将流匹配推广到齐性空间，通过李群提升实现更高效建模。
Flow matching on homogeneous spaces
- 通过数据分布提升到李群，将齐性空间问题转化为李代数上的欧氏流匹配。
- 无需定义预度量或测地线，计算更简单且完全内在化。
- 适用于需要几何不变性的生成建模任务，如3D形状生成。
我们提出一种通用框架，将流匹配扩展至齐性空间，即李群的商空间。该方法通过提升数据分布，将问题重新表述为底层李群上的流匹配任务。这一策略绕开了齐性空间复杂的几何结构，转而在李群上直接处理，进而将问题简化为李代数上的欧氏流匹配。与黎曼流匹配不同，本方法无需定义和计算预度量或测地线，从而实现更简单、更快且完全内在的框架。
原文摘要 · Abstract (English)
We propose a general framework to extend Flow Matching to homogeneous spaces, i.e. quotients of Lie groups. Our approach reformulates the problem as a flow matching task on the underlying Lie group by lifting the data distributions. This strategy avoids the potentially complicated geometry of homogeneous spaces by working directly on Lie groups, which in turn enables us reduce the problem to a Euclidean flow matching task on Lie algebras. In contrast to Riemannian Flow Matching, our method eliminates the need to define and compute premetrics or geodesics, resulting in a simpler, faster, and fully intrinsic framework.
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