Typical and generic ranks in matrix completion
Typical and generic ranks in matrix completion
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DOI：
10.1016/j.laa.2019.09.001
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发表时间：
2018-02
影响因子：
1.1
通讯作者：
D. Bernstein;Grigoriy Blekherman;Rainer Sinn
中科院分区：
文献类型：
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作者：
D. Bernstein;Grigoriy Blekherman;Rainer Sinn
我们从几何的角度考虑精确低阶矩阵完成问题：给定一个部分填充的矩阵M，我们保持指定项和未指定项的位置不变，并研究最小完成秩.如果矩阵的条目是复杂的，并且已知条目是根据连续分布随机选择的，则对于指定和未指定条目的固定位置模式，存在以概率1出现的唯一最小完成等级。我们称这个等级为通用完成等级。在实数上，可以有多个以正概率出现的等级；我们称它们为典型的完成等级。我们正式地引入了这些概念，并针对不同的已知和未知表项模式族，给出了一些典型的和一般的秩上的一些不等式和精确的结果。
We consider the problem of exact low-rank matrix completion from a geometric viewpoint: given a partially filled matrixM, we keep the positions of specified and unspecified entries fixed, and study the minimal completion rank. If the entries of the matrix are complex and the known entries are chosen randomly according to a continuous distribution, then for a fixed pattern of locations of specified and unspecified entries, there is a unique minimum completion rank which occurs with probability one. We call this rank the generic completion rank. Over the real numbers there can be multiple ranks that occur with positive probability; we call them typical completion ranks. We introduce these notions formally, and provide a number of inequalities and exact results on typical and generic ranks for different families of patterns of known and unknown entries.