This paper investigates the low-rank matrix completion (LRMC) problem from a generic view. Unlike most existing work which focused on numerically recovering exact or approximate missing matrix entries from the observed ones, the only available information herein is the pattern (structure) of observed/missing entries, and the observed entries are classified into two types, namely, fixed zero entries and unknown generic entries. The problem is whether there exists a matrix completion with a prescribed rank r for almost all values of the unknown generic entries except for a set of zero measure, which is called the generic low-rank matrix completion (GLRMC) problem. We first justify the existence of genericity in the complex field for this problem, that is, depending on the pattern of observed/missing entries, for almost all values of the unknown generic entries, either the answer to that problem is yes, or the answer is no. We then provide necessary and sufficient conditions for the feasibility of the GLRMC with the constraint that the rank of the completion reduces at least one. Afterward, we provide a sufficient condition and a necessary condition (which we conjecture to be sufficient) for the general case. Based on them, two randomized algorithms are proposed to determine upper and lower bounds for the generic minimum rank of the matrix completions. Our approaches are based on the algebraic geometry theory and the basis preservation principle discovered herein. Finally, numerical experiments are given to corroborate the theoretical findings and the effectiveness of the proposed algorithms.

翻译：本文从一般角度对低位矩阵完成问题进行了调查。 与大多数侧重于从观察到的条目中从数字上恢复准确或近似缺失的矩阵条目的现有工作不同,我们这里唯一的现有信息是观察/缺失条目的格局(结构),观察到的条目分为两类,即固定零条目和未知的通用条目。 问题是,除了一套称为通用低位矩阵完成(GLMC)问题的零度计量之外,对于几乎所有未知的通用条目来说,是否都有一个规定等级为r的矩阵完成,但几乎所有的通用条目都是如此。 与大多数现有的工作不同,我们首先证明在复杂领域存在关于该问题的通用性,即根据观察/缺失条目的格局,对几乎所有未知的通用条目的格局(结构),对该问题的答案是肯定的,或者答案是否定的。 然后,我们为GLRMC的可行性提供了必要和充分的条件,但制约是完成级别至少降低一个。 之后,我们为一般的低位矩阵完成率提供了充分的条件和必要条件(我们推测足够),对于一般案例来说,我们首先根据观察/ 缺失的输入模式,最后,我们提出的最起码的算算算的算的算的算。 最后,我们的拟议的矩阵是根据两个最下的结果,最后的矩阵,我们所研算的矩阵是最终的原理。 。