Bootstrapping persistent Betti numbers and other stabilizing statistics

Abstract

We investigate multivariate bootstrap procedures for general stabilizing statistics, with specific application to topological data analysis. The work relates to other general results in the area of stabilizing statistics, including central limit theorems for geometric and topological functionals of Poisson and binomial processes in the critical regime, where limit theorems prove difficult to use in practice, motivating the use of a bootstrap approach. A smoothed bootstrap procedure is shown to give consistent estimation in these settings. Specific statistics considered include the persistent Betti numbers of Čech and Vietoris–Rips complexes over point sets in Rd, along with Euler characteristics, and the total edge length of the k-nearest neighbor graph. Special emphasis is given to weakening the necessary conditions needed to establish bootstrap consistency. In particular, the assumption of a continuous underlying density is not required. Numerical studies illustrate the performance of the proposed method.