  Conventional matrix completion methods approximate the missing values by
assuming the matrix to be low-rank, which leads to a linear approximation of
missing values. It has been shown that enhanced performance could be attained
by using nonlinear estimators such as deep neural networks. Deep fully
connected neural networks (FCNNs), one of the most suitable architectures for
matrix completion, suffer from over-fitting due to their high capacity, which
leads to low generalizability. In this paper, we control over-fitting by
regularizing the FCNN model in terms of the $\ell_{1}$ norm of intermediate
representations and nuclear norm of weight matrices. As such, the resulting
regularized objective function becomes nonsmooth and nonconvex, i.e., existing
gradient-based methods cannot be applied to our model. We propose a variant of
the proximal gradient method and investigate its convergence to a critical
point. In the initial epochs of FCNN training, the regularization terms are
ignored, and through epochs, the effect of that increases. The gradual addition
of nonsmooth regularization terms is the main reason for the better performance
of the deep neural network with nonsmooth regularization terms (DNN-NSR)
algorithm. Our simulations indicate the superiority of the proposed algorithm
in comparison with existing linear and nonlinear algorithms.
