mlsanityData to 7 Oct 2026

On the Completeness of Invariant Geometric Deep Learning Models

arXiv 2402.04836 ↗First posted 7 Feb 2024cs.LG cs.AIcited by 11

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Abstract (arXiv)

Invariant models, one important class of geometric deep learning models, are capable of generating meaningful geometric representations by leveraging informative geometric features in point clouds. These models are characterized by their simplicity, good experimental results and computational efficiency. However, their theoretical expressive power still remains unclear, restricting a deeper understanding of the potential of such models. In this work, we concentrate on characterizing the theoretical expressiveness of a wide range of invariant models under fully-connected conditions. We first rigorously characterize the expressiveness of the most classic invariant model, message-passing neural networks incorporating distance (DisGNN), restricting its unidentifiable cases to be only highly symmetric point clouds. We then prove that GeoNGNN, the geometric counterpart of one of the simplest subgraph graph neural networks, can effectively break these corner cases' symmetry and thus achieve E(3)-completeness. By leveraging GeoNGNN as a theoretical tool, we further prove that: 1) most subgraph GNNs developed in traditional graph learning can be seamlessly extended to geometric scenarios with E(3)-completeness; 2) DimeNet, GemNet and SphereNet, three well-established invariant models, are also all capable of achieving E(3)-completeness. Our theoretical results fill the gap in the expressive power of invariant models, contributing to a rigorous and comprehensive understanding of their capabilities.

Key facts. Cited by 11 papers in the map (data through 7 Oct 2026). Builds on 57 of its 83 references that are in the map. First posted on arXiv 7 Feb 2024.

Builds on

57 of 83 references.

References · 57

  1. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,282
  2. 2017Inductive Representation Learning on Large Graphscited by 6,284
  3. 2018How Powerful are Graph Neural Networks?cited by 4,355
  4. 2017Neural Message Passing for Quantum Chemistrycited by 3,460
  5. 2019Fast Graph Representation Learning with PyTorch Geometriccited by 2,414
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  7. 2021E(n) Equivariant Graph Neural Networkscited by 951
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  9. 2020Learning to Simulate Complex Physics with Graph Networkscited by 867
  10. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 735
  11. 2021E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentialscited by 701
  12. 2020SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networkscited by 602
38 more
  1. 2019Deep Learning for 3D Point Clouds: A Surveycited by 557
  2. 2020Directional Message Passing for Molecular Graphscited by 556
  3. 2022MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fieldscited by 467
  4. 2021Equivariant message passing for the prediction of tensorial properties and molecular spectracited by 438
  5. 2019Provably Powerful Graph Networkscited by 385
  6. 2021GemNet: Universal Directional Graph Neural Networks for Moleculescited by 342
  7. 2018Invariant and Equivariant Graph Networkscited by 339
  8. 2021Vector Neurons: A General Framework for SO(3)-Equivariant Networkscited by 304
  9. 2022Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphscited by 258
  10. 2020Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Moleculescited by 254
  11. 2022Learning Local Equivariant Representations for Large-Scale Atomistic Dynamicscited by 254
  12. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 225
  13. 2021Identity-aware Graph Neural Networkscited by 197
  14. 2022TorchMD-NET: Equivariant Transformers for Neural Network based Molecular Potentialscited by 179
  15. 2021Spherical Message Passing for 3D Graph Networkscited by 166
  16. 2021Frame Averaging for Invariant and Equivariant Network Designcited by 166
  17. 2021Equivariant Subgraph Aggregation Networkscited by 154
  18. 2021From Stars to Subgraphs: Uplifting Any GNN with Local Structure Awarenesscited by 154
  19. 2021Nested Graph Neural Networkscited by 136
  20. 2021Scalars are universal: Equivariant machine learning, structured like classical physicscited by 127
  21. 2022Understanding and Extending Subgraph GNNs by Rethinking Their Symmetriescited by 112
  22. 2023On the Expressive Power of Geometric Graph Neural Networkscited by 89
  23. 2020On the Universality of Rotation Equivariant Point Cloud Networkscited by 80
  24. 2023FAENet: Frame Averaging Equivariant GNN for Materials Modelingcited by 72
  25. 2022ComENet: Towards Complete and Efficient Message Passing for 3D Molecular Graphscited by 69
  26. 2023A Complete Expressiveness Hierarchy for Subgraph GNNs via Subgraph Weisfeiler-Lehman Testscited by 64
  27. 2022GemNet-OC: Developing Graph Neural Networks for Large and Diverse Molecular Simulation Datasetscited by 61
  28. 2022Ordered Subgraph Aggregation Networkscited by 57
  29. 2022Geometrically Equivariant Graph Neural Networks: A Surveycited by 55
  30. 2023A new perspective on building efficient and expressive 3D equivariant graph neural networkscited by 46
  31. 2023Smooth, exact rotational symmetrization for deep learning on point cloudscited by 39
  32. 2022Low Dimensional Invariant Embeddings for Universal Geometric Learningcited by 38
  33. 2023Neural Injective Functions for Multisets, Measures and Graphs via a Finite Witness Theoremcited by 33
  34. 2021Learning 3D Representations of Molecular Chirality with Invariance to Bond Rotationscited by 29
  35. 2023Evaluation of the MACE Force Field Architecture: from Medicinal Chemistry to Materials Sciencecited by 25
  36. 2023Learning Probabilistic Symmetrization for Architecture Agnostic Equivariancecited by 25
  37. 2023Is Distance Matrix Enough for Geometric Deep Learning?cited by 22
  38. 2020Message Passing Networks for Molecules with Tetrahedral Chiralitycited by 14

50 most cited of 57 All 57 (JSON)

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 3; 2025: 5; 2026: 3.

11 papers.

  1. 2024A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applicationscited by 28
  2. 2024Geometric Representation Condition Improves Equivariant Molecule Generationcited by 13
  3. 2024On the Expressive Power of Sparse Geometric MPNNscited by 7
  4. 2026Rethinking Diffusion Models with Symmetries through Canonicalization with Applications to Molecular Graph Generationcited by 5
  5. 2025GeoRecon: Graph-Level Representation Learning for 3D Molecules via Reconstruction-Based Pretrainingcited by 3
  6. 2025Universally Invariant Learning in Equivariant GNNscited by 3
  7. 2025Geometric Mixture Models for Electrolyte Conductivity Predictioncited by 2
  8. 2025VDW-GNNs: Vector diffusion wavelets for geometric graph neural networkscited by 1
  9. 2025Toward bilipshiz geometric modelscited by 1
  10. 2026Quantitative Approximation Rates for Group Equivariant Learningcited by 1
  11. 2026Why Multi-Layer Message Passing Works: Completeness Theory for Graph Neural Network Interatomic Potentialscited by 0