Total Variation Rates for Riemannian Flow Matching

• URL: http://arxiv.org/abs/2602.05174v1
• Date: Thu, 05 Feb 2026 01:06:53 GMT
• Title: Total Variation Rates for Riemannian Flow Matching
• Authors: Yunrui Guan, Krishnakumar Balasubramanian, Shiqian Ma,
• Abstract summary: We develop a nonasymptotic Total Variation analysis for RFM samplers.<n>Our key technical ingredient is a differential inequality governing the evolution of TV between two manifold ODE flows.
• Score: 8.235086108564998
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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