Streaming Coreset Constructions for M-Estimators

We introduce a new method of maintaining a (k, )-coreset for clustering M -estimators over insertiononly streams. Let (P,w) be a weighted set (where w : P → [0,∞) is the weight function) of points in a ρ-metric space (meaning a set X equipped with a positive-semidefinite symmetric function D such that D(x, z) ≤ ρ(D(x, y) + D(y, z)) for all x, y, z ∈ X ). For any set of points C, we define COST(P,w,C) = ∑ p∈P w(p) minc∈C D(p, c). A (k, )-coreset for (P,w) is a weighted set (Q, v) such that for every set C of k points, (1 − )COST(P,w,C) ≤ COST(Q, v, C) ≤ (1 + )COST(P,w,C). Essentially, the coreset (Q, v) can be used in place of (P,w) for all operations concerning the COST function. Coresets, as a method of data reduction, are used to solve fundamental problems in machine learning of streaming and distributed data. M -estimators are functions D(x, y) that can be written as ψ(d(x, y)) where (X , d) is a true metric (i.e. 1-metric) space. Special cases of M -estimators include the well-known k-median (ψ(x) = x) and k-means (ψ(x) = x2) functions. Our technique takes an existing offline construction for an M -estimator coreset and converts it into the streaming setting, where n data points arrive sequentially. To our knowledge, this is the first streaming construction for any M -estimator that does not rely on the merge-and-reduce tree. For example, our coreset for streaming metric k-means uses O( −2k log k logn) points of storage. The previous state-of-the-art required storing at least O( −2k log k log4 n) points. 2012 ACM Subject Classification Theory of computation → Streaming models; Theory of computation → Facility location and clustering; Information systems → Query optimization