Sensitivity Sampling Over Dynamic Geometric Data Streams with Applications to $k$-Clustering

Sensitivity based sampling is crucial for constructing nearly-optimal coreset for $k$-means / median clustering. In this paper, we provide a novel data structure that enables sensitivity sampling over a dynamic data stream, where points from a high dimensional discrete Euclidean space can be either inserted or deleted. Based on this data structure, we provide a one-pass coreset construction for $k$-means %and M-estimator clustering using space $\widetilde{O}(k\mathrm{poly}(d))$ over $d$-dimensional geometric dynamic data streams. While previous best known result is only for $k$-median [Braverman, Frahling, Lang, Sohler, Yang' 17], which cannot be directly generalized to $k$-means to obtain algorithms with space nearly linear in $k$. To the best of our knowledge, our algorithm is the first dynamic geometric data stream algorithm for $k$-means using space polynomial in dimension and nearly optimal in $k$. We further show that our data structure for maintaining coreset can be extended as a unified approach for a more general classes of $k$-clustering, including $k$-median, $M$-estimator clustering, and clusterings with a more general set of cost functions over distances. For all these tasks, the space/time of our algorithm is similar to $k$-means with only $\mathrm{poly}(d)$ factor difference.