The Catastrophic Failure of The k-Means Algorithm in High Dimensions, and How Hartigan's Algorithm Avoids It

Lloyd's k-means algorithm is one of the most widely used clustering methods. We prove that in high-dimensional, high-noise settings, the algorithm exhibits catastrophic failure: with high probability, essentially every partition of the data is a fixed point. Consequently, Lloyd's algorithm simply returns its initial partition - even when the underlying clusters are trivially recoverable by other methods. In contrast, we prove that Hartigan's k-means algorithm does not exhibit this pathology. Our results show the stark difference between these algorithms and offer a theoretical explanation for the empirical difficulties often observed with k-means in high dimensions.

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

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.