چكيده به لاتين
Solving the matrix completion problem has always been researched as one of the practical issues in the field of signal processing. In recent years artificial neural networks in particular have been able to work well in solving many real-world problems such as computer vision, natural language processing, etc. The idea of this thesis is to use neural network to solve the matrix completion problem. Considering that the number of missing entries of a matrix in the problem of matrix completion is a large number compared to the total number of entries, then the matrix that we have to complete is sparse. On the other hand, the small amount of data for deep neural network training causes the network to lack generalization power and overfit. Our solution to control overfitting is to use the terms of the l1-norm of the output layers of the neural network and nuclear norm on all the weight matrices between the different layers of the network in addition to the main loss function of the network. Due to the non-smoothness of these terms, it is not possible to use conventional optimization methods such as SGD, ADAM, etc. The use of proximal algorithms is the same solution that approximates the effect of the non-smooth of the loss function with an operator. Due to the non-convexity of the network loss, the convergence of this algorithm to the critical points is proved under certain conditions. In the proposed DL-NLMC algorithm, we have achived this by limiting the network parameters and we have done the convergence proof. The proof is valid for any feed forward neural network with any number of non-smooth regularization terms. We have tested the presented algorithm on three types of matrix data, synthesis matrix, image and MovieLens dataset. The results are as follows: For the synthesis matrix, we have used PSNR and MSE criteria for different missing rates of entries and compared the results with 5 other algorithms. For the missing rate of 80%, the maximum improvement achieved with the PSNR criterion is 4.2 dB, which is a significant amount. The improvement rate with the MSE criterion was equal to 4:3377. We have considered two images with pixel missing rates of 30%, 40% and 50% and by applying the presented algorithm, we have measured the obtained results with PSNR and SSIM criteria. The improvement rate of the first criterion is 1.1068 dB for the first image and 0.9153 dB for the second image. These numbers are reported for a 50% missing rate. But the second criterion experienced a change of 0.0179 for the first image and 0.0081 for the second image. But for the data of the third matrix, recommender systems, and for two datasets of MovieLens 100k and MovieLens 1M, we have applied the algorithm. The improvement rate of the algorithm on the first and second datasets and for the missing rate of 30%, with the NMAE criterion is 9:8% and 1:18%, respectively, and for the missing rate of 50% equal to 5:8% and 3:49%.
