Statistical Rank Selection For Incomplete Low-Rank Matrices

We consider the problem of determining the rank in the low-rank matrix completion. We propose a statistical model for noisy observation. It is important for many existing algorithms and sometimes has practical meanings. We construct a test statistics for the low rank approximation problem. Under this model, we derive the distribution of the test statistics. By applying the test statistics, we propose a sequential rank test procedure to determine the rank with statistical inference. In the numerical section, we illustrate our theoretical results and give examples of our proposed rank test procedure.