Diffusion Schr\"odinger bridges (DSB) have recently emerged as a powerful framework for recovering stochastic dynamics via their marginal observations at different time points. Despite numerous successful applications, existing algorithms for solving DSBs have so far failed to utilize the structure of aligned data, which naturally arises in many biological phenomena. In this paper, we propose a novel algorithmic framework that, for the first time, solves DSBs while respecting the data alignment. Our approach hinges on a combination of two decades-old ideas: The classical Schr\"odinger bridge theory and Doob's $h$-transform. Compared to prior methods, our approach leads to a simpler training procedure with lower variance, which we further augment with principled regularization schemes. This ultimately leads to sizeable improvements across experiments on synthetic and real data, including the tasks of rigid protein docking and temporal evolution of cellular differentiation processes.

翻译：扩散薛定谔桥（DSB）近年来已成为一种强大的框架，用于通过随机过程在不同时间点的边缘观测来恢复其随机动力学。尽管已有众多成功应用，但现有求解DSB的算法至今未能利用对齐数据的结构，而这种结构在许多生物现象中自然存在。本文首次提出一种新颖的算法框架，在求解DSB的同时尊重数据对齐。我们的方法依赖于两种经典思想的结合：经典薛定谔桥理论与Doob的$h$-变换。与先前方法相比，我们的方法简化了训练过程并降低了方差，并通过原则性正则化方案进一步增强了该方法。最终，在包括刚性蛋白质对接和细胞分化过程的时间演化任务在内的合成数据与真实数据实验中，该方法取得了显著改进。