The matrix completion problem aims to reconstruct a low-rank matrix based on a revealed set of possibly noisy entries. Prior works consider completing the entire matrix with generalization error guarantees. However, the completion accuracy can be drastically different over different entries. This work establishes a new framework of partial matrix completion, where the goal is to identify a large subset of the entries that can be completed with high confidence. We propose an efficient algorithm with the following provable guarantees. Given access to samples from an unknown and arbitrary distribution, it guarantees: (a) high accuracy over completed entries, and (b) high coverage of the underlying distribution. We also consider an online learning variant of this problem, where we propose a low-regret algorithm based on iterative gradient updates. Preliminary empirical evaluations are included.

翻译：矩阵补全问题旨在基于一组可能含噪声的观测条目重构低秩矩阵。已有研究关注于在泛化误差保证下完成整个矩阵的补全，但不同条目的补全精度可能存在显著差异。本研究建立了局部矩阵补全的新框架，其目标是识别具有高置信度可补全的大量条目子集。我们提出了一种高效算法，具备以下可证明的保证：在未知任意分布中获取样本的条件下，该算法可保证：(a) 已补全条目具有高精度，以及(b) 对底层分布实现高覆盖率。此外，我们考虑了该问题的在线学习变体，并提出了一种基于迭代梯度更新的低遗憾算法。文中还包含了初步实验结果。