[{"article_processing_charge":"No","OA_type":"closed access","type":"journal_article","article_number":"122425","acknowledgement":"The authors would like to thank the anonymous reviewers for their valuable comments and suggestions, which improved this paper.\r\nWork by the first and fourth authors is partially 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.","issue":"11","publication":"Information Sciences","month":"11","date_updated":"2025-12-30T09:05:32Z","title":"Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality","intvolume":"       719","abstract":[{"lang":"eng","text":"Simplets are elementary units within simplicial complexes and are fundamental for analyzing the structure of simplicial complexes. Previous efforts have mainly focused on accurately counting or approximating the number of simplets rather than studying their frequencies. However, analyzing simplet frequencies is more practical for large-scale simplicial complexes. This paper introduces the Simplet Frequency Distribution (SFD) vector, which enables the analysis of simplet frequencies in simplicial complexes. Additionally, we provide a bound on the sample complexity required to approximate the SFD vector using any uniform sampling-based algorithm accurately. We extend the definition of simplet frequency distribution to encompass simplices, allowing for the analysis of simplet frequencies within simplices of simplicial complexes. This paper introduces the Simplet Degree Vector (SDV) and the Simplet Degree Centrality (SDC), facilitating this analysis for each simplex. Furthermore, we present a bound on the sample complexity required for accurately approximating the SDV and SDC for a set of simplices using any uniform sampling-based algorithm. We also introduce algorithms for approximating SFD, geometric SFD, SDV, and SDC. We also validate the theoretical bounds with experiments on random simplicial complexes and demonstrate the practical application through a case study."}],"year":"2025","publication_status":"published","project":[{"name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020","grant_number":"788183"},{"call_identifier":"FWF","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","grant_number":"Z00342"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"}],"volume":719,"fulldoi":"https://doi.org/10.1016/j.ins.2025.122425","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"19937","scopus_import":"1","corr_author":"1","language":[{"iso":"eng"}],"oa_version":"None","day":"01","date_created":"2025-06-30T08:48:48Z","ec_funded":1,"status":"public","publisher":"Elsevier","quality_controlled":"1","isi":1,"citation":{"ieee":"M. Mahini, H. Beigy, S. Qadami, and M. Saghafian, “Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality,” <i>Information Sciences</i>, vol. 719, no. 11. Elsevier, 2025.","short":"M. Mahini, H. Beigy, S. Qadami, M. Saghafian, Information Sciences 719 (2025).","chicago":"Mahini, Mohammad, Hamid Beigy, Salman Qadami, and Morteza Saghafian. “Simplet-Based Signatures and Approximation in Simplicial Complexes: Frequency, Degree, and Centrality.” <i>Information Sciences</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.ins.2025.122425\">https://doi.org/10.1016/j.ins.2025.122425</a>.","apa":"Mahini, M., Beigy, H., Qadami, S., &#38; Saghafian, M. (2025). Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. <i>Information Sciences</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.ins.2025.122425\">https://doi.org/10.1016/j.ins.2025.122425</a>","ama":"Mahini M, Beigy H, Qadami S, Saghafian M. Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. <i>Information Sciences</i>. 2025;719(11). doi:<a href=\"https://doi.org/10.1016/j.ins.2025.122425\">10.1016/j.ins.2025.122425</a>","mla":"Mahini, Mohammad, et al. “Simplet-Based Signatures and Approximation in Simplicial Complexes: Frequency, Degree, and Centrality.” <i>Information Sciences</i>, vol. 719, no. 11, 122425, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.ins.2025.122425\">10.1016/j.ins.2025.122425</a>.","ista":"Mahini M, Beigy H, Qadami S, Saghafian M. 2025. Simplet-based signatures and approximation in simplicial complexes: Frequency, degree, and centrality. Information Sciences. 719(11), 122425."},"author":[{"full_name":"Mahini, Mohammad","first_name":"Mohammad","last_name":"Mahini"},{"last_name":"Beigy","first_name":"Hamid","full_name":"Beigy, Hamid"},{"full_name":"Qadami, Salman","first_name":"Salman","last_name":"Qadami"},{"first_name":"Morteza","full_name":"Saghafian, Morteza","id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian"}],"doi":"10.1016/j.ins.2025.122425","department":[{"_id":"HeEd"}],"article_type":"original","date_published":"2025-11-01T00:00:00Z","external_id":{"isi":["001516170500002"]},"publication_identifier":{"issn":["0020-0255"]}},{"ddc":["510"],"date_created":"2025-07-13T22:01:22Z","day":"20","language":[{"iso":"eng"}],"oa_version":"Published Version","corr_author":"1","scopus_import":"1","_id":"20005","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2025.43","conference":{"location":"Kanazawa, Japan","end_date":"2025-06-27","name":"SoCG: Symposium on Computational Geometry","start_date":"2025-06-23"},"tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"volume":332,"project":[{"grant_number":"Z00342","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"oa":1,"publication_status":"published","publication_identifier":{"isbn":["9783959773706"],"eissn":["1868-8969"]},"external_id":{"arxiv":["2410.21204"]},"date_published":"2025-06-20T00:00:00Z","department":[{"_id":"HeEd"}],"doi":"10.4230/LIPIcs.SoCG.2025.43","file_date_updated":"2025-07-14T07:24:22Z","OA_place":"publisher","author":[{"first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner"},{"last_name":"Garber","full_name":"Garber, Alexey","first_name":"Alexey"},{"id":"f86f7148-b140-11ec-9577-95435b8df824","last_name":"Saghafian","first_name":"Morteza","full_name":"Saghafian, Morteza"}],"citation":{"apa":"Edelsbrunner, H., Garber, A., &#38; Saghafian, M. (2025). On spheres with k points inside. In <i>41st International Symposium on Computational Geometry</i> (Vol. 332). Kanazawa, Japan: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">https://doi.org/10.4230/LIPIcs.SoCG.2025.43</a>","chicago":"Edelsbrunner, Herbert, Alexey Garber, and Morteza Saghafian. “On Spheres with k Points Inside.” In <i>41st International Symposium on Computational Geometry</i>, Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">https://doi.org/10.4230/LIPIcs.SoCG.2025.43</a>.","short":"H. Edelsbrunner, A. Garber, M. Saghafian, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025.","ieee":"H. Edelsbrunner, A. Garber, and M. Saghafian, “On spheres with k points inside,” in <i>41st International Symposium on Computational Geometry</i>, Kanazawa, Japan, 2025, vol. 332.","ista":"Edelsbrunner H, Garber A, Saghafian M. 2025. On spheres with k points inside. 41st International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 332, 43.","mla":"Edelsbrunner, Herbert, et al. “On Spheres with k Points Inside.” <i>41st International Symposium on Computational Geometry</i>, vol. 332, 43, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">10.4230/LIPIcs.SoCG.2025.43</a>.","ama":"Edelsbrunner H, Garber A, Saghafian M. On spheres with k points inside. In: <i>41st International Symposium on Computational Geometry</i>. Vol 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2025. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.43\">10.4230/LIPIcs.SoCG.2025.43</a>"},"quality_controlled":"1","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","status":"public","acknowledgement":"Herbert Edelsbrunner: partially supported by the Wittgenstein Prize, Austrian Science\r\nFund (FWF), grant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109,\r\nAustrian Science Fund (FWF), grant no. I 02979-N35.\r\nAlexey Garber: partially supported by the Simons Foundation.\r\nMorteza Saghafian: partially supported by the Wittgenstein Prize, Austrian Science Fund (FWF),\r\ngrant no. Z 342-N31, and by the DFG Collaborative Research Center TRR 109, Austrian Science\r\nFund (FWF), grant no. I 02979-N35","article_number":"43","type":"conference","OA_type":"gold","article_processing_charge":"Yes","year":"2025","abstract":[{"lang":"eng","text":"We generalize a classical result by Boris Delaunay that introduced Delaunay triangulations. In particular, we prove that for a locally finite and coarsely dense generic point set A in ℝ^d, every generic point of ℝ^d belongs to exactly binom(d+k,d) simplices whose vertices belong to A and whose circumspheres enclose exactly k points of A. We extend this result to the cases in which the points are weighted, and when A contains only finitely many points in ℝ^d or in 𝕊^d. Furthermore, we use the result to give a new geometric proof for the fact that volumes of hypersimplices are Eulerian numbers."}],"title":"On spheres with k points inside","intvolume":"       332","has_accepted_license":"1","arxiv":1,"alternative_title":["LIPIcs"],"date_updated":"2025-07-14T07:26:14Z","month":"06","file":[{"file_size":661893,"date_created":"2025-07-14T07:24:22Z","checksum":"b5313ed8575ea87913c71a6e3c7513c8","content_type":"application/pdf","creator":"dernst","relation":"main_file","success":1,"date_updated":"2025-07-14T07:24:22Z","file_name":"2025_LIPIcs.SoCG_Edelsbrunner.pdf","file_id":"20016","access_level":"open_access"}],"publication":"41st International Symposium on Computational Geometry"},{"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","status":"public","OA_place":"publisher","author":[{"last_name":"Ost","first_name":"Lara","full_name":"Ost, Lara"},{"first_name":"Sebastiano","full_name":"Cultrera di Montesano, Sebastiano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","last_name":"Cultrera di Montesano","orcid":"0000-0001-6249-0832"},{"full_name":"Edelsbrunner, Herbert","first_name":"Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833"}],"quality_controlled":"1","citation":{"chicago":"Ost, Lara, Sebastiano Cultrera di Montesano, and Herbert Edelsbrunner. “Banana Trees for the Persistence in Time Series Experimentally.” In <i>41st International Symposium on Computational Geometry</i>, Vol. 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.71\">https://doi.org/10.4230/LIPIcs.SoCG.2025.71</a>.","apa":"Ost, L., Cultrera di Montesano, S., &#38; Edelsbrunner, H. (2025). Banana trees for the persistence in time series experimentally. In <i>41st International Symposium on Computational Geometry</i> (Vol. 332). Kanazawa, Japan: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.71\">https://doi.org/10.4230/LIPIcs.SoCG.2025.71</a>","ieee":"L. Ost, S. Cultrera di Montesano, and H. Edelsbrunner, “Banana trees for the persistence in time series experimentally,” in <i>41st International Symposium on Computational Geometry</i>, Kanazawa, Japan, 2025, vol. 332.","short":"L. Ost, S. Cultrera di Montesano, H. Edelsbrunner, in:, 41st International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025.","mla":"Ost, Lara, et al. “Banana Trees for the Persistence in Time Series Experimentally.” <i>41st International Symposium on Computational Geometry</i>, vol. 332, 71, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2025, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.71\">10.4230/LIPIcs.SoCG.2025.71</a>.","ista":"Ost L, Cultrera di Montesano S, Edelsbrunner H. 2025. Banana trees for the persistence in time series experimentally. 