将流匹配推广到黎曼对称空间，解决几何路径构建难题。
Cartan flow matching
- 利用李群对称性将流匹配转化为李代数上的线性问题。
- 在实格拉斯曼流形上实现高效训练，避免复杂测地线插值。
- 适合几何深度学习、流模型研究者参考。
我们提出卡坦流匹配（Cartan flow matching），一种在黎曼对称空间上训练流匹配模型的通用框架。这类空间具有在任意点存在测地对称性的性质，包含球面、双曲空间和格拉斯曼流形等。通过利用其代数结构，我们将对称空间上的流匹配问题转化为其同构群李代数子空间上的流匹配，实现了问题的线性化，无需在流形上构造测地线插值路径。作为应用，我们在实格拉斯曼流形 $\operatorname{SO}(n) / \operatorname{SO}(k) \times \operatorname{SO}(n-k)$ 上展示了该框架的有效性。
原文摘要 · Abstract (English)
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) $.
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