Pub Date : 2023-10-10DOI: 10.1007/s41468-023-00145-6
Sara Kališnik, Davorin Lešnik
Abstract A standard problem in applied topology is how to discover topological invariants of data from a noisy point cloud that approximates it. We consider the case where a sample is drawn from a properly embedded "Equation missing"-submanifold without boundary in a Euclidean space. We show that we can deformation retract the union of ellipsoids, centered at sample points and stretching in the tangent directions, to the manifold. Hence the homotopy type, and therefore also the homology type, of the manifold is the same as that of the nerve complex of the cover by ellipsoids. By thickening sample points to ellipsoids rather than balls, our results require a smaller sample density than comparable results in the literature. They also advocate using elongated shapes in the construction of barcodes in persistent homology.
{"title":"Finding the homology of manifolds using ellipsoids","authors":"Sara Kališnik, Davorin Lešnik","doi":"10.1007/s41468-023-00145-6","DOIUrl":"https://doi.org/10.1007/s41468-023-00145-6","url":null,"abstract":"Abstract A standard problem in applied topology is how to discover topological invariants of data from a noisy point cloud that approximates it. We consider the case where a sample is drawn from a properly embedded \"Equation missing\"<!-- image only, no MathML or LaTex -->-submanifold without boundary in a Euclidean space. We show that we can deformation retract the union of ellipsoids, centered at sample points and stretching in the tangent directions, to the manifold. Hence the homotopy type, and therefore also the homology type, of the manifold is the same as that of the nerve complex of the cover by ellipsoids. By thickening sample points to ellipsoids rather than balls, our results require a smaller sample density than comparable results in the literature. They also advocate using elongated shapes in the construction of barcodes in persistent homology.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136354168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-07DOI: 10.1007/s41468-023-00138-5
Facundo Mémoli, Anastasios Stefanou, Ling Zhou
Abstract One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and by utilizing cohomological ideas, e.g. the cohomological cup product. In this work, given a single parameter filtration, we investigate a certain 2-dimensional persistence module structure associated with persistent cohomology, where one parameter is the cup-length $$ell ge 0$$ ℓ≥0 and the other is the filtration parameter. This new persistence structure, called the persistent cup module , is induced by the cohomological cup product and adapted to the persistence setting. Furthermore, we show that this persistence structure is stable. By fixing the cup-length parameter $$ell $$ ℓ , we obtain a 1-dimensional persistence module, called the persistent $$ell $$ ℓ -cup module, and again show it is stable in the interleaving distance sense, and study their associated generalized persistence diagrams. In addition, we consider a generalized notion of a persistent invariant , which extends both the rank invariant (also referred to as persistent Betti number ), Puuska’s rank invariant induced by epi-mono-preserving invariants of abelian categories, and the recently-defined persistent cup-length invariant , and we establish their stability. This generalized notion of persistent invariant also enables us to lift the Lyusternik-Schnirelmann (LS) category of topological spaces to a novel stable persistent invariant of filtrations, called the persistent LS-category invariant .
一维持久同调是拓扑数据分析中最重要和最常用的计算工具。通过研究多维持久模块和利用上同调思想(如上同调杯积)，可以从数据集中提取额外的信息。在给定单参数过滤的情况下，我们研究了一类与持久上同调相关的二维持久化模块结构，其中一个参数为杯长$$ell ge 0$$，另一个参数为过滤参数。这种新的持久化结构称为持久化杯模块，由上同源杯产品引起，并适应于持久化设置。此外，我们还证明了这种持久性结构是稳定的。通过固定杯子长度参数$$ell $$，我们得到了一个一维的持久模，称为持久的$$ell $$ -杯子模，并再次证明了它在交错距离意义上是稳定的，并研究了它们相关的广义持久图。此外，我们考虑了持久不变量的广义概念，它扩展了秩不变量(也称为持久Betti数)、由阿贝尔范畴的外单保持不变量导出的Puuska秩不变量以及最近定义的持久杯长不变量，并建立了它们的稳定性。这种广义的持久不变量概念也使我们能够将拓扑空间的Lyusternik-Schnirelmann (LS)范畴提升到滤波的一种新的稳定的持久不变量，称为持久LS范畴不变量。
