Kernel of Partition Paths: A Unified Representation for Tree Ensembles

A new preprint introduces Kernel of Partition Paths (KPP), a unified geometric representation that indexes a tree ensemble’s feature map by its nodes rather than by individual splits, and wraps prediction, attribution, robustness, and risk bounds into a single non-diagonal Gram matrix [1]. The work, submitted in 2026, builds on a recent line of research that reframes individual decision trees as linear models on engineered split features [1]. Those formulations opened routes for oracle inequalities and feature-importance reinterpretation but left unresolved the question of what unified object a forest induces when the feature map is indexed by nodes instead of splits [2]. The paper studies that object directly. KPP weights each coordinate by a path metric on the tree, producing a squared-Euclidean path-isometric embedding in which distances recover the weighted path distance through the trees up to a normalization factor [3]. Unlike stump-based designs that yield a diagonal Gram matrix, and unlike leaf-based kernels that expose similarity without a metric on internal tree structure, the KPP Gram is non-diagonal and carries the path-metric structure explicitly [4]. The representation unifies four pillars under a single metric-carrying Gram: prediction, exact additive attribution, a deterministic Lipschitz robust radius measured in the KPP metric, and uniform Rademacher risk bounds for regression and classification [5]. The risk bounds are stated under three explicit conditioning regimes — fixed, honest, and cross-fit — and all probabilistic guarantees are conditional on the representation [2]. The robust-radius guarantee is deterministic in the KPP metric rather than in a norm on the raw input, a distinction that separates it from typical adversarial-robustness certificates [3]. The authors do not claim fast-rate refinements as theorems; they instead list conjectured fast-rate improvements for both regression and classification as open problems [4]. The paper appears on arXiv under the machine-learning category and is accompanied by experimental HTML and PDF versions [1][5]. The broader machine-learning community has long relied on curated datasets and standardized benchmarks to validate new theoretical constructs, a practice documented in public dataset registries [7], though the KPP manuscript focuses on representation-level guarantees rather than empirical benchmarks.