Trajectory inference seeks to recover the temporal dynamics of a population from snapshots of its (uncoupled) temporal marginals, i.e. where observed particles are not tracked over time. Lavenant et al. arXiv:2102.09204 addressed this challenging problem under a stochastic differential equation (SDE) model with a gradient-driven drift in the observed space, introducing a minimum entropy estimator relative to the Wiener measure. Chizat et al. arXiv:2205.07146 then provided a practical grid-free mean-field Langevin (MFL) algorithm using Schr\"odinger bridges. Motivated by the overwhelming success of observable state space models in the traditional paired trajectory inference problem (e.g. target tracking), we extend the above framework to a class of latent SDEs in the form of observable state space models. In this setting, we use partial observations to infer trajectories in the latent space under a specified dynamics model (e.g. the constant velocity/acceleration models from target tracking). We introduce PO-MFL to solve this latent trajectory inference problem and provide theoretical guarantees by extending the results of arXiv:2102.09204 to the partially observed setting. We leverage the MFL framework of arXiv:2205.07146, yielding an algorithm based on entropic OT between dynamics-adjusted adjacent time marginals. Experiments validate the robustness of our method and the exponential convergence of the MFL dynamics, and demonstrate significant outperformance over the latent-free method of arXiv:2205.07146 in key scenarios.

翻译：轨迹推断旨在从群体（非耦合）时间边际的快照中恢复其时间动态，即观测粒子未随时间被追踪的情况。Lavenant等人（arXiv:2102.09204）在观测空间中具有梯度驱动漂移的随机微分方程（SDE）模型下解决了这一具有挑战性的问题，引入了相对于维纳测度的最小熵估计器。随后，Chizat等人（arXiv:2205.07146）利用薛定谔桥提出了一种实用的无网格平均场朗之万（MFL）算法。受可观测状态空间模型在传统配对轨迹推断问题（例如目标跟踪）中巨大成功的启发，我们将上述框架扩展至一类以可观测状态空间模型形式存在的隐式随机微分方程。在此设定下，我们利用部分观测数据，在指定的动力学模型（例如目标跟踪中的恒定速度/加速度模型）下推断隐空间中的轨迹。我们提出了PO-MFL来解决这一隐式轨迹推断问题，并通过将arXiv:2102.09204的结果扩展至部分观测场景提供了理论保证。我们利用arXiv:2205.07146的MFL框架，得到了一种基于动力学调整的相邻时间边际之间熵正则化最优传输的算法。实验验证了我们方法的鲁棒性以及MFL动态的指数收敛性，并在关键场景中证明了其相对于arXiv:2205.07146的无隐式方法具有显著优越性。