# Wasserstein medians: robustness, PDE characterization and numerics

arXiv: https://arxiv.org/abs/2307.01765 · HTML: https://mlsanity.com/paper/2307.01765/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jul 2023.

## Built on by (most cited first, 1 of 1)

- [Robust Barycenters of Persistence Diagrams](https://mlsanity.com/paper/2509.14904/) (2025)

## Abstract (arXiv)

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 frac12. We give explicit constructions of Wasserstein medians in dimension one which enable us to obtain Lp estimates (which do not hold in higher dimensions). We also address dual and multimarginal reformulations. In convex subsets of ℝd, 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.