Abstract
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.
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Acknowledgements
The authors Sai Ji, Dachuan Xu and Dongmei Zhang are supported by National Natural Science Foundation of China (No. 11871081). The third author Longkun Guo is supported by National Natural Science Foundation of China (No. 61772005) and Natural Science Foundation of Fujian province (No. 2017J01753). The fourth author Min Li is supported by Higher Educational Science and Technology Program of Shandong Province (No. J17KA171) and Natural Science Foundation of Shandong Province (No. ZR2019MA032) of China.
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A preliminary version of this paper appeared in Proceedings of the 13th International Conference on Algorithmic Aspects in Information and Management, pp. 149–158, 2019.
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Ji, S., Xu, D., Guo, L. et al. The seeding algorithm for spherical k-means clustering with penalties. J Comb Optim 44, 1977–1994 (2022). https://doi.org/10.1007/s10878-020-00569-1
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DOI: https://doi.org/10.1007/s10878-020-00569-1