Distributing persistent homology via spectral sequences

We set up the theory for a distributed algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, together with an operation that leads to a method for obtaining images, kernels and cokernels of tame persis...

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Detalles Bibliográficos
Autor: Torras Casas, Álvaro
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2023
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/155790
Acceso en línea:https://hdl.handle.net/11441/155790
https://doi.org/10.1007/s00454-023-00549-2
Access Level:acceso abierto
Palabra clave:Spectral sequences
Distributed persistent homology
Mayer-Vietoris
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spelling Distributing persistent homology via spectral sequencesTorras Casas, ÁlvaroSpectral sequencesDistributed persistent homologyMayer-VietorisWe set up the theory for a distributed algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, together with an operation that leads to a method for obtaining images, kernels and cokernels of tame persistence morphisms. Our focus is on developing efficient methods for the computation of homology of chains of persistence modules. Later we give a brief, self-contained presentation of the Mayer–Vietoris spectral sequence. Then we study the Persistent Mayer-Vietoris spectral sequence and present a solution to the extension problem. This solution is given by finding coefficients that indicate gluings between bars on the same dimension. Finally, we review PERMAVISS, an algorithm that computes all pages in the spectral sequence and solves the extension problem. This procedure distributes computations on subcomplexes, while focusing on merging homological information. Additionally, some computational bounds are found which confirm the distribution of the method.and present a solution to the extension problem. This solution is given by finding coefficients that indicate gluings between bars on the same dimension. Finally, we review PerMaViss, an algorithm that computes all pages in the spectral sequence and solves the extension problem. This procedure distributes computations on subcomplexes, while focusing on merging homological information. Additionally, some computational bounds are found which confirm the distribution of the method.SpringerMatemática Aplicada IFQM369: Combinatorial Image AnalysisEngineering and Physical Sciences Research Council (UK)2023info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfapplication/pdfhttps://hdl.handle.net/11441/155790https://doi.org/10.1007/s00454-023-00549-2reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésDiscrete & Computational Geometry, 70, 580-619.EP/N509449/1https://link.springer.com/article/10.1007/s00454-023-00549-2info:eu-repo/semantics/openAccessoai:idus.us.es:11441/1557902026-06-17T12:51:07Z
dc.title.none.fl_str_mv Distributing persistent homology via spectral sequences
title Distributing persistent homology via spectral sequences
spellingShingle Distributing persistent homology via spectral sequences
Torras Casas, Álvaro
Spectral sequences
Distributed persistent homology
Mayer-Vietoris
title_short Distributing persistent homology via spectral sequences
title_full Distributing persistent homology via spectral sequences
title_fullStr Distributing persistent homology via spectral sequences
title_full_unstemmed Distributing persistent homology via spectral sequences
title_sort Distributing persistent homology via spectral sequences
dc.creator.none.fl_str_mv Torras Casas, Álvaro
author Torras Casas, Álvaro
author_facet Torras Casas, Álvaro
author_role author
dc.contributor.none.fl_str_mv Matemática Aplicada I
FQM369: Combinatorial Image Analysis
Engineering and Physical Sciences Research Council (UK)
dc.subject.none.fl_str_mv Spectral sequences
Distributed persistent homology
Mayer-Vietoris
topic Spectral sequences
Distributed persistent homology
Mayer-Vietoris
description We set up the theory for a distributed algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, together with an operation that leads to a method for obtaining images, kernels and cokernels of tame persistence morphisms. Our focus is on developing efficient methods for the computation of homology of chains of persistence modules. Later we give a brief, self-contained presentation of the Mayer–Vietoris spectral sequence. Then we study the Persistent Mayer-Vietoris spectral sequence and present a solution to the extension problem. This solution is given by finding coefficients that indicate gluings between bars on the same dimension. Finally, we review PERMAVISS, an algorithm that computes all pages in the spectral sequence and solves the extension problem. This procedure distributes computations on subcomplexes, while focusing on merging homological information. Additionally, some computational bounds are found which confirm the distribution of the method.and present a solution to the extension problem. This solution is given by finding coefficients that indicate gluings between bars on the same dimension. Finally, we review PerMaViss, an algorithm that computes all pages in the spectral sequence and solves the extension problem. This procedure distributes computations on subcomplexes, while focusing on merging homological information. Additionally, some computational bounds are found which confirm the distribution of the method.
publishDate 2023
dc.date.none.fl_str_mv 2023
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/11441/155790
https://doi.org/10.1007/s00454-023-00549-2
url https://hdl.handle.net/11441/155790
https://doi.org/10.1007/s00454-023-00549-2
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Discrete & Computational Geometry, 70, 580-619.
EP/N509449/1
https://link.springer.com/article/10.1007/s00454-023-00549-2
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,812455

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