  In this paper, the problem of matrix rank minimization under affine
constraints is addressed. The state-of-the-art algorithms can recover matrices
with a rank much less than what is sufficient for the uniqueness of the
solution of this optimization problem. We propose an algorithm based on a
smooth approximation of the rank function, which practically improves recovery
limits on the rank of the solution. This approximation leads to a non-convex
program; thus, to avoid getting trapped in local solutions, we use the
following scheme. Initially, a rough approximation of the rank function subject
to the affine constraints is optimized. As the algorithm proceeds, finer
approximations of the rank are optimized and the solver is initialized with the
solution of the previous approximation until reaching the desired accuracy.
  On the theoretical side, benefiting from the spherical section property, we
will show that the sequence of the solutions of the approximating function
converges to the minimum rank solution. On the experimental side, it will be
shown that the proposed algorithm, termed SRF standing for Smoothed Rank
Function, can recover matrices which are unique solutions of the rank
minimization problem and yet not recoverable by nuclear norm minimization.
Furthermore, it will be demonstrated that, in completing partially observed
matrices, the accuracy of SRF is considerably and consistently better than some
famous algorithms when the number of revealed entries is close to the minimum
number of parameters that uniquely represent a low-rank matrix.
