On hard c -means with cluster radius that makes the cluster partition homotopy equivalent to weighted $$\alpha$$α -complex

In data mining, a combination of discovery of connected components of data by cluster analysis and capture of topological structure based on persistent homology attracted attention in recent years is effective. Especially in cluster analysis, k-means (KM) also referred to as hard c-means (HCM) and fuzzy c-means (FCM), which are representative methods and called objective-based clustering, assign data of each cluster to one representative point. These methods have an aspect of data compression. As a result, a large amount of data can be compressed and handled as representative points, so that the structure of the entire data can be known by grasping the topological structure of the representative points. Therefore, HCM and FCM have high affinity with persistent homology. In order to consider data mining methods using persistent homology, it is necessary that the mathematical property of filtration holds. However, here, there is the big problem that “topological structure made from HCM and FCM does not mathematically guarantee filtration”. In this paper, we propose a new objective-based clustering method to classify a dataset into clusters with topological structure with filtration. The proposed algorithm makes the cluster partition homotopy equivalent to the weighted \n$$\\alpha$$\n\n-complex. This enables data mining using clustering and capture of the topological structure by persistent homology.