Computing persistence modules on commutative ladders of finite type

Emerson G. Escolar, Yasuaki Hiraoka

Research output: Chapter in Book/Report/Conference proceedingConference contribution

5 Citations (Scopus)

Abstract

Persistence modules on commutative ladders naturally arise in topological data analysis. It is known that all isomorphism classes of indecomposable modules, which are the counterparts to persistence intervals in the standard setting of persistent homology, can be derived for persistence modules on commutative ladders of finite type. Furthermore, the concept of persistence diagrams can be naturally generalized as functions defined on the Auslander-Reiten quivers of commutative ladders. A previous paper [4] presents an algorithm to compute persistence diagrams by inductively applying echelon form reductions to a given persistence module. In this work, we show that discrete Morse reduction can be generalized to this setting. Given a quiver complex double-struck X, we show that its persistence module H q(double-struck X) is isomorphic to the persistence module H q(double-struck A) of its Morse quiver complex double-struck A. With this preprocessing step, we reduce the computation time by computing H q(double-struck A) instead, since double-struck A is generally smaller in size. We also provide an algorithm to obtain such Morse quiver complexes.

Original languageEnglish
Title of host publicationMathematical Software, ICMS 2014 - 4th International Congress, Proceedings
PublisherSpringer-Verlag
Pages144-151
Number of pages8
ISBN (Print)9783662441985
DOIs
Publication statusPublished - 2014 Jan 1
Event4th International Congress on Mathematical Software, ICMS 2014 - Seoul, Korea, Republic of
Duration: 2014 Aug 52014 Aug 9

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume8592 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other4th International Congress on Mathematical Software, ICMS 2014
Country/TerritoryKorea, Republic of
CitySeoul
Period14/8/514/8/9

Keywords

  • Discrete Morse theory
  • Homology groups
  • Representation theory of quivers

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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