# Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators

arXiv: https://arxiv.org/abs/2006.11469 · HTML: https://mlsanity.com/paper/2006.11469/

## Key facts

- Cited by 85 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jun 2020.

## Built on by (most cited first, 30 of 85)

- [Neural Flows: Efficient Alternative to Neural ODEs](https://mlsanity.com/paper/2110.13040/) (2021)
- [On the Stability of Iterative Retraining of Generative Models on their own Data](https://mlsanity.com/paper/2310.00429/) (2023)
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- [Identifiability of deep generative models without auxiliary information](https://mlsanity.com/paper/2206.10044/) (2022)
- [Scalable Equilibrium Sampling with Sequential Boltzmann Generators](https://mlsanity.com/paper/2502.18462/) (2025)
- [Trumpets: Injective Flows for Inference and Inverse Problems](https://mlsanity.com/paper/2102.10461/) (2021)
- [Memory-Efficient Network for Large-scale Video Compressive Sensing](https://mlsanity.com/paper/2103.03089/) (2021)
- [Universal approximation property of invertible neural networks](https://mlsanity.com/paper/2204.07415/) (2022)
- [Free-form Flows: Make Any Architecture a Normalizing Flow](https://mlsanity.com/paper/2310.16624/) (2023)
- [Reliable amortized variational inference with physics-based latent distribution correction](https://mlsanity.com/paper/2207.11640/) (2022)
- [PatchNR: Learning from Very Few Images by Patch Normalizing Flow Regularization](https://mlsanity.com/paper/2205.12021/) (2022)
- [Deep Learning in Medical Image Registration: Magic or Mirage?](https://mlsanity.com/paper/2408.05839/) (2024)
- [Efficient Neural Network Approaches for Conditional Optimal Transport with Applications in Bayesian Inference](https://mlsanity.com/paper/2310.16975/) (2023)
- [On the Universality of Volume-Preserving and Coupling-Based Normalizing Flows](https://mlsanity.com/paper/2402.06578/) (2024)
- [Bridging Mean-Field Games and Normalizing Flows with Trajectory Regularization](https://mlsanity.com/paper/2206.14990/) (2022)
- [On the expressivity of bi-Lipschitz normalizing flows](https://mlsanity.com/paper/2107.07232/) (2021)
- [Flow-based Recurrent Belief State Learning for POMDPs](https://mlsanity.com/paper/2205.11051/) (2022)
- [Invertible Rescaling Network and Its Extensions](https://mlsanity.com/paper/2210.04188/) (2022)
- [Sparse Flows: Pruning Continuous-depth Models](https://mlsanity.com/paper/2106.12718/) (2021)
- [Universal Joint Approximation of Manifolds and Densities by Simple Injective Flows](https://mlsanity.com/paper/2110.04227/) (2021)
- [Diffeomorphically Learning Stable Koopman Operators](https://mlsanity.com/paper/2112.04085/) (2021)
- [Conditional Generative Models are Provably Robust: Pointwise Guarantees for Bayesian Inverse Problems](https://mlsanity.com/paper/2303.15845/) (2023)
- [Minimum Width of Leaky-ReLU Neural Networks for Uniform Universal Approximation](https://mlsanity.com/paper/2305.18460/) (2023)
- [Globally injective and bijective neural operators](https://mlsanity.com/paper/2306.03982/) (2023)
- [Flexible Tails for Normalizing Flows](https://mlsanity.com/paper/2406.16971/) (2024)
- [Rethinking Importance Weighting for Transfer Learning](https://mlsanity.com/paper/2112.10157/) (2021)
- [Learning reversible symplectic dynamics](https://mlsanity.com/paper/2204.12323/) (2022)
- [Gradual Domain Adaptation via Normalizing Flows](https://mlsanity.com/paper/2206.11492/) (2022)
- [Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees](https://mlsanity.com/paper/2402.08090/) (2024)
- [ASPIRE: Iterative Amortized Posterior Inference for Bayesian Inverse Problems](https://mlsanity.com/paper/2405.05398/) (2024)

## Abstract (arXiv)

Invertible neural networks based on coupling flows (CF-INNs) have various machine learning applications such as image synthesis and representation learning. However, their desirable characteristics such as analytic invertibility come at the cost of restricting the functional forms. This poses a question on their representation power: are CF-INNs universal approximators for invertible functions? Without a universality, there could be a well-behaved invertible transformation that the CF-INN can never approximate, hence it would render the model class unreliable. We answer this question by showing a convenient criterion: a CF-INN is universal if its layers contain affine coupling and invertible linear functions as special cases. As its corollary, we can affirmatively resolve a previously unsolved problem: whether normalizing flow models based on affine coupling can be universal distributional approximators. In the course of proving the universality, we prove a general theorem to show the equivalence of the universality for certain diffeomorphism classes, a theoretical insight that is of interest by itself.