On the Wasserstein median of probability measures

TL;DR

This paper introduces the Wasserstein median as a robust alternative to the Wasserstein barycenter for summarizing probability measures, establishing its theoretical properties and demonstrating its practical utility.

Contribution

It defines the Wasserstein median, proves its existence and robustness, and develops an iterative computational method using existing algorithms.

Findings

Wasserstein median exists and is consistent.

The method demonstrates robustness to outliers.

Applications show effectiveness on real and simulated data.

Abstract

The primary choice to summarize a finite collection of random objects is by using measures of central tendency, such as mean and median. In the field of optimal transport, the Wasserstein barycenter corresponds to the Fr\'{e}chet or geometric mean of a set of probability measures, which is defined as a minimizer of the sum of squared distances to each element in a given set with respect to the Wasserstein distance of order 2. We introduce the Wasserstein median as a robust alternative to the Wasserstein barycenter. The Wasserstein median corresponds to the Fr\'{e}chet median under the 2-Wasserstein metric. The existence and consistency of the Wasserstein median are first established, along with its robustness property. In addition, we present a general computational pipeline that employs any recognized algorithms for the Wasserstein barycenter in an iterative fashion and demonstrate its convergence. The utility of the Wasserstein median as a robust measure of central tendency is demonstrated using real and simulated data.