Gaussian processes are arguably the most important class of spatiotemporal models within machine learning. They encode prior information about the modeled function and can be used for exact or approximate Bayesian learning. In many applications, particularly in physical sciences and engineering, but also in areas such as geostatistics and neuroscience, invariance to symmetries is one of the most fundamental forms of prior information one can consider. The invariance of a Gaussian process' covariance to such symmetries gives rise to the most natural generalization of the concept of stationarity to such spaces. In this work, we develop constructive and practical techniques for building stationary Gaussian processes on a very large class of non-Euclidean spaces arising in the context of symmetries. Our techniques make it possible to (i) calculate covariance kernels and (ii) sample from prior and posterior Gaussian processes defined on such spaces, both in a practical manner. This work is split into two parts, each involving different technical considerations: part I studies compact spaces, while part II studies non-compact spaces possessing certain structure. Our contributions make the non-Euclidean Gaussian process models we study compatible with well-understood computational techniques available in standard Gaussian process software packages, thereby making them accessible to practitioners.

翻译：高斯过程无疑是机器学习中最重要的一类时空模型。它们能够对建模函数编码先验信息，并可用于精确或近似贝叶斯学习。在众多应用领域——特别是物理科学与工程领域，以及地统计学和神经科学等领域——对称性不变性是最基本的先验信息形式之一。高斯过程协方差对此类对称性的不变性，衍生出在这些空间上对平稳性概念最自然的推广。在本工作中，我们开发了面向一类具有对称性的非欧几里得空间的构造性且实用的平稳高斯过程构建技术。我们的方法能够：（i）计算协方差核，（ii）从定义在这些空间上的先验和后验高斯过程中进行采样，且两者均以实用方式实现。本文分为两个部分，各自涉及不同的技术考量：第一部分研究紧致空间，第二部分研究具有特定结构的非紧致空间。我们的贡献使得所研究的非欧几里得高斯过程模型能够与标准高斯过程软件包中成熟的计算技术兼容，从而便于实践者使用。