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.

翻译：低秩矩阵补全研究如何利用稀疏的观测条目来估计矩阵中未观测到的条目。我们考虑非均匀设置，其中观测条目以高度变化的概率进行采样，且这些概率可能具有不同的渐近尺度。我们证明，在结构化采样概率下，通常在较小子矩阵而非整个矩阵上运行估计算法更为有效，有时甚至是最优的。特别地，我们推导出针对每个条目定制的误差上界，这些上界在特定条件下与极小极大下界相匹配。我们的界刻画了每个条目估计的难度，该难度与局部化采样概率相关。我们提供了数值实验来证实理论发现。