  In semi-supervised fuzzy clustering, this paper extends the traditional
pairwise constraint (i.e., must-link or cannot-link) to fuzzy pairwise
constraint. The fuzzy pairwise constraint allows a supervisor to provide the
grade of similarity or dissimilarity between the implicit fuzzy vectors of a
pair of samples. This constraint can present more complicated relationship
between the pair of samples and avoid eliminating the fuzzy characteristics. We
propose a fuzzy discriminant clustering model (FDC) to fuse the fuzzy pairwise
constraints. The nonconvex optimization problem in our FDC is solved by a
modified expectation-maximization algorithm, involving to solve several
indefinite quadratic programming problems (IQPPs). Further, a diagonal block
coordinate decent (DBCD) algorithm is proposed for these IQPPs, whose
stationary points are guaranteed, and the global solutions can be obtained
under certain conditions. To suit for different applications, the FDC is
extended into various metric spaces, e.g., the Reproducing Kernel Hilbert
Space. Experimental results on several benchmark datasets and facial expression
database demonstrate the outperformance of our FDC compared with some
state-of-the-art clustering models.