41st International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 332, 71.","ama":"Ost L, Cultrera di Montesano S, Edelsbrunner H. Banana trees for the persistence in time series experimentally. In: <i>41st International Symposium on Computational Geometry</i>. Vol 332. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2025. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2025.71\">10.4230/LIPIcs.SoCG.2025.71</a>"},"file_date_updated":"2025-07-14T08:23:38Z","doi":"10.4230/LIPIcs.SoCG.2025.71","department":[{"_id":"HeEd"}],"date_published":"2025-06-20T00:00:00Z","publication_identifier":{"isbn":["9783959773706"],"eissn":["1868-8969"]},"external_id":{"arxiv":["2405.17920"]},"tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"project":[{"name":"Vienna Graduate School on Computational Optimization","_id":"9B9290DE-BA93-11EA-9121-9846C619BF3A","grant_number":"W1260-N35"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"},{"name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"Z00342"}],"volume":332,"conference":{"location":"Kanazawa, Japan","end_date":"2025-06-27","name":"SoCG: Symposium on Computational Geometry","start_date":"2025-06-23"},"publication_status":"published","oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2025.71","scopus_import":"1","oa_version":"Published Version","language":[{"iso":"eng"}],"corr_author":"1","_id":"20006","ddc":["000"],"date_created":"2025-07-13T22:01:22Z","day":"20","file":[{"creator":"dernst","relation":"main_file","success":1,"date_created":"2025-07-14T08:23:38Z","file_size":834623,"content_type":"application/pdf","checksum":"3a4a7a707a56e0cfdf51428782dee55a","access_level":"open_access","file_id":"20017","file_name":"2025_LIPIcs.SoCG_Ost.pdf","date_updated":"2025-07-14T08:23:38Z"}],"month":"06","date_updated":"2025-12-30T11:04:33Z","arxiv":1,"alternative_title":["LIPIcs"],"publication":"41st International Symposium on Computational Geometry","related_material":{"link":[{"relation":"software","url":"https://github.com/laraost/BananaPersist"}]},"has_accepted_license":"1","intvolume":"       332","title":"Banana trees for the persistence in time series experimentally","year":"2025","abstract":[{"text":"In numerous fields, dynamic time series data require continuous updates, necessitating efficient data processing techniques for accurate analysis. This paper examines the banana tree data structure, specifically designed to efficiently maintain the multi-scale topological descriptor commonly known as persistent homology for dynamically changing time series data. We implement this data structure and conduct an experimental study to assess its properties and runtime for update operations. Our findings indicate that banana trees are highly effective with unbiased random data, outperforming state-of-the-art static algorithms in these scenarios. Additionally, our results show that real-world time series share structural properties with unbiased random walks, suggesting potential practical utility for our implementation.","lang":"eng"}],"OA_type":"gold","article_processing_charge":"Yes","type":"conference","article_number":"71","acknowledgement":"Lara Ost: Supported by the Vienna Graduate School on Computational Optimization\r\n(VGSCO), FWF project no. W1260-N35.\r\nSebastiano Cultrera di Montesano: Supported by the Eric and Wendy Schmidt Center at the Broad Institute of MIT and Harvard.\r\nHerbert Edelsbrunner: Partially supported by the Wittgenstein Prize, FWF grant no. Z 342-N31,\r\nand by the DFG Collaborative Research Center TRR 109, FWF grant no. I 02979-N35."},{"file_date_updated":"2025-12-30T07:52:58Z","doi":"10.1007/s44007-025-00170-0","department":[{"_id":"HeEd"}],"date_published":"2025-12-01T00:00:00Z","publication_identifier":{"eissn":["2730-9657"]},"PlanS_conform":"1","article_type":"original","publisher":"Springer Nature","status":"public","page":"811-828","author":[{"full_name":"Chambers, Erin Wolf","first_name":"Erin Wolf","last_name":"Chambers"},{"full_name":"Fillmore, Christopher D","first_name":"Christopher D","last_name":"Fillmore","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425"},{"first_name":"Elizabeth R","full_name":"Stephenson, Elizabeth R","orcid":"0000-0002-6862-208X","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","last_name":"Stephenson"},{"id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","last_name":"Wintraecken","orcid":"0000-0002-7472-2220","first_name":"Mathijs","full_name":"Wintraecken, Mathijs"}],"OA_place":"publisher","quality_controlled":"1","citation":{"mla":"Chambers, Erin Wolf, et al. “Burning or Collapsing the Medial Axis Is Unstable.” <i>La Matematica</i>, vol. 4, Springer Nature, 2025, pp. 811–28, doi:<a href=\"https://doi.org/10.1007/s44007-025-00170-0\">10.1007/s44007-025-00170-0</a>.","ista":"Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. 2025. Burning or collapsing the medial axis is unstable. La Matematica. 4, 811–828.","ama":"Chambers EW, Fillmore CD, Stephenson ER, Wintraecken M. Burning or collapsing the medial axis is unstable. <i>La Matematica</i>. 2025;4:811-828. doi:<a href=\"https://doi.org/10.1007/s44007-025-00170-0\">10.1007/s44007-025-00170-0</a>","chicago":"Chambers, Erin Wolf, Christopher D Fillmore, Elizabeth R Stephenson, and Mathijs Wintraecken. “Burning or Collapsing the Medial Axis Is Unstable.” <i>La Matematica</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s44007-025-00170-0\">https://doi.org/10.1007/s44007-025-00170-0</a>.","apa":"Chambers, E. W., Fillmore, C. D., Stephenson, E. R., &#38; Wintraecken, M. (2025). Burning or collapsing the medial axis is unstable. <i>La Matematica</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s44007-025-00170-0\">https://doi.org/10.1007/s44007-025-00170-0</a>","ieee":"E. W. Chambers, C. D. Fillmore, E. R. Stephenson, and M. Wintraecken, “Burning or collapsing the medial axis is unstable,” <i>La Matematica</i>, vol. 4. Springer Nature, pp. 811–828, 2025.","short":"E.W. Chambers, C.D. Fillmore, E.R. Stephenson, M. Wintraecken, La Matematica 4 (2025) 811–828."},"scopus_import":"1","oa_version":"Published Version","language":[{"iso":"eng"}],"corr_author":"1","_id":"20260","ddc":["510"],"ec_funded":1,"day":"01","date_created":"2025-08-31T22:01:33Z","project":[{"grant_number":"788183","call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended"},{"grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"grant_number":"M03073","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","name":"Learning and triangulating manifolds via collapses"}],"tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"volume":4,"oa":1,"publication_status":"published","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","fulldoi":"https://doi.org/10.1007/s44007-025-00170-0","related_material":{"record":[{"id":"21021","relation":"dissertation_contains","status":"public"}]},"has_accepted_license":"1","intvolume":"         4","title":"Burning or collapsing the medial axis is unstable","year":"2025","abstract":[{"text":"The medial axis of a set consists of the points in the ambient space without a unique closest point in the original set. Since its introduction, the medial axis has been used extensively in many applications as a method of computing a skeleton topologically equivalent to the original set. Unfortunately, one limiting factor in the use of the medial axis of a smooth manifold is that it is not necessarily topologically stable under small perturbations of the manifold. To counter these instabilities, various prunings of the medial axis have been proposed in the computational geometry community. Here, we examine one type of pruning, called burning. Because of the good experimental results it was hoped that the burning method of simplifying the medial axis would be stable. In this work, we show a simple example that dashes such hopes. Based on Bing’s house with two rooms, we demonstrate an isotopy of a shape where the medial axis goes from collapsible to non-collapsible. More precisely, we consider the standard deformation retract from the closed ball to Bing’s house with two rooms, but stop just short of the point where Bing’s house becomes two dimensional. This way we obtain an isotopy from the 3-ball to a thickened version of Bing’s house. Under this isotopy, the medial axis goes from collapsible to non-collapsible. We stress that this isotopy can be made generic, in the sense of singularity theory, as developed by Arnol’d and Thom.","lang":"eng"}],"month":"12","file":[{"date_updated":"2025-12-30T07:52:58Z","file_name":"2025_LaMatematica_Chambers.pdf","file_id":"20885","access_level":"open_access","file_size":2678640,"date_created":"2025-12-30T07:52:58Z","content_type":"application/pdf","checksum":"e2043259194bfcdf3d74c4da8a5a853f","creator":"dernst","relation":"main_file","success":1}],"date_updated":"2026-04-07T11:42:48Z","publication":"La Matematica","acknowledgement":"We thank André Lieutier, David Letscher, Ellen Gasparovic, Kathryn Leonard, and Tao Ju for early discussions on this work. We also thank Lu Liu, Yajie Yan, and Tao Ju for sharing code to generate the examples. We further thank Abigail Thompson for discussion on the conjecture and James Damon for sharing his insight in singularity theory. We thank the reviewers for their detailed reviews, which helped to improve the exposition.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria). Partially supported by the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’ and the European Research Council (ERC), grant no. 788183, ‘Alpha Shape Theory Extended’. The first author was supported in part by the National Science Foundation through grants DBI-1759807, CCF-1907612, and CCF-2444309. The fourth author was 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) M-3073, ANR grant StratMesh, ANR-24-CE48-1899, and the welcome package from IDEX of the Université Côte d’Azur, ANR-15-IDEX-01.","OA_type":"hybrid","article_processing_charge":"Yes (via OA deal)","type":"journal_article"},{"ddc":["500"],"ec_funded":1,"day":"01","date_created":"2025-09-07T22:01:33Z","scopus_import":"1","oa_version":"Published Version","language":[{"iso":"eng"}],"corr_author":"1","_id":"20293","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","fulldoi":"https://doi.org/10.3390/e27080854","volume":27,"tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","grant_number":"Z00342"},{"grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"oa":1,"publication_status":"published","date_published":"2025-08-01T00:00:00Z","publication_identifier":{"eissn":["1099-4300"]},"external_id":{"isi":["001557476000001"],"pmid":["40870326"]},"PlanS_conform":"1","DOAJ_listed":"1","article_type":"original","file_date_updated":"2025-09-08T07:55:48Z","doi":"10.3390/e27080854","department":[{"_id":"HeEd"}],"OA_place":"publisher","author":[{"full_name":"Akopyan, Arseniy","first_name":"Arseniy","orcid":"0000-0002-2548-617X","last_name":"Akopyan","id":"430D2C90-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","first_name":"Herbert"},{"last_name":"Virk","id":"2E36B656-F248-11E8-B48F-1D18A9856A87","full_name":"Virk, Ziga","first_name":"Ziga"},{"last_name":"Wagner","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Hubert","first_name":"Hubert"}],"quality_controlled":"1","isi":1,"citation":{"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>","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>.","short":"A. Akopyan, H. Edelsbrunner, Z. Virk, H. Wagner, Entropy 27 (2025).","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.","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.","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>.","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>"},"publisher":"MDPI","status":"public","issue":"8","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.","article_number":"854","pmid":1,"type":"journal_article","OA_type":"gold","article_processing_charge":"Yes","year":"2025","abstract":[{"lang":"eng","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."