[{"isi":1,"file":[{"creator":"dernst","success":1,"checksum":"65c5399c4015d9c8abb8c7a96f3d7836","relation":"main_file","file_size":379340,"access_level":"open_access","file_name":"2025_Entropy_Akopyan.pdf","date_created":"2025-09-08T07:55:48Z","file_id":"20309","date_updated":"2025-09-08T07:55:48Z","content_type":"application/pdf"}],"citation":{"ieee":"A. Akopyan, H. Edelsbrunner, Z. Virk, and H. Wagner, “Tight bounds between the Jensen–Shannon divergence and the minmax divergence,” <i>Entropy</i>, vol. 27, no. 8. MDPI, 2025.","apa":"Akopyan, A., Edelsbrunner, H., Virk, Z., &#38; Wagner, H. (2025). Tight bounds between the Jensen–Shannon divergence and the minmax divergence. <i>Entropy</i>. MDPI. <a href=\"https://doi.org/10.3390/e27080854\">https://doi.org/10.3390/e27080854</a>","short":"A. Akopyan, H. Edelsbrunner, Z. Virk, H. Wagner, Entropy 27 (2025).","chicago":"Akopyan, Arseniy, Herbert Edelsbrunner, Ziga Virk, and Hubert Wagner. “Tight Bounds between the Jensen–Shannon Divergence and the Minmax Divergence.” <i>Entropy</i>. MDPI, 2025. <a href=\"https://doi.org/10.3390/e27080854\">https://doi.org/10.3390/e27080854</a>.","ista":"Akopyan A, Edelsbrunner H, Virk Z, Wagner H. 2025. Tight bounds between the Jensen–Shannon divergence and the minmax divergence. Entropy. 27(8), 854.","ama":"Akopyan A, Edelsbrunner H, Virk Z, Wagner H. Tight bounds between the Jensen–Shannon divergence and the minmax divergence. <i>Entropy</i>. 2025;27(8). doi:<a href=\"https://doi.org/10.3390/e27080854\">10.3390/e27080854</a>","mla":"Akopyan, Arseniy, et al. “Tight Bounds between the Jensen–Shannon Divergence and the Minmax Divergence.” <i>Entropy</i>, vol. 27, no. 8, 854, MDPI, 2025, doi:<a href=\"https://doi.org/10.3390/e27080854\">10.3390/e27080854</a>."},"publication_status":"published","ec_funded":1,"scopus_import":"1","DOAJ_listed":"1","author":[{"first_name":"Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","full_name":"Akopyan, Arseniy","last_name":"Akopyan"},{"last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833"},{"id":"2E36B656-F248-11E8-B48F-1D18A9856A87","first_name":"Ziga","last_name":"Virk","full_name":"Virk, Ziga"},{"full_name":"Wagner, Hubert","last_name":"Wagner","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","first_name":"Hubert"}],"day":"01","project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","grant_number":"788183","call_identifier":"H2020","name":"Alpha Shape Theory Extended"},{"name":"Mathematics, Computer Science","call_identifier":"FWF","grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425"},{"name":"Persistence and stability of geometric complexes","call_identifier":"FWF","grant_number":"I02979-N35","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"publisher":"MDPI","volume":27,"OA_place":"publisher","license":"https://creativecommons.org/licenses/by/4.0/","doi":"10.3390/e27080854","quality_controlled":"1","month":"08","article_type":"original","oa_version":"Published Version","status":"public","issue":"8","ddc":["500"],"article_number":"854","publication_identifier":{"eissn":["1099-4300"]},"acknowledgement":"This research received partial funding from the European Research Council (ERC) under\r\nthe European Union’s Horizon 2020 research and innovation programme, grant no. 788183, the\r\nWittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, the DFG Collaborative\r\nResearch Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35, and the 2022 Google Research Scholar Award for project ‘Algorithms for Topological Analysis of Neural Networks’. The APC was waived.","publication":"Entropy","date_created":"2025-09-07T22:01:33Z","date_updated":"2025-09-30T14:32:31Z","year":"2025","abstract":[{"text":"Motivated by questions arising at the intersection of information theory and geometry, we compare two dissimilarity measures between finite categorical distributions. One is the well-known Jensen–Shannon divergence, which is easy to compute and whose square root is a proper metric. The other is what we call the minmax divergence, which is harder to compute. Just like the Jensen–Shannon divergence, it arises naturally from the Kullback–Leibler divergence. The main contribution of this paper is a proof showing that the minmax divergence can be tightly approximated by the Jensen–Shannon divergence. The bounds suggest that the square root of the minmax divergence is a metric, and we prove that this is indeed true in the one-dimensional case. The general case remains open. Finally, we consider analogous questions in the context of another Bregman divergence and the corresponding Burbea–Rao (Jensen–Bregman) divergence.","lang":"eng"}],"PlanS_conform":"1","external_id":{"isi":["001557476000001"],"pmid":["40870326"]},"department":[{"_id":"HeEd"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","corr_author":"1","oa":1,"OA_type":"gold","language":[{"iso":"eng"}],"file_date_updated":"2025-09-08T07:55:48Z","title":"Tight bounds between the Jensen–Shannon divergence and the minmax divergence","article_processing_charge":"Yes","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"pmid":1,"date_published":"2025-08-01T00:00:00Z","intvolume":"        