A Rank-Corrected Procedure for Matrix Completion with Fixed Basis Coefficients

In this paper, we address low-rank matrix completion problems with fixed basis coefficients, which include the low-rank correlation matrix completion in various fields such as the financial market and the low-rank density matrix completion from the quantum state tomography. For this class of problems, the efficiency of the common nuclear norm penalized estimator for recovery may be challenged. Here, with a reasonable initial estimator, we propose a rank-corrected procedure to generate an estimator of high accuracy and low rank. For this new estimator, we establish a non-asymptotic recovery error bound and analyze the impact of adding the rank-correction term on improving the recoverability. We also provide necessary and sufficient conditions for rank consistency in the sense of Bach (JMLR 9:1019-1048, 2008), in which the concept of constraint nondegeneracy in matrix optimization plays an important role. As a byproduct, our results provide a theoretical foundation for the majorized penalty method of Gao and Sun (2010) and Gao (PhD thesis, 2010) for structured low-rank matrix optimization problems.