Probabilistic convergence and stability of random mapper graphs

article Probabilistic convergence and stability of random mapper graphs published yes Adam Brown author 70B7FDF6-608D-11E9-9333-8535E6697425 Omer Bobrowski author Elizabeth Munch author Bei Wang author HeEd department ISTplus - Postdoctoral Fellowships project 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. https://research-explorer.ista.ac.at/download/9111/9112/2020_JourApplCompTopology_Brown.pdf application/pdfno https://creativecommons.org/licenses/by/4.0/ Springer Nature2021 eng Journal of Applied and Computational Topology 2367-1726 2367-1734 1909.0348810.1007/s41468-020-00063-x 5199-140 Brown, A., Bobrowski, O., Munch, E., & Wang, B. (2021). Probabilistic convergence and stability of random mapper graphs. Journal of Applied and Computational Topology. Springer Nature. <a href="https://doi.org/10.1007/s41468-020-00063-x">https://doi.org/10.1007/s41468-020-00063-x</a> A. Brown, O. Bobrowski, E. Munch, and B. Wang, “Probabilistic convergence and stability of random mapper graphs,” Journal of Applied and Computational Topology, vol. 5, no. 1. Springer Nature, pp. 99–140, 2021. 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. A. Brown, O. Bobrowski, E. Munch, B. Wang, Journal of Applied and Computational Topology 5 (2021) 99–140. Brown A, Bobrowski O, Munch E, Wang B. Probabilistic convergence and stability of random mapper graphs. Journal of Applied and Computational Topology. 2021;5(1):99-140. doi:<a href="https://doi.org/10.1007/s41468-020-00063-x">10.1007/s41468-020-00063-x</a> Brown, Adam, et al. “Probabilistic Convergence and Stability of Random Mapper Graphs.” Journal of Applied and Computational Topology, 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>. Brown, Adam, Omer Bobrowski, Elizabeth Munch, and Bei Wang. “Probabilistic Convergence and Stability of Random Mapper Graphs.” Journal of Applied and Computational Topology. Springer Nature, 2021. <a href="https://doi.org/10.1007/s41468-020-00063-x">https://doi.org/10.1007/s41468-020-00063-x</a>. 91112021-02-11T14:41:02Z2025-04-14T07:43:51Z