An Error Analysis of Flow Matching for Deep Generative Modeling

Zhengyu Zhou, Weiwei Liu
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:78903-78932, 2025.

Abstract

Continuous Normalizing Flows (CNFs) have proven to be a highly efficient technique for generative modeling of complex data since the introduction of Flow Matching (FM). The core of FM is to learn the constructed velocity fields of CNFs through deep least squares regression. Despite its empirical effectiveness, theoretical investigations of FM remain limited. In this paper, we present the first end-to-end error analysis of CNFs built upon FM. Our analysis shows that for general target distributions with bounded support, the generated distribution of FM is guaranteed to converge to the target distribution in the sense of the Wasserstein-2 distance. Furthermore, the convergence rate is significantly improved under an additional mild Lipschitz condition of the target score function.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-zhou25l, title = {An Error Analysis of Flow Matching for Deep Generative Modeling}, author = {Zhou, Zhengyu and Liu, Weiwei}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {78903--78932}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/zhou25l/zhou25l.pdf}, url = {https://proceedings.mlr.press/v267/zhou25l.html}, abstract = {Continuous Normalizing Flows (CNFs) have proven to be a highly efficient technique for generative modeling of complex data since the introduction of Flow Matching (FM). The core of FM is to learn the constructed velocity fields of CNFs through deep least squares regression. Despite its empirical effectiveness, theoretical investigations of FM remain limited. In this paper, we present the first end-to-end error analysis of CNFs built upon FM. Our analysis shows that for general target distributions with bounded support, the generated distribution of FM is guaranteed to converge to the target distribution in the sense of the Wasserstein-2 distance. Furthermore, the convergence rate is significantly improved under an additional mild Lipschitz condition of the target score function.} }
Endnote
%0 Conference Paper %T An Error Analysis of Flow Matching for Deep Generative Modeling %A Zhengyu Zhou %A Weiwei Liu %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-zhou25l %I PMLR %P 78903--78932 %U https://proceedings.mlr.press/v267/zhou25l.html %V 267 %X Continuous Normalizing Flows (CNFs) have proven to be a highly efficient technique for generative modeling of complex data since the introduction of Flow Matching (FM). The core of FM is to learn the constructed velocity fields of CNFs through deep least squares regression. Despite its empirical effectiveness, theoretical investigations of FM remain limited. In this paper, we present the first end-to-end error analysis of CNFs built upon FM. Our analysis shows that for general target distributions with bounded support, the generated distribution of FM is guaranteed to converge to the target distribution in the sense of the Wasserstein-2 distance. Furthermore, the convergence rate is significantly improved under an additional mild Lipschitz condition of the target score function.
APA
Zhou, Z. & Liu, W.. (2025). An Error Analysis of Flow Matching for Deep Generative Modeling. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:78903-78932 Available from https://proceedings.mlr.press/v267/zhou25l.html.

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