  In this paper, we study the system identification problem for sparse linear
time-invariant systems. We propose a sparsity promoting block-regularized
estimator to identify the dynamics of the system with only a limited number of
input-state data samples. We characterize the properties of this estimator
under high-dimensional scaling, where the growth rate of the system dimension
is comparable to or even faster than that of the number of available sample
trajectories. In particular, using contemporary results on high-dimensional
statistics, we show that the proposed estimator results in a small element-wise
error, provided that the number of sample trajectories is above a threshold.
This threshold depends polynomially on the size of each block and the number of
nonzero elements at different rows of input and state matrices, but only
logarithmically on the system dimension. A by-product of this result is that
the number of sample trajectories required for sparse system identification is
significantly smaller than the dimension of the system. Furthermore, we show
that, unlike the recently celebrated least-squares estimators for system
identification problems, the method developed in this work is capable of
\textit{exact recovery} of the underlying sparsity structure of the system with
the aforementioned number of data samples. Extensive case studies on
synthetically generated systems, physical mass-spring networks, and multi-agent
systems are offered to demonstrate the effectiveness of the proposed method.
