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Beyond Canonicalization: How Tensorial Messages Improve Equivariant Message Passing
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Proposes tensorial messages between local canonicalization frames to make message passing O(d)-equivariant, instantiated as an equivariant PointNet++ that beats data augmentation and reaches state-of-the-art normal regression on ModelNet40.
Problem
Equivariant message passing on point clouds typically needs specialized architectures; local canonicalization is cheap and generic but scalar messages cannot communicate directional/tensorial geometric information between nodes with different local frames.
Approach
Predict equivariant local orthonormal frames per node, express features in local frames, and transform neighbor features into the receiving node's frame during message passing via representation matrices rho(R_i R_j^{-1}), so tensor/pseudotensor features are communicated consistently; frames are also iteratively refined per layer.
Key results
- On ModelNet40 normal regression under O(3)/O(3), learned frames with refining frames and tensor messages reach 0.88 cosine similarity, versus 0.86 for data augmentation and 0.83 for scalar messages.
- On ShapeNet segmentation (O(3)/O(3)), the refined tensor-message model reaches 80.2 IoU, versus 78.0 for data augmentation, though below VN-DGCNN's 81.4.
- Tensorial messages beat scalar messages even with random local frames (0.84 vs 0.80 cosine similarity on normal regression).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace scalar messages with tensor messages (ablation) | PointNet++ with learned and refined frames · ModelNet40 normal regression, O(3)/O(3) | cosine similarity: 0.83 -> 0.88 | Helped |
| replace scalar messages with tensor messages (ablation) | PointNet++ with learned and refined frames · ShapeNet segmentation, O(3)/O(3) | IoU: 79.0 -> 80.2 | Helped |
| replace scalar messages with tensor messages (ablation) | PointNet++ with learned and refined frames · ModelNet40 classification, O(3)/O(3) | accuracy: 87.3 -> 88.7 | Helped |
| replace scalar messages with tensor messages (ablation) | PointNet++ with learned and refined frames · ScanObjectNN classification, O(3)/O(3) | accuracy: 73.7 -> 81.0 | Helped |
| add iterative local-frame refinement (ablation) | PointNet++ with learned frames and tensor messages · ModelNet40 normal regression, O(3)/O(3) | cosine similarity: 0.87 -> 0.88 | Helped |
| add iterative local-frame refinement (ablation) | PointNet++ with learned frames and tensor messages · ShapeNet segmentation, O(3)/O(3) | IoU: 79.7 -> 80.2 | Helped |
| add iterative local-frame refinement (ablation) | PointNet++ with learned frames and tensor messages · ModelNet40 classification, O(3)/O(3) | accuracy: 88.0 -> 88.7 | Helped |
| add iterative local-frame refinement (ablation) | PointNet++ with learned frames and tensor messages · ScanObjectNN classification, O(3)/O(3) | accuracy: 79.8 -> 81.0 | Helped |
| replace learned local frames with random local frames (ablation) | PointNet++ with tensor messages, without frame refinement · ModelNet40 normal regression | cosine similarity: 0.87 -> 0.84 | Hurt |
| replace learned local frames with random local frames (ablation) | PointNet++ with scalar messages, without frame refinement · ModelNet40 normal regression | cosine similarity: 0.82 -> 0.80 | Hurt |
| replace learned local frames with PCA-based local frames (ablation) | PointNet++ with refined frames and tensor messages · ModelNet40 classification, O(3)/O(3) | accuracy: 88.7 -> 88.9 | Helped |
| replace learned local frames with PCA-based local frames (ablation) | PointNet++ with refined frames and tensor messages · ShapeNet segmentation, O(3)/O(3) | IoU: 80.2 vs. 80.2 | No clear effect |
Surprising findings
- With randomly chosen local frames, the tensor-message model still outperforms scalar-message models with learned frames.
- Data augmentation achieves higher cosine similarity than equivariant models in the z/z setting on normal regression and segmentation, and the error-vs-data curve shows that augmentation can be better at some dataset sizes.
Limitations
- Adapted PointNet++ does not reach state-of-the-art on classification or segmentation; segmentation trails VN-DGCNN.
- Equivariant frames are not fully continuous in highly symmetric cases, and the third frame vector is not learned directly.
- Equivariant models do not always beat data augmentation in accuracy at all dataset sizes.
Experimental setup
Datasets: ModelNet40 ShapeNet ScanObjectNN
Benchmarks: ModelNet40 normal vector regression ShapeNet segmentation ModelNet40 classification ScanObjectNN classification
Metrics: cosine similarity mIoU classification accuracy evaluation runtime
Baselines: PointNet++ with data augmentation RS-CNN DGCNN GC-Conv Luo et al. VN-DGCNN CRIN LGR-Net RI-Conv TFN VN-PointNet
Builds on: PointNet++ local canonicalization (learned local frames) tensor field networks
Closest papers by meaning
- Equivariant Point Cloud Analysis via Learning Orientations for Message Passing (CVPR 2022 · similarity 0.89)
- Equivariant Networks for Hierarchical Structures (NeurIPS 2020 · similarity 0.88)
- Fast, Expressive $\mathrm{SE}(n)$ Equivariant Networks through Weight-Sharing in Position-Orientation Space (ICLR 2024 · similarity 0.86)
- The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products (ICML 2025 · similarity 0.86)
- Probing Equivariance and Symmetry Breaking in Convolutional Networks (NeurIPS 2025 · similarity 0.85)
- Approximately Piecewise E(3) Equivariant Point Networks (ICLR 2024 · similarity 0.85)
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