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 Python 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.

翻译：核是机器学习中的基础技术原语。近年来，在不确定性量化至关重要的应用场景中，基于核的方法（如高斯过程）正变得日益重要。在处理定义于图、网格、流形或其他相关空间的结构化数据时，定义具有良好不确定性量化特性的核并对其进行数值计算，相比欧几里得空间情形更为复杂。为解决这一难题，我们提出了 GeometricKernels——一个 Python 软件包，它实现了经典欧几里得平方指数核（亦称热核）与 Matérn 核在几何空间上的类比，这两种核在重视不确定性的场景中被广泛使用。作为副产品，我们能够在广泛的几何空间上计算傅里叶特征型展开，这类展开本身亦被广泛使用。我们的实现通过后端无关的设计，同时支持当前所有主流框架中的自动微分功能。作为该软件包及其说明文档的配套论文，本文概述了该软件包的功能，并通过一个图示示例展示了其接口。文中还简要介绍了软件包所依据的理论基础，并在附录中提供了相关的历史背景。