  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.
