On k-means iterations and Gaussian clusters

Nowadays, k-means remains arguably the most popular clustering algorithm (Jain, 2010; Vouros et al., 2021). Two of its main properties are simplicity and speed in practice. Here, our main claim is that the average number of iterations k-means takes to converge (τ¯) is in fact very informative. We find this to be particularly interesting because τ¯ is always known when applying k-means but has never been, to our knowledge, used in the data analysis process. By experimenting with Gaussian clusters, we show that τ¯ is related to the structure of a data set under study. Data sets containing Gaussian clusters have a much lower τ¯ than those containing uniformly random data. In fact, we go considerably further and demonstrate a pattern of inverse correlation between τ¯ and the clustering quality. We illustrate the importance of our findings through two practical applications. First, we describe the cases in which τ¯ can be effectively used to identify irrelevant features present in a given data set or be used to improve the results of existing feature selection algorithms. Second, we show that there is a strong relationship between τ¯ and the number of clusters in a data set, and that this relationship can be used to find the true number of clusters it contains.