Statistical analysis of Mapper for stochastic and multivariate filters
نویسندگان
چکیده
Reeb spaces, as well their discretized versions called Mappers, are common descriptors used in topological data analysis, with plenty of applications various fields science, such computational biology and visualization, among others. The stability quantification the rate convergence Mapper to space has been studied a lot recent works (Brown et al. CoRR. arXiv:1909.03488 , 2019; Carrière Oudot Found Comput Math 18(6):1333–1396, 2017; J Mach Learn Res 19(12):1–39, 2018; Munch Wang in: 32nd international symposium on geometry (SoCG 2016), Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 51: 53:1–53:16, focusing case where scalar-valued filter is for computation Mapper. On other hand, much less known multivariate case, when codomain $${\mathbb {R}}^p$$ general it metric $$(\mathcal {Z},d_\mathcal {Z})$$ instead {R}}$$ . few results that available this setting (Dey 33rd 2017), 77, 36:1–36:16, Wang, 2016) can only handle continuous spaces cannot be finite representing data, point clouds distance matrices. In article, we introduce slight modification usual construction give risk bounds estimating using estimator. Our approach applies particular function compute also estimated from eigenfunctions PCA. given respect Gromov-Hausdorff distance, computed specific filter-based pseudometrics Mappers defined Dey (2017). We finally provide examples statistics machine learning different kinds target filters, numerical experiments demonstrate relevance our approach.
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ژورنال
عنوان ژورنال: Journal of applied and computational topology
سال: 2022
ISSN: ['2367-1726', '2367-1734']
DOI: https://doi.org/10.1007/s41468-022-00090-w