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

TL;DR

This paper introduces a nearly optimal dynamic streaming algorithm for high-dimensional $k$-means clustering, efficiently maintaining coresets with space polynomial in dimension and linear in the number of clusters.

Contribution

It presents the first dynamic geometric data stream algorithm for $k$-means with space polynomial in dimension and nearly linear in $k$, advancing streaming clustering methods.

Findings

Achieves one-pass coreset construction in dynamic streams.

Uses space $ ilde{O}(k ext{poly}(d, ext{log}\Delta))$, nearly optimal in $k$.

First such algorithm with polynomial space in dimension for $k$-means.

Abstract

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 $k$.