{
  "abstract": "We study the computational complexity of hypothesis testing in the spiked Wigner model, a prototypical model for detecting low-rank structure in a large random matrix. Below the \"BBP\" eigenvalue transition, it is expected that strong detection --- with both type I and II errors vanishing --- requires exponential time. Assuming this as a conjecture, we determine the limits of polynomial-time weak detection, exactly characterizing the possible tradeoffs between type I and II errors. Specifically, the optimal tradeoff is achieved by a particular linear spectral statistic. Thus, the question of weak detection is entirely reduced to that of strong detection. The proof builds on ideas of Nagda-Raghavendra (2025) and Moitra-Wein (2025). The low-degree likelihood ratio (LDLR) plays a key role: any test that slightly beats the LDLR can be boosted to have an even higher success probability. This leads us to establish a computational analogue of the Neyman-Pearson lemma for a subclass of additive Gaussian models: for a given super-polynomial runtime, the best possible tradeoff between type I and II errors is either the one achieved by thresholding the LDLR, or the trivial tradeoff that results from strong detection.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.36050",
  "author_total": 2,
  "authors": [
    {
      "name": "Ansh Nagda"
    },
    {
      "name": "Alexander S. Wein"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/00rjf9d6sxhdr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.36050",
  "fulltext_url": null,
  "id": "00rjf9d6sxhdr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.36050",
  "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.CC"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7214941364",
  "pdf_url": "https://arxiv.org/pdf/2609.36050",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/00rjf9d6sxhdr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.36050",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.36050"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "From Weak to Strong Testing in Gaussian Models",
  "title_text": "From Weak to Strong Testing in Gaussian Models",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:41Z",
  "url": "https://research.draconitic.com/paper/00rjf9d6sxhdr",
  "venue": "arXiv",
  "versions": []
}