{
  "abstract": "Approximate Message Passing (AMP) algorithms are attractive as they are computationally efficient and simultaneously admit a precise characterization in terms of the low-dimensional \"state-evolution\" recursion. In this work, we establish quantitative universality for AMP with centered rank-one sensing matrices $Z_i=(x_ix_i^\\top-I_d)/\\sqrt d$, where $x_i$ are independent standard Gaussian vectors. These matrices arise in quadratic regression and learning quadratic neural networks. Their normalized vectorizations have isotropic covariance in dimension $p=d(d+1)/2$, but strongly dependent coordinates. For linear observations with Gaussian noise and prescribed smooth, bounded spectral denoisers, we compare rank-one sensing with a covariance-matched isotropic Gaussian ensemble. At a fixed iteration horizon, with $n\\leq Cp$ and uniform bounds on the normalized signal energy, initialization, coefficients, and first four denoiser derivatives, normalized signal overlaps, cross-time overlaps, and smooth linear spectral statistics differ by $O(d^{-1/2})$ in every fixed $L^m$. This yields universality of the normalized mean-squared error, fixed spectral moments, and empirical spectral distributions. Whenever the Gaussian observables admit a joint state-evolution limit, the same predictions hold under rank-one sensing. The proof uses columnwise Lindeberg replacement, leave-one-out analysis, smooth spectral expansions, and high-moment concentration. For prescribed smooth spectral denoisers, the result establishes the AMP universality conjectured by Maillard et al. (arXiv:2408.03733).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05327",
  "author_total": 3,
  "authors": [
    {
      "name": "Duy Thuc Nguyen"
    },
    {
      "name": "Yue M. Lu"
    },
    {
      "name": "Subhabrata Sen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/491vh7qw3n1m2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05327",
  "fulltext_url": null,
  "id": "491vh7qw3n1m2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05327",
  "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",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "2206|Sparse and Compressive Sensing Techniques"
    ]
  ],
  "openalex": "W7220743054",
  "pdf_url": "https://arxiv.org/pdf/2610.05327",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/491vh7qw3n1m2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05327",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05327"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Quantitative Universality of Approximate Message Passing for Rank-One Quadratic Sensing",
  "title_text": "Quantitative Universality of Approximate Message Passing for Rank-One Quadratic Sensing",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:16Z",
  "url": "https://research.draconitic.com/paper/491vh7qw3n1m2",
  "venue": "arXiv",
  "versions": []
}