k-Means has polynomial smoothed complexity

David Arthur; Bodo Manthey; Heiko Röglin

doi:10.1109/FOCS.2009.14

k-Means has polynomial smoothed complexity

David Arthur, Bodo Manthey, Heiko Röglin

Discrete Mathematics and Mathematical Programming

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

67 Citations (Scopus)

91 Downloads (Pure)

Abstract

The k-means method is one of the most widely used clustering algorithms, drawing its popularity from its speed in practice. Recently, however, it was shown to have exponential worst-case running time. In order to close the gap between practical performance and theoretical analysis, the k-means method has been studied in the model of smoothed analysis. But even the smoothed analyses so far are unsatisfactory as the bounds are still super-polynomial in the number n of data points. In this paper, we settle the smoothed running time of the k-means method. We show that the smoothed number of iterations is bounded by a polynomial in n and 1/sigma, where sigma is the standard deviation of the Gaussian perturbations. This means that if an arbitrary input data set is randomly perturbed, then the k-means method will run in expected polynomial time on that input set.

Original language	Undefined
Title of host publication	Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009)
Editors	D.A. Spielman
Place of Publication	Los Alamitos
Publisher	IEEE Computer Society
Pages	405-414
Number of pages	10
ISBN (Print)	978-0-7695-3850-1
DOIs	https://doi.org/10.1109/FOCS.2009.14
Publication status	Published - 2009
Event	The 50th Annual IEEE Symposium on Foundations of Computer Science 2009 - Renaissance Hotel Downtown Atlanta, Atlanta, United States Duration: 24 Oct 2009 → 27 Oct 2009 Conference number: 50 https://www.cc.gatech.edu/focs2009/

Publication series

Name
Publisher	IEEE Computer Society Press

Conference

Conference	The 50th Annual IEEE Symposium on Foundations of Computer Science 2009
Abbreviated title	FOCS 2009
Country	United States
City	Atlanta
Period	24/10/09 → 27/10/09
Internet address	https://www.cc.gatech.edu/focs2009/

Keywords

METIS-266490
IR-70194
Smoothed analyis
Clustering
Probabilistic Analysis
k-Means
k-Means clustering
k-Means method
EWI-16979

Access to Document

10.1109/FOCS.2009.14

FOCS2009_ArthurEA_kMeans
open
FOCS2009_ArthurEA_kMeans
open

http://doi.ieeecomputersociety.org/10.1109/FOCS.2009.14

Cite this

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title = "k-Means has polynomial smoothed complexity",

abstract = "The k-means method is one of the most widely used clustering algorithms, drawing its popularity from its speed in practice. Recently, however, it was shown to have exponential worst-case running time. In order to close the gap between practical performance and theoretical analysis, the k-means method has been studied in the model of smoothed analysis. But even the smoothed analyses so far are unsatisfactory as the bounds are still super-polynomial in the number n of data points. In this paper, we settle the smoothed running time of the k-means method. We show that the smoothed number of iterations is bounded by a polynomial in n and 1/sigma, where sigma is the standard deviation of the Gaussian perturbations. This means that if an arbitrary input data set is randomly perturbed, then the k-means method will run in expected polynomial time on that input set.",

keywords = "METIS-266490, IR-70194, Smoothed analyis, Clustering, Probabilistic Analysis, k-Means, k-Means clustering, k-Means method, EWI-16979",

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language = "Undefined",

isbn = "978-0-7695-3850-1",

publisher = "IEEE Computer Society",

pages = "405--414",

editor = "D.A. Spielman",

booktitle = "Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009)",

address = "United States",

note = "null ; Conference date: 24-10-2009 Through 27-10-2009",

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Arthur, D, Manthey, B & Röglin, H 2009, k-Means has polynomial smoothed complexity. in DA Spielman (ed.), Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009)., 10.1109/FOCS.2009.14, IEEE Computer Society, Los Alamitos, pp. 405-414, The 50th Annual IEEE Symposium on Foundations of Computer Science 2009, Atlanta, United States, 24/10/09. https://doi.org/10.1109/FOCS.2009.14

k-Means has polynomial smoothed complexity. / Arthur, David; Manthey, Bodo; Röglin, Heiko.

Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009). ed. / D.A. Spielman. Los Alamitos : IEEE Computer Society, 2009. p. 405-414 10.1109/FOCS.2009.14.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

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AB - The k-means method is one of the most widely used clustering algorithms, drawing its popularity from its speed in practice. Recently, however, it was shown to have exponential worst-case running time. In order to close the gap between practical performance and theoretical analysis, the k-means method has been studied in the model of smoothed analysis. But even the smoothed analyses so far are unsatisfactory as the bounds are still super-polynomial in the number n of data points. In this paper, we settle the smoothed running time of the k-means method. We show that the smoothed number of iterations is bounded by a polynomial in n and 1/sigma, where sigma is the standard deviation of the Gaussian perturbations. This means that if an arbitrary input data set is randomly perturbed, then the k-means method will run in expected polynomial time on that input set.

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