Score-based diffusion models (SDMs) offer a flexible approach to sample from the posterior distribution in a variety of Bayesian inverse problems. In the literature, the prior score is utilized to sample from the posterior by different methods that require multiple evaluations of the forward mapping in order to generate a single posterior sample. These methods are often designed with the objective of enabling the direct use of the unconditional prior score and, therefore, task-independent training. In this paper, we focus on linear inverse problems, when evaluation of the forward mapping is computationally expensive and frequent posterior sampling is required for new measurement data, such as in medical imaging. We demonstrate that the evaluation of the forward mapping can be entirely bypassed during posterior sample generation. Instead, without introducing any error, the computational effort can be shifted to an offline task of training the score of a specific diffusion-like random process. In particular, the training is task-dependent requiring information about the forward mapping but not about the measurement data. It is shown that the conditional score corresponding to the posterior can be obtained from the auxiliary score by suitable affine transformations. We prove that this observation generalizes to the framework of infinite-dimensional diffusion models introduced recently and provide numerical analysis of the method. Moreover, we validate our findings with numerical experiments.

翻译：基于分数的扩散模型（SDMs）为从各种贝叶斯逆问题的后验分布中采样提供了一种灵活方法。现有文献中，先验分数被用于通过不同方法从后验中采样，这些方法需要多次评估前向映射才能生成单个后验样本。这些方法的设计目标通常是能够直接使用无条件先验分数，从而实现与任务无关的训练。本文聚焦于线性逆问题，此类问题中前向映射的计算成本高昂，且需要频繁针对新测量数据进行后验采样（例如医学成像场景）。我们证明，在后验样本生成过程中可以完全绕过前向映射的评估。取而代之的是，在不引入任何误差的前提下，计算负担可以转移至离线训练特定类扩散随机过程分数的任务中。特别地，该训练是任务相关的，需要前向映射信息但无需测量数据信息。研究表明，对应于后验的条件分数可通过适当的仿射变换从辅助分数中获得。我们证明这一观察可推广至近期提出的无限维扩散模型框架，并提供该方法的数值分析。此外，我们通过数值实验验证了研究结论。