{"title":"Persistent cup product structures and related invariants","authors":"Facundo Mémoli, Anastasios Stefanou, Ling Zhou","doi":"10.1007/s41468-023-00138-5","DOIUrl":"https://doi.org/10.1007/s41468-023-00138-5","url":null,"abstract":"Abstract One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and by utilizing cohomological ideas, e.g. the cohomological cup product. In this work, given a single parameter filtration, we investigate a certain 2-dimensional persistence module structure associated with persistent cohomology, where one parameter is the cup-length $$ell ge 0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and the other is the filtration parameter. This new persistence structure, called the persistent cup module , is induced by the cohomological cup product and adapted to the persistence setting. Furthermore, we show that this persistence structure is stable. By fixing the cup-length parameter $$ell $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ℓ</mml:mi> </mml:math> , we obtain a 1-dimensional persistence module, called the persistent $$ell $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ℓ</mml:mi> </mml:math> -cup module, and again show it is stable in the interleaving distance sense, and study their associated generalized persistence diagrams. In addition, we consider a generalized notion of a persistent invariant , which extends both the rank invariant (also referred to as persistent Betti number ), Puuska’s rank invariant induced by epi-mono-preserving invariants of abelian categories, and the recently-defined persistent cup-length invariant , and we establish their stability. This generalized notion of persistent invariant also enables us to lift the Lyusternik-Schnirelmann (LS) category of topological spaces to a novel stable persistent invariant of filtrations, called the persistent LS-category invariant .","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135252241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-30DOI: 10.1007/s41468-023-00142-9
Samir Chowdhury, Tom Needham, Ethan Semrad, Bei Wang, Youjia Zhou
{"title":"Hypergraph co-optimal transport: metric and categorical properties","authors":"Samir Chowdhury, Tom Needham, Ethan Semrad, Bei Wang, Youjia Zhou","doi":"10.1007/s41468-023-00142-9","DOIUrl":"https://doi.org/10.1007/s41468-023-00142-9","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"160 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136280508","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-30DOI: 10.1007/s41468-023-00140-x
Igor Wigman
Abstract We survey a decade worth of work pertaining to the nodal structures of random fields, with emphasis on the transformative techniques that shaped the field.
摘要:我们调查了十年来关于随机场节点结构的工作，重点是塑造该领域的变革技术。
{"title":"On the nodal structures of random fields: a decade of results","authors":"Igor Wigman","doi":"10.1007/s41468-023-00140-x","DOIUrl":"https://doi.org/10.1007/s41468-023-00140-x","url":null,"abstract":"Abstract We survey a decade worth of work pertaining to the nodal structures of random fields, with emphasis on the transformative techniques that shaped the field.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136279525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-16DOI: 10.1007/s41468-023-00134-9
Michael Farber
Abstract The paper surveys recent progress in understanding geometric, topological and combinatorial properties of large simplicial complexes, focusing mainly on ampleness, connectivity and universality (Even-Zohar et al. in Eur J Math 8(1):1–32, 2022; Farber and Mead in Topol Appl 272(22):107065, 2020; Farber et al. in J Appl Comput Topol 5(2):339–356, 2021). In the first part of the paper we concentrate on r -ample simplicial complexes which are high dimensional analogues of the r -e.c. graphs introduced originally by Erdős and Rényi (Acta Math Acad Sci Hungar 14:295–315, 1963), see also Bonato (Contrib Discrete Math 4(1):40–53, 2009). The class of r -ample complexes is useful for applications since these complexes allow extensions of subcomplexes of certain type in all possible ways; besides, r -ample complexes exhibit remarkable robustness properties. We discuss results about the existence of r -ample complexes and describe their probabilistic and deterministic constructions. The properties of random simplicial complexes in medial regime (Farber and Mead 2020) are important for this discussion since these complexes are ample, in certain range. We prove that the topological complexity of a random simplicial complex in the medial regime satisfies $$textsf{TC}(X)le 4$$ TC(X)≤4 , with probability tending to 1 as $$nrightarrow infty $$ n→∞ . There exists a unique (up to isomorphism) $$infty $$ ∞ -ample complex on countable set of vertexes (the Rado complex), and the second part of the paper surveys the results about universality, homogeneity, indestructibility and other important properties of this complex. The Appendix written by J.A. Barmak discusses connectivity of conic and ample complexes.
