On Clustering whose Cluster Partitioning Is Homotopy Equivalent with Weighted Alpha Complex

Abstract

Due to Information becomes enormous and complicated, data processing technology has de- veloped rapidly. In recent years, among the data mining methods which are means for extracting useful information from such enormous information, persistent homology which analyzes by focusing on the struc- ture of data with reference to topological geometry has attracted attention. On the other hand, clustering is one method of data mining of unsupervised learning method, and it is used in various fields including information science. By the way, in order to consider the data mining method using persistent homology, it is necessary that the mathematical property of filtration holds true. However, there is a big problem that “ topological structure made from Hard c-means and Fuzzy c-means, which are representative clustering methods, does not mathematically guarantee filtration ”. In this paper, we focus on the structure of the cluster, not the data, and propose a clustering algorithm whose clustering result is homotopy equivalent with weighted alpha complex. In addition, using the concept of alpha complex, we define the weighted alpha complex and show the mathematical properties of the proposed algorithm.Then we examine the effectiveness of the proposed method through numerical examples.