Acknowledgement
This work was supported by a 2-Year Research Grant of Pusan National University.
References
- 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
- 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
- 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
- H. B. Demuth, M. H. Beale and M. T. Hagan. Deep Learning Toolbox User's Guide. Natick, Massachusetts, United State, March 2021.
- J. E. Dennis and R. B. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Society for Industrial and Applied Mathematics, 1996.
- 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
- I. Goodfellow, Y. Bengio, and A. Courville. Deep Learning. MIT Press.
- L. F. Guilhoto. An overview of artificial neural networks for mathematicians. 2018.
- 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
- I. Guhring, G. Kutyniok and P. Petersen. Error bounds for approximations with deep ReLU neural networks in ws,p norms. 2019.
- 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
- 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
- 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
- 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
- T. Jaware, V. Patil and R. Badgujar. Artificial Neural Network. LAP Lambert Academic Publishing, 2019.
- 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
- 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
- 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
- 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
- J. E. McFarland. An iterative solution of the quadratic equation in banach space. American Mathematical Society, 9:824830, 1958.
- Y. Nesterov. Introductory lectures on convex optimization: a basic course; 1st ed. Applied optimization. Springer, Boston, 2004.
- J. M. Ortega and W. C. Rheinboldt. Iterative Solution of Nonlinear Equations in Several Variables. Society for Industrial and Applied Mathematics, 2000.
- I. Panageas and G. Piliouras. Gradient descent only converges to minimizers: Nonisolated critical points and invariant regions, 2016.
- 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
- A. Ravindran, K. M. Ragsdell and G. V. Reklaitis. Engineering Optimization: Methods and Applications, Second Edition. John Wiley & Sons, Inc., 2006.
- 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
- 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.