Free denoising via overlap measures and c-freeness techniques

• 成果类型: Article; Early Access
• 署名作者: Fevrier, Maxime; Nica, Alexandru; Szpojankowski, Kamil
• 署名单位: Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); University of Waterloo; Polish Academy of Sciences; Institute of Mathematics of the Polish Academy of Sciences; Warsaw University of Technology
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-026-01507-8
• 发表日期: 2026-06-13
• 关键词: FREE MULTIPLICATIVE CONVOLUTION; REGULARITY; atoms

摘要

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 E(a|a+b)\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}. To that end, we study a probability measure mu a,a+b(ov)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mu <^>{( extrm{ov} )}_{a,a+b} $$\end{document} on R2\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} which we call the overlap measure. We show that mu a,a+b(ov)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mu <^>{( extrm{ov} )}_{a,a+b} $$\end{document} is absolutely continuous with respect to the product measure mu a & times;mu a+b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _a imes \mu _{a+b}$$\end{document}. The Radon-Nikodym derivative gives direct access to E(a|a+b)\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}. We show that analogous results hold in the case of multiplicative noise when a, b are positive and the aim is to find E(a|a1/2ba1/2)\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}. In a parallel development we show that, for a general selfadjoint expression P(a, b) made with a and b, finding E(a|P(a,b))\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} is equivalent to finding the distribution of P(a, b) in a certain two-state probability space (A,phi,chi)\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}, where a, b are c-free with respect to (phi,chi)\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} in the sense of Bo & zdot;ejko-Leinert-Speicher. We discuss how free denoising (which is set in the framework of an abstract W & lowast;\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}-probability space) relates to the notion of matrix denoising previously discussed in the random matrix literature.