On the tractability of multivariate integration and approximation by neural networks
Let qgreater than or equal to1 be an integer, Q be a Borel subset of the Euclidean space R, it be a probability measure on Q, and F be a class of real valued, it-integrable functions on Q. The complexity problem of approximating integralf dmu using quasi-Monte Carlo methods is to estimate E(n) (F, mu) := inf(x1,...,xnis an element ofQ) sup(fis an element ofF) \integralfdmu - (1)/(n) Sigma(k=1)(n) f(x(k))\ The problem is said to be tractable if there exist constants c, alpha, beta independent of q (but possibly dependent on mu and JF) such that E(n) (F, mu) less than or equal to cq(alpha)n(-beta). We explore different regions (including manifolds), function classes, and measures for which this problem is tractable. Our results include tractability theorems for integration with respect to non-tensor product measures, and over unbounded and/or non-tensor product subsets, including the unit spheres of R(q) with respect to various norms. We discuss applications to approximation capabilities of neural and radial basis function networks. (C) 2003 Published by Elsevier Inc.
tractability neural networks radial basis function networks dimension independent bounds