The $k$-means is a popular clustering objective, although it is inherently non-robust and sensitive to outliers. Its popular seeding or initialization called $k$-means++ uses $D^{2}$ sampling and comes with a provable $O(\log k)$ approximation guarantee \cite{AV2007}. However, in the presence of adversarial noise or outliers, $D^{2}$ sampling is more likely to pick centers from distant outliers instead of inlier clusters, and therefore its approximation guarantees \textit{w.r.t.} $k$-means solution on inliers, does not hold. Assuming that the outliers constitute a constant fraction of the given data, we propose a simple variant in the $D^2$ sampling distribution, which makes it robust to the outliers. Our algorithm runs in $O(ndk)$ time, outputs $O(k)$ clusters, discards marginally more points than the optimal number of outliers, and comes with a provable $O(1)$ approximation guarantee. Our algorithm can also be modified to output exactly $k$ clusters instead of $O(k)$ clusters, while keeping its running time linear in $n$ and $d$. This is an improvement over previous results for robust $k$-means based on LP relaxation and rounding \cite{Charikar}, \cite{KrishnaswamyLS18} and \textit{robust $k$-means++} \cite{DeshpandeKP20}. Our empirical results show the advantage of our algorithm over $k$-means++~\cite{AV2007}, uniform random seeding, greedy sampling for $k$ means~\cite{tkmeanspp}, and robust $k$-means++~\cite{DeshpandeKP20}, on standard real-world and synthetic data sets used in previous work. Our proposal is easily amenable to scalable, faster, parallel implementations of $k$-means++ \cite{Bahmani,BachemL017} and is of independent interest for coreset constructions in the presence of outliers \cite{feldman2007ptas,langberg2010universal,feldman2011unified}.

翻译：k-means是一种流行的聚类目标函数，但其本质缺乏鲁棒性且对离群点敏感。其广泛使用的种子初始化方法k-means++采用$D^{2}$采样，并具有可证明的$O(\log k)$近似保证\cite{AV2007}。然而，在存在对抗性噪声或离群点时，$D^{2}$采样更倾向于从远距离离群点而非正常聚类簇中选择中心点，因此其关于正常数据点k-means解的近似保证不再成立。假设离群点占给定数据的恒定比例，我们提出$D^2$采样分布的简单变体，使其对离群点具备鲁棒性。本算法时间复杂度为$O(ndk)$，输出$O(k)$个聚类簇，丢弃的点数仅略多于最优离群点数量，并具有可证明的$O(1)$近似保证。该算法亦可修改为精确输出$k$个聚类簇（而非$O(k)$个），同时保持运行时间与$n$和$d$呈线性关系。相较于基于线性规划松弛和舍入的鲁棒k-means方法\cite{Charikar},\cite{KrishnaswamyLS18}及鲁棒k-means++\cite{DeshpandeKP20}，本文取得了改进。实验结果表明，在以往工作使用的标准真实和合成数据集上，本算法优于k-means++~\cite{AV2007}、均匀随机种子初始化、k-means贪婪采样~\cite{tkmeanspp}及鲁棒k-means++~\cite{DeshpandeKP20}。本方法易于适配k-means++的可扩展快速并行实现\cite{Bahmani,BachemL017}，且对存在离群点时的核心集构造具有独立研究价值\cite{feldman2007ptas,langberg2010universal,feldman2011unified}。