Provable Diffusion Posterior Sampling for Bayesian Inversion
This addresses the challenge of efficient and provable posterior sampling for Bayesian inverse problems, which is incremental as it builds on existing diffusion and plug-and-play methods with theoretical improvements.
The paper tackles the problem of sampling from complex, multi-modal posterior distributions in Bayesian inversion by proposing a diffusion-based posterior sampling method within a plug-and-play framework, achieving convergence with non-asymptotic error bounds that quantify errors from score estimation, initialization, and sampling.
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.