Involve: A Journal of Mathematics

  • Involve
  • Volume 12, Number 2 (2019), 301-319.

On the minimum of the mean-squared error in 2-means clustering

Bernhard G. Bodmann and Craig J. George

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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.

Article information

Involve, Volume 12, Number 2 (2019), 301-319.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]

$k$-means clustering performance guarantees mean-squared error


Bodmann, Bernhard G.; George, Craig J. On the minimum of the mean-squared error in 2-means clustering. Involve 12 (2019), no. 2, 301--319. doi:10.2140/involve.2019.12.301.

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