Taming Score-Based Diffusion Priors for Infinite-Dimensional Nonlinear Inverse Problems

This work introduces a sampling method capable of solving Bayesian inverse problems in function space. It does not assume the log-concavity of the likelihood, meaning that it is compatible with nonlinear inverse problems. The method leverages the recently defined infinite-dimensional score-based diffusion models as a learning-based prior, while enabling provable posterior sampling through a Langevin-type MCMC algorithm defined on function spaces. A novel convergence analysis is conducted, inspired by the fixed-point methods established for traditional regularization-by-denoising algorithms and compatible with weighted annealing. The obtained convergence bound explicitly depends on the approximation error of the score; a well-approximated score is essential to obtain a well-approximated posterior. Stylized and PDE-based examples are provided, demonstrating the validity of our convergence analysis. We conclude by presenting a discussion of the method's challenges related to learning the score and computational complexity.

Conditional score-based diffusion models for Bayesian inference in infinite dimensions

Since their first introduction, score-based diffusion models (SDMs) have been successfully applied to solve a variety of linear inverse problems in finite-dimensional vector spaces due to their ability to efficiently approximate the posterior distribution. However, using SDMs for inverse problems in infinite-dimensional function spaces has only been addressed recently and by learning the unconditional score. While this approach has some advantages, depending on the specific inverse problem at hand, in order to sample from the conditional distribution it needs to incorporate the information from the observed data with a proximal optimization step, solving an optimization problem numerous times. This may not be feasible in inverse problems with computationally costly forward operators. To address these limitations, in this work we propose a method to learn the posterior distribution in infinite-dimensional Bayesian linear inverse problems using amortized conditional SDMs. In particular, we prove that the conditional denoising estimator is a consistent estimator of the conditional score in infinite dimensions. We show that the extension of SDMs to the conditional setting requires some care because the conditional score typically blows up for small times contrarily to the unconditional score. We also discuss the robustness of the learned distribution against perturbations of the observations. We conclude by presenting numerical examples that validate our approach and provide additional insights.