|
|
梯度神经网络方法求解含有绝对值形式张量特征值
|
Abstract:
目前，关于特征值的研究主要集中在特征值互补、特征值估计和运用算法计算特征值等方向。受张量绝对值方程
启示，本文考虑一类新形式的特征值问题，并提出梯度神经网络方法求解新形式张量特征值和特征向量。数值实验表明了梯度神经网络方法求解该问题的可行性和有效性。
At present, research on eigenvalues mainly focuses on complementary eigenvalues, eigenvalue estimation, and the application of algorithms to calculate eigenvalues. Inspired by the tensor absolute value equation
, this paper considers a new form of eigenvalue problem and proposes a gradient neural network method to solve the eigenvalues and eigenvectors of the new form tensor. Numerical experiments have shown the feasibility and effectiveness of using gradient neural network methods to solve this problem.
| [1] | Qi, L. (2005) Eigenvalues of a Real Supersymmetric Tensor. Journal of Symbolic Computation, 40, 1302-1324. https://doi.org/10.1016/j.jsc.2005.05.007 |
| [2] | Li, H., Du, S. and Wang, Y. (2020) An Inexact Levenberg-Marquardt Method for Tensor Eigenvalue Complementarity Problem. Pacific Journal of Optimization, 16, 87-99. |
| [3] | Du, S., Zhang, L., Chen, C. and Qi, L. (2018) Tensor Absolute Value Equations. Science China Mathematics, 61, 1695-1710. https://doi.org/10.1007/s11425-017-9238-6 |
| [4] | Zhang, Y. (2006) A Set of Nonlinear Equations and Inequalities Arising in Robotics and Its Online Solution via a Primal Neural Network. Neurocomputing, 70, 513-524. https://doi.org/10.1016/j.neucom.2005.11.006 |
| [5] | Stanimirovic, P.S., Zivkovic, I.S. and Wei, Y. (2015) Recurrent Neural Network for Computing the Drazin Inverse. IEEE Transactions on Neural Networks and Learning Systems, 26, 2830-2843. https://doi.org/10.1109/tnnls.2015.2397551 |
| [6] | Zhang, Y., Chen, Z. and Chen, K. (2009) Convergence Properties Analysis of Gradient Neural Network for Solving Online Linear Equations. Acta Automatica Sinica, 35, 1136-1139. https://doi.org/10.3724/sp.j.1004.2009.01136 |
| [7] | Xiao, L. and Zhang, Y. (2014) Solving Time-Varying Inverse Kinematics Problem of Wheeled Mobile Manipulators Using Zhang Neural Network with Exponential Convergence. Nonlinear Dynamics, 76, 1543-1559. https://doi.org/10.1007/s11071-013-1227-7 |
| [8] | Chen, K. (2013) Implicit Dynamic System for Online Simultaneous Linear Equations Solving. Electronics Letters, 49, 101-102. https://doi.org/10.1049/el.2012.3501 |
| [9] | Yi, C. and Zhang, Y. (2008) Analogue Recurrent Neural Network for Linear Algebraic Equation Solving. Electronics Letters, 44, 1078-1080. https://doi.org/10.1049/el:20081390 |
| [10] | Zhang, Y. and Chen, K. (2008) Global Exponential Convergence and Stability of Wang Neural Network for Solving Online Linear Equations. Electronics Letters, 44, 145-146. https://doi.org/10.1049/el:20081928 |
| [11] | Ding, F. and Chen, T.W. (2005) Gradient Based Iterative Algorithms for Solving a Class of Matrix Equations. IEEE Transactions on Automatic Control, 50, 1216-1221. https://doi.org/10.1109/tac.2005.852558 |
| [12] | Wang, X., Che, M. and Wei, Y. (2020) Neural Network Approach for Solving Nonsingular Multi-Linear Tensor Systems. Journal of Computational and Applied Mathematics, 368, Article ID: 112569. https://doi.org/10.1016/j.cam.2019.112569 |
| [13] | Wilkinson, J.H. (1965) The Algebraic Eigenvalue Problem. Clarendon. |
| [14] | Chang, K.C., Pearson, K.J. and Zhang, T. (2013) Some Variational Principles for Z-Eigenvalues of Nonnegative Tensors. Linear Algebra and Its Applications, 438, 4166-4182. https://doi.org/10.1016/j.laa.2013.02.013 |
| [15] | Mo, C., Wang, X. and Wei, Y. (2020) Time-Varying Generalized Tensor Eigenanalysis via Zhang Neural Networks. Neurocomputing, 407, 465-479. https://doi.org/10.1016/j.neucom.2020.04.115 |
