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.
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