Supervised Guidance Training for Infinite-Dimensional Diffusion Models

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.

💡 Research Summary

This paper tackles the problem of conditioning score‑based diffusion models (SDMs) that live in infinite‑dimensional function spaces so that they can sample from posterior distributions arising in Bayesian inverse problems. While finite‑dimensional diffusion models can be conditioned via Doob’s h‑transform—splitting the conditional score into an unconditional score plus a guidance term—extending this machinery to Hilbert spaces is non‑trivial because Lebesgue densities do not exist, the Cameron‑Martin theorem imposes strict support constraints, and the Feldman‑Hájek theorem creates singularity issues.

The authors consider two broadly applicable settings for the prior distribution π: (1) π’s support lies inside a ball of the Cameron‑Martin space of the Wiener process used in the forward SDE, and (2) π is absolutely continuous with respect to a Gaussian measure. Under either assumption they prove that an infinite‑dimensional Doob h‑transform exists. The transform is defined as

 h_y(t,x)=ξ⁻¹ E