Approximation and Streaming Algorithms for Projective Clustering via Random Projections

Let P be a set of n points in R. In the projective clustering problem, given k, q and norm ρ ∈ [1,∞], we have to compute a set F of k q-dimensional flats such that ( ∑ p∈P d(p,F)) is minimized; here d(p,F) represents the (Euclidean) distance of p to the closest flat in F . We let f k (P, ρ) denote the minimal value and interpret f k (P,∞) to be maxr∈P d(r,F). When ρ = 1, 2 and ∞ and q = 0, the problem corresponds to the k-median, kmean and the k-center clustering problems respectively. For every 0 < ε < 1, S ⊂ P and ρ ≥ 1, we show that the orthogonal projection of P onto a randomly chosen flat of dimension O(((q + 1) log(1/ε)/ε) log n) will εapproximate f 1 (S, ρ). This result combines the concepts of geometric coresets and subspace embeddings based on the Johnson-Lindenstrauss Lemma. As a consequence, an orthogonal projection of P to an O(((q + 1) log((q + 1)/ε)/ε) log n) dimensional randomly chosen subspace ε-approximates projective clusterings for every k and ρ simultaneously. Note that the dimension of this subspace is independent of the number of clusters k. Using this dimension reduction result, we obtain new approximation and streaming algorithms for projective clustering problems. For example, given a stream of n points, we show how to compute an ε-approximate projective clustering for every k and ρ simultaneously using only O((n + d)((q + 1) log((q + 1)/ε))/ε log n) space. Compared to standard streaming algorithms with Ω(kd) space requirement, our approach is a significant improvement when the number of input points and their dimensions are of the same order of magnitude.