# Optimal Pre-Processing to Achieve Fairness and Its Relationship with Total Variation Barycenter

arXiv: https://arxiv.org/abs/2101.06811 · HTML: https://mlsanity.com/paper/2101.06811/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 26 references that are in the map.
- First posted on arXiv 18 Jan 2021.

## Builds on (most cited first, 5 of 5)

- [Certifying and removing disparate impact](https://mlsanity.com/paper/1412.3756/) (2014)
- [Fairness Beyond Disparate Treatment & Disparate Impact: Learning Classification without Disparate Mistreatment](https://mlsanity.com/paper/1610.08452/) (2016)
- [Fairness in Criminal Justice Risk Assessments: The State of the Art](https://mlsanity.com/paper/1703.09207/) (2017)
- [Empirical Risk Minimization under Fairness Constraints](https://mlsanity.com/paper/1802.08626/) (2018)
- [Wasserstein Fair Classification](https://mlsanity.com/paper/1907.12059/) (2019)

## Built on by (most cited first, 3 of 3)

- [How Robust is your Fair Model? Exploring the Robustness of Diverse Fairness Strategies](https://mlsanity.com/paper/2207.04581/) (2022)
- [Stay Fair! Ensuring Group Fairness in Diffusion Models Across Guidance Scales](https://mlsanity.com/paper/2605.28036/) (2026)
- [Intersectional Fairness via Mixed-Integer Optimization](https://mlsanity.com/paper/2601.19595/) (2026)

## Abstract (arXiv)

We use disparate impact, i.e., the extent that the probability of observing an output depends on protected attributes such as race and gender, to measure fairness. We prove that disparate impact is upper bounded by the total variation distance between the distribution of the inputs given the protected attributes. We then use pre-processing, also known as data repair, to enforce fairness. We show that utility degradation, i.e., the extent that the success of a forecasting model changes by pre-processing the data, is upper bounded by the total variation distance between the distribution of the data before and after pre-processing. Hence, the problem of finding the optimal pre-processing regiment for enforcing fairness can be cast as minimizing total variations distance between the distribution of the data before and after pre-processing subject to a constraint on the total variation distance between the distribution of the inputs given protected attributes. This problem is a linear program that can be efficiently solved. We show that this problem is intimately related to finding the barycenter (i.e., center of mass) of two distributions when distances in the probability space are measured by total variation distance. We also investigate the effect of differential privacy on fairness using the proposed the total variation distances. We demonstrate the results using numerical experimentation with a practice dataset.