Measuring dependence between random vectors via optimal transport

TL;DR

This paper introduces Wasserstein distance-based coefficients to measure dependence between random vectors, providing new estimators with explicit asymptotic properties and applications to EEG data analysis.

Contribution

It proposes novel dependence coefficients based on 2-Wasserstein distance and quasi-Gaussian assumptions, with explicit formulas and estimators that are asymptotically normal.

Findings

New dependence coefficients are normalized between 0 and 1.

Estimators are asymptotically normal with explicit variance formulas.

Application to EEG data demonstrates practical utility.

Abstract

To quantify the dependence between two random vectors of possibly different dimensions, we propose to rely on the properties of the 2-Wasserstein distance. We first propose two coefficients that are based on the Wasserstein distance between the actual distribution and a reference distribution with independent components. The coefficients are normalized to take values between 0 and 1, where 1 represents the maximal amount of dependence possible given the two multivariate margins. We then make a quasi-Gaussian assumption that yields two additional coefficients rooted in the same ideas as the first two. These different coefficients are more amenable for distributional results and admit attractive formulas in terms of the joint covariance or correlation matrix. Furthermore, maximal dependence is proved to occur at the covariance matrix with minimal von Neumann entropy given the covariance matrices of the two multivariate margins. This result also helps us revisit the RV coefficient by proposing a sharper normalisation. The two coefficients based on the quasi-Gaussian approach can be estimated easily via the empirical covariance matrix. The estimators are asymptotically normal and their asymptotic variances are explicit functions of the covariance matrix, which can thus be estimated consistently too. The results extend to the Gaussian copula case, in which case the estimators are rank-based. The results are illustrated through theoretical examples, Monte Carlo simulations, and a case study involving electroencephalography data.