A Constant-Factor Bi-Criteria Approximation Guarantee for k-means

This paper studies the $k$-means algorithm for clustering as well as the class of $D \ell$ sampling algorithms to which $k$-means belongs. It is shown that for any constant factor $\beta 1$, selecting $\beta k$ cluster centers by $D \ell$ sampling yields a constant-factor approximation to the optimal clustering with $k$ centers, in expectation and without conditions on the dataset. This result extends the previously known $O(\log k)$ guarantee for the case $\beta 1$ to the constant-factor bi-criteria regime. It also improves upon an existing constant-factor bi-criteria result that holds only with constant probability. Papers published at the Neural Information Processing Systems Conference.