Rank $2r$ iterative least squares: efficient recovery of ill-conditioned low rank matrices from few entries
This addresses matrix completion for applications like recommendation systems, but it is incremental as it builds on factorization-type iterative algorithms.
The paper tackles the problem of recovering ill-conditioned low-rank matrices from few entries by introducing R2RILS, a new iterative method that uses an over-parametrized rank 2r structure, achieving recovery near the information limit with stability to noise.
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.