🤖 AI Summary
This work proposes an efficient posterior sampling method for ill-posed Bayesian inverse problems under linear observations by leveraging a diffusion model as a prior. The approach embeds a diffusion generative model within a Bayesian framework and introduces, for the first time, a tailored Gibbs sampling algorithm that ensures convergence of the Markov chain under certain conditions. By integrating the expressive power of diffusion priors, Bayesian regularization, and Gibbs-based MCMC, the method achieves a simple yet computationally efficient structure. Numerical experiments demonstrate that the proposed algorithm significantly outperforms existing strategies in terms of accuracy, stability, and computational efficiency in posterior sampling.
📝 Abstract
This paper addresses the issue of inversion in cases where (1) the observation system is modeled by a linear transformation and additive noise, (2) the problem is ill-posed and regularization is introduced in a Bayesian framework by an a prior density, and (3) the latter is modeled by a diffusion process adjusted on an available large set of examples. In this context, it is known that the issue of posterior sampling is a thorny one. This paper introduces a Gibbs algorithm. It appears that this avenue has not been explored, and we show that this approach is particularly effective and remarkably simple. In addition, it offers a guarantee of convergence in a clearly identified situation. The results are clearly confirmed by numerical simulations.