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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

We study the decomposition of zero-dimensional persistence modules, viewed as functors valued in the category of vector spaces factorizing through sets. Instead of working directly at the level of vector spaces, we take a step back and first study the decomposition problem at the level of sets.
This approach allows us to define the combinatorial notion of rooted subsets. In the case of a filtered metric space M, rooted subsets relate the clustering behavior of the points of M with the decomposition of the associated persistence module. In particular, we can identify intervals in such a decomposition quickly. In addition, rooted subsets can be understood as a generalization of the elder rule, and are also related to the notion of constant conqueror of Cai, Kim, Mémoli and Wang. As an application, we give a lower bound on the number of intervals that we can expect in the decomposition of zero-dimensional persistence modules of a density-Rips filtration in Euclidean space: in the limit, and under very general circumstances, we can expect that at least 25% of the indecomposable summands are interval modules.

Ángel Javier Alonso and Michael Kerber. Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{alonso_et_al:LIPIcs.SoCG.2023.7, author = {Alonso, \'{A}ngel Javier and Kerber, Michael}, title = {{Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {7:1--7:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.7}, URN = {urn:nbn:de:0030-drops-178570}, doi = {10.4230/LIPIcs.SoCG.2023.7}, annote = {Keywords: Multiparameter persistent homology, Clustering, Decomposition of persistence modules, Elder Rule} }

Document

**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

For a finite set of balls of radius r, the k-fold cover is the space covered by at least k balls. Fixing the ball centers and varying the radius, we obtain a nested sequence of spaces that is called the k-fold filtration of the centers. For k = 1, the construction is the union-of-balls filtration that is popular in topological data analysis. For larger k, it yields a cleaner shape reconstruction in the presence of outliers. We contribute a sparsification algorithm to approximate the topology of the k-fold filtration. Our method is a combination and adaptation of several techniques from the well-studied case k = 1, resulting in a sparsification of linear size that can be computed in expected near-linear time with respect to the number of input points.

Mickaël Buchet, Bianca B. Dornelas, and Michael Kerber. Sparse Higher Order Čech Filtrations. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{buchet_et_al:LIPIcs.SoCG.2023.20, author = {Buchet, Micka\"{e}l and B. Dornelas, Bianca and Kerber, Michael}, title = {{Sparse Higher Order \v{C}ech Filtrations}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {20:1--20:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.20}, URN = {urn:nbn:de:0030-drops-178709}, doi = {10.4230/LIPIcs.SoCG.2023.20}, annote = {Keywords: Sparsification, k-fold cover, Higher order \v{C}ech complexes} }

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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

We propose an extension of the classical union-of-balls filtration of persistent homology: fixing a point q, we focus our attention to a ball centered at q whose radius is controlled by a second scale parameter. We discuss an absolute variant, where the union is just restricted to the q-ball, and a relative variant where the homology of the q-ball relative to its boundary is considered. Interestingly, these natural constructions lead to bifiltered simplicial complexes which are not k-critical for any finite k. Nevertheless, we demonstrate that these bifiltrations can be computed exactly and efficiently, and we provide a prototypical implementation using the CGAL library. We also argue that some of the recent algorithmic advances for 2-parameter persistence (which usually assume k-criticality for some finite k) carry over to the ∞-critical case.

Michael Kerber and Matthias Söls. The Localized Union-Of-Balls Bifiltration. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 45:1-45:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kerber_et_al:LIPIcs.SoCG.2023.45, author = {Kerber, Michael and S\"{o}ls, Matthias}, title = {{The Localized Union-Of-Balls Bifiltration}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {45:1--45:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.45}, URN = {urn:nbn:de:0030-drops-178953}, doi = {10.4230/LIPIcs.SoCG.2023.45}, annote = {Keywords: Topological Data Analysis, Multi-Parameter Persistence, Persistent Local Homology} }

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Complete Volume

**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

LIPIcs, Volume 224, SoCG 2022, Complete Volume

38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 1-1110, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Proceedings{goaoc_et_al:LIPIcs.SoCG.2022, title = {{LIPIcs, Volume 224, SoCG 2022, Complete Volume}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {1--1110}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022}, URN = {urn:nbn:de:0030-drops-160075}, doi = {10.4230/LIPIcs.SoCG.2022}, annote = {Keywords: LIPIcs, Volume 224, SoCG 2022, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Front Matter, Table of Contents, Preface, Conference Organization

