Abstract
A recent work by Lesnick and Wright proposed a visualisation of 2D persistence modules by using their restrictions onto lines, giving a family of 1D persistence modules. We give a constructive proof that any 1D persistence module with finite support can be found as a restriction of some indecomposable 2D persistence module with finite support. As consequences of our construction, we are able to exhibit indecomposable 2D persistence modules whose support has holes as well as an indecomposable 2D persistence module containing all 1D persistence modules with finite support as line restrictions. Finally, we also show that any finite-rectangle-decomposable nD persistence module can be found as a restriction of some indecomposable (n+1)D persistence module.
Highlights
In the theory of persistent homology (Edelsbrunner et al 2000), 1D persistence modules can be summarized and visualized using the so-called persistence diagrams, which led to successful applications of topological data analysis
Our main result is a constructive proof that any 1D persistence module V can be obtained via the restriction onto a line of an indecomposable 2D persistence module
#– In Sect. 4, we showed that for every 1D persistence module in rep An, we can build an indecomposable 2D persistence module containing it as a line restriction
Summary
In the theory of persistent homology (Edelsbrunner et al 2000), 1D persistence modules can be summarized and visualized using the so-called persistence diagrams, which led to successful applications of topological data analysis. Our main result is a constructive proof that any 1D persistence module V can be obtained via the restriction onto a line of an indecomposable 2D persistence module (of large enough, but finite, support). We are able to exhibit indecomposable 2D persistence modules whose support can have an arbitrary finite number of holes We believe that their existence was not obvious and that providing a concrete example will help deepen the understanding of 2D persistence modules. We build a single indecomposable 2D persistence module with infinite support that contains all possible 1D persistence modules with finite support as line restrictions. 6, we build the indecomposable 2D persistence module with infinite support containing all possible 1D persistence modules with finite support as line restrictions.
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