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Partially Observed Trajectory Inference using Optimal Transport and a Dynamics Prior
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Extends min-entropy trajectory inference with mean-field Langevin dynamics and Schrödinger-bridge OT to latent SDEs with a known dynamics prior (e.g. constant velocity), inferring latent trajectories from partial (position-only) observations.
Problem
Recovering latent trajectories and population dynamics from uncoupled time snapshots when only a function of the latent state is observed, so hidden states (e.g. velocity) must be inferred under a known dynamics model.
Approach
Extends the fully observed MFL/entropic-OT trajectory inference framework to latent SDEs with a known drift Xi, using a data-fit term in observation space via the observation map g and an Euler-Maruyama approximation of the Schrödinger bridge transition kernel with drift-shifted OT cost.
Key results
- On the constant velocity synthetic experiment, PO-MFL reconstructs trajectories while FO-MFL fails to converge.
- On Wikipedia traffic with m=3 particles, PO-MFL normalized W2 is 1.01±0.0395 versus 1.61±0.206 for FO-MFL.
- Empirical average y-velocity converges exponentially in the first 400 MFL iterations, consistent with the theory.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use PO-MFL with the autoregressive dynamics prior instead of FO-MFL | Trajectory inference on Wikipedia traffic, m=3 · Wikipedia traffic | normalized W2 distance: 1.61$\pm$0.206 -> 1.01$\pm $0.0395 | Helped |
| Use PO-MFL with the autoregressive dynamics prior instead of FO-MFL | Trajectory inference on Wikipedia traffic, m=6 · Wikipedia traffic | normalized W2 distance: 1.76$\pm$0.303 -> 1.07$\pm$0.0726 | Helped |
| Use PO-MFL with the autoregressive dynamics prior instead of FO-MFL | Trajectory inference on Wikipedia traffic, m=15 · Wikipedia traffic | normalized W2 distance: 1.91$\pm$0.443 -> 1.05$\pm$0.0956 | Helped |
| Add a constant-velocity dynamics prior and infer latent velocities | PO-MFL versus FO-MFL · Constant velocity synthetic experiment | trajectory reconstruction | Helped |
| Add a constant-velocity dynamics prior to distinguish crossing trajectories | PO-MFL versus FO-MFL · Crossing paths experiment | trajectory reconstruction | Helped |
Limitations
- Assumes the dynamics model Xi is known; robustness to misspecification of Xi is left to future work.
- Theory requires divergence-free drift, compact domain and technical conditions such as the torus assumption for convergence.
- Experiments are small synthetic and one real-data case study with few pages.
Experimental setup
Datasets: Wikipedia web traffic (train_2.csv, Kaggle)
Benchmarks: Wikipedia web traffic
Metrics: Wasserstein-2 distance normalized W2 distance
Baselines: FO-MFL
Builds on: FO-MFL minimum entropy estimator mean-field Langevin dynamics ensemble observability
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