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arXiv · 2506.14952

An Observation on Lloyd's k-Means Algorithm in High Dimensions

David Silva-Sánchez·Roy R. Lederman
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Abstract

Clustering and estimating cluster means are core problems in statistics and machine learning, with k-means and Expectation Maximization (EM) being two widely used algorithms. In this work, we provide a theoretical explanation for the failure of k-means in high-dimensional settings with high noise and limited sample sizes, using a simple Gaussian Mixture Model (GMM). We identify regimes where, with high probability, almost every partition of the data becomes a fixed point of the k-means algorithm. This study is motivated by challenges in the analysis of more complex cases, such as masked GMMs, and those arising from applications in Cryo-Electron Microscopy.

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David Silva-Sánchez, Roy R. Lederman. 2025-06-17. An Observation on Lloyd's k-Means Algorithm in High Dimensions. https://arxiv.org/abs/2506.14952

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