Critical Points to Determine Persistence Homology

Computation of simplicial complexes of a large point cloud often relies on extracting a sample, to reduce the associated computational burden. The sampling of the point cloud should minimally mutilate the features of the underlying object to enable effective "feature extraction" that lies at the center of modern data analysis techniques, e.g., machine learning. The study considers sampling critical points of a Morse function associated with a point cloud, to approximate the Vietoris-Rips complex and to compute persistence homology. The effectiveness of the approach is compared with the farthest point sampling (FPS), in the context of two classification problems. The empirical results suggest that sampling critical points of the Morse function can be more effective than FPS when determining the persistence homology for the cases where the critical points play a decisive role.