Fast and Accurate $k$-means++ via Rejection Sampling

$k$-means++ \cite{arthur2007k} is a widely used clustering algorithm that is easy to implement, has nice theoretical guarantees and strong empirical performance. Despite its wide adoption, $k$-means++ sometimes suffers from being slow on large data-sets so a natural question has been to obtain more efficient algorithms with similar guarantees. In this paper, we present such a near linear time algorithm for $k$-means++ seeding. Interestingly our algorithm obtains the same theoretical guarantees as $k$-means++ and significantly improves earlier results on fast $k$-means++ seeding. Moreover, we show empirically that our algorithm is significantly faster than $k$-means++ and obtains solutions of equivalent quality.