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2019 On the minimum of the mean-squared error in 2-means clustering
Bernhard G. Bodmann, Craig J. George
Involve 12(2): 301-319 (2019). DOI: 10.2140/involve.2019.12.301

Abstract

We study the minimum mean-squared error for 2-means clustering when the outcomes of the vector-valued random variable to be clustered are on two spheres, that is, the surface of two touching balls of unit radius in n-dimensional Euclidean space, and the underlying probability distribution is the normalized surface measure. For simplicity, we only consider the asymptotics of large sample sizes and replace empirical samples by the probability measure. The concrete question addressed here is whether a minimizer for the mean-squared error identifies the two individual spheres as clusters. Indeed, in dimensions n3, the minimum of the mean-squared error is achieved by a partition obtained from a separating hyperplane tangent to both spheres at the point where they touch. In dimension n=2, however, the minimizer fails to identify the individual spheres; an optimal partition is associated with a hyperplane that does not contain the intersection of the two spheres.

Citation

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Bernhard G. Bodmann. Craig J. George. "On the minimum of the mean-squared error in 2-means clustering." Involve 12 (2) 301 - 319, 2019. https://doi.org/10.2140/involve.2019.12.301

Information

Received: 6 November 2017; Revised: 9 February 2018; Accepted: 7 March 2018; Published: 2019
First available in Project Euclid: 25 October 2018

zbMATH: 06980503
MathSciNet: MR3864219
Digital Object Identifier: 10.2140/involve.2019.12.301

Subjects:
Primary: 62H30

Keywords: $k$-means clustering , mean-squared error , performance guarantees

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.12 • No. 2 • 2019
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