  Matrix sensing is the problem of reconstructing a low-rank matrix from a few
linear measurements. In many applications such as collaborative filtering, the
famous Netflix prize problem, and seismic data interpolation, there exists some
prior information about the column and row spaces of the ground-truth low-rank
matrix. In this paper, we exploit this prior information by proposing a
weighted optimization problem where its objective function promotes both rank
and prior subspace information. Using the recent results in conic integral
geometry, we obtain the unique optimal weights that minimize the required
number of measurements. As simulation results confirm, the proposed convex
program with optimal weights requires substantially fewer measurements than the
regular nuclear norm minimization.
