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.

翻译：自首次引入以来，得分扩散模型（SDMs）因其高效近似后验分布的能力，已被成功应用于解决有限维向量空间中的各类线性逆问题。然而，将SDMs应用于无限维函数空间的逆问题直至近期才得到关注，现有方法主要通过学习无条件得分实现。尽管该方法对某些逆问题具有优势，但其本质上属于启发式方法，且在后验采样过程中需要大量计算代价高昂的正向算子评估。为解决这些局限性，我们提出了一种基于摊销条件SDMs的理论框架，用于从无限维贝叶斯线性逆问题的后验分布中采样。具体而言，我们证明了有限维空间中最成功的条件得分估计方法——条件去噪估计器——同样适用于无限维空间。我们分析的一个重要部分旨在表明：将无限维SDMs扩展至条件设定时需要审慎处理，因为与无条件得分不同，条件得分在时间较小时通常会发散。最后，我们通过风格化与大尺度数值算例验证了所提方法的有效性，提供了额外见解，并证明我们的方法能够实现大规模、离散化不变的贝叶斯推断。