标题： 关于Whittle-Matérn过程在闭黎曼流形上的光滑度估计 显示英文标题

标题： Smoothness Estimation for Whittle-Matérn Processes on Closed Riemannian Manifolds

摘要： Matérn 核函数族在空间统计、函数逼近以及机器学习中的高斯过程方法中经常被使用。它们之所以受欢迎的一个原因是存在一个平滑参数，该参数可以控制例如克里金最优误差界和高斯过程回归后验收缩率等性质。 在闭黎曼流形上，我们证明了当数据来源于高斯过程的点评估时，可以通过最大化高斯似然函数来一致估计平滑参数，并且出人意料的是，即使数据来源于非高斯过程的评估时也可以做到这一点。观测该过程的点无需具有任何特定的空间结构，只需满足准均匀性即可。 我们的方法基于索博列夫空间尺度的逼近理论结果。此外，我们通过使用角谷定理将与 Matérn 核相关的著名等价测度现象推广到非高斯情形。 显示英文摘要

摘要： The family of Mat\'ern kernels are often used in spatial statistics, function approximation and Gaussian process methods in machine learning. One reason for their popularity is the presence of a smoothness parameter that controls, for example, optimal error bounds for kriging and posterior contraction rates in Gaussian process regression. On closed Riemannian manifolds, we show that the smoothness parameter can be consistently estimated from the maximizer(s) of the Gaussian likelihood when the underlying data are from point evaluations of a Gaussian process and, perhaps surprisingly, even when the data comprise evaluations of a non-Gaussian process. The points at which the process is observed need not have any particular spatial structure beyond quasi-uniformity. Our methods are based on results from approximation theory for the Sobolev scale of Hilbert spaces. Moreover, we generalize a well-known equivalence of measures phenomenon related to Mat\'ern kernels to the non-Gaussian case by using Kakutani's theorem.