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

The Wasserstein distance, rooted in optimal transport (OT) theory, is a popular discrepancy measure between probability distributions with various applications to statistics and machine learning. Despite their rich structure and demonstrated utility, Wasserstein distances are sensitive to outliers in the considered distributions, which hinders applicability in practice. We propose a new outlier-robust Wasserstein distance mathsfWp^varepsilon which allows for varepsilon outlier mass to be removed from each contaminated distribution. Under standard moment assumptions, mathsfWp^varepsilon is shown to achieve strong robust estimation guarantees under the Huber varepsilon-contamination model. Our formulation of this robust distance amounts to a highly regular optimization problem that lends itself better for analysis compared to previously considered frameworks. Leveraging this, we conduct a thorough theoretical study of mathsfWp^varepsilon, encompassing robustness guarantees, characterization of optimal perturbations, regularity, duality, and statistical estimation. In particular, by decoupling the optimization variables, we arrive at a simple dual form for mathsfWp^varepsilon that can be implemented via an elementary modification to standard, duality-based OT solvers. We illustrate the virtues of our framework via applications to generative modeling with contaminated datasets.