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 $\mathrm{TV}\le C_{\mathrm{Lip}},h + C_{\varepsilon},\varepsilon$ (with an additional higher-order $\varepsilon^2$ 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 \emph{explicit} polynomial iteration complexities on the hypersphere $S^d$, and on the SPD$(n)$ manifolds under mild moment conditions.

💡 Research Summary

This paper presents the first non‑asymptotic total‑variation (TV) convergence analysis for Riemannian Flow Matching (RFM), a deterministic generative modeling framework that transports a simple base distribution to a target distribution on a Riemannian manifold by integrating a learned time‑dependent tangent vector field. The authors derive a differential inequality that governs the evolution of TV between the true flow induced by the population optimal vector field v(t,·) and the flow induced by an estimated field \hat v(t,·). The inequality shows that the time‑derivative of TV is bounded by two terms: the divergence of the vector‑field mismatch and the inner product of the score (∇ log p_t) with the same mismatch. Controlling these terms requires careful handling of parallel transport, curvature, and the geometry of the manifold.

Under smoothness assumptions on v and either a uniform approximation guarantee (‖\hat v − v‖_∞ ≤ ε) for compact manifolds or a mean‑square guarantee (E