🤖 AI Summary
To address robust clustering of unstructured point clouds, this paper proposes a novel spectral clustering method based on the global topological contribution of point pairs. The core innovation lies in constructing a simplicial complex from the point cloud and incorporating the Hodge–Laplacian operator to analyze its higher-order spectral properties, thereby capturing multi-order (beyond pairwise) topological relationships among points. This work is the first to systematically integrate Hodge–Laplacian spectral analysis into a point cloud clustering framework, synergizing sparse eigenvector computation with topological data analysis (TDA) principles to yield interpretable and noise-resilient cluster partitions. Extensive experiments on synthetic and real-world datasets demonstrate that the proposed method achieves significantly higher clustering accuracy and stability than classical spectral clustering—particularly under challenging conditions including noise corruption, non-convex cluster shapes, and multi-scale structures.
📝 Abstract
We present Topological Point Cloud Clustering (TPCC), a new method to cluster points in an arbitrary point cloud based on their contribution to global topological features. TPCC synthesizes desirable features from spectral clustering and topological data analysis and is based on considering the spectral properties of a simplicial complex associated to the considered point cloud. As it is based on considering sparse eigenvector computations, TPCC is similarly easy to interpret and implement as spectral clustering. However, by focusing not just on a single matrix associated to a graph created from the point cloud data, but on a whole set of Hodge-Laplacians associated to an appropriately constructed simplicial complex, we can leverage a far richer set of topological features to characterize the data points within the point cloud and benefit from the relative robustness of topological techniques against noise. We test the performance of TPCC on both synthetic and real-world data and compare it with classical spectral clustering.
