Measuring Association on Topological Spaces Using Kernels and Geometric Graphs

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 geometric graphs 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 correlation coefficient proposed in Dette et al. [31], Chatterjee [22] and Azadkia and Chatterjee [7] can be seen as a special case of this general class of measures.

On boosting the power of Chatterjee's rank correlation

“…In contrast to the coefficient ξ n introduced in (2.3), in constructing ξ n,M we take the minimum instead of absolute difference. This is an idea pursued in Azadkia and Chatterjee (2021) and Deb et al (2020) as well; cf. the construction of the correlation coefficient T n in Azadkia and Chatterjee (2021) and Section 8.3.1 in Deb et al (2020).…”

“…This is an idea pursued in Azadkia and Chatterjee (2021) and Deb et al (2020) as well; cf. the construction of the correlation coefficient T n in Azadkia and Chatterjee (2021) and Section 8.3.1 in Deb et al (2020). However, it is worth pointing that, as M = 1,…”

“…Remark 2.4 (Relation to Deb et al ( 2020)'s proposal). Our idea to scale up M for boosting the power of rank-based tests is of course not new, and is particularly related to an earlier proposal made in Deb et al (2020); see, e.g., Equations (3.3) and (8.5) therein. It is hence helpful to point out our new discoveries.…”

“…Technically speaking, our analysis hinges on a careful (sharp up to some log n terms) calculation of the proposed correlation coefficients' means and variances under both null and local alternatives. A particular focus is on such M that can scale fast with and even at a rate close to n. Analogous results in Chatterjee (2021), Deb et al (2020), Auddy et al (2021), and Shi et al (2021a) are not quite helpful in this regime since they are focused on small M that is either fixed or scales to infinity at a sub-polynomial rate. More specifically, (i) we obtain explicit forms of the statistics' means and variances under the null (cf.…”

“…Remark 2.4 in Section 2). In addition, for guaranteeing a normal limiting null distribution, Deb et al (2020) required a very small M of order Poly-log n (cf. Deb et al (2020, Theorem 4.1)).…”