Partially Observed Trajectory Inference using Optimal Transport and a Dynamics Prior
This work addresses trajectory inference for populations in fields like target tracking, but it is incremental as it builds on prior SDE-based methods by incorporating latent dynamics.
The paper tackles the problem of inferring latent trajectories from partial observations by extending a stochastic differential equation framework to observable state space models, introducing the PO-MFL algorithm, which demonstrates significant outperformance over baselines in key scenarios.
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. Prior works 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 and a practical grid-free mean-field Langevin (MFL) algorithm using Schrödinger bridges. Motivated by the 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 the PO-MFL algorithm to solve this latent trajectory inference problem and provide theoretical guarantees to the partially observed setting. Experiments validate the robustness of our method and the exponential convergence of the MFL dynamics, and demonstrate significant outperformance over the latent-free baseline in key scenarios.