New Frameworks for Offline and Streaming Coreset Constructions

Let P be a set (called points), Q be a set (called queries) and a function f : P×Q→ [0,∞) (called cost). For an error parameter > 0, a set S ⊆ P with a weight function w : P → [0,∞) is an ε-coreset if ∑ s∈S w(s)f(s, q) approximates ∑ p∈P f(p, q) up to a multiplicative factor of 1 ± ε for every given query q ∈ Q. Coresets are used to solve fundamental problems in machine learning of streaming and distributed data. We construct coresets for the k-means clustering of n input points, both in an arbitrary metric space and d-dimensional Euclidean space. For Euclidean space, we present the first coreset whose size is simultaneously independent of both d and n. In particular, this is the first coreset of size o(n) for a stream of n sparse points in a d ≥ n dimensional space (e.g. adjacency matrices of graphs). We also provide the first generalizations of such coresets for handling outliers. For arbitrary metric spaces, we improve the dependence on k to k log k and present a matching lower bound. For M -estimator clustering (special cases include the well-known k-median and k-means clustering), we introduce a new technique for converting an offline coreset construction to the streaming setting. Our method yields streaming coreset algorithms requiring the storage of O(S+ k log n) points, where S is the size of the offline coreset. In comparison, the previous state-of-the-art was the merge-and-reduce technique that required O(S log n) points, where a is the exponent in the offline construction’s dependence on −1. For example, combining our offline and streaming results, we produce a streaming metric k-means coreset algorithm using O( −2k log k log n) points of storage. The previous state-of-the-art required O( −4k log k log n) points. ∗Department of Computer Science, Johns Hopkins University. This material is based upon work supported in part by the National Science Foundation under Grant No. 1447639, by the Google Faculty Award and by DARPA grant N660001-1-2-4014. Its contents are solely the responsibility of the authors and do not represent the official view of DARPA or the Department of Defense. †Department of Computer Science, University of Haifa ‡Department of Mathematics, Johns Hopkins University. This material is based upon work supported by the Franco-American Fulbright Commission. ar X iv :1 61 2. 00 88 9v 1 [ cs .D S] 2 D ec 2 01 6