Dynamic Gaussian Processes and the Vanilla-SPDE Exchange

2026-06-29

AI Summary

This work addresses the computational inefficiency of spatiotemporal Gaussian process (GP) inference, which suffers from cubic time complexity and high memory costs due to misalignment between observed and prediction locations. The authors propose the Vanilla-SPDE Exchange method, which establishes—for the first time—a direct equivalence between standard GPs and stochastic partial differential equation (SPDE)-based state-space models, thereby enabling a hybrid inference framework. By integrating sparse approximations with SPDE representations, the approach substantially reduces both computational and memory complexity while preserving predictive accuracy. Theoretical analysis and empirical evaluations demonstrate that the method achieves superior joint spatiotemporal inference efficiency, particularly on dense grids and under non-aligned observation settings.

Abstract

Gaussian process inference is often limited by cubic computational costs, a challenge that becomes more pronounced in spatio-temporal settings where posterior inference is required over dense grids. While state-space SPDE formulations enable linear complexity in time, exact inference remains cubic in space and deteriorates further when observation locations are disjoint from the prediction locations, which inflates the number of considered spatial points. To address this, we propose the Vanilla-SPDE Exchange, which exploits an equivalence between the standard and SPDE formulations of GP inference to construct a hybrid scheme with improved computational cost. We demonstrate these gains through complexity analysis and numerical experiments.

Problem

• Gaussian process
• computational complexity
• spatio-temporal inference
• SPDE
• posterior inference

Innovation

• Gaussian Processes
• SPDE
• computational complexity
• state-space models
• Vanilla-SPDE Exchange