{
  "abstract": "A novel, non-parametric clustering algorithm is developed, based on flows in the space of soft assignments $P_k^i$, representing the probability that point $x^i$ belongs to class $k$. These flows have two drivers: a nonlinear reaction component that relaxes $P$ to an evolving Bayesian-like posterior, and a diffusive component that relaxes $P$ to its local average, with a diffusivity $ν$ that adapts dynamically so as to balance the two components. The methodology extends to semi-supervised classification, where a subset of the labels are known. In the limit of infinitely many observations, it yields a set of non-standard reaction-diffusion equations, which produces sharp boundaries between species that separate spontaneously into well-balanced domains. Including more than one diffusive network opens the way to a broader class of applications, including the detection of regime changes in time series and a clustering procedure based on numerous features that defeats the curse of dimensionality. In the continuous limit, this extension results in a novel, non-local class of diffusive operators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11608",
  "author_total": 2,
  "authors": [
    {
      "name": "Paul A. Milewski"
    },
    {
      "name": "Esteban G. Tabak"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ak5548xrcpn42.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11608",
  "fulltext_url": null,
  "id": "ak5548xrcpn42",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11608",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Advanced Clustering Algorithms Research"
    ]
  ],
  "openalex": "W7212296288",
  "pdf_url": "https://arxiv.org/pdf/2609.11608",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ak5548xrcpn42.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11608",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11608"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Dynamical Clustering and its Limiting Differential Equations",
  "title_text": "Dynamical Clustering and its Limiting Differential Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:36Z",
  "url": "https://research.draconitic.com/paper/ak5548xrcpn42",
  "venue": "arXiv",
  "versions": []
}