A Gibbs posterior sampler for inverse problem based on prior diffusion model

TL;DR

This paper introduces a Gibbs sampling algorithm for inverse problems with linear observations, noise, and a prior modeled by a diffusion process, demonstrating effectiveness and convergence guarantees through simulations.

Contribution

It presents a novel Gibbs sampler for inverse problems with diffusion-based priors, a previously unexplored approach that simplifies sampling and provides convergence guarantees.

Findings

Effective Gibbs sampling algorithm for diffusion-based priors

Convergence guarantees under specific conditions

Validated by numerical simulations

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.