Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach
Journal:
arXiv
Published Date:
Jun 4, 2025
Abstract
Diffusion models (DMs) have proven to be effective in modeling
high-dimensional distributions, leading to their widespread adoption for
representing complex priors in Bayesian inverse problems (BIPs). However,
current DM-based posterior sampling methods proposed for solving common BIPs
rely on heuristic approximations to the generative process. To exploit the
generative capability of DMs and avoid the usage of such approximations, we
propose an ensemble-based algorithm that performs posterior sampling without
the use of heuristic approximations. Our algorithm is motivated by existing
works that combine DM-based methods with the sequential Monte Carlo (SMC)
method. By examining how the prior evolves through the diffusion process
encoded by the pre-trained score function, we derive a modified partial
differential equation (PDE) governing the evolution of the corresponding
posterior distribution. This PDE includes a modified diffusion term and a
reweighting term, which can be simulated via stochastic weighted particle
methods. Theoretically, we prove that the error between the true posterior
distribution can be bounded in terms of the training error of the pre-trained
score function and the number of particles in the ensemble. Empirically, we
validate our algorithm on several inverse problems in imaging to show that our
method gives more accurate reconstructions compared to existing DM-based
methods.
