Geometry-Grounded Flow Matching on Compact Manifolds

Riemannian Flow Matching extends flow-matching generative modeling to data that live on curved spaces such as spheres and tori, by learning a time-dependent vector field and generating samples through ordinary differential equation integration. This paper provides an end-to-end theoretical guarantee for the standard Riemannian Flow Matching pipeline on compact manifolds. Our analysis separates three sources of error: the statistical error from learning the conditional-mean velocity field produced by conditional flow matching, the approximation and optimization error arising from the chosen function class and empirical risk minimization, and the discretization error introduced by the numerical ODE solver. A key technical contribution is a flow-to-distribution stability result that is robust to geometry: curvature and injectivity radius influence only constants under standard boundedness and Lipschitz regularity conditions. Under metric-entropy assumptions, the learning rate is governed by the intrinsic manifold dimension rather than any ambient embedding dimension. Experiments on the circle, the sphere, and the two-torus support the predicted scaling behavior.