Expected Batch Optimal Transport Plans and Consequences for Flow Matching
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916005330026496 |
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| author | Boïté, Samuel Delon, Julie Nadjahi, Kimia |
| author_facet | Boïté, Samuel Delon, Julie Nadjahi, Kimia |
| contents | Solving optimal transport (OT) on random minibatches is a common surrogate for exact OT in large-scale learning. In flow matching (FM), this surrogate is used to obtain OT-like couplings that can straighten probability paths and reduce numerical integration cost. Yet, the population-level coupling induced by repeated minibatch OT remains only partially understood. We formalize this coupling as the expected batch OT plan $\overlineπ_{k}$, obtained by averaging empirical OT plans over independent minibatches of size $k$. We then establish its large-batch consistency and, in the semidiscrete case relevant to generative modeling, derive rates for both the transport-cost bias and the convergence of $\overlineπ_{k}$ to the OT plan. For FM, this yields a population coupling whose induced velocity field is regular enough to define a unique flow from the source to the discrete target. We finally quantify how OT batch size interacts with numerical integration in a tractable two-atom model and in synthetic and image experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12174 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Expected Batch Optimal Transport Plans and Consequences for Flow Matching Boïté, Samuel Delon, Julie Nadjahi, Kimia Machine Learning Probability Solving optimal transport (OT) on random minibatches is a common surrogate for exact OT in large-scale learning. In flow matching (FM), this surrogate is used to obtain OT-like couplings that can straighten probability paths and reduce numerical integration cost. Yet, the population-level coupling induced by repeated minibatch OT remains only partially understood. We formalize this coupling as the expected batch OT plan $\overlineπ_{k}$, obtained by averaging empirical OT plans over independent minibatches of size $k$. We then establish its large-batch consistency and, in the semidiscrete case relevant to generative modeling, derive rates for both the transport-cost bias and the convergence of $\overlineπ_{k}$ to the OT plan. For FM, this yields a population coupling whose induced velocity field is regular enough to define a unique flow from the source to the discrete target. We finally quantify how OT batch size interacts with numerical integration in a tractable two-atom model and in synthetic and image experiments. |
| title | Expected Batch Optimal Transport Plans and Consequences for Flow Matching |
| topic | Machine Learning Probability |
| url | https://arxiv.org/abs/2605.12174 |