On the tractability of multivariate integration and approximation by neural networks

The problem is said to be tractable if there exist constants c, α, β independent of q (but possibly dependent on μ and F) such that En(F , μ) ≤ cqαn−β. 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 Rq with respect to various norms. We discuss applications to approximation capabilities of neural and radial basis function networks.