双 Lipschitz 正则化流的表达性：基于分数扩散的视角
Expressivity of Bi-Lipschitz Normalizing Flows: A Score-Based Diffusion Perspective
AI总结：
本文从分数扩散模型的角度研究双 Lipschitz 正则化流的表达性，通过概率流ODE分析其分布逼近能力，并推导扩散运输的确定性收敛保证。
摘要：
许多正则化流架构施加了正则性约束，但其分布逼近性质尚未完全明确。我们通过分数扩散模型的视角研究双 Lipschitz 正则化流的表达性。对于方差保持扩散的概率流ODE，分数的Lipschitz正则性诱导了双 Lipschitz 偏微分运输映射的流。这种ODE桥梁使我们能够分析双 Lipschitz 正则化流的分布逼近能力，并反向推导基于扩散的运输的确定性收敛保证。我们的关键思想是利用概率流ODE将分数的正则性与诱导的运输映射的正则性联系起来。我们验证了广泛的目标密度的分数正则性，包括具有紧支撑的密度、高斯卷积的紧支撑测度以及有限高斯混合物。我们获得了一个通用的分布逼近结果：由双 Lipschitz 方差保持运输映射诱导的高斯拉回在所有概率密度中是L^1-密集的。对于高斯卷积目标，我们进一步获得无需早停的KL散度收敛性。
英文摘要：
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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