{
  "abstract": "Gaussian mixtures can have more modes than components. For three Gaussian components with a common covariance, four modes are attainable, whereas the best general upper bound was eight. We prove that four is the exact maximum in every dimension $d \\ge 2$, for arbitrary means and positive weights. We further show that the sharp bound drops to three when the whitened means are collinear or form a right or obtuse triangle. The result covers degenerate modes without genericity or separate finiteness assumptions. Our proof maps modes to local minima of a separable free-energy functional on the probability simplex, reduces the problem to counting scalar inverse-function branches, and controls competing branches through curvature and stationary-height comparisons.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15369",
  "author_total": 2,
  "authors": [
    {
      "name": "Rintaro Okahara"
    },
    {
      "name": "Kei Kobayashi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ejex0ev5wg2nm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15369",
  "fulltext_url": null,
  "id": "ejex0ev5wg2nm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15369",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Bayesian Methods and Mixture Models"
    ]
  ],
  "openalex": "W7221828481",
  "pdf_url": "https://arxiv.org/pdf/2610.15369",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ejex0ev5wg2nm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15369",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15369"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Three Homoscedastic Gaussian Components Have at Most Four Modes",
  "title_text": "Three Homoscedastic Gaussian Components Have at Most Four Modes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:26Z",
  "url": "https://research.draconitic.com/paper/ejex0ev5wg2nm",
  "venue": "arXiv",
  "versions": []
}