Certified Mapper: Repeated testing for acyclicity and obstructions to the nerve lemma
This addresses a methodological gap in topological data analysis for researchers using the Mapper algorithm, though it is incremental as it builds on existing statistical testing frameworks.
The paper tackles the problem of verifying whether the cover produced by the Mapper algorithm meets the nerve lemma requirements, proposing statistical methods like persistent nerve lemma and simulation testing to detect obstructions. It introduces Certified Mapper to generate certificates of non-obstruction or identify specific obstructions, with recommendations for appropriate statistical approaches.
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.