Predicting Covariate-Driven Spatial Deformation for Nonstationary Gaussian Processes

arXiv cs.LG / 5/1/2026

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Key Points

The paper addresses a limitation of the spatial deformation approach for nonstationary Gaussian processes by noting that the commonly used static warping fails to reflect covariate-driven changes in spatial correlation.
It proposes learning the spatial deformation itself as a function of covariates, by linking diffeomorphic deformations to Euclidean covariate vectors through velocity fields in a Lie algebra framework.
To stabilize estimation, the authors prove that high-order interactions among covariates in the Lie algebra can be truncated under a moderate physical assumption.
Leveraging these results, they derive a compact covariate-to-deformation functional form and develop an efficient estimation/inference algorithm for out-of-sample nonstationary GP prediction using limited covariate–deformation training pairs.
The method is validated through simulations and two application case studies in manufacturing and geostatistics, demonstrating both effectiveness and generalizability.

Abstract

Nonstationary Gaussian processes (GPs) are essential for modeling complex, locally heterogeneous spatial data. A common modeling approach is the spatial deformation method that warps the domain to recover isotropy. However, this static method does not account for changes in spatial correlation induced by covariates, limiting its ability to predict nonstationary GPs under new covariate conditions. To enable predictive modeling of the deformation method, we propose to model the spatial deformation as a function of covariates. The spaces of diffeomorphic deformations and Euclidean covariate vectors are connected by characterizing deformations as generated by velocity fields living in a Lie algebra. To overcome the estimation instability caused by high-order interactions between multiple covariates in a general Lie algebra, we prove that those interactions can be truncated with a moderate physical assumption. Based on the theoretical results, a concise functional form of deformations driven by multiple covariates can be established, and an efficient estimation-inference algorithm is developed for out-of-sample nonstationary GP prediction with limited covariate-deformation sample pairs. The effectiveness and generalizability of the method are demonstrated on a simulation study and two case studies, in the fields of manufacturing and geostatistics, respectively.