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.
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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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Vejdemo-Johansson, M., Leshchenko, A. (2020). Certified Mapper: Repeated Testing for Acyclicity and Obstructions to the Nerve Lemma. In: Baas, N., Carlsson, G., Quick, G., Szymik, M., Thaule, M. (eds) Topological Data Analysis. Abel Symposia, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-030-43408-3_19
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