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Noisy k-Means++ Revisited

Authors: Christoph Grunau, Ahmet Alper Özüdoğru, and Václav Rozhoň

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 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.

Cite as

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)

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  author =	{Grunau, Christoph and \"{O}z\"{u}do\u{g}ru, Ahmet Alper and Rozho\v{n}, V\'{a}clav},
  title =	{{Noisy k-Means++ Revisited}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{55:1--55:7},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.55},
  URN =		{urn:nbn:de:0030-drops-187080},
  doi =		{10.4230/LIPIcs.ESA.2023.55},
  annote =	{Keywords: clustering, k-means, k-means++, adversarial noise}
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