Expressivity of Bi-Lipschitz Normalizing Flows: A Score-Based Diffusion Perspective

TL;DR

This paper investigates the expressivity of bi-Lipschitz normalizing flows using score-based diffusion models, establishing their universal approximation capabilities and convergence properties.

Contribution

It introduces a novel analysis linking score regularity to transport map regularity, providing universal approximation results for bi-Lipschitz flows.

Findings

Gaussian pullbacks are $L^1$-dense among all densities.

Convergence in Kullback-Leibler divergence is achieved for Gaussian convolution targets.

Score regularity is verified for broad classes of target densities.

Abstract

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.