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. The analysis framework can extend to other first-order Wasserstein optimization schemes applied to flow-based generative models.

翻译：流式生成模型在计算数据生成与似然估计方面具有特定优势，近期展现出具有竞争力的实证表现。相较于相关基于得分的扩散模型领域日益积累的理论研究，在前向（数据→噪声）和逆向（噪声→数据）方向均具有确定性的流式模型理论分析仍然匮乏。本文为渐进式流模型——即JKO流模型（该模型在归一化流网络中实现Jordan-Kinderleherer-Otto (JKO)格式）生成数据分布提供了理论保证。利用Wasserstein空间中近端梯度下降（GD）的指数收敛性，我们证明当使用$N \lesssim \log (1/\varepsilon)$个JKO步（即流中的$N$个残差块）时，JKO流模型生成数据的Kullback-Leibler (KL)散度保证为$O(\varepsilon^2)$，其中$\varepsilon$表示每步一阶条件的误差。该假设仅要求数据密度具有有限二阶矩，且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形（此时可获得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泛函类，我们证明了JKO型$W_2$近端GD的非渐近收敛速率，该结果本身具有独立研究价值。该分析框架可推广至应用于流式生成模型的其他一阶Wasserstein优化方案。