27","_id":"20293","type":"journal_article","has_accepted_license":"1"},{"article_number":"637","issue":"8","ddc":["510"],"status":"public","oa_version":"Published Version","article_type":"original","month":"08","doi":"10.3390/e26080637","quality_controlled":"1","volume":26,"publisher":"MDPI","related_material":{"link":[{"relation":"software","url":"https://git.ista.ac.at/katharina.oelsboeck/wrap_2_3-public/"}]},"day":"01","author":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"orcid":"0000-0002-4672-8297","id":"4D4AA390-F248-11E8-B48F-1D18A9856A87","first_name":"Katharina","last_name":"Ölsböck","full_name":"Ölsböck, Katharina"},{"id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","first_name":"Hubert","last_name":"Wagner","full_name":"Wagner, Hubert"}],"scopus_import":"1","publication_status":"published","file":[{"checksum":"624a9e2c5b49d6c38b88b0f675467ba3","success":1,"creator":"dernst","content_type":"application/pdf","date_created":"2024-09-09T09:01:12Z","date_updated":"2024-09-09T09:01:12Z","file_id":"17948","file_name":"2024_Entropy_Edelsbrunner.pdf","access_level":"open_access","file_size":8025139,"relation":"main_file"}],"citation":{"ista":"Edelsbrunner H, Ölsböck K, Wagner H. 2024. Understanding higher-order interactions in information space. Entropy. 26(8), 637.","chicago":"Edelsbrunner, Herbert, Katharina Ölsböck, and Hubert Wagner. “Understanding Higher-Order Interactions in Information Space.” <i>Entropy</i>. MDPI, 2024. <a href=\"https://doi.org/10.3390/e26080637\">https://doi.org/10.3390/e26080637</a>.","mla":"Edelsbrunner, Herbert, et al. “Understanding Higher-Order Interactions in Information Space.” <i>Entropy</i>, vol. 26, no. 8, 637, MDPI, 2024, doi:<a href=\"https://doi.org/10.3390/e26080637\">10.3390/e26080637</a>.","ama":"Edelsbrunner H, Ölsböck K, Wagner H. Understanding higher-order interactions in information space. <i>Entropy</i>. 2024;26(8). doi:<a href=\"https://doi.org/10.3390/e26080637\">10.3390/e26080637</a>","apa":"Edelsbrunner, H., Ölsböck, K., &#38; Wagner, H. (2024). Understanding higher-order interactions in information space. <i>Entropy</i>. MDPI. <a href=\"https://doi.org/10.3390/e26080637\">https://doi.org/10.3390/e26080637</a>","short":"H. Edelsbrunner, K. Ölsböck, H. Wagner, Entropy 26 (2024).","ieee":"H. Edelsbrunner, K. Ölsböck, and H. Wagner, “Understanding higher-order interactions in information space,” <i>Entropy</i>, vol. 26, no. 8. MDPI, 2024."},"isi":1,"has_accepted_license":"1","type":"journal_article","_id":"17891","intvolume":"        26","date_published":"2024-08-01T00:00:00Z","pmid":1,"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"article_processing_charge":"Yes","title":"Understanding higher-order interactions in information space","language":[{"iso":"eng"}],"file_date_updated":"2024-09-09T09:01:12Z","oa":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","department":[{"_id":"HeEd"}],"external_id":{"isi":["001305543500001"],"pmid":["39202107"]},"abstract":[{"lang":"eng","text":"Abstract\r\nMethods used in topological data analysis naturally capture higher-order interactions in point cloud data embedded in a metric space. This methodology was recently extended to data living in an information space, by which we mean a space measured with an information theoretical distance. One such setting is a finite collection of discrete probability distributions embedded in the probability simplex measured with the relative entropy (Kullback–Leibler divergence). More generally, one can work with a Bregman divergence parameterized by a different notion of entropy. While theoretical algorithms exist for this setup, there is a paucity of implementations for exploring and comparing geometric-topological properties of various information spaces. The interest of this work is therefore twofold. First, we propose the first robust algorithms and software for geometric and topological data analysis in information space. Perhaps surprisingly, despite working with Bregman divergences, our design reuses robust libraries for the Euclidean case. Second, using the new software, we take the first steps towards understanding the geometric-topological structure of these spaces. In particular, we compare them with the more familiar spaces equipped with the Euclidean and Fisher metrics."}],"date_updated":"2025-09-08T09:13:44Z","year":"2024","publication":"Entropy","date_created":"2024-09-08T22:01:11Z","acknowledgement":"We thank Anton Nikitenko for first observing that the Wrap complex can be characterized as stated in Claim (ii) of the Wrap Complex Lemma, and Ondrej Draganov for correcting a critical mistake in one of our formulas in Section 2.","publication_identifier":{"eissn":["1099-4300"]}},{"publisher":"Carleton University","project":[{"grant_number":"I4887","_id":"0aa4bc98-070f-11eb-9043-e6fff9c6a316","name":"Persistent Homology, Algorithms and Stochastic