Optimal Clustering with Twofold Memberships.

This paper proposes two clustering algorithms of twofold memberships for each cluster. One uses a membership similar to that in K-means, while another membership is defined for a core of a cluster, which is compared to the lower approximation of a cluster in rough K-means. Two ideas for the lower approximation are proposed in this paper: one uses a neighborhood of a cluster boundary and another uses a simple circle from a cluster center. By using the two memberships, two alternate optimization algorithms are proposed. Numerical examples show the effectiveness of the proposed algorithms.