Flow Matching on Symmetric Spaces

TL;DR

This paper presents a unified framework for flow matching on Riemannian symmetric spaces, leveraging their algebraic structure to simplify modeling on manifolds like spheres, hyperbolic spaces, and Grassmannians.

Contribution

It reformulates flow matching on symmetric spaces as a linear problem in Lie algebra, enabling easier handling of geodesics and broadening applicability.

Findings

Framework successfully applied to real Grassmannians.

Simplifies flow matching by linearizing on Lie algebra subspaces.

Enhances modeling on complex manifolds using algebraic structure.

Abstract

We introduce a general framework for training flow matching models on Riemannian symmetric spaces, 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 greatly simplifying the handling of geodesics. As an application, we showcase our framework on the real Grassmannians $\operatorname{SO}(n) / \operatorname{SO}(k) \times \operatorname{SO}(n-k)$.