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

Lifeng Hao; Shaolin Ji

cs.LGSep 30, 2026

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

Authors: Lifeng Hao, Shaolin Ji

Organizations: Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, Shandong, China.

Abstract

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 Hölder 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.

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Oct 2, 2026, stat.MLJ/K move · Enter open · S save

Generalization Bounds for Flow-matching Generative Models for Intrinsically Low-dimensional Data

Saptarshi Chakraborty, Quentin Berthet, Peter L. Bartlett

Department of Statistics, University of Michigan · Google DeepMind · Department of Statistics, University of California, Berkeley+1

Riemannian Flow Matching

Open paper AskPDF arXiv

Despite the remarkable empirical success of flow-matching models, their statistical generalization guarantees remain underdeveloped. Existing analyses often impose restrictive assumptions on the estimated velocity field and yield convergence rates that fail to reflect the intrinsic low-dimensional structure common in real data, such as natural images and molecular geometries. In this work, we study the statistical generalization of flow-matching models for learning an unknown distribution

P_{\mathrm{data}}

from finitely many samples. We derive finite-sample error bounds on the learned generative distribution, measured in the Wasserstein-

p

distance, for all

p\geq 1

. Specifically, given

n

i.i.d. samples from

P_{\mathrm{data}}

, we show that, for every

d>d_p^\ast(P_{\mathrm{data}})

and appropriately chosen network architectures and hyperparameters, the learned distribution

\widehat{P}^{\mathrm{FM}}

satisfies

\mathbb{W}_p(\widehat{P}^{\mathrm{FM}},P_{\mathrm{data}}) \lesssim n^{-1/d}+n^{-1/(2p)}\bigl(\log(1/ξ)\bigr)^{1/(2p)}

with probability at least

1-ξ

, where

d_p^\ast(P_{\mathrm{data}})

denotes the Wasserstein-

p

dimension of the target measure. Our results demonstrate that flow matching naturally adapts to the intrinsic geometry of data and mitigates the curse of dimensionality, as the convergence exponent depends on the intrinsic rather than ambient dimension. These guarantees remain meaningful in high-dimensional regimes and provide a theoretical explanation for the empirical success of flow matching on structured data distributions under substantially more relaxed assumptions than those in existing analyses.

Abstract

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