Approximation algorithms for spherical k-means problem using local search scheme

In the spherical k-means problem (SKMP), which is a well-studied clustering problem in text mining, we are given an n-point set D in d-dimensional unit sphere S d, and an integer k ≤ n. The goal is to find a center subset S ⊂ S d with | S | ≤ k that minimizes the sum of cosine dissimilarity measure for each point in D to the nearest center. We prove that any γ-approximation algorithm for the k-means problem (KMP) can be adapted to the SKMP with 2γ-approximation ratio. It follows that there is a local search ( 18 + ϵ ) -approximation algorithm for the SKMP, by leveraging the classical local search ( 9 + ϵ ) -approximation algorithm for the KMP. Therefore, an interesting problem arises, that is whether there exists an approximation algorithm using local search scheme directly for the SKMP. In this paper, we present a local search approximation algorithm for the SKMP and prove its performance guarantee is ( 2 ( 4 + 7 ) + ϵ ). We also conduct numerical computation to show the efficiency of the local search approximation algorithm by single-swap operation in the end.

Authors

• Dongmei Zhang Peking University
• Yukun Cheng School of Business, Jiangnan University
• Min Li School of Computer Science and Engineering, Central South University, China
• Yishui Wang
• Dachuan Xu Beijing University of Technology, Beijing, China

Keywords

• Spherical k-means
• Local search
• Approximation algorithm
• Data mining