Planar Homeomorphic Embeddings of Decision Tree

Hang Zhang, Kai Ming Ting

Keywords: Decision Tree, Planar Homeomorphic Embedding, Dimensionality reduction

Abstract: Decision trees and their ensemble variants are widely celebrated for their accuracy, interpretability, and effectiveness on tabular data. Despite their intuitive structure, understanding the global geometric organization of the feature space partitions induced by these models remains challenging, particularly in high-dimensional settings. Traditional visualization techniques, such as node-link diagrams, fail to capture the topological relationships between decision regions, while standard dimensionality reduction methods prioritize data distribution over structural fidelity, often distorting adjacency and connectivity. To address this limitation, we propose a novel framework for embedding decision-tree-induced partitions into two-dimensional space while explicitly preserving adjacency relations among leaf regions. Our approach models the decision tree as a polyhedral complex and constructs a piecewise-linear (PL) embedding that maintains the combinatorial topology of the original high-dimensional partitioning. This adjacency-preserving visualization enables a more faithful interpretation of model behavior, revealing insights into decision boundary structure and data distribution. Our theoretical and experimental results demonstrate the feasibility of the proposed method and its ability to preserve the topological characteristics of the data.

Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

Submission Number: 7303