Provable Diffusion Posterior Sampling for Bayesian Inversion

TL;DR

This paper introduces a diffusion-based posterior sampling method for Bayesian inversion that leverages a Monte Carlo estimator and theoretical error bounds, improving sampling accuracy for complex distributions.

Contribution

It presents a novel diffusion-based PnP framework with a Monte Carlo score estimator and provides non-asymptotic error bounds for complex, multi-modal posteriors.

Findings

Effective sampling across complex inverse problems

Theoretical error bounds quantify convergence and errors

Model captures rich prior distribution features

Abstract

This paper proposes a novel diffusion-based posterior sampling method within a plug-and-play (PnP) framework. Our approach constructs a probability transport from an easy-to-sample terminal distribution to the target posterior, using a warm-start strategy to initialize the particles. To approximate the posterior score, we develop a Monte Carlo estimator in which particles are generated using Langevin dynamics, avoiding the heuristic approximations commonly used in prior work. The score governing the Langevin dynamics is learned from data, enabling the model to capture rich structural features of the underlying prior distribution. On the theoretical side, we provide non-asymptotic error bounds, showing that the method converges even for complex, multi-modal target posterior distributions. These bounds explicitly quantify the errors arising from posterior score estimation, the warm-start initialization, and the posterior sampling procedure. Our analysis further clarifies how the prior score-matching error and the condition number of the Bayesian inverse problem influence overall performance. Finally, we present numerical experiments demonstrating the effectiveness of the proposed method across a range of inverse problems.