Clustering Oligarchies

We investigate the extent to which clustering algorithms are robust to the addition of a small, potentially adversarial, set of points. Our analysis reveals radical differences in the robustness of popular clustering methods. k-means and several related techniques are robust when data is clusterable, and we provide a quantitative analysis capturing the precise relationship between clusterability and robustness. In contrast, common linkage-based algorithms and several standard objective-function-based clustering methods can be highly sensitive to the addition of a small set of points even when the data is highly clusterable. We call such sets of points oligarchies. Lastly, we show that the behavior with respect to oligarchies of the popular Lloyd’s method changes radically with the initialization technique.