The dynamic Schr\"odinger bridge problem provides an appealing setting for solving constrained time-series data generation tasks posed as optimal transport problems. It consists of learning non-linear diffusion processes using efficient iterative solvers. Recent works have demonstrated state-of-the-art results (eg. in modelling single-cell embryo RNA sequences or sampling from complex posteriors) but are limited to learning bridges with only initial and terminal constraints. Our work extends this paradigm by proposing the Iterative Smoothing Bridge (ISB). We integrate Bayesian filtering and optimal control into learning the diffusion process, enabling the generation of constrained stochastic processes governed by sparse observations at intermediate stages and terminal constraints. We assess the effectiveness of our method on synthetic and real-world data generation tasks and we show that the ISB generalises well to high-dimensional data, is computationally efficient, and provides accurate estimates of the marginals at intermediate and terminal times.

翻译：动态Schrödinger桥问题为求解受约束的时间序列数据生成任务提供了一个有吸引力的框架，该类任务本质上是最优输运问题。该问题涉及利用高效迭代求解器学习非线性扩散过程。近期研究已在多项任务中展现出最先进的性能（例如，对单细胞胚胎RNA序列建模或者从复杂后验分布中采样），但这些方法仅限于学习仅包含初始和终端约束的桥。我们的工作通过提出迭代平滑桥（ISB）拓展了这一范式。我们将贝叶斯滤波与最优控制融入扩散过程的学习，使得在中间阶段存在稀疏观测以及终端约束的条件下，能够生成受约束的随机过程。我们在合成数据和真实数据生成任务上评估了所提方法的有效性，结果表明ISB能够很好地泛化至高维数据，计算效率高，并且能够提供中间时刻与终端时刻边缘密度的精确估计。