On Gaussian kernels on Hilbert spaces and kernels on Hyperbolic spaces

TL;DR

This paper characterizes the universality and positive definiteness properties of Gaussian kernels on Hilbert and hyperbolic spaces, impacting kernel methods in machine learning and spatial statistics.

Contribution

It provides new characterizations of Gaussian kernels on Hilbert and hyperbolic spaces, extending known results to these geometries.

Findings

Characterization of Gaussian kernels on Hilbert spaces.

Extension of properties to kernels on hyperbolic spaces.

Implications for geostatistics and kernel methods.

Abstract

This paper describes the concepts of Universal/ Integrally Strictly Positive Definite/ $C_{0}$-Universal for the Gaussian kernel on a Hilbert space. As a consequence we obtain a similar characterization for an important family of kernels studied and developed by Schoenberg and also on a family of spatial-time kernels popular on geostatistics, the Gneiting class, and its generalizations. Either by using similar techniques, or by a direct consequence of the Gaussian kernel on Hilbert spaces, we characterize the same concepts for a family of kernels defined on a Hyperbolic space.