Dual-tree $k$-means with bounded iteration runtime

k-means is a widely used clustering algorithm, but for k clusters and a dataset size of N , each iteration of Lloyd’s algorithm costs O(kN) time. Although there are existing techniques to accelerate single Lloyd iterations, none of these are tailored to the case of large k, which is increasingly common as dataset sizes grow. We propose a dual-tree algorithm that gives the exact same results as standard k-means; when using cover trees, we use adaptive analysis techniques to, under some assumptions, bound the single-iteration runtime of the algorithm as O(N + k log k). To our knowledge these are the first subO(kN) bounds for exact Lloyd iterations. We then show that this theoretically favorable algorithm performs competitively in practice, especially for large N and k in low dimensions. Further, the algorithm is treeindependent, so any type of tree may be used.