Radial basis function methods for interpolation to functions of many variables

A review of interpolation to values of a function f(x), x 2 R d , by radial basis function methods is given. It addresses the nonsingularity of the interpolation equations, the inclusion of polynomial terms, and the accuracy of the approximation sf, where s is the interpolant. Then some numerical experiments investigate the situation when the data points are on a low dimensional nonlinear manifold in R d. They suggest that the number of data points that are necessary for good accuracy on the manifold is independent of d, even if d is very large. The experiments employ linear and multiquadric radial functions, because an iterative algorithm for these cases was developed at Cambridge recently. The algorithm is described brieey. Fortunately, the number of iterations is small when the data points are on a low dimensional manifold. We expect these ndings to be useful, because the manifold setting for large d is similar to typical distributions of interpolation points in data mining applications.