摘要本文综述了近年来在理解大型简单复合体的几何、拓扑和组合性质方面的研究进展，主要集中在丰度、连通性和普遍性方面(Even-Zohar et al. in Eur J Math 8(1):1 - 32,2022;中国生物医学工程学报，32 (2):444 - 444;Farber et al. [J] .计算机学报(英文版)5(2):339-356,2021)。在本文的第一部分，我们将重点放在r -样例简单复形上，它们是最初由Erdős和r nyi(匈牙利数学学术学报，1963年14:295-315)引入的r -e.c.图的高维类似物，也参见Bonato(贡献离散数学4(1):40 - 53,2009)。r -样本配合物类在应用中是有用的，因为这些配合物允许以所有可能的方式扩展特定类型的子配合物;此外，r -ample配合物具有显著的鲁棒性。讨论了r样本复合体存在性的结果，并描述了它们的概率和确定性结构。随机简单复合物在药物治疗中的性质(Farber and Mead 2020)对于本讨论很重要，因为这些复合物在一定范围内是丰富的。我们证明了一个随机简单复体在中间区域的拓扑复杂度满足$$textsf{TC}(X)le 4$$ TC (X)≤4，其概率趋于1为$$nrightarrow infty $$ n→∞。在可数顶点集上存在一个唯一(直至同构)$$infty $$∞-样复数(Rado复形)，本文第二部分讨论了该复形的普适性、同质性、不可破坏性等重要性质。由J.A. Barmak撰写的附录讨论了圆锥和充裕复合体的连通性。
{"title":"Large simplicial complexes: universality, randomness, and ampleness","authors":"Michael Farber","doi":"10.1007/s41468-023-00134-9","DOIUrl":"https://doi.org/10.1007/s41468-023-00134-9","url":null,"abstract":"Abstract The paper surveys recent progress in understanding geometric, topological and combinatorial properties of large simplicial complexes, focusing mainly on ampleness, connectivity and universality (Even-Zohar et al. in Eur J Math 8(1):1–32, 2022; Farber and Mead in Topol Appl 272(22):107065, 2020; Farber et al. in J Appl Comput Topol 5(2):339–356, 2021). In the first part of the paper we concentrate on r -ample simplicial complexes which are high dimensional analogues of the r -e.c. graphs introduced originally by Erdős and Rényi (Acta Math Acad Sci Hungar 14:295–315, 1963), see also Bonato (Contrib Discrete Math 4(1):40–53, 2009). The class of r -ample complexes is useful for applications since these complexes allow extensions of subcomplexes of certain type in all possible ways; besides, r -ample complexes exhibit remarkable robustness properties. We discuss results about the existence of r -ample complexes and describe their probabilistic and deterministic constructions. The properties of random simplicial complexes in medial regime (Farber and Mead 2020) are important for this discussion since these complexes are ample, in certain range. We prove that the topological complexity of a random simplicial complex in the medial regime satisfies $$textsf{TC}(X)le 4$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>TC</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> <mml:mo>≤</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , with probability tending to 1 as $$nrightarrow infty $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> . There exists a unique (up to isomorphism) $$infty $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>∞</mml:mi> </mml:math> -ample complex on countable set of vertexes (the Rado complex), and the second part of the paper surveys the results about universality, homogeneity, indestructibility and other important properties of this complex. The Appendix written by J.A. Barmak discusses connectivity of conic and ample complexes.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"133 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135306542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-04DOI: 10.1007/s41468-023-00133-w
Henry Kirveslahti, S. Mukherjee
{"title":"Representing fields without correspondences: the lifted Euler characteristic transform","authors":"Henry Kirveslahti, S. Mukherjee","doi":"10.1007/s41468-023-00133-w","DOIUrl":"https://doi.org/10.1007/s41468-023-00133-w","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"53194506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-17DOI: 10.1007/s41468-023-00126-9
R. Biswas, Sebastiano Cultrera di Montesano, H. Edelsbrunner, M. Saghafian
{"title":"Geometric characterization of the persistence of 1D maps","authors":"R. Biswas, Sebastiano Cultrera di Montesano, H. Edelsbrunner, M. Saghafian","doi":"10.1007/s41468-023-00126-9","DOIUrl":"https://doi.org/10.1007/s41468-023-00126-9","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44570458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-15DOI: 10.1007/s41468-023-00122-z
S. Majhi
{"title":"Vietoris–Rips complexes of metric spaces near a metric graph","authors":"S. Majhi","doi":"10.1007/s41468-023-00122-z","DOIUrl":"https://doi.org/10.1007/s41468-023-00122-z","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"7 1","pages":"741 - 770"},"PeriodicalIF":0.0,"publicationDate":"2023-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46073955","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-04DOI: 10.1007/s41468-023-00120-1
Thomas Kahl
Abstract Given a transition system with an independence relation on the alphabet of labels, one can associate with it a usually very large symmetric higher-dimensional automaton. The purpose of this paper is to show that by choosing an acyclic relation whose symmetric closure is the given independence relation, it is possible to construct a much smaller nonsymmetric HDA with the same homology language.
{"title":"On the homology language of HDA models of transition systems","authors":"Thomas Kahl","doi":"10.1007/s41468-023-00120-1","DOIUrl":"https://doi.org/10.1007/s41468-023-00120-1","url":null,"abstract":"Abstract Given a transition system with an independence relation on the alphabet of labels, one can associate with it a usually very large symmetric higher-dimensional automaton. The purpose of this paper is to show that by choosing an acyclic relation whose symmetric closure is the given independence relation, it is possible to construct a much smaller nonsymmetric HDA with the same homology language.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136375477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-17DOI: 10.1007/s41468-023-00116-x
J. Boissonnat, M. Wintraecken
{"title":"The reach of subsets of manifolds","authors":"J. Boissonnat, M. Wintraecken","doi":"10.1007/s41468-023-00116-x","DOIUrl":"https://doi.org/10.1007/s41468-023-00116-x","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":"7 1","pages":"619 - 641"},"PeriodicalIF":0.0,"publicationDate":"2023-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42619158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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