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We present a new, simple and computationally efficient iterative method forlow rank matrix completion. Our method is inspired by the class offactorization-type iterative algorithms, but substantially differs from them inthe way the problem is cast. Precisely, given a target rank $r$, instead ofoptimizing on the manifold of rank $r$ matrices, we allow our interim estimatedmatrix to have a specific over-parametrized rank $2r$ structure. Our algorithm,denoted R2RILS for rank $2r$ iterative least squares, has low memoryrequirements, and at each iteration it solves a computationally cheap sparseleast-squares problem. We motivate our algorithm by its theoretical analysisfor the simplified case of a rank-1 matrix. Empirically, R2RILS is able torecover ill conditioned low rank matrices from very few observations -- nearthe information limit, and it is stable to additive noise.

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