Linear and nonlinear approximation of spherical radial basis function networks

In this paper, the center-selection strategy of spherical radial basis function networks (SRBFNs) is considered. To approximate functions in the Bessel-potential Sobolev classes, we provide two lower bounds of nonlinear SRBFN approximation. In the first one, we prove that, up to a logarithmic factor, the lower bound of SRBFN approximation coincides with the Kolmogorov nn-width. In the other one, we prove that if a pseudo-dimension assumption is imposed on the activation function, then the logarithmic factor can even be omitted. These results together with the well known Jackson-type inequality of SRBFN approximation imply that the center-selection strategy does not affect the approximation capability of SRBFNs very much, provided the target function belongs to the Bessel-potential Sobolev classes. Thus, we can choose centers only for the algorithmic factor. Hence, a linear SRBFN approximant whose centers are specified before the training is recommended.