Geometry-Grounded Flow Matching on Compact Manifolds

Ali Baheri
Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling, PMLR 326:32-44, 2026.

Abstract

Riemannian Flow Matching extends Flow Matching generative modeling to data that lives on curved spaces such as spheres and tori by learning a time-dependent vector field and generating samples through ordinary differential equation integration. This paper provides an end-to-end theoretical guaranty for the standard Riemannian Flow Matching pipeline on compact manifolds. Our analysis separates three sources of error: the statistical error from learning the conditional-mean velocity field produced by conditional flow matching, the approximation and optimization error arising from the chosen function class and empirical risk minimization, and the discretization error introduced by the numerical ODE solver. A key technical contribution is a flow-to-distribution stability result that is robust to geometry: curvature and injectivity radius influence only constants under standard boundedness and Lipschitz regularity conditions. Under metric-entropy assumptions, the learning rate is governed by the intrinsic manifold dimension rather than any ambient embedding dimension. Experiments on the circle, the sphere, and the two-torus support the predicted scaling behavior.

Cite this Paper

BibTeX
@InProceedings{pmlr-v326-baheri26a, title = {{G}eometry-{G}rounded {F}low {M}atching on {C}ompact {M}anifolds}, author = {Baheri, Ali}, booktitle = {Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling}, pages = {32--44}, year = {2026}, editor = {Pouplin, Alison and Vadgama, Sharvaree and Bekkers, Erik and Kaba, Sékou-Oumar and Lawrence, Hannah and Lecha, Manuel and Baker, Elizabeth and Suk, Julian and Walters, Robin and Tomczak, Jakub and Jegelka, Stefanie}, volume = {326}, series = {Proceedings of Machine Learning Research}, month = {26 Apr}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v326/main/assets/baheri26a/baheri26a.pdf}, url = {https://proceedings.mlr.press/v326/baheri26a.html}, abstract = {Riemannian Flow Matching extends Flow Matching generative modeling to data that lives on curved spaces such as spheres and tori by learning a time-dependent vector field and generating samples through ordinary differential equation integration. This paper provides an end-to-end theoretical guaranty for the standard Riemannian Flow Matching pipeline on compact manifolds. Our analysis separates three sources of error: the statistical error from learning the conditional-mean velocity field produced by conditional flow matching, the approximation and optimization error arising from the chosen function class and empirical risk minimization, and the discretization error introduced by the numerical ODE solver. A key technical contribution is a flow-to-distribution stability result that is robust to geometry: curvature and injectivity radius influence only constants under standard boundedness and Lipschitz regularity conditions. Under metric-entropy assumptions, the learning rate is governed by the intrinsic manifold dimension rather than any ambient embedding dimension. Experiments on the circle, the sphere, and the two-torus support the predicted scaling behavior.} }
Endnote
%0 Conference Paper %T Geometry-Grounded Flow Matching on Compact Manifolds %A Ali Baheri %B Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling %C Proceedings of Machine Learning Research %D 2026 %E Alison Pouplin %E Sharvaree Vadgama %E Erik Bekkers %E Sékou-Oumar Kaba %E Hannah Lawrence %E Manuel Lecha %E Elizabeth Baker %E Julian Suk %E Robin Walters %E Jakub Tomczak %E Stefanie Jegelka %F pmlr-v326-baheri26a %I PMLR %P 32--44 %U https://proceedings.mlr.press/v326/baheri26a.html %V 326 %X Riemannian Flow Matching extends Flow Matching generative modeling to data that lives on curved spaces such as spheres and tori by learning a time-dependent vector field and generating samples through ordinary differential equation integration. This paper provides an end-to-end theoretical guaranty for the standard Riemannian Flow Matching pipeline on compact manifolds. Our analysis separates three sources of error: the statistical error from learning the conditional-mean velocity field produced by conditional flow matching, the approximation and optimization error arising from the chosen function class and empirical risk minimization, and the discretization error introduced by the numerical ODE solver. A key technical contribution is a flow-to-distribution stability result that is robust to geometry: curvature and injectivity radius influence only constants under standard boundedness and Lipschitz regularity conditions. Under metric-entropy assumptions, the learning rate is governed by the intrinsic manifold dimension rather than any ambient embedding dimension. Experiments on the circle, the sphere, and the two-torus support the predicted scaling behavior.
APA
Baheri, A.. (2026). Geometry-Grounded Flow Matching on Compact Manifolds. Proceedings of GRaM: the Second Edition of the Workshop on Geometry-grounded Representation Learning and Generative Modeling, in Proceedings of Machine Learning Research 326:32-44 Available from https://proceedings.mlr.press/v326/baheri26a.html.

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