Multiple typical ranks in matrix completion

IF 1 3区 数学 Q1 MATHEMATICS
Mareike Dressler , Robert Krone
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引用次数: 0

Abstract

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×n matrix such that n1 is a typical rank and fully determine the typical ranks of an n×n matrix with unspecified diagonal for n<9. Moreover, we study the asymptotic behavior of typical ranks and present results regarding unique matrix completions.
矩阵补全中的多个典型秩
低秩矩阵补全解决了从一组一般指定条目补全矩阵的问题。在复数上,具有给定输入模式的矩阵可以唯一地补全到一个特定的秩,称为一般补全秩。在现实中完成通常会有多个完成等级,称为典型等级。我们演示了证明许多特定条目集只有一个典型秩的技术,并展示了具有两个典型秩的其他族,特别关注由循环图表示的条目集。这概括了Bernstein、Blekherman和Sinn的结果。特别地,我们给出了n×n矩阵的未指定元素集合的完整表征,使得n−1是一个典型秩,并完全确定了对n<;9具有未指定对角线的n×n矩阵的典型秩。此外,我们还研究了典型秩的渐近行为,并给出了关于唯一矩阵补全的结果。
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来源期刊
CiteScore
2.20
自引率
9.10%
发文量
333
审稿时长
13.8 months
期刊介绍: Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
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