Sets Clustering

The input to the sets-k-means problem is an integer k 1 and a set P = fP1; Png of fixed sized sets in Rd. The goal is to compute a set C of k centers (points) in Rd that minimizes the sum P P2P minp2P;c2C kp-ck2 of squared distances to these sets. An "-core-set for this problem is a weighted subset of P that approximates this sum up to 1 " factor, for every set C of k centers in Rd. We prove that such a core-set of O(log2 n) sets always exists, and can be computed in O(n log n) time, for every input P and every fixed d; k 1 and " 2 (0; 1). The result easily generalized for any metric space, distances to the power of z 0, and M-estimators that handle outliers. Applying an inefficient but optimal algorithm on this coreset allows us to obtain the first PTAS (1 + " approximation) for the sets-k-means problem that takes time near linear in n. This is the first result even for sets-mean on the plane (k = 1, d = 2). Open source code and experimental results for document classification and facility locations are also provided.