Intrinsic Gaussian processes on complex constrained domains
This addresses the problem of smoothing on constrained domains for researchers and practitioners in spatial statistics and machine learning, offering a novel method for handling complex geometries.
The authors tackled the problem of Gaussian process interpolation, regression, and classification on complex constrained domains or irregular shaped spaces by proposing intrinsic Gaussian processes (in-GPs) that respect boundary conditions and intrinsic geometry, enabling practical application in great generality while existing methods are limited to simple cases.
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.