  Traditionally, practitioners initialize the {\tt k-means} algorithm with
centers chosen uniformly at random. Randomized initialization with uneven
weights ({\tt k-means++}) has recently been used to improve the performance
over this strategy in cost and run-time. We consider the k-means problem with
semi-supervised information, where some of the data are pre-labeled, and we
seek to label the rest according to the minimum cost solution. By extending the
{\tt k-means++} algorithm and analysis to account for the labels, we derive an
improved theoretical bound on expected cost and observe improved performance in
simulated and real data examples. This analysis provides theoretical
justification for a roughly linear semi-supervised clustering algorithm.

Machine Learning