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：高斯过程是机器学习中最重要的时空模型类别之一。它们对建模函数的先验信息进行编码，可用于精确或近似贝叶斯学习。在物理科学与工程学等众多应用中，以及在诸如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。高斯过程协方差函数对该类对称性的不变性，自然地将平稳性概念推广至此类空间。本研究针对对称性语境下出现的一大类非欧几里得空间，开发了构建平稳高斯过程的构造性与实用性技术。我们的技术能够：（i）计算协方差核；（ii）以实用方式从定义在所述空间上的先验与后验高斯过程中进行采样。本工作分为两部分，各自涉及不同的技术考量：第一部分研究紧致空间，第二部分研究具备特定结构的非紧致空间。我们的贡献在于，使所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成熟的计算技术兼容，从而为实践者提供可访问性。