HOTA: Hamiltonian framework for Optimal Transport Advection
This addresses the challenge of scalable and efficient trajectory optimization in generative models for researchers and practitioners, though it appears incremental as it builds on existing OT frameworks.
The paper tackles the problem of generating optimal transport trajectories that respect true optimality principles on manifolds, presenting HOTA, a Hamilton-Jacobi-Bellman method that outperforms baselines in benchmarks and non-differentiable cost datasets.
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.