Geometry"}],"volume":11,"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"first_name":"Ziga","id":"2E36B656-F248-11E8-B48F-1D18A9856A87","last_name":"Virk","full_name":"Virk, Ziga"},{"last_name":"Wagner","full_name":"Wagner, Hubert","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","first_name":"Hubert","orcid":"0009-0009-9111-8429"}],"day":"14","scopus_import":"1","file":[{"file_name":"2020_JournalOfComputationalGeometry_Edelsbrunner.pdf","relation":"main_file","access_level":"open_access","file_size":1449234,"content_type":"application/pdf","date_created":"2021-08-11T11:55:11Z","file_id":"9882","date_updated":"2021-08-11T11:55:11Z","creator":"asandaue","checksum":"f02d0b2b3838e7891a6c417fc34ffdcd","success":1}],"citation":{"apa":"Edelsbrunner, H., Virk, Z., &#38; Wagner, H. (2020). Topological data analysis in information space. <i>Journal of Computational Geometry</i>. Carleton University. <a href=\"https://doi.org/10.20382/jocg.v11i2a7\">https://doi.org/10.20382/jocg.v11i2a7</a>","short":"H. Edelsbrunner, Z. Virk, H. Wagner, Journal of Computational Geometry 11 (2020) 162–182.","ama":"Edelsbrunner H, Virk Z, Wagner H. Topological data analysis in information space. <i>Journal of Computational Geometry</i>. 2020;11(2):162-182. doi:<a href=\"https://doi.org/10.20382/jocg.v11i2a7\">10.20382/jocg.v11i2a7</a>","mla":"Edelsbrunner, Herbert, et al. “Topological Data Analysis in Information Space.” <i>Journal of Computational Geometry</i>, vol. 11, no. 2, Carleton University, 2020, pp. 162–82, doi:<a href=\"https://doi.org/10.20382/jocg.v11i2a7\">10.20382/jocg.v11i2a7</a>.","ista":"Edelsbrunner H, Virk Z, Wagner H. 2020. Topological data analysis in information space. Journal of Computational Geometry. 11(2), 162–182.","chicago":"Edelsbrunner, Herbert, Ziga Virk, and Hubert Wagner. “Topological Data Analysis in Information Space.” <i>Journal of Computational Geometry</i>. Carleton University, 2020. <a href=\"https://doi.org/10.20382/jocg.v11i2a7\">https://doi.org/10.20382/jocg.v11i2a7</a>.","ieee":"H. Edelsbrunner, Z. Virk, and H. Wagner, “Topological data analysis in information space,” <i>Journal of Computational Geometry</i>, vol. 11, no. 2. Carleton University, pp. 162–182, 2020."},"page":"162-182","publication_status":"published","status":"public","ddc":["510","000"],"issue":"2","month":"12","article_type":"original","oa_version":"Published Version","quality_controlled":"1","doi":"10.20382/jocg.v11i2a7","license":"https://creativecommons.org/licenses/by/3.0/","corr_author":"1","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","oa":1,"department":[{"_id":"HeEd"}],"abstract":[{"text":"Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms.  Importantly, instead of using the standard Euclidean distance, we look into dissimilarity measures with information-theoretic justification, and we develop the theory needed for applying topological data analysis in this setting. In doing so, we emphasize constructions that enable the usage of existing computational topology software in this context.","lang":"eng"}],"publication_identifier":{"eissn":["1920-180X"]},"acknowledgement":"This research is partially supported by the Office of Naval Research, through grant no. N62909-18-1-2038, and the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund (FWF).","date_created":"2021-07-04T22:01:26Z","publication":"Journal of Computational Geometry","year":"2020","date_updated":"2026-04-02T14:35:31Z","type":"journal_article","_id":"9630","has_accepted_license":"1","intvolume":"        11","article_processing_charge":"Yes","title":"Topological data analysis in information space","tmp":{"short":"CC BY (3.0)","name":"Creative Commons Attribution 3.0 Unported (CC BY 3.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/3.0/legalcode"},"date_published":"2020-12-14T00:00:00Z","file_date_updated":"2021-08-11T11:55:11Z","language":[{"iso":"eng"}]},{"quality_controlled":"1","doi":"10.4230/LIPICS.SOCG.2019.31","month":"06","oa_version":"Published Version","conference":{"location":"Portland, OR, United States","name":"SoCG 2019: Symposium on Computational Geometry","start_date":"2019-06-18","end_date":"2019-06-21"},"status":"public","ddc":["510"],"file":[{"creator":"dernst","checksum":"8ec8720730d4c789bf7b06540f1c29f4","file_name":"2019_LIPICS_Edelsbrunner.pdf","access_level":"open_access","relation":"main_file","file_size":1355179,"content_type":"application/pdf","date_created":"2019-07-24T06:40:01Z","file_id":"6666","date_updated":"2020-07-14T12:47:35Z"}],"citation":{"short":"H. Edelsbrunner, Z. Virk, H. Wagner, in:, 35th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019, p. 31:1-31:14.","apa":"Edelsbrunner, H., Virk, Z., &#38; Wagner, H. (2019). Topological data analysis in information space. In <i>35th International Symposium on Computational Geometry</i> (Vol. 129, p. 31:1-31:14). Portland, OR, United States: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPICS.SOCG.2019.31\">https://doi.org/10.4230/LIPICS.SOCG.2019.31</a>","mla":"Edelsbrunner, Herbert, et al. “Topological Data Analysis in Information Space.” <i>35th International Symposium on Computational Geometry</i>, vol. 129, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019, p. 31:1-31:14, doi:<a href=\"https://doi.org/10.4230/LIPICS.SOCG.2019.31\">10.4230/LIPICS.SOCG.2019.31</a>.","chicago":"Edelsbrunner, Herbert, Ziga Virk, and Hubert Wagner. “Topological Data Analysis in Information Space.” In <i>35th International Symposium on Computational Geometry</i>, 129:31:1-31:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. <a href=\"https://doi.org/10.4230/LIPICS.SOCG.2019.31\">https://doi.org/10.4230/LIPICS.SOCG.2019.31</a>.","ista":"Edelsbrunner H, Virk Z, Wagner H. 2019. Topological data analysis in information space. 