This paper is concerned with investigating the fundamental conditions on the locations of the sampled entries, i.e., sampling pattern, for finite completability of a matrix that represents the union of several subspaces with given ranks. In contrast with the existing analysis on Grassmannian manifold for the conventional matrix completion, we propose a geometric analysis on the manifold structure for the union of several subspaces to incorporate all given rank constraints simultaneously. In order to obtain the deterministic conditions on the sampling pattern, we characterizes the algebraic independence of a set of polynomials defined based on the sampling pattern, which is closely related to finite completion. We also give a probabilistic condition in terms of the number of samples per column, i.e., the sampling probability, which leads to finite completability with high probability. Furthermore, using the proposed geometric analysis for finite completability, we characterize sufficient conditions on the sampling pattern that ensure there exists only one completion for the sampled data.

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低秩子空间检索联合采样模式的基本条件

本文关注的是研究采样条目位置的基本条件，即采样模式，用于表示具有给定秩的几个子空间的并集的矩阵的有限可完备性。与现有的对传统矩阵完成的格拉斯曼流形分析相比，我们提出了对流形结构的几何分析，用于几个子空间的并集，以同时包含所有给定的秩约束。为了获得采样模式的确定性条件，我们刻画了一组基于采样模式定义的多项式的代数独立性，这与有限完备性密切相关。我们还给出了每列样本数的概率条件，即采样概率，这导致了高概率的有限可完成性。此外，使用所提出的有限可完成性几何分析，我们表征了采样模式的充分条件，以确保采样数据只存在一个完成。