Subsampling Methods for Persistent Homology

Abstract

Persistent homology is a multiscale method for analyzing the shape of sets\nand functions from point cloud data arising from an unknown distribution\nsupported on those sets. When the size of the sample is large, direct\ncomputation of the persistent homology is prohibitive due to the combinatorial\nnature of the existing algorithms. We propose to compute the persistent\nhomology of several subsamples of the data and then combine the resulting\nestimates. We study the risk of two estimators and we prove that the\nsubsampling approach carries stable topological information while achieving a\ngreat reduction in computational complexity.\n