A Multi-view & Multi-exemplar Fuzzy Clustering Approach: Theoretical Analysis and Experimental Studies

Multi-view & multi-exemplar fuzzy clustering aims at effectively integrating the fuzzy membership matrix of each individual view to search for a final partition of objects in which each cluster may well be represented by one and even multiple exemplars. However, how to integrate the corresponding fuzzy membership matrix of each view such that enhanced clustering performance can be theoretically guaranteed still keeps an open topic. In this study, with the proposed exemplar invariant assumption that an exemplar of a cluster in one view is always an exemplar of that cluster in each other view, we provably demonstrate that Multi-view & multi-exemplar fuzzy clustering has a theoretical guarantee of enhanced clustering performance. Based on the above theoretical result, we develop a novel multi-view & multi-exemplar fuzzy clustering approach (M2FC). The key features of the proposed approach are to (1) embed a quadratic penalization term into its objective function to minimize the discrepancy of exemplars across different views such that the exemplar invariant assumption can be met as much as possible; (2) optimize the proposed objective function of the proposed approach by applying the Lagrangian multiplier method and KKT conditions to assure non-negative fuzzy memberships. Extensive experimental results show that M2FC outperforms the existing state-of-the-art multi-view approaches in most cases.