The Gaussian kernel on the circle and spaces that admit isometric embeddings of the circle

TL;DR

This paper proves that the Gaussian kernel is not positive definite on the circle or any space that admits an isometric embedding of the circle, impacting its use in non-Euclidean geometry applications.

Contribution

It establishes a fundamental limitation of the Gaussian kernel's positive definiteness on the circle and related spaces, clarifying its applicability in non-Euclidean settings.

Findings

Gaussian kernel is not positive definite on the circle for any parameter

Implication for Gaussian kernel use on spheres, projective spaces, and Grassmannians

Limits kernel methods in certain non-Euclidean spaces

Abstract

On Euclidean spaces, the Gaussian kernel is one of the most widely used kernels in applications. It has also been used on non-Euclidean spaces, where it is known that there may be (and often are) scale parameters for which it is not positive definite. Hope remains that this kernel is positive definite for many choices of parameter. However, we show that the Gaussian kernel is not positive definite on the circle for any choice of parameter. This implies that on metric spaces in which the circle can be isometrically embedded, such as spheres, projective spaces and Grassmannians, the Gaussian kernel is not positive definite for any parameter.