Convex geometry and nonlinear approximation

A variety of properties for neural approximation follow from considerations of convexity. For example, if n and d are positive integers and X=L_p([0,1]^d) (with 1<p<∞) and if ε is any given positive constant, no matter how large, then it is not possible to have a continuous function φ 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 ||f-φ(f)|| exceeds the minimum possible error by more than ε. It is also shown that the additional multiplicative factor introduced into Barron´s bound (1992, 1993) by Kurkova, Savicky, and Hlavackova (1998) has an expected value of one half