A numerical approach for the filtered generalized Čech complex

Jesús F. Espinoza; Rosalía Hernández-Amador; Héctor A. Hernández-Hernández; Beatriz Ramonetti-Valencia

doi:10.3390/a13010011

A numerical approach for the filtered generalized Čech complex

Jesús F. Espinoza^*, Rosalía Hernández-Amador, Héctor A. Hernández-Hernández, Beatriz Ramonetti-Valencia

^*Autor correspondiente de este trabajo

Resultado de la investigación: Contribución a una revista › Artículo › revisión exhaustiva

Resumen

In this paper, we present an algorithm to compute the filtered generalized Čech complex for a finite collection of disks in the plane, which do not necessarily have the same radius. The key step behind the algorithm is to calculate the minimum scale factor needed to ensure rescaled disks have a nonempty intersection, through a numerical approach, whose convergence is guaranteed by a generalization of the well-known Vietoris-Rips Lemma, which we also prove in an alternative way, using elementary geometric arguments. We give an algorithm for computing the 2-dimensional filtered generalized Čech complex of a finite collection of d-dimensional disks in R^d, and we show the performance of our algorithm.

Idioma original	Inglés
Número de artículo	11
Publicación	Algorithms
Volumen	13
N.º	1
DOI	https://doi.org/10.3390/a13010011
Estado	Publicada - 1 ene 2020

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© 2019 by the authors.

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title = "A numerical approach for the filtered generalized {\v C}ech complex",

abstract = "In this paper, we present an algorithm to compute the filtered generalized {\v C}ech complex for a finite collection of disks in the plane, which do not necessarily have the same radius. The key step behind the algorithm is to calculate the minimum scale factor needed to ensure rescaled disks have a nonempty intersection, through a numerical approach, whose convergence is guaranteed by a generalization of the well-known Vietoris-Rips Lemma, which we also prove in an alternative way, using elementary geometric arguments. We give an algorithm for computing the 2-dimensional filtered generalized {\v C}ech complex of a finite collection of d-dimensional disks in Rd, and we show the performance of our algorithm.",

keywords = "Disk system, Generalized vietoris-rips lemma, Generalized {\v C}ech complex, Miniball problem, {\v C}ech scale",

author = "Espinoza, {Jes{\'u}s F.} and Rosal{\'i}a Hern{\'a}ndez-Amador and Hern{\'a}ndez-Hern{\'a}ndez, {H{\'e}ctor A.} and Beatriz Ramonetti-Valencia",

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AU - Espinoza, Jesús F.

AU - Hernández-Amador, Rosalía

AU - Hernández-Hernández, Héctor A.

AU - Ramonetti-Valencia, Beatriz

PY - 2020/1/1

Y1 - 2020/1/1

N2 - In this paper, we present an algorithm to compute the filtered generalized Čech complex for a finite collection of disks in the plane, which do not necessarily have the same radius. The key step behind the algorithm is to calculate the minimum scale factor needed to ensure rescaled disks have a nonempty intersection, through a numerical approach, whose convergence is guaranteed by a generalization of the well-known Vietoris-Rips Lemma, which we also prove in an alternative way, using elementary geometric arguments. We give an algorithm for computing the 2-dimensional filtered generalized Čech complex of a finite collection of d-dimensional disks in Rd, and we show the performance of our algorithm.

AB - In this paper, we present an algorithm to compute the filtered generalized Čech complex for a finite collection of disks in the plane, which do not necessarily have the same radius. The key step behind the algorithm is to calculate the minimum scale factor needed to ensure rescaled disks have a nonempty intersection, through a numerical approach, whose convergence is guaranteed by a generalization of the well-known Vietoris-Rips Lemma, which we also prove in an alternative way, using elementary geometric arguments. We give an algorithm for computing the 2-dimensional filtered generalized Čech complex of a finite collection of d-dimensional disks in Rd, and we show the performance of our algorithm.

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KW - Generalized vietoris-rips lemma

KW - Generalized Čech complex

KW - Miniball problem

KW - Čech scale

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JO - Algorithms

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A numerical approach for the filtered generalized Čech complex

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