# Robust normalizing flows using Bernstein-type polynomials

arXiv: https://arxiv.org/abs/2102.03509 · HTML: https://mlsanity.com/paper/2102.03509/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 44 references that are in the map.
- First posted on arXiv 6 Feb 2021.

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

- [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)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [WaveGlow: A Flow-based Generative Network for Speech Synthesis](https://mlsanity.com/paper/1811.00002/) (2018)
- [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)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Noise Flow: Noise Modeling with Conditional Normalizing Flows](https://mlsanity.com/paper/1908.08453/) (2019)
- [Sylvester Normalizing Flows for Variational Inference](https://mlsanity.com/paper/1803.05649/) (2018)
- [Targeted free energy estimation via learned mappings](https://mlsanity.com/paper/2002.04913/) (2020)
- [Sum-of-Squares Polynomial Flow](https://mlsanity.com/paper/1905.02325/) (2019)
- [Adversarial examples for generative models](https://mlsanity.com/paper/1702.06832/) (2017)
- [Leveraging exploration in off-policy algorithms via normalizing flows](https://mlsanity.com/paper/1905.06893/) (2019)
- [Improving Exploration in Soft-Actor-Critic with Normalizing Flows Policies](https://mlsanity.com/paper/1906.02771/) (2019)
- [Cubic-Spline Flows](https://mlsanity.com/paper/1906.02145/) (2019)
- [Deep Conditional Transformation Models](https://mlsanity.com/paper/2010.07860/) (2020)
- [Most Likely Transformations](https://mlsanity.com/paper/1508.06749/) (2015)
- [Universal audio synthesizer control with normalizing flows](https://mlsanity.com/paper/1907.00971/) (2019)
- [Woodbury Transformations for Deep Generative Flows](https://mlsanity.com/paper/2002.12229/) (2020)
- [Provably robust deep generative models](https://mlsanity.com/paper/2004.10608/) (2020)

## Built on by (most cited first, 6 of 6)

- [Smooth Normalizing Flows](https://mlsanity.com/paper/2110.00351/) (2021)
- [Short-Term Density Forecasting of Low-Voltage Load using Bernstein-Polynomial Normalizing Flows](https://mlsanity.com/paper/2204.13939/) (2022)
- [Embracing the chaos: analysis and diagnosis of numerical instability in variational flows](https://mlsanity.com/paper/2307.06957/) (2023)
- [Bernstein Flows for Flexible Posteriors in Variational Bayes](https://mlsanity.com/paper/2202.05650/) (2022)
- [How Inverse Conditional Flows Can Serve as a Substitute for Distributional Regression](https://mlsanity.com/paper/2405.05429/) (2024)
- [Transformation Models for Flexible Posteriors in Variational Bayes](https://mlsanity.com/paper/2106.00528/) (2021)

## Abstract (arXiv)

Modeling real-world distributions can often be challenging due to sample data that are subjected to perturbations, e.g., instrumentation errors, or added random noise. Since flow models are typically nonlinear algorithms, they amplify these initial errors, leading to poor generalizations. This paper proposes a framework to construct Normalizing Flows (NF), which demonstrates higher robustness against such initial errors. To this end, we utilize Bernstein-type polynomials inspired by the optimal stability of the Bernstein basis. Further, compared to the existing NF frameworks, our method provides compelling advantages like theoretical upper bounds for the approximation error, higher interpretability, suitability for compactly supported densities, and the ability to employ higher degree polynomials without training instability. We conduct a thorough theoretical analysis and empirically demonstrate the efficacy of the proposed technique using experiments on both real-world and synthetic datasets.