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：本文研究矩阵补全问题，其目标是通过少量观测元素重构低秩矩阵。常用方法涉及核范数最小化。为使该方法成功，观测元素的数量需至少与真实矩阵的秩及相干参数成比例增长。尽管通常仅有的先验信息是真实矩阵的低秩特性，但在众多实际场景中，我们可获得关于真实低秩矩阵的额外知识。例如，在协同过滤、Netflix问题及无线通信中的动态信道估计等应用中，我们可预先获得信号子空间的部分或完整信息。具体而言，我们已知某些与真实矩阵列空间和行空间构成多重角度的子空间。利用这一宝贵信息有望显著减少所需观测数量。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并融合可用子空间信息。所设计的权重通过独立惩罚先验子空间各基向量对应的角度来实现定制化处理。我们进一步提出通过最小化真实矩阵的相干参数来优化权重选择策略，这等价于最小化所需观测数量。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，与现有先进方法相比，我们提出的多权重优化问题显著降低了所需观测数量。