Some new results on the matching distance in biparameter persistent homology

Mosig García, Eloy (2023) Some new results on the matching distance in biparameter persistent homology. [Laurea magistrale], Università di Bologna, Corso di Studio in Matematica [LM-DM270]
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Abstract

When studyng monoparameter persistent homology, the bottleneck distance is the most standard choice in literature. The advantage of this choice is that persistence diagrams in the monoparameter setting are a complete invariant as shown by the Isometry Theorem. However, they are not a complete invariant in multiparameter persistent homology. This makes the multiparameter case much more harder to study. In this work we study some properties of the biparametric matching distance, which is an analogue of the bottleneck distance for higher dimensions, between persistence diagrams, along with the set of special values (a,b) in which the matching realising this distance may change abruptly. We prove a result that drastically reduces the cost of the computation of the matching distance, under suitable regularity conditions on the filtered space (X, φ) and on the special set. Moreover, we give a proof for an extension of the Position Theorem, which is a central step in the proof of our main theorem.

Abstract
Tipologia del documento
Tesi di laurea (Laurea magistrale)
Autore della tesi
Mosig García, Eloy
Relatore della tesi
Correlatore della tesi
Scuola
Corso di studio
Indirizzo
Curriculum Generale
Ordinamento Cds
DM270
Parole chiave
topological data analysis,persistent homology,extended Pareto grid,bottleneck distance,matching distance,Position Theorem
Data di discussione della Tesi
21 Luglio 2023
URI

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