Weighted Matrix Completion and Recovery with Prior Subspace Information

A low-rank matrix with “diffuse” entries can beefficiently reconstructed after observing a few of its entries,at random, and then solving a convex program. In manyapplications, in addition to these measurements, potentiallyvaluable prior knowledge about the column and row spaces ofthe matrix is also available to the practitioner. In this paper,we incorporate this prior knowledge in matrix completion—byminimizing a weighted nuclear norm—and precisely quantify anyimprovements. In particular, in theory, we find that reliable priorknowledge reduces the sample complexity of matrix completionby a logarithmic factor; the observed improvement is consider-ably more magnified in numerical simulations. We also presentsimilar results for the closely related problem of matrix recoveryfrom generic linear measurements.