{"scopus_import":1,"quality_controlled":"1","abstract":[{"lang":"eng","text":"We show that the framework of topological data analysis can be extended from metrics to general Bregman divergences, widening the scope of possible applications. Examples are the Kullback - Leibler divergence, which is commonly used for comparing text and images, and the Itakura - Saito divergence, popular for speech and sound. In particular, we prove that appropriately generalized čech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized čech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory. "}],"intvolume":"        77","citation":{"ista":"Edelsbrunner H, Wagner H. 2017. Topological data analysis with Bregman divergences. Symposium on Computational Geometry, SoCG, LIPIcs, vol. 77, 391–3916.","ama":"Edelsbrunner H, Wagner H. Topological data analysis with Bregman divergences. In: Vol 77. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2017:391-3916. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2017.39\">10.4230/LIPIcs.SoCG.2017.39</a>","short":"H. Edelsbrunner, H. Wagner, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017, pp. 391–3916.","chicago":"Edelsbrunner, Herbert, and Hubert Wagner. “Topological Data Analysis with Bregman Divergences,” 77:391–3916. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2017.39\">https://doi.org/10.4230/LIPIcs.SoCG.2017.39</a>.","mla":"Edelsbrunner, Herbert, and Hubert Wagner. <i>Topological Data Analysis with Bregman Divergences</i>. Vol. 77, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017, pp. 391–3916, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2017.39\">10.4230/LIPIcs.SoCG.2017.39</a>.","apa":"Edelsbrunner, H., &#38; Wagner, H. (2017). Topological data analysis with Bregman divergences (Vol. 77, pp. 391–3916). Presented at the Symposium on Computational Geometry, SoCG, Brisbane, Australia: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2017.39\">https://doi.org/10.4230/LIPIcs.SoCG.2017.39</a>","ieee":"H. Edelsbrunner and H. Wagner, “Topological data analysis with Bregman divergences,” presented at the Symposium on Computational Geometry, SoCG, Brisbane, Australia, 2017, vol. 77, pp. 391–3916."},"author":[{"full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Wagner, Hubert","last_name":"Wagner","first_name":"Hubert","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87"}],"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"department":[{"_id":"HeEd"},{"_id":"UlWa"}],"alternative_title":["LIPIcs"],"title":"Topological data analysis with Bregman divergences","status":"public","ddc":["514","516"],"month":"06","publication_identifier":{"issn":["18688969"]},"type":"conference","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","has_accepted_license":"1","date_published":"2017-06-01T00:00:00Z","file_date_updated":"2020-07-14T12:47:42Z","oa_version":"Published Version","date_created":"2018-12-11T11:47:56Z","conference":{"start_date":"2017-07-04","location":"Brisbane, Australia","name":"Symposium on Computational Geometry, SoCG","end_date":"2017-07-07"},"date_updated":"2021-01-12T08:09:26Z","_id":"688","publist_id":"7021","doi":"10.4230/LIPIcs.SoCG.2017.39","file":[{"content_type":"application/pdf","file_size":990546,"date_created":"2018-12-12T10:11:03Z","relation":"main_file","file_id":"4856","date_updated":"2020-07-14T12:47:42Z","checksum":"067ab0cb3f962bae6c3af6bf0094e0f3","access_level":"open_access","file_name":"IST-2017-895-v1+1_LIPIcs-SoCG-2017-39.pdf","creator":"system"}],"page":"391-3916","publication_status":"published","license":"https://creativecommons.org/licenses/by/4.0/","pubrep_id":"895","day":"01","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","volume":77,"year":"2017","oa":1,"language":[{"iso":"eng"}]}