Optimal transport (OT) theory and the related $p$-Wasserstein distance ($W_p$, $p\geq 1$) are widely-applied in statistics and machine learning. In spite of their popularity, inference based on these tools is sensitive to outliers or it can perform poorly when the underlying model has heavy-tails. To cope with these issues, we introduce a new class of procedures. (i) We consider a robust version of the primal OT problem (ROBOT) and show that it defines the {robust Wasserstein distance}, $W^{(\lambda)}$, which depends on a tuning parameter $\lambda > 0$. (ii) We illustrate the link between $W_1$ and $W^{(\lambda)}$ and study its key measure theoretic aspects. (iii) We derive some concentration inequalities for $W^{(\lambda)}$. (iii) We use $W^{(\lambda)}$ to define minimum distance estimators, we provide their statistical guarantees and we illustrate how to apply concentration inequalities for the selection of $\lambda$. (v) We derive the {dual} form of the ROBOT and illustrate its applicability to machine learning problems (generative adversarial networks and domain adaptation). Numerical exercises provide evidence of the benefits yielded by our methods.

翻译：最优传输（OT）理论与相关的$p$-瓦瑟斯坦距离（$W_p$, $p\geq 1$）在统计学和机器学习中应用广泛。尽管它们备受青睐，但基于这些工具的推断对异常值敏感，或在底层模型具有重尾特征时表现不佳。为解决这些问题，我们引入了一类新的方法：（i）我们考虑原始OT问题的鲁棒版本（ROBOT），并证明其定义了依赖于调整参数$\lambda > 0$的{鲁棒瓦瑟斯坦距离}$W^{(\lambda)}$；（ii）我们阐释$W_1$与$W^{(\lambda)}$之间的联系，并研究其关键的测度论性质；（iii）我们推导$W^{(\lambda)}$的若干浓度不等式；（iv）利用$W^{(\lambda)}$定义最小距离估计量，提供其统计保证，并说明如何应用浓度不等式来选择$\lambda$；（v）我们推导ROBOT的{对偶}形式，并阐述其在机器学习问题（生成对抗网络与领域自适应）中的适用性。数值实验表明了我们方法所取得的显著优势。