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

In many applications we want to find the number of clusters in a dataset. A common approach is to use the penalized k-means algorithm with an additive penalty term linear in the number of clusters. An open problem is estimating the value of the coefficient of the penalty term. Since estimating the value of the coefficient in a principled manner appears to be intractable for general clusters, we investigate "ideal clusters", i.e. identical spherical clusters with 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 the additive penalty. Unsurprisingly, the bounds depend on the correct number of clusters, which we want to find in the first place. We further show that additive penalty, even for this simplest case of ideal clusters, typically produces a weak and often ambiguous signature for the correct number of clusters. (b) As an alternative, we examine the k-means with multiplicative penalty, and show that this parameter-free formulation has a stronger, and less often ambiguous, signature for the correct number of clusters. We also empirically investigate certain types of deviations from ideal cluster assumption and show that combination of k-means with additive and multiplicative penalties can resolve ambiguous solutions.