Data stability in clustering: A closer look

Abstract

Abstract. We consider the model introduced by Bilu and Linial [12], who study problems for which the optimal clustering does not change when distances are perturbed. They show that even when a problem is NP-hard, it is sometimes possible to obtain efficient algorithms for instances resilient to certain multiplicative perturbations, e.g. on the order of O( √ n) for max-cut clustering. Awasthi et al.[6] consider centerbased objectives, and Balcan and Liang [9] analyze the k-median and min-sum objectives, giving efficient algorithms for instances resilient to certain constant multiplicative perturbations. Here, we are motivated by the question of to what extent these assumptions can be relaxed while allowing for efficient algorithms. We show there is little room to improve these results by giving NP-hardness lower bounds for both the k-median and min-sum objectives. On the other hand, we show that multiplicative resilience parameters, even only on the order of Θ(1), can be so strong as to make the clustering problem trivial, and we exploit these assumptions to present a simple one-pass streaming algorithm for the k-median objective. We also consider a model of additive perturbations and give a correspondence between additive and multiplicative notions of stability. Our results provide a close examination of the consequences of assuming, even constant, stability in data.

Clustering under Perturbation Stability in Near-Linear Time

“…There are many notions of clustering stability that have been considered in literature [1,8,15,19,20,24,37,49,56]. The exact definition of stability we study here was first introduced in Awasthi et al [16]; their definition in particular resembles the one of Bilu and Linial [25] for max-cut problem, which later has been adapted to other optimization problems [11,12,21,53,55].…”

“…On the hardness side, solving (3 − δ)-center proximal k-median instances in general metric spaces is NP-hard for any δ > 0 [16]. When restricted to Euclidean spaces in arbitrary dimensions, Ben-David and Reyzin [24] showed that for every δ > 0, solving discrete (2 − δ)-center proximal k-median instances is NP-hard. Similarly, the clustering problem for discrete k-center remains hard for α-stable instances when α < 2, assuming standard complexity assumption that NP = RP [22].…”

When is Clustering Perturbation Robust?

“…Previous work on perturbation robustness studies it from a computational perspective by identifying new efficient algorithms for robust instances Bilu and Linial [2010], Ackerman and Ben-David [2009], Awasthi et al [2012]. Ben-David and Reyzin Ben-David and Reyzin [2014] recently studied corresponding NP-hardness lower bounds.…”

On the Geometry of Stable Steiner Tree Instances

“…Instances that are γ-stable in the Bilu-Linial model have the property that the structure of the optimal solution is not only unique, but also does not change even when the underlying distances among the input points are perturbed by a multiplicative factor γ > 1. In their original paper, Bilu and Linial analyzed MAX-CUT clustering, and since their seminal work, other problems have been analyzed including center-based clustering [4,6,7], multi-way cut problems [15], and metric TSP [16]. 1 Here, we look at the metric Steiner tree problemand also the more restricted Euclidean version.…”