Cartan flow matching

Abstract:We introduce Cartan flow matching, a general framework for training flow matching models on Riemannian symmetric spaces, i.e. Riemannian manifolds with the property that at any point there exists a geodesic symmetry. This is a large class of manifolds that includes the sphere, hyperbolic space and Grassmannians. We exploit their algebraic structure to reformulate flow matching on symmetric spaces as flow matching on a subspace of the Lie algebra of their isometry group, thus linearizing the problem and avoiding the need to construct geodesic interpolation paths on the manifold. As an application, we showcase our framework on the real Grassmannians $ \operatorname{SO}(n) / \operatorname{SO}(k) \times \operatorname{SO}(n-k) $.

Major revision. Fixed a mistake in section 4. 14 pages, 1 figure

Machine Learning (cs.LG); Artificial Intelligence (cs.AI)

arXiv:2605.03588 [cs.LG]

(or arXiv:2605.03588v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2605.03588 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.