# Optimal Transport Flow Matching by Design

arXiv: https://arxiv.org/abs/2606.04092 · HTML: https://mlsanity.com/paper/2606.04092/

## Key facts

- Cited by 2 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 46 of its 57 references that are in the map.
- First posted on arXiv 2 Jun 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 46)

- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [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)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [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)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Cascaded Diffusion Models for High Fidelity Image Generation](https://mlsanity.com/paper/2106.15282/) (2021)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [Movie Gen: A Cast of Media Foundation Models](https://mlsanity.com/paper/2410.13720/) (2024)
- [AudioLM: a Language Modeling Approach to Audio Generation](https://mlsanity.com/paper/2209.03143/) (2022)
- [LTX-Video: Realtime Video Latent Diffusion](https://mlsanity.com/paper/2501.00103/) (2024)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024)
- [Voicebox: Text-Guided Multilingual Universal Speech Generation at Scale](https://mlsanity.com/paper/2306.15687/) (2023)
- [Pyramidal Flow Matching for Efficient Video Generative Modeling](https://mlsanity.com/paper/2410.05954/) (2024)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise](https://mlsanity.com/paper/2208.09392/) (2022)
- [Matcha-TTS: A fast TTS architecture with conditional flow matching](https://mlsanity.com/paper/2309.03199/) (2023)
- [AlphaFold Meets Flow Matching for Generating Protein Ensembles](https://mlsanity.com/paper/2402.04845/) (2024)
- [Consistency Flow Matching: Defining Straight Flows with Velocity Consistency](https://mlsanity.com/paper/2407.02398/) (2024)
- [Soft Diffusion: Score Matching for General Corruptions](https://mlsanity.com/paper/2209.05442/) (2022)

## Built on by (most cited first, 2 of 2)

- [One-Sided Quantile Coupling for Flow Matching](https://mlsanity.com/paper/2608.00978/) (2026)
- [Global Transport Couplings for Classifier-Free Guided Flows](https://mlsanity.com/paper/2610.07555/) (2026)

## Abstract (arXiv)

Flow matching models learn to transport samples from a simple prior distribution to a complex data distribution. When prior-data pairs are coupled via optimal transport (OT), the learned trajectories are straight and non-crossing, enabling fast, even single-step, generation. However, computing the OT coupling in high dimensions is intractable, and existing methods attempt to solve the OT problem, at the cost of persistent bias or significant overhead. Rather than solving for the OT coupling, we reformulate the problem. Once the prior is treated as a design choice rather than a fixed input, the OT coupling between prior and data is no longer unique. Many priors admit an OT-optimal identity coupling to the data, leaving us free to choose one that is also tractable to sample. We identify low-frequency projection of natural images as such a choice. The identity coupling between data and its low-frequency representation is empirically OT-optimal, the prior is structured enough to be sampled by a lightweight model at inference, and the remaining flow-matching task reduces to synthesizing high-frequency detail. Interpolating the prior with Gaussian noise further improves generation quality while preserving the OT coupling. The approach requires no modifications to the flow model itself, and integrates naturally with latent-space models, classifier-free guidance, and one-step generation frameworks. Across all benchmarks, our method reduces trajectory curvature by more than 2× compared to existing flow matching methods, yielding better generation quality in the few-step regime.