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The acquisition of channel state information (CSI) is essential in millimeter wave (mmWave) multiple-input multiple-output (MIMO) systems. The mmWave channel exhibits sparse scattering characteristics and a meaningful low-rank structure, which can be simultaneously employed to reduce the complexity of channel estimation. Most existing works recover the low-rank structure of channels using nuclear norm theory. However, solving the nuclear norm-based convex problem often leads to a suboptimal solution of the rank minimization problem, thus degrading the accuracy of channel estimation. Previous contributions recover the channel using over-complete dictionary with the assumption that the mmWave channel can be sparsely represented under some dictionary. While over-complete dictionary may increase the computational complexity. To address these problems, we propose a channel estimation framework based on non-convex low-rank approximation and dictionary learning by exploring the joint low-rank and sparse representations of wireless channels. We surrogate the widely used nuclear norm theory with non-convex low-rank approximation method and design a dictionary learning algorithm based on channel feature classification employing deep neural network (DNN). Our simulation results reveal the proposed scheme outperform the conventional dictionary learning algorithm, Bayesian framework algorithm, and compressed sensing-based algorithms.