HOTA: Hamiltonian framework for Optimal Transport Advection
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911164737257472 |
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| author | Buzun, Nazar Shlenskii, Daniil Bobrin, Maxim Dylov, Dmitry V. |
| author_facet | Buzun, Nazar Shlenskii, Daniil Bobrin, Maxim Dylov, Dmitry V. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17513 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | HOTA: Hamiltonian framework for Optimal Transport Advection Buzun, Nazar Shlenskii, Daniil Bobrin, Maxim Dylov, Dmitry V. Machine Learning Artificial Intelligence 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. |
| title | HOTA: Hamiltonian framework for Optimal Transport Advection |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2507.17513 |