This paper investigates the low-rank matrix completion (LRMC) problem from a generic vantage point. Unlike most existing work that has focused on recovering a low-rank matrix from a subset of the entries with specified values, the only information available here is just the pattern (i.e., positions) of observed entries. To be precise, given is an $n\times m$ pattern matrix ($n\le m$) whose entries take fixed zero, unknown generic values, and missing values that are free to be chosen from the complex field, the question of interest is whether there is a matrix completion with rank no more than $n-k$ ($k\ge 1$) for almost all values of the unknown generic entries, which is called the generic low-rank matrix completion (GLRMC) associated with $({\cal M},k)$, and abbreviated as GLRMC$({\cal M},k)$. Leveraging an elementary tool from the theory on systems of polynomial equations, we give a simple proof for genericity of this problem, which was first observed by $Kir\'aly et al.$, that is, depending on the pattern of the observed entries, the answer is either yes or no, for almost all realizations of the unknown generic entries. We provide necessary and sufficient conditions ensuring feasibility of GLRMC$({\cal M},1)$. Aso, we provide a sufficient condition and a necessary condition (which we conjecture to be sufficient too) for the general case (i.e., $k\ge 1$). In the following, two randomized algorithms are presented to determine upper and lower bounds for the generic minimum completion rank. Our approaches are based on the algebraic geometry theory, and a novel basis preservation principle discovered herein. Finally, numerical simulations are given to corroborate the theoretical findings and the effectiveness of the proposed algorithms.

翻译：本文从通用正方位点调查低位矩阵完成( LRCM) 问题 。 与大多数现有工作不同, 重点是从带有特定值的子条目中恢复一个低位矩阵的问题不同, 这里唯一的信息只是观察到的条目的图案( 位置 ) 。 确切地说, 给出的是 $\ time m$ 模式矩阵( 美元=le m$ ), 其条目为固定零, 未知的通用值, 以及从复杂字段中自由选择的缺失值 。 感兴趣的问题是, 对于几乎所有未知的通用条目的值来说, 是否有一个级别不超过 $- k$ (kge 1 g) 的矩阵完成。 与 $( gal M}, k) 相关的通用矩阵完成模式( GLRMC $ ( nn\ cal M), k) 。 在多位方位方位原则的理论中, 我们给出了一个基本工具, 简单证明这个问题的通用性。 由 $ Kir\\ g = flent real real deal deal deal deal deal deal) madeal deal deal deal deal.