Low-rank matrix completion concerns the problem of estimating unobserved
entries in a matrix using a sparse set of observed entries. We consider the
non-uniform setting where the observed entries are sampled with highly varying
probabilities, potentially with different asymptotic scalings. We show that
under structured sampling probabilities, it is often better and sometimes
optimal to run estimation algorithms on a smaller submatrix rather than the
entire matrix. In particular, we prove error upper bounds customized to each
entry, which match the minimax lower bounds under certain conditions. Our
bounds characterize the hardness of estimating each entry as a function of the
localized sampling probabilities. We provide numerical experiments that confirm
our theoretical findings.

低秩矩阵补全问题关注使用稀疏观测的一组观测条目来估计矩阵中未观测的条目。我们考虑非均匀设置，其中观测条目根据高度变化的概率进行采样，可能具有不同的渐近尺度。我们证明了在结构化采样概率下，使用较小的子矩阵而不是整个矩阵上运行估计算法通常更好，有时是最优的。特别地，在某些条件下，我们证明了适用于每个条目的错误上界，这些错误上界与最小化下界相匹配。我们提供了数值实验证实了我们的理论发现。