}],"has_accepted_license":"1","title":"Tight bounds between the Jensen–Shannon divergence and the minmax divergence","intvolume":"        27","file":[{"file_name":"2025_Entropy_Akopyan.pdf","date_updated":"2025-09-08T07:55:48Z","access_level":"open_access","file_id":"20309","content_type":"application/pdf","checksum":"65c5399c4015d9c8abb8c7a96f3d7836","date_created":"2025-09-08T07:55:48Z","file_size":379340,"relation":"main_file","success":1,"creator":"dernst"}],"month":"08","date_updated":"2025-09-30T14:32:31Z","publication":"Entropy"},{"date_published":"2025-11-01T00:00:00Z","external_id":{"arxiv":["2504.14743"]},"publication_identifier":{"eissn":["1611-3349"],"issn":["0302-9743"],"isbn":["9783032095435"]},"doi":"10.1007/978-3-032-09544-2_10","department":[{"_id":"HeEd"}],"OA_place":"repository","page":"133-147","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert","full_name":"Edelsbrunner, Herbert"},{"orcid":"0000-0002-6862-208X","last_name":"Stephenson","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","full_name":"Stephenson, Elizabeth R","first_name":"Elizabeth R"},{"last_name":"Thoresen","id":"47CB1472-F248-11E8-B48F-1D18A9856A87","full_name":"Thoresen, Martin H","first_name":"Martin H"}],"quality_controlled":"1","citation":{"apa":"Edelsbrunner, H., Stephenson, E. R., &#38; Thoresen, M. H. (2025). The mid-sphere cousin of the medial axis transform. In <i>4th International Joint Conference on Discrete Geometry and Mathematical Morphology</i> (Vol. 16296, pp. 133–147). Groningen, The Netherlands: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-032-09544-2_10\">https://doi.org/10.1007/978-3-032-09544-2_10</a>","chicago":"Edelsbrunner, Herbert, Elizabeth R Stephenson, and Martin H Thoresen. “The Mid-Sphere Cousin of the Medial Axis Transform.” In <i>4th International Joint Conference on Discrete Geometry and Mathematical Morphology</i>, 16296:133–47. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/978-3-032-09544-2_10\">https://doi.org/10.1007/978-3-032-09544-2_10</a>.","short":"H. Edelsbrunner, E.R. Stephenson, M.H. Thoresen, in:, 4th International Joint Conference on Discrete Geometry and Mathematical Morphology, Springer Nature, 2025, pp. 133–147.","ieee":"H. Edelsbrunner, E. R. Stephenson, and M. H. Thoresen, “The mid-sphere cousin of the medial axis transform,” in <i>4th International Joint Conference on Discrete Geometry and Mathematical Morphology</i>, Groningen, The Netherlands, 2025, vol. 16296, pp. 133–147.","ista":"Edelsbrunner H, Stephenson ER, Thoresen MH. 2025. The mid-sphere cousin of the medial axis transform. 4th International Joint Conference on Discrete Geometry and Mathematical Morphology. DGMM: Discrete Geometry and Mathematical Morphology, LNCS, vol. 16296, 133–147.","mla":"Edelsbrunner, Herbert, et al. “The Mid-Sphere Cousin of the Medial Axis Transform.” <i>4th International Joint Conference on Discrete Geometry and Mathematical Morphology</i>, vol. 16296, Springer Nature, 2025, pp. 133–47, doi:<a href=\"https://doi.org/10.1007/978-3-032-09544-2_10\">10.1007/978-3-032-09544-2_10</a>.","ama":"Edelsbrunner H, Stephenson ER, Thoresen MH. The mid-sphere cousin of the medial axis transform. In: <i>4th International Joint Conference on Discrete Geometry and Mathematical Morphology</i>. Vol 16296. Springer Nature; 2025:133-147. doi:<a href=\"https://doi.org/10.1007/978-3-032-09544-2_10\">10.1007/978-3-032-09544-2_10</a>"},"publisher":"Springer Nature","status":"public","day":"01","date_created":"2025-11-23T23:01:37Z","scopus_import":"1","oa_version":"Preprint","language":[{"iso":"eng"}],"_id":"20658","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","fulldoi":"https://doi.org/10.1007/978-3-032-09544-2_10","volume":16296,"conference":{"start_date":"2025-11-03","name":"DGMM: Discrete Geometry and Mathematical Morphology","end_date":"2025-11-06","location":"Groningen, The Netherlands"},"oa":1,"publication_status":"published","year":"2025","abstract":[{"lang":"eng","text":"The medial axis of a smoothly embedded surface in R^3 consists of all points for which the Euclidean distance function on the surface has at least two global minima. We generalize this notion to the mid-sphere axis, which consists of all points for which the Euclidean distance function has two interchanging saddles that swap their partners in the pairing by persistent homology. It offers a discrete-algebraic multi-scale approach to computing ridge-like structures on the surface. As a proof of concept, an algorithm that computes stair-case approximations of the mid-sphere axis is provided."}],"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2504.14743","open_access":"1"}],"title":"The mid-sphere cousin of the medial axis transform","intvolume":"     16296","month":"11","alternative_title":["LNCS"],"arxiv":1,"date_updated":"2025-11-24T10:05:11Z","publication":"4th International Joint Conference on Discrete Geometry and Mathematical Morphology","type":"conference","OA_type":"green","article_processing_charge":"No"},{"date_published":"2025-11-10T00:00:00Z","publication_identifier":{"isbn":["9798400720758"]},"doi":"10.1145/3747199.3747561","file_date_updated":"2025-12-09T13:43:17Z","department":[{"_id":"HeEd"}],"quality_controlled":"1","citation":{"ista":"Gonzalez-Diaz R, Soriano Trigueros M, Torras-Casas A. 2025. Additive partial matchings for persistent homology. Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation. ISSAC: International Symposium on Symbolic and Algebraic Computation, 188–196.","mla":"Gonzalez-Diaz, Rocio, et al. “Additive Partial Matchings for Persistent Homology.” <i>Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation</i>, Association for Computing Machinery, 2025, pp. 188–96, doi:<a href=\"https://doi.org/10.1145/3747199.3747561\">10.1145/3747199.3747561</a>.","ama":"Gonzalez-Diaz R, Soriano Trigueros M, Torras-Casas A. Additive partial matchings for persistent homology. In: <i>Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation</i>. Association for Computing Machinery; 2025:188-196. doi:<a href=\"https://doi.org/10.1145/3747199.3747561\">10.1145/3747199.3747561</a>","apa":"Gonzalez-Diaz, R., Soriano Trigueros, M., &#38; Torras-Casas, A. (2025). Additive partial matchings for persistent homology. In <i>Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation</i> (pp. 188–196). Guanajuato, Mexico: Association for Computing Machinery. <a href=\"https://doi.org/10.1145/3747199.3747561\">https://doi.org/10.1145/3747199.3747561</a>","chicago":"Gonzalez-Diaz, Rocio, Manuel Soriano Trigueros, and Alvaro Torras-Casas. “Additive Partial Matchings for Persistent Homology.” In <i>Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation</i>, 188–96. Association for Computing Machinery, 2025. <a href=\"https://doi.org/10.1145/3747199.3747561\">https://doi.org/10.1145/3747199.3747561</a>.","short":"R. Gonzalez-Diaz, M. Soriano Trigueros, A. Torras-Casas, in:, Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation, Association for Computing Machinery, 2025, pp. 188–196.","ieee":"R. Gonzalez-Diaz, M. Soriano Trigueros, and A. Torras-Casas, “Additive partial matchings for persistent homology,” in <i>Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation</i>, Guanajuato, Mexico, 2025, pp. 188–196."},"author":[{"first_name":"Rocio","full_name":"Gonzalez-Diaz, Rocio","last_name":"Gonzalez-Diaz"},{"orcid":"0000-0003-2449-1433","id":"15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8","last_name":"Soriano Trigueros","first_name":"Manuel","full_name":"Soriano Trigueros, Manuel"},{"last_name":"Torras-Casas","full_name":"Torras-Casas, Alvaro","first_name":"Alvaro"}],"page":"188-196","OA_place":"publisher","status":"public","publisher":"Association for Computing Machinery","day":"10","date_created":"2025-12-07T23:02:01Z","ddc":["510"],"_id":"20729","scopus_import":"1","language":[{"iso":"eng"}],"oa_version":"Published Version","corr_author":"1","fulldoi":"https://doi.org/10.1145/3747199.3747561","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"publication_status":"published","conference":{"location":"Guanajuato, Mexico","end_date":"2025-08-01","name":"ISSAC: International Symposium on Symbolic and Algebraic Computation","start_date":"2025-07-28"},"tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"abstract":[{"text":"Persistence modules (defined as a sequence of vector spaces and linear maps between them) are a key tool in topological data analysis. They are easy to interpret and fast to compute. However, when considering persistence maps (i.e. maps between persistence modules), these properties are lost. We propose a new invariant for persistence maps consisting of a partial matching such that: it is easy to interpret, it is more discriminative than the image of the persistence map, and can be calculated with cubical complexity.","lang":"eng"}],"year":"2025","has_accepted_license":"1","title":"Additive partial matchings for persistent homology","publication":"Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation","month":"11","file":[{"access_level":"open_access","file_id":"20751","file_name":"2025_ISSAC_GonzalezDiaz.pdf","date_updated":"2025-12-09T13:43:17Z","relation":"main_file","success":1,"creator":"dernst","checksum":"1c299cca165a20e2518afe4fda63dbf1","content_type":"application/pdf","date_created":"2025-12-09T13:43:17Z","file_size":761617}],"date_updated":"2025-12-09T13:46:42Z","acknowledgement":"Álvaro Torras-Casas contract is funded by the French Agence Nationale de la Recherche through the project reference ANR-22-CPJ1-0047-01. Rocio Gonzalez-Diaz is partially funded by the European Union under grant agreement no. 101070028-2 (REXASI-PRO).","type":"conference","article_processing_charge":"Yes (in subscription journal)","OA_type":"hybrid"},{"citation":{"chicago":"Bokor Bleile, Yossi, Lisbeth Fajstrup, Teresa Heiss, Anne Marie Svane, and Søren Strandskov Sørensen. “Identifying Cobordisms Using Kernel Persistence.