Decomposition of Covariance Matrix Using Cascade of Trees

We are looking at the statistical model approximation for jointly Gaussian random vectors. To do so, we are using a cascade of linear transformations that go beyond tree approximations. Here, we propose an algorithm which incorporates the Cholesky factorization method to compute the decomposition matrix and thus can approximate a simple graphical model using a cascade of the Cholesky factorization of the tree approximation transformations. The Cholesky decomposition keeps the sparsity pattern of the inverse decomposition and thus reduces computations for the tree structure linear transformation at each cascade stage of the algorithm. This is a different perspective on the approximation model, and algorithms such as Gaussian belief propagation can be used on this overall graph. We conclude with some simulation results.