Kernel Integrated $R^2$: A Measure of Dependence

TL;DR

This paper introduces kernel integrated R^2, a new dependence measure that captures complex relationships in multivariate and structured data, with proven theoretical properties and competitive empirical performance.

Contribution

It extends integrated R^2 to general spaces using RKHSs, introduces two estimators, and demonstrates their consistency and effectiveness in dependence testing.

Findings

Measure ranges in [0,1], equals zero iff independence, equals one iff response is a measurable function of covariates.

Proposed estimators are consistent with proven convergence rates.

Numerical experiments show competitive power, especially for non-linear and structured dependencies.

Abstract

We introduce kernel integrated $R^2$, 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 to measure dependence of multivariate, functional, and structured data, while remaining sensitive to tail behaviour and oscillatory dependence structures. We establish that (i) this new measure takes values in $[0,1]$, (ii) equals zero if and only if independence holds, and (iii) 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. We prove consistency and derive convergence rates 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 non-linear and structured relationships.