Stability of k-Means Clustering

Ben-David, Shai; Pál, Dávid; Simon, Hans-Ulrich

doi:10.1007/978-3-540-72927-3_4

Cited by 56 publications

(49 citation statements)

References 6 publications

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“…In particular, if all (1+α)-approximations to the objective are δ-close to the desired clustering in terms of how points are partitioned, they show one can efficiently get O(δ/α)-close to the desired clustering. Ben-David et al [10,9] consider a notion of stability of a clustering algorithm, which is called stable if it outputs similar clusters for different sets of n input points drawn from the same distribution. For k-means, the work of Meila [20] discusses the opposite directionclassifying instances where an approximated solution for k-means is close to the target clustering.…”

Section: Related Workmentioning

confidence: 99%

Center-based clustering under perturbation stability

Awasthi

Blum

Sheffet

2012

Information Processing Letters

157

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Clustering under most popular objective functions is NP-hard, even to approximate well, and so unlikely to be efficiently solvable in the worst case. Recently, Bilu and Linial [11] suggested an approach aimed at bypassing this computational barrier by using properties of instances one might hope to hold in practice. In particular, they argue that instances in practice should be stable to small perturbations in the metric space and give an efficient algorithm for clustering instances of the Max-Cut problem that are stable to perturbations of size O(n 1/2 ). In addition, they conjecture that instances stable to as little as O(1) perturbations should be solvable in polynomial time. In this paper we prove that this conjecture is true for any center-based clustering objective (such as k-median, k-means, and k-center). Specifically, we show we can efficiently find the optimal clustering assuming only stability to factor-3 perturbations of the underlying metric in spaces without Steiner points, and stability to factor 2 + √ 3 perturbations for general metrics. In particular, we show for such instances that the popular Single-Linkage algorithm combined with dynamic programming will find the optimal clustering. We also present NP-hardness results under a weaker but related condition.

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Section: Related Workmentioning

confidence: 99%

Center-based clustering under perturbation stability

Awasthi

Blum

Sheffet

2012

Information Processing Letters

157

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“…In the machine learning literature, several notions similar to differential privacy have been explored under the rubric of "algorithmic stability" [14,30,13,33,19,7]. The most closely related notion is change-one error stability, which measures how much the generalization error changes when an input are changed (see the survey [33]).…”

Section: Related Workmentioning

confidence: 99%

“…In contrast, differential privacy measures how the distribution over the entire output changes-a more complex measure of stability (in particular, differential privacy implies change-one error stability). A different notion, stability under resampling of the data from a given distribution [8,7], is connected to the sample-aggregate method of [37] but is not directly relevant to the techniques considered here.…”

Section: Related Workmentioning

confidence: 99%

What Can We Learn Privately?

Kasiviswanathan¹,

Lee²,

Nissim³

et al. 2011

SIAM J. Comput.

615

149

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Learning problems form an important category of computational tasks that generalizes many of the computations researchers apply to large real-life data sets. We ask: what concept classes can be learned privately, namely, by an algorithm whose output does not depend too heavily on any one input or specific training example? More precisely, we investigate learning algorithms that satisfy differential privacy, a notion that provides strong confidentiality guarantees in the contexts where aggregate information is released about a database containing sensitive information about individuals. We present several basic results that demonstrate general feasibility of private learning and relate several models previously studied separately in the contexts of privacy and standard learning.

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“…The software examined how positive or negative mood words were attached to the keywords. 9,10 For the opinion mining that we conducted in our study, positive sentiments were represented by the values of variables that showed a good or positive response towards a firm or its stock. For example, sentiment related to profits, improvements, innovativeness, new concepts, and about 200 other words were related to business progress.…”

Section: Context and Datamentioning

confidence: 99%