Varying-parameter Zhang neural network for approximating some expressions involving outer inverses. (English) Zbl 07333778
Summary: A varying-parameter ZNN (VPZNN) neural design is defined for approximating various generalized inverses and expressions involving generalized inverses of complex matrices. The proposed model is termed as \(\operatorname{CVPZNN}(A,F,G)\) and defined on the basis of the error function which includes three appropriate matrices \(A\), \(F\), \(G\). The \(\operatorname{CVPZNN}(A,F,G)\) evolution design includes so far defined VPZNN models for computing generalized inverses and also generates a number of matrix expressions involving these generalized inverses. Global and super-exponential convergence properties of the proposed model as well as behaviour of its equilibrium state are investigated. Main contribution of the defined model is its generality. Most important particular cases of the defined model are presented in order to show this fact explicitly. Presented simulation results illustrate generality and effectiveness of the discovered ZNN evolution design.
MSC:
| 65F20 | Numerical solutions to overdetermined systems, pseudoinverses |
| 15A09 | Theory of matrix inversion and generalized inverses |
| 65Y10 | Numerical algorithms for specific classes of architectures |
Keywords:
Zhang neural network; generalized inverses; super-exponential convergence; time-varying matrix; varying-parameter ZNN designReferences:
| [1] | Ben-Israel, A.; Greville, T. N.E., Generalized Inverses: Theory and Applications, 15 (2003), Springer Science & Business Media, New York · Zbl 1026.15004 |
| [2] | Campbell, S. L.; Meyer, C. D., Generalized Inverses of Linear Transformations, 56 (2008), SIAM, Philadelphia · Zbl 0732.15003 |
| [3] | Chen, Y.-L., A Cramer rule for solution of the general restricted linear equation, Linear Multilinear Algebra, 34, 177-186 (1993) · Zbl 0796.15005 · doi:10.1080/03081089308818219 |
| [4] | Chen, Y. L.; Chen, X., Representation and approximation of the outer inverse \(####\) of a matrix A, Linear Algebra Appl., 308, 1-3, 85-107 (2000) · Zbl 0957.15002 · doi:10.1016/S0024-3795(99)00269-4 |
| [5] | 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, 4, 387-394 (1992) · Zbl 0777.65024 |
| [6] | Djordjević, D. S.; Stanimirović, P. S.; Wei, Y., The representation and approximations of outer generalized inverses, Acta Math. Hung., 104, 1-2, 1-26 (2004) · Zbl 1071.65075 · doi:10.1023/B:AMHU.0000034359.98588.7b |
| [7] | Fa-Long, L.; Zheng, B., Neural network approach to computing matrix inversion, Appl. Math. Comput., 47, 109-120 (1992) · Zbl 0748.65025 |
| [8] | Getson, A.J. and Hsuan, F.C., \(####\)-inverses and their Statistical Application, Vol. 47. Lecture Notes in Statistics edition, Springer, Berlin, 1988. · Zbl 0671.62003 |
| [9] | Hsuan, F.; Langenberg, P.; Getson, A., The \(####\)-inverse with applications to statistics, Linear. Algebra Appl., 70, 241-248 (1985) · Zbl 0584.62078 · doi:10.1016/0024-3795(85)90055-2 |
| [10] | Jin, L.; Li, S.; Liao, B.; Zhang, Z., Zeroing neural networks: A survey, Neurocomputing, 267, 597-604 (2017) · doi:10.1016/j.neucom.2017.06.030 |
