Inference for Gaussian Processes with Matérn Covariogram on Compact Riemannian Manifolds

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. While Gaussian processes have been extensively studied for asymptotic inference on Euclidean spaces using well-defined (positive definite) covariograms, such results are relatively sparse on Riemannian manifolds. We undertake analogous developments for Gaussian processes constructed over compact Riemannian manifolds. Building upon the recently introduced Mat\'ern covariograms 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 microergodic parameters and formally establish the consistency of their maximum likelihood estimates as well as asymptotic optimality of the best linear unbiased predictor. The circle and sphere are studied as two specific examples of compact Riemannian manifolds with numerical experiments to illustrate and corroborate the theory.