# Stable Velocity: A Variance Perspective on Flow Matching

arXiv: https://arxiv.org/abs/2602.05435 · HTML: https://mlsanity.com/paper/2602.05435/

## Key facts

- Cited by 12 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 48 of its 67 references that are in the map.
- First posted on arXiv 5 Feb 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 48)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [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)
- [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)
- [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)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [GenEval: An Object-Focused Framework for Evaluating Text-to-Image Alignment](https://mlsanity.com/paper/2310.11513/) (2023)
- [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)
- [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Representation Alignment for Generation: Training Diffusion Transformers Is Easier Than You Think](https://mlsanity.com/paper/2410.06940/) (2024)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [UniPC: A Unified Predictor-Corrector Framework for Fast Sampling of Diffusion Models](https://mlsanity.com/paper/2302.04867/) (2023)
- [Perception Prioritized Training of Diffusion Models](https://mlsanity.com/paper/2204.00227/) (2022)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Generating Images with Sparse Representations](https://mlsanity.com/paper/2103.03841/) (2021)

## Built on by (most cited first, 12 of 12)

- [Hyper-DP3: Frequency-Aware Right-Sizing of 3D Diffusion Policies for Visuomotor Control](https://mlsanity.com/paper/2605.01581/) (2026)
- [StreamEdit: Training-Free Video Editing via Few-Step Streaming Video Generation](https://mlsanity.com/paper/2605.21466/) (2026)
- [NeuroSonic: Conditional Flow Matching for EEG-to-Speech Reconstruction](https://mlsanity.com/paper/2606.24087/) (2026)
- [On Variance Reduction in Learning Mean Flows](https://mlsanity.com/paper/2605.09235/) (2026)
- [Flow-Map Distillation on Relation Manifolds for Image Restoration](https://mlsanity.com/paper/2608.05769/) (2026)
- [Same Flow, Different Paths: Variance Reduction in Flow Matching](https://mlsanity.com/paper/2609.17287/) (2026)
- [AnyStep-WAM: Budget-Aligned Distillation and Adaptive Inference for World Action Models](https://mlsanity.com/paper/2609.33748/) (2026)
- [What Does a Stream Model Buy You in Flow Matching?](https://mlsanity.com/paper/2609.34123/) (2026)
- [Scaffold Then Internalize: Representation Injection for Diffusion Transformers](https://mlsanity.com/paper/2609.35292/) (2026)
- [Just Align bmx: Aligning Predictions, Not Representations](https://mlsanity.com/paper/2610.00600/) (2026)
- [Consistent Distribution Matching for Data-Free Diffusion Distillation](https://mlsanity.com/paper/2610.09221/) (2026)
- [Posterior-Averaged Bellman Flow Matching for Distributional Critics](https://mlsanity.com/paper/2610.17788/) (2026)

## Abstract (arXiv)

While flow matching is elegant, its reliance on single-sample conditional velocities leads to high-variance training targets that destabilize optimization and slow convergence. By explicitly characterizing this variance, we identify 1) a high-variance regime near the prior, where optimization is challenging, and 2) a low-variance regime near the data distribution, where conditional and marginal velocities nearly coincide. Leveraging this insight, we propose Stable Velocity, a unified framework that improves both training and sampling. For training, we introduce Stable Velocity Matching (StableVM), an unbiased variance-reduction objective, along with Variance-Aware Representation Alignment (VA-REPA), which adaptively strengthen auxiliary supervision in the low-variance regime. For inference, we show that dynamics in the low-variance regime admit closed-form simplifications, enabling Stable Velocity Sampling (StableVS), a finetuning-free acceleration. Extensive experiments on ImageNet 256×256 and large pretrained text-to-image and text-to-video models, including SD3.5, Flux, Qwen-Image, and Wan2.2, demonstrate consistent improvements in training efficiency and more than 2× faster sampling within the low-variance regime without degrading sample quality. Our code is available at https://github.com/linYDTHU/StableVelocity.