ON RADIAL BASIS FUNCTION NETS AND KERNAL REGRESSION - STATISTICAL CONSISTENCY, CONVERGENCE-RATES, AND RECEPTIVE-FIELD SIZE

Useful connections between radial basis function (RBF) nets and kernel regression estimators (KRE) are established. By using existing theoretical results obtained for KRE as tools, we obtain a number of interesting theoretical results for RBF nets. Upper bounds are presented for convergence rates of the approximation error with respect to the number of hidden units. The existence of a consistent estimator for RBF nets is proven constructively. Upper bounds are also provided for the pointwise and L2 convergence rates of the best consistent estimator for RBF nets as the numbers of both the samples and the hidden units tend to infinity. Moreover, the problem of selecting the appropriate size of the receptive field of the radial basis function is theoretically investigated and the way this selection is influenced by various factors is elaborated. In addition, some results are also given for the convergence of the empirical error obtained by the least squares estimator for RBF nets.