Experimental Positive Definiteness of the Gaussian Kernel

NeurIPS 2025 Workshop NeurReps Submission28 Authors

Keywords: kernel, Gaussian kernel, positive definite, curvature, metric space

TL;DR: We numerically show that the Gaussian kernel on cone spaces $Y_\alpha$ fails to be positive definite due to geometry when the cone angle $\alpha > 2\pi$ and due to numerical instability when $\alpha < 2\pi$.

Abstract:

We study positive definiteness of the Gaussian kernel $\exp(-\lambda d(x,y)^2)$ on two-dimensional cones.
Based on numerical experiments, we conjecture that the set of parameters $\lambda$ that make the kernel positive definite is related to the cone angle.
We relate our experimental results to existing positivity and non-positivity theorems in the literature.

Submission Number: 28