  Network modeling of high-dimensional time series data is a key learning task
due to its widespread use in a number of application areas, including
macroeconomics, finance and neuroscience. While the problem of sparse modeling
based on vector autoregressive models (VAR) has been investigated in depth in
the literature, more complex network structures that involve low rank and group
sparse components have received considerably less attention, despite their
presence in data. Failure to account for low-rank structures results in
spurious connectivity among the observed time series, which may lead
practitioners to draw incorrect conclusions about pertinent scientific or
policy questions. In order to accurately estimate a network of Granger causal
interactions after accounting for latent effects, we introduce a novel approach
for estimating low-rank and structured sparse high-dimensional VAR models. We
introduce a regularized framework involving a combination of nuclear norm and
lasso (or group lasso) penalty. Further, and subsequently establish
non-asymptotic upper bounds on the estimation error rates of the low-rank and
the structured sparse components. We also introduce a fast estimation algorithm
and finally demonstrate the performance of the proposed modeling framework over
standard sparse VAR estimates through numerical experiments on synthetic and
real data.
