Fixed-Domain Inference for Gausian Processes with Matérn Covariogram on Compact Riemannian Manifolds

Gaussian processes are widely employed as versatile modeling and predictive tools in spatial statistics, functional data analysis, computer modeling and in diverse applications of machine learning. Such processes have been widely studied over Euclidean spaces, where they are constructed using specified covariance functions or covariograms. These functions specify valid stochastic processes that can be used to model complex dependencies in spatial statistics and other machine learning contexts. Valid (positive definite) covariance functions have been extensively studied for Gaussian processes on Euclidean spaces. Such investigations have focused, among other aspects, on the identifiability and consistency of covariance parameters as well as the problem of spatial interpolation and prediction within the fixed-domain or infill paradigm of asymptotic inference. This manuscript undertakes analogous theoretical developments for Gaussian processes constructed over Riemannian manifolds. We begin by establishing formal notions and conditions for the equivalence of two Gaussian random measures on compact manifolds. We build upon recently introduced Mat\'ern covariograms on compact Riemannian manifold, derive the microergodic parameter and formally establish the consistency of maximum likelihood estimators and the asymptotic optimality of the best linear unbiased predictor (BLUP). The circle and sphere are studied as two specific examples of compact Riemannian manifolds with numerical experiments that illustrate the theory.