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

arXiv 2605.06172 ↗First posted 7 May 2026stat.ML cs.LG cs.NAcited by 0

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Abstract (arXiv)

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 L1-dense among all probability densities. For Gaussian convolution targets, we further obtain convergence in Kullback-Leibler divergence without early stopping.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 32 of its 48 references that are in the map. First posted on arXiv 7 May 2026.

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References · 32

  1. 2020Denoising Diffusion Probabilistic Modelscited by 21,076
  2. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 9,122
  3. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  4. 2018Neural Ordinary Differential Equationscited by 3,779
  5. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,200
  6. 2015Variational Inference with Normalizing Flowscited by 2,170
  7. 2018Spectral Normalization for Generative Adversarial Networkscited by 1,636
  8. 2018Glow: Generative Flow with Invertible 1x1 Convolutionscited by 1,545
  9. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,323
  10. 2014NICE: Non-linear Independent Components Estimationcited by 1,058
  11. 2022Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptionscited by 445
  12. 2018Invertible Residual Networkscited by 365
20 more
  1. 2023Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localizationcited by 235
  2. 2022Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptionscited by 219
  3. 2022Convergence of score-based generative modeling for general data distributionscited by 181
  4. 2023The probability flow ODE is provably fastcited by 146
  5. 2018A Hierarchical Latent Vector Model for Learning Long-Term Structure in Musiccited by 131
  6. 2019Noise Flow: Noise Modeling with Conditional Normalizing Flowscited by 108
  7. 2020Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximatorscited by 85
  8. 2023KL Convergence Guarantees for Score diffusion models under minimal data assumptionscited by 63
  9. 2019Approximation Capabilities of Neural ODEs and Invertible Residual Networkscited by 56
  10. 2024Convergence Analysis for General Probability Flow ODEs of Diffusion Models in Wasserstein Distancescited by 56
  11. 2020The Expressive Power of a Class of Normalizing Flow Modelscited by 35
  12. 2022Universal approximation property of invertible neural networkscited by 30
  13. 2023Improved Convergence of Score-Based Diffusion Models via Prediction-Correctioncited by 24
  14. 2025Advancing Wasserstein Convergence Analysis of Score-Based Models: Insights from Discretization and Second-Order Accelerationcited by 17
  15. 2021On the expressivity of bi-Lipschitz normalizing flowscited by 16
  16. 2025Diffusion models for inverse problemscited by 16
  17. 2021Universal Approximation of Residual Flows in Maximum Mean Discrepancycited by 7
  18. 2024Global Well-posedness and Convergence Analysis of Score-based Generative Models via Sharp Lipschitz Estimatescited by 6
  19. 2023On the Approximation of Bi-Lipschitz Maps by Invertible Neural Networkscited by 2
  20. 2025Learned Discrepancy Reconstruction and Benchmark Dataset for Magnetic Particle Imagingcited by 2

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