Gaussian kernels on non-simply-connected closed Riemannian manifolds are never positive definite

Abstract

We show that the Gaussian kernel $\exp \left\{-\lambda d_g^2(\bullet ,\bullet )\right\}$ on any non-simply-connected closed Riemannian manifold $(\mathcal{M},g)$, where $d_g$ is the geodesic distance, is not positive definite for any $\lambda >0$, combining analyses in the recent preprint~[9] by Da Costa--Mostajeran--Ortega and classical comparison theorems in Riemannian geometry.

9 pages

References in corpus (1)

• Kernel Methods on Riemannian Manifolds with Gaussian RBF Kernels

Cited by in corpus (1)

• Geometric Learning with Positively Decomposable Kernels