Flow matching on homogeneous spaces

TL;DR

This paper introduces a novel framework for flow matching on homogeneous spaces by leveraging Lie group structures, simplifying the process and avoiding complex geometric computations, thus enabling more efficient and intrinsic modeling.

Contribution

The authors extend Flow Matching to homogeneous spaces using Lie groups, eliminating the need for premetrics or geodesics, and simplifying the geometric complexity involved.

Findings

Reduces flow matching on homogeneous spaces to Euclidean tasks on Lie algebras.

Avoids complex geometric computations like geodesics and premetrics.

Provides a faster, simpler, and intrinsic framework for flow matching on Lie groups.

Abstract

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.