Kernel Integrated $R^2$: A Measure of Dependence
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866910034044125184 |
|---|---|
| author | Roudaki, Pouya Gavioli-Akilagun, Shakeel Kalinke, Florian Azadkia, Mona Szabó, Zoltán |
| author_facet | Roudaki, Pouya Gavioli-Akilagun, Shakeel Kalinke, Florian Azadkia, Mona Szabó, Zoltán |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_22985 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Kernel Integrated $R^2$: A Measure of Dependence Roudaki, Pouya Gavioli-Akilagun, Shakeel Kalinke, Florian Azadkia, Mona Szabó, Zoltán Machine Learning Information Theory 62H20 (Primary) 46E22, 62B10, 62G10, 94A17 (Secondary) G.3; H.1.1; I.2.6 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. |
| title | Kernel Integrated $R^2$: A Measure of Dependence |
| topic | Machine Learning Information Theory 62H20 (Primary) 46E22, 62B10, 62G10, 94A17 (Secondary) G.3; H.1.1; I.2.6 |
| url | https://arxiv.org/abs/2602.22985 |