Flow Matching on Symmetric Spaces
For researchers in generative modeling on manifolds, this provides a principled and simplified method for a broad class of symmetric spaces, though it is an incremental extension of flow matching to new domains.
The paper introduces a framework for flow matching on Riemannian symmetric spaces, linearizing the problem by reformulating it on a Lie algebra subspace. Applied to real Grassmannians, it simplifies geodesic handling and enables generative modeling on these manifolds.
We introduce a general framework for training flow matching models on Riemannian symmetric spaces, 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 greatly simplifying the handling of geodesics. As an application, we showcase our framework on the real Grassmannians $\operatorname{SO}(n) / \operatorname{SO}(k) \times \operatorname{SO}(n-k)$.