A Column-Wise Update Algorithm for Sparse Stochastic Matrix Factorization

Nonnegative matrix factorization arises widely in machine learning and data analysis. In this paper, for a given factorization of rank r, we consider the sparse stochastic matrix factorization (SSMF) of decomposing a prescribed m-by-n stochastic matrix V into a product of an m-by-r stochastic matrix W and an r-by-n stochastic matrix H, where both W and H are required to be sparse. With the prescribed sparsity level, we reformulate the SSMF as an unconstrained nonconvex-nonsmooth minimization problem and introduce a column-wise update algorithm for solving the minimization problem. We show that our algorithm converges globally. The main advantage of our algorithm is that the generated sequence converges to a special critical point of the cost function, which is nearly a global minimizer over each column vector of the W-factor and is a global minimizer over the H-factor as a whole if there is no sparsity requirement on H. Numerical experiments on both synthetic and real data sets are given to demonstrate the effectiveness of our proposed algorithm.