Flow matching on homogeneous spaces
This work addresses a technical challenge in generative modeling for researchers working with homogeneous spaces, offering an incremental improvement by simplifying computations.
The authors tackled the problem of extending Flow Matching to homogeneous spaces by proposing a framework that lifts data distributions to Lie groups, enabling Euclidean flow matching on Lie algebras. This approach eliminates the need for premetrics or geodesics, resulting in a simpler and faster method compared to Riemannian Flow Matching.
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.