Flow Matching, a promising approach in generative modeling, has recently gained popularity. Relying on ordinary differential equations, it offers a simple and flexible alternative to diffusion models, which are currently the state-of-the-art. Despite its empirical success, the mathematical understanding of its statistical power so far is very limited. This is largely due to the sensitivity of theoretical bounds to the Lipschitz constant of the vector field which drives the ODE. In this work, we study the assumptions that lead to controlling this dependency. Based on these results, we derive a convergence rate for the Wasserstein $1$ distance between the estimated distribution and the target distribution which improves previous results in high dimensional setting. This rate applies to certain classes of unbounded distributions and particularly does not require $\log$-concavity.

翻译：流匹配作为生成建模中一种前景广阔的方法，近期受到广泛关注。该方法基于常微分方程，为当前最先进的扩散模型提供了一种简单而灵活的替代方案。尽管其经验成果显著，但对其统计能力的数学理解迄今仍非常有限。这主要源于理论边界对驱动常微分方程的向量场Lipschitz常数的敏感性。在本工作中，我们研究了能够控制这种依赖关系的假设条件。基于这些结果，我们推导了估计分布与目标分布之间Wasserstein $1$ 距离的收敛速率，该结果在高维场景下改进了先前的研究结论。该速率适用于特定类别的无界分布，尤其不要求分布满足$\log$-凹性条件。