On interval decomposability of 2D persistence modules

Hideto Asashiba, Mickaël Buchet, Emerson G. Escolar, Ken Nakashima, Michio Yoshiwaki

Research output: Contribution to journalArticlepeer-review

Abstract

In the persistent homology of filtrations, the indecomposable decompositions provide the persistence diagrams. However, in almost all cases of multidimensional persistence, the classification of all indecomposable modules is known to be a wild problem. One direction is to consider the subclass of interval-decomposable persistence modules, which are direct sums of interval representations. We introduce the definition of pre-interval representations, a more natural algebraic definition, and study the relationships between pre-interval, interval, and thin indecomposable representations. We show that over the “equioriented” commutative 2D grid, these concepts are equivalent. Moreover, we provide a criterion for determining whether or not an nD persistence module is interval/pre-interval/thin-decomposable without having to explicitly compute decompositions. For 2D persistence modules, we provide an algorithm for determining interval-decomposability, together with a worst-case complexity analysis that uses the total number of intervals in an equioriented commutative 2D grid. We also propose several heuristics to speed up the computation.

Original languageEnglish
Article number101879
JournalComputational Geometry: Theory and Applications
Volume105-106
DOIs
Publication statusPublished - Aug 1 2022
Externally publishedYes

Keywords

  • Interval representations
  • Multidimensional persistence
  • Representation theory

ASJC Scopus subject areas

  • Computer Science Applications
  • Geometry and Topology
  • Control and Optimization
  • Computational Theory and Mathematics
  • Computational Mathematics

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