黎曼流匹配的总变差率

This paper presents the first study on the total variation (TV) rates for Riemannian Flow Matching (RFM), an emerging technique for generative modeling. We demonstrate that RFM achieves a convergence rate of O(1/N) in the Wasserstein distance, similar to diffusion models, where N is the number of samples. By establishing a connection between RFM and Wasserstein gradient flows, we derive explicit expressions for the TV rates and analyze their dependence on manifold geometry and data distribution. Specifically, we find that the TV rates are influenced by the Riemannian metric, curvature, and the gradient of the data distribution. Furthermore, we explore the performance of RFM on various manifolds and provide theoretical guarantees. These findings not only deepen the understanding of RFM's theoretical properties but also offer new perspectives for designing more efficient and stable generative models. Our analysis lays a solid theoretical foundation for the practical application of RFM and points towards future research directions, such as optimizing manifold structures to further enhance generation quality. The insights gained from this work are crucial for advancing the field of generative AI, particularly in scenarios where data resides on complex, non-Euclidean spaces. This comprehensive theoretical framework contributes significantly to the rigorous understanding and development of flow-based generative models on Riemannian manifolds.