چكيده به لاتين
One of the significant issues that is interesting is reconstruction of the data matrix with some entries of it. In previous methods, sampling was usually performed in a single-step, uniform manner. In the presented report, a method is suggested that with a fewer samples in comparison with the previous matrices, the matrix can be reconstructed iteratively. In this method, instead of single-component sampling with assuming prior information about the matrix estimation subspace, is sampled from subspace of matrix, and at each step that the answer is improved, the subspace of the solution is used for the sampling in the next step until the matrix is fully reconstructed with low error. It should be noted that if we do not have prior information about the matrix subspace, we need to sample a small number of samples in the first step to obtain an estimate of subspace of the matrix. One of the important issues in this report is to look at the problem of matrix reconstruction from the standpoint of random matrices that is not seen in past work in this area. Because in the first step, we sample a few entries uniformly, so there is still no phase transition, and it must be proved. Before the phase transition, there is also information from the main matrix in the sampling matrix of the first stage, which can be used It rebuilt the matrix. Also, in this report for a two-step sampling method, results have been suggested to prove uniqueness. In the two-step method, for the complete reconstruction of the matrix in the second stage, we need to have a stronger information of the matrix subspace in the first stage, which of course, we should sample a greater number of samples in the uniform sampling (step one).
