Improved Coresets for Euclidean k-Means

Given a set of n points in d dimensions, the Euclidean k-means problem consists of finding k centers such that the sum of squared distances from every point to its closest center is minimized. The arguably most popular way of dealing with this problem in the big data setting is to first compress the data by computing a weighted subset known as a coreset and then run any algorithm on this subset. The guarantee of the coreset is that for any candidate solution, the ratio between coreset cost and the cost of the original instance is less than a (1+epsilon) factor. The current state of the art coreset size for Euclidean k-means is O(k epsilon^{-2} min(k, epsilon^{-2})). This matches the lower bound of Omega(k epsilon^{-2}) up to the min(k, epsilon^{-2}) term. In this paper, we improve this bound to min(sqrt(k), epsilon^{-2}). In the regime where k = o(epsilon^{-2}), this is a strict improvement over the state of the art. In particular, ours is the first provable bound that breaks through the k^2 barrier while retaining an optimal dependency on epsilon.