Pub Date : 2023-11-13DOI: 10.1007/s41468-023-00147-4
Yicheng Fei, Xaq Pitkow
Abstract Probabilistic graphical models provide a powerful tool to describe complex statistical structure, with many real-world applications in science and engineering from controlling robotic arms to understanding neuronal computations. A major challenge for these graphical models is that inferences such as marginalization are intractable for general graphs. These inferences are often approximated by a distributed message-passing algorithm such as Belief Propagation, which does not always perform well on graphs with cycles, nor can it always be easily specified for complex continuous probability distributions. Such difficulties arise frequently in expressive graphical models that include intractable higher-order interactions. In this paper we define the Recurrent Factor Graph Neural Network (RF-GNN) to achieve fast approximate inference on graphical models that involve many-variable interactions. Experimental results on several families of graphical models demonstrate the out-of-distribution generalization capability of our method to different sized graphs, and indicate the domain in which our method outperforms Belief Propagation (BP). Moreover, we test the RF-GNN on a real-world Low-Density Parity-Check dataset as a benchmark along with other baseline models including BP variants and other GNN methods. Overall we find that RF-GNNs outperform other methods under high noise levels.
{"title":"Generalization of graph network inferences in higher-order graphical models","authors":"Yicheng Fei, Xaq Pitkow","doi":"10.1007/s41468-023-00147-4","DOIUrl":"https://doi.org/10.1007/s41468-023-00147-4","url":null,"abstract":"Abstract Probabilistic graphical models provide a powerful tool to describe complex statistical structure, with many real-world applications in science and engineering from controlling robotic arms to understanding neuronal computations. A major challenge for these graphical models is that inferences such as marginalization are intractable for general graphs. These inferences are often approximated by a distributed message-passing algorithm such as Belief Propagation, which does not always perform well on graphs with cycles, nor can it always be easily specified for complex continuous probability distributions. Such difficulties arise frequently in expressive graphical models that include intractable higher-order interactions. In this paper we define the Recurrent Factor Graph Neural Network (RF-GNN) to achieve fast approximate inference on graphical models that involve many-variable interactions. Experimental results on several families of graphical models demonstrate the out-of-distribution generalization capability of our method to different sized graphs, and indicate the domain in which our method outperforms Belief Propagation (BP). Moreover, we test the RF-GNN on a real-world Low-Density Parity-Check dataset as a benchmark along with other baseline models including BP variants and other GNN methods. Overall we find that RF-GNNs outperform other methods under high noise levels.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136283777","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 : 2023-11-13DOI: 10.1007/s41468-023-00150-9
Pierre Fraigniaud, Ran Gelles, Zvi Lotker
Studying distributed computing through the lens of algebraic topology has been the source of many significant breakthroughs during the last 2 decades, especially in the design of lower bounds or impossibility results. Despite hundred of results considering deterministic algorithms, none apply to randomized algorithms. This paper aims at studying randomized synchronous distributed computing through the lens of algebraic topology. We do so by studying the wide class of (input-free) symmetry-breaking tasks, e.g., leader election, in synchronous fault-free anonymous systems. We design a topological framework, which allows analyzing such tasks and determining their solvability. The pivotal technical observation is that, unlike in deterministic algorithm, where solvability means that the topological complex describing the protocol can be globally mapped into an output protocol, in our framework the solvability is determined “locally”, i.e., for each simplex of the protocol complex individually, without requiring any global consistency. As an interesting application, we derive necessary and sufficient conditions for solving leader election in shared-memory and message-passing models in which there might be correlations between the randomness provided to the nodes. We find that solvability of leader election relates to the number of parties that possess correlated randomness, either directly or via their greatest common divisor, depending on the specific communication model.
