Intrinsic Gaussian processes on complex constrained domains

TL;DR

This paper introduces intrinsic Gaussian processes tailored for complex constrained domains and manifolds, leveraging heat kernels and Brownian motion to enable flexible, geometry-aware interpolation, regression, and classification.

Contribution

The paper presents a novel approach to construct Gaussian processes on complex manifolds using heat kernels, extending applicability beyond simple constrained domains.

Findings

Effective in handling complex boundary conditions

Applicable to a wide range of irregular domains

Demonstrated through simulations and real data examples

Abstract

We propose a class of intrinsic Gaussian processes (in-GPs) for interpolation, regression and classification on manifolds with a primary focus on complex constrained domains or irregular shaped spaces arising as subsets or submanifolds of R, R2, R3 and beyond. For example, in-GPs can accommodate spatial domains arising as complex subsets of Euclidean space. in-GPs respect the potentially complex boundary or interior conditions as well as the intrinsic geometry of the spaces. The key novelty of the proposed approach is to utilise the relationship between heat kernels and the transition density of Brownian motion on manifolds for constructing and approximating valid and computationally feasible covariance kernels. This enables in-GPs to be practically applied in great generality, while existing approaches for smoothing on constrained domains are limited to simple special cases. The broad utilities of the in-GP approach is illustrated through simulation studies and data examples.