APPROXIMATION BY RIDGE FUNCTIONS AND NEURAL NETWORKS WITH ONE HIDDEN LAYER

We describe the configuration of an infinite set V of vectors in Rs, s ≥ 1, for which the closure with respect to C(K) of the algebraic span of {f{hook}(〈v, ·〉):v ε{lunate} V, f{hook} ε{lunate} C(R)} is all of C(K), where K is any compact set in Rs. This configuration also guarantees that for any sigmoidal function σ ge C(R), the span of {σ(m〈v, · 〉 +k):v ε{lunate} V;m, k ε{lunate} Z} is already dense in C(K). In particular, neural networks with one hidden layer of the form ∑(I,k) ε{lunate} J c(i,k) σ(〈i,x〉+k), where k ε{lunate} Z, c(i, k) ε{lunate} R, and i ε{lunate} Zs, can be designed to approximate any continuous functions in s variables. © 1992.