Finding Wasserstein Ball Center: Efficient Algorithm and The Applications in Fairness

Wasserstein Barycenter (WB) is a fundamental geometric optimization problem in machine learning, 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 towards underrepresented groups in some applications (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 Center (WBC)''. Specifically, WBC is a distribution that encompasses all input distributions within the minimum Wasserstein distance, which can be formulated as a ``minmax'' optimization problem. We show that the WBC problem with fixed support is equivalent to solving a large-scale linear programming (LP) instance, which is quite different from the previously studied LP model for WB. By incorporating some novel observations on the induced normal equation, we propose an efficient algorithm that accelerates the interior point method by $O(\min(N^2m, Nm^2, m^4))$ 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 efficiency of our algorithm, and showing its ability to provide more fairness for input distributions.

In this paper, we present an efficient algorithm to compute Wasserstein ball center, an alternative of Wasserstein barycenter with emphasis on minority probability distributions, which makes it more suitable for the data fusion tasks that are sensitive to fairness.