  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 definite kernel on a geodesic metric
space if the space is flat. As a result, for data on a Riemannian manifold, the
geodesic Gaussian kernel is only positive definite 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 definite distances the geodesic Laplacian kernel can be
generalized while retaining positive definiteness. This implies that geodesic
Laplacian kernels can be generalized to some curved spaces, including spheres
and hyperbolic spaces. Our theoretical results are verified empirically.

Machine Learning;Computer Vision and Pattern Recognition