38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 0:i-0:xx, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{goaoc_et_al:LIPIcs.SoCG.2022.0, author = {Goaoc, Xavier and Kerber, Michael}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {0:i--0:xx}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.0}, URN = {urn:nbn:de:0030-drops-160087}, doi = {10.4230/LIPIcs.SoCG.2022.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

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

Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow larger when r increases or k decreases, called the multicover bifiltration. Motivated by the problem of computing the homology of this bifiltration, we introduce two closely related combinatorial bifiltrations, one polyhedral and the other simplicial, which are both topologically equivalent to the multicover bifiltration and far smaller than a Čech-based model considered in prior work of Sheehy. Our polyhedral construction is a bifiltration of the rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using a variant of an algorithm given by these authors as well. Using an implementation for dimension 2 and 3, we provide experimental results. Our simplicial construction is useful for understanding the polyhedral construction and proving its correctness.

René Corbet, Michael Kerber, Michael Lesnick, and Georg Osang. Computing the Multicover Bifiltration. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{corbet_et_al:LIPIcs.SoCG.2021.27, author = {Corbet, Ren\'{e} and Kerber, Michael and Lesnick, Michael and Osang, Georg}, title = {{Computing the Multicover Bifiltration}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {27:1--27:17}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.27}, URN = {urn:nbn:de:0030-drops-138260}, doi = {10.4230/LIPIcs.SoCG.2021.27}, annote = {Keywords: Bifiltrations, nerves, higher-order Delaunay complexes, higher-order Voronoi diagrams, rhomboid tiling, multiparameter persistent homology, denoising} }

Document

**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

In topological data analysis, the matching distance is a computationally tractable metric on multi-filtered simplicial complexes. We design efficient algorithms for approximating the matching distance of two bi-filtered complexes to any desired precision ε>0. Our approach is based on a quad-tree refinement strategy introduced by Biasotti et al., but we recast their approach entirely in geometric terms. This point of view leads to several novel observations resulting in a practically faster algorithm. We demonstrate this speed-up by experimental comparison and provide our code in a public repository which provides the first efficient publicly available implementation of the matching distance.

Michael Kerber and Arnur Nigmetov. Efficient Approximation of the Matching Distance for 2-Parameter Persistence. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 53:1-53:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kerber_et_al:LIPIcs.SoCG.2020.53, author = {Kerber, Michael and Nigmetov, Arnur}, title = {{Efficient Approximation of the Matching Distance for 2-Parameter Persistence}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {53:1--53:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.53}, URN = {urn:nbn:de:0030-drops-122116}, doi = {10.4230/LIPIcs.SoCG.2020.53}, annote = {Keywords: multi-parameter persistence, matching distance, approximation algorithm} }

Document

**Published in:** Dagstuhl Reports, Volume 9, Issue 5 (2019)

This report documents the program and the outcomes of Dagstuhl Seminar 19212 "Topology, Computation and Data Analysis". The seminar brought together researchers with mathematical and computational backgrounds in addressing emerging directions within computational topology for data analysis in practice. This seminar was designed to be a followup event after a very successful Dagstuhl Seminar (17292; July 2017). The list of topics and participants were updated to keep the discussions diverse, refreshing, and engaging. This seminar facilitated close interactions among the attendees with the aim of accelerating the convergence between mathematical and computational thinking in the development of theories and scalable algorithms for data analysis.

Michael Kerber, Vijay Natarajan, and Bei Wang. Topology, Computation and Data Analysis (Dagstuhl Seminar 19212). In Dagstuhl Reports, Volume 9, Issue 5, pp. 110-131, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Article{kerber_et_al:DagRep.9.5.110, author = {Kerber, Michael and Natarajan, Vijay and Wang, Bei}, title = {{Topology, Computation and Data Analysis (Dagstuhl Seminar 19212)}}, pages = {110--131}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2019}, volume = {9}, number = {5}, editor = {Kerber, Michael and Natarajan, Vijay and Wang, Bei}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.9.5.110}, URN = {urn:nbn:de:0030-drops-113832}, doi = {10.4230/DagRep.9.5.110}, annote = {Keywords: computational topology, topological data analysis, Topology based visualization} }

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

The extension of persistent homology to multi-parameter setups is an algorithmic challenge. Since most computation tasks scale badly with the size of the input complex, an important pre-processing step consists of simplifying the input while maintaining the homological information. We present an algorithm that drastically reduces the size of an input. Our approach is an extension of the chunk algorithm for persistent homology (Bauer et al., Topological Methods in Data Analysis and Visualization III, 2014). We show that our construction produces the smallest multi-filtered chain complex among all the complexes quasi-isomorphic to the input, improving on the guarantees of previous work in the context of discrete Morse theory. Our algorithm also offers an immediate parallelization scheme in shared memory. Already its sequential version compares favorably with existing simplification schemes, as we show by experimental evaluation.

