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

摘要: For the problems of low-rank matrix completion, efficiency widely-used nuclear norm technique may be challenged under many circumstances, especially when certain basis coefficients are fixed, for example, correlation completion in various fields such as financial market and density from quantum state tomography. To seek a solution high recovery quality beyond reach norm, this paper, we propose rank-corrected procedure using semi-norm to generate new estimator. estimator, establish non-asymptotic error bound. More importantly, quantify reduction bound procedure. Compared with one obtained penalized least squares can substantial (around 50%). We also provide necessary sufficient conditions rank consistency sense Bach (2008). Very interestingly, these highly related concept constraint nondegeneracy optimization. As byproduct, our results theoretical foundation majorized penalty method Gao Sun (2010) structured optimization problems. Extensive numerical experiments demonstrate that proposed simultaneously achieve accuracy capture structure.