As a kind of important soft clustering model, the fuzzy C-means method is widely applied in many fields. In this method, instead of the strict distributive ability in the classical k-means method, all the sample points are endowed with degrees of membership to each center to depict the fuzzy clustering. In this paper, we show that the fuzzy C-means++ algorithm, which introduces the k-means++ algorithm as a seeding strategy, gives a solution for which the approximation guarantee is $O(k^2\ln k)$. A novel seeding algorithm is then designed based on the contribution of the fuzzy potential function, which improves the approximation ratio to $O(k\ln k)$. Preliminary numerical experiments are proposed to support the theoretical results of this paper.

模糊C- means方法作为一种重要的软聚类模型，在很多领域都有广泛的应用。在该方法中，代替经典k- means方法中的严格分配能力，所有样本点都被赋予了每个中心的隶属度来描述模糊聚类。在本文中，我们展示了引入k- means++ 算法作为种子策略的模糊C- means++ 算法，给出了近似保证为$哦({克}^2\mathrm{输入}克)$. 然后基于模糊势函数的贡献设计了一种新的播种算法，该算法将逼近比提高到$哦(克\mathrm{输入}克)$. 初步数值实验被提出来支持本文的理论结果。