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

arXiv: https://arxiv.org/abs/2402.06578 · HTML: https://mlsanity.com/paper/2402.06578/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 52 references that are in the map.
- First posted on arXiv 9 Feb 2024.

## Builds on (most cited first, 26 of 26)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Design](https://mlsanity.com/paper/1902.00275/) (2019)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Unconstrained Monotonic Neural Networks](https://mlsanity.com/paper/1908.05164/) (2019)
- [Disentanglement by Nonlinear ICA with General Incompressible-flow Networks (GIN)](https://mlsanity.com/paper/2001.04872/) (2020)
- [Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators](https://mlsanity.com/paper/2006.11469/) (2020)
- [Latent Normalizing Flows for Discrete Sequences](https://mlsanity.com/paper/1901.10548/) (2019)
- [Sum-of-Squares Polynomial Flow](https://mlsanity.com/paper/1905.02325/) (2019)
- [Approximation Capabilities of Neural ODEs and Invertible Residual Networks](https://mlsanity.com/paper/1907.12998/) (2019)
- [Cubic-Spline Flows](https://mlsanity.com/paper/1906.02145/) (2019)
- [Universal Approximation Property of Neural Ordinary Differential Equations](https://mlsanity.com/paper/2012.02414/) (2020)
- [Universal approximation property of invertible neural networks](https://mlsanity.com/paper/2204.07415/) (2022)
- [Representational aspects of depth and conditioning in normalizing flows](https://mlsanity.com/paper/2010.01155/) (2020)
- [A Review of Change of Variable Formulas for Generative Modeling](https://mlsanity.com/paper/2308.02652/) (2023)
- [Whitening Convergence Rate of Coupling-based Normalizing Flows](https://mlsanity.com/paper/2210.14032/) (2022)
- [On the Convergence Rate of Gaussianization with Random Rotations](https://mlsanity.com/paper/2306.13520/) (2023)
- [Fixed-kinetic Neural Hamiltonian Flows for enhanced interpretability and reduced complexity](https://mlsanity.com/paper/2302.01955/) (2023)
- [Universality of parametric Coupling Flows over parametric diffeomorphisms](https://mlsanity.com/paper/2202.02906/) (2022)

## Built on by (most cited first, 18 of 18)

- [FARMER: Flow AutoRegressive Transformer over Pixels](https://mlsanity.com/paper/2510.23588/) (2025)
- [Temperature-Annealed Boltzmann Generators](https://mlsanity.com/paper/2501.19077/) (2025)
- [Normalizing Flows are Capable Models for Continuous Control](https://mlsanity.com/paper/2505.23527/) (2025)
- [Variational Bayesian Optimal Experimental Design with Normalizing Flows](https://mlsanity.com/paper/2404.13056/) (2024)
- [Flexible Tails for Normalizing Flows](https://mlsanity.com/paper/2406.16971/) (2024)
- [Probabilistic Bayesian optimal experimental design using conditional normalizing flows](https://mlsanity.com/paper/2402.18337/) (2024)
- [SimFlow: Simplified and End-to-End Training of Latent Normalizing Flows](https://mlsanity.com/paper/2512.04084/) (2025)
- [Improving the Accuracy of Amortized Model Comparison with Self-Consistency](https://mlsanity.com/paper/2508.20614/) (2025)
- [Markov Balance Satisfaction Improves Performance in Strictly Batch Offline Imitation Learning](https://mlsanity.com/paper/2408.09125/) (2024)
- [VeriFlow: Modeling Distributions for Neural Network Verification](https://mlsanity.com/paper/2406.14265/) (2024)
- [Neural Conjugate Flows: Physics-informed architectures with flow structure](https://mlsanity.com/paper/2411.08326/) (2024)
- [Empirical evaluation of normalizing flows in Markov Chain Monte Carlo](https://mlsanity.com/paper/2412.17136/) (2024)
- [Uncertainty-Aware Surrogate-based Amortized Bayesian Inference for Computationally Expensive Models](https://mlsanity.com/paper/2505.08683/) (2025)
- [Analytic Bijections for Smooth and Interpretable Normalizing Flows](https://mlsanity.com/paper/2601.10774/) (2026)
- [MIMFlow: Integrating Masked Image Modeling with Normalizing Flows for End-to-End Image Generation](https://mlsanity.com/paper/2606.26016/) (2026)
- [On Uniformly Scaling Flows: A Density-Aligned Approach to Deep One-Class Classification](https://mlsanity.com/paper/2510.09452/) (2025)
- [Why the Counterintuitive Phenomenon of Likelihood Rarely Appears in Tabular Anomaly Detection with Deep Generative Models?](https://mlsanity.com/paper/2602.09593/) (2026)
- [Multivariate Time Series Anomaly Detection via Dual-Branch Reconstruction and Autoregressive Flow-based Residual Density Estimation](https://mlsanity.com/paper/2604.08582/) (2026)

## 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.