  Network estimation from multi-variate point process or time series data is a
problem of fundamental importance. Prior work has focused on parametric
approaches that require a known parametric model, which makes estimation
procedures less robust to model mis-specification, non-linearities and
heterogeneities. In this paper, we develop a semi-parametric approach based on
the monotone single-index multi-variate autoregressive model (SIMAM) which
addresses these challenges. We provide theoretical guarantees for dependent
data and an alternating projected gradient descent algorithm. Significantly we
do not explicitly assume mixing conditions on the process (although we do
require conditions analogous to restricted strong convexity) and we achieve
rates of the form $O(T^{-\frac{1}{3}} \sqrt{s\log(TM)})$ (optimal in the
independent design case) where $s$ is the threshold for the maximum in-degree
of the network that indicates the sparsity level, $M$ is the number of actors
and $T$ is the number of time points. In addition, we demonstrate the superior
performance both on simulated data and two real data examples where our SIMAM
approach out-performs state-of-the-art parametric methods both in terms of
prediction and network estimation.
