# Flexible Tails for Normalizing Flows

arXiv: https://arxiv.org/abs/2406.16971 · HTML: https://mlsanity.com/paper/2406.16971/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 50 references that are in the map.
- First posted on arXiv 22 Jun 2024.

## Builds on (most cited first, 23 of 23)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Sequential Neural Likelihood: Fast Likelihood-free Inference with Autoregressive Flows](https://mlsanity.com/paper/1805.07226/) (2018)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators](https://mlsanity.com/paper/2006.11469/) (2020)
- [Sum-of-Squares Polynomial Flow](https://mlsanity.com/paper/1905.02325/) (2019)
- [Yes, but Did It Work?: Evaluating Variational Inference](https://mlsanity.com/paper/1802.02538/) (2018)
- [Transformation Autoregressive Networks](https://mlsanity.com/paper/1801.09819/) (2018)
- [On the Universality of Volume-Preserving and Coupling-Based Normalizing Flows](https://mlsanity.com/paper/2402.06578/) (2024)
- [Marginal Tail-Adaptive Normalizing Flows](https://mlsanity.com/paper/2206.10311/) (2022)
- [Fat-Tailed Variational Inference with Anisotropic Tail Adaptive Flows](https://mlsanity.com/paper/2205.07918/) (2022)
- [Variational auto-encoders with Student's t-prior](https://mlsanity.com/paper/2004.02581/) (2020)
- [COMET Flows: Towards Generative Modeling of Multivariate Extremes and Tail Dependence](https://mlsanity.com/paper/2205.01224/) (2022)
- [Modeling of spatial extremes in environmental data science: Time to move away from max-stable processes](https://mlsanity.com/paper/2401.17430/) (2024)
- [Stereographic Markov Chain Monte Carlo](https://mlsanity.com/paper/2205.12112/) (2022)
- [An Extreme Value Bayesian Lasso for the Conditional Left and Right Tails](https://mlsanity.com/paper/2010.07164/) (2020)
- [A Heavy-Tailed Algebra for Probabilistic Programming](https://mlsanity.com/paper/2306.09262/) (2023)
- [Flexible Tails for Normalising Flows, with Application to the Modelling of Financial Return Data](https://mlsanity.com/paper/2311.00580/) (2023)

## Built on by (most cited first, 12 of 12)

- [Generative Machine Learning for Multivariate Angular Simulation](https://mlsanity.com/paper/2504.21505/) (2025)
- [Simulation of Multivariate Extremes: a Wasserstein-Aitchison GAN approach](https://mlsanity.com/paper/2504.21438/) (2025)
- [Uncertainty Propagation Networks for Neural Ordinary Differential Equations](https://mlsanity.com/paper/2508.16815/) (2025)
- [Fine-Tuning Generative Models for Extreme Events via CVaR-Penalized Wasserstein Gradient Flows](https://mlsanity.com/paper/2608.11544/) (2026)
- [Comparing Normalizing Flows with Kernel Density Estimation in Estimating Risk of Automated Driving Systems](https://mlsanity.com/paper/2507.22429/) (2025)
- [Stick-Breaking Mixture Normalizing Flows with Component-Wise Tail Adaptation for Variational Inference](https://mlsanity.com/paper/2510.07965/) (2025)
- [Adapting Noise to Data: Generative Flows from 1D Processes](https://mlsanity.com/paper/2510.12636/) (2025)
- [Correcting Source Mismatch in Flow Matching with Radial-Angular Transport](https://mlsanity.com/paper/2604.04291/) (2026)
- [Tail Annealing for Heavy-Tailed Flow Matching](https://mlsanity.com/paper/2605.20068/) (2026)
- [Decomposing Ensemble Spread in Lorenz '96 With Learned Stochastic Parameterizations](https://mlsanity.com/paper/2605.22242/) (2026)
- [Heavy-Tailed Flow Matching via Random Clocks](https://mlsanity.com/paper/2607.13841/) (2026)
- [Radial Tail Repair: Post-Training Correction for Diffusion and Flow Models](https://mlsanity.com/paper/2610.16851/) (2026)

## Abstract (arXiv)

Normalizing flows are a flexible class of probability distributions, expressed as transformations of a simple base distribution. A limitation of standard normalizing flows is representing distributions with heavy tails, which arise in applications to both density estimation and variational inference. A popular current solution to this problem is to use a heavy tailed base distribution. We argue this can lead to poor performance due to the difficulty of optimising neural networks, such as normalizing flows, under heavy tailed input. We propose an alternative, "tail transform flow" (TTF), which uses a Gaussian base distribution and a final transformation layer which can produce heavy tails. Experimental results show this approach outperforms current methods, especially when the target distribution has large dimension or tail weight.