论文标题
低等级矩阵完成和Schur补充的独特性
Uniqueness of Low Rank Matrix Completion and Schur Complement
论文作者
论文摘要
在本文中，我们使用Schur补充工具研究了低级矩阵完成问题。我们提供了足够的必要条件，以使完整的矩阵在给定的数据上在全球范围内独特。我们假设矩阵的观察到的条目遵循特殊的“楼梯”结构。在此假设下，矩阵完成问题是全球唯一的，或者具有无限的解决方案（因此不包括局部唯一性）。实际上，矩阵完成问题的唯一性完全取决于“楼梯”角落的子膜的排名。定理的证据广泛使用了Schur补充。
In this paper we study the low rank matrix completion problem using tools from Schur complement. We give a sufficient and necessary condition such that the completed matrix is globally unique with given data. We assume the observed entries of the matrix follow a special "staircase" structure. Under this assumption, the matrix completion problem is either globally unique or has infinitely many solutions (thus excluding local uniqueness). In fact, the uniqueness of the matrix completion problem totally depends on the rank of the submatrices at the corners of the "staircase". The proof of the theorems make extensive use of the Schur complement.