35th International Symposium on Computational Geometry. SoCG 2019: Symposium on Computational Geometry, LIPIcs, vol. 129, 31:1-31:14.","ama":"Edelsbrunner H, Virk Z, Wagner H. Topological data analysis in information space. In: <i>35th International Symposium on Computational Geometry</i>. Vol 129. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2019:31:1-31:14. doi:<a href=\"https://doi.org/10.4230/LIPICS.SOCG.2019.31\">10.4230/LIPICS.SOCG.2019.31</a>","ieee":"H. Edelsbrunner, Z. Virk, and H. Wagner, “Topological data analysis in information space,” in <i>35th International Symposium on Computational Geometry</i>, Portland, OR, United States, 2019, vol. 129, p. 31:1-31:14."},"page":"31:1-31:14","publication_status":"published","alternative_title":["LIPIcs"],"scopus_import":1,"author":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert"},{"full_name":"Virk, Ziga","last_name":"Virk","first_name":"Ziga"},{"last_name":"Wagner","full_name":"Wagner, Hubert","first_name":"Hubert","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87"}],"day":"01","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","project":[{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"}],"volume":129,"file_date_updated":"2020-07-14T12:47:35Z","language":[{"iso":"eng"}],"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"title":"Topological data analysis in information space","date_published":"2019-06-01T00:00:00Z","intvolume":"       129","type":"conference","_id":"6648","has_accepted_license":"1","publication_identifier":{"isbn":["9783959771047"]},"arxiv":1,"date_updated":"2024-10-09T20:58:55Z","year":"2019","publication":"35th International Symposium on Computational Geometry","date_created":"2019-07-17T10:36:09Z","abstract":[{"text":"Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms. Importantly, instead of using the standard Euclidean distance, we look into dissimilarity measures with information-theoretic justification, and we develop the theory\r\nneeded for applying topological data analysis in this setting. In doing so, we emphasize constructions that enable the usage of existing computational topology software in this context.","lang":"eng"}],"external_id":{"arxiv":["1903.08510"]},"department":[{"_id":"HeEd"}],"oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","corr_author":"1"},{"department":[{"_id":"HeEd"}],"oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"PP is grateful to Julian Borill from the Planck consortium for providing the data, and for the illuminating discussions and inputs. PP also thanks Hans Kristen Eriksen, Anne Ducout, and Francois R. Bouchet for significantly helpful discussions at various stages. The authors collectively thank the anonymous referee for the invaluable comments and suggestions that have added significant value to the contents of the manuscript. PP and RA acknowledge the support of ERC advanced grant Understanding Random Systems through Algebraic Topology (URSAT) (no: 320422, PI: RA). This work is also part of a project that has received funding for PP and TB from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement ERC advanced grant 740021– Advances in Research on THeories of the dark UniverSe (ARTHUS), PI: TB). HE and HW acknowledge the support by the Office of Naval Research, through grant N62909-18-1-2038, and by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics”, through grant I02979-N35 of the Austrian Science Fund (FWF). PP acknowledges the support and use of resources at the NERSC computing center.","arxiv":1,"publication_identifier":{"eissn":["1432-0746"],"issn":["0004-6361"]},"year":"2019","date_updated":"2025-05-20T08:01:55Z","date_created":"2019-08-04T21:59:18Z","publication":"Astronomy and Astrophysics","abstract":[{"text":"We study the topology generated by the temperature fluctuations of the cosmic microwave background (CMB) radiation, as quantified by the number of components and holes, formally given by the Betti numbers, in the growing excursion sets. We compare CMB maps observed by the Planck satellite with a thousand simulated maps generated according to the ΛCDM paradigm with Gaussian distributed fluctuations. The comparison is multi-scale, being performed on a sequence of degraded maps with mean pixel separation ranging from 0.05 to 7.33°. The survey of the CMB over 𝕊2 is incomplete due to obfuscation effects by bright point sources and other extended foreground objects like our own galaxy. To deal with such situations, where analysis in the presence of “masks” is of importance, we introduce the concept of relative homology. The parametric χ2-test shows differences between observations and simulations, yielding p-values