---
res:
  bibo_abstract:
  - '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.@eng'
  bibo_authorlist:
  - foaf_Person:
      foaf_givenName: Adam
      foaf_name: Brown, Adam
      foaf_surname: Brown
      foaf_workInfoHomepage: http://www.librecat.org/personId=70B7FDF6-608D-11E9-9333-8535E6697425
  - foaf_Person:
      foaf_givenName: Omer
      foaf_name: Bobrowski, Omer
      foaf_surname: Bobrowski
  - foaf_Person:
      foaf_givenName: Elizabeth
      foaf_name: Munch, Elizabeth
      foaf_surname: Munch
  - foaf_Person:
      foaf_givenName: Bei
      foaf_name: Wang, Bei
      foaf_surname: Wang
  bibo_doi: 10.1007/s41468-020-00063-x
  bibo_issue: '1'
  bibo_volume: 5
  dct_date: 2021^xs_gYear
  dct_isPartOf:
  - http://id.crossref.org/issn/2367-1726
  - http://id.crossref.org/issn/2367-1734
  dct_language: eng
  dct_publisher: Springer Nature@
  dct_title: Probabilistic convergence and stability of random mapper graphs@
...