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：连接最优传输与变分推断，我们提出了一个以路径空间散度为核心的、用于采样和生成建模的原则性且系统性的框架。该工作最终发展了用于贝叶斯计算的受控蒙特卡洛扩散采样器（CMCD），这是一种分数退火技术，关键性地同时调整了扩散模型中的正向和反向动力学。在此过程中，我们澄清了EM算法与薛定谔桥的迭代比例拟合（IPF）之间的关系，并推导出一个正则化目标函数，从而绕过了标准IPF更新的迭代瓶颈。最后，我们证明CMCD在统计物理中的Jarzinsky和Crooks恒等式下具有坚实基础，并在大量实验中显著优于竞争方法。