Taming Score-Based Diffusion Priors for Infinite-Dimensional Nonlinear Inverse Problems
<|reference_start|>Taming Score-Based Diffusion Priors for Infinite-Dimensional Nonlinear Inverse Problems: 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.<|reference_end|>
@article{baldassari2024taming,
    title={Taming Score-Based Diffusion Priors for Infinite-Dimensional Nonlinear
  Inverse Problems},
    author={Lorenzo Baldassari, Ali Siahkoohi, Josselin Garnier, Knut Solna,
  Maarten V. de Hoop},
    journal={arXiv preprint arXiv:2405.15676},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.15676},
    primaryClass={stat.ML cs.LG cs.NA math.NA}
}