On the Wasserstein median of probability measures

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.