported to show the efficiency of the proposed methods.
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iram based method for Eigenpairs and their derivatives of large matrix valued functions
Numerical Linear Algebra With Applications, 2011Co-Authors: Huiqing XieAbstract:Based on the implicitly restarted Arnoldi method for Eigenpairs of large matrix, a new method is presented for the computation of a few Eigenpairs and their derivatives of large matrix-valued functions. Eigenpairs and their derivatives are calculated simultaneously. Equation systems that are solved for eigenvector derivatives are greatly reduced from the original matrix size. The left eigenvectors are not required. Hence, the computational cost is saved. The convergence theory of the proposed method is established. Finally, numerical experiments are given to illustrate the efficiency of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.
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Davidson method for Eigenpairs and their partial derivatives of generalized eigenvalue problems
Communications in Numerical Methods in Engineering, 2005Co-Authors: Huiqing Xie, Hua DaiAbstract:Based on Davidson method for solving generalized eigenvalue problems, a new method for synchro calculation of Eigenpairs and their partial derivatives of generalized eigenvalue problems is presented. Eigenpairs and their partial derivatives are computed simultaneously. The equation systems that are solved for eigenvector partial derivatives can be greatly reduced from the original matrix sizes, thus the efficiency of computing eigenvector partial derivatives is improved. Numerical results show that the proposed method is efficient, especially for the large-scale symmetric generalized eigenvalue problems. Copyright © 2005 John Wiley & Sons, Ltd.
Xiurui Geng - One of the best experts on this subject based on the ideXlab platform.
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the real Eigenpairs of symmetric tensors and its application to independent component analysis
IEEE Transactions on Systems Man and Cybernetics, 2021Co-Authors: Lei Wang, Xiurui GengAbstract:It has been proved that the determination of independent components (ICs) in the independent component analysis (ICA) can be attributed to calculating the Eigenpairs of high-order statistical tensors of the data. However, previous works can only obtain approximate solutions, which may affect the accuracy of the ICs. In addition, the number of ICs would need to be set manually. Recently, an algorithm based on semidefinite programming (SDP) has been proposed, which utilizes the first-order gradient information of the Lagrangian function and can obtain all the accurate real Eigenpairs. In this article, for the first time, we introduce this into the ICA field, which tends to further improve the accuracy of the ICs. Note that the number of Eigenpairs of symmetric tensors is usually larger than the number of ICs, indicating that the results directly obtained by SDP are redundant. Thus, in practice, it is necessary to introduce second-order derivative information to identify local extremum solutions. Therefore, originating from the SDP method, we present a new modified version, called modified SDP (MSDP), which incorporates the concept of the projected Hessian matrix into SDP and, thus, can intellectually exclude redundant ICs and select true ICs. Some cases that have been tested in the experiments demonstrate its effectiveness. Experiments on the image/sound blind separation and real multi/hyperspectral image also show its superiority in improving the accuracy of ICs and automatically determining the number of ICs. In addition, the results on hyperspectral simulation and real data also demonstrate that MSDP is also capable of dealing with cases, where the number of features is less than the number of ICs.
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npsa nonorthogonal principal skewness analysis
IEEE Transactions on Image Processing, 2020Co-Authors: Xiurui Geng, Lei WangAbstract:Principal skewness analysis (PSA) has been introduced for feature extraction in hyperspectral imagery. As a third-order generalization of principal component analysis (PCA), its solution of searching for the local maximum skewness direction is transformed into the problem of calculating the Eigenpairs (the eigenvalues and the corresponding eigenvectors) of a coskewness tensor. By combining a fixed-point method with an orthogonal constraint, the new Eigenpairs are prevented from converging to the same previously determined maxima. However, in general, the eigenvectors of the supersymmetric tensor are not inherently orthogonal, which implies that the results obtained by the search strategy used in PSA may unavoidably deviate from the actual Eigenpairs. In this paper, we propose a new nonorthogonal search strategy to solve this problem and the new algorithm is named nonorthogonal principal skewness analysis (NPSA). The contribution of NPSA lies in the finding that the search space of the eigenvector to be determined can be enlarged by using the orthogonal complement of the Kronecker product of the previous eigenvector with itself, instead of its orthogonal complement space. We also give a detailed theoretical proof on why we can obtain the more accurate Eigenpairs through the new search strategy by comparison with PSA. In addition, after some algebraic derivations, the complexity of the presented algorithm is also greatly reduced. Experiments with both simulated data and real multi/hyperspectral imagery demonstrate its validity in feature extraction.
Jackson R. Mayo - One of the best experts on this subject based on the ideXlab platform.
