# Piecewise Normalizing Flows

arXiv: https://arxiv.org/abs/2305.02930 · HTML: https://mlsanity.com/paper/2305.02930/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 38 references that are in the map.
- First posted on arXiv 4 May 2023.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [A Survey on Bias and Fairness in Machine Learning](https://mlsanity.com/paper/1908.09635/) (2019)
- [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)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Design](https://mlsanity.com/paper/1902.00275/) (2019)
- [Parallel WaveNet: Fast High-Fidelity Speech Synthesis](https://mlsanity.com/paper/1711.10433/) (2017)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Stochastic Normalizing Flows](https://mlsanity.com/paper/2002.06707/) (2020)
- [Identifying and Correcting Label Bias in Machine Learning](https://mlsanity.com/paper/1901.04966/) (2019)
- [Semi-Supervised Learning with Normalizing Flows](https://mlsanity.com/paper/1912.13025/) (2019)
- [Targeted free energy estimation via learned mappings](https://mlsanity.com/paper/2002.04913/) (2020)
- [Densely connected normalizing flows](https://mlsanity.com/paper/2106.04627/) (2021)
- [Resampling Base Distributions of Normalizing Flows](https://mlsanity.com/paper/2110.15828/) (2021)
- [Stabilizing Invertible Neural Networks Using Mixture Models](https://mlsanity.com/paper/2009.02994/) (2020)
- [Semi-Discrete Normalizing Flows through Differentiable Tessellation](https://mlsanity.com/paper/2203.06832/) (2022)
- [Marginal Post Processing of Bayesian Inference Products with Normalizing Flows and Kernel Density Estimators](https://mlsanity.com/paper/2205.12841/) (2022)
- [Nested sampling with any prior you like](https://mlsanity.com/paper/2102.12478/) (2021)

## Built on by (most cited first, 1 of 1)

- [Are Euler angles a useful rotation parameterisation for pose estimation with Normalizing Flows?](https://mlsanity.com/paper/2511.02277/) (2025)

## Abstract (arXiv)

Normalizing flows are an established approach for modelling complex probability densities through invertible transformations from a base distribution. However, the accuracy with which the target distribution can be captured by the normalizing flow is strongly influenced by the topology of the base distribution. A mismatch between the topology of the target and the base can result in a poor performance, as is typically the case for multi-modal problems. A number of different works have attempted to modify the topology of the base distribution to better match the target, either through the use of Gaussian Mixture Models (Izmailov et al., 2020; Ardizzone et al., 2020; Hagemann & Neumayer, 2021) or learned accept/reject sampling (Stimper et al., 2022). We introduce piecewise normalizing flows which divide the target distribution into clusters, with topologies that better match the standard normal base distribution, and train a series of flows to model complex multi-modal targets. We demonstrate the performance of the piecewise flows using some standard benchmarks and compare the accuracy of the flows to the approach taken in Stimper et al. (2022) for modelling multi-modal distributions. We find that our approach consistently outperforms the approach in Stimper et al. (2022) with a higher emulation accuracy on the standard benchmarks.