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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.00397 |
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| _version_ | 1866912737999716352 |
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| author | Dagdoug, Mehdi Dombry, Clement Duchamps, Jean-Jil |
| author_facet | Dagdoug, Mehdi Dombry, Clement Duchamps, Jean-Jil |
| contents | Random Forests and Gradient Boosting are among the most effective algorithms for supervised learning on tabular data. Both belong to the class of tree-based ensemble methods, where predictions are obtained by aggregating many randomized regression trees. In this paper, we develop a theoretical framework for analyzing such methods through Reproducing Kernel Hilbert Spaces (RKHSs) constructed on tree ensembles -- more precisely, on the random partitions generated by randomized regression trees. We establish fundamental analytical properties of the resulting Random Forest kernel, including boundedness, continuity, and universality, and show that a Random Forest predictor can be characterized as the unique minimizer of a penalized empirical risk functional in this RKHS, providing a variational interpretation of ensemble learning. We further extend this perspective to the continuous-time formulation of Gradient Boosting introduced by Dombry and Duchamps, and demonstrate that it corresponds to a gradient flow on a Hilbert manifold induced by the Random Forest RKHS. A key feature of this framework is that both the kernel and the RKHS geometry are data-dependent, offering a theoretical explanation for the strong empirical performance of tree-based ensembles. Finally, we illustrate the practical potential of this approach by introducing a kernel principal component analysis built on the Random Forest kernel, which enhances the interpretability of ensemble models, as well as GVI, a new geometric variable importance criterion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_00397 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An RKHS Perspective on Tree Ensembles Dagdoug, Mehdi Dombry, Clement Duchamps, Jean-Jil Machine Learning Random Forests and Gradient Boosting are among the most effective algorithms for supervised learning on tabular data. Both belong to the class of tree-based ensemble methods, where predictions are obtained by aggregating many randomized regression trees. In this paper, we develop a theoretical framework for analyzing such methods through Reproducing Kernel Hilbert Spaces (RKHSs) constructed on tree ensembles -- more precisely, on the random partitions generated by randomized regression trees. We establish fundamental analytical properties of the resulting Random Forest kernel, including boundedness, continuity, and universality, and show that a Random Forest predictor can be characterized as the unique minimizer of a penalized empirical risk functional in this RKHS, providing a variational interpretation of ensemble learning. We further extend this perspective to the continuous-time formulation of Gradient Boosting introduced by Dombry and Duchamps, and demonstrate that it corresponds to a gradient flow on a Hilbert manifold induced by the Random Forest RKHS. A key feature of this framework is that both the kernel and the RKHS geometry are data-dependent, offering a theoretical explanation for the strong empirical performance of tree-based ensembles. Finally, we illustrate the practical potential of this approach by introducing a kernel principal component analysis built on the Random Forest kernel, which enhances the interpretability of ensemble models, as well as GVI, a new geometric variable importance criterion. |
| title | An RKHS Perspective on Tree Ensembles |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2512.00397 |