{
  "id": "2106.13589",
  "version": "v1",
  "published": "2021-06-25T12:40:19.000Z",
  "updated": "2021-06-25T12:40:19.000Z",
  "title": "$\\ell^p$-Distances on Multiparameter Persistence Modules",
  "authors": [
    "Håvard Bakke Bjerkevik",
    "Michael Lesnick"
  ],
  "comment": "43 pages",
  "categories": [
    "math.AT",
    "cs.CG"
  ],
  "abstract": "Motivated both by theoretical and practical considerations in topological data analysis, we generalize the $p$-Wasserstein distance on barcodes to multiparameter persistence modules. For each $p\\in [1,\\infty]$, we in fact introduce two such generalizations $d_{\\mathcal I}^p$ and $d_{\\mathcal M}^p$, such that $d_{\\mathcal I}^\\infty$ equals the interleaving distance and $d_{\\mathcal M}^\\infty$ equals the matching distance. We show that $d_{\\mathcal M}^p\\leq d_{\\mathcal I}^p$ for all $p\\in [1,\\infty]$, extending an observation of Landi in the $p=\\infty$ case. We observe that the distances $d_{\\mathcal M}^p$ can be efficiently approximated. Finally, we show that on 1- or 2-parameter persistence modules over prime fields, $d_{\\mathcal I}^p$ is the universal (i.e., largest) metric satisfying a natural stability property; our result extends a stability result of Skraba and Turner for the $p$-Wasserstein distance on barcodes in the 1-parameter case, and is also a close analogue of a universality property for the interleaving distance given by the second author. In a companion paper, we apply some of these results to study the stability of ($2$-parameter) multicover persistent homology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-25T12:40:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "multiparameter persistence modules",
      "wasserstein distance",
      "interleaving distance",
      "natural stability property",
      "multicover persistent homology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}