{
  "abstract": "Consider Gaussian denoising with a fixed prior and varying noise variance. When the prior has a density, maximum a posteriori estimation uses its negative logarithm as a fixed regularizer; only the weight of the quadratic data term changes with the noise variance. We ask whether a fixed regularizer can also recover the posterior mean. Although a suitable regularizer exists at each variance, we prove that no single regularizer works at two distinct positive variances for any compactly supported prior other than a point mass. This holds even for nonconvex and nonsmooth regularizers. A residual computed from two denoiser evaluations gives positive lower bounds on the uniform approximation error and excess mean squared error of estimators using a common regularizer. Numerical examples illustrate these bounds and the cost of reusing a regularizer across noise levels.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13926",
  "author_total": 1,
  "authors": [
    {
      "name": "Kang Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ap256k62za8pv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13926",
  "fulltext_url": null,
  "id": "ap256k62za8pv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13926",
  "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",
      "eess.SP"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "math.ST"
    ],
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      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1707|Image and Signal Denoising Methods"
    ]
  ],
  "openalex": "W7213232459",
  "pdf_url": "https://arxiv.org/pdf/2609.13926",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ap256k62za8pv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13926",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13926"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Can one regularizer reproduce posterior means at two noise levels?",
  "title_text": "Can one regularizer reproduce posterior means at two noise levels?",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:36Z",
  "url": "https://research.draconitic.com/paper/ap256k62za8pv",
  "venue": "arXiv",
  "versions": []
}