[Paper Review] Inference for Gaussian Processes with Matérn Covariogram on Compact Riemannian Manifolds

Didong Li, Wenpin Tang+1 more|PubMed|Apr 8, 2021

Soil Geostatistics and Mapping53 references4 citations

TL;DR

This paper establishes asymptotic inference theory for Gaussian processes with Matérn covariograms on compact Riemannian manifolds, leveraging spectral properties of the Laplace–Beltrami operator. It proves consistency of maximum likelihood estimators and asymptotic optimality of best linear unbiased predictors, with formal identification of the microergodic parameter and theoretical validation on the circle manifold.

ABSTRACT

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.

Motivation & Objective

• To develop a rigorous asymptotic inference framework for Gaussian processes with Matérn covariograms on compact Riemannian manifolds, where traditional Euclidean assumptions do not apply.
• To identify the microergodic parameter in the Matérn covariogram under equivalence conditions of Gaussian measures on manifolds.
• To formally establish consistency of maximum likelihood estimators and asymptotic optimality of the best linear unbiased predictor (BLUP) in this non-Euclidean setting.
• To validate theoretical results through numerical experiments on the circle, a canonical compact Riemannian manifold.
• To bridge the gap between numerical approximations and formal statistical inference for Matérn processes on manifolds, especially in non-flat geometries.

Proposed method

• The authors use the spectral decomposition of the Laplace–Beltrami operator on compact Riemannian manifolds to define a valid Matérn covariogram via eigenfunctions and eigenvalues.
• They apply formal conditions for equivalence of Gaussian measures to derive the microergodic parameter, which governs asymptotic identifiability.
• Consistency of maximum likelihood estimators is established using almost sure convergence of sample variance and measure equivalence under regularity conditions.
• Asymptotic optimality of the BLUP is proven by analyzing the prediction error variance and showing convergence to the theoretical minimum.
• A truncated series representation of the covariogram is used for numerical approximation, with error bounds derived via $L^2$-norm control of tail sums.
• Theoretical bounds on the minimal eigenvalue of the covariance matrix are derived using results from Narcowich et al. (1998), ensuring positive definiteness of the truncated covariance matrix.

Experimental results

Research questions

• RQ1 Which parameters in the Matérn covariogram on a compact Riemannian manifold are identifiable under asymptotic sampling?
• RQ2 Under what conditions is the maximum likelihood estimator of the Matérn parameters consistent on a compact manifold?
• RQ3 Is the best linear unbiased predictor asymptotically optimal for Gaussian processes on non-Euclidean domains?
• RQ4 How can the Matérn covariogram be approximated numerically on a manifold while preserving statistical properties?
• RQ5 What is the role of the Laplace–Beltrami spectrum in defining and inferring the Matérn process on manifolds?

Key findings

• The microergodic parameter in the Matérn covariogram on a compact Riemannian manifold is formally identified as the product of the variance and the inverse of the smoothness parameter, under measure equivalence conditions.
• The maximum likelihood estimator of the microergodic parameter is consistent, with almost sure convergence of the sample variance to the true variance under the true measure.
• The best linear unbiased predictor achieves asymptotic optimality, as its prediction error variance converges to the theoretical minimum under the true model.
• For the circle manifold, numerical experiments confirm the theoretical consistency and optimality, with prediction errors decreasing as sample size increases.
• The truncated series approximation of the covariogram converges uniformly with error bounded by $O(L^{-2 u})$, where $L$ is the truncation level and $ u$ is the smoothness parameter.
• The minimal eigenvalue of the covariance matrix remains bounded away from zero for sufficiently large $L$, ensuring positive definiteness of the approximate covariance matrix.