Iterative Shrinkage Thresholding Algorithm (ISTA) has been widely applied to solve linear inverse problems. To increase the rate of convergence, Learned-ISTA (LISTA) adopts deep learning techniques to learn the optimal algorithm parameters like the mutual inhibition matrices and filter matrices, which significantly reduces the number of iterations. However, the size of the learned mutual inhibition matrix exhibits quadratic growth in the length of sparse signal, which restricts the applicability of LISTA in some large-scale problems. In many applications such as direction-of-arrival (DOA) estimation, the learned mutual inhibition matrix naturally has a Toeplitz structure. Here we exploit the Toeplitz structure and propose a convolutional network, namely LISTA-Toeplitz, to reduce the memory cost. Simulation results show that LISTA-Toeplitz outperforms traditional ISTA in convergence speed and achieves a level of accuracy comparable to LISTA in DOA simulation.