Eignets for function approximation on manifolds

Abstract

Let X be a compact, smooth, connected, Riemannian manifold without boundary, G : X × X → R be a kernel. Analogous to a radial basis function network, an eignet is an expression of the form ∑ j = 1 M a j G ( ○ , y j ) , where a j ∈ R , y j ∈ X , 1 ⩽ j ⩽ M . We describe a deterministic, universal algorithm for constructing an eignet for approximating functions in L p ( μ ; X ) for a general class of measures μ and kernels G. Our algorithm yields linear operators. Using the minimal separation among the centers y j as the cost of approximation, we give modulus of smoothness estimates for the degree of approximation by our eignets, and show by means of a converse theorem that these are the best possible for every individual function. We also give estimates on the coefficients a j in terms of the norm of the eignet. Finally, we demonstrate that if any sequence of eignets satisfies the optimal estimates for the degree of approximation of a smooth function, measured in terms of the minimal separation, then the derivatives of the eignets also approximate the corresponding derivatives of the target function in an optimal manner.