A Close Look at Matrix Completion with Noisy Data

Matrix completion is a technique that utilizes the low rank characteristic of a matrix to recover unknown elements of the matrix with some observed elements. In practical applications, the observed elements are often contaminated by noise, which makes the matrix completion more difficult. In this paper, we first establish the mathematical models of the matrix completion with noise, and investigate three classical methods including SVT, ADM and OptSpace. Particularly, we propose to add convergence criterion and restrain divergence criterion to strengthen the constraint of the stopping criterion, solving the problem of non-convergence caused by noise data when the relative recovery error is used as the stopping criterion. Several experiments are designed to estimate the performance of these three algorithms in the task of matrix completion with noise on the synthesized dataset and Jester Joke dataset. Experimental results show that it is risky to arbitrarily assume that one of these methods is optimal. Specifically, SVT has the advantage in cost, OptSpace has the advantage in anti-noise, and ADM has the better robustness to rank. Furthermore, we apply ADM algorithm in the image recovery task, and obtain excellent results.