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

2025-06-04

AI Summary

This work addresses the limitation of heuristic approximations in diffusion model (DM)-based posterior sampling for Bayesian inverse problems (BIPs). We propose an exact, theoretically grounded posterior sampling framework that avoids all approximations. Methodologically, we couple a pre-trained score function with weighted particle filtering to precisely solve a Fokker–Planck-type PDE governing posterior evolution—explicitly incorporating data-driven correction of the diffusion coefficient and particle reweighting. Theoretically, we establish the first rigorous posterior sampling error bound, quantifying its dependence on score estimation error and particle count; no heuristic assumptions are introduced. Empirically, our method achieves significantly higher reconstruction accuracy than state-of-the-art diffusion-based approaches on imaging inverse problems—including computed tomography and super-resolution—demonstrating superior efficacy, robustness, and theoretical soundness.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Probabilistic Inference
• Machine Learning: Bayesian Learning

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

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.

Problem

• Avoid heuristic approximations in diffusion-based posterior sampling
• Develop ensemble algorithm for accurate Bayesian inverse problems
• Improve reconstruction accuracy in imaging inverse problems

Innovation

• Ensemble sampling without heuristic approximations
• Modified PDE for posterior distribution evolution
• Stochastic weighted particle simulation method