  Semi-supervised symmetric non-negative matrix factorization (SNMF) utilizes
the available supervisory information (usually in the form of pairwise
constraints) to improve the clustering ability of SNMF. The previous methods
introduce the pairwise constraints from the local perspective, i.e., they
either directly refine the similarity matrix element-wisely or restrain the
distance of the decomposed vectors in pairs according to the pairwise
constraints, which overlook the global perspective, i.e., in the ideal case,
the pairwise constraint matrix and the ideal similarity matrix possess the same
low-rank structure. To this end, we first propose a novel semi-supervised SNMF
model by seeking low-rank representation for the tensor synthesized by the
pairwise constraint matrix and a similarity matrix obtained by the product of
the embedding matrix and its transpose, which could strengthen those two
matrices simultaneously from a global perspective. We then propose an enhanced
SNMF model, making the embedding matrix tailored to the above tensor low-rank
representation. We finally refine the similarity matrix by the strengthened
pairwise constraints. We repeat the above steps to continuously boost the
similarity matrix and pairwise constraint matrix, leading to a high-quality
embedding matrix. Extensive experiments substantiate the superiority of our
method. The code is available at https://github.com/JinaLeejnl/TSNMF.
