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Noisy k-Means++ Revisited

Authors Christoph Grunau , Ahmet Alper Özüdoğru, Václav Rozhoň

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Subject Classification

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

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Abstract

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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.ESA.2023.55

Author Details

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

Funding

  • Grunau, Christoph: Supported by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 853109).
  • Rozhoň, Václav: Supported by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 853109).

Acknowledgements

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

References

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