Title: Horizontal Diffusion Models: Riemannian Score-based Generative Modeling via Frame-Connection Geometry
Primary area: learning on graphs and other geometries & topologies
Keywords: Riemannian generative model, frame bundle, geometric score-based generative model

Abstract: Modeling data supported on curved manifolds poses significant challenges due to the need for geometric operations such as geodesic computations, parallel transport, and geodesic distance, which are often intractable or ill-defined on general Riemannian manifolds. To address this, we propose a novel framework for generative modeling on manifolds that bypasses these limitations by operating directly on the orthonormal frame bundle, a geometric space that retains manifold structure while offering computational compatibility with Euclidean learning. Our method introduces horizontal diffusion processes whose dynamics and score fields respect the underlying geometry without requiring manifold-specific neural architectures. A key insight is that standard Euclidean score networks can be lifted into the frame bundle to yield geometry-consistent vector fields, enabling seamless integration of manifold constraints with modern generative modeling techniques. Through theoretical analysis and experiments on complex curved domains, including the parametric surfaces and celestial bodies, we demonstrate that our approach achieves high-quality generation while preserving geometric fidelity. This work provides a general and scalable pathway for bridging differential geometry and score-based generative models.