The GeometricKernels Package: Heat and Matérn Kernels for Geometric Learning on Manifolds, Meshes, and Graphs

TL;DR

The paper presents GeometricKernels, a software package that extends kernel methods to non-Euclidean domains by implementing geometric analogs of the heat and Matérn kernels on graphs, meshes, and manifolds. It integrates Fourier-feature-based expansions, supports multiple backends with automatic differentiation, and provides approximate feature maps for scalable Gaussian-process-like modeling. The work details a modular design (spaces, kernels, feature maps) and demonstrates practical use through an illustrative example, while situating the approach within the broader historical development of kernels on structured domains. This enables principled uncertainty quantification in geometric learning tasks and facilitates plug-and-play use within existing GP ecosystems. The package thus broadens the applicability of kernel methods to complex geometries common in robotics, neuroscience, and related fields, with GPU acceleration and backend flexibility as key practical advantages.

Abstract

Kernels are a fundamental technical primitive in machine learning. In recent years, kernel-based methods such as Gaussian processes are becoming increasingly important in applications where quantifying uncertainty is of key interest. In settings that involve structured data defined on graphs, meshes, manifolds, or other related spaces, defining kernels with good uncertainty-quantification behavior, and computing their value numerically, is less straightforward than in the Euclidean setting. To address this difficulty, we present GeometricKernels, a software package which implements the geometric analogs of classical Euclidean squared exponential - also known as heat - and Matérn kernels, which are widely-used in settings where uncertainty is of key interest. As a byproduct, we obtain the ability to compute Fourier-feature-type expansions, which are widely used in their own right, on a wide set of geometric spaces. Our implementation supports automatic differentiation in every major current framework simultaneously via a backend-agnostic design. In this companion paper to the package and its documentation, we outline the capabilities of the package and present an illustrated example of its interface. We also include a brief overview of the theory the package is built upon and provide some historic context in the appendix.

Figures (1)

• Figure 1: Left: GeometricKernels-computed samples from a set of Gaussian processes, with covariance given by the heat kernel, on three spaces: the Stanford bunny mesh, the unit sphere $\mathbb{S}_2$, and the hyperbolic space $\mathbb{H}_2$. Right: a GeometricKernels-computed Matérn kernel $k(\mathbin{},\.)$ on a graph.