{"title":"The topology of randomized symmetry-breaking distributed computing","authors":"Pierre Fraigniaud, Ran Gelles, Zvi Lotker","doi":"10.1007/s41468-023-00150-9","DOIUrl":"https://doi.org/10.1007/s41468-023-00150-9","url":null,"abstract":"Studying distributed computing through the lens of algebraic topology has been the source of many significant breakthroughs during the last 2 decades, especially in the design of lower bounds or impossibility results. Despite hundred of results considering deterministic algorithms, none apply to randomized algorithms. This paper aims at studying randomized synchronous distributed computing through the lens of algebraic topology. We do so by studying the wide class of (input-free) symmetry-breaking tasks, e.g., leader election, in synchronous fault-free anonymous systems. We design a topological framework, which allows analyzing such tasks and determining their solvability. The pivotal technical observation is that, unlike in deterministic algorithm, where solvability means that the topological complex describing the protocol can be globally mapped into an output protocol, in our framework the solvability is determined “locally”, i.e., for each simplex of the protocol complex individually, without requiring any global consistency. As an interesting application, we derive necessary and sufficient conditions for solving leader election in shared-memory and message-passing models in which there might be correlations between the randomness provided to the nodes. We find that solvability of leader election relates to the number of parties that possess correlated randomness, either directly or via their greatest common divisor, depending on the specific communication model.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134993015","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 : 2023-11-10DOI: 10.1007/s41468-023-00148-3
Ron Rosenthal
{"title":"Simplicial branching random walks","authors":"Ron Rosenthal","doi":"10.1007/s41468-023-00148-3","DOIUrl":"https://doi.org/10.1007/s41468-023-00148-3","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135138044","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 : 2023-11-04DOI: 10.1007/s41468-023-00149-2
Donald Woukeng, Damian Sadowski, Jakub Leśkiewicz, Michał Lipiński, Tomasz Kapela
Abstract Motivated by the theoretical results of Mrozek et al. (Commun Nonlinear Sci Numer Simul 108:106–226, 2022) we present an algorithmic construction of a transversal cellular decomposition for a planar ODE. We then use the associated combinatorial multivector field to algorithmically detect the existence of an isolated invariant set with the Conley index of a periodic orbit and admitting a combinatorial Poincaré section. This construction combined with the theoretical results of Mrozek et al. (2022) leads to a method for automatized computer assisted proofs of the existence of periodic solutions in ODE’s.
受Mrozek等人(common Nonlinear Sci numerical Simul 108:106 - 226,2022)的理论结果的启发，我们提出了一种平面ODE的横向元胞分解算法。然后，我们使用相关的组合多向量场算法检测周期轨道的Conley指数的孤立不变量集的存在性，并允许一个组合poincarcarr截面。这种结构与Mrozek等人(2022)的理论结果相结合，产生了一种自动化计算机辅助证明ODE周期解存在性的方法。
{"title":"Rigorous computation in dynamics based on topological methods for multivector fields","authors":"Donald Woukeng, Damian Sadowski, Jakub Leśkiewicz, Michał Lipiński, Tomasz Kapela","doi":"10.1007/s41468-023-00149-2","DOIUrl":"https://doi.org/10.1007/s41468-023-00149-2","url":null,"abstract":"Abstract Motivated by the theoretical results of Mrozek et al. (Commun Nonlinear Sci Numer Simul 108:106–226, 2022) we present an algorithmic construction of a transversal cellular decomposition for a planar ODE. We then use the associated combinatorial multivector field to algorithmically detect the existence of an isolated invariant set with the Conley index of a periodic orbit and admitting a combinatorial Poincaré section. This construction combined with the theoretical results of Mrozek et al. (2022) leads to a method for automatized computer assisted proofs of the existence of periodic solutions in ODE’s.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135774489","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 : 2023-10-28DOI: 10.1007/s41468-023-00139-4
Bruno Benedetti, Crystal Lai, Davide Lofano, Frank H. Lutz
Abstract We implement an algorithm RSHT (random simple-homotopy) to study the simple-homotopy types of simplicial complexes, with a particular focus on contractible spaces and on finding substructures in higher-dimensional complexes. The algorithm combines elementary simplicial collapses with pure elementary expansions . For triangulated d -manifolds with $$dle 6$$ d≤6 , we show that RSHT reduces to (random) bistellar flips. Among the many examples on which we test RSHT, we describe an explicit 15-vertex triangulation of the Abalone, and more generally, $$(14k+1)$$ (14k+1) -vertex triangulations of a new series of Bing’s houses with k rooms, $$kge 3$$ k≥3 , which all can be deformed to a point using only six pure elementary expansions.
