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The Wasserstein distance, rooted in optimal transport (OT) theory, is apopular discrepancy measure between probability distributions with variousapplications to statistics and machine learning. Despite their rich structureand demonstrated utility, Wasserstein distances are sensitive to outliers inthe considered distributions, which hinders applicability in practice. Wepropose a new outlier-robust Wasserstein distance $\mathsf{W}_p^\varepsilon$which allows for $\varepsilon$ outlier mass to be removed from eachcontaminated distribution. Under standard moment assumptions,$\mathsf{W}_p^\varepsilon$ is shown to achieve strong robust estimationguarantees under the Huber $\varepsilon$-contamination model. Our formulationof this robust distance amounts to a highly regular optimization problem thatlends itself better for analysis compared to previously considered frameworks.Leveraging this, we conduct a thorough theoretical study of$\mathsf{W}_p^\varepsilon$, encompassing robustness guarantees,characterization of optimal perturbations, regularity, duality, and statisticalestimation. In particular, by decoupling the optimization variables, we arriveat a simple dual form for $\mathsf{W}_p^\varepsilon$ that can be implementedvia an elementary modification to standard, duality-based OT solvers. Weillustrate the virtues of our framework via applications to generative modelingwith contaminated datasets.

Outlier-Robust Optimal Transport: Duality, Structure, and Statistical Analysis