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（Diffusion Schrödinger Bridges，简称DSB）近期已成为一种强大的框架，用于通过不同时间点的边际观测恢复随机动力学。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特性，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时同时尊重数据对齐。我们的方法融合了两项数十年历史的经典思想：标准的薛定谔桥理论与Doob的$h$-变换。与先前方法相比，我们的方法带来了更简化的训练流程和更低的方差，并通过原理性正则化方案进一步增强了性能。最终，在合成数据和真实数据（包括刚性蛋白质对接任务和细胞分化过程的时间演化）的实验中，我们取得了显著改进。