Lower estimation of approximation rate for neural networks

Let SFd and Pi(phi,n,d) = {Sigma(n)(j=1)b(j)phi(omega j.x+theta j) :b(j),theta(j)is an element of R,omega(j)is an element of R-d} be the set of periodic and Lebesgue's square-integrable functions and the set of feedforward neural network (FNN) functions, respectively. Denote by dist (SFd, Pi(phi,n,d)) the deviation of the set SFd from the set pi(phi,n,d). A main purpose of this paper is to estimate the deviation. In particular, based on the Fourier transforms and the theory of approximation, a lower estimation for dist (SFd, Pi(phi,n,d)) is proved. That is, dist(SFd,Pi(phi,n,d)) >= C/(nlog(2)(n))(1/2). The obtained estimation depends only on the number of neuron in the hidden layer, and is independent of the approximated target functions and dimensional number of input. This estimation also reveals the relationship between the approximation rate of FNNs and the topology structure of hidden layer.