MATRIX COMPLETION MODELS WITH FIXED BASIS COEFFICIENTS AND RANK REGULARIZED PROBLEMS WITH HARD CONSTRAINTS

摘要: The problems with embedded low-rank structures arise in diverse areas such as engineering, statistics, quantum information, finance and graph theory. This thesis is devoted to dealing the structure via techniques beyond widely-used nuclear norm for achieving better performance. In first part, we propose a rank-corrected procedure matrix completion fixed basis coefficients. We establish non-asymptotic recovery error bounds provide necessary sufficient conditions rank consistency. obtained results, together numerical experiments, indicate superiority of our pro- posed rank-correction step over penalization. second an adaptive semi-nuclear regularization approach address regularized hard constraints solving their nonconvex but con- tinuous approximation instead. overcomes difficulty extending iterative reweighted l1 minimization from vector case case. Numerical experiments show that scheme pose has advantages both low-rank-structure-preserving ability computational efficiency.