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：最优运输（OT）理论及相关$p$-Wasserstein距离（$W_p$，$p\geq 1$）在统计学和机器学习中广泛应用。尽管这些工具备受青睐，但基于它们的推理对异常值敏感，或当底层模型具有重尾分布时表现不佳。为应对这些问题，我们引入了一类新方法：（i）考虑原始OT问题的鲁棒版本（ROBOT），并证明其定义了依赖于调优参数$\lambda > 0$的{鲁棒Wasserstein距离}$W^{(\lambda)}$；（ii）阐明$W_1$与$W^{(\lambda)}$之间的联系，并研究其关键测度论性质；（iii）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，给出其统计保证，并说明如何应用浓度不等式选择$\lambda$；（v）推导ROBOT的{对偶}形式，并展示其在机器学习问题（生成对抗网络和领域自适应）中的适用性。数值实验证明了我们方法所带来的优势。