Measures of Dependence based on Wasserstein distances
Marta Catalano, Hugo Lavenant
TL;DR
This paper surveys Wasserstein- and optimal-transport-based measures of dependence, outlining two core strategies: quantifying how far the joint law is from independence and how far the conditional law is from the marginal, leading to the joint and conditional indices $D_{\otimes}$ and $D_{|\bullet}$. It develops practical index formulations $I_{\otimes}$ and $I_{|\bullet}$ via upper bounds $U$, discusses invariances and maximal-dependence perspectives, and provides analytic expressions in 1D and Gaussian settings. The work also covers statistical aspects, including robustness, adaptation-sensitive continuity, and sample complexity, and introduces entropic regularization to mitigate the curse of dimensionality, along with computational considerations and extensions to conditional independence and multivariate settings. Overall, it highlights both the promise and the open questions in applying OT-based dependence measures to general metric spaces and complex stochastic objects. The discussions point toward future work on principled maximal-dependence definitions, scalable estimation in high dimensions, and broadening the framework to multi-variable and process-level dependencies.
Abstract
Measuring dependence between random variables is a fundamental problem in Statistics, with applications across diverse fields. While classical measures such as Pearson's correlation have been widely used for over a century, they have notable limitations, particularly in capturing nonlinear relationships and extending to general metric spaces. In recent years, the theory of Optimal Transport and Wasserstein distances has provided new tools to define measures of dependence that generalize beyond Euclidean settings. This survey explores recent proposals, outlining two main approaches: one based on the distance between the joint distribution and the product of marginals, and another leveraging conditional distributions. We discuss key properties, including characterization of independence, normalization, invariances, robustness, sample, and computational complexity. Additionally, we propose an alternative perspective that measures deviation from maximal dependence rather than independence, leading to new insights and potential extensions. Our work highlights recent advances in the field and suggests directions for further research in the measurement of dependence using Optimal Transport.