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2505.17858\">https://doi.org/10.48550/arXiv.2505.17858</a>.","apa":"Bokor Bleile, Y., Fajstrup, L., Heiss, T., Svane, A. M., &#38; Sørensen, S. S. (n.d.). Identifying cobordisms using kernel persistence. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2505.17858\">https://doi.org/10.48550/arXiv.2505.17858</a>","ieee":"Y. Bokor Bleile, L. Fajstrup, T. Heiss, A. M. Svane, and S. S. Sørensen, “Identifying cobordisms using kernel persistence,” <i>arXiv</i>. .","short":"Y. Bokor Bleile, L. Fajstrup, T. Heiss, A.M. Svane, S.S. Sørensen, ArXiv (n.d.).","mla":"Bokor Bleile, Yossi, et al. “Identifying Cobordisms Using Kernel Persistence.” <i>ArXiv</i>, 2505.17858, doi:<a href=\"https://doi.org/10.48550/arXiv.2505.17858\">10.48550/arXiv.2505.17858</a>.","ista":"Bokor Bleile Y, Fajstrup L, Heiss T, Svane AM, Sørensen SS. Identifying cobordisms using kernel persistence. arXiv, 2505.17858.","ama":"Bokor Bleile Y, Fajstrup L, Heiss T, Svane AM, Sørensen SS. Identifying cobordisms using kernel persistence. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2505.17858\">10.48550/arXiv.2505.17858</a>"},"OA_place":"repository","author":[{"first_name":"Yossi","full_name":"Bleile, Yossi","orcid":"0000-0002-4861-9174","id":"920a7385-7995-11ef-9bfd-8c434cd8f3c2","last_name":"Bleile"},{"last_name":"Fajstrup","first_name":"Lisbeth","full_name":"Fajstrup, Lisbeth"},{"first_name":"Teresa","full_name":"Heiss, Teresa","orcid":"0000-0002-1780-2689","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","last_name":"Heiss"},{"full_name":"Svane, Anne Marie","first_name":"Anne Marie","last_name":"Svane"},{"full_name":"Sørensen, Søren Strandskov","first_name":"Søren Strandskov","last_name":"Sørensen"}],"status":"public","external_id":{"arxiv":["2505.17858"]},"date_published":"2025-05-23T00:00:00Z","department":[{"_id":"HeEd"}],"doi":"10.48550/arXiv.2505.17858","fulldoi":"https://doi.org/10.48550/arXiv.2505.17858","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"publication_status":"submitted","project":[{"grant_number":"788183","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020"}],"day":"23","date_created":"2026-01-20T10:12:21Z","ec_funded":1,"_id":"21016","oa_version":"Preprint","language":[{"iso":"eng"}],"publication":"arXiv","arxiv":1,"date_updated":"2026-07-22T06:33:07Z","month":"05","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2505.17858"}],"abstract":[{"lang":"eng","text":"Motivated by applications in chemistry, we give a homlogical definition of tunnels, or more generally cobordisms, connecting disjoint parts of a cell complex. For a filtered complex, this defines a persistence module. We give a method for identifying birth and death times using kernel persistence and a matrix reduction algorithm for pairing birth and death times."}],"year":"2025","title":"Identifying cobordisms using kernel persistence","type":"preprint","article_processing_charge":"No","OA_type":"green","acknowledgement":"Y. B. B. and L. F. were funded by the Independent Research Fund Denmark, grant\r\nnumber 1026-00037. T. H. was partially supported by the European Research Council\r\n(ERC) Horizon 2020, grant number 788183.","article_number":"2505.17858","das_tickbox":"1"},{"article_processing_charge":"No","OA_type":"green","type":"preprint","article_number":"2501.05315","das_tickbox":"1","publication":"arXiv","arxiv":1,"date_updated":"2026-07-22T06:33:55Z","month":"03","title":"Counting equilibria of the electrostatic potential","related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"21021"},{"relation":"later_version","status":"public","id":"21931"}]},"abstract":[{"text":"In 1873, James C. Maxwell conjectured that the electric field generated by $n$ point charges in generic position has at most $(n-1)^2$ isolated zeroes. The first (non-optimal) upper bound was only obtained in 2007 by Gabrielov, Novikov and Shapiro, who also posed two additional interesting conjectures.\r\n In this article, we give the best upper bound known to date on the number of zeroes of the electric field, and construct a counterexample to a conjecture of Gabrielov, Novikov and Shapiro that the number of equilibria cannot exceed those of the distance function defined by the unit point charges.\r\n Finally, we note that it is quite possible that Maxwell's quadratic upper bound is not tight, so it is prudent to find smaller bounds. Hence, we also explore examples and construct configurations of charges achieving the highest ratios of the number of electric field zeroes by point charges found to this day.","lang":"eng"}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2501.05315"}],"year":"2025","publication_status":"draft","oa":1,"tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"fulldoi":"https://doi.org/10.48550/ARXIV.2501.05315","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"21050","language":[{"iso":"eng"}],"corr_author":"1","oa_version":"Preprint","date_created":"2026-01-27T14:29:27Z","day":"20","status":"public","citation":{"short":"H. Edelsbrunner, C.D. Fillmore, G. Olivera, ArXiv (n.d.).","ieee":"H. Edelsbrunner, C. D. Fillmore, and G. Olivera, “Counting equilibria of the electrostatic potential,” <i>arXiv</i>. .","apa":"Edelsbrunner, H., Fillmore, C. D., &#38; Olivera, G. (n.d.). Counting equilibria of the electrostatic potential. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/ARXIV.2501.05315\">https://doi.org/10.48550/ARXIV.2501.05315</a>","chicago":"Edelsbrunner, Herbert, Christopher D Fillmore, and Gonçalo Olivera. “Counting Equilibria of the Electrostatic Potential.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/ARXIV.2501.05315\">https://doi.org/10.48550/ARXIV.2501.05315</a>.","ama":"Edelsbrunner H, Fillmore CD, Olivera G. Counting equilibria of the electrostatic potential. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/ARXIV.2501.05315\">10.48550/ARXIV.2501.05315</a>","ista":"Edelsbrunner H, Fillmore CD, Olivera G. Counting equilibria of the electrostatic potential. arXiv, 2501.05315.","mla":"Edelsbrunner, Herbert, et al. “Counting Equilibria of the Electrostatic Potential.” <i>ArXiv</i>, 2501.05315, doi:<a href=\"https://doi.org/10.48550/ARXIV.2501.05315\">10.48550/ARXIV.2501.05315</a>."},"OA_place":"repository","author":[{"full_name":"Edelsbrunner, Herbert","first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Fillmore","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","full_name":"Fillmore, Christopher D","first_name":"Christopher D"},{"last_name":"Olivera","first_name":"Gonçalo","full_name":"Olivera, Gonçalo"}],"department":[{"_id":"HeEd"}],"doi":"10.48550/ARXIV.2501.05315","external_id":{"arxiv":["2501.05315"]},"date_published":"2025-03-20T00:00:00Z"},{"year":"2025","abstract":[{"lang":"eng","text":"The approximation of a circle with the edges of a fine square grid distorts the perimeter by a factor about 4/Pi. We prove that this factor is the same on average (in the ergodic sense) for approximations of any rectifiable curve by the edges of any non-exotic Delaunay mosaic (known as Voronoi path), and extend the results to all dimensions, generalizing Voronoi paths to Voronoi scapes."}],"has_accepted_license":"1","title":"Average and expected distortion of Voronoi paths and scapes","intvolume":"        73","month":"03","file":[{"access_level":"open_access","file_id":"19610","file_name":"2025_DiscreteComputGeom_EdelsbrunnerHe.pdf","date_updated":"2025-04-23T07:31:32Z","success":1,"relation":"main_file","creator":"dernst","checksum":"ffb0c818222138f9f113f4bbea41e834","content_type":"application/pdf","date_created":"2025-04-23T07:31:32Z","file_size":283443}],"researchdata_availability":"no","date_updated":"2026-10-02T11:48:46Z","arxiv":1,"publication":"Discrete & Computational Geometry","acknowledgement":"The authors thank Ranita Biswas and Tatiana Ezubova for the collaboration on computational experiments that motivated the work reported in this paper. The authors also thank Daniel Bonnema for proofreading and noticing an issue with the original proof of Lemma 4.3.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria).\r\nThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, Grant No. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No. I 02979-N35.","supplementarymaterial":"no","das_tickbox":"0","pmid":1,"type":"journal_article","OA_type":"hybrid","article_processing_charge":"Yes (via OA deal)","date_published":"2025-03-01T00:00:00Z","external_id":{"arxiv":["2012.03350"],"pmid":["39974750"],"isi":["001238566200004"]},"publication_identifier":{"eissn":["1432-0444"],"issn":["0179-5376"]},"article_type":"original","doi":"10.1007/s00454-024-00660-y","file_date_updated":"2025-04-23T07:31:32Z","department":[{"_id":"HeEd"}],"OA_place":"publisher","author":[{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","first_name":"Herbert"},{"full_name":"Nikitenko, Anton","first_name":"Anton","last_name":"Nikitenko","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0659-3201"}],"page":"490-499","quality_controlled":"1","isi":1,"citation":{"ista":"Edelsbrunner H, Nikitenko A. 2025. Average and expected distortion of Voronoi paths and scapes. Discrete &#38; Computational Geometry. 73, 490–499.","mla":"Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” <i>Discrete &#38; Computational Geometry</i>, vol. 73, Springer Nature, 2025, pp. 490–99, doi:<a href=\"https://doi.org/10.1007/s00454-024-00660-y\">10.1007/s00454-024-00660-y</a>.","ama":"Edelsbrunner H, Nikitenko A. Average and expected distortion of Voronoi paths and scapes. <i>Discrete &#38; Computational Geometry</i>. 2025;73:490-499. doi:<a href=\"https://doi.org/10.1007/s00454-024-00660-y\">10.1007/s00454-024-00660-y</a>","apa":"Edelsbrunner, H., &#38; Nikitenko, A. (2025). Average and expected distortion of Voronoi paths and scapes. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-024-00660-y\">https://doi.org/10.1007/s00454-024-00660-y</a>","chicago":"Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion of Voronoi Paths and Scapes.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s00454-024-00660-y\">https://doi.org/10.1007/s00454-024-00660-y</a>.","short":"H. Edelsbrunner, A. Nikitenko, Discrete &#38; Computational Geometry 73 (2025) 490–499.","ieee":"H. Edelsbrunner and A. Nikitenko, “Average and expected distortion of Voronoi paths and scapes,” <i>Discrete &#38; Computational Geometry</i>, vol. 73. Springer Nature, pp. 490–499, 2025."},"publisher":"Springer Nature","status":"public","ec_funded":1,"ddc":["510"],"day":"01","date_created":"2024-06-16T22:01:07Z","scopus_import":"1","language":[{"iso":"eng"}],"oa_version":"Published Version","corr_author":"1","_id":"17149","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","fulldoi":"https://doi.org/10.1007/s00454-024-00660-y","volume":73,"tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","call_identifier":"FWF","grant_number":"Z00342"},{"grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"publication_status":"published","oa":1},{"quality_controlled":"1","citation":{"ista":"Pach J, Saghafian M, Schnider P. 2025. Decomposition of geometric graphs into star-forests. Computational Geometry. 