论文标题
矩阵完成中的多个典型等级
Multiple typical ranks in matrix completion
论文作者
论文摘要
低率矩阵完成解决了从一组通用指定条目中完成矩阵的问题。在复数上，具有给定入口模式的矩阵可以独特地完成为特定等级，称为“通用完成”等级。对实数的完成可能通常具有多个完成等级，称为典型等级。我们证明了证明许多指定条目的技术仅具有一个典型等级，并向其他具有两个典型等级的家庭展示，特别是专注于以循环图表为代表的进入集。这概括了Bernstein，Blekherman和Sinn的结果。特别是，我们提供了$ n \ times n $矩阵的一组未指定条目的完整表征，使得$ n-1 $是典型的等级，并充分确定了$ n <9 $的条目集合$ g（n，1）$的典型等级。此外，我们研究了典型等级的渐近行为，并在唯一的矩阵完成方面进行了结果。
Low-rank matrix completion addresses the problem of completing a matrix from a certain set of generic specified entries. Over the complex numbers a matrix with a given entry pattern can be uniquely completed to a specific rank, called the generic completion rank. Completions over the reals may generically have multiple completion ranks, called typical ranks. We demonstrate techniques for proving that many sets of specified entries have only one typical rank, and show other families with two typical ranks, specifically focusing on entry sets represented by circulant graphs. This generalizes the results of Bernstein, Blekherman, and Sinn. In particular, we provide a complete characterization of the set of unspecified entries of an $n\times n$ matrix such that $n-1$ is a typical rank and fully determine the typical ranks for entry set $G(n,1)$ for $n<9$. Moreover, we study the asymptotic behaviour of typical ranks and present results regarding unique matrix completions.