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an adaptive shifted power method for computing generalized tensor Eigenpairs
SIAM Journal on Matrix Analysis and Applications, 2014Co-Authors: Tamara G. Kolda, Jackson R. MayoAbstract:Several tensor eigenpair definitions have been put forth in the past decade, but these can all be unified under generalized tensor eigenpair framework, introduced by Chang, Pearson, and Zhang [J. Math. Anal. Appl., 350 (2009), pp. 416--422]. Given mth-order, n-dimensional real-valued symmetric tensors ${\mathscr{A}}$ and $\boldsymbol{\mathscr{B}}$, the goal is to find $\lambda \in \mathbb{R}$ and $\mathbf{x} \in \mathbb{R}^{n}, \mathbf{x} \neq 0$ such that ${\mathscr{A}}\mathbf{x}^{m-1} = \lambda {\mathscr{B}}\mathbf{x}^{m-1}$. Different choices for ${\mathscr{B}}$ yield different versions of the tensor eigenvalue problem. We present our generalized eigenproblem adaptive power (GEAP) method for solving the problem, which is an extension of the shifted symmetric higher-order power method (SS-HOPM) for finding Z-Eigenpairs. A major drawback of SS-HOPM is that its performance depended on choosing an appropriate shift, but our GEAP method also includes an adaptive method for choosing the shift automatically.
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an adaptive shifted power method for computing generalized tensor Eigenpairs
arXiv: Numerical Analysis, 2014Co-Authors: Tamara G. Kolda, Jackson R. MayoAbstract:Several tensor eigenpair definitions have been put forth in the past decade, but these can all be unified under generalized tensor eigenpair framework, introduced by Chang, Pearson, and Zhang (2009). Given mth-order, n-dimensional real-valued symmetric tensors A and B, the goal is to find $\lambda \in R$ and $x \in R^n$, $x \neq 0$, such that $Ax^{m-1} = \lambda Bx^{m-1}$. Different choices for B yield different versions of the tensor eigenvalue problem. We present our generalized eigenproblem adaptive power method (GEAP) method for solving the problem, which is an extension of the shifted symmetric higher-order power method (SS-HOPM) for finding Z-Eigenpairs. A major drawback of SS-HOPM was that its performance depended in choosing an appropriate shift, but our GEAP method also includes an adaptive method for choosing the shift automatically.
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shifted power method for computing tensor Eigenpairs
SIAM Journal on Matrix Analysis and Applications, 2011Co-Authors: Tamara G. Kolda, Jackson R. MayoAbstract:Recent work on eigenvalues and eigenvectors for tensors of order $m \ge 3$ has been motivated by applications in blind source separation, magnetic resonance imaging, molecular conformation, and more. In this paper, we consider methods for computing real symmetric-tensor Eigenpairs of the form $\boldsymbol{\mathscr{A}}\mathbf{x}^{m-1} = \lambda \mathbf{x}$ subject to $\|\mathbf{x}\|=1$, which is closely related to optimal rank-1 approximation of a symmetric tensor. Our contribution is a shifted symmetric higher-order power method (SS-HOPM), which we show is guaranteed to converge to a tensor eigenpair. SS-HOPM can be viewed as a generalization of the power iteration method for matrices or of the symmetric higher-order power method. Additionally, using fixed point analysis, we can characterize exactly which Eigenpairs can and cannot be found by the method. Numerical examples are presented, including examples from an extension of the method to finding complex Eigenpairs.
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Shifted Power Method for Computing Tensor Eigenpairs
SIAM Journal on Matrix Analysis and Applications, 2011Co-Authors: Tamara G. Kolda, Jackson R. MayoAbstract:Recent work on eigenvalues and eigenvectors for tensors of order m >= 3 has been motivated by applications in blind source separation, magnetic resonance imaging, molecular conformation, and more. In this paper, we consider methods for computing real symmetric-tensor Eigenpairs of the form Ax^{m-1} = \lambda x subject to ||x||=1, which is closely related to optimal rank-1 approximation of a symmetric tensor. Our contribution is a shifted symmetric higher-order power method (SS-HOPM), which we show is guaranteed to converge to a tensor eigenpair. SS-HOPM can be viewed as a generalization of the power iteration method for matrices or of the symmetric higher-order power method. Additionally, using fixed point analysis, we can characterize exactly which Eigenpairs can and cannot be found by the method. Numerical examples are presented, including examples from an extension of the method to finding complex Eigenpairs.
Siyu Zhan - One of the best experts on this subject based on the ideXlab platform.
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iteration on single vector for extracting two extremal Eigenpairs of symmetric matrices
Neurocomputing, 2019Co-Authors: Ying Tang, Yuan Tang, Siyu ZhanAbstract:Abstract It has been shown that the first two extremal Eigenpairs of symmetric matrices are involved a lot in spectral clustering, dimensionality reduction, image segmentation, and graph theory. We also know that the eigengap directly affects the stability of principal eigenvector related algorithms, such as the Hyperlink-Induced Topic Search (HITS). To extract two extremal Eigenpairs, conventional methods are generally iterated on two orthogonal vectors, i.e., 2-dimensional subspace. This paper introduces an approach that is iterated on single vector but can simultaneously extract the largest or smallest two Eigenpairs of general symmetric matrices, which reduces the solution scale by half. The complete stability analysis is also presented here. Numerical experiments demonstrate the superior performance of the proposed algorithm.