Non-Convex Matrix Completion Against a Semi-Random Adversary

Matrix completion is a well-studied problem with many machine learning applications. In practice, the problem is often solved by non-convex optimization algorithms. However, the current theoretical analysis for non-convex algorithms relies heavily on the assumption that every entry is observed with exactly the same probability $p$, which is not realistic in practice. In this paper, we investigate a more realistic semi-random model, where the probability of observing each entry is at least $p$. Even with this mild semi-random perturbation, we can construct counter-examples where existing non-convex algorithms get stuck in bad local optima. In light of the negative results, we propose a pre-processing step that tries to re-weight the semi-random input, so that it becomes "similar" to a random input. We give a nearly-linear time algorithm for this problem, and show that after our pre-processing, all the local minima of the non-convex objective can be used to approximately recover the underlying ground-truth matrix.