Inference via robust optimal transportation: theory and methods

TL;DR

This paper introduces a robust version of optimal transportation, the robust Wasserstein distance, addressing issues like outlier sensitivity and infinite moments, and applies it to statistical inference and machine learning tasks.

Contribution

It develops a robust Wasserstein distance, analyzes its properties, derives concentration inequalities, and demonstrates its application in statistical estimation and machine learning.

Findings

Robust Wasserstein distance improves outlier resistance.

Theoretical guarantees for minimum distance estimators.

Enhanced performance in GANs and domain adaptation.

Abstract

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.