{
  "abstract": "This paper proposes fuzzy entropic triple k-means (FE3KM), an entropy-regularized fuzzy partitioning method for three-way data arrays that simultaneously clusters objects, variables, and occasions. Unlike standard fuzzy clustering, which controls fuzziness through a nonlinear exponent m, FE3KM embeds memberships linearly within a Least-Squares (LS) objective, using row-stochastic membership matrices and entropy regularization to represent uncertainty in cluster assignments. This linear structure is what enables an exact decomposition of total deviance into within- and between-cluster components, further attributable to each mode (objects, variables, occasions) and to individual clusters within each mode. This is an interpretive property unavailable to exponent-m fuzzy models. We derive Alternating Least-Squares algorithms for FE3KM, including an accelerated variant for large arrays, and formally relate the method to tandem clustering procedures and to Tucker3/three-mode partitioning models. A simulation study assesses recovery accuracy, robustness to noise and fuzziness mis-specification, and performance against tandem and three-way clustering baselines. An application to a benchmark three-way TV ratings dataset demonstrates how FE3KM recovers interpretable object-variable-occasion structures and quantifies the contribution of each cluster and mode to overall data variability.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21538",
  "author_total": 3,
  "authors": [
    {
      "name": "Mariaelena Bottazzi Schenone"
    },
    {
      "name": "Roberto Rocci"
    },
    {
      "name": "Maurizio Vichi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dm6x0tf5s8sac.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21538",
  "fulltext_url": null,
  "id": "dm6x0tf5s8sac",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21538",
  "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",
      "stat.ME"
    ],
    [
      "openalex",
      "1702|Advanced Clustering Algorithms Research"
    ]
  ],
  "openalex": "W7213933210",
  "pdf_url": "https://arxiv.org/pdf/2609.21538",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dm6x0tf5s8sac.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:12Z",
      "native_id": "2609.21538",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21538"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ME",
      "subfield": "statistics"
    }
  ],
  "title": "Fuzzy entropic triple k-means",
  "title_text": "Fuzzy entropic triple k-means",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:53Z",
  "url": "https://research.draconitic.com/paper/dm6x0tf5s8sac",
  "venue": "arXiv",
  "versions": []
}