{
  "abstract": "Low-rank matrix denoising is a central primitive in patch-based image restoration and many other inverse problems. Classical SVD-based image denoising methods often choose a truncation rank by matching residual singular-value energy with an estimated noise energy, but this rule is not a finite-sample risk principle because a fitted low-rank approximation inevitably absorbs part of the noise. This paper develops a mathematically rigorous alternative based on Stein's unbiased risk estimate (SURE). Since singular value hard thresholding is discontinuous and does not satisfy the hypotheses of Stein's lemma, we introduce a logistic smooth hard-threshold spectral estimator. We prove that the smooth shrinker satisfies the regularity conditions required by a spectral-estimator version of Stein's lemma, and therefore admits an exactly unbiased fixed-threshold risk estimate under Gaussian noise. For a fixed observed matrix and a finite set of candidate thresholds separated from the observed singular values, the ordering of the fixed-threshold smooth SURE objective eventually agrees with a simple limiting score. The limiting score has the same algebraic form as the biased hard-threshold SURE formula, but here it is used only as a computational device for ranking finite candidates. Selecting the minimizing threshold is a data-adaptive tuning step; the selected SURE value should not be interpreted as an unbiased risk estimate of the finally selected estimator.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.06632",
  "author_total": 3,
  "authors": [
    {
      "name": "Guanzhong Yang"
    },
    {
      "name": "Zi Wang"
    },
    {
      "name": "Guang Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2sxf4dn3pptkf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.06632",
  "fulltext_url": null,
  "id": "2sxf4dn3pptkf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.06632",
  "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.NA"
    ],
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      "arxiv",
      "eess.IV"
    ],
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      "arxiv",
      "eess.SP"
    ],
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      "arxiv",
      "math.NA"
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      "arxiv",
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      "arxiv",
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    [
      "openalex",
      "1707|Image and Signal Denoising Methods"
    ]
  ],
  "openalex": "W7163902716",
  "pdf_url": "https://arxiv.org/pdf/2606.06632",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2sxf4dn3pptkf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2606.06632",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.06632"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Smooth Hard-Thresholding for Singular Values with Stein's Unbiased Risk Estimate",
  "title_text": "Smooth Hard-Thresholding for Singular Values with Stein's Unbiased Risk Estimate",
  "type": "preprint",
  "updated_at": "2026-10-30T12:30:00Z",
  "url": "https://research.draconitic.com/paper/2sxf4dn3pptkf",
  "venue": "arXiv",
  "versions": []
}