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The Flood Complex: Large-Scale Persistent Homology on Millions of Points
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Introduces the Flood complex, a landmark-based filtered simplicial complex on Delaunay triangulations of landmarks that is aware of the full point cloud, enabling GPU-accelerated persistent homology on millions of points.
Problem
Persistent homology on large-scale Euclidean point clouds is infeasible with Vietoris-Rips, Alpha, and subsampling-based approaches because complexes become too large or subsampling loses topological information.
Approach
Build a filtration on the Delaunay triangulation of a small landmark subset, where each simplex enters when its convex hull is fully covered by balls of radius r centered on the entire point cloud (flooding), with GPU-based computation of filtration values via enclosing-ball masking.
Key results
- On swisscheese, Flood PH reaches 0.98 balanced accuracy in about 1.1 s per point cloud, versus 0.85 for Alpha† PH and about 0.5 for PointNet++, PointMLP and PVT.
- On rocks classification, Flood PH achieves 0.88 balanced accuracy, while neural baselines are near chance (0.51-0.54).
- Flood PH computes H0-H2 on point clouds with millions of points (12M-point RV-A89) in seconds, while full Alpha PH needs more than 15 minutes and over 90 GB RAM.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Flood PH instead of Alpha† PH | PH-based classification · swisscheese | balanced accuracy: 0.85 ± 0.04 -> 0.98 ± 0.01 | Helped |
| Use Flood PH instead of Alpha† PH | PH-based classification · rocks classification | balanced accuracy: 0.78 ± 0.03 -> 0.88 ± 0.03 | Helped |
| Use Flood PH instead of PointMLP | rocks regression · rocks regression | MSE: (8.8 ± 1.0) × 1e-3 -> (0.5 ± 0.2) × 1e-3 | Helped |
| Use Flood PH instead of PointMLP | PH-based classification · corals | balanced accuracy: 0.59 ± 0.15 -> 0.77 ± 0.09 | Helped |
| Use Flood PH instead of Alpha† PH | PH-based classification · mcb-c | balanced accuracy: 0.66 ± 0.03 -> 0.74 ± 0.02 | Helped |
| Use Flood PH instead of PVT | point-cloud classification · mcb-c | balanced accuracy: 0.77 ± 0.03 -> 0.74 ± 0.02 | Hurt |
| Use Flood PH instead of PVT | point-cloud classification · modelnet10 | balanced accuracy: 0.94 ± 0.01 -> 0.72 ± 0.02 | Hurt |
| Replace farthest point sampling with uniform random landmark sampling (ablation) | Flood PH · RV-A89 | bottleneck distance H0: 1.7 ± 0.1 -> 3.2 ± 0.1 | Hurt |
| Replace farthest point sampling with uniform random landmark sampling (ablation) | Flood PH · RV-A89 | bottleneck distance H1: 3.9 ± 0.8 -> 3.3 ± 0.5 | Helped |
| Replace farthest point sampling with uniform random landmark sampling (ablation) | Flood PH · RV-A89 | bottleneck distance H2: 5.7 ± 1.1 -> 5.4 ± 1.6 | Helped |
| Replace farthest point sampling with uniform random landmark sampling (ablation) | Flood PH · Leptoseris paschalensis | bottleneck distance H0: 0.9 ± 0.1 -> 0.9 ± 0.1 | No clear effect |
| Replace farthest point sampling with uniform random landmark sampling (ablation) | Flood PH · Leptoseris paschalensis | bottleneck distance H1: 1.0 ± 0.1 -> 1.2 ± 0.2 | Hurt |
Surprising findings
- On ModelNet10, neural networks outperform all PH-based methods by more than 20 percentage points, despite the PH approach being strong on topologically complex data.
- Full-data Alpha PH on swisscheese reaches only 0.94 accuracy at 1M points, below Flood PH, suggesting the size of persistence diagrams hampers vectorization.
Limitations
- Theoretical homotopy equivalence proven only for points in R^2; higher-dimensional case is conjectured.
- Each k-simplex in Del(L) can detect at most one k-dimensional hole, so resolution is limited by landmark placement.
- On simple object datasets such as ModelNet10, PH-based methods trail neural networks substantially.
Experimental setup
Datasets: ModelNet10 MCB-c (subset of MCB-A) corals (Smithsonian 3D Digitization) swisscheese (synthetic) rocks (synthetic, PoreSpy) RV-A89 cryo-EM point cloud Leptoseris paschalensis coral scan
Benchmarks: swisscheese rocks ModelNet10 MCB-c corals RV-A89 bottleneck distance vs Alpha PH (full)
Metrics: balanced accuracy mean squared error bottleneck distance runtime (s)
Baselines: Alpha PH (full) Alpha PH subsampled (Alpha†) Alpha PH (avg.) PointNet++ PointMLP PVT
Builds on: Alpha complex Witness complex Vietoris-Rips complex Ritter's heuristic
Closest papers by meaning
- Persistent Homology for High-dimensional Data Based on Spectral Methods (NeurIPS 2024 · similarity 0.88)
- Point-Level Topological Representation Learning on Point Clouds (ICML 2025 · similarity 0.87)
- Towards Scalable Topological Regularizers (ICLR 2025 · similarity 0.86)
- Differentiability and Optimization of Multiparameter Persistent Homology (ICML 2024 · similarity 0.85)
- Graphcode: Learning from multiparameter persistent homology using graph neural networks (NeurIPS 2024 · similarity 0.84)
- Wasserstein convergence of Cech persistence diagrams for samplings of submanifolds (NeurIPS 2024 · similarity 0.84)
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