Nearly Optimal Dynamic $k$-Means Clustering for High-Dimensional Data.

We consider the $k$-means clustering problem in the dynamic streaming setting, where points from a discrete Euclidean space ${1, 2, ldots, Delta}^d$ can be dynamically inserted to or deleted from the dataset. For this problem, we provide a one-pass coreset construction algorithm using space $tilde{O}(kcdot mathrm{poly}(d, logDelta))$, where $k$ is the target number of centers. To our knowledge, this is the first dynamic geometric data stream algorithm for $k$-means using space polynomial in dimension and nearly optimal (linear) in $k$.