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Point-Level Topological Representation Learning on Point Clouds
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Proposes TOPF, a method that extracts point-level topological features from point clouds by projecting persistent homology generators onto harmonic subspaces of the Hodge Laplacian, and introduces a topological clustering benchmark suite.
Problem
Global topological descriptions from persistent homology give a single description of a point cloud, but machine learning tasks such as classification and clustering need point-level features that encode topological information.
Approach
Select significant persistent homology classes, embed their Z/3Z generators into real vectors, project them onto the harmonic space of the simplicial Hodge Laplacian at a chosen scale, and average the normalised simplex values to each vertex to obtain per-point features.
Key results
- TOPF with spectral clustering reaches mean ARI 0.86 across seven benchmark datasets, versus 0.58 for TPCC and 0.64 for DGCNN.
- TOPF has mean running time of 15.1 s versus 298.9 s for TPCC on the benchmark.
- Ablation: removing the harmonic projection drops mean ARI from 0.86 to 0.47.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use TOPF clustering instead of TPCC | 4spheres clustering · 4spheres | ARI: 0.52$\pm$0.17 -> 0.81 | Helped |
| use TOPF clustering instead of Geo | Ellipses clustering · Ellipses | ARI: 0.81 -> 0.95 | Helped |
| use TOPF clustering instead of DBSCAN | 4Circles+Grid clustering · 4Circles+Grid | ARI: 0.92 -> 0.70 | Hurt |
| use TOPF clustering instead of DGCNN | Halved Circle clustering · Halved Circle | ARI: 0.39$\pm$0.07 -> 0.71 | Helped |
| use TOPF clustering instead of TPCC | 2Spheres2Circle clustering · 2Spheres2Circle | ARI: 0.97$\pm$0.01 -> 0.94 | Hurt |
| use TOPF clustering instead of TPCC | SphereinCircle clustering · SphereinCircle | ARI: 0.98$\pm$<0.1 -> 0.97 | Hurt |
| use TOPF clustering instead of Geo | Spaceship clustering · Spaceship | ARI: 0.87 -> 0.92 | Helped |
| add harmonic projection (ablation) | TOPF · 4spheres | ARI: 0.08 -> 0.81 | Helped |
| add harmonic projection (ablation) | TOPF · Ellipses | ARI: 0.53 -> 0.95 | Helped |
| add harmonic projection (ablation) | TOPF · 4Circles+Grid | ARI: 0.11 -> 0.70 | Helped |
| add harmonic projection (ablation) | TOPF · Halved Circle | ARI: 0.15 -> 0.71 | Helped |
| add harmonic projection (ablation) | TOPF · 2Spheres2Circle | ARI: 0.95 -> 0.94 | Hurt |
Surprising findings
- Node2Vec achieves near-zero ARI on the benchmark (mean 0.03), indicating it captures little topological information.
- Without harmonic projection, TOPF performs better on 2Spheres2Circle and SphereinCircle, so the projection is not always beneficial.
Limitations
- Only produces meaningful output on point clouds with topological structure detectable by persistent homology.
- Runtime grows on very large point clouds, especially in higher dimensions where alpha filtrations are infeasible.
- Feature selection heuristic (smallest persistence quotient) is hard and relies on domain knowledge.
Experimental setup
Datasets: ShapeNetPart 3DMatch
Benchmarks: Topological Clustering Benchmark Suite
Metrics: adjusted rand index running time
Baselines: TPCC spectral clustering k-means OPTICS DBSCAN agglomerative clustering ToMATo Geo (pgeof) node2vec PointNet WSDesc DGCNN
Builds on: persistent homology Hodge Laplacian TPCC alpha complex
Closest papers by meaning
- Diffeomorphic interpolation for efficient persistence-based topological optimization (NeurIPS 2024 · similarity 0.88)
- The Flood Complex: Large-Scale Persistent Homology on Millions of Points (NeurIPS 2025 · similarity 0.87)
- Cover learning for large-scale topology representation (ICML 2025 · similarity 0.87)
- Towards Scalable Topological Regularizers (ICLR 2025 · similarity 0.86)
- Rethinking Point Cloud Data Augmentation: Topologically Consistent Deformation (ICML 2025 · similarity 0.86)
- Topological Blindspots: Understanding and Extending Topological Deep Learning Through the Lens of Expressivity (ICLR 2025 · similarity 0.86)
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