Semi-Supervised Algorithms for Approximately Optimal and Accurate Clustering

Buddhima Gamlath, Sangxia Huang, Ola Svensson, Michael Wagner
2018 International Colloquium on Automata, Languages and Programming  
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 − δ), simultaneously has a cost of at most (1 + ) times the optimal cost and an accuracy of at least (1 − )? We show how to achieve such a clustering on n points with O((k ) oracle
more » ... queries, when the k clusters can be learned with an error and a failure probability δ using m(Q, , δ ) labeled samples in the supervised setting, where Q is the set of candidate cluster centers. We show that m(Q, , δ ) 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 We also show that the number of queries needed for (1− )-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.
doi:10.4230/lipics.icalp.2018.57 dblp:conf/icalp/GamlathHS18 fatcat:yme5qmwocjb2focyhtnfyozzq4