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$\mathbb{R}^d$中同时包含数据点插入和删除操作的私有聚类问题。具体而言，针对持续观测下的$k$-均值目标，我们提出了一种$\varepsilon$-差分隐私聚类机制。这是首个针对该问题的近似算法，其加性误差仅与更新次数$T$呈对数关系，乘性误差与非私有近似算法几乎一致。为实现这一目标，我们展示了如何在持续观测下进行降维处理，并将其与差分隐私的贪心近似算法相结合用于$k$-均值问题。此外，我们的结果还可部分推广至$k$-中位数问题。