Stochastic Ml Simplex-Structured Matrix Factorization Under The Dirichlet Mixture Model

Simplex-structured matrix factorization (SSMF) is a problem of recovering a basis matrix and the corresponding coefficient vectors from data, where the coefficient vectors are constrained to lie in the unit simplex. SSMF has attracted growing attention in recent years, with numerous applications such as hyperspectral unmixing and document clustering. In this work, we develop a maximum likelihood (ML) approach for SSMF. Specifically, by modeling the coefficient vectors as random variables following a Dirichlet mixture distribution which allows us to model more complex data distributions in real-life data, a probabilistic model for SSMF is employed. We consider a marginalized likelihood with respect to the coefficient vectors, and use ML estimation to learn the basis matrix and unknown Dirichlet mixture parameters. The marginalized likelihood does not admit a closed form and is non-concave, and this makes the problem challenging to solve. To handle this challenge, an effective algorithm using sample average approximation and block successive upper-bound minimization is proposed. We consider the aforementioned two real-world applications by simulations. Numerical results show that the proposed algorithm delivers appealing performance in both applications.