Provable Matrix Sensing using Alternating Minimization

Alternating minimization has emerged as a popular heuristic for large-scale machine learning problems involving low-rank matrices. However, there have been few (if any) theoretical guarantees on its performance. In this work, we investigate the natural alternating minimization algorithm for the popular matrix sensing problem first formulated in [RFP07]; this problem asks for the recovery of an unknown low-rank matrix from a small number of linear measurements thereof. We show that under suitable RIP conditions, alternating minimization linearly converges to the true matrix. Our result can be extended to matrix completion from randomly sampled entries. Our analysis uses only elementary linear algebra and exploits the fact that, under RIP, alternating minimization can be viewed as a noisy version of orthogonal iteration (which is used to compute the top singular vectors of a matrix).