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Shadow Cones: A Generalized Framework for Partial Order Embeddings
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Introduces shadow cones, a physics-inspired framework that models partial orders as shadow subset relations in hyperbolic space (umbral and penumbral cones in Poincaré ball and half-space), outperforming entailment cones on WordNet, MCG, Hearst, and Mammal hierarchies.
Problem
Embedding partial orders (hierarchies such as DAGs and taxonomies) in hyperbolic space; existing entailment cones suffer from an epsilon-hole at the origin of the Poincaré ball and optimization difficulties.
Approach
Model partial orders as subset relations between shadows cast by a light source onto opaque objects in hyperbolic space; four formulations (umbral/penumbral in Poincaré ball and half-space) reduce region-region subset tests to point-in-cone tests, with a distance-based shadow loss, and the half-space penumbral/umbral variants avoid the origin hole by placing the light source at infinity or at a ho
Key results
- On the Mammal subgraph, Umbral-half-space reaches 91.8% F1 at dimension 5 with 90% non-basic edges versus 82.9% for the entailment cone baseline.
- On WordNet Noun at dimension 10 with 50% non-basic edges, Umbral-half-space achieves 97.0% F1 versus 95.2% for the entailment cone.
- Half-space shadow cone variants consistently outperform Poincaré-ball variants, which the authors attribute to the epsilon-hole at the origin.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace Entailment Cone with Umbral-half-space (d=10) | Hyperbolic partial-order embeddings · WordNet Noun, MCG, and Hearst | F1 (%) | Helped |
| Use Umbral-half-space instead of Entailment Cone (d=10) | WordNet Noun, 0% non-basic edges · WordNet Noun | F1 (%): 32.1 -> 52.2 | Helped |
| Use Umbral-half-space instead of Entailment Cone (d=10) | MCG, 25% non-basic edges · MCG | F1 (%): 55.5 -> 87.5 | Helped |
| Use Umbral-half-space instead of Entailment Cone (d=10) | Hearst, 5% non-basic edges · Hearst | F1 (%): 58.2 -> 86.9 | Helped |
| Use Umbral-half-space instead of Poincaré-ball shadow cone formulations | Shadow cone embeddings · Mammal sub-graph | F1 (%) | Mixed |
| Use Umbral-half-space instead of Penumbral-half-space | Shadow cone embeddings · Mammal sub-graph | F1 (%) | Mixed |
Surprising findings
- Penumbral-half-space cones have a height limit and underperform umbral-half-space cones on large datasets, despite penumbral cones being geodesically convex.
- The angle-based max-margin energy of entailment cones leads to vanishing gradients for misclassified negatives, which the distance-based shadow loss avoids.
Limitations
- Penumbral-half-space cones have a height limit, underperforming umbral-half-space on large datasets
- Experiments use fixed radius r and horosphere level h as constants rather than learned per-node parameters in main results
- Umbral cones are not geodesically convex
Experimental setup
Datasets: WordNet Noun Mammal subgraph MCG Hearst patterns
Benchmarks: WordNet Noun Mammal subgraph MCG Hearst
Metrics: F1 score
Baselines: Entailment Cone GBC-box VBC-box
Builds on: entailment cones Poincaré embeddings box embeddings
Closest papers by meaning
- Hyperbolic Entailment Cones for Learning Hierarchical Embeddings (ICML 2018 · similarity 0.91)
- Modeling Heterogeneous Hierarchies with Relation-specific Hyperbolic Cones (NeurIPS 2021 · similarity 0.87)
- Compositional Entailment Learning for Hyperbolic Vision-Language Models (ICLR 2025 · similarity 0.84)
- Low-distortion and GPU-compatible Tree Embeddings in Hyperbolic Space (ICML 2025 · similarity 0.84)
- Hyperbolic Embeddings of Supervised Models (NeurIPS 2024 · similarity 0.83)
- SPHINX: Structural Prediction using Hypergraph Inference Network (ICML 2025 · similarity 0.83)
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