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**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

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.

Herbert Edelsbrunner and János Pach. Maximum Betti Numbers of Čech Complexes. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 53:1-53:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{edelsbrunner_et_al:LIPIcs.SoCG.2024.53, author = {Edelsbrunner, Herbert and Pach, J\'{a}nos}, title = {{Maximum Betti Numbers of \v{C}ech Complexes}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {53:1--53:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.53}, URN = {urn:nbn:de:0030-drops-199981}, doi = {10.4230/LIPIcs.SoCG.2024.53}, annote = {Keywords: Discrete geometry, computational topology, \v{C}ech complexes, Delaunay mosaics, Alpha complexes, Betti numbers, extremal questions} }

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**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Generalizing Lee’s inductive argument for counting the cells of higher order Voronoi tessellations in ℝ² to ℝ³, we get precise relations in terms of Morse theoretic quantities for piecewise constant functions on planar arrangements. Specifically, we prove that for a generic set of n ≥ 5 points in ℝ³, the number of regions in the order-k Voronoi tessellation is N_{k-1} - binom(k,2)n + n, for 1 ≤ k ≤ n-1, in which N_{k-1} is the sum of Euler characteristics of these function’s first k-1 sublevel sets. We get similar expressions for the vertices, edges, and polygons of the order-k Voronoi tessellation.

Ranita Biswas, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner, and Morteza Saghafian. Counting Cells of Order-k Voronoi Tessellations in ℝ³ with Morse Theory. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{biswas_et_al:LIPIcs.SoCG.2021.16, author = {Biswas, Ranita and Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Saghafian, Morteza}, title = {{Counting Cells of Order-k Voronoi Tessellations in \mathbb{R}³ with Morse Theory}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {16:1--16:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.16}, URN = {urn:nbn:de:0030-drops-138152}, doi = {10.4230/LIPIcs.SoCG.2021.16}, annote = {Keywords: Voronoi tessellations, Delaunay mosaics, arrangements, convex polytopes, Morse theory, counting} }

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**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functions that facilitates the efficient search for new materials and material properties. We prove invariance under isometries, continuity, and completeness in the generic case, which are necessary features for the reliable comparison of crystals. The proof of continuity integrates methods from discrete geometry and lattice theory, while the proof of generic completeness combines techniques from geometry with analysis. The fingerprint has a fast algorithm based on Brillouin zones and related inclusion-exclusion formulae. We have implemented the algorithm and describe its application to crystal structure prediction.

Herbert Edelsbrunner, Teresa Heiss, Vitaliy Kurlin, Philip Smith, and Mathijs Wintraecken. The Density Fingerprint of a Periodic Point Set. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 32:1-32:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{edelsbrunner_et_al:LIPIcs.SoCG.2021.32, author = {Edelsbrunner, Herbert and Heiss, Teresa and Kurlin, Vitaliy and Smith, Philip and Wintraecken, Mathijs}, title = {{The Density Fingerprint of a Periodic Point Set}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {32:1--32:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.32}, URN = {urn:nbn:de:0030-drops-138310}, doi = {10.4230/LIPIcs.SoCG.2021.32}, annote = {Keywords: Lattices, periodic sets, isometries, Dirichlet-Voronoi domains, Brillouin zones, bottleneck distance, stability, continuity, crystal database} }

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**Published in:** LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms. Importantly, instead of using the standard Euclidean distance, we look into dissimilarity measures with information-theoretic justification, and we develop the theory needed for applying topological data analysis in this setting. In doing so, we emphasize constructions that enable the usage of existing computational topology software in this context.

