ICML 2025

Finding Wasserstein Ball Center: Efficient Algorithm and The Applications in
Fairness
Yuntao Wang * 1 Yuxuan li * 2 Qingyuan Yang 1 Hu Ding 1
Abstract
Wasserstein Barycenter (WB) is a fundamental
geometric optimization problem in machine learn-
ing, whose objective is to find a representative
probability measure that minimizes the sum of
Wasserstein distances to given distributions. WB
has a number of applications in various areas.
However, WB may lead to unfair outcome to-
wards underrepresented groups in some applica-
tions (e.g., a “minority” distribution may be far
away from the obtained WB under Wasserstein
distance). To address this issue, we propose an
alternative objective called “Wasserstein Ball Cen-
ter (WBC)”. Specifically, WBC is a distribution
that encompasses all input distributions within
the minimum Wasserstein distance, which can
be formulated as a “minmax” optimization prob-
lem. We show that the WBC problem with fixed
support is equivalent to solving a large-scale lin-
ear programming (LP) instance, which is quite
different from the previously studied LP model
for WB. By incorporating some novel observa-
tions on the induced normal equation, we propose
an efficient algorithm that accelerates the inte-
rior point method by O(min{N 2m, Nm2, m4})
times (“N” is the number of distributions and “m”
is the support size). Finally, we conduct a set
of experiments on both synthetic and real-world
datasets, demonstrating the computational effi-
ciency of our algorithm, and showing its ability
to provide more fairness for input distributions.
*Equal contribution 1School of Computer Science and Technol-
ogy, University of Science and Technology of China, Hefei, China
2School of Artificial Intelligence and Data Science, University of
Science and Technology of China, Hefei, China. Correspondence
to: Hu Ding <huding@ustc.edu.cn>.
Proceedings of the 42 nd International Conference on Machine
Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025
by the author...