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Geometry Awakening: Cross-Geometry Learning Exhibits Superiority over Individual Structures
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Proposes a cross-geometry graph knowledge distillation framework that uses Euclidean and hyperbolic GCN/HGCN teachers, selects hint embeddings per node by local subgraph hyperbolicity, and aligns them via a geometric embedding optimization module to train a Euclidean GCN student.
Problem
Real-world graphs have geometrically heterogeneous local structures, so a single geometry (Euclidean, hyperbolic, or spherical) may not capture them; existing graph KD methods rely on one geometry for hint embeddings.
Approach
Compute Gromov δ-hyperbolicity of each node's k-hop subgraph to choose between hyperbolic and Euclidean teacher embeddings per node (structure-wise knowledge transfer), then fuse and optimize the mixed-geometry embeddings with a triplet-loss GEO network before distilling into a Euclidean student.
Key results
- Average NC F1 improvement of about 1.11% over the best baseline; 2.66% on Cora and 1.37% on Wiki-CS.
- Average LP ROC AUC improvement of about 1.58% over baselines, with the cross-geometry method reaching 86.63% on Wiki-CS.
- Student models achieve about 232x average inference speed-up over teacher models (1422 ms teacher vs 4.22 ms student).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use cross-geometry distillation with Euclidean and hyperbolic teachers instead of the strongest KD baseline | GCN student · Cora | F1 score: 83.39±1.22 -> 86.05±0.60 | Helped |
| Use cross-geometry distillation with Euclidean and hyperbolic teachers instead of the strongest KD baseline | GCN student · Cora | ROC AUC: 84.38±1.04 -> 86.95±0.43 | Helped |
| Use cross-geometry distillation with Euclidean and hyperbolic teachers instead of the strongest KD baseline | GCN student · Wiki-CS | F1 score: 72.73±0.02 -> 74.17±0.50 | Helped |
| Use cross-geometry distillation with Euclidean and hyperbolic teachers instead of the strongest KD baseline | GCN student · Wiki-CS | ROC AUC: 84.71±0.60 -> 86.63±0.31 | Helped |
| Use cross-geometry distillation instead of a single-geometry teacher | GCN student · Arxiv | F1 score: 73.21 ±0.19 -> 70.89 ±0.53 | Hurt |
| Use cross-geometry distillation instead of a single-geometry teacher | GCN student · Proteins | F1 score: 72.83 ±0.09 -> 71.22 ±0.41 | Hurt |
| Use only the Euclidean teacher rather than both geometric teachers (ablation) | Our GCN student · Wiki-CS | F1 score: 72.84 ± 1.66 -> 74.17 ± 0.50 | Hurt |
| Use only the hyperbolic teacher rather than both geometric teachers (ablation) | Our GCN student · Wiki-CS | F1 score: 72.38 ± 1.83 -> 74.17 ± 0.50 | Hurt |
| Add the SWKT module (ablation) | Our GCN student · Wiki-CS | F1 score: 73.40 ± 1.26 -> 74.17 ± 0.50 | Helped |
| Add the SWKT module (ablation) | Our GCN student · Wiki-CS | ROC AUC: 85.08 ± 0.55 -> 86.63 ± 0.31 | Helped |
| Add the GEO module (ablation) | Our GCN student · Wiki-CS | F1 score: 73.39 ± 1.27 -> 74.17 ± 0.50 | Helped |
| Add the GEO module (ablation) | Our GCN student · Wiki-CS | ROC AUC: 85.49 ± 0.97 -> 86.63 ± 0.31 | Helped |
Surprising findings
- Hyperbolic LSP performed worse than Euclidean LSP on Wiki-CS (about 3.05% drop), suggesting a teacher's advantage in one geometry may not transfer to a student in another geometry.
- Naively combining Euclidean and hyperbolic teacher embeddings (the 'Cross' baselines) often underperforms single-geometry teachers.
Limitations
- Integrating multiple geometries introduces hyperparameters (λ threshold, β weight) and results depend on their selection, affecting stability.
- Distillation requires pre-trained teacher models in multiple geometries, increasing resource and time requirements.
Experimental setup
Datasets: Cora Citeseer Pubmed Wiki-CS Coauthor-Physics Ogbn-Arxiv Ogbn-Proteins
Benchmarks: Cora Citeseer Pubmed Wiki-CS Coauthor-Physics Ogbn-Arxiv Ogbn-Proteins
Metrics: F1 score ROC AUC inference time
Baselines: FitNet AT LSP MSKD VQG
Builds on: FitNet GCN HGCN triplet loss LSP
Closest papers by meaning
- Distilling Graph Geometry: Knowledge Gap from GNNs to MLPs (NeurIPS 2026 · similarity 0.89)
- Geometry-Aware Edge Pooling for Graph Neural Networks (NeurIPS 2025 · similarity 0.87)
- Toward a Unified Geometry Understanding : Riemannian Diffusion Framework for Graph Generation and Prediction (NeurIPS 2025 · similarity 0.87)
- Simplicial Representation Learning with Neural $k$-Forms (ICLR 2024 · similarity 0.87)
- Contextual Structure Knowledge Transfer for Graph Neural Networks (AAAI 2025 · similarity 0.86)
- What do Graph Neural Networks learn? Insights from Tropical Geometry (NeurIPS 2024 · similarity 0.86)
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