Measuring Association on Topological Spaces Using Kernels and Geometric Graphs

TL;DR

This paper introduces a new class of nonparametric, interpretable measures of association for variables in topological spaces, capable of testing independence with favorable theoretical and computational properties.

Contribution

It develops a novel framework for measuring dependence using kernels and geometric graphs applicable to general topological spaces, including consistent estimation and independence testing.

Findings

Measures are 0 if and only if variables are independent

Some estimators adapt to intrinsic data dimensionality

Empirical measures can be computed in near linear time

Abstract

In this paper we propose and study a class of simple, nonparametric, yet interpretable measures of association between two random variables $X$ and $Y$ taking values in general topological spaces. These nonparametric measures -- defined using the theory of reproducing kernel Hilbert spaces -- capture the strength of dependence between $X$ and $Y$ and have the property that they are 0 if and only if the variables are independent and 1 if and only if one variable is a measurable function of the other. Further, these population measures can be consistently estimated using the general framework of graph functionals which include $k$-nearest neighbor graphs and minimum spanning trees. Moreover, a sub-class of these estimators are also shown to adapt to the intrinsic dimensionality of the underlying distribution. Some of these empirical measures can also be computed in near linear time. Under the hypothesis of independence between $X$ and $Y$, these empirical measures (properly normalized) have a standard normal limiting distribution. Thus, these measures can also be readily used to test the hypothesis of mutual independence between $X$ and $Y$. In fact, as far as we are aware, these are the only procedures that possess all the above mentioned desirable properties. Furthermore, when restricting to Euclidean spaces, we can make these sample measures of association finite-sample distribution-free, under the hypothesis of independence, by using multivariate ranks defined via the theory of optimal transport. The recent correlation coefficient proposed in Dette et al. (2013), Chatterjee (2019), and Azadkia and Chatterjee (2019) can be seen as a special case of this general class of measures.