Matrix Completion under Low-Rank Missing Mechanism
Matrix Completion under Low-Rank Missing Mechanism
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DOI：
10.5705/ss.202019.0196
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发表时间：
2018-12
期刊：
影响因子：
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通讯作者：
Xiaojun Mao;Raymond K. W. Wong;Songxi Chen
中科院分区：
文献类型：
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作者：
Xiaojun Mao;Raymond K. W. Wong;Songxi Chen
矩阵完备化是一个现代的缺失数据问题，其中缺失的结构和底层参数都是高维的。虽然缺失结构是任何缺失数据问题的关键组成部分，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的矩阵完成损坏的数据。通过高维低秩矩阵估计过程估计观测的概率矩阵，并进一步通过逆概率加权来完成目标矩阵。由于高维和极端（即，非常小）的性质，逆概率加权的效果需要仔细研究。我们得到的最佳渐近收敛速度的估计的观测概率和目标矩阵。
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.