Learning a Small Mixture of Trees

The problem of approximating a given probability distribution using a simpler distribution plays an important role in several areas of machine learning, e.g. variational inference and classification. Within this context, we consider the task of learning a mixture of tree distributions. Although mixtures of trees can be learned by minimizing the KL-divergence using an EM algorithm, its success depends heavily on the initialization. We propose an efficient strategy for obtaining a good initial set of trees that attempts to cover the entire observed distribution by minimizing the $\alpha$-divergence with $\alpha = \infty$. We formulate the problem using the fractional covering framework and present a convergent sequential algorithm that only relies on solving a convex program at each iteration. Compared to previous methods, our approach results in a significantly smaller mixture of trees that provides similar or better accuracies. We demonstrate the usefulness of our approach by learning pictorial structures for face recognition.