Semi-Random Matrix Completion via Flow-Based Adaptive Reweighting

Part of Advances in Neural Information Processing Systems 37 (NeurIPS 2024) Main Conference Track

Authors

Jonathan A. Kelner, Jerry Li, Allen Liu, Aaron Sidford, Kevin Tian

Abstract

We consider the well-studied problem of completing a rank-$r$, $\mu$-incoherent matrix $\mathbf{M} \in \mathbb{R}^{d \times d}$ from incomplete observations. We focus on this problem in the semi-random setting where each entry is independently revealed with probability at least $p = \frac{\textup{poly}(r, \mu, \log d)}{d}$. Whereas multiple nearly-linear time algorithms have been established in the more specialized fully-random setting where each entry is revealed with probablity exactly $p$, the only known nearly-linear time algorithm in the semi-random setting is due to [CG18], whose sample complexity has a polynomial dependence on the inverse accuracy and condition number and thus cannot achieve high-accuracy recovery. Our main result is the first high-accuracy nearly-linear time algorithm for solving semi-random matrix completion, and an extension to the noisy observation setting.Our result builds upon the recent short-flat decomposition framework of [KLLST23a, KLLST23b] and leverages fast algorithms for flow problems on graphs to solve adaptive reweighting subproblems efficiently.