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

Pouya Roudaki, Shakeel Gavioli-Akilagun, Florian Kalinke, Mona Azadkia, Zoltán Szabó
Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, PMLR 337:5764-5789, 2026.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v337-roudaki26a, title = {Kernel Integrated $R^2$: A Measure of Dependence}, author = {Roudaki, Pouya and Gavioli-Akilagun, Shakeel and Kalinke, Florian and Azadkia, Mona and Szab\'{o}, Zolt\'{a}n}, booktitle = {Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence}, pages = {5764--5789}, year = {2026}, editor = {Perković, Emilija and Malinsky, Daniel}, volume = {337}, series = {Proceedings of Machine Learning Research}, month = {17--21 Aug}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v337/main/assets/roudaki26a/roudaki26a.pdf}, url = {https://proceedings.mlr.press/v337/roudaki26a.html}, 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.} }
Endnote
%0 Conference Paper %T Kernel Integrated $R^2$: A Measure of Dependence %A Pouya Roudaki %A Shakeel Gavioli-Akilagun %A Florian Kalinke %A Mona Azadkia %A Zoltán Szabó %B Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2026 %E Emilija Perković %E Daniel Malinsky %F pmlr-v337-roudaki26a %I PMLR %P 5764--5789 %U https://proceedings.mlr.press/v337/roudaki26a.html %V 337 %X 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.
APA
Roudaki, P., Gavioli-Akilagun, S., Kalinke, F., Azadkia, M. & Szabó, Z.. (2026). Kernel Integrated $R^2$: A Measure of Dependence. Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 337:5764-5789 Available from https://proceedings.mlr.press/v337/roudaki26a.html.

Related Material