# Fair Normalizing Flows

arXiv: https://arxiv.org/abs/2106.05937 · HTML: https://mlsanity.com/paper/2106.05937/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- Builds on 52 of its 93 references that are in the map.
- First posted on arXiv 10 Jun 2021.

## Builds on (most cited first, 30 of 52)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Equality of Opportunity in Supervised Learning](https://mlsanity.com/paper/1610.02413/) (2016)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (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)
- [Conditional Image Generation with PixelCNN Decoders](https://mlsanity.com/paper/1606.05328/) (2016)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Learning Adversarially Fair and Transferable Representations](https://mlsanity.com/paper/1802.06309/) (2018)
- [Actionable Recourse in Linear Classification](https://mlsanity.com/paper/1809.06514/) (2018)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [Algorithmic Recourse: from Counterfactual Explanations to Interventions](https://mlsanity.com/paper/2002.06278/) (2020)
- [FACE: Feasible and Actionable Counterfactual Explanations](https://mlsanity.com/paper/1909.09369/) (2019)
- [Pareto Multi-Task Learning](https://mlsanity.com/paper/1912.12854/) (2019)
- [The Variational Fair Autoencoder](https://mlsanity.com/paper/1511.00830/) (2015)
- [Flexibly Fair Representation Learning by Disentanglement](https://mlsanity.com/paper/1906.02589/) (2019)
- [Censoring Representations with an Adversary](https://mlsanity.com/paper/1511.05897/) (2015)
- [Adversarial Removal of Demographic Attributes from Text Data](https://mlsanity.com/paper/1808.06640/) (2018)
- [A Theory of Usable Information Under Computational Constraints](https://mlsanity.com/paper/2002.10689/) (2020)
- [Minimax Pareto Fairness: A Multi Objective Perspective](https://mlsanity.com/paper/2011.01821/) (2020)
- [On the Fairness of Disentangled Representations](https://mlsanity.com/paper/1905.13662/) (2019)
- [Inherent Tradeoffs in Learning Fair Representations](https://mlsanity.com/paper/1906.08386/) (2019)
- [Controllable Invariance through Adversarial Feature Learning](https://mlsanity.com/paper/1705.11122/) (2017)
- [Invariant Representations without Adversarial Training](https://mlsanity.com/paper/1805.09458/) (2018)
- [Generative Probabilistic Novelty Detection with Adversarial Autoencoders](https://mlsanity.com/paper/1807.02588/) (2018)
- [Learning Controllable Fair Representations](https://mlsanity.com/paper/1812.04218/) (2018)
- [Integer Discrete Flows and Lossless Compression](https://mlsanity.com/paper/1905.07376/) (2019)

## Built on by (most cited first, 30 of 38)

- [Bias Mitigation for Machine Learning Classifiers: A Comprehensive Survey](https://mlsanity.com/paper/2207.07068/) (2022)
- [Efficient fair PCA for fair representation learning](https://mlsanity.com/paper/2302.13319/) (2023)
- [FARE: Provably Fair Representation Learning with Practical Certificates](https://mlsanity.com/paper/2210.07213/) (2022)
- [Latent Space Smoothing for Individually Fair Representations](https://mlsanity.com/paper/2111.13650/) (2021)
- [Unraveling Privacy Risks of Individual Fairness in Graph Neural Networks](https://mlsanity.com/paper/2301.12951/) (2023)
- [Don't Throw it Away! The Utility of Unlabeled Data in Fair Decision Making](https://mlsanity.com/paper/2205.04790/) (2022)
- [CuTS: Customizable Tabular Synthetic Data Generation](https://mlsanity.com/paper/2307.03577/) (2023)
- [Certifying Some Distributional Fairness with Subpopulation Decomposition](https://mlsanity.com/paper/2205.15494/) (2022)
- [Normalizing Flow-based Differentiable Particle Filters](https://mlsanity.com/paper/2403.01499/) (2024)
- [Scalable Infomin Learning](https://mlsanity.com/paper/2302.10701/) (2023)
- [Monitoring Algorithmic Fairness](https://mlsanity.com/paper/2305.15979/) (2023)
- [Towards Fair In-Context Learning with Tabular Foundation Models](https://mlsanity.com/paper/2505.09503/) (2025)
- [Monitoring Algorithmic Fairness under Partial Observations](https://mlsanity.com/paper/2308.00341/) (2023)
- [Toward Operationalizing Pipeline-aware ML Fairness: A Research Agenda for Developing Practical Guidelines and Tools](https://mlsanity.com/paper/2309.17337/) (2023)
- [10 Years of Fair Representations: Challenges and Opportunities](https://mlsanity.com/paper/2407.03834/) (2024)
- [FairNorm: Fair and Fast Graph Neural Network Training](https://mlsanity.com/paper/2205.09977/) (2022)
- [From Principle to Practice: Vertical Data Minimization for Machine Learning](https://mlsanity.com/paper/2311.10500/) (2023)
- [Benchmarking the Fairness of Image Upsampling Methods](https://mlsanity.com/paper/2401.13555/) (2024)
- [Efficient Fairness-Performance Pareto Front Computation](https://mlsanity.com/paper/2409.17643/) (2024)
- [Counterfactual Reasoning for Bias Evaluation and Detection in a Fairness under Unawareness setting](https://mlsanity.com/paper/2302.08204/) (2023)
- [Weighted Risk Invariance: Domain Generalization under Invariant Feature Shift](https://mlsanity.com/paper/2407.18428/) (2024)
- [Do Fairness Interventions Come at the Cost of Privacy: Evaluations for Binary Classifiers](https://mlsanity.com/paper/2503.06150/) (2025)
- [Underrepresentation, Label Bias, and Proxies: Towards Data Bias Profiles for the EU AI Act and Beyond](https://mlsanity.com/paper/2507.08866/) (2025)
- [On Optimal Steering to Achieve Exact Fairness](https://mlsanity.com/paper/2509.15759/) (2025)
- [Learning Fair Representations with High-Confidence Guarantees](https://mlsanity.com/paper/2310.15358/) (2023)
- [AdapFair: Ensuring Adaptive Fairness for Machine Learning Operations](https://mlsanity.com/paper/2409.15088/) (2024)
- [SoK: Data Minimization in Machine Learning](https://mlsanity.com/paper/2508.10836/) (2025)
- [Face4FairShifts: A Large Image Benchmark for Fairness and Robust Learning across Visual Domains](https://mlsanity.com/paper/2509.00658/) (2025)
- [Counterfactual Fair Opportunity: Measuring Decision Model Fairness with Counterfactual Reasoning](https://mlsanity.com/paper/2302.08158/) (2023)
- [A Fair Classifier Embracing Triplet Collapse](https://mlsanity.com/paper/2306.04400/) (2023)

## Abstract (arXiv)

Fair representation learning is an attractive approach that promises fairness of downstream predictors by encoding sensitive data. Unfortunately, recent work has shown that strong adversarial predictors can still exhibit unfairness by recovering sensitive attributes from these representations. In this work, we present Fair Normalizing Flows (FNF), a new approach offering more rigorous fairness guarantees for learned representations. Specifically, we consider a practical setting where we can estimate the probability density for sensitive groups. The key idea is to model the encoder as a normalizing flow trained to minimize the statistical distance between the latent representations of different groups. The main advantage of FNF is that its exact likelihood computation allows us to obtain guarantees on the maximum unfairness of any potentially adversarial downstream predictor. We experimentally demonstrate the effectiveness of FNF in enforcing various group fairness notions, as well as other attractive properties such as interpretability and transfer learning, on a variety of challenging real-world datasets.