Flow Matching on Symmetric Spaces

2026-05-05

AI Summary

This work extends flow matching models to general Riemannian symmetric spaces—such as spheres, hyperbolic spaces, and Grassmann manifolds—for the first time. By leveraging the algebraic structure inherent in symmetric spaces, the flow matching problem is reformulated as a linear problem on a subspace of the Lie algebra of the isometry group, thereby simplifying geodesic computations and enabling efficient generative modeling. This approach provides a unified framework for handling diverse non-Euclidean geometries. The authors validate the efficacy and generality of their method on the real Grassmann manifold $\operatorname{SO}(n)/\operatorname{SO}(k) \times \operatorname{SO}(n-k)$, establishing a novel paradigm for generative modeling in non-Euclidean settings.

Abstract

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 $\operatorname{SO}(n) / \operatorname{SO}(k) \times \operatorname{SO}(n-k)$.

Problem

• flow matching
• symmetric spaces
• Riemannian manifolds
• Grassmannians
• geodesics

Innovation

• flow matching
• symmetric spaces
• Lie algebra
• Riemannian manifolds
• geodesics