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On the Completeness of Invariant Geometric Deep Learning Models

ICLR 2025 · Point clouds › Point cloud classification · approach: Equivariant point cloud networks · 1 citations (OpenAlex)

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Characterizes the expressiveness of invariant geometric models on point clouds under fully-connected conditions, showing DisGNN is nearly E(3)-complete and that GeoNGNN, geometric subgraph GNNs, DimeNet, GemNet and SphereNet achieve E(3)-completeness.

Problem

The theoretical expressive power (E(3)-completeness) of invariant geometric deep learning models on point clouds remains unclear, especially for DisGNN and related invariant architectures.

Approach

Introduces the notion of "identify" (finer than pairwise distinguish), characterizes DisGNN's unidentifiable point clouds as a measure-zero highly symmetric set via center-distance and DisGNN-based symmetry notions, and proposes GeoNGNN (geometric nested GNN with node marking) as a theoretical tool to break remaining symmetry and prove completeness for a family of geometric subgraph GNNs and exist

Key results

• Unidentifiable point clouds for DisGNN are restricted to a measure-zero symmetric set; only 6 of ~130K QM9 molecules are D-symmetric at the strictest tolerance.
• On ModelNet40, no D-symmetric point clouds are found across all deviation errors, and at most one C-symmetric point cloud under deviation error below 1e-2.
• On the 17-case counterexample dataset, DimeNet, SphereNet, GemNet and GeoNGNN separate 100% of pairs while SchNet and DisGNN separate 0%.

Measured effects reported in the paper

Surprising findings

• Geometric subgraph GNNs and DimeNet, which are weaker than 2-FWL in traditional graph learning, are nonetheless E(3)-complete under fully-connected conditions.
• Equivariant models PaiNN and MACE also separate all counterexample pairs, though their completeness is not proven.

Limitations

• Completeness results hold only for fully-connected graphs; sparse graphs remain open.
• Theory covers invariant models only; vector/high-order equivariant models like PaiNN and MACE lack characterization.
• Practical implementations do not satisfy the theoretical injectivity and full-connectivity conditions.

Experimental setup

Datasets: QM9 ModelNet40

Benchmarks: QM9 ModelNet40 geometric expressiveness dataset (17 counterexamples)

Metrics: proportion of symmetric point clouds separation rate on counterexample pairs

Baselines: DisGNN SchNet DimeNet SphereNet GemNet GeoNGNN PaiNN MACE

Builds on: DisGNN Nested GNN SchNet DimeNet SphereNet

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