Abstract
For approximate interpolation, a type of single-hidden layer feedforward neural networks with the inverse multiquadric activation function is presented in this paper. We give a new and quantitative proof of the fact that a single layer neural networks with n + 1 hidden neurons can learn n + 1 distinct samples with zero error. Based on this result, approximate interpolants are given. They can approximate interpolate, with arbitrary precision, any set of distinct data in one or several dimensions. They can uniformly approximate any C 1 continuous function of one variable.
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References
Huang, G.B., Babri, H.A.: Feedforward neural networks with arbitrary bounded nonlinear activation functions. IEEE Trans. Neural Networks 9(1), 224–229 (1998)
Sartori, M.A., Antsaklis, P.J.: A simple method to derive bounds on the size and to train multilayer neural networks. IEEE Trans. Neural Networks 2(4), 467–471 (1991)
Tamura, S., Tateishi, M.: Capabilities of a four-layered feedforward neural network. IEEE Trans. Neural Networks 8(2), 251–255 (1997)
Barhen, J., Cogswell, R., Protopopescu, V.: Single iteration training algorithm for multilayer feedforward neural networks. Neural Process. Lett. 11, 113–129 (2000)
Li, X.: Interpolation by ridge polynomials and its application in neural networks. J. Comput. Appl. Math. 144, 197–209 (2002)
Sontag, E.D.: Feedforward nets for interpolation and classification. J. Comp. Syst. Sci. 45, 20–48 (1992)
Chui, C.K., Li, X., Mhaskar, H.N.: Neural networks for localized approximation. Mathematics of Computation 63, 607–623 (1994)
Chui, C.K., Li, X., Mhaskar, H.N.: Limitations of the approximation capabilities of neural networks with one hidden layer. Adv. Comput. Math. 5, 233–243 (1996)
Debao, C.: Degree of approximation by superpositions of a sigmoidal function. Approx. Theory & its Appl. 9, 17–28 (1993)
Mhaskar, H.N., Michelli, C.A.: Approximation by superposition of sigmoidal and radial basis functions. Adv. Appl. Math. 13, 350–373 (1992)
Lianas, B., Sainz, F.J.: Constructive approximate interpolation by neural networks. J. Comput. Appl. Math. 188, 283–308 (2006)
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Han, X. (2007). Approximate Interpolation by Neural Networks with the Inverse Multiquadric Functions. In: Kang, L., Liu, Y., Zeng, S. (eds) Advances in Computation and Intelligence. ISICA 2007. Lecture Notes in Computer Science, vol 4683. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74581-5_32
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DOI: https://doi.org/10.1007/978-3-540-74581-5_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74580-8
Online ISBN: 978-3-540-74581-5
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