It is a crucial challenge to reconstruct population dynamics using unlabeled samples from distributions at coarse time intervals. Recent approaches such as flow-based models or Schr\"odinger Bridge (SB) models have demonstrated appealing performance, yet the inferred sample trajectories either fail to account for the underlying stochasticity or are $\underline{D}$eep $\underline{M}$omentum Multi-Marginal $\underline{S}$chr\"odinger $\underline{B}$ridge(DMSB), a novel computational framework that learns the smooth measure-valued spline for stochastic systems that satisfy position marginal constraints across time. By tailoring the celebrated Bregman Iteration and extending the Iteration Proportional Fitting to phase space, we manage to handle high-dimensional multi-marginal trajectory inference tasks efficiently. Our algorithm outperforms baselines significantly, as evidenced by experiments for synthetic datasets and a real-world single-cell RNA sequence dataset. Additionally, the proposed approach can reasonably reconstruct the evolution of velocity distribution, from position snapshots only, when there is a ground truth velocity that is nevertheless inaccessible.

翻译：利用粗时间间隔下未标记样本重构群体动力学是一项关键挑战。近期基于流模型或薛定谔桥模型的方法展现出优异性能，但推断出的样本轨迹要么未能考虑潜在随机性，要么存在局限性。本文提出深度动量多边缘薛定谔桥（DMSB），这是一种新颖的计算框架，能够学习满足时间位置边缘约束的随机系统的光滑测度值样条。通过定制经典的布雷格曼迭代并将迭代比例拟合扩展至相空间，我们成功高效处理了高维多边缘轨迹推断任务。在合成数据集和真实单细胞RNA序列数据集上的实验表明，我们的算法显著优于基线方法。此外，当存在不可获取的真实速度时，所提方法仅通过位置快照即可合理重构速度演化的分布过程。