# Constrained Reweighting of Distributions: an Optimal Transport Approach

arXiv: https://arxiv.org/abs/2310.12447 · HTML: https://mlsanity.com/paper/2310.12447/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 72 references that are in the map.
- First posted on arXiv 19 Oct 2023.

## Builds on (most cited first, 15 of 15)

- [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018)
- [Maximum Entropy RL (Provably) Solves Some Robust RL Problems](https://mlsanity.com/paper/2103.06257/) (2021)
- [Fair Regression: Quantitative Definitions and Reduction-based Algorithms](https://mlsanity.com/paper/1905.12843/) (2019)
- [Wasserstein Fair Classification](https://mlsanity.com/paper/1907.12059/) (2019)
- [On Formalizing Fairness in Prediction with Machine Learning](https://mlsanity.com/paper/1710.03184/) (2017)
- [Fair and Optimal Classification via Post-Processing](https://mlsanity.com/paper/2211.01528/) (2022)
- [Variational Tempering](https://mlsanity.com/paper/1411.1810/) (2014)
- [A General Framework for Fair Regression](https://mlsanity.com/paper/1810.05041/) (2018)
- [Achieving Fairness via Post-Processing in Web-Scale Recommender Systems](https://mlsanity.com/paper/2006.11350/) (2020)
- [Connectivity Inference from Neural Recording Data: Challenges, Mathematical Bases and Research Directions](https://mlsanity.com/paper/1708.01888/) (2017)
- [FORML: Learning to Reweight Data for Fairness](https://mlsanity.com/paper/2202.01719/) (2022)
- [Fair quantile regression](https://mlsanity.com/paper/1907.08646/) (2019)
- [Bayesian Estimation and Comparison of Moment Condition Models](https://mlsanity.com/paper/1606.02931/) (2016)
- [Fully Bayesian Estimation Under Informative Sampling](https://mlsanity.com/paper/1710.00019/) (2017)
- [Robust probabilistic inference via a constrained transport metric](https://mlsanity.com/paper/2303.10085/) (2023)

## Abstract (arXiv)

We commonly encounter the problem of identifying an optimally weight adjusted version of the empirical distribution of observed data, adhering to predefined constraints on the weights. Such constraints often manifest as restrictions on the moments, tail behaviour, shapes, number of modes, etc., of the resulting weight adjusted empirical distribution. In this article, we substantially enhance the flexibility of such methodology by introducing a nonparametrically imbued distributional constraints on the weights, and developing a general framework leveraging the maximum entropy principle and tools from optimal transport. The key idea is to ensure that the maximum entropy weight adjusted empirical distribution of the observed data is close to a pre-specified probability distribution in terms of the optimal transport metric while allowing for subtle departures. The versatility of the framework is demonstrated in the context of three disparate applications where data re-weighting is warranted to satisfy side constraints on the optimization problem at the heart of the statistical task: namely, portfolio allocation, semi-parametric inference for complex surveys, and ensuring algorithmic fairness in machine learning algorithms.