{
  "abstract": "Approximate message passing (AMP) provides fast iterative algorithms for high-dimensional signal recovery with Gaussian design matrices, while the Convex Gaussian Min--max Theorem (CGMT) gives a static optimization framework for obtaining sharp asymptotic characterizations of convex estimators. Although these two frameworks often lead to the same scalar state-evolution equations, their connection is usually indirect. In this paper, we establish a direct connection between the two for regularized linear regression in the proportional high-dimensional regime. When the CGMT Auxiliary Optimization (AO) and Primary Optimization (PO) give the same primal-dual solution, we show that the CGMT framework recovers the AMP fixed-point equations, including the Onsager correction. We further identify the AO Gaussian vectors with the Gaussian perturbations in the primal and residual AMP channels. For regularized $M$-estimation, the same viewpoint recovers the fixed point of scalar-variance max-sum Generalized AMP (GAMP). These results show that the AMP (and GAMP) fixed-point equations can be recovered from the CGMT framework and may suggest AMP-like algorithms in settings where standard derivations are unavailable.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.27769",
  "author_total": 2,
  "authors": [
    {
      "name": "Vikrant Malik"
    },
    {
      "name": "Babak Hassibi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9dtr2b9cexp2c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.27769",
  "fulltext_url": null,
  "id": "9dtr2b9cexp2c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.27769",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2606.27769",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9dtr2b9cexp2c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2606.27769",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.27769"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Deriving Approximate Message Passing from the Convex Gaussian Min-Max Theorem",
  "title_text": "Deriving Approximate Message Passing from the Convex Gaussian Min-Max Theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/9dtr2b9cexp2c",
  "venue": "arXiv",
  "versions": []
}