Abstract

This paper considers the problem of learning, from samples, the dependency structure of a system of linear stochastic differential equations, when some of the variables are latent. We observe the time evolution of some variables, and never observe other variables; from this, we would like to find the dependency structure of the observed variables – separating out the spurious interactions caused by the latent variables’ time series. We develop a new convex optimization based method to do so in the case when the number of latent variables is smaller than the number of observed ones. For the case when the dependency structure between the observed variables is sparse, we theoretically establish a high-dimensional scaling result for structure recovery. We verify our theoretical result with both synthetic and real data (from the stock market).