  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.
