A Multiview and Multiexemplar Fuzzy Clustering Approach: Theoretical Analysis and Experimental Studies

Abstract

Multiview and multiexemplar 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 demonstrate that multiview & multiexemplar fuzzy clustering has a theoretical guarantee of enhanced clustering performance. Based on the above-mentioned theoretical result, we develop a novel multiview & multiexemplar fuzzy clustering approach (M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> FC). The key features of the proposed approach are: first, 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; and second, optimize the proposed objective function of the proposed approach by applying the Lagrangian multiplier method and Karush-Kuhn-Tuchker conditions to assure nonnegative fuzzy memberships. Extensive experimental results show that M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> FC outperforms the existing state-of-the-art multiview approaches in most cases.