Reducing the cost of posterior sampling in linear inverse problems via task-dependent score learning

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.