紧黎曼流形上具有Matérn协方差函数的高斯过程的推断
Inference for Gaussian Processes with Matérn Covariogram on Compact Riemannian Manifolds
摘要：
高斯过程被广泛用作空间统计、函数数据分析、计算机建模和机器学习的各种应用中的通用建模和预测工具。它们已经在欧几里得空间上被广泛研究，在欧几里得空间中，它们被指定使用协方差函数或协方差图来建模复杂的依赖关系。有越来越多的文献在黎曼流形上的高斯过程，以发展更丰富和更灵活的非欧数据的推理框架。虽然通过图形表示的数值逼近已经很好地研究了Matérn协方差函数和热内核，渐近推断的协方差函数的参数的行为得到了相对较少的关注。我们重点研究紧黎曼流形上构造的高斯过程的渐近推断。最近推出的Matérn协方差函数的紧黎曼流形上，我们采用正式的概念和条件的等价性的两个Matérn高斯随机措施的紧流形上导出的参数是可识别的，也被称为微遍历参数，并正式建立的最大似然估计的一致性和最佳线性无偏预测的渐近最优性。作为紧致黎曼流形的一个具体例子，对圆进行了研究，并用数值实验来说明和证实理论。
英文摘要：
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 the asymptotic inference for Gaussian processes constructed over compact Riemannian manifolds. Building upon the 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.
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