Abstract

We consider kernel methods on general geodesic metric spaces and provide both negative and positive results. First we show that the common Gaussian kernel can only be generalized to a positive defifinite kernel on a geodesic metric space if the space is flflat. As a result, for data on a Riemannian manifold, the geodesic Gaussian kernel is only positive defifinite if the Riemannian manifold is Euclidean. This implies that any attempt to design geodesic Gaussian kernels on curved Riemannian manifolds is futile. However, we show that for spaces with conditionally negative defifi- nite distances the geodesic Laplacian kernel can be generalized while retaining positive defifiniteness. This implies that geodesic Laplacian kernels can be generalized to some curved spaces, including spheres and hyperbolic spaces. Our theoretical results are verifified empirically