Geodesic Exponential Kernels: When Curvature and Linearity Conflict
This addresses a fundamental limitation in kernel methods for non-Euclidean data, with implications for machine learning on curved manifolds.
The paper demonstrates that Gaussian kernels cannot be generalized to curved geodesic spaces while retaining positive definiteness, but shows that Laplacian kernels can be adapted to some curved spaces like spheres and hyperbolic spaces, with empirical verification.
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.