A nonconvex splitting method for symmetric nonnegative matrix factorization: Convergence analysis and optimality

Symmetric non-negative matrix factorization (SymNMF) has important applications in data analytics problems such as document clustering, community detection and image segmentation. In this paper, we propose a novel nonconvex variable splitting method for solving SymNMF. Different from the existing works, we prove that the algorithm converges to the set of Karush-Kuhn-Tucker (KKT) points of the nonconvex SymNMF problem with a global sublinear convergence rate. We also show that the algorithm can be efficiently implemented in a distributed manner. Further, we provide sufficient conditions that guarantee the global and local optimality of the obtained solutions. Extensive numerical results performed on both synthetic and real data sets suggest that the proposed algorithm yields high quality of the solutions and converges quickly to the set of local minimum solutions compared with other algorithms.