Convergence Rate of Fourier Neural Networks

Abstract

The paper investigates a convergence rate for 2-layer feedforward Fourier Neural Network (FNN). Such networks are motivated by the approximation properties of wellknown Fourier series. Several implementations of FNNs were proposed since 1990’s: by Gallant and White; A. Silvescu; Tan, Zuo and Cai; Liu. The main focus of this research is Silvescu’s FNN, because such activation function does not fit into the category of networks, where the linearly transformed input is exposed to activation. The latter ones were extensively described by Hornik in 1989. In regard to non-trivial Silvescu’s FNN, its convergence rate is proven to be of order 𝑂(1/𝑛). The paper continues investigating classes of functions approximated by Silvescu FNN, which appeared to be from Schwartz space and space of positive definite functions.