Flow Matching on Symmetric Spaces

• URL: http://arxiv.org/abs/2605.03588v1
• Date: Tue, 05 May 2026 10:03:12 GMT
• Title: Flow Matching on Symmetric Spaces
• Abstract summary: We reformulate flow matching on symmetric spaces as flow matching on a subspace of the Lie algebra of their isometry group.<n>As an application, we showcase our framework on the real Grassmannians.
• Score: 29.28717229244793
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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 $\operatorname{SO}(n) / \operatorname{SO}(k) \times \operatorname{SO}(n-k)$.

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