# Total Variation Rates for Riemannian Flow Matching

arXiv: https://arxiv.org/abs/2602.05174 · HTML: https://mlsanity.com/paper/2602.05174/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 10 of its 44 references that are in the map.
- First posted on arXiv 5 Feb 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 10 of 10)

- [A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models](https://mlsanity.com/paper/2408.02320/) (2024)
- [Convergence of the Riemannian Langevin Algorithm](https://mlsanity.com/paper/2204.10818/) (2022)
- [On the Convergence and Straightness of Rectified Flow](https://mlsanity.com/paper/2410.14949/) (2024)
- [Statistical Properties of Rectified Flow](https://mlsanity.com/paper/2511.03193/) (2025)
- [On Flow Matching KL Divergence](https://mlsanity.com/paper/2511.05480/) (2025)
- [Polynomial Convergence of Riemannian Diffusion Models](https://mlsanity.com/paper/2601.02499/) (2026)
- [Low-Dimensional Adaptation of Rectified Flow: A Diffusion and Stochastic Localization Perspective](https://mlsanity.com/paper/2601.15500/) (2026)
- [Curvature-Dependant Global Convergence Rates for Optimization on Manifolds of Bounded Geometry](https://mlsanity.com/paper/2008.02517/) (2020)
- [Riemannian Neural Geodesic Interpolant](https://mlsanity.com/paper/2504.15736/) (2025)
- [Finite-Time Convergence Analysis of ODE-based Generative Models for Stochastic Interpolants](https://mlsanity.com/paper/2508.07333/) (2025)

## Built on by (most cited first, 2 of 2)

- [Particle Dynamics of Flow Matching and Classifier-Free Guidance from a Stagewise Geometry Perspective](https://mlsanity.com/paper/2609.06947/) (2026)
- [Counterfactual Generation via Flow Matching: Coupling-Sensitive End-to-End Rates](https://mlsanity.com/paper/2610.01193/) (2026)

## Abstract (arXiv)

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.