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Autoencoder recommenders have recently shown state-of-the-art performance in the recommendation task due to their ability to model non-linear item relationships effectively. However, existing autoencoder recommenders use fully-connected neural network layers and do not employ structure learning. This can lead to inefficient training, especially when the data is sparse as commonly found in collaborative filtering. The aforementioned results in lower generalization ability and reduced performance. In this paper, we introduce structure learning for autoencoder recommenders by taking advantage of the inherent item groups present in the collaborative filtering domain. Due to the nature of items in general, we know that certain items are more related to each other than to other items. Based on this, we propose a method that first learns groups of related items and then uses this information to determine the connectivity structure of an auto-encoding neural network. This results in a network that is sparsely connected. This sparse structure can be viewed as a prior that guides the network training. Empirically we demonstrate that the proposed structure learning enables the autoencoder to converge to a local optimum with a much smaller spectral norm and generalization error bound than the fully-connected network. The resultant sparse network considerably outperforms the state-of-the-art methods like \textsc{Mult-vae/Mult-dae} on multiple benchmarked datasets even when the same number of parameters and flops are used. It also has a better cold-start performance.