{
  "abstract": "We prove a sharp lower bound for growing-degree polynomial estimation in the symmetric rank-one matrix estimation problem. The observation is \\[ \\boldsymbol{Y}=\\frac{1}{\\sqrt{n}} \\boldsymbolθ \\boldsymbolθ^{\\top}+\\boldsymbol{W}, \\] where the coordinates of $\\boldsymbolθ$ are independent with a common, fixed sub-Gaussian distribution $π_Θ$ satisfying $\\mathbb E[Θ]\\neq0$, and $\\boldsymbol{W}$ is symmetric with independent standard Gaussian upper-triangular entries. For every such prior, we give an explicit finite-dimensional bound implying that every sequence of polynomial estimators of degree $D(n)=o(n^{1/60})$ has normalized mean-square error with limit inferior at least $\\mathbb E[Θ^2]-q_{\\mathsf{amp}}$, the limiting error of Bayes approximate message passing (AMP). This extends the constant-degree equivalence of Montanari and Wein~\\cite{montanari2025equivalence} to growing degree under the same prior assumptions. Combined with their polynomial approximation of fixed-iteration AMP, the bound identifies the exact limiting low-degree MMSE whenever $D(n)\\to\\infty$ within this range. It thereby answers the growing-degree AMP-equivalence question discussed in~\\cite{wein2025computational, maleki2026high} for nonzero-mean sub-Gaussian priors in the stated degree range. We also obtain a moment-dependent finite-dimensional lower bound for any nonzero-mean prior with all moments finite. The proof bounds low-degree MMSE using \\emph{conditional} joint cumulants and a refined application of Jensen's inequality. The conditional cumulants are defined using an auxiliary Gaussian channel $\\boldsymbol{R}$ calibrated to the AMP fixed point. This retains signal dependence that is lost in unconditional cumulant bounds and produces the cancellations needed for quantitative control as the degree grows.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06988",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhangsong Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9ftzy9v5s21wx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06988",
  "fulltext_url": null,
  "id": "9ftzy9v5s21wx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06988",
  "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.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2206|Sparse and Compressive Sensing Techniques"
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  ],
  "openalex": "W7212034061",
  "pdf_url": "https://arxiv.org/pdf/2609.06988",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9ftzy9v5s21wx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2609.06988",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06988"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Almost Sharp Equivalence between Approximate Message Passing and Low-Degree Polynomials",
  "title_text": "Almost Sharp Equivalence between Approximate Message Passing and Low-Degree Polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:38Z",
  "url": "https://research.draconitic.com/paper/9ftzy9v5s21wx",
  "venue": "arXiv",
  "versions": []
}