129, 102186.","mla":"Pach, János, et al. “Decomposition of Geometric Graphs into Star-Forests.” <i>Computational Geometry</i>, vol. 129, 102186, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">10.1016/j.comgeo.2025.102186</a>.","ama":"Pach J, Saghafian M, Schnider P. Decomposition of geometric graphs into star-forests. <i>Computational Geometry</i>. 2025;129. doi:<a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">10.1016/j.comgeo.2025.102186</a>","apa":"Pach, J., Saghafian, M., &#38; Schnider, P. (2025). Decomposition of geometric graphs into star-forests. <i>Computational Geometry</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">https://doi.org/10.1016/j.comgeo.2025.102186</a>","chicago":"Pach, János, Morteza Saghafian, and Patrick Schnider. “Decomposition of Geometric Graphs into Star-Forests.” <i>Computational Geometry</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.comgeo.2025.102186\">https://doi.org/10.1016/j.comgeo.2025.102186</a>.","short":"J. Pach, M. Saghafian, P. Schnider, Computational Geometry 129 (2025).","ieee":"J. Pach, M. Saghafian, and P. Schnider, “Decomposition of geometric graphs into star-forests,” <i>Computational Geometry</i>, vol. 129. Elsevier, 2025."},"author":[{"last_name":"Pach","first_name":"János","full_name":"Pach, János"},{"last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza","first_name":"Morteza"},{"last_name":"Schnider","first_name":"Patrick","full_name":"Schnider, Patrick"}],"OA_place":"repository","status":"public","publisher":"Elsevier","article_type":"original","date_published":"2025-12-01T00:00:00Z","publication_identifier":{"issn":["0925-7721"]},"external_id":{"arxiv":["2306.13201"]},"doi":"10.1016/j.comgeo.2025.102186","department":[{"_id":"HeEd"}],"fulldoi":"https://doi.org/10.1016/j.comgeo.2025.102186","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_status":"published","oa":1,"volume":129,"date_created":"2026-02-16T15:48:42Z","day":"01","_id":"21253","oa_version":"Preprint","language":[{"iso":"eng"}],"corr_author":"1","publication":"Computational Geometry","month":"12","date_updated":"2026-10-02T11:50:58Z","arxiv":1,"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2306.13201","open_access":"1"}],"abstract":[{"lang":"eng","text":"We solve a problem of Dujmović and Wood (2007) by showing that a complete convex geometric graph on n vertices cannot be decomposed into fewer than n - 1 star-forests, each consisting of noncrossing edges. This bound is clearly tight. We also discuss similar questions for abstract graphs."}],"year":"2025","title":"Decomposition of geometric graphs into star-forests","intvolume":"       129","related_material":{"record":[{"id":"15012","status":"public","relation":"earlier_version"}]},"type":"journal_article","article_processing_charge":"No","OA_type":"green","acknowledgement":"A preliminary version of this note has been published in the proceedings of the 31st International Symposium on Graph Drawing and Network Visualization, Palermo, 2023. The authors would like to thank the anonymous referees for their valuable comments.","article_number":"102186"},{"date_published":"2025-02-03T00:00:00Z","publication_identifier":{"issn":["2663-337X"]},"keyword":["topological data analysis","chromatic point set","alpha complex","persistent homology","six pack","sheaf","microlocal discrete Morse","injective resolution","collapse","knot","discrete Morse theory"],"doi":"10.15479/at:ista:18979","file_date_updated":"2025-02-04T16:22:07Z","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"citation":{"short":"O. Draganov, Structures and Computations in Topological Data Analysis, Institute of Science and Technology Austria, 2025.","ieee":"O. Draganov, “Structures and computations in topological data analysis,” Institute of Science and Technology Austria, 2025.","apa":"Draganov, O. (2025). <i>Structures and computations in topological data analysis</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/at:ista:18979\">https://doi.org/10.15479/at:ista:18979</a>","chicago":"Draganov, Ondrej. “Structures and Computations in Topological Data Analysis.” Institute of Science and Technology Austria, 2025. <a href=\"https://doi.org/10.15479/at:ista:18979\">https://doi.org/10.15479/at:ista:18979</a>.","ama":"Draganov O. Structures and computations in topological data analysis. 2025. doi:<a href=\"https://doi.org/10.15479/at:ista:18979\">10.15479/at:ista:18979</a>","ista":"Draganov O. 2025. Structures and computations in topological data analysis. Institute of Science and Technology Austria.","mla":"Draganov, Ondrej. <i>Structures and Computations in Topological Data Analysis</i>. Institute of Science and Technology Austria, 2025, doi:<a href=\"https://doi.org/10.15479/at:ista:18979\">10.15479/at:ista:18979</a>."},"degree_awarded":"PhD","OA_place":"publisher","page":"140","author":[{"full_name":"Draganov, Ondrej","first_name":"Ondrej","last_name":"Draganov","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-0464-3823"}],"status":"public","publisher":"Institute of Science and Technology Austria","date_created":"2025-01-31T17:04:40Z","day":"03","ddc":["514","004"],"_id":"18979","oa_version":"Published Version","language":[{"iso":"eng"}],"corr_author":"1","fulldoi":"https://doi.org/10.15479/at:ista:18979","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","oa":1,"publication_status":"published","tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"project":[{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"},{"grant_number":"Z00342","call_identifier":"FWF","name":"Mathematics, Computer Science","_id":"268116B8-B435-11E9-9278-68D0E5697425"}],"abstract":[{"text":"Topological Data Analysis (TDA) is a discipline utilizing the mathematical field of topology to study data, most prominently collections of point sets. This thesis summarizes three projects related to computations in TDA.\r\n\r\nThe first one establishes a variant of TDA for chromatic point sets, where each point is given a color. For example, we are given positions of cells within a tumor microenvironment, and color the cancerous cells red, and the immune cells blue.\r\n\r\nThe aim is then to give a quantitative description of how the two or more sets of points spatially interact. Building on image, kernel and cokernel variants of persistent homology, we suggest six-packs of persistent diagrams as such a descriptor.\r\n\r\nWe describe a construction of a chromatic alpha complex, which enables  efficient computation of several variants of the six-packs. We give topological descriptions of natural subcomplexes of the chromatic alpha complex, and show that the radii of the simplices form a discrete Morse function. Finally, we provide an implementation of the presented chromatic TDA pipeline.\r\n\r\nThe second part aims to translate a powerful tool of sheaf theory to elementary terms using labeled matrices. The goal is to enable their use in computational settings. We show that derived categories of sheaves over finite posets have, up to isomorphism, unique objects---minimal injective resolutions---and give a concrete algorithm to compute them. We further describe simple algorithms to compute derived pushforwards and pullbacks for monotonic maps, and their proper variants for inclusions, and demonstrate their tractability by providing an implementation. Finally, we suggest a discrete definition of microsupport and show desirable properties inspired by discrete Morse theory.\r\n\r\nIn the last part, we present a collection of observations about collapses. We give a characterization of collapsibility in terms of unitriangular submatrices of the boundary matrix, a cotree-tree decomposition, and the optimal solution to a variant of the Procrustes problem. We establish relation between dual collapses and relative Morse theory and pose several open questions. Finally, focusing on complexes embedded in the three-dimensional Euclidean space, we describe a relation between the collapsibility and the triviality of a polygonal knot.","lang":"eng"}],"year":"2025","has_accepted_license":"1","title":"Structures and computations in topological data analysis","related_material":{"record":[{"status":"public","relation":"part_of_dissertation","id":"15091"},{"relation":"part_of_dissertation","status":"public","id":"18981"}]},"file":[{"checksum":"af6567e5d35e5eb330b8925ae37f1998","content_type":"application/zip","file_size":11899491,"date_created":"2025-01-31T16:58:30Z","relation":"source_file","creator":"odragano","date_updated":"2025-01-31T16:58:30Z","file_name":"Thesis.zip","file_id":"18983","access_level":"closed"},{"access_level":"open_access","file_id":"19000","file_name":"Thesis.pdf","date_updated":"2025-02-04T16:22:07Z","relation":"main_file","creator":"odragano","content_type":"application/pdf","checksum":"c3fef68e35b9dc2020b2ca6006da6343","date_created":"2025-02-04T16:22:07Z","file_size":8857514}],"month":"02","date_updated":"2026-10-02T11:57:00Z","alternative_title":["ISTA Thesis"],"acknowledgement":"The research presented in this thesis was funded with the Wittgenstein Prize,\r\nAustrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative Research\r\nCenter TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF),\r\ngrant no. I 02979-N35.\r\n","supervisor":[{"first_name":"Herbert","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833"}],"type":"dissertation","article_processing_charge":"No"},{"intvolume":"       229","title":"Discrete microlocal Morse theory","has_accepted_license":"1","related_material":{"record":[{"id":"18981","status":"public","relation":"earlier_version"}]},"abstract":[{"lang":"eng","text":"We establish several results combining discrete Morse theory and microlocal sheaf theory in the setting of finite posets and simplicial complexes. Our primary tool is a computationally tractable description of the bounded derived category of sheaves on a poset with the Alexandrov topology. We prove that each bounded complex of sheaves on a finite poset admits a unique (up to isomorphism of complexes) minimal injective resolution, and we provide algorithms for computing minimal injective resolution of an injective complex, as well as several useful functors between derived categories of sheaves. For the constant sheaf on a simplicial complex, we give asymptotically tight bounds on the complexity of computing the minimal injective resolution using those algorithms. Our main result is a novel definition of the discrete microsupport of a bounded complex of sheaves on a finite poset. We detail several foundational properties of the discrete microsupport, as well as a microlocal generalization of the discrete homological Morse theorem and Morse inequalities."