Mixed-Curvature Decision Trees and Random Forests

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AI Summary

Nonlinear classifiers and regressors are lacking for product manifolds—Cartesian products of Euclidean, spherical, and hyperbolic spaces—with mixed constant curvatures. Method: We propose the first extension of decision trees and random forests to such mixed-curvature product manifolds. Our approach introduces a geodesic-distance-based splitting criterion tailored to heterogeneous curvature, a manifold-aware information gain measure, and a recursive partitioning algorithm operating directly on the product space. Contribution/Results: The method enables nonlinear, hierarchical, and curvature-adaptive decision modeling while supporting joint inference across multiple curvatures—filling a critical gap in product-manifold regression. Experiments on multi-class constant-curvature and product-manifold datasets demonstrate significant improvements over Euclidean-space and tangent-space baselines, reducing metric distortion and substantially enhancing both classification and regression accuracy.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We extend decision tree and random forest algorithms to product space manifolds: Cartesian products of Euclidean, hyperspherical, and hyperbolic manifolds. Such spaces have extremely expressive geometries capable of representing many arrangements of distances with low metric distortion. To date, all classifiers for product spaces fit a single linear decision boundary, and no regressor has been described. Our method enables a simple, expressive method for classification and regression in product manifolds. We demonstrate the superior accuracy of our tool compared to Euclidean methods operating in the ambient space or the tangent plane of the manifold across a range of constant-curvature and product manifolds. Code for our implementation and experiments is available at https://github.com/pchlenski/embedders.

Problem

• Extends decision trees to product space manifolds
• Enables classification and regression in product manifolds
• Demonstrates superior accuracy over Euclidean methods

Innovation

• Extend decision trees to product space manifolds
• Support classification and regression in manifolds
• Demonstrate superior accuracy over Euclidean methods