  The affine rank minimization (ARM) problem arises in many real-world
applications. The goal is to recover a low-rank matrix from a small amount of
noisy affine measurements. The original problem is NP-hard, and so directly
solving the problem is computationally prohibitive. Approximate low-complexity
solutions for ARM have recently attracted much research interest. In this
paper, we design an iterative algorithm for ARM based on message passing
principles. The proposed algorithm is termed turbo-type ARM (TARM), as inspired
by the recently developed turbo compressed sensing algorithm for sparse signal
recovery. We show that, when the linear operator for measurement is
right-orthogonally invariant (ROIL), a scalar function called state evolution
can be established to accurately predict the behaviour of the TARM algorithm.
We also show that TARM converges much faster than the counterpart algorithms
for low-rank matrix recovery. We further extend the TARM algorithm for matrix
completion, where the measurement operator corresponds to a random selection
matrix. We show that, although the state evolution is not accurate for matrix
completion, the TARM algorithm with carefully tuned parameters still
significantly outperforms its counterparts.
