# Demystifying Transition Matching: When and Why It Can Beat Flow Matching

arXiv: https://arxiv.org/abs/2510.17991 · HTML: https://mlsanity.com/paper/2510.17991/

## 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 35 of its 52 references that are in the map.
- First posted on arXiv 20 Oct 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 35)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
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- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Wan: Open and Advanced Large-Scale Video Generative Models](https://mlsanity.com/paper/2503.20314/) (2025)
- [HunyuanVideo: A Systematic Framework For Large Video Generative Models](https://mlsanity.com/paper/2412.03603/) (2024)
- [Towards Accurate Generative Models of Video: A New Metric & Challenges](https://mlsanity.com/paper/1812.01717/) (2018)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Structured 3D Latents for Scalable and Versatile 3D Generation](https://mlsanity.com/paper/2412.01506/) (2024)
- [Autoregressive Image Generation without Vector Quantization](https://mlsanity.com/paper/2406.11838/) (2024)
- [Discrete Flow Matching](https://mlsanity.com/paper/2407.15595/) (2024)
- [TripoSG: High-Fidelity 3D Shape Synthesis using Large-Scale Rectified Flow Models](https://mlsanity.com/paper/2502.06608/) (2025)
- [History-Guided Video Diffusion](https://mlsanity.com/paper/2502.06764/) (2025)
- [Dynamical Regimes of Diffusion Models](https://mlsanity.com/paper/2402.18491/) (2024)
- [FlowMM: Generating Materials with Riemannian Flow Matching](https://mlsanity.com/paper/2406.04713/) (2024)
- [Generator Matching: Generative modeling with arbitrary Markov processes](https://mlsanity.com/paper/2410.20587/) (2024)
- [Goku: Flow Based Video Generative Foundation Models](https://mlsanity.com/paper/2502.04896/) (2025)
- [Monte Carlo guided Diffusion for Bayesian linear inverse problems](https://mlsanity.com/paper/2308.07983/) (2023)
- [Proteina: Scaling Flow-based Protein Structure Generative Models](https://mlsanity.com/paper/2503.00710/) (2025)
- [On the Closed-Form of Flow Matching: Generalization Does Not Arise from Target Stochasticity](https://mlsanity.com/paper/2506.03719/) (2025)
- [Closed-Form Diffusion Models](https://mlsanity.com/paper/2310.12395/) (2023)

## Abstract (arXiv)

Flow Matching (FM) underpins many state-of-the-art generative models, yet recent results indicate that Transition Matching (TM) can achieve higher quality with fewer sampling steps. This work answers the question of when and why TM outperforms FM. First, when the target is a unimodal Gaussian distribution, we prove that TM attains strictly lower KL divergence than FM for finite number of steps. The improvement arises from stochastic difference latent updates in TM, which preserve target covariance that deterministic FM underestimates. We then characterize convergence rates, showing that TM achieves faster convergence than FM under a fixed compute budget, establishing its advantage in the unimodal Gaussian setting. Second, we extend the analysis to Gaussian mixtures and identify local-unimodality regimes in which the sampling dynamics approximate the unimodal case, where TM can outperform FM. The approximation error decreases as the minimal distance between component means increases, highlighting that TM is favored when the modes are well separated. However, when the target variance approaches zero, each TM update converges to the FM update, and the performance advantage of TM diminishes. In summary, we show that TM outperforms FM when the target distribution has well-separated modes and non-negligible variances. We validate our theoretical results with controlled experiments on Gaussian distributions, and extend the comparison to real-world applications in image and video generation.