Partially Observed Trajectory Inference using Optimal Transport and a Dynamics Prior

2024-06-11

arXiv.org

AI Summary

This work addresses latent state trajectory inference under partial observability—crucial for applications such as single-cell developmental biology and target tracking where continuous observation is infeasible. We propose PO-MFL, the first method to integrate observable state-space modeling into partially observed trajectory estimation. PO-MFL unifies optimal transport, the Schrödinger bridge framework, implicit stochastic differential equations (SDEs), and physics-informed dynamical priors (e.g., constant-velocity or constant-acceleration models) within a mesh-free mean-field Langevin optimization framework, jointly learning latent dynamics and observation mappings. We establish theoretical convergence guarantees for the algorithm. Experiments demonstrate that PO-MFL significantly outperforms baseline methods lacking latent variables in both robustness and dynamic reconstruction accuracy. Moreover, the learned MFL dynamics exhibit exponential convergence—a theoretically grounded property enabling stable long-term trajectory estimation.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

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 Schr""odinger 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.

Problem

• Infer latent trajectories from partial observations
• Extend SDE framework to observable state space models
• Propose PO-MFL algorithm with theoretical guarantees

Innovation

• Optimal Transport
• Latent SDEs
• PO-MFL Algorithm