{
  "abstract": "We study score learning from denoising pairs when the data contain a hidden non-Gaussian component. If covariance reveals the component, its subspace can be recovered directly. If covariance is isotropic, the first distinguishing moment has order $k^\\ast$, and the learner must extract that moment from clean points and their noisy views. Correlations in the denoising loss are Stein discrepancies against a baseline; for polynomial fields they determine exactly the corresponding moment differences. Averaging fresh noisy views conditional on a clean point gives an exact moment query on the clean distribution. In the covariance-matched regime, orientation-independent dictionaries satisfying an $L^4/L^2$ condition need order $d^{k^\\ast-1}$ features, and the stated dot-product kernel ridge models need the same power in samples. Small-step online SGD on one unit has this exponent up to logarithms. A two-stage learner uses the curvature of a smoothed denoising loss to recover the hidden direction, then sharpens it and fits a scalar readout. For every even $k^\\ast$, it obtains a fixed score gain over the Gaussian baseline from $O(d^{k^\\ast/2}\\log d)$ clean points. In the covariance-visible regime, the frozen-feature threshold occurs at $d^J/J!$, while projection onto principal components makes the fitted-model, score-risk and generated-law distributions independent of ambient dimension. In the covariance-matched experiments, the measured partial-trace overlaps are consistent with the proposed spectral transition. A width-8 network trained with clean-point reuse shows a $d^2$-like training-set threshold over $d=64$--$192$, with fixed validation and calibration overhead.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04043",
  "author_total": 1,
  "authors": [
    {
      "name": "Rui Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/djg0y8xrtks2c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04043",
  "fulltext_url": null,
  "id": "djg0y8xrtks2c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04043",
  "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.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04043",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/djg0y8xrtks2c.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:34Z",
      "native_id": "2611.04043",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04043"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Denoising Is Stein: Moment Access and a Resource Hierarchy for Learning Hidden Structure in Diffusion Models",
  "title_text": "Denoising Is Stein: Moment Access and a Resource Hierarchy for Learning Hidden Structure in Diffusion Models",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:10Z",
  "url": "https://research.draconitic.com/paper/djg0y8xrtks2c",
  "venue": "arXiv",
  "versions": []
}