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.

本次研究提出一种使用摊销条件受限 SDMs 学习无限维贝叶斯线性反问题的后验分布的方法，并且证明有条件的去噪估计器是无限维条件得分的一致估计，指出引入条件得分需要特别小心处理，同时证明所学习的分布对于观测的微小扰动是稳健的。