论文标题
近乎最佳的加权矩阵完成
Near-Optimal Weighted Matrix Completion
论文作者
论文摘要
矩阵完成文献中的最新工作表明，可以将矩阵行和列空间的先验知识成功地纳入重建程序中，从而实质上使矩阵恢复受益。本文提出了一种新的方法，该方法利用了基于子空间的已知矩阵结构的更一般形式。这项工作得出了在实践中提供信息的重建误差界限，从而为文献中的先前方法提供了见解，同时引入了新颖的程序，从而严重降低了采样复杂性。主要结果表明，包括$ n \ times n $矩阵（其中$ n \ times n $矩阵（其中$ m_1 \ geq 1 $）的结构属性的一组加权核规范最小化计划，该计划结合了$ n \ times n $ n $ n $ n $ n $的子空间，允许准确地近似级别的rith $ rith rith y Interial intries intries tocress inter-nork of-norbe of-norke toceptimir norkermir norkermir norke norke norke norke norke norke norke toceptimir norke norke norke toarmi。 $ m_1 r）$。结果是可靠的，其中误差与测量噪声成正比，适用于全等级矩阵，并在使用错误的先验信息时反映了降级的输出。提出了数值实验，以验证为几个示例加权程序得出的理论行为。
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.
