[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81676-en":3,"doc-seo-81676-105":30,"detail-sidebar-cat-0-en-105":83},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},81676,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Kernel of Partition Paths A Unified Representation for Tree Ensembles","A unified geometric view is developed for forests by indexing a forest’s feature map by its nodes rather than by split rules. Kernel of Partition Paths (KPP) assigns each coordinate a path-metric weight to obtain a squared-Euclidean, path-isometric embedding. Its Gram geometry is non-diagonal and directly carries the tree path metric. The framework unifies prediction, exact additive attribution, deterministic KPP-metric Lipschitz robustness, and uniform Rademacher generalization bounds for regression and classification under fixed, honest, or cross-fit conditioning.","arXiv :2606 . 18853v3 [ stat .ML] 10 Jul 2026  \nKernel of Partition Paths:  \nA Unified Representation for Tree Ensembles  \nNicolas Mahler [nicolas.mahler@datapred. com](nicolas.mahler@datapred. com)  \nDatapred SAS  \n23 rue Mirabeau  \n94300 Vincennes, France  \nDatapred SA  \nEPFL Innovation Park – Bâtiment A 1015 Lausanne, Switzerland  \nAbstract  \nA recent line of work has reframed individual decision trees as linear models on engineered features associated with their splits, opening routes for oracle inequalities and featureimportance reinterpretation, but leaving open the question of what unified geometric object a forest induces when one indexes its feature map by nodes rather than by splits. The present paper studies that object. KPP indexes the feature map by the nodes of the forest, weighted by a path metric that turns each coordinate into a component of a squared-Euclidean pathisometric embedding. KPP unifies four pillars under a single node-indexed representation whose Gram is non-diagonal and carries a metric: prediction, exact additive attribution, deterministic Lipschitz robust radius in the KPP metric, and uniform Rademacher risk bounds for regression and classification under fixed, honest, or cross-fit conditioning. All probabilistic guarantees are conditional on the representation and are stated under three explicit conditioning regimes; the robust-radius guarantee is deterministic in the KPP metric rather than in a norm on the raw input. Conjectured fast-rate refinements for both regression and classification are stated as open problems and are not claimed as theorems.  \n1 Introduction  \nRandom forests (Breiman, 2001) have remained one of the standard predictive tools for tabular data, and have also served as a substrate for the theoretical study of ensemble methods and for the design of modellevel interpretation tools. A recent line of work has reframed individual CART trees as linear models on engineered features associated with their splits (Klusowski & Tian, 2024), and has used this reframing to extend forests with explicit regularisation and generalised linear link functions (Agarwal et al., 2025) . This line of work shows that the partitioning power of forests and the analytical tractability of linear regression can be combined within a single estimator, and that the resulting linear view opens routes for oracle inequalities, feature-importance reinterpretation, and inductive-bias analysis. The present paper sits within this broad direction and asks a related but distinct question: what geometric object does a forest induce when one indexes its feature map not by splits but by nodes, and what guarantees does that object support.  \nThe recent literature on linear and geometric views of forests splits along two structurally distinct directions, surveyed in section 2 . One direction, descending from rule ensembles (Friedman & Popescu, 2008) and refined in recent work on linear views of CART, indexes features by splits and obtains a diagonal feature Gram through the orthogonality of per-split decision stumps. Another direction, descending from forest proximities and from the view of forests as adaptive kernels (Scornet, 2016), indexes similarity by leaf comembership and exposes a kernel but no explicit metric over the internal structure of the trees. The first direction supports linear-model machinery on top of the forest at the price of a representation whose geometry is degenerate (a diagonal feature Gram cannot encode path distance) . The second direction supports kernel  \nmachinery at the price of losing the additive split-level decomposition that the stump-based view affords. These two directions, taken individually, operate within narrower scopes than a single representation that simultaneously supports prediction, attribution, robustness, and generalisation.  \nKPP indexes its feature map by the nodes of the forest, with each coordinate weighted by a path metric on the tree (section 3.3) . The resulting em","cbCaipsNKSG3OBSb","https://ap.wps.com/l/cbCaipsNKSG3OBSb","pdf",760780,2,1,38,"English","en",105,"# Introduction\n## Linear and geometric views of forests\n## KPP node-indexed representation\n## Conditioning regimes and guarantees","[{\"question\":\"What guarantees does the paper provide for prediction and generalization?\",\"answer\":\"The paper establishes prediction, exact additive attribution, a deterministic Lipschitz robust radius measured in the KPP metric, and uniform Rademacher risk bounds for regression and classification. These probabilistic guarantees are conditional under three regimes: fixed, honest, and cross-fit.\"}]",1784175356,96,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":78,"head_meta":80,"extra_data":82,"updated_unix":28},"kernel-of-partition-paths-a-unified-representation-for-tree-ensembles","",{"@graph":36,"@context":77},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/kernel-of-partition-paths-a-unified-representation-for-tree-ensembles/81676/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"What guarantees does the paper provide for prediction and generalization?","Question",{"text":75,"@type":76},"The paper establishes prediction, exact additive attribution, a deterministic Lipschitz robust radius measured in the KPP metric, and uniform Rademacher risk bounds for regression and classification. 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