Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach

TL;DR

This paper introduces an ensemble sampling algorithm for Bayesian inverse problems that leverages diffusion models without heuristic approximations, providing more accurate reconstructions validated through imaging experiments.

Contribution

It develops a novel approximation-free ensemble sampling method based on diffusion models and PDE analysis, improving posterior sampling accuracy in inverse problems.

Findings

The proposed method outperforms existing DM-based approaches in image reconstruction tasks.

Theoretical error bounds relate posterior approximation accuracy to score function training error and particle count.

Empirical results demonstrate improved reconstruction quality across multiple inverse imaging problems.

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.