# Same Flow, Different Paths: Variance Reduction in Flow Matching

arXiv: https://arxiv.org/abs/2609.17287 · HTML: https://mlsanity.com/paper/2609.17287/

## 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 21 of its 31 references that are in the map.
- First posted on arXiv 15 Sep 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, 21 of 21)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
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- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [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)
- [Wan: Open and Advanced Large-Scale Video Generative Models](https://mlsanity.com/paper/2503.20314/) (2025)
- [StarGAN v2: Diverse Image Synthesis for Multiple Domains](https://mlsanity.com/paper/1912.01865/) (2019)
- [Qwen-Image Technical Report](https://mlsanity.com/paper/2508.02324/) (2025)
- [Improved Precision and Recall Metric for Assessing Generative Models](https://mlsanity.com/paper/1904.06991/) (2019)
- [WorldSimBench: Towards Video Generation Models as World Simulators](https://mlsanity.com/paper/2410.18072/) (2024)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [The Principles of Diffusion Models](https://mlsanity.com/paper/2510.21890/) (2025)
- [Kernel Conditional Moment Test via Maximum Moment Restriction](https://mlsanity.com/paper/2002.09225/) (2020)
- [Instrumental Variable Regression via Kernel Maximum Moment Loss](https://mlsanity.com/paper/2010.07684/) (2020)
- [Stable Velocity: A Variance Perspective on Flow Matching](https://mlsanity.com/paper/2602.05435/) (2026)
- [Adaptive Primal-Dual Stochastic Gradient Method for Expectation-constrained Convex Stochastic Programs](https://mlsanity.com/paper/2012.14943/) (2020)
- [Temporal Pair Consistency for Variance-Reduced Flow Matching](https://mlsanity.com/paper/2602.04908/) (2026)
- [Preconditioned Flow Matching](https://mlsanity.com/paper/2603.02337/) (2026)

## Abstract (arXiv)

In flow matching (FM), a velocity model vθ is trained using a predefined path gt that connects data and noise samples (e.g., gt(x0, x1) = (1 - t) x0 + t x1). In this work, we study the choice of this path from an optimization perspective by analyzing the variance of stochastic gradients. We consider the class G(pt,v^start) of paths that induce the same marginal distributions pt and marginal velocity field v^start, and therefore the same FM objective. Our main finding is that the choice of path gt can fundamentally change the convergence rate of SGD, even when the FM objective remains exactly the same. (i) For a linear velocity model and one-dimensional Gaussian data, we derive a tight bound on the SGD iteration complexity up to logarithmic factors and find an analytically optimal path that minimizes this bound among linear paths inducing the same FM problem. (ii) We then extend the variance analysis to general FM problems and formulate path selection at a fixed θ as the variance-minimization problem PathOptθ, constrained to gt∈ G(pt,v^start). We show that this constraint is essential: reducing variance without it can lead to slower convergence. (iii) Since the constraint gt ∈ G(pt,v^start) cannot generally be verified directly, we derive an equivalent formulation with constraints that can be estimated from samples, allowing paths to be found numerically. Our theoretical results are supported by experiments with Gaussian data, Gaussian mixture models, and real datasets.