Some persistent cohomology invariants and an axiomatic version of persistent homology
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Data
2021-12-15
Autores
Contessoto, Marco Antônio de Freitas
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Universidade Estadual Paulista (Unesp)
Resumo
Neste trabalho encontramos dois grandes capítulos que têm como foco duas das mais importantes ferramentas da Análise Topológica de Dados (TDA): homologia de persistência e cohomologia de persistência. As abordagens dadas a essas duas ferramentas são de natureza e objetivos muito distintos. Com inúmeras aplicações nas mais variadas áreas, a homologia de persistência já se mostrou uma ferramenta muito poderosa, porém pouco se estudou a respeito de uma abordagem axiomática sobre a mesma. Definimos adaptações persistentes dos axiomas de Eilenberg-Steenrod, com os quais podemos desenvolver e construir as propriedades da mesma. Para concluir, provamos um teorema de unicidade, mostrando a total caracterização de nossa teoria por meio desses axiomas. Considerando a ferramenta dual da anterior, temos a cohomologia de persistência. Muito estudada em artigos recentes, a cohomologia vem como uma forma alternativa, mais rápida e de mesma eficiência que a homologia de persistência, já que devido às dualidades temos construções semelhantes. Porém, pouquíssima abordada nesses trabalhos, a estrutura de anel que se ganha ao trabalhar com cohomologia não teve desenvolvimento relevante em TDA. Nesse trabalho, definiremos dois invariantes totalmente relacionados a essa estrutura de anel, que surge através dos produtos cup. Calcularemos vários exemplos desses invariantes, mostrando situações em que eles são capazes de nos dar informações mais completas que as antigas ferramentas.
In this work we find two chapters focussing on two of the most important tools for topological data analysis: persistent homology and persistent cohomology. The approaches given to these two tools are very different in nature and objectives. With numerous applications in the most varied areas, the persistent homology has already proved to be a very powerful tool, but there are no study about an axiomatic approach of it. We define persistent versions of the Eilenberg-Sttenrod axioms, with which we can develop and construct its properties. To conclude, we prove a uniqueness theorem, showing the full characterization of our theory through these axioms. Considering the dual tool of the previous one, we have the persistence cohomology. Much studied in recent articles, cohomology comes as an alternative form, faster and with the same efficiency than persistence homology, since due to the dualities, we have similar constructions. However, very little addressed in these works, the ring structure that is gained by working with cohomology did not have relevant development in TDA. In this work, we will define two invariants totally related to this ring structure, which arises through the cup products. We will calculate several examples of these invariants, showing situations in which they are able to give us more information than the old tools.
In this work we find two chapters focussing on two of the most important tools for topological data analysis: persistent homology and persistent cohomology. The approaches given to these two tools are very different in nature and objectives. With numerous applications in the most varied areas, the persistent homology has already proved to be a very powerful tool, but there are no study about an axiomatic approach of it. We define persistent versions of the Eilenberg-Sttenrod axioms, with which we can develop and construct its properties. To conclude, we prove a uniqueness theorem, showing the full characterization of our theory through these axioms. Considering the dual tool of the previous one, we have the persistence cohomology. Much studied in recent articles, cohomology comes as an alternative form, faster and with the same efficiency than persistence homology, since due to the dualities, we have similar constructions. However, very little addressed in these works, the ring structure that is gained by working with cohomology did not have relevant development in TDA. In this work, we will define two invariants totally related to this ring structure, which arises through the cup products. We will calculate several examples of these invariants, showing situations in which they are able to give us more information than the old tools.
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Palavras-chave
Análise topológica de dados, Homologia de persistência, Axiomatização, Cohomologia de persistência, Invariantes persistentes, Topological data analysis, Persistent homology, Axiomatization, Persistent cohomology, Persistent invariants