HOTA: Hamiltonian framework for Optimal Transport Advection

TL;DR

HOTA introduces a Hamiltonian framework for optimal transport that improves trajectory optimization in generative models, especially on complex manifolds, without relying on density estimation or smooth cost functions.

Contribution

It presents a scalable Hamilton-Jacobi-Bellman based method for dual dynamical OT, explicitly solving the problem via Kantorovich potentials, handling non-smooth costs effectively.

Findings

Outperforms baselines on standard benchmarks.

Handles non-differentiable cost functions effectively.

Demonstrates improved feasibility and optimality in trajectory generation.

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.