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

2026-02-26

AI Summary

This work proposes a kernel ensemble $R^2$ method to address the challenge of measuring statistical dependence in multivariate, functional, and structured data under nonlinear, tail, or oscillatory dependency scenarios. By integrating local normalization with the flexibility of reproducing kernel Hilbert spaces (RKHS), the method extends ensemble $R^2$ to general kernel settings for the first time, rigorously satisfying properties such as boundedness in [0,1] and necessary and sufficient conditions for independence and deterministic functional relationships. Leveraging k-nearest neighbor graphs and conditional mean embeddings, the authors establish consistency of the graph-based estimator and derive adaptive convergence rates with respect to the intrinsic dimensionality. Experiments on both synthetic and real-world media annotation datasets demonstrate that the proposed approach significantly outperforms state-of-the-art methods, particularly in detecting nonlinear and structured dependencies with higher statistical power.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Graphical Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• statistical dependence
• kernel methods
• structured data
• nonlinear relationships
• dependence measure

Innovation

• kernel integrated R²
• statistical dependence
• reproducing kernel Hilbert space
• conditional mean embedding
• intrinsic dimensionality