Georgia Institute of Technology College of Sciences

We study approximation properties of Gaussian reproducing kernel Hilbert spaces restricted to low-dimensional manifolds embedded in Euclidean space. Using only ambient Gaussian kernels, and without assuming any smooth ambient extensions or estimating geometric quantities of the manifold, we show that intrinsically defined Hölder functions on the manifold can be approximated at rates governed by intrinsic dimension and smoothness. The construction is based on a small-scale expansion in real space rather than a spectral representation. As an application, we obtain adaptive nonparametric convergence rates for Gaussian process regression on manifolds, where the regression procedure itself is unchanged and intrinsic adaptivity results from the approximation analysis.