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 probability 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.<\/p>\n","protected":false},"excerpt":{"rendered":"
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 $p = \\frac{\\textup{poly}(r, \\mu, \\log d)}{d}$. 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