Matrix completion is a modern missing data problem where both the missing structure and the underlying parameter are high dimensional. Although missing structure is a key component to any missing data problems, existing matrix completion methods often assume a simple uniform missing mechanism. In this work, we study matrix completion from corrupted data under a novel low-rank missing mechanism. The probability matrix of observation is estimated via a high dimensional low-rank matrix estimation procedure, and further used to complete the target matrix via inverse probabilities weighting. Due to both high dimensional and extreme (i.e., very small) nature of the true probability matrix, the effect of inverse probability weighting requires careful study. We derive optimal asymptotic convergence rates of the proposed estimators for both the observation probabilities and the target matrix.

中文翻译：

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低秩缺失机制下的矩阵补全

矩阵补全是一种现代缺失数据问题，其中缺失结构和基础参数都是高维的。尽管缺失结构是任何缺失数据问题的关键组成部分，但现有的矩阵补全方法通常假设一个简单的统一缺失机制。在这项工作中，我们研究了一种新的低秩缺失机制下损坏数据的矩阵补全。观测概率矩阵通过高维低秩矩阵估计过程进行估计，并进一步通过逆概率加权完成目标矩阵。由于真实概率矩阵的高维和极端（即非常小）性质，逆概率加权的影响需要仔细研究。