Recent work in the matrix completion literature has shown that prior knowledge of a matrix's row and column spaces can be successfully incorporated into reconstruction programs to substantially benefit matrix recovery. This paper proposes a novel methodology that exploits more general forms of known matrix structure in terms of subspaces. The work derives reconstruction error bounds that are informative in practice, providing insight to previous approaches in the literature while introducing novel programs that severely reduce sampling complexity. The main result shows that a family of weighted nuclear norm minimization programs incorporating a $M_1 r$-dimensional subspace of $n\times n$ matrices (where $M_1\geq 1$ conveys structural properties of the subspace) allow accurate approximation of a rank $r$ matrix aligned with the subspace from a near-optimal number of observed entries (within a logarithmic factor of $M_1 r)$. The result is robust, where the error is proportional to measurement noise, applies to full rank matrices, and reflects degraded output when erroneous prior information is utilized. Numerical experiments are presented that validate the theoretical behavior derived for several example weighted programs.

翻译：近期矩阵补全文献中的研究表明，矩阵行空间和列空间的先验知识可成功融入重构算法，从而显著提升矩阵恢复性能。本文提出一种利用子空间形式下更广义已知矩阵结构的新方法。研究工作推导出具有实践指导意义的重构误差界，既为文献中现有方法提供了理论洞见，又引入了能大幅降低采样复杂度的新型算法。主要结果表明：一类融合了$n\times n$矩阵$M_1 r$维子空间（其中$M_1\geq 1$表征子空间结构特性）的加权核范数最小化程序，能够从近最优观测条目数（在$M_1 r$的对数因子范围内）实现对与该子空间对齐的秩$r$矩阵的精确逼近。该结果具有鲁棒性——误差与测量噪声成正比，适用于满秩矩阵，并在先验信息错误时呈现退化输出。通过数值实验验证了多个加权程序实例的理论性能特征。