Semi-Random Matrix Completion via Flow-Based Adaptive Reweighting
Keywords: matrix completion, semi-random model, flow solver, short-flat decomposition, adaptive reweighting
TL;DR: We give the first nearly-linear time algorithm for solving semi-random matrix completion to high accuracy and with noisy observations.
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.
Primary Area: Learning theory
Submission Number: 21541
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