Learning Latent Networks in Vector Auto Regressive Models

We study the problem of learning the dependency graph between random processes in a Vector Autoregressive (VAR) model from samples when a subset of the variables are latent. We show that the dependencies among the observed processes can be identified successfully under some conditions on the VAR model. Moreover, we can recover the length of all directed paths between any two observed processes which pass through latent part. By utilizing this information, we reconstruct the latent subgraph with minimum number of nodes uniquely if its topology is a directed tree. Furthermore, we propose an algorithm that finds all possible minimal latent networks if there exists at most one directed path of each length between any two observed nodes through the latent part. Experimental results on various synthetic and real-world datasets validate our theoretical results.