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 $\tilde{O}(k\cdot \mathrm{poly}(d, \log\Delta))$, 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 $k$.