We consider the problem of clustering privately a dataset in $\mathbb{R}^d$ that undergoes both insertion and deletion of points. Specifically, we give an $\varepsilon$-differentially private clustering mechanism for the $k$-means objective under continual observation. This is the first approximation algorithm for that problem with an additive error that depends only logarithmically in the number $T$ of updates. The multiplicative error is almost the same as non privately. To do so we show how to perform dimension reduction under continual observation and combine it with a differentially private greedy approximation algorithm for $k$-means. We also partially extend our results to the $k$-median problem.

翻译：我们研究在$\mathbb{R}^d$空间中数据集经历点插入和删除操作时的隐私聚类问题。具体而言，我们提出了一种针对持续观测下$k$-means目标的$\varepsilon$-差分隐私聚类机制。这是该问题的首个近似算法，其加性误差仅与更新次数$T$呈对数关系。该算法的乘性误差几乎与非隐私情形相当。为此，我们展示了如何在持续观测下进行维度降维，并将其与差分隐私贪心近似算法相结合以解决$k$-means问题。此外，我们还将部分结果扩展到了$k$-median问题。