Certified Mapper: Repeated Testing for Acyclicity and Obstructions to the Nerve Lemma

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 propose 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.