  We propose a new Riemannian geometry for fixed-rank matrices that is
specifically tailored to the low-rank matrix completion problem. Exploiting the
degree of freedom of a quotient space, we tune the metric on our search space
to the particular least square cost function. At one level, it illustrates in a
novel way how to exploit the versatile framework of optimization on quotient
manifold. At another level, our algorithm can be considered as an improved
version of LMaFit, the state-of-the-art Gauss-Seidel algorithm. We develop
necessary tools needed to perform both first-order and second-order
optimization. In particular, we propose gradient descent schemes (steepest
descent and conjugate gradient) and trust-region algorithms. We also show that,
thanks to the simplicity of the cost function, it is numerically cheap to
perform an exact linesearch given a search direction, which makes our
algorithms competitive with the state-of-the-art on standard low-rank matrix
completion instances.
