# Scaling Online Optimal Transport Minibatches for Flow Matching

arXiv: https://arxiv.org/abs/2610.18838 · HTML: https://mlsanity.com/paper/2610.18838/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 44 of its 58 references that are in the map.
- First posted on arXiv 13 Oct 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 44)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [SGD: General Analysis and Improved Rates](https://mlsanity.com/paper/1901.09401/) (2019)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022)
- [Wasserstein-2 Generative Networks](https://mlsanity.com/paper/1909.13082/) (2019)
- [Overparameterized Nonlinear Learning: Gradient Descent Takes the Shortest Path?](https://mlsanity.com/paper/1812.10004/) (2018)
- [Learning with minibatch Wasserstein : asymptotic and gradient properties](https://mlsanity.com/paper/1910.04091/) (2019)
- [Optimal Flow Matching: Learning Straight Trajectories in Just One Step](https://mlsanity.com/paper/2403.13117/) (2024)
- [On Efficient Optimal Transport: An Analysis of Greedy and Accelerated Mirror Descent Algorithms](https://mlsanity.com/paper/1901.06482/) (2019)
- [Continuous Wasserstein-2 Barycenter Estimation without Minimax Optimization](https://mlsanity.com/paper/2102.01752/) (2021)
- [Scalable Computations of Wasserstein Barycenter via Input Convex Neural Networks](https://mlsanity.com/paper/2007.04462/) (2020)
- [Neural Optimal Transport with General Cost Functionals](https://mlsanity.com/paper/2205.15403/) (2022)
- [Entropic Optimal Transport: Convergence of Potentials](https://mlsanity.com/paper/2104.11720/) (2021)
- [On the Complexity of Approximating Multimarginal Optimal Transport](https://mlsanity.com/paper/1910.00152/) (2019)
- [An explicit analysis of the entropic penalty in linear programming](https://mlsanity.com/paper/1806.01879/) (2018)
- [Gradients should stay on Path: Better Estimators of the Reverse- and Forward KL Divergence for Normalizing Flows](https://mlsanity.com/paper/2207.08219/) (2022)
- [Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm](https://mlsanity.com/paper/2110.06798/) (2021)
- [Flow Matching with Semidiscrete Couplings](https://mlsanity.com/paper/2509.25519/) (2025)
- [Path-Gradient Estimators for Continuous Normalizing Flows](https://mlsanity.com/paper/2206.09016/) (2022)
- [On Fitting Flow Models with Large Sinkhorn Couplings](https://mlsanity.com/paper/2506.05526/) (2025)
- [FlashSinkhorn: IO-Aware Entropic Optimal Transport on GPU](https://mlsanity.com/paper/2602.03067/) (2026)
- [Adaptive Non-uniform Timestep Sampling for Accelerating Diffusion Model Training](https://mlsanity.com/paper/2411.09998/) (2024)

## Abstract (arXiv)

A central challenge in scaling flow models is maintaining stability in training and integration. In flow matching, this stability depends critically on how source samples are paired with data. Independent random pairing introduces substantial stochasticity, motivating minibatch optimal transport (OT) to construct online source-target couplings that reduce this randomness (Tong et al., 2024). However, as applications extend to larger and higher-dimensional datasets, small minibatches can limit these benefits, while scaling the coupling to substantially larger minibatches at every iteration is computationally prohibitive. Moreover, how OT coupling affects the full stochastic gradient and integration remains theoretically unresolved. In this work, we develop a scalable online entropic OT approximation that enables substantially larger source-target couplings while preserving the benefits of OT. We close the theoretical gap for both unregularized and entropic OT by proving strict reductions in full stochastic-gradient variance and integration error relative to independent pairing, and further characterize how these improvements strengthen as the online coupling minibatch grows. Experiments on single-cell gene interpolation-flow and image-flow training validate these predictions, showing that larger online couplings consistently improve training stability and final generation performance.