Expected Batch Optimal Transport Plans and Consequences for Flow Matching

• URL: http://arxiv.org/abs/2605.12174v1
• Date: Tue, 12 May 2026 14:19:09 GMT
• Title: Expected Batch Optimal Transport Plans and Consequences for Flow Matching
• Abstract summary: Solving optimal transport on random minibatches is a common surrogate for exact OT in large-scale learning.<n>We formalize this coupling as the expected batch OT plan $overline_k$, obtained by averaging empirical OT plans over independent minibatches of size $k$.<n>We quantify how OT batch size interacts with numerical integration in a tractable two-atom model and in synthetic and image experiments.
• Score: 6.30004266133706
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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