  In this paper, we develop a parameter estimation method for factorially
parametrized models such as Factorial Gaussian Mixture Model and Factorial
Hidden Markov Model. Our contributions are two-fold. First, we show that the
emission matrix of the standard Factorial Model is unidentifiable even if the
true assignment matrix is known. Secondly, we address the issue of
identifiability by making a one component sharing assumption and derive a
parameter learning algorithm for this case. Our approach is based on a
dictionary learning problem of the form $X = O R$, where the goal is to learn
the dictionary $O$ given the data matrix $X$. We argue that due to the specific
structure of the activation matrix $R$ in the shared component factorial
mixture model, and an incoherence assumption on the shared component, it is
possible to extract the columns of the $O$ matrix without the need for
alternating between the estimation of $O$ and $R$.
