Tree-structured statistical modeling via convex optimization

We develop a semidefinite-programming-based approach to stochastic modeling with multiscale autoregressive (MAR) processes—a class of stochastic processes indexed by the vertices of a tree. Given a tree and the covariance matrix of the variables corresponding to the leaves of the tree, our procedure aims to construct an MAR process with small state dimensions at each vertex that approximately realizes the given covariance at the leaves. Our method does not require prior specification of the state dimensions at each vertex. Furthermore, we establish a large class of MAR processes for which, given only the index tree and the leaf covariance of the process, our method can recover a parametrization that matches the leaf-covariance and has the correct state dimensions. Finally we demonstrate, using synthetic examples, that given i.i.d. samples of the leaf variables our method can recover the correct state dimensions of an underlying MAR process.