Ulderico Fugacci and Michael Kerber. Chunk Reduction for Multi-Parameter Persistent Homology. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 37:1-37:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fugacci_et_al:LIPIcs.SoCG.2019.37, author = {Fugacci, Ulderico and Kerber, Michael}, title = {{Chunk Reduction for Multi-Parameter Persistent Homology}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {37:1--37:14}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.37}, URN = {urn:nbn:de:0030-drops-104413}, doi = {10.4230/LIPIcs.SoCG.2019.37}, annote = {Keywords: Multi-parameter persistent homology, Matrix reduction, Chain complexes} }

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

The matching distance is a pseudometric on multi-parameter persistence modules, defined in terms of the weighted bottleneck distance on the restriction of the modules to affine lines. It is known that this distance is stable in a reasonable sense, and can be efficiently approximated, which makes it a promising tool for practical applications. In this work, we show that in the 2-parameter setting, the matching distance can be computed exactly in polynomial time. Our approach subdivides the space of affine lines into regions, via a line arrangement. In each region, the matching distance restricts to a simple analytic function, whose maximum is easily computed. As a byproduct, our analysis establishes that the matching distance is a rational number, if the bigrades of the input modules are rational.

Michael Kerber, Michael Lesnick, and Steve Oudot. Exact Computation of the Matching Distance on 2-Parameter Persistence Modules. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kerber_et_al:LIPIcs.SoCG.2019.46, author = {Kerber, Michael and Lesnick, Michael and Oudot, Steve}, title = {{Exact Computation of the Matching Distance on 2-Parameter Persistence Modules}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {46:1--46:15}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.46}, URN = {urn:nbn:de:0030-drops-104505}, doi = {10.4230/LIPIcs.SoCG.2019.46}, annote = {Keywords: Topological Data Analysis, Multi-Parameter Persistence, Line arrangements} }

Document

**Published in:** Dagstuhl Reports, Volume 7, Issue 7 (2018)

This report documents the program and the outcomes of Dagstuhl Seminar 17292 "Topology, Computation and Data Analysis".
This seminar was the first of its kind in bringing together researchers with mathematical and computational backgrounds in addressing emerging directions within computational topology for data analysis in practice. The seminar connected pure and applied mathematicians, with theoretical and applied computer scientists with an interest in computational topology. It helped to facilitate interactions among data theorist and data practitioners from several communities to address challenges in computational topology, topological data analysis, and topological visualization.

Hamish Carr, Michael Kerber, and Bei Wang. Topology, Computation and Data Analysis (Dagstuhl Seminar 17292). In Dagstuhl Reports, Volume 7, Issue 7, pp. 88-109, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@Article{carr_et_al:DagRep.7.7.88, author = {Carr, Hamish and Kerber, Michael and Wang, Bei}, title = {{Topology, Computation and Data Analysis (Dagstuhl Seminar 17292)}}, pages = {88--109}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2018}, volume = {7}, number = {7}, editor = {Carr, Hamish and Kerber, Michael and Wang, Bei}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.7.7.88}, URN = {urn:nbn:de:0030-drops-84258}, doi = {10.4230/DagRep.7.7.88}, annote = {Keywords: computational topology, topological data analysis, Topological data visualization} }

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**Published in:** LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)

Rips complexes are important structures for analyzing topological features of metric spaces. Unfortunately, generating these complexes constitutes an expensive task because of a combinatorial explosion in the complex size. For n points in R^d, we present a scheme to construct a 4.24-approximation of the multi-scale filtration of the Rips complex in the L-infinity metric, which extends to a O(d^{0.25})-approximation of the Rips filtration for the Euclidean case. The k-skeleton of the resulting approximation has a total size of n2^{O(d log k)}. The scheme is based on the integer lattice and on the barycentric subdivision of the d-cube.

