We introduce an algorithm to decompose any finite-type persistence module with coefficients in a field into what we call an {em interval basis}. This construction yields both the standard persistence pairs of Topological Data Analysis (TDA), as well as a special set of generators inducing the interval decomposition of the Structure theorem. The computation of this basis can be distributed over the steps in the persistence module. This construction works for general persistence modules on a field $mathbb{F}$, not necessarily deriving from persistent homology. We subsequently provide a parallel algorithm to build a persistent homology module over $mathbb{R}$ by leveraging the Hodge decomposition, thus providing new motivation to explore the interplay between TDA and the Hodge Laplacian.

Parallel decomposition of persistence modules through interval bases / DE GREGORIO, Alessandro; Guerra, Marco; Scaramuccia, Sara; Vaccarino, Francesco. - ELETTRONICO. - (2021).

Parallel decomposition of persistence modules through interval bases

Alessandro De Gregorio;Marco Guerra;Sara Scaramuccia;Francesco Vaccarino
2021

Abstract

We introduce an algorithm to decompose any finite-type persistence module with coefficients in a field into what we call an {em interval basis}. This construction yields both the standard persistence pairs of Topological Data Analysis (TDA), as well as a special set of generators inducing the interval decomposition of the Structure theorem. The computation of this basis can be distributed over the steps in the persistence module. This construction works for general persistence modules on a field $mathbb{F}$, not necessarily deriving from persistent homology. We subsequently provide a parallel algorithm to build a persistent homology module over $mathbb{R}$ by leveraging the Hodge decomposition, thus providing new motivation to explore the interplay between TDA and the Hodge Laplacian.
File in questo prodotto:
File Dimensione Formato  
2106.11884 (1).pdf

accesso aperto

Descrizione: Articolo principale
Tipologia: 1. Preprint / submitted version [pre- review]
Licenza: PUBBLICO - Tutti i diritti riservati
Dimensione 676.94 kB
Formato Adobe PDF
676.94 kB Adobe PDF Visualizza/Apri
Pubblicazioni consigliate

Caricamento pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2942232