Inter-Domain Gaussian Processes with Arbitrary Mean Functions in Factor Graphs

Alex Ledbetter, Hoang Minh Huu Nguyen, Lucas Carolus van Laake, Irene A. Kuling, Yoeri van de Burgt, Thijs van de Laar
Proceedings of the 2nd International Conference on Probabilistic Numerics, PMLR 341:90-101, 2026.

Abstract

The modeling flexibility and expressivity of inter-domain Gaussian processes, enabled by applying arbitrary linear operators to a latent Gaussian process, is significant and desirable in robotics and probabilistic numerics communities. However, inter-domain Gaussian processes have not yet been available in a probabilistic message-passing formulation. In this paper, we formalize how to enable inter-domain observations for Gaussian processes in a variational message-passing framework. We develop a decoupled inter-domain variational sparse Gaussian process (dID-VSGP) model for univariate latent Gaussian processes with arbitrary mean functions, and derive the mean-field variational message-passing update rules that allow inference in this model. We validate our derivations in a shape exploration and modeling task, and by solution to a linear stochastic partial differential equation representative of physics-informed exploration. We confirm that our message-passing implementation maintains the same scaling complexity as the VSGP analytical solution. Our results further unify the analytical and message-passing approaches to variational inference and enable inter-domain observations in factor-graph Gaussian processes.

Cite this Paper

BibTeX
@InProceedings{pmlr-v341-ledbetter26a, title = {Inter-Domain {G}aussian Processes with Arbitrary Mean Functions in Factor Graphs}, author = {Ledbetter, Alex and Nguyen, Hoang Minh Huu and van Laake, Lucas Carolus and Kuling, Irene A. and van de Burgt, Yoeri and van de Laar, Thijs}, booktitle = {Proceedings of the 2nd International Conference on Probabilistic Numerics}, pages = {90--101}, year = {2026}, editor = {Karvonen, Toni and Bosch, Nathanael and Cockayne, Jon and Gessner, Alexandra and Hennig, Philipp and Kouw, Wouter}, volume = {341}, series = {Proceedings of Machine Learning Research}, month = {09--11 Sep}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v341/main/assets/ledbetter26a/ledbetter26a.pdf}, url = {https://proceedings.mlr.press/v341/ledbetter26a.html}, abstract = {The modeling flexibility and expressivity of inter-domain Gaussian processes, enabled by applying arbitrary linear operators to a latent Gaussian process, is significant and desirable in robotics and probabilistic numerics communities. However, inter-domain Gaussian processes have not yet been available in a probabilistic message-passing formulation. In this paper, we formalize how to enable inter-domain observations for Gaussian processes in a variational message-passing framework. We develop a decoupled inter-domain variational sparse Gaussian process (dID-VSGP) model for univariate latent Gaussian processes with arbitrary mean functions, and derive the mean-field variational message-passing update rules that allow inference in this model. We validate our derivations in a shape exploration and modeling task, and by solution to a linear stochastic partial differential equation representative of physics-informed exploration. We confirm that our message-passing implementation maintains the same scaling complexity as the VSGP analytical solution. Our results further unify the analytical and message-passing approaches to variational inference and enable inter-domain observations in factor-graph Gaussian processes.} }
Endnote
%0 Conference Paper %T Inter-Domain Gaussian Processes with Arbitrary Mean Functions in Factor Graphs %A Alex Ledbetter %A Hoang Minh Huu Nguyen %A Lucas Carolus van Laake %A Irene A. Kuling %A Yoeri van de Burgt %A Thijs van de Laar %B Proceedings of the 2nd International Conference on Probabilistic Numerics %C Proceedings of Machine Learning Research %D 2026 %E Toni Karvonen %E Nathanael Bosch %E Jon Cockayne %E Alexandra Gessner %E Philipp Hennig %E Wouter Kouw %F pmlr-v341-ledbetter26a %I PMLR %P 90--101 %U https://proceedings.mlr.press/v341/ledbetter26a.html %V 341 %X The modeling flexibility and expressivity of inter-domain Gaussian processes, enabled by applying arbitrary linear operators to a latent Gaussian process, is significant and desirable in robotics and probabilistic numerics communities. However, inter-domain Gaussian processes have not yet been available in a probabilistic message-passing formulation. In this paper, we formalize how to enable inter-domain observations for Gaussian processes in a variational message-passing framework. We develop a decoupled inter-domain variational sparse Gaussian process (dID-VSGP) model for univariate latent Gaussian processes with arbitrary mean functions, and derive the mean-field variational message-passing update rules that allow inference in this model. We validate our derivations in a shape exploration and modeling task, and by solution to a linear stochastic partial differential equation representative of physics-informed exploration. We confirm that our message-passing implementation maintains the same scaling complexity as the VSGP analytical solution. Our results further unify the analytical and message-passing approaches to variational inference and enable inter-domain observations in factor-graph Gaussian processes.
APA
Ledbetter, A., Nguyen, H.M.H., van Laake, L.C., Kuling, I.A., van de Burgt, Y. & van de Laar, T.. (2026). Inter-Domain Gaussian Processes with Arbitrary Mean Functions in Factor Graphs. Proceedings of the 2nd International Conference on Probabilistic Numerics, in Proceedings of Machine Learning Research 341:90-101 Available from https://proceedings.mlr.press/v341/ledbetter26a.html.

Related Material