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Neural Snowflakes: Universal Latent Graph Inference via Trainable Latent Geometries
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Introduces neural snowflakes, trainable fractal-like metrics on R^d built from snowflake activations, proves MLP+neural snowflake can isometrically embed any finite weighted graph, and uses them as differentiable geometry in DGM latent graph inference.
Problem
Latent graph inference relies on choosing a latent geometry, usually by costly random search over non-differentiable geometries, and lacks theoretical embedding guarantees for those geometries.
Approach
A trainable neural network of snowflake activations (monotone, concave, non-negative-weighted layers) that transforms Euclidean distances into quasi-metrics, so an MLP plus this snowflake can isometrically embed any finite weighted graph and the geometry is learned by backpropagation.
Key results
- Neural Snowflake reaches 87.07% on Cora and 74.76% on CiteSeer with the original graph, the top results in that table.
- Without the input graph, Neural Snowflake gets 71.22% on Cora and 67.80% on CiteSeer, with Tadpole 90.02% and Aerothermodynamics 88.65%, competitive with DGM variants.
- In synthetic embedding, neural snowflake in R^100 with 847 parameters reaches errors around 1e-5 to 3e-3, versus MLP errors up to 1.32 with 5422 parameters.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Neural Snowflake (+ MLP) instead of MLP for metric embedding | Synthetic graph embedding · ||x-y||^0.5 log(1+||x-y||)^0.5 metric | Mean square embedding error: 1.3196 -> 0.0034 | Helped |
| Use Neural Snowflake (+ MLP) instead of MLP for metric embedding | Synthetic graph embedding · ||x-y||^0.1 log(1+||x-y||)^0.9 metric | Mean square embedding error: 1.2555 -> 0.0032 | Helped |
| Use Neural Snowflake (+ MLP) instead of MLP for metric embedding | Synthetic graph embedding · 1 - 1/(1+||x-y||^0.5) metric | Mean square embedding error: 0.1738 -> 0.00004 | Helped |
| Use Neural Snowflake (+ MLP) instead of MLP for metric embedding | Synthetic graph embedding · 1 - exp(-(||x-y||-1)/log(||x-y||)) metric | Mean square embedding error: 0.3249 -> 0.00008 | Helped |
| Use Neural Snowflake (+ MLP) instead of MLP for metric embedding | Synthetic graph embedding · 1 - 1/(1+||x-y||)^0.2 metric | Mean square embedding error: 0.0315 -> 0.00005 | Helped |
| Use Neural Snowflake (+ MLP) instead of MLP for metric embedding | Synthetic graph embedding · 1 - 1/(1+||x-y||^0.2+||x-y||^0.5) metric | Mean square embedding error: 0.2484 -> 0.00002 | Helped |
| Use Neural Snowflake instead of DGM (Poincare) | Latent graph inference · Cora (original input graph) | Accuracy: 86.07 ± 3.53 -> 87.07 ± 3.45 | Helped |
| Use Neural Snowflake instead of DGM (Hyperboloid × Hyperboloid) | Latent graph inference · CiteSeer (original input graph) | Accuracy: 74.42 ± 1.83 -> 74.76 ± 1.74 | Helped |
| Use Neural Snowflake instead of DGM (Hyperboloid) | Latent graph inference · Cora (no input graph) | Accuracy: 70.00 ± 4.08 -> 71.22 ± 4.27 | Helped |
| Use Neural Snowflake instead of DGM (Hyperboloid) | Latent graph inference · CiteSeer (no input graph) | Accuracy: 68.34 ± 1.59 -> 67.80 ± 2.44 | Hurt |
| Use Neural Snowflake instead of DGM (Snowflake) | Latent graph inference · Tadpole (no input graph) | Accuracy: 91.61 ± 4.38 -> 90.02 ± 3.51 | Hurt |
| Use Neural Snowflake instead of DGM (Hyperboloid × Hyperboloid) | Latent graph inference · Aerothermodynamics (no input graph) | Accuracy: 89.55 ± 2.52 -> 88.65 ± 3.10 | Hurt |
Surprising findings
- Riemannian (smooth) manifolds cannot isometrically embed a particular 5-node graph, so smooth geometries are too rigid for universal graph embedding.
Limitations
- Gumbel top-k edge sampling is suboptimal and dampens the gains of the learned metric in latent graph inference.
- Gains over DGM variants on graph benchmarks are modest and often within standard deviations.
- Scalability to large graphs is not addressed; NodeFormer-style scalable architectures are left to future work.
Experimental setup
Datasets: Cora CiteSeer Tadpole Aerothermodynamics
Benchmarks: Cora CiteSeer Tadpole Aerothermodynamics
Metrics: accuracy mean square embedding error
Baselines: DGM Euclidean DGM Hyperboloid DGM Poincare DGM Spherical DGM product manifolds GCN MLP
Builds on: Differentiable Graph Module (DGM) metric snowflakes Gumbel top-k edge sampling input convex neural networks
Closest papers by meaning
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