Herbert Edelsbrunner, Žiga Virk, and Hubert Wagner. Topological Data Analysis in Information Space. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{edelsbrunner_et_al:LIPIcs.SoCG.2019.31, author = {Edelsbrunner, Herbert and Virk, \v{Z}iga and Wagner, Hubert}, title = {{Topological Data Analysis in Information Space}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {31:1--31:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.31}, URN = {urn:nbn:de:0030-drops-104357}, doi = {10.4230/LIPIcs.SoCG.2019.31}, annote = {Keywords: Computational topology, persistent homology, information theory, entropy} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Given a locally finite X subseteq R^d and a radius r >= 0, the k-fold cover of X and r consists of all points in R^d that have k or more points of X within distance r. We consider two filtrations - one in scale obtained by fixing k and increasing r, and the other in depth obtained by fixing r and decreasing k - and we compute the persistence diagrams of both. While standard methods suffice for the filtration in scale, we need novel geometric and topological concepts for the filtration in depth. In particular, we introduce a rhomboid tiling in R^{d+1} whose horizontal integer slices are the order-k Delaunay mosaics of X, and construct a zigzag module from Delaunay mosaics that is isomorphic to the persistence module of the multi-covers.

Herbert Edelsbrunner and Georg Osang. The Multi-cover Persistence of Euclidean Balls. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{edelsbrunner_et_al:LIPIcs.SoCG.2018.34, author = {Edelsbrunner, Herbert and Osang, Georg}, title = {{The Multi-cover Persistence of Euclidean Balls}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {34:1--34:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.34}, URN = {urn:nbn:de:0030-drops-87471}, doi = {10.4230/LIPIcs.SoCG.2018.34}, annote = {Keywords: Delaunay mosaics, hyperplane arrangements, discrete Morse theory, zigzag modules, persistent homology} }

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**Published in:** LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Smallest enclosing spheres of finite point sets are central to methods in topological data analysis. Focusing on Bregman divergences to measure dissimilarity, we prove bounds on the location of the center of a smallest enclosing sphere. These bounds depend on the range of radii for which Bregman balls are convex.

Herbert Edelsbrunner, Ziga Virk, and Hubert Wagner. Smallest Enclosing Spheres and Chernoff Points in BregmanGeometry. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 35:1-35:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{edelsbrunner_et_al:LIPIcs.SoCG.2018.35, author = {Edelsbrunner, Herbert and Virk, Ziga and Wagner, Hubert}, title = {{Smallest Enclosing Spheres and Chernoff Points in BregmanGeometry}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {35:1--35:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.35}, URN = {urn:nbn:de:0030-drops-87487}, doi = {10.4230/LIPIcs.SoCG.2018.35}, annote = {Keywords: Bregman divergence, smallest enclosing spheres, Chernoff points, convexity, barycenter polytopes} }

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**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

We show that the framework of topological data analysis can be extended from metrics to general Bregman divergences, widening the scope of possible applications. Examples are the Kullback-Leibler divergence, which is commonly used for comparing text and images, and the Itakura-Saito divergence, popular for speech and sound. In particular, we prove that appropriately generalized Cech and Delaunay (alpha) complexes capture the correct homotopy type, namely that of the corresponding union of Bregman balls. Consequently, their filtrations give the correct persistence diagram, namely the one generated by the uniformly growing Bregman balls. Moreover, we show that unlike the metric setting, the filtration of Vietoris-Rips complexes may fail to approximate the persistence diagram. We propose algorithms to compute the thus generalized Cech, Vietoris-Rips and Delaunay complexes and experimentally test their efficiency. Lastly, we explain their surprisingly good performance by making a connection with discrete Morse theory.

Herbert Edelsbrunner and Hubert Wagner. Topological Data Analysis with Bregman Divergences. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 39:1-39:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{edelsbrunner_et_al:LIPIcs.SoCG.2017.39, author = {Edelsbrunner, Herbert and Wagner, Hubert}, title = {{Topological Data Analysis with Bregman Divergences}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {39:1--39:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.39}, URN = {urn:nbn:de:0030-drops-71985}, doi = {10.4230/LIPIcs.SoCG.2017.39}, annote = {Keywords: Topological data analysis, Bregman divergences, persistent homology, proximity complexes, algorithms} }

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