  Weighted nuclear norm minimization has been recently recognized as a
technique for reconstruction of a low-rank matrix from compressively sampled
measurements when some prior information about the column and row subspaces of
the matrix is available. In this work, we study the recovery conditions and the
associated recovery guarantees of weighted nuclear norm minimization when
multiple weights are allowed. This setup might be used when one has access to
prior subspaces forming multiple angles with the column and row subspaces of
the ground-truth matrix. While existing works in this field use a single weight
to penalize all the angles, we propose a multi-weight problem which is designed
to penalize each angle independently using a distinct weight. Specifically, we
prove that our proposed multi-weight problem is stable and robust under weaker
conditions for the measurement operator than the analogous conditions for
single-weight scenario and standard nuclear norm minimization. Moreover, it
provides better reconstruction error than the state of the art methods. We
illustrate our results with extensive numerical experiments that demonstrate
the advantages of allowing multiple weights in the recovery procedure.
