提出带利普希茨保证的流匹配方法，提升高维分布估计精度。
Distribution estimation via Flow Matching with Lipschitz guarantees
- 基于可控利普希茨常数的向量场设计，增强理论稳定性。
- 在高维场景下实现更优的Wasserstein-1距离收敛速度。
- 适用于非对数凹分布，拓展了应用范围。
流匹配作为一种生成建模的有前景方法，近年来受到广泛关注。它依赖于常微分方程，为当前最先进的扩散模型提供了一种简单且灵活的替代方案。尽管其在实验中表现成功，但对其统计性能的数学理解仍十分有限，这主要源于理论界面对驱动ODE的向量场利普希茨常数的高度敏感性。本文研究了使该依赖关系可控制的假设条件，并据此推导出估计分布与目标分布之间Wasserstein-1距离的收敛速率，在高维设置下优于先前结果。该速率适用于某些无界分布类，尤其不要求对数凹性。
原文摘要 · Abstract (English)
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.
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