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Two finite-time convergent Zhang neural network models for time-varying complex matrix Drazin inverse. (English) Zbl 1415.15007

Summary: This paper concerns the computation of the Drazin inverse of a complex time-varying matrix. Based on two Zhang functions constructed from two limit representations of the Drazin inverse, we present two complex Zhang neural network (ZNN) models with the Li activation function for computing the Drazin inverse of a complex time-varying square matrix. We prove that our ZNN models globally converge in finite time. In addition, upper bounds of the convergence time are derived analytically via the Lyapunov theory. Our simulation results verify the theoretical analysis and demonstrate the superiority of our ZNN models over the gradient-based GNN models.

MSC:

15A09 Theory of matrix inversion and generalized inverses
65F20 Numerical solutions to overdetermined systems, pseudoinverses
92B20 Neural networks for/in biological studies, artificial life and related topics
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[1] Avrachenkov, K. E.; Lasserre, J. B., Analytic perturbation of generalized inverses, Linear Algebra Appl., 438, 1793-1813 (2013) · Zbl 1347.15006
[2] Bhat, S. P.; Bernstein, D. S., Finite-time stability of continuous autonomous systems, SIAM J. Control Optim., 38, 751-766 (2000) · Zbl 0945.34039
[3] Booker, S. M., A family of optimal excitations for inducing complex dynamics in planar dynamical systems, Nonlinearity, 13, 145-163 (2000), (19 pp.) · Zbl 1024.34039
[4] Campbell, S. L., Singular Systems of Differential Equations, Res. Notes Math., vol. 5, 97-107 (1980), Pitman Advanced Publishing Program: Pitman Advanced Publishing Program Boston, Mass.-London · Zbl 0634.34006
[5] Campbell, S. L.; Meyer, C. D., Generalized Inverses of Linear Transformations (1991), Dover Publications, Inc.: Dover Publications, Inc. New York, corrected reprint of the 1979 original · Zbl 0732.15003
[6] Castro González, N.; Koliha, J. J.; Wei, Y., Perturbation of the Drazin inverse for matrices with equal eigenprojections at zero, Linear Algebra Appl., 312, 181-189 (2000) · Zbl 0963.15002
[7] Chen, Y., Representation and approximation for the Drazin inverse \(a^{(d)}\), Appl. Math. Comput., 119, 147-160 (2001) · Zbl 1023.65032
[8] Cichocki, A.; Kaczorek, T.; Stajniak, A., Computation of the Drazin inverse of a singular matrix making use of neural networks, Bull. Pol. Acad. Sci., Tech. Sci., 40, 387-394 (1992) · Zbl 0777.65024
[9] Cline, R. E.; Greville, T. N.E., A Drazin inverse for rectangular matrices, Linear Algebra Appl., 29, 53-62 (1980) · Zbl 0433.15002
[10] Golub, G. H.; Hansen, P. C.; O’Leary, D. P., Tikhonov regularization and total least squares, SIAM J. Matrix Anal. Appl., 21, 185-194 (1999) · Zbl 0945.65042
[11] González, N. C.; Koliha, J.; Wei, Y., Error bounds for perturbation of the Drazin inverse of closed operators with equal spectral projections, Appl. Anal., 81, 915-928 (2002) · Zbl 1044.47010
[12] Hirose, A.; Yoshida, S., Generalization characteristics of complex-valued feedforward neural networks in relation to signal coherence, IEEE Trans. Neural Netw. Learn. Syst., 23, 541-551 (2012)
[13] Hong, X.; Chen, S., Modeling of complex-valued wiener systems using B-spline neural network, IEEE Trans. Neural Netw., 22, 818-825 (2011)
[14] Hopfield, J. J., Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Natl. Acad. Sci., 81, 3088-3092 (1984) · Zbl 1371.92015
[15] Hu, J.; Wang, J., Global stability of complex-valued recurrent neural networks with time-delays, IEEE Trans. Neural Netw. Learn. Syst., 23, 853-865 (2012)
[16] Ilyashenko, Y., Some open problems in real and complex dynamical systems, Nonlinearity, 21, T101-T107 (2008) · Zbl 1183.37016
[17] Jang, J. S.; Lee, S. Y.; Shin, S. Y.; Jang, J. S.; Shin, S. Y., An optimization network for matrix inversion, Neural Inf. Process. Syst., 397-401 (1987)
[18] Li, S.; Chen, S.; Liu, B., Accelerating a recurrent neural network to finite-time convergence for solving time-varying Sylvester equation by using a sign-bi-power activation function, Neural Process. Lett., 37, 189-205 (2013)
