Computing Persistence Modules on Commutative Ladders of Finite Type

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 \(\mathbb{X}\), we show that its persistence module \(H_q(\mathbb{X})\) is isomorphic to the persistence module \(H_q(\mathbb{A})\) of its Morse quiver complex \(\mathbb{A}\). With this preprocessing step, we reduce the computation time by computing \(H_q(\mathbb{A})\) instead, since \(\mathbb{A}\) is generally smaller in size. We also provide an algorithm to obtain such Morse quiver complexes.KeywordsHomology groupsRepresentation theory of quiversDiscrete Morse theory