  This work introduces a sampling method capable of solving Bayesian inverse
problems in function space. It does not assume the log-concavity of the
likelihood, meaning that it is compatible with nonlinear inverse problems. The
method leverages the recently defined infinite-dimensional score-based
diffusion models as a learning-based prior, while enabling provable posterior
sampling through a Langevin-type MCMC algorithm defined on function spaces. A
novel convergence analysis is conducted, inspired by the fixed-point methods
established for traditional regularization-by-denoising algorithms and
compatible with weighted annealing. The obtained convergence bound explicitly
depends on the approximation error of the score; a well-approximated score is
essential to obtain a well-approximated posterior. Stylized and PDE-based
examples are provided, demonstrating the validity of our convergence analysis.
We conclude by presenting a discussion of the method's challenges related to
learning the score and computational complexity.
