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

  • Mikael Vejdemo-JohanssonEmail author
  • Alisa Leshchenko
Conference paper
  • 63 Downloads
Part of the Abel Symposia book series (ABEL, volume 15)

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.

Notes

Acknowledgements

The authors would like to acknowledge and thank: Sayan Mukherjee for invaluable advice and help designing Method 4; Anthea Monod and Kate Turner for helpful conversations; Dana Sylvan, Leo Carlsson and Nathaniel Saul for giving feedback and advice on the manuscript; The MAA for a travel grant; The Abel Symposium for a participation and travel grant.

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.CUNY College of Staten IslandDepartment of MathematicsStaten IslandUSA
  2. 2.CUNY Baccalaureate of Unique and Interdisciplinary StudiesNew YorkUSA

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