Approximation properties of some two-layer feedforward neural networks

• Autorzy: Nowak M. A.
• Języki publikacji: EN

Abstrakty

EN

In this article, we present a multiyariate two-layer feedforward neural networks that approximate continuos functions defined on [0, 1]d. We show that the L1 error of approximation is asymptotically proportional to the modulus of continuity of the underlying function taken at √d/n, where n is the number of function values used.

Słowa kluczowe

EN

neural networks; approximation of functions; sigmoidal function

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2007
• Tom: Vol. 27, no. 1
• Strony: 59--72
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Nowak M. A.

• AGH University of Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Cracow, Poland,

Bibliografia

• [1] G. Anastassiou, Quantitative Approximations, Chapman & Hall/CRC, Boca Raton, 2001.
• [2] G. Anastassiou, Rate of convergence of some neural network operators to the unit-univariate case, Journal of Mathematical Analysis and Applications 212 (1997), 237-262.
• [3] A.R. Barron, Universal approximation bounds for superpositions of sigmoid function, IEEE Transactions on Information Theory 3 (1993), 930-945.
• [4] P. Cardaliaguet, G. Euvrard, Approximation of a function and its derivative with a neural network, Neural Networks 5 (1992), 207-220.
• [5] G. Cybenko, Approximations by superpositions of a sigmoidal function, Mathematics of Control, Signals and Systems 2 (1989), 303-314.
• [6] K.I. Funahashi, On the approximate realization of continuos mappings by neural networks, Neural Networks 2 (1989), 183-192.
• [7] K. Hornik, M. Stinchcombe, H. White, Multilayer feedforward networks are universal approximators, Neural Networks 2 (1989), 359-366.
• [8] K. Hornik, M. Stinchcombe, H. White, Universal approximation of an unknown mapping and its derivatives, Neural Networks 3 (1989), 551-560.
• [9] M.A. Kon, L. Plaskota, Information complexity of neural networks, Neural Networks 13 (2000), 365-375.
• [10] B. Lenze, How to make sigma-pi neural networks perform perfectly on regular training sets, Neural Networks 7 (1994), 1285-1293.
• [11] B. Lenze, One-sided approximation and interpolation operators generating hyperbolic sigma-pi neural networks, In: Multivariate Approximation and Splines, G. Nurnberger, J. W. Schmidt, G. Walz (eds.), Birkhauser Verlag, Basel 1997, 99-112.
• [12] Leshno M., Lin V., Pinkus A., Schocken S., Multilayer feedforward networks with non-polynomial activation functions can approximate any continuos function, Neural Networks 6 (1993), 861-867.
• [13] Scarselli F., Ah Chung Tsoi, Universal approximation using feedforward neural networks: a survey of some existing methods, and some new results, Neural Networks 11 (1998), 15-37.
• [14] Xi Min Zhanga, Yan Qiu Chenb, Nirwan Ansaria, Yun Q. Shia, Mini-max initialization for function approximation, Neurocomputing 57 (2004), 389-409.