Subsampling Methods for Persistent Homology

TL;DR

This paper introduces a subsampling approach for persistent homology that reduces computational complexity while maintaining stable topological information, enabling analysis of large point cloud datasets.

Contribution

It proposes a novel subsampling method for persistent homology, providing theoretical guarantees of stability and efficiency improvements over existing algorithms.

Findings

Subsampling preserves essential topological features.

The method significantly reduces computational costs.

Theoretical analysis confirms estimator stability.

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.