The seeding algorithm for spherical k -means clustering with penalties

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 2max{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 2max{3,M+1} with high probability, where M is the ratio of the maximal and the minimal penalty cost of the given data set.