On the Wasserstein median of probability measures

Measures of central tendency such as the mean and the median are a primary way to summarize a given collection of random objects. In the field of optimal transport, the Wasserstein barycenter corresponds to the Fr\'echet or geometric mean of a set of probability measures, which is defined as a minimizer of the sum of its squared distances to each element of the set when the order is 2. We present the Wasserstein median, an equivalent of the Fr\'echet median under the 2- Wasserstein metric, as a robust alternative to the barycenter. The Wasserstein median is shown to be well defined and exist under mild conditions. We also propose an algorithm that makes use of any established routine for the Wasserstein barycenter in an iterative manner and prove its convergence. Our proposal is validated with simulated and real data examples when the objects of interest are univariate distributions, centered Gaussian distributions, and discrete measures on regular grids.