Flow-based generative models enjoy certain advantages in computing the data generation and the likelihood, and have recently shown competitive empirical performance. Compared to the accumulating theoretical studies on related score-based diffusion models, analysis of flow-based models, which are deterministic in both forward (data-to-noise) and reverse (noise-to-data) directions, remain sparse. In this paper, we provide a theoretical guarantee of generating data distribution by a progressive flow model, the so-called JKO flow model, which implements the Jordan-Kinderleherer-Otto (JKO) scheme in a normalizing flow network. Leveraging the exponential convergence of the proximal gradient descent (GD) in Wasserstein space, we prove the Kullback-Leibler (KL) guarantee of data generation by a JKO flow model to be $O(\varepsilon^2)$ when using $N \lesssim \log (1/\varepsilon)$ many JKO steps ($N$ Residual Blocks in the flow) where $\varepsilon $ is the error in the per-step first-order condition. The assumption on data density is merely a finite second moment, and the theory extends to data distributions without density and when there are inversion errors in the reverse process where we obtain KL-$W_2$ mixed error guarantees. The non-asymptotic convergence rate of the JKO-type $W_2$-proximal GD is proved for a general class of convex objective functionals that includes the KL divergence as a special case, which can be of independent interest.

翻译：基于流的生成模型在数据生成和似然计算方面具有特定优势，近期已展现出具有竞争力的实证表现。与相关基于分数的扩散模型不断积累的理论研究相比，流模型在正向（数据到噪声）和反向（噪声到数据）方向上均为确定性过程，其理论分析仍较为稀缺。本文为渐进式流模型（即JKO流模型）生成数据分布提供了理论保证，该模型在归一化流网络中实现了Jordan-Kinderleherer-Otto（JKO）格式。利用Wasserstein空间中近端梯度下降（GD）的指数收敛性，我们证明了JKO流模型的数据生成Kullback-Leibler（KL）误差保证为$O(\varepsilon^2)$，其中使用$N \lesssim \log (1/\varepsilon)$步JKO步进（即流中的$N$个残差块），$\varepsilon$为每步一阶条件的误差。数据密度的假设仅要求存在有限二阶矩，且该理论可扩展至无密度形式的数据分布，以及逆向过程中存在反演误差的情况，此时我们获得KL-$W_2$混合误差保证。对于包含KL散度作为特例的一般凸目标函数类，本文证明了JKO型$W_2$-近端GD的非渐近收敛率，该结论可独立应用于其他研究领域。