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A Linear Time and Space Local Point Cloud Geometry Encoder via Vectorized Kernel Mixture (VecKM)
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Proposes VecKM, a linear time and space local point cloud geometry encoder that vectorizes a Gaussian kernel mixture via random Fourier features, enabling dense per-point encodings from all neighbors without explicit grouping.
Problem
Local geometry encoders for point clouds require n^2 neighbor grouping and nK MLP evaluations, causing memory and compute bottlenecks that force downsampling of neighborhoods and yield less descriptive local encodings.
Approach
Represent the local point cloud as a Gaussian kernel mixture and vectorize it with random complex exponential features exp(i x A); the factorization exp(i(x_k - x_0)A) = exp(i x_k A)/exp(i x_0 A) plus a random-feature approximation of the adjacency matrix lets all local encodings be computed in O(npd) without explicit neighborhood grouping.
Key results
- On PCPNet normal estimation, VecKM achieves 17.34 average RMSE versus 20.17 for the best PointNet encoder (>16% lower error) and is over 100x faster; it takes about 150 ms on 100k points.
- On ModelNet40, PointNet with VecKM preprocessing reaches 92.9% instance accuracy vs 90.8% for PointNet; VecKM replacing PCT's embedding raises accuracy to 93.1% and runs 5.98x faster (21.44 ms vs 149.72 ms).
- On S3DIS, VecKM with PointNet++ improves instance mIoU from 64.05 to 67.48, and VecKM with Point Transformer is 20% faster (447 ms vs 559 ms) with slightly higher mIoU (69.53 vs 69.29).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace dense/grouped local encoder with VecKM (dense factorized kernel encoding using all neighbors) | normal estimation encoder · PCPNet normal estimation | average RMSE (lower better): 17.34 vs 20.17 best baseline (PointNet) | Helped |
| add VecKM as preprocessing before PointNet | PointNet · ModelNet40 | instance accuracy: +2.1% (90.8% to 92.9%), runtime 3.04 to 14.32 ms | Helped |
| replace first set abstraction of PointNet++ with VecKM | PointNet++ · ModelNet40 | instance accuracy and inference time: +0.3% accuracy (92.7 to 93.0); 117.13 to 65.78 ms | Helped |
| replace input embedding of PCT with VecKM | PCT · ModelNet40 | instance accuracy and inference time: +0.2% accuracy (92.9 to 93.1); 149.72 to 21.44 ms | Helped |
| replace first dense local encoder of PCT with VecKM | PCT · ShapeNet part segmentation | instance mIoU: +0.1% (85.7 to 85.8); 2.11x faster | Helped |
| replace first layer of PointNet++ with VecKM | PointNet++ · ShapeNet part segmentation | instance mIoU: +0.3% (85.0 to 85.3) | Helped |
| add VecKM before PointNet++ first layer | PointNet++ · S3DIS | instance mIoU: +3.43 (64.05 to 67.48) | Helped |
| replace Point Transformer's first dense local encoder with VecKM | Point Transformer · S3DIS | instance mIoU and inference time: mIoU +0.24 (69.29 to 69.53); inference 559 to 447 ms | Mixed |
| smaller beta (beta=4) vs beta=9 in VecKM->PN (ablation) | VecKM->PN · ModelNet40 | instance accuracy: 91.73% vs 92.95% at alpha=30 | Hurt |
| use all neighbors (no downsampling) via factorized VecKM encoding | local geometry encoder · PCPNet normal estimation | runtime: VecKM >100x faster than compared encoders; 150 ms at 100k points | Helped |
| number of fully connected layers on VecKM encoding: two layers vs more/fewer (ablation) | VecKM normal estimation · PCPNet normal estimation | average RMSE | No clear effect |
Surprising findings
- Replacing Point Transformer's local encoder with VecKM improves speed but gives only marginal accuracy gains, suggesting the heavy backbone already reasons well about geometry.
- The authors conjecture VecKM may be the only possible linear local geometry encoder, a claim not formally proven.
Limitations
- Uses approximations (random features with finite d and p) whose guarantees hold only as d, p grow; theory relies on asymptotic limits and parameters alpha, beta are task-tuned.
- Added VecKM as a preprocessing module increases runtime and parameter counts (e.g., +7.61M parameters for PointNet).
- Accuracy gains on segmentation with Point Transformer are marginal; the 'only linear encoder' conjecture is unproven.
Experimental setup
Datasets: PCPNet ModelNet40 ShapeNet S3DIS
Benchmarks: PCPNet ModelNet40 ShapeNet S3DIS
Metrics: RMSE (degrees) instance accuracy avg class accuracy instance mIoU avg class mIoU overall accuracy inference time (ms)
Baselines: PointNet KPConv DGCNN PointNet++ PCT PointMLP Point Transformer
Builds on: PointNet PointNet++ Point Transformer Random Fourier features (Bochner's theorem) Performer (random feature attention)
Closest papers by meaning
- LCM: Locally Constrained Compact Point Cloud Model for Masked Point Modeling (NeurIPS 2024 · similarity 0.89)
- PointCFormer: A Relation-Based Progressive Feature Extraction Network for Point Cloud Completion (AAAI 2025 · similarity 0.87)
- LinNet: Linear Network for Efficient Point Cloud Representation Learning (NeurIPS 2024 · similarity 0.87)
- Point-Cache: Test-time Dynamic and Hierarchical Cache for Robust and Generalizable Point Cloud Analysis (CVPR 2025 · similarity 0.87)
- On Geometry-Enhanced Parameter-Efficient Fine-Tuning for 3D Scene Segmentation (NeurIPS 2025 · similarity 0.87)
- FreePoint: Unsupervised Point Cloud Instance Segmentation (CVPR 2024 · similarity 0.87)
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