{
  "id": "2206.08818",
  "version": "v1",
  "published": "2022-06-17T14:52:15.000Z",
  "updated": "2022-06-17T14:52:15.000Z",
  "title": "Projected distances for multi-parameter persistence modules",
  "authors": [
    "Nicolas Berkouk",
    "Francois Petit"
  ],
  "comment": "52 pages",
  "categories": [
    "math.AT",
    "cs.CG"
  ],
  "abstract": "We introduce the new notions of projected distances and projected barcodes for multi-parameter persistence modules. Projected barcodes are defined as derived pushforward of persistence modules onto $\\mathbb{R}$. Projected distances come in two flavors: the integral sheaf metrics (ISM) and the sliced convolution distances (SCD). We conduct a systematic study of the stability of projected barcodes and show that the fibered barcode is a particular instance of projected barcodes. We prove that the ISM and the SCD provide lower bounds for the convolution distance. Furthermore, we show that the $\\gamma$-linear ISM and the $\\gamma$-linear SCD which are projected distances tailored for $\\gamma$-sheaves can be computed using TDA software dedicated to one-parameter persistence modules. Moreover, the time and memory complexity required to compute these two metrics are advantageous, since our approach does not require computing nor storing an entire $n$-persistence module.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-17T14:52:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "multi-parameter persistence modules",
      "projected distances",
      "projected barcodes",
      "integral sheaf metrics",
      "one-parameter persistence modules"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}