eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-07-04
57:1
57:14
10.4230/LIPIcs.ICALP.2018.57
article
Semi-Supervised Algorithms for Approximately Optimal and Accurate Clustering
Gamlath, Buddhima
1
Huang, Sangxia
2
Svensson, Ola
1
École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
Sony Mobile Communications, Lund, Sweden
We study k-means clustering in a semi-supervised 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 k-means instances in Euclidean space and for those in finite metric spaces. We further show that, for the Euclidean k-means 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 k-means must linearly depend on the dimension of the underlying Euclidean space, and for finite metric space k-means, 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol107-icalp2018/LIPIcs.ICALP.2018.57/LIPIcs.ICALP.2018.57.pdf
Clustering
Semi-supervised Learning
Approximation Algorithms
k-Means
k-Median