New Coresets for Projective Clustering and Applications

TL;DR

This paper introduces novel coreset algorithms for $(j,k)$-projective clustering and general $M$-estimator regression, enabling efficient approximation of complex clustering and regression tasks in high-dimensional data.

Contribution

It presents the first polynomial-size $L_ abla$ coreset for $(j,k)$-projective clustering and the first strong coreset construction for various $M$-estimator regressions, advancing scalable data analysis.

Findings

Coreset algorithms significantly reduce data size for clustering and regression.

Experimental results demonstrate practical efficiency and accuracy.

New methods outperform existing approaches on real-world datasets.

Abstract

$(j,k)$-projective clustering is the natural generalization of the family of $k$-clustering and $j$-subspace clustering problems. Given a set of points $P$ in $\mathbb{R}^d$, 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 $L_\infty$ 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, $L_1-L_2$, 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.