| [11] | Jin, L.; Zhang, Y.; Li, S., Integration-enhanced Zhang neural network for real-time-varying matrix inversion in the presence of various kinds of noises, IEEE Trans. Neural. Netw. Learn. Syst., 27, 12, 2615-2627 (2016) · doi:10.1109/TNNLS.2015.2497715 |
| [12] | Jin, L.; Zhang, Y.; Li, S.; Zhang, Y., Noise-tolerant ZNN models for solving time-varying zero-finding problems: A control-theoretic approach, IEEE Trans. Automat. Control, 62, 2, 992-997 (2017) · Zbl 1364.65111 · doi:10.1109/TAC.2016.2566880 |
| [13] | Levine, J.; Hartwig, R. E., Applications of the Drazin inverse to the hill cryptographic system, Part I, Cryptologia, 4, 71-85 (1980) · Zbl 0427.94015 · doi:10.1080/0161-118091854906 |
| [14] | Li, W.; Li, Z., A family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non-square matrix, Appl. Math. Comput., 215, 3433-3442 (2010) · Zbl 1185.65057 |
| [15] | Li, S.; Wang, H.; Rafique, M. U., A novel recurrent neural network for manipulator control with improved noise tolerance, IEEE Trans. Neural. Netw. Learn. Syst., 29, 1908-1918 (2018) · doi:10.1109/TNNLS.2017.2672989 |
| [16] | Li, S.; Zhang, Y.; Jin, L., Kinematic control of redundant manipulators using neural networks, IEEE Trans. Neural. Netw. Learn. Syst., 28, 10, 2243-2254 (2017) · doi:10.1109/TNNLS.2016.2574363 |
| [17] | Li, X.; Yu, J.; Li, S.; Ni, L., A nonlinear and noise-tolerant ZNN model solving for time-varying linear matrix equation, Neurocomputing, 317, 70-78 (2018) · doi:10.1016/j.neucom.2018.07.067 |
| [18] | Li, S.; Zhou, M. C.; Luo, X., Modified primal-dual neural networks for motion control of redundant manipulators with dynamic rejection of harmonic noises, IEEE Trans. Neural. Netw. Learn. Syst., 29, 10, 4791-4801 (2018) · doi:10.1109/TNNLS.2017.2770172 |
| [19] | 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) · doi:10.1109/TNNLS.2013.2271779 |
| [20] | 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) · doi:10.1016/j.neucom.2013.12.001 |
| [21] | Mao, M.; Li, J.; Jin, L.; Li, S.; Zhang, Y., Enhanced discrete-time Zhang neural network for time-variant matrix inversion in the presence of bias noises, Neurocomputing, 207, 220-230 (2016) · doi:10.1016/j.neucom.2016.05.010 |
| [22] | Nashed, Z. M., Generalized Inverses and Applications (1976), Academic Press, New York · Zbl 0346.15001 |
| [23] | Qiao, S.; Wang, X.-Z.; Wei, Y., Two finite-time convergent Zhang neural network models for time-varying complex matrix Drazin inverse, Linear Algebra Appl., 542, 101-117 (2018) · Zbl 1415.15007 · doi:10.1016/j.laa.2017.03.014 |
| [24] | Stanimirović, P. S.; Katsikis, V. N.; Li, S., Integration enhanced and noise tolerant ZNN for computing various expressions involving outerinverses, Neurocomputing, 329, 129-143 (2019) · doi:10.1016/j.neucom.2018.10.054 |
| [25] | Stanimirović, P. S.; Petković, M. D., Gauss-Jordan elimination method for computing outer inverses, Appl. Math. Comput., 219, 4667-4679 (2013) · Zbl 06447273 |
| [26] | Stanimirović, P. S.; Petković, M. D.; Gerontitis, D., Gradient neural network with nonlinear activation for computing inner inverses and the Drazin inverse, Neural Process. Lett., 48, 109-133 (2018) · doi:10.1007/s11063-017-9705-4 |
