UNCONSTRAINED POSSIBILISTIC FUZZY C-MEANS ALGORITHM

Abstract

Possibilistic Fuzzy C-Means (PFCM) jointly models fuzzy memberships alongside typicality values. It handles noisy data well and avoids the coincident-cluster trap, but it often converges to a suboptimal local minimum. This problem becomes particu larly severe in high-dimensional or noisy data. MMPFCM rederives PFCM using the majorization-minimization method. It drops the explicit centroids and introduces a lower-rank surrogate variable that smooths out the optimization landscape. Separately, unconstrained fuzzy clustering relaxes the row-sum-to-one membership constraint by substituting the analytical membership solution into the objective, converting the con strained problem into an unconstrained one and opening the door to more flexible, more stable optimization. We unify both ideas – unconstrained optimization and the typicality mechanism - into a single framework. Our method, Unconstrained Possibilistic Fuzzy C-Means (UC-PFCM), substitutes the closed-form membership solution directly into the PFCM cost while keeping the typicality terms unchanged, and then minimises the resulting cost via gradient descent with momentum, updating centroids directly. The per-iteration cost stays the same as PFCM and MMPFCM. UC-PFCM converges to lower objective values than its competitors in almost every case on twelve UCI datasets and records the best ranks across four standard clustering metrics.