Measuring Association on Topological Spaces Using Kernels and Geometric Graphs

Understand

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.