HOTA: Hamiltonian framework for Optimal Transport Advection

2025-07-23

AI Summary

Existing generative models struggle to satisfy the geometric and optimality requirements of optimal transport (OT) on non-Euclidean manifolds, often relying on Euclidean assumptions and strong density estimation—leading to trajectories that deviate from true OT solutions. To address this, we propose Hamiltonian Optimal Transport Guidance (HOTA), a novel framework that explicitly solves the dual dynamic OT problem via the Hamilton–Jacobi–Bellman equation, bypassing explicit density modeling and accommodating nonsmooth cost functions. By coupling Kantorovich potentials with Hamiltonian dynamics, HOTA enables manifold-adaptive probability flow guidance and trajectory optimization. Experiments demonstrate that HOTA consistently outperforms state-of-the-art methods on standard benchmarks and datasets with nondifferentiable costs, achieving superior performance in solution feasibility, optimality, and scalability.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Planning, Routing, and Scheduling: Planning with Language Models
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Optimal transport (OT) has become a natural framework for guiding the probability flows. Yet, the majority of recent generative models assume trivial geometry (e.g., Euclidean) and rely on strong density-estimation assumptions, yielding trajectories that do not respect the true principles of optimality in the underlying manifold. We present Hamiltonian Optimal Transport Advection (HOTA), a Hamilton-Jacobi-Bellman based method that tackles the dual dynamical OT problem explicitly through Kantorovich potentials, enabling efficient and scalable trajectory optimization. Our approach effectively evades the need for explicit density modeling, performing even when the cost functionals are non-smooth. Empirically, HOTA outperforms all baselines in standard benchmarks, as well as in custom datasets with non-differentiable costs, both in terms of feasibility and optimality.

Problem

• Overcoming Euclidean geometry limitations in optimal transport
• Avoiding strong density-estimation assumptions in generative models
• Optimizing trajectories with non-smooth cost functionals efficiently

Innovation

• Hamilton-Jacobi-Bellman based dual dynamical OT
• Efficient trajectory optimization via Kantorovich potentials
• No explicit density modeling, handles non-smooth costs