Aruni Choudhary, Michael Kerber, and Sharath Raghvendra. Improved Approximate Rips Filtrations with Shifted Integer Lattices. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{choudhary_et_al:LIPIcs.ESA.2017.28, author = {Choudhary, Aruni and Kerber, Michael and Raghvendra, Sharath}, title = {{Improved Approximate Rips Filtrations with Shifted Integer Lattices}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {28:1--28:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.28}, URN = {urn:nbn:de:0030-drops-78259}, doi = {10.4230/LIPIcs.ESA.2017.28}, annote = {Keywords: Persistent homology, Rips filtrations, Approximation algorithms, Topological Data Analysis} }

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

Given a polytope P in R^d and a subset U of its vertices, is there a triangulation of P using d-simplices that all contain U? We answer this question by proving an equivalent and easy-to-check combinatorial criterion for the facets of P. Our proof relates triangulations of P to triangulations of its "shadow", a projection to a lower-dimensional space determined by U. In particular, we obtain a formula relating the volume of P with the volume of its shadow. This leads to an exact formula for the volume of a polytope arising in the theory of unit equations.

Michael Kerber, Robert Tichy, and Mario Weitzer. Constrained Triangulations, Volumes of Polytopes, and Unit Equations. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kerber_et_al:LIPIcs.SoCG.2017.46, author = {Kerber, Michael and Tichy, Robert and Weitzer, Mario}, title = {{Constrained Triangulations, Volumes of Polytopes, and Unit Equations}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {46:1--46:15}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.46}, URN = {urn:nbn:de:0030-drops-71812}, doi = {10.4230/LIPIcs.SoCG.2017.46}, annote = {Keywords: constrained triangulations, simplotopes, volumes of polytopes, projections of polytopes, unit equations, S-integers} }

Document

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

A tower is a sequence of simplicial complexes connected by simplicial maps. We show how to compute a filtration, a sequence of nested simplicial complexes, with the same persistent barcode as the tower. Our approach is based on the coning strategy by Dey et al. (SoCG 2014). We show that a variant of this approach yields a filtration that is asymptotically only marginally larger than the tower and can be efficiently computed by a streaming algorithm, both in theory and in practice. Furthermore, we show that our approach can be combined with a streaming algorithm to compute the barcode of the tower via matrix reduction. The space complexity of the algorithm does not depend on the length of the tower, but the maximal size of any subcomplex within the tower. Experimental evaluations show that our approach can efficiently handle towers with billions of complexes.

Michael Kerber and Hannah Schreiber. Barcodes of Towers and a Streaming Algorithm for Persistent Homology. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 57:1-57:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kerber_et_al:LIPIcs.SoCG.2017.57, author = {Kerber, Michael and Schreiber, Hannah}, title = {{Barcodes of Towers and a Streaming Algorithm for Persistent Homology}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {57:1--57:16}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.57}, URN = {urn:nbn:de:0030-drops-71936}, doi = {10.4230/LIPIcs.SoCG.2017.57}, annote = {Keywords: Persistent Homology, Topological Data Analysis, Matrix reduction, Streaming algorithms, Simplicial Approximation} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Classical methods to model topological properties of point clouds, such as the Vietoris-Rips complex, suffer from the combinatorial explosion of complex sizes. We propose a novel technique to approximate a multi-scale filtration of the Rips complex with improved bounds for size: precisely, for n points in R^d, we obtain a O(d)-approximation with at most n2^{O(d log k)} simplices of dimension k or lower. In conjunction with dimension reduction techniques, our approach yields a O(polylog (n))-approximation of size n^{O(1)} for Rips filtrations on arbitrary metric spaces. This result stems from high-dimensional lattice geometry and exploits properties of the permutahedral lattice, a well-studied structure in discrete geometry.
Building on the same geometric concept, we also present a lower bound result on the size of an approximate filtration: we construct a point set for which every (1+epsilon)-approximation of the Cech filtration has to contain n^{Omega(log log n)} features, provided that epsilon < 1/(log^{1+c}n) for c in (0,1).

Aruni Choudhary, Michael Kerber, and Sharath Raghvendra. Polynomial-Sized Topological Approximations Using the Permutahedron. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 31:1-31:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{choudhary_et_al:LIPIcs.SoCG.2016.31, author = {Choudhary, Aruni and Kerber, Michael and Raghvendra, Sharath}, title = {{Polynomial-Sized Topological Approximations Using the Permutahedron}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {31:1--31:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.31}, URN = {urn:nbn:de:0030-drops-59236}, doi = {10.4230/LIPIcs.SoCG.2016.31}, annote = {Keywords: Persistent Homology, Topological Data Analysis, Simplicial Approximation, Permutahedron, Approximation Algorithms} }

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