at percent to less than permil levels roughly between 2 and 7°, with the difference in the number of components and holes peaking at more than 3σ sporadically at these scales. The highest observed deviation between the observations and simulations for b0 and b1 is approximately between 3σ and 4σ at scales of 3–7°. There are reports of mildly unusual behaviour of the Euler characteristic at 3.66° in the literature, computed from independent measurements of the CMB temperature fluctuations by Planck’s predecessor, the Wilkinson Microwave Anisotropy Probe (WMAP) satellite. The mildly anomalous behaviour of the Euler characteristic is phenomenologically related to the strongly anomalous behaviour of components and holes, or the zeroth and first Betti numbers, respectively. Further, since these topological descriptors show consistent anomalous behaviour over independent measurements of Planck and WMAP, instrumental and systematic errors may be an unlikely source. These are also the scales at which the observed maps exhibit low variance compared to the simulations, and approximately the range of scales at which the power spectrum exhibits a dip with respect to the theoretical model. Non-parametric tests show even stronger differences at almost all scales. Crucially, Gaussian simulations based on power-spectrum matching the characteristics of the observed dipped power spectrum are not able to resolve the anomaly. Understanding the origin of the anomalies in the CMB, whether cosmological in nature or arising due to late-time effects, is an extremely challenging task. Regardless, beyond the trivial possibility that this may still be a manifestation of an extreme Gaussian case, these observations, along with the super-horizon scales involved, may motivate the study of primordial non-Gaussianity. Alternative scenarios worth exploring may be models with non-trivial topology, including topological defect models.","lang":"eng"}],"external_id":{"arxiv":["1812.07678"],"isi":["000475839300003"]},"intvolume":"       627","type":"journal_article","_id":"6756","has_accepted_license":"1","OA_type":"hybrid","language":[{"iso":"eng"}],"file_date_updated":"2020-07-14T12:47:39Z","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"article_processing_charge":"No","title":"Unexpected topology of the temperature fluctuations in the cosmic microwave background","date_published":"2019-07-17T00:00:00Z","author":[{"last_name":"Pranav","full_name":"Pranav, Pratyush","first_name":"Pratyush"},{"first_name":"Robert J.","full_name":"Adler, Robert J.","last_name":"Adler"},{"first_name":"Thomas","full_name":"Buchert, Thomas","last_name":"Buchert"},{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"first_name":"Bernard J.T.","last_name":"Jones","full_name":"Jones, Bernard J.T."},{"last_name":"Schwartzman","full_name":"Schwartzman, Armin","first_name":"Armin"},{"id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","first_name":"Hubert","full_name":"Wagner, Hubert","last_name":"Wagner"},{"last_name":"Van De Weygaert","full_name":"Van De Weygaert, Rien","first_name":"Rien"}],"day":"17","publisher":"EDP Sciences","project":[{"grant_number":"M62909-18-1-2038","_id":"265683E4-B435-11E9-9278-68D0E5697425","name":"Toward Computational Information Topology"},{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"}],"volume":627,"citation":{"ama":"Pranav P, Adler RJ, Buchert T, et al. Unexpected topology of the temperature fluctuations in the cosmic microwave background. <i>Astronomy and Astrophysics</i>. 2019;627. doi:<a href=\"https://doi.org/10.1051/0004-6361/201834916\">10.1051/0004-6361/201834916</a>","mla":"Pranav, Pratyush, et al. “Unexpected Topology of the Temperature Fluctuations in the Cosmic Microwave Background.” <i>Astronomy and Astrophysics</i>, vol. 627, A163, EDP Sciences, 2019, doi:<a href=\"https://doi.org/10.1051/0004-6361/201834916\">10.1051/0004-6361/201834916</a>.","ista":"Pranav P, Adler RJ, Buchert T, Edelsbrunner H, Jones BJT, Schwartzman A, Wagner H, Van De Weygaert R. 2019. Unexpected topology of the temperature fluctuations in the cosmic microwave background. Astronomy and Astrophysics. 627, A163.","chicago":"Pranav, Pratyush, Robert J. Adler, Thomas Buchert, Herbert Edelsbrunner, Bernard J.T. Jones, Armin Schwartzman, Hubert Wagner, and Rien Van De Weygaert. “Unexpected Topology of the Temperature Fluctuations in the Cosmic Microwave Background.” <i>Astronomy and Astrophysics</i>. EDP Sciences, 2019. <a href=\"https://doi.org/10.1051/0004-6361/201834916\">https://doi.org/10.1051/0004-6361/201834916</a>.","short":"P. Pranav, R.J. Adler, T. Buchert, H. Edelsbrunner, B.J.T. Jones, A. Schwartzman, H. Wagner, R. Van De Weygaert, Astronomy and Astrophysics 627 (2019).","apa":"Pranav, P., Adler, R. J., Buchert, T., Edelsbrunner, H., Jones, B. J. T., Schwartzman, A., … Van De Weygaert, R. (2019). Unexpected topology of the temperature fluctuations in the cosmic microwave background. <i>Astronomy and Astrophysics</i>. EDP Sciences. <a href=\"https://doi.org/10.1051/0004-6361/201834916\">https://doi.org/10.1051/0004-6361/201834916</a>","ieee":"P. Pranav <i>et al.