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