As one of the most commonly used clustering methods, fuzzy clustering technique such as the Fuzzy C-means (FCM) has undergone a rapid development. In this paper, a general FCM clustering algorithm based on contraction mapping (cGFCM) is proposed for more general cases of using Minkowski metric ($L_p$-norm distance) as the similarity measure, and the analytical method for calculating the parameters of the proposed algorithm is given. The core of the proposed cGFCM algorithm lies on constructing a contraction mapping to update the prototypes when an arbitrary Minkowski metric is used to measure the closeness of data points. Subsequently, mainly guided by the Banach contraction mapping principle, the algorithm and implementation approaches are discussed in detail, and the correctness and feasibility of the proposed method are proved. Moreover, the convergence of the proposed algorithm is also discussed. Experimental studies carried out on both synthetic data sets and real-world data sets show that the proposed cGFCM algorithm extends FCM to more general cases without extra time and space costs. Compared with another generalized FCM clustering strategy and other five state-of-the-art clustering methods, the proposed algorithm can not only reach better performance in both clustering accuracy and stability, but reduce the running time several-fold.

作为最常用的聚类方法之一，模糊C均值（FCM）等模糊聚类技术得到了快速发展。在本文中，针对使用 Minkowski 度量的更一般情况，提出了一种基于收缩映射（cGFCM）的通用 FCM 聚类算法（${升}_{磷}$-范数距离）作为相似性度量，并给出了计算该算法参数的解析方法。所提出的 cGFCM 算法的核心在于构建收缩映射以在使用任意 Minkowski 度量来测量数据点的接近度时更新原型。随后，主要以Banach收缩映射原理为指导，详细讨论了算法和实现方法，证明了所提出方法的正确性和可行性。此外，还讨论了所提出算法的收敛性。在合成数据集和真实数据集上进行的实验研究表明，所提出的 cGFCM 算法将 FCM 扩展到更一般的情况，而无需额外的时间和空间成本。