# Huber-Wasserstein barycenters for robust distribution-valued data

arXiv: https://arxiv.org/abs/2608.13131 · HTML: https://mlsanity.com/paper/2608.13131/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 70 references that are in the map.
- First posted on arXiv 13 Aug 2026.

## Builds on (most cited first, 6 of 6)

- [On the breakdown point of transport-based quantiles](https://mlsanity.com/paper/2410.16554/) (2024)
- [On the robustness of semi-discrete optimal transport](https://mlsanity.com/paper/2410.19596/) (2024)
- [Wasserstein Spatial Depth](https://mlsanity.com/paper/2411.10646/) (2024)
- [Robust Wasserstein barycenter](https://mlsanity.com/paper/2603.07563/) (2026)
- [Breakdown properties of optimal transport maps: general transportation costs](https://mlsanity.com/paper/2603.16005/) (2026)
- [The Influence Function of Transport-based Quantiles](https://mlsanity.com/paper/2607.19080/) (2026)

## Abstract (arXiv)

We propose a robust barycenter for distribution-valued data by incorporating the Huber loss directly into the optimal transport cost. In contrast to metric-space Huber means, which apply the Huber loss to the Wasserstein distance after optimization, our construction acts on individual transport displacements, preserving quadratic behavior locally while limiting the influence of large displacements. The resulting Huber-Wasserstein barycenters form a natural interpolation between Wasserstein means and L1-type Wasserstein medians. We establish the analytical and statistical foundations of this construction. For optimal transport with Huber loss, we prove regularity and uniqueness properties of dual potentials, existence of optimal transport maps, and stability as the Huber parameter varies. For the associated barycenter problem, we prove existence and characterization results, consistency of empirical plug-in estimators, and a finite-sample breakdown point essentially equal to 1/2. In dimension one, we further derive the pointwise influence function and asymptotic distribution, quantify the associated robustness-efficiency trade-off, and show that displacement-wise Huberization can retain first-order information that is lost by distance-based Huberization under localized shape contamination. Numerical experiments on contaminated distribution-valued data demonstrate the robustness of the proposed barycenters and illustrate their interpolation between mean- and median-like behavior.