Flow Matching (FM) (also referred to as stochastic interpolants or rectified flows) stands out as a class of generative models that aims to bridge in finite time the target distribution $\nu^\star$ with an auxiliary distribution $\mu$, leveraging a fixed coupling $\pi$ and a bridge which can either be deterministic or stochastic. These two ingredients define a path measure which can then be approximated by learning the drift of its Markovian projection. The main contribution of this paper is to provide relatively mild assumptions on $\nu^\star$, $\mu$ and $\pi$ to obtain non-asymptotics guarantees for Diffusion Flow Matching (DFM) models using as bridge the conditional distribution associated with the Brownian motion. More precisely, we establish bounds on the Kullback-Leibler divergence between the target distribution and the one generated by such DFM models under moment conditions on the score of $\nu^\star$, $\mu$ and $\pi$, and a standard $L^2$-drift-approximation error assumption.

本研究解决了扩散流匹配模型在有限时间内实现目标分布与辅助分布之间的桥接问题。通过提出相对温和的假设，研究为扩散流匹配模型提供了非渐近性的理论保障，确立了目标分布与生成分布之间的Kullback-Leibler散度的界限，具有重要的应用潜力。

扩散流匹配中的KL理论保障