{
  "abstract": "Over-fitted Gaussian mixtures can contain many parameter configurations with nearly the same density, making the order of density cancellation central to their parameter-learning rates. Ho and Nguyen encoded these cancellations in a polynomial system and conjectured its exact order as a function of cluster size. The key to resolving the conjecture is a common positive-variance shift that turns the coefficient equations into Gaussian moment matching. For every cluster size $\\Mhn\\ge2$, we prove that the real, nonzero-weight system has order $2\\Mhn$: a sharp sign-change bound forces rigidity at that order, while positive Gauss--Hermite quadrature constructs a non-trivial solution through order $2\\Mhn-1$. Population-density calculations at 110-digit working precision recover the predicted leading term for two to six components; finite multistart searches recover the same observed powers for two to five components, with smaller finite-scale constants for three to five components. For parameter learning, the exact order gives an explicit pointwise maximum-likelihood exponent and an exact local minimax order on specified shrinking neighbourhoods of a fixed known mixing measure. The same coefficient viewpoint separates high-order cancellation from exact parameter symmetries in mixtures of experts. Synchronized replication preserves the conditional density while keeping the corresponding parameter losses positive in two broad standard-softmax classes, showing why statistical rate arguments must account for exact parameter symmetries.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27265",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuyang Zhang"
    },
    {
      "name": "Qihuang Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/af6kec7ejhewb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27265",
  "fulltext_url": null,
  "id": "af6kec7ejhewb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27265",
  "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.LG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Bayesian Methods and Mixture Models"
    ]
  ],
  "openalex": "W7223097163",
  "pdf_url": "https://arxiv.org/pdf/2610.27265",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/af6kec7ejhewb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27265",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27265"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Solving the Ho--Nguyen Conjecture for Gaussian Mixtures: Applications to Mixtures of Experts",
  "title_text": "Solving the Ho--Nguyen Conjecture for Gaussian Mixtures: Applications to Mixtures of Experts",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:13Z",
  "url": "https://research.draconitic.com/paper/af6kec7ejhewb",
  "venue": "arXiv",
  "versions": []
}