</i>, “Unexpected topology of the temperature fluctuations in the cosmic microwave background,” <i>Astronomy and Astrophysics</i>, vol. 627. EDP Sciences, 2019."},"file":[{"access_level":"open_access","file_size":14420451,"relation":"main_file","file_name":"2019_AstronomyAstrophysics_Pranav.pdf","date_created":"2019-08-05T08:08:59Z","date_updated":"2020-07-14T12:47:39Z","file_id":"6766","content_type":"application/pdf","creator":"dernst","checksum":"83b9209ed9eefbdcefd89019c5a97805"}],"isi":1,"publication_status":"published","scopus_import":"1","article_type":"original","month":"07","oa_version":"Published Version","status":"public","article_number":"A163","ddc":["520","530"],"OA_place":"publisher","doi":"10.1051/0004-6361/201834916","quality_controlled":"1"},{"month":"06","conference":{"end_date":"2018-06-14","start_date":"2018-06-11","name":"SoCG: Symposium on Computational Geometry","location":"Budapest, Hungary"},"oa_version":"Published Version","status":"public","ddc":["000"],"quality_controlled":"1","doi":"10.4230/LIPIcs.SoCG.2018.35","author":[{"full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert"},{"last_name":"Virk","full_name":"Virk, Ziga","first_name":"Ziga"},{"id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","first_name":"Hubert","last_name":"Wagner","full_name":"Wagner, Hubert"}],"day":"11","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","project":[{"_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes"}],"volume":99,"file":[{"checksum":"7509403803b3ac1aee94bbc2ad293d21","creator":"dernst","date_updated":"2020-07-14T12:45:20Z","file_id":"5724","date_created":"2018-12-17T16:31:31Z","content_type":"application/pdf","relation":"main_file","file_size":489080,"access_level":"open_access","file_name":"2018_LIPIcs_Edelsbrunner.pdf"}],"citation":{"ieee":"H. Edelsbrunner, Z. Virk, and H. Wagner, “Smallest enclosing spheres and Chernoff points in Bregman geometry,” presented at the SoCG: Symposium on Computational Geometry, Budapest, Hungary, 2018, vol. 99, p. 35:1-35:13.","chicago":"Edelsbrunner, Herbert, Ziga Virk, and Hubert Wagner. “Smallest Enclosing Spheres and Chernoff Points in Bregman Geometry,” 99:35:1-35:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2018.35\">https://doi.org/10.4230/LIPIcs.SoCG.2018.35</a>.","ista":"Edelsbrunner H, Virk Z, Wagner H. 2018. Smallest enclosing spheres and Chernoff points in Bregman geometry. SoCG: Symposium on Computational Geometry, Leibniz International Proceedings in Information, LIPIcs, vol. 99, 35:1-35:13.","ama":"Edelsbrunner H, Virk Z, Wagner H. Smallest enclosing spheres and Chernoff points in Bregman geometry. In: Vol 99. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2018:35:1-35:13. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2018.35\">10.4230/LIPIcs.SoCG.2018.35</a>","mla":"Edelsbrunner, Herbert, et al. <i>Smallest Enclosing Spheres and Chernoff Points in Bregman Geometry</i>. Vol. 99, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018, p. 35:1-35:13, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2018.35\">10.4230/LIPIcs.SoCG.2018.35</a>.","apa":"Edelsbrunner, H., Virk, Z., &#38; Wagner, H. (2018). Smallest enclosing spheres and Chernoff points in Bregman geometry (Vol. 99, p. 35:1-35:13). Presented at the SoCG: Symposium on Computational Geometry, Budapest, Hungary: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2018.35\">https://doi.org/10.4230/LIPIcs.SoCG.2018.35</a>","short":"H. Edelsbrunner, Z. Virk, H. Wagner, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018, p. 35:1-35:13."},"publication_status":"published","alternative_title":["Leibniz International Proceedings in Information, LIPIcs"],"page":"35:1 - 35:13","scopus_import":1,"intvolume":"        99","_id":"188","type":"conference","has_accepted_license":"1","language":[{"iso":"eng"}],"file_date_updated":"2020-07-14T12:45:20Z","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"title":"Smallest enclosing spheres and Chernoff points in Bregman geometry","date_published":"2018-06-11T00:00:00Z","department":[{"_id":"HeEd"}],"publist_id":"7733","oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"This research is partially supported by the Office of Naval Research, through grant no. N62909-18-1-2038, and the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, through grant no. I02979-N35 of the Austrian Science Fund","year":"2018","date_updated":"2021-01-12T06:53:48Z","date_created":"2018-12-11T11:45:05Z","abstract":[{"lang":"eng","text":"Smallest enclosing spheres of finite point sets are central to methods in topological data analysis. Focusing on Bregman divergences to measure dissimilarity, we prove bounds on the location of the center of a smallest enclosing sphere. These bounds depend on the range of radii for which Bregman balls are convex."