从扩散模型视角揭示双 Lipschitz 流的表达能力，实现通用分布逼近。
Expressivity of Bi-Lipschitz Normalizing Flows: A Score-Based Diffusion Perspective
- 通过概率流 ODE 将得分函数正则性映射为双 Lipschitz 变换映射
- 证明双 Lipschitz 流可 $L^1$ 稠密逼近任意概率密度
- 对高斯卷积目标可保证 KL 散度收敛，无需早停
许多归一化流架构施加了正则性约束，但其分布逼近性质尚未完全刻画。本文从基于得分的扩散模型视角研究双 Lipschitz 归一化流的表达能力。对于方差保持扩散的概率流 ODE，得分函数的 Lipschitz 正则性诱导出双 Lipschitz 微分同胚变换映射。该 ODE 桥接使我们能分析双 Lipschitz 归一化流的分布逼近能力，并反向推导出基于扩散的传输的确定性收敛保证。核心思想是利用概率流 ODE 将得分正则性与诱导变换映射的正则性关联。我们验证了广泛目标密度（包括紧支撑密度、紧支撑测度的高斯卷积及有限高斯混合）的得分正则性。得到一个普适分布逼近结果：由双 Lipschitz 方差保持变换诱导的高斯拉回在所有概率密度中 $L^1$ 稠密。对于高斯卷积目标，进一步获得无需早停的 KL 散度收敛性。
原文摘要 · Abstract (English)
Many normalizing flow architectures impose regularity constraints, yet their distributional approximation properties are not fully characterized. We study the expressivity of bi-Lipschitz normalizing flows through the lens of score-based diffusion models. For the probability flow ODE of a variance-preserving diffusion, Lipschitz regularity of the score induces a flow of bi-Lipschitz diffeomorphic transport maps. This ODE bridge allows us to analyze the distributional approximation power of bi-Lipschitz normalizing flows and, conversely, derive deterministic convergence guarantees for diffusion-based transport. Our key idea is to use the probability flow ODE to link regularity of the score to regularity of the induced transport maps. We verify score regularity for broad target densities, including compactly supported densities, Gaussian convolutions of compactly supported measures and finite Gaussian mixtures. We obtain a universal distributional approximation result: Gaussian pullbacks induced by bi-Lipschitz variance-preserving transport maps are $L^1$-dense among all probability densities. For Gaussian convolution targets, we further obtain convergence in Kullback-Leibler divergence without early stopping.
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