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 R2RILS for rank 2r iterative least squares, 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, R2RILS is able to recover ill conditioned low rank matrices from very few observations -- near the information limit, and it is stable to additive noise.