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Mixed-curvature decision trees and random forests
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Extends decision trees and random forests to product manifolds (mixed hyperbolic, hyperspherical, Euclidean components) using homogeneous hyperplanes and angular midpoints, benchmarked on synthetic Gaussian mixtures.
Problem
Classification and regression in non-Euclidean product space manifolds lack expressive tree-based tools; the only prior product-space classifier is a single linear decision boundary.
Approach
Fit CART-style decision trees whose splits are homogeneous hyperplanes through the ambient origin, chosen via angular midpoints on each component manifold, and let one tree span all components of a product space by iterating over dimensions with a shared special dimension.
Key results
- On constant-curvature Gaussian mixtures, product/component trees and forests outperform Euclidean baselines increasingly as |K| grows; all three decision trees are identical at K=0.
- On (H5)^2, the product DT reached 97.6% accuracy versus 92.8% for the ambient Euclidean DT.
- On (S5)^2 all methods perform similarly (about 60-64%), so the product method gives no clear advantage there.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use product-manifold DT instead of ambient-space Euclidean DT | Decision tree · (H5)2, four-Gaussian mixtures | accuracy: 92.8 ± 2.3 -> 97.6 ± 1.0 | Helped |
| Use product-manifold DT instead of tangent-plane DT | Decision tree · (H5)2, four-Gaussian mixtures | accuracy: 96.1 ± 1.4 -> 97.6 ± 1.0 | Helped |
| Use product-manifold RF instead of ambient-space Euclidean RF | Random forest · (H5)2, four-Gaussian mixtures | accuracy: 94.0 ± 1.9 -> 98.0 ± 1.0 | Helped |
| Use product-manifold RF instead of tangent-plane RF | Random forest · (H5)2, four-Gaussian mixtures | accuracy: 97.3 ± 1.2 -> 98.0 ± 1.0 | Helped |
| Use product-manifold DT instead of the best Euclidean DT baseline | Decision tree · (S5)2, four-Gaussian mixtures | accuracy: 62.5 ± 3.5 -> 61.9 ± 3.5 | Hurt |
| Use product-manifold RF instead of the best Euclidean RF baseline | Random forest · (S5)2, four-Gaussian mixtures | accuracy: 64.6 ± 3.4 -> 64.1 ± 3.6 | Hurt |
| Use product-manifold DT instead of ambient-space Euclidean DT | Decision tree · H5 × S5, four-Gaussian mixtures | accuracy: 92.8 ± 2.2 -> 97.6 ± 1.0 | Helped |
| Use product-manifold RF instead of ambient-space Euclidean RF | Random forest · H5 × S5, four-Gaussian mixtures | accuracy: 92.4 ± 2.2 -> 98.0 ± 1.0 | Helped |
| Use product-manifold RF instead of tangent-plane RF | Random forest · H5 × S5, four-Gaussian mixtures | accuracy: 96.7 ± 1.2 -> 98.0 ± 1.0 | Helped |
| Use product-manifold DT instead of the best Euclidean DT baseline | Decision tree · (H2)5, four-Gaussian mixtures | accuracy: 80.3 ± 4.4 -> 80.9 ± 4.5 | Helped |
| Use product-manifold RF instead of the best Euclidean RF baseline | Random forest · (H2)5, four-Gaussian mixtures | accuracy: 82.6 ± 4.2 -> 82.0 ± 4.1 | Hurt |
| Use product-manifold DT instead of the best Euclidean DT baseline | Decision tree · (S2)5, four-Gaussian mixtures | accuracy: 58.9 ± 4.6 -> 59.5 ± 4.6 | Helped |
Surprising findings
- On the (S5)^2 signature all methods score near 60-64% accuracy with no significant advantage for the product-space method.
Limitations
- Discontinuity in distance functions across negative, zero, and positive curvature; Euclidean embeddings lifted arbitrarily to x0=1.
- Benchmarks are small and synthetic (1,000 points, depth-3 trees, 10 mixtures per signature); no real data or regression experiments.
- Only Gaussian mixture classification evaluated; regression variant described but not benchmarked.
Experimental setup
Datasets: synthetic Gaussian mixtures sampled on product and constant-curvature manifolds
Benchmarks: synthetic Gaussian mixture classification on constant-curvature manifolds (K from -4 to 4) synthetic Gaussian mixture classification on product manifold signatures (10 dimensions)
Metrics: micro-averaged F1 classification accuracy with 95% confidence intervals statistical significance p<0.05
Baselines: Euclidean decision tree (ambient) Euclidean decision tree (tangent plane) Euclidean random forest (ambient) Euclidean random forest (tangent plane)
Builds on: CART Random forests Hyperbolic decision trees (Chlenski et al., 2024) Product space manifolds (Gu et al., 2018) Tabaghi et al. 2021 product space classifiers
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