Statistical Analysis and Parameter Selection for Mapper

TL;DR

This paper establishes the statistical convergence of Mapper to the Reeb graph, demonstrating its optimality as an estimator and providing methods for automatic parameter tuning and confidence region computation for topological features.

Contribution

It introduces a theoretical framework proving Mapper's optimality and offers practical tools for automatic parameter selection and confidence estimation in topological data analysis.

Findings

Mapper converges to the Reeb graph with statistical guarantees

Provides a method for automatic parameter tuning

Enables confidence region computation for topological features

Abstract

In this article, we study the question of the statistical convergence of the 1-dimensional Mapper to its continuous analogue, the Reeb graph. We show that the Mapper is an optimal estimator of the Reeb graph, which gives, as a byproduct, a method to automatically tune its parameters and compute confidence regions on its topological features, such as its loops and flares. This allows to circumvent the issue of testing a large grid of parameters and keeping the most stable ones in the brute-force setting, which is widely used in visualization, clustering and feature selection with the Mapper.