Consider the nonnegative matrix factorization (NMF) : min_{W,H} || V – W||_F, where V ∈ R^{m×n} is a given nonnegative matrix, W ∈ R^{m×r} and H ∈ R^{r×n} are unknown nonnegative matrices, and ||・||_F represents the Frobenius norm of the corresponding matrix. Here, r ≪ min(m,n) is assumed. Therefore, the NMF problem seeks a nonnegative low rank approximation of a given nonnegative matrix. NMF arises in many scientific computing and engineering applications, e.g., image processing, spectral data analysis, audio signal separation, text mining, document clustering, recommender system, etc. For the solution of NMF, we propose a new alternating nonnegative least squares method by utilizing the modulus method [2, 3] for solving the nonnegative constrained least squares (NNLS) problems in each iteration. The method employs the modulus transform H = Z + |Z| and W = Y + |Y| for each subproblem to transform the NNLS problem to a sequence of unconstrained least squares problems, which can be solved by a CGLS method for matrix variables. Numerical experiments on random problems and ORL face image problems show the efficiency of the proposed method compared to the multiplicative update method [4] and gradient-type methods.

This is joint work with Dr. Ning Zheng and Dr. Nobutaka Ono