  Since their initial 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, primarily through methods that learn the unconditional score. While
this approach is advantageous for some inverse problems, it is mostly heuristic
and involves numerous computationally costly forward operator evaluations
during posterior sampling. To address these limitations, we propose a
theoretically grounded method for sampling from the posterior of
infinite-dimensional Bayesian linear inverse problems based on amortized
conditional SDMs. In particular, we prove that one of the most successful
approaches for estimating the conditional score in finite dimensions - the
conditional denoising estimator - can also be applied in infinite dimensions. A
significant part of our analysis is dedicated to demonstrating that extending
infinite-dimensional SDMs to the conditional setting requires careful
consideration, as the conditional score typically blows up for small times,
contrarily to the unconditional score. We conclude by presenting stylized and
large-scale numerical examples that validate our approach, offer additional
insights, and demonstrate that our method enables large-scale,
discretization-invariant Bayesian inference.
