Free denoising via overlap measures and c-freeness techniques

[2412.20792] Free denoising via overlap measures and c-freeness techniques Skip to main content Search Submit Donate Log in Mathematics > Operator Algebras arXiv:2412.20792 (math) [Submitted on 30 Dec 2024 ( v1 ), last revised 23 May 2025 (this version, v3)] Title: Free denoising via overlap measures and c-freeness techniques Authors: Maxime Fevrier , Alexandru Nica , Kamil Szpojankowski View PDF HTML (experimental) 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 $E(a|a+b)$. To that end, we study a probability measure $\mu^{( \mathrm{ov} )}_{a,a+b}$ on $\mathbb{R}^2$ which we call the overlap measure. We show that $\mu^{( \mathrm{ov} )}_{a,a+b}$ is absolutely continuous with respect to the product measure $\mu_a\times \mu_{a+b}$. The Radon-Nikodym derivative gives direct access to $E(a|a+b)$. We show that analogous results hold in the case of multiplicative noise when $a,b$ are positive and the aim is to find $E(a|a^{1/2}ba^{1/2})$. In a parallel development we show that, for a general selfadjoint expression $P(a,b)$ made with $a$ and $b$, finding $E(a|P(a,b))$ is equivalent to finding the distribution of $P(a,b)$ in a certain two-state probability space $(\mathcal{A},\varphi,\chi)$, where $a,b$ are c-free with respect to $(\varphi,\chi)$ in the sense of Bożejko-Leinert-Speicher. We discuss how free denoising (which is set in the framework of an abstract $W^{*}$-probability space) relates to the notion of ''matrix denoising'' previously discussed in the random matrix literature. Comments: Version 3: Framework of Borel non-commutative probability space introduced, relation to matrix denoising updated Subjects: Operator Algebras (math.OA) ; Probability (math.PR) Cite as: arXiv:2412.20792 [math.OA] (or arXiv:2412.20792v3 [math.OA] for this version) https://doi.org/10.48550/arXiv.2412.20792 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Probab. Theory Relat. Fields (2026) Related DOI : https://doi.org/10.1007/s00440-026-01507-8 Focus to learn more DOI(s) linking to related resources Submission history From: Kamil Szpojankowski [ view email ] [v1] Mon, 30 Dec 2024 08:18:56 UTC (35 KB) [v2] Sun, 16 Mar 2025 01:16:08 UTC (38 KB) [v3] Fri, 23 May 2025 06:29:34 UTC (55 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.OA < prev | next > new | recent | 2024-12 Change to browse by: math math.PR References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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