A Nonconvex Splitting Method for Symmetric Nonnegative Matrix Factorization: Convergence Analysis and Optimality
This work addresses SymNMF for applications like document clustering and community detection, but it is incremental as it builds on existing methods with convergence guarantees.
The authors tackled the symmetric nonnegative matrix factorization (SymNMF) problem by proposing a nonconvex variable splitting method, which is guaranteed to converge to KKT points with a global sublinear rate and shows quick convergence to local minima in numerical tests.
Symmetric nonnegative 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. The proposed algorithm is guaranteed to converge to the set of Karush-Kuhn-Tucker (KKT) points of the nonconvex SymNMF problem. Furthermore, it achieves a global sublinear convergence rate. We also show that the algorithm can be efficiently implemented in parallel. Further, sufficient conditions are provided which 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 converges quickly to a local minimum solution.