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Wasserstein medians: robustness, PDE characterization and numerics

Guillaume Carlier, Enis Chenchene, Katharina Eichinger

TL;DR

The paper investigates Wasserstein medians as robust Fréchet medians in the $W_1$ space, offering a rigorous foundation for their existence, stability, and practical computation. It develops both one-dimensional constructions (vertical and horizontal median selections) and rich higher-dimensional theory via multimarginal reformulations and a Beckenmann/MK PDE framework, including a $p$-Laplace approximation. A central result is that Wasserstein medians have a breakdown point of approximately $\tfrac{1}{2}$, highlighting robustness to outliers, with explicit 1D properties preserved under density bounds. Numerically, the authors introduce a Douglas–Rachford splitting method on the Beckmann flow formulation, achieving scalable and stable computation of medians on grids, and they compare against sorting, LP, and Sinkhorn approaches. Overall, the work provides a cohesive theory and practical solver that enable robust aggregation and interpolation of probability measures beyond the standard Wasserstein barycenter.

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 à 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.

Wasserstein medians: robustness, PDE characterization and numerics

TL;DR

The paper investigates Wasserstein medians as robust Fréchet medians in the space, offering a rigorous foundation for their existence, stability, and practical computation. It develops both one-dimensional constructions (vertical and horizontal median selections) and rich higher-dimensional theory via multimarginal reformulations and a Beckenmann/MK PDE framework, including a -Laplace approximation. A central result is that Wasserstein medians have a breakdown point of approximately , highlighting robustness to outliers, with explicit 1D properties preserved under density bounds. Numerically, the authors introduce a Douglas–Rachford splitting method on the Beckmann flow formulation, achieving scalable and stable computation of medians on grids, and they compare against sorting, LP, and Sinkhorn approaches. Overall, the work provides a cohesive theory and practical solver that enable robust aggregation and interpolation of probability measures beyond the standard Wasserstein barycenter.

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 . We give explicit constructions of Wasserstein medians in dimension one which enable us to obtain estimates (which do not hold in higher dimensions). We also address dual and multimarginal reformulations. In convex subsets of , we connect Wasserstein medians to a minimal (multi) flow problem à la Beckmann and a system of PDEs of Monge-Kantorovich-type, for which we propose a -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.
Paper Structure (23 sections, 17 theorems, 183 equations, 6 figures, 1 algorithm)

This paper contains 23 sections, 17 theorems, 183 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Definition, existence and basic examples
  3. Setting.
  4. Wasserstein medians.
  5. Stability and robustness
  6. Compactness in $W_1$ distance and stability with respect to data
  7. Robustness of Wasserstein medians
  8. One dimensional Wasserstein medians
  9. Multi-marginal and dual formulations
  10. Multi-marginal formulation
  11. A counterexample to linear $L^{\infty}$ density bounds in several dimensions.
  12. Dual Formulation
  13. Beckmann minimal flow formulation
  14. The Beckmann problem
  15. A system of PDEs for Wasserstein medians
  16. ...and 8 more sections

Key Result

Lemma 2.4

Let $N\geq1$, $\boldsymbol{\nu}=(\nu_1,\dots,\nu_N) \in \mathcal{P}_1(\mathcal{X})^N$ and $\boldsymbol{\lambda}\in \Delta_N$, then there exists a minimizer of eq:Medianprob and the set $\mathop{\mathrm{\mathsf{Med}}}\nolimits_{\boldsymbol{\lambda}}(\nu_1,\dots,\nu_N)$ is a convex and narrowly compac

Figures (6)

  • Figure 1: Superposition of a Wasserstein median (blue), a Wasserstein barycenter (black) and the corresponding sample of $N = 9, \ 29, \ 39, \ 59,\ 81$ one-dimensional histograms. Each histogram represents the daily attendance frequency of some London underground stations. Second row: the corresponding cumulative distribution functions.
  • Figure 2: Comparison between a Wasserstein barycenter and a Wasserstein median for a sample of five measures computed with Sinkhorn (cf., Section \ref{['sec:numerics']}) in $1000$ iterations.
  • Figure 3: Some Wasserstein medians on a $420\times420$ grid computed with Douglas--Rachford up to $2000$ iterations, with a final residual of about $\sim 10^{-7}$, cf., Section \ref{['sec:numerics']}.
  • Figure 4: Comparison between different Wasserstein median selections. In the second line we displayed the corresponding cumulative distribution functions.
  • Figure 5: Counterexample to linear $L^\infty$ bounds in dimension two. The support of the four given uniformly distributed measures are indicated in gray. The support of any Wasserstein median is contained in the green area. Confer Example \ref{['ex:counterex-reg']} for further details.
  • ...and 1 more figures

Theorems & Definitions (47)

  • Example 2.1: Medians on the real line
  • Example 2.2: Torricelli--Fermat--Weber points
  • Definition 2.3: Wasserstein medians
  • Lemma 2.4: Existence of Wasserstein medians
  • proof
  • Example 2.5: Medians of Dirac masses
  • Example 2.6: Threshold effect
  • Example 2.7: Medians of two measures
  • Example 2.8: Medians of translated measures
  • Lemma 3.1
  • ...and 37 more