The problem of completing a large low rank matrix using a subset of revealed entries has received much attention in the last ten years. The main result of this paper gives a necessary and sufficient condition, stated in the language of graph limit theory, for a sequence of matrix completion problems with arbitrary missing patterns to be asymptotically solvable. It is then shown that a small modification of the Candès–Recht nuclear norm minimization algorithm provides the required asymptotic solution whenever the sequence of problems is asymptotically solvable. The theory is fully deterministic, with no assumption of randomness. A number of open questions are listed.

中文翻译：

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低秩矩阵补全的确定性理论

使用显示条目的子集完成大型低秩矩阵的问题在过去十年中受到了很多关注。本文的主要结果给出了一个充要条件，用图极限理论的语言表述，对于具有任意缺失模式的矩阵完成问题序列是渐近可解的。然后表明，只要问题序列是渐近可解的，对 Candès-Recht 核范数最小化算法的一个小的修改就提供了所需的渐近解。该理论是完全确定性的，没有随机性假设。列出了许多未解决的问题。