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

arXiv: https://arxiv.org/abs/2302.10623 · HTML: https://mlsanity.com/paper/2302.10623/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 26 references that are in the map.
- First posted on arXiv 21 Feb 2023.

## Builds on (most cited first, 8 of 8)

- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [The Signature Kernel is the solution of a Goursat PDE](https://mlsanity.com/paper/2006.14794/) (2020)
- [Geodesic Exponential Kernels: When Curvature and Linearity Conflict](https://mlsanity.com/paper/1411.0296/) (2014)
- [Kernel Methods on Riemannian Manifolds with Gaussian RBF Kernels](https://mlsanity.com/paper/1412.0265/) (2014)
- [Bayesian Optimization Meets Riemannian Manifolds in Robot Learning](https://mlsanity.com/paper/1910.04998/) (2019)
- [Gaussians on Riemannian Manifolds: Applications for Robot Learning and Adaptive Control](https://mlsanity.com/paper/1909.05946/) (2019)
- [Dimension reduction in recurrent networks by canonicalization](https://mlsanity.com/paper/2007.12141/) (2020)
- [Reservoir kernels and Volterra series](https://mlsanity.com/paper/2212.14641/) (2022)

## Built on by (most cited first, 5 of 5)

- [Invariant kernels on Riemannian symmetric spaces: a harmonic-analytic approach](https://mlsanity.com/paper/2310.19270/) (2023)
- [Gaussian kernels on non-simply-connected closed Riemannian manifolds are never positive definite](https://mlsanity.com/paper/2303.06558/) (2023)
- [Geometric Learning with Positively Decomposable Kernels](https://mlsanity.com/paper/2310.13821/) (2023)
- [Memory Capacity of Nonlinear Recurrent Networks: Is it Informative?](https://mlsanity.com/paper/2502.04832/) (2025)
- [RW-Flow: One-Step Generation on Compact Manifolds via Wasserstein Gradient Flows](https://mlsanity.com/paper/2609.39271/) (2026)

## Abstract (arXiv)

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.