Provably Robust Score-Based Diffusion Posterior Sampling for
  Plug-and-Play Image Reconstruction
Xingyu Xu Yuejie Chi
In a great number of tasks in science and engineering, the goal is to infer
an unknown image from a small number of measurements collected from a known
forward model describing certain sensing or imaging modality. Due to resource
constraints, this task is often extremely ill-posed, which necessitates the
adoption of expressive prior information to regularize the solution space.
Score-based diffusion models, due to its impressive empirical success, have
emerged as an appealing candidate of an expressive prior in image
reconstruction. In order to accommodate diverse tasks at once, it is of great
interest to develop efficient, consistent and robust algorithms that
incorporate {\em unconditional} score functions of an image prior distribution
in conjunction with flexible choices of forward models.
  This work develops an algorithmic framework for employing score-based
diffusion models as an expressive data prior in general nonlinear inverse
problems. Motivated by the plug-and-play framework in the imaging community, we
introduce a diffusion plug-and-play method (\textsf{DPnP}) that alternatively
calls two samplers, a proximal consistency sampler based solely on the
likelihood function of the forward model, and a denoising diffusion sampler
based solely on the score functions of the image prior. The key insight is that
denoising under white Gaussian noise can be solved {\em rigorously} via both
stochastic (i.e., DDPM-type) and deterministic (i.e., DDIM-type) samplers using
the unconditional score functions. We establish both asymptotic and
non-asymptotic performance guarantees of \textsf{DPnP}, and provide numerical
experiments to illustrate its promise in solving both linear and nonlinear
image reconstruction tasks. To the best of our knowledge, \textsf{DPnP} is the
first provably-robust posterior sampling method for nonlinear inverse problems
using unconditional diffusion priors.