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Since their initial introduction, score-based diffusion models (SDMs) havebeen successfully applied to solve a variety of linear inverse problems infinite-dimensional vector spaces due to their ability to efficientlyapproximate the posterior distribution. However, using SDMs for inverseproblems in infinite-dimensional function spaces has only been addressedrecently, primarily through methods that learn the unconditional score. Whilethis approach is advantageous for some inverse problems, it is mostly heuristicand involves numerous computationally costly forward operator evaluationsduring posterior sampling. To address these limitations, we propose atheoretically grounded method for sampling from the posterior ofinfinite-dimensional Bayesian linear inverse problems based on amortizedconditional SDMs. In particular, we prove that one of the most successfulapproaches for estimating the conditional score in finite dimensions - theconditional denoising estimator - can also be applied in infinite dimensions. Asignificant part of our analysis is dedicated to demonstrating that extendinginfinite-dimensional SDMs to the conditional setting requires carefulconsideration, as the conditional score typically blows up for small times,contrarily to the unconditional score. We conclude by presenting stylized andlarge-scale numerical examples that validate our approach, offer additionalinsights, and demonstrate that our method enables large-scale,discretization-invariant Bayesian inference.

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