TY - GEN

T1 - Computing persistence modules on commutative ladders of finite type

AU - Escolar, Emerson G.

AU - Hiraoka, Yasuaki

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

KW - Discrete Morse theory

KW - Homology groups

KW - Representation theory of quivers

UR - http://www.scopus.com/inward/record.url?scp=84905851679&partnerID=8YFLogxK

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U2 - 10.1007/978-3-662-44199-2_25

DO - 10.1007/978-3-662-44199-2_25

M3 - Conference contribution

AN - SCOPUS:84905851679

SN - 9783662441985

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 144

EP - 151

BT - Mathematical Software, ICMS 2014 - 4th International Congress, Proceedings

PB - Springer-Verlag

T2 - 4th International Congress on Mathematical Software, ICMS 2014

Y2 - 5 August 2014 through 9 August 2014

ER -