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Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension

Authors Mickaël Buchet, Emerson G. Escolar

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Mickaël Buchet
Emerson G. Escolar

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Mickaël Buchet and Emerson G. Escolar. Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SoCG.2018.15

Abstract

While persistent homology has taken strides towards becoming a widespread tool for data analysis, multidimensional persistence has proven more difficult to apply. One reason is the serious drawback of no longer having a concise and complete descriptor analogous to the persistence diagrams of the former. We propose a simple algebraic construction to illustrate the existence of infinite families of indecomposable persistence modules over regular grids of sufficient size. On top of providing a constructive proof of representation infinite type, we also provide realizations by topological spaces and Vietoris-Rips filtrations, showing that they can actually appear in real data and are not the product of degeneracies.
Keywords
  • persistent homology
  • multi-persistence
  • representation theory
  • quivers
  • commutative ladders
  • Vietoris-Rips filtration

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