Towards Variational Flow Matching on General Geometries

2025-02-18

AI Summary

This work addresses generative modeling on structured manifolds—particularly Riemannian manifolds admitting closed-form geodesics. We propose Riemannian Gaussian Variational Flow Matching (RG-VFM), the first systematic extension of the flow matching framework to general Riemannian manifolds. RG-VFM introduces the Riemannian Gaussian distribution as the foundational probabilistic model and rigorously derives a curvature-aware variational objective grounded in manifold geometry. Compared to Euclidean flow matching and leading baselines, RG-VFM achieves substantial improvements in generation quality and geometric fidelity on curvature-sensitive tasks—e.g., spherical-wrapped checkerboard distributions—demonstrating its capacity to accurately capture intrinsic manifold structure. Our core contribution is the development of the first analytically tractable, geometrically consistent flow matching framework for Riemannian manifolds, enabling principled, curvature-informed generative modeling.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Diffusion Models for Vision
• Reasoning under Uncertainty: Relational Probabilistic Models

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
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• Extends Variational Flow Matching to structured manifolds
• Uses Riemannian Gaussian distributions for generative modeling
• Improves geometric structure capture in generative models

Innovation

• Riemannian Gaussian Variational Flow Matching
• Generative modeling on manifolds
• Closed-form geodesics variational objective