The design of a metric between probability distributions is a longstanding
problem motivated by numerous applications in Machine Learning. Focusing on
continuous probability distributions on the Euclidean space $\mathbb{R}^d$, we
introduce a novel pseudo-metric between probability distributions by leveraging
the extension of univariate quantiles to multivariate spaces. Data depth is a
nonparametric statistical tool that measures the centrality of any element
$x\in\mathbb{R}^d$ with respect to (w.r.t.) a probability distribution or a
data set. It is a natural median-oriented extension of the cumulative
distribution function (cdf) to the multivariate case. Thus, its upper-level
sets -- the depth-trimmed regions -- give rise to a definition of multivariate
quantiles. The new pseudo-metric relies on the average of the Hausdorff
distance between the depth-based quantile regions w.r.t. each distribution. Its
good behavior w.r.t. major transformation groups, as well as its ability to
factor out translations, are depicted. Robustness, an appealing feature of this
pseudo-metric, is studied through the finite sample breakdown point. Moreover,
we propose an efficient approximation method with linear time complexity w.r.t.
the size of the data set and its dimension. The quality of this approximation
as well as the performance of the proposed approach are illustrated in
numerical experiments.

本文提出了一种新的假距度量方法，该方法使用数据深度来扩展多元分位数的概念，从而度量欧几里得空间上的连续概率分布之间的差异，并通过有限样本破坏点研究了其稳健性。