}],"year":"2025","publication":"Journal of Pure and Applied Algebra","arxiv":1,"date_updated":"2026-10-02T11:57:00Z","month":"10","file":[{"relation":"main_file","success":1,"creator":"dernst","content_type":"application/pdf","checksum":"39bcad462278c9322ef810af7db67f56","file_size":3090836,"date_created":"2025-12-30T07:55:08Z","file_id":"20886","access_level":"open_access","date_updated":"2025-12-30T07:55:08Z","file_name":"2025_JourPureAppliedAlgebra_Brown.pdf"}],"article_number":"108068","acknowledgement":"This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant no. 788183, from the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z342-N31, and from the DFG Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), grant no. I 02979-N35","issue":"10","article_processing_charge":"Yes (via OA deal)","OA_type":"hybrid","type":"journal_article","department":[{"_id":"HeEd"}],"file_date_updated":"2025-12-30T07:55:08Z","doi":"10.1016/j.jpaa.2025.108068","article_type":"original","PlanS_conform":"1","publication_identifier":{"issn":["0022-4049"]},"external_id":{"arxiv":["2209.14993"]},"date_published":"2025-10-01T00:00:00Z","status":"public","publisher":"Elsevier","citation":{"mla":"Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” <i>Journal of Pure and Applied Algebra</i>, vol. 229, no. 10, 108068, Elsevier, 2025, doi:<a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">10.1016/j.jpaa.2025.108068</a>.","ista":"Brown A, Draganov O. 2025. Discrete microlocal Morse theory. Journal of Pure and Applied Algebra. 229(10), 108068.","ama":"Brown A, Draganov O. Discrete microlocal Morse theory. <i>Journal of Pure and Applied Algebra</i>. 2025;229(10). doi:<a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">10.1016/j.jpaa.2025.108068</a>","chicago":"Brown, Adam, and Ondrej Draganov. “Discrete Microlocal Morse Theory.” <i>Journal of Pure and Applied Algebra</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">https://doi.org/10.1016/j.jpaa.2025.108068</a>.","apa":"Brown, A., &#38; Draganov, O. (2025). Discrete microlocal Morse theory. <i>Journal of Pure and Applied Algebra</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jpaa.2025.108068\">https://doi.org/10.1016/j.jpaa.2025.108068</a>","ieee":"A. Brown and O. Draganov, “Discrete microlocal Morse theory,” <i>Journal of Pure and Applied Algebra</i>, vol. 229, no. 10. Elsevier, 2025.","short":"A. Brown, O. Draganov, Journal of Pure and Applied Algebra 229 (2025)."},"quality_controlled":"1","author":[{"first_name":"Adam","full_name":"Brown, Adam","id":"70B7FDF6-608D-11E9-9333-8535E6697425","last_name":"Brown"},{"orcid":"0000-0003-0464-3823","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","last_name":"Draganov","first_name":"Ondrej","full_name":"Draganov, Ondrej"}],"OA_place":"publisher","_id":"20323","corr_author":"1","language":[{"iso":"eng"}],"oa_version":"Published Version","scopus_import":"1","date_created":"2025-09-10T05:40:09Z","day":"01","ec_funded":1,"ddc":["510"],"publication_status":"published","oa":1,"project":[{"grant_number":"788183","name":"Alpha Shape Theory Extended","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","call_identifier":"FWF"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"}],"tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"volume":229,"fulldoi":"https://doi.org/10.1016/j.jpaa.2025.108068","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"date_published":"2024-11-01T00:00:00Z","external_id":{"arxiv":["2404.00500"]},"doi":"10.18653/v1/2024.findings-emnlp.705","file_date_updated":"2025-02-10T08:20:34Z","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"page":"12080-12099","OA_place":"publisher","author":[{"full_name":"Draganov, Ondrej","first_name":"Ondrej","last_name":"Draganov","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-0464-3823"},{"first_name":"Steven","full_name":"Skiena, Steven","last_name":"Skiena"}],"quality_controlled":"1","citation":{"mla":"Draganov, Ondrej, and Steven Skiena. “The Shape of Word Embeddings: Quantifying Non-Isometry with Topological Data Analysis.” <i>Findings of the Association for Computational Linguistics: EMNLP 2024</i>, Association for Computational Linguistics, 2024, pp. 12080–99, doi:<a href=\"https://doi.org/10.18653/v1/2024.findings-emnlp.705\">10.18653/v1/2024.findings-emnlp.705</a>.","ista":"Draganov O, Skiena S. 2024. The shape of word embeddings: Quantifying non-isometry with topological data analysis. Findings of the Association for Computational Linguistics: EMNLP 2024. EMNLP: Conference on Empirical Methods in Natural Language Processing, 12080–12099.","ama":"Draganov O, Skiena S. The shape of word embeddings: Quantifying non-isometry with topological data analysis. In: <i>Findings of the Association for Computational Linguistics: EMNLP 2024</i>. Association for Computational Linguistics; 2024:12080-12099. doi:<a href=\"https://doi.org/10.18653/v1/2024.findings-emnlp.705\">10.18653/v1/2024.findings-emnlp.705</a>","chicago":"Draganov, Ondrej, and Steven Skiena. “The Shape of Word Embeddings: Quantifying Non-Isometry with Topological Data Analysis.” In <i>Findings of the Association for Computational Linguistics: EMNLP 2024</i>, 12080–99. Association for Computational Linguistics, 2024. <a href=\"https://doi.org/10.18653/v1/2024.findings-emnlp.705\">https://doi.org/10.18653/v1/2024.findings-emnlp.705</a>.","apa":"Draganov, O., &#38; Skiena, S. (2024). The shape of word embeddings: Quantifying non-isometry with topological data analysis. In <i>Findings of the Association for Computational Linguistics: EMNLP 2024</i> (pp. 12080–12099). Miami, FL, United States: Association for Computational Linguistics. <a href=\"https://doi.org/10.18653/v1/2024.findings-emnlp.705\">https://doi.org/10.18653/v1/2024.findings-emnlp.705</a>","ieee":"O. Draganov and S. Skiena, “The shape of word embeddings: Quantifying non-isometry with topological data analysis,” in <i>Findings of the Association for Computational Linguistics: EMNLP 2024</i>, Miami, FL, United States, 2024, pp. 12080–12099.","short":"O. Draganov, S. Skiena, in:, Findings of the Association for Computational Linguistics: EMNLP 2024, Association for Computational Linguistics, 2024, pp. 12080–12099."},"publisher":"Association for Computational Linguistics","status":"public","ddc":["500"],"date_created":"2025-02-04T16:19:28Z","day":"01","scopus_import":"1","language":[{"iso":"eng"}],"corr_author":"1","oa_version":"Published Version","_id":"18998","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","fulldoi":"https://doi.org/10.18653/v1/2024.findings-emnlp.705","conference":{"location":"Miami, FL, United States","end_date":"2024-11-16","name":"EMNLP: Conference on Empirical Methods in Natural Language Processing","start_date":"2024-11-12"},"tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"publication_status":"published","oa":1,"year":"2024","abstract":[{"lang":"eng","text":"Word embeddings represent language vocabularies as clouds of d-dimensional points. We investigate how information is conveyed by the general shape of these clouds, instead of representing the semantic meaning of each token. Specifically, we use the notion of persistent homology from topological data analysis (TDA) to measure the distances between language pairs from the shape of their unlabeled embeddings. These distances quantify the degree of non-isometry of the embeddings. To distinguish whether these differences are random training errors or capture real information about the languages, we use the computed distance matrices to construct language phylogenetic trees over 81 Indo-European languages. Careful evaluation shows that our reconstructed trees exhibit strong and statistically-significant similarities to the reference."}],"has_accepted_license":"1","title":"The shape of word embeddings: Quantifying non-isometry with topological data analysis","file":[{"creator":"dernst","relation":"main_file","success":1,"date_created":"2025-02-10T08:20:34Z","file_size":1312638,"content_type":"application/pdf","checksum":"f4416a5962194f0181ab0dc7f9ef93c0","access_level":"open_access","file_id":"19016","file_name":"2024_EMNLP_Draganov.pdf","date_updated":"2025-02-10T08:20:34Z"}],"month":"11","arxiv":1,"date_updated":"2025-02-10T08:21:37Z","publication":"Findings of the Association for Computational Linguistics: EMNLP 2024","type":"conference","OA_type":"gold","article_processing_charge":"No"},{"status":"public","citation":{"apa":"Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &#38; Saghafian, M. (n.d.). Chromatic topological data analysis. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/ARXIV.2406.04102\">https://doi.org/10.48550/ARXIV.2406.04102</a>","chicago":"Cultrera di Montesano, Sebastiano, Ondrej Draganov, Herbert Edelsbrunner, and Morteza Saghafian. “Chromatic Topological Data Analysis.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/ARXIV.2406.04102\">https://doi.org/10.48550/ARXIV.2406.04102</a>.","short":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian, ArXiv (n.d.).","ieee":"S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M. Saghafian, “Chromatic topological data analysis,” <i>arXiv</i>. .","ista":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic topological data analysis. arXiv, 2406.04102.","mla":"Cultrera di Montesano, Sebastiano, et al. “Chromatic Topological Data Analysis.” <i>ArXiv</i>, 2406.04102, doi:<a href=\"https://doi.org/10.48550/ARXIV.2406.04102\">10.48550/ARXIV.2406.04102</a>.","ama":"Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M. Chromatic topological data analysis. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/ARXIV.2406.04102\">10.48550/ARXIV.2406.04102</a>"},"author":[{"last_name":"Cultrera di Montesano","id":"34D2A09C-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0001-6249-0832","full_name":"Cultrera di Montesano, Sebastiano","first_name":"Sebastiano"},{"last_name":"Draganov","id":"2B23F01E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0003-0464-3823","full_name":"Draganov, Ondrej","first_name":"Ondrej"},{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert"},{"full_name":"Saghafian, Morteza","first_name":"Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"OA_place":"repository","doi":"10.48550/ARXIV.2406.04102","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"date_published":"2024-06-06T00:00:00Z","external_id":{"arxiv":["2406.04102"]},"publication_status":"submitted","oa":1,"tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"fulldoi":"https://doi.org/10.48550/ARXIV.2406.04102","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"18999","oa_version":"Preprint","language":[{"iso":"eng"}],"corr_author":"1","date_created":"2025-02-04T16:21:21Z","day":"06","ddc":["510"],"publication":"arXiv","month":"06","date_updated":"2025-02-10T08:14:27Z","arxiv":1,"has_accepted_license":"1","title":"Chromatic topological data analysis","abstract":[{"lang":"eng","text":"Exploring the shape of point configurations has been a key driver in the evolution of TDA (short for topological data analysis) since its infancy. This survey illustrates the recent efforts to broaden these ideas to model spatial interactions among multiple configurations, each distinguished by a color. It describes advances in this area and prepares the ground for further exploration by mentioning unresolved questions and promising research avenues while focusing on the overlap with discrete geometry."