Cartan flow matching

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) $.