  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.
