A Penalty Function for Computing Orthogonal Non-negative Matrix Factorizations

Nonnegative matrix factorization (NMF) is a widely-used method for multivariate analysis of nonnegative data to obtain a reduced representation of data matrix only using a basis matrix and a encoding variable matrix having non-negative elements. A NMF of a data matrix can be obtained by finding a solution of a nonlinear optimization problem over a specified cost function. In this paper we investigate the formulation and then the computational techniques to obtain orthogonal NMF, when the orthogonal constraint on the columns of the basis is added. We propose a penalty objective function to be minimized on the intersection of the set of non-negative matrices and the Stiefel manifold in order to derive a projected gradient flow whose solutions preserve both the orthogonality and the non-negativity.