Topological data analysis is a new theoretical trend using topological techniques to mine data. This approach helps determine topological data structures. It focuses on investigating the global shape of data rather than on local information of high-dimensional data. The Mapper algorithm is considered as a sound representative approach in this area. It is used to cluster and identify concise and meaningful global topological data structures that are out of reach for many other clustering methods. In this article, we propose a new method called the Shape Fuzzy C-Means (SFCM) algorithm, which is constructed based on the Fuzzy C-Means algorithm with particular features of the Mapper algorithm. The SFCM algorithm can not only exhibit the same clustering ability as the Fuzzy C-Means but also reveal some relationships through visualizing the global shape of data supplied by the Mapper. We present a formal proof and include experiments to confirm our claims. The performance of the enhanced algorithm is demonstrated through a comparative analysis involving the original algorithm, Mapper, and the other fuzzy set based improved algorithm, F-Mapper, for synthetic and real-world data. The comparison is conducted with respect to output visualization in the topological sense and clustering stability.

中文翻译：

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SFCM：一种提取数据形状信息的模糊聚类算法

拓扑数据分析是利用拓扑技术挖掘数据的一种新的理论趋势。这种方法有助于确定拓扑数据结构。它侧重于研究数据的全局形状，而不是高维数据的局部信息。Mapper 算法被认为是该领域的一种有代表性的方法。它用于聚类和识别许多其他聚类方法无法实现的简洁而有意义的全局拓扑数据结构。在本文中，我们提出了一种称为形状模糊 C 均值 (SFCM) 算法的新方法，该算法是基于模糊 C 均值算法构建的，具有 Mapper 算法的特定特征。SFCM 算法不仅可以表现出与 Fuzzy C-Means 相同的聚类能力，还可以通过可视化 Mapper 提供的数据的全局形状来揭示一些关系。我们提出了一个正式的证明并包括实验来证实我们的主张。通过对原始算法 Mapper 和其他基于模糊集的改进算法 F-Mapper 的比较分析，证明了增强算法的性能，用于合成数据和真实世界数据。比较是针对拓扑意义上的输出可视化和聚类稳定性进行的。以及另一个基于模糊集的改进算法 F-Mapper，用于合成和真实世界的数据。比较是针对拓扑意义上的输出可视化和聚类稳定性进行的。以及另一个基于模糊集的改进算法 F-Mapper，用于合成和真实世界的数据。比较是针对拓扑意义上的输出可视化和聚类稳定性进行的。