# Flow Matching with Semidiscrete Couplings

arXiv: https://arxiv.org/abs/2509.25519 · HTML: https://mlsanity.com/paper/2509.25519/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 42 of its 60 references that are in the map.
- First posted on arXiv 29 Sep 2025.

## 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 42)

- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [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)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Improved Precision and Recall Metric for Assessing Generative Models](https://mlsanity.com/paper/1904.06991/) (2019)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Open-Sora: Democratizing Efficient Video Production for All](https://mlsanity.com/paper/2412.20404/) (2024)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Improved Techniques for Training Consistency Models](https://mlsanity.com/paper/2310.14189/) (2023)
- [InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generation](https://mlsanity.com/paper/2309.06380/) (2023)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Guiding a Diffusion Model with a Bad Version of Itself](https://mlsanity.com/paper/2406.02507/) (2024)
- [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Computational Optimal Transport: Complexity by Accelerated Gradient Descent Is Better Than by Sinkhorn's Algorithm](https://mlsanity.com/paper/1802.04367/) (2018)
- [Asymmetric LSH (ALSH) for Sublinear Time Maximum Inner Product Search (MIPS)](https://mlsanity.com/paper/1405.5869/) (2014)
- [Flow map matching with stochastic interpolants: A mathematical framework for consistency models](https://mlsanity.com/paper/2406.07507/) (2024)
- [Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts](https://mlsanity.com/paper/2503.02819/) (2025)
- [Minimizing Trajectory Curvature of ODE-based Generative Models](https://mlsanity.com/paper/2301.12003/) (2023)
- [Align Your Flow: Scaling Continuous-Time Flow Map Distillation](https://mlsanity.com/paper/2506.14603/) (2025)
- [Classifier-Free Guidance is a Predictor-Corrector](https://mlsanity.com/paper/2408.09000/) (2024)

## Built on by (most cited first, 13 of 13)

- [Test-time scaling of diffusions with flow maps](https://mlsanity.com/paper/2511.22688/) (2025)
- [Diffusion Models in Simulation-Based Inference: A Tutorial Review](https://mlsanity.com/paper/2512.20685/) (2025)
- [Counterfactual Identifiability via Dynamic Optimal Transport](https://mlsanity.com/paper/2510.08294/) (2025)
- [The Coupling Within: Flow Matching via Distilled Normalizing Flows](https://mlsanity.com/paper/2603.09014/) (2026)
- [Generalized Wasserstein Flow Matching: Transport Plans, Everywhere, All at Once](https://mlsanity.com/paper/2605.08424/) (2026)
- [Optimal Transport Flow Matching by Design](https://mlsanity.com/paper/2606.04092/) (2026)
- [Reinforcement Learning for Flow-Matching Policies with Density Transport](https://mlsanity.com/paper/2606.08602/) (2026)
- [Quantifying the noise sensitivity of the Wasserstein metric for images](https://mlsanity.com/paper/2510.01015/) (2025)
- [Generative Modeling of Discrete Data Using Geometric Latent Subspaces](https://mlsanity.com/paper/2601.21831/) (2026)
- [Gromov-Monge Flow Matching for Equivariant Graph Generation](https://mlsanity.com/paper/2608.26961/) (2026)
- [Beyond Straightness: Non-Crossing Flow Matching via Quantile AlignTree Coupling](https://mlsanity.com/paper/2609.03443/) (2026)
- [Dual-guided Hierarchical Edge Localization for Large-scale Optimal Transport Across Dimensions](https://mlsanity.com/paper/2609.13010/) (2026)
- [Scaling Online Optimal Transport Minibatches for Flow Matching](https://mlsanity.com/paper/2610.18838/) (2026)

## Abstract (arXiv)

Flow models parameterized as time-dependent velocity fields can generate data from noise by integrating an ODE. These models are often trained using flow matching, i.e. by sampling random pairs of noise and target points (x0,x1) and ensuring that the velocity field is aligned, on average, with x1-x0 when evaluated along a segment linking x0 to x1. While these pairs are sampled independently by default, they can also be selected more carefully by matching batches of n noise to n target points using an optimal transport (OT) solver. Although promising in theory, the OT flow matching (OT-FM) approach is not widely used in practice. Zhang et al. (2025) pointed out recently that OT-FM truly starts paying off when the batch size n grows significantly, which only a multi-GPU implementation of the Sinkhorn algorithm can handle. Unfortunately, the costs of running Sinkhorn can quickly balloon, requiring O(n2/varepsilon2) operations for every n pairs used to fit the velocity field, where varepsilon is a regularization parameter that should be typically small to yield better results. To fulfill the theoretical promises of OT-FM, we propose to move away from batch-OT and rely instead on a semidiscrete formulation that leverages the fact that the target dataset distribution is usually of finite size N. The SD-OT problem is solved by estimating a dual potential vector using SGD; using that vector, freshly sampled noise vectors at train time can then be matched with data points at the cost of a maximum inner product search (MIPS). Semidiscrete FM (SD-FM) removes the quadratic dependency on n/varepsilon that bottlenecks OT-FM. SD-FM beats both FM and OT-FM on all training metrics and inference budget constraints, across multiple datasets, on unconditional/conditional generation, or when using mean-flow models.