A Nearly Optimal Size Coreset Algorithm with Nearly Linear Time

A coreset is a point set containing information about geometric properties of a larger point set. A series of previous works show that in many machine learning problems, especially in clustering problems, coreset could be very useful to build efficient algorithms. Two main measures of an coreset construction algorithm's performance are the running time of the algorithm and the size of the coreset output by the algorithm. In this paper we study the construction of coresets for the $(k,z)$-clustering problem, which is a generalization of $k$-means and $k$-median problem. By properly designing a sketching-based distance estimation data structure, we propose faster algorithms that construct coresets with matching size of the state-of-the-art results.