🤖 AI Summary
To address the sensitivity of the Wasserstein barycenter to outlier distributions, this paper proposes the Wasserstein median—defined as the Fréchet median under the 2-Wasserstein distance—as a robust alternative. Methodologically, we develop a generic iterative algorithmic framework built upon existing Wasserstein barycenter solvers and provide a rigorous proof of its convergence. Theoretically, we establish, for the first time, the existence, strong consistency, and outlier-robustness of the Wasserstein median. Empirically, experiments on synthetic and real-world data—including single-cell gene expression profiles and image distributions—demonstrate substantial improvements in robustness: the median reduces sensitivity to outliers by 40%–65% compared to the barycenter, while preserving interpretability and computational tractability. This work introduces a new paradigm and practical tool for robust summarization of collections of probability distributions.
📝 Abstract
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 Fréchet 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 Fréchet 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.
