(Maths-IA) Wasserstein medians for outlier detection and robustness

The presence of outliers is a recurrent concern in statistics and learning. In some cases, they appear as observations of interest that we want to detect. In others, we want to estimate predictive models without being too affected by them. In this talk, we present a new method for detecting outliers in a data set. The method combines the notion of Wasserstein median in the space of probability measures with a divide-and-conquer approach, in the spirit of the median-of-means estimator. The resulting measure-valued estimator can be viewed as a cleaned version of the empirical distribution. To illustrate this intuition we provide statistical and robustness guarantees under arbitrary corruptions of a fixed proportion of the sample. We will also discuss computational aspects of approximating discrete Wasserstein medians, including a subgradient method based on Sinkhorn iterations and the resulting approximation error.