Betti numbers in multidimensional persistent homology are stable functions
type: Item
instance of: scholarly article
title: Betti numbers in multidimensional persistent homology are stable functions
zbMATH Open document ID: 1278.55012
author: Andrea Cerri
author: Barbara Di Fabio
author: Massimo Ferri
author: Patrizio Frosini
author: C. Landi
published in: Mathematical Methods in the Applied Sciences
publication date: 2013-11-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/0908.0064
full work available at URL: mp101107.pdf
Mathematics Subject Classification ID: 55N35
Mathematics Subject Classification ID: 55N05
Mathematics Subject Classification ID: 68U05
zbMATH DE Number: 6222224
zbMATH Keywords: multidimensional persistence
zbMATH Keywords: persistence diagram
zbMATH Keywords: Čech homology
zbMATH Keywords: foliation
zbMATH Keywords: continuous filtering function
zbMATH Keywords: natural pseudo-distance
zbMATH Keywords: persistent Betti numbers
Wikidata QID: Q59442781
MaRDI profile type: Publication
OpenAlex ID: W2014209644
arXiv ID: 0908.0064
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DOI: 10.1002/MMA.2704
review text: Define the relation \(\prec\) on \({\mathbb R}^{n}\) by \(u \prec v\) iff \(u_{i} < v_{i}\) for all \(i = 1,\ldots,n\). Given a function \(\phi : X \rightarrow {\mathbb R}^{n}\) for any \(u \prec v\) one has an induced mapping of Čech homology \(\pi_{k}^{(u,v) }: H_{k}(X(\phi \prec u) \rightarrow H_{k}(X(\phi \prec v))\). The image of this map is called the multidimensional \(k\)th persistent homology group of \((X,\phi)\) at \((u,v)\). The authors show that if \(X\) is a triangulable space then these Betti numbers are finite. One can associate to the maps \(\phi\) a diagram in \(\Delta^{+} = \{(u,v) \in {\mathbb R}^{n} \times {\mathbb R}^{n}: u \prec v\}\) that represents the persistent Betti numbers. If the map \(\phi\) is perturbed then the diagram shifts and a distance between the diagrams can be defined. The main result of the paper is that these multidimensional persistent Betti numbers are stable in the sense that small perturbations in the function \(\phi\) induce only small changes in the Betti numbers. This is significant for the applications of persistent homology to the study of large data sets.
reviewed by: Jonathan Hodgson
description: scientific article; zbMATH DE number 6222224