Bootstrapping Persistent Betti Numbers and Other Stabilizing Statistics

TL;DR

This paper develops and validates a smoothed bootstrap method for stabilizing statistics in topological data analysis, enabling reliable confidence intervals without requiring continuous underlying densities, with applications to cosmic web data.

Contribution

It introduces a smoothed bootstrap procedure for stabilizing statistics like persistent Betti numbers, relaxing density assumptions, and demonstrates its effectiveness through simulations and real data applications.

Findings

Smoothed bootstrap provides consistent estimation for stabilizing statistics.

The method performs well in finite samples according to simulation results.

Application to SDSS cosmic web data illustrates practical utility.

Abstract

The present contribution investigates multivariate bootstrap procedures for general stabilizing statistics, with specific application to topological data analysis. Existing limit theorems for topological statistics prove difficult to use in practice for the construction of confidence intervals, motivating the use of the bootstrap in this capacity. However, the standard nonparametric bootstrap does not directly provide for asymptotically valid confidence intervals in some situations. A smoothed bootstrap procedure, instead, is shown to give consistent estimation in these settings. The present work relates to other general results in the area of stabilizing statistics, including central limit theorems for functionals of Poisson and Binomial processes in the critical regime. Specific statistics considered include the persistent Betti numbers of \v{C}ech and Vietoris-Rips complexes over point sets in $\mathbb R^d$, along with Euler characteristics, and the total edge length of the $k$-nearest neighbor graph. Special emphasis is made throughout to weakening the necessary conditions needed to establish bootstrap consistency. In particular, the assumption of a continuous underlying density is not required. A simulation study is provided to assess the performance of the smoothed bootstrap for finite sample sizes, and the method is further applied to the cosmic web dataset from the Sloan Digital Sky Survey (SDSS). Source code is available at github.com/btroycraft/stabilizing_statistics_bootstrap.