Convex geometry and nonlinear approximation

Abstract

A variety of properties for neural approximation follow from considerations of convexity. For example, if n and d are positive integers and X = Lp([0,lld) (with 1 < p < CO) and if E is any given positive constant, no matter how large, then it is not possible to have a continuous function 4 which associates to each element in X an input-output function of a one-hidden-layer neural network with n hidden units and one linear output units unless for some f in X the error Ilf -4(f)ll exceeds the minimum possible error by more than E . It is also shown that the additional multiplicative factor introduced into Barron's bound by KtirkovA, Savicki, and Hlavikkovi has an expected value of one half.