Exact Recovery of Community Detection in dependent Gaussian Mixture Models

[2209.14859] Exact Recovery of Community Detection in dependent Gaussian Mixture Models Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Statistics Theory arXiv:2209.14859 (math) [Submitted on 23 Sep 2022 ( v1 ), last revised 2 May 2026 (this version, v3)] Title: Exact Recovery of Community Detection in dependent Gaussian Mixture Models Authors: Zhongyang Li , Sichen Yang View a PDF of the paper titled Exact Recovery of Community Detection in dependent Gaussian Mixture Models, by Zhongyang Li and Sichen Yang View PDF HTML (experimental) Abstract: We study exact recovery for community detection in a Gaussian mixture model with dependent and heterogeneous Gaussian noise. The noise covariance matrix $\Sigma$ may be non-diagonal and, in the general formulation, singular. In the singular case, we write the Gaussian likelihood on the support of the induced measure and show that the maximum likelihood estimator (MLE) is a constrained quadratic optimization problem involving the Moore--Penrose inverse. For general covariance structures, we obtain sufficient conditions for exact recovery of the MLE when the community sizes are unknown and when they are known. These conditions are driven by the $\Sigma$-whitened separation $L_\Sigma(x,y)$ together with local one-step comparison inequalities in the near-truth regime. Under the additional assumption that $\Sigma$ is invertible, we derive converse results showing failure of exact recovery when a large family of local perturbations has sufficiently nondegenerate Gaussian comparison statistics. We then analyze a full-rank non-diagonal block-covariance model, prove a sharp exact-recovery threshold in the unknown-size setting, and identify a general no-gap mechanism under which the sufficient and necessary conditions coincide asymptotically. Subjects: Statistics Theory (math.ST) ; Probability (math.PR); Machine Learning (stat.ML) MSC classes: 91D30 Cite as: arXiv:2209.14859 [math.ST] (or arXiv:2209.14859v3 [math.ST] for this version) https://doi.org/10.48550/arXiv.2209.14859 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Zhongyang Li [ view email ] [v1] Fri, 23 Sep 2022 14:26:19 UTC (32 KB) [v2] Thu, 9 Apr 2026 18:25:03 UTC (31 KB) [v3] Sat, 2 May 2026 20:58:46 UTC (39 KB) Full-text links: Access Paper: View a PDF of the paper titled Exact Recovery of Community Detection in dependent Gaussian Mixture Models, by Zhongyang Li and Sichen Yang View PDF HTML (experimental) TeX Source view license Current browse context: math.ST < prev | next > new | recent | 2022-09 Change to browse by: math math.PR stat stat.ML stat.TH References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: 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) Major funding support from

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