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We consider kernel methods on general geodesic metric spaces and provide bothnegative and positive results. First we show that the common Gaussian kernelcan only be generalized to a positive definite kernel on a geodesic metricspace if the space is flat. As a result, for data on a Riemannian manifold, thegeodesic Gaussian kernel is only positive definite if the Riemannian manifoldis Euclidean. This implies that any attempt to design geodesic Gaussian kernelson curved Riemannian manifolds is futile. However, we show that for spaces withconditionally negative definite distances the geodesic Laplacian kernel can begeneralized while retaining positive definiteness. This implies that geodesicLaplacian kernels can be generalized to some curved spaces, including spheresand hyperbolic spaces. Our theoretical results are verified empirically.

Geodesic Exponential Kernels: When Curvature and Linearity Conflict