Li Didong, Tang Wenpin, Banerjee Sudipto
Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA.
Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027, USA.
J Mach Learn Res. 2023 Mar;24.
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 Matérn covariogram and heat kernel, the behaviour of asymptotic inference on the parameters of the covariogram has received relatively scant attention. We focus on asymptotic behaviour for Gaussian processes constructed over compact Riemannian manifolds. Building upon a recently introduced Matérn covariogram on a compact Riemannian manifold, we employ formal notions and conditions for the equivalence of two Matérn 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.
高斯过程在空间统计、函数数据分析、计算机建模以及机器学习的各种应用中被广泛用作通用的建模和预测工具。它们在欧几里得空间中已得到广泛研究，在该空间中，通过协方差函数或协方差图来指定它们以对复杂的依赖性进行建模。为了为非欧几里得数据开发更丰富、更灵活的推理框架，关于黎曼流形上的高斯过程的文献越来越多。虽然通过图表示的数值近似对于马特恩协方差图和热核已经得到了很好的研究，但协方差图参数的渐近推断行为受到的关注相对较少。我们关注在紧致黎曼流形上构建的高斯过程的渐近行为。基于最近在紧致黎曼流形上引入的马特恩协方差图，我们使用紧致流形上两个马特恩高斯随机测度等价的形式概念和条件来推导可识别的参数，也称为微遍历参数，并正式建立最大似然估计的一致性以及最佳线性无偏预测器的渐近最优性。以圆作为紧致黎曼流形的一个具体例子进行研究，并通过数值实验来说明和证实该理论。
