A Riemannian geometry for low-rank matrix completion

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 algorithms can be considered as improved versions 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.