# zbMATH — the first resource for mathematics

Neural-network-based approach for extracting eigenvectors and eigenvalues of real normal matrices and some extension to real matrices. (English) Zbl 1266.65062
Summary: This paper introduces a novel neural-network-based approach for extracting some eigenpairs of real normal matrices of order $$n$$. Based on the proposed algorithm, the eigenvalues that have the largest and smallest modulus, real parts, or absolute values of imaginary parts can be extracted, respectively, as well as the corresponding eigenvectors. Although the ordinary differential equation on which our proposed algorithm is built is only $$n$$-dimensional, it can succeed to extract $$n$$-dimensional complex eigenvectors that are indeed $$2n$$-dimensional real vectors. Moreover, we show that extracting eigenpairs of general real matrices can be reduced to those of real normal matrices by employing the norm-reducing skill. Numerical experiments verified the computational capability of the proposed algorithm.
##### MSC:
 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 92B20 Neural networks for/in biological studies, artificial life and related topics
Full Text:
##### References:
 [1] S. Attallah and K. Abed-Meraim, “A fast adaptive algorithm for the generalized symmetric eigenvalue problem,” IEEE Signal Processing Letters, vol. 15, pp. 797-800, 2008. [2] T. Laudadio, N. Mastronardi, and M. Van Barel, “Computing a lower bound of the smallest eigenvalue of a symmetric positive-definite Toeplitz matrix,” IEEE Transactions on Information Theory, vol. 54, no. 10, pp. 4726-4731, 2008. · Zbl 1322.65050 [3] J. Shawe-Taylor, C. K. I. Williams, N. Cristianini, and J. Kandola, “On the eigenspectrum of the Gram matrix and the generalization error of kernel-PCA,” IEEE Transactions on Information Theory, vol. 51, no. 7, pp. 2510-2522, 2005. · Zbl 1310.15076 [4] D. Q. Wang and M. Zhang, “A new approach to multiple class pattern classification with random matrices,” Journal of Applied Mathematics and Decision Sciences, no. 3, pp. 165-175, 2005. · Zbl 1151.62336 [5] M. R. Bastian, J. H. Gunther, and T. K. Moon, “A simplified natural gradient learning algorithm,” Advances in Artificial Neural Systems, vol. 2011, Article ID 407497, 9 pages, 2011. · Zbl 05946086 [6] T. H. Le, “Applying artificial neural networks for face recognition,” Advances in Artificial Neural Systems, vol. 2011, Article ID 673016, 16 pages, 2011. · Zbl 06015495 [7] Y. Xia, “An extended projection neural network for constrained optimization,” Neural Computation, vol. 16, no. 4, pp. 863-883, 2004. · Zbl 1097.68609 [8] Y. Xia and J. Wang, “A recurrent neural network for solving nonlinear convex programs subject to linear constraints,” IEEE Transactions on Neural Networks, vol. 16, no. 2, pp. 379-386, 2005. [9] T. Voegtlin, “Recursive principal components analysis,” Neural Networks, vol. 18, no. 8, pp. 1051-1063, 2005. [10] J. Qiu, H. Wang, J. Lu, B. Zhang, and K.-L. Du, “Neural network implementations for PCA and its extensions,” ISRN Artificial Intelligence, vol. 2012, Article ID 847305, 19 pages, 2012. [11] H. Liu and J. Wang, “Integrating independent component analysis and principal component analysis with neural network to predict Chinese stock market,” Mathematical Problems in Engineering, vol. 2011, Article ID 382659, 15 pages, 2011. · Zbl 06021076 [12] F. L. Luo, R. Unbehauen, and A. Cichocki, “A minor component analysis algorithm,” Neural Networks, vol. 10, no. 2, pp. 291-297, 1997. [13] G. H. Golub and C. F. Van Loan, Matrix Computations, vol. 3 of Johns Hopkins Series in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, Md, USA, 1983. · Zbl 0559.65011 [14] C. Chatterjee, V. P. Roychowdhury, J. Ramos, and M. D. Zoltowski, “Self-organizing algorithms for generalized eigen-decomposition,” IEEE Transactions on Neural Networks, vol. 8, no. 6, pp. 1518-1530, 1997. [15] H. Kakeya and T. Kindo, “Eigenspace separation of autocorrelation memory matrices for capacity expansion,” Neural Networks, vol. 10, no. 5, pp. 833-843, 1997. [16] Y. Liu, Z. You, and L. Cao, “A functional neural network for computing the largest modulus eigenvalues and their corresponding eigenvectors of an anti-symmetric matrix,” Neurocomputing, vol. 67, no. 1-4, pp. 384-397, 2005. · Zbl 02224110 [17] Y. Liu, Z. You, and L. Cao, “A functional neural network computing some eigenvalues and eigenvectors of a special real matrix,” Neural Networks, vol. 18, no. 10, pp. 1293-1300, 2005. · Zbl 1101.68774 [18] Y. Liu, Z. You, and L. Cao, “A recurrent neural network computing the largest imaginary or real part of eigenvalues of real matrices,” Computers & Mathematics with Applications, vol. 53, no. 1, pp. 41-53, 2007. · Zbl 1135.15006 [19] E. Oja, “Principal components, minor components, and linear neural networks,” Neural Networks, vol. 5, no. 6, pp. 927-935, 1992. [20] R. Perfetti and E. Massarelli, “Training spatially homogeneous fully recurrent neural networks in eigenvalue space,” Neural Networks, vol. 10, no. 1, pp. 125-137, 1997. · Zbl 1067.68589 [21] L. Xu, E. Oja, and C. Y. Suen, “Modified Hebbian learning for curve and surface fitting,” Neural Networks, vol. 5, no. 3, pp. 441-457, 1992. [22] Q. Zhang and Y. W. Leung, “A class of learning algorithms for principal component analysis and minor component analysis,” IEEE Transactions on Neural Networks, vol. 11, no. 2, pp. 529-533, 2000. [23] Y. Zhang, Y. Fu, and H. J. Tang, “Neural networks based approach for computing eigenvectors and eigenvalues of symmetric matrix,” Computers & Mathematics with Applications, vol. 47, no. 8-9, pp. 1155-1164, 2004. · Zbl 1067.65038 [24] L. Mirsky, “On the minimization of matrix norms,” The American Mathematical Monthly, vol. 65, pp. 106-107, 1958. [25] C. P. ] Huang and R. T. Gregory, A Norm-Reducing Jacobi-Like Algorithm for the Eigenvalues of Non-Normal Matrices, Colloquia Mathematica Societatis Janos Bolyai, Keszthely, Hungary, 1977. · Zbl 0455.65029 [26] I. Kiessling and A. Paulik, “A norm-reducing Jacobi-like algorithm for the eigenvalues of non-normal matrices,” Journal of Computational and Applied Mathematics, vol. 8, no. 3, pp. 203-207, 1982. · Zbl 0484.65019 [27] R. Sacks-Davis, “A real norm-reducing Jacobi-type eigenvalue algorithm,” The Australian Computer Journal, vol. 7, no. 2, pp. 65-69, 1975. · Zbl 0315.65022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.