Total Variation Rates for Riemannian Flow Matching

TL;DR

This paper provides a nonasymptotic Total Variation convergence analysis for Riemannian flow matching (RFM) models on manifolds, establishing explicit bounds and iteration complexities under certain smoothness and approximation assumptions.

Contribution

It introduces a novel TV convergence analysis for RFM on manifolds, accounting for curvature and parallel transport, with explicit bounds and complexity results.

Findings

Derived a differential inequality for TV evolution on manifolds.

Established explicit bounds separating discretization and learning errors.

Provided polynomial iteration complexity results for hypersphere and SPD manifolds.

Abstract

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.