Persistence stability for geometric complexes

Frédéric Chazal; Vin de Silva; Steve Oudot

doi:10.1007/s10711-013-9937-z

Persistence stability for geometric complexes

Frédéric Chazal
, Vin de Silva
, Steve Oudot

INRIA
Pomona College

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris–Rips, Čech and witness complexes) built on top of totally bounded metric spaces. Using recent developments in the theory of topological persistence, we provide simple and natural proofs of the stability of the persistent homology of such complexes with respect to the Gromov–Hausdorff distance. We also exhibit a few noteworthy properties of the homology of the Rips and Čech complexes built on top of compact spaces.

Original language	English
Pages (from-to)	193-214
Number of pages	22
Journal	Geometriae Dedicata
Volume	173
Issue number	1
DOIs	https://doi.org/10.1007/s10711-013-9937-z
Publication status	Published - 1 Dec 2014
Externally published	Yes

Keywords

Cech complex
Gromov–Hausdorff distance
Persistent homology
Vietoris–Rips complex

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10.1007/s10711-013-9937-z

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title = "Persistence stability for geometric complexes",

abstract = "In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris–Rips, {\v C}ech and witness complexes) built on top of totally bounded metric spaces. Using recent developments in the theory of topological persistence, we provide simple and natural proofs of the stability of the persistent homology of such complexes with respect to the Gromov–Hausdorff distance. We also exhibit a few noteworthy properties of the homology of the Rips and {\v C}ech complexes built on top of compact spaces.",

keywords = "Cech complex, Gromov–Hausdorff distance, Persistent homology, Vietoris–Rips complex",

author = "Fr{\'e}d{\'e}ric Chazal and \{de Silva\}, Vin and Steve Oudot",

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year = "2014",

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doi = "10.1007/s10711-013-9937-z",

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AU - Chazal, Frédéric

AU - de Silva, Vin

AU - Oudot, Steve

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Y1 - 2014/12/1

N2 - In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris–Rips, Čech and witness complexes) built on top of totally bounded metric spaces. Using recent developments in the theory of topological persistence, we provide simple and natural proofs of the stability of the persistent homology of such complexes with respect to the Gromov–Hausdorff distance. We also exhibit a few noteworthy properties of the homology of the Rips and Čech complexes built on top of compact spaces.

AB - In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris–Rips, Čech and witness complexes) built on top of totally bounded metric spaces. Using recent developments in the theory of topological persistence, we provide simple and natural proofs of the stability of the persistent homology of such complexes with respect to the Gromov–Hausdorff distance. We also exhibit a few noteworthy properties of the homology of the Rips and Čech complexes built on top of compact spaces.

KW - Cech complex

KW - Gromov–Hausdorff distance

KW - Persistent homology

KW - Vietoris–Rips complex

UR - https://www.scopus.com/pages/publications/84888795758

U2 - 10.1007/s10711-013-9937-z

DO - 10.1007/s10711-013-9937-z

M3 - Article

AN - SCOPUS:84888795758

SN - 0046-5755

VL - 173

SP - 193

EP - 214

JO - Geometriae Dedicata

JF - Geometriae Dedicata

IS - 1

ER -

Persistence stability for geometric complexes

Abstract

Keywords

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