Davis-Kahan Theorem in the two-to-infinity norm and its application to perfect clustering

[2411.11728] Davis-Kahan Theorem in the two-to-infinity norm and its application to perfect clustering Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Statistics > Methodology arXiv:2411.11728 (stat) [Submitted on 18 Nov 2024 ( v1 ), last revised 14 Aug 2026 (this version, v3)] Title: Davis-Kahan Theorem in the two-to-infinity norm and its application to perfect clustering Authors: Marianna Pensky View a PDF of the paper titled Davis-Kahan Theorem in the two-to-infinity norm and its application to perfect clustering, by Marianna Pensky View PDF HTML (experimental) Abstract: Many statistical applications, such as the Principal Component Analysis, matrix completion, tensor regression and many others, rely on accurate estimation of leading eigenvectors of a matrix. The Davis-Kahan theorem is known to be instrumental for bounding above the distances between matrices $U$ and $\widehat{U}$ of population eigenvectors and their sample versions. While those distances can be measured in various metrics, the recent developments have shown advantages of evaluation of the deviation in the two-to-infinity norm. The purpose of this paper is to develop a toolbox for derivation of upper bounds for the distances between $U$ and $\widehat{U}$ in the two-to-infinity norm for a variety of possible scenarios. Although this problem has been studied by several authors, the difference between this paper and its predecessors is that the upper bounds are obtained under various sets of assumptions. The upper bounds are initially derived with no or mild probabilistic assumptions on the error, and are subsequently refined, when some generic probabilistic assumptions on the errors hold. The paper also provides rectification of the upper bounds in the cases of heavy-tailed or exponentially fast decaying errors. In addition, the paper suggests alternative methods for evaluation of $\widehat{U}$ and, therefore, enables one to compare the resulting accuracies. As an example of an application of the techniques in the paper, we derive sufficient conditions for perfect clustering in a generic setting, and then employ them in various scenarios. Comments: 45 pages Subjects: Methodology (stat.ME) Cite as: arXiv:2411.11728 [stat.ME] (or arXiv:2411.11728v3 [stat.ME] for this version) https://doi.org/10.48550/arXiv.2411.11728 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Marianna Pensky [ view email ] [v1] Mon, 18 Nov 2024 16:58:35 UTC (49 KB) [v2] Tue, 7 Apr 2026 22:27:04 UTC (60 KB) [v3] Fri, 14 Aug 2026 21:09:46 UTC (63 KB) Full-text links: Access Paper: View a PDF of the paper titled Davis-Kahan Theorem in the two-to-infinity norm and its application to perfect clustering, by Marianna Pensky View PDF HTML (experimental) TeX Source view license Current browse context: stat.ME < prev | next > new | recent | 2024-11 Change to browse by: stat References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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