A Gibbs posterior sampler for inverse problem based on prior diffusion model
It addresses a thorny issue in Bayesian inversion for researchers in computational statistics or inverse problems, though it appears incremental as it builds on existing frameworks.
The paper tackles the problem of posterior sampling in ill-posed inverse problems with a prior modeled by a diffusion process, introducing a Gibbs algorithm that is shown to be effective, simple, and convergent in specific cases, as confirmed by numerical simulations.
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.