# Universality of parametric Coupling Flows over parametric diffeomorphisms

arXiv: https://arxiv.org/abs/2202.02906 · HTML: https://mlsanity.com/paper/2202.02906/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 44 references that are in the map.
- First posted on arXiv 7 Feb 2022.

## Builds on (most cited first, 24 of 24)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [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)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [A Tutorial on Thompson Sampling](https://mlsanity.com/paper/1707.02038/) (2017)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [Parallel WaveNet: Fast High-Fidelity Speech Synthesis](https://mlsanity.com/paper/1711.10433/) (2017)
- [Multiplicative Normalizing Flows for Variational Bayesian Neural Networks](https://mlsanity.com/paper/1703.01961/) (2017)
- [Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators](https://mlsanity.com/paper/2006.11469/) (2020)
- [Semi-Supervised Learning with Normalizing Flows](https://mlsanity.com/paper/1912.13025/) (2019)
- [Sylvester Normalizing Flows for Variational Inference](https://mlsanity.com/paper/1803.05649/) (2018)
- [Sum-of-Squares Polynomial Flow](https://mlsanity.com/paper/1905.02325/) (2019)
- [The Expressive Power of a Class of Normalizing Flow Models](https://mlsanity.com/paper/2006.00392/) (2020)
- [A Neural Networks Committee for the Contextual Bandit Problem](https://mlsanity.com/paper/1409.8191/) (2014)
- [Continuous-Time Flows for Efficient Inference and Density Estimation](https://mlsanity.com/paper/1709.01179/) (2017)
- [Random Forest for the Contextual Bandit Problem - extended version](https://mlsanity.com/paper/1504.06952/) (2015)
- [Deep Diffeomorphic Normalizing Flows](https://mlsanity.com/paper/1810.03256/) (2018)
- [Offline Contextual Bayesian Optimization for Nuclear Fusion](https://mlsanity.com/paper/2001.01793/) (2020)

## Built on by (most cited first, 2 of 2)

- [Universal approximation property of invertible neural networks](https://mlsanity.com/paper/2204.07415/) (2022)
- [On the Universality of Volume-Preserving and Coupling-Based Normalizing Flows](https://mlsanity.com/paper/2402.06578/) (2024)

## Abstract (arXiv)

Invertible neural networks based on Coupling Flows CFlows) have various applications such as image synthesis and data compression. The approximation universality for CFlows is of paramount importance to ensure the model expressiveness. In this paper, we prove that CFlows can approximate any diffeomorphism in C^k-norm if its layers can approximate certain single-coordinate transforms. Specifically, we derive that a composition of affine coupling layers and invertible linear transforms achieves this universality. Furthermore, in parametric cases where the diffeomorphism depends on some extra parameters, we prove the corresponding approximation theorems for our proposed parametric coupling flows named Para-CFlows. In practice, we apply Para-CFlows as a neural surrogate model in contextual Bayesian optimization tasks, to demonstrate its superiority over other neural surrogate models in terms of optimization performance.