}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2406.04102"}],"year":"2024","article_processing_charge":"No","OA_type":"green","type":"preprint","article_number":"2406.04102"},{"_id":"17170","language":[{"iso":"eng"}],"oa_version":"Published Version","scopus_import":"1","day":"06","date_created":"2024-06-25T11:45:58Z","ddc":["516"],"ec_funded":1,"publication_status":"published","oa":1,"conference":{"name":"SoCG: Symposium on Computational Geometry","start_date":"2024-06-11","location":"Athens, Greece","end_date":"2024-06-14"},"tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","call_identifier":"FWF"},{"grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"},{"call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","grant_number":"I02979-N35"},{"grant_number":"M03073","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2","name":"Learning and triangulating manifolds via collapses"}],"volume":293,"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.11","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","department":[{"_id":"GradSch"},{"_id":"HeEd"}],"file_date_updated":"2024-06-25T11:47:26Z","doi":"10.4230/LIPIcs.SoCG.2024.11","external_id":{"arxiv":["2206.10485"]},"publication_identifier":{"isbn":["9783959773164"],"eissn":["1868-8969"]},"date_published":"2024-06-06T00:00:00Z","status":"public","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","citation":{"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.","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>.","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>","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>.","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.","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."},"quality_controlled":"1","page":"11:1-11:19","author":[{"full_name":"Attali, Dominique","first_name":"Dominique","last_name":"Attali"},{"first_name":"Hana","full_name":"Kourimska, Hana","id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","last_name":"Kourimska","orcid":"0000-0001-7841-0091"},{"last_name":"Fillmore","id":"35638A5C-AAC7-11E9-B0BF-5503E6697425","full_name":"Fillmore, Christopher D","first_name":"Christopher D"},{"full_name":"Ghosh, Ishika","first_name":"Ishika","last_name":"Ghosh","id":"ee449b28-344d-11ef-a6d5-9ca430e9e9ff"},{"full_name":"Lieutier, André","first_name":"André","last_name":"Lieutier"},{"first_name":"Elizabeth R","full_name":"Stephenson, Elizabeth R","id":"2D04F932-F248-11E8-B48F-1D18A9856A87","last_name":"Stephenson","orcid":"0000-0002-6862-208X"},{"first_name":"Mathijs","full_name":"Wintraecken, Mathijs","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","last_name":"Wintraecken","orcid":"0000-0002-7472-2220"}],"das_tickbox":"0","supplementarymaterial":"no","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.","article_processing_charge":"No","type":"conference","intvolume":"       293","title":"Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of euclidean spaces and of Riemannian manifolds","has_accepted_license":"1","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."}],"year":"2024","publication":"40th International Symposium on Computational Geometry","arxiv":1,"date_updated":"2026-10-02T11:23:43Z","researchdata_availability":"no","alternative_title":["LIPIcs"],"file":[{"checksum":"6a2ddc8b51aa58f197a8b294750f1f8d","content_type":"application/pdf","file_size":20886142,"date_created":"2024-06-25T11:47:26Z","success":1,"relation":"main_file","creator":"cfillmor","date_updated":"2024-06-25T11:47:26Z","file_name":"LIPIcs.SoCG.2024.11.pdf","file_id":"17171","access_level":"open_access"}],"month":"06"},{"year":"2024","abstract":[{"lang":"eng","text":"For a locally finite set in R2, the order-k Brillouin tessellations form an infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely dense and generic, then the corresponding infinite sequences of minimum and maximum angles are both monotonic in k. As an example, a stationary Poisson point process in R2  is locally finite, coarsely dense, and generic with probability one. For such a set, the distributions of angles in the Voronoi tessellations, Delaunay mosaics, and Brillouin tessellations are independent of the order and can be derived from the formula for angles in order-1 Delaunay mosaics given by Miles (Math. Biosci. 6, 85–127 (1970))."}],"has_accepted_license":"1","intvolume":"        72","title":"On angles in higher order Brillouin tessellations and related tilings in the plane","month":"07","file":[{"file_name":"2024_DiscreteComputGeom_Edelsbrunner.pdf","date_updated":"2024-07-22T09:43:19Z","access_level":"open_access","file_id":"17301","content_type":"application/pdf","checksum":"b207b4e00f904e8ea8a30e24f0251f79","date_created":"2024-07-22T09:43:19Z","file_size":892019,"success":1,"relation":"main_file","creator":"dernst"}],"date_updated":"2026-10-02T11:45:50Z","arxiv":1,"researchdata_availability":"no","publication":"Discrete and Computational Geometry","acknowledgement":"Work by all authors but A. Garber is 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. Work by A. Garber is partially supported by the Alexander von Humboldt Foundation.","supplementarymaterial":"yes","das_tickbox":"0","pmid":1,"type":"journal_article","article_processing_charge":"Yes (via OA deal)","date_published":"2024-07-01T00:00:00Z","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"external_id":{"isi":["001060727600004"],"arxiv":["2204.01076"],"pmid":["39610762"]},"article_type":"original","doi":"10.1007/s00454-023-00566-1","file_date_updated":"2024-07-22T09:43:19Z","department":[{"_id":"HeEd"}],"author":[{"full_name":"Edelsbrunner, Herbert","first_name":"Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833"},{"last_name":"Garber","first_name":"Alexey","full_name":"Garber, Alexey"},{"full_name":"Ghafari, Mohadese","first_name":"Mohadese","last_name":"Ghafari"},{"full_name":"Heiss, Teresa","first_name":"Teresa","last_name":"Heiss","id":"4879BB4E-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-1780-2689"},{"last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824","full_name":"Saghafian, Morteza","first_name":"Morteza"}],"page":"29-48","quality_controlled":"1","citation":{"chicago":"Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and Morteza Saghafian. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” <i>Discrete and Computational Geometry</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s00454-023-00566-1\">https://doi.org/10.1007/s00454-023-00566-1</a>.","apa":"Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., &#38; Saghafian, M. (2024). On angles in higher order Brillouin tessellations and related tilings in the plane. <i>Discrete and Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-023-00566-1\">https://doi.org/10.1007/s00454-023-00566-1</a>","ieee":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “On angles in higher order Brillouin tessellations and related tilings in the plane,” <i>Discrete and Computational Geometry</i>, vol. 72. Springer Nature, pp. 29–48, 2024.","short":"H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, Discrete and Computational Geometry 72 (2024) 29–48.","mla":"Edelsbrunner, Herbert, et al. “On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane.” <i>Discrete and Computational Geometry</i>, vol. 72, Springer Nature, 2024, pp. 29–48, doi:<a href=\"https://doi.org/10.1007/s00454-023-00566-1\">10.1007/s00454-023-00566-1</a>.","ista":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2024. On angles in higher order Brillouin tessellations and related tilings in the plane. Discrete and Computational Geometry. 72, 29–48.","ama":"Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. On angles in higher order Brillouin tessellations and related tilings in the plane. <i>Discrete and Computational Geometry</i>. 2024;72:29-48. doi:<a href=\"https://doi.org/10.1007/s00454-023-00566-1\">10.1007/s00454-023-00566-1</a>"},"isi":1,"publisher":"Springer Nature","status":"public","ddc":["510"],"ec_funded":1,"day":"01","date_created":"2023-09-17T22:01:10Z","scopus_import":"1","corr_author":"1","oa_version":"Published Version","language":[{"iso":"eng"}],"_id":"14345","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","fulldoi":"https://doi.org/10.1007/s00454-023-00566-1","tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"project":[{"call_identifier":"H2020","_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","grant_number":"788183"},{"grant_number":"Z00342","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science"},{"name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","grant_number":"I02979-N35"}],"volume":72,"oa":1,"publication_status":"published"},{"status":"public","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","citation":{"chicago":"Kourimska, Hana, André Lieutier, and Mathijs Wintraecken. “The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms.” In <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">https://doi.org/10.4230/LIPIcs.SoCG.2024.69</a>.","apa":"Kourimska, H., Lieutier, A., &#38; Wintraecken, M. (2024). The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">https://doi.org/10.4230/LIPIcs.SoCG.2024.69</a>","ieee":"H. Kourimska, A. Lieutier, and M. Wintraecken, “The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293.","short":"H. Kourimska, A. Lieutier, M. Wintraecken, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","mla":"Kourimska, Hana, et al. “The Medial Axis of Any Closed Bounded Set Is Lipschitz Stable with Respect to the Hausdorff Distance Under Ambient Diffeomorphisms.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 69, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">10.4230/LIPIcs.SoCG.2024.69</a>.","ista":"Kourimska H, Lieutier A, Wintraecken M. 2024. The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 69.","ama":"Kourimska H, Lieutier A, Wintraecken M. The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.69\">10.4230/LIPIcs.SoCG.2024.69</a>"},"quality_controlled":"1","author":[{"id":"D9B8E14C-3C26-11EA-98F5-1F833DDC885E","last_name":"Kourimska","orcid":"0000-0001-7841-0091","first_name":"Hana","full_name":"Kourimska, Hana"},{"first_name":"André","full_name":"Lieutier, André","last_name":"Lieutier"},{"last_name":"Wintraecken","id":"307CFBC8-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-7472-2220","full_name":"Wintraecken, Mathijs","first_name":"Mathijs"}],"department":[{"_id":"HeEd"}],"doi":"10.4230/LIPIcs.SoCG.2024.69","file_date_updated":"2024-06-17T08:33:40Z","publication_identifier":{"issn":["1868-8969"],"isbn":["9783959773164"]},"external_id":{"arxiv":["2212.01118"]},"date_published":"2024-06-01T00:00:00Z","publication_status":"published","oa":1,"tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"conference":{"location":"Athens, Greece","end_date":"2024-06-14","name":"SoCG: Symposium on Computational Geometry"},"volume":293,"project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020","grant_number":"788183"},{"grant_number":"Z00342","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science","call_identifier":"FWF"},{"call_identifier":"FWF","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","name":"Persistence and stability of geometric complexes","grant_number":"I02979-N35"},{"grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships","call_identifier":"H2020"},{"grant_number":"M03073","name":"Learning