Streaming coreset constructions for M-estimators

We introduce a new method of maintaining a (k, ϵ)-coreset for clustering M-estimators over insertion-only 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) min_c∈C D(p, c). A (k, ϵ)-coreset for (P, w) is a weighted set (Q, v) such that for every set C of k points, (1 − ϵ)COST(P, w, C) ≤ COST(Q, v, C) ≤ (1 + ϵ)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(ϵ⁻²k log k log n) points of storage. The previous state-of-the-art required storing at least O(ϵ⁻²k log k log⁴ n) points.