  In this work, we consider the matrix completion problem, where the objective
is to reconstruct a low-rank matrix from a few observed entries. A commonly
employed approach involves nuclear norm minimization. For this method to
succeed, the number of observed entries needs to scale at least proportional to
both the rank of the ground-truth matrix and the coherence parameter. While the
only prior information is oftentimes the low-rank nature of the ground-truth
matrix, in various real-world scenarios, additional knowledge about the
ground-truth low-rank matrix is available. For instance, in collaborative
filtering, Netflix problem, and dynamic channel estimation in wireless
communications, we have partial or full knowledge about the signal subspace in
advance. Specifically, we are aware of some subspaces that form multiple angles
with the column and row spaces of the ground-truth matrix. Leveraging this
valuable information has the potential to significantly reduce the required
number of observations. To this end, we introduce a multi-weight nuclear norm
optimization problem that concurrently promotes the low-rank property as well
the information about the available subspaces. The proposed weights are
tailored to penalize each angle corresponding to each basis of the prior
subspace independently. We further propose an optimal weight selection strategy
by minimizing the coherence parameter of the ground-truth matrix, which is
equivalent to minimizing the required number of observations. Simulation
results validate the advantages of incorporating multiple weights in the
completion procedure. Specifically, our proposed multi-weight optimization
problem demonstrates a substantial reduction in the required number of
observations compared to the state-of-the-art methods.
