Two Novel Algorithms for Low-Rank Matrix Completion Problem

Low-rank matrix completion has been exploited in many signal processing problems over the past few years. It has been shown that nuclear norm minimization can retrieve such low-rank matrices via uniform sampling under coherence conditions. However, in many applications such as the Netflix problem, the low-rank matrix is not coherent, hence nuclear norm minimization is unable to solve the problem because of uniform sampling. To overcome this issue, we propose two algorithms to recover both coherent and incoherent low-rank matrices. In the first algorithm, it is assumed that we have a limited sampling budget that is associated with the available number of samples. Using a portion of the sampling budget, we propose an Uncertainty Information (UI) metric to quantify the uncertainty that we have for each element and consider the elements with the most uncertainty as the most informative elements to sample. Finally, the simulations results are performed to investigate the performance of the proposed algorithms and show their superiority to conventional approaches such as nuclear norm minimization Candès and Recht, 2009 and two-phase sampling algorithm Chen et al. 2015.