[2309.00578] Consistency of Lloyd's Algorithm Under Perturbations Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Computer Science > Machine Learning arXiv:2309.00578 (cs) [Submitted on 1 Sep 2023 ( v1 ), last revised 25 Apr 2026 (this version, v2)] Title: Consistency of Lloyd's Algorithm Under Perturbations Authors: Dhruv Patel , Hui Shen , Shankar Bhamidi , Yufeng Liu , Vladas Pipiras View a PDF of the paper titled Consistency of Lloyd's Algorithm Under Perturbations, by Dhruv Patel and Hui Shen and Shankar Bhamidi and Yufeng Liu and Vladas Pipiras View PDF HTML (experimental) Abstract: In the context of unsupervised learning, Lloyd's algorithm is one of the most widely used clustering algorithms. It has inspired a plethora of work investigating the correctness of the algorithm under various settings with ground truth clusters. In particular, in 2016, Lu and Zhou have shown that the mis-clustering rate of Lloyd's algorithm on $n$ independent samples from a sub-Gaussian mixture is exponentially bounded after $O(\log(n))$ iterations, assuming proper initialization of the algorithm. However, in many applications, the true samples are unobserved and need to be learned from the data via pre-processing pipelines such as spectral methods on appropriate data matrices. We show that the mis-clustering rate of Lloyd's algorithm on perturbed samples from a sub-Gaussian mixture is also exponentially bounded after $O(\log(n))$ iterations under the assumptions of proper initialization and that the perturbation is small relative to the sub-Gaussian noise. In canonical settings with ground truth clusters, we derive bounds for algorithms such as $k$-means$++$ to find good initializations and thus leading to the correctness of clustering via the main result. We show the implications of the results for pipelines measuring the statistical significance of derived clusters from data such as SigClust. We use these general results to derive implications in providing theoretical guarantees on the misclustering rate for Lloyd's algorithm in a host of applications, including high-dimensional time series, multi-dimensional scaling, and community detection for sparse networks via spectral clustering. Subjects: Machine Learning (cs.LG) ; Statistics Theory (math.ST) MSC classes: 62E20, 60C05 Cite as: arXiv:2309.00578 [cs.LG] (or arXiv:2309.00578v2 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2309.00578 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Hui Shen [ view email ] [v1] Fri, 1 Sep 2023 16:45:52 UTC (633 KB) [v2] Sat, 25 Apr 2026 15:06:11 UTC (648 KB) Full-text links: Access Paper: View a PDF of the paper titled Consistency of Lloyd's Algorithm Under Perturbations, by Dhruv Patel and Hui Shen and Shankar Bhamidi and Yufeng Liu and Vladas Pipiras View PDF HTML (experimental) TeX Source view license Current browse context: cs.LG < prev | next > new | recent | 2023-09 Change to browse by: cs math math.ST stat stat.TH References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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