Geodesic Exponential Kernels: When Curvature and Linearity Conflict

TL;DR

This paper investigates the limitations and possibilities of defining positive definite kernels on geodesic metric spaces, showing Gaussian kernels are limited to flat spaces while Laplacian kernels can be extended to some curved spaces.

Contribution

It establishes theoretical boundaries for geodesic Gaussian kernels and introduces conditions under which geodesic Laplacian kernels can be generalized to curved spaces.

Findings

Gaussian kernels only positive definite on flat spaces

Laplacian kernels can be generalized to spheres and hyperbolic spaces

Empirical verification supports theoretical results

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 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.