| [27] | Stanimirović, P. S.; Soleymani, F., A class of numerical algorithms for computing outer inverses, J. Comput. Appl. Math., 263, 236-245 (2014) · Zbl 1301.65032 · doi:10.1016/j.cam.2013.12.033 |
| [28] | Stanimirović, P. S.; Zivković, 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 · doi:10.1162/NECO_a_00771 |
| [29] | Stanimirović, P. S.; Zivković, I. S.; Wei, Y., Recurrent neural network for computing the Drazin inverse, IEEE Trans. Neural. Netw. Learn. Syst., 26, 2830-2843 (2015) · doi:10.1109/TNNLS.2015.2397551 |
| [30] | Stanimirović, P. S.; Zivković, I. S.; Wei, Y., Neural network approach to computing outer inverses based on the full rank representation, Linear Algebra Appl., 501, 344-362 (2016) · Zbl 1334.15014 · doi:10.1016/j.laa.2016.03.035 |
| [31] | Wang, G.; Wei, Y.; Qiao, S., Generalized Inverses: Theory and Computations (2018), Springer Nature Singapore Pte Ltd and Science Press, Beijing · Zbl 1395.15002 |
| [32] | Wang, J., Recurrent neural networks for solving linear matrix equations, Comput. Math. Appl., 26, 23-34 (1993) · Zbl 0796.65058 · doi:10.1016/0898-1221(93)90003-E |
| [33] | Wang, J., A recurrent neural network for real-time matrix inversion, Appl. Math. Comput., 55, 89-100 (1993) · Zbl 0772.65015 |
| [34] | Wang, J., Recurrent neural networks for computing pseudoinverses of rank-deficient matrices, SIAM J. Sci. Comput., 18, 1479-1493 (1997) · Zbl 0891.93034 · doi:10.1137/S1064827594267161 |
| [35] | Wang, X.-Z.; Ma, H.; Stanimirović, P. S., Nonlinearly activated recurrent neural network for computing the Drazin inverse, Neural Process. Lett., 46, 195-217 (2017) · doi:10.1007/s11063-017-9581-y |
| [36] | Wang, X.-Z.; Ma, H.; Stanimirović, P. S., Recurrent neural network for computing the W-weighted Drazin inverse, Appl. Math. Comput., 300, 1-20 (2017) · Zbl 1411.65064 |
| [37] | Wang, X.-Z.; Stanimirović, P. S.; Wei, Y., Complex ZFs for computing time-varying complex outer inverses, Neurocomputing, 275, 983-1001 (2018) · doi:10.1016/j.neucom.2017.09.034 |
| [38] | Wang, X.-Z.; Wei, Yimin; Stanimirović, P. S., Complex neural network models for time-varying Drazin inverse, Neural Comput., 28, 2790-2824 (2016) · Zbl 1474.68157 · doi:10.1162/NECO_a_00866 |
| [39] | Wei, Y., A characterization and representation of the generalized inverse \(####\) and its applications, Linear Algebra Appl., 280, 87-96 (1998) · Zbl 0934.15003 · doi:10.1016/S0024-3795(98)00008-1 |
| [40] | Wei, Y., Recurrent neural networks for computing weighted Moore Penrose inverse, Appl. Math. Comput., 116, 279-287 (2000) · Zbl 1023.65030 · doi:10.1016/S0377-0427(00)00313-7 |
| [41] | Xia, Y.; Zhang, S.; Stanimirović, P. S., Neural network for computing pseudoinverses and outer inverses of complex-valued matrices, Appl. Math. Comput., 273, 1107-1121 (2016) · Zbl 1410.65084 |
| [42] | Xiang, Q.; Liao, B.; Xiao, L.; Jin, L., A noise-tolerant z-type neural network for time-dependent pseudoinverse matrices, Optik, 165, 16-28 (2018) · doi:10.1016/j.ijleo.2018.03.078 |
| [43] | Yu, Y.; Wei, Y., Determinantal representation of the generalized inverse \(####\) over integral domains and its applications, Linear Multilinear Algebra, 57, 547-559 (2009) · Zbl 1182.15007 · doi:10.1080/03081080701871665 |
