Covariance Matrix Decomposition Using Cascade of Linear Tree Transformations

Abstract

The tree model can be computed efficiently using the Chow-Liu algorithm to minimize the Kullback-Leibler (KL) divergence. This paper goes beyond tree approximations by systematically forming a cascade of linear transformations where each linear transformation represents a tree structure. The linear transformation is found via a Cholesky factorization to provide sparsity to the inverse covariance matrix. We show that each successive additional cascade linear transformation improves the approximation with respect to the KL divergence. We conclude by showing some simulation results on synthetic data examining the quality of tree and non-tree approximations.