摘要:我们实现了一个RSHT(随机简单同伦)算法来研究简单复合体的简单同伦类型，特别关注高维复合体的可收缩空间和寻找子结构。该算法结合了初等简单坍缩和纯初等展开。对于$$dle 6$$ d≤6的三角化d流形，我们证明RSHT减少为(随机)双星翻转。在我们测试RSHT的许多例子中，我们描述了Abalone的显式15顶点三角剖分，更一般地说，$$(14k+1)$$ (14 k + 1)顶点三角剖分，包含k个房间的新系列Bing的房子，$$kge 3$$ k≥3，它们都可以变形为一个点，仅使用六个纯初等展开。
{"title":"Random simple-homotopy theory","authors":"Bruno Benedetti, Crystal Lai, Davide Lofano, Frank H. Lutz","doi":"10.1007/s41468-023-00139-4","DOIUrl":"https://doi.org/10.1007/s41468-023-00139-4","url":null,"abstract":"Abstract We implement an algorithm RSHT (random simple-homotopy) to study the simple-homotopy types of simplicial complexes, with a particular focus on contractible spaces and on finding substructures in higher-dimensional complexes. The algorithm combines elementary simplicial collapses with pure elementary expansions . For triangulated d -manifolds with $$dle 6$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> , we show that RSHT reduces to (random) bistellar flips. Among the many examples on which we test RSHT, we describe an explicit 15-vertex triangulation of the Abalone, and more generally, $$(14k+1)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>14</mml:mn> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -vertex triangulations of a new series of Bing’s houses with k rooms, $$kge 3$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , which all can be deformed to a point using only six pure elementary expansions.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136157610","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}
Abstract We make use of the 3D nature of knots and links to find savings in computational complexity when computing knot invariants such as the linking number and, in general, most finite type invariants. These savings are achieved in comparison with the 2D approach to knots using knot diagrams.
{"title":"Yarn ball knots and faster computations","authors":"Dror Bar-Natan, Itai Bar-Natan, Iva Halacheva, Nancy Scherich","doi":"10.1007/s41468-023-00144-7","DOIUrl":"https://doi.org/10.1007/s41468-023-00144-7","url":null,"abstract":"Abstract We make use of the 3D nature of knots and links to find savings in computational complexity when computing knot invariants such as the linking number and, in general, most finite type invariants. These savings are achieved in comparison with the 2D approach to knots using knot diagrams.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136157611","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 : 2023-10-21DOI: 10.1007/s41468-023-00146-5
Tadas Temčinas, Vidit Nanda, Gesine Reinert
Abstract Motivated by open problems in applied and computational algebraic topology, we establish multivariate normal approximation theorems for three random vectors which arise organically in the study of random clique complexes. These are: the vector of critical simplex counts attained by a lexicographical Morse matching, the vector of simplex counts in the link of a fixed simplex, and the vector of total simplex counts. The first of these random vectors forms a cornerstone of modern homology algorithms, while the second one provides a natural generalisation for the notion of vertex degree, and the third one may be viewed from the perspective of U -statistics. To obtain distributional approximations for these random vectors, we extend the notion of dissociated sums to a multivariate setting and prove a new central limit theorem for such sums using Stein’s method.
{"title":"Multivariate central limit theorems for random clique complexes","authors":"Tadas Temčinas, Vidit Nanda, Gesine Reinert","doi":"10.1007/s41468-023-00146-5","DOIUrl":"https://doi.org/10.1007/s41468-023-00146-5","url":null,"abstract":"Abstract Motivated by open problems in applied and computational algebraic topology, we establish multivariate normal approximation theorems for three random vectors which arise organically in the study of random clique complexes. These are: the vector of critical simplex counts attained by a lexicographical Morse matching, the vector of simplex counts in the link of a fixed simplex, and the vector of total simplex counts. The first of these random vectors forms a cornerstone of modern homology algorithms, while the second one provides a natural generalisation for the notion of vertex degree, and the third one may be viewed from the perspective of U -statistics. To obtain distributional approximations for these random vectors, we extend the notion of dissociated sums to a multivariate setting and prove a new central limit theorem for such sums using Stein’s method.","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135510925","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 : 2023-10-21DOI: 10.1007/s41468-023-00141-w
Nikolas C. Schonsheck, Stefan C. Schonsheck
{"title":"Spherical coordinates from persistent cohomology","authors":"Nikolas C. Schonsheck, Stefan C. Schonsheck","doi":"10.1007/s41468-023-00141-w","DOIUrl":"https://doi.org/10.1007/s41468-023-00141-w","url":null,"abstract":"","PeriodicalId":73600,"journal":{"name":"Journal of applied and computational topology","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135511557","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 : 2023-10-10DOI: 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":null,"pages":null},"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":null,"pages":null},"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}
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