Transport Dependency: Optimal Transport Based Dependency Measures

Abstract

Finding meaningful ways to determine the dependency between two random variables $\xi$ and $\zeta$ is a timeless statistical endeavor with vast practical relevance. In recent years, several concepts that aim to extend classical means (such as the Pearson correlation or rank-based coefficients like Spearman's $\rho$) to more general spaces have been introduced and popularized, a well-known example being the distance correlation. In this article, we propose and study an alternative framework for measuring statistical dependency, the transport dependency $\tau \ge 0$, which relies on the notion of optimal transport and is applicable in general Polish spaces. It can be estimated consistently via the corresponding empirical measure, is versatile and adaptable to various scenarios by proper choices of the cost function. Notably, statistical independence is characterized by $\tau = 0$, while large values of $\tau$ indicate highly regular relations between $\xi$ and $\zeta$. Indeed, for suitable base costs, $\tau$ is maximized if and only if $\zeta$ can be expressed as 1-Lipschitz function of $\xi$ or vice versa. Based on sharp upper bounds, we exploit this characterization and define three distinct dependency coefficients (a-c) with values in 1$, each of which emphasizes different functional relations. These transport correlations attain the value $ if and only if $\zeta = \varphi(\xi)$, where $\varphi$ is a) a Lipschitz function, b) a measurable function, c) a multiple of an isometry. The properties of coefficient c) make it comparable to the distance correlation, while coefficient b) is a limit case of a) that was recently studied independently by Wiesel (2021). Numerical results suggest that the transport dependency is a robust quantity that efficiently discerns structure from noise in simple settings, often out-performing other commonly applied coefficients of dependency.