Gaussian kernels on non-simply-connected closed Riemannian manifolds are never positive definite
This result addresses a theoretical limitation in kernel methods for machine learning on non-simply-connected manifolds, showing incremental progress by extending known results to broader geometric settings.
The paper proves that Gaussian kernels based on geodesic distance are never positive definite on any non-simply-connected closed Riemannian manifold, for all positive scaling parameters, using a combination of recent preprint analyses and classical Riemannian geometry theorems.
We show that the Gaussian kernel $\exp\left\{-λd_g^2(\bullet, \bullet)\right\}$ on any non-simply-connected closed Riemannian manifold $(\mathcal{M},g)$, where $d_g$ is the geodesic distance, is not positive definite for any $λ> 0$, combining analyses in the recent preprint~[9] by Da Costa--Mostajeran--Ortega and classical comparison theorems in Riemannian geometry.