ICML 2026
Learning Manifold Data with Flow Matching
Flow-matching theory that follows the data's intrinsic dimension, rather than the ambient space.
Abstract
We study the statistical efficiency of flow-matching transformers when data lie in a low-dimensional linear subspace. The key idea is to decompose the velocity field into motion along the data subspace and motion toward it. This separates the nonlinear learning problem from a simple normal contraction. We derive guarantees for velocity approximation, velocity estimation, and distribution estimation whose statistical rates depend on the intrinsic dimension. The decomposition also extends to higher-order flow matching. These results explain how a suitable model structure can exploit low-dimensional data and avoid the ambient-dimensional rates of a full-space analysis.
Key results
Separate tangent and normal motion
Decompose the velocity field under a linear latent-subspace assumption to isolate the part of the dynamics that must be learned.
Intrinsic-dimension rates
Establish approximation, estimation, and Wasserstein distribution guarantees that reflect low-dimensional data structure.
Beyond first-order flows
Extend the velocity decomposition to higher-order flow matching and obtain corresponding intrinsic-dimension statistical rates.
BibTeX
Official BibTeX from PMLR.
@InProceedings{pmlr-v306-pi26a,
title = {Learning Manifold Data with Flow Matching},
author = {Pi, Sophia and Lu, Mingcheng and Su, Maojiang and Wu, Weimin and Hu, Jerry Yao-Chieh and Liu, Han},
booktitle = {Proceedings of the 43rd International Conference on Machine Learning},
pages = {98666--98695},
year = {2026},
editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto},
volume = {306},
series = {Proceedings of Machine Learning Research},
month = {06--11 Jul},
publisher = {PMLR},
pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/pi26a/pi26a.pdf},
url = {https://proceedings.mlr.press/v306/pi26a.html},
abstract = {We study flow-matching transformers when data lie on a low-dimensional manifold. Our key insight is a flow decomposition that splits motion along the manifold from motion off the manifold. The scheme works for first and higher-order flow matching and ties model complexity to the intrinsic manifold dimension. Building on these, we establish tighter sample-complexity bounds for velocity approximation, velocity estimation, and distribution estimation. Our results show how flow-matching transformers escape the curse of dimensionality by utilizing intrinsic data structure.}
}