  A semi-parametric, non-linear regression model in the presence of latent
variables is introduced. These latent variables can correspond to unmodeled
phenomena or unmeasured agents in a complex networked system. This new
formulation allows joint estimation of certain non-linearities in the system,
the direct interactions between measured variables, and the effects of
unmodeled elements on the observed system. The particular form of the model
adopted is justified, and learning is posed as a regularized empirical risk
minimization. This leads to classes of structured convex optimization problems
with a "sparse plus low-rank" flavor. Relations between the proposed model and
several common model paradigms, such as those of Robust Principal Component
Analysis (PCA) and Vector Autoregression (VAR), are established. Particularly
in the VAR setting, the low-rank contributions can come from broad trends
exhibited in the time series. Details of the algorithm for learning the model
are presented. Experiments demonstrate the performance of the model and the
estimation algorithm on simulated and real data.
