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.

翻译：本研究提出一种能够解决函数空间中贝叶斯反问题的采样方法。该方法不要求似然函数满足对数凹性，这意味着其适用于非线性反问题。该方法利用近期定义的无限维分数扩散模型作为基于学习的先验，同时通过在函数空间上定义的朗之万型MCMC算法实现可证明的后验采样。受传统去噪正则化算法中定点方法的启发，我们提出了一种兼容加权退火策略的新型收敛性分析。所得收敛界显式依赖于分数逼近误差；精确逼近的分数对于获得精确逼近的后验至关重要。本文提供了风格化示例和基于偏微分方程的案例，验证了收敛分析的有效性。最后，我们讨论了该方法在分数学习和计算复杂度方面面临的挑战。