NDLI: Clustering under Perturbation Resilience

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Clustering under Perturbation Resilience

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Liang, Yingyu Balcan, Maria Florina
Copyright Year	2016
Abstract	Motivated by the fact that distances between data points in many real-world clustering instances are often based on heuristic measures, Bilu and Linial [Proceedings of the Symposium on Innovations in Computer Science, 2010] proposed analyzing objective based clustering problems under the assumption that the optimum clustering to the objective is preserved under small multiplicative perturbations to distances between points. The hope is that by exploiting the structure in such instances, one can overcome worst case hardness results. In this paper, we provide several results within this framework. For center-based objectives, we present an algorithm that can optimally cluster instances resilient to perturbations of factor $(1 + \sqrt{2})$, solving an open problem of Awasthi, Blum, and Sheffet [Proceedings of the IEEE Annual Symposium on Foundations of Computer Science, 2010]. For $k$-median, a center-based objective of special interest, we additionally give algorithms for a more relaxed assumption in which we allow the optimal solution to change in a small $\epsilon$ fraction of the points after perturbation. We give the first bounds known for $k$-median under this more realistic and more general assumption. We also provide positive results for min-sum clustering, which is typically a harder objective than center-based objectives from an approximability standpoint. Our algorithms are based on new linkage criteria that may be of independent interest. Additionally, we give sublinear-time algorithms, showing algorithms that can return an implicit clustering from access to only a small random sample.
Sponsorship	DOD \| Office of Naval Research (ONR). Google. Microsoft. National Science Foundation (NSF). DOD \| Air Force Office of Scientific Research (AFOSR)
Starting Page	102
Ending Page	155
Page Count	54
File Format	PDF
ISSN	00975397
DOI	10.1137/140981575
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	1
Volume Number	45
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2016-01-28
Access Restriction	Subscribed
Subject Keyword	min-sum clustering Learning and adaptive systems Analysis of algorithms and problem complexity Approximation algorithms clustering Computational learning theory $k$-median clustering perturbation resilience
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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