A Dictionary Learning Approach for Factorial Gaussian Models
This work solves a specific identifiability problem in factorial models for researchers in machine learning, but it is incremental as it builds on existing dictionary learning methods.
The paper tackled the identifiability issue in factorial Gaussian models by showing that the emission matrix is unidentifiable even with known assignments, and addressed this with a one component sharing assumption to derive a parameter learning algorithm based on dictionary learning.
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 $R$.