Supervised Guidance Training for Infinite-Dimensional Diffusion Models

TL;DR

This paper develops a novel method for fine-tuning diffusion models in infinite-dimensional spaces to accurately sample from posterior distributions in Bayesian inverse problems, using a new supervised guidance training approach.

Contribution

It introduces a theoretical framework for conditioning diffusion models in function spaces and proposes a practical, simulation-free training method for posterior sampling.

Findings

Proves that models can be conditioned via an infinite-dimensional Doob's h-transform.

Develops a guidance term decomposition for conditional score functions.

Demonstrates the method's effectiveness on Bayesian inverse problems in function spaces.

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.