Towards Marginal Fairness Sliced Wasserstein Barycenter
Khai Nguyen, Hai Nguyen, Nhat Ho
TL;DR
The paper tackles marginal fairness in sliced Wasserstein barycenters by formulating Marginal Fairness Sliced Wasserstein Barycenter (MFSWB) as a constrained optimization problem and proposing three hyperparameter-free surrogates—$\text{s-}MFSWB$, $\text{us-}MFSWB$, and $\text{es-}MFSWB$—that avoid tuning while promoting balanced distances to marginals. It establishes theoretical connections to sliced multi-marginal Wasserstein distance with a maximal ground metric and shows that these surrogates provide computable lower bounds with unbiased gradients in practice. Through experiments on Gaussian mixtures, 3D point-cloud averaging, color harmonization, and class-fair SWAE, the authors demonstrate consistent improvements in marginal fairness relative to standard SWB baselines, with es-MFSWB frequently achieving the best trade-off between fairness and centrality. The work offers scalable, robust tools for fair averaging in high-dimensional measure spaces and outlines directions for extending to generalized sliced distances and broader fairness settings.
Abstract
The sliced Wasserstein barycenter (SWB) is a widely acknowledged method for efficiently generalizing the averaging operation within probability measure spaces. However, achieving marginal fairness SWB, ensuring approximately equal distances from the barycenter to marginals, remains unexplored. The uniform weighted SWB is not necessarily the optimal choice to obtain the desired marginal fairness barycenter due to the heterogeneous structure of marginals and the non-optimality of the optimization. As the first attempt to tackle the problem, we define the marginal fairness sliced Wasserstein barycenter (MFSWB) as a constrained SWB problem. Due to the computational disadvantages of the formal definition, we propose two hyperparameter-free and computationally tractable surrogate MFSWB problems that implicitly minimize the distances to marginals and encourage marginal fairness at the same time. To further improve the efficiency, we perform slicing distribution selection and obtain the third surrogate definition by introducing a new slicing distribution that focuses more on marginally unfair projecting directions. We discuss the relationship of the three proposed problems and their relationship to sliced multi-marginal Wasserstein distance. Finally, we conduct experiments on finding 3D point-clouds averaging, color harmonization, and training of sliced Wasserstein autoencoder with class-fairness representation to show the favorable performance of the proposed surrogate MFSWB problems.