Optimal transportation 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 has some issues. For instance, it is sensitive to outliers and it 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}, $W^{(\lambda)}$, depending on a tuning parameter $\lambda > 0$. Second, we illustrate the link between $W_1$ and $W^{(\lambda)}$ and study its key measure theoretic aspects. Third, we derive some concentration inequalities for $W^{(\lambda)}$. Fourth, we use $W^{(\lambda)}$ to define minimum distance estimators, we provide their statistical guarantees and we illustrate how to apply the derived concentration inequalities for a data driven selection of $\lambda$. Fifth, we provide the {dual} form of the robust optimal transportation problem and we apply it to machine learning problems (generative adversarial networks and domain adaptation). Numerical exercises provide evidence of the benefits yielded by our novel methods.

翻译：最优传输理论及相关的$p$-Wasserstein距离（$W_p$，$p\geq 1$）在统计学和机器学习中广泛应用。然而，基于这些工具的推断存在若干问题，例如对异常值敏感，且当底层模型具有无限矩时可能无法定义。为解决这些问题，我们首先考虑原始传输问题的稳健版本，并证明其所定义的{稳健Wasserstein距离}$W^{(\lambda)}$依赖于调节参数$\lambda > 0$。其次，我们阐明$W_1$与$W^{(\lambda)}$之间的联系，并研究其关键的测度理论性质。第三，我们推导$W^{(\lambda)}$的相关集中不等式。第四，利用$W^{(\lambda)}$定义最小距离估计量，给出统计保证，并说明如何应用所获集中不等式进行数据驱动的$\lambda$选择。第五，我们给出稳健最优传输问题的{对偶}形式，并将其应用于机器学习问题（生成对抗网络和领域自适应）。数值实验证明了新方法的有效性。