Gaussian processes are arguably the most important model class in spatial statistics. They encode prior information about the modeled function and can be used for exact or approximate Bayesian inference. 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：高斯过程可以说是空间统计学中最重要的模型类别。它们编码了关于被建模函数的先验信息，并可用于精确或近似贝叶斯推断。在众多应用中，尤其是在物理科学与工程领域，以及地质统计学和神经科学等领域，对称性不变性是可以考虑的最基本形式的先验信息之一。高斯过程协方差对此类对称性的不变性，催生了平稳性概念在这些空间上最自然的推广。在本研究中，我们开发了构建性且实用的技术，用于在由对称性衍生出的一大类非欧几里得空间上建立平稳高斯过程。我们的技术能够以实用方式实现：（i）计算协方差核，以及（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。本工作分为两部分，每部分涉及不同的技术考量：第一部分研究紧致空间，而第二部分研究具有特定结构的非紧致空间。我们的贡献使我们研究的非欧几里得高斯过程模型与标准高斯过程软件包中易于理解的计算技术相容，从而为实践者所用。