  In this paper, we develop a variant of the well-known Gauss-Newton (GN)
method to solve a class of nonconvex optimization problems involving low-rank
matrix variables. As opposed to the standard GN method, our algorithm allows
one to handle general smooth convex objective function. We show, under mild
conditions, that the proposed algorithm globally and locally converges to a
stationary point of the original problem. We also show empirically that the GN
algorithm achieves higher accurate solutions than the alternating minimization
algorithm (AMA). Then, we specify our GN scheme to handle the symmetric case
and prove its convergence, where AMA is not applicable. Next, we incorporate
our GN scheme into the alternating direction method of multipliers (ADMM) to
develop an ADMM-GN algorithm. We prove that, under mild conditions and a proper
choice of the penalty parameter, our ADMM-GN globally converges to a stationary
point of the original problem. Finally, we provide several numerical
experiments to illustrate the proposed algorithms. Our results show that the
new algorithms have encouraging performance compared to existing methods.
