A Pseudo-Metric between Probability Distributions based on Depth-Trimmed Regions

TL;DR

This paper introduces a new pseudo-metric for probability distributions in Euclidean space using depth-trimmed regions, offering robustness, invariance properties, and an efficient approximation method demonstrated through numerical experiments.

Contribution

It proposes a novel depth-based pseudo-metric between probability distributions, extending univariate quantiles to multivariate spaces with robust and computationally efficient features.

Findings

The pseudo-metric exhibits good invariance and robustness properties.

An efficient linear-time approximation method is developed.

Numerical experiments validate the effectiveness of the proposed approach.

Abstract

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.