Rates of Convex Approximation

This paper deals with sparse approximations by means of convex combinations of elements from a predetermined \basis" subset S of a function space. Speciically, the focus is on the rate at which the lowest achievable error can be reduced as larger subsets of S are allowed when constructing an approximant. The new results extend those given for Hilbert spaces by Jones and Barron, including in particular a computationally attractive in-cremental approximation scheme. Bounds are derived for broad classes of Banach spaces; in particular, for Lp spaces with 1 < p < 1, the O(n ?1=2) bounds of Barron and Jones are recovered when p = 2. One motivation for the questions studied here arises from the area of \artiicial neural networks," where the problem can be stated in terms of the growth in the number of \neurons" (the elements of S) needed in order to achieve a desired error rate. The focus on non-Hilbert spaces is due to the desire to understand approximation in the more \robust" (resistant to exemplar noise) Lp, 1 p < 2 norms. The techniquesused borrow from results regardingmoduli of smoothness in functional analysis as well as from the theory of stochastic processes on function spaces.