}]},{"oa_version":"Published Version","month":"01","article_type":"original","ddc":["500"],"main_file_link":[{"url":"https://doi.org/10.1016/j.jsc.2016.03.008","open_access":"1"}],"status":"public","quality_controlled":"1","doi":"10.1016/j.jsc.2016.03.008","day":"01","related_material":{"record":[{"relation":"earlier_version","status":"public","id":"10894"}]},"author":[{"last_name":"Bauer","full_name":"Bauer, Ulrich","first_name":"Ulrich"},{"first_name":"Michael","full_name":"Kerber, Michael","last_name":"Kerber"},{"first_name":"Jan","last_name":"Reininghaus","full_name":"Reininghaus, Jan"},{"last_name":"Wagner","full_name":"Wagner, Hubert","first_name":"Hubert","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","orcid":"0009-0009-9111-8429"}],"volume":78,"publisher":"Academic Press","project":[{"name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425","grant_number":"318493","call_identifier":"FP7"}],"page":"76 - 90","publication_status":"published","citation":{"ama":"Bauer U, Kerber M, Reininghaus J, Wagner H. Phat - Persistent homology algorithms toolbox. <i>Journal of Symbolic Computation</i>. 2017;78:76-90. doi:<a href=\"https://doi.org/10.1016/j.jsc.2016.03.008\">10.1016/j.jsc.2016.03.008</a>","mla":"Bauer, Ulrich, et al. “Phat - Persistent Homology Algorithms Toolbox.” <i>Journal of Symbolic Computation</i>, vol. 78, Academic Press, 2017, pp. 76–90, doi:<a href=\"https://doi.org/10.1016/j.jsc.2016.03.008\">10.1016/j.jsc.2016.03.008</a>.","chicago":"Bauer, Ulrich, Michael Kerber, Jan Reininghaus, and Hubert Wagner. “Phat - Persistent Homology Algorithms Toolbox.” <i>Journal of Symbolic Computation</i>. Academic Press, 2017. <a href=\"https://doi.org/10.1016/j.jsc.2016.03.008\">https://doi.org/10.1016/j.jsc.2016.03.008</a>.","ista":"Bauer U, Kerber M, Reininghaus J, Wagner H. 2017. Phat - Persistent homology algorithms toolbox. Journal of Symbolic Computation. 78, 76–90.","short":"U. Bauer, M. Kerber, J. Reininghaus, H. Wagner, Journal of Symbolic Computation 78 (2017) 76–90.","apa":"Bauer, U., Kerber, M., Reininghaus, J., &#38; Wagner, H. (2017). Phat - Persistent homology algorithms toolbox. <i>Journal of Symbolic Computation</i>. Academic Press. <a href=\"https://doi.org/10.1016/j.jsc.2016.03.008\">https://doi.org/10.1016/j.jsc.2016.03.008</a>","ieee":"U. Bauer, M. Kerber, J. Reininghaus, and H. Wagner, “Phat - Persistent homology algorithms toolbox,” <i>Journal of Symbolic Computation</i>, vol. 78. Academic Press, pp. 76–90, 2017."},"isi":1,"scopus_import":"1","ec_funded":1,"intvolume":"        78","type":"journal_article","_id":"1433","language":[{"iso":"eng"}],"OA_type":"free access","date_published":"2017-01-01T00:00:00Z","article_processing_charge":"No","title":"Phat - Persistent homology algorithms toolbox","department":[{"_id":"HeEd"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","corr_author":"1","oa":1,"publist_id":"5765","date_created":"2018-12-11T11:51:59Z","publication":"Journal of Symbolic Computation","date_updated":"2026-06-18T17:35:16Z","year":"2017","publication_identifier":{"issn":[" 0747-7171"]},"acknowledgement":"Michael Kerber acknowledges support by the Max Planck Center for Visual Computing and Communications (FKZ-01IMC01 and FKZ-01IM10001). Ulrich Bauer, Jan Reininghaus, and Hubert Wagner acknowledge support by the EU Project TOPOSYS (FP7-ICT-318493-STREP).","external_id":{"isi":["000384396000005"]},"abstract":[{"lang":"eng","text":"Phat is an open-source C. ++ library for the computation of persistent homology by matrix reduction, targeted towards developers of software for topological data analysis. We aim for a simple generic design that decouples algorithms from data structures without sacrificing efficiency or user-friendliness. We provide numerous different reduction strategies as well as data types to store and manipulate the boundary matrix. We compare the different combinations through extensive experimental evaluation and identify optimization techniques that work well in practical situations. We also compare our software with various other publicly available libraries for persistent homology."}]},{"language":[{"iso":"eng"}],"editor":[{"full_name":"Felsberg, Michael","last_name":"Felsberg","first_name":"Michael"},{"last_name":"Heyden","full_name":"Heyden, Anders","first_name":"Anders"},{"full_name":"Krüger, Norbert","last_name":"Krüger","first_name":"Norbert"}],"date_published":"2017-07-28T00:00:00Z","article_processing_charge":"No","title":"Streaming algorithm for Euler characteristic curves of multidimensional images","intvolume":"     10424","type":"conference","_id":"833","date_created":"2018-12-11T11:48:45Z","year":"2017","date_updated":"2025-06-04T09:54:22Z","arxiv":1,"publication_identifier":{"issn":["0302-9743"]},"external_id":{"isi":["000432085900032"],"arxiv":["1705.02045"]},"abstract":[{"text":"We present an efficient algorithm to compute Euler characteristic curves of gray scale images of arbitrary dimension. In various applications the Euler characteristic curve is used as a descriptor of an image. Our algorithm is the first streaming algorithm for Euler characteristic curves. The usage of streaming removes the necessity to store the entire image in RAM. Experiments show that our implementation handles terabyte scale images on commodity hardware. Due to lock-free parallelism, it scales well with the number of processor cores. Additionally, we put the concept of the Euler characteristic