Abstract
We study kmeans clustering in a semisupervised setting. Given an oracle that returns whether two given points belong to the same cluster in a fixed optimal clustering, we investigate the following question: how many oracle queries are sufficient to efficiently recover a clustering that, with probability at least (1  delta), simultaneously has a cost of at most (1 + epsilon) times the optimal cost and an accuracy of at least (1  epsilon)?
We show how to achieve such a clustering on n points with O{((k^2 log n) * m{(Q, epsilon^4, delta / (k log n))})} oracle queries, when the k clusters can be learned with an epsilon' error and a failure probability delta' using m(Q, epsilon',delta') labeled samples in the supervised setting, where Q is the set of candidate cluster centers. We show that m(Q, epsilon', delta') is small both for kmeans instances in Euclidean space and for those in finite metric spaces. We further show that, for the Euclidean kmeans instances, we can avoid the dependency on n in the query complexity at the expense of an increased dependency on k: specifically, we give a slightly more involved algorithm that uses O{(k^4/(epsilon^2 delta) + (k^{9}/epsilon^4) log(1/delta) + k * m{({R}^r, epsilon^4/k, delta)})} oracle queries.
We also show that the number of queries needed for (1  epsilon)accuracy in Euclidean kmeans must linearly depend on the dimension of the underlying Euclidean space, and for finite metric space kmeans, we show that it must at least be logarithmic in the number of candidate centers. This shows that our query complexities capture the right dependencies on the respective parameters.
BibTeX  Entry
@InProceedings{gamlath_et_al:LIPIcs:2018:9061,
author = {Buddhima Gamlath and Sangxia Huang and Ola Svensson},
title = {{SemiSupervised Algorithms for Approximately Optimal and Accurate Clustering}},
booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
pages = {57:157:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770767},
ISSN = {18688969},
year = {2018},
volume = {107},
editor = {Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9061},
URN = {urn:nbn:de:0030drops90612},
doi = {10.4230/LIPIcs.ICALP.2018.57},
annote = {Keywords: Clustering, Semisupervised Learning, Approximation Algorithms, kMeans, kMedian}
}
Keywords: 

Clustering, Semisupervised Learning, Approximation Algorithms, kMeans, kMedian 
Collection: 

45th International Colloquium on Automata, Languages, and Programming (ICALP 2018) 
Issue Date: 

2018 
Date of publication: 

04.07.2018 