# Flow Matching under Noisy Latent Structure: Beyond Exact Low-Dimensional Support

arXiv: https://arxiv.org/abs/2609.38918 · HTML: https://mlsanity.com/paper/2609.38918/

## 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 16 of its 38 references that are in the map.
- First posted on arXiv 30 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, 16 of 16)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [A note on the evaluation of generative models](https://mlsanity.com/paper/1511.01844/) (2015)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Convergence of denoising diffusion models under the manifold hypothesis](https://mlsanity.com/paper/2208.05314/) (2022)
- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [Error Bounds for Flow Matching Methods](https://mlsanity.com/paper/2305.16860/) (2023)
- [Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions](https://mlsanity.com/paper/2409.18804/) (2024)
- [GSURE-Based Diffusion Model Training with Corrupted Data](https://mlsanity.com/paper/2305.13128/) (2023)
- [Flow matching achieves almost minimax optimal convergence](https://mlsanity.com/paper/2405.20879/) (2024)
- [Generalization error bound for denoising score matching under relaxed manifold assumption](https://mlsanity.com/paper/2502.13662/) (2025)
- [Gaussian Interpolation Flows](https://mlsanity.com/paper/2311.11475/) (2023)
- [On the minimax optimality of Flow Matching through the connection to kernel density estimation](https://mlsanity.com/paper/2504.13336/) (2025)
- [Flow Matching is Adaptive to Manifold Structures](https://mlsanity.com/paper/2602.22486/) (2026)
- [Low-Dimensional Adaptation of Rectified Flow: A Diffusion and Stochastic Localization Perspective](https://mlsanity.com/paper/2601.15500/) (2026)

## Abstract (arXiv)

Flow Matching (FM) learns a velocity field whose ODE transports a simple source distribution to a target law. Existing finite-sample theory largely treats ambient-space regularity or data supported exactly on low-dimensional sets. We study linear FM under a noisy latent-generator model, where a low-dimensional H"older map is perturbed by nondegenerate ambient Gaussian noise, so the target law is full-dimensional despite its latent structure. We construct a spatially regular ReLU velocity class and establish non-asymptotic high-probability approximation and estimation bounds whose leading sample-size exponent is governed by the latent dimension rather than the ambient dimension, with ambient and noise dependence kept explicit. Fixed positive target noise keeps the interpolation nondegenerate over the full time interval. The same spatial regularity propagates the learned velocity error through the transport ODE, yielding a corresponding Wasserstein convergence guarantee. These results show that exact low-dimensional support is not necessary for Flow Matching to retain latent-dimensional statistical behavior.