Certified Mapper: Repeated testing for acyclicity and obstructions to the nerve lemma

TL;DR

Certified Mapper introduces statistical methods to verify the conditions of the nerve lemma in Mapper, enabling the detection of obstructions or certification of non-obstruction through repeated testing.

Contribution

The paper develops a framework combining multiple statistical approaches to check for nerve lemma conditions in Mapper, addressing a gap in existing topological data analysis methods.

Findings

Certified Mapper can reliably identify obstructions to the nerve lemma.

The proposed methods provide certificates of non-obstruction in topological data analysis.

Simulation-based tests are effective for verifying nerve conditions.

Abstract

The Mapper algorithm does not include a check for whether the cover produced conforms to the requirements of the nerve lemma. To perform a check for obstructions to the nerve lemma, statistical considerations of multiple testing quickly arise. In this paper, we propose several statistical approaches to finding obstructions: through a persistent nerve lemma, through simulation testing, and using a parametric refinement of simulation tests. We suggest Certified Mapper -- a method built from these approaches to generate certificates of non-obstruction, or identify specific obstructions to the nerve lemma -- and we give recommendations for which statistical approaches are most appropriate for the task.