Coresets for Discrete Integration and Clustering

Given a set P of n points on the real line and a (potentially infinite) family of functions, we investigate the problem of finding a small (weighted) subset S C P, such that for any f ∈ F, we have that f(P) is a (1 ± e)-approximation to f(S). Here, f(Q) = Σ qeQ w(q)f(q) denotes the weighted discrete integral of f over the point set Q, where w(q) is the weight assigned to the point q. We study this problem, and provide tight bounds on the size S for several families of functions. As an application, we present some coreset constructions for clustering.

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