Computing Persistent Homology

We show that the persistent homology of a filtered d-dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enable...

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Bibliographic Details
Published in	Discrete & computational geometry Vol. 33; no. 2; pp. 249 - 274
Main Authors	Zomorodian, Afra, Carlsson, Gunnar
Format	Journal Article
Language	English
Published	New York Springer Nature B.V 01.02.2005
Subjects	Computers Geometry
Online Access	Get full text
ISSN	0179-5376 1432-0444 1432-0444
DOI	10.1007/s00454-004-1146-y

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Abstract	We show that the persistent homology of a filtered d-dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field. This result generalizes and extends the previously known algorithm that was restricted to subcomplexes of S3 and Z2 coefficients. Finally, our study implies the lack of a simple classification over non-fields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary principal ideal domain in any dimension. [PUBLICATION ABSTRACT]
AbstractList	We show that the persistent homology of a filtered d-dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field. This result generalizes and extends the previously known algorithm that was restricted to subcomplexes of S3 and Z2 coefficients. Finally, our study implies the lack of a simple classification over non-fields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary principal ideal domain in any dimension. [PUBLICATION ABSTRACT]
Author	Carlsson, Gunnar Zomorodian, Afra
Author_xml	– sequence: 1 givenname: Afra surname: Zomorodian fullname: Zomorodian, Afra – sequence: 2 givenname: Gunnar surname: Carlsson fullname: Carlsson, Gunnar
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Snippet	We show that the persistent homology of a filtered d-dimensional simplicial complex is simply the standard homology of a particular graded module over a...
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Title	Computing Persistent Homology
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