Descartes' Rule of Signs for Radial Basis Function Neural Networks

We establish versions of Descartes' rule of signs for radial basis function (RBF) neural networks. The RBF rules of signs provide tight bounds for the number of zeros of univariate networks with certain parameter restrictions. Moreover, they can be used to infer that the Vapnik-Chervonenkis (VC) dimension and pseudo-dimension of these networks are no more than linear. This contrasts with previous work showing that RBF neural networks with two and more input nodes have superlinear VC dimension. The rules give rise also to lower bounds for network sizes, thus demonstrating the relevance of network parameters for the complexity of computing with RBF neural networks.