| [44] | 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) · doi:10.1109/TNN.2005.857946 |
| [45] | Zhang, Y.; Li, Z.; Li, K., Complex-valued Zhang neural network for online complex-valued time-varying matrix inversion, Appl. Math. Comput., 217, 10066-10073 (2011) · Zbl 1220.65036 |
| [46] | Zhang, Y.; Mu, B.; Zheng, H., Link between and comparison and combination of Zhang neural network and quasi-Newton BFGS method for time-varying quadratic minimization, Trans. Cybern., 43, 490-503 (2013) · doi:10.1109/TSMCB.2012.2210038 |
| [47] | 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, Inf. Process. Lett., 115, 703-706 (2015) · Zbl 1342.65250 · doi:10.1016/j.ipl.2015.03.007 |
| [48] | 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 · doi:10.1007/s00607-010-0133-9 |
| [49] | Zhang, Y.; Yi, C.; Guo, D.; Zheng, J., Comparison on Zhang neural dynamics and gradient-based neural dynamics for online solution of nonlinear time-varying equation, Neural Comput., 20, 1-7 (2011) · doi:10.1007/s00521-010-0452-y |
| [50] | Zhang, Y.; Yi, C.; Ma, W., Simulation and verification of Zhang neural network for online time-varying matrix inversion, Simul. Model. Pract. Theory, 17, 1603-1617 (2009) · doi:10.1016/j.simpat.2009.07.001 |
| [51] | Zhijun, Zhang; Siyuan, Chen; Shuai, Li, Compatible convex-nonconvex constrained QP-based dual neural networks for motion planning of redundant robot manipulators, IEEE Trans. Control Syst. Technol., 1-9 (2018) |
| [52] | Zhang, Z.; Fu, T.; Yan, Z.; Jin, L.; Xiao, L.; Sun, Y.; Yu, Z.; Li, Y., A varying-parameter convergent-differential neural network for solving joint-angular-drift problems of redundant robot manipulators, IEEE Trans. Mechatron., 23, 679-689 (2018) · doi:10.1109/TMECH.2018.2799724 |
| [53] | Zhang, Z.; Lu, Y.; Zheng, L.; Li, S.; Yu, Z.; Li, Y., A new varying-parameter convergent-differential neural-network for solving time-varying convex QP problem constrained by linear-equality, IEEE Trans. Automat. Control, 63, 4110-4125 (2018) · Zbl 1423.90170 · doi:10.1109/TAC.2018.2810039 |
| [54] | Zhang, Z.; Zheng, L., A complex varying-parameter convergent-differential neural-network for solving online time-varying complex Sylvester equation, IEEE Trans. Cybern., 1-13 (2018) |
| [55] | Zhang, Z.; Zheng, L.; Guo, Q., A varying-parameter convergent neural dynamic controller of multirotor UAVs for tracking time-varying tasks, IEEE Trans. Vehicular Technol., 67, 4793-4805 (2018) · doi:10.1109/TVT.2018.2802909 |
| [56] | Zhang, Z.; Zheng, L.; Weng, J.; Mao, Y.; Lu, W.; Xiao, L., A new varying-parameter recurrent neural-network for online solution of time-varying Sylvester equation, IEEE Trans. Cybern., 48, 3135-3148 (2018) · doi:10.1109/TCYB.2017.2760883 |
| [57] | Zheng, B.; Bapat, R. B., Generalized inverse \(####\) and a rank equation, Appl. Math. Comput., 155, 407-415 (2004) · Zbl 1057.15008 |
| [58] | Zielke, G., Report on test matrices for generalized inverses, Computing, 36, 105-162 (1986) · Zbl 0566.65026 · doi:10.1007/BF02238196 |
| [59] | Zivković, I. S.; Stanimirović, P. S.; Wei, Y., Recurrent neural network for computing outer inverse, Neural Comput., 28, 970-998 (2016) · Zbl 1472.92050 · doi:10.1162/NECO_a_00821 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
