{"has_accepted_license":"1","external_id":{"arxiv":["2206.10485"]},"project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020"},{"call_identifier":"FWF","grant_number":"Z00342","name":"The Wittgenstein Prize","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"754411"},{"name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35","call_identifier":"FWF"},{"name":"Learning and triangulating manifolds via collapses","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","grant_number":"M03073"}],"volume":293,"acknowledgement":"This research has been supported by the European Research Council (ERC), grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund (FWF), grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant No. I 02979-N35.\r\nWintraecken, Mathijs: Supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411, the Austrian science fund (FWF) grant No. M-3073, and the welcome package from IDEX of the Université Côte d'Azur.","ec_funded":1,"intvolume":"       293","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"date_updated":"2024-06-27T06:27:47Z","status":"public","abstract":[{"lang":"eng","text":"In this article we extend and strengthen the seminal work by Niyogi, Smale, and Weinberger on the learning of the homotopy type from a sample of an underlying space. In their work, Niyogi, Smale, and Weinberger studied samples of C² manifolds with positive reach embedded in ℝ^d. We extend their results in the following ways: - As the ambient space we consider both ℝ^d and Riemannian manifolds with lower bounded sectional curvature. - In both types of ambient spaces, we study sets of positive reach - a significantly more general setting than C² manifolds - as well as general manifolds of positive reach. - The sample P of a set (or a manifold) 𝒮 of positive reach may be noisy. We work with two one-sided Hausdorff distances - ε and δ - between P and 𝒮. We provide tight bounds in terms of ε and δ, that guarantee that there exists a parameter r such that the union of balls of radius r centred at the sample P deformation-retracts to 𝒮. We exhibit their tightness by an explicit construction. We carefully distinguish the roles of δ and ε. This is not only essential to achieve tight bounds, but also sensible in practical situations, since it allows one to adapt the bound according to sample density and the amount of noise present in the sample separately."}],"day":"06","quality_controlled":"1","_id":"17170","doi":"10.4230/LIPIcs.SoCG.2024.11","publication_identifier":{"eissn":["1868-8969"],"isbn":["9783959773164"]},"month":"06","title":"Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds","oa_version":"Published Version","citation":{"ieee":"D. Attali <i>et al.</i>, “Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293, p. 11:1-11:19.","mla":"Attali, Dominique, et al. “Tight Bounds for the Learning of Homotopy à La Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, p. 11:1-11:19, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">10.4230/LIPIcs.SoCG.2024.11</a>.","chicago":"Attali, Dominique, Hana Kourimska, Christopher D Fillmore, Ishika Ghosh, André Lieutier, Elizabeth R Stephenson, and Mathijs Wintraecken. “Tight Bounds for the Learning of Homotopy à La Niyogi, Smale, and Weinberger for Subsets of Euclidean Spaces and of Riemannian Manifolds.” In <i>40th International Symposium on Computational Geometry</i>, 293:11:1-11:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">https://doi.org/10.4230/LIPIcs.SoCG.2024.11</a>.","ista":"Attali D, Kourimska H, Fillmore CD, Ghosh I, Lieutier A, Stephenson ER, Wintraecken M. 2024. Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 11:1-11:19.","ama":"Attali D, Kourimska H, Fillmore CD, et al. Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024:11:1-11:19. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">10.4230/LIPIcs.SoCG.2024.11</a>","apa":"Attali, D., Kourimska, H., Fillmore, C. D., Ghosh, I., Lieutier, A., Stephenson, E. R., &#38; Wintraecken, M. (2024). Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293, p. 11:1-11:19). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.11\">https://doi.org/10.4230/LIPIcs.SoCG.2024.11</a>","short":"D. Attali, H. Kourimska, C.D. Fillmore, I. Ghosh, A. Lieutier, E.R. Stephenson, M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, p. 11:1-11:19."},"type":"conference","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","scopus_import":"1","oa":1,"conference":{"end_date":"2024-06-14","start_date":"2024-06-11","location":"Athens, Greece","name":"SoCG: Symposium on Computational Geometry"},"publication_status":"published","file_date_updated":"2024-06-25T11:47:26Z","language":[{"iso":"eng"}],"author":[{"last_name":"Attali","full_name":"Attali, Dominique","first_name":"Dominique"},{"orcid":"0000-0001-7841-0091","first_name":"Hana","full_name":"Kourimska, Hana","last_name":"Kourimska","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E"},{"full_name":"Fillmore, Christopher D","first_name":"Christopher D","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","last_name":"Fillmore"},{"first_name":"Ishika","full_name":"Ghosh, Ishika","id":"ee449b28-344d-11ef-a6d5-9ca430e9e9ff","last_name":"Ghosh"},{"first_name":"André","full_name":"Lieutier, André","last_name":"Lieutier"},{"id":"2D04F932-F248-11E8-B48F-1D18A9856A87","last_name":"Stephenson","full_name":"Stephenson, Elizabeth R","first_name":"Elizabeth R","orcid":"0000-0002-6862-208X"},{"full_name":"Wintraecken, Mathijs","first_name":"Mathijs","orcid":"0000-0002-7472-2220","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","last_name":"Wintraecken"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"11:1-11:19","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"alternative_title":["LIPIcs"],"publication":"40th International Symposium on Computational Geometry","date_created":"2024-06-25T11:45:58Z","year":"2024","ddc":["516"],"file":[{"success":1,"access_level":"open_access","file_id":"17171","creator":"cfillmor","checksum":"6a2ddc8b51aa58f197a8b294750f1f8d","date_created":"2024-06-25T11:47:26Z","file_name":"LIPIcs.SoCG.2024.11.pdf","file_size":20886142,"date_updated":"2024-06-25T11:47:26Z","content_type":"application/pdf","relation":"main_file"}],"article_processing_charge":"No","date_published":"2024-06-06T00:00:00Z"}