curve in the wider context of computational topology. In particular, we explain the connection with persistence diagrams.","lang":"eng"}],"department":[{"_id":"HeEd"}],"corr_author":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"publist_id":"6815","quality_controlled":"1","doi":"10.1007/978-3-319-64689-3_32","conference":{"name":"CAIP: Computer Analysis of Images and Patterns","start_date":"2017-08-22","end_date":"2017-08-24","location":"Ystad, Sweden"},"oa_version":"Submitted Version","month":"07","status":"public","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1705.02045"}],"publication_status":"published","page":"397 - 409","alternative_title":["LNCS"],"citation":{"ieee":"T. Heiss and H. Wagner, “Streaming algorithm for Euler characteristic curves of multidimensional images,” presented at the CAIP: Computer Analysis of Images and Patterns, Ystad, Sweden, 2017, vol. 10424, pp. 397–409.","ama":"Heiss T, Wagner H. Streaming algorithm for Euler characteristic curves of multidimensional images. In: Felsberg M, Heyden A, Krüger N, eds. Vol 10424. Springer; 2017:397-409. doi:<a href=\"https://doi.org/10.1007/978-3-319-64689-3_32\">10.1007/978-3-319-64689-3_32</a>","chicago":"Heiss, Teresa, and Hubert Wagner. “Streaming Algorithm for Euler Characteristic Curves of Multidimensional Images.” edited by Michael Felsberg, Anders Heyden, and Norbert Krüger, 10424:397–409. Springer, 2017. <a href=\"https://doi.org/10.1007/978-3-319-64689-3_32\">https://doi.org/10.1007/978-3-319-64689-3_32</a>.","ista":"Heiss T, Wagner H. 2017. Streaming algorithm for Euler characteristic curves of multidimensional images. CAIP: Computer Analysis of Images and Patterns, LNCS, vol. 10424, 397–409.","mla":"Heiss, Teresa, and Hubert Wagner. <i>Streaming Algorithm for Euler Characteristic Curves of Multidimensional Images</i>. Edited by Michael Felsberg et al., vol. 10424, Springer, 2017, pp. 397–409, doi:<a href=\"https://doi.org/10.1007/978-3-319-64689-3_32\">10.1007/978-3-319-64689-3_32</a>.","apa":"Heiss, T., &#38; Wagner, H. (2017). Streaming algorithm for Euler characteristic curves of multidimensional images. In M. Felsberg, A. Heyden, &#38; N. Krüger (Eds.) (Vol. 10424, pp. 397–409). Presented at the CAIP: Computer Analysis of Images and Patterns, Ystad, Sweden: Springer. <a href=\"https://doi.org/10.1007/978-3-319-64689-3_32\">https://doi.org/10.1007/978-3-319-64689-3_32</a>","short":"T. Heiss, H. Wagner, in:, M. Felsberg, A. Heyden, N. Krüger (Eds.), Springer, 2017, pp. 397–409."},"isi":1,"scopus_import":"1","day":"28","author":[{"first_name":"Teresa","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-1780-2689","last_name":"Heiss","full_name":"Heiss, Teresa"},{"id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","first_name":"Hubert","full_name":"Wagner, Hubert","last_name":"Wagner"}],"volume":10424,"publisher":"Springer"},{"intvolume":"        77","type":"conference","_id":"688","has_accepted_license":"1","file_date_updated":"2020-07-14T12:47:42Z","language":[{"iso":"eng"}],"title":"Topological data analysis with Bregman divergences","article_processing_charge":"No","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"date_published":"2017-06-01T00:00:00Z","department":[{"_id":"HeEd"},{"_id":"UlWa"}],"publist_id":"7021","corr_author":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"publication_identifier":{"issn":["1868-8969"]},"date_created":"2018-12-11T11:47:56Z","date_updated":"2025-07-10T11:53:56Z","year":"2017","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. "}],"month":"06","conference":{"end_date":"2017-07-07","name":"Symposium on Computational Geometry, SoCG","start_date":"2017-07-04","location":"Brisbane, Australia"},"oa_version":"Published Version","status":"public","ddc":["514","516"],"doi":"10.4230/LIPIcs.SoCG.2017.39","quality_controlled":"1","author":[{"first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"last_name":"Wagner","full_name":"Wagner, Hubert","first_name":"Hubert","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87"}],"day":"01","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","volume":77,"citation":{"short":"H. Edelsbrunner, H. Wagner, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017, pp. 391–3916.","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>","ista":"Edelsbrunner H, Wagner H. 2017. Topological data analysis with Bregman divergences. Symposium on Computational Geometry, SoCG, LIPIcs, vol. 77, 391–3916.","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>.","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>.","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>","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."},"file":[{"creator":"system","checksum":"067ab0cb3f962bae6c3af6bf0094e0f3","file_name":"IST-2017-895-v1+1_LIPIcs-SoCG-2017-39.pdf","file_size":990546,"relation":"main_file","access_level":"open_access","content_type":"application/pdf","file_id":"4856","date_updated":"2020-07-14T12:47:42Z","date_created":"2018-12-12T10:11:03Z"}],"alternative_title":["LIPIcs"],"page":"391-3916","publication_status":"published","pubrep_id":"895","scopus_import":"1"}]