Coreset for Line-Sets Clustering

The input to the line-sets k-median problem is an integer k ≥ 1, and a set L = {L₁,..., L_n} that contains n sets of lines in R^d. The goal is to compute a set C of k centers (points in R^d) that minimizes the sum Σ_L∈L min_ℓ∈L,c∈_C dist(ℓ, c) of Euclidean distances from each set to its closest center, where dist(ℓ, c):= min_x∈_ℓ ∥x - c∥₂. An ε-coreset for this problem is a weighted subset of sets in L that approximates this sum up to 1 ± ε multiplicative factor, for every set C of k centers. We prove that every such input set L has a small ε-coreset, and provide the first coreset construction for this problem and its variants. The coreset consists of O(log² n) weighted line-sets from L, and is constructed in O(n log n) time for every fixed d, k ≥ 1 and ε ∈ (0, 1). The main technique is based on a novel reduction to a “fair clustering” of colored points to colored centers. We then provide a coreset for this coloring problem, which may be of independent interest. Open source code and experiments are also provided.