# Subsampling Methods for Persistent Homology

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Frederic Chazal,
Brittany Fasy,
Fabrizio Lecci,
Bertrand Michel,
Alessandro Rinaldo,
Larry Wasserman
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Proceedings of the 32nd International Conference on Machine Learning, PMLR 37:2143-2151, 2015.

#### Abstract

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

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