{
  "abstract": "Persistence diagrams do not naturally live in a Hilbert space. Carrière, Cuturi, and Oudot embedded them in kernel methods through the Sliced Wasserstein kernel, built from the Sliced Wasserstein distance $\\mathrm{SW}_1$, which is conditionally negative definite (CND). We study the projected ($\\mathrm{SW}_p$) and reflected ($\\mathrm{RW}_p$) Sliced Wasserstein distances for every order $p\\ge1$, and compare them with the $p$-Wasserstein distance $d_p$ and with a reflected variant $d_p^r$, in which projection onto the diagonal is replaced by reflection across it. We show that $2^{1/p}d_p\\le d_p^r\\le 2d_p$ with sharp constants, so that $d_1^r=2d_1$ and $d_p^r$ inherits the stability of $d_p$, although it is a metric only for $p=1$. The distances $\\mathrm{SW}_1$ and $\\mathrm{RW}_1$ are CND metrics, and $\\mathrm{RW}_1$ gives rise to a new positive definite kernel, stable with respect to $d_1$. For $p>1$, neither $\\mathrm{SW}_p$ nor $\\mathrm{RW}_p$ satisfies the triangle inequality, and rigorous interval-arithmetic certificates prove that $\\mathrm{SW}_2$ and $\\mathrm{RW}_2$ are not CND; numerical experiments suggest that the CND property fails for every $p$ above some $p^*\\in(1,1.2]$. On diagrams with at most $N$ points, $\\mathrm{SW}_p$ and $\\mathrm{RW}_p$ are metrically equivalent to $d_p$ and $d_p^r$, with sharp upper constants. On an orbit recognition benchmark, the kernels built from $\\mathrm{SW}_1$ and $\\mathrm{RW}_1$ reach about 85% accuracy, outperform the persistence scale space kernel, perform on par with the persistence weighted Gaussian kernel, and are about one percentage point below the kernel built from $\\mathrm{SW}_2$, which is not positive definite.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06619",
  "author_total": 2,
  "authors": [
    {
      "name": "Julián Uribe Castañeda"
    },
    {
      "name": "Mircea Petrache"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cg9ppnys9jdaq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06619",
  "fulltext_url": null,
  "id": "cg9ppnys9jdaq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06619",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CG"
    ],
    [
      "arxiv",
      "math.AT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06619",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cg9ppnys9jdaq.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06619",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06619"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Projected and Reflected Sliced Wasserstein Kernels for Persistence Diagrams",
  "title_text": "Projected and Reflected Sliced Wasserstein Kernels for Persistence Diagrams",
  "type": "preprint",
  "updated_at": "2026-11-07T18:20:08Z",
  "url": "https://research.draconitic.com/paper/cg9ppnys9jdaq",
  "venue": "arXiv",
  "versions": []
}