New Coresets for Projective Clustering and Applications

(j, k)-projective clustering is the natural generalization of the family of k-clustering and j-subspace clustering problems. Given a set of points P in R^d, the goal is to find k flats of dimension j, i.e., affine subspaces, that best fit P under a given distance measure. In this paper, we propose the first algorithm that returns an L_∞ coreset of size polynomial in d. Moreover, we give the first strong coreset construction for general M-estimator regression. Specifically, we show that our construction provides efficient coreset constructions for Cauchy, Welsch, Huber, Geman-McClure, Tukey, L₁ − L₂, and Fair regression, as well as general concave and power-bounded loss functions. Finally, we provide experimental results based on real-world datasets, showing the efficacy of our approach.