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De-coupled NeuroGF for Shortest Path Distance Approximations on Large Terrain Graphs
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Proposes decoupled-NeuroGF, which trains a GAT Siamese L1 embedding first (on coarsened terrain graphs in the M-CTR variant), freezes it, and then trains only an MLP distance head on full-resolution terrains to answer approximate shortest-path-distance queries at large scale.
Problem
Answering approximate shortest-path-distance (geodesic) queries on very large terrain DEMs (millions of nodes) is too costly for existing algorithmic data structures and for end-to-end neural approaches such as NeuroGF and GeGNN.
Approach
Decouple NeuroGF training into two stages: a GAT embedding trained with L1 distance, frozen, then an MLP distance head fine-tuned on full-resolution terrain; combined with coarse-to-refined training (M-CTR) that trains the costly embedding on a downsampled grid.
Key results
- On Norway-250, decoupled NeuroGF achieves 0.84% relative error versus 1.32% for GAT+MLP (NeuroGF) and 13.10% for GeGNN.
- M-CTR reduces relative error by 15.4% on LA (16M nodes) over Coarse-GAT+L1 while improving accuracy from 85.51% to 94.41%, with similar training time (183 vs 137 min).
- On weighted terrains, M-CTR reduces relative error on LA from 3.55% to 1.91% (-46.2%) relative to Coarse-GAT+L1.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add a distance MLP after the GAT+L1 embedding and train it separately (ablation) | NeuroGF on Norway-250 · Norway-250 (downsampled) | relative error (%): 1.32 ± 2.03 -> 0.84 ± 1.66 | Helped |
| Use a distance MLP instead of L1 distance in the latent space (ablation) | GAT embedding on Norway-250 · Norway-250 (downsampled) | relative error (%): 0.92 ± 0.95 -> 1.32 ± 2.03 | Hurt |
| Use M-CTR instead of coarse embeddings with L1 distance alone | NeuroGF on Norway · Norway (4M nodes) | relative error (%): 1.05 ± 0.99 -> 0.90 ± 1.49 | Helped |
| Use M-CTR instead of coarse embeddings with L1 distance alone | NeuroGF on Los Angeles · Los Angeles (16M nodes) | relative error (%): 1.04 ± 1.09 -> 0.88 ± 3.47 | Helped |
| Use M-CTR instead of coarse embeddings with L1 distance alone | NeuroGF on Philadelphia · Philadelphia (1M nodes) | relative error (%): 2.07 ± 1.47 -> 0.51 ± 0.71 | Helped |
| Use M-CTR instead of coarse embeddings with L1 distance alone | NeuroGF on Holland · Holland (1M nodes) | relative error (%): 2.06 ± 1.54 -> 0.86 ± 2.29 | Helped |
| Use M-CTR instead of coarse embeddings with L1 distance alone | NeuroGF on weighted terrain graphs · Los Angeles, weighted terrain | relative error (%): 3.55 ± 2.76 -> 1.91 ± 4.66 | Helped |
| Use M-CTR instead of coarse embeddings with L1 distance alone | NeuroGF on weighted terrain graphs · Philadelphia, weighted terrain | relative error (%): 2.43 ± 4.62 -> 1.32 ± 2.96 | Helped |
| Use M-CTR instead of coarse embeddings with L1 distance alone | NeuroGF on weighted terrain graphs · Holland, weighted terrain | relative error (%): 2.48 ± 6.20 -> 0.82 ± 1.69 | Helped |
| Reuse embeddings and retrain only the distance MLP for terrain instantiations | NeuroGF on uncertain Norway terrain · Norway, uncertain terrain | relative error (%): 19.4 ± 3.04 -> 1.76 ± 2.30 | Helped |
| Reuse embeddings and retrain only the distance MLP for terrain instantiations | NeuroGF on uncertain Los Angeles terrain · Los Angeles, uncertain terrain | relative error (%): 2.27 ± 2.06 -> 0.77 ± 1.04 | Helped |
| Reuse embeddings and retrain only the distance MLP for terrain instantiations | NeuroGF on uncertain Philadelphia terrain · Philadelphia, uncertain terrain | relative error (%): 29.8 ± 6.84 -> 6.14 ± 7.88 | Helped |
Surprising findings
- GAT+L1 (Siamese with L1 latent distance) outperforms both GeGNN and the original end-to-end NeuroGF (GAT+MLP) on the downsampled Norway terrain.
- Full-resolution GAT+L1 is better than M-CTR on Philadelphia and Holland, though much slower to train.
Limitations
- Full-resolution GAT+L1 training on LA did not finish after 30+ hours, so the LA full-model comparison is extrapolated.
- Relative error on several datasets has high variance (e.g., 0.88 ± 3.47 on LA), and M-CTR is worse than full-resolution training on Philadelphia and Holland.
- Approximates geodesic distance by shortest paths on the grid graph rather than the true surface geodesic.
Experimental setup
Datasets: Synthetic Gaussian terrains (50x50 grid) Norway DEM (4M nodes) Los Angeles DEM (16M nodes) Holland, Indiana DEM (1M nodes) Philadelphia DEM (1M nodes)
Benchmarks: Synthetic Gaussian terrains Norway-250 downsampled LA Norway Holland Philadelphia weighted terrains uncertainty terrains
Metrics: average relative error (%) accuracy within 2% relative error (%) training time (min)
Baselines: NeuroGF (GAT+MLP) GeGNN GAT+L1 Coarse-GAT+L1 GCN+MLP
Builds on: NeuroGF graph attention network graph convolutional network metric learning Siamese network
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