Abstract
This paper investigates the simulation of a gradient-based recurrent neural network for online solution of the matrix-inverse problem. Several important techniques are employed as follows to simulate such a neural system. 1) Kronecker product of matrices is introduced to transform a matrix-differential-equation (MDE) to a vector-differential-equation (VDE); i.e., finally, a standard ordinary-differential-equation (ODE) is obtained. 2) MATLAB routine “ode45” is introduced to solve the transformed initial-value ODE problem. 3) In addition to various implementation errors, different kinds of activation functions are simulated to show the characteristics of such a neural network. Simulation results substantiate the theoretical analysis and efficacy of the gradient-based neural network for online constant matrix inversion.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Zhang, Y.: Towards Piecewise-Linear Primal Neural Networks for Optimization and Redundant Robotics. In: Proceedings of IEEE International Conference on Networking, Sensing and Control, pp. 374–379. IEEE Computer Society Press, Los Alamitos (2006)
Steriti, R.J., Fiddy, M.A.: Regularized Image Reconstruction Using SVD and a Neural Network Method for Matrix Inversion. IEEE Transactions on Signal Processing 41, 3074–3077 (1993)
Sarkar, T., Siarkiewicz, K., Stratton, R.: Survey of Numerical Methods for Solution of Large Systems of Linear Equations for Electromagnetic Field Problems. IEEE Transactions on Antennas and Propagation 29, 847–856 (1981)
Sturges Jr., R.H.: Analog Matrix Inversion (Robot Kinematics). IEEE Journal of Robotics and Automation 4, 157–162 (1988)
Yeung, K.S., Kumbi, F.: Symbolic Matrix Inversion with Application to Electronic Circuits. IEEE Transactions on Circuits and Systems 35, 235–238 (1988)
El-Amawy, A.: A Systolic Architecture for Fast Dense Matrix Inversion. IEEE Transactions on Computers 38, 449–455 (1989)
Neagoe, V.E.: Inversion of the Van Der Monde Matrix. IEEE Signal Processing Letters 3, 119–120 (1996)
Wang, Y.Q., Gooi, H.B.: New Ordering Methods for Space Matrix Inversion via Diagonaliztion. IEEE Transactions on Power Systems 12, 1298–1305 (1997)
Koc, C.K., Chen, G.: Inversion of All Principal Submatrices of a Matrix. IEEE Transactions on Aerospace and Electronic Systems, 30, 280–281 (1994)
Zhang, Y., Leithead, W.E., Leith, D.J.: Time-Series Gaussian Process Regression Based on Toeplitz Computation of O(N 2) Operations and O(N)-Level Storage. In: Proceedings of the 44th IEEE Conference on Decision and Control, pp. 3711–3716. IEEE Computer Society Press, Los Alamitos (2005)
Leithead, W.E., Zhang, Y.: O(N 2)-Operation Approximation of Covariance Matrix Inverse in Gaussian Process Regression Based on Quasi-Newton BFGS Methods. Communications in Statistics - Simulation and Computation 36, 367–380 (2007)
Manherz, R.K., Jordan, B.W., Hakimi, S.L.: Analog Methods for Computation of the Generalized Inverse. IEEE Transactions on Automatic Control 13, 582–585 (1968)
Jang, J., Lee, S., Shin, S.: An Optimization Network for Matrix Inversion. Neural Information Processing Systems, American Institute of Physics, NY 397–401 (1988)
Wang, J.: A Recurrent Neural Network for Real-Time Matrix Inversion. Applied Mathematics and Computation, 55, 89–100 (1993)
Zhang, Y.: Revisit the Analog Computer and Gradient-Based Neural System for Matrix Inversion. In: Proceedings of IEEE International Symposium on Intelligent Control, pp. 1411–1416. IEEE Computer Society Press, Los Alamitos (2005)
Zhang, Y., Jiang, D., Wang, J.: A Recurrent Neural Network for Solving Sylvester Equation with Time-Varying Coefficients. IEEE Transactions on Neural Networks 13, 1053–1063 (2002)
Zhang, Y., Ge, S.S.: A General Recurrent Neural Network Model for Time-Varying Matrix Inversion. In: Proceedings of the 42nd IEEE Conference on Decision and Control, pp. 6169–6174. IEEE Computer Society Press, Los Alamitos (2003)
Zhang, Y., Ge, S.S.: Design and Analysis of a General Recurrent Neural Network Model for Time-Varying Matrix Inversion. IEEE Transactions on Neural Networks, 16, 1477–1490 (2005)
Carneiro, N.C.F., Caloba, L.P.: A New Algorithm for Analog Matrix Inversion. In: Proceedings of the 38th Midwest Symposium on Circuits and Systems, vol. 1, pp. 401–404 (1995)
Mead, C.: Analog VLSI and Neural Systems. Addison-Wesley, Reading, MA (1989)
Zhang, Y., Li, Z., Fan, Z., Wang, G.: Matrix-Inverse Primal Neural Network with Application to Robotics. Dynamics of Continuous, Discrete and Impulsive Systems, Series B 14, 400–407 (2007)
Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Zhang, Y., Chen, K., Ma, W., Li, XD. (2007). MATLAB Simulation of Gradient-Based Neural Network for Online Matrix Inversion. In: Huang, DS., Heutte, L., Loog, M. (eds) Advanced Intelligent Computing Theories and Applications. With Aspects of Artificial Intelligence. ICIC 2007. Lecture Notes in Computer Science(), vol 4682. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74205-0_12
Download citation
DOI: https://doi.org/10.1007/978-3-540-74205-0_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74201-2
Online ISBN: 978-3-540-74205-0
eBook Packages: Computer ScienceComputer Science (R0)