Title: Optimal Transport Dependency

Abstract: 

Finding meaningful ways to determine the dependency between two random variables 𝜉 and 𝜁 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 𝜌) to more general spaces have been introduced and popularized, a well-known example being the distance correlation. In this talk, we propose and study an alternative framework for measuring statistical dependency, the transport dependency 𝜏 ≥ 0 (TD), which relies on the notion of optimal transport and is applicable in general Polish spaces. It can be estimated via the corresponding empirical measure, is versatile and adaptable to various scenarios by proper choices of the cost function. It intrinsically respects metric and geometric properties of the ground spaces. Notably, statistical independence is characterized by 𝜏 = 0, while large values of 𝜏 indicate highly regular relations between 𝜉 and 𝜁 . Based on sharp upper bounds, we exploit three distinct dependency coefficients with values in [0, 1], each of which emphasizes different functional relations: These transport correlations attain the value 1 if and only if 𝜁 = 𝜑(𝜉), where 𝜑 is a) a Lipschitz function, b) a measurable function, c) a multiple of an isometry. Besides a conceptual discussion of transport dependency, we address numerical issues and its ability to adapt automatically to the potentially low intrinsic dimension of the ground space. Monte Carlo results suggest that TD is a robust quantity that efficiently discerns dependency structure from noise for data sets with complex internal metric geometry. The use of TD for inferential tasks is illustrated for independence testing on a data set of trees from cancer genetics.
Joint work with Giacomo Nies and Thomas Staudt.