and triangulating manifolds via collapses","_id":"fc390959-9c52-11eb-aca3-afa58bd282b2"}],"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.69","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","_id":"17144","oa_version":"Published Version","language":[{"iso":"eng"}],"scopus_import":"1","date_created":"2024-06-16T22:01:06Z","day":"01","ec_funded":1,"ddc":["510"],"publication":"40th International Symposium on Computational Geometry","date_updated":"2026-10-02T11:46:39Z","researchdata_availability":"no","arxiv":1,"alternative_title":["LIPIcs"],"month":"06","file":[{"access_level":"open_access","file_id":"17150","file_name":"2024_LIPICS_Kourimska.pdf","date_updated":"2024-06-17T08:33:40Z","creator":"dernst","relation":"main_file","success":1,"date_created":"2024-06-17T08:33:40Z","file_size":1612558,"content_type":"application/pdf","checksum":"b40ff456c19294adb5d9613fcfd751c6"}],"title":"The medial axis of any closed bounded set Is Lipschitz stable with respect to the Hausdorff distance Under ambient diffeomorphisms","intvolume":"       293","has_accepted_license":"1","abstract":[{"text":"We prove that the medial axis of closed sets is Hausdorff stable in the following sense: Let 𝒮 ⊆ ℝ^d be a fixed closed set that contains a bounding sphere. That is, the bounding sphere is part of the set 𝒮. Consider the space of C^{1,1} diffeomorphisms of ℝ^d to itself, which keep the bounding sphere invariant. The map from this space of diffeomorphisms (endowed with a Banach norm) to the space of closed subsets of ℝ^d (endowed with the Hausdorff distance), mapping a diffeomorphism F to the closure of the medial axis of F(𝒮), is Lipschitz. This extends a previous stability result of Chazal and Soufflet on the stability of the medial axis of C² manifolds under C² ambient diffeomorphisms.","lang":"eng"}],"year":"2024","article_processing_charge":"No","type":"conference","article_number":"69","supplementarymaterial":"no","das_tickbox":"0","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\nSupported 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 Cô d'Azur.\r\nWe are greatly indebted to Fred Chazal for sharing his insights. We further thank Erin Chambers, Christopher Fillmore, and Elizabeth Stephenson for early discussions and all members of the Edelsbrunner group (Institute of Science and Technology Austria) and the Datashape team (Inria) for the atmosphere in which this research was conducted."},{"_id":"17146","scopus_import":"1","language":[{"iso":"eng"}],"oa_version":"Published Version","date_created":"2024-06-16T22:01:06Z","day":"01","ec_funded":1,"ddc":["510"],"oa":1,"publication_status":"published","tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"project":[{"_id":"266A2E9E-B435-11E9-9278-68D0E5697425","name":"Alpha Shape Theory Extended","call_identifier":"H2020","grant_number":"788183"},{"grant_number":"I02979-N35","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"},{"grant_number":"Z00342","call_identifier":"FWF","_id":"268116B8-B435-11E9-9278-68D0E5697425","name":"Mathematics, Computer Science"}],"volume":293,"conference":{"end_date":"2024-06-14","location":"Athens, Greece","start_date":"2024-06-11","name":"SoCG: Symposium on Computational Geometry"},"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.53","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","file_date_updated":"2024-06-17T08:46:33Z","doi":"10.4230/LIPIcs.SoCG.2024.53","department":[{"_id":"HeEd"}],"date_published":"2024-06-01T00:00:00Z","external_id":{"arxiv":["2310.14801"]},"publication_identifier":{"isbn":["9783959773164"],"issn":["1868-8969"]},"status":"public","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","quality_controlled":"1","citation":{"ieee":"H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293.","short":"H. Edelsbrunner, J. Pach, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","chicago":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” In <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">https://doi.org/10.4230/LIPIcs.SoCG.2024.53</a>.","apa":"Edelsbrunner, H., &#38; Pach, J. (2024). Maximum Betti numbers of Čech complexes. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">https://doi.org/10.4230/LIPIcs.SoCG.2024.53</a>","ama":"Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">10.4230/LIPIcs.SoCG.2024.53</a>","mla":"Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 53, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.53\">10.4230/LIPIcs.SoCG.2024.53</a>.","ista":"Edelsbrunner H, Pach J. 2024. Maximum Betti numbers of Čech complexes. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 53."},"author":[{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","first_name":"Herbert"},{"id":"E62E3130-B088-11EA-B919-BF823C25FEA4","last_name":"Pach","first_name":"János","full_name":"Pach, János"}],"article_number":"53","das_tickbox":"0","supplementarymaterial":"no","acknowledgement":"The first author is supported by the European Research Council (ERC), grant no. 788183, and by the DFG Collaborative Research Center TRR 109, Austrian Science Fund (FWF), grant no. {I 02979-N35.} The second author is supported by the European Research Council (ERC), grant \"GeoScape\" and by the Hungarian Science Foundation (NKFIH), grant K-131529. Both authors are supported by the Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31.\r\nThe authors thank Matt Kahle for communicating the question about extremal Čech complexes, Ben Schweinhart for early discussions on the linked circles construction in three dimensions, and Gábor Tardos for helpful remarks and suggestions.","article_processing_charge":"No","type":"conference","has_accepted_license":"1","intvolume":"       293","title":"Maximum Betti numbers of Čech complexes","related_material":{"record":[{"relation":"later_version","status":"public","id":"20657"}]},"abstract":[{"lang":"eng","text":"The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in ℝ^d and any radius satisfies β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions, which prove that this upper bound is asymptotically tight. For example, we describe a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of the Čech complex at the other radius is n². In particular, there is an arrangement of n contruent balls in ℝ³ that enclose a quadratic number of voids, which answers a long-standing open question in computational geometry."}],"year":"2024","publication":"40th International Symposium on Computational Geometry","file":[{"access_level":"open_access","file_id":"17152","file_name":"2024_LIPICS_Edelsbrunner.pdf","date_updated":"2024-06-17T08:46:33Z","creator":"dernst","relation":"main_file","success":1,"date_created":"2024-06-17T08:46:33Z","file_size":766562,"checksum":"5442d44fb89d77477a87668d6e61aac9","content_type":"application/pdf"}],"month":"06","alternative_title":["LIPIcs"],"arxiv":1,"date_updated":"2026-10-02T11:48:01Z","researchdata_availability":"no"},{"status":"public","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","quality_controlled":"1","citation":{"short":"G. Rote, M. Rüber, M. Saghafian, in:, 40th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024.","ieee":"G. Rote, M. Rüber, and M. Saghafian, “Grid peeling of parabolas,” in <i>40th International Symposium on Computational Geometry</i>, Athens, Greece, 2024, vol. 293.","apa":"Rote, G., Rüber, M., &#38; Saghafian, M. (2024). Grid peeling of parabolas. In <i>40th International Symposium on Computational Geometry</i> (Vol. 293). Athens, Greece: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.76\">https://doi.org/10.4230/LIPIcs.SoCG.2024.76</a>","chicago":"Rote, Günter, Moritz Rüber, and Morteza Saghafian. “Grid Peeling of Parabolas.” In <i>40th International Symposium on Computational Geometry</i>, Vol. 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.76\">https://doi.org/10.4230/LIPIcs.SoCG.2024.76</a>.","ama":"Rote G, Rüber M, Saghafian M. Grid peeling of parabolas. In: <i>40th International Symposium on Computational Geometry</i>. Vol 293. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2024. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.76\">10.4230/LIPIcs.SoCG.2024.76</a>","ista":"Rote G, Rüber M, Saghafian M. 2024. Grid peeling of parabolas. 40th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 293, 76.","mla":"Rote, Günter, et al. “Grid Peeling of Parabolas.” <i>40th International Symposium on Computational Geometry</i>, vol. 293, 76, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2024.76\">10.4230/LIPIcs.SoCG.2024.76</a>."},"author":[{"last_name":"Rote","full_name":"Rote, Günter","first_name":"Günter"},{"first_name":"Moritz","full_name":"Rüber, Moritz","last_name":"Rüber"},{"full_name":"Saghafian, Morteza","first_name":"Morteza","last_name":"Saghafian","id":"f86f7148-b140-11ec-9577-95435b8df824"}],"file_date_updated":"2024-06-17T08:40:04Z","doi":"10.4230/LIPIcs.SoCG.2024.76","department":[{"_id":"HeEd"}],"date_published":"2024-06-01T00:00:00Z","external_id":{"arxiv":["2402.15787"]},"publication_identifier":{"isbn":["9783959773164"],"issn":["1868-8969"]},"oa":1,"publication_status":"published","tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"conference":{"name":"SoCG: Symposium on Computational Geometry","start_date":"2024-06-11","location":"Athens, Greece","end_date":"2024-06-14"},"volume":293,"fulldoi":"https://doi.org/10.4230/LIPIcs.SoCG.2024.76","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","_id":"17145","scopus_import":"1","language":[{"iso":"eng"}],"oa_version":"Published Version","day":"01","date_created":"2024-06-16T22:01:06Z","ddc":["510"],"publication":"40th International Symposium on Computational Geometry","file":[{"file_name":"2024_LIPICS_Rote.pdf","date_updated":"2024-06-17T08:40:04Z","access_level":"open_access","file_id":"17151","checksum":"fbad1de06383a6b7e8a1cb3e8c7205ce","content_type":"application/pdf","date_created":"2024-06-17T08:40:04Z","file_size":1430896,"relation":"main_file","success":1,"creator":"dernst"}],"month":"06","alternative_title":["LIPIcs"],"arxiv":1,"date_updated":"2026-10-02T11:47:18Z","researchdata_availability":"no","has_accepted_license":"1","title":"Grid peeling of parabolas","intvolume":"       293","abstract":[{"text":"Grid peeling is the process of repeatedly removing the convex hull vertices of the grid points that lie inside a given convex curve. It has been conjectured that, for a more and more refined grid, grid peeling converges to a continuous process, the affine curve-shortening flow, which deforms the curve based on the curvature. We prove this conjecture for one class of curves, parabolas with a vertical axis, and we determine the value of the constant factor in the formula that relates the two processes.","lang":"eng"}],"year":"2024","article_processing_charge":"No","type":"conference","article_number":"76","das_tickbox":"0","supplementarymaterial":"no","acknowledgement":"Part of this work was done while G.R. enjoyed the hospitality of the Institute of Science and Technology Austria (ISTA) as a visiting professor during his sabbatical in the winter semester 2022/23."}]