Connecting optimal transport and variational inference, we present a principled and systematic framework for sampling and generative modelling centred around divergences on path space. Our work culminates in the development of the \emph{Controlled Monte Carlo Diffusion} sampler (CMCD) for Bayesian computation, a score-based annealing technique that crucially adapts both forward and backward dynamics in a diffusion model. On the way, we clarify the relationship between the EM-algorithm and iterative proportional fitting (IPF) for Schr{\"o}dinger bridges, deriving as well a regularised objective that bypasses the iterative bottleneck of standard IPF-updates. Finally, we show that CMCD has a strong foundation in the Jarzinsky and Crooks identities from statistical physics, and that it convincingly outperforms competing approaches across a wide array of experiments.

翻译：将最优传输与变分推断相结合，我们提出了一套以路径空间散度为核心的生成式建模与采样原则性系统框架。本文工作的集大成者是面向贝叶斯计算的受控蒙特卡洛扩散采样器（CMCD）——一种在扩散模型中关键性地同时调整正向与反向动力学的基于分数的退火技术。在此过程中，我们厘清了薛定谔桥问题中EM算法与迭代比例拟合（IPF）的关系，并推导出一个可规避标准IPF更新迭代瓶颈的正则化目标。最后，我们证明CMCD以统计物理学中的Jarzinsky恒等式和Crooks涨落关系为坚实基础，且在一系列广泛实验中显著优于竞争方法。