Optimal Transport with Huber Loss: Barycentres and Robustness

We introduce a robust framework for distributional data analysis based on optimal transport with Huber loss. Wasserstein barycentres provide a natural notion of centrality for probability measures, but they are known to be sensitive to outliers. To address this, we study barycentres induced by a Huber optimal transport cost, combining the local quadratic behaviour of the 2-Wasserstein geometry with the linear growth that underlies robust estimation.

We define a Huber-loss barycentre through a convex objective built from the optimal transport cost associated with the Huber loss. This construction yields a robust alternative to the classical Wasserstein barycentre while preserving desirable stability and convexity properties. We establish duality results for the Huber transport problem, prove stability of the transport cost and dual potentials, and study the existence of Huber optimal transport maps. We further show the existence and stability of the resulting barycentres under weak assumptions, provide a characterisation of minimisers, and analyse their finite-sample breakdown point.

This work is currently in preparation.