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.

翻译：我们提出拓扑点云聚类（TPCC），这是一种基于点对全局拓扑特征贡献程度对任意点云中点进行聚类的新方法。TPCC融合了谱聚类与拓扑数据分析的优良特性，其核心思想是考察与给定点云相关联的单纯复形的谱性质。由于基于稀疏特征向量计算，TPCC在可解释性和实现便捷性上与谱聚类相当。然而，不同于仅关注由点云数据生成的单一图矩阵，该算法通过考虑适构单纯复形对应的一组霍奇-拉普拉斯算子，能够利用更丰富的拓扑特征刻画点云中的数据点，并受益于拓扑技术对噪声的相对鲁棒性。我们在合成数据与真实数据上测试了TPCC的性能，并将其与经典谱聚类进行了对比。