Most of the existing works on provable guarantees for low-rank matrix completion algorithms rely on some unrealistic assumptions such that matrix entries are sampled randomly or the sampling pattern has a specific structure. In this work, we establish theoretical guarantee for the exact and approximate low-rank matrix completion problems which can be applied to any deterministic sampling schemes. For this, we introduce a graph having observed entries as its edge set, and investigate its graph properties involving the performance of the standard constrained nuclear norm minimization algorithm. We theoretically and experimentally show that the algorithm can be successful as the observation graph is well-connected and has similar node degrees. Our result can be viewed as an extension of the works by Bhojanapalli and Jain [2014] and Burnwal and Vidyasagar [2020], in which the node degrees of the observation graph were assumed to be the same. In particular, our theory significantly improves their results when the underlying matrix is symmetric.

翻译：现有关于低秩矩阵补全算法可证明保证的大部分工作依赖于一些不切实际的假设，例如矩阵条目被随机采样或采样模式具有特定结构。在本工作中，我们建立了适用于任意确定性采样方案的精确与近似低秩矩阵补全问题的理论保证。为此，我们引入一个以观测条目为边集的图，并研究了其与标准约束核范数最小化算法性能相关的图论性质。我们从理论上和实验上证明，当观测图具有良好连通性且节点度数相近时，算法可以成功。我们的结果可视为Bhojanapalli与Jain [2014]以及Burnwal与Vidyasagar [2020]工作的推广——这些工作假设观测图的节点度数相同。特别地，当底层矩阵为对称矩阵时，我们的理论显著改进了他们的结果。