Approximation by approximate interpolation neural networks with single hidden layer

A bounded function φ defined on (−∞, +∞) is called general sigmoidal function if it satisfies lim<inf>x→+∞</inf>φ(x) = M, lim<inf>x→−∞</inf>φ(x) = m. Using the general sigmoidal function as the activation function, a type of neural networks with single hidden layer and n + 1 hidden neurons is constructed. These networks are called approximate interpolation networks, which can approximately interpolate, with arbitrary precision, any set of distinct data in one dimension. By using the modulus of continuity of function as metric, the errors of approximation by the constructed networks is estimated.

[1]  George Cybenko,et al.  Approximation by superpositions of a sigmoidal function , 1992, Math. Control. Signals Syst..

[2]  Ron Meir,et al.  Approximation bounds for smooth functions in C(Rd) by neural and mixture networks , 1998, IEEE Trans. Neural Networks.

[3]  Allan Pinkus,et al.  Approximation theory of the MLP model in neural networks , 1999, Acta Numerica.

[4]  Hong Chen,et al.  Approximation capability in C(R¯n) by multilayer feedforward networks and related problems , 1995, IEEE Trans. Neural Networks.

[5]  Ken-ichi Funahashi,et al.  On the approximate realization of continuous mappings by neural networks , 1989, Neural Networks.

[6]  George Cybenko,et al.  Approximation by superpositions of a sigmoidal function , 1989, Math. Control. Signals Syst..

[7]  Andrew R. Barron,et al.  Universal approximation bounds for superpositions of a sigmoidal function , 1993, IEEE Trans. Inf. Theory.

[8]  S. Dasgupta,et al.  Neural networks for exact matching of functions on a discrete domain , 1990, 29th IEEE Conference on Decision and Control.

[9]  Y. Makovoz.  Uniform Approximation by Neural Networks , 1998 .

[10]  Tianping Chen,et al.  Approximation capability to functions of several variables, nonlinear functionals and operators by radial basis function neural networks , 1993, Proceedings of 1993 International Conference on Neural Networks (IJCNN-93-Nagoya, Japan).

[11]  C. Micchelli,et al.  Degree of Approximation by Neural and Translation Networks with a Single Hidden Layer , 1995 .

[12]  Hong Chen,et al.  Approximation capability to functions of several variables, nonlinear functionals, and operators by radial basis function neural networks , 1993, IEEE Trans. Neural Networks.

[13]  F. J. Sainz,et al.  Constructive approximate interpolation by neural networks , 2006 .

[14]  Kurt Hornik,et al.  Multilayer feedforward networks are universal approximators , 1989, Neural Networks.

[15]  Zongben Xu,et al.  Simultaneous Lp-approximation order for neural networks , 2005, Neural Networks.

[16]  Allan Pinkus,et al.  Multilayer Feedforward Networks with a Non-Polynomial Activation Function Can Approximate Any Function , 1991, Neural Networks.

[17]  R. Kress.  Numerical Analysis , 1998 .

[18]  Hong Chen,et al.  Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems , 1995, IEEE Trans. Neural Networks.

[19]  Nahmwoo Hahm,et al.  Degree of approximation by neural networks , 1996 .

[20]  Shin Suzuki,et al.  Constructive function-approximation by three-layer artificial neural networks , 1998, Neural Networks.

[21]  Eduardo D. Sontag,et al.  Feedforward Nets for Interpolation and Classification , 1992, J. Comput. Syst. Sci..