Preprint

PUBDB-2026-01497

http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.png Supervised Guidance Training for Infinite-Dimensional Diffusion Models

Baker, E. L. (Corresponding author) ; Denker, A.DESY* ; Frellsen, J.

2026

This record in other databases:  

Please use a persistent id in citations: doi:10.3204/PUBDB-2026-01497

Report No.: arXiv:2601.20756

Abstract: Score-based diffusion models have recently been extended to infinite-dimensional function spaces, with uses such as inverse problems arising from partial differential equations. In the Bayesian formulation of inverse problems, the aim is to sample from a posterior distribution over functions obtained by conditioning a prior on noisy observations. While diffusion models provide expressive priors in function space, the theory of conditioning them to sample from the posterior remains open. We address this, assuming that either the prior lies in the Cameron-Martin space, or is absolutely continuous with respect to a Gaussian measure. We prove that the models can be conditioned using an infinite-dimensional extension of Doob's $h$-transform, and that the conditional score decomposes into an unconditional score and a guidance term. As the guidance term is intractable, we propose a simulation-free score matching objective (called Supervised Guidance Training) enabling efficient and stable posterior sampling. We illustrate the theory with numerical examples on Bayesian inverse problems in function spaces. In summary, our work offers the first function-space method for fine-tuning trained diffusion models to accurately sample from a posterior.

Note: Submission to The 43rd International Conference on Machine Learning: acceptedFunding information: Alexander Denker acknowledges support from the EPSRC (EP/V026259/1) and support from DESY (Hamburg, Germany), a member of the Helmholtz Association HGF. Elizabeth Baker and Jes Frellsen were supported by funding from Villum Foundation Synergy project number 50091 entitled “Physics-aware machine learning” as well as the Center for Basic Machine Learning Research in Life Science (MLLS) through the Novo Nordisk Foundation (NNF20OC0062606). Jes Frellsen was further supported by funding from the Reinholdt W. Jorck og Hustrus Fond.

---

Contributing Institute(s):

1. Computational Imaging (FS-CI)

Research Program(s):

1. 623 - Data Management and Analysis (POF4-623) (POF4-623)

Experiment(s):

1. No specific instrument

Appears in the scientific report 2026

---

Database coverage:
;

---