Rank $2r$ iterative least squares: efficient recovery of ill-conditioned low rank matrices from few entries

We present a new, simple and computationally efficient iterative method for low rank matrix completion. Our method is inspired by the class of factorization-type iterative algorithms, but substantially differs from them in the way the problem is cast. Precisely, given a target rank $r$, instead of optimizing on the manifold of rank $r$ matrices, we allow our interim estimated matrix to have a specific over-parametrized rank $2r$ structure. Our algorithm, denoted \texttt{R2RILS}, for rank $2r$ iterative least squares, thus has low memory requirements, and at each iteration it solves a computationally cheap sparse least-squares problem. We motivate our algorithm by its theoretical analysis for the simplified case of a rank-1 matrix. Empirically, \texttt{R2RILS} is able to recover, with machine precision, ill conditioned low rank matrices from very few observations -- near the information limit. Finally, \texttt{R2RILS} is stable to corruption of the observed entries by additive zero mean Gaussian noise.