Generalization Bounds for Flow-matching Generative Models for Intrinsically Low-dimensional Data
Abstract
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 Pdata from finitely many samples. We derive finite-sample error bounds on the learned generative distribution, measured in the Wasserstein-p distance, for all p≥ 1. Specifically, given n i.i.d. samples from Pdata, we show that, for every d>dp∗(Pdata) and appropriately chosen network architectures and hyperparameters, the learned distribution P̂FM 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 dp∗(Pdata) 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.
Related
- From Privacy to Generalization: Linear Max-Information Bounds for Differentially Private Learning Algorithms
- How Many Independent Samples Does a Satellite Image Contain? Generalization Bounds for Spatially Dependent Data
- Residual-Information Generalization Bounds Under Selective Privacy: Extended Version
- A Short Proof and Generalization of the Rainbow Cycle Theorem and Improved Bounds for Binary Even-Query LCCs
- Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs
BibTeX
@misc{chakraborty2026generalization,
title = {{Generalization Bounds for Flow-matching Generative Models for Intrinsically Low-dimensional Data}},
author = {Saptarshi Chakraborty and Quentin Berthet and Peter L. Bartlett},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02663},
eprint = {2610.02663},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02663},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Generalization Bounds for Flow-matching Generative Models for Intrinsically Low-dimensional Data
AU - Saptarshi Chakraborty
AU - Quentin Berthet
AU - Peter L. Bartlett
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.02663
UR - https://arxiv.org/abs/2610.02663
AB - 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_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≥ 1. Specifically, given n i.i.d. samples from P_data, we show that, for every d>d_p^∗(P_data) and appropriately chosen network architectures and hyperparameters, the learned distribution P̂^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^∗(P_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.
ER -