Local Gaussian Processes on Compact Lie Groups

Gaussian processes are a cornerstone of modern probabilistic machine learning, providing principled uncertainty quantification and flexible nonparametric modeling. While the Gaussian kernel is widely used in Euclidean spaces, many real-world problems involve data residing on non-Euclidean domains, particularly compact Lie groups that naturally encode \emph{symmetries}. In this work, we construct a local Gaussian kernel on arbitrary compact Lie groups by decomposing the group into a \emph{maximal torus} and its orthogonal complement. For the prominent Lie groups $\mathbf{SO}(3)$ and $\mathbf{SU}(2)$, we derive closed-form kernel expressions using the Rodrigues formula. Extensive experiments verify the positive definiteness of the proposed kernel and demonstrate its effectiveness in Gaussian process regression, achieving accurate predictions across the entire manifold.