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：流生成模型在计算数据生成和似然度方面具有特定优势，近期展现出具有竞争力的实证性能。相较于相关基于分数的扩散模型不断累积的理论研究，对前向（数据到噪声）和反向（噪声到数据）过程均为确定性的流模型分析仍然匮乏。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，该模型在标准化流网络中实现了Jordan-Kinderleherer-Otto（JKO）格式。通过利用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步骤（流中的$N$个残差块）时，JKO流模型生成数据的Kullback-Leibler（KL）误差保证为$O(\varepsilon^2)$，其中$\varepsilon$为每步一阶条件误差。对数据密度的假设仅需有限二阶矩，该理论可推广至无密度数据分布及反向过程存在逆误差的情形，此时我们获得KL-$W_2$混合误差保证。针对包含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，该结果具有独立的理论价值。本分析框架可扩展至应用于流生成模型的其他一阶Wasserstein优化方案。