{
  "abstract": "We prove that every finite multivariate Gaussian mixture density has only finitely many modes. Our approach combines an algebraic formulation of the ridgeline theory of Ray and Lindsay (2005) with a transcendence-degree argument based on Ax's functional-transcendence theorem to bound the cardinality of the set of critical points. Our techniques extend recent work by Wang (2026), who used Ax's theorem together with real-analytic curve selection to prove finiteness of the critical set of homoscedastic Gaussian mixtures. We introduce the ridgeline correspondence and use it to obtain a finiteness result that applies to arbitrary heteroscedastic Gaussian mixtures. Our framework also establishes finiteness of the number of modes for additional classes of polynomial-exponential mixtures and generalized Gaussian mixtures.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28558",
  "author_total": 2,
  "authors": [
    {
      "name": "Carlos Améndola"
    },
    {
      "name": "Jose Israel Rodriguez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9ghssc72tzp38.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28558",
  "fulltext_url": null,
  "id": "9ghssc72tzp38",
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  "landing_url": "https://arxiv.org/abs/2608.28558",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
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      "arxiv",
      "math.ST"
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      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Bayesian Methods and Mixture Models"
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  "openalex": "W7204754676",
  "pdf_url": "https://arxiv.org/pdf/2608.28558",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9ghssc72tzp38.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.28558",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28558"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "A ridgeline correspondence criterion: the number of modes of a Gaussian mixture is finite",
  "title_text": "A ridgeline correspondence criterion: the number of modes of a Gaussian mixture is finite",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:34Z",
  "url": "https://research.draconitic.com/paper/9ghssc72tzp38",
  "venue": "arXiv",
  "versions": []
}