On Clustering Based on Optimization of Objective Function whose Cluster Partitioning Is Homotopy Equivalent with Weighted Alpha Complex

Abstract

Due to Information becomes enormous and complicated, data processing technology has developed rapidly. In recent years, among the data analysis methods which are means for extracting useful information from such enormous information, topological data analysis which analyzes by focusing on the structure of data with reference to topological geometry has attracted attention. On the other hand, clustering is one method of data analysis of unsupervised learning method, and it is used in various fields including information science. 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, we show the mathematical properties of the proposed algorithm and examine the effectiveness of the proposed method through numerical examples.