Persistence Diagrams as Diagrams: A Categorification of the Stability Theorem

Persistence Diagrams as Diagrams: A Categorification of the Stability Theorem. Bauer, U. & Lesnick, M.

Persistent homology, a central tool of topological data analysis, provides invariants of data called barcodes (also known as persistence diagrams). A barcode is simply a multiset of real intervals. Recent work of Edelsbrunner, Jablonski, and Mrozek suggests an equivalent description of barcodes as functors R -$>$ Mch, where R is the poset category of real numbers and Mch is the category whose objects are sets and whose morphisms are matchings (i.e., partial injective functions). Such functors form a category Mch\^R whose morphisms are the natural transformations. Thus, this interpretation of barcodes gives us a hitherto unstudied categorical structure on barcodes. The aim of this note is to show that this categorical structure leads to surprisingly simple reformulations of both the well-known stability theorem for persistent homology and a recent generalization called the induced matching theorem.

@article{bauerPersistenceDiagramsDiagrams2016,
  archivePrefix = {arXiv},
  eprinttype = {arxiv},
  eprint = {1610.10085},
  primaryClass = {cs, math},
  title = {Persistence {{Diagrams}} as {{Diagrams}}: {{A Categorification}} of the {{Stability Theorem}}},
  url = {http://arxiv.org/abs/1610.10085},
  shorttitle = {Persistence {{Diagrams}} as {{Diagrams}}},
  abstract = {Persistent homology, a central tool of topological data analysis, provides invariants of data called barcodes (also known as persistence diagrams). A barcode is simply a multiset of real intervals. Recent work of Edelsbrunner, Jablonski, and Mrozek suggests an equivalent description of barcodes as functors R -{$>$} Mch, where R is the poset category of real numbers and Mch is the category whose objects are sets and whose morphisms are matchings (i.e., partial injective functions). Such functors form a category Mch\^R whose morphisms are the natural transformations. Thus, this interpretation of barcodes gives us a hitherto unstudied categorical structure on barcodes. The aim of this note is to show that this categorical structure leads to surprisingly simple reformulations of both the well-known stability theorem for persistent homology and a recent generalization called the induced matching theorem.},
  urldate = {2019-03-19},
  date = {2016-10-31},
  keywords = {Computer Science - Computational Geometry,Mathematics - Algebraic Topology,Mathematics - Category Theory,13P20; 55U99},
  author = {Bauer, Ulrich and Lesnick, Michael},
  file = {/home/dimitri/Nextcloud/Zotero/storage/9YK34HBQ/Bauer and Lesnick - 2016 - Persistence Diagrams as Diagrams A Categorificati.pdf;/home/dimitri/Nextcloud/Zotero/storage/25LJ568T/1610.html}
}

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