  The problem of low-rank matrix reconstruction arises in various applications
in communications and signal processing. The state of the art research largely
focuses on the recovery techniques that utilize affine maps satisfying the
restricted isometry property (RIP). However, the affine map design and
reconstruction under a priori information, i.e., column or row subspace
information, has not been thoroughly investigated. To this end, we present
designs of affine maps and reconstruction algorithms that fully exploit the
low-rank matrix subspace information. Compared to the randomly generated affine
map, the proposed affine map design permits an enhanced reconstruction. In
addition, we derive an optimal representation of low-rank matrices, which is
exploited to optimize the rank and subspace of the estimate by adapting them to
the noise level in order to achieve the minimum mean square error (MSE).
Moreover, in the case when the subspace information is not a priori available,
we propose a two-step algorithm, where, in the first step, it estimates the
column subspace of a low-rank matrix, and in the second step, it exploits the
estimated information to complete the reconstruction. The simulation results
show that the proposed algorithm achieves robust performance with much lower
complexity than existing reconstruction algorithms.
