mlsanityData to 7 Oct 2026

On the Universality of Volume-Preserving and Coupling-Based Normalizing Flows

arXiv 2402.06578 ↗First posted 9 Feb 2024cs.LG stat.MLcited by 18

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Abstract (arXiv)

We present a novel theoretical framework for understanding the expressive power of normalizing flows. Despite their prevalence in scientific applications, a comprehensive understanding of flows remains elusive due to their restricted architectures. Existing theorems fall short as they require the use of arbitrarily ill-conditioned neural networks, limiting practical applicability. We propose a distributional universality theorem for well-conditioned coupling-based normalizing flows such as RealNVP. In addition, we show that volume-preserving normalizing flows are not universal, what distribution they learn instead, and how to fix their expressivity. Our results support the general wisdom that affine and related couplings are expressive and in general outperform volume-preserving flows, bridging a gap between empirical results and theoretical understanding.

Key facts. Cited by 18 papers in the map (data through 7 Oct 2026). Builds on 26 of its 52 references that are in the map. First posted on arXiv 9 Feb 2024.

Builds on

26 of 52 references.

References · 26

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  2. 2015Variational Inference with Normalizing Flowscited by 2,161
  3. 2016Density estimation using Real NVPcited by 2,029
  4. 2018Glow: Generative Flow with Invertible 1x1 Convolutionscited by 1,539
  5. 2019Neural Spline Flowscited by 590
  6. 2019Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Designcited by 258
  7. 2018Analyzing Inverse Problems with Invertible Neural Networkscited by 239
  8. 2018Neural Autoregressive Flowscited by 207
  9. 2018Neural Importance Samplingcited by 154
  10. 2019Flow-based generative models for Markov chain Monte Carlo in lattice field theorycited by 147
  11. 2019Hamiltonian Generative Networkscited by 136
  12. 2019Unconstrained Monotonic Neural Networkscited by 91
14 more
  1. 2020Disentanglement by Nonlinear ICA with General Incompressible-flow Networks (GIN)cited by 91
  2. 2020Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximatorscited by 85
  3. 2019Latent Normalizing Flows for Discrete Sequencescited by 66
  4. 2019Sum-of-Squares Polynomial Flowcited by 62
  5. 2019Approximation Capabilities of Neural ODEs and Invertible Residual Networkscited by 56
  6. 2019Cubic-Spline Flowscited by 34
  7. 2020Universal Approximation Property of Neural Ordinary Differential Equationscited by 32
  8. 2022Universal approximation property of invertible neural networkscited by 30
  9. 2020Representational aspects of depth and conditioning in normalizing flowscited by 25
  10. 2023A Review of Change of Variable Formulas for Generative Modelingcited by 12
  11. 2022Whitening Convergence Rate of Coupling-based Normalizing Flowscited by 6
  12. 2023On the Convergence Rate of Gaussianization with Random Rotationscited by 5
  13. 2023Fixed-kinetic Neural Hamiltonian Flows for enhanced interpretability and reduced complexitycited by 4
  14. 2022Universality of parametric Coupling Flows over parametric diffeomorphismscited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 7; 2025: 7; 2026: 4.

18 papers.

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  1. 2025FARMER: Flow AutoRegressive Transformer over Pixelscited by 17
  2. 2025Temperature-Annealed Boltzmann Generatorscited by 14
  3. 2024Variational Bayesian Optimal Experimental Design with Normalizing Flowscited by 13
  4. 2025Normalizing Flows are Capable Models for Continuous Controlcited by 13
  5. 2024Flexible Tails for Normalizing Flowscited by 11
  6. 2024Probabilistic Bayesian optimal experimental design using conditional normalizing flowscited by 10
  7. 2025SimFlow: Simplified and End-to-End Training of Latent Normalizing Flowscited by 6
  8. 2025Improving the Accuracy of Amortized Model Comparison with Self-Consistencycited by 5
  9. 2024Markov Balance Satisfaction Improves Performance in Strictly Batch Offline Imitation Learningcited by 3
  10. 2024VeriFlow: Modeling Distributions for Neural Network Verificationcited by 1
  11. 2024Neural Conjugate Flows: Physics-informed architectures with flow structurecited by 1
  12. 2024Empirical evaluation of normalizing flows in Markov Chain Monte Carlocited by 1
6 more
  1. 2025Uncertainty-Aware Surrogate-based Amortized Bayesian Inference for Computationally Expensive Modelscited by 1
  2. 2026Analytic Bijections for Smooth and Interpretable Normalizing Flowscited by 1
  3. 2026MIMFlow: Integrating Masked Image Modeling with Normalizing Flows for End-to-End Image Generationcited by 1
  4. 2025On Uniformly Scaling Flows: A Density-Aligned Approach to Deep One-Class Classificationcited by 0
  5. 2026Why the Counterintuitive Phenomenon of Likelihood Rarely Appears in Tabular Anomaly Detection with Deep Generative Models?cited by 0
  6. 2026Multivariate Time Series Anomaly Detection via Dual-Branch Reconstruction and Autoregressive Flow-based Residual Density Estimationcited by 0