In this work, we consider the matrix completion problem, where the objective is to reconstruct a low-rank matrix from a few observed entries. A commonly employed approach involves nuclear norm minimization. For this method to succeed, the number of observed entries needs to scale at least proportional to both the rank of the ground-truth matrix and the coherence parameter. While the only prior information is oftentimes the low-rank nature of the ground-truth matrix, in various real-world scenarios, additional knowledge about the ground-truth low-rank matrix is available. For instance, in collaborative filtering, Netflix problem, and dynamic channel estimation in wireless communications, we have partial or full knowledge about the signal subspace in advance. Specifically, we are aware of some subspaces that form multiple angles with the column and row spaces of the ground-truth matrix. Leveraging this valuable information has the potential to significantly reduce the required number of observations. To this end, we introduce a multi-weight nuclear norm optimization problem that concurrently promotes the low-rank property as well the information about the available subspaces. The proposed weights are tailored to penalize each angle corresponding to each basis of the prior subspace independently. We further propose an optimal weight selection strategy by minimizing the coherence parameter of the ground-truth matrix, which is equivalent to minimizing the required number of observations. Simulation results validate the advantages of incorporating multiple weights in the completion procedure. Specifically, our proposed multi-weight optimization problem demonstrates a substantial reduction in the required number of observations compared to the state-of-the-art methods.

翻译：本文研究了矩阵补全问题，其目标是从少量观测条目中重构低秩矩阵。常用方法涉及核范数最小化。为使该方法成功，所需观测条目数量需至少与真实矩阵的秩及相干性参数成比例缩放。尽管通常仅有真实矩阵的低秩性质作为先验信息，但在多种实际场景中，可获取关于真实低秩矩阵的额外知识。例如，在协同过滤、Netflix问题以及无线通信中的动态信道估计中，我们可预先获知信号子空间的部分或全部信息。具体而言，我们了解某些子空间，它们与真实矩阵的列空间和行空间形成多个夹角。利用这一宝贵信息有望显著减少所需观测次数。为此，我们提出一种多权重核范数优化问题，该问题同时促进低秩特性及可用子空间信息。所提出的权重专门设计为独立惩罚与先验子空间每个基对应的每个夹角。我们进一步提出通过最小化真实矩阵的相干性参数（等价于最小化所需观测数量）的最优权重选择策略。仿真结果验证了在补全过程中引入多重权重的优势。具体而言，与现有最优方法相比，我们提出的多权重优化问题在所需观测数量上实现了大幅减少。