DOI QR코드

DOI QR Code

DEEP LEARNING APPROACH FOR SOLVING A QUADRATIC MATRIX EQUATION

  • Kim, Garam (Department of Mathematics, Pusan National University) ;
  • Kim, Hyun-Min (Department of Mathematics, Pusan National University)
  • 투고 : 2021.12.24
  • 심사 : 2022.01.12
  • 발행 : 2022.01.31

초록

In this paper, we consider a quadratic matrix equation Q(X) = AX2 + BX + C = 0 where A, B, C ∈ ℝn×n. A new approach is proposed to find solutions of Q(X), using the novel structure of the information processing system. We also present some numerical experimetns with Artificial Neural Network.

키워드

과제정보

This work was supported by a 2-Year Research Grant of Pusan National University.

참고문헌

  1. A. R. Barron. Universal approximation bounds for superpositions of a sigmoidal function. IEEE Transactions on Information Theory, 39(3):930-945, 1993. https://doi.org/10.1109/18.256500
  2. G. Cybenko. Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals and Systems, 2(4):303-314, 1989. https://doi.org/10.1007/BF02551274
  3. G. J. Davis. Numerical solution of a quadratic matrix equation. SIAM J. Sci. Stat. Comput., 2(2):164175, June 1981. https://doi.org/10.1137/0902014
  4. H. B. Demuth, M. H. Beale and M. T. Hagan. Deep Learning Toolbox User's Guide. Natick, Massachusetts, United State, March 2021.
  5. J. E. Dennis and R. B. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Society for Industrial and Applied Mathematics, 1996.
  6. J. Eisenfeld. Operator equations and nonlinear eigenparameter problems. Journal of Functional Analysis, 12(4):475-490, 1973. https://doi.org/10.1016/0022-1236(73)90007-4
  7. I. Goodfellow, Y. Bengio, and A. Courville. Deep Learning. MIT Press.
  8. L. F. Guilhoto. An overview of artificial neural networks for mathematicians. 2018.
  9. C. Guo. On a quadratic matrix equation associated with an M-matrix. IMA Journal of Numerical Analysis, 23(1):11-27, 2003. https://doi.org/10.1093/imanum/23.1.11
  10. I. Guhring, G. Kutyniok and P. Petersen. Error bounds for approximations with deep ReLU neural networks in ws,p norms. 2019.
  11. N. J. Higham. Computing real square roots of a real matrix. Linear Algebra and its Applications, 88-89:405-430, 1987. https://doi.org/10.1016/0024-3795(87)90118-2
  12. N. J. Higham and H.-M. Kim. Numerical analysis of a quadratic matrix equation . IMA Journal of Numerical Analysis, 20(4):499-519, 10 2000. https://doi.org/10.1093/imanum/20.4.499
  13. N. J. Higham and H.-M. Kim. Solving a quadratic matrix equation by Newton's method with exact line searches. SIAM Journal on Matrix Analysis and Applications, 23(2):303-316, 2001. https://doi.org/10.1137/S0895479899350976
  14. K. Hornik. Approximation capabilities of multilayer feedforward networks. Neural Networks, 4(2):251 - 257, 1991. https://doi.org/10.1016/0893-6080(91)90009-T
  15. T. Jaware, V. Patil and R. Badgujar. Artificial Neural Network. LAP Lambert Academic Publishing, 2019.
  16. H.-M. Kim. Convergence of Newtons method for solving a class of quadratic matrix equations. Honam Mathematical Journal, 30(2):399-409. https://doi.org/10.5831/HMJ.2008.30.2.399
  17. W. Kratz and E. Stickel. Numerical Solution of Matrix Polynomial Equations by Newton's Method. IMA Journal of Numerical Analysis, 7(3):355-369, 1987. https://doi.org/10.1093/imanum/7.3.355
  18. P. Lancaster and J. G. Rokne. Solutions of nonlinear operator equations. SIAM Journal on Mathematical Analysis, 8(3):448-457, 1977. https://doi.org/10.1137/0508033
  19. M. Leshno, V. Ya. Lin, A. Pinkus and S. Schocken. Multilayer feedforward networks with a nonpolynomial activation function can approximate any function. Neural Networks, 6(6):861 - 867, 1993. https://doi.org/10.1016/S0893-6080(05)80131-5
  20. J. E. McFarland. An iterative solution of the quadratic equation in banach space. American Mathematical Society, 9:824830, 1958.
  21. Y. Nesterov. Introductory lectures on convex optimization: a basic course; 1st ed. Applied optimization. Springer, Boston, 2004.
  22. J. M. Ortega and W. C. Rheinboldt. Iterative Solution of Nonlinear Equations in Several Variables. Society for Industrial and Applied Mathematics, 2000.
  23. I. Panageas and G. Piliouras. Gradient descent only converges to minimizers: Nonisolated critical points and invariant regions, 2016.
  24. R. G. Pratt, C. Shin and G. J. Hick. GaussNewton and full Newton methods in frequencyspace seismic waveform inversion. Geophysical Journal International, 133(2):341-362, 05 1998. https://doi.org/10.1046/j.1365-246X.1998.00498.x
  25. A. Ravindran, K. M. Ragsdell and G. V. Reklaitis. Engineering Optimization: Methods and Applications, Second Edition. John Wiley & Sons, Inc., 2006.
  26. L. C. G. Rogers. Fluid models in queueing theory and wiener-hopf factorization of markov chains. The Annals of Applied Probability, 4(2):390-413, 1994. https://doi.org/10.1214/aoap/1177005065
  27. F. Santosa. W. W. Symes and G. Raggio. Inversion of band-limited reflection seismograms using stacking velocities as constraints. IOP Publishing Ltd, 3(3):448-457, 1977.