A Homotopy Optimization Method for Orthogonal Non-Negative Matrix Factorization

Data clustering is a key problem in data science and machine learning. In this paper, we consider orthogonal nonnegative matrix factorization (ONMF) for scaled data clustering. The non-convex orthogonality constraint of ONMF raises a great challenge from an optimization viewpoint. We study a convex-constrained transformation of ONMF that allows us to control the approximation accuracy and problem difficulty through a parameter. We then apply a homotopy strategy in which we trace the solution path of a sequence of the aforementioned transformed problems, gradually moving from easy problems to near-ONMF problems. Intuitively, doing so may allow us to avoid local minima. Numerical results show that our homotopy method yields competitive clustering performance in synthetic data experiments and in a real-data hyperspectral clustering experiment.