A1633
Title: Kernel integrated $R^2$: A measure of dependence
Authors: Shakeel Gavioli-Akilagun - London School of Economics (United Kingdom) [presenting]
Mona Azadkia - London School of Economics (United Kingdom)
Florian Kalinke - Karlsruhe Institute of Technology (Germany)
Zoltan Szabo - LSE (United Kingdom)
Seyedpouya Mirrezaeiroudaki - London School of Economics (United Kingdom)
Abstract: The kernel integrated $R^2$ is a new measure of statistical dependence that combines the local normalization principle of the recently introduced integrated $R^2$ with the flexibility of reproducing kernel Hilbert spaces (RKHSs). The proposed measure extends integrated $R^2$ from scalar responses to responses taking values on general spaces equipped with a characteristic kernel, allowing measurement of dependence for multivariate, functional, and structured data while remaining sensitive to tail behaviour and oscillatory dependence structures. This measure takes values in $[0,1]$, equals zero if and only if independence holds, and equals one if and only if the response is almost surely a measurable function of the covariates. Two estimators are proposed: a graph-based method using $K$-nearest neighbours and an RKHS-based method built on conditional mean embeddings. Consistency is established and convergence rates are derived for the graph-based estimator, showing its adaptation to intrinsic dimensionality. Numerical experiments on simulated data and a real data experiment in the context of dependency testing for media annotations demonstrate competitive power against state-of-the-art dependence measures, particularly in settings involving nonlinear and structured relationships.
