We consider the problem of clustering data points coming from sub-Gaussian mixtures. Existing methods that provably achieve the optimal mislabeling error, such as the Lloyd algorithm, are usually vulnerable to outliers. In contrast, clustering methods seemingly robust to adversarial perturbations are not known to satisfy the optimal statistical guarantees. We propose a simple algorithm that obtains the optimal mislabeling rate even when we allow adversarial outliers to be present. Our algorithm achieves the optimal error rate in constant iterations when a weak initialization condition is satisfied. In the absence of outliers, in fixed dimensions, our theoretical guarantees are similar to that of the Lloyd algorithm. Extensive experiments on various simulated data sets are conducted to support the theoretical guarantees of our method.

翻译：我们考虑来自亚高斯混合模型的数据点聚类问题。现有能够达到最优误标注率的算法（如Lloyd算法）通常对异常值敏感。相反，看似能抵抗对抗性扰动的聚类方法尚未被证明能达到最优统计保证。我们提出一种简单算法，即使在允许存在对抗异常值的情况下，也能获得最优误标注率。当满足弱初始化条件时，我们的算法在常数次迭代内达到最优错误率。在无异常值且固定维度的情形下，我们的理论保证与Lloyd算法类似。我们在各种模拟数据集上进行了大量实验，以支撑我们方法的理论保证。