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Mathematics > Algebraic Topology

arXiv:1803.05722 (math)
[Submitted on 15 Mar 2018 (v1), last revised 19 Dec 2018 (this version, v3)]

Title:Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension

Authors:Mickaël Buchet, Emerson G. Escolar
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Abstract:While persistent homology has taken strides towards becoming a wide-spread 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.
Comments: 18 pages. Expanded version of the conference paper that appeared in SoCG 2018
Subjects: Algebraic Topology (math.AT)
Cite as: arXiv:1803.05722 [math.AT]
  (or arXiv:1803.05722v3 [math.AT] for this version)
  https://doi.org/10.48550/arXiv.1803.05722

Submission history

From: Mickaël Buchet [view email]
[v1] Thu, 15 Mar 2018 13:12:24 UTC (29 KB)
[v2] Tue, 20 Mar 2018 02:43:10 UTC (29 KB)
[v3] Wed, 19 Dec 2018 14:23:26 UTC (31 KB)
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