[19] Li, S.; Li, Y.; Wang, Z., A class of finite-time dual neural networks for solving quadratic programming problems and its k-winners-take-all application, Neural Networks, 39, 27-39 (2013) · Zbl 1338.90291
[20] Liao, B.; Zhang, Y., Different complex ZFs leading to different complex ZNN models for time-varying complex generalized inverse matrices, IEEE Trans. Neural Netw. Learn. Syst., 25, 1621-1631 (2014)
[21] Liao, B.; Zhang, Y., From different ZFs to different ZNN models accelerated via Li activation functions to finite-time convergence for time-varying matrix pseudoinversion, Neurocomputing, 133, 512-522 (2014)
[22] Lippmann, R. P., An introduction to computing with neural nets, IEEE ASSP Mag., 4, 4-22 (1987)
[23] Liu, Q.; Wang, J., Finite-time convergent recurrent neural network with a hard-limiting activation function for constrained optimization with piecewise-linear objective functions, IEEE Trans. Neural Netw., 22, 601-613 (2011)
[24] Luo, F. L.; Bao, Z., Neural network approach to computing matrix inversion, Appl. Math. Comput., 47, 109-120 (1992) · Zbl 0748.65025
[25] Soleymani, F.; Stanimirović, P. S., A higher order iterative method for computing the Drazin inverse, Sci. World J., 2013, 206-232 (2013)
[26] Song, J.; Yam, Y., Complex recurrent neural network for computing the inverse and pseudo-inverse of the complex matrix, Appl. Math. Comput., 93, 195-205 (1998) · Zbl 0943.65040
[27] Stanimirović, P. S.; Živković, I. S.; Wei, Y., Recurrent neural network approach based on the integral representation of the Drazin inverse, Neural Comput., 27, 2107-2131 (2015) · Zbl 1472.92042
[28] Stanimirović, P. S.; Živković, I. S.; Wei, Y., Recurrent neural network for computing the Drazin inverse, IEEE Trans. Neural Netw. Learn. Syst., 26, 2830-2843 (2015)
[29] Wang, J., A recurrent neural network for real-time matrix inversion, Appl. Math. Comput., 55, 89-100 (1993) · Zbl 0772.65015
[30] Wang, J., Recurrent neural networks for solving linear matrix equations, Comput. Math. Appl., 26, 23-34 (1993) · Zbl 0796.65058
[31] Wang, J., Recurrent neural networks for computing pseudoinverses of rank-deficient matrices, SIAM J. Sci. Comput., 18, 1479-1493 (1997) · Zbl 0891.93034
[32] Wang, X. Z.; Ma, H.; Stanimirović, P. S., Nonlinearly activated recurrent neural network for computing the Drazin inverse, Neural Process. Lett. (2017)
[33] Wang, X. Z.; Wei, Y.; Stanimirović, P. S., Complex neural network models for time-varying Drazin inverse, Neural Comput., 28, 2790-2824 (2016) · Zbl 1474.68157
[34] Wei, Y., Recurrent neural networks for computing weighted Moore-Penrose inverse, Appl. Math. Comput., 116, 279-287 (2000) · Zbl 1023.65030
[35] Wei, Y., The Drazin inverse of a modified matrix, Appl. Math. Comput., 125, 295-301 (2002) · Zbl 1025.15005
[36] Wei, Y.; Wang, G., The perturbation theory for the Drazin inverse and its applications, Linear Algebra Appl., 1, 118-120 (1996) · Zbl 0860.15006
[37] Wei, Y.; Wu, H., Challenging problems on the perturbation of Drazin inverse, Ann. Oper. Res., 103, 371-378 (2001) · Zbl 0993.65047
[38] Xia, Y.; Jelfs, B.; Van Hulle, M. M.; Principe, J. C.; Mandic, D. P., An augmented echo state network for nonlinear adaptive filtering of complex noncircular signals, IEEE Trans. Neural Netw., 22, 74-83 (2011)
[39] Xiao, L., A finite-time convergent neural dynamics for online solution of time-varying linear complex matrix equation, Neurocomputing, 167, 254-259 (2015)
[40] Zhang, Y.; Ge, S. S., Design and analysis of a general recurrent neural network model for time-varying matrix inversion, IEEE Trans. Neural Netw., 16, 1477-1490 (2005)
[41] Zhang, Y.; Qiu, B.; Jin, L.; Guo, D.; Yang, Z., Infinitely many Zhang functions resulting in various ZNN models for time-varying matrix inversion with link to Drazin inverse, Inform. Process. Lett., 115, 703-706 (2015) · Zbl 1342.65250
[42] Zhang, Y.; Yang, Y.; Tan, N.; Cai, B., Zhang neural network solving for time-varying full-rank matrix Moore-Penrose inverse, Computing, 92, 97-121 (2011) · Zbl 1226.93075
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