Research map / Point clouds / Point cloud classification
Fast, Expressive $\mathrm{SE}(n)$ Equivariant Networks through Weight-Sharing in Position-Orientation Space
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Derives bijective invariant edge attributes for weight sharing over point-pair equivalence classes on SE(n) homogeneous spaces, and builds PΘNITA, a fast fully convolutional SE(3)-equivariant network on position-orientation space R^3×S^2.
Problem
Equivariant networks for 3D point clouds are either computationally expensive (full SE(3) group convolutions, tensor field networks) or limited in directional expressivity (R^3 methods like SchNet); a principled way to define weight-sharing attributes for SE(n) message passing is needed.
Approach
Weight sharing is formalized as conditioning message functions on bijective invariant attributes of equivalence classes of point pairs; for position-orientation space R^3×S^2 these attributes are (o_i^T(p_j-p_i), ||orthogonal displacement||, arccos o_i^T o_j). The resulting kernel is factorized into separable spatial, spherical and channel-mixing convolutions inside a ConvNeXt-style block.
Key results
- On rMD17, PΘNITA achieves the best energy MAE on Aspirin (1.7 vs 2.2 for MACE) and force MAE on Aspirin (5.8 vs 6.6 for ViSNet), matching or beating SOTA on several molecules.
- PΘNITA trains about 3.5x faster per epoch than NequIP (6.1 s vs 20.7 s), and the PNITA R^3 baseline is faster still (5.7 s).
- Equivariant diffusion with PΘNITA in the EDM framework on QM9 improves atom and molecule stability over baselines such as G-Schnet, E-NF and GDM.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use position-orientation feature maps instead of position-only feature maps (ablation) | PΘNITA (compared with PNITA) · rMD17, ten molecules | energy and force MAE: Aspirin E 4.7 -> 1.7; Aspirin F 16.3 -> 5.8; Azobenzene E 3.2 -> 0.7; Azobenzene F 12.2 -> 2.3 | Helped |
| Use PΘNITA instead of NequIP | rMD17 · rMD17 training | training time per epoch (seconds): 20.7 compared to 6.1 | Helped |
Surprising findings
- The R^3 baseline PNITA performs much worse on rMD17 (e.g., 16.3 vs 5.8 force MAE on Aspirin), showing the gain comes from directional information.
Limitations
- Discretization of the orientation sphere makes equivariance approximate rather than exact, as the authors acknowledge.
- Regular group convolution requires a fixed orientation grid, so the separable implementation applies only when the grid is shared across positions.
- Reported gains are mixed: PΘNITA does not win on every molecule or metric in rMD17.
Experimental setup
Datasets: rMD17 QM9 charged N-body particle system Superpixel MNIST
Benchmarks: rMD17 QM9 charged N-body particle system Superpixel MNIST
Metrics: MAE energy MAE force atom stability molecule stability MSE training time per epoch
Baselines: UNiTE GemNet NequIP BOTNet Allegro MACE ViSNet PNITA SEGNN EGNN SE(3)-Transformer Tensor Field Networks
Builds on: ConvNeXt LieConv DimeNet SchNet EDM
Closest papers by meaning
- E2PN: Efficient SE(3)-Equivariant Point Network (CVPR 2023 · similarity 0.90)
- Equivariant Point Network for 3D Point Cloud Analysis (CVPR 2021 · similarity 0.89)
- Probing Equivariance and Symmetry Breaking in Convolutional Networks (NeurIPS 2025 · similarity 0.89)
- Learning symmetries via weight-sharing with doubly stochastic tensors (NeurIPS 2024 · similarity 0.88)
- Approximately Piecewise E(3) Equivariant Point Networks (ICLR 2024 · similarity 0.87)
- Lie Group Decompositions for Equivariant Neural Networks (ICLR 2024 · similarity 0.87)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.