Inference via Robust Optimal Transportation: Theory and Methods
针对最优运输对异常值敏感且模型有无穷矩时无法定义的问题，提出鲁棒Wasserstein距离，给出其集中不等式，并用于定义最小距离估计器，在机器学习中展示优势。
Summary 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, , depending on a tuning parameter . Second, we illustrate the link between 1‐Wasserstein distance and and study its key measure theoretic aspects. Third, we derive some concentration inequalities for . Fourth, we use to define minimum distance estimators, provide their statistical guarantees and illustrate how to apply the concentration inequalities for a selection of . 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.