Spherical k-means clustering as a known NP-hard variant of the k-means problem has broad applications in data mining. In contrast to k-means, it aims to partition a collection of given data distributed on a spherical surface into k sets so as to minimize the within-cluster sum of cosine dissimilarity. In the paper, we introduce spherical k-means clustering with penalties and give a \(2\max \{2,M\}(1+M)(\ln k+2)\)-approximation algorithm. Moreover, we prove that when against spherical k-means clustering with penalties but on separable instances, our algorithm is with an approximation ratio \(2\max \{3,M+1\}\) with high probability, where M is the ratio of the maximal and the minimal penalty cost of the given data set.

球形ķ -means聚类作为一个已知的NP-硬变体ķ -means问题在数据挖掘广阔的应用领域。与k-均值相反，它旨在将分布在球形表面上的给定数据集合划分为k个集合，以最小化余弦相似度的簇内和。在本文中，我们介绍了具有惩罚的球面k-均值聚类，并给出了\（2 \ max \ {2，M \}（1 + M）（\ ln k + 2）\）逼近算法。此外，我们证明了当反对带有惩罚的球形k-均值聚类但在可分实例上时，我们的算法的逼近比为\（2 \ max \ {3，M + 1 \} \）极有可能，其中M是给定数据集的最大和最小代价成本之比。