A Bayesian perspective for determinant minimization based robust structured matrix factorization

Abstract

We introduce a Bayesian perspective for the structured matrix factorization problem. The proposed framework provides a probabilistic interpretation for existing geometric methods based on determinant minimization. We model input data vectors as linear transformations of latent vectors drawn from a distribution uniform over a particular domain reflecting structural assumptions, such as the probability simplex in Nonnegative Matrix Factorization and polytopes in Polytopic Matrix Factorization. We represent the rows of the linear transformation matrix as vectors generated independently from a normal distribution whose covariance matrix is inverse Wishart distributed. We show that the corresponding maximum a posteriori estimation problem boils down to the robust determinant minimization approach for structured matrix factorization, providing insights about parameter selections and potential algorithmic extensions.