Low Rank Matrix Completion via Robust Alternating Minimization in Nearly Linear Time

Given a matrix $M\in \mathbb{R}^{m\times n}$, the low rank matrix completion problem asks us to find a rank-$k$ approximation of $M$ as $UV^\top$ for $U\in \mathbb{R}^{m\times k}$ and $V\in \mathbb{R}^{n\times k}$ by only observing a few entries masked by a binary matrix $P_{\Omega}\in \{0, 1 \}^{m\times n}$. As a particular instance of the weighted low rank approximation problem, solving low rank matrix completion is known to be computationally hard even to find an approximate solution [RSW16]. However, due to its practical importance, many heuristics have been proposed for this problem. In the seminal work of Jain, Netrapalli, and Sanghavi [JNS13], they show that the alternating minimization framework provides provable guarantees for low rank matrix completion problem whenever $M$ admits an incoherent low rank factorization. Unfortunately, their algorithm requires solving two exact multiple response regressions per iteration and their analysis is non-robust as they exploit the structure of the exact solution. In this paper, we take a major step towards a more efficient and robust alternating minimization framework for low rank matrix completion. Our main result is a robust alternating minimization algorithm that can tolerate moderate errors even though the regressions are solved approximately. Consequently, we also significantly improve the running time of [JNS13] from $\widetilde{O}(mnk^2 )$ to $\widetilde{O}(mnk )$ which is nearly linear in the problem size, as verifying the low rank approximation takes $O(mnk)$ time. Our core algorithmic building block is a high accuracy regression solver that solves the regression in nearly linear time per iteration.