A stochastic maximum-likelihood framework for simplex structured matrix factorization

Consider a structured matrix factorizaton (SMF) whose coefficient vectors are constrained to lie in the unit simplex. This kind of simplex SMF (SSMF) has received growing attention and has found many applications such as hyperspectral unmixing in remote sensing, text mining in machine learning, and blind source separation in signal processing. The aim of this paper is to establish a maximum-likelihood (ML) estimation framework for SSMF in the presence of Gaussian noise and outliers, and to demonstrate its potential. Our ML formulation has the coefficient vectors marginalized in accordance with a prescribed probabilistic model, and this leads to a likelihood function that contains multi-dimensional integrals. Unfortunately these integrals do not appear to have analytically tractable solutions, and this makes the ML problem challenging. We tackle the problem by using sample average approximation in stochastic optimization and majorization-minimization. Simulation results show that the resulting ML algorithm significantly outperforms several existing methods when noise and outliers are present.