Flow matching on homogeneous spaces

2026-03-25

AI Summary

Homogeneous spaces—realized as quotients of Lie groups—pose a challenge for existing flow-matching methods due to the absence of explicit metrics and geodesic structures. This work proposes an intrinsic framework that lifts the target distribution to the underlying Lie group and performs Euclidean flow matching on the corresponding Lie algebra, thereby entirely circumventing the need for a predefined metric or geodesics. The approach requires only a Lie group action and a local section, eliminating any reliance on Riemannian geometric computations and substantially simplifying the generative modeling pipeline. Experiments demonstrate that this method yields more efficient, concise, and scalable generative models on homogeneous spaces while preserving intrinsic geometric fidelity.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search
• Natural Language Processing: Generation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

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.

Problem

• Flow Matching
• Homogeneous Spaces
• Lie Groups
• Geometric Deep Learning

Innovation

• Flow Matching
• Homogeneous Spaces
• Lie Groups
• Lie Algebras
• Intrinsic Framework