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Noisy k-Means++ Revisited

Authors Christoph Grunau , Ahmet Alper Özüdoğru, Václav Rozhoň

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Author Details

Christoph Grunau
  • ETH Zürich, Switzerland
Ahmet Alper Özüdoğru
  • ETH Zürich, Switzerland
Václav Rozhoň
  • ETH Zürich, Switzerland


We would like to thank Mohsen Ghaffari for many helpful comments.

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Christoph Grunau, Ahmet Alper Özüdoğru, and Václav Rozhoň. Noisy k-Means++ Revisited. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 55:1-55:7, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


The k-means++ algorithm by Arthur and Vassilvitskii [SODA 2007] is a classical and time-tested algorithm for the k-means problem. While being very practical, the algorithm also has good theoretical guarantees: its solution is O(log k)-approximate, in expectation. In a recent work, Bhattacharya, Eube, Roglin, and Schmidt [ESA 2020] considered the following question: does the algorithm retain its guarantees if we allow for a slight adversarial noise in the sampling probability distributions used by the algorithm? This is motivated e.g. by the fact that computations with real numbers in k-means++ implementations are inexact. Surprisingly, the analysis under this scenario gets substantially more difficult and the authors were able to prove only a weaker approximation guarantee of O(log² k). In this paper, we close the gap by providing a tight, O(log k)-approximate guarantee for the k-means++ algorithm with noise.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Unsupervised learning and clustering
  • clustering
  • k-means
  • k-means++
  • adversarial noise


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