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In many applications we want to find the number of clusters in a dataset. Acommon approach is to use the penalized k-means algorithm with an additivepenalty term linear in the number of clusters. An open problem is estimatingthe value of the coefficient of the penalty term. Since estimating the value ofthe coefficient in a principled manner appears to be intractable for generalclusters, we investigate "ideal clusters", i.e. identical spherical clusterswith no overlaps and no outlier background noise. In this paper: (a) We derive,for the case of ideal clusters, rigorous bounds for the coefficient of theadditive penalty. Unsurprisingly, the bounds depend on the correct number ofclusters, which we want to find in the first place. We further show thatadditive penalty, even for this simplest case of ideal clusters, typicallyproduces a weak and often ambiguous signature for the correct number ofclusters. (b) As an alternative, we examine the k-means with multiplicativepenalty, and show that this parameter-free formulation has a stronger, and lessoften ambiguous, signature for the correct number of clusters. We alsoempirically investigate certain types of deviations from ideal clusterassumption and show that combination of k-means with additive andmultiplicative penalties can resolve ambiguous solutions.

Penalized k-means algorithms for finding the correct number of clusters in a dataset