Total Variation Rates for Riemannian Flow Matching

Riemannian flow matching (RFM) extends flow-based generative modeling to data supported on manifolds by learning a time-dependent tangent vector field whose flow-ODE transports a simple base distribution to the data law. We develop a nonasymptotic Total Variation (TV) convergence analysis for RFM samplers that use a learned vector field together with Euler discretization on manifolds. Our key technical ingredient is a differential inequality governing the evolution of TV between two manifold ODE flows, which expresses the time-derivative of TV through the divergence of the vector-field mismatch and the score of the reference flow; controlling these terms requires establishing new bounds that explicitly account for parallel transport and curvature. Under smoothness assumptions on the population flow-matching field and either uniform (compact manifolds) or mean-square (Hadamard manifolds) approximation guarantees for the learned field, we obtain explicit bounds of the form TV≤ CLip,h + Cvarepsilon,varepsilon (with an additional higher-order varepsilon2 term on compact manifolds), cleanly separating numerical discretization and learning errors. Here, h is the step-size and varepsilon is the target accuracy. Instantiations yield explicit polynomial iteration complexities on the hypersphere Sd, and on the SPD(n) manifolds under mild moment conditions.