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.

通过在深度神经网络中引入非光滑正则化项，我们控制过拟合，并在矩阵填充任务中实现更好的性能。我们提出了一个变种的近端梯度法，并研究其收敛性，通过逐步添加非光滑正则化项来改善深度神经网络的性能。通过模拟实验，我们证明了所提算法相对于线性和非线性算法的优越性。

基于完全连接神经网络的非光滑正则化矩阵补全