Learning Paths for Dynamic Measure Transport: A Control Perspective

2025-11-05

AI Summary

This paper addresses the suboptimal performance of conventional metric paths in Dynamic Measure Transport (DMT). Methodologically, it formulates path optimization as a constrained optimal control problem, jointly learning both the tilted metric path and the associated velocity field, while incorporating a smoothness regularizer to enhance numerical stability. Theoretically, it establishes an intrinsic connection between DMT and mean-field games, providing a principled foundation for the proposed framework. Algorithmically, Gaussian processes are employed to efficiently solve the resulting partial differential equations governing the transport dynamics. Experiments demonstrate that the learned paths significantly improve sampling efficiency, convergence speed, and generative quality over standard reference paths—thereby empirically validating the critical role of velocity field smoothness in transport performance.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning with Manifolds
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success

Abstract

We bring a control perspective to the problem of identifying paths of measures for sampling via dynamic measure transport (DMT). We highlight the fact that commonly used paths may be poor choices for DMT and connect existing methods for learning alternate paths to mean-field games. Based on these connections we pose a flexible family of optimization problems for identifying tilted paths of measures for DMT and advocate for the use of objective terms which encourage smoothness of the corresponding velocities. We present a numerical algorithm for solving these problems based on recent Gaussian process methods for solution of partial differential equations and demonstrate the ability of our method to recover more efficient and smooth transport models compared to those which use an untilted reference path.

Problem

• Identifying optimal measure paths for dynamic transport sampling
• Connecting existing path learning methods to mean-field games
• Developing smooth velocity optimization for tilted transport paths

Innovation

• Optimizing tilted measure paths via control theory
• Using Gaussian processes for solving PDEs numerically
• Encouraging velocity smoothness in transport models