[{"file":[{"date_created":"2021-02-11T14:43:59Z","file_name":"2020_JourApplCompTopology_Brown.pdf","access_level":"open_access","relation":"main_file","creator":"dernst","content_type":"application/pdf","checksum":"3f02e9d47c428484733da0f588a3c069","file_id":"9112","success":1,"file_size":2090265,"date_updated":"2021-02-11T14:43:59Z"}],"page":"99-140","external_id":{"arxiv":["1909.03488"]},"volume":5,"year":"2021","publisher":"Springer Nature","publication":"Journal of Applied and Computational Topology","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"quality_controlled":"1","date_updated":"2025-04-14T07:43:51Z","oa_version":"Published Version","doi":"10.1007/s41468-020-00063-x","publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]},"oa":1,"fulldoi":"https://doi.org/10.1007/s41468-020-00063-x","arxiv":1,"article_processing_charge":"Yes (via OA deal)","project":[{"call_identifier":"H2020","_id":"260C2330-B435-11E9-9278-68D0E5697425","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships"}],"ec_funded":1,"language":[{"iso":"eng"}],"author":[{"id":"70B7FDF6-608D-11E9-9333-8535E6697425","first_name":"Adam","last_name":"Brown","full_name":"Brown, Adam"},{"first_name":"Omer","last_name":"Bobrowski","full_name":"Bobrowski, Omer"},{"first_name":"Elizabeth","last_name":"Munch","full_name":"Munch, Elizabeth"},{"first_name":"Bei","last_name":"Wang","full_name":"Wang, Bei"}],"article_type":"original","ddc":["510"],"intvolume":"         5","publication_status":"published","month":"03","date_created":"2021-02-11T14:41:02Z","citation":{"chicago":"Brown, Adam, Omer Bobrowski, Elizabeth Munch, and Bei Wang. “Probabilistic Convergence and Stability of Random Mapper Graphs.” <i>Journal of Applied and Computational Topology</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s41468-020-00063-x\">https://doi.org/10.1007/s41468-020-00063-x</a>.","ama":"Brown A, Bobrowski O, Munch E, Wang B. Probabilistic convergence and stability of random mapper graphs. <i>Journal of Applied and Computational Topology</i>. 2021;5(1):99-140. doi:<a href=\"https://doi.org/10.1007/s41468-020-00063-x\">10.1007/s41468-020-00063-x</a>","ista":"Brown A, Bobrowski O, Munch E, Wang B. 2021. Probabilistic convergence and stability of random mapper graphs. Journal of Applied and Computational Topology. 5(1), 99–140.","apa":"Brown, A., Bobrowski, O., Munch, E., &#38; Wang, B. (2021). Probabilistic convergence and stability of random mapper graphs. <i>Journal of Applied and Computational Topology</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s41468-020-00063-x\">https://doi.org/10.1007/s41468-020-00063-x</a>","short":"A. Brown, O. Bobrowski, E. Munch, B. Wang, Journal of Applied and Computational Topology 5 (2021) 99–140.","ieee":"A. Brown, O. Bobrowski, E. Munch, and B. Wang, “Probabilistic convergence and stability of random mapper graphs,” <i>Journal of Applied and Computational Topology</i>, vol. 5, no. 1. Springer Nature, pp. 99–140, 2021.","mla":"Brown, Adam, et al. “Probabilistic Convergence and Stability of Random Mapper Graphs.” <i>Journal of Applied and Computational Topology</i>, vol. 5, no. 1, Springer Nature, 2021, pp. 99–140, doi:<a href=\"https://doi.org/10.1007/s41468-020-00063-x\">10.1007/s41468-020-00063-x</a>."},"type":"journal_article","title":"Probabilistic convergence and stability of random mapper graphs","day":"01","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","scopus_import":"1","issue":"1","department":[{"_id":"HeEd"}],"date_published":"2021-03-01T00:00:00Z","status":"public","abstract":[{"lang":"eng","text":"We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space X equipped with a continuous function f:X→R. We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line R. We then introduce a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium on point-based graphics, 2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of (X,f) when it is applied to points randomly sampled from a probability density function concentrated on (X,f). Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (In: 32nd international symposium on computational geometry, volume 51 of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany, pp 53:1–53:16, 2016), we first show that the mapper graph of (X,f), a constructible R-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of (X,f) to the mapper of a super-level set of a probability density function concentrated on (X,f). Finally, building on the approach of Bobrowski et al. (Bernoulli 23(1):288–328, 2017b), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data."}],"_id":"9111","has_accepted_license":"1","file_date_updated":"2021-02-11T14:43:59Z","acknowledgement":"AB was supported in part by the European Union’s Horizon 2020 research and innovation\r\nprogramme under the Marie Sklodowska-Curie GrantAgreement No. 754411 and NSF IIS-1513616. OB was supported in part by the Israel Science Foundation, Grant 1965/19. BW was supported in part by NSF IIS-1513616 and DBI-1661375. EM was supported in part by NSF CMMI-1800466, DMS-1800446, and CCF-1907591.We would like to thank the Institute for Mathematics and its Applications for hosting a workshop titled Bridging Statistics and Sheaves in May 2018, where this work was conceived.\r\nOpen Access funding provided by Institute of Science and Technology (IST Austria)."}]