Learning Manifold Data with Flow Matching

Sophia Pi, Mingcheng Lu, Maojiang Su, Weimin Wu, Jerry Yao-Chieh Hu, Han Liu
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:98666-98695, 2026.

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.

Cite this Paper

BibTeX
@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.} }
Endnote
%0 Conference Paper %T Learning Manifold Data with Flow Matching %A Sophia Pi %A Mingcheng Lu %A Maojiang Su %A Weimin Wu %A Jerry Yao-Chieh Hu %A Han Liu %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-pi26a %I PMLR %P 98666--98695 %U https://proceedings.mlr.press/v306/pi26a.html %V 306 %X 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.
APA
Pi, S., Lu, M., Su, M., Wu, W., Hu, J.Y. & Liu, H.. (2026). Learning Manifold Data with Flow Matching. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:98666-98695 Available from https://proceedings.mlr.press/v306/pi26a.html.

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