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Topology"],"published-print":{"date-parts":[[2021,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb {X}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>X<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> equipped with a continuous function <jats:inline-formula><jats:alternatives><jats:tex-math>$$f: {\\mathbb {X}}\\rightarrow \\mathbb {R}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>:<\/mml:mo>\n                    <mml:mi>X<\/mml:mi>\n                    <mml:mo>\u2192<\/mml:mo>\n                    <mml:mi>R<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. 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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {R}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>R<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We then introduce a variant of the classic mapper graph of Singh et al.\u00a0(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 <jats:inline-formula><jats:alternatives><jats:tex-math>$$({\\mathbb {X}}, f)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>X<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> when it is applied to points randomly sampled from a probability density function concentrated on <jats:inline-formula><jats:alternatives><jats:tex-math>$$({\\mathbb {X}}, f)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>X<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. 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\u201353:16, 2016), we first show that the mapper graph of <jats:inline-formula><jats:alternatives><jats:tex-math>$$({\\mathbb {X}}, f)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>X<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, a constructible <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {R}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>R<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of <jats:inline-formula><jats:alternatives><jats:tex-math>$$({\\mathbb {X}},f)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>X<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> to the mapper of a super-level set of a probability density function concentrated on <jats:inline-formula><jats:alternatives><jats:tex-math>$$({\\mathbb {X}}, f)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>X<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Finally, building on the approach of Bobrowski et al.\u00a0(Bernoulli <jats:bold>23<\/jats:bold>(1):288\u2013328, 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.<\/jats:p>","DOI":"10.1007\/s41468-020-00063-x","type":"journal-article","created":{"date-parts":[[2020,12,17]],"date-time":"2020-12-17T05:03:52Z","timestamp":1608181432000},"page":"99-140","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Probabilistic convergence and stability of random mapper graphs"],"prefix":"10.1007","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0955-0119","authenticated-orcid":false,"given":"Adam","family":"Brown","sequence":"first","affiliation":[]},{"given":"Omer","family":"Bobrowski","sequence":"additional","affiliation":[]},{"given":"Elizabeth","family":"Munch","sequence":"additional","affiliation":[]},{"given":"Bei","family":"Wang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,12,17]]},"reference":[{"key":"63_CR1","unstructured":"Alagappan, M.: From 5 to 13: redefining the positions in basketball. 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