{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,23]],"date-time":"2026-02-23T02:51:45Z","timestamp":1771815105848,"version":"3.50.1"},"reference-count":18,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2021,6,14]],"date-time":"2021-06-14T00:00:00Z","timestamp":1623628800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,6,14]],"date-time":"2021-06-14T00:00:00Z","timestamp":1623628800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000643","name":"Daphne Jackson Trust","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000643","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2022,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The reach of a submanifold is a crucial regularity parameter for manifold learning and geometric inference from point clouds. This paper relates the reach of a submanifold to its convexity defect function. Using the stability properties of convexity defect functions, along with some new bounds and the recent submanifold estimator of Aamari and Levrard (Ann. Statist. <jats:bold>47<\/jats:bold>(1), 177\u2013204 (2019)), an estimator for the reach is given. A uniform expected loss bound over a <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathscr {C}}^k$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>C<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mi>k<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> model is found. Lower bounds for the minimax rate for estimating the reach over these models are also provided. The estimator almost achieves these rates in the <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathscr {C}}^3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>C<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mn>3<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathscr {C}}^4$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>C<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mn>4<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> cases, with a gap given by a logarithmic factor.<\/jats:p>","DOI":"10.1007\/s00454-021-00290-8","type":"journal-article","created":{"date-parts":[[2021,6,14]],"date-time":"2021-06-14T15:03:11Z","timestamp":1623682991000},"page":"403-438","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Estimating the Reach of a Manifold via its Convexity Defect Function"],"prefix":"10.1007","volume":"67","author":[{"given":"Cl\u00e9ment","family":"Berenfeld","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9211-0060","authenticated-orcid":false,"given":"John","family":"Harvey","sequence":"additional","affiliation":[]},{"given":"Marc","family":"Hoffmann","sequence":"additional","affiliation":[]},{"given":"Krishnan","family":"Shankar","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,6,14]]},"reference":[{"issue":"1","key":"290_CR1","doi-asserted-by":"publisher","first-page":"1359","DOI":"10.1214\/19-EJS1551","volume":"13","author":"E Aamari","year":"2019","unstructured":"Aamari, E., Kim, J., Chazal, F., Michel, B., Rinaldo, A., Wasserman, L.: Estimating the reach of a manifold. 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