Gaussian processes are widely employed as versatile modelling and predictive tools in spatial statistics, functional data analysis, computer modelling and diverse applications of machine learning. They have been widely studied over Euclidean spaces, where they are specified using covariance functions or covariograms for modelling complex dependencies. There is a growing literature on Gaussian processes over Riemannian manifolds in order to develop richer and more flexible inferential frameworks for non-Euclidean data. While numerical approximations through graph representations have been well studied for the Mat\'ern covariogram and heat kernel, the behaviour of asymptotic inference on the parameters of the covariogram has received relatively scant attention. We focus on the asymptotic inference for Gaussian processes constructed over compact Riemannian manifolds. Building upon the recently introduced Mat\'ern covariogram on a compact Riemannian manifold, we employ formal notions and conditions for the equivalence of two Mat\'ern Gaussian random measures on compact manifolds to derive the parameter that is identifiable, also known as the microergodic parameter, and formally establish the consistency of the maximum likelihood estimate and the asymptotic optimality of the best linear unbiased predictor. The circle is studied as a specific example of compact Riemannian manifolds with numerical experiments to illustrate and corroborate the theory.

翻译：高斯过程在空间统计、函数数据分析、计算机建模及机器学习多领域应用中，被广泛用作灵活的建模与预测工具。在欧氏空间框架下，可通过协方差函数或协方差结构对复杂依赖性进行建模，相关研究已十分充分。为针对非欧氏数据发展更丰富、更灵活的推断框架，关于黎曼流形上高斯过程的研究文献正日益增多。尽管通过图表示进行数值近似的方法在Matérn协方差函数与热核中已得到充分研究，但针对协方差参数渐近推断行为的研究相对匮乏。本文聚焦于紧致黎曼流形上构造的高斯过程的渐近推断问题。基于新近提出的紧致黎曼流形上Matérn协方差函数，我们利用紧流形上两个Matérn高斯随机测度等价的正式概念与条件，推导出可识别参数（即微遍历参数），并严格建立了极大似然估计的一致性以及最优线性无偏预测的渐近最优性。最后以圆流形作为紧致黎曼流形的具体案例进行数值实验，以阐释与验证理论结果。