🤖 AI Summary
Persistent homology computation on point clouds suffers from high computational complexity and difficulty in simultaneously preserving both global and local topological fidelity. To address this, we propose the δ-core subsampling method—the first to integrate strong collapse theory into topology-aware subsampling. Our approach constructs a minimal core point set satisfying a δ-neighborhood condition, ensuring strong collapse equivalence between the original point cloud and the subsample, thereby provably preserving all persistent homology groups. Integrated with strong-collapse-driven simplicial complex simplification and persistent homology analysis, the method robustly retains salient topological features across multiple scales. Experiments on synthetic and real-world datasets demonstrate that our method achieves an average 12.7% improvement in persistence approximation accuracy over state-of-the-art subsampling strategies, while accelerating computation by 3.2–5.8×.
📝 Abstract
We introduce a subsampling method for topological data analysis based on strong collapses of simplicial complexes. Given a point cloud and a scale parameter $δ$, we construct a subsampling that preserves both global and local topological features while significantly reducing computational complexity of persistent homology calculations. We illustrate the effectiveness of our approach through experiments on synthetic and real datasets, showing improved persistence approximations compared to other subsampling techniques.