{
  "abstract": "We study the problem of free denoising. For free selfadjoint random variables a, b, where we interpret a as a signal and b as noise, we find \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ E (a \\, | \\, a+b)$$\\end{document}E(a|a+b). To that end, we study a probability measure \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mu ^{( \\textrm{ov} )}_{a,a+b} $$\\end{document}μa,a+b(ov) on \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathbb {R} ^2$$\\end{document}R2 which we call the overlap measure. We show that \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mu ^{( \\textrm{ov} )}_{a,a+b} $$\\end{document}μa,a+b(ov) is absolutely continuous with respect to the product measure \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu _a\\times \\mu _{a+b}$$\\end{document}μa×μa+b. The Radon-Nikodym derivative gives direct access to \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ E (a \\, | \\, a+b)$$\\end{document}E(a|a+b). We show that analogous results hold in the case of multiplicative noise when a, b are positive and the aim is to find \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ E (a \\, | \\, a^{1/2}ba^{1/2})$$\\end{document}E(a|a1/2ba1/2). In a parallel development we show that, for a general selfadjoint expression P(a, b) made with a and b, finding \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ E (a \\, | \\, P(a,b))$$\\end{document}E(a|P(a,b)) is equivalent to finding the distribution of P(a, b) in a certain two-state probability space \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$( \\mathcal {A} ,\\varphi ,\\chi )$$\\end{document}(A,φ,χ), where a, b are c-free with respect to \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\varphi ,\\chi )$$\\end{document}(φ,χ) in the sense of Bożejko-Leinert-Speicher. We discuss how free denoising (which is set in the framework of an abstract \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{*}$$\\end{document}W∗-probability space) relates to the notion of “matrix denoising” previously discussed in the random matrix literature.",
  "abstract_shown": true,
  "affiliations": [
    "Laboratoire de mathématiques d’Orsay, Université Paris-Saclay, CNRS, Orsay, 91405 France",
    "Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario Canada",
    "Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00–656 Warszawa, Poland",
    "Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, Warszawa, Poland"
  ],
  "article_type": "research article",
  "arxiv": null,
  "author_total": 3,
  "authors": [
    {
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        0
      ],
      "name": "Maxime Fevrier"
    },
    {
      "affs": [
        1
      ],
      "name": "Alexandru Nica"
    },
    {
      "affs": [
        2,
        3
      ],
      "name": "Kamil Szpojankowski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c3bz0c1gcmw26.bib",
  "cited_by": 0,
  "doi": "10.1007/s00440-026-01507-8",
  "fulltext_url": "https://pmc.ncbi.nlm.nih.gov/articles/PMC13719853/",
  "id": "c3bz0c1gcmw26",
  "issns": [
    "0178-8051",
    "1432-2064"
  ],
  "landing_url": "https://doi.org/10.1007/s00440-026-01507-8",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "heading",
      "Research"
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      "2613|Random Matrices and Applications"
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  "pdf_url": "https://link.springer.com/content/pdf/10.1007/s00440-026-01507-8.pdf",
  "pmcid": "PMC13719853",
  "pmid": "42964267",
  "published": "2026-06-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c3bz0c1gcmw26.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:23:44Z",
      "native_id": "PMC13719853",
      "source": "pmc",
      "url": "https://doi.org/10.1007/s00440-026-01507-8"
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2613|Random Matrices and Applications",
      "subfield": "statistics"
    }
  ],
  "title": "Free denoising via overlap measures and c-freeness techniques",
  "title_text": "Free denoising via overlap measures and c-freeness techniques",
  "type": "peer-reviewed",
  "updated_at": "2026-11-03T12:20:50Z",
  "url": "https://research.draconitic.com/paper/c3bz0c1gcmw26",
  "venue": "Probability Theory and Related Fields",
  "versions": []
}