  We study the problem of learning the support of transition matrix between
random processes in a Vector Autoregressive (VAR) model from samples when a
subset of the processes are latent. It is well known that ignoring the effect
of the latent processes may lead to very different estimates of the influences
among observed processes, and we are concerned with identifying the influences
among the observed processes, those between the latent ones, and those from the
latent to the observed ones. We show that the support of transition matrix
among the observed processes and lengths of all latent paths between any two
observed processes can be identified successfully under some conditions on the
VAR model. From the lengths of latent paths, we reconstruct the latent subgraph
(representing the influences among the latent processes) with a minimum number
of variables uniquely if its topology is a directed tree. Furthermore, we
propose an algorithm that finds all possible minimal latent graphs under some
conditions on the lengths of latent paths. Our results apply to both
non-Gaussian and Gaussian cases, and experimental results on various synthetic
and real-world datasets validate our theoretical results.
