Towards Variational Flow Matching on General Geometries

Olga Zaghen, Floor Eijkelboom, Alison Pouplin, Erik J Bekkers

Track: tiny / short paper (up to 4 pages)

Keywords: generative modeling, flow matching, variational inference, general manifolds

TL;DR: We derive a variational objective for flow matching on manifolds with closed-form geodesics and test it on a checkerboard dataset on the sphere.

Abstract: We introduce Riemannian Gaussian Variational Flow Matching (RG-VFM), an extension of Variational Flow Matching (VFM) that leverages Riemannian Gaussian distributions for generative modeling on structured manifolds. We derive a variational objective for probability flows on manifolds with closed-form geodesics, making RG-VFM comparable -- though fundamentally different to Riemannian Flow Matching (RFM) in this geometric setting. Experiments on a checkerboard dataset wrapped on the sphere demonstrate that RG-VFM captures geometric structure more effectively than Euclidean VFM and baseline methods, establishing it as a robust framework for manifold-aware generative modeling.

Submission Number: 95