# Optimal Transport under Group Fairness Constraints

arXiv: https://arxiv.org/abs/2601.07144 · HTML: https://mlsanity.com/paper/2601.07144/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 41 of its 83 references that are in the map.
- First posted on arXiv 12 Jan 2026.

## Builds on (most cited first, 30 of 41)

- [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)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Gradient-based Hyperparameter Optimization through Reversible Learning](https://mlsanity.com/paper/1502.03492/) (2015)
- [Approximation Methods for Bilevel Programming](https://mlsanity.com/paper/1802.02246/) (2018)
- [Forward and Reverse Gradient-Based Hyperparameter Optimization](https://mlsanity.com/paper/1703.01785/) (2017)
- [Inherent Tradeoffs in Learning Fair Representations](https://mlsanity.com/paper/1906.08386/) (2019)
- [Fair Regression with Wasserstein Barycenters](https://mlsanity.com/paper/2006.07286/) (2020)
- [A framework for bilevel optimization that enables stochastic and global variance reduction algorithms](https://mlsanity.com/paper/2201.13409/) (2022)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)
- [Entropic estimation of optimal transport maps](https://mlsanity.com/paper/2109.12004/) (2021)
- [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019)
- [Statistical and Topological Properties of Sliced Probability Divergences](https://mlsanity.com/paper/2003.05783/) (2020)
- [Smooth and Sparse Optimal Transport](https://mlsanity.com/paper/1710.06276/) (2017)
- [The Price of Fair PCA: One Extra Dimension](https://mlsanity.com/paper/1811.00103/) (2018)
- [Fair and Optimal Classification via Post-Processing](https://mlsanity.com/paper/2211.01528/) (2022)
- [Sparsity-Constrained Optimal Transport](https://mlsanity.com/paper/2209.15466/) (2022)
- [Structured Optimal Transport](https://mlsanity.com/paper/1712.06199/) (2017)
- [A Unified Framework for Implicit Sinkhorn Differentiation](https://mlsanity.com/paper/2205.06688/) (2022)
- [One-step differentiation of iterative algorithms](https://mlsanity.com/paper/2305.13768/) (2023)
- [Learning to Match via Inverse Optimal Transport](https://mlsanity.com/paper/1802.03644/) (2018)
- [A Lower Bound and a Near-Optimal Algorithm for Bilevel Empirical Risk Minimization](https://mlsanity.com/paper/2302.08766/) (2023)
- [Asymptotics for semi-discrete entropic optimal transport](https://mlsanity.com/paper/2106.11862/) (2021)
- [Fairness in Multi-Task Learning via Wasserstein Barycenters](https://mlsanity.com/paper/2306.10155/) (2023)
- [Optimal Transport of Classifiers to Fairness](https://mlsanity.com/paper/2202.03814/) (2022)
- [Generalized conditional gradient: analysis of convergence and applications](https://mlsanity.com/paper/1510.06567/) (2015)
- [The derivatives of Sinkhorn-Knopp converge](https://mlsanity.com/paper/2207.12717/) (2022)
- [Fair Wasserstein Coresets](https://mlsanity.com/paper/2311.05436/) (2023)
- [SISTA: learning optimal transport costs under sparsity constraints](https://mlsanity.com/paper/2009.08564/) (2020)
- [On Sinkhorn's Algorithm and Choice Modeling](https://mlsanity.com/paper/2310.00260/) (2023)

## Abstract (arXiv)

Ensuring fairness in matching algorithms is a key challenge in allocating scarce resources and positions. Focusing on Optimal Transport (OT), we introduce a novel notion of group fairness requiring that the probability of matching two individuals from any two given groups in the OT plan satisfies a predefined target. We first propose a modified Sinkhorn algorithm to compute perfectly fair transport plans efficiently. Since exact fairness can significantly degrade matching quality in practice, we then develop two relaxation strategies. The first one involves solving a penalized OT problem, for which we derive novel finite-sample complexity guarantees. Our second strategy leverages bilevel optimization to learn a ground cost that induces a fair OT solution, and we establish a bound on the deviation of fairness when matching unseen data. Finally, we present empirical results illustrating the performance of our approaches and the trade-off between fairness and transport cost.