In the spherical k-means problem (SKMP), which is a well-studied clustering problem in text mining, we are given an n-point set $\mathcal{D}$ in d-dimensional unit sphere ${\mathbb{S}}^d$, and an integer $k\le n$. The goal is to find a center subset $S\subset {\mathbb{S}}^d$ with $|S|\le k$ that minimizes the sum of cosine dissimilarity measure for each point in $\mathcal{D}$ 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 $(2(4+\sqrt{7})+ϵ)$. We also conduct numerical computation to show the efficiency of the local search approximation algorithm by single-swap operation in the end.

在文本挖掘中经过充分研究的聚类问题球形k-均值问题（SKMP）中，我们得到了一个n点集$\mathcal{d}$在d维单位球体内${\mathbb{小号}}^d$和一个整数 $ķ\le ñ$。目标是找到一个中心子集$\mathrm{小号}\subset {\mathbb{小号}}^d$ 与 $|\mathrm{小号}|\le ķ$ 可以最大程度地减少每个点的余弦相异度测量值之和 $\mathcal{d}$到最近的中心。我们证明了任何γ为近似算法ķ -means问题（KMP）可以适应于与SKMP 2 γ -近似比例。因此，存在本地搜索$（\mathrm{18岁}+ϵ）$利用经典局部搜索的SKMP近似算法 $（9+ϵ）$-KMP的近似算法。因此，出现一个有趣的问题，即是否存在直接针对SKMP使用本地搜索方案的近似算法。本文提出了一种针对SKMP的局部搜索近似算法，证明了其性能保证是$（2（4+\sqrt{7}）+ϵ）$。最后，我们还进行了数值计算，以显示通过单次交换操作进行局部搜索近似算法的效率。