# Comparison of Affine and Rational Quadratic Spline Coupling and Autoregressive Flows through Robust Statistical Tests

arXiv: https://arxiv.org/abs/2302.12024 · HTML: https://mlsanity.com/paper/2302.12024/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 89 references that are in the map.
- First posted on arXiv 23 Feb 2023.

## Builds on (most cited first, 28 of 28)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [Anomaly Detection with Density Estimation](https://mlsanity.com/paper/2001.04990/) (2020)
- [Continual Repeated Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2201.13117/) (2022)
- [Mining gold from implicit models to improve likelihood-free inference](https://mlsanity.com/paper/1805.12244/) (2018)
- [CaloFlow: Fast and Accurate Generation of Calorimeter Showers with Normalizing Flows](https://mlsanity.com/paper/2106.05285/) (2021)
- [Neural Adaptive Sequential Monte Carlo](https://mlsanity.com/paper/1506.03338/) (2015)
- [CaloFlow II: Even Faster and Still Accurate Generation of Calorimeter Showers with Normalizing Flows](https://mlsanity.com/paper/2110.11377/) (2021)
- [Inference Compilation and Universal Probabilistic Programming](https://mlsanity.com/paper/1610.09900/) (2016)
- [i-flow: High-dimensional Integration and Sampling with Normalizing Flows](https://mlsanity.com/paper/2001.05486/) (2020)
- [Complete parameter inference for GW150914 using deep learning](https://mlsanity.com/paper/2008.03312/) (2020)
- [Generative Networks for Precision Enthusiasts](https://mlsanity.com/paper/2110.13632/) (2021)
- [Learning new physics efficiently with nonparametric methods](https://mlsanity.com/paper/2204.02317/) (2022)
- [Inference Networks for Sequential Monte Carlo in Graphical Models](https://mlsanity.com/paper/1602.06701/) (2016)
- [An unfolding method based on conditional Invertible Neural Networks (cINN) using iterative training](https://mlsanity.com/paper/2212.08674/) (2022)
- [Gravitational-wave parameter estimation with autoregressive neural network flows](https://mlsanity.com/paper/2002.07656/) (2020)
- [Testing the boundaries: Normalizing Flows for higher dimensional data sets](https://mlsanity.com/paper/2202.09188/) (2022)
- [HMC with Normalizing Flows](https://mlsanity.com/paper/2112.01586/) (2021)
- [Machine Learning Trivializing Maps: A First Step Towards Understanding How Flow-Based Samplers Scale Up](https://mlsanity.com/paper/2112.15532/) (2021)
- [Exhaustive Neural Importance Sampling applied to Monte Carlo event generation](https://mlsanity.com/paper/2005.12719/) (2020)
- [Normalizing Flows for Hierarchical Bayesian Analysis: A Gravitational Wave Population Study](https://mlsanity.com/paper/2211.09008/) (2022)

## Built on by (most cited first, 5 of 5)

- [Learning to Validate Generative Models: a Goodness-of-Fit Approach](https://mlsanity.com/paper/2511.09118/) (2025)
- [The NFLikelihood: an unsupervised DNNLikelihood from Normalizing Flows](https://mlsanity.com/paper/2309.09743/) (2023)
- [Variational Transdimensional Inference](https://mlsanity.com/paper/2506.04749/) (2025)
- [Generative Models and Statistical Validation](https://mlsanity.com/paper/2605.30453/) (2026)
- [BUSSARD: Normalizing Flows for Bijective Universal Scene-Specific Anomalous Relationship Detection](https://mlsanity.com/paper/2603.16645/) (2026)

## Abstract (arXiv)

Normalizing flows have emerged as a powerful brand of generative models, as they not only allow for efficient sampling of complicated target distributions but also deliver density estimation by construction. We propose here an in-depth comparison of coupling and autoregressive flows, both based on symmetric (affine) and non-symmetric (rational quadratic spline) bijectors, considering four different architectures: real-valued non-Volume preserving (RealNVP), masked autoregressive flow (MAF), coupling rational quadratic spline (C-RQS), and autoregressive rational quadratic spline (A-RQS). We focus on a set of multimodal target distributions of increasing dimensionality ranging from 4 to 400. The performances were compared by means of different test statistics for two-sample tests, built from known distance measures: the sliced Wasserstein distance, the dimension-averaged one-dimensional Kolmogorov--Smirnov test, and the Frobenius norm of the difference between correlation matrices. Furthermore, we included estimations of the variance of both the metrics and the trained models. Our results indicate that the A-RQS algorithm stands out both in terms of accuracy and training speed. Nonetheless, all the algorithms are generally able, without too much fine-tuning, to learn complicated distributions with limited training data and in a reasonable time of the order of hours on a Tesla A40 GPU. The only exception is the C-RQS, which takes significantly longer to train, does not always provide good accuracy, and becomes unstable for large dimensionalities. All algorithms were implemented using textscTensorFlow2 and textscTensorFlow Probability and have been made available on hrefhttps://github.com/NF4HEP/NormalizingFlowsHDGitHub.