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A Bayesian Perspective for Determinant Minimization Based Robust Structured Matrix Factorizatio
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