Wasserstein medians: robustness, PDE characterization and numerics

TL;DR

This paper explores Wasserstein medians as a robust alternative to barycenters, providing theoretical analysis, explicit constructions, PDE characterizations, and a new numerical method based on a Douglas-Rachford scheme.

Contribution

It introduces the concept of Wasserstein medians, analyzes their robustness, provides explicit 1D constructions, links them to PDEs and flow problems, and develops a novel numerical algorithm.

Findings

Wasserstein medians have a breakdown point of approximately 1/2.

Explicit constructions of Wasserstein medians in one dimension enable $L^p$ estimates.

A new numerical method using Douglas-Rachford scheme effectively computes Wasserstein medians.

Abstract

We investigate the notion of Wasserstein median as an alternative to the Wasserstein barycenter, which has become popular but may be sensitive to outliers. In terms of robustness to corrupted data, we indeed show that Wasserstein medians have a breakdown point of approximately $\frac{1}{2}$. We give explicit constructions of Wasserstein medians in dimension one which enable us to obtain $L^p$ estimates (which do not hold in higher dimensions). We also address dual and multimarginal reformulations. In convex subsets of $\mathbb{R}^d$, we connect Wasserstein medians to a minimal (multi) flow problem \`a la Beckmann and a system of PDEs of Monge-Kantorovich-type, for which we propose a $p$-Laplacian approximation. Our analysis eventually leads to a new numerical method to compute Wasserstein medians, which is based on a Douglas-Rachford scheme applied to the minimal flow formulation of the problem.