Orthogonal Nonnegative Matrix Factorization via Minimization over the Null Space

Abstract. This paper gives a necessary and sufficient condition for a nonnegative matrix that has an orthogonal nonnegative matrix factorization (ONMF) via characterization of the null space. We propose an optimization model to minimize the Frobenius norm of the product of a given nonnegative matrix and a variable matrix subject to the constraints defined by the necessary and sufficient condition. Moreover, we present an augmented Lagrangian algorithm for solving this minimization model and prove the global convergence to a stationary point. Two factor matrices for the ONMF of the given matrix can be easily obtained by the outputs of the algorithm. Preliminary numerical results using synthetic and real-world data with applications in clustering show that our approach outperforms some existing ONMF methods regarding accuracy and robustness. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as recognition that the authors have followed reproducibility principles valued by SIMAX and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/Qilun-Luo/ONMF . [Formula: see text]