Peer-reviewedCC BY 4.0Probability Theory and Related Fields13 Jun 2026

Free denoising via overlap measures and c-freeness techniques

Maxime Fevrier¹
Alexandru Nica²
Kamil Szpojankowski^3,4

4 affiliations

¹Laboratoire de mathématiques d’Orsay, Université Paris-Saclay, CNRS, Orsay, 91405 France
²Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario Canada
³Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00–656 Warszawa, Poland
⁴Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, Warszawa, Poland

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}μ ^{( 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}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}μ ^{( 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}μ _a× μ _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^1/2ba^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}( 𝒜 ,φ ,χ )\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}(φ ,χ )\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.

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Cite

BibTeX RIS

BibTeX

@article{fevrier2026free,
  title = {{Free denoising via overlap measures and c-freeness techniques}},
  author = {Maxime Fevrier and Alexandru Nica and Kamil Szpojankowski},
  journal = {Probability Theory and Related Fields},
  year = {2026},
  month = jun,
  doi = {10.1007/s00440-026-01507-8},
  url = {https://doi.org/10.1007/s00440-026-01507-8},
  note = {CC BY 4.0},
}

RIS

TY  - JOUR
TI  - Free denoising via overlap measures and c-freeness techniques
AU  - Maxime Fevrier
AU  - Alexandru Nica
AU  - Kamil Szpojankowski
PY  - 2026
DA  - 2026/06/13
JO  - Probability Theory and Related Fields
DO  - 10.1007/s00440-026-01507-8
UR  - https://doi.org/10.1007/s00440-026-01507-8
AB  - 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}μ ^( 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}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}μ ^( 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}μ _a× μ _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^1/2ba^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}( 𝒜 ,φ ,χ )\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}(φ ,χ )\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.
SN  - 0178-8051
SN  - 1432-2064
ER  -