Riemannian Flow Matching on General Geometries

We propose Riemannian Flow Matching (RFM), a simple yet powerful framework for training continuous normalizing flows on manifolds. Existing methods for generative modeling on manifolds either require expensive simulation, inherently cannot scale to high dimensions, or use approximations to limiting quantities that result in biased objectives. Riemannian Flow Matching bypasses these inconveniences and exhibits multiple benefits over prior approaches: It is completely simulation-free on simple geometries, it does not require divergence computation, and its target vector field is computed in closed form even on general geometries. The key ingredient behind RFM is the construction of a simple kernel function for defining per-sample vector fields, which subsumes existing Euclidean cases. Extending to general geometries, we rely on the use of spectral decompositions to efficiently compute kernel functions. Our method achieves state-of-the-art performance on real-world non-Euclidean datasets, and we showcase, for the first time, tractable training on general geometries, including on triangular meshes and maze-like manifolds with boundaries.