Inference via Robust Optimal Transportation: Theory and Methods

Abstract

Optimal transportation (OT) is widely applied in statistics and machine learning. Despite its popularity, inference based on OT has some issues. For instance, it is sensitive to outliers and may not be even defined when the underlying model has infinite moments. To cope with these problems, first, we consider a robust version of the primal transportation problem and show that it defines the robust Wasserstein distance, $W^{\left(\lambda \right)}$, depending on a tuning parameter $\lambda >0$. Second, we illustrate the link between 1-Wasserstein distance $W_1$ and $W^{\left(\lambda \right)}$ and study its key measure theoretic aspects. Third, we derive some concentration inequalities for $W^{\left(\lambda \right)}$. Fourth, we use $W^{\left(\lambda \right)}$ to define minimum distance estimators, provide their statistical guarantees and illustrate how to apply the concentration inequalities for a selection of $\lambda$. Fifth, we provide the dual form of the robust optimal transportation (ROBOT) and apply it to machine learning problems. We review, in a unified perspective, the key aspects of OT and ROBOT, while complementing the existing results with our new methodological findings. Numerical exercises provide evidence of the benefits of our novel methods.