Semi-supervised fuzzy c-means regularized with pairwise constraints

Abstract

Clustering with pairwise constraints has been a popular method in semi-supervised clustering not only due to effectiveness but also the wide application domain. This paper works on semi-supervised fuzzy clustering with pairwise constraints. Specifically, we formulate two variants of Pairwise Constrained Fuzzy C-Means (PCFCM), referred to as PCFCM-vio and PCFCM-dist, by incorporating different penalty terms into the objective function of FCM. For PCFCM-vio, the penalt term (i.e., - vio) is defined as the total violation of assignments in terms of fuzzy memberships regarding to the constrained pairs. We show that PCFCM-vio is a more generalized form of existing pairwise constrained fuzzy clustering approaches. For the second approach PCFCM-dist, the penalty term (i.e., - dist) is the total Euclidean distance of memberships for those constrained pairs. Through theoretical analysis, we find the connection between the two forms of regularization, based on which we present a unified frame work of algorithm for the two approaches. Experimental results with both simulated data and real world data are provided, which demonstrate and compare the performance of the two proposed PCFCM variants.