Abstract
An approximation result is given concerning Gaussian radial basis functions in a general inner product space. Applications are described concerning the classification of the elements of disjoint sets of signals, and also the approximation of continuous real functions defined on all of ℝn using radial basis function (RBF) networks. More specifically, it is shown that an important large class of classification problems involving signals can be solved using a structure consisting of only a generalized RBF network followed by a quantizer. It is also shown that Gaussian radial basis functions defined on ℝn can uniformly approximate arbitrarily well over all of ℝn any continuous real functionalf on ℝn that meets the condition that |f(x)|→0 as ‖x‖→∞.
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References
J. Park and I. W. Sandberg, Universal approximation using radial basis-function networks,Neural Comput., 3(2), 246–257, 1991.
J. Park and I. W. Sandberg, Approximation and radial-basis function networks,Neural Comput., 5(2), 305–316, March 1993.
E. J. Hartman, J. D. Keeler, and J. M. Kowalski, Layered neural networks with Gaussian hidden units as universal approximators,Neural Comput., 2(2), 210–215, 1990.
I. W. Sandberg, General structures for classification,IEEE Trans. Circuits Syst. I, 41(5), 372–376, May 1994.
I. W. Sandberg, J. T. Lo, C. Francourt, J. Principe, S. Katagiri, and S. Haykin,Nonlinear Dynamical Systems: Feedforward Neural Network Perspectives, John Wiley, New York, 2001.
I. W. Sandberg, Structure theorems for nonlinear systems,Multidimen. Syst. Signal Process., 2(3), 267–286, 1991, See also Errata, 3(1), 101, 1992. A conference version of the paper appears inIntegral Methods in Science and Engineering-90, (Proceedings of the International Conference on Integral Methods in Science and Engineering), edited by A. H. Haji-Sheikh, Hemisphere Publishing, New York, pp. 92–110, 1991.
R. A. DeVore and G. G. Lorentz,Constructive Approximation, Springer, New York, 1993.
M. H. Stone, A generalized Weierstrass approximation theorem, inStudies in Modern Analysis, MAA Studies in Mathematics, vol. 1, edited by R. C. Buck, Prentice-Hall, Englewood Cliffs, NJ, pp. 30–87, March 1962.
W. A. Sutherland,Introduction to Metric and Topological Spaces, Clarendon Press, Oxford, 1975.
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Sandberg, I.W. Gaussian radial basis functions and inner product spaces. Circuits Systems and Signal Process 20, 635–642 (2001). https://doi.org/10.1007/BF01270933
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DOI: https://doi.org/10.1007/BF01270933