Geometric foundations of possibilistic clustering: A hard possibilistic clustering algorithm
James C. Bezdek, Thomas A. Runkler · Fuzzy Sets and Systems · 2026
Possibilistic c-means (PCM) clustering began in 1993, and has been used since then in many applications. In this article we discuss the geometric foundations of PCM and introduce a new hard possibilistic c-means (HPCM) clustering algorithm. We use limit theory to prove that the extended set of possibilistic c-partitions is the unit hypercube in R c n ; and that its vertices are exactly the hard possibilistic c-partitions on n objects defined herein. This enables completion of the geometric description of the domain of possibilistic clustering algorithms. We give examples that compare the results of clustering with Hard c-means (HCM) to HPCM on three small synthetic data sets. Our proof-of-concept examples show that the new algorithm performs as expected, and provides much more realistic interpretation of clusters than HCM when the data contain bridge points or noise.