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Approximately Piecewise E(3) Equivariant Point Networks
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Proposes APEN, a point network framework for approximately piecewise E(3)-equivariant processing whose equivariance approximation error is bounded by partition uncertainty and the probability of a bad partition, evaluated on human body part segmentation and room-scan one-shot segmentation.
Problem
Piecewise E(3) equivariance for point clouds requires knowing the partition into independently moving parts, which is unknown and must be predicted, so partition errors cause equivariance approximation errors that are usually uncontrolled.
Approach
Build a layered encoder where each layer is piecewise E(3)-equivariant over a predicted partition (argmax of a GMM-based soft assignment with merge regularization); the partition goes from fine to coarse across layers, so equivariance error is bounded by partition uncertainty and the probability of drawing a non-proper sub-partition.
Key results
- Human body part segmentation on DFAUST unseen random sequences: 92.2% mean IoU vs. 78.1% for FA and 77.8% for EPN.
- One-shot room scan segmentation: outperforms baselines trained on synthetic data in 7 of 8 scenes (e.g., Scene 2: 98.2 vs 97.3 PointNet synthetic).
- Subject classification accuracy 71.4% vs. 32.1% for DGCNN and 18.5% for PointNet.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use APEN instead of EPN | Human body parts segmentation · Human body parts segmentation, random split | Mean IoU(%): 89.6 -> 94.2 | Helped |
| Use APEN instead of EPN | Human body parts segmentation · Human body parts segmentation, unseen random seq. | Mean IoU(%): 77.8 -> 92.2 | Helped |
| Use APEN instead of EPN | Human body parts segmentation · Human body parts segmentation, unseen seq. | Mean IoU(%): 84.1 -> 93.5 | Helped |
| Train APEN on a single scan rather than use synthetic-data baselines | One-shot room scan segmentation · DynLab room scans, Scenes 1–8 | Mean IoU(%) | Mixed |
| Use APEN instead of DGCNN (Synthetic) | Room scan segmentation · DynLab Scene 6 | Mean IoU(%): 95.6±1.1 -> 93.4±2.8 | Hurt |
| Use APEN instead of DGCNN | Subject classification · AMASS DFaust, PosePrior test | Accuracy (%): 32.1 -> 71.4 | Helped |
Surprising findings
- Baselines trained on a large synthetic dataset are outperformed in 7 of 8 room scenes by a model trained on only a single scan.
Limitations
- Equivariance error bounds are presented as initial insights and not analyzed in depth.
- Theoretical bound relies on the assumed partition-uncertainty functional delta and the lambda term, which is not analyzed in closed form for Q_simple.
- Requires ground-truth segmentation to supervise part-center votes.
Experimental setup
Datasets: SMPL DFAUST AMASS PosePrior DynLab ShapeNet
Benchmarks: SMPL human body part segmentation DFAUST human body part segmentation DynLab room scans one-shot segmentation AMASS/PosePrior subject classification
Metrics: mean IoU accuracy
Baselines: PointNet DGCNN Vector Neurons VN-Transformer Frame Averaging EPN
Builds on: Frame Averaging PointNet Vector Neurons Equivariant Point Network
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