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16976 Improved smoothed analysis of the $k$ -means method

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Manthey, B. and Röglin, H. (2009) Improved smoothed analysis of the $k$ -means method. In: Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, 4-6 Jan 2009, New York, NY, USA. pp. 461-470. SIAM. ISBN not assigned

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Official URL: http://www.siam.org/proceedings/soda/2009/SODA09_051_mantheyb.pdf

Abstract

The $k$ -means method is a widely used clustering algorithm. One of its distinguished features is its speed in practice. Its worst-case running-time, however, is exponential, leaving a gap between practical and theoretical performance. Arthur and Vassilvitskii aimed at closing this gap, and they proved a bound of poly( $n^k, 1/\sigma$ ) on the smoothed runningtime of the $k$ -means method, where $n$ is the number of data points and $\sigma$ is the standard deviation of the Gaussian perturbation. This bound, though better than the worstcase bound, is still much larger than the running-time observed in practice.
We improve the smoothed analysis of the $k$ -means method by showing two upper bounds on the expected running-time of $k$ -means. First, we prove that the expected running-time is bounded by a polynomial in $n^{\sqrt{k}}$ and $1/\sigma.$ Second, we prove an upper bound of $k^{kd}$ poly( $n, 1/\sigma$ ), where $d$ is the dimension of the data space. The polynomial is independent of $k$ and $d$ , and we obtain a polynomial bound for the expected running-time for $k, d \in O(\sqrt{\log n/ \log\log n}).$
Finally, we show that $k$ -means runs in smoothed polynomial time for one-dimensional instances.

Item Type:	Conference or Workshop Paper (Full Paper, Talk)
Research Group:	EWI-DMMP: Discrete Mathematics and Mathematical Programming
Research Program:	CTIT-IE&ICT: Industrial Engineering and ICT
Uncontrolled Keywords:	Smoothed analysis, $k$ -means method, Clustering
ID Code:	16976
Status:	Published
Deposited On:	10 December 2009
Refereed:	Yes
International:	Yes
More Information:	statistics metis

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